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Federal Highway Administration Research and Technology
Coordinating, Developing, and Delivering Highway Transportation Innovations
This report is an archived publication and may contain dated technical, contact, and link information 

Publication Number: FHWAHRT04127 Date: January 2006 
Figure 1. Flowchart. Hiperpav earlyage behavior framework showing improved models in Hiperpav 2. The flowchart depicts a text box on the top with text: Temperature Prediction*, PCC Hydration*, and Environmental Effects*. This text box flows downward into two other sets of text boxes. The left set is the Critical Stress Models with three text boxes: Axial Stress and Axial Restraint*, Curling Stress and Vertical Restraint*, Shrinkage Stress*. The right set is the Mechanical Properties with three text boxes: Coefficient of Thermal Expansion, RelaxationCreep Model, and Tensile Strength. The top two text boxes from the right set flow toward the top two text boxes from the left set. These sets of text boxes flow downward to one text box containing the text Stress (on the left) and Strength (on the right) connected by a bidirectional arrow. The asterisk represents improved models.
Figure 2. Sketch. Heat transfer mechanisms between pavement and its surroundings. The schematic shows a pavement, Hydrating Concrete on top of a Base Layer on top of a Subbase Layer. There is some sun and cloud cover above, with some wind close to the pavement. Solar radiation is being transmitted from the sun to the hydrating concrete causing solar adsorption. The concrete layer is putting out irradiation. The wind causes convection in the concrete layer. There is conduction happening between the concrete layer and the base layer and between the base layer and the subbase layer.
Figure 3. Graph. Concrete specific heat as influenced by the mixture constituents, temperature, and degree of hydration. Graph depicts Specific Heat on the Yaxis and Degree of Hydration on the Xaxis. There are two sets of lines in the graph, the higher set is for limestone and the lower set is for siliceous river gravel. Each set contains three straight solid lines starting from the same point. The highest line is for temperature of 35 degrees Celsius, the middle line is for temperature of 20 degrees Celsius, and the lowest line is for temperature of 5 degrees Celsius. All lines decrease from left to right.
Figure 4. Graph. Comparison of different convection coefficients as influenced by the windspeed. The graph depicts Convection on the Yaxis and Windspeed on the Xaxis. There are three lines starting from the same point on the left and increase as they go to the right. The top linear line is from Equation 21 and represents the vertical plate. The middle log line is from Equation 21 with T subscript lowercase C equal to 30 degrees Celsius and T subscript lowercase A equal to 20 degrees Celsius. The lower log line is from equations 22 and 23 with T subscript lowercase C equal to 30 degrees Celsius and T subscript lowercase A equal to 25 degrees Celsius. These two lower lines represent the Hiperpav horizontal plate.
Figure 5. Sketch. Radiant energy exchanges between the sky and an exposed thermally black plate. The sketch depicts a thermally black plate at temperature T subscript lowercase S on top of thermal insulation. This black plate is represented in equation form by sigma time T subscript lowercase S raised to the fourth power. On top is the atmospheric radiation represented by A subscript R. To the side is the total solar radiation represented by lowercase Q subscript lowercase R. The equation for radiation loss rate (R) is R equal to sigma times T subscript lowercase S minus A subscript R.
Figure 6. Graph. Emissivity of moist air at a total pressure of 1 atmosphere and a temperature of 20 degrees Celsius. The graph shows Total Emissivity on the Yaxis and WaterVapor Density Length Product on the Xaxis. There are three lines in the graph. The smooth Sshape line increasing from left to right represents water vapor. Another line that traces this line except for the upward bump along the middle represents mixture of water vapor and carbon dioxide. The small mountain shape line at the bottom that looks like the bump is the second line represents the correction due to the presence of carbon dioxide (P subscript C divided P subscript W equals 0.10).
Figure 7. Graph. Sensitivity of the apparent surrounding temperature to changes in climatic conditions, atmospheric pressure. The graph has the Apparent Surrounding Temperature on the Yaxis and DryBulb Temperature on the Xaxis. It has three straight lines, increasing from left to right. The lines are, from top to bottom, for P subscript Z of 1000 millibars, 750 millibars, and 500 millibars, respectively.
Figure 8. Graph. Sensitivity of the apparent surrounding temperature to changes in climatic conditions, relative humidity. The graph has the Apparent Surrounding Temperature on the Yaxis and DryBulb Temperature on the Xaxis. It has three straight lines, increasing from left to right. The lines are, from top to bottom, for relative humidity (RH)of 100 percent, 60 percent, and 20 percent, respectively.
Figure 9. Graph. Sensitivity of the apparent surrounding temperature to changes in climatic conditions, ratio of carbon dioxide to water vapor. The graph has the Apparent Surrounding Temperature on the Yaxis and DryBulb Temperature on the Xaxis. It has three lines that are virtually on top of each other, except for the lowest line, which starts out lower and curves into the other two lines. The lines are for P subscript C divided by P subscript W that equal 1.0, 0.5, and 0.1, respectively, and increase from left to right.
Figure 10. Sketch. Surface layer zone subjected to drying shrinkage for a slipformed pavement. The sketch depicts a shaded area that is in the shape of a bench. The thickness of this shaded area is depicted as L subscript SD, and within the shaded area is the symbol alpha subscript SD.
Figure 11. Graph. Influence of water to cementitious materials ratio on total shrinkage predicted by the Jonasson model. The graph shows Total Shrinkage on the Yaxis and Time on the Xaxis. There are five lines that start out at zero and stay horizontal until the time reaches approximately 25 hours. The lines then branch out to five logwavy shaped lines, increasing from left to right. The lines are, from bottom to top, for watertocement ratios that equal 0.2, 0.25, 0.3, 0.35, and 0.399, respectively.
Figure 12. Graph. Effect of watertocement ratio on total shrinkage predicted by the BaźantPanula model. The graph shows Total Shrinkage on the Yaxis and Time on the Xaxis. There are three wavy lines close to each other and have local peaks at around times of 15 hours, 32 hours, and 55 hours. The lines are for, from bottom to top, watertocement ratios that equal 0.4, 0.5, and 0.6, respectively.
Figure 13. Graph. Comparison of the BaźantPanula and JonassonHedlund shrinkage models. The graph shows Total Shrinkage on the Yaxis and Time on the Xaxis. There are three wavy lines. The top line, with significant waves and is much higher than the other two lines, represents the BazantPanula's watertocement ratio that equals 0.4. The middle line, with small waves, represents the Jonasson's watertocement ratio that equals 0.399, TS equal to 1. The bottom line, with small waves, represents the Jonasson's watertocement ratio that equal 0.399, TS equal to set time. All three seem to have local peaks at time of 32 hours and 55 hours.
Figure 14. Two Graphs. Timedependent deformation at time T, for a loading at time T subscript 0. The top graph has Applied Stress on the Yaxis and Time on the Xaxis. It has a vertical line (at time T subscript 0) connecting a horizontal line, forming a sideways L. The height of the vertical line represents the Stress at time T subscript 0. The bottom graph has Strain on the Yaxis and Time on the Xaxis. It has a vertical line (at time T subscript 0) connected by a log line increasing from left to right and leveling out at time T. The vertical line is the instantaneous elastic strain and the log line is the creep strain. The height of the vertical line is represented by the equation Instantaneous Elastic Strain at time T subscript 0 equal to the quotient of the Stress at time T subscript 0 and the Instantaneous Modulus of Elasticity at time T subscript 0. The height of the log line is represented by the equation Creep Strain from time T subscript 0 to time T equal to the creep coefficient times the Instantaneous Elastic Strain at time T subscript 0. The total height of both is equal to the creep compliance defined as the response at time T after loading at time 0 times the applied stress at time T subscript 0.
Figure 15. Graph. A schematic of the additional creep compliance functions to extend the Triple Power Law for the earlyage creep response. The graph shows Creep Compliance on the Yaxis and Time on the Xaxis. There are two vertical lines, the left representing time T subscript 1and the right representing T subscript 0. The T subscript 1 line meets two inverse exponential lines, decreasing from left to right. These two lines start at different heights and merge together when they level out. The height between these two lines where they intersect with the T subscript 0 vertical line is represented by the equation psi subscript 1 at time T subscript 0 divided by the initial Modulus of Elasticity. The T subscript 0 line meets three log lines, increasing from left to right. The distance between the top log line and the middle log line is represented by the equation psi subscript 2 at time T, T subscript 0 divided by initial Modulus of Elasticity.
Figure 16. Graph. Decomposition of stress history into stress steps. The graph shows Stress on the Yaxis and Time on the Xaxis. The relationship is represented by diagonal line increasing from left to right meeting up with a horizontal line. There are little vertical lines representing the stress increment at time T subscript 0.
Figure 17. Graphs. Discreet subdivision of time for numerical creep analysis. The top graph has Strain on the Yaxis and Time and lowercase R on the Xaxis. It shows a vertical line (at time T subscript 0, lowercase R equal to 0 and 1) meeting up with a horizontal line as lowercase R increases by increments of 1. The increment represents the change in T subscript lowercase R. The bottom graph has Stress on the Yaxis and Time and lowercase R on the Xaxis. It is similar to the strain graph, except instead of a horizontal line, the vertical line meets up with an inverse exponential line, decreasing from left to right. The incremental height in the inverse exponential line represents the change is sigma subscript lowercase R.
Figure 18. Graph. Superposition of various strains intensities: Loading. The figure has epsilon subscript 1 on the Yaxis and Time and lowercase R on Xaxis. It shows a hillshaped curve (peaking around lowercase R equals 7) with the incremental height representing the change in epsilon subscript lowercase R as lowercase R goes from 3 to 4.
Figure 19. Graph. Superposition of various strains intensities: Unloading. The figure has epsilon subscript 2 on the Yaxis and Time and lowercase R on Xaxis. It shows a hillshaped curve (peaking around lowercase R equals 7) with the incremental height representing the change in epsilon subscript lowercase R as lowercase R goes from 11 to 12.
Figure 20. Graph. Superposition of various strains intensities: Net applied strains. The figure has an equation epsilon subscript T equal to epsilon subscript 1 minus epsilon subscript 2 on the Yaxis and Time and lowercase R on Xaxis. It shows a hillshaped curve (peaking around lowercase R equals 7) that is the difference of epsilon subscript 1 minus epsilon subscript 2.
Figure 21. Graph. Comparison of the results of the relaxation model and model without relaxation. The graph shows Total Stress on the left Yaxis, Total Strain on the right Yaxis, and Concrete Age on the Xaxis. The graph has three wavy lines, the bottom most represents the Baźant and Westman model, the one that starts out in the middle but ends up as the highest represents the No Relaxation model, and the one that starts out highest but ends up in the middle represents the Strain model. All three seem to peak at concrete age of 42 hours and seem to dip at concrete age of 28 hours.
Figure 22. Graph. Monthly moisture content variation for lean clay (CL), scenarios 14. The graph depicts Moisture Content on the Yaxis and Month on the Xaxis. The graph shows four scenarios, represented by four wavy lines. Scenarios 1 and 2 are closer to each other and are lower than scenarios 3 and 4, with scenario 2 having higher and lower peaks than scenario 1. Scenarios 3 and 4 are close to each other with scenario 4 having higher and lower peaks than scenario 3. All scenarios seem to have local peaks in May, July, and November and have low points in March, June, and September.
Figure 23. Graph. Monthly moisture content variation for wellgraded silty gravel (GWGM), scenarios 58. The graph depicts Moisture Content on the Yaxis and Month on the Xaxis. The graph shows four scenarios, represented by four wavy lines. Scenarios 5 and 6 are close to each other and are lower than scenarios 7 and 8, with scenario 6 having higher and lower peaks than scenario 5. Scenarios 7 and 8 are close to each other with scenario 8 having higher and lower peaks than scenario 7. All scenarios seem to have local peaks in May, July, and November and have low points in March, June, and September.
Figure 24. Graph. Monthly variation for wellgraded gravel (GW), scenarios 912. The graph depicts Moisture Content on the Yaxis and Month on the Xaxis. The graph shows four scenarios, represented by four wavy lines. All scenarios (9, 10, 11, and 12) are very close to each other with scenarios 9 and 10 having lower peaks at times. There are really no peaks, but there are low points in March, June, and September.
Figure 25. Graph. Monthly variation for lean clay (CL) in five U.S. cities. The graph depicts Moisture Content on the Yaxis and Month on the Xaxis. The graph shows five wavy lines that are widely spread representing conditions for five cities. The cities are, from highest moisture to lowest, Atlanta (peak in March), Chicago (peak in August), Oklahoma (peak in May), San Antonio (peak in May), and Reno (highest in January). There are periods where the Atlanta line falls below the Chicago line.
Figure 26. Graph. Monthly variation for wellgraded silty gravel (GWGM) in five U.S. cities. The graph depicts Moisture Content on the Yaxis and Month on the Xaxis. The graph shows five wavy lines that are closely spaced representing conditions for five cities. The cities are, from highest moisture to lowest, Atlanta (peak in March), Chicago (peak in August), Oklahoma (peak in May), San Antonio (peak in May), and Reno (highest in January). There are periods where the Atlanta line falls below the Chicago line and meets the Oklahoma line.
Figure 27. Graph. Comparison of predicted and measured values from the AASHO road test. The graph depicts Moisture Content on the Yaxis and Month on the Xaxis. The graph shows three wavy lines and six dotted lines. The three wavy lines represented the predicted moisture content for embankment (highest), subbase (middle), and base (lowest). The subbase and base wavy lines are close to each other while the embankment line is much higher. All three wavy lines are lowest in January and February and peak in July and September. Each wavy line has two dotted horizontal lines under them representing the measured moisture content. The measured moisture content from March to June is close to the predicted moisture content, while the measured moisture from June to September is much lower than the predicted moisture content.
Figure 28. Graph. Texas LTPP site comparison. The graph depicts Moisture Content on the Yaxis and Month on the Xaxis. It shows two wavy lines. The higher one, averaging around 20 percent moisture content, represents the subgrade. The lower one, averaging around 12 percent moisture content, represents the subbase. Both lines have local peaks around May, July, and November, and low points around March, June, and September. There is a horizontal line closely under the subbase wavy line representing the LTPP Value for Subbase moisture content.
Figure 29. Graph. Maine LTPP site comparison. The graph depicts Moisture Content on the Yaxis and Month on the Xaxis. It shows two relatively straight solid lines. The higher one, averaging around 15 percent moisture content, represents the subgrade. The lower one, averaging around 8 percent moisture content, represents the subbase. There is a relatively straight dotted line closely under each of the above lines, representing the LTPP Values for Subgrade and Subbase moisture content, respectively.
Figure 30. Graph. Comparison of test results with the correlation proposed in equation 122. The graph shows Modulus of Rupture on the Yaxis and Compressive Strength on the Xaxis. It has many data points representing data from Gonnerman and Shuman, Walker and Bloem, Houk, and Grieb and Werner increasing from left to right and is best fitted by a line that has the equation Modulus of Rupture is equal to 2.3 times the Compressive Strength raised to the twothirds power.
Figure 31. Graph. Modulus of rupture versus compressive strength for Texas data. The graph shows Modulus of Rupture on the Yaxis and Compressive Strength on the Xaxis. It has many data points representing data from the Texas DOT quality control/quality assurance project increasing from left to right and is best fitted by Raphael's relationship, which is from the equation the Modulus of Rupture equal to 2.3 times the Compressive Strength raised to the twothirds power. The ACI relationship, which comes from the equation the Modulus of Rupture equal to 0.7 times the square root of the Compressive Strength falls below the data points.
Figure 32. Graph. Tensile strength and elastic modulus values calculated using the CEBFIP equation. The graph has Tensile Strength on the left Yaxis and Elastic Modulus on the right Yaxis and Age of PCC on the Xaxis. The Xaxis is on a log scale. On the Yaxis, 1 megapascal in tensile strength equals 5000 megapascals in elastic modulus. There are two log shape lines increasing from left to right then leveling out, one for tensile strength and one for elastic modulus. The tensile strength line starts out at about 1.7 and levels out at about 6.1. The elastic modulus line starts out at about 16,000 and levels out at about 31,000.
Figure 33. Sketch. Schematic of deflection load transfer (LTE subscript delta) for a doweled JCP. The schematic shows to pieces of slabs jointed by a dowel with the load on top of the left slab. Each slab has a delta divided by 2 symbol underneath representing the same deflection for each.
Figure 34. Flowchart. Schematic of LTE model logic. The flowchart starts out with two text boxes, Dowel Influence and Aggregate Interlock, on the left. These two then flow into a Joint Stiffness textbox. This text box flows to two text boxes, Unloaded Edge Deflection and Unloaded Edge Stress. These two flow to two more text boxes, Loaded Edge Deflection and Stress and Load Transfer Efficiency (LTE). There are two text boxes, Free Edge Stress and Free Edge Deflection that are on top and flowing downward toward the Loaded Edge Deflection and Stress textbox.
Figure 35. Graph. Relationship between the calculated aggregate interlock parameter and measured JTE. The graph shows Dimensionless Aggregate Interlock on the Yaxis and Joint Transfer Efficiency (JTE) on the Xaxis. The Yaxis is on a log scale. There are data points that are in the shape of an exponential line starting at 40 percent and seem to go vertical at around 98 percent.
Figure 36. Graph. Relationship between measured crack opening and the calculated aggregate interlock parameter. The graph depicts Dimensionless Aggregate Interlock on the Yaxis and Crack Opening on the Xaxis. The Yaxis is on a log scale. There are two sets of data, one for the 229millimeter data and one for the 178millimeter data. Each is best fitted by a straight line decreasing from left to right. The line for the 229millimeter data is higher than the line for the 178millimeter data.
Figure 37. Sketch. Free edge loading of JCP. The sketch shows a slab on top of a subbase on top of a subgrade. The slab is being loaded at a free edge and is deflecting. That deflection, the freeedge deflection, is represented by delta subscript F.
Figure 38. Sketch. Loaded and unloaded deflections of a JCP. The sketch shows a jointed slab on top of a subbase on top of a subgrade. The left edge is being load and is deflecting a little more than the right edge. The loaded deflection is represented by delta subscript l and the unloaded deflection is represented by delta subscript U.
Figure 39. Graph. Relationship between JTE and LTE subscript delta. The graph depicts Deflection Load Transfer Efficiency (LTE) on the Yaxis and Joint Transfer Efficiency (JTE) on the Xaxis. There is a straight line with a slope of 1 increasing from left to right representing the line of equality. The actual line is a curve that is under this line touching the two ends of the straight line forming a thin D shape.
Figure 40. Graph. Relationship between LTE subscript sigma and LTE subscript delta. Deflection Load Transfer Efficiency (LTE) is on the Yaxis and Stress Load Transfer Efficiency (LTE) on the Xaxis. There are six logshaped curves increasing from left to right representing different AtoL ratios. The curves are for, from highest and most curvy to lowest and flattest, AtoL ratios of 0.02, 0.05, 0.1, 0.2, 0.5, and 1.0.
Figure 41. Graph. JTE results obtained from the experimental work performed by Colley and Humphrey as a function of loading cycles of a 4086kilogram load. Percent Effectiveness is on the Yaxis and Loading Cycles from 0 to 1,000,000 are on the Xaxis. There are five lines that depict different joint openings at 0.6 millimeters, 0.9 millimeters, 1.1 millimeters, 1.7 millimeters, and 2.2 millimeters. As the size of the openings increases, their effectiveness decreases. All but the 2.2millimeter joint openings decrease steadily in effectiveness as loading cycles increase, eventually leveling out. The 2.2millimeter opening also decreases as loading cycles increase, but effectiveness drops to 0 percent at 500,000 loading cycles.
Figure 42. Graph. Influence of joint opening on JTE, 229millimeter concrete slab, 152millimeter gravel subbase. Effectiveness is on the Yaxis and Loading Cycles are on the Xaxis. There are five curves decreasing from left to right representing different joint openings. The top line is an inverse exponential line that remains straight after the first initial drop and is for joint opening of 0.6 millimeter. The next three lines are inverse exponential lines and are for, from top to bottom, joint openings of 0.9, 1.1, and 1.7 millimeters, respectively. The fifth line, in an S shape, far below the others, is for a joint opening of 2.2 millimeters.
Figure 43. Influence of joint opening on JTE, 178millimeter concrete slab, 152millimeter gravel subbase. Effectiveness is on the Yaxis and Loading Cycles are on the Xaxis. There are five curves decreasing from left to right representing different joint openings. The top line is a linear line and is for joint opening of 0.4 millimeter. The next three lines are inverse exponential lines that have low slopes and are for, from top to bottom, joint openings of 0.6, 0.9, and 1.1 millimeters, respectively. The fifth line, also an exponential line but with a very steep drop, is for a joint opening of 2.2 millimeters.
