Enhanced Analysis of Falling Weight Deflectometer Data for Use With Mechanistic-Empirical Flexible Pavement Design and Analysis and Recommendations for Improvements to Falling Weight Deflectometers

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Figure 1. Diagram. FWD testing schematic. This diagram shows a gray box labeled falling mass. The falling mass is centered halfway up a pole; the bottom of the pole sits on a base that is labeled load cell. Arrows are shown above the falling mass to indicate its downward direction. Two rubber buffers are attached on the bottom of the falling mass on both sides of the pole. Beneath the load cell is a base plate. To the right of the base plate at ground level are five deflection sensors evenly spaced and shown as blue boxes. Centered beneath the base plate is a half circle with colors radiating outward beginning at the center with red, then orange, and the outside is yellow.

Figure 2. Photo. Grontmij Pavement Consultants FWDs. This photo shows a trailer-mounted falling weight deflectometer (FWD) on the left. The trailer is double axled. In the center of the photo is a van-mounted FWD. A white van is shown from the side with FWD shown inside. On the right of the photo is a portable lightweight FWD.

Figure 3. Drawings. Comparison of two Dynatest® FWDs. Two photos are shown. The top photo shows a drawing of a Dynatest® Model 8000 falling weight deflectometer. This photo shows a trailer with two wheels. Centered between the trailer wheels is the sensor device. Near the front of the trailer near the hitch is another wheel. The second photo shows a drawing of a Dynatest® Model 8082 heavy weight deflectometer. This photo shows a double-axle trailer. Centered between the wheels is the sensor device. At the front of the trailer is another smaller wheel.

Figure 4. Photo. KUAB FWD. This photo shows a trailer-mounted dynamic impulse loading device. A red metal housing contains the device. The trailer has two wheels and is being pulled by a vehicle down the road.

Figure 5. Equation. Objective function for the search algorithm. f is equal to the summation, with the lower bound j equals 1 and the upper bound of m, of a subscript j times the quantity parenthesis w subscript jm minus w subscript jc, end parenthesis end quantity squared.

Figure 6. Equation. Search method using a set of equations. Bracket F end bracket, raised to the k power, bracket d end bracket, raised to the k power, equals bracket r end bracket, raised to the k power.

Figure 7. Equation. Parseval theorem. The summation, with the lower bound n equals 0 and the upper bound N minus 1, of the absolute value of x times n, squared, end absolute value, equals 1 over N times the summation, with the lower bound k equals 0 and the upper bound N minus 1, of the absolute value of X times k, squared, end absolute value; or the summation, with the lower bound n equals 0 and the upper bound N minus 1, of the absolute value of x times n, squared, end absolute value, equals the summation, with the lower bound k equals 0 and the upper bound N minus 1, of the absolute value of the quantity X times k divided by the square root of N, end quantity, end absolute value, squared.

Figure 8. Equation. DFT of a nonzero-mean function at zero frequency. X bracket 0 equals the quantity 1 over the square root of N end quantity times the summation, with the lower bound n equals 0 and the upper bound N minus 1, of x bracket n, end bracket.

Figure 9. Drawing and Graph. Plot of inverse of deflection offset versus measured deflection. A drawing and a graph are shown. The drawing is a box with three horizontal layers dividing it. An arrow points downward on the top of the box causing a deformation, and an arrow points horizontally to the first layer on the box with the label r axis. The graph is a plot with 1/r (inverse of deflection) on the x-axis and Dr (measured deflection) on the y‑axis. One line is shown on the graph that begins just before r subscript o and above 0 measured deflection. The line increases in a convex shape until just past r subscript o, where it begins to increase linearly. The curved part of the line is labeled nonlinear behavior due to stress in the subgrade. The line increases linearly, and near the top of the graph, it becomes nonlinear again, curving in a concave shape. Here it is labeled Nonlinear due to stiff upper layer. There is a dotted line on the graph that shows that the linear portion of the line, if extended down to the x-axis, intersects the point (r_o, 0).

Figure 10. Equation. Calculation of the depth to stiff layer using the modified Roesset’s equations. Two equations are shown. The first equation is for saturated subgrade with bedrock: D subscript b equals the quotient of the convolution of V subscript s and T subscript d, divided by 1.35. The second equation is for nonsaturated subgrade with bedrock or groundwater table: D subscript b equals the quotient of the convolution of V subscript s with T subscript d divided by parenthesis pi minus the convolution of 2.24 and u end parenthesis.

Figure 11. Graph. Natural period T_d from sensor deflection time histories. This graph shows time in ms, ranging from 0 to 140 in intervals of 20, on the x-axis, and the deflection in mil, ranging from –10 to 30, on the y-axis. The graph shows the results of seven sensors. For each sensor, the line has a wave shape with one maximum peak in deflection at approximately 17 ms. After this large peak, the wave shape continues with decreasing amplitude with increasing time until the graph ends at 150 ms. The amplitude of the second peak is approximately 2 for all sensors at 45 ms. The max deflections for sensors 1 through 7 are 25, 17, 15, 11, 7, 5, and 2 mil respectively. The graph also has a period labeled T subscript d that ranges from 45 to 75 ms.

Figure 12. Equation. Shear wave velocity. V subscript s equals the square root of G divided by rho, which is also equal to the square root of E subscript sg divided by 2 times parenthesis 1 plus v end parenthesis, all divided by rho, end square root.

Figure 13. Graph. Example of time histories showing dynamic behavior for LTPP section 161020, station 1. The graph shows time in ms, ranging from 0 to 60, on the x-axis, and the deflection in mil, ranging from –6 to 30, on the left y-axis, and the stress in psi, ranging from –35 to 165, on the right y-axis. The graph has 10 lines, 9 of which represent D1 through D9 and the tenth line represents the stress. Each line is shown to have a parabolic shape that begins at a time and deflection of 0 and a stress of 10 psi. After reaching their peak deflections, the lines decrease to a negative deflection and then increase to a second smaller positive peak in deflection before decreasing toward 0 when the graph ends at 60 ms. For D1, the first peak has a deflection of 27mil at 145 psi stress at 16 ms, the negative peak is at –5 mil and 32 ms, and the second positive peak is at 4 mil and 20 psi stress at approximately 45 s. For D2, the first peak has a deflection of 22 mil and 115 psi stress at 16 ms, the negative peak is at –5 mil and 32 ms, and the second positive peak occurs at 2 mil and 11 psi stress at approximately 48 s. For D3, the first peak has a deflection of 18 mil and 95 psi stress at approximately 16 ms, and the negative and second positive peaks follow the same trend as D2. For D4, the first peak occurs at 13 mil and 65psi stress at approximately 16 ms, and the negative and second positive peaks follow the same trend as D2. For D5, the first peak occurs at 10 mil and 50 psi stress at approximately 17 ms, and the negative and second positive peaks follow the same trend as D2. For D6, the first peak occurs at 6 mil and 28 psi stress at approximately 18 ms, the negative peak is at –4 mil and 33 ms, and the second positive peak has a trend similar to D2. For D7, the first peak occurs at 4 mil and 21psi stress at approximately 19 ms, the negative peak is at –2 mil and 33 ms, and the second positive peak has a trend similar to D2. For D8, the first peak has a deflection of 2 mil and 14 psi stress at approximately 20 ms, the negative peak is at –2 mil deflection and 32 ms, and the second positive peak has a trend similar to D2. For D9, the first peak occurs at 18 mil and 90 psi stress at approximately 17 ms, the negative peak is at –5 mil and 32 ms, and the second positive peak has a trend similar to D2. The last line represents stress; it has a peak deflection at 27 mil and 145 psi stress at approximately 12 ms, and after this peak it falls to 0 deflection at 27 ms and remains at 0 until the x-axis ends at 60 ms.

Figure 14. Graph. Example of time histories showing no dynamic behavior for LTPP section 169034, station 3. The graph shows time in ms, ranging from 0 to 60 in intervals of 10, on the x‑axis, and the deflection in mil, ranging from –2 to 16, on the left y-axis. On the y-axis on the right is stress in psi, ranging from –20 to 160. The graph has 10 lines, 9 of which represent D1 through D9 and the tenth line representing the stress. Each line begins at 0 deflection and 0stress and has a parabolic shape with one large peak in deflection, after which the peak deflection of all of the lines decreases to approximately 0 mil at 60 ms. The line representing stress has a double peak, the first peak occurs at 11 mil and 110 psi at 8 ms, and the second peak occurs at 15 mil and 150 psi stress at approximately 10 ms. The line representing D1 has a peak occurring at 13mil and 130 psi stress at approximately 17 ms. D2 has a peak occurring at 12 mil and 118 psi stress at approximately 17 ms. D3 has a peak occurring at 11 mil and 105 psi stress at approximately 17 ms. D4 has a peak occurring at 9 mil and 90 psi stress at approximately 17ms. D5 has a peak occurring at 7.5 mil and 72 psi stress at approximately 19 ms. D6 has a peak occurring at 5 mil and 50 psi stress at approximately 19 ms. D7 has a peak occurring at 3.5mil and 32 psi stress at approximately 20 ms. D8 has a peak occurring at 2.2 mil and 25 psi stress at approximately 23 ms. D9 and D3 are over each other.

Figure 15. Graph. Example of stiffening behavior for LTPP section 81053, station 3. This graph shows load level in psi, ranging from 55 to 155 in intervals of 25, on the x-axis, and the load-to-deflection ratio in thousands of pci, ranging from 0 to 60, on the y-axis. The graph contains lines for eight different load cases. The first line, Load/Def 8, has a linear increasing trend beginning at a load-to-deflection ratio of 50 at approximately 60 psi, which increases to a load-to-deflection ratio of 52 at approximately 150 psi. The second line, Load/Def 7, has a linear increasing trend beginning at a load-to-deflection ratio of 36 at approximately 60 psi, which increases to a load-to-deflection ratio of 38 at approximately 150 psi. The third line, Load/Def 6, has a linear increasing trend beginning at a load-to-deflection ratio of 28 at approximately 60 psi, which increases to a load-to-deflection ratio of 29 at approximately 150 psi. The fourth line, Load/Def5, has a linear increasing trend beginning at a load-to-deflection ratio of 19 at approximately 60 psi, which increases to a load-to-deflection ratio of 20 at approximately 150psi. The fifth line, Load/Def 4, has a linear increasing trend beginning at a load-to-deflection ratio of 17 at approximately 60 psi, which increases to a load-to-deflection ratio of 18 at approximately 150 psi. The sixth line, Load/Def 3, has a linear increasing trend beginning at a load-to-deflection ratio of 13 at approximately 60 psi, which increases to a load-to-deflection ratio of 14 at approximately 150 psi. The seventh line, Load/Def 2, has a linear increasing trend beginning at a load-to-deflection ratio of 12 at approximately 60 psi, which increases to a load-to-deflection ratio of 13 at approximately 150 psi. The bottom line, Load/Def 1, has a linear increasing trend beginning at a load-to-deflection ratio of 11 at approximately 60 psi, which increases to a load-to-deflection ratio of 12 at approximately 150 psi.

Figure 16. Graph. Example of softening behavior LTPP for section 87781, station 3. This graph shows load level in psi, ranging from 55 to 155 in intervals of 25, on the x-axis, and the load-to-deflection ratio in thousands of pci, ranging from 0 to 60, on the y-axis. The graph contains lines for eight different load cases. The first line, Load/Def 8, has a linear decreasing trend beginning at a load-to-deflection ratio of 58 at approximately 60 psi, which decreases to load-to-deflection ratio of 54at 150 psi. The second line, Load/Def 7, has a linear decreasing trend beginning at a load-to-deflection ratio of 45 at 60 psi, which decreases to a load-to-deflection ratio of 40 at 150psi. The third line, Load/Def 6, has a linear decreasing trend beginning at a load-to-deflection ratio of 33at 60 psi, which decreases to a load-to-deflection ratio of 30 at 150 psi. The fourth line, Load/Def 5, has a linear decreasing trend beginning at a load-to-deflection ratio of 23at 60 psi, which decreases to a load-to-deflection ratio of 22 at 150psi. The fifth line, Load/Def 4, has a linear decreasing trend beginning at a load-to-deflection ratio of 19 at 60 psi, which decreases to a load-to-deflection ratio of 18 at 150 psi. The sixth line, Load/Def 3, has a linear decreasing trend beginning at a load-to-deflection ratio of 16 at 60 psi, which decreases to a load-to-deflection ratio of 15 at 150 psi. The seventh line, Load/Def 2, has a linear decreasing trend beginning at a load-to-deflection ratio of 14 at 60 psi, which decreases to a load-to-deflection ratio of 13 at 150 psi. The bottom line, Load/Def 1, has a linear decreasing trend beginning at a load-to-deflection ratio of 11 at 60 psi, which decreases to a load-to-deflection ratio of 10 at 150 psi.

Figure 17. Graphs. Preliminary Results—evidence of dynamic behavior by climatic information: classification by season (top), temperature (middle), and climate zone (bottom). Three graphs are shown. The top graph is a bar graph classifying the sections by season; on the x-axis are the four seasons (fall, winter, spring, and summer), and the y-axis is the percent of total sections. For each season, the percent of the total section is divided into two categories; dynamic and no dynamic. For fall, 74 percent of the total section is dynamic and the remaining 26 percent is not dynamic. For winter, 69 percent of the total section is dynamic and the remaining 31 percent is not dynamic. For spring, 55 percent of the total section is dynamic and the remaining 45 percent is not dynamic. For summer, 89 percent of the total section is dynamic and the remaining 11percent is not dynamic.

The middle graph is a bar graph classifying the sections by temperature; on the x-axis is the temperature in °F and on the y-axis is the percent of total sections. For each temperature, the percent of the total section is divided into two categories, dynamic and no dynamic. At 36.1 °F, 100 percent is dynamic. At 41.5 °F, 0 percent is dynamic and not dynamic. At 46.9 °F 61 percent is dynamic and 39 percent is not dynamic. At 52.3 °F, 58 percent is dynamic and 42 percent is not dynamic. At 57.7 °F, 86 percent is dynamic and 14 percent is not dynamic. At 63.1 °F, 67 percent is dynamic and 33 percent is not dynamic. At 68.5 °F, 60 percent is dynamic and 40 percent is not dynamic. At 73.9 °F, 75 percent is dynamic and 25 percent is not dynamic. At 79.3 °F, 72 percent is dynamic and 28 percent is not dynamic. At 84.7 °F, 56 percent is dynamic and 44 percent is not dynamic. At 90.1 °F, 100percent is dynamic. At 95.5 °F, 61 percent is dynamic and 39 percent is not dynamic. At 100.9 °F, 66 percent is dynamic and 34 percent is not dynamic. At 106.3 °F, 33percent is dynamic and 67 percent is not dynamic. At 111.7 °F, 40 percent is dynamic and 60percent is not dynamic. At 117.1 °F, 100 percent is not dynamic. At 122.5 °F, 75 percent is dynamic and 25percent is not dynamic. At 127.9 °F, 33 percent is dynamic and 67 percent is not dynamic. At 133.3 °F, 100 percent is dynamic.

The bottom graph classifies the sections by climate zone. The graph is a bar graph with the percent of total tests on the y-axis. There are three bars shown on the graph. The first bar shows that 65 percent of the total tests were dynamic testing and the remaining 35 percent were not dynamic testing. The next bar on the graph shows that of the dynamic testing, 22 percent of tests were wet and 43 percent were dry. The last bar on the graph shows that of the dynamic sections, 36 percent of tests were freeze and 29 percent were no freeze.

Figure 18. Graph. Mean and standard deviation of percent of sections with dynamics for wet/dry and freeze/no freeze. This graph shows percent of total sections on the y-axis, and on the x-axis, there are two categories, wet/dry and freeze/no freeze. The wet/dry line is a vertical line extending from 16 to 102 percent representing the standard deviation range. The line has two dots indicating that the dry mean is at 85 percent with a standard deviation from 68 to 102percent and the wet mean is 45 percent with a standard deviation from 16 to 78 percent. The freeze/no freeze line is a vertical line extending from 22 to 98 percent representing the standard deviation range. The line also has two dots indicating that the mean value for freeze is 79 percent with a standard deviation from 60 to 98 percent and the mean value for no freeze is 55 percent with a standard deviation from 22 to 90 percent.

Figure 19. Graphs. Distribution of load-deflection slope by sensor. Nine graphs are shown. The first eight are for sensors 1 through 8. Each has percent slope on the x‑axis, percent frequency on the y-axis on the left, and on the y-axis on the right is cumulative percent. Each plot contains a bar graph in which the bars represent the frequency. On each graph, the bars show a Gaussian distribution for the different slope percentages. There is also a line on each graph that has an S shape representing the cumulative percent.

On the first graph, labeled Sensor 1, the maximum frequency is 25 percent at a slope of 5percent. The end limits for frequency are 3 percent at a slope of –15 percent and 0.1 percent at a slope of 35percent. On the second graph, labeled Sensor 2, the maximum frequency is 25percent at a slope of 0 percent. The end limits for frequency are 5 percent at a slope of –15percent and 1percent at a slope of 30 percent. On the third graph, labeled Sensor 3, the maximum frequency is 23percent at a slope of 0 percent. The end limits for frequency are 2 percent at a slope of –20 percent and 1percent at a slope of 30 percent. On the fourth graph, labeled Sensor 4, the maximum frequency is 25 percent at a slope of 0 percent. The end limits for frequency are 1 percent at a slope of –25 percent and 0.5percent at a slope of 25 percent. On the fifth graph, labeled Sensor 5, the maximum frequency is 23 percent at a slope of 0 percent. The end limits for frequency are 2percent at a slope of –25 percent and 1 percent at a slope of 20 percent. On the sixth graph, labeled Sensor 6, the maximum frequency is 27 percent at a slope of –5 percent. The end limits for frequency are 1percent at a slope of –30percent and 0.5 percent at a slope of 20 percent. On the seventh graph, labeled Sensor 7, the maximum frequency is 23 percent at a slope of –5 percent. The end limits for frequency are 3 percent at a slope of –25 percent and 0.5 percent at a slope of 25percent. On the eighth graph, labeled Sensor 8, the maximum frequency is 23 percent at a slope of –5 percent. The end limits for frequency are 3 percent at a slope of –25 percent and 0.1 percent at a slope of 40percent. On each of the graphs, the cumulative percent line begins at 0 percent frequency at –50 percent slope, until about –25 percent slope when the line begins to increase until it reaches a slope percent at about 20. The line then becomes a horizontal line at 100 percent until the graph ends at a slope percent of 50.

The last graph shows the load on the x-axis and the load-to-deflection ratio on the y-axis. Three lines are shown—all with linear trends. The first line represents the softening, the second line represents a straight linear horizontal line, and the third line represents stiffening. The three lines begin at the same point; the linear line is drawn horizontally on the graph, the softening line increases linearly from that point, and the stiffening line decreases linearly from that point. The softening and stiffening lines have the same magnitude of slope. A label on the graph states: delta P divided by delta is less than 0: softening, and delta P divided by delta is greater than 0: stiffening.

Figure 20. Graphs. Distribution of linear versus nonlinear behavior for a 5- to 7-percent threshold load-to-deflection slope. Six graphs are shown. For each of them, the x-axis represents the sensor, ranging from S1 to S8, and the y-axis represents the percent of stations. On each graph, two bars are shown for each sensor; the bars represent linear and nonlinear or stiffening and softening.

The first graph is labeled Percent Linear versus Nonlinear (5 percent threshold). For sensor S1, the percent linear is 48 percent, and the percent nonlinear is 52 percent. For sensor S2, the percent linear is 47 percent, and the percent nonlinear is 53 percent. For sensor S3, the percent linear is 43percent, and the percent nonlinear is 57 percent. For sensor S4, the percent linear is 41 percent, and the percent nonlinear is 59 percent. For sensor S5, the percent linear is 39 percent, and the percent nonlinear is 61 percent. For sensor S6, the percent linear is 35 percent, and the percent nonlinear is 65 percent. For sensor S7, the percent linear is 36 percent, and the percent nonlinear is 64 percent. For the last sensor S8, the percent linear is 38 percent, and the percent nonlinear is 62 percent.

The second graph is labeled Percent Stiffening versus Softening (5 percent threshold). For sensor S1, the percent stiffening is 45 percent, and the percent softening is 55 percent. For sensor S2, the percent stiffening is 43 percent, and the percent softening is 57 percent. For sensor S3, the percent stiffening is 37 percent, and the percent softening is 63 percent. For sensor S4, the percent stiffening is 31 percent, and the percent softening is 69 percent. For sensor S5, the percent stiffening is 23 percent, and the percent softening is 77 percent. For sensor S6, the percent stiffening is 15 percent, and the percent softening is 85 percent. For sensor S7, the percent stiffening is 15 percent, and the percent softening is 85 percent. For the last sensor S8, the percent stiffening is 15 percent, and the percent softening is 85 percent.

The third graph is labeled Percent Linear versus Nonlinear (6 percent threshold). For sensor S1, the percent linear is 55 percent, and the percent nonlinear is 45 percent. For sensor S2, the percent linear is 53 percent, and the percent nonlinear is 47 percent. For sensor S3, the percent linear is 50percent, and the percent nonlinear is 50 percent. For sensor S4, the percent linear is 49percent, and the percent nonlinear is 51 percent. For sensor S5, the percent linear is 47 percent, and the percent nonlinear is 53 percent. For sensor S6, the percent linear is 42 percent, and the percent nonlinear is 58 percent. For sensor S7, the percent linear is 44 percent, and the percent nonlinear is 56 percent. For the last sensor S8, the percent linear is 43 percent, and the percent nonlinear is 57 percent.

The fourth graph is labeled Percent Stiffening versus Softening (6 percent threshold). For sensor S1, the percent stiffening is 47 percent, and the percent softening is 53 percent. For sensor S2, the percent stiffening is 43 percent, and the percent softening is 57 percent. For sensor S3, the percent stiffening is 37 percent, and the percent softening is 63 percent. For sensor S4, the percent stiffening is 29 percent, and the percent softening is 71 percent. For sensor S5, the percent stiffening is 20 percent, and the percent softening is 80 percent. For sensor S6, the percent stiffening is 15 percent, and the percent softening is 85 percent. For sensor S7, the percent stiffening is 12 percent, and the percent softening is 88 percent. For the last sensor S8, the percent stiffening is 14 percent, and the percent softening is 86 percent.

The fifth graph is labeled Percent Linear versus Nonlinear (7 percent threshold). For sensor S1, the percent linear is 63 percent, and the percent nonlinear is 37 percent. For sensor S2, the percent linear is 59 percent, and the percent nonlinear is 41 percent. For sensor S3, the percent linear is 56 percent, and the percent nonlinear is 44 percent. For sensor S4, the percent linear is 55 percent, and the percent nonlinear is 45 percent. For sensor S5, the percent linear is 54percent, and the percent nonlinear is 46 percent. For sensor S6, the percent linear is 49 percent, and the percent nonlinear is 51 percent. For sensor S7, the percent linear is 50percent, and the percent nonlinear is 50 percent. For the last sensor S8, the percent linear is 50percent, and the percent nonlinear is 50 percent.

The sixth graph is labeled Percent Stiffening versus Softening (7 percent threshold). For sensor S1, the percent stiffening is 51 percent, and the percent softening is 49 percent. For sensor S2, the percent stiffening is 44 percent, and the percent softening is 56 percent. For sensor S3, the percent stiffening is 37 percent, and the percent softening is 63 percent. For sensor S4, the percent stiffening is 28 percent, and the percent softening is 72 percent. For sensor S5, the percent stiffening is 19 percent, and the percent softening is 81 percent. For sensor S6, the percent stiffening is 13 percent, and the percent softening is 87 percent. For sensor S7, the percent stiffening is 9 percent, and the percent softening is 91 percent. For the last sensor S8, the percent stiffening is 12 percent, and the percent softening is 88 percent.

Figure 21. Graphs. Distribution of linear versus nonlinear behavior for 8- to 10-percent threshold load-to-deflection slope. Six graphs are shown. For each of them, the x-axis represents the sensor, ranging from S1 to S8, and the y-axis represents the percent of stations. For each sensor, there are two bars that represent linear and nonlinear data or stiffening and softening data.

The first graph is labeled Percent Linear versus Nonlinear (8% threshold). For sensor S1, the percent linear is 69 percent, and the percent nonlinear is 31 percent. For sensor S2, the percent linear is 65 percent, and the percent nonlinear is 35 percent. For sensor S3, the percent linear is 61percent, and the percent nonlinear is 39 percent. For sensor S4, the percent linear is 58percent, and the percent nonlinear is 42 percent. For sensor S5, the percent linear is 58percent, and the percent nonlinear is 42 percent. For sensor S6, the percent linear is 57percent, and the percent nonlinear is 43 percent. For sensor S7, the percent linear is 58percent, and the percent nonlinear is 42 percent. For the last sensor S8, the percent linear is 56percent and the percent nonlinear is 44 percent.

The second graph is labeled Percent Stiffening versus Softening (8% threshold). For sensor S1, the percent stiffening is 54 percent, and the percent softening is 46 percent. For sensor S2, the percent stiffening is 43 percent, and the percent softening is 57 percent. For sensor S3, the percent stiffening is 36 percent, and the percent softening is 64 percent. For sensor S4, the percent stiffening is 29 percent, and the percent softening is 71 percent. For sensor S5, the percent stiffening is 19 percent, and the percent softening is 81 percent. For sensor S6, the percent stiffening is 11 percent, and the percent softening is 89 percent. For sensor S7, the percent stiffening is 8 percent, and the percent softening is 92 percent. For the last sensor S8, the percent stiffening is 12 percent, and the percent softening is 88 percent.

The third graph is labeled Percent Linear versus Nonlinear (9% threshold). For sensor S1, the percent linear is 72 percent, and the percent nonlinear is 28 percent. For sensor S2, the percent linear is 69 percent, and the percent nonlinear is 31 percent. For sensor S3, the percent linear is 66 percent, and the percent nonlinear is 34 percent. For sensor S4, the percent linear is 63percent, and the percent nonlinear is 37 percent. For sensor S5, the percent linear is 62 percent, and the percent nonlinear is 38 percent. For sensor S6, the percent linear is 66 percent, and the percent nonlinear is 34 percent. For sensor S7, the percent linear is 64 percent, and the percent nonlinear is 36 percent. For the last sensor S8, the percent linear is 62percent, and the percent nonlinear is 38 percent.

The fourth graph is labeled, Percent Stiffening vs. Softening (9% threshold). For sensor S1 the percent stiffening is 55 percent and the percent softening is 45 percent. For sensor S2 the percent stiffening is 46 percent and the percent softening is 54 percent. For sensor S3 the percent stiffening is 37 percent and the percent softening is 63 percent. For sensor S4 the percent stiffening is 28 percent and the percent softening is 72 percent. For sensor S5 the percent stiffening is 18 percent and the percent softening is 82 percent. For sensor S6 the percent stiffening is 7 percent and the percent softening is 93 percent. For sensor S7 the percent stiffening is 9 percent and the percent softening is 91 percent. For the last sensor S8, the percent stiffening is 13 percent and the percent softening is 87 percent.

The fifth graph is labeled Percent Linear versus Nonlinear (10% threshold). For sensor S1, the percent linear is 76 percent, and the percent nonlinear is 24 percent. For sensor S2, the percent linear is 74 percent, and the percent nonlinear is 26 percent. For sensor S3, the percent linear is 69 percent, and the percent nonlinear is 31 percent. For sensor S4, the percent linear is 68percent, and the percent nonlinear is 32 percent. For sensor S5, the percent linear is 68percent, and the percent nonlinear is 32 percent. For sensor S6, the percent linear is 69percent, and the percent nonlinear is 31 percent. For sensor S7, the percent linear is 69 percent, and the percent nonlinear is 31 percent. For sensor S8, the percent linear is 67 percent, and the percent nonlinear is 33 percent.

The sixth graph is labeled Percent Stiffening vs. Softening (10% threshold). For sensor S1, the percent stiffening is 54 percent, and the percent softening is 46 percent. For sensor S2, the percent stiffening is 46 percent, and the percent softening is 54 percent. For sensor S3, the percent stiffening is 36 percent, and the percent softening is 64 percent. For sensor S4, the percent stiffening is 25 percent, and the percent softening is 75 percent. For sensor S5, the percent stiffening is 14 percent, and the percent softening is 86 percent. For sensor S6, the percent stiffening is 6 percent, and the percent softening is 94 percent. For sensor S7, the percent stiffening is 7 percent, and the percent softening is 93 percent. For sensor S8, the percent stiffening is 11 percent, and the percent softening is 89 percent.

Figure 22. Graphs. Percent of sections by season where nonlinear behavior was prevalent. Eight graphs are shown, one each for sensors 1 to 8. Each graph has the seasons fall, winter, spring, and summer on the x-axis and percent of total sections on the y-axis. For each season, the percent of the total section is divided into two categories, linear and nonlinear.

The first graph is labeled Sensor 1. For fall, 54 percent of the total section is linear, and the remaining 46 percent is nonlinear. For winter, 28 percent of the total section is linear, and the remaining 72 percent is nonlinear. For spring 54 percent of the total section is linear and the remaining 46 percent is nonlinear. For summer, 69 percent of the total section is linear, and the remaining 31 percent is nonlinear.

The second graph is labeled Sensor 2. For fall, 55 percent of the total section is linear, and the remaining 45 percent is nonlinear. For winter 27, percent of the total section is linear, and the remaining 73 percent is nonlinear. For spring, 55 percent of the total section is linear, and the remaining 45 percent is nonlinear. For summer, 77 percent of the total section is linear, and the remaining 23 percent is nonlinear.

The third graph is labeled Sensor 3. For fall, 54 percent of the total section is linear, and the remaining 46 percent is nonlinear. For winter, 28 percent of the total section is linear, and the remaining 72 percent is nonlinear. For spring, 61 percent of the total section is linear, and the remaining 39 percent is nonlinear. For summer, 86 percent of the total section is linear, and the remaining 14 percent is nonlinear.

The fourth graph is labeled Sensor 4. For fall, 55 percent of the total section is linear, and the remaining 45 percent is nonlinear. For winter, 34 percent of the total section is linear, and the remaining 66 percent is nonlinear. For spring, 62 percent of the total section is linear, and the remaining 38 percent is nonlinear. For summer, 88 percent of the total section is linear, and the remaining 12 percent is nonlinear.

The fifth graph is labeled Sensor 5. For fall, 54 percent of the total section is linear, and the remaining 46 percent is nonlinear. For winter, 40 percent of the total section is linear, and the remaining 60 percent is nonlinear. For spring, 65 percent of the total section is linear, and the remaining 35 percent is nonlinear. For summer, 88 percent of the total section is linear, and the remaining 12 percent is nonlinear.

The sixth graph is labeled Sensor 6. For fall, 75 percent of the total section is linear, and the remaining 25 percent is nonlinear. For winter, 42 percent of the total section is linear, and the remaining 58 percent is nonlinear. For spring, 66 percent of the total section is linear, and the remaining 34 percent is nonlinear. For summer, 77 percent of the total section is linear, and the remaining 23 percent is nonlinear.

The seventh graph is labeled Sensor 7. For fall, 75 percent of the total section is linear, and the remaining 25 percent is nonlinear. For winter, 43 percent of the total section is linear, and the remaining 57 percent is nonlinear. For spring, 64 percent of the total section is linear, and the remaining 36 percent is nonlinear. For summer, 71 percent of the total section is linear, and the remaining 29 percent is nonlinear.

The last graph is labeled Sensor 8. For fall, 68 percent of the total section is linear, and the remaining 32 percent is nonlinear. For winter, 56 percent of the total section is linear, and the remaining 44 percent is nonlinear. For spring, 62 percent of the total section is linear, and the remaining 38 percent is nonlinear. For summer, 56 percent of the total section is linear, and the remaining 44 percent is nonlinear.

Figure 23. Graphs. Percent of sections by temperature where nonlinear behavior was prevalent. Eight graphs are shown, one each for sensors 1 to 8. Each graph is a bar graph that has the temperature in °F on the x-axis and the percent of total section on the y-axis. For each temperature, the percent of the total section is divided into two categories, linear and nonlinear.

The first graph is labeled Sensor 1. At 36 °F, 100 percent is linear. At 41.5 °F, 0 percent is linear and nonlinear. At 47 °F, 61 percent is nonlinear, and 39 percent is linear. At 52.3 °F, 22 percent is nonlinear, and 78 percent is linear. At 57.7 °F, 52 percent is nonlinear, and 48 percent is linear. At 63.1 °F, 64 percent is nonlinear, and 36 percent is linear. At 68.5 °F, 68 percent is nonlinear, and 32 percent is linear. At 73.9 °F, 22 percent is nonlinear, and 78 percent is linear. At 79.3 °F, 41percent is nonlinear, and 59 percent is linear. At 84.7 °F, 48 percent is nonlinear, and 52percent is linear. At 90.1 °F, 62 percent is nonlinear, and 38 percent is linear. At 95.5 °F, 50percent is nonlinear, and 50 percent is linear. At 100.9 °F, 83 percent is nonlinear, and 17 percent is linear. At 106.3 °F, 72 percent is nonlinear, and 28 percent is linear. At 111.7 °F, 31 percent is nonlinear, and 69 percent is linear. At 117.1 °F, 36 percent is nonlinear, and 64 percent is linear. At 122.5 °F, 42 percent is nonlinear, and 58 percent is linear. At 127.9 °F, 60 percent is nonlinear, and 40percent is linear, and at 133.3 °F, 100 percent is nonlinear.

The second graph is labeled Sensor 2. At 36 °F, 8 percent is nonlinear and 92 percent linear. At 41.5 °F, 0 percent is linear and nonlinear,. At 47 °F, 54 percent is nonlinear, and 46 percent is linear. At 52 °F, 28 percent is nonlinear, and 72 percent is linear. At 57.7 °F, 53 percent is nonlinear, and 47 percent is linear. At 63.1 °F, 62 percent is nonlinear, and 38 percent is linear. At 68.5 °F, 76 percent is nonlinear, and 24 percent is linear. At 73.9 °F, 22 percent is nonlinear, and 78 percent is linear. At 79.3 °F, 44 percent is nonlinear, and 56 percent is linear. At 84.7 °F, 21percent is nonlinear, and 79 percent is linear. At 90.1 °F, 57 percent is nonlinear, and 43 percent is linear. At 95.5 °F, 48 percent is nonlinear, and 52 percent is linear. At 100.9 °F, 83 percent is nonlinear, and 17 percent is linear. At 106.3 °F, 72 percent is nonlinear, and 28 percent is linear. At 111.7 °F, 58 percent is nonlinear, and 42 percent is linear. At 117.1 °F, 48 percent is nonlinear, and 52 percent is linear. At 122.5 °F, 56 percent is nonlinear, and 44 percent is linear. At 127.9 °F, 39 percent is nonlinear, and 61 percent is linear, and at 133.3 °F, 100 percent is nonlinear.

The third graph is labeled Sensor 3. At 36 °F, 100 percent is linear. At 41.5 °F, 0 percent is linear and nonlinear. At 47 °F, 73 percent is nonlinear, and 27 percent is linear. At 52 °F, 33 percent is nonlinear, and 67 percent is linear. At 57.7 °F, 55 percent is nonlinear, and 45 percent is linear. At 63.1 °F, 56 percent is nonlinear, and 44 percent is linear. At 68.5 °F, 74 percent is nonlinear, and 26 percent is linear. At 73.9 °F, 38 percent is nonlinear, and 62 percent is linear. At 79.3 °F, 42percent is nonlinear, and 58 percent is linear. At 84.7 °F, 21 percent is nonlinear, and 79 percent is linear. At 90.1 °F, 53 percent is nonlinear, and 47 percent is linear. At 95.5 °F, 58 percent is nonlinear, and 42 percent is linear. At 100.9 °F, 82 percent is nonlinear, and 18 percent is linear. At 106.3 °F, 90 percent is nonlinear, and 10 percent is linear. At 111.7 °F, 60 percent is nonlinear, and 40 percent is linear. At 117.1 °F, 66 percent is nonlinear, and 34 percent is linear. At 122.5 °F, 77 percent is nonlinear, and 23 percent is linear. At 127.9 °F, 64 percent is nonlinear, and 36percent is linear, and at 133.3 °F, 100 percent is nonlinear.

The fourth graph is labeled Sensor 4. At 36 °F, 9 percent is nonlinear, and 91 percent is linear. At 41.5 °F, 0 percent is linear and nonlinear. At 47 °F, 80 percent is nonlinear, and 20 percent is linear. At 52 °F, 44 percent is nonlinear, and 56 percent is linear. At 57.7 °F, 50 percent is nonlinear, and 50 percent is linear. At 63.1 °F, 65 percent is nonlinear, and 35 percent is linear. At 68.5 °F, 79 percent is nonlinear, and 21 percent is linear. At 73.9 °F, 37 percent is nonlinear, and 63 percent is linear. At 79.3 °F, 40 percent is nonlinear, and 60 percent is linear. At 84.7 °F, 18percent is nonlinear, and 82 percent is linear. At 90.1 °F, 57 percent is nonlinear, and 43 percent is linear. At 95.5 °F, 66 percent is nonlinear, and 34 percent is linear. At 100.9 °F, 85 percent is nonlinear, and 15 percent is linear. At 106.3 °F, 94 percent is nonlinear, and 6 percent is linear. At 111.7 °F, 60 percent is nonlinear, and 40 percent is linear. At 117.1 °F, 75 percent is nonlinear, and 25 percent is linear. At 122.5 °F, 77 percent is nonlinear, and 23 percent is linear. At 127.9 °F, 69percent is nonlinear, and 31 percent is linear, and at 133.3 °F, 64 percent is nonlinear, and 36percent is linear.

The fifth graph is labeled Sensor 5. At 36 °F, 9 percent is nonlinear, and 91 percent is linear. At 41.5 °F, 0 percent is linear and nonlinear. At 47 °F, 69 percent is nonlinear, and 31 percent is linear. At 52 °F, 47 percent is nonlinear, and 53 percent is linear. At 57.7 °F, 55 percent is nonlinear, and 45 percent is linear. At 63.1 °F, 66 percent is nonlinear, and 34 percent is linear. At 68.5 °F, 70 percent is nonlinear, and 30 percent is linear. At 73.9 °F, 48 percent is nonlinear, and 52 percent is linear. At 79.3 °F, 43 percent is nonlinear, and 57 percent is linear. At 84.7 °F, 36 percent is nonlinear, and 54 percent is linear. At 90.1 °F, 67 percent is nonlinear, and 33percent is linear. At 95.5 °F, 64 percent is nonlinear, and 34 percent is linear. At 100.9 °F, 94percent is nonlinear, and 6 percent is linear. At 106.3 °F, 82 percent is nonlinear, and 18percent is linear. At 111.7 °F, 75 percent is nonlinear, and 25 percent is linear. At 117.1 °F, 82 percent is nonlinear, and 18 percent is linear. At 122.5 °F, 62 percent is nonlinear, and 38percent is linear. At 127.9 °F, 76 percent is nonlinear, and 24 percent is linear, and at 133.3°F, 18 percent is nonlinear, and 82 percent is linear.

The sixth graph is labeled Sensor 6. At 36 °F, 18 percent is nonlinear, and 82 percent is linear. At 41.5 °F, 0 percent is linear and nonlinear. At 47 °F, 69 percent is nonlinear, and 31 percent is linear. At 52 °F, 55 percent is nonlinear, and 45 percent is linear. At 57.7 °F, 53 percent is nonlinear, and 47 percent is linear. At 63.1 °F, 72 percent is nonlinear, and 28 percent is linear. At 68.5 °F, 57 percent is nonlinear, and 43 percent is linear. At 73.9 °F, 42 percent is nonlinear, and 58 percent is linear. At 79.3 °F, 52 percent is nonlinear, and 48 percent is linear. At 84.7 °F, 54 percent is nonlinear, and 46 percent is linear. At 90.1 °F, 78 percent is nonlinear, and 22percent is linear. At 95.5 °F, 80 percent is nonlinear, and 20 percent is linear. At 100.9 °F, 90percent is nonlinear, and 10 percent is linear. At 106.3 °F, 88 percent is nonlinear, and 12percent is linear. At 111.7 °F, 78 percent is nonlinear, and 22 percent is linear. At 117.1 °F, 88 percent is nonlinear, and 12 percent is linear. At 122.5 °F, 66 percent is nonlinear, and 34percent is linear. At 127.9 °F, 72 percent is nonlinear, and 24 percent is linear, and at 133.3°F, 100 percent is nonlinear.

The seventh graph is labeled Sensor 7. At 36 °F, 18 percent is nonlinear, and 82 percent is linear. At 41.5 °F, 0 percent is linear and nonlinear. At 47 °F, 62 percent is nonlinear, and 38percent is linear. At 52 °F, 50 percent is nonlinear, and 50 percent is linear. At 57.7 °F, 45percent is nonlinear, and 55 percent is linear. At 63.1 °F, 65 percent is nonlinear, and 35percent is linear. At 68.5 °F, 68 percent is nonlinear, and 32 percent is linear. At 73.9 °F, 40percent is nonlinear, and 60 percent is linear. At 79.3 °F, 54 percent is nonlinear, and 46percent is linear. At 84.7 °F, 72percent is nonlinear, and 28 percent is linear. At 90.1 °F, 79percent is nonlinear, and 21 percent is linear. At 95.5 °F, 74 percent is nonlinear, and 26 percent is linear. At 100.9 °F, 82 percent is nonlinear, and 18 percent is linear. At 106.3 °F, 88percent is nonlinear, and 12 percent is linear. At 111.7 °F, 62 percent is nonlinear, and 38 percent is linear. At 117.1 °F, 82 percent is nonlinear, and 18 percent is linear. At 122.5 °F, 72 percent is nonlinear, and 28 percent is linear. At 127.9 °F, 69 percent is nonlinear, and 31 percent is linear, and at 133.3 °F, 100 percent is nonlinear.

The last graph is labeled Sensor 8. At 36 °F, 36 percent is nonlinear, and 64 percent is linear. At 41.5 °F, 0 percent is linear and nonlinear. At 47 °F, 67 percent is nonlinear, and 33 percent is linear. At 52 °F, 52 percent is nonlinear, and 48 percent is linear. At 57.7 °F, 61 percent is nonlinear, and 39 percent is linear. At 63.1 °F, 64 percent is nonlinear, and 36 percent is linear. At 68.5 °F, 61 percent is nonlinear, and 39 percent is linear. At 73.9 °F, 44 percent is nonlinear, and 56 percent is linear. At 79.3 °F, 49 percent is nonlinear, and 51 percent is linear. At 84.7 °F, 72percent is nonlinear, and 28 percent is linear. At 90.1 °F, 83 percent is nonlinear, and 17 percent is linear. At 95.5 °F, 76 percent is nonlinear, and 24 percent is linear. At 100.9 °F, 60 percent is nonlinear, and 40 percent is linear. At 106.3 °F, 79 percent is nonlinear, and 21 percent is linear. At 111.7 °F, 62 percent is nonlinear, and 38 percent is linear. At 117.1 °F, 72 percent is nonlinear, and 28 percent is linear. At 122.5 °F, 64 percent is nonlinear, and 36 percent is linear. At 127.9 °F, 69 percent is nonlinear, and 31 percent is linear, and at 133.3 °F, 82 percent is nonlinear, and 18percent is linear.

Figure 24. Graphs. Percent of sections by climate where nonlinear behavior was prevalent. Eight bar graphs are shown, one each for sensors 1 to 8. The y-axis is the percent of the total stations. Each graph shows three bars. The first bar shows the percent of total stations that are nonlinear versus linear, the second bar shows the percent of total stations that are wet versus dry, and the last bar in each graph shows the percent of total stations that are freeze versus no freeze.

The first graph is labeled Sensor 1. In the graph, the first bar shows 52 percent of the total tests were nonlinear, and the remaining 48 percent were linear. The second bar on the graph shows that 23 percent of tests were wet, and 29 percent were dry. The third bar shows that 24 percent of tests were freeze, and 28 percent were no freeze.

The second graph is labeled Sensor 2. In the graph, the first bar shows 54 percent of the total tests were nonlinear, and the remaining 46 percent were linear. The second bar on the graph shows that 24 percent of tests were wet, and 30 percent were dry. The third bar shows that 25percent of tests were freeze, and 29 percent were no freeze.

The third graph is labeled Sensor 3. In the graph, the first bar shows 57 percent of the total tests were nonlinear, and the remaining 43 percent were linear. The second bar on the graph shows that 28 percent of tests were wet, and 29 percent were dry. The third bar shows that 26percent of tests were freeze, and 31 percent were no freeze.

The fourth graph is labeled Sensor 4. In the graph, the first bar shows 59 percent of the total tests were nonlinear, and the remaining 41 percent were linear. The second bar on the graph shows that 31 percent of tests were wet, and 28 percent were dry. The third bar shows that 26percent of tests were freeze, and 33 percent were no freeze.

The fifth graph is labeled Sensor 5. In the graph, the first bar shows 61 percent of the total tests were nonlinear, and the remaining 39 percent were linear. The second bar on the graph shows that 32 percent of tests were wet, and 29 percent were dry. The third bar shows that 28 percent of tests were freeze, and 33 percent were no freeze.

The sixth graph is labeled Sensor 6. In the graph, the first bar shows 66 percent of the total tests were nonlinear, and the remaining 34 percent were linear. The second bar on the graph shows that 33 percent of tests were wet, and 33 percent were dry. The third bar shows that 32percent of tests were freeze, and 34 percent were no freeze.

The seventh graph is labeled Sensor 7. In the graph, 64 percent of the total tests were nonlinear, and the remaining 36 percent were linear. The second bar on the graph shows that 32percent of tests were wet, and 32 percent were dry. The third bar shows that 30 percent of tests were freeze, and 34 percent were no freeze.

The eighth graph is labeled Sensor 8. In the graph, 63 percent of the total tests were nonlinear, and the remaining 37 percent were linear. The second bar on the graph shows that 32 percent of tests were wet, and 31 percent were dry. The third bar shows that 28 percent of tests were freeze, and 35 percent were no freeze.

Figure 25. Graphs. Examples of measurement issues. Six graphs are shown for different sections. The time in ms is on the x-axis, the deflection in mil is on the y-axis on the left, and the load in psi is on the y-axis on the right. Each graph contains 10 lines, labeled D1 through D9 and a line representing the stress.

The first graph is labeled Section 220125 station 3. All of the lines have a parabolic shape with one peak, except for the line representing stress, which has a double peak. The line for sensor D1 has a peak value in deflection of 5 mil and 150 psi load at 20 ms. After this peak, the line decreases to 1.2 mil and 36 psi at 60 ms. The next line, for sensor D2, has a peak value in deflection of 4 mil and 125 psi load, at 20 ms. After this peak, the line decreases to 0 deflection and –8 psi at 60 ms; the remainder of the lines all end around these values for deflection and load at 60 ms. The third line, for sensor D3, has a peak value in deflection of 3.8 mil and 114 psi load at 20 ms. After this peak, the line decreases to 0 mil and −4 psi at 60 ms. The line for sensor D9 has the same trend as this line. The fourth line, for sensor D4, has a peak value in deflection of 3.5 mil and 106 psi load at 20 ms. The fifth line, for sensor, D5 has a peak value in deflection of 3.2 mil and 97 psi load at 20 ms. The sixth line, for sensor D6, has a peak value in deflection of 2.8 mil and 86 psi load at 20 ms. The seventh line, for sensor D7, has a peak value in deflection of 2.4 mil and 72 psi load at 20 ms. The eighth line, for sensor D8, has a peak value in deflection of 2 mil and 58 psi load at 20 ms. The line representing stress has its largest peak value of 135psi load at 13 ms, decreases to 0 load at 30 ms, and remains there for the duration of time.

