U.S. Department of Transportation
Federal Highway Administration
1200 New Jersey Avenue, SE
Washington, DC 20590
2023664000
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:
FHWARD03089

Figure 1. Diagram. Parameter identification system input and output options.
The flowchart shows typical excitations and outputs of a parameter identification system. The inputs listed on the left side are either controlled (impact loads, shakers, known truck loads, or free decay) or operating (earthquakes or unknown loads). Combined within the parameter identification system (explained in figure 2), these inputs produce outputs of either structural model estimates expressed as omega (frequency), phi (phase), or xi (damping) or parameter estimates (structural element stiffness).
Figure 2. Diagram. Structural parameteridentification system tree.
The structural parameter identification system classifies problems with or without mathematical models. The left side of the tree shows the mathematical models subdivided into static and dynamic models. Static models include displacement and rotational or strain measurements while dynamic models have time or frequency domains. On the right side of the tree, structural parameter identification without mathematical models subdivides into neural network, signal processing, pattern recognition, and expert systems.
Figure 3. Photo. Relief structure 4 of the Trinity River Bridge with the Vibroseis truck over bent 2.
This image shows relief structure 4 of the Trinity River Relief Bridge in the middle of the picture. On top of the bridge, at bent 2, is a Vibroseis truck, which is a large truck with a servohydraulic vibrator mounted on top.
Figure 4. Photo. Excessive differential settlements on relief structure 4, Trinity River Relief Bridge.
This image is a closeup of relief structure 4 of the Trinity River Relief Bridge shown from an upper angle where the excessive differential settlements of the flat concrete slab are evident.
Figure 5. Diagram. Geometric details of bent 2, structure 4, of the Trinity River Relief Bridge.
This cross section shows the elements and dimensions of bent 2 as detailed in the text, except that the concrete footing is not shown and the excavation of the piles described in chapter 4 is shown. The top deck is 7.68 meters (25 feet 2 inches) wide. The asphalt curb is 0.191 meters (7.5 inches) high. The concrete slab is 0.39 meters (1 foot 3.5 inches) high. The beam cap is 7.17 meters (23 feet 6 inches) long and 0.71 meters (2 feet 5 inches) high. The four piles are 0.35 meters (1 foot 2 inches) in diameter and 1.93 meters (6 feet 4 inches) apart. The drawing also shows the excitation point in the middle of the concrete slab, nodes 13 (second column) and 15 (broken fourth column), and the 1.83 meters (6 feet) of exposed column with the 2.75meter (9foot) excavation between columns 3 and 4.
Figure 6. Diagram. SASW method.
This drawing shows the setup of the Spectral Analysis of Surface Waves (SASW) method used to measure the seismic properties of the subsurface soils of all three bridges. The drawing has three components. From left to right they are the source, receiver R1, and receiver R2. Receivers R1 and R2 are Freedom NDT PCs from Olsen Instruments. The distance, which is divided into two segments, is labeled from left to right as D/2 and D. The D/2 segment from the source to the first receiver R1 is about a third of the distance. The D segment from the first receiver R1 to the second receiver R2, is about twothirds of the distance.
Figure 7. Photo. SASW tests at bent 12, structure 4, Trinity River Relief Bridge.
This image shows tests conducted at bent 12 of relief structure 4 of the Trinity River Relief Bridge using the spectral analysis of surface wave method. It shows a worker digging beneath the bridge, with wires hanging from the bridge.
Figure 8. Photo. Western pier of the southwestern span of the Woodville Road Bridge.
This image shows the western pier of the southwestern span of the Woodville Road Bridge in Bryan, Texas. The photo was taken from the grassy shoulder of Texas State Highway 6 looking forward and slightly up at the bridge. The photo shows three of the four prestressed concrete I girders resting directly on the beam caps through polystyrene bearing pads. A truck used for dynamic testing is parked under the bridge.
Figure 9. Photo. Vibroseis truck over the east pier, Old Reliance Bridge.
This image is taken from the northeast at eye level with the Old Reliance Bridge. A Vibroseis truck is parked on the bridge at about the onethird mark. Also parked near the Vibroseis truck are a dark minivan and a white pickup truck. The bottom half of the picture shows a green landscape.