Figure 44. Graph. Asymptote dimensionless joint stiffness as a function of joint opening. Aggregate interlock parameter is on the Yaxis and 1 divided by joint opening in millimeters is on the Xaxis. The Yaxis is on a log scale. There are two sets of data, the one to the left is for aggregate interlock parameter asymptote (229 millimeter) and the one on the right is for the aggregate interlock parameter asymptote (178 millimeter). Each set of data is best fitted by a straight line increasing from left to right.
Figure 45. Graph. Sensitivity of faulting to joint opening and cumulative traffic loading for nondoweled JPCP (MESAL equals 1,000,000 ESAL). Faulting is on the Yaxis and MESALs is on the Xaxis. There are seven log shape lines that are widely spaced from each other increasing from left to right representing joint openings of, from top to bottom, 12.7, 5.1, 2.5, 1.3, 0.5, 0.3, and 0.1 millimeter, respectively.
Figure 46. Graph. Sensitivity of faulting to joint opening and cumulative traffic loading for doweled JPCP (MESAL equals 1,000,000 ESAL). Faulting is on the Yaxis and MESALs is on the Xaxis. There are seven log shape lines that are very closely spaced to each other increasing from left to right representing joint openings of, from top to bottom, 12.7, 5.1, 2.5, 1.3, 0.5, 0.3, and 0.1 millimeter, respectively.
Figure 47. Graph. Relationship between joint opening model predictions. Hiperpav Model Joint Opening is on the Yaxis and FHWA Model Joint Opening is on the Xaxis. There is a 45degree line representing the line of equality. The data points left of this line suggest that the Hiperpav model predicts greater joint openings than the FHWA model.
Figure 48. Graph. Relationship between revised joint opening model predictions. Hiperpav Model Joint Opening is on the Yaxis and FHWA Model Joint Opening is on the Xaxis. There is a 45degree line representing the line of equality. The data points falling on the line suggest that the Hiperpav model's predictions of joint openings are virtually the same as the FHWA model's predictions.
Figure 49. Graph. Allowable loads versus stresstostrength ratio. Allowable Load Applications is on the Yaxis and StressStrength Ratio is on the Xaxis. There are five lines decreasing from left to right representing five different models. The steep linear line represents the ZeroMaintenance Model. The flat, close to linear line represents the ARE Model. The three inverse exponential lines represent, from bottom to top, the NCHRP Model 126 at 5 percent failure, the ERESCOE Model, and the NCHRP Model 126 at 50 percent failure, respectively.
Figure 50. Graph. Fatigue cracking model with associated data. Percent Slab Cracked is on the Yaxis and Fatigue Damage is on the Xaxis. There are scattered data points that are mostly on the right side of the graph and are best fitted by an Sshaped curve increasing from left to right.
Figure 51. Flowchart. Flowchart of stress, damage, and cracking module. Flowchart starts with a "Loop over two times of day" text box that flows to a "Loop over six axle groups" text box that flows to a "Calculate transverse edge stress adjustment" text box that flows to a "Calculate longitudinal corner stress adjustment" text box that flows to a "Loop over 45 load groups" text box that flows to a "Calculate Westergaard transverse edge stress adjustment" text box that flows to a "Calculate Westergaard longitudinal edge stress adjustment" text box that flows to a "Adjust Westergaard stresses" text box that flows to a "Calculate midslab fatigue" text box that flows to a "Accumulate midslab damage" text box that flows to a "Have all load groups been processed?" text box. If the answer is yes, it flows to a "Have all axle groups been processed?" text box. If the answer is no, it flows back to the "Loop over 45 load groups" text box. If the answer to the "Have all axle groups been processed?" text box is yes, it flows to a "Have all times of day been processed?" text box. If the answer is no, it flows back to the "Loop over six axle groups" text box. If the answer to the text box "Have all times of day been processed?" is yes, it flows to a "Compute cracking for season" text box. If the answer is no, then it flows back to the "Loop over two times of day" text box.
Figure 52. Graph. Amount of each individual distress required to reach a PSI of 2.0. The graph is a bar graph describing JCP distresses. The left bar, going past 2000, is the cumulative transverse joint faulting in millimeters per kilometer. The middle bar, reaching 500, is the number of deteriorated (medium and high severity) transverse joints per kilometer. The right bar, almost reaching 1000, is the number of transverse cracks (all severities) per kilometer.
Figure 53. Graph. Shape of IRI versus PSI curve. IRI is on the Yaxis and PSI is on the Xaxis. The relationship is represented by a linear line decreasing from left to right that corresponds to the equation IRI (meters per kilometer) equal to 1.0602 times the difference of 5 minus psi.
Figure 54. Graph. Effect of faulting on IRI. IRI (meters per kilometer) is on the Yaxis and Total Faulting per Section (millimeter) is on the Xaxis. The relationship is represented by a linear line increasing from left to right that corresponds to the equation IRI (meters per kilometer) equal to 0.0098 times the total faulting per section plus 1.268.
Figure 55. Graph. Comparison of faulting influence between the various IRI models. IRI is on the Yaxis and Faulting is on the Xaxis. There are five linear lines increasing from left to right representing five different IRI versus faulting models. The slopes of all the lines tend to be similar. One of the major distinctions is the high initial IRI subscript 0 of the FHWARD97147 model. The FHWARD00130 model and the Transtec model are very similar, as are the two 137A models.
Figure 56. Graph. Effect of percent cracked slabs on FHWARD00130, 137A Guide, and contractor IRI models. IRI is on the Yaxis and Percent Cracking is on the Xaxis. There are four linear lines increasing from left to right representing four different IRI versus cracking models. Increasing percent cracked slabs causes a marked increase in IRI according to the 137A model presented in 2000 at the Transportation Research Board (TRB) conference. Its effect on the other three models is not as severe, with the FHWARD00130 model and the 137A model presented in 2001 at TRB being virtually the same.
Figure 57. Graph. Effect of percent spalling on FHWARD00130, 137A Guide, and contractor IRI models. IRI is on the Yaxis and Percent Spalling is on the Xaxis. There are four linear lines increasing from left to right representing four different IRI versus spalling models. Increasing the percent spalling causes a slight increase in IRI for all four models. The slopes of the FHWARD00130 and 137A IRI models are approximately the same while the slope of the Transtec model is slightly higher.
Figure 58. Graph. Effect of percent patching on IRI for the 137A models. IRI is on the Yaxis and Percent Spalling is on the Xaxis. There are two linear lines increasing from left to right representing the 137A models. The 137A model presented in 2000 is more sensitive to patching than is the one presented in 2001.
Figure 59. Graph. Effect of initial IRI subscript 0 on FHWARD00130, 137A, and contractor IRI models. IRI is on the Yaxis and Initial IRI is on the Xaxis. There are four linear lines representing four different IRI versus spalling models. The Transtec model does not account for initial IRI and thus it produced a straight line. The slopes of all the other models are very similar with the two most similar models being the FHWARD00130 model and the 137A model presented in 2001.
Figure 60. Sketch. Schematic representation of analysis of concrete and steel stresses in CRCP8. The sketch shows a schematic representation of the one dimensional model for analysis of concrete and steel stresses in a CRCP basic analysis unit bounded by two adjacent transverse cracks and longitudinal joints. The schematic also shows a free body diagram depicting different forces at work.
Figure 61. Sketch. Simplified coordinate system for development of bond stress distribution functions. The sketch depicts a slab from midslab to a crack. Bond slippage begins twothirds of the way from midslab. From midslab to that point is the fully bonded zone. From that point to the crack is the bond development zone.
Figure 62. Graph. Compressive strength as a function of watertocement ratio and cement content. Compressive strength is on the Yaxis and watertocement ratio is on the Xaxis. There are two lines mostly decreasing from left to right representing the Basic Model and the Augmented Model. These two models are very close to each other except at low watertocement ratios (less than 0.3); the compressive strength continues to increase according to the basic model. This relationship would be correct for fully compacted concrete, but at lower watertocement ratios, it is difficult to achieve full compaction. The augmented model, used in COMET, provides a more reasonable prediction of compressive strength, since it drops slightly at watertocement ratios of less 0.3.
Figure 63. Sketch. Schematic of dowel deformation and loading at the joint. The sketch depicts two slabs jointed by a dowel curling upward, causing the dowel bar to curl downward. The dowel is subjected to moments M subscript 0.
Figure 64. Sketch. Schematic of dowel deformation without concrete compliance and with concrete compliance at the joint. Two cases are shown this sketch. In the first case, the gray dowel depicts behavior at the joint assuming that the concrete underneath it does not deform, or that there is no concrete compliance. In the second case, the concrete has deformed underneath the black dowel. The gray dowel bar deforms more than the black dowel bar.
Figure 65. Graph. Effect of varying the dowel diameter (lowercase D subscript D) on the bearing stress. Bearing Stress is on the Yaxis and Dowel Diameter is on the Xaxis. As the dowel diameter increases from about 16 millimeters to about 38 millimeters, the Bearing Stress increases linearly from about 4.5 megapascals to about 8.8 megapascals.
Figure 66. Graph. Effect of varying the effective modulus of dowel support (K subscript D) on the bearing stress. Bearing Stress is on the Yaxis and Effective Modulus of Dowel Support (KD) is on the Xaxis. As K subscript D increases from about 280,000 megapascals per meter to about 830,000 megapascals per meter, the Bearing Stress increases linearly from about 4.5 megapascals to about 10.5 megapascals.
Figure 67. Graph. Effect of varying the joint opening on the bearing stress. Bearing Stress is on the Yaxis and Joint Opening is on the Xaxis. As the joint opening increases from 0 millimeters to 25 millimeters, the Bearing Stress decreases linearly from about 7.8 megapascals to about 5.8 megapascals.
Figure 68. Graph. Effect of varying the concrete modulus (E subscript C) on the bearing stress. Bearing Stress is on the Yaxis and Concrete Modulus is on the Xaxis. As the concrete modulus increases from about 21,000 megapascals to about 42,000 megapascals, the Bearing Stress increases linearly from about 7 megapascals to about 8.9 megapascals.
Figure 69. Graph. Effect of varying the concrete CTE (alpha subscript C) on the bearing stress. Bearing Stress is on the Yaxis and Concrete Coefficient of Thermal Expansion is on the Xaxis. As the coefficient increases from about 7.2 to about 12.7, the Bearing Stress increases linearly from about 5 megapascals to about 9 megapascals.
Figure 70. Graph. Effect of varying the slab length (L) on the bearing stress. Bearing Stress is on the Yaxis and Slab Length is on the Xaxis. As the slab length increases from 3 meters to 4.7 meters, the bearing stress decreases linearly from 100 megapascals to 75 megapascals. From then on, an increase in slab length doesn't seem to affect the bearing stress.
Figure 71. Graph. Effect of varying the slab thickness (lowercase H) on the bearing stress. Bearing Stress is on the Yaxis and slab thickness is on the Xaxis. As the slab thickness increases from about 200 millimeters to about 360 millimeters, the Bearing Stress increases very close to linearly from about 62 megapascals to about 115 megapascals.
Figure 72. Graph. Effect of varying the modulus of subgrade reaction (lowercase K) on the bearing stress. Bearing Stress is on the Yaxis and the modulus of subgrade reaction (lowercase K) is on the Xaxis. As k increases, the Bearing Stress decreases. The bearing stress is most sensitive to the change in lowercase K if lowercase K is less than 50 megapascals per meter. Changes in lowercase K when lowercase K is more than 50 megapascals per meter does not have significant effect on the bearing stress.
Figure 73. Graph. Effect of varying the linear temperature gradient (T) on the bearing stress. Bearing Stress is on the Yaxis and the linear temperature gradient (T) is on the Xaxis. The relationship is represented by a V originating from 0, meaning the bearing stress increases with the increase in the absolute value of the linear temperature gradient.
Figure 74. Sketch. Schematic representation of slabs loaded in shear. The sketch depicts joints in the pavements being loaded in shear and the dowel bars resisting this load. The free edge deflection, lowercase Y subscript 0 (in meters), is restrained by twice the value of lowercase Y subscript 1 (in meters) due to the dowels.
Figure 75. Screen Shot. Hiperpav 2 screen shot showing typical output from the dowel bar module. The screen shot shows the dowel analysis module in Hiperpav 2. The output shows a graph in which the dowel bearing strength and stress is on the Yaxis and time of day is on the Xaxis. The graph depicts relationships between bearing strength, curlcurl stress, and flatcurl stress and time of day. The bearing strength increased in a shape of a log line over the total elapsed time while the stresses go through cycles represented by little moundshaped curves. Both the stresses seem to be greatest at 6 AM.
Figure 76. Graph. Pavement temperature profiles for Illinois JPCP evaluation. Depth is on the Yaxis and Pavement Temperature is on the Xaxis. There are three sets of data points representing three different times of day. In the morning, the pavement temperature is slightly higher with increasing depth. During midday and evening times, the pavement temperature is much higher closer to the surface.
Figure 77. Graph. Joint movement for section AA in Illinois JPCP evaluation. Movement is on the Yaxis and Mean Pavement Temperature is on the Xaxis. The downward trend is indicative of joints closing under higher temperatures and opening under cooler temperatures. After 36 degrees Celsius, no further movement is observed, possibly because of joint closure after that temperature.
Figure 78. Graph. Longitudinal profiles for section AA, 253millimeter thickness. Deviation from Grade is on the Yaxis and Distance is on the Xaxis. The data points can be best fitted by wavy curves relating to a curling pattern at every 6meter interval corresponding to the slab length for this section.
Figure 79. Sketch. Ticuman bypass project location. The illustration shows the general location and layout of the road of the Ticuman bypass section, which is located south of Mexico City.
Figure 80. Graph. Mean concrete compressive and flexural strength gain curves for the Ticuman bypass. Strength is on the Yaxis and Time is on the Xaxis. The compressive strength increases greatly in the first 3 days and then steadily from there to 28 days, reaching a value of 30 megapascals. The flexural strength increases gradually over a period of 14 days and then seems to stay the same from there to 28 days, reaching a value of 5 megapascals.
Figure 81. Graph. Number of distressed slabs per kilometer for the Ticuman bypass. Number of Distresses per Kilometer is on the Yaxis and Date is on the Xaxis. There are four lines increasing from left to right representing four different distresses. It can be observed from the graph that longitudinal cracking is the predominant distress type, reaching 35 per kilometer. Also a considerably large number of slabs have a mix of longitudinal, transverse, and/or corner cracking. These slabs were reported as shattered slabs, reaching 18 per kilometer.
Figure 82. Graph. Faulting distribution for the Ticuman bypass. Frequency is on the left Yaxis, Cumulative is on the right Yaxis, and Faulting is on the Xaxis. The frequencies are represented by bars while the cumulative is represented by a line. Faulting of 0 and 1 millimeter have the highest frequency (48 and 53, respectively), followed by faulting of 0.5, 1.5, and 2 millimeters. The cumulative line increases slightly in a linear fashion from negative 3.0 to negative 0.5 millimeters, then it takes the form of a log line increasing from left to right.
Figure 83. Graph. Measured joint opening during August 22 to 24. Number of Joints is on the left Yaxis, Cumulative is on the right Yaxis, and Joint Movement is on the Xaxis. The frequencies are represented by bars while the cumulative is represented by a line. Joint movement of negative 0.033 millimeter per degree Celsius has the highest frequency (4 joints) followed by joint movement of negative 0.035 millimeter per degree Celsius (3 joints). There are no joints that experience movement equal to or less than negative 0.041 millimeter per degree Celsius or equal to or more than negative 0.015 millimeter per degree Celsius. The cumulative line seems to take on the form of a slanted S shape with some minor variations in the middle.
Figure 84. Graph. Comparison of PSI ratings on the southbound direction (summer 2001 versus summer 1995). PSI is on the Yaxis and Station is on the Xaxis. There are three bars at each station representing readings done in the morning of 2001, afternoon of 2001, and 1995. The morning readings are higher than the afternoon readings are closer to the readings in 1995. Almost all of the readings fall between 2.0 and 3.0 PSI.
Figure 85. Graph. Comparison of PSI ratings on the northbound direction (summer 2001 versus summer 1995). PSI is on the Yaxis and Station is on the Xaxis. There are three bars at each station representing readings done in the morning of 2001, afternoon of 2001, and 1995. The afternoon readings are mostly higher than the morning readings while both take turn being close to the 95 readings, depending on the station. Most of the readings fall between 2.0 and 3.0 PSI.
Figure 86. Drawing. Typical cross section for the CRCP instrumented section. The sketch shows the typical cross section at the construction site of part of one of the access ramps on the IH35/IH30 interchange. The pavement is 26 feet wide and consists of 8 inches continuously reinforced concrete on top of 4 inches of hotmix asphalt, type D underlayment on top of 8 inches of lime treated subgrade. For metric conversion, 1 foot equals 0.3048 meters, 1 inch equals 25.4 millimeters.
Figure 87. Photo. Instrumented section delineated by crack inducers. The photo shows the position of the crack inducers on the instrumented area. The crack inducers were spaced at approximately 4.9 meters due to a survey of previous sections that found that the average crack spacing measured was approximately 4.9 meters.
Figure 88. Photo. Strain gages in PCC and on reinforcing steel. The photo shows the strain gages in concrete and on the reinforcing steel at the corner location. Roctest embedment vibrating wire gages type EM5 were installed on the instrumented area at three different depths at midslab, edge, and corner (150 millimeters from the crack). Type "T" thermocouples were also installed at seven different depths throughout the pavement depth. The thermocouple located at the PCC surface was installed just after the slip form paver passed by the instrumented area. In addition, resistance gages type CEA06125UN120 were epoxied to the steel rebar at different distances from the expected crack along with embedment gages to monitor the steel and concrete strain.
Figure 89. Photo. Shear failure as a result of pushoff test. The photo depicts a plane of failure in the asphalt along the border of the slab against which the load was applied. This plane of failure originated from shear forces that developed in the asphalt due to the strong bond between the concrete and asphalt.
Figure 90. Drawing. Position of strain gages and thermocouples as constructed. The drawing shows the sensor locations with respect to the observed cracks as constructed. The thermal couple is placed in the middle while most of the concrete and steel strain gages are place on the right side with some placed on the left side.
Figure 91. Drawing. Typical cross section for I29, South Dakota (from opposite direction to traffic). The typical cross section at the construction site on IH29 is designed to accommodate a 4lane divided road. Each direction is 12.2 meters wide with a 1.8meter left shoulder, two 3.65meter lanes, and a 3meter right shoulder. The central portion of the pavement was being constructed with CRCP pavement to conform a 3.65meterwide right lane with 0.6meter PCC widened shoulder and a 3.65meterwide left lane. The rest of the shoulders were being paved with hotmix asphalt. A longitudinal joint was being cut between lanes. The CRCP pavement is 279 millimeters thick on top of a 102millimeter granular subbase.
Figure 92. Photo. Strain gages in PCC and on reinforcing steel. The photo shows the positions of gages in the concrete and on the reinforcing steel. To position the Roctest vibrating wire gages type EM5, two wood dowels were driven into the subbase (granular subbase) separated approximately 102 millimeters for each set of gages. The gage was then fixed on the dowels at the proper height with plastic zip ties. Wood dowels were also used to install type "T" thermocouples at seven different depths throughout the pavement depth. The thermocouple located at the PCC surface was installed just after the slip form paver passed by the instrumented area.
Figure 93. Drawing. Cracking pattern on instrumented section. The drawing shows cracking patterns observed at 18 hours and at 34 hours. After 18 hours, three cracks were observed; one left, one right, and one in the middle. The left and right cracks are close to the respective crack transducer. The middle crack is close to the sensors. After 34 hours, two more cracks were observed between the left and middle crack.
Figure 94. Drawing. Location of the sensors with respect to the pavement edge and cracks as constructed. The drawing shows the sensor locations with respect to the observed cracks as constructed. According to the drawing, the PCC and steel strain gages were placed very close to the cracks, with the farthest distance being 191 millimeters from a PCC strain gage, and 356 millimeters from a steel strain gage.
Figure 95. Graph. Time growth of Texas compressive strength. Average Compressive Strength is on the Yaxis and Test Age is on the Xaxis. The strength increases with time. The strength rate of increase is high in the first day, moderate from then to day 7, and low from day 7 to day 14, reaching an average compressive strength of 29 megapascals.
Figure 96. Graph. Time growth of Texas splitting tensile strength. Average Splitting Tensile Strength is on the Yaxis and Test Age is on the Xaxis. The strength increases with time. The strength rate of increase is high in the first day and low from then up to day 14, reaching an average tensile strength of 2.3 megapascals.