The second graph is labeled Section 220125 station 6. All of the lines have a parabolic shape with one maximum peak, except for the line representing stress, which has a double peak. After the peak, all of the lines decrease until the graph ends at 60 ms. The line for sensor D1 has a peak at 3.5 mil and 125 psi load at 20 ms and decreases to a deflection of 0.75 mil at 60 ms. The next line, for sensor D2, has a peak at 2.4 mil and 93 psi load at 20 ms. It then decreases to a deflection of –0.2 mil. The third line, for sensor D3, has a peak at 2.7 mil and 96 psi load at 22ms. It then decreases to a deflection of 0.7 mil. The fourth line, for sensor D4, has a peak at 2.2 mil and 80 psi load at 20 ms. It then decreases to a deflection of –0.3 mil. The fifth line, for sensor D5, has a peak at 2.7 mil and 97 psi load at 27 ms and then decreases to a deflection of 1.3 mil. The sixth line, for sensor D6, has a peak at 2 mil and 75 psi load at 25 ms and then decreases to a deflection of 0.03 mil. The seventh line, for sensor D7, has a peak at 1.9 mil and 67 psi load at 28 ms. It then decreases to a deflection of 0.08 mil. The eighth line, for sensor D8, has a peak at 1.6 mil and 56 psi load at 27 ms and then decreases to a deflection of 0.04 mil. The ninth line, for sensor D9, has a peak at 2.6 mil and 92 psi load at 21 ms. It then decreases to a deflection of 0.4 mil. The line representing stress has a small peak occurring at 90 psi at 12 ms before it reaches its large peak at 110 psi load at 19 ms. It then decreases to 0 at 30 ms and remains there until 60 ms.

The third graph is labeled Section 130508 station 1. All of the lines have a parabolic shape that is either positive or negative and begins at 0 deflection except the line for stress. The line for sensor D1 has a peak occurring at 3 mil and 107 psi load at 18 ms. It then decreases to 0deflection at approximately 35 ms before increasing to 1.2 mil at 60 ms. The next line, for sensor D2, has a peak occurring at –0.07 mil and 25 psi load at 12 ms. After the peak, the line then decreases to –2 mil at −5.7 psi and 30 ms, and then increases back toward 0 until the graph ends at 60 ms. The third line, for sensor D3, has a peak occurring at 2.1 mil and 86 psi load at 19ms. It then decreases to –0.4 mil before increasing again to 0.6 mil at 60 ms. The fourth line, for sensor D4, has a peak occurring at 2.2 mil and 82 psi load at 19 ms. It then decreases to
–0.4 mil and then up to just above 0 mil at 60 ms. The fifth line, for sensor D5, has a peak occurring at 1.8 mil and 80 psi load at 20 ms. It then decreases to –0.4 mil and oscillates slightly around this value until it reaches 60 ms. The sixth line, for sensor D6, has a peak occurring at 1.6mil and 75 psi load at 19 ms. It then decreases to a negative deflection and then up to
–0.02 mil deflection at 60 ms. The seventh line, for sensor D7, has a peak occurring at 1.3 mil and 70 psi load at 20 ms and then decreases to 0 deflection at 60 ms. The eighth line, for sensor D8, has a peak occurring at –1.2 mil and 8.5 psi load at 15 ms. Then the deflection increases to 0.03 mil at 20 ms before decreasing to −0.8 mil at 60 ms. The ninth line, for sensor D9, has a peak occurring at 1.6 mil and 76 psi load at 18 ms. It then decreases to a negative deflection before increasing to 0.7 mil at 60 ms. The line representing stress begins at 0 psi and increases to a small peak of 97 psi at 12 ms and then to a large peak of 115 psi at 17 ms. It then decreases to 0load at 30 ms and remains here until the line stops at 60 ms.

The fourth graph is labeled Section 130566 station 1. All of the lines begin at 0 deflection, except the line for stress, and have a parabolic shape. After reaching the peak deflection and load, all of the lines begin to decrease until the graph ends at 60 ms. The line for sensor D1 has a peak value of 4.6 mil and 145 psi load at 19 ms. The line then decreases to a negative deflection and unlike the other lines, it then increases to a deflection of 2.4 mil at 100 psi at 60 ms. The next line, for sensor D2, has a peak value of 4.8 mil and 146 psi load at 20 ms. After reaching the peak value, the line decreases to −0.8 mil and 40 psi at 60 ms. The third line, for sensor D3, has a peak value of 4.6 mil and 145 psi load at 19 ms. The line then decreases to –0.8 mil at 60 ms. The fourth line, for sensor D4, has a peak value of 4.4 mil and 140 psi load at 20 ms. The line then decreases to −0.4 mil and 15 psi at 60 ms. The fifth line, for sensor D5, has a peak value of 4.2 mil and 140 psi load at 20 ms. The line then decreases to –0.8 mil and 36 psi at 60 ms. The sixth line, for sensor D6, is not shown on the graph. The seventh line, for sensor D7, has a peak value of 4.1 mil and 122 psi load at 25 ms. After reaching this peak, the line decreases to 0.4 mil and 58 psi at 60 ms. The eighth line, for sensor D8, has a peak value of 2.6 mil and 103 psi load at 18 ms. After reaching this peak, the line decreases to –1.6 mil and 18 psi at 60 ms. The ninth line, for sensor D9, has a peak value of 4.8 mil and 146 psi load at 18 ms. It then decreases to
–3.4 mil at 60 ms. The line representing stress begins at 0 load; it increases to its peak value of 145 psi load at 12 ms. After this peak, it decreases to and 0 KPa at 30 ms and remains there until the line ends at 60 ms.

The fifth graph is labeled Section 260116 station 3. All of the lines have a parabolic shape and begin at 0 deflection except the line for stress. After reaching the peak deflections and loads, the lines decrease until the graph ends at 60 ms. The line for sensor D1 has a peak of 2.4 mil and 54psi, at 27 ms. It then decreases to a deflection at about 0.3 mil and 25 psi at 60 ms. The next line, for sensor D2, has a peak of 2 mil and 52 psi at 27 ms. It decreases to 0 deflection and 23psi at 60 ms. The third line, for sensor D3, has a peak of 2 mil and 47 psi, at 27 ms. It decreases to 0 deflection and 23 psi at 60 ms. The fourth line, for sensor D4, has a peak of 1.6mil and 45 psi at 28 ms. The line then decreases to 0 deflection around 45 ms and then back up to 0.2 mil and 26 psi at 60 ms. The fifth line, for sensor D5, has a peak of 1.2 mil and 40 psi at 27 ms. It then decreases to 0 deflection and 23 psi at 60 ms. The sixth line, for sensor D6, has a peak of 1 mil and 36 psi at 28 ms. It then decreases to –0.2 mil and 26 psi at 60 ms. The seventh line, for sensor D7, has a peak of 0.7 mil and 33 psi at 29 ms. It then decreases to .2 mil and 24psi at 60 ms. The eighth line, for sensor D8, has a peak of 2 mil and 52 psi, at 27 ms. It decreases to 0.4 mil and 28 psi at 60 ms. The ninth line, for sensor D9, is the same as the third line for sensor 3. The last line, representing stress, begins at 0 load and increases to 56 psi at 19ms, decreases and then increases to 56 psi at 25 ms. After this, it decreases back to 0 psi at 35ms until the graph ends at 60 ms.

The sixth graph is labeled Section 482108 station 1. All of the lines have a parabolic shape and begin at 0 deflection except the line for stress. After each line reaches its peak point in deflection, it decreases to a deflection between 0.04 and 0.2 mil, and a stress around 3.5 psi at 60ms, except the line representing stress. The line for sensor D1 has a peak value of 3 mil and 60 psi at 35 ms. The next line, for sensor D2, has a peak of 1.6 mil and 53 psi, at 35 ms. The third line, for sensor D3, has a peak of 2.7 mil and 51 psi at 35 ms. The fourth line, for sensor D4, has a peak of 2.3 mil and 47 psi at 35 ms. The fifth line, for sensor D5, has a peak of 2.1 mil and 44 psi at 36 ms. The sixth line, for sensor D6, has a peak of 1.8 mil and 37 psi load, at 37 ms. The seventh line, for sensor D7, has a peak of 1.6 mil and 44 psi load at 39 ms. The eighth line, for sensor D8, has a peak of 1.4 mil and 24 psi at 40 ms. The ninth line, for sensor D9, has a peak of 1.6 mil and 52 psi at 37 ms. The line representing stress begins at –0.04 mil and 0 load and increases to 3 mil and 61 psi at 24 ms. It then decreases back to 0 load and −0.04 mil from 35 ms until the graph ends at 60 ms.

Figure 26. Equation. Boltzman’s superposition principle. The linear viscoelastic response at coordinates (x,y,z) and time t: R superscript ve, as a function of x, y, z, and t, equals the integral with respect to tau, from tau equals 0 to t of the following: R subscript H superscript ve, parenthesis x, y, z, t minus tau end parenthesis times the derivative of I as a function of tau with respect to tau.

Figure 27. Equation. Quasi-elastic approximation of a unit response function such as the creep compliance. The unit viscoelastic response of the pavement system: R subscript H, superscript ve, as a function of x, y, z, t, is approximately equal to R subscript H, superscript e, parenthesis x, y, z, E parenthesis t end parenthesis, end parenthesis.

Figure 28. Equation. Hereditary integral using the quasi-elastic approximation of a unit response function such as the creep compliance. R superscript ve, as a function of x, y, z, and t, equals the integral with respect to tau from tau equals 0 to t subscript R of the following: R subscript H, superscript e, parenthesis x, y, z, E parenthesis t subscript R minus tau end parenthesis, end parenthesis times the derivative of I as a function of tau with respect to tau.

Figure 29. Equation. Hereditary integral using the quasi-elastic approximation of unit vertical deflection at the surface. u subscript vertical, superscript ve, as a function of r, z, and t, equals the integral from tau equals 0 to t subscript R of the following: u subscript H minus vertical, superscript e, parenthesis E, parenthesis t subscript R minus tau end parenthesis, comma r, comma z, end parenthesis times the derivative of sigma as a function of tau with respect to tau.

Figure 30. Diagram. Typical flexible pavement geometry for analysis. This diagram shows a rectangle divided into four horizontal layers. The bottom layer is the semi-infinite layer labeled subgrade E subscript 3, nu subscript 3. The depth is labeled h subscript 3 equals Inf. The second layer from the bottom is blank, which represents optional possible layers. The next layer is labeled base E subscript 2, nu subscript 2. Its depth is labeled h subscript 2. The top layer is labeled AC, E parenthesis t end parenthesis, nu subscript 1. Its depth is labeled h subscript 1. Above the rectangle is a distributive load represented by arrows pointing downward onto the surface of the top layer. This distributive load is labeled sigma parenthesis t end parenthesis. The distance from the right side to the center, of the distributive load is labeled r equals 5.9 inches. Six points on the surface of the top layer are evenly spaced. The first point is located at the center of the distributed load and is labeled 1. There are five other points on the top of the AC surface to going to the right of the distributive load and labeled 2 through 6.

Figure 31. Graph. Discretization of stress history in forward analysis. This graph has time in s on the x-axis, ranging from 0 to 0.1, and the stress in psi, ranging from 0 to 120, on the y‑axis. The graph shows a line with a haversine distribution that begins at the origin and increases to 100 psi at 0.025 s. The stress then decreases back down to 0 at 0.05 s and remains at 0 for the remaining time. The graph is labeled number of time intervals: N subscript s equals 50.

Figure 32. Equation. Sigmoid form of relaxation modulus master curve. The log of E as a function of t equals c subscript 1 plus the quotient c subscript 2 divided by parenthesis 1 plus exp parenthesis negative c subscript 3 minus c subscript 4 times the log of t subscript R end parenthesis, end parenthesis.

Figure 33. Equation. Shift factor coefficient polynomial. The log of a subscript T as a function of T equals a subscript 1 parenthesis T squared minus T subscript ref squared, end parenthesis, plus a subscript 2 parenthesis T minus T subscript ref end parenthesis.

Figure 34. Graph. Discretization of the relaxation modulus master curve. This graph has reduced time in s on the x-axis and relaxation modulus, E parenthesis t end parenthesis, in psi on the y-axis. The graph has one curve that begins at 1.1 times 10⁶ psi at a reduced time of 10^-8 s. It decreases in a concave and then convex manner until the line ends at 1.05 times 10³ psi at a reduced time of 10⁸ s. There is also a label on the graph, number of time interval: NE equals100.

Figure 35. Equation. Quasi-elastic approximation of unit vertical deflection at the surface. u subscript H superscript ve, as a function of t subscript i is approximately equal to u subscript H, superscript e.

Figure 36. Graph. Deflections calculated under unit stress for points at different distances from the centerline of the circular load at the surface. This graph has time in s, ranging from 10^-10 to 10¹⁰, on the x-axis, and vertical surface displacements in inches, ranging from 10^-5 to 10^-2, on the y-axis. The graph contains eight curves representing response under unit stress at different radial distances, r subscript c. All the curves are obtained over time, ranging from 10^-8 s to 10⁸ s. The first curve is for r subscript c equals 0. It begins at 10^-8 s at 2.0 times 10^-4 inches and ends at 10⁸ s and 4.5 times 10^-3 inches. The second curve represents r subscript c equals 8 inches. It begins at 10^-8 s just below the first curve and ends at 10⁸ s and 8 times 10^-4 inches. The third curve represents r subscript c equals 12 inches. It begins at 10^-8 s just below the second curve and ends at 10⁸ s and 3 times 10^-4. The fourth line represents r subscript c equals 18 inches. It begins at 10^-8 s just below the third curve and ends at 10⁸ s and 2 times 10^-4 inches. The fifth curve represents r subscript c equals 24. It begins at 10^-8 s at just below the fourth curve and ends at 10⁸s and 1.9 times 10^-4 inches. The sixth, seventh, and eighth curves represent r subscript c equal 36 inches, 48 inches, and 60 inches, respectively. The three curves are approximately parallel and appear in that order, one below the next.

Figure 37. Equation. Discrete formulation. u superscript ve as a function of t subscript i equals the summation, with the lower bound j equals 0 and the upper bound i, of the following: u subscript H, superscript e parenthesis t subscript i minus tau subscript j end parenthesis multiplied by the derivative of sigma as a function of tau subscript j.

Figure 38. Graph. dσ( τ_j ) for each time step τ_j. This graph has time in s on the x-axis, ranging from 0 to 0.1, and the stress in psi on the y-axis, ranging from 0to 120. The graph contains a line with a haversine distribution; it begins at the origin and increases to a peak at 100psi and 0.025s. The stress then decreases back to 0 at 0.05 s and remains at 0 for the remainder of the time.

Figure 39. Diagram. Example problem geometry. This diagram shows a rectangle divided into three horizontal layers. The bottom layer is the semi-infinite layer. It is labeled: subgrade E subscript 3 equals 8,000 psi, nu subscript 3 equals 0.35. The depth is labeled H subscript 3 equals infinity. The middle layer is labeled base E subscript 2 equals 10,000 psi, nu subscript 2 equals 0.35. Its depth is labeled H subscript 2 equals 10 inches. The top layer is labeled AC, E parenthesis t end parenthesis equal to figure 34, and nu subscript 1 equals 0.35. Its depth is labeled H subscript 1 equals 6 inches. Above this layer is a distributive load that is represented by arrows pointing downward onto the surface of the AC layer. The distributive load is labeled sigma parenthesis t end parenthesis from figure 38. The distance from the right side to the center of the distributive load is labeled r equals 5.9 inches. Eight points are shown on the surface of the top layer representing deflection sensors. The first point is located at the center of the distributed load and is labeled 1. There are seven other points on the top of the AC surface extending to the right of the distributive load and labeled 2 through 8.

Figure 40. Graph. Examples of computed viscoelastic surface deflections at different radial distances from the centerline of the load. This graph has time in s, ranging from 0 to 0.1, on the x-axis, and vertical surface deflections in inches, ranging from 0 to 0.025, on the y-axis. Nine curves are shown on the double y-axis graph. The first eight curves are the deflection histories calculated at radial distances 0, 8, 12, 18, 24, 36, 48, and 60 inches from the center of center of loading. All the curves have a haversine shape and begin at 0. All the curves reach peak vertical deflection at approximately 17.5 s and gradually decrease to approximately 0inches at 0.035 s. The curves are shown up to 0.1 s. The peak vertical deflections of the eight curves are approximately 0.0245, 0.022, 0.021, 0.018, 0.0155, 0.012, 0.08, 0.065 inches, respectively.

Figure 41. Graphs. Comparison of dynamic solutions (time delay removed) and viscoelastic solution for case 116. Two graphs are shown. In both graphs, the x-axis has time in s, ranging from 0 to 0.12, and the y-axis has deflection in mil, ranging from –4 to 24. Twelve curves are shown on the graph, and each curve begins at 0 and has a parabolic shape. After reaching the maximum deflections, the curves decrease toward 0.

The first graph is labeled VE solution versus SAPSI for Case 116. The first line, which represents data for VE sensor 0, has a peak value of 21 mil at 0.025 s. It decreases to a deflection of approximately 0.04 mil at 0.097 s. The line for SAPSI sensor 0 has the same trend except that after the peak, it has slightly larger values for deflection. The next line represents data for VE sensor 1; it has a peak value of 14 mil at 0.025 s. It then decreases to a deflection of approximately 0.04 mil at 0.0.097 s. The line for SAPSI sensor 1 has the same trend except it is delayed by approximately 0.0001 s. The next line represents data for VE sensor 2; it has a peak value of 10 mil at 0.025 s. It decreases to a deflection of approximately 0.04 mil at 0.0.097 s. The line for SAPSI sensor 2 has the same trend except it is approximately 0.0001 s ahead of the VE sensor. The next line represents data for VE sensor 3; it has a peak value of 6 mil at 0.025 s. It then decreases to a deflection of approximately 0.04 mil at 0.0.097 s. The line for SAPSI sensor 3 has the same trend except it is approximately 0.0001 s ahead of the VE sensor. The next line represents data for VE sensor 4; it has a peak value of 4 mil at 0.025 s. It then decreases to a deflection of approximately 0.04 mil at 0.0.097 s. The line for SAPSI sensor 4 has the same trend except it is approximately 0.0001 s ahead of the VE sensor. The next line represents data for VE sensor 5; it has a peak value of 2.8 mil at 0.025 s. It then decreases to a deflection of approximately 0.04 mil at 0.0.097 s. The line for SAPSI sensor 3 has the same trend except it is approximately 0.0002 s ahead of the VE sensor.

The second graph is labeled VE solution VS. Lamda for Case 116. The first line represents data for VE sensor 0; it has a peak value of 21.6 mil at 0.025 s. It then decreases to a deflection of approximately 0.04 mil at 0.0.097 s. The line for Lamda sensor 0 has the same trend except it has a slightly smaller peak deflection and around 0.05 s, its deflection values become 0.12 mil smaller than the VE sensor. The second line represents data for VE sensor 1; it has a peak value of 14 mil at 0.025 s. It then decreases to a deflection of approximately 0.04 mil at 0.097 s. The line for Lamda sensor 1 has the same trend except it has a smaller peak deflection of 13.2 mil. The next line represents data for VE sensor 2; it has a peak value of 10 mil at 0.025 s. It then decreases to a deflection of approximately 0.04 mil at 0.097 s. The line for Lamda sensor 2 has the same trend except it has a smaller peak deflection of 9.6 mil. The next line represents data for VE sensor 3; it has a peak value of 6 mil at 0.025 s. It then decreases to a deflection of approximately 0.04 mil at 0.0.097 s. The line for Lamda sensor 3 has the same trend except it has a smaller peak deflection of 5.6 mil. The next line represents data for VE sensor 4; it has a peak value of 4.4 mil at 0.025 s. It then decreases to a deflection of approximately 0.04 mil at 0.097 s. The line for Lamda sensor 4 has the same trend except it has a smaller peak deflection of 4 mil. The next line represents data for VE sensor 5; it has a peak value of 2.8 mil at 0.025 s. It then decreases to a deflection of approximately 0.04 mil at 0.097 s. The line for Lamda sensor 5 has the same trend except it has a slightly smaller peak deflection.

Figure 42. Graphs. Comparison of dynamic solutions (time delay removed) and viscoelastic solution for case 120. Two graphs are shown. The x-axis is the time in s, ranging from 0to 0.12, and the y-axis is the deflection in mil, ranging from –4 to 28 mil. Twelve lines are shown on each graph; the lines all begin at 0 and have a parabolic shape. After reaching the maximum deflection, the lines then decrease toward 0.

The first graph is labeled VE solution VS. SAPSI for Case 120. The first line represents data for VE sensor 0; it has a peak value of 22.4 mil at 0.025 s. It then decreases to a deflection of approximately 0.04 mil at 0.095 s. The line for SAPSI sensor 0 has the same trend except that after the peak, it has slightly larger values for deflection. The second line represents data for VE sensor 1; it has a peak value of 14.8 mil at 0.025 s. It then decreases to a deflection of approximately 0.12 mil at 0.095 s. The line for SAPSI sensor 1 has the same trend except its peak value is 15.2 mil, and decreases at a slower rate than the VE sensor data. The next line represents data for VE sensor 2; it has a peak value of 10 mil at 0.025 s. It then decreases to a deflection of approximately 0 at 0.095 s. The line for SAPSI sensor 2 has the same trend except it decreases at a slightly slower rate than the VE sensor data. The next line represents data for VE sensor 3; it has a peak value of 6 mil at 0.025 s. It then decreases to a deflection of approximately 0 at 0.095 s. The line for SAPSI sensor 3 has the same trend except it occurs approximately 0.0001 s ahead of the VE sensor. The next line represents data for VE sensor 4; it has a peak value of 4.4 mil at 0.025 s. It then decreases to a deflection of approximately 0 at 0.095 s. The line for SAPSI sensor 4 has the same trend except it is approximately 0.0001 s ahead of the VE sensor. The next line represents data for VE sensor 5; it has a peak value of 2.8 mil at 0.025 s. It then decreases to a deflection of approximately 0 at 0.095 s. The line for SAPSI sensor 5 has the same trend except it is approximately 0.0001 s ahead of the VE sensor.

The second graph is labeled VE solution VS Lamda for case 120. The first line represents data for VE sensor 0; it has a peak value of 22.4 mil at 0.025 s. It then decreases to a deflection of approximately 0.04 mil at 0.1 s. The line for Lamda sensor 0 has the same trend except it has a slightly larger peak deflection, is delayed very minutely, and at around 0.05 s, its deflection values become 0.12 mil smaller than the VE sensor. The second line represents data for VE sensor 1; it has a peak value of 14.4 mil at 0.025 s. It then decreases to a deflection of approximately 0.04 mil at 0.0.097 s. The line for Lamda sensor 1 has the same trend except it has a smaller peak deflection of 13.6 mil and its deflection increases slower before the peak and decreases faster after the peak. The next line represents data for VE sensor 2; it has a peak value of 10 mil at 0.025 s. It then decreases to a deflection of approximately 0 at 0.1 s. The line for Lamda sensor 2 has the same trend except it is has a smaller peak deflection of 9.6 mil. The next line represents data for VE sensor 3; it has a peak value of 6 mil at 0.025 s. It then decreases to a deflection of approximately 0 at 0.1 s. The line for Lamda sensor 3 has the same trend except it has a smaller peak deflection of 5.2 mil. The next line represents data for VE sensor 4; it has a peak value of 4.2 mil at 0.025 s. It then decreases to a deflection of approximately 0 at 0.1 s. The line for Lamda sensor 4 has the same trend except it has a smaller peak deflection of 4 mil. The next line represents data for VE sensor 5; it has a peak value of 3.2 mil at 0.025 s. It then decreases to a deflection of approximately 0 at 0.1 s. The line for Lamda sensor 5 has the same trend except it has a slightly smaller peak deflection of 2.8 mil.

Figure 43. Diagram. Schematic of temperature profile. This diagram shows the asphalt concrete (AC) layer represented as a rectangle. To the right of the rectangle, six temperature profile curves are shown for the AC layer. The first curve is a linear line with a decreasing slope, labeled 1. To the right of the rectangle are three steps that extend up to the left. The top step is at the top of the AC layer, and the ground level of the AC layer is at bottom of the first step. The right-to-left three-step temperature profile is labeled 2. The third temperature profile is a concave curve, which begins at the corner of the top step and extends to the bottom of the first step (at floor level). To the right of this set of steps is another set that is identical to the first set extending up to the right.

Figure 44. Graphs. Comparison of response calculated using (T-profile LAVA) LAVAP and original LAVA. Three graphs are shown. Each of them has the time in s on the x‑axis and viscoelastic deflection in inches on the y-axis. Sixteen lines are shown on each graph representing the T-profile LAVA (LAVAP) and LAVA for sensors 1 through 8. The lines for LAVA and T-profile LAVA are exactly the same for each sensor. All of the lines begin at 0deflection and 0 time and increase to peak maximum value before decreasing toward 0until the graph ends at 0.035 s.

The graph at top left is labeled Temperature = 32 °F, 32 °F, 32 °F. The maximum deflection occurs at 0.0105 s for each line. The peak deflection is 0.0115 inches for sensor 1, 0.0105 inches for sensor 2, 0.01 inches for sensor 3, 0.008 for sensor 4, 0.007 for sensor 5, 0.006for sensor 6, 0.0045 for sensor 7, and 0.003 for sensor 8. All of the lines return to approximately 0 viscoelastic deflection at 0.035 s.

The second graph at top right is labeled Temperature = 86 °F, 86 °F, 86 °F. The peak in deflection occurs at 0.015 s for each line. The peak deflection occurs at 0.017 inches for sensor1 and then decreases to 0.002 inches. The peak deflection for sensor 2 occurs at 0.013inches and then decreases to 0.001 inches. The peak deflection for sensor 3 occurs at 0.012 inches and then decreases to 0.001 inches. The peak deflection for sensor 4 occurs at 0.01inches and then decreases to 0.0005 inches. The peak deflection for sensor 5 occurs at 0.008inches and then decreases to 0.0005 inches. The peak deflection for sensor 6 occurs at 0.006 inches and then decreases to 0.0001 inches. The peak deflection for sensor 7 occurs at 0.004 inches and then decreases to 0.0001 inches. The peak deflection for sensor 8 occurs at 0.003 inches and then decreases to 0.

The third graph on the bottom is labeled Temperature = 122 °F, 122 °F, 122 °F. The peak in deflection occurs at 0.015 s for each line, and all of the lines except for the one representing sensor 1 decrease to approximately 0 deflection. The peak deflection is 0.026 inches for sensor 1; the line then decreases to 0.004 inches. The peak deflection for sensor 2 occurs at 0.017 inches; for sensor 3, it occurs at 0.014 inches; for sensor 4, it occurs at 0.01 inches and then decreases to 0.0005 inches; for sensor 5, it occurs at 0.008 inches; for sensor 6, it occurs at 0.006 inches; for sensor 7, it occurs at 0.004 inches; and for sensor 8, it occurs at 0.003inches.

Figure 45. Graph. Comparison of responses calculated using (T-profile LAVA) LAVAP at temperature profile {104, 86, 68} °F and original LAVA at constant 104°F temperature. This graph has time in s on the x-axis and the viscoelastic deflection in inches on the y‑axis. Sixteen lines are shown on the graph representing the T-profile LAVA at the temperature profile {104, 86, 68} °F and LAVA at 104 °F for sensors 1 through 8. All of the lines begin at 0deflection and 0 time and increase to a maximum deflection at 0.015 s. After reaching this peak, the lines then decrease until they end at 0.035 s.

The first line represents LAVA for sensor 1; the line has a maximum deflection of 0.021 and then decreases to 0.003 inches. The next line represents the T-profile for sensor 1; the line has a maximum deflection of 0.017 and then decreases to 0.002 inches. The next line represents LAVA for sensor 2; the line has a maximum deflection of 0.016 and then decreases to 0.001inches. The next line represents the T-profile for sensor 2; the line has a maximum deflection of 0.014 and then decreases to 0.001 inches. The next line represents LAVA for sensor three; the line has a maximum deflection of 0.013 and then decreases to 0.0005 inches. The next line represents the T-profile for sensor 3; the line has a maximum deflection of 0.0135 and then decreases to 0.0005 inches. The next line represents LAVA for sensor 4; the line has a maximum deflection of 0.01, and then decreases to 0 deflection. The line representing the T-profile for sensor 4 follows exactly the same trend. The next line represents LAVA for sensor 5; the line has a maximum deflection of 0.008 and then decreases to 0 deflection. The line representing the T‑profile for sensor 5 follows exactly the same trend. The next line represents LAVA for sensor6; the line has a maximum deflection of 0.006 and then decreases to 0 deflection. The line representing the T-profile for sensor 6 follows exactly the same trend. The next line represents LAVA for sensor 7; the line has a maximum deflection of 0.004 and then decreases to 0deflection. The line representing the T-Profile for seven follows exactly the same trend. The next line represents LAVA for sensor 8; the line has a maximum deflection of 0.003 and then decreases to 0 deflection. The line representing the T-profile for sensor 8 follows exactly the same trend.

Figure 46. Graph. Comparison of responses calculated using (T-profile LAVA) LAVAP at temperature profile {104, 86, 68} °F and original LAVA at a constant temperature of 86 °F. This graph has time in s on x-axis and the viscoelastic deflection in inches on the y-axis. Sixteen lines are shown on the graph representing the T-profile LAVA at the temperature profile {104, 86, 68} °F and LAVA at 86 °F, for sensors 1 through 8. All of the lines begin at 0deflection and 0 time and increase to a maximum deflection occurring at 0.015 s. After reaching this peak, the lines then decrease until they end at 0.035 s.

The first line represents the T-profile for sensor 1; the line has a maximum deflection of 0.017and then decreases to 0.002 inches. The next line represents the LAVA for sensor 1; the line has a maximum deflection of 0.0165 and then decreases to 0.002 inches. The next line represents the T-profile for sensor 2; the line has a maximum deflection of 0.014 and then decreases to 0.001inches. The next line represents the LAVA for sensor 2; the line has a maximum deflection of 0.0135 and then decreases to 0.001 inches. The next line represents the T-profile for sensor 3; the line has a maximum deflection of 0.0125 and then decreases to 0.0005inches. The next line represents the LAVA for sensor 3; the line has a maximum deflection of 0.4 and then decreases to 0.0005 inches. The next line represents the T-profile for sensor 4; the line has a maximum deflection of 0.01 and then decreases to 0 deflection. The line representing the LAVA for sensor 4 follows exactly the same trend. The next line represents T‑profile for sensor 5; the line has a maximum deflection of 0.008 and then decreases to 0deflection. The line representing the LAVA for sensor 5 follows exactly the same trend. The next line represents the T-profile for sensor 6; the line has a maximum deflection of 0.006 and then decreases to 0 deflection. The line representing the LAVA for sensor 6 follows exactly the same trend. The next line represents the T-profile for sensor 7; the line has a maximum deflection of 0.004 and then decreases to 0deflection. The line representing the LAVA for sensor 7 follows exactly the same trend. The next line represents the T-profile for sensor 8; the line has a maximum deflection of 0.003 and then decreases to 0 deflection. The line representing the LAVA for sensor 8 follows exactly the same trend.

Figure 47. Graph. Comparison of responses calculated using (T-profile LAVA) LAVAP at temperature profile {104, 86, 68} °F and original LAVA at a constant temperature of 68 °F. This graph has time in s on x-axis and the viscoelastic deflection in inches the y-axis. Sixteen lines are shown representing the T-profile LAVA at temperature profile {104, 86, 68} °F and LAVA at 68 °F, for sensors 1 through 8. All of the lines begin at 0deflection and 0 time and end at 0.035 s, with a maximum deflection occurring at 0.015 s.

The first line represents the T-profile for sensor 1; the line has a maximum deflection of 0.017and then decreases to 0.002 inches. The next line represents the T-profile for sensor 2; the line has maximum deflection of 0.014 and then decreases to 0.001 inches. The next line represents the LAVA for sensor 1; this line has a maximum deflection just below 0.014 and then decreases to 0.001 inches. The next line represents the LAVA for sensor 2; the line has a maximum deflection at 0.013 and then decreases to 0.001 inches. The line that represents the T‑profile for sensor 3 also has a maximum deflection of 0.013 and then decreases to 0.0001inches. The next line represents the LAVA for sensor 3; the line has a maximum deflection of 0.011 and then decreases to 0.0005 inches. The next line represents the T-profile for sensor 4; the line has a maximum deflection of 0.01 and then decreases to 0 deflection. The line representing the LAVA for sensor 4 has a maximum deflection of 0.0095 inches and then decreases to 0 deflection. The next line represents the T-profile for sensor 5; the line has a maximum deflection of 0.008 and then decreases to 0 deflection. The line representing the LAVA for sensor 5 follows has a maximum deflection just below 0.008. The next line represents the T-profile for sensor 6; the line has a maximum deflection of 0.006 and then decreases to 0deflection. The line representing the LAVA for sensor 6 follows exactly the same trend. The next line represents the T-profile for sensor 7; the line has a maximum deflection of 0.0045 and then decreases to 0 deflection. The line representing the LAVA for sensor 7 follows exactly the same trend. The next line represents the T-profile for sensor 8; the line has a maximum deflection of 0.004 and then decreases to 0deflection. The line representing the LAVA for sensor 8 follows exactly the same trend.

Figure 48. Graph. Region of E(t) master curve (at 66.2 °F reference temperature) used by (T-profile LAVA) LAVAP for calculating response at temperature profile {104, 86, 68} °F. The x-axis is time in s at 66.2 °F reference temperature, and the y-axis is E parenthesis t end parenthesis in psi. Shown on the graph is a sigmoid curve representing the entire E parenthesis t end parenthesis that contains data points from different temperatures. The curve begins at 5 times 10⁶ psi at 10^-5 s and curves very slightly in a concave fashion until it reaches 1.1 times 10⁴ psi at 10⁴ s. The data point representing 68 °F ranges from 5 times 10⁶ psi at 10^-5 s to 2 times 10⁶ psi at 0.02 s, data representing 86 °F ranges from 4 times 10⁶ psi at 10^-4 s to 9 times 10⁵ psi at 0.3 s, and data representing 104 °F ranges from 5 times 10⁶ psi at 10^-5 s to 6 times 10⁵ psi at 2 s.

Figure 49. Graphs. Relaxation modulus and shift factor for master curves at a reference of temperature of 66 °F. Two graphs are shown. The left graph has reduced time in s on the x-axis and relaxation modulus, E parenthesis t end parenthesis, in psi on the y-axis. Two sigmoid curves are shown on the graph, each with a similar trend. The first curve represents terpolymer, and the second curve represents SBS 64-40. The two curves follow the same trend except the relaxation modulus for the terpolymer is larger. The curves has a concave and then convex shape. For the terpolymer, the curve beings at 10^-8 s and 3 times 10⁶ psi and ends at 10⁸ s and 4times 10³ psi. The curve for the SBS 64-40 begins at 2 times 10⁶ psi and ends at 3 times 10³ psi.

The graph on the right has temperature in °F on the x-axis and the shift factor, a times T, on the y-axis. Two curves are shown on the graph, each with a similar quadratic trend. The first curve represents terpolymer, and the second curve represents SBS 64-40. Both curves begin at 32 °F and end at 140 °F. The curve for terpolymer begins at about a shift factor of 550 and decreases to a shift factor just below 10³. The curve for SBS64-40 begins at a shift factor of 200 and decreases to a shift factor of 10^-3.

Figure 50. Graph. Comparison between LAVAP and ABAQUS at a temperature profile of {66, 86} °F (terpolymer). This graph has time in s on the x-axis and the viscoelastic deflection in inches times 10^-3 on the y-axis. The data on the graph represent the surface deflection measured at different radial distances for LAVAP and ABAQUS. The graph has a haversine distribution, and the radial distances that were measured using both methods were the following: 0, 7.99, 12.01, 17.99, 24.02, 35.98, 47.99, and 60 inches. For each of these distances, the results of LAVAP and ABAQUS were very similar. All of the curves begin at 0 deflection and 0 time and end at 0.035 s, with a maximum deflection occurring at about 0.017 s.

The first set of curves represents a radial distance of 0; the deflections for LAVA are slightly larger than those for ABAQUS. The maximum deflection is about 0.82 times 10^-3 inches, and the lines decrease to 0.15 times 10^-3 inches. The next set of curves represents a radial distance of 7.99. The maximum deflection is about 0.7times 10^-3 inches for LAVA and 0.68 times 10^-3 for ABAQUS, and the lines decrease to 0.007 times 10^-3 inches. The next set of curves represents a radial distance of 12.01; the deflections for LAVA are slightly larger than those for ABAQUS. The maximum deflection for LAVA is 0.6 times 10^-3 inches, and the maximum deflection for ABAQUS is 0.58 times 10^-3 inches. Both curves then decrease to 0.005 times 10^-3 inches. The next set of curves represents a radial distance of 17.99; the maximum deflection for LAVA is 0.5times 10^-3 inches, and the maximum deflection for ABAQUS is 0.48 times 10^-3 inches. Both lines then decrease to 0.1 times 10^-3 inches. The next set of curves represents a radial distance of 24.02; the maximum deflection for LAVA is 0.45 times 10^-3 inches, and the maximum deflection for ABAQUS is 0.42 times 10^-3 inches. Both lines then decrease to 0. The next set of curves represents a radial distance of 35.98; the maximum deflection for LAVA is 0.3 times 10^-3 inches, and the maximum deflection for ABAQUS is 0.28 10^-3 inches. Both lines then decrease to 0. The next set of curves represents a radial distance of 47.99; the maximum deflection for LAVA is 0.2times 10^-3 inches, and the maximum deflection for ABAQUS is 0.18 times 10^-3 inches. Both lines then decrease to 0. The last set of curves represents a radial distance of 60; the maximum deflection for LAVA is 0.19 times 10^-3 inches, and the maximum deflection for ABAQUS is 0.48 times 10^-3 inches. Both curves then decrease to 0inches at 0.035 s.

Figure 51. Graph. Comparison between LAVAP and ABAQUS at a temperature profile of {66, 86} °F (SBS 64-40). This graph has time in s on the x-axis and the viscoelastic deflection in inches times 10^-3 on the y-axis. The data on the graph represent the surface deflection measured at different radial distances for LAVA and ABAQUS. The graph has a haversine distribution, and the radial distances that were measured using both methods were the following: 0, 7.99, 12.01, 17.99, 24.02, 35.98, 47.99, and 60 inches. For each of these distances, the results of LAVA and ABAQUS were very similar. All of the curves begin at 0 deflection and 0 time and end at 0.035 s, with a maximum deflection occurring at about 0.017 s.

The first set of curves represents a radial distance of 0; the deflections for LAVA are very slightly larger than those for ABAQUS. The maximum deflection is about 1.02 times 10^-3 inches for LAVA and 10^-3 inches for ABAQUS. The curves then decrease to 0.18 times 10^-3 inches. The next set of curves represents a radial distance of 7.99. The maximum deflection is about 0.8 times 10^-3 inches for LAVA and 0.79 times 10^-3 for ABAQUS. The curves decrease to 10^-3 inches. The next set of curves represents a radial distance of 12.01; the maximum deflection for LAVA is 0.7times 10^-3 inches, and the maximum deflection for ABAQUS is 0.68 times 10^-3 inches. Both curves then decrease to 0.05 times 10³ inches. The next set of curves represents a radial distance of 17.99; the maximum deflection for LAVA is 0.5 times 10^-3 inches, and the maximum deflection for ABAQUS is 0.48 times 10^-3 inches. Both curves then decrease to 0.01 times 10^-3 inches. The next set of curves represents a radial distance of 24.02; the maximum deflection for LAVA is 0.42 times 10^-3 inches, and the maximum deflection for ABAQUS is 0.4 times 10^-3 inches. Both curves then decrease to 0. The next set of curves represents a radial distance of 35.98; the maximum deflection for LAVA is 0.25 times 10^-3 inches, and the maximum deflection for ABAQUS is 0.23 times 10^-3 inches. Both curves then decrease to 0. The next set of curves represents a radial distance of 48; the maximum deflection for LAVA is 0.18times 10^-3 inches, and the maximum deflection for ABAQUS is 0.2 times 10^-3 inches. Both curves then decrease to 0. The last set of curves represents a radial distance of 60; the maximum deflection for LAVA is 0.17 times 10^-3 inches and the maximum deflection for ABAQUS is 0.17times 10^-3 inches. Both curves then decrease to 0 inches.

Figure 52. Equation. Resilient modulus. M subscript R equals the quotient sigma subscript d divided by epsilon subscript r

Figure 53. Equation. Resilient modulus as a function of stress invariant. M subscript R equals k subscript 1 parenthesis theta end parenthesis raised to the k subscript 2 power.

Figure 54. Equation. Uzan’s nonlinearity model. M subscript R equals k subscript 1 parenthesis the quotient theta divided by P subscript a end quotient, end parenthesis raised to k subscript 2 power times parenthesis the quotient sigma subscript d divided by p subscript a end quotient, end parenthesis raised to k subscript 3 power.

Figure 55. Equation. Witczak and Uzan’s nonlinearity model. M subscript R equals k subscript 1 times parenthesis the quotient theta divided by P subscript a end quotient, end parenthesis raised to k subscript 2 power times parenthesis the quotient tau subscript oct divided by p subscript a end quotient end parenthesis raised to k subscript 3 power.

Figure 56. Equation. Generalized Uzan’s model. M subscript R equals k subscript 1 times P subscript a, times parenthesis, the quotient theta minus 3 times k subscript 6, divided by P subscript a end quotient end parenthesis raised to k subscript 2 power times parenthesis the quotient tau subscript oct divided by P subscript a, plus k subscript 7, end quotient, end parenthesis, raised to k subscript 3 power.

Figure 57. Equation. MEPDG model for resilient modulus. M subscript R equals k subscript 1 times P subscript a times parenthesis the quotient theta divided by P subscript a end quotient end parenthesis raised to k subscript 2 power, times parenthesis the quotient tau subscript oct divided by P subscript a end quotient, plus one end parenthesis raised to k subscript 3 power.

Figure 58. Equation. Elasticity constitutive equation. Sigma subscript ij equals the quotient v times E divided by parenthesis 1 plus v end parenthesis times parenthesis 1 minus 2 times v end parenthesis, end quotient, multiplied by epsilon subscript kk times delta subscript ij plus the quotient E divided by 1 plus v end quotient times epsilon subscript ij.

Figure 59. Equation. Nonlinear viscoelastic formulation for stress when relaxation modulus is a function of strain. Sigma as a function of t equals the integral from 0 to t with respect to tau, of the following: E parenthesis t minus tau comma epsilon end parenthesis times the derivative of epsilon as a function of tau with respect to tau.

Figure 60. Equation. Nonlinear viscoelastic formulation for strain. Epsilon as a function of t equals the integral from 0 to t with respect to tau of the following: D parenthesis t minus tau comma sigma end parenthesis times the derivative of sigma as a function of tau with respect to tau.

Figure 61. Equation. Nonlinear creep compliance formulation. D as a function of t and sigma equals g as a function of parenthesis sigma end parenthesis, times D subscript t as a function of t.

Figure 62. Equation. Nonlinear relaxation modulus formulation. E parenthesis t, epsilon end parenthesis equals f parenthesis epsilon end parenthesis, times E subscript t parenthesis t end parenthesis.

Figure 63. Equation. Nonlinear viscoelastic formulation for stress when relaxation modulus is separated from strain dependence function. Sigma as a function of t equals the integral with respect to tau from 0 to t subscript R of the following: E subscript t parenthesis t subscript R minus tau end parenthesis multiplied by the derivative of f as a function of epsilon, parenthesis tau end parenthesis, with respect to epsilon multiplied by the derivative of epsilon parenthesis tau end parenthesis with respect to tau.

Figure 64. Equation. Nonlinear viscoelastic formulation for stress when relaxation modulus is separated from strain dependence function and when formulation is applied to a multilayered pavement structure. Sigma parenthesis t end parenthesis equals the integral with respect to tau from 0 to t subscript R of the following: E subscript t parenthesis x, y, z, t subscript R minus tau end parenthesis times the derivative of f parenthesis x, y, z, epsilon parenthesis tau end parenthesis, end parenthesis, with respect to epsilon multiplied by the derivative of epsilon parenthesis tau end parenthesis, with respect to tau.

Figure 65. Equation. Nonlinear viscoelastic formulation for deflection. The surface displacement: u superscript ve parenthesis t end parenthesis equals the integral with respect to tau, from tau equals 0 to t subscript R of the following: g parenthesis, sigma end parenthesis, u subscript H minus t, superscript e times parenthesis t subscript R minus tau comma sigma equals 1 end parenthesis times the differential of the function sigma of tau.

Figure 66. Equation. Nonlinear viscoelastic formulation. The function of stress: g parenthesis sigma end parenthesis equals the quotient u subscript H superscript e, parenthesis t subscript R comma sigma end parenthesis divided by u subscript H minus t, superscript e, parenthesis t subscript R comma sigma equals 1 end parenthesis.

Figure 67. Diagram. Flexible pavement cross section for nonlinear viscoelastic pavement analysis. This diagram shows a rectangle divided into three horizontal layers. The bottom layer is the semi-infinite layer, and it is labeled Subgrade (Linear elastic layer). The middle layer has a depth labeled h subscript base, equals 9.84 inches. This layer is labeled Granular Base (Stress dependent, Nonlinear). From the center of this layer a line extends two-thirds of the way to the right, and it is labeled r equals 3.5a. The center point is labeled Stress state at r equals 0, and the right point is labeled Stress state at r equals 3.5a. The top layer has a depth labeled h subscript AC, equals 5.9 inches. This layer is labeled Asphalt Concrete (Linear Viscoelastic). Centered over the top of this layer are four arrows, two arrows on either side pointing downward toward the surface. The distance from the centerline to the last arrow is labeled r equals a.

Figure 68. Graph. Variation of g(σ) with stress and E(t) of AC layer. The x-axis represents relaxation modulus E parenthesis t end parenthesis in psi, the y-axis represents g parenthesis sigma end parenthesis. Variation of g(σ) with E(t) of AC layer are shown by nine curves corresponding to stress levels of 5, 10, 15, 20, 25, 35, 50, 70, and 140 psi. For each stress level, the relaxation modulus varies from 2000 psi to 4 times 10⁶. All nine curves follow a similar pattern, g parenthesis sigma end parenthesis reduces with increase in relaxation modulus and reaches a minimum value at 10,000 psi and then increases and reaches maximum value at 4times 10⁶ psi. At stress level 5 psi, the g parenthesis sigma end parenthesis starts at 0.91, reaches a minimum of 0.88 and ends at 0.99. At stress level 140 psi, the g parenthesis sigma end parenthesis starts at 0.73, reaches a minimum of 0.6, and ends at 0.88. The variation for other stress levels is between 5 psi and 140 psi.

Figure 69. Equation. Generalized nonlinear viscoelastic formulation. The nonlinear viscoelastic response at coordinates (x,y,z) and time t: R superscript ve parenthesis x, y, z, t end parenthesis equals the integral with respect to tau from 0 to t subscript R of the following: R subscript H, superscript e, parenthesis x, y, z , I parenthesis tau end parenthesis, t subscript R minus tau end parenthesis, times the derivative of I as a function of tau with respect to tau.

Figure 70. Equation. Generalized nonlinear viscoelastic formulation for deflection. The vertical deflection at time t and location (z, r): u subscript vertical superscript ve parenthesis z comma r comma t end parenthesis equals the integral with respect to tau from tau equals 0 to t subscript r of the following: u subscript H minus vertical superscript e parenthesis z comma r comma sigma parenthesis tau end parenthesis comma t subscript R minus tau end parenthesis multiplied by the derivative of sigma as a function of tau, with respect to tau.

Figure 71. Equation. Discretized nonlinear formulation. The vertical deflection at time t subscript i and location (z, r): u subscript vertical superscript ve parenthesis z comma r comma t subscript I end parenthesis equals the summation with the lower bound j equals 1 and the upper bound N of the following: u subscript H minus vertical superscript e parenthesis z comma r comma t subscript Ri minus tau subscript j, end parenthesis, end parenthesis multiplied by delta sigma parenthesis tau subscript j end parenthesis.