Figure 10. Photo. Vibroseis truck over Woodville Road Bridge pier.
This image shows the Vibroseis truck parked on the bridge. The large truck has a servohydraulic vibrator mounted on its load bed. The photo is taken from the street that passes under the bridge at a sightline slightly below bridge level.
Figure 11. Photo. Vertical vibrator mechanism of Vibroseis truck.
This image shows the vertical vibrator located directly on the pavement behind the truck. The vibrator is separated from the Vibroseis truck through a system of air springs.
Figure 12. Photo. Seismic accelerometer and power supply unit.
This image shows the PCB Piezotronics Model 393C seismic accelerometer that contains a builtin microelectronic amplifier that converts the high impedance voltage signals from the quartz crystals to a low impedance voltage. The accelerometer is connected to a power unit by a coaxial cable.
Figure 13. Photo. DP420 Dynamic Signal Analyzer in portable PC.
This image shows the type of portable field computer used as the data acquisition system for the project. Test data were recorded and saved in the field on the hard drive of the computer, which was linked with the vibrator and seismic accelerometers by lead cables.
Figure 14. Photo. Drilling concrete holes to mount seismic accelerometers on bent 2 of structure 4, Trinity River Relief Bridge.
This image taken under the bridge shows a worker drilling small holes at bent 2 of relief structure 4 of the Trinity River Relief Bridge. The holes are for concrete anchors where seismic accelerometers are to be attached.
Figure 15. Photo. Seismic accelerometers on blocks bolted to piles of bent 2 of structure 4, Trinity River Relief Bridge.
This image taken beneath the bridge shows seismic accelerometers attached to aluminum triaxial mounting blocks that are bolted to piles of bent 2 of relief structure 4 of the Trinity River Relief Bridge.
Figure 16. Photo. Onehundredpound vibrator with dynamic load cell on bent 2.
This image, taken at slightly above pavement level, shows a 100pound vibrator with dynamic load cell on bent 2. The 100pound vibrator was one of several sources of dynamic force tested to find a source that produced the frequency range of interest, which was 3 to 100 hertz.
Figure 17. Photo. Twelvepound impulse sledgehammer with soft gray rubber tip.
This image shows the hand of a worker holding a 12pound impulse sledgehammer with a soft gray rubber tip for low frequency excitation. The 12pound hammer was one of several sources of dynamic force tested to find a source that produced the frequency range of interest, which was 3 to 100 hertz.
Figure 18. Photo. Closeup of the vibrator frame of the Vibroseis truck.
This image shows a closeup view of the Vibroseis truck vibrator frame above the pavement.
Figure 19. Photo. Dualwheel depressions in asphalt overlay on deck over bent 2.
This image shows depressions of the Vibroseis truck’s dual wheels in the asphalt overlay of the deck over bent 2.
Figure 20. Diagram. Loading points and accelerometer receiver locations on bents 2 and 12, Trinity River Relief Bridge.
This schematic shows the bent 2 and bent 12 loading and receiver locations with seven nodes, numbered 1 through 7, running horizontally on the top deck. A down arrow points to a midpoint loading at node 4. Each of the top deck uneven numbered nodes has a vertical line running downward to two additional nodes. The nodes connected by vertical lines are: 1, 8, and 12; 3, 9, and 13; 5, 10, and 14; and 7, 11, and 15.
Figure 21. Photo. Vibroseis singlepointplate loading system.
This image taken from just above pavement level shows the lower portion of the Vibroseis truck with a singlepoint plate in contact with the pavement. A round steel plate with dimensions of 0.3meter (1foot) diameter and 25.4millimeter (1inch) thickness is placed between the bridge deck and a Beowulf load cell.
Figure 22. Photo. Demolition of structure 4, Trinity River Relief Bridge.
This image, taken from the bridge roadway, shows a fullsize tracked excavator machine with crane arm demolishing relief structure 4 of the Trinity River Relief Bridge in September 1997.
Figure 23. Photo. Backhoe excavation at bent 12, Trinity River Relief Bridge.
This image shows a backhoe digging a 3meter (10foot)deep excavation around the most southern piles of bent 12 of the Trinity River Relief Bridge.