Figure 97. Graph. Time growth of Texas modulus of elasticity. Average Modulus of Elasticity is on the Yaxis and Test Age is on the Xaxis. Both the ACI and the ACI 318predicted values are virtually the same, increasing with time, with the modulus increasing significantly after the first day and then less and less as the test age goes up to day 14. The test data points do not increase with time. The modulus increases from 0.5 to 3 days (the actual values being higher than the ACI values, reaching 30,000 megapascals), and then drops at 7 and 14 days (the actual values being lower than the ACI values, about 20,000 megapascals).
Figure 98. Graph. Time growth of South Dakota compressive strength. Average Compressive Strength is on the Yaxis and Test Age is on the Xaxis. The strength increases with time. The strength rate of increase is moderate from day 3 to day 7, and is slightly lower from then to day 28, reaching an average compressive strength of 37 megapascals.
Figure 99. Graph. Time growth of South Dakota splitting tensile strength. Average Splitting Tensile Strength is on the Yaxis and Test Age is on the Xaxis. The strength increases with time. The strength rate of increase is high from day 1 to day 3 (measured in the field), and is moderate from day 3 to day 28 (measured in the lab), reaching an average tensile strength of 4 megapascals.
Figure 100. Graph. Time growth of South Dakota modulus of elasticity. Average Modulus of Elasticity is on the Yaxis and Test Age is on the Xaxis. Both the ACI and the ACI 318predicted values are virtually the same, increasing with time, with the rate of increase is moderate from day 3 to day 7, and is slightly lower from then to day 28. The test data points are higher, reaching 34,000 megapascals but seem to follow the same pattern, although no data points exist for day 3.
Figure 101. Graph. Time growth of Mexico compressive strength. Average Compressive Strength is on the Yaxis and Test Age is on the Xaxis. The strength increases with time. The strength rate of increase is moderate from day 1 to day 28, and is slightly lower from day 28 to approximately day 6000, reaching an average compressive strength of 42 megapascals.
Figure 102. Graph. Time growth of Mexico splitting tensile strength. Average Splitting Tensile Strength is on the Yaxis and Test Age is on the Xaxis. The graph shows one linear line representing the steady increase in strength with time, with the average tensile strength reaching 3.8 megapascals after approximately 6000 days.
Figure 103. Graph. Time growth of Mexico modulus of elasticity. Average Modulus of Elasticity is on the Yaxis and Test Age is on the Xaxis. Both the ACI and the ACI 318predicted values are virtually the same, increasing with time, with the rate of increase is moderate from day 1 to day 28, and is slightly lower from day 28 to approximately day 6000. The test data points show a higher rate of increase from day 28, with the average modulus of elasticity reaching 40,000 megapascals. No data points exist for earlier periods.
Figure 104. Graph. Time growth of Illinois compressive strength. Average Compressive Strength is on the Yaxis and Test Age is on the Xaxis. The graph shows data points of measured compressive strength and also data points of estimated compressive strength from flexural strength. After 14 days, the data shows a minimal increase in compressive strength, except for the 90day calculated value, which increases significantly from the 28day value. According to the graph, the final average compressive strength is about 40 megapascals.
Figure 105. Graph. Time growth of Illinois modulus of elasticity. Modulus of Elasticity is on the Yaxis and Test Age is on the Xaxis. Both the ACIpredicted values and the measured values are basically the same, except for the 90day calculated value as mentioned above. Modulus of elasticity is relatively stable after 14 days, reaching 28,000 megapascals.
Figure 106. Graph. Drying shrinkage of South Dakota concrete. Drying Shrinkage is on the Yaxis and T minus T subscript 0 in days is on the Xaxis. The ultimate shrinkage predicted is represented by a logshape line increasing as time increases. The ultimate shrinkage set is the same as the predicted line, except higher. Higher still are the data points with the highest value being 410 at 13 days. The ultimate shrinkage set line seems to level out at about 360, and the predicted line seems to level out at about 275.
Figure 107. Graph. Drying shrinkage of Texas concrete. Drying Shrinkage is on the Yaxis and T minus T subscript 0 in days is on the Xaxis. The ultimate shrinkage predicted is represented by a logshape line increasing as time increases. The ultimate shrinkage set is the same as the predicted line, except just slightly higher. The data points seem to fall along these two lines with the highest value being 410 at 13 days. The ultimate shrinkage set line seems to level out at about 380, and the predicted line seems to level out at about 350.
Figure 108. Graph. Setting times of concrete for Texas and South Dakota concretes. Penetration Resistance is on the Yaxis and Elapsed Time is on the Xaxis. The set time data for Texas is best fit using a power function with R squared equal to 0.99 and the set time data for South Dakota is best fit using a power function with R squared equal to 0.97. The total elapsed time is much longer for the South Dakota data set since the ambient temperature for set time measurements is much lower in South Dakota.
Figure 109. Graph. Concrete and ambient air temperatures for the South Dakota concrete. Temperature is on the Yaxis and Time is on the Xaxis. The air temperature stays around 17.5 degrees Celsius. The concrete temperature decreases sharply from 22.5 to 18.5 degrees Celsius in the first 3 hours and then increases slightly to 19.5 degrees Celsius over the next 5 hours then gradually decreases to coincide with the air temperature after 35 hours.
Figure 110. Graph. Concrete and ambient air temperatures for the Texas concrete. Temperature is on the Yaxis and Time is on the Xaxis. The air temperature stays around 36 degrees Celsius. The concrete temperature increases sharply from 22.6 to 40.2 degrees Celsius in the first 6 hours then gradually decreases to coincide with the air temperature after 23 hours.
Figure 111. Graph. Measured versus predicted joint opening. Predicted ER is on the Yaxis and Measured ER is on the Xaxis. There is a linear line with a slope of 1 representing equality. In general, it can be observed that the Hiperpav 2 joint opening model slightly overpredicts joint movement for most of the sections, since all but one data point fall above this line.
Figure 112. Graph. Typical plot of PCC temperature versus joint LTE, section 370201. LTE is on the Yaxis and Temperature is on the Xaxis. There is a trend line increasing close to linearly from left to right. For PCC temperatures below the threshold temperature of 27 degrees Celsius, the joint LTE values are scattered and fall below 50 percent. For PCC temperatures above the threshold temperature, the joint LTE values are more closely packed and are all above 80 percent.
Figure 113. Graph. Computed LTE versus PCC temperature for section 493011. LTE is on the Yaxis and Temperature is on the Xaxis. For PCC temperatures below the threshold temperature of 25 degrees Celsius, the joint LTE values are scattered and fall below 20 percent. For PCC temperatures above the threshold temperature, the joint LTE values are more closely packed and are all above 80 percent. Below freezing (PCC temperature less than 0 degrees Celsius), LTE is slightly higher than at PCC temperatures immediately above freezing.
Figure 114. Graph. Average LTE above freezing and below 25 degrees Celsius, section 493011. Average LTE is on the Yaxis and Joint ID and Date of Testing is on the Xaxis. For each joint, the computed LTE is grouped into two categories: tests performed before 1996 (B96); and tests performed in 1996 and thereafter (A96). The average LTEs are shown in bar graphs. An evident decrease in LTE is noted for the tests performed after 1996 as compared to the tests performed in previous years. The highest LTE is about 76 percent and corresponds to Joint 5, 5.35, B96. The lowest LTE is about 36 percent and corresponds to Joint 4, 14.15, A96.
Figure 115. Graph. Variability of LTE for individual joints, section 493011. Measured LTE is on the Yaxis and Temperature is on the Xaxis. There are two sets of data points, one for joint negative 9.25 and one joint 8.95. These data points are very scattered ranging from 20 percent to 100 percent. The data points for joint negative 9.25 are best fitted by a linear line increasing from left to right with R squared equal to 0.5555. The data points for joint 8.95 are best fitted by a linear line increasing from left to right with R squared equal to 0.1698.
Figure 116. Graph. Effect of joint spacing on LTE, section 493011. LTE is on the Yaxis and Spacing is on the Xaxis. There are four sets of data points representing four different tests. The data points for approach test before 1996 are best fitted by a linear line decreasing from left to right with an r squared equal to 0.57. The data points for approach test after 1996 are best fitted by a linear line decreasing from left to right with an r squared equal to 0.37. The data points for leave test before 1996 are best fitted by a linear line decreasing from left to right with an r squared equal to 0.19. The data points for leave test after 1996 are best fitted by a linear line decreasing from left to right with an r squared equal to 0.03. The before 1996 values are higher than the after 1996 values for both tests.
Figure 117. Graph. Computed LTE versus PCC temperature for nondoweled section 313018. LTE is on the Yaxis and Temperature is on the Xaxis. For PCC temperatures below the threshold temperature of 30 degrees Celsius, the joint LTE values are scattered and fall below 20 percent. For PCC temperatures above the threshold temperature, the joint LTE values are more closely packed and are all above 90 percent.
Figure 118. Graph. Computed LTE versus PCC temperature for nondoweled section 063042. LTE is on the Yaxis and Temperature is on the Xaxis. For PCC temperatures below the threshold temperature of 25 degrees Celsius, the joint LTE values are scattered and fall below 20 percent. For PCC temperatures above the threshold temperature, the joint LTE values are more closely packed and are all above 90 percent.
Figure 119. Graph. Computed LTE versus PCC temperature for nondoweled section 833802. LTE is on the Yaxis and Temperature is on the Xaxis. For PCC temperatures below the threshold temperature of 22 degrees Celsius, the joint LTE values are scattered and fall below 20 percent. For PCC temperatures above the threshold temperature, the joint LTE values are more closely packed and are all above 90 percent. It is also interesting to note that for temperatures below negative 10 degrees Celsius, the joint LTE values are also closely packed and stay mostly above 80 percent.
Figure 120. Graph. Computed LTE versus PCC temperature for nondoweled section 533813. LTE is on the Yaxis and Temperature is on the Xaxis. For PCC temperatures below the threshold temperature of 20 degrees Celsius, the joint LTE values are scattered and fall as low as 40 percent. For PCC temperatures above the threshold temperature, the joint LTE values are more closely packed and are all above 85 percent.
Figure 121. Graph. LTE versus PCC temperature for doweled section 040215. LTE is on the Yaxis and Temperature is on the Xaxis. For PCC temperatures below the threshold temperature of 35 degrees Celsius, the joint LTE values are fairly scattered and fall as low as 40 percent. For PCC temperatures above the threshold temperature, the joint LTE values are more closely packed and are all above 80 percent.
Figure 122. Graph. LTE versus PCC temperature for doweled section 183002. LTE is on the Yaxis and Temperature is on the Xaxis. The joint LTE values are all above 60 percent and are not that scattered, although for temperatures higher than 25 degrees Celsius, the joint LTE values are above 80 percent.
Figure 123. Graph. LTE versus PCC temperature for doweled section 133019. LTE is on the Yaxis and Temperature is on the Xaxis. For PCC temperatures below the threshold temperature of 18 degrees Celsius, the joint LTE values are fairly scattered and fall below 60 percent. For PCC temperatures above the threshold temperature, the joint LTE values are more closely packed and are all above 80 percent.
Figure 124. Graph. LTE versus PCC temperature for doweled section 320204 . LTE is on the Yaxis and Temperature is on the Xaxis. The joint LTE values are all above 80 percent and are closely packed.
Figure 125. Graph. LTE versus PCC temperature for doweled section 893015. LTE is on the Yaxis and Temperature is on the Xaxis. The joint LTE values are mostly above 80 percent and are closely packed, even for temperatures below freezing. There are a few values that fall below 80 percent for temperatures at and right below 5 degrees Celsius.
Figure 126. Graph. LTE versus PCC temperature for doweled section 390204. LTE is on the Yaxis and Temperature is on the Xaxis. The joint LTE values are all above 80 percent and are closely packed.
Figure 127. Graph. LTE versus joint opening, for section 493011, joint at 5.5 meters from start of section. LTE is on the Yaxis and Joint Opening is on the Xaxis. There are two sets of data points, for LTE before 1996 and LTE after 1996. The LTE before 1996 data points are scattered and fall between 40 percent and 100 percent, and are best fitted by a linear line decreasing from right to left. The LTE after 1996 data points are also scattered and fall between 20 percent and 100 percent, and are best fitted by a linear line decreasing from right to left that is of similar slope but is below the before 1996 line.
Figure 128. Graph. Predicted versus computed LTE for section 313018, joint 5.5 meters from start of section. LTE is on the Yaxis and Joint Opening is on the Xaxis. There are two sets of data points, one for LTE on 1996 and before and one for LTE after 1996. At the same joint opening, tests performed before 1996 show higher LTE values than those performed after 1996. However, a significant variability is noted that makes it slightly difficult to determine a clear difference between testing dates. Also shown in the figure are the 4 lines predicting LTE values for the following loads: 0, 250,000, 500,000, and 12,400,000 ESALs. The predicted LTE at 250,000 ESALs seems to better match the computed LTE from tests at the end of 1996. After 500,000 ESALs the predicted LTE seems to have very little variation with cumulative traffic.
Figure 129. Graph. Predicted versus computed LTE for section 493011, joint 5.5 meters from start of section. LTE is on the Yaxis and Joint Opening is on the Xaxis. There are two sets of data points, one for LTE on 1996 and before and one for LTE after 1996. At the same joint opening, tests performed before 1996 show higher LTE values than those performed after 1996. However, there is still variability that makes it slightly difficult to determine a clear difference between testing dates. Also shown in the figure are the 4 lines predicting LTE values for the following loads: 0, 280,000, 500,000, and 8,088,000 ESALs. The predicted LTE at 280,000 ESALs seems to better match the computed LTE from tests at the end of 1996. After 500,000 ESALs the predicted LTE seems to have very little variation with cumulative traffic.
Figure 130. Graph. Predicted versus computed LTE for section 063042, joint 148.1 meters from start of section. LTE is on the Yaxis and Joint Opening is on the Xaxis. There are two sets of data points, one for LTE on 1996 and before and one for LTE after 1996. At the same joint opening, tests performed before 1996 show higher LTE values than those performed after 1996. Also shown in the figure are the 4 lines predicting LTE values for the following loads: 0, 150,000, 500,000, and 29,066,000 ESALs. The predicted LTE at 150,000 ESALs seems to better match the computed LTE from tests at the end of 1996. After 500,000 ESALs the predicted LTE seems to have very little variation with cumulative traffic.
Figure 131. Graph. Predicted versus computed LTE for section 833802, joint 149.7 meters from start of section. LTE is on the Yaxis and Joint Opening is on the Xaxis. There are two sets of data points, one for LTE on and before 1996 and one for LTE after 1996. At the same joint opening, tests performed before 1996 show lower LTE values than those performed after 1996. Also shown in the figure are the 5 lines predicting LTE values for the following loads: 0, 200,000, 300,000, 500,000, and 1,975,000 ESALs. The predicted LTE at 200,000 ESALs seems to better match the computed LTE from tests after 1996. The predicted LTE at 300,000 ESALs seems to better match the computed LTE from tests on and before 1996. After 500,000 ESALs the predicted LTE seems to have very little variation with cumulative traffic.
Figure 132. Graph. Predicted versus computed LTE for section 533813, joint 0.0 meters from start of section. LTE is on the Yaxis and Joint Opening is on the Xaxis. There are two sets of data points, one for LTE on 1996 and before and one for LTE after 1996. At the same joint opening, tests performed before 1996 show higher LTE values than those performed after 1996. Also shown in the figure are the 3 lines predicting LTE values for the following loads: 0, 100,000, and 3,790,000 ESALs. The predicted LTE at 100,000 ESALs seems to better match the computed LTE from tests at the end of 1996.
Figure 133. Graph. Sensitivity analysis of LTE model for nondoweled pavements. LTE is on the Yaxis and Parameter is on the Xaxis. According to this graph, the most predominant factors affecting LTE are joint opening (difference between 9 percent for 5 millimeters and 98 percent for 0.5 millimeter), cumulative load applications (difference between 45 percent for 270,000 and 93 percent for 0), and slab thickness (difference between 83 percent for 178 millimeters and 97 percent for 330 millimeters). No major effects from PCC stiffness, wheel load, pressure, or kvalue were observed on LTE.
Figure 134. Graph. LTE model sensitivity for doweled sections. LTE is on the Yaxis and Parameter is on the Xaxis. According to this graph, the most predominant factors affecting LTE are, from most to least are: dowel looseness, load weight, K value, slab thickness, PCC stiffness, joint opening, and cumulative load applications, respectively. No major effects from dowel diameter, dowel diameter, and load pressure were observed on LTE.
Figure 135. Graph. Predicted versus computed LTE for section 370201, joint 145.1 meters from start of section. LTE is on the Yaxis and Joint Opening is on the Xaxis. There are two sets of data points, one for LTE on 1996 and before and one for LTE after 1996. At the same joint opening, tests performed before 1996 show higher LTE values than those performed after 1996. However, there is still variability that makes it slightly difficult to determine a clear difference between testing dates. Also shown in the figure are four lines predicting LTE values for four different conditions. The line for the condition ESAL's factor equal to 0.048, looseness equal to 0.11 millimeter seems to better match the computed LTE from tests at the end of 1996.
Figure 136. Graph. Predicted versus computed LTE for section 040215, joint 144.5 meters from start of section. LTE is on the Yaxis and Joint Opening is on the Xaxis. There are two sets of data points, one for LTE on 1996 and before and one for LTE after 1996. The data is so variable that it is difficult to determine the difference between testing dates. Also shown in the figure are four lines predicting LTE values for four different conditions. The line for the condition ESAL's factor equal to 0.074, looseness equal to 0.13 millimeter seems to better match the computed LTE from the tests.
Figure 137. Graph. Predicted versus computed LTE for section 133019, joint 21.3 meters from start of section. LTE is on the Yaxis and Joint Opening is on the Xaxis. There are two sets of data points, one for LTE on 1996 and before and one for LTE after 1996. The data is so variable that it is difficult to determine the difference between testing dates. Also shown in the figure are five lines predicting LTE values for five different conditions. The line for the condition ESAL's factor equal to 0.09 seems to better match the computed LTE from the tests.
Figure 138. Graph. Predicted versus computed LTE for section 320204, joint 8.5 meters from start of section. LTE is on the Yaxis and Joint Opening is on the Xaxis. There are two sets of data points, one for LTE on 1996 and before and one for LTE after 1996. At the same joint opening, tests performed before 1996 show higher LTE values than those performed after 1996. However, there is still variability that makes it slightly difficult to determine a clear difference between testing dates. Also shown in the figure are five lines predicting LTE values for five different conditions. The line for the condition ESAL's factor equal to 0.048, looseness equal to 0.000 millimeter seems to better match the computed LTE from tests at the end of 1996.
Figure 139. Graph. Predicted versus computed LTE for section 893015, joint 36.3 meters from start of section. LTE is on the Yaxis and Joint Opening is on the Xaxis. There are two sets of data points, one for LTE on 1996 and before and one for LTE after 1996. The data is so variable that it is difficult to determine the difference between testing dates. Also shown in the figure are three lines predicting LTE values for three different conditions. The line for the condition ESAL's factor equal to 0.048, looseness equal to 0.000 millimeter seems to better match the computed LTE from the tests.
Figure 140. Graph. Predicted versus computed LTE for section 390204, joint 0.0 meters from start of section. LTE is on the Yaxis and Joint Opening is on the Xaxis. There are two sets of data points, one for LTE on 1996 and before and one for LTE after 1996. The data is so variable that it is difficult to determine the difference between testing dates. Also shown in the figure are four lines predicting LTE values for four different conditions. The line for the condition ESAL's factor equal to 0.048, looseness equal to 0.000 millimeter seems to better match the computed LTE from the tests.
Figure 141. Graph. Crack spacing history for summer sections, SH6. Crack Spacing is on the Yaxis and Pavement Age is on the left axis. A typical trend of reduction in crack spacing with age can be observed, however, for most sections, higher average crack spacing were reported for the surveying date at 4700 days than for previous dates. This may be due to omitted cracks during crack spacing measurements since there is no recorded history of rehabilitation.
Figure 142. Graph. Crack spacing history for winter sections, SH6. Crack Spacing is on the Yaxis and Pavement Age is on the left axis. A typical trend of reduction in crack spacing with age can be observed, however, for most sections, higher average crack spacing were reported for the surveying date at 4700 days than for previous dates. This may be due to omitted cracks during crack spacing measurements since there is no recorded history of rehabilitation.
Figure 143. Graph. Preliminary longterm crack spacing prediction. Predicted Average Crack Spacing is on the Yaxis and Observed Average Crack Spacing is on the Xaxis. The 45degree line represents equality. Most of data points fall slightly above this line, suggesting a slight overprediction of crack spacing. However, a reasonably good R squared of 0.82 was obtained with respect to the 45degree line, which has an R squared of 0.71.