Figure 72. Equation. Resilient modulus. E subscript base is equal to M subscript R, which is also equal to k subscript 1 parenthesis the quotient theta divided by P subscript a end quotient, end parenthesis, raised to k subscript 2 power times parenthesis the quotient tau subscript oct divided by P subscript a, end quotient, plus 1 end parenthesis, raised to k subscript 3 power.

Figure 73. Graph. Relaxation moduli of mixes used in LAVAN validation. This graph has reduced time at 66 °F in s on the x-axis and relaxation modulus in psi on the y-axis. Two sigmoid curves are displayed on the graph, the first representing control 70-22 and the second representing crumb rubber terminal blend (CRTB). The curves follow a similar trend with the control 70-22 always larger than CRTB. The curve for control 70-22 begins at 5 times 10⁶ psi and 10^-8 s. The curve is slightly concave and then slightly convex until it reaches 7 times 10³ psi at 10⁸ s. The curve for CRTB begins at 2.5 times 10⁶ psi and times 10^-8 s and follows the same trend. It ends at 2 times 10³ psi at 10⁸ s.

Figure 74. Equation. ABAQUS Jacobian formulation. The Jacobian matrix reads: J subscript ijkl equals the partial derivative of sigma subscript ij with respect to epsilon subscript kl.

Figure 75. Equation. ABAQUS stress update formulation. The updated stress: sigma subscript ij superscript n plus 1, equals sigma subscript ij superscript n plus J subscript ijkl times delta times epsilon subscript kl superscript n plus 1.

Figure 76. Graphs. Surface deflection comparison of ABAQUS and LAVAN for the control mix. Two graphs are shown. Both graphs have time in ms on the x-axis and surface deflection in mil on the y-axis. On each graph are eight curves labeled AS1 through AS8 that represent the surface deflection for sensors 1 through 8 using ABAQUS, and eight curves labeled LS1 through LS8 that represent the surface deflection for sensors 1 through 8 using LAVAN. All of the curves have a haversine shape that begins at 0 and end at 35 s, with the peak deflection occurring at 17 ms.

The first graph, labeled LAVA-nonlin, uses stress at r equals 0. The curve with the highest peak is for sensors AS1 and LS1, with a peak deflection 33 mil. The next curve represents sensors AS2 and LS2, with the peak at 30 mil. The next curve represents sensor AS3, with the peak at 28mil, and the curve for sensor LS3 is just below it with the peak at 27.8 mil. The curve for sensor AS4 has a peak at 25.2mil, and the peak of the LS4 sensor curve is at 24.4 mil. The curve for sensor AS5 has a peak deflection at 22.4 mil, and the peak for the LS5 sensor curve is at 20.8mil. Next is the curve for sensor AS6, which has a peak at 18.6 mil, and the LS6 sensor peak is at 15.8 mil. Next is the curve for sensor AS7, which has a peak at 13.4 mil, and the LS7 sensor peak is at 11.6 mil. Finally, the peak deflection for the curve for sensor AS8 is at 10mil and for sensor LS8 at 8mil.

The second graph, labeled LAVA-nonlin, uses stress at r equals 3.5a. The curve with the highest peak is for sensors AS1 and LS1, with a peak deflection at 33 mil. The next curve represents sensors AS2 and LS2 with the peak at 30 mil. The next curve represents sensors AS3 and LS3 with the peak at 28 mil. The curve for sensor AS4 has a peak at 25. 6 mil, and the peak for the LS4 sensor curve is at 25.2 mil. The curve for sensor AS5 has a peak deflection at 22.6 mil, and the peak for the LS5 sensor curve is at 22 mil. Next is the curve for sensor AS6, with its peak at 17.2 mil, and the peak for the LS6 sensor curve is at 15.8 mil. Next is the curve for sensor AS7, with its peak at 13.2 mil, and the peak for the LS7 sensor curve is at 11.6mil. Finally, the peak deflection for the curve for sensor AS8 is at 9.8 mil and for sensor LS8 at 8.6 mil.

Figure 77. Graphs. Surface deflection comparison of ABAQUS and LAVA for the CRTB mix. Two graphs are shown. Both graphs have time in ms on the x-axis and surface deflection in mil on the y-axis. On each graph are eight curves labeled AS1 through AS8 that represent the surface deflection for sensors 1 through 8 using ABAQUS, and eight curves that represent LS1 through LS8 that represent the surface deflection for sensors 1 through 8 using LAVAN. All of the curves have a haversine shape that begins at 0 and end at 35 s, with the peak deflection occurring at 17 ms.

The first graph, labeled LAVA-nonlin, uses stress at r equals 0. The curve with the highest peak is for sensor LS1 at 44 mil and then sensor AS1, with a peak deflection of 43.2 mil. The next curve represents sensors AS2 and LS2 with the peak at 38 mil. The next curve represents AS3 and LS3 with the peak at 33.6 mil. The line for AS4 has a peak at 30 mil, and the peak for the LS4 curve is at 28.8 mil. The curve for AS5 has a peak deflection at 24.6 mil, and the peak for the LS5 curve is at 24 mil. Next is the curve for AS6 with a peak at 17.6 mil, and the peak for LS6, with its peak is at 16mil. Next is the curve for AS7, which has a peak at 12 mil, and the LS7 peak curve is at 10 mil. Finally, the peak deflection for the curve for sensor AS8 is at 8.4mil and for sensor LS8 at 7.6 mil.

The second graph, labeled LAVA-nonlin, uses stress at r equals 3.5a. The curve with the highest peak is for sensor LS1 at 44.6 mil and then sensor AS1 with a peak deflection 43.2 mil. The next curve represents sensor LS2 with a peak of 40 mil and sensor AS2 with the peak at 38 mil. The next curve represents sensor LS3 with a peak of 36 mil and sensor AS3 with the peak at 35.2 mil. The curve for sensors AS4 and LS4 has a peak at 30 mil. The curve for sensor AS5 has a peak deflection at 24.6 mil, and the peak for sensor LS5 is at 24.4 mil. Next is the curve for sensor AS6 with a peak at 17.2, and the LS6 sensor peak at 16 mil. Next is the curve for sensor AS7 with its peak at 12 mil and the LS7 sensor peak at 10 mil. Finally, the peak deflection for the sensor AS8 is at 8.4 mil and for sensor LS8 at 7.6 mil.

Figure 78. Equation. Error in peak deflection. PE subscript peak equals the quotient the absolute value of delta subscript ABAQUS superscript peak minus delta subscript LAVAN superscript peak end absolute value, divided by delta subscript ABAQUS superscript peak, end quotient, multiplied by 100.

Figure 79. Equation. Average error in normalized deflection history. PE subscript avg equals the quotient one divided by N end quotient times the summation, with the lower bound i equals 1and the upper bound N, of the following: the quotient of the absolute value of delta subscript ABAQUS times parenthesis t subscript i end parenthesis divided by delta subscript ABAQUS superscript peak end quotient minus the quotient delta subscript LAVAN times parenthesis t subscript i end parenthesis divided by delta subscript LAVAN superscript peak, end quotient, end absolute value, all divided by the quotient delta subscript ABAQUS times parenthesis t subscript i end parenthesis divided by delta subscript ABAQUS superscript peak, end quotient, all multiplied by 100.

Figure 80. Graph. Percent error (PE_peak) calculated using the peaks of the deflections for LAVAN-ABAQUS comparison (control mix). This bar graph has bars representing the data for the control mix using stress state at r equals 0 and r equals 3.5a. The x-axis represents sensors numbered S1 through S8, and the y-axis is the percent error. At S1, the percent error is 1 for r equals 0, and 1.5 for r equals 3.5a. At S2, the percent error is 2 for r equals 0 and 0 for r equals 3.5a. At S3, the percent error is 3 for r equals 0 and 1 for r equals 3.5a. At S4, the percent error is 4 for r equals 0 and 2.5 for r equals 3.5a. At S5, the percent error is 6 for r equals 0 and 3.5 for r equals 3.5a. At S6, the percent error is 9 for r equals 0 and 7.5 for r equals 3.5a. At S7, the percent error is 13 for r equals 0 and 11.5 for r equals 3.5a. At S8, the percent error is 16.5 for r equals 0 and 14.5 for r equals 3.5a.

Figure 81. Graph. Average percent error (PE_avg) calculated using the entire time history for LAVAN-ABAQUS comparison (control mix). This bar graph has bars representing the data for the control mix using stress state at r equals 0 and r equals 3.5a. The x-axis represents sensors numbered S1 through S8, and the y-axis is the average percent error. At S1, the percent error is 1.5 for r equals 0 and 2.1 for r equals 3.5a. At S2, the percent error is 1.4 for requals0 and 1.9for r equals 3.5a. At S3, the percent error is 1.4 for r equals 0 and 1.6 for r equals 3.5a. At S4, the percent error is 1.3 for r equals 0 and 1.4 for r equals 3.5a. At S5, the percent error is 1.4 for r equals 0 and 1.3 for r equals 3.5a. At S6, the percent error is 1.8 for r equals 0 and 1.5 for r equals 3.5a. At S7, the percent error is 2.2 for r equals 0 and 1.9 for r equals 3.5a. At S8, the percent error is 2.3 for r equals 0 and 2.1 for r equals 3.5a.

Figure 82. Graph. Percent error (PE_peak) calculated using the peaks of the deflections for LAVAN-ABAQUS comparison (CRTB mix). This bar graph has bars representing the data for the crumb rubber terminal blend (CRTB) mix using stress state at r equals 0, and the CRTB mix r equals 3.5a. The x-axis represents sensors numbered S1 through S8, and the y-axis is the average percent error. At S1, the percent error is 3 for r equals 0 and 2 for r equals 6.5a. At S2, the percent error is 1 for r equals 0 and 4 for r equals 3.5a. At S3, the percent error is 1 for r equals 0 and 3 for requals3.5a. At S4, the percent error is 3.5 for r equals 0 and 0 for r equals 3.5a. At S5, the percent error is 7for r equals 0 and 3 for r equals 3.5a. At S6, the percent error is 12.5 for r equals 0 and 9 for r equals 3.5a. At S7, the percent error is 16.5 for r equals 0 and 14for r equals 3.5a. At S8, the percent error is 18 for r equals 0 and 19for r equals 3.5a.

Figure 83. Graph. Average percent error (PE_avg) calculated using the entire time history for LAVAN-ABAQUS comparison (CRTB mix). This bar graph has bars representing the data for the crumb rubber terminal blend (CRTB) mix r equals 0 and the CRTB mix r equals 3.5a. The x‑axis represents sensors numbered S1 through S8, and the y-axis is the average percent error. At S1, the percent error is 1.5 for r equals 0 and 2 for r equals 3.5a. At S2, the percent error is 1.4for r equals 0 and 1.8 for r equals 3.5a. At S3, the percent error is 1.4 for r equals 0 and 1.6for r equals 3.5a. At S4, the percent error is 1.5 for r equals 0 and 1.4 for r equals 3.5a. At S5, the percent error is 2 for r equals 0 and 1.5 for r equals 3.5a. At S6, the percent error is 2.6 for r equals 0 and 2 for r equals 3.5a. At S7, the percent error is 2.5 for r equals 0, and 2.2 for r equals 3.5a. At S8, the percent error is 1.5 for r equals 0 and 1.8 for requals3.5a.

Figure 84. Equation. Sigmoid form of relaxation modulus curve. Log of E parenthesis t end parenthesis equals c subscript 1 plus the quotient C subscript 2 divided by 1 plus the exponential function of parenthesis negative c subscript 3 minus c subscript 4 times the log of t subscript R, end parenthesis, end parenthesis, end quotient.

Figure 85. Equation. Optimization model. E subscript r equals the summation, with the lower bound k equals 1 and the upper bound m, of 100, multiplied by the summation, with the lower bound i,o equals 1 and the upper bound n, of the following: the quotient the absolute value of parenthesis d subscript i superscript k minus d subscript o superscript k end parenthesis, end absolute value, divided by d subscript i superscript k, end quotient.

Figure 86. Equation. Average error in backcalculated moduli of base and subgrade layers. Xi subscript unbound equals the absolute value of parenthesis the quotient E subscript act minus E subscript bc, divided by E lower subscript act, end parenthesis, end quotient, end absolute value, multiplied by 100.

Figure 87. Equation. Error in backcalculated relaxation moduli at different reduced times. Xi subscript AC, parenthesis t, subscript i, end parenthesis equals the quotient E subscript act, parenthesis t, subscript i, end parenthesis, minus E subscript bc, parenthesis t, subscript i, end parenthesis, divided by E subscript act, parenthesis t, subscript i, end parenthesis, end quotient, multiplied by 100.

Figure 88. Equation. Average error in backcalculated relaxation moduli. Xi subscript AC, superscript avg equals the quotient 1 divided by n, end quotient, multiplied by parenthesis the summation, with the lower bound i equals 1 and the upper bound n of the following: the absolute value of Xi subscript AC, parenthesis t, subscript i, end parenthesis end absolute value, end parenthesis.

Figure 89. Graph. Error in unbound layer modulus in optimal number of sensor analysis. This graph has sensor range on the x-axis and error percent on the y-axis. For each sensor range, there is a bar representing layer 2 and a bar representing layer 3. When the range equals 1, layer 2 has an error percent of 12.26, and layer 3 has an error percent of 6.15. When the range is 1 to 2, layer 2 has an error percent of 4.55, and layer 3 has an error percent of 9.91. When the range is 1to 3, layer 2 has an error percent of 9.12, and layer 3 has an error percent of 7.7. When the range is 1 to 4, layer 2 has an error percent of 3.32, and layer 3 has an error percent of 7.7. When the range is 1 to 5, layer 2 has an error percent of 3.32, and layer 3 has an error percent of 3.94. When the range is 1 to 7, layer 2 has an error percent of 1.33, and layer 3 has an error percent of 0.24. When the range is 1 to 8, layer 2 has an error percent of 0.17, and layer 3 has an error percent of 0.41. Finally, when the range is 1 to 9, layer 2 has an error percent of 0.18, and
layer 3 has an error percent of 0.41.

Figure 90. Graph. Backcalculated and actual E(t) master curve at the reference temperature of 66°F using FWD data from only sensor 1. This graph has time in s on the x‑axis and E parenthesis t end parenthesis in psi on the y-axis. The graph has two sigmoid curves, one representing the actual E parenthesis t end parenthesis and the other representing backcalculated E parenthesis t end parenthesis. The curve for actual E parenthesis t end parenthesis begins at 3.5 times 10⁶ psi at 10^-7 s and ends at 1.05 times 10³ psi at 10⁶ s. The two curves follow each other almost exactly starting horizontally and then decreasing until about 10⁴ s, then the curve for backcalculated drops below the actual line and ends at 10³ psi.

Figure 91. Graph. Variation of error when using FWD data from only sensor 1. This graph has time in s on the x-axis and percent error on the y-axis. The graph has a curve that has two colors shown on it; blue hatching representing the entire backcalculated E parenthesis t end parenthesis and green circles representing E parenthesis t end parenthesis used in back calculation. The line begins at 0 at 10^-7 s and then decreases in a concave manner to –7-percent error at 10 s. Then the line increases to a percent error of 23 at 10⁷ s. The first part of the curve is blue hatched until 10^-5 s, and transitions to green circles, and then remains green circles until 10⁵s when it changes back to blue hatched.

Figure 92. Graph. Error in unbound layer modulus using FWD data from only farther sensors. This graph has the sensor range on the x-axis and error percent on the y-axis. For each sensor range, there is a bar representing layer 2 and a bar representing layer 3. When the range is 5 to 9, layer 2 has an error percent of 1.55, and layer 3 has an error percent of 0.27. When the range is 6 to 9, layer 2 has an error percent of 6.95, and layer 3 has an error percent of 0.28. When the range is 7 to 9, layer 2 has an error percent of 4.55, and layer 3 has an error percent of 1.43. When the range is 8 to 9, layer 2 has an error percent of 1.0, and layer 3 has an error percent of 0.41. When the range is 9, layer 2 has an error percent of 8.96, and layer 3 has an error percent of 0.41.

Figure 93. Graph. Variation of error in backcalculated unbound layer moduli when FWD data run at different sets of pavement temperatures are used. This graph has the temperature range in °F on the x-axis and error percent on the y-axis. For each sensor range, there is a bar representing layer 2, and a bar representing layer 3. When the temperature range is from 32 to 50, layer 2 has an error percent of 0.12, and layer 3 has an error percent of 0.06. When the range is 50 to 68, layer 2 has an error percent of 0.05, and layer 3 has an error percent of 0.02. When the range is 50 to 86, layer 2 has an error percent of 0.08, and layer 3 has an error percent of 0.03. When the range is 86 to 104, layer 2 has an error percent of 0.06, and layer 3 has an error percent of 0.02. When the range is 104 to 122, layer 2 has an error percent of 0.03, and
layer 3 has an error percent of 0.01. When the range is 122 to 140, layer 2 has an error percent of 0.02, and layer 3 has an error percent of 0.

Figure 94. Graph. Error in backcalculated E(t) curve in optimal backcalculation temperature set analysis minimizing percent error. This bar graph has the temperature range in °F on the x-axis and error percent on the y-axis. For the temperature range from 32 to 50, the percent error is 25.6. For the temperature range from 50 to 68, the percent error is 26.02. For the temperature range from 50 to 86, the percent error is 7.73. For the temperature range from 50 to 68 to 86, the percent error is 4.24. For the temperature range from 68 to 86 to 104, the percent error is 5.97. For the temperature range from 86 to 104, the percent error is 5.63. For the temperature range from 86 to 104 to 122, the percent error is 1.02. For the temperature range from 104 to 122, the percent error is 22.86. For the temperature range from 104 to 122 to 140, the percent error is 10.77. For the temperature range from 122 to 140, the percent error is 72.58percent. For the temperature range from 140 to 158, the percent error is16.09.

Figure 95. Graphs. Results for backcalculation at {50,86} °F temperature set: left side—only GA used, right side—GA+fminsearch used. Four graphs are shown. The graph on the top left is labeled backcalculated E parenthesis t end parenthesis curve using GA only. The graph has time in s on the x-axis and E parenthesis t end parenthesis in psi on the y-axis. Two sigmoid curves are shown on the graph; the first represents actual E parenthesis t end parenthesis, and the other represents background E parenthesis t end parenthesis. The line for backcalculated E parenthesis t end parenthesis begins at 4 times 10⁶ psi at 10^-8 s, and the line for actual E parenthesis t end parenthesis begins at 3 times 10⁶ psi at 10^-8 s. The two lines follow similar paths, bending concavely and then convexly until the backcalculated E parenthesis t end parenthesis reaches 1.05 times 10³ psi at 10⁸ s and actual E parenthesis t end parenthesis reaches 1.02 times 10³ psi at 10⁸ s.

The graph on the bottom left is labeled variation of error using GA only. The graph has time in s on the x-axis and error percent on the y-axis. The graph has a curve that has two colors shown on it; blue hatching representing the entire backcalculated E parenthesis t end parenthesis and green circles representing E parenthesis t end parenthesis used in back calculation. The curve begins at –17-percent error at 10^-8 s. It has a normal distribution with the peak of 23-percent error at 10 s. After reaching the peak the line then decreases, ending at –19-percent error at 10⁸ s. Almost the entire line is blue hatching except for the two areas on the ascending side of the peak, from –2- to 7-percent error, and again from 16 to 21-percent error.

The graph on top right is labeled backcalculated E parenthesis t end parenthesis curve using GA plus fminsearch. The graph has time in s on the x-axis and E parenthesis t end parenthesis in psi on the y-axis. Two curves are shown on the graph. The first represents actual E parenthesis t end parenthesis and the other represents backcalculated E parenthesis t end parenthesis. Both curves begin at 3 times 10⁶ psi at 10^-8 s, and then follow similar paths, bending concavely and then convexly until the backcalculated E parenthesis t end parenthesis reaches 2 times 10³ psi at 10⁸ s and actual E parenthesis t end parenthesis reaches 1.05 times 10³ psi at 10⁸ s.

The graph on the bottom right is labeled variation of error using GA plus fminsearch. The graph has time in s on the x-axis and error percent on the y-axis. The graph has a line that has two colors shown on it, blue hatching representing the entire backcalculated E parenthesis t end parenthesis and green circles representing E parenthesis t end parenthesis used in back calculation. The line on the graph begins at 0-percent error at 10^-8 s and continues horizontally until about 1 s when the line begins to decrease linearly until it ends at –33-percent error at 10⁸ s. The entire line is blue hatched except for the area from 10^-3 s to 10 s.

Figure 96. Graphs. Results for backcalculation at {86, 104} °F temperature set. Four graphs are shown. The graph on the top left is labeled backcalculated E parenthesis t end parenthesis curve using GA. The graph has time in s on the x-axis and E parenthesis t end parenthesis in psi on the y-axis. Two sigmoid curves are shown on the graph. The first curve represents actual E parenthesis t end parenthesis, and the other curve represents backcalculated E parenthesis t end parenthesis. The two curves begins at 3 times 10⁶ psi at 10⁸ s, the two lines then follow similar paths bending concavely then convexly until they reach 1.05 times 10³ psi at 10⁸ s. At this point, E parenthesis t end parenthesis backcalculated is slightly less than the actual results.

The graph on the bottom left is labeled variation of error using GA only. The graph has time in s on the x-axis and error percent on the y-axis. The graph has a curve that has two colors shown on it; blue hatching representing the entire backcalculated E parenthesis t end parenthesis and green circles representing E parenthesis t end parenthesis used in backcalculation. The line on the graph begins at 0-percent error at 10^-8 s, it increases slightly in a linear fashion until it reaches an error percent of 1 at 10^‑4 s. The curve then begins to decrease until it reaches a percent error of –17 at 10³s. After reaching this negative peak, the curve begins to increase linearly until it reaches a percent error of 13 at 10⁸ s. The entire curve is blue hatched except for an area on the decreasing portion of the line; from -6-percent error to the peak of –17-percent error, the curve is green circles.

The graph on the top right is labeled backcalculated E parenthesis t end parenthesis curve using GA plus fminsearch. The graph has time in s on the x-axis and E parenthesis t end parenthesis in psi on the y-axis. Two sigmoid curves are shown on the graph. The first represents actual E parenthesis t end parenthesis, and the other represents backcalculated E parenthesis t end parenthesis. Both curves begin at 3 times 10⁶ psi at 10^-8 s, and then follow similar paths bending concavely then convexly until the backcalculated E parenthesis t end parenthesis reaches 10³ psi at 10⁸ s and actual E parenthesis t end parenthesis reaches 1.05 times 10³ psi at 10⁸ s.

The graph on bottom right is labeled variation of error using GA plus fminsearch. The graph has time in s on the x-axis and error percent on the y-axis. The graph has a curve that has two colors shown on it, blue hatching representing the entire backcalculated E parenthesis t end parenthesis and green circles representing E parenthesis t end parenthesis used in backcalculation. The curve on the graph begins at –4-percent error at 10^-8 s, the curve then increases in a linear manner to a percent error of 1 at .1 s. The curve then continues to increase but only slightly until reaching a percent error of 2 at 10³ s. The curve then begins to increase linearly to a percent error of 20 at 10⁸ s. The entire curve is blue hatched except for the portion of the curve between 1 and 10³ s where the curve is green circles.

Figure 97. Graphs. Results for backcalculation at {86, 104, 122} °F temperature set. Four graphs are shown. The graph on the top left is labeled backcalculated E parenthesis t end parenthesis curve using GA only. The graph has time in s on the x-axis and E parenthesis t end parenthesis in psi on the y-axis. Two sigmoid curves are shown on the graph. The first represents actual E parenthesis t end parenthesis and the other represents backcalculated E parenthesis t end parenthesis. The two curves begin at 3 times 10⁶ psi at 10^-8 s. Both lines then follow similar paths bending concavely then convexly until the actual E parenthesis t end parenthesis reaches 1.05times 10³ psi at 10⁸ s and the E parenthesis t end parenthesis backcalculated reaches 10³ psi at 10⁸ s.

The graph on the bottom left is labeled variation of error using GA only. The graph has time in s on the x-axis and error percent on the y-axis. The graph contains a curve that has two colors shown on it, blue hatching representing the entire backcalculated E parenthesis t end parenthesis and green circles representing E parenthesis t end parenthesis used in backcalculation. The curve shown on the graph begins at a percent error of –4.5 at 10^-8 s, the curve then begins to decrease to a percent error of –5 at 10^-5 s. The curve then increases in a linear manner until it reaches an error percent of 39 at 10³ s. The curve then decreases to a percent error of 17 at 10⁸ s. The entire curve is shown as blue hatched except for the portion of the curve that is between 1 s and 10³ s; this region is shown as green circles.

The graph on the top right is labeled backcalculated E parenthesis t end parenthesis curve using GA plus fminsearch. The graph has time in s on the x-axis and E parenthesis t end parenthesis in psi on the y-axis. Two sigmoid curves are shown on the graph, the first represents Actual E parenthesis t end parenthesis and the other represents backcalculated E parenthesis t end parenthesis. Both curves begin at 3 times 10⁶ psi at 10^-8 s, and then follow similar paths bending concavely then convexly until the two curves reach 1.05 times 10³ psi at 10⁸ s.

The graph on the bottom right is labeled variation of error using GA plus fminsearch. The graph has time in s on the x-axis and error percent on the y-axis. The graph contains a line that has two colors shown on it; blue hatching representing the entire backcalculated E parenthesis t end parenthesis and green circles representing E parenthesis t end parenthesis used in backcalculation. The curve on the graph begins at –1.4-percent error at 10^-8 s and then increases to a percent error of –0.25 at .1 s. The curve then decreases to a percent error of –1.75 at 10³ s before increasing linearly to a percent error of 1.25 at 10⁸ s. The entire curve is shown as blue hatched except for the portion of the curve that is between 1 and 10³ s; this region is green circles.

Figure 98. Equation. Normalized error in deflection history. E subscript r equals the summation with the lower bound k equals 1 and the upper bound m, of 100, multiplied by the summation with the lower bound i,o equals 1 and the upper bound n, of the following: the quotient of the absolute value of parenthesis d subscript i superscript k minus d subscript o superscript k end parenthesis, end absolute value, divided by d subscript max superscript k, end quotient.

Figure 99. Equation. Constraints in optimization model. c subscript 1 plus c subscript 2 is less than or equal to s subscript 1 and c subscript 1 plus c subscript 2 is greater than or equal to s subscript 2.

Figure 100. Equation. Sigmoid variables in optimization model. Four equations are shown. The top line reads: var 1 equals c subscript 1. The next line reads: var 2 equals x, which also equals c subscript 1 plus c subscript 2 and x superscript 1 is less than or equal to x, which is less than or equal to x superscript u. The third line reads: var 3 equals c subscript 3, and the fourth line reads: var 4 equals c subscript 4.

Figure 101. Equation. Dynamic modulus in complex form. Dynamic modulus E star equals the real part of dynamic modulus E star plus i times the imaginary part of dynamic modulus E star.

Figure 102. Equation. Real component of dynamic modulus. The real part of dynamic modulus E star equals the absolute value dynamic modulus E star, end absolute value, multiplied by the cosine of phi, which also equals E subscript o plus the summation, with the lower bound I equals 1and the upper bound N of the following: E subscript i multiplied by the quotient of parenthesis 2times pi times f times rho subscript i end parenthesis squared, divided by 1 plus parenthesis 2times pi times f times rho subscript i end parenthesis squared, end parenthesis, end quotient.

Figure 103. Equation. Imaginary component of dynamic modulus. The imaginary part of dynamic modulus E star equals the absolute value of the dynamic modulus E star, end absolute value, multiplied by the sine of phi, which also equals the summation, with the lower bound iequals1 and the upper bound N of the following: E subscript i multiplied by the quotient of 2times pi times f times rho subscript i, divided by 1 plus parenthesis 2 times pi times f times rho subscript i end parenthesis squared, end parenthesis, end quotient.

Figure 104. Equation. Dynamic modulus and phase angle. Two equations are shown. The first equation states: the absolute value of the complex modulus E star equals the square root of the real part of the dynamic modulus E star squared plus the imaginary part of dynamic modulus E star squared, end square root. The second equation states: the phase angle phi equals the inverse tangent of the imaginary part of dynamic modulus E star divided by the real part of the E star.

Figure 105. Graph. Backcalculated |E*| master curve using FWD data at temperature set {50, 86} °F minimizing normalized error. This graph has reduced frequency at 66 °F, in inverse s on the x-axis and dynamic modulus in psi on the y-axis. Three sigmoid curves are shown on the graph. They represent the actual absolute value of E star, the backcalculated absolute value of E star at 50-86 °F, and the backcalculated absolute value of E star at 50-68-86 °F. All three curves follow a very similar path, all beginning at about 1.05 times 10³ psi at 10^-9 inverse s. At this point, the actual and backcalculated at 50-86 °F have the same values and are slightly larger than the backcalculated at 50-68-86 °F. The three curves then increase in a convex and then concave shape until they reach 10⁶ inverse s at 3 times 10⁶ psi. The curves then continue almost horizontally at 3 times 10⁶ psi until they reach 10⁹ inverse s. At this point, the dynamic modulus is slightly larger for the actual and backcalculated at 50-86 °F than for the backcalculated at 50-68-86 °F.

Figure 106. Graph. Backcalculated E(t) master curve using FWD data at temperature set {50-68-86} °F minimizing normalized error. This graph has reduced time at 66 °F, in inverse s on the x-axis and relaxation modulus in psi on the y-axis. Three sigmoid curves are shown on the graph. They represent the actual E parenthesis t end parenthesis, the backcalculated E parenthesis t end parenthesis at 50-86 °F, and the backcalculated E parenthesis t end parenthesis at 50-68-86 °F. The three curves follow a very similar path; they all begin at about 3times 10⁶ psi at 10^-8 inverse s. Here the actual and backcalculated E parenthesis t end parenthesis at 50-86 °F have the same value and are slightly larger than the backcalculated E parenthesis t end parenthesis at 50-68-86 °F. The three curves then decrease bending concave and then convex until they reach 1.05times 10³ psi at 10⁸ inverse s. At this point, the actual and backcalculated E parenthesis t end parenthesis at 50-86 °F have the same value for the relaxation modulus and are slightly larger than the backcalculated E parenthesis t end parenthesis at 50-68-86 °F.

Figure 107. Graph. Variation of ξ^avg_AC at different FWD temperature sets This bar graph has the temperature range in °F on the x-axis and E parenthesis t end parenthesis error percent on the y-axis. For the temperature range from 32 to 50, the percent error is 20.58percent. For the temperature range from 50 to 68, the percent error is 10.98 percent. For the temperature range from 50 to 86, the percent error is 3.28 percent. For the temperature range from 50 to 68 to 86, the percent error is 5.89 percent. For the temperature range from 68 to 86 to 104, the percent error is 1.01percent. For the temperature range from 86 to 104, the percent error is 21.5percent. For the temperature range from 86 to 104 to 122, the percent error is 0.81 percent. For the temperature range from 104 to 122, the percent error is 8.01 percent. For the temperature range 104-122-140, the percent error is 32.90 percent. For the temperature range from 122to 140, the percent error is 65.02 percent. For the temperature range from 140 to 158, the percent error is 71.02 percent.

Figure 108. Graph. Variation of ξ _unbound at different FWD temperature sets. This graph has temperature range in °F on the x-axis and error percent on the y-axis. For each sensor range, there is a bar representing layer 2 and a bar representing layer 3. When the temperature range is from 32 to 50, layer 2 has an error percent of 4.37, and layer 3 has an error percent of 1.42. When the range is 50 to 68, layer 2 has an error percent of 2.91, and layer 3 has an error percent of 1.20. When the range is 50 to 86, layer 2 has an error percent of 1.35, and layer 3 has an error percent of 0.41. When the range is 86 to 104, layer 2 has an error percent of 1.14, and layer 3 has an error percent of 1.46. When the range is 104 to 122, layer 2 has an error percent of 0.18, and layer 3 has an error percent of 0.41. When the range is 122 to140, layer 2 has an error percent of 0.17, and layer 3 has an error percent of 0.09.

Figure 109. Graphs. Backcalculation results obtained using modified sigmoid variables. Four graphs are shown. The graph on the top left is backcalculated E parenthesis t end parenthesis curve using GA (at temperature {50, 86} °F). The graph has time in s on the x-axis and E parenthesis t end parenthesis in psi on the y-axis. Two sigmoid curves are shown on the graph. The first represents actual E parenthesis t end parenthesis, and the other represents backcalculated E parenthesis t end parenthesis. The two curves begin at 3 times 10⁶ psi at 10^-8 s, and then both lines follow the same path, bending concavely then convexly until they reach 1.05times 10³ psi at 10⁸ s.

The graph on the bottom left is labeled variation of error E parenthesis t end parenthesis (result at temperature {50,86} °F). The graph has time in s on the x-axis and error percent on the y‑axis. The graph contains a curve that has two colors shown on it; blue hatching representing the entire backcalculated E parenthesis t end parenthesis and green circles representing E parenthesis t end parenthesis used in backcalculation. The curve starts at 0-percent error at 10^-8 s and begins to decrease slightly in linear fashion to an error percent of –1 at 10^-4 s. The curve then begins to increase to a peak percent error of 9.5 at 10³ s and then decreases until it reaches a percent error of –2.5 at 10⁸ s. The entire line is blue hatched except for a portion of the curve that extends from 0- to 8-percent error.

The graph on the top right is labeled backcalculated E parenthesis t end parenthesis curve using GA (at temperature {68,86,104} °F). The graph has time in s on the x-axis and E parenthesis t end parenthesis in psi on the y-axis. Two sigmoid curves are shown on the graph. The first represents actual E parenthesis t end parenthesis, and the other represents backcalculated E parenthesis t end parenthesis. Both curves begin at 3 times 10⁶ psi at 10^-8 s and then follow similar paths, bending concavely then convexly until they reach 1.05 times 10³ psi at 10⁸ s.

The graph on the bottom right is labeled variation of error (result at temperature {68,86,104} °F). The graph has time in s on the x-axis and error percent on the y-axis. The graph contains a curve that has two colors shown on it, blue hatching representing the entire backcalculated E parenthesis t end parenthesis and green circles representing E parenthesis t end parenthesis used in backcalculation. The line starts at 0.5-percent error at 10^-8 s and begins to decrease to a percent error of –2.5 at 1 s. The curve then increases to a percent error of ‑0.1 at 10⁵ s, before decreasing to a percent error of –0.75 at 10⁸ s. The entire curve is blue hatched except for an area that extends from 10^-2 to 10³ s; this area is green circles.

Figure 110. Graphs. Viscoelastic properties of field mix in optimal temperature analysis. Two graphs are shown. The graph at the top is labeled relaxation modulus at 66 ° F. The graph has reduced time in s on the x-axis and the relaxation modulus in psi on the y-axis. Nine sigmoid curves are shown on the graph that represent nine typical mixtures. All the curves follow a similar trend, bending concavely then convexly. All nine curves begin at reduced time of 10^-8 s and end at a reduced time of 10⁸ s. The uppermost curve represents control 70-22. It has an initial relaxation modulus of 3 times 10⁶ psi and final relaxation modulus of 1.05 times 10³ psi. The next curve represents SBS64-40. It has an initial relaxation modulus of 2 times 10⁶ psi and final relaxation modulus of 4 times 10³. The next curve represents air blown. It has an initial relaxation modulus of 3 times 10⁶ psi and final relaxation modulus of 2 times 10³ psi. The next curve represents PPA + Elvaloy. It has an initial relaxation modulus of 10⁶ psi and final relaxation modulus of 6 times 10³ psi. The next curve represents crumb rubber terminal blend (CRTB). It has an initial relaxation modulus of 3 times 10⁶ psi and final relaxation modulus of 2times 10³ psi. The next curve represents warm asphalt mix (WAM) foam. It has an initial relaxation modulus of 2 times 10⁶ psi and final relaxation modulus of 2 times 10³ psi. The next curve represents SBS LG. It has an initial relaxation modulus of 3 times 10⁶ psi and final relaxation modulus of 2 times 10³. The next curve represents terapolymer. It has an initial relaxation modulus of 2 times 10⁶ psi and final relaxation modulus of 5 times 10³ psi. The next curve represents PG64-22(12GTR). It has an initial relaxation modulus of 2 times 10⁶ psi and final relaxation modulus of 1.05 times 10³ psi.

The graph at the bottom is labeled time-temperature shift factor. The graph has temperature in °F on the x-axis and the shift factor, aT, on the y-axis. Nine curves are shown on the graph that represent the nine typical mixtures. All the curves follow a similar trend, with a linearly decreasing shift factor. All nine curves begin at a temperature of 32 °F and end at a temperature of 140 °F. The curves are all very close together and begin between the shift factors of 10² and 10³ and end between the shift factors 10^-4 and 10^-3. Beginning at a temperature of 32°F and going from the highest to lowest shift factor values are the curves: SBS64-40, SBS LG, control 70-22, CR-TB, PPA + Elvaloy, PG64-22(12GTR), and then air blown. All the curves decrease linearly, and at the end going from highest to lowest shift factor values are the lines: PG64‑22(12GTR), terapolymer, PPA + Elvaloy, SBS LG, air blown, CR-TB, WAM foam, and then SBS64-40.

Figure 111. Graphs. Variation of error calculated over three ranges of reduced time—top =
10^-5 to 1 s, middle= 10^-5 to 10² s, and bottom = 10^-5 to 10³ s. Six graphs are shown. For each, the temperature set in °F is shown on the x-axis, and the average percent error is on the y-axis. Nine sets of data are shown on each of the graphs that represent the nine typical mixtures. The top two graphs represent the average percent error when t, subscript i, ranges from 10^-5 s to 1 s. For a temperature range of {50-68} the average percent error is 1.2 for an air-blown mixture, 1for crumb rubber terminal blend (CR-TB), 6 for PG64-22(12GTR), 3 for PPA + Elvaloy, 1 for control 70-22, 1.3 for SBS LG, 2for SBS 64-40, 1.3 for terapolymer, and 0.5 for warm asphalt mix (WAM) foam. For a temperature range of {68-86}, the average percent error is 1 for an air-blown mixture, 0.9 for CR-TB, 4.5 for PG64-22(12GTR), 2 for PPA + Elvaloy, 1 for control 70‑22, 2for SBS LG, 8 for SBS 64-40, 2.5 for terapolymer, and 2.8 for WAM foam. For a temperature range of {86-104}, the average percent error is 3 for an air-blown mixture, 6 for CR‑TB, 6 for PG64-22(12GTR), 4.5for PPA + Elvaloy, 3 for control 70-22, 11.5 for SBS LG, 16 for SBS 64-40, 10.5 for terapolymer, and 7 for WAM foam. For a temperature range of {104-122}, the average percent error is 6 for an air-blown mixture, 16 for CR-TB, 22 for PG64‑22(12GTR), 5for PPA + Elvaloy, 5.2 for control 70-22, 13 for SBS LG, 26 for SBS 64‑40, 14 for terapolymer, and 10.5for WAM foam.

The middle two graphs represent the average percent error when t, subscript i, range from 10^-5 to 10² s. For a temperature range of {50-68}, the average percent error is 1.5 for an air-blown mixture, 2 for CR-TB, 7 for PG64-22(12GTR), 3 for PPA + Elvaloy, 2.5 for control 70-22, 8 for SBS LG, 8.5 for SBS 64-40, 8 for terapolymer, and 4 for WAM foam. For a temperature range of {68-86}, the average percent error is 7 for an air-blown mixture, 0.5 for CR-TB, 4.5 for PG64-22(12GTR), 5 for PPA + Elvaloy, 1 for control 70-22, 2for SBS LG, 15.5 for SBS 64-40, 1.5 for terapolymer, and 7.5 for WAM foam. For a temperature range of {86-104}, the average percent error is 4.5 for an air-blown mixture, 7 for CR-TB, 8 for PG64-22(12GTR), 7 for PPA + Elvaloy, 7 for control 70-22, 10.5 for SBS LG, 18for SBS 64-40, 13 for terapolymer, and 7 for WAM foam. For a temperature range of {104-122} the average percent error is 16 for an air-blown mixture, 18 for CR-TB, 23 for PG64-22(12GTR), 4.5 for PPA + Elvaloy, 4.5 for control 70-22, 9.5 for SBS LG, 23 for SBS 64-40, 12for terapolymer, and 9 for WAM foam.

The bottom two graphs represent the average percent error when t, subscript i, range from 10^-5 to 10³ s. For a temperature range of {50-68}, the average percent error is 2.5 for an air-blown mixture, 2 for CR-TB, 6.5 for PG64-22(12GTR), 5 for PPA + Elvaloy, 3.5 for control 70-22, 13for SBS LG, 13.5 for SBS 64-40, 12 for terapolymer, and 8.5for WAM foam. For a temperature range of {68-86}, the average percent error is 8.5 for an air-blown mixture, 0.5 for CR-TB, 4 for PG64-22(12GTR), 8.5 for PPA + Elvaloy, 1 for control 70-22, 4 for SBS LG, 21.5for SBS 64-40, 10.5 for terapolymer, and 12 for WAM foam. For a temperature range of {86-104}, the average percent error is 4 for an air-blown mixture, 7 for CR-TB, 11 for PG64‑22(12GTR), 9.5 for PPA + Elvaloy, 9 for control 70-22, 11 for SBS LG, 22for SBS 64-40, 16.5 for terapolymer, and 6.5 for WAM foam. For a temperature range of {104-122}, the average percent error is 17 for an air-blown mixture, 19 for CR-TB, 24.5 for PG64-22(12GTR), 4.5 for PPA + Elvaloy, 5 for control 70-22, 9.5 for SBS LG, 24 for SBS 64-40, 10.5for terapolymer, and 9 for WAM foam.

Figure 112. Graphs. Applied stress and resulting deflection basin for multiple pulse loading analysis. Two graphs are shown. The graph on left has time in s on the x-axis and stress in psi on the y-axis. The loading curve consists of series of eight haversine load pulse, ranging from 0 s to 45 s. The first four pulses each have a loading period of approximately 0.035 s followed by a rest period of 0.035 s. This is followed by next four haversine pulses, each approximately 15 s followed by 5 s of rest period. The first four pulses are zoomed in for more clarity.

The graph on the right has time in s on the x-axis and deflection in inches on the y-axis. The graph shows six curves, representing deflection histories obtained from sensors at radial distance 0, 13, 21, 35, 49, and 63 inches from the loading center. All six curves follow the same trend, similar to the loading stress. The curves start at time equal 0 s and end at 45 s. All six curves have eight haversine pulses. The first four pulses in the curve have an approximately 0.035 s pulse period and equal peaks. The next four pulses have an approximately 15 s pulse period and equal peaks. The first curve represents sensor 1; its peak value for the first four pulses is 0.064inches, and 0.085 inches for the following four pulses. For sensor 2, the peak values are 0.04 and 0.042 inches. From sensor 3, 4, 5, and 6, all eight pulses have equal peaks. For sensor 3, it is 0.03 inches, for sensor 4, it is 0.017 inches, for sensor 5, it is 0.01 inches, and for sensor 6, it is 0.009 inches.

Figure 113. Graph. Backcalculated E(t) and deflection histories using the multiple stress pulses. This graph shows backcalculated E(t) curve using fminsearch (using deflection history obtained from multiple stress pulses). The graph has time in s on the x-axis and E(t) in psi on the y-axis. Two sigmoid curves are shown on the graph. The first represents Actual E(t) and the second represents backcalculated E(t). The two curves begins at 9 times 10⁵psi at 10^-3 s, and the lines follow similar paths bending slight concavely then slight convexly until they reach 1.05times 10⁴ psi at 10² s. At this point, the E(t) backcalculated is slightly less than the actual results.

Figure 114. Graphs. Comparison of actual and backcalculated values in backcalculation using temperature profile. Three graphs are shown. The graph on the top left is labeled case1 deflection history. The graph has time in s on the x-axis and deflection times 10^-3 inches on the y-axis. The graph has curves that represent the measured and backcalculated values at sensors 1 through 9. All of the lines have a haversine shape beginning at 0deflection. The lines then increase, reaching a peak value at 0.015 s, and then decrease to approximately 0 at 0.035 s. For each sensor, the measured and backcalculated results are the same. The first line represents sensor 1; its peak value in deflection is fourteen times 10^-3 inches. For sensor 2, the peak value in deflection is 12 times 10^-3 inches. For sensor 3, it is 10^-3 inches. For sensor 4, it is 9 times 10^-3 inches. For sensor 5, it is 7 times 10^-3 inches. For sensor 6, it is 4 times 10^-3 inches. For sensor 7, it is 3 times 10^-3 inches. For sensor 8, it is 2times 10^-3 inches, and for sensor 1, it is 10^-3 inches. There is also a label on the graph: T subscript 1 superscript AC equals 68 °F, T subscript 2 superscript AC equals 59 °F, and T subscript 3 superscript AC equals 50 °F.

The graph on the top right is labeled case 2 deflection history. The graph has time in s on the x‑axis and the deflection times 10^-3 inches on the y-axis. The graph has curves that represent the measured and backcalculated values at sensors 1 through 9. All of the curves have a haversine shape beginning at 0 deflection. The curves then increase, reaching their peak value at 0.015 s, and then decrease to approximately 0 at 0.034 s. For each sensor, the measured and backcalculated results are the same. The first curve represents sensor 1; its peak value in deflection is 17 times 10^-3 inches. For sensor 2, the peak value in deflection is 14 times 10^-3 inches. For sensor 3, it is 12 times 10^-3 inches. For sensor 4, it is 9 times 10^-3 inches. For sensor 5, it is 7 times 10^-3 inches. For sensor 6, it is 4 times 10^-3 inches. For sensor 7 it is 2 times 10^-3 inches. For sensor 8, it is 1.5 times 10^-3 inches, and for sensor 1, it is 10^-3 inches. There is also a label on the graph: T subscript 1 superscript AC equals 86 °F, T subscript 2 superscript AC equals 77 °F, and T subscript 3 superscript AC equals 68 °F.

The graph at the bottom is labeled relaxation modulus. This graph has reduced time in s on the x‑axis and relaxation modulus E parenthesis t end parenthesis in psi on the y-axis. The graph contains three curves representing actual, backcalculated case 1 (error equals 5.2 percent), and backcalculated case 2 (error equals 4.4 percent). All three curves begin at a reduced time of 10^-8 s at 3 times 10⁶ psi; the lines then decrease in a concave then convex shape until reaching 10⁸ s, where case 2 has a relaxation modulus of 2 times 10³ psi and the actual and case 1 have a relaxation modulus of 1.05 times 10³ psi.

Figure 115. Graph. Error in backcalculated E(t) curve for a three-temperature profile. This graph has temperature set in °F on the x-axis and E parenthesis t end parenthesis error percent on the y-axis. For each sensor range, there is a bar representing reduced time 1 (10^-8 to 10^-8) and a bar representing reduced time 2 (10^-5 to 10⁵⁾. When the temperature set is 68-59-50, the percent error for reduced time 1 is 14.5 percent, and for reduced time 2 is 15 percent. When the temperature set is 86-77-68, the percent error for reduced time 1 is 5.5 percent, and for reduced time 2 is 6 percent. When the temperature set is 104-95-86, the percent error for reduced time 1is 28 percent, and for reduced time 2 is 30 percent. When the temperature set is 122-113-104, the percent error for reduced time 1 is 50percent, and for reduced time 2 is 62 percent.