Figure 24. Photo. Ground water in excavation of bent 2, Trinity River Relief Bridge.
This image shows ground water just below the strip footing in the excavation of the south and centersouth piles of bent 2 of the Trinity River Relief Bridge.
Figure 25. Photo. Pile of bent 2 just after excavation.
This image shows a pile of bent 2 of the Trinity River Relief Bridge just after excavation, with the upper 2.7 meters (9 feet) around the pile exposed on three sides. Soil remains against the fourth side.
Figure 26. Photo. Piles of bent 12 just after excavation.
This image shows the south and centersouth piles of bent 12 of the Trinity River Relief Bridge just after excavation. The upper 2.7 meters (9 feet) of the south pile fully exposed and the upper 9 feet (2.7 meters) of the centersouth pile are exposed on the south, east, and west sides.
Figure 27. Photo. Shearing of pile of bent 2.
This image shows the south pile/column of bent 2 of the Trinity River Relief Bridge being sheared and broken using a vibrating breaker point on a backhoe. The water level in the excavation is near the bottom of the strip footing.
Figure 28. Photo. Shearing of pile of bent 12.
This image shows a cloud of dust as the south pile/column of bent 12 of the Trinity River Relief Bridge is being sheared and broken by a vibrating breaker point on a backhoe.
Figure 29. Photo. Sheared south pile of bent 2 with ground water seepage.
Figure 29. Photo. Sheared south pile of bent 2 with ground water seepage.
Figure 30. Photo. Bent and broken rebars of south column of bent 2.
This image shows a closeup of the sheared south pile/column of bent 2, Trinity River Relief Bridge, with one of the four exposed number 8 steel rebars completely broken and the others bent. Ground water is visible under the strip footing.
Figure 31. Photo. Sheared south pile of bent 12.
This image shows the sheared south pile/column of bent 12, Trinity River Relief Bridge, at about 2.4 meters (8 feet) below the pile beam cap. Four number 8 steel rebars are exposed for a distance of 0.3 meters (1 foot).
Figure 32. Photo. Rebars of south column of bent 12.
This image shows a closeup of the sheared south pile/column of bent 12 at about 2.4 meters (8 feet) below the pile beam cap. Approximately 0.3 meters (1 foot) of concrete has been removed, exposing the number 8 steel rebars, none of which are broken.
Figure 33. Photo. Vibroseis truck over west pier of Woodville Road Bridge.
This image shows a Vibroseis truck parked over a bent in the eastbound lane on the deck of the Woodville Road Bridge.
Figure 34. Diagram. Woodville Road Bridge, loading points and accelerometer receiver locations.
This cross section of the Woodville Road Bridge shows accelerometer receiver positions 1 through 8 running horizontally on the beam cap. Positions 9, 11, 13, and 15 run vertically from top to bottom on the left pier. Positions 10, 12, 14, and 16 run vertically from top to bottom on the right pier.
Figure 35. Photo. Accelerometers on column and beam cap of west pier, Woodville Road Bridge.
This image shows the south column and beam cap of the west pier of the Woodville Road Bridge. A ladder reaches from the ground to the beam cap. The ladder was used to place a seismic accelerometer receiver on the beam cap. Another accelerometer is located at the base of the column.
Figure 36. Photo. Biaxial accelerometer mount with vertical and horizontal accelerometers, west pier, Woodville Road Bridge.
This image shows a biaxial accelerometer mount with vertical and horizontal seismic accelerometers installed on the beam cap of the west pier, Woodville Road Bridge. A ladder is leaning against the beam cap next to the mount.
Figure 37. Photo. Vertical accelerometer at base of column of west pier, Woodville Road Bridge.
This image shows a vertical seismic accelerometer installed at the base of a column of the west pier of the Woodville Road Bridge.
Figure 38. Photo. Horizontal accelerometer on side of column of west pier, Woodville Road Bridge.
This image shows a horizontal seismic accelerometer installed on the side of a column of the west pier of the Woodville Road Bridge.
Figure 39. Diagram. Old Reliance Road Bridge, loading points and accelerometer receiver locations.