Figure 144. Graph. Crack spacing prediction at 3 days from construction. Predicted Average Crack Spacing is on the Yaxis and Observed Average Crack Spacing is on the Xaxis. The 45degree line represents equality. Most of the data points fall far from this line, suggesting poor prediction of crack spacing at early ages.
Figure 145. Graph. Measured versus predicted crack widths, SH6. Predicted Average Crack Width is on the Yaxis and Observed Average Crack Width is on the Xaxis. The 45degree line represents equality. Although a fair trend in crack width prediction is observed, most of data points fall far above this line, suggesting a large overprediction of crack width.
Figure 146. Graph. Conceptual representation of residual drying shrinkage effect (adapted from Otero et al.). Shrinkage is on the Yaxis and Pavement Age is on the Xaxis. The relationship is represented by a log curve. According to the graph, the drying shrinkage after crack formation is dependent on the concrete age when the crack occurs.
Figure 147. Graph. Comparison of measured and predicted strength, Illinois site. Flexural Strength is on the Yaxis and Years is on the Xaxis. The graph shows flexural strength at 14, 28, and 90 days along with the estimated flexural strength after 16 years from the splitting tensile strength obtained from cores. The graph also shows the earlyage and longterm strength predicted with Hiperpav 2. In general, although the 90day strength is slightly underpredicted, a reasonably good prediction of the measured longterm strength is obtained.
Figure 148. Graph. Comparison of measured and predicted modulus of elasticity, Illinois site. Elastic Modulus is on the Yaxis and Years is on the Xaxis. The graph shows the measured modulus of elasticity at 28 days and the one obtained from concrete cores extracted from the field visit in 2001. The graph also shows the earlyage and longterm modulus predicted with Hiperpav 2. While the predicted elastic modulus shows an increasing trend, the elastic modulus obtained from field cores in 2001 shows a slight decrease from the 28day value. A significant difference in predicted versus measured elastic modulus is thus observed after 16 years.
Figure 149. Graph. Comparison of measured versus predicted LTE. LTE is on the Yaxis and Years is on the Xaxis. The graph shows the measured and predicted LTE for the sections with a 216millimeter thickness. The predicted LTE is shown as continuous lines for four times of day: 2 AM, 8 AM, 2 PM, and 8 PM. The fluctuating trend observed is due to seasonal effects. In general, a slight overprediction of LTE is observed for all testing ages.
Figure 150. Graph Earlyage analysis for section AA, for placement at 2 PM. Strength or Stress is on the Yaxis and PCC Age is on the Xaxis. The strength line increases with time in a log trend. In general, the stress line also increases with time, although in a fluctuating trend. The stress line stays under the strength line.
Figure 151. Graph. Earlyage analysis for section IA, for placement at 2 PM. Strength or Stress is on the Yaxis and PCC Age is on the Xaxis. The strength line increases with time in a log trend. In general, the stress line also increases with time, although in a fluctuating trend. The stress line stays under the strength line.
Figure 152. Graph. Earlyage analysis for section NA for placement at 2 PM. Strength or Stress is on the Yaxis and PCC Age is on the Xaxis. The strength line increases with time in a log trend. In general, the stress line also increases with time, although in a fluctuating trend. The stress line at times fluctuates over the strength line, suggesting high stresses.
Figure 153. Graph. Predicted faulting for sections MA and NA (191 millimeters thick). Faulting is on the Yaxis and Years is on the Xaxis. The prediction (at 90 percent reliability) is represented by a sort of step function, 0 faulting up to 2 years, 0.25 millimeters from 2 to 12 years, and 0.5 millimeters from 12 to 16 years. No faulting was observed after 16 years as indicated by the data point.
Figure 154. Graph. Comparison of measured and predicted transverse cracking (sections NA and MA). Transverse Cracking is on the Yaxis and Years is on the Xaxis. The prediction (at 90 percent reliability) is a horizontal line along the Xaxis. Most of section MA data points fall on this line, while the section NA data points are higher than this line showing an increasing trend up to 12 percent.
Figure 155. Graph. Comparison of measured and predicted transverse cracking for sections (IA, JA, KA, LA). Transverse Cracking is on the Yaxis and Years is on the Xaxis. The prediction (at 90 percent reliability) is a horizontal line along the Xaxis. Most of the section KA data points fall on this line, while the other sections data points are higher than this line, with the highest being sections JA and LA at around 6 to 8 percent.
Figure 156. Graph. Transverse cracking for section AA, thickness equals 241 millimeters. Transverse cracking is on the Yaxis and Years is on the Xaxis. The prediction (at 90 percent reliability) is a horizontal line along the Xaxis. All of the data points fall on this line.
Figure 157. Graph. Comparison of measured and predicted longitudinal cracking (sections MA and NA). Transverse cracking is on the Yaxis and Years is on the Xaxis. The prediction (at 90 percent reliability) is a horizontal line along the Xaxis. All of section MA data points fall on this line, and all but 1 data point from section NA fall on this line, and that data point was only slightly above, at 1 percent.
Figure 158. Graph. Comparison of measured and predicted IRI (section NA). IRI is on the Yaxis and Years is on the Xaxis. There are three linear lines increasing from left to right representing three prediction lines at different reliability levels. The line at 90 percent reliability is the highest ranging from 1.4 to 1.6 meters per kilometer for IRI, followed by the lines at 50 percent and 10 percent reliability, respectively. Half of the measured IRIs falls along the lines, while half goes up to 1.8 meters per kilometer.
Figure 159. Graph. Comparison of measured versus predicted flexural strength, Ticuman, Mexico. Flexural Strength is on the Yaxis and Years (log scale) is on the Xaxis. The graph shows flexural strength at 1, 3, 7, 14, and 28 days along with the estimated flexural strength after 8 years from compressive and indirect tensile strength test results obtained from cores. The graph also shows the earlyage and longterm strength predicted with Hiperpav 2. In general, a reasonably good prediction of the strength is achieved.
Figure 160. Graph. Comparison of measured versus predicted modulus of elasticity, Ticuman, Mexico. Modulus Elastic is on the Yaxis and Years (log scale) is on the Xaxis. The graph shows modulus at 28 days and 8 years of age. The graph also shows the earlyage and longterm strength predicted with Hiperpav 2. A significant underprediction of the elastic modulus at 8 years of age is observed. While the predicted modulus of elasticity shows an increase of 12 percent from 28 days to 8 years, the measured value shows an increase of 55 percent.
Figure 161. Graph. Comparison of measured versus predicted LTE, Ticuman, Mexico. LTE is on the Yaxis and Years is on the Xaxis. The graph shows the measured (during the first few months of construction and after 8 years) and predicted LTE. The predicted LTE is shown as continuous lines for four times of day: 2 AM, 8 AM, 2 PM, and 8 PM. The fluctuating trend observed is due to seasonal effects. Compared to the LTE predicted with Hiperpav 2, the LTE predicted is higher than the measured in 1993 and slightly lower than the one measured in 2001.
Figure 162. Graph. Analysis for 229millimeter slab at different construction times and builtin gradient conditions, Ticuman bypass. Longitudinal Cracking is on the Yaxis and Years is on the Xaxis. The graph shows the effect of builtin curling for longitudinal cracking on placement times at 10 AM and 6 PM. It is observed that for the 10 AM placement, the gradient at time of loading with no gradient builtin at set produces higher cracking (up to 4 percent after 8 years) than the condition with builtin curling. In contrast, the placement at 6 PM shows the opposite trend with the condition with builtin curling showing higher longitudinal cracking (at 3 percent after 8 years).
Figure 163. Graph. Comparison of measured versus predicted joint faulting, Ticuman, Mexico. Joint Faulting is on the Yaxis and Years is on the Xaxis. There are three lines close to the log, increasing from left to right representing three prediction lines at different reliability levels, with the line at 90 percent reliability being the highest, followed by lines at 50 percent and 10 percent reliability, respectively. The average measured faulting of 1.0 millimeters is lower than the faulting predicted at 50 percent reliability and above the faulting predicted at 10 percent reliability.
Figure 164. Graph. Comparison of measured versus predicted transverse cracking, Ticuman, Mexico. Transverse Cracking is on the Yaxis and Years is on the Xaxis. There are three closetolinear lines increasing from left to right representing three prediction lines at different reliability levels, with the line at 90 percent reliability being the highest (going past 30 percent after 8 years), followed by lines at 50 percent and 10 percent reliability (which stayed under 5 percent after 8 years), respectively. The early measured data points were on and slightly higher than the 90 percent reliability line, while the measured transverse cracking after 8 years is slightly above the 50 percent reliability line at 8 percent.
Figure 165. Graph. Comparison of measured versus predicted longitudinal cracking, Ticuman, Mexico. Longitudinal Cracking is on the Yaxis and Years is on the Xaxis. There are three closetolinear lines representing three prediction lines at different reliability levels, with the line at 90 percent reliability being the highest (going past 30 percent after 8 years), followed by lines at 50 percent (which stayed under 3 percent after 8 years) and 10 percent reliability (which remained horizontal along the Xaxis). The early measured data points were slightly higher than the 90 percent reliability line, while the measured longitudinal cracking after 8 years is between the 90 percent and 50 percent reliability line at 15 percent.
Figure 166. Graph. Comparison of measured versus predicted present serviceability index, Ticuman, Mexico. Serviceability is on the Yaxis and Years is on the Xaxis. There are three closetolinear lines decreasing from left to right representing three prediction lines at different reliability levels, with the line at 10 percent reliability being the highest, followed by lines at 50 percent and 90 percent reliability, respectively. The early measured data points were lower than the 90 percent reliability line, while the measured serviceability after 8 years is slightly above the 90 percent reliability line at 2.7 PSI.
Figure 167. Graph. Restraint at the slab/subbase interface. Restraint Stress is on the Yaxis and Displacement is on the Xaxis. The graph shows a plot of the measured restraint values, which begins low and then rises, coinciding with the best fitted nonlinear restraint power function with ultimate displacement of 3.8 millimeters, critical restraint stress of 58.6 kilopascals, and power coefficient of 3.5.
Figure 168. Graph. Determination of set time with pulse velocity equipment. PV Time is on the Yaxis and Age is on the Xaxis. The relationship is represented by an inverse exponential line with a significant decrease in wave travel between 2.5 and 5.0 hours, which indicates a significant stiffening of the concrete and initial set time occurrence. The estimated set times are indicated in this figure to be within the transition zone between the plastic and hardened state of concrete.
Figure 169. Graph. PCC strains during PCC set time as a function of temperature changes. PCC Strain is on the Yaxis and Temperature is on the Xaxis. When the strain gage is initially embedded in the concrete cylinder, relative strain readings do not follow a definitive trend with change in temperature as the concrete is still plastic, it is much lower. As time progresses and the concrete stiffens, the strains tend to follow the temperature changes in the concrete. The transition as observed in the graph occurs between 2.1 and 4.5 hours, corresponding to the set times from laboratory testing.
Figure 170. Graph. Determination of PCC CTE with the use of PCC strains on an unconfined concrete cylinder. PCC Strain is on the Yaxis and Temperature is on the Xaxis. The relationship between strain and temperature from the graph (Y equals 6.5703X minus 307.87) results in a CTE of 6.6 times 10 raised to the power of 6 meters per meter per degree Celsius. This is lower than the laboratory testing, which reported a CTE of 7.48 times raised to the power of 6 meters per meter per degree Celsius.
Figure 171. Graph. Steel strains at various distances from the crack location. Steel Measured Strain is on the left Yaxis, PCC Temperature is on the right Yaxis, and Age is on the Xaxis. Strains closer to the crack are generally higher than strains away from the crack. Steel strain at the crack seems to increase with time and at lower temperatures. Steel strain 127 millimeters from the crack seems to increase with time but follow the temperature fluctuations. Steel strains 254 millimeters from the crack and further seem to follow the temperature fluctuations but are not affected by time.
Figure 172. Graph. Steel strain along the slab length at different ages. Steel Strain is on the Yaxis and Distance from Crack is on the Xaxis. There are three inverse exponential lines representing strain peaks measured at different hours. Strain at 21 hours begins lower but flattens out above the other two. Strain at 44.8 hours begins slightly lower than the strain at 68.5 hours but coincides with it as it flattens out. The distance from the crack to where the strains become horizontal is the bond development length.
Figure 173. Graph. Concrete strains at middepth along slab length. PCC Strains is on the left Yaxis, PCC Temperature is on the right Yaxis, and Age is on the Xaxis. The graph shows the strains observed in the concrete at different distances from the crack. The time of crack formation can be identified clearly by the sharp increase in strains (almost vertical) on the gages at 127 millimeters from the crack. All the strains seem to decrease with time in general and follow the temperature fluctuations.
Figure 174. Graph. Drying shrinkage observed in the field on an unrestrained PCC cylinder. Shrinkage Strain is on the Yaxis and Age is on the Xaxis. The shrinkage strain seems to decrease with time in general (equaling 0 at 5 hours and ending up equaling negative 50 at 76 hours), although there are fluctuations.
Figure 175. Graph. Strains in concrete and steel at 68.5 hours after construction. Strains are on the Yaxis and Distance from Crack is on the Xaxis. The strains in the concrete appear almost constant along the slab at approximately negative 40 microstrains. The strains in the steel also appear within the same order of magnitude along most of the slab, except at the distance farther than 2.2 meters from midslab where they start to increase significantly.
Figure 176. Graph. Displacements in steel and concrete along slab length at 68.5 hours. Displacement is on the Yaxis and Distance from Midslab is on the Xaxis. The displacements in steel and concrete are the same, decreasing linearly at a rate of 0.04 millimeters per meter increase in distance from midslab. When the distance from midslab reaches about 2.2 meters, the displacement in steel increases sharply.
Figure 177. Graph. Steel stress along the slab at 68.5 hours. Stress is on the Yaxis and Distance from Crack is on the Xaxis. The graph shows predicted steel stress and the stress from measured strains. The predicted steel stress starts out at about 390 megapascals at the crack and decreases close to linearly to about negative 30 megapascals at just above 0.6 meters from the crack, then stays constant. The steel stress from measured strains starts out at about 245 megapascals at the crack and decreases close to linearly to just below 0 megapascals at just above 0.2 meters from the crack, then stays constant.
Figure 178. Graph. Measured versus predicted crack spacing at 3 days of age, Fort Worth, Texas. Cumulative Frequency is on the Yaxis and Crack Spacing is on the Xaxis. The measured line increases close to linearly while the predicted line increases in a log form. There is overprediction frequency for crack spacing smaller than 4.5 meters and underprediction for crack spacing larger than 4.5 meters.
Figure 179. Graph. PCC strains during PCC set time as a function of temperature changes. Strains are on the Yaxis and Temperature is on the Xaxis. When the strain gage is initially embedded in the concrete cylinder, relative strain readings do not follow a definitive trend with change in temperature, as the concrete is still plastic. As time progresses and the concrete stiffens, the strains tend to follow the temperature changes in the concrete. The transition as observed in the graph occurs between 2.8 hours and 6.5 hours.
Figure 180. Graph. Determination of PCC CTE with the use of PCC strains on an unconfined concrete cylinder. Strains are on the Yaxis and Temperature is on the Xaxis. The relationship between strain and temperature from the graph (Y equals 11.61X minus 347.01) results in a CTE of 11.6 times 10 raised to the power of 6 meters per meter per degree Celsius. This is higher than the laboratory testing, which reported a CTE of 10.9 times 10 raised to the power of 6 meters per meter per degree Celsius.
Figure 181. Graph. Steel strains at various distances from the crack location. Steel Measured Strain is on the left Yaxis, Temperature is on the right Yaxis, and Age is on the Xaxis. Strains closer to the crack are generally higher than strains away from the crack. Strains closer to the crack also seem to increase slightly with time while the strains away from the crack do not, although there are fluctuations.
Figure 182. Graph. Steel strain along the slab length at different ages. Steel Strain is on the Yaxis and Distance from Crack is on the Xaxis. Steel strains decrease with the increase in distance from crack. Steel strains at 20.3 hours are significantly lower than steel strains at 43.2 and 67.2 hours, which are very close to each other. Extrapolating the steel strains to estimate the strain at the crack yields 550 microstrains at 20.3 hours and 1200 to 1500 microstrains for the other two ages.
Figure 183. Graph. Concrete strains at middepth along slab length. PCC Strains are on the left Yaxis, Temperature is on the right Yaxis, and Age is on the Xaxis. The graph shows the strains observed in the concrete at different distances from the crack. The time of crack formation can be clearly identified by the sharp increase in strains (almost vertical). All the strains seem to decrease with time in general and follow the temperatures fluctuations.
Figure 184. Graph. Drying shrinkage observed in the field on an unrestrained PCC cylinder. Shrinkage Strain is on the Yaxis and Age is on the Xaxis. The shrinkage strain seems to decrease with time in general (equal to 0 at 5 hours and ending up equal to negative 130 at 74 hours), although there are fluctuations.
Figure 185. Graph. Strains in concrete and steel at 67.2 hours after construction. Strains are on the Yaxis and Distance from Midslab is on the Xaxis. The strains in the concrete appear almost constant along the slab at approximately negative 40 microstrains. The strains in the steel also appear within the same order of magnitude along most of the slab, except at the distance farther than 1.7 meters from the midslab where they start to increase significantly.
Figure 186. Graph. Comparison of measured and predicted bond development length. Bond Development Length is on the Yaxis and Age is on the Xaxis. The bond development length predicted with Hiperpav 2 for this site for the first 3 days of age varies from 0.48 meter to 0.66 meter, which is a good prediction when compared to the measured bond development length, which varies from 0.38 meter to 0.64 meter.
Figure 187. Graph. Steel stress along the slab at 67.2 hours. Stress is on the Yaxis and Distance from Crack is on the Xaxis. The graph shows predicted steel stress and the stress from measured strains. The predicted steel stress begins at about 290 megapascals at the crack and decreases close to linearly to right below 0 megapascals at near 0.6 meter from the crack, then stays constant. The steel stress from measured strains begins at about 270 (extrapolated) megapascals at the crack and decreases close to linearly to about negative 20 megapascals at near 0.5 meter from the crack, then stays constant.
Figure 188. Graph. Measured versus predicted crack spacing at 3 days of age, Sioux Falls, South Dakota. Cumulative Frequency is on the Yaxis and Crack Spacing is on the Xaxis. The measured line increases close to linearly while the predicted line increases in an S shape. There is underprediction frequency for crack spacing smaller than 4.5 meters and overprediction for crack spacing larger than that.
Figure 189. Graph. Drying shrinkage results for North Carolina site. Strains are on the Yaxis and Time is on the Xaxis. The data line and BaźantPanula strains stay close to each other and decreases with time in general, although there are fluctuations. Both the strains start out at 0 and end up near negative 75 microstrains after 70 hours, with the data strains being positive from 10 to 20 hours.
Figure 190. Graph. Drying shrinkage results for Texas site. Strains are on the Yaxis and Time is on the Xaxis. The data line and BaźantPanula line fluctuates away from each other at times but both slightly decreases with time in general. Both the strains start out at 0 and end up near negative 5 microstrains after 120 hours, with the lowest being negative 30 microstrains at 75 hours, with the data strains being positive from 10 to 20 hours and the BaźantPanula strains going positive from 90 to 100 hours.
Figure 191. Graph. Drying shrinkage results for Arizona site. Strains are on the Yaxis and Time is on the Xaxis. The data line and BaźantPanula line stay close to each other and decrease with time in general, although there are fluctuations. Both the strains start out at 0 and end up near negative 90 microstrains after 75 hours.
Figure 192. Graph. Drying shrinkage results for Nebraska site. Strains are on the Yaxis and Time is on the Xaxis. The data line and BaźantPanula line fluctuates away from each other at times, but both slightly decrease with time in general. The data strains start out at 0 and end up near negative 25 microstrains after 70 hours, positive from 10 to 20 hours, with the highest being 35 microstrains at 12 hours. The BaźantPanula strains start out at 0 and end up near negative 12 microstrains after 70 hours, with the lowest being negative 20 microstrains at 52 hours.
Figure 193. Graph. Drying shrinkage results for Minnesota site. Strains are on the Yaxis and Time is on the Xaxis. The data line and BaźantPanula line fluctuates away from each other at times but both slightly decreases with time in general. The data strains start out at 0 and end up at negative 42 microstrains after 120 hours, being positive from 15 to 20 hours and from 35 to 38 hours. The BaźantPanula strains start out at 0 and end up near negative 30 microstrains after 120 hours, with the lowest being negative 38 microstrains at 80 hours.