Figure 116. Graphs. Backcalculated and measured deflection time histories for LTPP sections 10101 and 350801. Two graphs are shown. The top graph is labeled section 010101. The time in s is shown on the x-axis, and the deflection in inches times 10^-3 is on the y-axis. The curves shown represent the measured and backcalculated results for sensors 1 through 9. All of the curves follow the same trend, a parabolic shape with a double peak. The first smaller peak occurs at 0.015 s, and the large peak occurs at 0.0225 s. After the larger peak, all of the lines decrease in deflection. For sensor 1, the backcalculated results have a peak at 4.9 times 10^-3 inches and the measured results at 4.75 times 10^-3 inches. For sensor 2, the backcalculated results have a peak at 3.75 times 10^-3 inches and the measured at 3.6 times 10^-3 inches. For sensor 3, the backcalculated results have a peak at 3 times 10^-3 inches and the measured at 2.9 times 10^-3 inches. For sensor 4, the backcalculated results have a peak at 2.25 times 10^-3 inches and the measured at 2.1 times 10^-3 inches. For sensor 5, the backcalculated results have a peak at 1.7 times 10^-3 inches and the measured at 1.6 times 10^-3 inches. For sensor 6, the backcalculated results have a peak at 10^-3 inches and the measured at 0.8 times 10^-3 inches. For sensor 7, the backcalculated results have a peak at 0.55 times 10^-3 inches and the measured at 0.5 times 10^-3 inches. For sensor 8, the backcalculated results have a peak at 0.3 times 10^-3 inches and the measured at 0.3 times 10^-3 inches.

The bottom graph is labeled section 350801. The time in s is shown on the x-axis, and the deflection in inches times 10^-3 on the y-axis. The lines represent the measured and backcalculated results for sensors 1 through 9. All of the lines for the backcalculated results follow the same trend, a parabolic shape with a double peak. The first smaller peak occurs at 0.01 s, and the large peak occurs at 0.0175 s. The lines for the measured results all have a parabolic shape with the peak deflection occurring at 0.0175 s. After each of the lines reaches its peak value, there is a decrease in deflection. For sensor 1, the backcalculated results have a peak at 10^-3 inches and the measured at 9.5 times 10^-3 inches. For sensor 2, the backcalculated results have a peak at 7.9times 10^-3 inches and the measured at 7.5 times 10^-3 inches. For sensor 3, the backcalculated results have a peak at 6.5 times 10^-3 inches and the measured at 6 times 10^-3 inches. For sensor 4, the backcalculated results have a peak at 4.5 times 10^-3 inches and the measured at 4.4 times 10^-3 inches. For sensor 5, the backcalculated and measured results have a peak at 3.5 times 10^-3 inches. For sensor 6, the backcalculated and measured results have a peak at 2 times 10^-3 inches. For sensor 7, the backcalculated results have a peak at 1.75 times 10^-3 inches and the measured at 1.7 times 10^-3 inches. For sensor 8, the backcalculated results have a peak at 10^-3 inches and the measured at 0.9 times 10^-3 inches.

Figure 117. Equation. Creep compliance power law. D parenthesis t end parenthesis equals D subscript 1 multiplied by t raised to the n power.

Figure 118. Equation. Relaxation modulus and creep compliance relationship. E parenthesis t end parenthesis multiplied by D parenthesis t end parenthesis equals the quotient sine of n times pi, divided by n times pi.

Figure 119. Graphs. Comparison of measured and backcalculated E(t) and aT(T) for LTPP section 010101. Two graphs are shown. The graph on the left is labeled relaxation modulus. It has reduced time in s on the x-axis and relaxation modulus in psi on the y-axis. The graph is a scatter plot with differently shaped dots representing drop 1, drop 2, drop 3, drop 4, measured 1, and measured 2. The dots on the scatter plot create a decreasing sigmoid trend. The data begin at a reduced time of 10^-2 s and extend to 10⁴ s. At 10^-2 s, in order of increasing relaxation modulus is the dot representing measured 2 results at 2 times 10⁶ psi, then the measured 1, drop 2, drop 3, drop 1, and drop 4 results, with each of these dots overlapping with next. The ending values of the curves, in order of increasing relaxation modulus begin at 4 times 10³ psi for drop 4, then drop 3, drop 2, drop 1, and measured 1, ending at 5 times 10³, with each of these overlapping the next. Ending above the other lines is the highest relaxation modulus for measured 2 results at 10⁴ psi.

The graph on the right is labeled shift factor. This graph has temperature in °F on the x-axis and the shift factor, aT, on the y-axis. The graph contains quadratic curves representing drop 1, drop 2, drop 3, drop 4, and measured results. All of the curves have a linear decreasing trend. The curve representing measured begins at a shift factor of 1.05 times 10² at 32 °F and ends at a shift factor of 10^-5 at 122 °F. The next curve represents drop 4 results and begins at a shift factor just above 10² at 32 °F and ends at a shift factor just below 10^-2 at 122 °F. The curve representing drop 1 results begins at a shift factor just below drop 4, 10² at 32 °F, and ends at the same shift factor as that of drop 4. The curve representing drop 3 results begins just below 10^-2 and ends at a shift factor of 10^-3 at 122 °F. The last curve represents drop2 results and begins at a shift factor just above 10 at 32 °F and ends at the same shift factor as drops 1 and 4.

Figure 120. Graphs. Comparison of measured and backcalculated E(t) and aT(T) for LTPP section 06A805. Two graphs are shown. The graph on the left is labeled relaxation modulus. This graph has reduced time in s on the x-axis and relaxation modulus in psi on the y-axis. The graph is a scatter plot with differently shaped dots representing drop 1, drop 2, drop 3, drop 4, measured 1, and measured 2 results. The dots on the scatter plot create a decreasing sigmoid trend. The data begin at a reduced time of approximately 10^-2 s and extend to 10³ s. At 10^-2 s, relaxation modulus is approximately 3 times 10⁶. The ending values of relaxation modulus at 10³s are close to 1.5 times 10⁴ psi. The measured 2 curve does not follow sigmoid shape and shows significant deviations from rest of the curves at 1 s.

The graph on the right is labeled shift factor coefficients. This graph has temperature in °F on the x-axis and shift factor, aT, on the y-axis. The graph contains curves representing drop 1, drop 2, drop 3, drop 4, measured, and ANNACAP results. All of the lines have a linear decreasing trend up to 86 °F. The curve representing ANNACAP results starts at 2times 10³ at 32°F and ends at 4 times 10^-3 at 122 °F. The curve is linear in trend. The curve representing measured results starts at 2 times 10² at 32 °F and ends at 4 times 10^-6 at 122 °F. The curve is linear in trend. The curves representing drop 1, drop 2, drop 3, and drop 4 results almost overlap each other. The trend is quadratic and slightly concave after 86 °F. The curves start at 50 at 32 °F and end at 9.9at the end.

Figure 121. Graphs. Comparison of measured and backcalculated E(t) and aT(T) for LTPP section 06A806. Two graphs are shown. The graph on the left is labeled relaxation modulus. This graph has the reduced time in s on the x-axis and relaxation modulus in psi on the y‑axis. The graph is a scatter plot with differently shaped dots representing drop 1, drop 2, drop 3, drop 4, measured 1, and measured 2 results. The dots on the scatter plot create a decreasing sigmoid trend. The data begin at a reduced time of approximately 10^-3 s and extend to 10³ s. At 10^-3 s, relaxation modulus is approximately 3 times 10⁶ for all the curves. The curves diverge significantly after 10^-2 s. Ending values of relaxation modulus are at approximately 10³ s for drop 1, drop 2, drop 3, drop 4, measured 1, and measured 2 results are 10⁴ psi, 2 times 10³ psi, 2 times 10³ psi, 2 times 10³ psi, 2 times 10³ psi, 2 times 10⁴ psi, and 1.5 times 10⁴ psi, respectively. The measured 2 curve does not follow sigmoid shape and deviates from rest of the curves at 1 s.

The graph on the right is labeled shift factor coefficients. This graph has temperature in °F on the x-axis and the shift factor, aT, on the y-axis. The graph contains curves representing drop 1, drop 2, drop 3, drop 4, measured, and ANNACAP results. All of the lines have a linear decreasing trend up to 86 °F. The curves for drop 2, measured, and ANNACAP continue to drop with temperature beyond 86 °F, whereas drop 1, drop 3 and drop 4 results are slightly concave after 86 °F. All the curves start at 32 °F, with ANNACAP at the top followed by drop 2, measured, drop 3, drop 4 and drop 1. The values of aT for ANNACAP and drop 1 at 32 °F are 2times
10³ and 20, respectively. All the curves end at 122 °F, with drop1 at the top and measured at the bottom. The values of aT for drop 1 and measured at 122°F are 8 times 10^-1 and 2 to times 10^-5, respectively.

Figure 122. Graphs. Comparison of measured and backcalculated E(t) and aT(T) for LTPP section 300113. Two graphs are shown. The graph on the left is labeled relaxation modulus. This graph has reduced time in s on the x-axis and relaxation modulus in psi on the y-axis. The graph is a scatter plot with differently shaped dots representing drop 1, drop 2, drop 3, drop 4, measured 1, and measured 2 results. The dots on the scatter plot create a decreasing sigmoid trend. The data begin at a reduced time of approximately 10^-4 s and extend to 10³ s. At 10^-3 s, relaxation modulus is approximately 10⁶ for all the curves. The curves follow each other closely until the end at approximately 10³ s and 4 times 10³ psi.

The graph on the right is labeled shift factor coefficients. This graph has temperature in °F on the xaxis and the shift factor, aT, on the y-axis. The graph contains curves representing drop 1, drop 2, drop 3, drop 4, measured, and ANNACAP results. All of the curves have a linear decreasing trend. The data begin at 32 °F with ANNACAP at the top followed by drop 2, measured, drop 3, drop 4, and drop 1. The values of aT for ANNACAP and drop 1 at 32°F are 2times 10⁴ and 90, respectively. All the curves end at 122 °F, with drop 1 at the top and drop 4 at the bottom. The values of aT for drop 1 and drop 4 at 122 °F are 2 times 10^-2 and 3 times 10^-5, respectively.

Figure 123. Graphs. Comparison of measured and backcalculated E(t) and aT(T) for LTPP section 340801. Two graphs are shown. The graph on the left is labeled relaxation modulus. This graph has reduced time in s on the x-axis and relaxation modulus in psi on the y-axis. The graph is a scatter plot with differently shaped dots representing drop 1, drop 2, drop 3, drop 4, measured 1, and measured 2 results. The dots on the scatter plot create a decreasing sigmoid trend. The data begin at a reduced time of 10^-3 s and extend to 2 times 10² s. All the curves almost overlap, starting between 3 times 10⁶ psi and 4 times 10⁶ psi and ending between 2 times 10³ psi and 4times 10³ psi.

The graph on the right is labeled shift factor coefficients. This graph has temperature in °F on the x-axis and the shift factor, aT, on the y-axis. The graph contains lines representing drop 1, drop 2, drop 3, drop 4, and measured results. All of the lines have a linear decreasing trend. All the curves start at 32 °F and aT approximately equal to 10² with the exception of ANNACAP. All the curves almost overlap.

Figure 124. Graphs. Comparison of measured and backcalculated E(t) and aT(T) for LTPP section 340802. Two graphs are shown. The graph on the left is labeled relaxation modulus. This graph has reduced time in s on the x-axis and relaxation modulus in psi on the y-axis. The graph is a scatter plot with differently shaped dots representing drop 1, drop 2, drop 3, drop 4, measured 1, and measured 2 results. The dots on the scatter plot create a decreasing trend. The data begin at a reduced time of 10^-4 s and extend to 10³ s. All the curves almost overlap, starting between 3 times 10⁶ psi and 5 times 10⁶ psi and ending between 10³ psi and 3 times 10³ psi.

The graph on the right is labeled shift factor coefficients. This graph has temperature in °F on the x-axis and the shift factor, aT, on the y-axis. The graph contains lines representing drop 1, drop 2, drop 3, drop 4, and measured results. All of the lines have a linear decreasing trend, starting at 10³ at 32 °F and ending at 10^-4 at 122°F. All the curves match well except ANNACAP, which deviates after 86 °F.

Figure 125. Graphs. Comparison of measured and backcalculated E(t) and aT(T) for LTPP section 350801. Two graphs are shown. The graph on the left is labeled relaxation modulus. This graph has reduced time in s on the x-axis and relaxation modulus in psi on the y-axis. The graph is a scatter plot with differently shaped dots representing drop 1, drop 2, drop 3, drop 4, measured 1, and measured 2 results. The dots on the scatter plot create a decreasing sigmoid trend. The data begin at a reduced time of 10^-3 s and extend to 10³ s. Drop 1, drop 3, and drop 4 results overlap. The curves start at 2 times 10⁶ psi and end at 2times 10³ psi. Measured 2 and measured 1 results overlap. The curves start at 1.5 times 10⁶ psi and end at 1.5 times 10⁴ psi. Drop 2 results match measured 1 and measured 2 results after 10 s.

The graph on the right is labeled shift factor coefficients. This graph has temperature in °F on the x-axis and the shift factor, aT, on the y-axis. The graph contains lines representing drop 1, drop 2, drop 3, drop 4, and measured results. All of the lines have a linear decreasing trend. All the curves match well from 50 °F at aT value of 8 to 86 °F at aT value of50.

Figure 126. Graphs. Comparison of measured and backcalculated E(t) and aT(T) for LTPP section 350802. Two graphs are shown. The graph on the left is labeled relaxation modulus. This graph has reduced time in s on the x-axis and relaxation modulus in psi on the y-axis. The graph is a scatter plot with differently shaped dots representing drop 1, drop 2, drop 3, drop 4, measured 1, and measured 2 results. The dots on the scatter plot create a decreasing sigmoid trend. The data begin at a reduced time of 10^-2 s and extend to 10² s. At 10^-2 s, drop 1, drop 2, drop 3, and drop 4 results are 3 times 10⁶ psi, whereas measure 2 results are slightly higher than measured 1 results at 2 times 10⁶psi. At 10² s, drop 1 results are 2 times 10⁵ psi, drop 2 results are 9 times 10⁵ psi, measured 1 results are 2 times 10⁴psi, measured 2 results are 1.5 times 10⁴psi, and drop 3 and drop 4 results are 10⁴ psi.

The graph on the right is labeled shift factor coefficients. This graph has temperature in °F on the x-axis and the shift factor, aT, on the y-axis. The graph contains quadratic curves representing drop 1, drop 2, drop 3, drop 4, and measured results. Drop 4, measured, and ANNACAP results have a linear decreasing trend, whereas drop 1, drop 2, and drop 3 results become concave at 95 °F. At the start of the curve (32 °F), ANNACAP, measured, drop 4, drop 1, drop 2, and drop 3 results are 3times 10³, 2 times 10², 19, 11, 11, and 11, respectively. At the end of the curve (122 °F), drop 1, drop 2, drop 3, drop 4, ANNACAP, and measured results are 1, 1, 1, 1.05 times 10^-2, 1.1 times 10^-3, and 1.1times 10^-3, respectively.

Figure 127. Graphs. Comparison of measured and backcalculated E(t) and aT(T) for LTPP section 460804. Two graphs are shown. The graph on the left is labeled relaxation modulus. This graph has reduced time in s on the x-axis and relaxation modulus in psi on the y-axis. The graph is a scatter plot with differently shaped dots representing drop 1, drop 2, drop 3, drop 4, measured 1 and measured 2 results. The dots on the scatter plot create a decreasing trend. The data begin at a reduced time of 10^-3 s and extend to 2 times 10² s. At 10^-3, drop 1, drop 2, drop 3, drop4, measured 2, and measured 1 results are 3times 10⁶ psi, 3 times 10⁶ psi, 3 times 10⁶ psi, 3times 10⁶ psi, 3 times 10⁶ psi, and 2 times 10⁶ psi, respectively. At 2 times 10² , drop 1, drop 2, drop 3, drop 4, measured 2, and measured 1 results are very close, i.e., between 2 times 10³ psi and 3 times 10³psi.

The graph on the right is labeled shift factor coefficients. This graph has temperature in °F on the x-axis and the shift factor, aT, on the y-axis. The graph contains lines representing drop 1, drop 2, drop 3, drop 4, and measured results. All of the lines have a linear decreasing trend except drop 1, which flattens after 95 °F. At 32 °F, ANNACAP, measured, drop 4, drop 1, drop 2, and drop 3 results are 3 times 10³, 4 times 10², 3 times 10², 10², 90, and 60, respectively. At the end of the curves (122 °F), drop 1, drop 2, drop 3, drop 4, measured, and ANNACAP results are 9times 10^-1, 2 times 10² , 10^-2, 10^-2, 5 times 10^-3 and 5 times 10^-3, respectively.

Figure 128. Graphs. Comparison of measured and backcalculated |E*| and phase angle for LTPP section 010101. Two graphs are shown. The graph on the left is labeled dynamic modulus. The graph has reduced frequency in inverse s on the x-axis and dynamic modulus in psi on the y-axis. The graph is a scatter plot with differently shaped dots representing drop 1, drop 2, drop 3, drop 4, measured 1, measured 2, and ANNACAP results. The dots all follow the same sigmoidal trend, linearly increasing until about 1 Hz and then curving into a horizontal line at 10²Hz until the graph ends at 10⁶ Hz. At the beginning of the graph, the dynamic modulus is about 2times 10³psi for both measured 1 and 2 results overlapping with drop 1 and drop 2 results, followed by drop 3 and drop 4 results overlapping at dynamic modulus 10³ psi. At the end of the graph, all of the dots overlap each other at 2times 10⁶ psi.

The graph on the right is labeled phase angle. The graph has reduced frequency in inverse seconds on the x-axis and phase angle in degrees on the y-axis. The graph is a scatter plot containing dots that represent drop 1, drop 2, drop 3, drop 4, measured 1, and measured 2 results. All of the dots create a trend that has a parabolic shape, with the maximum phase angle occurring at 10^-3 Hz and decreasing down to a 0 phase angle at 10⁴ Hz. In general, drop 1, drop 2, drop 3, and drop 4 results overlap each other. The curves start at 15degrees and reach the peak value of 55 degrees at 10^-3 Hz. Measured 1 results do not follow the same trend and reach the peak value of 40 degrees at 5 times 10^-3 Hz. Measured 2 results start at 10 degrees and increase to 48degrees before overlapping with the drop 1 curve.

Figure 129. Graphs. Comparison of measured and backcalculated |E*| and phase angle for LTPP section 06A805. Two graphs are shown. The graph on the top is labeled dynamic modulus. The graph has reduced frequency in inverse s on the x-axis and dynamic modulus in psi on the y‑axis. The graph is a scatter plot with differently shaped dots representing drop 1, drop 2, drop 3, drop 4, measured 1, and measured 2 results. The dots all follow the same sigmoidal trend linearly increasing until about 1 s and then curving into a horizontal line at 10² Hz until the graph ends at 10⁶ Hz. At the beginning of the graph, the dynamic modulus is 10⁴ psi for measured 2 and ANNACAP results and 9 times 10⁴ psi for drop1 results. Below the drop 1 curve and overlapping it slightly are curves for drop 2, drop 3, and drop 4. At the end of the graph, all of the dots are between 3 times 10⁶ psi and 6times 10⁶ psi and overlap each other.

The graph on the right is labeled phase angle. The graph has reduced frequency in inverse s on the x-axis and phase angle in degrees on the y-axis. The graph is a scatter plot containing dots that represent drop 1, drop 2, drop 3, drop 4, measured 1, and measured 2 results. All of the dots create a trend that has a parabolic shape, with the maximum phase angle occurring at 10⁶ Hz and decreasing down to a 0 phase angle at 10⁴ Hz. At the beginning of the plot, measured 1 results have the largest phase angle of 22 degrees; it then increases to its peak of 49degrees before decreasing to 0. Drop 1, drop 2, drop 3, and drop 4 results overlap each other, starting from 10degrees at 10^-6 Hz to 50 degrees at the peak. Finally the dots representing measured 2 results begin at 2 degrees, increase to 60 degrees at 10^-2 Hz, and then decrease to 0 at 10² Hz.

Figure 130. Graphs. Comparison of measured and backcalculated |E*| and phase angle for LTPP section 06A806. Two graphs are shown. The graph on the top is labeled dynamic modulus. The graph has reduced frequency in inverse s on the x-axis and dynamic modulus in psi on the y-axis. The graph is a scatter plot with differently shaped dots representing drop 1, drop 2, drop 3, drop 4, measured 1, measured 2, and ANNACAP results. The dots all follow the same sigmoidal trend linearly increasing until about 1 Hz and then curving into a horizontal line at 10²Hz until the graph ends at 10⁶ Hz. The curves for drop 2, drop 3, and drop 4 overlap over the entire frequency. All the curves closely match at frequencies higher than 1 Hz. At frequencies lower than 1 Hz, drop 1, measured, and ANNACAP curves deviate significantly. ANNACAP results present the highest value followed by measured 1, measured 2, drop 1, drop2, drop3, and drop 4 results.

The graph on the right is labeled phase angle. The graph has reduced frequency in inverse s on the x-axis and phase angle in degrees on the y-axis. The graph is a scatter plot with differently shaped dots that represent drop 1, drop 2, drop 3, drop 4, measured 1, and measured2 results. All of the dots create a trend that has a parabolic shape, with the maximum phase angle occurring at 10^-3 Hz and decreasing to a 0 phase angle at 10⁴ Hz. In general drop 1, measured 1, and measured 2 results match, reaching peak phase angle of 45 to 50 degrees. Similarly, drop 2, drop3, and drop 4 results overlap each other, reaching peak phase angle of 67 degrees.

Figure 131. Graphs. Comparison of measured and backcalculated |E*| and phase angle for LTPP section 300113. Two graphs are shown. The graph has reduced frequency in inverse s on the x-axis and dynamic modulus in psi on the y-axis. The graph is a scatter plot with differently shaped dots representing drop 1, drop 2, drop 3, drop 4, measured 1, measured 2, and ANNACAP results. The dots all follow the same sigmoidal trend linearly increasing until about 1 Hz and then curving into a horizontal line at 10² Hz until the graph ends at 10⁶ Hz. All the curves follow each other closely over the entire frequency domain except for drop 1 and ANNACAP. The curves start at 2 times 10³ psi and end at 2 times 10⁶ psi. Drop 1 results deviate at frequencies greater than 10² Hz, and ANNACAP results deviate at frequency less than 10^-4 Hz from the rest of the curves.

The graph on the right is labeled phase angle. The graph has reduced frequency in inverse s on the x-axis and phase angle in degrees on the y-axis. The graph is a scatter plot with differently shaped dots that represent drop 1, drop 2, drop 3, drop 4, measured 1, and measured 2 results. All of the dots create a trend that has a parabolic shape, with the maximum phase angle occurring at 10^-2 Hz and decreasing to a 0 phase angle at 10⁶ Hz. Drop 1 results reach the peak phase angle of 45 degrees followed by measured 1, drop3, drop 2, drop 4, and measured 2 results, varying between 40 to 35 degrees.

Figure 132. Graphs. Comparison of measured and backcalculated |E*| and phase angle at LTPP section 340801. Two graphs are shown. The graph on the left is labeled dynamic modulus. The graph has reduced frequency in inverse s on the x-axis and dynamic modulus in psi on the y-axis. The graph is a scatter plot with differently shaped dots representing drop 1, drop 2, drop 3, drop 4, measured 1, measured 2, and ANNACAP results. The dots all follow the same sigmoidal trend, linearly increasing until about 1 Hz and then curving into a horizontal line at 10²Hz until the graph ends at 10⁶ Hz. Drop 1, drop 2, drop 3, and drop 4 results overlap each other. The curves start at 10³ psi at 10^-6 Hz and reach 7 times 10⁶ psi at 10⁶ Hz. Measured 1, measured 2, and ANNACAP results match well over frequencies less than 10^-1 Hz and reach 2times 10⁶ psi at 10⁶ Hz.

The graph on the right is labeled phase angle. The graph has reduced frequency in inverse s on the x-axis and phase angle in degrees on the y-axis. The graph is a scatter plot with differently shaped dots that represent drop 1, drop 2, drop 3, drop 4, measured 1, and measured2 results. All of the dots create a trend that has a parabolic shape, with the maximum phase angle occurring at 10^-2 Hz and decreasing to a 0 phase angle at 10⁶ Hz. Drop 1, drop 2, drop 3, and drop 4 results match well. The curves start at approximately 12 degrees and reach a maximum value of 65degrees at 10^-2 Hz. Measure 1 results start at 40 degrees and reach the peak value of 48degrees at 10^-3 Hz. Measure 2 results start at 18 degrees and reach the peak value of 48degrees at 10^-2 Hz.

Figure 133. Graphs. Comparison of measured and backcalculated |E*| and phase angle at LTPP section 340802. Two graphs are shown. The graph on the left is labeled dynamic modulus. The graph has reduced frequency in inverse s on the x-axis and dynamic modulus in psi on the y-axis. The graph is a scatter plot with differently shaped dots representing drop 1, drop 2, drop 3, drop 4, measured 1, measured 2, and ANNACAP results. The dots all follow the same sigmoidal trend, linearly increasing until about 1 Hz and then curving into a horizontal line at 10²Hz until the graph ends at 10⁶ Hz. Drop 1, drop 2, drop 3, and drop 4 results overlap each other. The curves start at 10³ psi at 10^-6 Hz and reach 7 times 10⁶ psi at 10⁶ Hz. Measured 1, measured 2, and ANNACAP results match well over frequencies less than 10^-1 Hz and reach 2times 10⁶ psi at 10⁶ Hz.

The graph on the right is labeled phase angle. The graph has reduced frequency in inverse s on the x-axis and phase angle in degrees on the y-axis. The graph is a scatter plot with differently shaped dots that represent drop 1, drop 2, drop 3, drop 4, measured 1, and measured2 results. All of the dots create a trend that has a parabolic shape, with the maximum phase angle occurring at 10^-2 Hz and decreasing down to a 0 phase angle at 10⁶ Hz. Drop 1, drop2, drop 3, and drop 4 results match well. The curves start at approximately 12 degrees and reach a maximum value of 65degrees at 10^-2 Hz. Measure 1 result starts at 40 degrees and reach the peak value of 48degrees at 10^-3 Hz. Measure 2 results start at 18 degrees and reach the peak value of 48degrees at 10^-3 Hz.

Figure 134. Graphs. Comparison of measured and backcalculated |E*| and phase angle at LTPP section 350801. Two graphs are shown. The graph on the left is labeled dynamic modulus. The graph has reduced frequency in inverse s on the x-axis and dynamic modulus in psi on the y-axis. The graph is a scatter plot with differently shaped dots representing drop 1, drop 2, drop 3, drop 4, measured 1, measured 2, and ANNACAP results. The dots all follow the same sigmoidal trend, linearly increasing until about 1 Hz and then curving into a horizontal line at 10²Hz until the graph ends at 10⁶ Hz. At the beginning of the graph the dynamic modulus is 10⁴psi for ANNACAP results, 9 times 10³ psi for drop 2 results, 7 times 10³ psi for measured 1 results, 2times 10³ psi for measured 2 results, 10³ psi for drop 3 results overlapping with drop 4 and drop 1.

The graph on the right is labeled phase angle. The graph has reduced frequency in inverse s on the x-axis and phase angle in degrees on the y-axis. The graph is a scatter plot with differently shaped dots that represent drop 1, drop 2, drop 3, drop 4, and measured 1 and measured 2 results. All of the dots create a trend that has a parabolic shape, with the maximum phase angle occurring just before 10^-2 Hz and decreasing down to a 0 phase angle at 10⁴ Hz. Drop 4 results have the highest peak phase angle of 63 degrees, followed by drop 1 results overlapping with drop 3 with peak phase angle 60degrees. Next are drop 2, measured 2, and measured 1 results with peak phase angles of 50, 45, and 35 degrees, respectively.

Figure 135. Graphs. Comparison of measured and backcalculated |E*| and phase angle at LTPP section 350802. Two graphs are shown. The graph on the left is labeled dynamic modulus. The graph has reduced frequency in inverse s on the x-axis and dynamic modulus in psi on the y-axis. The graph is a scatter plot with differently shaped dots representing drop 1, drop 2, drop 3, drop 4, measured 1, measured 2, and ANNACAP results. The dots all follow the same sigmoidal trend. Drop 3 and drop 4 results match each other over the entire frequency range of 10^-6 Hz to 10⁶ Hz. Measured 1 and ANNACAP results match over frequencies higher than 1 Hz. However, in general the curves do not match. Drop 1 results have the highest dynamic modulus value, 3 times 10⁴ psi at the start. Next are drop 2 and ANACAP results, 10⁴psi, followed by measured 1, measured 2, drop 4, and drop 3 results, varying from 2 times 10³ psi to 10³ psi. Curves for drop 1, drop 2, drop 3, drop 4, and measured 2 results end at 8 times 10⁶ psi, and measured 1 and ANNACAP results end at 10⁶ psi.

The graph on the right is labeled phase angle. The graph has reduced frequency in inverse s on the x-axis and phase angle in degrees on the y-axis. The graph is a scatter plot with differently shaped dots that represent drop 1, drop 2, drop 3, drop 4, and measured 1, and measured 2 results. All of the dots create a trend that has a parabolic shape, with the maximum phase angle occurring just before 10^-2 Hz and decreasing down to a 0 phase angle at 10⁴ Hz. Curves for drop 3 and drop 4 results match. The curves start at 18 degrees at 10⁶ Hz and reach peak at 10^-3 Hz. Drop 2 results start at 11 degrees and reach peak value of 50 degrees close to 10^-2 Hz. Measured 1 and measured 2 results start at 30 degrees and reach peak values of 45 and 40 degrees 10^-3 Hz and 10^-4 Hz, respectively. Drop 1 results start at 20 degrees and reach peak value of 25 degrees at 10^-4 Hz.

Figure 136. Graphs. Comparison of measured and backcalculated |E*| and phase angle at LTPP section 46804. Two graphs are shown. The graph on the left is labeled dynamic modulus. The graph has reduced frequency in inverse s on the x-axis and dynamic modulus in psi on the yaxis. The graph is a scatter plot with differently shaped dots representing drop 1, drop 2, drop 3, drop 4, measured 1, measured 2, and ANNACAP results. The dots all follow the same sigmoidal trend, linearly increasing until about 1 Hz and then curving into a horizontal line at 10²Hz until the graph ends at 10⁶ Hz. Drop 1, drop 2, drop 3, drop 4, and measured 2 results overlap each other. The curves start at 10³ psi at 10⁶ Hz and reach 7 times 10⁶ psi at 10⁶ Hz. Measured 1 and ANNACAP results match well over frequencies less than 10^-1 Hz and reach 2times 10⁶ psi at 10⁶ Hz.

The graph on the right is labeled phase angle. The graph has reduced frequency in inverse s on the x-axis and phase angle in degrees on the y-axis. The graph is a scatter plot with differently shaped dots that represent drop 1, drop 2, drop 3, drop 4, measured 1, and measured2 results. All of the dots create a trend that has a parabolic shape, with the maximum phase angle occurring at 10^-6 Hz and decreasing to a 0 phase angle at 10⁶ Hz. At the beginning of the plot, measured 1 results have the largest phase angle of 22 degrees. It then increases to its peak of 52before decreasing to 0. Drops 1 to 4 results overlap each other, starting from 9 degrees at 10^-6 Hz to 55to 65 degrees at the peak. Finally the curve representing measured 2 beginning at 15degrees increases to 55 degrees at 10^-2 Hz and then decreases to 0 at 10⁶ Hz.

Figure 137. Graph. Elastic backcalculation of two-step temperature profile FWD data, assuming AC as a single layer in two-stage backcalculation. This bar graph has temperature profile in °F on the x-axis and average modulus in psi on the y-axis. It contains two bars for each temperature profile; the first represents E base and the second represents E subgrade. Two horizontal lines run across the entire graph showing the average backcalculated base and subgrade modulus values are 25,000 psi and 11,000 psi, respectively.

The average moduli for the temperature profile {50-32} are 25,000 psi for E base and 11,000psi for E subgrade. The average moduli for the temperature profile {59-50} are 24,800psi for E base and 11,000 psi for E subgrade. The average moduli for the temperature profile {68-50} are 24,900 psi for E base and 11,000 psi for E subgrade. The average moduli for the temperature profile {68-59} are 24,500 psi for E base and 11,000 psi for E subgrade. The average moduli for the temperature profile {77-68} are 24,500 psi for E base and 11,000 psi for E subgrade. The average moduli for the temperature profile {86-68} are 24,600 psi for E base and 11,000 psi for E subgrade. The average moduli for the temperature profile {86-77} are 23,500 psi for E base and 11,000 psi for E subgrade. The average moduli for the temperature profile {95-86} are 24,500 psi for E base and 11,000 psi for E subgrade. The average moduli for the temperature profile {104-86} are 24,000 psi for E base and 11,000 psi for E subgrade. The average moduli for the temperature profile {104-95} are 24,900 psi for E base and 11,000 psi for E subgrade.

Figure 138. Graph. Elastic backcalculation of three-step temperature profile FWD data, assuming AC as a single layer in two-stage backcalculation. This bar graph has the temperature profile in °F on the x-axis and average modulus in psi on the y-axis. It contains two bars for each temperature profile; the first represents E base and the second represents E subgrade. Two horizontal lines run across the entire graph showing the average backcalculated base and subgrade modulus values are 25,500 psi and 11,000 psi, respectively. The average moduli for the temperature profile {59-50-41} are 25,500 psi for E base and 11,000psi for E subgrade. The average moduli for the temperature profile {68-59-50} are 24,500psi for E base and 11,000 psi for E subgrade. The average moduli for the temperature profile {77-68-59} are 25,500 psi for E base and 11,000 psi for E subgrade. The average moduli for the temperature profile {86-77-68} are 25,500 psi for E base and 11,000 psi for E subgrade. The average moduli for the temperature profile {95-86-77} are 24,000 psi for E base and 11,000psi for E subgrade. The average moduli for the temperature profile {104-95-86} are 25,500 psi for E base and 11,000 psi for E subgrade.

Figure 139. Graph. Elastic backcalculation of two-step temperature profile FWD data, assuming two AC sublayers in two-stage backcalculation. This bar graph has the temperature profile in °F on the x-axis and average modulus in psi on the y-axis. It contains two bars for each temperature profile; the first represents E base and the second represents E subgrade. Two horizontal lines show the average backcalculated base and subgrade modulus values are 25,500psi and 11,500 psi, respectively.

The average moduli for the temperature profile {50-32} are 24,500 psi for E base and 11,500 psi for E subgrade. The average moduli for the temperature profile {59-50} are 25,500 psi for E base and 11,500 psi for E subgrade. The average moduli for the temperature profile {68-50} are 25,000 psi for E base and 11,500 psi for E subgrade. The average moduli for the temperature profile {68-59} are 24,000 psi for E base and 11,500 psi for E subgrade. The average moduli for the temperature profile {77-68} are 24,500 psi for E base and 11,500 psi for E subgrade. The average moduli for the temperature profile {86-68} are 24,000 psi for E base and 11,500 psi for E subgrade. The average moduli for the temperature profile {86-77} are 25,000 psi for E base and 11,500 psi for E subgrade. The average moduli for the temperature profile {95-86} are 26,000 psi for E base and 11,500 psi for E subgrade. The average moduli for the temperature profile {104-86} are 25,500 psi for E base and 11,500 psi for E subgrade. The average moduli for the temperature profile {104-95} are 24,000 psi for E base and 11,500 psi for E subgrade.

Figure 140. Graph. Elastic backcalculation of three-step temperature profile FWD data, assuming three AC sublayers in two-stage backcalculation. This bar graph has the temperature profile in °F on the x-axis and average modulus in psi on the y-axis. It contains two bars for each temperature profile; the first represents E base and the second represents E subgrade. Two horizontal lines drawn across the entire graph show that the average backcalculated base and subgrade modulus values are 25,500 psi and 11,500, psi, respectively.

The average moduli for the temperature profile {59-50-41} are 25,000 psi for E base and 11,500psi for E subgrade. The average moduli for the temperature profile {68-59-50} are 25,500psi for E base and 11,500 psi for E subgrade. The average moduli for the temperature profile {77-68-59} are 23,000 psi for E base and 11,500 psi for E subgrade. The average moduli for the temperature profile {86-77-68} are 24,000 psi for E base and 11,500 psi for E subgrade. The average moduli for the temperature profile {95-86-77} are 25,000 psi for E base and 11,500psi for E subgrade. The average moduli for the temperature profile {104-95-86} are 25,000 psi for E base and 11,500 psi for E subgrade.

Figure 141. Graphs. Error in backcalculated E(t) curve from two-step temperature profile FWD test data in two-stage backcalculation. Two graphs are shown. The first graph has the temperature profile set in °F on the x-axis and the average percent error on the y-axis. For the temperature profile set {50-32} and {68-50}, there is an 8.5-percent error. For the temperature profile set {68-59} and {59-50}, there is an 11.5-percent error. For the temperature profile set {68-50} and {86-68}, there is a 12-percent error. For the temperature profile set {86-77} and {77-68}, there is a 9-percent error. For the temperature profile set {95-86} and {86-77} there is a 3.5-percent error. For the temperature profile set {86-68} and {104-86} there is a 17-percent error. Finally, for the temperature profile set {104-95} and {95-86}, there is a 6‑percent error. There is an arrow pointing to the bar for the {86-68} and {104-86} profile set that leads to the other graph. This graph has the time in seconds on the x-axis and the E parenthesis t end parenthesis in psi on the y-axis. Two lines are shown on the graph representing the actual E parenthesis t end parenthesis and the backcalculated E parenthesis t end parenthesis results. The two lines begin about 10^-8 s at 3 times 10⁶ psi and decrease first concavely then convexly until they reach 10⁸ s at 1.05 times 10³ psi. The actual E parenthesis t end parenthesis values are slightly larger than the backcalculated E parenthesis t end parenthesis values at all times.

Figure 142. Graphs. Error in backcalculated E(t) curve from three-step temperature profile FWD test data in two-stage backcalculation. Three graphs are shown. The main graph has the temperature profile set in °F on the x-axis and the average percent error on the y-axis. For the temperature profile set {68-59-50} and {59-50-41}, there is a 4.5-percent error. For the temperature profile set {77-68-59} and {68-59-50}, there is a 5.5-percent error. For the temperature profile set {86-77-68} & {77-68-59} there is a 16-percent error. For the temperature profile set {86-77-68} and {95-86-77}, there is a 3-percent error. For the temperature profile set {104-95-86} and {95-86-77}, there is a 2.5-percent error.

Two smaller graphs are inset into the main graph. The smaller graph on the upper left has an arrow pointing to the {68-59-50} and {59-50-41} temperature profile on the main graph. The smaller graph has the time in seconds on the x-axis and the E parenthesis t end parenthesis in psi on the y-axis. Two lines are shown on the graph representing the actual E parenthesis t end parenthesis and the backcalculated E parenthesis t end parenthesis results . The two lines begin about 10^-8 s at 3times 10⁶ psi and decrease first concavely then convexly until they reach 10⁸ s at 1.05times 10³ psi. From 10^-8 s until 1 s, the two lines overlap each other; after 1 s, the values of E parenthesis t end parenthesis actual are larger than the backcalculated values. The smaller graph on the upper right has an arrow pointing to the {86-77-68} and {77-68-59} temperature profile on the main graph. This smaller graph also has the time in seconds on the x-axis and the E parenthesis t end parenthesis in psi on the y-axis. Two lines are shown on the graph representing the actual E parenthesis t end parenthesis and the backcalculated E parenthesis t end parenthesis results . The two lines begin about 10^-8 s at 3 times 10⁶ psi and decrease first concavely then convexly until they reach 10⁸ s and the actual E parenthesis t end parenthesis results are 1.05times 10³ psi and the backcalculated results are just above 10³ psi. The values for E parenthesis t end parenthesis actual are slightly larger than those for the backcalculated E parenthesis t end parenthesis from 10^-8 s until around 10² s. After 10² s, the lines begin decreasing faster and overlap. Then at about 1 s, the actual E parenthesis t end parenthesis values again become larger than the backcalculated E parenthesis t end parenthesis values.

Figure 143. Graph. Nonlinear elastic backcalculated AC modulus for control and CRTB mixes using FWD data at different test temperatures. This bar graph has the temperature in °F on the x-axis and the asphalt concrete modulus, Eac, in psi on the y-axis. For each temperature, there are two bars, one for the control mix and the other for the crumb rubber terminal blend (CRTB). At 50 °F, the asphalt concrete modulus is 1.7 times 10⁶ psi for the control mix and 1.05 times 10⁶ psi for the CRTB. At 68 °F, the asphalt concrete modulus is 1.05times 10⁶ psi for the control mix and 6.05 times 10⁵ psi for the CRTB. At 86 °F, the asphalt concrete modulus is 5.08 times 10⁵ psi for the control mix and 3.0 times 10⁵ psi for the CRTB. At 104°F, the asphalt concrete modulus is 2.5 times 10⁵ psi for the control mix and 1.5 times 10⁵ psi for the CRTB. Finally, at 50 °F, the asphalt concrete modulus is 10⁵ psi for the control mix and 0.8times 10⁵ psi for the CRTB.

Figure 144. Graphs. Nonlinear elastic backcalculated unbound layer properties for control and CRTB mix using FWD data at different test temperatures. Four graphs are shown. The graph on top left shows the data for k1. This graph has the temperature in °F on the x-axis and k1in psi on the y-axis. The graph is a bar graph, with two different types of bars, the first representing control mix and the other crumb rubber terminal blend (CRTB). There is a dashed line drawn horizontally across the graph at 3,650 psi. At 50 °F, the control mix has a value of 3,600psi, and the CRTB has a value of 3,500 psi. At 68 °F, the control mix has a value of 3,600psi, and the CRTB has a value of 3,200 psi. At 86 °F, the control mix has a value of 2,900psi, and the CRTB has a value of 3,200 psi. At 104 °F, the control mix has a value of 2,800psi, and the CRTB has a value of 3,000 psi. Finally, at 122 °F, the control mix has a value of 3,000 psi, and the CRTB has a value of 3,000 psi.

The graph on the top right shows the data for k2. The graph has the temperature in °F on the x‑axis and k2 on the y-axis. The graph is a bar graph, with two different types of bars, the first representing the control mix and the other CRTB. There is a dashed line drawn horizontally across the graph at 0.5. For 50 °F, the control mix has a value of 0.45 and the CRTB has a value of 0.45. At 68 °F, the control mix has a value of 0.45, and the CRTB has a value of 0.41. At 86°F, the control mix has a value of 0.46, and the CRTB has a value of 0.46. At 104 °F, the control mix has a value of 0.42, and the CRTB has a value of 0.42. Finally, at a temperature of 122 °F, the control mix has a value of 0.4, and the CRTB has a value of 0.41.

The graph on bottom left shows the data for k3. This graph has the temperature in °F on the x‑axis and k3 on the y-axis. The graph is a bar graph, with two different types of bars, the first representing control mix and the other CRTB. The bars on this graph begin at the top of the graph extending downward, and there is a dashed line drawn horizontally across the graph at
–0.5. For 50 °F, the control mix has a value of -0.45, and the CRTB has a value of –0.56. At 68°F, the control mix has a value of -0.5, and the CRTB has a value of –0.5. At 86 °F, the control mix has a value of -0.61, and the CRTB has a value of –0.55. At 104 °F, the control mix has a value of -0.53, and the CRTB has a value of –0.62. Finally, at 122 °F, the control mix has a value of –0.62, and the CRTB has a value of –0.63.

The graph on the bottom right shows the data for subgrade modulus. This graph has the temperature in °F on the x-axis and subgrade modulus in psi on the y-axis. The graph is a bar graph, with two different types of bars, the first representing the control mix and the other CRTB. There is a dashed line drawn horizontally across the graph at 10,000 psi. For 50 °F, the control mix has a value of 9,500 psi, and the CRTB has a value of 9,600 psi. At 68 °F, the control mix has a value of 9,600 psi, and the CRTB has a value of 9,700 psi. At 86 °F, the control mix has a value of 9,800 psi, and the CRTB has a value of 9,700 psi. At 104 °F, the control mix has a value of 9,800 psi, and the CRTB has a value of 9,800 psi. Finally, at 122 °F, the control mix has a value of 9,900 psi, and the CRTB has a value of 9,900 psi.

Figure 145. Graphs. Control mix backcalculation results from two-stage nonlinear viscoelastic backcalculation. Two graphs are shown. The graph on the left is the average percent error, PE_avg curves. The falling weight deflectometer (FWD) temperature set in °F is on the x-axis, and the average percent error is on the y-axis. Four bars are shown for each temperature set, representing the four time ranges. The first is the time range 10^-5 to 10 s, the second is 10^-5 to 10² s, the third is 10⁵ to 10³ s, and the fourth is 10^-5 to 10⁵ s. For the FWD temperature set {50, 68}, the average percent error is 7 for the first time range, 8.8 for the second time range, 9 for the third time range, and 12 for the fourth time range. For the FWD temperature set {68, 86}, the average percent error is 9 for the first and second time ranges, 8.5 for the third time range, and 8 for the fourth time range. For the FWD temperature set {86, 104}, the average percent error is 2 for the first time range, 1.5 for the second time range, 1.4 for the third time range, and 2 for the fourth time range. For the FWD temperature set {104, 122}, the average percent error is 8 for the first time range, 7.5 for the second time range, 8 for the third time range, and 7 for the fourth time range.

The graph on the right is labeled backcalculated E parenthesis t end parenthesis master curves. This graph has reduced time at 66 °F in s on the x-axis and relaxation modulus in psi on the y‑axis. Five lines are shown on the graph representing actual, 50 and 68 °F, 68 and 86 °F, 86and 104 °F, and 104 and 50 °F. All of the lines start at 3 times 10⁶ psi at 10^-8 s. The lines are horizontal until about 10^-5 psi when they begin to decrease linearly until about 10³ s when the linear behavior stops and the lines become slightly convex ending at 10⁸ s. After the linear behavior stops, the lines no longer have the same relaxation modulus. At 10⁸ s, the relaxation modulus is 4 times 10³ psi for 50, 68 °F, 2 times 10³ psi for 86, 104 °F, and between 10³ psi and 2 times 10³ psi are the lines for actual, 68, 86 °F, and 104, 122 °F in decreasing order.

Figure 146. Graphs. CRTB mix backcalculation results from two-stage nonlinear viscoelastic backcalculation. Two graphs are shown. The graph on the left is the average percent error, PEavg curves. The FWD temperature set in °F is on the x-axis, and the average percent error is on the y-axis. Four bars are shown for each temperature set; they represent the four time ranges. The first is the time range 10 s, the second is 10^-5 to 10² s, the third is 10^-5 to 10³ s, and the fourth is 10^-5 to 10⁵ s. For the FWD temperature set {50, 68}, the average percent error is 8 for the first and second time ranges, 8.5 for the third time range, and 10.5 for the fourth time range. For the FWD temperature set {68, 86}, the average percent error is 11 for the first and second time range, 10.8 for the third time range, and 10.5 for the fourth time range. For the FWD temperature set {86, 104}, the average percent error is 12 for the first time range, 12.8 for the second time range, 12.5 for the third time range, and 11 for the fourth time range. For the FWD temperature set {104, 122}, the average percent error is 14 for the first time range, 13 for the second time range, 12 for the third time range, and 10 for the fourth time range.

The second graph on right is labeled Backcalculated E(t) master curves. This graph has the reduced time at 66 °F in s on the x-axis and the relaxation modulus in psi on the y-axis. Five lines are shown on the graph representing; actual, 50 and 68 °F, 68 and 86 °F, 86 and 104°F, and 104 and 122 °F. The lines all have the same trend beginning at 10^-8 s and bend concavely and then convexly until 10⁸ s. The lines for actual, 50 and 68 °F, and 68 and 30 °F all begin at 3times 10⁶ psi. The lines for 86 and 104 °F and 104 and 122 °F begin at 2 times 10⁶ psi. The lines decrease to about 2 times 10³ psi and have values that are very close together. In order of decreasing relaxation modulus are the lines for 68 and 86 °F, actual, and then the last three are stacked upon each other.

Figure 147. Graphs. Comparison of nonlinear viscoelastic backcalculated and measured E(t) and aT(T) for LTPP section 10101. Two graphs are shown. The graph on the left is labeled relaxation modulus. This graph has reduced time at 66 °F in s on the x-axis and relaxation modulus in psi on the y-axis. Three lines are shown on the graph representing measured, backcalculated 1, and backcalculated 2 results. The lines all begin just past 10^-4 s at about 2 times 10⁶psi; the largest relaxation modulus at this point is for the measured results, then the backcalculated 1 results, and then the backcalculated 2 results. The lines all decrease until they reach 10³ s. At this point, the measured results have a relaxation modulus of 10⁴ psi, and both of the backcalculated results have a relaxation modulus of 7 times 10³ psi.