This cross section of the Old Reliance Road Bridge shows accelerometer receiver positions 1 through 8 running horizontally on the beam cap. Positions 9, 11, 13, and 15 run vertically from top to bottom on the left pier. Positions 10, 12, 14, and 16 run vertically from top to bottom on the right pier.
Figure 40. Diagram. Node locations on bent 2.
This diagram is a cross section of bent 2 of the Trinity River Relief Bridge. It shows 27 node locations for the vibration measurements and analyses. The nodes are located on the four vertical piles and the horizontal beam on top of the piles. The nodes on the beam are numbered, from left to right, 1 through 11. The piles are below nodes 3, 5, 7, and 9. The distance between nodes 3 and 5 is 1.98 meters (6.5 feet). The distances between nodes 5 and 7 and between nodes 7 and 9 are also 1.98 meters (6.5 feet). Nodes 6, 7, and 8 are source and receiver locations. The distance between nodes 1 and 3 is 0.70 meters (2.3 feet). Nodes 24, 25, 26, and 27, receiver locations, are placed 0.31 meters (2 feet) below nodes 3, 5, 7, and 9. Nodes 12, 13, 14, and 15 are 1.53 meters (5 feet) below, respectively, nodes 3, 5, 7, and 9. Nodes 16, 17, 18, and 19 are at ground level on top of a concrete footing. The four nodes are 2.14 meters (7 feet) below, respectively, nodes 3, 5, 7, and 9. Nodes 20, 21, 22, and 23 are 4.58 meters (15 feet) below ground level and, respectively, nodes 16, 17, 18, and 19.
Figure 41. Diagram. Node locations on bent 12.
This diagram is a cross section of bent 12 of the Trinity River Relief Bridge. It shows 27 node locations for the vibration measurements and analyses. The nodes are located on the four vertical piles and the horizontal beam on top of the piles. The nodes on the beam are numbered, from left to right, 1 through 11. The piles are below nodes 3, 5, 7, and 9. The distance between nodes 3 and 5 is 1.98 meters (6.5 feet). The distances between nodes 5 and 7 and between nodes 7 and 9 are also 1.98 meters (6.5 feet). Nodes 6, 7, and 8 are source and receiver locations. The distance between nodes 1 and 3 is 0.70 meters (2.3 feet). Nodes 3, 5, 7, and 9, all receiver locations, are placed 0.31 meters (2 feet) respectively above nodes 24, 25, 26, and 27. Nodes 12, 13, 14, and 15 are 1.53 meters (5 feet) below, respectively, nodes 3, 5, 7, and 9. Nodes 16, 17, 18, and 19 are at ground level. The four nodes are 2.14 meters (7 feet) below, respectively, nodes 3, 5, 7, and 9. Nodes 20, 21, 22, and 23 are 4.58 meters (15 feet) below ground level and, respectively, nodes 16, 17, 18, and 19.
Figure 42. Graph. Vibroseis at node 8 of bent 2.
The graph is a screen dump—a captured computer screen—from a computer program. The graph plots the time domain data of the TR210 vibrator force at node 8 of bent 2. The vertical axis is in poundforce ranging from negative 16K to 16K. The horizontal axis is time in seconds ranging from 1 to 6. Peaktopeak force equals 123K newtons (27K poundforce) at about 0.5 seconds with a second peak loading force of about 80K newtons (18K poundforce) at 2.5 seconds. The solid infill of the chart starting at about 2 seconds reflects the increase excitation from 3 hertz at the start of the chirp pulse to 80 hertz.
Figure 43. Graph. Vertical accelerometer at node 27 of bent 2.
The graph is a screen dump—a captured computer screen—from a computer program. The graph plots the calibrated acceleration responses of the vertical seismic accelerometer at node 27 (TR211 receiver). The vertical axis is inches per second per second (inches per second squared) ranging from negative 100 to 100. The horizontal axis is time in seconds ranging from 1 to 6. This chart shows the peaktopeak greatest acceleration of about 4.24 meters (167 inches) per second per second at 0.5 and 2.5 seconds because the accelerometer was located directly below the excitation force at node 8.
Figure 44. Graph. Horizontal accelerometer at node 27 of bent 2.