Figure 194. Graph. Calibration of drying shrinkage factor for Houston, Texas, sections constructed in summer. Drying Shrinkage is on the Yaxis and Size Factor (effective D) is on the Xaxis. Three drying shrinkage curves are plot for 3 days, 28 days, and 1 year of age. The predicted drying shrinkage increases with increase in time and decreases with increase in size factor. At 28 days, the measured drying shrinkage was 171 microstrains, matching up with a size factor of 0.2.
Figure 195. Graph. Calibration of drying shrinkage factor for Houston, Texas, sections constructed in winter. Drying Shrinkage is on the Yaxis and Size Factor (effective D) is on the Xaxis. Three drying shrinkage curves are plot for 3 days, 28 days, and 1 year of age. The predicted drying shrinkage increases with increase in time and decreases with increase in size factor. At 28 days, the measured drying shrinkage was 165 microstrains, matching up with a size factor of 0.2.
Figure 196. Graph. Measured concrete and air temperatures for Minnesota, Slab 1. Temperature is on the Yaxis and Concrete Age is on the Xaxis. The temperatures for concrete (25 millimeters from top, at middepth, and 25 millimeters from bottom) all fluctuate with the air temperature except during the early ages and are generally higher than the air temperature, with the biggest difference of about 15 degrees Celsius occurring from 18 to 24 hours.
Figure 197. Graph. Measured versus predicted temperatures 25 millimeters from top of slab for Minnesota, Slab 1. Temperature is on the Yaxis and Concrete Age is on the Xaxis. Most of the predicted temperatures are close to the measured temperatures ranging from 12 to 22 degrees Celsius, with the biggest difference being about 3 degrees Celsius at 1 hour.
Figure 198. Graph. Measured versus predicted temperatures 25 millimeters from bottom of slab for Minnesota, Slab 1. Temperature is on the Yaxis and Concrete Age is on the Xaxis. Most of the predicted temperatures are close to the measured temperatures ranging from 13 to 22 degrees Celsius, with the biggest difference being about 2 degrees Celsius from 27 to 35 hours.
Figure 199. Graph. Measured versus predicted temperature gradient for Minnesota, Slab 1. Temperature Gradient is on the Yaxis and Concrete Age is on the Xaxis. The predicted temperature gradients are mostly close to the measured temperature gradients ranging from negative 5 degrees Celsius to 5 degrees Celsius, with the biggest difference being about 2 degrees Celsius at 8 hours.
Figure 200. Graph. Measured concrete and air temperatures for Minnesota, Slab 4. Temperature is on the Yaxis and Concrete Age is on the Xaxis. The temperature in concrete (25 millimeters from top) fluctuates with the air temperature and both increase to about 18 degrees Celsius at 42 hours and stay close to constant from there. The temperatures in concrete (at midslab and 25 millimeters from bottom) increase to about 18 degrees Celsius at 20 hours and stay close to constant from there.
Figure 201. Graph. Measured versus predicted temperatures 25 millimeters from top of slab for Minnesota, Slab 4. Temperature is on the Yaxis and Concrete Age is on the Xaxis. Most of the predicted temperatures are close to the measured temperatures ranging from 12 to 18 degrees Celsius, with the biggest difference being about 2 degrees Celsius at 40 hours.
Figure 202. Graph. Measured versus predicted temperatures 25 millimeters from bottom of slab for Minnesota, Slab 4. Temperature is on the Yaxis and Concrete Age is on the Xaxis. The predicted temperatures are mostly very close to the measured temperatures ranging from 10 to 18 degrees Celsius, with the biggest difference being about 1 degree Celsius at 48 hours.
Figure 203. Graph. Measured versus predicted temperature gradient for Minnesota, Slab 4. Temperature Gradient is on the Yaxis and Concrete Age is on the Xaxis. The predicted temperature gradients are mostly close to the measured temperature gradients ranging from negative 3 degrees Celsius to 5 degrees Celsius, with the biggest difference being about 2 degrees Celsius at 40 hours.
Figure 204. Graph. Measured concrete and air temperatures for Arizona, Slab 1. Temperature is on the Yaxis and Concrete Age is on the Xaxis. The temperatures for concrete (25 millimeters from top, at middepth, and 25 millimeters from bottom) all fluctuate with the air temperature except at early ages and are generally higher than the air temperature with the biggest difference of about 15 degrees Celsius occurring from 12 to 24 hours.
Figure 205. Graph. Measured versus predicted temperatures 25 millimeters from top of slab for Arizona, Slab 1. Temperature is on the Yaxis and Concrete Age is on the Xaxis. The predicted temperatures are mostly very close to the measured temperatures ranging from 6 to 23 degrees Celsius, with the biggest difference being about 3 degrees Celsius at 1 hour.
Figure 206. Graph. Measured versus predicted temperatures 25 millimeters from bottom of slab for Arizona, Slab 1. Temperature is on the Yaxis and Concrete Age is on the Xaxis. Most of the predicted temperatures are close to the measured temperatures ranging from 10 to 23 degrees Celsius, with the biggest difference being about 2 degrees Celsius at 25 hours.
Figure 207. Graph. Measured versus predicted temperature gradient for Arizona, Slab 1. Temperature Gradient is on the Yaxis and Concrete Age is on the Xaxis. The predicted temperature gradients are mostly close to the measured temperature gradients ranging from negative 7 degrees Celsius to 3 degrees Celsius, with the biggest difference being about 2.5 degrees Celsius at 46 hours.
Figure 208. Graph. Measured concrete and air temperatures for Arizona, Slab 3. Temperature is on the Yaxis and Concrete Age is on the Xaxis. The temperatures for concrete (25 millimeters from top, at middepth, and 25 millimeters from bottom) all fluctuate with the air temperature except at early ages and are generally higher than the air temperature by about 10 degrees Celsius on average.
Figure 209. Graph. Measured versus predicted temperatures 25 millimeters from top of slab for Arizona, Slab 3. Temperature is on the Yaxis and Concrete Age is on the Xaxis. Most of the predicted temperatures are close to the measured temperatures ranging from 10 to 22 degrees Celsius, with the biggest difference being about 2 degrees Celsius at 9 hours.
Figure 210. Graph. Measured versus predicted temperatures 25 millimeters from bottom of slab for Arizona, Slab 3. Temperature is on the Yaxis and Concrete Age is on the Xaxis. The predicted temperatures are mostly very close to the measured temperatures ranging from 12 to 20 degrees Celsius, with the biggest difference being about 1 degree Celsius at 24 hours.
Figure 211. Graph. Measured versus predicted temperature gradient for Arizona, Slab 3. Temperature Gradient is on the Yaxis and Concrete Age is on the Xaxis. The predicted temperature gradients are mostly close to the measured temperature gradients ranging from negative 5 degrees Celsius to 5 degrees Celsius, with the biggest difference being about 2 degrees Celsius at 10 hours.
Figure 212. Graph. Measured concrete and air temperatures for Arizona, Slab 6. Temperature is on the Yaxis and Concrete Age is on the Xaxis. The temperatures for concrete (25 millimeters from top, at middepth, and 25 millimeters from bottom) all fluctuate with the air temperature except at early ages and are generally higher than the air temperature, with the biggest difference of about 15 degrees Celsius.
Figure 213. Graph. Measured versus predicted temperatures 25 millimeters from top of slab for Arizona, Slab 6. Temperature is on the Yaxis and Concrete Age is on the Xaxis. Most of the predicted temperatures are close to the measured temperatures, ranging from 13 to 23 degrees Celsius, with the biggest difference being about 2 degrees Celsius at 50 hours.
Figure 214. Graph. Measured versus predicted temperatures 25 millimeters from bottom of slab for Arizona, Slab 6. Temperature is on the Yaxis and Concrete Age is on the Xaxis. Most of the predicted temperatures are very close to the measured temperatures, ranging from 15 to 22 degrees Celsius, with the biggest difference being about 1 degree Celsius at various ages.
Figure 215. Graph. Measured versus predicted temperature gradient for Arizona, Slab 6. Temperature Gradient is on the Yaxis and Concrete Age is on the Xaxis. Most of the predicted temperature gradients are close to the measured temperature gradients, ranging from negative 4 degrees Celsius to 4 degrees Celsius, with the biggest difference being about 3 degrees Celsius at 26 hours.
Figure 216. Graph. Measured concrete and air temperatures for Lufkin, Texas, Slab 2. Temperature is on the Yaxis and Concrete Age is on the Xaxis. The temperatures for concrete (25 millimeters from top and 25 millimeters from bottom) fluctuate with the air temperature, except between 12 and 24 hours, and are generally higher than the air temperature, with the biggest difference of about 15 degrees Celsius.
Figure 217. Graph. Measured versus predicted temperatures 25 millimeters from top of slab for Lufkin, Texas, Slab 2. Temperature is on the Yaxis and Concrete Age is on the Xaxis. Most of the predicted temperatures are close to the measured temperatures, ranging from 20 to 33 degrees Celsius, with the biggest difference being about 2 degrees Celsius at various ages.
Figure 218. Graph. Measured versus predicted temperatures 25 millimeters from bottom of slab for Lufkin, Texas, Slab 2. Temperature is on the Yaxis and Concrete Age is on the Xaxis. Most of the predicted temperatures are close to the measured temperatures, ranging from 20 to 35 degrees Celsius, with the biggest difference being about 3 degrees Celsius at 15 hours.
Figure 219. Graph. Measured versus predicted temperature gradient for Lufkin, Texas, Slab 2. Temperature Gradient is on the Yaxis and Concrete Age is on the Xaxis. Most of the predicted temperature gradients are close to the measured temperature gradients, ranging from negative 8 degrees Celsius to 8 degrees Celsius, with the biggest difference being about 3 degrees Celsius at various ages.
Figure 220. Graph. Measured concrete and air temperatures for Lufkin, Texas, Slab 3. Temperature is on the Yaxis and Concrete Age is on the Xaxis. The temperatures for concrete (25 millimeters from top and 25 millimeters from bottom) fluctuate with the air temperature and are generally higher than the air temperature, with the biggest difference of about 20 degrees Celsius.
Figure 221. Graph. Measured versus predicted temperatures 25 millimeters from top of slab for Lufkin, Texas, Slab 3. Temperature is on the Yaxis and Concrete Age is on the Xaxis. Most of the predicted temperatures are close to the measured temperatures, ranging from 22 to 47 degrees Celsius, with the biggest difference being about 5 degrees Celsius at 24 hours.
Figure 222. Graph. Measured versus predicted temperatures 25 millimeters from bottom of slab for Lufkin, Texas, Slab 3. Temperature is on the Yaxis and Concrete Age is on the Xaxis. Most of the predicted temperatures are close to the measured temperatures, ranging from 20 to 48 degrees Celsius, with the biggest difference being about 3 degrees Celsius at 15 hours.
Figure 223. Graph. Measured versus predicted temperature gradient for Lufkin, Texas, Slab 3. Temperature Gradient is on the Yaxis and Concrete Age is on the Xaxis. The predicted temperature gradients are not that close to the measured temperature gradients, ranging from negative 10 degrees Celsius to 13 degrees Celsius, while the measured temperature ranges from negative 5 to 11 degrees Celsius, with the biggest difference being about 6 degrees Celsius at 18 hours.
Figure 224. Graph. Measured concrete and air temperatures for North Carolina, Slab 2. Temperature is on the Yaxis and Concrete Age is on the Xaxis. The temperatures for concrete (25 millimeters from top and 25 millimeters from bottom) fluctuate with the air temperature, except at early ages, and are generally higher than the air temperature, with the biggest difference of about 25 degrees Celsius.
Figure 225. Graph. Measured versus predicted temperatures 25 millimeters from top of slab for North Carolina, Slab 2. Temperature is on the Yaxis and Concrete Age is on the Xaxis. Most of the predicted temperatures are close to the measured temperatures, ranging from 20 to 45 degrees Celsius, with the biggest difference being about 3 degrees Celsius at 12 hours.
Figure 226. Graph. Measured versus predicted temperatures 25 millimeters from bottom of slab for North Carolina, Slab 2. Temperature is on the Yaxis and Concrete Age is on the Xaxis. Most of the predicted temperatures are close to the measured temperatures, ranging from 25 to 40 degrees Celsius, with the biggest difference being about 3 degrees Celsius at 12 hours.
Figure 227. Graph. Measured versus predicted temperature gradient for North Carolina, Slab 2. Temperature Gradient is on the Yaxis and Concrete Age is on the Xaxis. Most of the predicted temperature gradients are close to the measured temperature gradients, ranging from negative 10 degrees Celsius to 8 degrees Celsius, with the biggest difference being about 3 degrees Celsius at 25 hours.
Figure 228. Graph. Measured concrete and air temperatures for North Carolina, Slab 3. Temperature is on the Yaxis and Concrete Age is on the Xaxis. The temperatures for concrete (25 millimeters from top and 25 millimeters from bottom) fluctuate with the air temperature and are generally higher than the air temperature, with the biggest difference of about 15 degrees Celsius.
Figure 229. Graph. Measured versus predicted temperatures 25 millimeters from top of slab for North Carolina, Slab 3. Temperature is on the Yaxis and Concrete Age is on the Xaxis. Most of the predicted temperatures are close to the measured temperatures, ranging from 20 to 45 degrees Celsius, with the biggest difference being about 3 degrees Celsius at 4 hours.
Figure 230. Graph. Measured versus predicted temperatures 25 millimeters from bottom of slab for North Carolina, Slab 3. Temperature is on the Yaxis and Concrete Age is on the Xaxis. Most of the predicted temperatures are close to the measured temperatures, ranging from 25 to 40 degrees Celsius, with the biggest difference being about 3 degrees Celsius at various ages.
Figure 231. Graph. Measured versus predicted temperature gradient for North Carolina, Slab 3. Temperature Gradient is on the Yaxis and Concrete Age is on the Xaxis. Most of the predicted temperature gradients are close to the measured temperature gradients, ranging from negative 10 degrees Celsius to 10 degrees Celsius, with the biggest difference being about 3 degrees Celsius at various ages.
Figure 232. Graph. Measured concrete and air temperatures for Minnesota, Slab 2. Temperature is on the Yaxis and Concrete Age is on the Xaxis. The temperatures for concrete (25 millimeters from top, at middepth, and 25 millimeters from bottom) all fluctuate with the air temperature, except at early ages, and are generally higher than the air temperature, with the biggest difference of about 20 degrees Celsius.
Figure 233. Graph. Measured versus predicted temperatures 25 millimeters from top of slab for Minnesota, Slab 2. Temperature is on the Yaxis and Concrete Age is on the Xaxis. Most of the predicted temperatures are close to the measured temperatures, ranging from 12 to 22 degrees Celsius, with the biggest difference being about 2 degrees Celsius at 15 hours.
Figure 234. Graph. Measured versus predicted temperatures 25 millimeters from bottom of slab for Minnesota, Slab 2. Temperature is on the Yaxis and Concrete Age is on the Xaxis. Most of the predicted temperatures are close to the measured temperatures, ranging from 14 to 22 degrees Celsius, with the biggest difference being about 2 degrees Celsius at 24 hours.
Figure 235. Graph. Measured versus predicted temperature gradient for Minnesota, Slab 2. Temperature Gradient is on the Yaxis and Concrete Age is on the Xaxis. Most of the predicted temperature gradients are close to the measured temperature gradients, ranging from negative 5 degrees Celsius to 5 degrees Celsius, with the biggest difference being about 3 degrees Celsius at 28 hours.
Figure 236. Graph. Measured concrete and air temperatures for Minnesota, Slab 3. Temperature is on the Yaxis and Concrete Age is on the Xaxis. The temperature in concrete (25 millimeters from top) fluctuates with the air temperature, and both increase to about 15 degrees Celsius at 42 hours and stay near constant from there. The temperatures in concrete (at midslab and 25 millimeters from bottom) increase to about 15 degrees Celsius at 20 hours and stay near constant from there.
Figure 237. Graph. Measured versus predicted temperatures 25 millimeters from top of slab for Minnesota, Slab 3. Temperature is on the Yaxis and Concrete Age is on the Xaxis. Most of the predicted temperatures are close to the measured temperatures, ranging from 10 to 18 degrees Celsius, with the biggest difference being about 2 degrees Celsius at 42 hours.
Figure 238. Graph. Measured versus predicted temperatures 25 millimeters from bottom of slab for Minnesota, Slab 3. Temperature is on the Yaxis and Concrete Age is on the Xaxis. Most of the predicted temperatures are very close to the measured temperatures, ranging from 9 to 18 degrees Celsius, with the biggest difference being about 1 degree Celsius at 48 hours.
Figure 239. Graph. Measured versus predicted temperature gradient for Minnesota, Slab 3. Temperature Gradient is on the Yaxis and Concrete Age is on the Xaxis. Most of the predicted temperature gradients are close to the measured temperature gradients, ranging from negative 3 degrees Celsius to 5 degrees Celsius, with the biggest difference being about 2 degrees Celsius at 6 hours.
Figure 240. Graph. Measured concrete and air temperatures for Arizona, Slab 4. Temperature is on the Yaxis and Concrete Age is on the Xaxis. The temperatures for concrete (25 millimeters from top, at middepth, and 25 millimeters from bottom) all fluctuate with the air temperature, except at early ages, and are generally higher than the air temperature, with the biggest difference of about 20 degrees Celsius.
Figure 241. Graph. Measured versus predicted temperatures 25 millimeters from top of slab for Arizona, Slab 4. Temperature is on the Yaxis and Concrete Age is on the Xaxis. Most of the predicted temperatures are close to the measured temperatures, ranging from 10 to 22 degrees Celsius, with the biggest difference being about 3 degrees Celsius at 27 hours.
Figure 242. Graph. Measured versus predicted temperatures 25 millimeters from bottom of slab for Arizona, Slab 4. Temperature is on the Yaxis and Concrete Age is on the Xaxis. Most of the predicted temperatures are very close to the measured temperatures, ranging from 12 to 20 degrees Celsius, with the biggest difference being about 1 degree Celsius at various ages.
Figure 243. Graph. Measured versus predicted temperature gradient for Arizona, Slab 4. Temperature Gradient is on the Yaxis and Concrete Age is on the Xaxis. Most of the predicted temperature gradients are close to the measured temperature gradients, ranging from negative 5 degrees Celsius to 3 degrees Celsius, with the biggest difference being about 3 degrees Celsius at 48 hours.
Figure 244. Graph. Measured concrete and air temperatures for Arizona, Slab 5. Temperature is on the Yaxis and Concrete Age is on the Xaxis. The temperatures for concrete (25 millimeters from top, at middepth, and 25 millimeters from bottom) all fluctuate with the air temperature, except at early ages, and are generally higher than the air temperature, with the biggest difference of about 17 degrees Celsius.
Figure 245. Graph. Measured versus predicted temperatures 25 millimeters from top of slab for Arizona, Slab 5. Temperature is on the Yaxis and Concrete Age is on the Xaxis. Most of the predicted temperatures are very close to the measured temperatures, ranging from 12 to 20 degrees Celsius, with the biggest difference being about 1 degree Celsius at various ages.
Figure 246. Graph. Measured versus predicted temperatures 25 millimeters from bottom of slab for Arizona, Slab 5. Temperature is on the Yaxis and Concrete Age is on the Xaxis. Most of the predicted temperatures are very close to the measured temperatures, ranging from 12 to 20 degrees Celsius, with the biggest difference being about 1 degree Celsius at various ages.
Figure 247. Graph. Measured versus predicted temperature gradient for Arizona, Slab 5. Temperature Gradient is on the Yaxis and Concrete Age is on the Xaxis. Most of the predicted temperature gradients are close to the measured temperature gradients, ranging from negative 4 degrees Celsius to 4 degrees Celsius, with the biggest difference being about 2 degrees Celsius at 24 hours.
Figure 248. Graph. Measured concrete and air temperatures for Lufkin, Texas, Slab 1. Temperature is on the Yaxis and Concrete Age is on the Xaxis. The temperatures for concrete (25 millimeters from top and 25 millimeters from bottom) fluctuate with the air temperature and are generally higher than the air temperature, with the biggest difference of about 20 degrees Celsius.
Figure 249. Graph. Measured versus predicted temperatures 25 millimeters from top of slab for Lufkin, Texas, Slab 1. Temperature is on the Yaxis and Concrete Age is on the Xaxis. Most of the predicted temperatures are close to the measured temperatures, ranging from 18 to 35 degrees Celsius, with the biggest difference being about 2 degrees Celsius at various ages.
Figure 250. Graph. Measured versus predicted temperatures 25 millimeters from bottom of slab for Lufkin, Texas, Slab 1. Temperature is on the Yaxis and Concrete Age is on the Xaxis. Most of the predicted temperatures are close to the measured temperatures, ranging from 18 to 37 degrees Celsius, with the biggest difference being about 3 degrees Celsius at 15 hours.