The graph on the right is labeled shift factor at 66 °F. This graph has temperature in °F on the x‑axis and shift factor, aT, on the y-axis. Three lines are shown on the graph representing measured, backcalculated 1, and backcalculated 2 results. The three lines all begin at a shift factor just above 10² at 32 °F and decrease in a linear trend until they reach 131 °F. At this point, the shift factor for both backcalculated results is 10^-3 and 10^-5 for the measured.

Figure 148. Graphs. Comparison of nonlinear viscoelastic backcalculated and measured |E*| and phase angle for LTPP section 10101. Two graphs are shown. The graph on the left is labeled dynamic modulus. This graph has reduced frequency at 66 °F in inverse s on the x-axis and dynamic modulus in psi on the y-axis. Three curves are shown on the graph representing measured, backcalculated 1, and backcalculated 2 results. All of the curves have a similar trend beginning at 10^-6 s and increasing in a concave and convex manner until 10⁶ s. At the beginning, the dynamic modulus is 2 times⁶ psi for the measured results and just below this are the backcalculated 1 results. At the end of graph, the dynamic modulus for the measured results is 10⁴ psi, and the backcalculated 1 and backcalculated 2 results are just below this value.

The graph on the right is labeled phase angle. This graph has the reduced frequency at 66 °F in inverse s on the x-axis and the phase angle on the y-axis. Three lines are shown on the graph representing measured, backcalculated 1, and backcalculated 2 results. The lines begin at 10^-6 s and at a phase angle of 20 for backcalculated 2 results, and 14 for the measured and backcalculated 1 results. The lines all follow a parabolic trend, with the maximum phase angle occurring at 50 and 10^-3 s for backcalculated 2 results, 45 and 10^-2 s for backcalculated 1 results, and 42 and 10^-2 s for measured results. Then the lines all decrease down to 0 phase angle at 10⁶ s.

Figure 149. Graphs. Comparison of surface deflections of a layered elastic structure using ViscoWave-II and LAMDA. Three graphs are shown. The graph on the top left is labeled ViscoWave-II. Time in s is on the x-axis, and the deflection in mil on the y-axis. Seven lines are shown on the graph representing; r equal to 0, 8, 12, 18, 24, 36, and 60 inches. The lines all begin at 0 deflection and have a parabolic shape. After reaching a peak value, they decrease back to 0 deflection. The line for r = 0 inches has a peak deflection of 21 at 0.015 s, and for r = 8 inches, the peak deflection is 15 at 0.015 s. For r = 12 inches, the peak is at 12 and 0.015 s; for r = 18 inches, it is at 9 and 0.0175 s; for r = 24 inches, it is at 7 and 0.175 s; for r = 36 inches, it is at 5 and 0.02 s; and for r = 60 inches, it is 3 at 0.025 s.

The graph on the top right is labeled LAMDA. Time in s is on the x-axis and the deflection in mil on the y-axis. Seven lines are shown on the graph representing r equal to 0, 8, 12, 18, 24, 36, and 60 inches. This graph shows exactly the same results as those in the top left graph.

The bottom graph is labeled peak deflection. The sensor location in ft is on the x-axis, and the peak deflection in mil is on the y-axis. Two lines are shown on the graph representing LAMDA and ViscoWave. At a sensor location of 0, LAMDA has a peak deflection of 21 mil and ViscoWave has a peak deflection of 20.5 mil. After this point, the two lines have the same values in deflection. At sensor 1, the peak is 12, at sensor 2 it is 7, at sensor 3 it is 5, and at sensor 5, it is 4.

Figure 150. Graph. Simulated FWD load. This graph has time in s on the x-axis and the load in psi on the y-axis. The line on the graph has a parabolic shape with a double peak. The line begins at 0 psi and 0 s, the load remains at 0 until about 0.005 s when the load begins to increase quickly up to its first peak at 60 psi and 0.01 s. From here, the load begins to increase again until it reaches its peak at 80 psi and 0.016 s. It then decreases to 0 psi at 0.033 s and remains at 0 until 0.05 s.

Figure 151. Graph. AC layer master curve for viscoelastic simulation. This graph has time in s on the x-axis and the absolute value of E star on the y-axis in psi. The line on the graph begins at 5 million psi at 10^-8 s and remains at this value until about 10^-6 s; the line then begins to decrease concavely then convexly until it ends at 1,100 psi at 10⁸ psi.

Figure 152. Graphs. Results from ViscoWave-II for viscoelastic simulations of thin (top), medium (middle), and thick (bottom) pavements. Three graphs are shown. The top graph represents thin pavement. The graph has time in ms on the x-axis and the vertical deflections in mil on the y-axis. Nine lines are shown on the graph representing r equals 0, 8 12, 18, 24, 36, 48, 60, and 72 inches. All of the lines begin at 0 deflection and have a parabolic shape. After reaching a peak value, the lines decrease. The line for r equals 0 has a peak deflection of 27at 0.02 s; for r equals 8, the peak is 20 at 0.021 s; for r equals 12, it is 16 at 0.022 s; for r equals 18, it is 13 at 0.024 s; for r equals 24, it is 10 at 0.0026s, for r equals 36, it is 8 at 0.028 s; for r equals 48, it is 6 at 0.03 s; for r equals 60, it is 5at 0.038 s; and for r equals 72 it, is 3 at 0.04 s.

The graph in the middle represents medium pavement. The graph has time in s on the x‑axis and deflection in mil on the y-axis. Nine lines are shown on the graph representing r equals 0, 8, 12, 18, 24, 36, 48, 60, and 72 inches. All of the lines begin at 0 deflection and have a parabolic shape. After reaching a peak value the lines decrease. The line for r equals 0 has a peak deflection of 19 at 0.02 s; for r equals 8, the peak is 14 at 0.021s; for r equals 12, it is 13 at 0.022s; for r equals 18, it is 11 at 0.023 s; for r equals 24, it is 9 at 0.0025 s; for r equals 36, it is 7 at 0.028 s; for r equals 48, it is 5 at 0.03 s; for requals 60, it is 4 at 0.032 s; and for r equals 72, it is 3 at 0.038 s.

The graph at the bottom represents thick pavement. The graph has time in s on the x-axis and deflection in mil on the y-axis. Nine lines are shown on the graph representing r equals 0inches, requals 8, 12, 18, 24, 36, 48, 60, and 72 inches. All of the lines begin at 0 deflection and have a parabolic shape. After reaching a peak value, the lines decrease. The line for r equals 0 has a peak deflection of 13 at 0.02 s; for r equals 8, the peak is 9.5 at 0.022 s, for r equals 12 it is 9 at 0.023 s, for r equals 18, it is 8 at 0.024s; for r equals 24, it is 7 at 0.0025 s; for r equals 36, it is 6at 0.028 s; for r equals 48, it is 4.5 at 0.03 s; for r equals 60, it is 3.5 at 0.032 s; and for r equals 72, it is 2.5 at 0.035 s.

Figure 153. Graphs. Low- (left) and high- (right )temperature AC creep compliance curves used for ViscoWave II simulation. Two graphs are shown. The graph on the left is labeled low temperature. This graph has time in s on the x-axis and creep compliance in inverse psi times 10^-6 on the y-axis. The line has a logarithmic shape beginning at 0.3 inverse psi at 0 s and increasing to 1 inverse psi at 0.8 s. The graph on the right is labeled high temperature. The graph has time in s on the x-axis and the creep compliance in inverse psi times 10^-5 on the y‑axis. The line has a logarithmic shape beginning at 0.2 inverse psi at 0 s and increasing to 2.2inverse psi at 0.8 s.

Figure 154. Diagrams. Schematic of the pavement structure with half-space (left) and bedrock (right). Two diagrams are shown. The diagram on the left is labeled half-space. It shows a rectangle divided into three horizontal layers. The bottom layer is the thickest and is labeled subgrade. The depth is labeled h subscript 3 equals infinity. The middle layer is labeled base, and its depth is labeled h subscript 2 equals 10 inches. The top layer is labeled AC, and its depth is labeled h subscript 1 equals 6 inches. Above the rectangle is a distributive load represented by arrows pointing downward onto to the surface of the top layer. The distributive load is labeled 80psi and has an arrow showing that it is 1 ft long.

The diagram on the right is labeled bedrock. It shows a rectangle divided into four horizontal layers. The bottom layer is the thickest layer, and it is labeled bedrock. The next layer up is labeled subgrade; its depth is labeled h subscript 3 equals 120 inches. The next layer is labeled base; its depth is labeled h subscript 2 equals 10 inches. The top layer is labeled AC; its depth is labeled h subscript 1 equals 6 inches. Above the rectangle is a distributive load represented by arrows pointing downward onto to the surface of the top layer. The distributive load is labeled 80psi and has an arrow showing that it is 1 ft long.

Figure 155. Diagrams. Axisymmetric FEM geometry (top) and FEM mesh (bottom) used for simulation of pavement response under FWD loading. Two diagrams are shown. The top diagram is labeled axisymmetric finite element geometry. It shows a rectangle divided into three layers horizontally. Lining the two sides of the rectangle are small circles, and lining the bottom of the rectangle are small triangles. The length of the rectangle is labeled 20 ft. The bottom layer is the thickest and is labeled subgrade; its depth is shown as 480 inches for half-space simulation and 100 inches for bedrock simulation. The next layer up is labeled base, and its depth is labeled 6inches. At the left end of the rectangle is a line extending up vertically with an arrow indicating a counterclockwise direction, and it is labeled axisymmetric.

The bottom diagram is labeled finite element mesh. It shows a rectangle with a pattern of lines drawn across it creating tiny blocks. This crisscross type pattern is close together near the upper left corner and spreads out toward the lower right corner. The lining on the two sides of the rectangle are small circles, and lining the bottom of the rectangle are small triangles. On the top surface is a small distributive load with its length labeled 6 inches beginning at the left end of the rectangle. Also at the left end of the rectangle is a line extending up vertically with an arrow indicating a counterclockwise direction, and it is labeled axisymmetric.

Figure 156. Graphs. Surface deflections of a layered viscoelastic structure with a half-space at low temperature simulated using ViscoWave-II and ADINA. Two graphs are shown. The left graph is labeled ViscoWave-II. Time in s is on the x-axis and deflection in mil on the y-axis. There are seven lines shown on the graph representing r equals 0, 8, 12, 18, 24, 36, and 60 inches. All of the lines begin at 0 deflection and have a parabolic shape, After reaching a peak value, the lines decrease toward 0. The line for r equals 0 has a peak deflection of 9 at 0.017 s; for r equals 8, the peak is 8.5 at 0.017 s; for r equals 12, it is 8 at 0.017 s, for r equals 8, it is 7 at 0.018 s; for r equals 24, it is 6 at 0.0018 s; for r equals 36, it is a 4.5 at 0.02 s; and for r equals 60, it is 2.5 at 0.025 s.

The right graph is labeled ADINA. Time in s is on the x-axis and deflection in mil on the y-axis. Seven lines are shown on the graph representing; r equals 0, 8, 12, 18, 24, 36, and 60inches. All of the lines begin at 0 deflection and have a parabolic shape, after reaching a peak value, the lines decrease. The line for r equals 0 has a peak deflection of 9 at 0.017 s; for r equals 8, the peak is 8.5 at 0.017 s; for r equals 12, it is 7.8 at 0.017 s; for r equals 18, it is 6.8 at 0.018 s; for r equals 24, it is 5.8 at 0.019 s, for r equals 36, it is 4.2 and 0.02 s; and for r equals 60, it is 2.5at 0.025 s.

Figure 157. Graphs. Surface deflections of a layered viscoelastic structure with a bedrock at low temperature simulated using ViscoWave-II and ADINA. Two graphs are shown. The left graph is labeled ViscoWave-II. Time in s is on the x-axis and deflection in mil on the y‑axis. Seven lines are shown on the graph representing r equals 0 8, 12, 18, 24, 36, and 60inches. All of the lines begin at 0 deflection and have a parabolic shape. After reaching their peak value in deflection, the lines decrease to a negative deflection of 2 mil at about 0.04 s. The lines then increase back to a positive deflection between 1 and 2 mil at 0.06 s before beginning to decrease again until the graph ends at 0.07 s. The line for r equals 0 has a peak deflection of 9 at 0.017 s; for r equals 8, the peak is 7.5 at 0.017 s; for r equals 12, it is 7 at 0.017 s; for r equals 18, it is 6.2at 0.018 s; for r equals 24, it is 5.5 at 0.019 s; for r equals 36, it is 3.8 at 0.019 s; and for r equals 60, it is 2 at 0.02 s.

The right graph is labeled ADINA. Time in s is on the x-axis and deflection in mil on the y-axis. Seven lines are shown on the graph representing r equals 0 8, 12, 18, 24, 36, and 60inches. All of the lines begin at 0 deflection and have a parabolic shape. After reaching their peak value in deflection, the lines decrease to a negative deflection of 2 mil at about 0.038 s, except for r equals 60, which decreases to 1.5 mil. The lines then increase to a positive deflection of 1.5 mil at 0.057 s, except for r equals 60, which increases to 1 mil. After this smaller peak, the lines decrease again until the graph ends at 0.07 s. The line for r equals 0 has a peak deflection of 9 at 0.017 s; for r equals 8, the peak is at 8.5 and 0.017 s; for r equals 12, it is 7.5 at 0.018 s; for r equals 18, it is 6.5 at 0.018 s; for r equals 24, it is 4.5 at 0.019 s; for r equals 36, it is 4 at 0.02 s, and for r equals 60, it is 2 at 0.022 s.

Figure 158. Graphs. Surface deflections of a layered viscoelastic structure with a half-space at high temperature simulated using ViscoWave-II and ADINA. Two graphs are shown. The left graph is labeled ViscoWave-II. Time in s is on the x-axis and deflection in mil on the y-axis. Seven lines are shown on the graph representing r equals 0, 8, 12, 18, 24, 36, and 60inches. All of the lines begin at 0 deflection and have a parabolic shape. After reaching their peak value in deflection, the lines decrease to about 1 mil at 0.05 s. The line for r equals 0 has a peak deflection of 15.5 at 0.018 s; for r equals 8 the peak is 12.5 at 0.018 s; for r equals 12, the peak is 11 at 0.019 s, for r equals 18, it is 8.5 at 0.019 s; for r equals 24, it is 7 at 0.02 s, for r equals 36, it is 4.5 at 0.022 s; and for r equals 60, it is 2.5 at 0.028 s.

The right graph is labeled ADINA. Time in s is on the x-axis and deflection in mil on the y-axis. Seven lines are shown on the graph representing r equals 0, 8, 12, 18, 24, 36, and 60inches. All of the lines begin at 0 deflection and have a parabolic shape. After reaching their peak value in deflection, the lines decrease to about 1 mil at 0.05 s. The line for r equals 0 has a peak deflection of 15 at 0.017 s; for r equals 8, the peak is 12.5 at 0.017 s; for r equals 12, the peak is 11 at 0.017 s, for r equals 18, it is 8.5 at 0.018 s; for r equals 24, it is 7 at 0.018 s, for r equals 36, it is 4.5 at 0.02 s; and for r equals 60, it is 2.5 at 0.027 s.

Figure 159. Graphs. Surface deflections of a layered viscoelastic structure with a bedrock at high temperature simulated using ViscoWave-II and ADINA. Two graphs are shown. The left graph is labeled ViscoWave-II. Time in s is on the x-axis and deflection in mil on the y-axis. Seven lines are shown on the graph representing r equals 0, 8, 12, 18, 24, 36, and 60inches. All of the lines begin at 0 deflection and have a parabolic shape. After reaching their peak value in deflection, the lines decrease to a negative deflection between 0.5 and 2 at about 0.038 s. The lines then increase to a positive deflection between 0.5 and 2 at 0.06 s. After this smaller peak, the lines begin to decrease again until the graph ends at 0.07 s. The line for r equals 0 has a peak deflection of 14.5 at 0.018 s; for r equals 8, the peak is 12 at 0.018 s; for r equals 12, it is 10 at 0.018 s; for r equals 18, it is 8 at 0.018 s; for r equals 24, it is 6 at 0.018 s; for r equals 36, it is 4 at 0.02 s; and for r equals 60, it is 1.5 at 0.025 s.

The right graph is labeled ADINA. Time in s is on the x-axis and deflection in mil on the y-axis. Seven lines are shown on the graph representing r equals 0, 8, 12, 18, 24, 36, and 60inches. All of the lines begin at 0 deflection and have a parabolic shape. After reaching their peak value in deflection, the lines decrease to a negative deflection between 1 and 3 at about 0.037 s. The lines then increase to a positive deflection between 1 and 3 at 0.06 s. After this smaller peak, the lines begin to decrease again until the graph ends at 0.07 s. The line for r equals 0 has a peak deflection of 15 at 0.017 s; for r equals 8, the peak is 12 at 0.017 s; for r equals 12, the peak is 10.5 at 0.017 s, for r equals 18, it is 8 at 0.017 s; for r equals 24, it is 6.5 at 0.017 s; for r equals 36, it is 4 at 0.018 s; and for r equals 60, it is 2 at 0.025 s.

Figure 160. Diagram. Pavement structure with soils having E-values increasing with depth. This diagram shows a rectangle divided into six equal horizontal layers. The bottom layer is labeled E subscript subgrade 10 equals 2 million psi; the depth of this layer is labeled h subscript 10 equals 4.8 ft. The next layer up has no labels and is blank. The layer on top of that is labeled E subscript subgrade 2 equals 13,500 + the convolution of h subscript 5 and slope; its depth is labeled h subscript 5 equals 4.8 ft. There is an arrow pointing to these bottom three layers labeled subgrade. The layer above these layers is labeled E subscript subgrade 1 equals 13,500 psi; its depth is labeled h subscript 4 equals 2 ft. The next layer (second from the top) is labeled E subscript base equals 20,000 psi; its depth is labeled h subscript 3 equals 6 inches. The top layer is labeled AC; its depth is labeled h subscript 1 equals 4 inches. Above the top layer is a distributive load represented by arrows pointing downward onto the surface of the top layer. The distributive load is labeled 80 psi and has an arrow showing that it is 1 ft long.

Figure 161. Graph and Diagram. AC layer parameters. A graph and a diagram are shown. The graph at the top is labeled master curve. It has the time in s on the x-axis and the absolute value of the complex conjugate E in psi on the y-axis. There are two lines indicated on the graph representing the sigmoidal and Prony series. The two lines overlap and begin at 10^-8 s and 1,200,000 psi. The lines then decrease in a concave then convex manor ending at 10⁷ s at 5,000psi.

The diagram at the bottom is labeled temperature profile. It has two parallel horizontal lines with the area between them labeled AC. The distance between these lines is indicated with two arrows showing that the distance to the center from each of the two lines is 2 inches. Next to this is the temperature profile, which is shown as two steps. The bottom line, or bottom step has a temperature of 79.3 °F, and the next step is at the same height as the half distance between the two horizontal lines. This step has a temperature of 86.9 °F.

Figure 162. Graphs. Surface deflections of pavement structure with shallow stiff layer and soils having E-values increasing with depth. Two graphs are shown. The left graph is labeled shallow stiff layer. Time in s is on the x-axis and deflection in mil on the y-axis. Nine lines are shown on the graph representing r equals 0, 8, 12, 18, 24, 36, 48, 60, and 72 inches. All of the lines begin at 0 deflection and have a parabolic shape. After reaching their peak value in deflection, the lines decrease to a negative deflection between 1 and 3 at about 0.04 s. The lines then increase to a deflection of 0 at 0.05 s. The line for r equals 0 has a peak deflection of 23 at 0.02 s; for r equals 8, the peak is 16 at 0.02 s; for r equals 12, the peak is 13 at 0.02 s; for r equals 18, it is 10 at 0.021 s; for r equals 24, it is 7 at 0.022 s; for r equals 36, it is 4 at 0.024 s; for r equals 48, it is 3 at 0.026 s; for r equals 60, it is 2at 0.026 s; and for r equals 72, it is 1 at 0.03 s.

The second graph is labeled soils having E-values increasing with depth. Time in s is on the x‑axis and the deflection in mil on the y-axis. Nine lines are shown on the graph representing r equals 0, 8, 12, 18, 24, 36, 48, 60, and 72 inches. All of the lines begin at 0 deflection and have a parabolic shape. After reaching their peak value in deflection, the lines decrease to a deflection between 0 and -2 at about 0.035 s. The lines then increase to a deflection between 0 and 1 at 0.05s. The line for r equals 0 has a peak deflection of 21 at 0.018s; for r equals 8, the peak is 13at 0.019 s; for r equals 12, the peak is 11 at 0.019 s; for r equals 18, it is 7 at 0.02 s, for r equals 24, it is 5 at 0.02 s; for r equals 36, it is 2 at 0.022 s; for r equals 48, it is 1 at 0.027 s; for r equals 60, it is 0.5 at 0.028 s; and for r equals 72, it is 0.25 at 0.029 s.

Figure 163. Photos. FWD used during the field tests. Two photos are shown. The larger photo shows a yellow four-wheel trailer attached to the back of a van. There is a metal box that encases the falling weight deflectometer (FWD) sitting on the trailer. There are doors on either side of the box that are open. The smaller second photo shows a close-up of the FWD loading system and the deflection sensors that are inside the door on the metal box.

Figure 164. Photo. Illustration of temperature measurement at different depths of the pavement. This photo shows five holes that were drilled into the pavement at equal distances along a white painted line. The pavement also has the label 11:06 a.m. Extending from each of the holes is a cord runs to a box. The FWD test loading system is shown in the background. The photo also shows a man using a handheld temperature sensor to evaluate the temperature in the fifth hole.

Figure 165. Graphs. Comparison of deflection response from ViscoWave-II and LAVA predictions for station 1. Eight graphs are shown displaying the results for sensors 1 through 8. The time in s is on the x-axis and the deflection in mil is on the y-axis. There are three lines on each graph representing LAVE, ViscoWave-II, and field data.

The first graph is labeled Sensor 1. The three lines on the graph all begin at 0 deflection, and at 0.005 s, the deflection begins to increase until it reaches a maximum value. The line representing LAVA has a double peak; the smaller peak is at 15 mil at 0.01 s, and the larger peak is at 22 at 0.017 s. After this peak, the line decreases sharply to 2 at 0.034 s, and then decreases further to 0.5 mil at 0.05 s. The line for ViscoWave-II has a peak deflection at 22.5 mil at 0.02 s, and the line for field data has the same peak at 0.021 s. After this point, both lines decrease to –2 at 0.04s and then increase to 0 and 1, respectively, at 0.05 s.

The second graph is labeled Sensor 2. The three lines on the graph all begin at 0 deflection, and about 0.005 s, the deflection begins to increase until it reaches a maximum value. The line representing LAVA has a double peak; the smaller peak is at 11.5 mil at 0.02 s, and the larger peak is at 16 at 0.027 s. After this peak, the line decreases to 1 at 0.033 s and then decreases further to 0 mil at 0.05 s. The line for ViscoWave-II has a peak deflection at 17 mil at 0.02 s, and the line for field data has a peak at 16.5 at 0.021 s. After this point, both lines decrease to –2 at 0.04 s and then increase to 0 at 0.05 s.

The third graph is labeled Sensor 3. The three lines on the graph all begin at 0 deflection, and at 0.005 s, the deflection begins to increase until it reaches a maximum value. The line representing LAVA has a double peak; the smaller peak is at 9 mil at 0.01 s, and the larger peak is at 12.5 at 0.017 s. After this peak, the line decreases to 0 at 0.035 s and remains at 0 until it reaches 0.5 s. The line for ViscoWave-II has a peak deflection at 13.5 mil at 0.019 s, and the line for field data has a peak at 13.4 mil at 0.02 s. After this point, both lines decrease to –2 at 0.04 s and then increase to 0 at 0.05 s.

The fourth graph is labeled Sensor 4. The line representing LAVA begins at 0 deflection until it reaches 0.005 s, when the deflection begins to increase. This line has a double peak; the smaller peak is at 6 mil at 0.01 s, and the larger peak is at 8.2 mil at 0.017 s. After this peak, the line decreases to 0 at 0.034 s and remains at 0 until it reaches the end of the graph at 0.05 s. The two lines representing ViscoWave-II and field data both begin at 0 and remain so until they reach 0.008 s. The two lines then increase until they reach their peak deflections. The line for ViscoWave-II has a peak deflection of 9.5mil at 0.019 s, and the line for field data has a peak of 16.5 mil at 0.021 s. After these two lines reach their peaks, they decrease to a negative deflection of –2 at 0.04 s and then increase to –1 at 0.05 s.

The fifth graph is labeled Sensor 5. The line representing LAVA begins at 0 deflection until it reaches 0.005 s, when the deflection begins to increase. This line has a double peak; the smaller peak is at 4.2 mil at 0.01 s, and the larger peak is at 5.5 mil at 0.015 s. After this peak, the line decreases to –1 at 0.032 s and remains there until the graph ends at 0.05 s. The two lines representing ViscoWave-II and field data both begin at 0 and remain so until they reach 0.008 s. The two lines then increase until they reach their peak deflections. The two lines have a peak deflection of 6.7mil at 0.02 s. After these two lines reach their peaks, they decrease to a negative deflection of –2 at 0.04 s and then increase to –0.5 at 0.05 s.

The sixth graph is labeled Sensor 6. The line representing LAVA begins at 0 deflection until it reaches 0.005 s, and then the deflection begins to increase. This line has a double peak; the smaller peak is at 1.8 mil at 0.012 s, and the larger peak is at 2.5 mil at 0.017 s. After the larger peak, the line decreases to 0 at 0.032 s and remains at 0 until the graph ends at 0.05 s. The two lines representing ViscoWave-II and field data both begin at 0 deflection and remain so until they reach about 0.009s, when the two lines decrease slightly to –0.2 mil. The point where the lines have negative deflection is circled on the graph. At 0.01 s, the two lines begin to increase until they reach their peak deflections of 3.8 mil at 0.025 s. After the lines reach their peak, they decrease to a negative deflection at 0.04 s; for the ViscoWave-II the negative deflection is –1.7, and for the field data it is –1.5. The two lines then increase again, and at 0.05 s, their deflections are –0.7 and –0.5 for the ViscoWave-II and the field data, respectively.

The seventh graph is labeled Sensor 7. The line representing LAVA begins at 0 deflection until it reaches 0.005 s, and then the deflection begins to increase. This line has a double peak; the smaller peak is at 0.9 mil at 0.01 s, and the larger peak is at 1.2 mil at 0.017 s. After the larger peak, the line decreases to 0 at 0.032 s and remains at 0 until the graph ends at 0.05 s. The two lines representing ViscoWave-II and field data both begin at 0 deflection and remain so until they reach about 0.01 s, when the two lines decrease slightly to –0.25 mil. The point where the lines have negative deflection is circled on the graph. At 0.015 s, the two lines begin to increase until they reach their peak deflections of 2.4 mil at 0.027 s. After the lines reach their peak, they decrease to a negative deflection at 0.043 s; for the ViscoWave-II the negative deflection is –1.5, and for the field data it is –1.3. The two lines then increase again, and at 0.05 s, their deflections are –0.8 and –0.6 for the ViscoWave-II and the field data, respectively.

The eighth graph is labeled Sensor 8. The line representing LAVA begins at 0 deflection until it reaches 0.005 s, and then the deflection begins to increase. This line has a double peak; the smaller peak is at 0.4 mil at 0.01 s, and the larger peak is at 0.6 mil at 0.017 s. After the larger peak, the line decreases to 0 at 0.032 s and remains at 0 until the graph ends at 0.05 s. The two lines representing ViscoWave-II and field data both begin at 0 deflection and remain so until they reach about 0.01 s, when the two lines decrease slightly to –0.25 mil. The point where the lines have negative deflection is circled on the graph. The circled portion is magnified in the inset image in the graph. At 0.015 s, the two lines begin to increase until they reach their peak deflections at 0.028s. The peak deflection for the ViscoWave-II is 16.5 mil, and for the field data is 18 mil. After the lines reach their peak, they decrease to a negative deflection at 0.045 s; for the ViscoWave-II the negative deflection is –1.2, and for the field data it is –1. The two lines then increase again, and at 0.05 s, their deflections are –0.8 and –0.6 for the ViscoWave-II and the field data, respectively.

Figure 166. Graphs. Comparison of deflection response from ViscoWave-II and LAVA predictions for station 3. Eight graphs are shown displaying the results for sensors 1 through 8. The time in s is on the x-axis and the deflection in mil is on the y-axis. There are three lines on each graph representing LAVE, ViscoWave-II, and field data.

The first graph is labeled Sensor 1. The three lines on the graph all begin at 0 deflection, and at 0.005 s, the deflections begin to increase until they reach a maximum value. The line representing LAVA has a double peak; the smaller peak is at 17 mil at 0.01 s, and the larger peak is at 24 mil at 0.017 s. After this peak, the line decreases sharply to 2 mil at 0.033 s, and then decreases further to 0.5 mil at 0.05 s. The lines for ViscoWave-II and field data have a peak deflection at 25 mil at 0.02 s. After this peak, the lines both decrease to –2 at 0.04 s and then increase again to –0.5 mil for the ViscoWave-II and –1 for the field data at 0.05 s.

The second graph is labeled Sensor 2. The three lines on the graph all begin at 0 deflection, and at about 0.005 s, the deflections begin to increase until they reach a maximum value. The line representing LAVA has a double peak; the smaller peak is at 13 mil at 0.01 s, and the larger peak is at 17 mil at 0.018 s. After this peak, the line decreases to 1 at 0.033 s, and then decreases further to 0 mil at 0.05 s. The lines for ViscoWave-II and field data have a peak deflection at 19mil at 0.02 s. After this point, both lines decrease to –2 mil at 0.04 s and then increase to
–1 mil for the ViscoWave-II and 0 mil for the field data at 0.05 s.

The third graph is labeled Sensor 3. The three lines on the graph all begin at 0 deflection, and at 0.005 s, the deflection begins to increase until it reaches a maximum value. The line representing LAVA has a double peak; the smaller peak is at 10 mil at 0.01 s, and the larger peak is at 13.5mil at 0.018 s. After this peak, the line decreases to 0 at 0.035 s and remains at 0 until it reaches 0.5 s. The line for ViscoWave-II has a peak deflection at 15 mil at 0.021 s, and the line for field data has a peak at 16 mil at 0.02 s. After this point, both lines decrease to –3 mil at 0.04s and then increase to –1 mil for the ViscoWave-II and 0 mil for the field data at 0.05 s.

The fourth graph is labeled Sensor 4. The line representing LAVA begins at 0 deflection until it reaches 0.005 s, when the deflection begins to increase. This line has a double peak; the smaller peak is at 7 mil at 0.01 s, and the larger peak is at 9 mil at 0.015 s. After this peak, the line decreases to 0 at 0.034 s and remains at 0 until the graph ends at 0.05 s. The two lines representing ViscoWave-II and field data both begin at 0 and remain so until 0.008 s. The two lines then increase until they reach their peak deflections. The line for ViscoWave-II has a peak deflection of 10.5 mil at 0.02 s, and the line for field data has a peak of 11 mil at 0.02 s. After these two lines reach their peaks, they decrease to a negative deflection of –2.5 for the ViscoWave-II and –3 for the field data at 0.04 s; then the two lines increase again to –1 and –0.5 mil, respectively, at 0.05 s.

The fifth graph is labeled Sensor 5. The line representing LAVA begins at 0 deflection until it reaches 0.005 s, when the deflection begins to increase. This line has a double peak; the smaller peak is at 4.5 mil at 0.01 s, and the larger peak is at 6 mil at 0.017 s. After this peak, the line decreases to 0 at 0.032 s and remains there until the graph ends at 0.05 s. The two lines representing ViscoWave-II and field data both begin at 0 and remain so until 0.009 s. The two lines then increase until they reach their peak deflections. The line for ViscoWave-II has a peak deflection of 7.5 mil at 0.023 s, and the line for field data has a peak of 8 mil at 0.023 s. After these two lines reach their peaks, they decrease to a negative deflection of –2.5 for the ViscoWave-II and –3 for the field data at 0.042 s; then the two lines increase again to –1 and –0.5 mil, respectively, at 0.05 s.

The sixth graph is labeled Sensor 6. The line representing LAVA begins at 0 deflection and remains so until it reaches 0.005 s, and then the deflection begins to increase. This line has a double peak; the smaller peak is at 2 mil at 0.01 s and the larger peak is at 2.7 mil at 0.018 s. After the larger peak, the line decreases to 0 at 0.032 s and remains at 0 until the graph ends at 0.05 s. The two lines representing ViscoWave-II and field data both begin at 0 deflection and remain so until about 0.009 s, when the two lines decrease slightly to –0.2 mil. The point where the lines have negative deflection is circled on the graph. At 0.012 s, the two lines begin to increase until they reach their peak deflections of 4.5 mil at 0.025 s. After the lines reach their peak, they decrease to a negative deflection at 0.04 s; for the ViscoWave-II, the negative deflection is 2, and for the field data, it is 2.2. The two lines then increase again, and at 0.05 s, their deflections are –0.8 and –1.2 for the ViscoWave-II and the field data, respectively.

The seventh graph is labeled Sensor 7. The line representing LAVA begins at 0 deflection and remains so until it reaches 0.005 s, and then the deflection begins to increase. This line has a double peak; the smaller peak is at 1 mil at 0.01 s, and the larger peak is at 1.3 mil at 0.017 s. After the larger peak, the line decreases to 0 at 0.032 s and remains at 0 until the graph ends at 0.05 s. The two lines representing ViscoWave-II and field data both begin at 0 deflection and remain so until about 0.01 s, when the two lines decrease slightly to –0.25 mil. The point where the lines have negative deflection is circled on the graph. At 0.015 s, the two lines begin to increase until they reach their peak deflection of 2.7 mil at 0.027 s. After the lines reach their peak, they decrease to a negative deflection of –1.7 at 0.045 s. The two lines then increase again, and at 0.05 s, their deflections are –1 and –0.7 for the ViscoWave-II and the field data, respectively.

The eighth graph is labeled Sensor 8. The line representing LAVA begins at 0 deflection and remains so until it reaches 0.005 s, and then the deflection begins to increase. This line has a double peak; the smaller peak is at 0.4 mil at 0.01 s, and the larger peak is at 0.6 mil at 0.018 s. After the larger peak, the line decreases to 0 at 0.032 s and remains at 0 until the graph ends at 0.05 s. The two lines representing ViscoWave-II and field data both begin at 0 deflection and remain so until about 0.01 s, when the two lines begin to decrease to –0.35 mil. The point where the lines have negative deflection is circled on the graph. At 0.017 s, the two lines begin to increase until they reach their peak deflection of 1.8 at 0.03 s. After the lines reach their peak, they decrease to a negative deflection at 0.045 s; for the ViscoWave-II, the negative deflection is –1.35 mil, and for the field data it is –1.4 mil. The two lines then increase again, and at 0.05 s, their deflections are –1 and –0.8 for the ViscoWave-II and the field data, respectively.

Figure 167. Graphs. Comparison of ViscoWave-II and LAVA solutions with measured deflections for station 1. Two graphs are shown. The first graph is a bar graph labeled Peak Deflection. This graph has sensor number, D1 through D8, on the x-axis and the peak deflection in mil on the y-axis. There are three bars for each sensor representing ViscoWave, LAVA, and measured results. For sensor D1, the peak deflection found for the ViscoWave is 22, for LAVA is 21, and for the measured is 22. For sensor D2, the peak deflection found for the ViscoWave is 16, for LAVA is 15, and for the measured is 16. For sensor D3, the peak deflection found for the ViscoWave is 12, for LAVA is 11, and for the measured is 13. For sensor D4, the peak deflection found for the ViscoWave is 7, for LAVA is 6, and for the measured is 8. For sensor D5, the peak deflection found for the ViscoWave is 6, for LAVA is 4, and for the measured is 6. For sensor D6, the peak deflection found for the ViscoWave is 3, for LAVA is 2, and for the measured is 3. For sensor D7, the peak deflection found for the ViscoWave is 2, for LAVA is 1, and for the measured is 2. Lastly, for sensor D8, the peak deflection found for the ViscoWave is 1, for LAVA is 0.5, and for the measured is 1.

The second graph is labeled Ratio of Predicted to Measured Deflection. The graph is a line graph that has the sensor numbers 1 through 8, on the x-axis and the predicted-to-measured ratio on the y-axis. Two lines are shown on the graph representing ViscoWave and LAVA. The two lines begin at a ratio of 1 for sensor 1. At sensor 2, the ratio for the ViscoWave is 1.025 and for LAVA is 0.95. At sensor 3, the ratio for the ViscoWave is 1.02 and for LAVA is 0.9. For sensor 4, the ratio for the ViscoWave is 1.025 and for LAVA is 0.85. For sensor 5, the ratio for the ViscoWave is 1 and for LAVA is 0.8. For sensor 6, the ratio for the ViscoWave is 1 and for LAVA is 0.65. For sensor 7, the ratio for the ViscoWave is 1.02 and for LAVA is 0.5. For sensor 8, the ratio for the ViscoWave is 0.9 and for LAVA is 0.35.

Figure 168. Graphs. Example time histories from ViscoWave-II with decreasing stiff layer modulus and measured sensor deflections for station 1. Eight graphs are shown for different stiff layer moduli. Each graph has the time in s on the x-axis and the deflection in mil on the y‑axis. Nine lines are shown on each graph representing D1 through D9.

The first graph is labeled Estiff equals 2 times 10⁶ psi. The lines on the graph all have a parabolic shape, and after reaching their peak deflections, the lines all decrease to a negative deflection between –1 and –3 mil at about 0.04 s. After the negative deflection, the lines all begin to increase again toward 0 deflection until the graph ends at 0.05 s. The peak in deflection for D1 is 23 mil at 0.02 s; for D2, it is 17 mil at 0.02 s; for D3, it is 13.5 mil at 0.02 s; for D4, it is 9.5 mil at 0.021 s; for D5, it is 7 mil at 0.022 s; for D6, it is 4 mil at 0.024 s; for D7, it is 3 mil at 0.025 s; for D8, it is 2.5 mil at 0.029 s; and for D9, it is 2 mil at 0.03 s.

The second graph is labeled Estiff equals 200,000 psi. This graph is identical to the first graph. The third graph is labeled Estiff equals 100,000 psi. This graph is identical to the first two graphs. The fourth graph is labeled Estiff equals 60,000 psi. This graph is identical to the first three graphs.

The fifth graph is labeled Estiff equals 40,000 psi. This graph is identical to the first four graphs except the peak deflection for D7 occurs at 0.028 s, for D8, it occurs at 0.03 s, and for D9, it occurs at 0.032 s. The lines also all decrease to a negative deflection between –2 and –1 mil.

The sixth graph is labeled Estiff equals 20,000 psi. This graph is identical to the fifth graph except that after the peak deflections, the lines decrease to a deflection between –1 and 1 mil.

The seventh graph is labeled Estiff equals 10,000 psi. This graph is identical to the fifth graph except the line for D6 has a peak deflection of 4.5 mil, the line for D7 has a peak deflection of 3.5 mil, and the lines for D8 and D9 have a peak of 3 mil. After the peak deflections, the lines decrease to a deflection between 0 and 1 mil.

The eighth graph is labeled Measured Deflections. This graph is identical to the fifth graph except the line for D2 has a peak deflection of 16.5 mil, the lines for D7 and D8 have a peak deflection of 3 mil, and the line for D9 remains at a constant 0 deflection throughout the graph. After the peak deflections, the lines decrease to a deflection between 0 and –3 mil.

Figure 169. Graph. Effect of stiff layer modulus on ratio of predicted to measured sensor deflections for station 1. The graph is a bar graph. It has the stiff layer modulus in psi, ranging from 2 million to 10,000 psi, on the x-axis, and the predicted-to-measured ratio on the y-axis. For each stiff layer modulus, there are eight bars representing sensors D1 through D8.

For sensor D1, the predicted-to-measured ratio is constantly 0.98; for sensor D2 the predicted-to-measured ratio is constantly 1; for sensor D3, the predicted-to-measured ratio is constantly 0.99; and for sensor D4, the predicted to measured ratio is constantly 1. For sensor D5, the ratio begins at 0.97 and slowly increases with decreasing stiffness layer modulus until it is 1.05 at 10,000 psi. For sensor D6, the ratio begins at 0.96 and increases with decreasing stiffness layer modulus until it is 1.05 at 10,000 psi. For sensor D7, the ratio begins at 0.98 and slowly increases with decreasing stiffness layer modulus until it is 1.3 at 10,000 psi. For sensor D8, the ratio begins at 0.9 and slowly increases with decreasing stiffness layer modulus until it is 1.55 at 10,000 psi.

Figure 170. Graph. Effect of stiff layer modulus on predicted sensor deflection amplification for station 1. The graph is a bar graph. It has the stiff layer modulus in psi, ranging from 2 million to 10,000 psi, on the x-axis, and amplification on the y-axis. For each stiff layer modulus, there are eight bars representing sensors D1 through D9.

For sensors D1 through D5, the predicted-to-measured ratio is constantly 0.95 for all values of the stiff layer modulus. For sensor D6, the amplification begins at 0.95 and slowly increases with decreasing stiffness layer modulus until it is 1 at 10,000 psi. For sensor D7, the ratio begins at 0.95 and increases with decreasing stiffness layer modulus until it is 1.3 at 10,000 psi. For sensor D8, the ratio begins at 0.95 and slowly increases with decreasing stiffness layer modulus until it is 1.7 at 10,000 psi. For sensor D9, the ratio begins at 0.95 and slowly increases with decreasing stiffness layer modulus until it is 2.1 at 10,000 psi.

Figure 171. Equation. Fitting steps of the Prony series. Five equations are shown. The first line is labeled step 1. It states that E parenthesis t subscript n end parenthesis equals c subscript 1 plus the quotient of c subscript 2 divided by parenthesis 1 plus e raised to the negative c subscript 3 minus c subscript 4 times the log of t subscript n end parenthesis power, end quotient. Next to the equation is the note: n equals 1, up to any value N.

The second line is labeled step 2. It states that log of t subscript n equals the log of parenthesis t end parenthesis, minus the log of parenthesis a times T end parenthesis.

The third line is labeled step 3. It states the log of parenthesis a times T end parenthesis equals a subscript 1 multiplied by T squared plus a subscript 2 multiplied by T, plus a subscript 3.

The fourth line is labeled step 4. It states that E parenthesis t subscript n end parenthesis minus E subscript infinity equals the summation, with the lower bound m equals 0 and the upper bound 14, of e raised to parenthesis, negative t subscript n divided by T times K subscript m end parenthesis power, multiplied by E subscript m. Next to the equation is the note: a n equals 1, up to any value N. In this equation, the part of the equation on the left side of the equal sign is labeled A. The part of the equation pm the right side of the equation is divided into two portions, the portion labeled B is the summation, with the lower bound m equal 0 and the upper bound 14, of: e raised to parenthesis, negative t subscript n divided by T times K subscript m end parenthesis power. The part labeled C is E subscript m.

The fifth line is labeled step 5. It states minimize the absolute value of B multiplied by C minus A end absolute value, such that C is greater or equal to 0.

Figure 172. Equation. Optimization problem. The first line is labeled objective function. It states that E subscript r equals the summation with the lower bound k equals 1 and the upper bound m, of 100, multiplied by the summation with the lower bound i,o equal 1 and the upper bound n, of the following: the quotient of the absolute value of parenthesis d subscript i superscript k minus d subscript o superscript k end parenthesis, end absolute value, divided by the quantity d superscript k end quantity subscript max.

The next line is labeled bound constraints. There are nine different cases that are divided into three subgroups. The first group is the limits for c. The first line states: c subscript 1, superscript l is less than or equal to c subscript 1, which is less than or equal to c subscript 1, superscript u. The second line states: c lower subscript 2, superscript l is less than or equal to c subscript 2, which is less than or equal to c subscript 2, superscript u. The third line states: c subscript 3, superscript l is less than or equal to c subscript 3, which is less than or equal to c subscript 3, superscript u. The fourth line states: c subscript 4, superscript l is less than or equal to c subscript 4, which is less than or equal to c subscript 4, superscript u.

The second group is the limits for a. The first line states a subscript 1, superscript l is less than or equal to a subscript 1, which is less than or equal to a subscript 1, superscript u. The second line states: a subscript 2, superscript l is less than or equal to a subscript 2, which is less than or equal to a subscript 2, superscript u. The third line states: a subscript 3, superscript l is less than or equal to a subscript 3, which is less than or equal to a subscript 3, superscript u.

The third group is the limits for E: The first line states E subscript b, superscript l is less than or equal to E subscript b, which is less than or equal to E subscript b, superscript u. The second line states: E subscript sg, superscript l is less than or equal to E subscript sg, which is less than or equal to E subscript sg, superscript u.

Figure 173. Graph. Backcalculated master curve for different population/generation combinations optimization problem. This graph has the time in s on the x-axis and the absolute value of the complex conjugate E in psi on the y-axis. Six lines are shown on the graph representing the population-generations: E parenthesis t end parenthesis, 25, 70, 100, 200, and 300. The lines all have the same shape; they begin at 10^-8 s and decrease concavely then convexly until they reach 10⁸ s. The population-generation line for 35 begins at 6 million psi and ends at 3,000 psi, the line for 300 begins at 3 million and ends at 8,000 psi, the line for E parenthesis t end parenthesis also begins at 6 million psi but ends at 5,000 psi, the line for 200begins at 1.5 million psi and ends at 7,000 psi, the line for 100 begins at 1 million psi and ends at 4,000 psi, and the line for 70 begins at 700,000 psi and ends at 4,000 psi.

Figure 174. Diagrams. Schematic of the pavement structure with stiff soils. Two diagrams are shown. The left diagram shows a rectangle divided into four horizontal layers. The bottom layer is labeled bedrock, h subscript 4 equals infinity, E subscript 4 equals 2,000 ksi, nu equals 0.25, and rho equals 125 pcf. The layer on top of this is labeled subgrade, h subscript 3 equals 96inches, E subscript 3 equals 13.5 ksi, nu equals 0.45, and rho equals 100 pcf. The next layer is labeled base, h subscript 2 equals 6 inches, E subscript 2 equals 20 ksi, nu equals 0.4, and rho equals 125 pcf. The top layer is labeled AC, h subscript 1 equals 4 inches, E subscript 1 parenthesis Figure 2 end parenthesis, nu equals 0.35, and rho equals 145 pcf. Above the top layer is a distributive load represented by arrows pointing downward toward the surface of the top layer. The distributive load has the label 80 psi and its length is shown as 1 ft.

The right diagram is labeled pavement structure with soils having E-values increasing with depth. The illustration shows a rectangle divided into six equal horizontal layers. The bottom layer is labeled E subscript subgrade 11 equals 2,000 ksi, and the depth of this layer is labeled h subscript 13 equals infinite depth. The next layer up has no labels and is blank. The layer on top of that is labeled E subscript subgrade 2 equals 13.5 + the convolution of h subscript 5 and slope; its depth is labeled h subscript 4 equals 4.8 feet. There is an arrow pointing to these bottom three layers labeled subgrade. The layer above these layers is labeled E subscript subgrade 1 equals 13.5 psi; its depth is labeled h subscript 3 equals 2 ft. The next layer is second from the top; it is labeled E subscript base equals 20 ksi; its depth is labeled h subscript 2 equals 6 inches. The top layer is labeled AC; its depth is labeled h subscript 1 equals 4 inches. Above the top layer is a distributive load represented by arrows pointing downward onto the surface of the top layer. The distributive load is labeled 80 psi and has an arrow showing that it is 1 ft long.