The graph is a screen dump—a captured computer screen—from a computer program. The graph plots the calibrated acceleration responses of the horizontal seismic accelerometer at node 27 (TR212 receiver). The vertical axis is inches per second per second (inches per second squared) ranging from negative 100 to 100. The horizontal axis is time in seconds ranging from 1 to 6. This chart shows the peaktopeak greatest acceleration of about 0.89 meters (35 inches) per second per second at approximately 0.7 seconds and about 1.49 meters (57 inches) per second per second at approximately 2.75 seconds, which approximately correspond to the times of the greatest excitation force.
Figure 45. Graph. Vertical accelerometer at node 24 of bent 2.
The graph is a screen dump—a captured computer screen—from a computer program. The graph plots the calibrated acceleration responses of the vertical seismic accelerometer at node 24 (TR213 receiver). The vertical axis is inches per second per second (inches per second squared) ranging from negative 100 to 100. The horizontal axis is time in seconds ranging from 1 to 6. This chart shows the peaktopeak greatest acceleration of about 0.94 meter (37 inches) per second per second at approximately 0.7 seconds and 3 seconds, which approximately correspond to the times of the greatest excitation force.
Figure 46. Graphs. Average spectra of figures 42 through 45 for bent 2.
The graphs are screen dumps—captured computer screens—from a computer program. Four graphs plot the average spectra for figures 42 through 45. The horizontal scale is frequency in hertz that ranges from 0 to 55. The G11 spectrum (2TR202) corresponds to the node 8 Vibroseis in poundforce ranging from 0 to 700 (0 to 3,115 newtons) on the vertical scale. The maximum of 606.299 (2,698 newtons) occurs at 15 hertz and levels off at about 300 (1,335 newtons) from 20 to 50 hertz. G22 (2TR202) shows a vertical scale range up to 50.8 millimeters (2 inches) per second per second (inches per second squared) for node 27V. Between 15 hertz to 45 hertz, the range varies from 25.4 to 114.3 millimeters (1 to 1.5 inches) with a steep increase between 46 hertz to 55 hertz, where the maximum of 47.02 millimeters (1.851 inches) per second per second occurs. G33 (2TR203) shows a vertical scale range of 25.4 to 35.56 millimeters (1 to 1.4 inches) per second per second for node 27H. The frequency climbs steadily from 0 hertz to 45 hertz, where the maximum is 29.41 millimeters (1.158 inches), then it drops sharply to 5.08 millimeters (0.2 inch) at 51 hertz. G44 (2TR204) shows a vertical range of 0 to 22.86 millimeters (0 to 0.9 inches) per second per second for node 24V. Except for dips to 0.05 inch (1.27 millimeters) at 18 hertz, 0.15 inch (3.81 millimeters) at 27 hertz, and 0.3 inch (7.62 millimeters) at 48 hertz, the average climbs steadily to its maximum of 0.708918 inch (18.01 millimeters) at 55 hertz.
Figure 47. Equation. Accelerance TF.
The input force times the acceleration divided by the input force equals the acceleration, where the acceleration divided by the input force is the accelerance transfer function (TF).
Figure 48. Equation. Mobility TF.
The input force times the velocity divided by the input force equals the velocity, where the velocity divided by the input force is the mobility transfer function.
Figure 49. Equation. Flexibility TF.
The input force times the displacement divided by the input force equals the displacement, where the displacement divided by the input force is the flexibility transfer function.
Figure 50. Graphs. Accelerance TF for figures 42 through 45 for bent 2.
The graphs are screen dumps—captured computer screens—from a computer program. Three graphs show the accelerance transfer functions for data in figures 42 through 45. Each graph has a frequency range of 0 to 55 hertz on the horizontal scale. The vertical scales are labeled inches per second squared pounds, which is not a standard measurement; it is simply the combination of the units of measurement in the transfer functions. For nodes 27V/8V (H12: TR225), the vertical scale ranges from 0 to 0.014. The plot is fairly steady at about 0.006 for 15 to 40 hertz, then rises to a maximum of 0.011520 inches per second squared pounds. For nodes 27H/8V (H13: TR226) the vertical scale ranges from 0 to 0.004. The plot climbs steadily to a maximum of 0.003671 at 55 hertz. For nodes 24V/8V (H14: TR227) the vertical scale ranges from 0 to 0.003. The plot is erratic with peaks of about 0.0015 at 17 hertz, 32 hertz, and 45 hertz with the maximum at 0.002578 at 55 hertz. To convert accelerance transfer functions to metric units, inches per second squared pounds can be multiplied by 0.005708 to give meters per second squared newtons.