Figure 251. Graph. Measured versus predicted temperature gradient for Lufkin, Texas, Slab 1. Temperature Gradient is on the Yaxis and Concrete Age is on the Xaxis. Most of the predicted temperature gradients are close to the measured temperature gradients, ranging from negative 8 degrees Celsius to 9 degrees Celsius, with the biggest difference being about 5 degrees Celsius at 16 hours.
Figure 252. Graph. Measured concrete and air temperatures for Lufkin, Texas, Slab 4. Temperature is on the Yaxis and Concrete Age is on the Xaxis. The temperatures for concrete (25 millimeters from top and 25 millimeters from bottom) fluctuate with the air temperature and are generally higher than the air temperature, with the biggest difference of about 20 degrees Celsius.
Figure 253. Graph. Measured versus predicted temperatures 25 millimeters from top of slab for Lufkin, Texas, Slab 4. Temperature is on the Yaxis and Concrete Age is on the Xaxis. Most of the predicted temperatures are close to the measured temperatures, ranging from 27 to 45 degrees Celsius, with the biggest difference being about 2 degrees Celsius at various ages.
Figure 254. Graph. Measured versus predicted temperatures 25 millimeters from bottom of slab for Lufkin, Texas, Slab 4. Temperature is on the Yaxis and Concrete Age is on the Xaxis. Most of the predicted temperatures are close to the measured temperatures, ranging from 25 to 45 degrees Celsius, with the biggest difference being about 5 degrees Celsius at 12 hours.
Figure 255. Graph. Measured versus predicted temperature gradient for Lufkin, Texas, Slab 4. Temperature Gradient is on the Yaxis and Concrete Age is on the Xaxis. The predicted temperature gradients are not that close to the measured temperature gradients, ranging from negative 12 degrees Celsius to 12 degrees Celsius, while the measured temperature ranges from negative 5 to 9 degrees Celsius, with the biggest difference being about 7 degrees Celsius at 12 hours.
Figure 256. Graph. Measured concrete and air temperatures for North Carolina, Slab 1. Temperature is on the Yaxis and Concrete Age is on the Xaxis. The temperatures for concrete (25 millimeters from top and 25 millimeters from bottom) fluctuate with the air temperature and are generally higher than the air temperature, with the biggest difference of about 20 degrees Celsius.
Figure 257. Graph. Measured versus predicted temperatures 25 millimeters from top of slab for North Carolina, Slab 1. Temperature is on the Yaxis and Concrete Age is on the Xaxis. Most of the predicted temperatures are close to the measured temperatures, ranging from 20 to 45 degrees Celsius, with the biggest difference being about 3 degrees Celsius at 28 hours.
Figure 258. Graph. Measured versus predicted temperatures 25 millimeters from bottom of slab for North Carolina, Slab 1. Temperature is on the Yaxis and Concrete Age is on the Xaxis. Most of the predicted temperatures are very close to the measured temperatures, ranging from 27 to 40 degrees Celsius, with the biggest difference being about 1 degree Celsius at various ages.
Figure 259. Graph. Measured versus predicted temperature gradient for North Carolina, Slab 1. Temperature Gradient is on the Yaxis and Concrete Age is on the Xaxis. Most of the predicted temperature gradients are close to the measured temperature gradients, ranging from negative 13 degrees Celsius to 12 degrees Celsius, with the biggest difference being about 4 degrees Celsius at various ages.
Figure 260. Graph. Measured concrete and air temperatures for North Carolina, Slab 4. Temperature is on the Yaxis and Concrete Age is on the Xaxis. The temperatures for concrete (25 millimeters from top and 25 millimeters from bottom) fluctuate with the air temperature and are generally higher than the air temperature, with the biggest difference of about 20 degrees Celsius.
Figure 261. Graph. Measured versus predicted temperatures 25 millimeters from top of slab for North Carolina, Slab 4. Temperature is on the Yaxis and Concrete Age is on the Xaxis. Most of the predicted temperatures are close to the measured temperatures, ranging from 20 to 45 degrees Celsius, with the biggest difference being about 3 degrees Celsius at 8 hours.
Figure 262. Graph. Measured versus predicted temperatures 25 millimeters from bottom of slab for North Carolina, Slab 4. Temperature is on the Yaxis and Concrete Age is on the Xaxis. Most of the predicted temperatures are close to the measured temperatures, ranging from 27 to 40 degrees Celsius, with the biggest difference being about 2 degrees Celsius at 8 hours.
Figure 263. Graph. Measured versus predicted temperature gradient for North Carolina, Slab 4. Temperature Gradient is on the Yaxis and Concrete Age is on the Xaxis. Most of the predicted temperature gradients are close to the measured temperature gradients, ranging from negative 8 degrees Celsius to 8 degrees Celsius, with the biggest difference being about 3 degrees Celsius at 36 hours.
Figure 264. Graph. Measured concrete and air temperatures for Fort Worth, Texas, Slab 1. Temperature is on the Yaxis and Concrete Age is on the Xaxis. The temperatures for concrete (25 millimeters from top and 25 millimeters from bottom) fluctuate with the air temperature and are higher than the air temperature, with the biggest difference of about 20 degrees Celsius.
Figure 265. Graph. Measured versus predicted temperatures 25 millimeters from top of slab for Fort Worth, Texas, Slab 1. Temperature is on the Yaxis and Concrete Age is on the Xaxis. Most of the predicted temperatures are close to the measured temperatures, ranging from 33 to 60 degrees Celsius, with the biggest difference being about 3 degrees Celsius at 18 hours.
Figure 266. Graph. Measured versus predicted temperatures 25 millimeters from bottom of slab for Fort Worth, Texas, Slab 1. Temperature is on the Yaxis and Concrete Age is on the Xaxis. Most of the predicted temperatures are close to the measured temperatures, ranging from 33 to 55 degrees Celsius, with the biggest difference being about 2 degrees Celsius at 24 hours.
Figure 267. Graph. Measured versus predicted temperature gradient for Fort Worth, Texas, Slab 1. Temperature Gradient is on the Yaxis and Concrete Age is on the Xaxis. Most of the predicted temperature gradients are close to the measured temperature gradients, ranging from negative 8 degrees Celsius to 8 degrees Celsius, with the biggest difference being about 3 degrees Celsius at 18 hours.
Equation 1. The Degree of Hydration. Alpha as a function of T subscript E equals the product of alpha subscript U and the exponential function of the following: the negative quotient of tau divided by T subscript E with that quotient raised to the beta.
Equation 2. The Rate of Heat Liberation. Q subscript H as a function of lowercase T subscript E equals the product of six terms. The first is H subscript U. The second is uppercase C subscript lowercase C. The third is the quotient of tau divided by lowercase T subscript E with that quotient raised to the beta. The fourth is beta divided by lowercase T subscript E. The fifth is alpha as a function of lowercase T subscript E. The sixth is the exponential function of E divided by R times the sum of 1 divided by the sum of 273 plus uppercase T subscript R minus 1 divided by the sum of 273 plus uppercase T subscript lowercase C.
Equation 3. The Total Heat of Hydration of Cementitious Materials at 100 percent R Hydration. H subscript U equals the sum of three terms. The first is H subscript CEM times P subscript CEM. The second is 461 times P subscript SLAG. The third is 1800 times P subscript FAcalcium oxide times P subscript FA.
Equation 4. Heat of Hydration of the Cement. H subscript CEM equals the sum of seven products. The first product is 500 times P subscript tricalcium silicate. The second product is 260 times P subscript dicalcium silicate. The third product is 866 times P subscript tricalcium silicate. The fourth product is 420 times P subscript tetracalcium aluminoferrite. The fifth product is 624 times P subscript sulphur trioxide. The sixth product is 1186 times P subscript FREECA. The seventh product is 850 times P subscript magnesium oxide.
Equation 5. Hydration Time Parameter. Tau equals the product of 66.78 times P subscript C_{3}A raised to the negative 0.154 times P subscript tricalcium silicate raised to the negative 0.401 times Blaine raised to the negative 0.804 times P subscript sulphur trioxide raised to the negative 0.758 times the exponential function of the following: 2.187 times P subscript SLAG plus 9.50 times P subscript FA times P subscript FACalcium oxide.
Equation 6. Hydration Shape Parameter. Beta equals the product of 181.4 times P subscript C_{3}A raised to the 0.146 times P subscript tricalcium silicate raised to the 0.227 times Blaine raised to the negative 0.535 times P subscript sulphur trioxide raised to the 0.558 times the exponential function of the following: negative 0.647 times P subscript SLAG.
Equation 7. Ultimate Degree of Hydration. alpha subscript U equals the quotient of the following: the numerator consists of 1.031 times W divided by CM, and the denominator consists of 0.194 plus W divided by CM, end quotient, plus 0.5 times P subscript FA plus 0.3 times P subscript SLAG. That sum is less than or equal to 1.0.
Equation 8. The Recommended Activation Energy Model. E equals the product of 22,100 times F subscript E times P subscript tricalcium aluminate raised to the 0.30 times P subscript tetracalcium aluminoferrite raised to the 0.25 times Blaine raised to the 0.35.
Equation 9. Activation Energy Modification Factor for Mineral Admixtures. F equals 1 minus the product of 1.05 times P subscript FA times the following: 1 minus P subscript FA minus calcium oxide all over 0.40, end of product, plus 0.40 times P subscript SLAG.
Equation 10. Degree of Hydration at Initial Set. Alpha subscript I equals 0.15 times the quotient of W divided by CM.
Equation 11. Degree of Hydration at Final Set. Alpha subscript F equals 0.26 times the quotient of W divided by cm.
Equation 12. Equivalent Age at Initial Set. T subscript EI equals tau times the negative natural log of the following: 0.14 times W divided by CM all over alpha subscript U, end of natural log, raised to the negative 1 over beta.
Equation 13. Equivalent Age at Final Set. T subscript EF equals tau times the negative natural log of the following: 0.26 times W divided by CM all over alpha subscript U, end of natural log, raised to the negative 1 over beta.
Equation 14. Adjusted Hydration Time Parameter to Include the Effect of Retarder or Accelerators. Tau subscript CHEM equals the sum of T subscript EI plus delta subscript CHEM times the negative natural log of the following: 0.14 times W divided by CM all over alpha subscript U, end of natural log, raised to the negative 1 over beta.
Equation 15. Heat Transfer Model. The derivative with respect to X of K times the derivative of uppercase T with respect to X plus Q subscript H equals rho times C subscript P times the derivative of uppercase T with respect to lowercase T.
Equation 16. Current Specific Heat of the Concrete Mixture. C subscript P equals one over rho times uppercase W subscript C times alpha times C subscript CEF plus uppercase W subscript C times the sum of quantity 1 minus alpha end quantity times C subscript C plus uppercase W subscript A times C subscript A plus uppercase W subscript lowercase W times C subscript lowercase W.
Equation 17. Current Thermal Conductivity of the Concrete. K subscript I equals K subscript infinity times the sum of 1.33 minus 0.33 times alpha
Equation 18. Heat Flux. Q equals H subscript 0 times the sum of T subscript S minus T subscript A.
Equation 19. Overall Heat Transfer Coefficient. Q subscript 0 equals the sum of the series of D subscript 1 divided by K subscript 1 plus D subscript 2 divided by K subscript 2 and so forth, plus D subscript N divided by K subscript N, all raised to the negative one.
Equation 20. Heat Transferred Due to Convection. Q subscript C equals H subscript C times the sum of T subscript S minus T subscript A.
Equation 21. Surface Convection Coefficient. H subscript lowercase C equals the product of 3.727 times uppercase C times the sum of T subscript S minus T subscript A raised to the 0.266 times the square root of one plus 2.857 times W, end square root, times the following: 0.9 times the sum of T subscript S plus T subscript A, end product, plus 32, all raised to the negative 0.181.
Equation 22. Surface Convection Coefficient. H subscript C equals 20 plus 14 times W when W is less than or equal to 5 meters per second.
Equation 23. H subscript C equals 25.6 times 0.78 times w when W is greater than 5 meters per second.
Equation 24. Surface Convection Coefficient. H subscript C equals 20.24 plus 14.08 times W when W is less than or equal to 4.87 meters per second.
Equation 25. H subscript C equals 25.82 times 0.78 times W when W is greater than 4.87 meters per second.
Equation 26. Overall Heat Transfer Coefficient. H subscript 0 equals the sum of the following series: 1 divided by H subscript C plus D subscript 1 divided by K subscript 1 plus D subscript 2 divided by K subscript 2 plus and so forth, D subscript N divided by K subscript N. That sum raised to the negative one.
Equation 27. Amount of Energy Transferred Through Evaporative Cooling. Q subscript EVAP equals the product of negative E subscript R times H subscript LAT.
Equation 28. Latent Heat of Vaporization. H subscript LAT equals the sum of 2,500 plus 1.859 times T subscript A.
Equation 29. Solar Absorption. Q subscript S equals the product of beta subscript epsilon times I subscript F times Q subscript solar.
Equation 30. Heat Flux of Heat Emission from the Surface. Q subscript R equals the product of epsilon times sigma times the sum of T subscript C raised to the 4 minus T subscript infinity raised to the 4.
Equation 31. Total Atmospheric Emissivity. Epsilon subscript ATM equals the sum of two terms. The first term is the sum of 1.009 plus 0.2191 divided by the square root of M subscript W plus 1.57 times 10 raised to the negative 5 all over M subscript W raised to the 1.5, and that sum raised to the negative 1. The second term is the product of 0.185 times the difference between of two exponential functions. The first exponential function is the product of negative 3.060 times M subscript W. The second exponential function is the product of negative 1.8 times M subscript W times the sum of 1.7 plus 50 times rho subscript C divided by rho subscript W.
Equation 32. The Variation of Pressure with Height Above Ground Level. P subscript Z equals P subscript I times the exponential function of the following: Negative 1.2 times 10 raised to the negative 4 times Z, end exponential function.
Equation 33. Adjusted DensityLength Product of the Water Vapor. M prime subscript W equals M subscript W times P subscript Z divided by P subscript 0.
Equation 34. The Variation of Temperature with Height. T subscript Z equals the sum of T subscript I minus 0.006 times Z
Equation 35. The Temperature Adjustment Factor. T subscript F equals the quotient of two terms raised to the fourth power. The first term is T subscript I plus 273. The second term is T subscript 0 plus 273.
Equation 36. Total Precipitable Water Contained Below a Certain Height. The integral from 0 to Z of the change in M prime subscript W equals the product of two terms. The first term is the quotient of P subscript WI divided by the product of 5.7 times 10 raised to the negative 6. The second term is the sum of 1 minus the exponential function of the following product: negative Z times 5.7 times 10 raised to the negative 6.
Equation 37. The Water Vapor Saturation Pressure For DewPoint Range of Negative 100 to 0 Degrees Celsius. The natural log of P subscript WS equals 0.68 times the sum of the following: C subscript 1 divided by T subscript R plus C subscript 2 plus C subscript 3 times T subscript R plus C subscript 4 times T squared subscript R plus C subscript 5 times T raised to the 3 subscript R plus C subscript 6 times T raised to the 4 subscript R plus C subscript 7 times the natural log of T subscript R.
Equation 38. The Water Vapor Saturation Pressure For DewPoint Range of 0 to 200 Degrees Celsius. The natural log of P subscript WS equals 0.68 times the sum of the following: C subscript 8 divided by T subscript R plus C subscript 9 plus C subscript 10 times T subscript R plus C subscript 11 times T squared subscript R plus C subscript 12 times T raised to the 3 subscript R plus C subscript 13 times natural log of T subscript R.
Equation 39. The Water Vapor Pressure of the Moist Air. P subscript W equals RH times P subscript WS.
Equation 40. The Intensity of Atmospheric Radiation. A subscript R equals epsilon subscript APP times sigma times T raised to the fourth subscript A times W divided by M squared.
Equation 41. The Apparent Surrounding Air Temperature. T subscript infinity equals the sum of the quotient of A subscript R divided by sigma raised to the 4 minus 273 degrees Celsius.
Equation 42. The 24hour Periodic Temperature. T equals T subscript M plus the product of three terms. The first term is T subscript V. The second term is the quotient of the product of H times the exponential function of negative X times C divided by the square root of the sum of H plus C squared plus C squared, end of square root. The second term is the sine function of the sum of pi over 12 times T minus X times C minus the arc tangent of C divided by the sum of C plus H.
Equation 43. The Total Shrinkage Strain For a Concrete Cross Section. Epsilon subscript CS as a function of T equals the sum of epsilon subscript CS0 as a function of T plus epsilon subscript CSD as a function of T. This sum is less than 0.
Equation 44. Autogenous Shrinkage. Epsilon subscript CS0 as a function of T equals the product of epsilon subscript S 0 infinity times beta subscript S0 as a function of T.
Equation 45. The Time Distribution of Autogenous Shrinkage. Beta subscript S0 as a function of T equals the exponential function of the negative quotient of T subscript S0 divided by the sum of T minus T subscript start, with that quotient raised to the 0.3.
Equation 46. The Ultimate Autogenous Shrinkage. Epsilon subscript S 0 infinity equals the sum of negative 0.65 plus the quotient of 1.3 times W divided by B, with that sum multiplied by 10 raised to the negative 3.
Equation 47. Surface Layer Drying. Alpha subscript SD equals U times LL subscript SD divided by A subscript C which is less than or equal to 1.
Equation 48. Typical Length of Surface for Water Exchange. L subscript SD equals the quotient of L subscript SD,REF divided by the sum of 0.5 minus W divided by B.
Equation 49. Drying Shrinkage Plus Autogenous Shrinkage When the Whole Specimen is Affected by Drying. Epsilon subscript SD,all as a function of T equals epsilon subscript SD, TOT times beta subscript SD as a function of t times beta subscript SD, RH.
Equation 50. Time Development of Drying Shrinkage. Beta subscript SD as a function of T equals the quotient of two terms raised to the 0.5. The first term is the sum of T minus T subscript S. The second term is the sum of T subscript SD plus T minus T subscript S.
Equation 51. The Total Shrinkage. Epsilon subscript SD, TOT equals the product of the sum of negative 0.1 minus 0.8 times W divided by B all times 10 raised to the negative 3.
Equation 52. Additional Strain Due to Drying / Wetting of the Concrete with the Environment. Epsilon subscript CSD as a function of T equals alpha subscript SD times epsilon subscript SD,all as a function of T
Equation 53. Additional Strain Due to Drying / Wetting of the Concrete with the Environment. Epsilon subscript CSD as a function of T equals alpha subscript SD times epsilon subscript SD,TOT times beta subscript SD as a function of T times beta subscript SD,RH.
Equation 54. Final External Strain Due to Drying Shrinkage. Epsilon subscript SD infinity equals alpha subscript SD times epsilon subscript SD, TOT
Equation 55. Additional Strain Due to Drying / Wetting of the Concrete with the Environment. Epsilon subscript CSD as a function of T equals epsilon subscript SD infinity times beta subscript SD as a function of T times beta subscript SD, RH.
Equation 56. The Shrinkage of HPC as a Function of Relative Humidity. Beta subscript SD,RH equals the quotient of two terms. The first term is the sum of RH subscript 0 minus RH. The second term is the absolute value of RH subscript REF minus RH subscript 0.
Equation 57. The Shrinkage of HPC as a Function of Relative Humidity. Beta subscript SD,RH equals 1 when RH is less than or equal to RH subscript REF.
Equation 58. The Drying Shrinkage Strain. Epsilon subscript SH as a function of T hat and T subscript 0 equals the product of epsilon subscript SH infinity times K subscript H times S as a function of T hat.
Equation 59. The Time Dependence. S as a function of T hat equals the square root of T hat divided by the sum of tau subscript SH plus T hat end square root.
Equation 60. The Size Dependence of the Diffusion Type. Tau subscript SH equals the product of three terms. The first term is 600. The second term is the squared quotient of K subscript S divided by 150 all times D. The third term is the quotient of C subscript 1 superscript REF divided by C subscript 1 as a function of T subscript 0.
Equation 61. Age Dependence. C subscript 1 as a function of T equals the product of C subscript 7 times K prime subscript T times the sum of 0.05 plus the square root of the quotient of 6.3 divided by T.
Equation 62. Temperature Dependence Coefficient. K prime subscript T equals the quotient of T divided by T subscript 0 times the exponential function of the sum of 5000 divided by T subscript 0 minus 5000 divided by T.