Figure 175. Graph and Diagram. AC layer master curve and temperature profile. A graph and a diagram are shown. The graph is labeled master curve. It has the time in seconds on the xaxis and E parenthesis t end parenthesis in psi on the y-axis. There are two lines on the graph representing the sigmoidal and Prony series. The two lines start at 3 million psi at 10^-8 s and decrease concavely then convexly until they reach 5,000 psi at 10⁷ s.

The diagram is labeled temperature profile. It has two horizontal parallel lines; the area between them is labeled AC. The distance between these lines is indicated with two arrows; they show that the distance to the center from each of the two lines is 2 inches. Next to this is the temperature profile, which is shown as two steps. The bottom line, or bottom step, has a temperature of 79.3 °F. It is at the same level as the bottom line, and the next step is at the same height as the middle between the two horizontal lines. This step has a temperature of 86.9 °F.

Figure 176. Graphs. Error in the backcalculated time histories by sensor—backcalculation of layer moduli only. Nine graphs are shown representing the data for sensors 1 through 9, with the time in s on the x-axis and the error percent on the y-axis. On each graph there are several peaks in the percent error. The top left graph represents the data for sensor 1; its maximum percent error is 2, occurring at about 0.02 s. The top middle graph for sensor 2 has a maximum percent error of 1.4 at 0.01 s. The top right graph is for sensor 3; its maximum percent error is 1.5 at 0.01 s. The left graph in the middle row is for sensor 4; its maximum percent error is 1 at 0.02 s. The middle graph in the middle row is for sensor 5; it has a maximum percent error of 2at 0.02 s. The left graph in the middle row is for sensor 6; it has a maximum percent error of 1at 0.015 s. The bottom left graph is for sensor 7; it has a maximum percent error of 1at 0.015 s. The bottom middle graph has a maximum percent error of 1.9 at 0.04 s. The bottom right graph is for sensor 9; it has a maximum percent error of 3 at 0.015 s.

Figure 177. Graphs. Backcalculation results of the master curve—backcalculation of layer moduli only. Two graphs are shown. The top graph is labeled master curve. This graph has time in s on the x-axis and the relaxation modulus in psi on the y-axis. Two lines are displayed on the graph representing the simulated and backcalculated data. Both of the lines begin at 10⁶ s and 2times 10⁶ psi. The two lines have similar trends, bending concavely then convexly until the backcalculated line reaches 6 times 10³ psi at 10⁸ s and the simulated line reaches 2 times 10³ psi at 10⁸ s.

The bottom graph is labeled error. Three errors are shown on the graph. The overall error is a line that begins at 0 percent at 10^-8 s and then decreases to -10 percent before increasing up to 95‑percent error at 10⁸ s. The error at 86.9 °F is shown on the line over the range from 0 to 10‑percent error. The error at 79.3 °F is shown on the line as well, ranging from –10- to 0‑percent error.

Figure 178. Graphs. Error in the backcalculated time histories by sensor—backcalculation of layer moduli and subgrade thickness. Nine graphs are shown representing the data for sensors 1through 9. For each graph, the time in s is on the x-axis and error percent on the y-axis. For each graph, there are several peaks in the percent error. The top left graph represents sensor 1; its maximum percent error is 2 occurring about 0.01 s. The top middle graph for sensor 2 has a peak of 0.7-percent error at 0.01 s and a second peak of 0.7 percent at 0.025 s. The top right graph is for sensor 3; it has a maximum percent error of 1.2 at 0.01 s. The left graph in the middle row is for sensor 4; it has a peak in percent error of 0.8 at 0.02 s. The middle graph in the middle row for sensor 5 has two large peaks at 1.2-percent error at 0.01 s and 1-percent error at about 0.02 s. The right graph in the middle row for sensor 6 has one peak at 2.2 error at 0.02 s. The left bottom graph for sensor 7 has a maximum peak of 1.2-percent error at 0.015. The bottom middle graph has a maximum percent error of 1 at 0.015 s. The bottom right graph is for sensor 9; its maximum percent error is 1.4 at 0.015 s.

Figure 179. Graphs. Backcalculation results of the AC master curve—backcalculation of layer moduli and subgrade thickness. Two graphs are shown. The top graph is labeled master curve. This graph has reduced time in s on the x-axis and relaxation modulus in psi on the y-axis. Two lines are displayed on the graph representing data for the simulation and backcalculated E parenthesis t end parenthesis. The lines are very close to each other; both begin at about 2 times 10⁶ psi at 10^-8 s and extend concavely then convexly until the simulation line reaches 3 times 10³ psi and the backcalculated line reaches 4 time 10³ psi at 10⁸ s. Throughout the graph, the relaxation modulus for the simulation is greater than for the backcalculated until 1s is reached, and then it becomes less.

The second graph is labeled error. Reduced time in s is on the x-axis and relaxation modulus in psi on the y-axis. The graph has a wave shape; it begins at about 11-percent error at 10^-8 s. The error then decreases to -17 percent at 10^-2 s, then increases to 37 percent at 10⁴ s, and finally decreases to 22 percent at 10⁸ s.

Figure 180. Equation. New and old shift factor equations. Two equations are shown. The first is labeled old shift factor: the log of a times T equals a subscript 1 times T squared plus a subscript 2 times T plus a subscript 3. The second equation is labeled new shift factor: the log of a times T equals a subscript 1 times parenthesis T minus T subscript ref end parenthesis squared plus a subscript 2 times parenthesis T minus T subscript ref end parenthesis.

Figure 181. Equation. New optimization problem. The first line is labeled objective function. It states that E subscript r equals the summation with the lower bound k equals 1 and the upper bound m of 100, multiplied by the summation with the lower bound i,o equals 1 and the upper bound n, of the following: the quotient of the absolute value of parenthesis d subscript i superscript k minus d subscript o superscript k end parenthesis, end absolute value, divided by the quantity d superscript k end quantity subscript max.

The next line is labeled bound constraints. There are eight different cases, which are divided into three subgroups. The first group is the limits for c: The first line states: c subscript 1, superscript l is less than or equal to c subscript 1, which is less than or equal to c subscript 1, superscript u. The second line states: C upper subscript l is less than or equal to C, which is equal to c subscript 1 plus c subscript 2, which is less than or equal to C superscript u. The third line states: c subscript 3, superscript l is less than or equal to c subscript 3, which is less than or equal to c subscript 3, superscript u. The fourth line states: c subscript 4, superscript l is less than or equal to c subscript 4, which is less than or equal to c subscript 4, superscript u.

The second group is the limits for a: The first line states: a subscript 1, superscript l is less than or equal to a subscript 1, which is less than or equal to a subscript 1, superscript u. The second line states: a subscript 2, superscript l is less than or equal to a subscript 2, which is less than or equal to a subscript 2, superscript u.

The third group is the limits for E: The first line states: E subscript b, superscript l is less than or equal to E subscript b, which is less than or equal to E subscript b, superscript u. The second line states: E subscript sg, superscript l is less than or equal to E subscript sg, which is less than or equal to E subscript sg, superscript u.

Figure 182. Diagrams. Waverly Road section 1 temperature profile at 9 a.m. and 1 p.m. Two diagrams are shown. The left diagram represents the temperature profile at 9 a.m. It has two horizontal parallel lines; the area between them is labeled AC. The distance between these lines is indicated with two arrows showing that the upper and lower distances to the center between the two lines are 2 inches. Next to this is the temperature profile, which is shown as two steps. The bottom line, or bottom step has a temperature of 60.5 °F, and the next step is at the same height as the half distance between the two horizontal lines. This step has a temperature of 62 °F.

The right diagram is labeled 1 p.m. This diagram has two horizontal parallel lines, and the area between them it is labeled AC. The distance between these lines is indicated with arrows showing that the distance to the center between the two lines is 2 inches. Next to this is the temperature profile, which is shown as two steps. The bottom line, or bottom step has a temperature of 79.3 °F, and the next step is at the same height as the half distance between the two horizontal lines. This step has a temperature of 86.9 °F.

Figure 183. Graphs. Waverly Road FWD time histories for section 1 collected at 9 a.m. and 1p.m. Two graphs are shown. Time in ms is on the x-axis and vertical deflection in mil on the y‑axis. Nine lines are shown on each graph representing r equals 0, 8, 12, 18, 24, 36, 48, 60, and 72 inches. The top graph is labeled 9 a.m. All of the lines begin at 0 deflection and have a parabolic shape. After reaching their peak value in deflection, the lines decrease to a negative deflection between 0 and 2 mil at 0.04 s. The lines then increase to a deflection between 0 and 1at 0.05 s. The line for r equals 0 has a peak deflection of 20 at 0.02 s; for r equals 8, the peak is 16 at 0.02 s; for r equals 12, it is 13 at 0.02 s; for r equals 18, it is 9.5 at 0.02 s; for r equals 24, it is 7 at 0.021 s; for r equals 36; it is 4 at 0.022 s, for r equals 48, it is 3 at 0.026 s; for r equals 60, it is 2 at 0.026 s; and for r equals 72, it is 1 at 0.03 s.

The bottom graph is labeled 1 p.m. All of the lines begin at 0 deflection and have a parabolic shape. After reaching their peak value in deflection, the lines decrease to a negative deflection between 1 and 3 mil at 0.04 s. The lines then increase to a deflection between –1 and 1 at 0.05 s. The line for r equals 0 has a peak deflection of 23 at 0.02 s; for r equals 8, the peak is 16 at 0.02s; for r equals 12, it is 13 at 0.02 s; for r equals 18, it is 9 at 0.021 s, for r equals 24, it is 7at 0.021 s; for r equals 36, it is 4 at 0.022 s; for r equals 48, it is 2 at 0.026 s; for r equals 60, it is 1.5 at 0.026s; and for r equals 72, the line remains constant at 0 deflection throughout the entire graph.

Figure 184. Graph. Backcalculated master curve for Waverly Road. This graph has reduced time in s on the x-axis and relaxation modulus in psi on the y-axis. Two lines are displayed on the graph representing data for the average laboratory and backcalculated E parenthesis t end parenthesis. The lines are very close to each other, both beginning at about 1.4times 10⁶ psi at 10^-8 s and bend concavely then convexly until they reach 5 times 10³ psi at 10⁸s. The backcalculated E parenthesis t end parenthesis is slightly greater than the average laboratory results at this point

Figure 185. Graph. Error in the Backcalculated master curve for Waverly Road. This graph has reduced time in s on the x-axis and the percent error in the relaxation modulus on the y-axis. The graph begins at a 4-percent error at 10^-8 s; the error then increases to 6 percent at 1.8times 10^-2 s. The line then decreases to 3.8 percent at about 10 s before it finally increases to 16 percent at 10⁸ s.

Figure 186. Graphs. Predicted versus measured deflection time histories by sensor for 1p.m. test for Waverly Road. Eight graphs are shown labeled sensor 1 to 8. Each graph has the time in s on the x-axis and the deflection in mil on the y-axis. There are two lines on each of graph representing the measured and backcalculated results. All of the lines have the same shape; they begin at 0 deflection and remain at 0 until about 0.01 s; this is followed by one large peak in deflection, then a decrease in deflection, followed by a small increase where the lines then level out.

The first graph is labeled Sensor 1. The peak deflection is at 15 mil at 0.02 s; then the line decreases to 0 at 0.04 s, followed by an increase to 1 mil at 0.07 s.

The second graph is labeled Sensor 2. The peak deflection is at 11 mil at 0.02 s; then the line decreases to –1 mil at 0.045 s, followed by an increase to 1 mil for the backcalculated and 0 for measured at 0.07 s.

The third graph is labeled Sensor 3. The peak deflection is at 8 mil at 0.02 s; then the line decreases to –1 at 0.04 s, followed by an increase to 0.2 mil for the backcalculated and 0 for measured at 0.07 s.

The fourth graph is labeled Sensor 4. The peak deflection is at 5.5 mil at 0.02 s, then the line decreases to –1 at 0.04 s, followed by an increase to 0.2 mil for the backcalculated and 0 for measured at 0.07 s.

The fifth graph is labeled Sensor 5. The peak deflection is at 4 mil at 0.02 s; then the line decreases to –1 at 0.04 s, followed by an increase to 0.5 mil for the backcalculated and 0 for measured at 0.07 s.

The sixth graph is labeled Sensor 6. The peak deflection is at 2.4 mil at 0.025 s; then the line decreases to –1 for the backcalculated at –0.75 for the measured at 0.045 s, followed by an increase to 0.5 mil for the backcalculated and 0 for measured at 0.07 s.

The seventh graph is labeled Sensor 7. The peak deflection is at 1.5 mil at 0.03 s; then the line decreases to –0.75 for the backcalculated and –0.5 mil for the measured at 0.045 s, followed by an increase to 0.5 mil for the backcalculated and 0 for measured at 0.07 s.

The eighth graph is labeled Sensor 8. The peak deflection is at 1 mil at 0.03 s; then the line decreases to –0.6 for the backcalculated at –0.5 mil for the measured at 0.05 s, followed by an increase to 0.4 mil for the backcalculated and 0 for measured at 0.07 s.

Figure 187. Graphs. Error in the backcalculated deflection time histories by sensor for 1 p.m. test for Waverly Road. Eight graphs are shown representing data from sensors 1 through 8. Each graph has the time in s on the x-axis and the error percent on the y-axis. The values of error percent oscillate up and down on each of the graphs. For sensor 1, the maximum percent error is 14 at 0.025 s; for sensor 2, the maximum percent error is 12 at 0.025 s; for sensor 3, the maximum percent error is 12 at 0.025 s; for sensor 4, the maximum percent error is 11 at 0.05 s; for sensor 5, the maximum percent error is 15 at 0.045 s; for sensor 6, the maximum percent error is 18 at 0.045 s; for sensor 7, the maximum percent error is 20 at 0.05 s, and for sensor 8, the maximum percent error is 20 at 0.03 s and again at 0.06 s.

Figure 188. Diagram. Temperature profile for LTPP section 350801. This diagram shows two horizontal parallel lines; the area between them is labeled AC. The distance between these lines is indicated with two arrows showing that the distance to the center is 2 inches from each line. Next to this is the temperature profile, which is shown as two steps. The bottom line horizontal line or bottom step has a temperature of 63.3 °F, and the next step is at the same height as the half distance between the two horizontal lines. This step has a temperature of
70.5 °F.

Figure 189. Graph. Measured FWD time histories for LTPP section 350801. This graph has the time in s on the x-axis and deflection in mil on the y-axis. Nine lines are shown on the graph representing r equals 0, 8, 12, 18, 24, 36, 48, 60, and 72 inches. The lines all begin at 0deflection until about 0.01 s when the lines all increase to a peak deflection and then decrease to a deflection at about 0 at 0.04 s and remain about 0 until the graph ends at 0.06 s. The line for r equals 0 has a peak deflection of 13.5 at 0.02 s; for r equals 8, the peak is 11.5 at 0.02 s; for r equals 12, it is 9 at 0.02 s; for r equals 18, it is 7 at 0.02 s; for r equals 24, it is 5.5 at 0.021 s; for r equals 36, it is 4 at 0.022 s; for r equals 48, it is 2.5 at 0.023 s; for r equals 60, it is 2 at 0.024 s; and for r equals 72, it is 8 at 0.02 s.

Figure 190. Graph. Backcalculation results of the AC master curve for LTPP section 350801.This graph has the reduced time in s on the x-axis and the relaxation modulus in psi on the y‑axis. There are four lines shown on the graph representing E(t) back run 2, E(t) back run 1, average lab, and average lab fit. All of the lines begin at 3 million psi at 10^-8 s and decrease until they reach 10⁷ s. For back run 2, the final relaxation modulus is 2,500 psi; for back run, 1 it is 2,000 psi; for the average lab, the line stops at 10,000 psi at 10⁴ s; and for the average lab fit, the final relaxation modulus is 1,500 psi.

Figure 191. Graph. Error in the backcalculated master curve for LTPP section 350801. This graph has reduced time in s on the x-axis and error percent on the y-axis. The line on the graph begins just below 0 error at 10^-8 s; the line then decreases to –37-percent error at about 10³ s, and then increases to 40-percent error just past 10⁷ s.

Figure 192. Graphs. Error in the backcalculated time histories by sensor for LTPP section 350801. Eight graphs are shown representing the data for sensors 1 through 8. For each graph, time in s is on the x-axis and error percent is on the y-axis. The percent errors oscillate up and down on each graph. The graph for sensor 1 has a maximum percent error of 3.8 at 0.04 s. The graph for sensor 2 has a maximum percent error of 14 at 0.015 s. The graph for sensor 3 has a maximum percent error of 9 at 0.02 s. graph for sensor 4 has a maximum percent error of 9.9 at 0.04 s. The graph for sensor 5 has a maximum percent error of 11 at 0.02 s. The graph for sensor 6 has a maximum percent error of 28 at 0.03 s. The graph for sensor 7 has a maximum percent error of 35 at 0.025 s. The graph for sensor 8 has a maximum percent error of 40 at 0.025 s.

Figure 193. Graphs. Backcalculated versus measured time histories by sensor for LTPP section 350801, station 1. Eight graphs are shown labeled sensor 1 through 8. Each graph has the time in s on the x-axis and the deflection in mil on the y-axis. There are two lines on each of graph representing the measured and backcalculated results. All of the graphs have the same shape; they begin at 0 deflection and remain at 0 for about 0.01 s; this is followed by one large peak in deflection, then a drop in deflection, followed by a small increase where the lines then level out.

The first graph is labeled Sensor 1. The peak deflection is at 14.5 mil at 0.02 s; then the line decreases to 0 at 0.04 s, followed by an increase to 1 mil at 0.06 s.

The second graph is labeled Sensor 2. The peak deflection is at 11 mil for measured and 10 mil for backcalculated at 0.02 s; the lines then decrease to –1 mil for the backcalculated and 0 mil for the measured at 0.04 s. This is followed by an increase to 0.5 mil for the measured and just below 0.5 mil for backcalculated at 0.06 s.

The third graph is labeled Sensor 3. The peak deflection is at 9 mil for measured and 8.5 mil for backcalculated at 0.02 s; the lines then decrease to –1 mil for the backcalculated and 0 mil for measured at 0.04 s, followed by an increase to 0.4 mil for the backcalculated and 0.5 for measured at 0.06 s.

The fourth graph is labeled Sensor 4. The peak deflection is at 7 mil for measured and 6.5 mil for backcalculated at 0.02 s; the lines then decrease to –1 mil for the backcalculated and 0 mil for measured at 0.04 s, followed by an increase to 0.2 mil for the backcalculated and 0.5 mil for measured at 0.06 s.

The fifth graph is labeled Sensor 5. The peak deflection is at 5.5 mil for measured and 5 mil for backcalculated at 0.02 s; the lines then decrease to –1.7 mil for the backcalculated and –1 mil for measured at 0.04 s, followed by an increase to 0 mil for the backcalculated and measured at 0.06s.

The sixth graph is labeled Sensor 6. The peak deflection is at 4 mil for measured and 3 mil for backcalculated 0.02 s; the lines then decrease to –1 mil for the backcalculated and –0.5 mil for measured at 0.04 s, followed by an increase to 0.2 mil for the backcalculated and 0.5 mil for measured at 0.06 s.

The seventh graph is labeled Sensor 7. The peak deflection is at 2.7 mil for measured and 2 mil for backcalculated 0.025 s; the lines then decrease to –1 mil for the backcalculated and –0.8 mil for measured just after 0.04 s, followed by an increase to 0.3 mil for the backcalculated and 0 for the measured at 0.06 s.

The eighth graph is labeled Sensor 8. The peak deflection is at 2 mil for measured and 1.3 mil for backcalculated 0.03 s, the lines then decrease to –0.7 mil for the backcalculated and –0.8 mil for measured at 0.045 s, followed by an increase to 0.3 mil for the backcalculated and measured at 0.06 s.

Figure 194. Diagram. Temperature profile for LTPP section 350801. This diagram has three horizontal parallel lines; the area between them is labeled AC. The distance between these lines is indicated with three arrows showing that the upper, center, and lower distances between the two lines are 1.4 inches each. Next to this is the temperature profile, which is shown as three steps. The bottom line, or bottom step, has a temperature of 63 °F. The second step, or the center step, has a temperature of 65.1 °F. The third step, or the upper step, has a temperature of 74.1 °F.

Figure 195. Graphs. Measured FWD load and deflection time histories for LTPP section 350801. Two graphs are shown. The first graph is labeled FWD load history. The graph has the time in s on the x-axis and the load magnitude on the y-axis. The line reaches its peak magnitude at 0.026 s and decreases to 0 at 0.06 s.

The second graph is labeled deflection time histories. The graph has the time in s on the x‑axis and the deflection in mil on the y-axis. Nine lines are shown representing r equals 0, 8, 12, 18, 24, 36, 48, and 60 inches. All of the lines begin at 0 deflection and have a parabolic shape. After reaching their peak value in deflection, the lines decrease to about 1 mil at 0.06 s. The line for r equals 0 has a peak deflection of 9.5 at 0.022 s; for r equals 8, the peak is 7.5 at 0.023 s; for r equals 12, the peak is 5.5 at 0.023 s; for r equals 18, it is 4 at 0.024 s; for r equals 24, it is 3 at 0.025 s; for r equals 36, it is 2 at 0.026 s; for r equals 48, it is 1.2 at 0.028 s; and for r equals 60, it is 0.8 at 0.03 s.

Figure 196. Graph. Backcalculation results of the AC master curve for LTPP section 350801. This graph has reduced time in s on the x-axis and relaxation modulus in psi on the y–axis. Two lines are shown on the graph representing the average lab fitting and the backcalculated E parenthesis t, end parenthesis. Both lines begin at approximately 2 million psi at 10^-8 s and decrease to 400 psi for the average lab fitting curve and 1,000 psi for backcalculated master curve at 10⁸ s.

Figure 197. Graph. Error in the backcalculated master curve for LTPP section 350801. This graph has reduced time in s is on the x-axis, and percent error in the relaxation modulus is on the y-axis. The graph begins at a –1 percent error at 10^-8 s; the error then decreases to –10 percent at 1.8 times 10^-2 s. The line then increases to –1 percent at about 10² s before it finally decreases to ‑25 percent at 10⁴ s.

Figure 198. Graphs. Backcalculated versus measured time histories by sensor for LTPP section 350801, station 8. Eight graphs are shown labeled sensor 1 through 8. Each graph has the time in s on the x-axis and the deflection in mil on the y-axis. There are two lines on each of graph representing the measured and backcalculated results. All of the graphs have the same shape; they begin at 0 deflection and remain at 0 for about 0.01 s; this is followed by one large peak in deflection, then a drop in deflection, followed by a small increase where the lines then level out.

The first graph is labeled Sensor 1. The peak deflection is at 9.5 mil at 0.025 s; the lines then decrease to –1 mil for the backcalculated and 0 mil for the measured at 0.05 s, followed by an increase to 0 mil for the measured and the backcalculated at 0.06 s.

The second graph is labeled Sensor 2. The peak deflection is at 7.5 mil; the lines then decrease to –1 mil for the backcalculated and 0 mil for the measured at 0.05 s, followed by an increase to just below –0.5 mil for backcalculated at 0.06 s.

The third graph is labeled Sensor 3. The peak deflection is at 6 mil for both measured and backcalculated deflections at 0.025 s; the lines then decrease to –0.5 mil for the backcalculated and 0 mil for measured at 0.05 s, followed by an increase to 0 mil for the backcalculated at 0.06s.

The fourth graph is labeled Sensor 4. The peak deflection is at 4.5 mil for both measured and backcalculated deflections at 0.028 s; the lines then decrease to –1 mil for the backcalculated and 0 mil for measured at 0.055 s, followed by an increase to –0.5 mil for the backcalculated at 0.06s.

The fifth graph is labeled Sensor 5. The peak deflection is at 3 mil for both measured and backcalculated deflections at 0.03 s; the lines then decrease to –0.7 mil for the backcalculated and -0.2 mil for measured at 0.05 s, followed by an increase to 0 mil for the backcalculated and measured at 0.06 s.

The sixth graph is labeled Sensor 6. The peak deflection is at 2.2 mil for measured at 0.025 s and 2 mil for backcalculated at 0.03 s; the lines then decrease to -0.7 mil for the backcalculated and –0.2 mil for measured at 0.04 s, followed by an increase to –0.5 mil for the backcalculated and 0mil for measured at 0.06 s.

The seventh graph is labeled Sensor 7. The peak deflection is at 1.4 mil for measured at 0.025 s and 1.5 mil for backcalculated 0.03 s; the lines then decrease to –0.5 mil for the backcalculated at 0.055 s and –0.3 mil for measured just after 0.04 s.

The eighth graph is labeled Sensor 8. The peak deflection is at 1 mil for measured at 0.03 s and 1.2 mil for backcalculated 0.04 s; the lines then decrease to –0.5 mil for the backcalculated at 0.055 s and –0.1 mil for measured at 0.045 s.

Figure 199. Graph. Simulated FWD load pulses with various durations. This graph has time in s on the x-axis and normalized load magnitude on the y-axis. Four lines are shown on the graph representing 35, 40, 45, and 50 ms. All of the lines begin at 0 and increase to a magnitude of 1. After this point, the lines decrease to 0 and remain there until the graph ends at 0.06 s. The 35-ms line reaches its peak magnitude at 0.018 s and decreases to 0 at 0.035 s. The line for 40 ms reaches its peak magnitude at 0.021 s and decreases to 0 at 0.04 s. The line for 45ms reaches its peak magnitude at 0.024 s and decreases to 0 at 0.035 s. The line for 50 ms reaches its peak magnitude at 0.026 s and decreases to 0 at 0.05 s.

Figure 200. Diagram. Schematic of the pavement structure with bedrock. This diagram shows a rectangle divided into four horizontal layers. The bottom layer is labeled bedrock, h subscript 4 equals infinity, E subscript 4 equals 2,000 ksi, nu equals 0.25, and rho equals 125 pcf. The layer on top of this is labeled subgrade, h subscript 3 equals 96 inches, E subscript 3 equals 13.5 ksi, nu equals 0.45, and rho equals 100 pcf. The next layer is labeled base, h subscript 2 equals 6 inches, E subscript 2 equals 20 ksi, nu equals 0.4, and rho equals 125 pcf. The top layer is labeled AC, h subscript 1 equals 4 inches, E subscript 1 parenthesis Figure 11 end parenthesis, nu equals 0.35, and rho equals 145 pcf. Above the top layer is a distributive load represented by arrows pointing downward toward the surface of the top layer. The distributive load is labeled 80psi and has an arrow showing that it is 1 ft long.

Figure 201. Graph and Diagram. AC layer parameters. A graph and a diagram are shown. The graph is labeled master curve. It has time in s on the x-axis and E parenthesis t end parenthesis in psi on the y-axis. Two lines are shown on the graph representing the sigmoidal and Prony series. Both lines begin at 10^-8 s at 3 million psi and decrease to 10⁷ s at 5,000 psi. The end point of the line for the Prony series is slightly below the sigmoidal.

The diagram is labeled temperature profile. It has two horizontal parallel lines, and the area between them is labeled AC. The distance between these lines is indicated with two arrows; they show that the distance to the center from each of the two lines is 2 inches. Next to this is the temperature profile, which is shown as two steps. The bottom line, or bottom step has a temperature of 79.3 °F, and the next step is at the height directly between the two horizontal lines. This step has a temperature of 86.9 °F.

Figure 202. Graphs. Surface deflections of pavement structure for different width of load pulses. Four graphs are shown. Each graph has time in s on the x-axis and deflection in mil on the y-axis. Nine lines are shown on each graph representing; r equals 0, 8, 12, 18, 24, 36, 48, 60, and 72 inches.

The top left graph is labeled 35 ms. The lines all begin at 0 deflection and have a parabolic shape. After reaching their peak deflections, the lines decrease to a negative deflection between ‑2 and 0 mil at about 0.045 s and then increase to a deflection between 0 and 1 at 0.06 s. The line for r equals 0 has a peak deflection of 26 at 0.02 s; for r equals 8, the peak is 17 at 0.02s; for r equals 12, it is 13 at 0.021 s; for r equals 18, it is 9.5 at 0.022 s; for r equals 24, it is 7 at 0.023 s; for r equals 36, it is 3.5 at 0.024 s; for r equals 48, it is 2 at 0.026 s; for r equals 60, it is 1.5 at 0.028 s; and for r equals 72, it is 1 at 0.03 s.

The top right graph is labeled 40 ms. The lines all begin at 0 deflection and have a parabolic shape. After reaching their peak deflections, the lines decrease to a negative deflection between ‑2 and 0 mil at about 0.045 s and then increase to a deflection between 0 and 1 at 0.06 s. The line for r equals 0 has a peak deflection of 26 at 0.024 s; for r equals 8, the peak is 17 at 0.025 s; for r equals 12, it is 13 at 0.025 s; for r equals 18, it is 9.5 at 0.026 s; for r equals 24, it is 6 at 0.026 s; for r equals 36, it is 3.5 at 0.027 s; for r equals 48, it is 2 at 0.028 s; for r equals 60, it is 1.5 at 0.03 s; and for r equals 72, it is 1 and 0.032 s.

The bottom left graph is labeled 45 ms. The lines all begin at 0 deflection and have a parabolic shape. After reaching their peak deflections, the lines then decrease to a negative deflection between 0 and -3 mil at about 0.05 s and then increase to a deflection between 0 and 1 at 0.06 s. The line for r equals 0 has a peak deflection of 26 at 0.025 s; for r equals 8, the peak is 17 at 0.026 s; for r equals 12, it is 13 at 0.027 s; for r equals 18, it is 9.5 at 0.028 s; for r equals 24, it is 7 at 0.029 s; for r equals 36, it is 3.5 at 0.03 s; for r equals 48, it is 2 at 0.032 s; for r equals 60, it is 1 at 0.034 s; and for r equals 72, it is 0.5 at 0.036 s.

The bottom right graph is labeled 50 ms. The lines all begin at 0 deflection and have a parabolic shape. After reaching their peak deflections, the lines then decrease to a deflection between 1 and -2 mil at about 0.052 s and then increase to a deflection between 0 and 1 at 0.06 s. The values for peak deflections and times are exactly the same as those for the bottom left graph.

Figure 203. Graphs. Error in the backcalculated time histories by sensor for a pulse width of 35 ms. Eight graphs are shown representing the data for sensors 1 through 8. For each graph, time in s is on the x-axis and error percent on the y-axis. The percent errors oscillate up and down on each graph. The top left graph, which represents sensor 1, has a maximum percent error of 12at 0.02 s. The top right graph for sensor 2 has a maximum percent error of 10 at 0.02s. The second row left graph for sensor 3 has a maximum percent error of 10 at 0.02 s. The second row right graph for sensor 4 has a maximum percent error of 9 at 0.025 s. The third row left graph for sensor 5 has a maximum percent error of 8.5 at 0.025 s. The third row right graph for sensor 6 has a maximum percent error of 7 at 0.03 s. The bottom left graph for sensor 7 has a maximum percent error of 6.5 at 0.048 s. The bottom right graph for sensor 8 has a maximum percent error of 6.5 at 0.049 s.

Figure 204. Graphs. Error in the backcalculated time histories by sensor for a pulse width of 40 ms. Eight graphs are shown representing the data for sensors 1 through 8. For each graph, time in s is on the x-axis and error percent is on the y-axis. The percent errors oscillate up and down on each graph. The top left graph, which represents sensor 1, has a maximum percent error of 12 at 0.025 s. The top right graph for sensor 2 has a maximum percent error of 8 at 0.025s. The second row left graph for sensor 3 has a maximum percent error of 13 at 0.025 s. The second row right for sensor 4 has a maximum percent error of 13 at 0.025 s. The third row left graph for sensor 5 has a maximum percent error of 14 at 0.025 s. The third row right graph for sensor 6 has a maximum percent error of 10 at 0.025 s. The bottom row left for sensor 7 has a maximum percent error of 9 at 0.04 s. The bottom row left graph for sensor 8 has a maximum percent error of 9.5 at 0.02 s.

Figure 205. Graphs. Error in the backcalculated time histories by sensor for a pulse width of 45 ms. Eight graphs are shown representing data from sensors 1 through 8. Each graph has time in s on the x-axis and error percent on the y-axis. The values of error percent oscillate up and down on each of the graphs. For sensor 1, the maximum percent error is 18 at 0.025 s; for sensor two, the maximum percent error is 17 at 0.025 s; for sensor 3, the maximum percent error is 18 at 0.025 s; for sensor 4, the maximum percent error is 18 at 0.025 s; for sensor 5, the maximum percent error is 17 at 0.025 s; for sensor 6, the maximum percent error is 14 at 0.025 s; for sensor 7, the maximum percent error is 12 at 0.045 s; and for sensor 8, the maximum percent error is 11 at 0.02.

Figure 206. Graphs. Error in the backcalculated time histories by sensor for a pulse width of 50 ms. Eight graphs are shown representing data from sensors 1 through 8. Each graph has time in s on the x-axis and error percent on the y-axis. The values of error percent oscillate up and down on each of the graphs. For sensor 1, the maximum percent error is 4.5 at 0.025 s; for sensor 2, the maximum percent error is 4 at 0.025 s; for sensor 3, the maximum percent error is 4.5 at 0.025 s; for sensor 4, the maximum percent error is 3.8 at 0.025 s; for sensor 5, the maximum percent error is 3.6 at 0.025 s; for sensor 6, the maximum percent error is 2 at 0.025 s; for sensor 7, the maximum percent error is 2.3 at 0.045 s; and for sensor 8, the maximum percent error is 6 at 0.04 s.

Figure 207. Graph. Backcalculation results of the AC master curve for different pulse widths. This graph has the reduced time in s on the x-axis and the relaxation modulus in psi on the y-axis. Five lines are shown on the graph representing back-35 ms, back-40 ms, back-45 ms, back-50 ms, and original. The lines all begin at approximately 2 million psi at 10^-8 s and decrease to 10⁸ s and 2,000 psi for original, 4,000 psi for back-50 ms, 4,500 psi for back 45-ms, 5,000 psi for back-40 ms, and the last line for back -35 ms ends at 6,000 psi.

Figure 208. Graph. Error in the backcalculated master curve for different pulse widths. This graph has the reduced time in s on the x-axis and the error percent on the y-axis. Four lines are displayed on the graph representing back-35 ms, back-40 ms, back-45 ms, and back-50 ms. The line for back-50 ms begins at an error of 2 percent at 10^-8 s, it decreases to –5 percent error at 10 s, and then increases to 18-percent error at 10⁷ s. The line for back-45 ms begins at an error of –11 percent at 10^-8 s; it increases to 35 percent error at 10 s, and then increases to 98 percent error at 10⁷ s. The line for back-40 ms begins at an error of –17 percent at 10^-8 s; it increases to 25 percent error and then decreases to 20 percent error before increasing to 58 percent error at 10⁷ s. The line for back-35 ms begins at an error of –24 percent at 10^-8 s; it increases to 20 percent error and then decreases again to 5 percent error before increasing to 26percent error at 10⁷ s.

Figure 209. Graph and Diagram. AC layer master curve and temperature profile. A graph and a diagram are shown. The graph is labeled master curve. It has the time in s on the x-axis and E parenthesis t end parenthesis in psi on the y-axis. There are two lines on the graph representing the sigmoidal and Prony series. Both lines begin at 10^-8 s at 2 million psi and decrease to 5,000psi at 10⁷ s; here the line for the Prony series is slightly below the sigmoidal.

The diagram is labeled temperature profile. It has two horizontal parallel lines, and the area between them it is labeled AC. The distance between these lines is indicated with two arrows; they show that the distance to the center from each of the two lines is 2 inches. Next to this is the temperature profile, which is shown as two steps. The bottom line, or bottom step has a temperature of 79.3 °F, and the next step is at the height directly between the two horizontal lines, 2 inches from the bottom. This step has a temperature of 86.9 °F.

Figure 210. Graphs. Simulated FWD pulse and deflection time histories. Two graphs are shown. The first graph is labeled simulated FWD pulse. Time in ms is on the x-axis and pressure in psi on the y-axis. The line on the graph begins at a time and pressure of 0. The line increases to 100 psi at 17 ms and then decreases to 0 at 34 ms and remains at 0 until the graph ends at 50 ms.

The second graph is labeled simulated deflections. The x-axis is time in ms, and the yaxis is vertical deflection in mil. Nine lines are shown on the graph representing r equals 0, 8, 12, 18, 24, 36, 48, 60, and 72 inches. The lines all begin at 0 deflection and have a parabolic shape. After the lines increase to a peak deflection, they decrease to a negative deflection between 0 and –4 at 40 ms and then increase toward 0 until the graph ends at 50 ms. The line for r equals 0 has a peak deflection of 29.5 at 0.02 s; for r equals 8, the peak is 20.5 at 0.02 s; for r equals 12, it is 15 at 0.02 s; for r equals 18, it is 11 at 0.022 s, for r equals 24, it is 8 at 0.023 s; for r equals 36, it is 5 at 0.026 s; for r equals 48, it is 3 at 0.028 s; for r equals 60, it is 2 at 0.03 s; and for r equals 72, it is 1 at 0.032 s.

Figure 211. Graph. Average error in the backcalculated AC layer master curve for all runs in LM method. This bar graph has number of iterations on the x-axis and average error in the relaxation modulus percent on the y-axis. For 100 iterations, there is a 50-percent error; for 150iterations, there is a 150-percent error; for 200 iterations, there is a 100-percent error; for 250iterations, there is a 60-percent error; for 300 iterations, there is a 50-percent error; for 350iterations, there is a 50-percent error; for 400 iterations, there is a 240-percent error; for 450iterations, there is a 260-percent error. For 500 iterations, there is a 60-percent error; for 550iterations, there is a 340-percent error; for 600 iterations, there is a 65-percent error, for 650iterations, there is a 50-percent error; for 700 iterations, there is an 80-percent error; for 750iterations, there is a 50-percent error; for 800 iterations, there is a 65-percent error; for 850iterations, there is a 60-percent error; for 900 iterations, there is a 750-percent error; for 950iterations, there is a 100-percent error; and at 1,000 iterations, there is a 70-percent error.

Figure 212. Graph. Average error in the backcalculated AC layer master curve for all runs. This bar graph has run number on the x-axis and average error percent on the y-axis. The graph is divided into four sections. The first section, labeled LM, is for runs 1 through 19. Run 1 has a 46-percent error, and run 2 has a 20-percent error. After run 2, the error decreases steadily and ends with run 19 at 2.5-percent error. The second section, labeled GA+LM, is for runs 20 through 44. Run 20 has a 29-percent error, and the error decreases to 3 percent at run 44. The third section, labeled GA with five generations, is for runs 45 through 52. Run 45 has a 25‑percent error, and the error decreases to 10 percent at run 52. The last section, labeled GA with 15generations, is for runs 53 through 60. Run 53 has an 11-percent error, and the error decreases to 3.5 percent at run 60.

Figure 213. Graph. Average error in the backcalculated base layer modulus for all runs. This bar graph has run number on the x-axis and average error percent on the y-axis. The graph is divided into four sections. The first section, labeled LM, is for runs 1 through 19. Run 1 has 100‑percent error. The error decreases steadily and ends with run 19 at 6-percent error. The second section, labeled GA+LM, is for runs 20 through 44. Run 20 has a 40-percent error, and the error decreases to 0.5 percent at run 44. The third section, labeled GA with five generations, is for runs 45 through 52. Run 45 has a 37-percent error, and the error decreases to 3 percent at run 52. The last section, labeled GA with 15 generations, is for runs 53 through 60. Run 53 has a 37‑percent error, and the error decreases to 1 percent at run 60.

Figure 214. Graph. Average error in the backcalculated subgrade modulus for all runs. This bar graph has run number on the x-axis and average error percent on the y-axis. The graph is divided into four sections. The first section, labeled LM, is for runs 1 through 19. Run 1 has 100‑percent error. The error decreases steadily and ends with run 19 at 2-percent error. The second section, labeled GA+LM, is for runs 20 through 44. Run 20 has a 9-percent error, and the error decreases to 0.5 percent at run 44. The third section, labeled GA with five generations, is for runs 45 through 52. Run 45 has a 10-percent error, and the error decreases to 0.5 percent at run 52. The last section, labeled GA with 15 generations, is for runs 53 through 60. Run 53 has a 49‑percent error, and the error decreases to 0.5 percent at run 60.

Figure 215. Graph. Average error in the backcalculated stiff layer modulus for all runs. This a bar graph has run number on the x-axis and average error percent on the y-axis. The graph is divided into four sections. The first section, labeled LM, is for runs 1 through 19. Run 1 has 82‑percent error. The error decreases steadily and ends with run 19 at 2-percent error. The second section, labeled GA+LM, is for runs 20 through 44. Run 20 has a 100-percent error, and the error decreases to 0.5 percent at run 44. The third section, labeled GA with five generations, it for runs 45 through 52. Run 45 has a 55-percent error, and the error decreases to 10 percent at run 52. The last section, labeled GA with 15 generations, is for runs 53 through 60. Run 53 has a 52‑percent error, and the error decreases to 0.5 percent at run 60.

Figure 216. Graph. Average error in the backcalculated depth to stiff layer for all runs. This a bar graph has run number on the x-axis and average error percent on the y-axis. The graph is divided into four sections. The first section, labeled LM, is for runs 1 through 19. Run 1 has 100‑percent error. The error decreases steadily and ends with run 19 at 3-percent error. The second section, labeled GA+LM, is for runs 20 through 44. Run 20 has a 17-percent error, and the error decreases to 0.25% at run 44. The third section, labeled GA with five generations, is for runs 45 through 52. Run 45 has a 20-percent error, and the error decreases to 1-percent at run 52. The last section, labeled GA with 15 generations, is for runs 53 through 60. Run 53 has a 10‑percent error, and the error decreases to 1 percent at run 60.

Figure 217. Graphs. Error in the backcalculated deflections for run 30. Nine graphs are shown representing the data for sensors 1 through 9. Each graph has time in s on the x‑axis and percent error on the y-axis. The values of error percent oscillate up and down on each of the graphs. For sensor 1, the maximum percent error is 0.9 at 0.04 s; for sensor 2, the maximum percent error is 1.2 at 0.04 s; for sensor 3, the maximum percent error is 2 at 0.03 s; for sensor 4, the maximum percent error is 3.5 at 0.03 s; for sensor 5, the maximum percent error is 4.8 at 0.025 s; for sensor 6 the maximum percent error is 5.5 at 0.025 s; for sensor 7, the maximum percent error is 4 at 0.045 s; for sensor 8, the maximum percent error is 3.5 at 0.045 s; and for sensor 9, the maximum percent error is 3.2 at 0.02 s.

Figure 218. Graphs. Error in the backcalculated deflections for run 35. Nine graphs are shown representing the data for sensors 1 through 9; each graph has time in s on the x-axis and percent error on the y-axis. The values of error percent oscillate up and down on each of the graphs. For sensor 1, the maximum percent error is 1 at 0.035 s; for sensor 2, the maximum percent error is 0.8 at 0.035 s; for sensor 3, the maximum percent error is 0.7 at 0.03 s; for sensor 4, the maximum percent error is 0.75 at 0.04 s; for sensor 5, the maximum percent error is 1.25 at 0.04 s; for sensor 6, the maximum percent error is 2 at 0.04 s; for sensor 7, the maximum percent error is 4 at 0.035 s; for sensor 8, the maximum percent error is 3.5 at 0.04 s; and for sensor 9, the maximum percent error is 4.5 at 0.04 s.

Figure 219. Graph. Backcalculated master curves for runs 30 and 35. This graph has the reduced time in s on the x-axis and the relaxation modulus in psi on the y-axis. There are three lines shown on the graph representing simulated, GA100/5-LM100, and GA150/5-LM100. The three lines all begin at 2 million psi at 10^-8 s. The lines for simulated and GA150 both decrease to 3,000 psi at 10⁸ s, and the line for GA100 decreases to 2,500 psi at 10⁸ s.

Figure 220. Graph. Percent error in the backcalculated master curves for all combinations. This graph has the reduced time in s on the x-axis and the relaxation modulus in psi on the y‑axis. Two lines are shown on the graph representing GA100/5-LM100 and GA150/5-LM100. The line for GA100/5-LM100 begins at 3 psi at 10^-8 s; it decreases to –10 psi and then increases to 13.5 psi before decreasing to –20 psi at 10⁸ s. The line for GA150/5-LM100 begins at 5.5 psi; it then decreases to –9 psi and then increases back to 3 psi at 10⁸ s.

Figure 221. Graphs. Measured FWD load and time histories for LTPP section 10101. Four pairs of two graphs each are shown representing the falling weight deflectometer (FWD) data for four drops. The top left graph is labeled FWD load history for drop 1. The graph has the time in ms on the x-axis and the stress in psi on the y-axis. The line reaches its peak magnitude of 50 psi at 24 ms and decreases to 0 at 60 ms. The top right graph is labeled FWD deflection histories drop 1. The graph has the time in ms on the x-axis and the vertical deflection in mil on the y-axis. Nine lines are shown representing r equals 0, 8, 12, 18, 24, 36, 48, and 60 inches. All of the lines begin at 0 deflection and have a parabolic shape. After reaching their peak value in deflection, the lines decrease to about 0.5 mil at 60 ms. The line for r equals 0 has a peak deflection of 4.5 at 0.025 s; for r equals 8, the peak is 3.7 at 0.025 s; for r equals 12, the peak is 3at 0.026 s; for r equals 18, it is 2 at 0.026 s; for r equals 24, it is 1.5 at 0.027 s; for r equals 36, it is 1 at 0.028 s; for r equals 48, it is 0.7 at 0.029 s; and for r equals 60, it is 0.4 at 0.03 s.

The second row left graph is labeled FWD load history for drop 2. The graph has the time in ms on the x-axis and the stress in psi on the y-axis. The line reaches its peak magnitude of 80 psi at 19 ms and decreases to 0 at 60 ms. The second row right graph is labeled FWD deflection histories drop 2. The graph has the time in ms on the x-axis and the vertical deflection in mil on the y-axis. Nine lines are shown representing r equals 0, 8, 12, 18, 24, 36, 48, and 60 inches. All of the lines begin at 0 deflection and have a parabolic shape. After reaching their peak value in deflection, the lines decrease to about 0.5 mil at 60 ms. The line for r equals 0 has a peak deflection of 7 at 0.020 s; for r equals 8, the peak is 5.5 at 0.022 s; for r equals 12, the peak is 4.5at 0.023 s; for r equals 18, it is 3 at 0.024 s; for r equals 24, it is 2 at 0.025 s; for r equals 36, it is 1.5 at 0.026 s; for r equals 48, it is 1 at 0.027 s; and for r equals 60, it is 0.7 at 0.028 s.

The third row left graph is labeled FWD load history for drop 3. The graph has the time in ms on the x-axis and the stress in psi on the y-axis. The line reaches its peak magnitude of 110 psi at 16ms and decreases to 0 at 60 ms. The third row right graph is labeled FWD deflection histories drop 3. The graph has the time in ms on the x-axis and the vertical deflection in mil on the y‑axis. Nine lines are shown representing r equals 0, 8, 12, 18, 24, 36, 48, and 60 inches. All of the lines begin at 0 deflection and have a parabolic shape. After reaching their peak value in deflection, the lines decrease to about 0.5 mil at 60 ms. The line for r equals 0 has a peak deflection of 10 at 0.019 s; for r equals 8, the peak is 8 at 0.020 s; for r equals 12, the peak is 6 at 0.020 s; for r equals 18, it is 4.5 at 0.021 s; for r equals 24, it is 3.5 at 0.022 s; for r equals 36, it is 2 at 0.023 s; for r equals 48, it is 1.5 at 0.024 s; and for r equals 60, it is 1 at 0.025 s.