Figure 51. Graphs. Flexibility TFs with coherences for test configuration identical to bent 2.
The graphs are screen dumps—captured computer screens—from a computer program. Six graphs show the flexibility (top three) and coherence (bottom three) for the accelerance plots in figure 49, which in turn are based on the data of figures 44 to 48. The horizontal scales are frequency in hertz ranging from 0 to 55. The vertical coherence scale ranges from 0 to 1. Most of the coherence data exceeded 0.9, which means the input was not contaminated with noise. The vertical scales for the flexibility transfer functions range from 0 to 800 nanoinches per poundforce. For nodes 27V/8V the maximum of 623.354 nanoinches per poundforce occurs at approximately 16 hertz. For nodes 27H/8V the maximum of 214.861 nanoinches per poundforce occurs at about 15 hertz. For nodes 24V/8V the maximum of 141.114 nanoinches per poundforce occurs at about 15 hertz. To convert flexibility transfer functions to metric units, inches per poundforce can be multiplied by 0.005708 to give meters per newtons.
Figure 52. Graphs. Bent 12 flexibility transfer functions with coherences for test configuration identical to bent 2.
The graphs are screen dumps—captured computer screens—from a computer program. Six graphs show the flexibility (top three) and coherence (bottom three) for data from nodes on bent 12. The nodes on bent 12 were in the same test configuration as the nodes on bent 2. The horizontal scales are frequency in hertz ranging from 0 to 55. The vertical coherence scale ranges from zero to one. Most of the coherence data exceeded 0.9, which means the input was not contaminated with noise. The vertical scales for the flexibility transfer functions range from 0 to 800 nanoinches per poundforce. For nodes 27V/8V, the maximum of 316.716 nanoinches per poundforce occurs at approximately 18 hertz. For nodes 27H/8V, the maximum of 441.495 nanoinches per poundforce appears to occur at zero hertz. For nodes 24V/8V the maximum of 96.949 nanoinches per poundforce occurs at about 4 hertz. To convert flexibility transfer functions to metric units, inches per poundforce can be multiplied by 0.005708 to give meters per newtons.
Figure 53. Graph. Curvefit for accelerance TF, bent 2, node 27V/node 8V.
The graph is a screen dump—a captured computer screen—from a computer program. The program is PCMODAL. The program used the data that produced the accelerance transfer function of the top graph of figure 50 to produce a curvefitted accelerance transfer function. The axes are not identified, but the horizontal axis, which ranges from 5.00 to 55.0, appears to be frequency in hertz, and the vertical axis, which ranges from 0.00 to 11.5, appears to be the accelerance transfer function in milliinches per second squared pounds. To convert accelerance transfer functions to metric units, inches per second squared pounds can be multiplied by 0.005708 to give meters per second squared newtons. The curve of the graph rises generally from left to right. To the right of the graph, but still part of the screen dump, is a table that apparently lists the frequency peaks and the associated damping ratios. The table has 18 rows but only the first two are filled. Row 1: frequency 16.24; damping 15.97 percent. Second row: frequency 27.698; damping 5.36 percent.
Figure 54. Graph. Curve fit for accelerance TF, bent 2, node 27H/node 8V.
The graph is a screen dump—a captured computer screen—from a computer program. The program is PCMODAL. The program used the data that produced the accelerance transfer function of the middle graph of figure 50 to produce a curvefitted accelerance transfer function. The axes are not identified, but the horizontal axis, which ranges from 5.00 to 55.0, appears to be frequency in hertz, and the vertical axis, which ranges from 0.00 to 3.76, appears to be the accelerance transfer function in milliinches per second squared pounds. To convert accelerance transfer functions to metric units, inches per second squared pounds can be multiplied by 0.005708 to give meters per second squared newtons. The curve of the graph rises generally from left to right. To the right of the graph, but still part of the screen dump, is a table that apparently lists the frequency peaks and the associated damping ratios. The table has 18 rows but only the first two are filled. First row: frequency, 16.744; damping, 17.44 percent. Second row: frequency 35.257; damping, 10.27 percent.