Equation 63. Empirical Shrinkage. Epsilon subscript SH infinity equals epsilon subscript S infinity times the quotient of two terms. The first term is E times the sum of 7 plus 600. The second term is E times the sum of T subscript 0 plus tau subscript SH.
Equation 64. Modulus of Elasticity as a Function of Time. E as a function of T equals E as a function of 28 times the square root of the quotient of T divided by the sum of 4 plus 0.85 times T.
Equation 65. Humidity Dependence. K subscript H equals 1 minus H raised to the 3 for H less than or equal to 0.98, or negative 0.2 for H equal to 1, or linear interpolation for 0.98 less than or equal to H less than or equal to 1.
Equation 66. The Reference Diffusivity. Uppercase C subscript 7 equals the sum of the product of one divided by 8 times W divided by lowercase C times lowercase C (variable) minus 12.
Equation 67. Final Shrinkage. Epsilon subscript S infinity equals 1,210 minus 880 times Y.
Equation 68. Parameter Y. Y equals the sum raised to the negative 1 of 390 times Z raised to the negative 4 plus 1.
Equation 69. Parameter Z. Z equals the product of three terms minus 12 for Z greater than or equal to 0. The first term is the sum of the product of 1.25 times the square root of the quotient of A divided by C plus 0.5 times the quotient of the square of G divided by S. The second term is the cube root of the quotient of 1 plus S divided by C divided by W divided by C. The third term is the square root of 0.145 times F prime subscript C. Z equals 0 for all other cases.
Equation 70. Empirical Shrinkage. Epsilon subscript S infinity equals the product of alpha subscript 1 times alpha subscript 2 times the sum of 1,210 minus 880 times Y.
Equation 71. Alpha subscript 1 equals 1.0 for Type one cement, or 0.85 for Type two cement, or 1.1 for Type 3 cement.
Equation 72. Alpha subscript 2 equals 0.75 for steamcured specimens, or 1.0 for watercured specimens, or 1.2 for sealed specimens.
Equation 73. Nonlinear Model. Tau subscript F equals the product of uppercase C subscript 2 times the quotient of U subscript lowercase C divided by delta subscript F, and that quotient raised to the 1 over N.
Equation 74. Equivalent Linear Temperature Gradient. Delta T subscript EQ equals the quotient of negative 12 times M superscript asterisk divided by alpha times H squared.
Equation 75. Constant Dependent on the Temperature Distribution. M superscript asterisk equals the integral from negative H divided by 2 to H divided by 2 of epsilon as a function of Z times Z with respect to Z.
Equation 76. Strain Profile. Epsilon as a function of Z equals the product of alpha times the difference of T subscript Z minus T subscript Z, SET.
Equation 77. The Total Linear Time Dependent Deformation. Epsilon as a function of T equals the product of J as a function of T and T subscript 0 times sigma as a function of T subscript 0.
Equation 78. Creep Compliance. J as a function of T and T subscript 0 equals the quotient of two terms which equals 1 divided by E subscript EFF. The first term is the sum of 1 plus phi as a function of T and T subscript 0. The second term is E as a function of T subscript 0.
Equation 79. Creep Compliance According to the Extended Triple Power Law. J as a function of T and T subscript 0 equals the sum of four terms. The first is 1 divided by E subscript 0. The second term is the product of phi subscript 1 divided by E subscript 0 times the product of two terms: the first term is the sum of T subscript 0 raised to the negative M plus alpha, the second term is the difference of T minus T subscript 0 that difference raised to the N minus B as a function of T, T subscript 0, and N. The third term is the quotient of PSI subscript 1 as a function of T subscript 0 divided by E subscript 0. The fourth term is the quotient of PSI subscript 2 as a function of T and T subscript 0 divided by E subscript 0.
Equation 80. Parameter alpha. Alpha equals the products of 1 divided by 40 times W divided by C.
Equation 81. Parameter phi subscript 1. Phi subscript 1 equals the product of two terms. The first term is the quotient of 10 raised to the 3 times N divided by 2. The second term is the sum of 28 raised to the negative M plus alpha.
Equation 82. Parameter M. M equals the sum of 0.28 plus the product of 0.145 times F prime subscript C, and that product raised to the negative 2.
Equation 83. Parameter N. N equals the sum of 0.12 plus the quotient of two terms when X is greater than 4. The first term is 0.07 times X raised to the 6. The second term is 5130 plus X raised to the 6. N equals 0.12 if X is less than or equal to 4.
Equation 84. Parameter X. X equals the product of A subscript 1 times the sum of two terms minus 4. The first term is 2.1 times A divided by C divided by the quotient of S divided by C raised to the 1.4. The second term is 0.1 times F prime subscript C raised to the 1.5 times the quotient of W divided by C raised to the 1 divided by 3 times the quotient of A divided by G raised to the 22.
Equation 85. Binomial integral B as a function of T and T subscript 0 and N. B as a function of T and T subscript 0 and N equals the product of N times T subscript 0 raised to the N times the sum of three terms. The first term is the quotient of beta raised to the negative N minus 1 divided by N. The second term is the natural log of beta. The third term is the summation from K equal 1 to infinity of the product of N subscript K raised to the negative 1 times negative one raised to the K times the sum of the quotient of beta raised to the difference of K minus N, minus 1 divided by N minus K minus the quotient of 1 minus beta raised to the K divided by K.
Equation 86. Parameter beta. Beta equals T subscript 0 divided by T and K equals the summation times step.
Equation 87. Function psi subscript 1. If T subscript 0 is less than or equal to T subscript 1 then psi subscript 1 as a function of T subscript 0 equals gamma subscript 1 times the quotient raised to the A subscript 1 of T subscript 1 minus T subscript 0 divided by T subscript 1 minus T subscript S.
Equation 88. Function psi subscript 1 when T subscript 0 is greater than T subscript 1. If T subscript 0 is greater than T subscript 1 then psi subscript 1 as a function of T subscript 0 equals 0.
Equation 89. Function psi subscript 2. If T subscript 0 is less than or equal to T subscript 3 then psi subscript 2 as a function of T and T subscript 0 equals gamma subscript 2 times the difference of 1 minus the exponential function of the negative quotient of T minus T subscript 0 divided by T subscript 2 raised to the A subscript 2 times the quotient of T subscript 1 minus T subscript 0 divided by T subscript 1 minus T subscript S raised to the A subscript 3.
Equation 90. Function psi subscript 2 when T subscript 0 is greater than T subscript 3. If T subscript 0 is greater than T subscript 3 then psi subscript 2 as a function of T and T subscript 0 equals 0
Equation 91. Parameter N subscript T. N subscript T equals the product of beta subscript T times N
Equation 92. Parameter beta subscript T. Beta subscript T equals the quotient of 0.25 divided by the sum of 1 plus the quotient of 74 divided by T minus 253.2 raised to the 7 plus 1
Equation 93. Parameter Phi subscript T. Phi subscript T equals the product of phi subscript 1 times 1 plus C subscript T
Equation 94. Parameter C subscript T. C subscript T equals the product of C subscript T1 times tau subscript T times C subscript 0
Equation 95. Parameter C subscript T1. C subscript T1 equals the quotient of 0.25 divided by the sum of 1 plus the quotient of 100 divided by the difference of T minus 253.2 raised to the 3.5 minus 1.
Equation 96. Parameter Tau subscript T. Tau subscript T equals the quotient of 1 divided by the sum of 1 plus the quotient of 60 divided by lowercase T subscript 0 uppercase T raised to the 0.69 plus 0.78.
Equation 97. Parameter C subscript 0. Uppercase C subscript 0 equals 0.125 times the quotient of W divided by lowercase C squared times the quotient of A divided by lowercase C times A subscript 1.
Equation 98. Equivalent Hydration Period. T prime equals T prime subscript E equals the integral from 0 to T prime of beta prime subscript uppercase T as a function of T double prime with respect to T double prime.
Equation 99. Parameter beta prime subscript T. Beta prime subscript T equals the exponential function of the difference of 4,000 divided by T subscript 0 minus 4,000 divided by T
Equation 100. The Strain History. Epsilon as a function of T equals the integral form 0 to T of J as a function of T and T subscript 0 with respect to sigma as a function of T subscript 0 plus epsilon subscript 0 as a function of T
Equation 101. The Incremental Strain. Delta epsilon double prime subscript R equals the summation from S equals 1 to R minus 1 of delta J subscript R times delta sigma subscript S plus delta epsilon subscript R superscript 0.
Equation 102. The Stress Increment. Delta sigma subscript R equals the product of E double prime subscript R times the difference of delta epsilon subscript R minus delta epsilon double prime subscript R.
Equation 103. Iteration interval. At R equal to zero, delta sigma subscript R is equal to zero and T is equal to T subscript zero days.
Equation 104. Iteration interval. At R equal to one, delta epsilon double prime subscript R is equal to zero, T is equal to the sum of T subscript zero plus 0.01 divided by 24 (days), E double prime subscript R is equal to one divided by the product of J as a function of R and R minus 0.1, and delta sigma subscript R is equal to the product of E double prime subscript R and delta epsilon subscript R.
Equation 105. Reference suction value. If the percent of clay is less or equal to 6 percent, and THMI is greater or equal to negative 12.5, SUC subscript REF is equal to 1.86
Equation 106. Reference suction value. If the percent of clay is less or equal to 6 percent, and THMI is less than negative 12.5, SUC subscript REF is equal to negative 0.0141 times THMI plus 1.69
Equation 107. Reference suction value. If the percent clay is greater than 6 percent but less than 40 percent and THMI is less than or equal to negative 0.5, SUC subscript REF is equal to negative 0.0251 times THMI plus 2.5.
Equation 108. Reference suction value. If the percent clay is greater than 6 percent but less than 40 percent and THMI is greater than negative 0.5, SUC subscript REF is equal to negative 0.0062 times THMI plus 2.46.
Equation 109. Reference suction value. If the percent clay is greater than or equal to 40 percent and THMI is less than or equal than negative 5.98, SUC subscript REF is equal to negative 0.043 times THMI plus 3.23.
Equation 110. Reference suction value. If the percent clay is greater than or equal to 40 percent and THMI is greater than negative 5.98, SUC subscript REF is equal to negative 0.0118 times THMI plus 3.4.
Equation 111. Average monthly rainfall. Rain subscript AVG is equal to the sum of rain subscript MAX and rain subscript MIN divided by 2.
Equation 112. Monthly soil suction calculation. If rain subscript MO is greater than rain subscript AVG, SUC subscript MO is equal to SUC subscript REF minua from the quotient of rain subscript MO subtracted by rain subscript AVG and rain subscript max subtracted by rain subscript AVG multiplied by delta.
Equation 113. Monthly soil suction. If rain subscript MO is less than rain subscript AVG, SUC subscript MO is equal to SUC subscript REF plus the quotient of rain subscript AVG subtracted by rain subscript MO and rain subscript AVG subtracted by rain subscript MIN multiplied by delta.
Equation 114. Soil suction parameter. If rain subscript MO is equal to rain subscript AVG, then SUC subscript MO is equal to SUC subscript REF.
Equation 115. Moisture content value. If SUC subscript MO is greater than or equal to SUC subscript PL, then W subscript MO is equal to the quotient of SUC subscript MO minus 5.5 and SUC subscript PL minus 5.5 multiplied by PL.
Equation 116. Moisture content value. If SUC subscript MO is less than SUC subscript PL, and W subscript SAT is less than LL, then w subscript MO is equal to PL plus the quotient of SUC subscript MO minus SUC subscript PL and 1.0 minus SUC subscript PL multiplied by PI.
Equation 117. Moisture content value. If SUC subscript MO is less than SUC subscript PL, and W subscript sat is greater than or equal to LL, then w subscript MO is equal to PL plus the quotient of SUC subscript MO minus SUC subscript PL and 1.0 minus SUC subscript PL multiplied by the difference of W subscript SAT minus PL.
Equation 118. The Transformation Equation . The ratio of F subscript ST divided by F subscript RC equals 0.62
Equation 119. Parameter F subscript R. F subscript R equals 0.864 times F subscript RC.
Equation 120. Ratio between the splitting tensile strength and the third point loaded modulus of rupture. The ratio of F subscript ST divided by F subscript R equals 0.72
Equation 121. Relationship Between the Modulus of Rupture and Compressive Strength. F subscript R equals the product of 0.7 and the square root of F prime subscript C.
Equation 122. The Regression Analysis. F subscript R equals 2.3 times F subscript C raised to the 2 divided by 3
Equation 123. Modulus of rupture. F subscript R equals 2.28 times F subscript C raised to the 2 divided by 3.
Equation 124. Elastic Modulus of Concrete. E subscript C equals W subscript C raised to the 1.5 times 0.043 times the square root of F prime subscript C.
Equation 125. The Concrete Unit Weight. W subscript C equals the product of 16.02 times the sum of 33 plus SG times the difference of 45 minus 0.6 times Air.
Equation 126. The Tensile Strength at Time T. F subscript T as a function of T equals F subscript T times 28 times beta as a function of T.
Equation 127. Parameter Beta as a function of T. Beta as a function of T equals the exponential function of S times the difference of 1 minus the quotient of 28 divided by T over T subscript 1 raised to the 1 divided by 2.
Equation 128. The Radius of the Contact Area. Alpha equals the square root of the quotient of P divided by the product of pi times P subscript T
Equation 129. The Radius of Relative Stiffness. L equals the fourth root of a quotient. The numerator is the product of E subscript C times H raised to the 3. The denominator is 12 times the squared quantity of the difference of 1 minus mu subscript C, and quantity times K.
Equation 130. The Bending Moment of Inertia. I subscript D equals the quotient of the products of pi times phi subscript D raised to the 4 all over 64.
Equation 131. The CrossSectional Area of a Circular Dowel. A subscript D equals the quotient of pi times phi subscript D squared divided by 4.
Equation 132. Effective CrossSectional Area. A subscript DZ equals the product of 0.9 times A subscript D.
Equation 133. The Shear Modulus of the Dowel Bar. G subscript D equals the quotient of E subscript D over the product of 2 times the sum of 1 plus mu subscript D.
Equation 134. Correction Factor for DeepBeam Shear Deformation. Phi equals the quotient of the quantity 12 times E subscript D times I subscript D all over the product of G subscript D times A subscript DZ times W squared.
Equation 135. The Relative Stiffness. Beta equals the fourth root of the quotient of K subscript D times phi subscript D all over 4 times E subscript D times I subscript D.
Equation 136. The Shear Stiffness of the Dowel. C equals the quotient of E subscript D times I subscript D all over W raised to the 3 times the sum of 1 plus phi.
Equation 137. The DowelConcrete Interaction Parameter. DCI equals the quantity of the quotient of 4 times beta raised to the 3 times E subscript D times I subscript D all over the sum of 2 plus beta times W.
Equation 138. Dowel Shear Stiffness. D prime equals the quotient of 1 all over the sum of 1 divided by DCI plus 1 divided by the product of 12 times C.
Equation 139. The Dimensionless Aggregate Interlock Parameter. AGG divided by KL equals gamma times the exponential function of the product of negative 1.5 times W.
Equation 140. Aggregate Interlock Model Intercept. For H minus T subscript SAW less than or equal to 178 millimeters, then gamma equals the product of 1.388 times 10 raised to the negative 6 times the quantity of the difference of H minus T subscript SAW and quantity raised to the 3.5309.
Equation 141. H minus T subscript SAW greater than 178 millimeter, then gamma equals the difference of 2.425 times the quantity of the difference of H minus T subscript SAW and minus 309.
Equation 142. Aggregate Interlock. AGG equals AGG divided by KL times KL.
Equation 143. Dimensionless Joint Stiffness Due to Dowels. J subscript D equals the quotient of D prime divided by the product S subscript D times KL.
Equation 144. Dimensionless Joint Stiffness Due to Aggregate Interlock. J subscript AGG equals AGG divided by KL.
Equation 145. Overall Dimensionless Joint Stiffness. J equals the sum of J subscript D plus J subscript AGG.
Equation 146. Dimensionless FreeEdge Deflection. Delta subscript F superscript asterisk equals B subscript 3 minus the product of B subscript 4 times A divided by L plus the product of 0.5 B subscript 5 times the squared quantity of A divided by L.
Equation 147. Westergaard Coefficients. B subscript 3 equals the quotient of the square root of the sum of 2 plus 1.2 times mu subscript C all over the square root of 12 times the difference of 1 minus mu squared subscript C.
Equation 148. Westergaard Coefficients. B subscript 4 equals the product of B subscript 3 times the sum of 0.74 plus 0.4 times mu subscript C,
Equation 149. Westergaard Coefficients. B subscript 5 equals the quotient of B subscript 4 squared divided by 2 times B subscript 3.
Equation 150. The FreeEdge Deflection. Delta subscript F equals the quotient of delta subscript F super script asterisk times P all over KL squared.
Equation 151. Dimensionless FreeEdge Stress. Sigma subscript F superscript asterisk equals sigma bar superscript asterisk plus sigma subscript C superscript asterisk.
Equation 152. First Component. Sigma bar super script asterisk equals the product of two terms. The first term is a quotient with a numerator of 12 times the sum of 1 plus mu subscript C and a denominator of pi times the sum of 3 plus mu subscript C. The second term is the sum of 0.1159 minus the natural log of A divided by L plus the quotient of 1 minus mu subscript C all over 8.
Equation 153. Second Component. Sigma subscript C super script asterisk equals the product of three terms. The fist terms is the quotient with a numerator of 12 times the sum of 1 plus mu and a denominator of pi times 3 plus U subscript C. The second term is the quantity of the difference of 3 minus mu subscript C all over 4 and quantity minus 0.9544 plus 0.3822 times A divided by L. The third term equals 1.30259 plus 2.98398 times the quantity A divided by L squared.
Equation 154. FreeEdge Stress. Sigma subscript F equals the quotient of sigma subscript F super script asterisk times P all over H squared.
Equation 155. Dimensionless Unloaded Edge Deflection. Delta subscript U superscript asterisk equals B prime subscript 3 minus B prime subscript 4 times the quantity A divided by L and quantity plus 0.008 times the quantity A divided by L squared and quantity times the sum of 1 plus log of J.
Equation 156. Westergaard Coefficients. B prime subscript 3 equals the quotient of the sum of uppercase J minus 0.6367 times the log of 1 plus uppercase J divided by 4.6516 times lowercase J plus 1.8210.
Equation 157. Westergaard Coefficients. B prime subscript 4 equals B prime subscript 3 times the quantity of the sum of 0.6984 plus the product of 0.0441 times the log of 1 plus J minus 0.00655 times J raised to the 0.24.
Equation 158. Unloaded Edge Deflection. Delta subscript U equals the quotient of delta subscript U superscript asterisk times P all over KL squared.
Equation 159. Dimensionless FreeEdge Stress. Sigma subscript U superscript asterisk equals the quotient of 24 times the difference of 1 minus mu subscript C squared all over pi times the quantity of the sum of B prime subscript 1 minus B prime subscript 2 times A divided by L plus 0.5 times B prime subscript 9 times the quantity A divided by L squared.
Equation 160. Westergaard Coefficients. B prime subscript 1 equals 0.03316 plus 0.07205 times log of 1 plus uppercase J all plus the quantity of the quotient of 0.00773 times the square root of uppercase J minus 0.03360 all over uppercase J plus 1.
Equation 161. Westergaard Coefficients. B prime subscript 2 equals B prime subscript 1 times the quantity of the sum of 0.08281 times J plus 0.4790 minus the quotient of 0.000149 divided by J, all raised to the 1 over 4.209.
Equation 162. Westergaard Coefficients. B prime subscript 9 equals the exponential function of the sum of negative 5.0908 plus 2.2805 times the log of J, minus 0.1577 times the log squared of J.
Equation 163. Unloaded Edge Stress. Sigma subscript U equals the quotient of sigma subscript U superscript asterisk times P all over H squared.
Equation 164. Loaded Edge Deflection. Delta subscript 1 equals delta subscript F minus delta subscript U.
Equation 165. Loaded Edge Stress. Sigma subscript 1 equals sigma subscript F minus sigma subscript U.
Equation 166. Joint Transfer Efficiency. JTE equals the quotient of 2 times delta subscript U all over delta subscript 1 plus delta subscript U.
Equation 167. Deflection Load Transfer Efficiency. LTE subscript delta equals delta subscript U divided by delta subscript L.
Equation 168. Stress Load Transfer Efficiency. LTE subscript sigma equals sigma subscript U divided by sigma subscript L.
Equation 169. Deflection Load Transfer Efficiency. LTE subscript delta equals the quotient of 0.5 times JTE all over the difference of 1 minus 0.5 times JTE.
Equation 170. Dimensionless Aggregate Interlock Parameter. AGG divided by KL subscript wear equals AGG divided by KL subscript ASYMP plus AGG divided by KL subscript INIT times the exponential function of negative N divided by S.