The bottom row left graph is labeled FWD load history for drop 4. The graph has the time in ms on the x-axis and the stress in psi on the y-axis. The line reaches its peak magnitude of 140 psi at 15 ms and decreases to 0 at 60 ms. The bottom row right graph is labeled FWD deflection histories drop 4. The graph has the time in ms on the x-axis and the vertical deflection in mil on the y-axis. Nine lines are shown representing r equals 0, 8, 12, 18, 24, 36, 48, and 60 inches. All of the lines begin at 0 deflection and have a parabolic shape. After reaching their peak value in deflection, the lines decrease to about 0.5 mil at 60 ms. The line for r equals 0 has a peak deflection of 13 at 0.019 s; for r equals 8, the peak is 10 at 0.020 s; for r equals 12, the peak is 8 at 0.020 s; for r equals 18, it is 6 at 0.021 s; for r equals 24, it is 4.5 at 0.022 s; for r equals 36, it is 2.5 at 0.023 s; for r equals 48, it is 1.9 at 0.024 s; and for r equals 60, it is 1.7 at 0.025 s.

Figure 222. Graph. Backcalculated master curves for LTPP section 10101 from all the drops. This graph has the reduced time in s on the x-axis and the relaxation modulus in psi on the y–axis. There are five lines shown on the graph representing lab fitting, backcalculated from drop 1, backcalculated from drop 2, backcalculated from drop 3, and backcalculated from drop 4. The five lines all begin at about 2 million psi at 10^-8 s and decrease to 600 psi at 10⁸s.

Figure 223. Graph. Backcalculated shift factors for LTPP section 10101 from all the drops. This graph has the temperature in °F on the x-axis and the shift factor on the y-axis. There are five lines shown on the graph representing lab fitting, backcalculated from drop 1, backcalculated from drop 2, backcalculated from drop 3, and backcalculated from drop 4. The lab fitting shift factor curve starts at about 500 at 32 °F and decreases linearly to 3.5 × 10^-5 at 132°F. All the backcalculated curves have parabolic shapes. The backcalculated from drop 1 curve starts at about 60at 32 °F and decreases to about 0.1 at 132 °F. The backcalculated from drop 2 curve starts at about 30 at 32 °F and decreases to about 0.1 at 122 °F. The backcalculated from drop 3 curve starts at about 40 at 32 °F and decreases to about 0.1 at 132 °F. The backcalculated from drop 4 curve starts at about 10 at 32 °F and decreases to about 0.5 at 132 °F. All five curves intersect at about 1at 70 °F.

Figure 224. Graph. Softening behavior for LTPP section 10101. This graph shows load level in psi, ranging from 50 to 150 in intervals of 20, on the x-axis, and load-to-deflection ratio in pci times 1,000, ranging from 0 to 100, on the y-axis. The graph contains lines for eight different sensors. The first line is Load/Def 8; it has a linear decreasing trend beginning at a load-to-deflection ratio of 97 at approximately 55 psi and decreases to a load-to-deflection ratio of 90 at 140 psi. The second line is Load/Def 7; it has a linear decreasing trend beginning at a load-to-deflection ratio of 82 at 55 psi and decreases to a load-to-deflection ratio of 73 at 140 psi. The third line is Load/Def 6; it has a linear decreasing trend beginning at a load-to-deflection ratio of 58 at 55 psi and decreases to a load-to-deflection ratio of 50 at 140 psi. The fourth line is Load/Def 5; it has a linear decreasing trend beginning at a load-to-deflection ratio of 34 at 55 psi and decreases to a load-to-deflection ratio of 29 at 140 psi. The fifth line is Load/Def 4; it has a linear decreasing trend beginning at a load-to-deflection ratio of 26 at 55 psi and decreases to a load-to-deflection ratio of 22 at 140 psi. The sixth line is Load/Def 3, and has a linear decreasing trend beginning at a load-to-deflection ratio of 19 at 55 psi and decreases to a load-to-deflection ratio of 16 at 140 psi. The seventh line is Load/Def 2; it has a linear decreasing trend beginning at a load-to-deflection ratio of 15 at 55 psi and decreases to a load-to-deflection ratio of 13 at 140psi. The bottom line is Load/Def 1; it has a linear decreasing trend beginning at a load-to-deflection ratio of 11 at 55 psi and decreases to a load-to-deflection ratio of 10 at 140 psi.

Figure 225. Graphs. Measured FWD load and time histories for LTPP section 6A805. Two graphs are shown representing the falling weight deflectometer (FWD) data for drop 1. The first graph is labeled FWD load history for drop 1. Time in ms is on the x-axis and pressure in psi is on the y-axis. The line on the graph begins at a time and pressure of 0. The line has two peaks. One is 45 psi at 17 ms; the line then decreases to 40 psi at 19 ms and then increases to a second peak of 56 psi at 24 ms. It then decreases to 0 psi at 40 ms, remains slightly less than 0 until 56ms, and then increases to slightly above 0 at 60 ms.

The second graph is labeled FWD deflection histories. The x-axis is time in ms, and the y-axis is vertical deflection in mil. Eight lines are shown on the graph representing; r equals 0, 8, 12, 18, 24, 36, 48, and 60 inches. The lines all begin at 0 deflection and have a parabolic shape. After the lines increase to a peak deflection, they decrease to 0 deflection at 40 ms and then to negative deflection between 0 and –1 mil until the graph ends at 60 ms. The line for r equals 0 has a peak deflection of 7.1 mil at 0.025 s; for r equals 8, the peak is 6.4 at 0.027 s; for r equals 12, it is 5.5at 0.03 s; for r equals 18, it is 4.2 at 0.031 s; for r equals 24, it is 3.8 at 0.032 s; for r equals 36, it is 2 at 0.034 s; for r equals 48, it is 1.5 at 0.035 s; and for r equals 60, it is 1 at 0.036s.

Figure 226. Graph. Backcalculated master curves for LTPP section 6A805. This graph has reduced time in s on the x-axis and relaxation modulus in psi on the y-axis. Two lines are shown on the graph representing lab fitting and backcalculated master curves. They both have a sigmoidal shape. The two lines both begin at about 3 million psi at 10^-8 s. The lab fitting curve decreases to 4,000 psi at 10⁸ s. The backcalculated master curve decreases to about 6,000 psi at 10⁸ s. The curves agree until a reduced time of 10⁴ s.

Figure 227. Graph. Backcalculated shift factors for section LTPP 6A805. This graph has temperature in °F on the x-axis and shift factor on the y-axis. Two lines are shown on the graph representing lab fitting and backcalculated results from drop 1. Both curves have parabolic shapes. The lab fitting shift factor curve starts at about 100 at 32 °F and decreases linearly to 3.5at 10^-8 132 °F. The backcalculated shift factor curve starts at about 10 at 32 °F and decreases to about 5 at 132 °F. The curves intersect at about 1 at 70 °F.

Figure 228. Graph. Load-to-deflection ratio for LTPP section 6A805. This graph shows load level in psi, ranging from 50 to 150 in intervals of 20, on the x-axis, and load-to-deflection ratio in pci times 1,000, ranging from 0 to 60, on the y-axis. The graph contains lines for eight different sensors. The first line is Load/Def 8; it has a linear decreasing trend beginning at a load-to-deflection ratio of 43 at about 60 psi and decreases to a load-to-deflection ratio of 38 at 146 psi. The second line is Load/Def 7; it has a linear decreasing trend beginning at a load-to-deflection ratio of 31 at 60 psi and decreases to a load-to-deflection ratio of 29 at 146 psi. The third line is Load/Def 6; it has a linear decreasing trend beginning at a load-to-deflection ratio of 24 at 60 psi and decreases to a load-to-deflection ratio of 22 at 146 psi. The fourth line is Load/Def 5; it has a linear decreasing trend beginning at a load-to-deflection ratio of 16 at 60 psi and decreases to a load-to-deflection ratio of 15 at 146 psi. The fifth through eighth lines are Load/Def 4, Load/Def 3, Load/Def 2, and Load/Def 1. They have a constant trend at load-to-deflection ratios of 13, 10, 9, and 8, respectively.

Figure 229. Graphs. Measured FWD load and time histories for LTPP section 6A806. Two graphs are shown representing the falling weight deflectometer (FWD) data for drop 1. The first graph is labeled FWD load history for drop 1. Time in ms is on the x-axis, and pressure in psi is on the y-axis. The line on the graph begins at a time and pressure of 0. The line has two peaks. One is 43 psi at 17 ms; the line then decreases to 40 psi at 19 ms and then increases to a second peak of 56 psi at 24 ms. It then decreases to 0 psi at 40 ms, remains slightly less than 0until 56ms, and then increases to slightly above 0 at 60 ms.

The second graph is labeled FWD deflection histories. The x-axis is time in ms, and the y-axis is vertical deflection in mil. Eight lines are shown on the graph representing r equals 0, 8, 12, 18, 24, 36, 48, and 60 inches. The lines all begin at 0 deflection and have a parabolic shape. After the lines increase to a peak deflection, they decrease to 0 deflection at 40 ms and then to negative deflection between 0 and –1 mil until the graph ends at 60 ms. The line for r equals 0has a peak deflection of 4.8 mil at 0.027 s; for r equals 8, the peak is 4.2 at 0.028 s; for r equals 12, it is at 3.8 at 0.03 s; for r equals 18, it is 3.3 at 0.031 s; for r equals 24, it is 2.8 at 0.032 s; for r equals 36, it is 2 at 0.034 s; for r equals 48, it is 1.5 at 0.035 s; and for r equals 60, it is 1.2 at 0.036 s.

Figure 230. Graph. Backcalculated master curves for LTPP section 6A806. This graph has reduced time in s on the x-axis and relaxation modulus in psi on the y-axis. Two lines are shown on the graph representing lab fitting and backcalculated master curves. Both curves have a sigmoidal shape. The two lines begin at about 4 million psi at 10^-8 s and decrease to about 3,000psi at 10⁸s. The curves agree until a reduced time of 10^-2 s and from 10⁵ to 10⁸ s.

Figure 231. Graph. Backcalculated shift factors for LTPP section 6A806. This graph has temperature in °F on the x-axis and shift factor on the y-axis. Two lines are shown on the graph representing lab fitting and backcalculated results from drop 1. Both curves have parabolic shapes. The lab fitting shift factor curve starts at about 200 at 32 °F and decreases to 10^-10 at 132°F. The backcalculated shift factor curve starts at about 800 at 32 °F and decreases to about 4 times 10^-7 at 132 °F. The curves intersect at a temperature range of 10 at 70 °F, where the shift factors are between 2 and 1.

Figure 232. Graph. Load-to-deflection ratio for LTPP section 6A806. This graph shows load level in psi, ranging from 50 to 150 in intervals of 20, on the x-axis, and load-to-deflection ratio in pci times 1,000, ranging from 0 to 60, on the y-axis. The graph contains lines for eight different sensors. The first line is Load/Def 8; it has a linear decreasing trend beginning at a load-to-deflection ratio of 48 at about 60 psi and decreases to load-to-deflection ratio of 43 at 146 psi. The second line is Load/Def 7; it has a linear decreasing trend beginning at a load-to-deflection ratio of 38 at 60 psi and decreases to a load-to-deflection ratio of 34 at 146 psi. The third line is Load/Def 6; it has a linear decreasing trend beginning at a load-to-deflection ratio of 28 at 60 psi and decreases to a load-to-deflection ratio of 26 at 146 psi. The fourth line is Load/Def 5; it has a linear decreasing trend beginning at a load-to-deflection ratio of 20 at 60 psi and decreases to a load-to-deflection ratio of 19 at 146 psi. The fifth through eighth lines are Load/Def 4, Load/Def3, Load/Def 2, and Load/Def 1. They have a constant trend at load-to-deflection ratios of 17, 15, 14, and 13, respectively.

Figure 233. Graphs. Measured FWD load and time histories for LTPP section 300113. Two graphs are shown representing the falling weight deflectometer (FWD) data for drop 1. The first graph is labeled FWD load history for drop 1. Time in ms is on the x-axis, and pressure in psi is on the y-axis. The line on the graph begins at a time and pressure of 0. The line has two peaks. One is 45 psi at 17 ms; the line then increases to second peak of 56 psi at 24 ms. It then decreases to 0 psi at 40 ms, remains slightly less than 0 until it reaches 56 ms, and then increases to slightly above 0 at 60 ms.

The second graph is labeled FWD deflection histories. The x-axis is time in ms, and the y-axis is vertical deflection in mil. Eight lines are shown on the graph representing r equals 0, 8, 12, 18, 24, 36, 48, and 60 inches. The lines all begin at 0 deflection and have a parabolic shape. After the lines increase to a peak deflection, they decrease to 0 deflection at 40 ms and then to negative deflection between 0 and –0.3 mil until the graph ends at 60 ms. The line for r equals 0has a peak deflection of 10.5 mil at 0.025 s; for r equals 8, the peak is 8 and 0.027 s; for r equals 12, it is at 6 at 0.03 s; for r equals 18, it is 4 at 0.031 s; for r equals 24, it is 3 at 0.032 s; for r equals 36, it is 1 at 0.034 s; for r equals 48, it is 0.6 at 0.035 s; and for r equals 60, it is 0.5 at 0.036 s.

Figure 234. Graph. Backcalculated master curves for LTPP section 300113. This graph has reduced time in s on the x-axis and relaxation modulus in psi on the y-axis. Two lines are shown on the graph representing lab fitting and backcalculated master curves. Both curves have sigmoidal shapes. Both lines begin at about 4 million psi at 10^-8 s. The lab fitting curve decreases to 1,000psi at 10⁸ s. The backcalculated master curve decreases to about 2,000 psi at 10⁸ s. The curves agree until a reduced time of 1 s.

Figure 235. Graph. Backcalculated shift factors for LTPP section 300113. This graph has temperature in °F on the x-axis and shift factor on the y-axis. Two lines are shown on the graph representing lab fitting and backcalculated from drop 1. Both curves have parabolic shapes. The lab fitting shift factor curve starts at about 600 at 32 °F and decreases to 5 times 10^-5 at 132 °F. The backcalculated shift factor curve starts at about 10 at 32 °F and decreases to about 2 at 132°F. The curves intersect at 1 at 70 °F.

Figure 236. Graph. Load-to-deflection ratio for LTPP section 300113. This graph shows load level in psi, ranging from 50 to 150 in intervals of 20, on the x-axis, and the load-to-deflection ratio in pci times 1,000, ranging from 0 to 80, on the y-axis. The graph contains lines for eight different sensors. The first line is Load/Def 8; it has a linear decreasing trend beginning at a load-to-deflection ratio of 60 at about 60 psi and decreases to a load-to-deflection ratio of 58 at 146psi. The second line is Load/Def 7; it has a linear decreasing trend beginning at a load-to-deflection ratio of 53 at 60 psi and decreases to a load-to-deflection ratio of 47 at 146 psi. The third line is Load/Def 6; it has a linear decreasing trend beginning at a load-to-deflection ratio of 36 at 60 psi and decreases to a load-to-deflection ratio of 33 at 146 psi. The fourth line is Load/Def 5; it has a linear decreasing trend beginning at a load-to-deflection ratio of 19 at 60 psi and decreases to a load-to-deflection ratio of 18 at 146 psi. The fifth through eighth lines are Load/Def 4, Load/Def 3, Load/Def 2, and Load/Def 1. They have a constant trend at load-to-deflection ratios of 13, 9, 7, and 5, respectively.

Figure 237. Graphs. Frequency response of geophone components. Two graphs are shown. The left graph is labeled mechanical system. It has frequency in Hz on the x-axis and mechanical sensitivity in ft/ft/s² on the y-axis. The line on the graph begins at a sensitivity of 8times 10^-3 ft/ft/s² at 10^-2 Hz. It remains at this sensitivity until about 1Hz when the line begins to decrease slightly; then at 10^-1 Hz, the line decreases quickly to 10^-4 ft/ft/s² at 10² Hz.

The right graph is labeled electrical transducer. The graph has frequency in Hz on the x-axis and electrical sensitivity in V/yd on the y-axis. The line has a linear trend beginning at 1V/yd at 10^-2 Hz and increases to 2 times 10⁴ V/yd at 10² Hz.

Figure 238. Diagram and Graph. A mechanical inertial seismometer with a natural frequency of 1 Hz. A diagram and a graph are shown. The diagram is labeled mechanical inertial seismometer. It shows a spring hanging from a bar; attached to the bottom of the spring is a square labeled mass. A line extending from the left side of the mass leads to a damping device on the ground. Extending from the right side of the mass is an arrow that points to a ruler positioned next to the system. The ruler is labeled measure of mass displacement.

The graph is labeled amplitude response function for a seismometer with a natural frequency of 1Hz. It has frequency in Hz on the x-axis and the amplitude (normalized to 1 at 1 Hz) on the y‑axis. Five lines are displayed on the graph representing different values of h. Each line begins at a frequency and amplitude of approximately 0.1 and increases to an amplitude of 1 at 10 Hz. The line for h equals ¼ has a peak amplitude of 2 at 1 Hz; the line for h equals ½ has a tiny peak of 1at 1 Hz; and for the lines h equals 1, 2, 3, and 4, there are no peak values, just an increasing trend.

Figure 239. Graphs. Time-frequency content of the load for LTPP section 60565. Two graphs are shown. The first graph is labeled load and deflection time histories. The time in ms is on the x-axis, and the stress in psi and deflection in mil are on the two y-axes. Eight lines are shown on the graph representing sensors 1 through 7 and a line representing the stress. All of the lines begin at 0 and have a parabolic shape. After reaching their peaks, the lines then decrease back to 0. The line for sensor 1 has a peak of 28 at 22 ms; for sensor 2, the peak is 24 at 22 ms; for sensor 3, the peak is at 20 at 23 ms; for sensor 4, it is at 16 and 25 ms; for sensor 5, it is 12 at 26 ms; for sensor 6, it is 6 at 30 ms; and for sensor 7, the peak is at 3 at 38 ms. The line for stress has a double peak; the first peak is at 90 at 10 ms, and the second peak is at 105 at 18 ms.

The second graph is labeled time-frequency representation of the load. The time in s is on the x‑axis, ranging from 0 to 0.06 s, and the frequency in Hz, ranging 0 to 100, is on the y-axis. The graph shows the distribution of the energy contained in the signal in different shades of gray. The left bottom corner of the graph (from 0 to 0.01 s and 0 to 10 Hz) is white, representing 80 dB. Surrounding this area, the brightness reduces gradually, radiating outward to gray from the bottom left corner, representing lower dB. All the way to the right of the graph at 0.06 s, the colors change to dark gray, representing 0 dB, and then to black, representing −40 dB for all values of frequency.

Figure 240. Graphs. Time-frequency content of the load for LTPP section 320101. Two graphs are displayed. The first graph is labeled load and deflection time histories. The time in ms is on the x-axis, and the stress in psi and deflection in mil are on the two y-axes. Eight lines are shown on the graph representing sensors 1 through 7 and a line representing the stress. All of the lines begin at 0 and have a parabolic shape. After reaching their peaks, the lines decrease to 0. The line for sensor 1 has a peak of 5 at 24 ms; for sensor 2, the peak is 4 at 25 ms; for sensor 3, the peak is at 3.4 at 25 ms; for sensor 4, it is at 3 at 25 ms; for sensor 5 it is 2 at 25ms; for sensor 6, it is 1 at 27 ms; and for sensor 7, the peak is at 0.8 at 31 ms. The line for stress has a double peak; the first peak is at 50 at 15 ms, and the second peak is at 65 at 23 ms.

The second graph is labeled time-frequency representation of the load. The time in s is on the x‑axis, ranging from 0 to 0.06 s, and the frequency in Hz, ranging from 0 to 100, is on the y‑axis. The graph shows the distribution of the energy contained in the signal in different shades of gray, ranging from black at −40 to white at 80. The left bottom corner of the graph (at 0.01 s and 0 to 20 Hz), is white, representing 80 dB. Surrounding this area, the brightness reduces gradually, radiating outward to gray from the bottom left corner, representing lower dB. All the way to the right of the graph (from 0.05 to 0.06 s), the colors change to dark gray, representing 0 dB and then to black, representing −40 dB for all values of frequency.

Figure 241. Graphs. Time-frequency content of the load for LTPP section 400113. Two graphs are shown. The first graph is labeled load and deflection time histories. The time in ms is on the x-axis, and the stress in psi and deflection in mil are on the two y-axes. Eight lines are shown on the graph representing sensors 1 through 7 and a line representing the stress. All of the lines begin at 0 and have a parabolic shape. After reaching their peaks, the lines decrease to a deflection between 2 and –2 at 40 ms. The line for sensor 1 has a peak of 32 at 20m; for sensor2, the peak is 24 at 20 ms; for sensor 3, the peak is at 19 at 20 ms; for sensor 4, it is at 14at 20ms; for sensor 5, it is 9 at 20 ms; for sensor 6, it is 4 at 21 ms; and for sensor 7, the peak is at 3 at 22 ms. The line for stress has a peak at 110 at 17 ms.

The second graph is labeled time-frequency representation of the load. The time in s is on the x‑axis, ranging from 0 to 0.06 s, and the frequency in Hz, ranging from 0 to 100, is on the y‑axis. The graph shows the distribution of the energy contained in the signal in different colors, ranging from black at −40 to white at 80 dB. The left bottom corner of the graph (for 0 to 0.01 s and 0 to 20 Hz), is white, representing 80 dB. Surrounding this area, the brightness reduces gradually, radiating outward to gray from the bottom left corner, representing lower dB. All the way to the right of the graph (from 0.055 to 0.06 s), the colors change to dark gray, representing 0 dB and then to black representing −40 dB for all values of frequency.

Figure 242. Graphs. Spectrum of deflection at each sensor for LTPP section 60565. Six graphs are shown. The time in s is on the x-axis, ranging from 0 to 0.06 s, and the frequency in Hz, ranging from 0 to 100, is on the y-axis. The graph shows the distribution of the energy contained in the deflection time histories on each sensor in different shades of gray, ranging from black to white.

The first graph is labeled time-frequency representation of the deflection in sensor 1. The title above the graph reads spectrogram of the deflection in sensor 1 (dB). The left bottom corner of the graph (at 0.01 s and from 0 to 50 Hz) is the white area on the graph, representing 80 dB. Surrounding this area, the brightness reduces gradually, radiating outward to gray from the bottom left corner, representing lower dB and then transitioning to dark gray (0), and then to black (−40) all the way to the right.

The second graph is labeled time-frequency representation of the deflection in sensor 2. The title above the graph reads spectrogram of the deflection in sensor 2 (dB). This graph is almost identical to the first graph except the white area is slightly smaller and the black area represents ‑20 dB.

The third graph is labeled time-frequency representation of the deflection in sensor 3. The title above the graph reads spectrogram of the deflection in sensor 3 (dB). This graph is identical to second graph.

The fourth graph is labeled time-frequency representation of the deflection in sensor 4. The title above the graph reads spectrogram of the deflection in sensor 4 (dB). This graph is identical to second graph, except that the white area is slightly smaller and is centered over 0.012 s.

The fifth graph is labeled time-frequency representation of the deflection in sensor 5. The title above the graph reads spectrogram of the deflection in sensor 5 (dB). This graph is identical to fourth graph, except that white area is wider and the gray area extends further.

The sixth graph is labeled time-frequency representation of the deflection in sensor 6. The title above the graph reads spectrogram of the deflection in sensor 6 (dB). This graph is identical to the fifth graph, except that the white area is centered over 0.025 s and represents 70 dB. The black area represents a −30 dB.

Figure 243. Graphs. Spectrum of deflection at each sensor for LTPP section 320101. Six graphs are shown. The time in s is on the x-axis, ranging from 0 to 0.06 s, and the frequency in Hz, ranging from 0 to 100, is on the y-axis. The graph shows the distribution of the energy contained in the deflection time histories on each sensor in different shades of gray, ranging from black to white.

The first graph is labeled time-frequency representation of the deflection in sensor 1. The title above the graph reads spectrogram of the deflection in sensor 1 (dB). Near the left bottom corner of the graph (at 0.015 s and from 0 to 40 Hz) is the white area on the graph, representing 70 dB. Surrounding this area, the brightness reduces gradually, radiating outward to gray from the bottom left corner, representing lower dB, and then transitioning to dark gray (0) and then to black (−40) all the way to right.

The second graph is labeled time-frequency representation of the deflection in sensor 2. The title above the graph reads spectrogram of the deflection in sensor 2 (dB). This graph is identical to first graph.

The third graph is labeled time-frequency representation of the deflection in sensor 3. The title above the graph reads spectrogram of the deflection in sensor 3 (dB). This graph is identical to first graph, except that the right side of the graph turns black 0.005 s sooner, and the gray area is smaller.

The fourth graph is labeled time-frequency representation of the deflection in sensor 4. The title above the graph reads spectrogram of the deflection in sensor 4 (dB). This graph is identical to first graph, except the white area is slightly smaller and the gray area is large.

The fifth graph is labeled time-frequency representation of the deflection in sensor 5. The title above the graph reads spectrogram of the deflection in sensor 5 (dB). This graph is identical to fourth graph, except that the gray area extends up for a larger range of frequencies.

The sixth graph is labeled time-frequency representation of the deflection in sensor 6. The title above the graph reads spectrogram of the deflection in sensor 6 (dB). This graph is identical to fifth graph, except that the white area is centered over 0.02 s.

Figure 244. Graphs. Spectrum of deflection at each sensor for LTPP section 400113. Six graphs are shown. The time in s is on the x-axis, ranging from 0 to 0.06 s, and the frequency in Hz, ranging from 0 to 100, is on the y-axis. The graph shows the distribution of the energy contained in the deflection time histories on each sensor in different shades of gray colors ranging from black to white.

The first graph is labeled time-frequency representation of the deflection in sensor 1. The title above the graph reads spectrogram of the deflection in sensor 1 (dB). Near the left bottom corner of the graph (centered over 0.007 s and from 0 to 40 Hz), is the white area on the graph representing 80 dB. Surrounding this area, brightness reduces gradually, radiating outward to gray from the bottom left corner, representing lower dB, and then transitioning to light gray (20) all the way to right.

The second graph is labeled time-frequency representation of the deflection in sensor 2. The title above the graph reads spectrogram of the deflection in sensor 2 (dB). This graph is identical to first graph, except the white area is smaller and gray area does not extend as far. Also, all the way to right, the gray area transitions to dark gray (0) and then to black (−20).

The third graph is labeled time-frequency representation of the deflection in sensor 3. The title above the graph reads spectrogram of the deflection in sensor 3 (dB). This graph is identical to second graph, except that the gray area is smaller and the colors transition to dark gray and then black 0.005 s faster.

The fourth graph is labeled time-frequency representation of the deflection in sensor 4. The title above the graph reads spectrogram of the deflection in sensor 4 (dB). This graph is identical to second graph, except the white (70) area is slightly smaller and gray area is larger before the colors transition to dark gray and then to black (−40).

The fifth graph is labeled time-frequency representation of the deflection in sensor 5. The title above the graph reads spectrogram of the deflection in sensor 5 (dB). This graph is identical to fourth graph, except that the white area is smaller and the gray area extends up for a larger range of frequencies. Also, the colors transition to dark gray and then black faster.

The sixth graph is labeled time-frequency representation of the deflection in sensor 6. The title above the graph reads spectrogram of the deflection in sensor 6 (dB). This graph is identical to fifth graph, except that the white area is centered over 0.01 s and there is a second white area centered over 0.03 s. Also, to the right of the graph, the colors transition to black quickly.

Figure 245. Photos. Geophone (left), seismometer (center), and high-accuracy piezoelectric accelerometer (right). Three photos are shown. The left photo shows a geophone. The right end of the geophone has a tip that is to be put into the ground, and extending from the front is a cord. The middle photo shows a metallic frame that houses the seismometer. The right photo shows the accelerometer next to a penny to show that they are approximately the same size. The accelerometer has a box shape.

Figure 246. Photos. Setup attached to the FWD system at TFHRC. Two photos are shown. The right photo shows system mounted onto the back of a truck. Two wires extend out from the back of the device. The left photo shows this same system from a different angle.

Figure 247. Graphs. Sample measured signal (left) and frequency content (right). Two graphs are shown. The first graph is labeled sample measured signal. Time in s is on the x-axis and laser output in mil on the y-axis. The line on the graph shows an oscillating laser output. The values oscillate up and down around 0, and the overall trend of the peaks decreases with time until the line goes to 0. The maximum peak is at a laser output of 16 mil at 2.2 s.

The second graph is labeled frequency content. It has frequency in Hz on the x-axis and Fast Fourier Transform (FFT) on the y-axis. The line on the graph begins at 0.47 FFT at 0 Hz and oscillates up and down, with an overall decreasing trend until the FFT becomes 0 at 55 Hz.

Figure 248. Photo. LK-H008 laser head for deflection measurement. The photo shows an LK-H008 laser head for deflection measurement. It is a black square with a cord extending from the left side.

Figure 249. Diagram. Schematic of the designed fixture. This diagram shows the pieces of the designed fixture. A box-shaped mounting device has two bars connected to one side and the deflectometer and laser connected to the other side. The accelerometer is mounted on top of the box, and the geophone is mounted on the bottom.

Figure 250. Photo. Geophone placed directly on beam next to laser sensor. The photo shows the geophone and laser mounted onto a beam system.

Figure 251. Photos. Test setup for mounted geophone and laser sensor. Three photos are shown. The top photo shows the MTS® loading machine. There is a laser head mounted to a fixed reference, the wall. There is a metal beam placed between two supports. On top of the beam are the geophone and a device that simulates deflection. The middle photo shows a close-up of the geophone, which rests on top of the metal beam. The bottom photo shows a close-up of the device that simulates the deflection on the beam.

Figure 252. Graph. Example of raw data from geophone. This graph has data samples on the x-axis and voltage on the y-axis. The graph shows the voltage oscillating very quickly around 0. The amplitude of the waves ranges from –0.3 to 0.3 V for data samples 10 through 10.6. After this point, the voltage drastically jumps down to -0.35 V and then up to 0.25 V. The voltage then decreases to –0.1 V for data sample 10.9, before returning to its initial range of oscillation for data samples 11.1 through 14.

Figure 253. Graph. Filtered geophone data with multiple replications. This graph has time in s on the x-axis and velocity in inches/s on the y-axis. The graph begins at a velocity of about 0until 0.1 s when the velocity decreases to –0.42 and then increases to 0.35 at 0.2 s. From this point, the velocity decreases to –0.1 at 0.4 s. This decrease is circled on the graph. From
–0.1 inches/s the velocity increases to 0 at 0.55 s, decreases to –0.02 at 0.6 s and then increases to 0.02 at 0.7 s. At 0.9 s, the velocity returns to 0. There is another circle around the portion of the line on the graph from 0.5 s to 0.9 s.

Figure 254. Graph. Laser data with multiple replications. This graph has time in s on the x‑axis and displacement in mil on the y-axis. Two lines are shown on the graph. They begin at 0displacement until about around 0.1 s when the displacement decreases to –17 at 0.15 s. The displacement then returns to 0 at 0.25 s and remains at 0 all the way to 1 s.

Figure 255. Graphs. Comparison of filtered geophone velocity data with the laser derivative output for test series 1. Two graphs are shown. They have time in s on the x‑axis and velocity in mil/s on the y-axis. Both graphs have two lines representing the geophone and laser derivative. The data for the geophone are exactly the same on each graph; however, the results for the laser vary slightly. For both graphs, the line representing the geophone begins at about a velocity of 0 until 0.1 s when the velocity decreases to –0.45 then increases to 0.35 at 0.2s. From this point, the velocity decreases to –0.01 at 0.4 s. On the both graphs there is a circle around the peak velocity. The circled area includes all the positive velocity values for the geophone. After 0.4 s, the velocity increases to 0 at 0.55 s and then decreases –0.04 at 0.6 s. It then increases to 0.04 at about 0.7 s, and at 0.9 s, the velocity returns to 0.

The second line represents the laser. For both graphs, the same peak values are shown; however, in the first graph, there is a slight variation in the results. The line representing the laser has a small wave throughout the entire graph with an amplitude ranging from –0.5 to 0.5 while the second graph shows the laser having a smoother line. For both graphs, the velocity begins at about 0 for 0.1 s. The velocity then decreases to –0.7 at 0.15 s. It then increases to a velocity of 0.2 until about 0.25 s and then decreases to 0 for the remained of the time.

Figure 256. Graphs. Comparison of integrated geophone data with direct laser displacement output for test series 1. Two graphs are shown. They have time in s on the x-axis and displacement in mil on the y-axis. Both graphs have two lines representing direct measurement from the laser and the integration of geophone data. The data for the two lines are almost exactly the same on each graph. In the first graph, the direct measurement line begins at a displacement of 0 and the integration data are very slightly above 0. They remain at about 0 until 0.1 s when the displacement decreases to –17 for the direct measurement and –18 for the integration data. The lines then increase; the integration data increases to 12 mil at 0.3 s, and this peak is circled on the graph, and the direct measurement line increases to 0 at 0.25 s. The direct measurement line remains at 0 for the remainder of the test time while the integration line drops to –4 mil at 0.65 s and then increases to a displacement of 1 mil at 0.9 s before dropping to 0 at 1s.

In the second graph, the only differences from the first graph are that the direct measurement line begins exactly at 0 up to 0.1 s, and the drop for the integration data to –18 mil occurs 0.01 s before it does in the first graph.

Figure 257. Graphs. Comparison of integrated geophone data with direct laser displacement output for test series 2. Two graphs are shown. They have time in s on the x-axis and displacement in mil on the y-axis. Both graphs have two lines representing direct measurement from the laser and the integration of geophone data. The data for both lines are almost exactly the same on each graph.

For the first graph, both lines begin at a displacement of 0 until 0.05 s; the displacement then increases to 5.5 mil for the direct measurement and 4.4 for the integration data at 0.08 s. The two lines then decrease; the integration data decrease to –4.2 mil at 0.13 s, and this drop is circled on the graph. The direct measurement line decreases to 0 at 0.13 s and remains at 0 for the remainder of the test time. The integration line increases to 1 mil at 0.023 s and then decreases to 0displacement at 0.3 s.

For the second graph, the only difference from the first graph is that the integration data line increases slightly slower to its peak values of 4.4 mm.

Figure 258. Photos. Test setup. Three photos are shown. The left photo shows a geophone mounted inside the falling weight deflectometer. It is attached to a metal bar, and there are wires extending from the geophone to a computer. The middle and right photos show the geophone at different angles.

Figure 259. Graphs. Sample of recorded raw geophone data. Four graphs are shown representing data for two different tests. The top left graph is labeled Test 1/5493 lbf load. It has a range of 0 to approximately 16 on the x-axis, and the y-axis represents the voltage in volts. The line on the graph begins at 0 V and oscillates very slightly around 0 from 0 to 8 on the x-axis. The line then jumps to 0.24 V, decreases to -0.17 V, increases to 0.1 V, and decreases to ‑0.12 V, and then oscillates back and forth with increasingly smaller amplitude until it reaches 10 on the x‑axis. It then returns to oscillating around 0 V.

The top right graph is labeled Test 1/5493 lbf load. It has a range of 0 to approximately 16 on the x-axis, and velocity in inches times 10^-2 per second on the y-axis. The line on the graph begins at a velocity of 0 and oscillates very slightly around 0 from 0 to 9 on the x-axis. The line then jumps to a velocity of 24 and decreases to –32, then increases to 16 and decreases to –12, and then oscillates back and forth with increasingly smaller amplitude until it reaches 10 on the x‑axis. It then returns to oscillating around a velocity 0.

The bottom left graph is labeled Test 4/18841 lbf load. It has a range of 0 to 14 on the x-axis, and the voltage in volts on the y-axis. The line on the graph begins at 0 V from 0 to 6.5 on the x‑axis. The line then jumps to a voltage of 1 and decreases to –.7, and then oscillates back and forth with increasingly smaller amplitude until it reaches 9 on the x-axis and returns to 0 V.

The bottom right graph is labeled Test 4/18841 lbf load. It has a range of 0 to 14 on the x-axis and velocity in inches per second on the y-axis. The line on the graph begins at a velocity of 0from 0 to 6.5 on the x-axis. The line then jumps to a velocity of 1 and then decreases to a velocity of –1.4, and then oscillates back and forth with increasingly smaller amplitude until it reaches 9on the x-axis and then returns to a velocity of 0.

Figure 260. Graph. Example of numerical drift due to integration of raw geophone data. This graph is labeled Test 4/18841 lbf load. It has a range of 0 to 3,000 on the x-axis, and deflection in mil on the y-axis. The line on the graph begins at 0 and increases lineally to a deflection of 2 mil at 900 on the x-axis; the slope of this portion of the line is shown as a dashed line continuing throughout the rest of the graph. The line then jumps up to 26 mil and then decreases to –15 mil before returning to 0 at 1,000 on the x-axis. The line then continues to have an increasing and decreasing deflection with increasingly smaller amplitudes, oscillating around the dashed line. The only outlier of the oscillation pattern is at 1,400 on the x-axis when the deflection increases to 14 and then decreases to –6.

Figure 261. Graphs. Illustration of the windowing and filtering procedure and the observed effects on the raw velocity data: frequency content of the velocity signal (left) and effect of the selected cutoff frequency on the signal magnitude as a source of errors (right). Two graphs are shown. The left graph shows the frequency content of the velocity signal and is labeled test 4/18841 lbf load. This graph has frequency in Hz on the x-axis and velocity in inches times 10^-2 per second on the y-axis. The graph begins at a velocity of 0.3 at 0 Hz. The line on the graph continually oscillates up and down, but the general trend of the line increases up to a maximum velocity of 4.8 at 10 Hz then back down to about a velocity of 0.4 at 25 Hz and remains at a velocity of about that value until 40 Hz when the velocity begins to decrease to 0 at 100 Hz.

The right graph shows the effect of the selected cutoff frequency on the signal magnitude as a source of errors and is labeled Test 4, 18841 lbf load. The x-axis has a range of 450 to 800, and the y-axis has velocity in inches times 10^-2 per second. Four lines are shown on the graph representing 2, 5, 10, and 15 Hz. The line representing 2 Hz begins at 0 velocity and increases to a peak velocity of 30 at 505; it then decreases to a velocity of –50 at 540. After this point, it increases to a velocity of 30 at 570, and from there, it slowly decreases in velocity until it reaches 0 at 800. The line representing 5 Hz begins at a velocity of 0 and increases to a peak velocity of 100 at 500; it then decreases to a velocity of -145 at 530. After this point, it increases up to a velocity of 30 at 570, and from there it slowly decreases in velocity until it reaches 0 at 800. The line representing 10 Hz begins at a velocity of 0 and increases to a peak velocity of 80 at 495; it then decreases to a velocity of –120 at 550. After this point, it increases to a velocity of 30 at 580, and from there it slowly decreases in velocity until it reaches 0 at 800. The last line, representing 15 Hz, begins at a velocity of 10; it increases to 120 before decreasing to a velocity of –150 at 550. After this point, it increases to a velocity of 30 at 625, and from there it slowly decreases in velocity until it reaches 0 at 800.

Figure 262. Graphs. Comparison between the filtered and treated seismometer data rendered by the device software and the integrated unfiltered geophone data (left) and integrated and filtered geophone data showing post-peak effects due to propagation of cumulative errors (right).Two graphs are displayed. The left graph shows a comparison between the filtered and treated seismometer data rendered by the device software, and the integrated unfiltered geophone data, and it is labeled Test 1/5493 lbf load. The graph has time in ms on the x-axis and deflection in mil on the y-axis. Two lines are shown on the graph representing KUAB data and geophone data. The two lines begin at 0 deflection and remain so until 30 ms when they both begin increasing. The geophone line increases to 6.5 mil at 60 ms, and the KUAB line increases to 4.5 at 55 ms. Both lines begin decreasing after reaching these peak values; the geophone line decreases to –5 mil at 105 ms and then increases to –1 at 150 ms. The KUAB line decreases to –1 mil at 100 ms and remains at this deflection until the line ends at 120ms.

The right graph presents integrated and filtered geophone data showing post-peak effects due to propagation of cumulative errors, and it is labeled Test 1/5493 lbf load. The graph has time in ms on the x-axis and deflection in mil on the y-axis. The line on the graph represents geophone data. It begins at 0 deflection and remains so until 30 ms when it increases to 6.5 mil at 60 ms; the line then decreases to –5 at 105 ms and then it increases to 2.2 mil at 180 ms before decreasing to 0.7mil at 200 ms when the graph ends.

Figure 263. Graphs. Comparison between the filtered and treated seismometer data rendered by the device software and the integrated unfiltered geophone data at different load levels. Three graphs are shown. For each graph, time in ms is on the x-axis, and deflection in mil is on the y-axis. There are two lines shown on each graph representing KUAB data and geophone data. All three graphs show very similar results.

The top left graph is labeled Test 2/9441 lbf load. The two lines begin at 0 deflection and remain so until 30 ms when they both begin increasing. The geophone line increases to 12 at 60 ms, and the KUAB line increases to 8 mil at 55 ms. Both lines begin decreasing after reaching these peak values; the geophone line decreases to –8 mil at 95 ms and then increases to –0.5 at 150 ms. The KUAB line decreases to –0.5 at 90 ms and remains at this deflection until the line ends at 120 ms.

The top right graph is labeled Test 3/14603 lbf load. The two lines begin at 0 deflection and remain so until 30ms when they both begin increasing. The geophone line increases to 20 mil at 60 ms, and the KUAB line increases to 13 at 55 ms. Both lines begin decreasing after reaching these peak values. The geophone line decreases to –12.3 mil at 95 ms and then increases to –1 at 150 ms. The KUAB line decreases to -3 mil at 85 ms and then decreases to –3.5 when line ends at 120ms.

The bottom graph is labeled Test 4/18841 lbf load. The two lines begin at 0 deflection and remain so until 30ms when they both begin increasing. The geophone line increases to 27 mil at 60 ms, and the KUAB line increases to 18 mil at 55 ms. Both lines begin decreasing after reaching these peak values. The geophone line decreases to –12 at 90 ms and then increases to –2 at 150 ms. The KUAB line decreases to –3.6 mil at 85 ms and then decreases to –4 mil when line stops at 120ms.

Figure 264. Photos. FWD test setup: view of the beam used for mounting the laser (top left), (top right) close-up view of mounted laser, and view of laser sensor setup (bottom). Three photos are shown. The top left photo shows a view of the beam used for mounting the laser. Two white sticks are shown extending up vertically from the ground. The second photo shows a close-up of the laser mounted in the apparatus. Several wires are shown extending off the device. The third shows a different angle of the laser sensor setup.

Figure 265. Graph. Comparison between the seismometer data rendered by the KUAB software and the filtered and treated laser data measured for a 9,500-lbf load. The graph has time in ms on the x-axis and deflection in mil on the y-axis. Two lines are shown on each graph representing KUAB data and laser signal-filtered/treated. The two lines begin at 0deflection and remain so until 35 ms when they both begin increasing. Both lines increase to 8.4 mil at 55 ms, and the laser signal line has a slightly larger deflection at this point. Both lines then begin decreasing after reaching these peak values; the laser signal line decreases to –2.4 mil at 95 ms and then increases to 0.5 at 120 ms. It then decreases one more time to –1 mil at 135ms. The KUAB line decreases to –2.2 mil at 85 ms and then decreases to –2.5 mil when line ends at 120ms.

Figure 266. Equation. Relationship between resilient modulus and stress invariants. E equals k subscript 1 multiplied by theta raised to k subscript 2 power.

Figure 267. Graph. Results for multilayer nonlinear structure surface deflection at the center of the load (r = 0 inches). The x-axis represents E parenthesis t end parenthesis in psi, and the y-axis is the deflection in inches. Twelve lines are displayed on the graph that represent different contact pressures, they are labeled: CP = 10_Algorithm1, CP = 10_Algorithm2, CP = 10_Michpave, CP = 30_Algorithm1, CP = 30_Algorithm2, CP = 30_Michpave, CP = 60_Algorithm1, CP = 60_Algorithm2, CP = 60_Michpave, CP = 80_Algorithm1, CP = 80_Algorithm2, and CP = 80_Michpave. All of the lines have a convex decreasing shape.

The lines for CP = 80_Algorithm1, CP = 80_Algorithm2, and CP = 80_Michpave overlap and begin at 2.15 times 10^-1 inches at 10³ psi and decrease to 10^-2 inches. The lines for CP = 60_Algorithm1, CP = 60_Algorithm2, and CP = 60_Michpave overlap and begin at 1.6 times 10^-1 inches and decrease to 10^-2 inches. The lines for CP = 30_Algorithm1, CP = 30_Algorithm2, and CP = 30_Michpave overlap and begin at 8 time 10^1- inches and decrease to 10^-2 inches. The lines for CP = 10_Algorithm1, CP = 10_Algorithm2, and CP = 10_Michpave overlap and begin at 3 times 10^-2 and decrease to 0.

Figure 268. Equation. Resilient modulus. M subscript R equals, parenthesis quotient of theta divided P subscript a end quotient, end parenthesis, raised to k subscript 2 power.

Figure 269. Graph. Variation of g(σ) with stress and E(t) of AC layer. The x-axis represents relaxation modulus E parenthesis t end parenthesis in psi, and the y-axis represents g parenthesis sigma end parenthesis. Variation of g(σ) with E(t) of AC layer are shown by nine curves corresponding to stress levels of 5, 10, 15, 20, 25, 35, 50, 70, and 140 psi. For each stress level, the relaxation modulus varies from 2,000 to 4 times 10⁶ psi. The nine curves follow a similar pattern. g parenthesis sigma end parenthesis reduces with increase in relaxation modulus and reaches a minimum value at 10,000 psi and then increases and reaches maximum value at 4times 10⁶ psi. At stress level 5 psi, the g parenthesis sigma end parenthesis starts at 0.91, reaches a minimum of 0.88, and ends at 0.99. At stress level 140 psi, the g parenthesis sigma end parenthesis starts at 0.73, reaches a minimum of 0.6, and ends at 0.88. The variation for other stress levels is between 5 and 140 psi.

Figure 270. Graphs. Comparison of ABAQUS and LAVA for nonlinear viscoelastic structure for the control mix where (top) LAVAN uses stress at r = 0, and (bottom) LAVAN uses stress at r = 3.5a. Two graphs are shown. Both graphs have time in ms on the x‑axis and surface deflection in mil on the y-axis. Each graph shows eight lines labeled AS1 through AS8, which represent the surface deflection for sensors 1 through 8 using ABAQUS, and eight lines labeled LS1 through LS8, which represent the surface deflection for sensors 1 through 8 using LAVAN. All of the lines have a haversine shape with the peak deflection occurring at 17 ms. Each line begins at 0 and ends at 35 ms with a surface deflection ranging from 0 to 36 mil.

The top graph shows results from LAVAN using stress at r equals 0. The line with the highest peak is for sensors AS1 and LS1, with a peak deflection 32.8 mil. The next line represents sensors AS2 and LS2; the peak is at 30 mil. The next line represents AS3; its peak is at 28 mil, and the line for LS3 is just below it at 27.8 mil. The line for AS4 has a peak at 25.4, and LS4 is at 24.4. The line for AS5 has a peak deflection at 22.4, and the LS5 peak is at 20.8. Next is the line for AS6; its peak is at 17.8, and the LS6 peak is at 15.8. Next is the line for AS7; its peak is at 13.2, and the LS7 peak is at 12.8. Finally, the peak deflections for the line AS8 is at 10, and LS8 is at 8.4.

The bottom graph shows results for LAVAN using stress at r equals 3.5a. The line with the highest peak is for sensors AS1 and LS1, with a peak deflection 32.8 mil. The next line represents sensors AS2 and LS2; the peak is at 30 mil. The next line represents AS3 and LS3; the peak is at 28. The line for AS4 has a peak at 25.6, and LS4 is at 25.4. The line for AS5 has a peak deflection at 22.6, and the LS5 peak is at 22. Next is the line for AS6; its peak is at 17.6, and the LS6 peak is at 15.8. Next is the line for AS7; its peak is at 13.4, and the LS7 peak is at 11.6. Finally, the peak deflections for the line AS8 is at 9.8 and LS8 is at 8.6.

Figure 271. Graphs. Comparison of ABAQUS and LAVA for nonlinear viscoelastic structure for the CRTB mix where (top) LAVAN uses stress at r = 0 and (bottom) LAVAN uses stress at r = 3.5a. Two graphs are shown. Both graphs have time in ms on the x-axis and surface deflection in mil on the y-axis. On each graph, there are eight lines labeled AS1 through AS8, which represent the surface deflection for sensors 1 through 8 using ABAQUS, and eightlines labeled LS1 through LS8, which represent the surface deflection for sensors 1 through 8 using LAVAN. All of the lines have a haversine shape with the peak deflection occurring at 17ms. Each line begins at 0 and ends at 35 ms with a surface deflection ranging from 0 to 48mil.

The top graph corresponds to LAVAN using stress at r equals 0. The highest peak is for lines LS1 and AS1, with a peak surface deflection of 43.4 mil. The next line represents sensors AS2 and LS2; the peak is at 38. The next line represents AS3 and LS3; its peak is at 33.6. The line for AS4 has a peak at 30, and LS4 is at 28.6. The line for AS5 has a peak deflection at 24.6, and the LS5 peak is at 23.8. Next is the line for AS6; its peak is at 17.6, and the LS6 peak is at 15.6. Next is the line for AS7, its peak is at 12, and the LS7 peak is at 10. Finally, the peak deflections for the line AS8 is at 8.6, and LS8 is at 7.6.

The bottom graph corresponds to LAVAN using stress at r=3.5a. The line with the highest peak is sensor LS1 at 44.6; next is AS1 with a peak deflection of 43.2. The next line represents sensor LS2 with a peak of 40, and then AS2 with a peak of 38. The next line represents LS3 with a peak of 36 and then AS3 with a peak of 35.2. The line for AS4 and LS4 has a peak at 30. The line for AS5 has a peak deflection at 24.6, and the LS5 peak is at 24.4. Next is the line for AS6; its peak is at 17.4, and the LS6 peak is at 16. Next is the line for AS7; its peak is at 12, and the LS7 peak is at 10. Finally, the peak deflection for the line for AS8 is at 8.4, and LS8 is at 7.6.

Figure 272. Graph. Percent error (PE_peak) calculated using the peaks of the deflections for LAVAN-ABAQUS comparison (control mix). This bar graph has with bars representing, control mix r equals 0 and control mix r equals 3.5a. The x-axis represents sensors numbered S1 through S8, and the y-axis is the percent error. At S1, the percent error is 1.5 for r equals 0 and 0.5 for r equals 3.5a. At S2, the percent error is 2.5 for r equals 0 and 0.25 for r equals 3.5a. At S3, the percent error is 3 for r equals 0 and 1 for r equals 3.5a. At S4, the percent error is 4.75 for r equals 0 and 2.5 for r equals 3.5a. At S5, the percent error is 6.2 for r equals 0 and 4 for r equals 3.5a. At S6, the percent error is 9.8 for r equals 0 and 7.8 for r equals 3.5a. At S7, the percent error is 12.5 for r equals 0 and 11.5 for r equals 3.5a. At S8, the percent error is 16.2 for r equals 0 and 15 for r equals 3.5a.

Figure 273. Graph. Percent error (PE_peak) calculated using the peaks of the deflections for LAVAN-ABAQUS comparison (CRTB mix). This bar graph has bars representing data for the crumb rubber terminal blend (CRTB) r equals 0, and the CRTB r equals 3.5a. The x-axis represents sensors numbered S1 through S8, and the y-axis is the average percent error. At S1, the percent error is 2 for r equals 0 and 5.8 for r equals 6.5a. At S2, the percent error is 0 for r equals 0 and 3.8 for r equals 3.5a. At S3, the percent error is 1.5 for r equals 0 and 2.2 for r equals 3.5a. At S4, the percent error is 4.5 for r equals 0 and 0.5 for r equals 3.5a. At S5 the percent error is 7.5 for r equals 0 and 3.5 for r equals 3.5a. At S6, the percent error is 13 for r equals 0 and 9.5 for r equals 3.5a. At S7, the percent error is 16.6 for r equals 0 and 14.1 for r equals 3.5a. At S8, the percent error is 19 for r equals 0 and 17.5 for r equals 3.5a.

Figure 274. Graph. Average percent error (PE_avg) calculated using the entire time history for LAVAN-ABAQUS comparison (control mix). This bar graph has bars representing the control r equals 0 and the control r equals 3.5a. The x-axis represents sensors numbered S1 through S8, and the y-axis is the average percent error. At S1 the percent error is 1.5 for r equals 0, and 2 for r equals 3.5a. At S2, the percent error is 1.4 for r equals 0 and 1.8 for r equals 3.5a. At S3, the percent error is 1.3 for r equals 0 and 1.6 for r equals 3.5a. At S4, the percent error is 1.3 for r equals 0 and 1.4 for r equals 3.5a. At S5, the percent error is 1.5 for r equals 0 and 1.3for r equals 3.5a. At S6, the percent error is 1.9 for r equals 0 and 1.7 for r equals 3.5a. At S7, the percent error is 2.3 for r equals 0 and 2 for r equals 3.5a. At S8, the percent error is 2.2 for r equals 0 and 2.1 for r equals 3.5a.

Figure 275. Graph. Average percent error (PE_avg) calculated using the entire time history for LAVAN-ABAQUS comparison (CRTB mix). This bar graph has bars representing the crumb rubber terminal blend (CRTB) r equals 0 and the control r equals 3.5a. The x-axis represents sensors numbered S1 through S8, and the y-axis is the average percent error. At S1, the percent error is 1.5 for r equals 0 and 2 for r equals 3.5a. At S2, the percent error is 1.3 for r equals 0 and 1.6 for r equals 3.5a. At S3, the percent error is 1.4 for r equals 0 and 1.5 for r equals 3.5a. At S4, the percent error is 1.7 for r equals 0 and 1.4 for r equals 3.5a. At S5, the percent error is 2.2 for r equals 0 and 1.5 for r equals 3.5a. At S6, the percent error is 2.8 for r equals 0 and 2.1 for r equals 3.5a. At S7, the percent error is 2.6 for r equals 0 and 2.4 for r equals 3.5a. At S8, the percent error is 1.4 for r equals 0 and 1.7 for r equals 3.5a.

Figure 276. Graph. E(t) used to compute the deflection basin in examples 1 and 2. The graph shows sigmoid E parenthesis t end parenthesis curve. It has reduced time in s on the x-axis and relaxation modulus, E parenthesis t end parenthesis in psi on the y-axis. The curve has a concave and then convex shape. The curve beings at 10^-3 s and 8.5 times 10⁵ psi and ends at 10² s and 1.6times 10⁴ psi.

Figure 277. Graphs. FWD deflection history for example 1. Two graphs are shown. Both graphs have time in ms on the x-axis. The graph on left shows stress history. The graph has time in s on the x-axis, ranging from 0 to 0.07, and the stress in psi on the y-axis, ranging from 0 to 120. The graph contains a curve with a haversine trend; it begins at the origin and increases to peak at 100 psi at 0.017 s. The stress then decreases back to 0 at 0.035 s and remains at 0 for the remainder of the time.

The graph on the right has time in s on the x-axis and deflection in inches on the y-axis. The figure shows six curves, representing deflection histories obtained from sensors at radial distance 0, 13, 21, 35, 49, and 63 inches from the loading center. All six curves follow the same trend, similar to the loading stress. The curves start at time equal 0 s and end at 0.07 s. The pulses in the curves have a period of approximately 0.035 s. The first curve represents sensor 1; its peak value is 0.06 inches. For sensor 2, the peak values are 0.04 inches. For sensor 3, it is 0.03 inches; for sensor 4; it is 0.017 inches; for sensor 5, it is 0.01 inches; and for sensor 6, it is 0.009 inches.

Figure 278. Graphs. Initial (left) and final (right) backcalculation stage in example 1. Two graphs are shown. The graphs show two sigmoid curves, with time in x-axis varying from 10^-3 s to 10³ s, and E parenthesis t end parenthesis in y-axis varying from 10³ to 10⁶ psi. Continuous lines in the graph represent the backcalculation results, whereas the squares represent the actual values.

In the graph on the left, the actual curve starts at 10⁶ psi at 10^-3 s and ends at 10⁴psi at 10³ s. The backcalculated curve labeled initial starts at 9 times 10³ psi at 10^-3 s and ends at 2 times 10³ psi at 10³ s. In summary, the two curves are completely different.

In the graph on right, the backcalculated curve starts at 10⁶ psi at 10^-3 s and ends at 10⁴ psi at 10³s. The actual curve starts at 10⁶ psi at 10^-3 s and ends at 10⁴ psi at 10³ s. In summary, the two curves almost overlap each other. Sigmoid coefficients for E parenthesis t end parenthesis are presented as 5.0125, −5.0328, −0.26098, and 0.28614.

Figure 279. Graphs. Comparison of backcalculated and actual |E*| and phase angle master curves for example 1. Two graphs are shown. The top graph has reduced frequency in Hz on the x-axis and dynamic modulus in psi on the y-axis. The graph is a scatter plot containing dots that represent the backcalculated curve and squares representing the actual curve. They follow the same sigmoidal trend. At the beginning of the backcalculated curve, the dynamic modulus is 1psi at 1 times 10^-6 Hz, and the curve ends at 2 times 10³ psi at 10⁶ Hz. At the beginning of the actual curve, the dynamic modulus is 4 psi at 10^-8 s, and the curve ends at 2 times 10³ psi at 10⁶ Hz. The two curves match well over 10^-3 Hz to 10⁶ Hz.

The bottom graph has reduced frequency in Hz on the x-axis, and phase angle in degrees on the y-axis. The graph is a scatter plot containing crosses that represent the backcalculated curve and squares representing actual curve. They follow the same convex trend. Both curves have a parabolic shape, with the maximum phase angle occurring at 10^-1 Hz and decreasing to close to a 10-degree angle at 10⁴ Hz. The backcalculated curve starts at 24 degrees at 5 times 10^-5 Hz, the and actual curve starts at 8 degrees, narrowing down the difference to 0 at 10 Hz.

Figure 280. Graphs. Applied stress and resulting deflection basin for example 2. Two pairs of graphs are shown. Each pair is a full graph with a zoomed-in graph shown above it. The graph on the left has time in s on the x-axis and stress in psi on the y-axis. Loading, consisting of a series of eight haversine load pulses from 0 to 45 s, are shown. The first four pulses each have a loading period of approximately 0.035 s followed by a rest period of 0.07 s. This is followed by the next four haversine pulses, each approximately 10 s . The first four pulses are shown in the zoomed-in graph shown above the main graph for more clarity.

The graph on the right has time in s on the x-axis and deflection in inches on the y-axis. The graph shows six curves, representing deflection histories obtained from sensors at radial distance of 0, 13, 21, 35, 49 and 63 inches from the loading center. All six curves follow a similar trend to the loading stress. The curves start at time equal 0 s and end at 45 s. All six curves have eight haversine pulses. The first four pulses in the curve have an approximately 0.035 s pulse period. The next four pulses have an approximately 10 s pulse period. The first four pulses are shown in the zoomed-in graph above the main graph for more clarity. The first curve represents sensor 1; its peak value for the first four pulses gradually increases from 0.013 to 0.014 inches and 0.023 to 0.027 inches for the following four pulses. For sensor 2, the peak values are 0.012 inches and 0.0145 inches deflection, respectively, for the first four and the following four pulses. For sensor 3, 4, 5 and 6, all eight pulses have equal peaks. For sensor 3, it is 0.009 inches; for sensor 4, it is 0.004 inches; for sensor 5, it is 0.0025 inches; and for sensor 6, it is 0.002 inches.

Figure 281. Graph. Backcalculated E(t) using multiple stress pulses. This graph shows backcalculated E(t) curve using fminsearch (using deflection history obtained from multiple stress pulses). The graph has time in s on the x-axis and E parenthesis t end parenthesis in psi on the y-axis. Two sigmoid curves are shown on the graph. The first represents Actual E(t) and the other represents backcalculated E(t). The two curves begins at 9 times 10⁵ psi at 10^-3 s; both lines follow similar paths bending slight concavely then slight convexly until they reach 1.05 times 10⁴psi at 10² s. At this point, the E(t) backcalculated is slightly less than the actual results.

Figure 282. Graphs. Comparison of backcalculated and actual |E*| and phase angle master curves for example 2. Two graphs are shown. The top graph has reduced frequency in Hz on the x-axis and dynamic modulus in psi on the y-axis. The graph is a scatter plot that containing dots that represent the backcalculated curve and squares that represent the actual curve. They follow the same sigmoidal trend. At the beginning of the backcalculated curve, the dynamic modulus is 2 psi at 10^-6 Hz; the curve ends at 2 times 10³ psi at 10⁶ Hz. At the beginning of the actual curve, the dynamic modulus is 4 psi at 10^-8 s; the curve ends at 2 times 10³ psi at 10⁶ Hz. The two curves match well over 10^-3 Hz to 10⁶ Hz.

The bottom graph has reduced frequency in Hz on the x-axis and phase angle in degrees on the y-axis. The graph is a scatter plot containing crosses that represent the backcalculated curve and squares representing the actual curve. They follow the same convex trend. Both curves have a parabolic shape, with the maximum phase angle occurring at 4 times 10^-1 Hz and decreasing to close to a 10-degree angle at 10⁴ Hz. The backcalculated curve starts at 16 degrees at 5 times 10^-5 Hz, and the actual curve starts at 12 degrees. They match reasonably well until the end of the curve.

Figure 283. Graph. Stress history used in the constant frequency multiple pulse analysis. This graph has time in s on the x-axis, ranging from 0 to 1 s, and stress in psi on the y-axis, ranging from 0 to 200 psi. The graph contains a curve with three haversine pulses each of 0.07 s. The first pulse starts at 0 s and reaches a peak value of 100 psi at 0.035 s. The second pulse starts at the end of first pulse at 0.07 s at 55 psi. The second pulse reaches a peak value of 155 psi at 0.105 s. The third pulse starts at the end of second pulse at 0.14 s at 155 psi. It reaches a peak value of 200 psi at 1.85 s and ends at 0.24 s at 0 psi, with the 0 stress continuous up to 1 s.

Figure 284. Graphs. Deflection at different sensors at different temperatures in example 1. The figure has four graphs labeled 50 °F, 86 °F, 104 °F, and 140 °F. Each graph has nine curves, representing surface deflection at increasing radial distance from the center of loading. The graphs have time in s on the x-axis, ranging from 0 to 1 s, and the deflection in inches on the y‑axis. The graphs contain curves with three haversine pulses each of 0.07 s. The first pulse starts at 0 s and reaches peak value at 0.035 s. The second pulse starts at the end of first pulse at 0.07 s. The second pulse reaches peak value at 0.105 s. The third pulse starts at the end of second pulse at 0.14 s. They all reach peak value at 1.85 s and continuously decrease until the end of the graph to approximately 0 at 1 s. The maximum peak values of the curves in graph labeled 50 °F are 0.027, 0.024, 0.021, 0.017, 0.014, 0.009, 0.006, 0.004, and 0.003 inches. The maximum peak values of the curves in the graph labeled 86 °F are 0.048, 0.036, 0.029, 0.021, 0.015, 0.009, 0.006, 0.004, and 0.003 inches. The maximum peak values of the curves in the graph labeled 104°F are 0.068, 0.045, 0.032, 0.021, 0.015, 0.009, 0.006, 0.004, and 0.003 inches. The maximum peak values of the curves in the graph labeled 140 °F are 1.28, 0.05, 0.035, 0.021, 0.015, 0.009, 0.006, 0.004, and 0.003 inches.

Figure 285. Graph. Result for back calculated E(t) curve in example 1. This graph has reduced time in inverse s on the x-axis and relaxation modulus in psi on the y-axis. Two sigmoid curves are shown on the graph; they represent the actual E parenthesis t end parenthesis and the backcalculated E parenthesis t end parenthesis data. The two curves follow a very similar path; they begin at about 3.5 times 10⁶ psi at 10^-15 s. Here the actual and backcalculated curves have the same value and match each other up to 10⁵ s. The curves then decrease bending convexly and then concavely until they reach 10¹⁵ s., where the actual curve is 10³ psi and the backcalculated curve is 1.9 times 10³ psi.

Figure 286. Graph. Result for backcalculated E(t) curve for example 2. This graph has reduced time in inverse s on the x-axis and relaxation modulus in psi on the y-axis. Two sigmoid curves are shown. They represent the actual E parenthesis t end parenthesis and the backcalculated E parenthesis t end parenthesis data. The curves follow a very similar path. They begin at about 3.5times 10⁶ psi at 10^-15 s. After 10⁵ s the curves decrease bending convexly, and bend concavely until they reach 10¹⁵ s, where the actual curve and backcalculated curve both have approximately the same value of 1.1 times 10³ psi.

Figure 287. Graphs. Deflections at different sensors at different temperatures for example 2. The figure has four graphs labeled 32 °F, 50 °F, 68 °F, and 86 °F. Each graph has nine curves, representing surface deflection at increasing radial distance from the center of loading. The graphs have time in s on the x-axis, ranging from 0 to 1 s, and the deflection in inches on the yaxis. The graphs contain curves with three haversine pulses each of 0.07 s. The first pulse starts at 0 s and reaches peak value at 0.035 s. The second pulse starts at the end of first pulse at 0.07 s. The second pulse reaches peak value at 0.105 s. The third pulse starts at the end of second pulse at 0.14 s. They all reach peak value at 1.85 s and continuously decrease until the end of the third pulse at 0.24 s. The maximum peak values of the curves in the graph labeled 32 °F are 0.023, 0.020, 0.018, 0.016, 0.013, 0.008, 0.006, 0.004, and 0.003 inches. The maximum peak values of the curves in the graph labeled 50 °F are 0.027, 0.024, 0.021, 0.017, 0.014, 0.009, 0.006, 0.004, and 0.003 inches. The maximum peak values of the curves in graph labeled 68 °F are 0.035, 0.029, 0.025, 0.020, 0.014, 0.008, 0.005, 0.004, and 0.003 inches. The maximum peak values of the curves in the graph labeled 86 °F are 0.048, 0.036, 0.029, 0.021, 0.015, 0.009, 0.006, 0.004, and 0.003 inches.

Figure 288. Equation. Equation of motion for a continuous medium. The dot product of nabla and sigma plus b equals rho multiplied by the second derivate of u.

Figure 289. Equation. Stress-strain relationship for a linear, homogenous, and isotropic material. The stress tensor: Sigma equals lambda times t times r parenthesis epsilon end parenthesis multiplied by I plus 2 times mu times epsilon.

Figure 290. Equation. Strain-displacement relationship for a linear, homogenous, and isotropic material. The strain tensor, epsilon equals one half times parenthesis nabla times u plus parenthesis nabla times u end parenthesis raised to the T power, end parenthesis.

Figure 291. Equation. Stress-strain relationship for a viscoelastic material. The stress, sigma, equals the convolution of lambda and t, times r parenthesis epsilon end parenthesis, multiplied by I equals 2 times the convolution of mu and epsilon.

Figure 292. Equation. Stieltjes convolution integral. The convolution of alpha and beta equals the integral from 0 to t with respect to tau of the following: alpha, parenthesis t minus tau end parenthesis, times the partial derivative of beta as a function of tau with respect to tau.

Figure 293. Equation. Equation of motion in terms of displacements. Parenthesis lambda plus mu end parenthesis, convolved with nabla times parenthesis the dot product of nabla and u end parenthesis, plus the convolution of mu and nabla squared times u, all of which equals rho multiplied by the second derivate of u.

Figure 294. Equation. Displacement vector in terms of potentials. The displacement vector u equals nabla times phi plus the cross product of nabla and H.

Figure 295. Diagram. Coordinate system for axisymmetric layers on a half-space. The diagram has an x, y, and z-axes, and the reader is facing the x and z plane. Horizontal lines running in the x-direction are equally spaced representing the different layers. The distances between each of the layers are labeled h subscript n, where n is the layer number. A vertical line in the z-direction intersects the center of the layers. At the top of the diagram on the first line is a circle drawn over the x-y plane to show it is axisymmetric. The right side of this line ends with an arrow labeled r, and the left end has an arrow labeled x. The angle between x and r is labeled theta. The different components of each layer are shown starting with the first layer on the top line. There is an arrow extending to the right in the r-direction labeled u subscript rn and an arrow pointing downward in the z-direction labeled u subscript zn, where n is the layer number.

Figure 296. Equation. Equation of motion in terms of the scalar potential. Parenthesis lambda plus 2 times mu end parenthesis, convolved with nabla squared, times phi, all of which equals parenthesis lambda plus 2 times mu end parenthesis, convolved with bracket the second partial derivative of phi with respect to r, plus the partial derivative of phi with respect to r, divided by r, plus the second partial derivative of phi with respect to z, end bracket, all of which equals rho times the second partial derivative of phi with respect to t.

Figure 297. Equation. Equation of motion in terms of the vector potential. The convolution of mu with parenthesis nabla squared times H subscript theta, minus the quotient of H subscript theta divided by r squared, end quotient, end parenthesis, equals the convolution of mu and bracket the second partial derivative of H subscript theta with respect to r, plus the partial derivative of H subscript theta with respect to r, divided by r, plus the second partial derivative of H subscript theta with respect to z, minus the quotient H subscript theta divided by r squared, end quotient, end bracket, all of which equals, rho times the second partial derivative of H subscript theta with respect to t.

Figure 298. Equation. Vector potential H_θ. The vector potential, H subscript theta equals the negative partial derivative of psi with respect to r.

Figure 299. Equation. Scalar potential ψ. The convolution of mu and nabla squared times psi equals the convolution of mu and bracket the second partial derivative of psi with respect to r, plus the partial derivative of psi with respect to r, divided by r, plus the second partial derivative of psi with respect to z, end bracket, all of which equals rho times the second partial derivative of psi with respect to t.

Figure 300. Equation. Relationship between radial displacement and potentials. u subscript r equals the partial derivative of phi with respect to r, plus the second partial derivative of phi with respect to r and z.

Figure 301. Equation. Relationship between vertical displacement and potentials. u subscript z equals the partial derivative of phi with respect to z, minus the second partial derivative of psi with respect to r, minus the partial derivative of psi with respect to r, divided byr.

Figure 302. Equation. Relationship between shear stress and potentials. Sigma subscript rz equals the convolution of mu and the partial derivative with respect to r of bracket 2 times the partial derivative of phi with respect to z, minus the second partial derivative of psi with respect to r, minus the partial derivative of psi with respect to r, divided by r, plus the second partial derivative of psi with respect to z, end bracket.

Figure 303. Equation. Relationship between vertical stress and potentials. Sigma subscript z equals the convolution of lambda and bracket the second partial derivative of phi with respect to r, plus the partial derivative of phi with respect to r, divided by r, plus the second partial derivative of phi with respect to z, end bracket, plus 2 times the convolution of mu and the partial derivative with respect to z times bracket the partial derivative of phi with respect to z, minus the second partial derivative of psi with respect to r, minus the partial derivative of psi with respect to r, divided by r, end bracket.

Figure 304. Equation. Equation of motion in terms of the scalar potential Φ in the Laplace domain. s parenthesis lambda with a hat plus 2 times mu with a hat, end parenthesis, dot product bracket the second partial derivative of phi with respect to r, plus the partial derivative of phi with respect to r, divided by r, plus the second partial derivative of phi with respect to z, end bracket, all of which equals rho times s squared times phi with a hat.

Figure 305. Equation. Equation of motion in terms of the scalar potential Φ in the Laplace-Hankel domain. s parenthesis lambda with a hat plus 2 times mu with a hat, end parenthesis dot product bracket, the second partial derivative of phi with a bar over top, with respect to z, minus k squared times phi with a bar over top, end bracket, equals rho times s squared times phi with a bar over top.

Figure 306. Equation. Simplified form of the equation in figure 305. Two lines of equations are shown. The first line reads: the second partial derivative of phi with a bar over top, with respect to z, minus bracket k squared plus the quotient s divided by c with a hat superscript 2, subscript 1, end quotient, end bracket, times phi with a bar over top, equals 0. The second line reads: c with a hat superscript 2, subscript 1 equals parenthesis, lambda with a hat, plus 2 times mu with a hat, end parenthesis, all divided by rho.

Figure 307. Equation. General form solution of the equation in figure 306. Phi with a bar over top equals A times e raised to the power: negative z times the square root of k squared plus the quotient of s divided by c with a hat superscript 2, subscript 1, end quotient, end square root, equals A times e raised to the power: negative z times f as a function of k and s.

Figure 308. Equation. Equation of motion in terms of the scalar potential Φ in the Laplace-Hankel domain. Two lines of equations are shown. The first line reads: the second partial derivative of psi with a bar over top, with respect to z, minus bracket k squared plus the quotient s divided by c with a hat superscript 2, subscript 2, end quotient, end bracket, times psi with a bar over top, equals 0. The second line reads: c with a hat superscript 2, subscript 2 equals the quotient mu with a hat divided by rho.

Figure 309. Equation. General form solution of the equation in figure 306. Psi with a bar over top equals C times e raised to the power: negative z times the square root of k squared plus the quotient s divided by c with a hat superscript 2, subscript 2, end square root, end parenthesis, equals C times e raised to the power: negative z times g as a function of k and s.

Figure 310. Graph. Bessel functions of the first kind. This graph has values for r on x-axis that range from 0 to 20 and the 0 and first order Bessel functions as a function of r on the y-axis. Two lines are shown on the graph representing the 0 order, J subscript 0 and first order J subscript 1. Both lines have a wave shape with increasingly smaller amplitudes. The line for the 0 order function begins at 1 when r equals 0; it then decreases to -0.4 when r equals 4 and then continues increasing and decreasing until the graph ends at 0.2 when r equals 20. The line for the first order function begins at 0 when r equals 0; it then increases to 0.6 at r equals 3 and then decreases to -0.35 when r equals 5. The line continues increasing and decreasing until it stops at 0.1 when r equals 20.

Figure 311. Equation. Relationship between radial displacement and potentials in the Laplace-Hankel domain. u with a bar over top subscript r equals negative k times phi with a bar over top, minus k times the partial derivative of psi with a bar over top with respect to z.

Figure 312. Equation. Relationship between vertical displacement and potentials in the Laplace-Hankel domain. u with a bar over top subscript z equals the partial derivative of psi with a bar over top with respect to z, plus k squared times phi with a bar over top.

Figure 313. Equation. Hankel transform of a function’s derivative. The integral from 0 to infinity with respect to r of: the partial derivative of f with a hat as a function of r, with respect to r, times J subscript 1 parenthesis k times r end parenthesis times r, all of which equals negative k times the integral from 0 to infinity with respect to r of: f with a hat parenthesis r end parenthesis, times J subscript 0 parenthesis k times r end parenthesis, times r, equals the negative dot product of k and f with a bar over top parenthesis k end parenthesis.

Figure 314. Equation. Relationship between shear stress and potentials in the Laplace-Hankel domain. Sigma subscript rz equals negative s times k times mu with a hat times bracket 2 times the partial derivative of phi with a bar over top, with respect to z, plus 2 times the second partial derivative of psi with a bar over top, with respect to z, minus the quotient s divided by c with a hat superscript 2, subscript 2, end quotient, end bracket, equals negative s times k times mu with a hat times bracket 2 times the partial derivative of phi with a bar over top, with respect to z, plus 2 times k squared times psi with a bar over top, plus the quotient s times psi with a bar over top, divided by c with a hat superscript 2, subscript 2,end quotient, end bracket.

Figure 315. Equation. Relationship between vertical stress and potentials in the Laplace-Hankel domain. Sigma with a bar over top subscript z equals s times the dot product of lambda with a hat and bracket negative k squared times phi with a bar over top plus the second partial derivative of phi with a bar over top with respect to z, end bracket; plus 2 times s times mu with a hat, times the partial derivative with respect to z times bracket the partial derivative of phi with a bar over top with respect to z, plus k squared times psi with a bar over top, end bracket, all of which equals mu with a hat times the quotient of s squared, divided by c with a hat superscript 2, subscript 2 times phi with a bar, plus 2 times s times k squared times mu with a hat times parenthesis, phi with a bar over top plus the partial derivative of psi bar with respect to z.

Figure 316. Equation. Scalar potential Φ in the Laplace-Hankel domain. Phi with a bar over top equals A times e raised to the negative z times f power, plus B times e raised to the negative parenthesis h minus z end parenthesis times f power.

Figure 317. Equation. Scalar potential ψ in the Laplace-Hankel domain. Psi with a bar over top equals C times e raised to the negative z times g power, plus D times e raised to the negative parenthesis h minus z end parenthesis times g power.

Figure 318. Equation. Radial displacement in the Laplace-Hankel domain. u with a bar over top, subscript r, equals negative A times k times e raised to negative z times f power, minus B times k times e raised to the negative parenthesis h minus z end parenthesis, times f, power minus C times k times g times e raised to negative z times g power, minus D times k times g times e raised to negative parenthesis h minus z end parenthesis times g power.

Figure 319. Equation. The vertical displacement in the Laplace-Hankel domain. u with a bar over top, subscript z, equals negative A times f times e raised to negative z times f power, plus B times f times e raised to negative parenthesis h minus z end parenthesis, times f power, plus C times k squared times e raised to negative z times g power, plus D times k squared times e raised to negative parenthesis h minus z end parenthesis times g power.

Figure 320. Equation. Relationship between shape factors and boundary conditions. A four-by-one matrix (starting at the top of the column and moving down the matrix) reads: u with a bar over top subscript r1, u with a bar over top subscript z1, u with a bar over top subscript r2, and u with a bar over top subscript z2. This matrix equals a four-by-four matrix (reading each row from left to right): negative k, negative k times e raised to the negative hf power, k times g, and negative k times g times e raised to the negative hg power. The next row reads: negative f, f times e raised to the negative hf power, k squared, and k squared times e raised to the negative hg power. The next row reads: negative k times e raised to the negative hf power, negative k, k times g times e raised to the negative hg power, and negative k times g. The last row reads: negative f times e raised to the negative hf power, f, k squared times e raised to the negative hg power, and k squared. This matrix is multiplied by a four-by-one matrix that reads: A, B, C, and D. This is equal to the dot product of S subscript 1 with a four-by-one matrix: A, B, C, and D.

Figure 321. Equation. Shear and vertical stress in the Laplace-Hankel domain. Three equations are shown. The first equation reads: Sigma with a bar over top subscript rz equals negative s times mu with a hat, times k, multiplied by bracket negative 2 times A times f times e raised to the negative zf power, plus 2 times B times f times e raised to the negative parenthesis h minus z end parenthesis times f power, plus C times K times e raised to the negative zg power, plus D times K times e raised to the negative parenthesis h minus z end parenthesis times g power, end bracket.

The second equation reads: Sigma with a bar over top subscript z equals s times mu with a hat multiplied by bracket A times K times e raised to the negative zf power, plus B times K times e raised to the negative parenthesis h minus z end parenthesis times f power, minus 2 times C times K squared times g times e raised to the negative zg power, plus 2 times D times k squared times g times e raised to the negative parenthesis h minus z end parenthesis times g power, end bracket.

The third equation reads: K equals 2 times k squared plus the quotient s divided by c with a hat subscript 2 superscript 2.

Figure 322. Equation. Relationship between stresses and shape factors. A four-by-one matrix (starting at the top of the column and moving down the matrix) reads: sigma with a bar over top subscript rz1, sigma with a bar over top subscript z1, sigma with a bar over top subscript rz2, and sigma with a bar over top subscript z2. This equals s times the dot product of mu with a hat and a four-by-four matrix. Reading that matrix from left to right: 2 times f, negative 2 times f times e raised to the negative hf power, negative K, and negative K times e raised to the negative hg power. The second row reads: K, K times e raised to the negative hf power, negative 2 times k squared times g, and 2 times k squared times g times e raised to the negative hg power. The third row reads: 2 times f times e raised to the negative hf power, negative 2 times f, negative K times e raised to the negative hg power, and negative K. The last row reads: K times e raised to the negative hf power, K, negative 2 times k squared times g times e raised to the negative hg power, and 2 times k squared times g. This matrix is multiplied by a four-by-one matrix that reads: A, B, C, and D. This is equal to s times the dot product of mu with a hat and S subscript 2, times a four-by-one matrix: A, B, C, and D.

Figure 323. Equation. Stress-displacement relationship in the Laplace-Hankel domain. A four-by-one matrix reads: sigma with a bar over top subscript rz1, sigma with a bar over top subscript z1, sigma with a bar over top subscript rz2, and sigma with a bar over top subscript z2. This matrix equals s times the dot product of mu with a hat and S subscript 2 and with the inverse of S subscript 1 times a four-by-one matrix that reads: u with a bar over top subscript r1, u with a bar over top subscript z1, u with a bar over top subscript r2, and u with a bar over top subscript z2.

Figure 324. Equation. Relationship between the tractions, stresses, and displacements. A four-by-one matrix reads: T with a bar over top subscript r1, T with a bar over top subscript z1, T with a bar over top subscript r2, and T with a bar over top subscript z2. This equals a four-by-one matrix, which reads: negative sigma with a bar over top subscript rz1, negative sigma with a bar over top subscript z1, sigma with a bar over top subscript rz2, and sigma with a bar over top subscript z2. This is equal to the dot product of S subscript 2-noded with a four-by-one matrix that reads: u with a bar over top subscript r1, u with a bar over top subscript z1, u with a bar over top subscript r2, and u with a bar over top subscript z2.

Figure 325. Equation. Local stiffness matrix of the two-noded layer element. S subscript 2-noded equals s times mu times N times S subscript 2 times the inverse of S subscript 1, where N equals the four-by-four matrix: first row reads 1, 0, 0, 0, second row reads 0, -1, 0, 0, third row read 0, 0, 1, 0, and fourth row reads 0, 0, 0, 1.

Figure 326. Equation. Radial displacement of a one-noded layer element. u with a bar over top subscript r equals negative A times k times e raised to the negative zf power minus C times k times g times e raised to the negative zg power.

Figure 327. Equation. Vertical displacement of a one-noded layer element. u with a bar over top subscript z equals negative A times f times e raised to the negative zf power plus C times k squared times e raised to the negative zg power.

Figure 328. Equation. Displacements at the boundary of a one-noded layer element. A two-by-one matrix reads: u with a bar over top subscript r1, and u with a bar over top subscript z1, which equals a two-by-two matrix that reads (from left to right): negative k, k times g, second row: negative f, and k squared, which is multiplied by a two-by-one matrix: A, C. Equals S subscript 3 times a two-by-one matrix: A, C.

Figure 329. Equation. Shear stress of a one-noded layer element. Sigma with a bar over top subscript rz equals negative s times mu with a hat multiplied by bracket, negative 2 times A times k times f times e raised to the negative zf power, plus C times k times parenthesis k squared plus g squared end parenthesis times e raised to the negative zg power, end bracket.

Figure 330. Equation. Vertical stress of a one-noded layer element. Sigma with a bar over top subscript z equals s times mu with a hat multiplied by bracket A parenthesis k squared plus g squared end parenthesis, times e raised to the negative zf power minus 2 times C times k squared times g times e raised to the negative zg power, end bracket.

Figure 331. Equation. Stresses at the boundary of a one-noded layer element. A two-by-one matrix reads: sigma with a bar over top subscript r1, sigma with a bar over top subscript z1. This matrix equals s times the dot product of, mu with a hat and a two-by-two matrix. The first line of the two-by-two matrix reads: two times k times f and negative k times parenthesis k squared plus g squared end parenthesis; the second line reads: parenthesis k squared plus g squared end parenthesis and negative two times k squared times g. This matrix is multiplied by a two-by-one matrix: A, C. This is equal to s times mu with a hat times S subscript 4 times a two-by-one matrix: A, C.

Figure 332. Equation. Stress-displacements relationship at the boundary of a one-noded layer element. A two-by-one matrix reads: sigma with a bar over top subscript rz1, sigma with a bar over top subscript z1, which equals s times the dot product of mu with a hat, with S subscript 4 and the inverse of S subscript 3 times a two-by-one matrix that reads: u with a bar over top subscript r1, u with a bar over top subscript z1.

Figure 333. Equation. Relationship between the tractions, stresses, and displacements at the boundary of a one-noded layer element. A two-by-one matrix reads: T with a bar over top subscript r1, T with a bar over top subscript z1, which equals a two-by-one matrix that reads: negative sigma with a bar over top subscript rz1, negative sigma with a bar over top subscript z1, which equals the dot product of S subscript 1-noded with a two-by-one matrix that reads: u with a bar over top subscript r1, u with a bar over top subscript z1.

Figure 334. Equation. Local stiffness matrix of the one-noded layer element. S subscript 1-noded equals negative s times the dot product of mu with a hat with S subscript 3 and the inverse of S subscript 4.

Figure 335. Equation. Relationship between the elastic modulus (E) and the lamé constant (μ) for homogenous, isotropic, elastic material. Mu equals E divided by two times parenthesis 1 plus nu, end parenthesis.

Figure 336. Equation. Laplace transform of the equation in figure 335. Mu with a hat parenthesis s end parenthesis equals mu divided by s, which also equals the dot product of E divided by 2 times parenthesis 1 plus nu end parenthesis, with 1 divided by s.

Figure 337. Equation. Lamé constant for elastic material. Mu with a hat parenthesis s end parenthesis equals the dot product of mu with a hat parenthesis s end parenthesis with parenthesis 1 plus the dot product of zeta with s, end parenthesis.

Figure 338. Equation. Prony series. Three lines of equations are shown. The first equation states: E parenthesis t end parenthesis, equals E subscript infinity plus the summation from 1 to 14 of the following: parenthesis E subscript m times e raised to parenthesis negative t subscript r divided by T times K subscript m power end parenthesis, end parenthesis. The second equation states: The log of t subscript r, equals the log of t minus the log of a times T. The third equation states: The log a times T equals a subscript 1 times parenthesis T minus Tref squared end parenthesis plus a subscript 2 times parenthesis T minus Tref end parenthesis.

Figure 339. Equation. Prony series in the Laplace domain. Three equations are shown. The first equation reads: E parenthesis s end parenthesis equals E subscript infinity plus the summation from 1 to 14 of the following: parenthesis the cross product of E subscript m with the quotient convolution of T times K subscript m with a times T divided by the cross product of T times K subscript m times s subscript r plus one end quotient, end parenthesis. The second equation reads: s subscript r equals the cross product or s with a times T. The third equation reads: the log of a times T, equals the cross product of a subscript 1 with parenthesis T minus Tref squared end parenthesis plus the cross product of a subscript 2 times parenthesis T minus Tref end parenthesis.

Figure 340. Equation. Relationship between the elastic modulus (E) and the lamé constant (μ) for viscoelastic material. Mu with a hat parenthesis s end parenthesis equals E with a hat parenthesis s end parenthesis, divided by 2 times parenthesis 1 plus nu, end parenthesis.

Figure 341. Equation. Construction of the global stiffness matrix for structures with a half-space. The equation reads: S subscript global equals a global matrix whose diagonal is constructed of local matrixes. Within the global matrix, starting in the upper left corner, is a matrix with the label S lower subscript 2-noded superscript 1. The next matrix in the diagonal has the label, S subscript 2-noded superscript 2. There are dots indicating the pattern repeats, and then in the bottom right corner is the local matrix with the label, S subscript 2-noded superscript n–1 and within this matrix is another matrix with the label S subscript 1-noded superscript n.

Figure 342. Equation. The displacement at the system nodes. Three equations are shown. The first equation reads: U with a bar over top equals the dot product of the inverse of S subscript Global with P with a bar over top.

The second equation reads: U with a bar over top equals bracket, U with a bar over top subscript r1, U with a bar over top subscript z1…U with a bar over top subscript ri, U with a bar over top subscript zi…U with a bar over top subscript rn, U with a bar over top subscript zn… end bracket.

The third equation reads: P with a bar over top equals bracket, P with a bar over top subscript r1, P with a bar over top subscript z1….P with a bar over top subscript ri, P with a bar over top subscript zi…P with a bar over top subscript rn, P with a bar over top subscript zn… end bracket.

Figure 343. Equation. Boundary conditions. Two equations are shown. The first equation reads: P subscript z1 as a function of r and t equals the dot product of R parenthesis r end parenthesis, with delta parenthesis t end parenthesis. The second equation reads: R parenthesis r end parenthesis equals 1 if r is greater than 0 and less than or equal to a, or it is equal to 0 if r is greater than a.

Figure 344. Equation. Boundary conditions in the Laplace-Hankel domain. P with a bar over top, subscript z1 as a function of k and s equals the quotient a divided by k end quotient times J subscript 1 times parenthesis k times a, end parenthesis.

Figure 345. Equation. Inverse Hankel transform of the vertical displacement. U with a hat subscript zi, parenthesis r end parenthesis, equals the integral from 0 to infinity with respect to k of the following: U with a bar over top subscript zi times parenthesis k end parenthesis times J subscript 0 times parenthesis k times r end parenthesis, times k times d times k.

Figure 346. Equation. Inverse Hankel transform as a series of integrals. U with a hat subscript zi, parenthesis r end parenthesis equals the integral from b subscript 1 to b subscript 2, with respect to k of the following: U with a bar over top subscript zi times parenthesis k end parenthesis times J subscript 0 times parenthesis k times r end parenthesis, times k, plus the integral from b subscript 2 to b subscript 3, with respect to k of the following: U with a bar over top subscript zi times parenthesis k end parenthesis times J subscript 0 times parenthesis k times r end parenthesis, times k, plus the integral from b subscript n to b subscript n +1, with respect to k of the following: U with a bar over top subscript zi times parenthesis k end parenthesis times J subscript 0 times parenthesis k times r end parenthesis, times k, where n is any number.

Figure 347. Equation. Evaluation of the inverse Hankel transform using six-point Gaussian quadrature scheme. Two equations are shown. The first equation reads: the integral from b subscript n to b subscript n +1, with respect to k of the following: U with a bar over top subscript zi times parenthesis k end parenthesis times J subscript 0 times parenthesis k times r end parenthesis, times k, equals the quotient b subscript n +1 minus b subscript n, divided by 2 end quotient times the summation with the lower bound p = 1 and the upper bound 6 of the following: w subscript p times U with a bar over top subscript zi times parenthesis beta subscript p end parenthesis, times J subscript 0 times parenthesis beta subscript p end parenthesis times beta subscript p.

The second equation reads: beta subscript p equals parenthesis, the quotient b subscript n+1 minus b subscript n divided by 2, end quotient, end parenthesis, multiplied by x subscript p, plus parenthesis the quotient b subscript n +1 plus b subscript n divided by two, end quotient, end parenthesis.

Figure 348. Equation. Bromwich integral. U subscript zi, parenthesis t end parenthesis equals the quotient one divided by 2 times pi times j, end quotient times the integral from B, with respect to s of the following: e raised to the ts power times U with a hat subscript zi times parenthesis s end parenthesis.

Figure 349. Equation. The contour for the Bromwich integral. Two equations are shown. The first equation reads: S parenthesis theta end parenthesis equals alpha times theta times parenthesis cotangent of theta plus j end parenthesis, when theta is greater than negative pi and less than positive pi. The second equation reads: alpha equals 2 times M divided by 5 times t.

Figure 350. Equation. Bromwich integral along the chosen contour path. Two equations are shown. The first equation reads: U subscript zi, parenthesis t end parenthesis, equals the quotient of alpha divided by pi end quotient multiplied by the integral from 0 to pi with respect to theta of the following: R times e, bracket, e raised to the ts parenthesis theta end parenthesis power, times U with a hat subscript zi times parenthesis s parenthesis theta, end parenthesis, end parenthesis multiplied by parenthesis 1 plus j times gamma parenthesis theta end parenthesis, end parenthesis, end bracket.

The second equation reads: Gamma parenthesis theta end parenthesis equals theta plus parenthesis cotangent of theta minus 1 end parenthesis, multiplied by cotangent theta.

Figure 351. Equation. Evaluation of Bromwich integral through the trapezoidal rule. Two equations are shown. The first equation reads: U subscript zi, parenthesis t end parenthesis equals the quotient alpha divided by M end quotient multiplied by bracket one-half times U with a hat, subscript zi, parenthesis alpha end parenthesis times e raised to the alpha times t power plus the summation, with the lower bound q equals 1 and the upper bound M minus 1 of the following: R times e, bracket, e raised to the ts parenthesis theta subscript q end parenthesis power multiplied by U with a hat subscript zi times parenthesis s parenthesis theta subscript q, end parenthesis, end parenthesis multiplied by parenthesis 1 plus j times gamma parenthesis theta subscript q end parenthesis, end parenthesis, end bracket. The second equation reads: theta subscript q equals q times pi divided by M.

Figure 352. Equation. Convolution integral for a continuous function. The vertical displacement at node i: y subscript zi parenthesis t end parenthesis equals the convolution of U subscript zi, parenthesis t end parenthesis, with T parenthesis t end parenthesis, which also equals the integral from 0 to t with respect to tau of the following: U subscript zi, multiplied by parenthesis t minus tau end parenthesis, multiplied by T parenthesis tau end parenthesis.

Figure 353. Equation. Numerical evaluation of the convolution integral. The vertical displacement at node i: y subscript zi parenthesis t subscript n end parenthesis equals the summation, with the lower bound t subscript p equals 1 and the upper bound of t subscript n of the following: U subscript zi, parenthesis t subscript n minus t subscript p end parenthesis, times T parenthesis t subscript p end parenthesis multiplied by delta t.

Figure 354. Graph. Laboratory-measured dynamic modulus at station 1 using 1.5- by 3.94‑inch samples. The x-axis is labeled reduced frequency f subscript r equals the convolution of f with aT. The y-axis is the absolute value of the complex conjugate E in psi. Three sets of data are shown on the graph representing the bottom sample, top sample, and average. The average data are a line on the graph beginning just below 10⁴ psi at 10^-4, and increasing up to 10⁶ psi at 10⁴. The bottom sample and top sample are shown on the graph as dots that sit above or below the average line. From the beginning of the line until reaching a reduced frequency of 10^-1, the majority of dots for the top sample sit above the average line. Then, from 10^-1 to 1, the dots representing the top sample sit above the average line, and the dots representing the bottom sample sit below the average line. Then, from 1 to the end of the graph, the dots for the top sample sit below the average line, and the dots for bottom sample sit above the average line.

Figure 355. Graph. Laboratory-measured dynamic modulus at station 1, using 3.94- by 5.9‑inch samples. The x-axis is labeled reduced frequency f subscript r equals the convolution of f with aT. The y-axis is the absolute value of the complex conjugate E in psi. Three sets of data are shown on the graph representing sample A, sample B, and average. The average sample is a line on the graph beginning just below 10⁴ psi at 10^-5, and increasing up to 10⁴ psi at 10⁵. Samples A and B are shown on the graph as dots that sit above or below the average line. From the beginning of the line until reaching a reduced frequency of 10^-3, the dots for samples A and B are overlapping and sit above the average line. Then, from 10^-3 to 10^-1, the dots representing samples A and B sit above and on the average line. Then from 10^-1 to the end of the graph, the dots for samples A and B sit on the average line.

Figure 356. Graph. Laboratory-measured dynamic modulus at station 3, using 1.5- by 3.94‑inch samples. The x-axis is labeled reduced frequency f subscript r equals the convolution of f with aT. The y-axis is the absolute value of the complex conjugate E in psi. Three sets of data are shown on the graph representing the top sample, bottom sample, and average. The average sample is a line on the graph beginning just below 10⁴ psi at 10^-5, and increasing up to 10⁶ psi at 10⁴. The top sample and bottom sample are shown on the graph as dots that sit above or below the average line. From the beginning of the line until reaching a reduced frequency of 10^-3, the dots for the bottom sample sit above the average line. Then, from 10^-3 to 10^-1, the dots representing the top sample are sitting directly above the average line, and the dots representing the bottom sample are sitting above those for the top sample. Dots for the top and bottom samples are shown directly on the average line here as well. Then from 10^-1 to the end of the graph, the dots for the top and bottom samples are very closely spaced together and sit on the average line. A few dots for the bottom sample are also sitting below the average line.