Equation 171. Parameter AGG divided by KL subscript asymptote. AGG divided by KL subscript ASYMPTOTE equals the product of xi times the exponential function of 3.63 times W.
Equation 172. The Resulting Model for The xi Intercept. For H minus T subscript SAW less than or equal to 178 mm, xi equals the product of 1.935 times 10 raised to the negative 8 times the sum of H minus T subscript SAW raised to the 6.9814.
Equation 173. The Resulting Model for the xi Intercept. For H minus T subscript SAW greater than 178 millimeters, then xi equals the difference of the product of 0.01533 times the sum of H minus T subscript SAW plus negative 0.092.
Equation 174. The Faulting Equation. FAULT equals ESAL raised to the 0.25 times the sum of 0.000038 plus 0.01830 times the quantity of 100 times OPENING end quantity raised to the 0.5585 plus 0.000619 times the quantity of 100 times DEFLAMI end quantity raised to the 1.7229 plus 0.04 times the quotient of FI divided by 1000 raised to the 1.9840 plus 0.00565 times BTERM minus 0.00770 times EDGESUP minus 0.00263 times STYPE minus 0.00891 times DRAIN.
Equation 175. Ioannides' Pavement Corner Deflection. DEFLAMI equals P times the quotient of the difference of 1.2 minus the quotient of 0.88 times 1.4142 times A all over L in the numerator all over KSTAT times L squared in the denominator.
Equation 176. Base Type Factor. BSTERM equals 10 times the quantity ESAL raised to the 0.2076 times the sum of 0.045456 plus 0.05115 times GB plus 0.007279 times CTB plus 0.003183 times ATB minus 0.003714 times OGB minus 0.006441 times LCB.
Equation 177. The MechanisticEmpirical Faulting Equation . Fault equals ESAL raised to the 0.5280 times the quantity of the sum of the following: the sum of 0.1204 plus 0.04048 times the quotient of BSTRESS divided by 1000 this quotient raised to the 0.3388 plus 0.007353 times AVJSPACE divided by 10 this quotient raised to the 0.6725 minus 0.1492 times the quotient of KSTAT divided by 100 this quotient raised to the 0.05911, minus 0.01868 times DRAIN minus 0.00879 times EDGESUP minus 0.00959 times STYPE.
Equation 178. Maximum Dowel Bearing Stress. BSTRESS equals the product of F subscript D times P times T times the quantity of K subscript D times the quotient of 2 plus BETA times OPENING all over 4 times E subscript S times I times BETA raised to the 3.
Equation 179. Joint Opening. OPENING equals CON times AVJSPACE times 12 times the quantity of the quotient of ALPHA times TRANGE all over 2 and quantity plus epsilon.
Equation 180. Joint Opening. OPENING equals CON times AVJSPACE times 12 times epsilon subscript free.
Equation 181. Regression Analysis. OPENING subscript FHWA divided by OPENING subscript HP equals 0.84241 minus 0.018459 divided by H minus 2.3010 times 10 raised to the negative 4 times F subscript max divided by delta subscript CRIT plus 57923 divided by E subscript C.
Equation 182. ERES / COE Model. The log of N equals 2.13 times SR raised to the negative 1.2.
Equation 183. National Cooperative Highways Research Program (NCHRP) 126 Model. The log of N equals the quotient of negative SR raised to the negative 5.367 times the log of 1 minus P all over 0.0032, with that quotient raised to the 0.2276.
Equation 184. ZeroMaintenance Model. The log of N equals 17.61 minus 17.61 times SR.
Equation 185. ARE Model. N equals the product of 23440 times SR raised to the negative 3.21.
Equation 186. Percent Cracking. Percent Cracking equals 100 divided by the sum of 1 plus 1.41 times FD raised to the negative 1.66.
Equation 187. Accumulated Fatigue Damage. FD equals the summation of lowercase N subscript lowercase I divided by uppercase N subscript lowercase I.
Equation 188. Westergaard Edge Loading and Deflection Solutions. The quotient of sigma times H squared divided by P.
Equation 189. Westergaard Edge Loading and Deflection Solutions. The quotient of delta times K times L squared divided by P.
Equation 190. Westergaard Edge Loading and Deflection Solutions. The quotient of Q times L squared divided by P.
Equation 191. Equivalent edge stress. Westergaard edge stress adjusted for the specific design and conditions relating to the pavement in question. Sigma subscript E equals the sum of sigma subscript WE times R subscript 1 times R subscript 2 times R subscript 3 times R subscript 4 times R subscript 5 and sigma subscript CE times R subscript T, and that sum multiplied by F subscript 1.
Equation 192. Westergaard ClosedForm Edge Stress Equation. Sigma subscript W equals the quotient of 3 times P times the sum of 1 plus mu all over pi times H squared times the sum of 3 plus mu, and that quotient multiplied by the sum of the log of the quotient of E times H raised to the 3 divided by 100 times KA raised to the 4 plus 1.84 minus 4 divided by 3 times mu plus 1 minus mu divided by 2 plus 1.18 times A divided by L times the sum of 1 plus 2 times mu.
Equation 193. Westergaard/Bradbury Curling Stress Equation. Sigma subscript lowercase C equals the quotient of uppercase C times E times alpha times delta T all over 2.
Equation 194. Curling Stress Coefficient. C equals the sum of 1 minus the quotient of 2 times the cosine function of lamda times the hyperbolic cosine function of lamda all over the sine function of 2 times lamda times the hyperbolic sine function of 2 times lamda. The result of which is times the sum of the tangent function of lamda minus the hyperbolic tangent function of lamda.
Equation 195. Parameter lambda. Lamda equals the quotient of W divided by L times the product of L and the square root of 8.
Equation 196. Dimensionless Mechanistic Variables. Loading  only condition equals the function of the quotient of A over lowercase L, the quotient of uppercase L over lowercase L, the quotient of W over lowercase L, the quotient of S over lowercase L, the quotient of T over lowercase L, the quotient of D subscript O over lowercase L, the quotient of AGG over the product of K and lowercase L, all times the squared product of H subscript EFF divided by H.
Equation 197. Dimensionless Mechanistic Variables. Loading plus Thermal Curling equals the function of the quotient of A over lowercase L, the quotient of L over lowercase L, the quotient of W over lowercase L, alpha times delta T, the quotient of H squared times gamma over the product of lowercase KL squared, and P times H over lowercase KL raised to the 4.
Equation 198. Effective Thickness of Two Unbonded Layers. H subscript EFF equals the square root of the sum of H subscript 1 squared and the quotient of H subscript 2 squared times E subscript 2 times H subscript 2 over E subscript 1 times H subscript 1.
Equation 199. PSI of Rigid Pavements. PSI equals 5.41 minus the product of 1.71 and the logarithm of the sum of 1 plus SV bar minus 0.09 times the square root of the sum of C plus P.
Equation 200. PSI Regression Model. PSI equals 4.536 minus 0.0182 times TFAULT minus 0.00313 times SPALL minus 0.00162 times TCRKS minus 0.00317 times FDR.
Equation 201. The Model Incorporated in HIPERPAV II. PSI equals 4.536 minus 0.0182 times TFAULT minus 0.00313 times SPALL minus 0.00162 times TCRKS
Equation 202. A Function of the PSI. IRI in meters per kilometer equals 1.0602 times the differences of 5 minus PSI.
Equation 203. IRI. IRI equals a function of PSI, Joint Spacing, Season, and Location.
Equation 204. IRI with Initial Roughness. IRI equals IRI subscript O plus delta IRI.
Equation 205. IRI Related to Faulting. IRI in meters per kilometer equals 0.0098 times TFAULT plus 1.268
Equation 206. IRI Model Based on Pavement Distress. IRI equals IRI subscript 0 plus 0.013 times percent CRACKED plus 0.007 times percent SPALL plus 0.001 times TFAULT plus 0.03 times SITE.
Equation 207. Site Factor. SITE equals the product of AGE times the sum of 1 plus FI raised to the 1.5 times the sum of 1 plus P subscript 0.075 times 10 raised to the negative 6.
Equation 208. IRI model presented at the 2000 TRB session for the 137A Guide for Mechanistic Design of Pavements. IRI equals IRI subscript 0 plus 0.0337 times TRANSCK plus 0.0083 times JT SPALL plus 0.0997 times PATCH plus 0.062 times CORBREAK plus 0.01255 times AGE plus 0.001547 times TOTFAULT.
Equation 209. Pavement Smoothness. S as a function of T equals S subscript 0 plus A subscript 1 times D as a function of T subscript 1 plus and so forth...plus A subscript N times D as a function of T subscript N plus B times SF plus C subscript J times M subscript J.
Equation 210. The Model Regression Coefficients. IRI equals IRI subscript I plus 0.0137 times CRK plus 0.07 times SPALL plus 0.005 times PATCH plus 0.0015 times TFAULT plus 0.04 times SF.
Equation 211. Site Factor. SITE equals the products of AGE superscript asterisk multiplied by 10 raised to the negative 6 and multiplied by two terms; the first term is sum of 1 and FI raised to the 1.5 power and the second term is the sum of 1 and P subscript 0.075.
Equation 212. CRCP stress equilibrium relationship. Sum of sigma subscript SX plus the quotient of sigma subscript CX divided by P subscript S minus sigma subscript SC minus the quotient of the integral from X to L of F subscript I with respect to X all over P subscript S times D equals 0.
Equation 213. Stresses in the Concrete at any Location. Sigma subscript CX equals the quotient of sigma subscript SX divided by N plus E subscript C times the sum of epsilon subscript SH and delta T times the difference of alpha subscript C minus alpha subscript S.
Equation 214. Steel Stress at the Crack. Sigma subscript SC equals the sum of 1 and 1 over N times P subscript S times sigma subscript SX plus E subscript C divided by P subscript S, and that quotient times the sum of alpha subscript C minus alpha subscript S times delta T plus epsilon subscript SH, minus the quotient of the integral from X to L of F subscript I with respect to X over P subscript S times D.
Equation 215. The Rate of Steel Stress as a Function of the Bond Stress. The derivative of sigma subscript SX with respect to X equals the quotient of negative F subscript I divided by the product of D and the difference of P subscript S minus 1 over N.
Equation 216. The Rate of Concrete Stress as a Function of the Bond Stress. The derivative of sigma subscript CX with respect to X equals the quotient of negative F subscript I divided by the product of N times D times the sum of P subscript S minus 1 over N.
Equation 217. Derivative of Stress in the Steel at any Location. The derivative of sigma subscript SX with respect to X equals the quotient of 4 times F subscript B as a function of X all over phi.
Equation 218. Derivative of Stress in the Concrete at any Location. The derivative of sigma subscript CX with respect to X equals the difference of the quotient of negative 4 times lowercase F subscript B as a function of X times P subscript S all over phi minus the quotient of uppercase F subscript I divided by D.
Equation 219. The Steel Stress for a slab Length. The integral from 0 to lowercase X of sigma subscript S lowercase X with respect to lowercase X equals the product of E subscript S and alpha subscript S and delta T and uppercase X subscript BAR.
Equation 220. Bond Stress From the Location Where Bond Slippage Begins. F subscript B as a function of S equals the sum of K times W as a function of S plus C times S squared plus D plus E times the cosine function of pi times S over 2 times B.
Equation 221. Bond Slip at Location S. W as a function of S equals the sum of A times the exponential function of CS over lowercase B plus uppercase B times the exponential function of negative CS over lowercase B minus 1 over K times the sum of three terms. The first is C times S squared. The second is 2 times C over alpha times K, and the third is D. All this is minus alpha times E divided by alpha times K plus pi squared over 4 times B squared times the cosine function of pi times S over 2 times lowercase B minus beta times the sum of S plus A divided by alpha uppercase K.
Equation 222. Double integral of bond stress. The quotient of negative 4 divided by phi times the double integral from negative A to B and from X to B of F subscript B as a function of S with respect to S and X equals sigma subscript SC times L minus E subscript S times alpha subscript S times delta T times L.
Equation 223. Parameter B. B equals the quotient of phi divided by the product of 4 times 266, and that quotient times the sum of the product of sigma subscript SC times open parentheses the difference of 1 minus 1 over C subscript 1 close parentheses plus open parentheses the quotient of C subscript 2 over C subscript 1 minus the quotient of the integral from negative A to B of F subscript I with respect to X divided by the product of P subscript S times D times C subscript 1, and that quotient plus the quotient of the integral from negative A to 0 of F subscript I with respect to X all over D times the sum of P subscript S plus 1 divided by N.
Equation 224. Parameter C subscript 1. C subscript 1 equals 1 plus the quotient of 1 divided by N times P subscript S.
Equation 225. Parameter C subscript 2. Uppercase C subscript 2 equals the quotient of E subscript lowercase C divided by P subscript S, and that quotient times the sum of epsilon subscript SH and delta T times the difference of alpha subscript C minus alpha subscript S.
Equation 226. "Facecentered design" methodology. NB equals the sum of 2 raised to the K plus 2 times K plus X equals FP plus AP plus CP.
Equation 227. Relates Compressive Strength Directly to WatertoCement Ratio. F subscript lowercase C equals A subscript A over B subscript A raised to the W divided by lowercase C all plus uppercase C subscript A.
Equation 228. Compressive Strength at 28 Days. F subscript C for 28 day in kips per square inch equals the quotient of 27.122 divided by 23.07 raised to the W divided by C.
Equation 229. Compressive Strength at 28 Days that Accounts Directly for Cement Content. F subscript C for 28 day in kips per square inch equals the quotient of 51290 divided by 23.66 raised to the following: W divided by C plus 0.000378 times C.
Equation 230. The Relative Change in Strength. F subscript REL equals the quotient of F divided by F subscript 1 which equals the product the quotient A subscript A divided by A subscript A subscript 1 times the quotient of B subscript A subscript 1 raised to the W divided by C subscript one all over B subscript A raised to the W divided by C.
Equation 231. Concrete Strength Related to W over C. F subscript REL equals the quotient of B subscript A subscript 1 raised to the W over C subscript 1all over B subscript A raised to the W over C, which equals RATIO.
Equation 232. Cement Replaced by Fly Ash. B subscript A equals the sum of the product of 5.5 times 6.3 raised to the P over C, and 6.
Equation 233. 28day Compressive Strength Related to its EarlyAge Values for Type I Cement. B subscript A equals the sum of 46 divided by T, and 6.
Equation 234. 28day Compressive Strength Related to its EarlyAge Values for Type III Cement. B subscript A equals the sum of 27 divided by T, and 6.
Equation 235. The Effect of Aggregate Ratio. Ratio subscript Aggregate equals the quotient of FA divided by CA all raised to the 0.17.
Equation 236. Combining the effects for Compressive Strength. F prime subscript C equals the product of Ratio subscript Age times Ratio subscript Pozzolan times Ratio subscript Aggregate times F subscript C.
Equation 237. The Radius of Relative Stiffness. L equals the quotient of E subscript C times H raised to the 3 all over 12 times the squared difference of 1 minus V subscript C, all times K, with the entire quotient raised to the 0.25
Equation 238. The Dowel Moment of Inertia. I subscript uppercase D equals the product of 1 divided by 64 times pi times lowercase D subscript uppercase D raised to the 4.
Equation 239. The Relative Stiffness of the DowelConcrete System. Beta equals the 4^{th} root of the quotient of K subscript D times lowercase D subscript uppercase D all over the product of 4 times E subscript uppercase D times I subscript uppercase D.
Equation 240. The Angles of the Dowels. The tangent function of alpha minus alpha prime equals the derivative of Y subscript O with respect to X.
Equation 241. The Deflection of the Dowel at the Slab Face. y subscript O for X equals 0 equal the product of the quotient of negative M subscript O times the exponential function of negative beta times X all over 2 times beta squared times E subscript D times I subscript D with that quotient times the difference of the cosine function of beta times X minus the sine function of beta times X all for X equals 0 which equals the quotient of negative M subscript O all over 2 times beta squared times E subscript D times I subscript D.
Equation 242. The Slope of the Dowel without Concrete Compliance. Z equals negative Z subscript O times the quotient of 2 times the cosine function of lamda times the hyperbolic cosine function of lamda all over the sine function of 2 times lamda minus the hyperbolic sine of 2 times lamda all times the sum of negative tangent function of lamda and the hyperbolic tangent of lamda times the cosine function of V times the hyperbolic cosine function of V plus the sum of the tangent function of lamda plus the hyperbolic tangent function of lamda times the sine function of V times the hyperbolic sine function of V.
Equation 243. Parameter lambda. Lambda equals uppercase L divided by the product of lowercase L times the square root of 8.
Equation 244. Parameter Z subscript 0. Z subscript 0 equals the product of the sum of 1 plus V subscript C times alpha subscript C times the gradient of T times L squared.
Equation 245. Parameter V. V equals the quotient of X subscript COORD divided by the product of L times the square root of 2
Equation 246. Parameter X subscript COORD. X subscript COORD equals negative L divided by 2.
Equation 247. The Angle of the Dowel. Alpha equals negative 1 divided by the product of L times the square root of 2, all times Z subscript O times the quotient of 2 times the cosine function of lamda times the hyperbolic cosine function of lamda all over the difference of the sine function of 2 times lamda minus the hyperbolic sine function of 2 times lamda times the sum of the sum of the negative tangent function of lamda plus the hyperbolic tangent of lamda times the negative sine function of V times the hyperbolic cosine function of V plus the cosine function of V times the hyperbolic sine function of V, all plus the sum of the tangent function of lamda plus the hyperbolic tangent function of lamda times the cosine function of V times the hyperbolic sine function of V plus the sine function of V times the hyperbolic cosine function of V.
Equation 248. Angle of the Deflected Dowel Shape. Alpha prime equals the quotient of M subscript O times W all over 2 times E subscript D times I subscript D.
Equation 249. The Moment Causing the Dowel to Deflect. M subscript O equals the product of 2 times E subscript D times I subscript D times alpha all over the sum of 2 divided by beta plus W.
Equation 250. The Resulting Dowel Bar Bearing Stress. F subscript B equals the absolute value of Y subscript O times K subscript D.
Equation 251. Dowel Load. P subscript dowel is equal to C multiplied by the difference of Y subscript 0 minus 2Y subscript 1.
Equation 252. Slab load. P subscript slab is equal to 2 multiplied by the quotient of K all over 2 lambda subscript B times Y subscript 1 which is equal to the quotient of K divided by lambda subscript B multiplied by Y subscript 1 which is equal to K times L times the square root of 2 times Y subscript 1, since lambda subscript B is equal to the quotient of 1 divided by L times the square root of 2.
Equation 253. Slab load. P subscript slab is equal to 2 multiplied by the quotient of KL divided by the square root of 2 multiplied by Y subscript 1 which is equal to KL times the square root of 2 times Y subscript 1.
Equation 254. Load equivalence. P subscript dowel is equal to P subscript slab.
Equation 255. Deflection. Y subscript 1 is equal to the quotient of C times Y subscript 0 all over KL times the square root of 2 plus 2C.
Equation 256. Dowel load. P subscript dowel is equal to C times the difference of Y subscript 0 minus 2 times Y subscript 1 multiplied by Dowel Spacing.
Equation 257. Dowel deflection at the joint face. Delta subscript face is equal to P subscript dowel times the quotient of 2 plus beta times W all over 4 times beta to the third power times EI.
Equation 258. Dowel bearing stress. Gamma subscript B as a function of bearing stress is equal to delta subscript face times K subscript dowel.
Equation 259. ACI Correlations Between Compressive Strength and Concrete Modulus of Elasticity, ACI 318 Code(1). E equals the product of 4,700 times F subscript C raised to the 0.5 times megapascals.
Equation 260. ACI Correlations Between Compressive Strength and Concrete Modulus of Elasticity, ACI 318 Code(2). E equals the product of W subscript C raised to the 1.5 times 0.043 times F subscript C raised to the 0.5 times megapascals when W subscript C equals 2340 kilograms per cubic meter.
Equation 261. The Early Age Compressive Strength. CS equals the quotient of 0.864 times CNT PT FLEX STR all over 2.3 with that quotient raised to the 1.5.
Equation 262. The LTE Across Joints. LTE equals the quotient of 2 times W subscript U all over the sum of W subscript U plus W subscript I, all times 100.
Equation 263. Dowel Looseness after Construction. DL subscript 0 equals the sum of 0.0016 plus the quotient of .0032 all over the sum of 1 plus the quotient of phi divided by 20.32 with that quotient raised to the 6.4.
Equation 264. Dowel Looseness after N Cumulative Loading. DL subscript N equals DL subscript 0 plus the product of 0.00019 times ESALS raised to the 0.185.