Figure 55. Graph. Curve fit for accelerance TF, bent 2, node 24V/node 8V.
The graph is a screen dump—a captured computer screen—from a computer program. The program is PCMODAL. The program used the data that produced the accelerance transfer function of the bottom graph of figure 49 to produce a curvefitted accelerance transfer function. The axes are not identified, but the horizontal axis, which ranges from 5.00 to 55.0, appears to be frequency in hertz, and the vertical axis, which ranges from 0.00 to 2.58, appears to be the accelerance transfer function in milliinches per second squared pounds. To convert accelerance transfer functions to metric units, inches per second squared pounds can be multiplied by 0.005708 to give meters per second squared newtons. The curve of the graph rises generally from left to right, but with significant peaks and valleys. To the right of the graph, but still part of the screen dump, is a table that apparently lists the frequency peaks and the associated damping ratios. The table has 18 rows but only the first two are filled. First row: frequency 15.134; damping, 8.28 percent. Second row: frequency, 28.263; damping, 7.13 percent.
Figure 56. Diagram. First mode shape, frequency, and damping for bent 2.
The diagram is a screen dump—a captured computer screen—from a computer program. The program is PCMODAL. On the left side of the screen dump are six unlabeled lines. Four of the lines are vertical and parallel to each other, and of roughly equal length. Each of the three rightmost lines is a bit higher on the screen than the line to its immediate left. The remaining two lines are parallel to each other and rise diagonally from left to right. The topmost line touches the top of each of the four vertical lines. The bottom line intersects the four vertical lines just above their midpoints. The right side of the screen dump contains five categories of data in roughly tabular form: model, zoom, move, animate, and rotate. Most of the data appear to be computer settings for the program. The two most important data items appear to be those labeled Freq and Damp. Freq is 1.5127E plus 01, and Damp is 1.65E minus 01.
Figure 57. Diagram. Node points and geometry of bent 2.
This cross section shows nodes 1 through 15 with 1 through 7 on the concrete slab having downloading arrows. The bent 2 components include the 7.68meter (25foot 2inch) top deck total dimension, the asphalt curb with a height of 0.19 meters (7.5 inches), the concrete slab with a height to the bottom of the lip of 0.49 meters (1 foot 7.5 inches), and the beam cap with a thickness of 0.73 meters (2 feet 5 inches). The drawing also shows the 2.14 meters (7 feet) of exposed column, the footing with a thickness of 0.61 meters (2 feet), and the 2.75meter (9foot) excavation between columns 3 and 4. The space between columns is 1.94 meters (6 feet 4 inches).
Figure 58. Diagram. Node points and geometry of bent 12.
This cross section shows nodes 1 through 15 with 1 through 7 on the concrete slab having downloading arrows. The excitation point is in the center of the slab. The bent 12 components include the 7.68 meters (25foot 2inch) wide top deck, the asphalt curb (0.19 meters in height or7.5 inches), the concrete slab, with a height to the bottom of the lip of 0.49 meters (1 foot 7.5 inches) and lip thickness of 0.20 meters (6.5 inches), and the beam cap with a lip width of 0.46 meters (1 foot 6 inches ) and lip height of 0.41 meters (1 foot 4 inches). The drawing also shows the 1.93 meters (6 feet) of exposed column with the 2.75meter (9foot) excavation between columns 3 and 4. The space between columns is 1.94 meters (6 feet 4 inches), and the width of a column is 0.35 meters (1 foot 2 inches).
Figure 59. Graph. Bent 12 flexibility TFs at node 2 for intact (I), excavated (E), and broken (N) piles.
This graph shows the flexibility transfer functions at node 2 of bent 12 for intact (I), excavated (E), and broken (N) piles. The horizontal axis is frequency and ranges from 10 to 30 hertz. The vertical axis shows flexibility (inches per poundforce) and ranges from 0.00E plus 00 to 8.00E minus 07. All three curves plot similar configurations with intersecting values, and the intact curve shows the lowest values. The values range between 3.00E minus 07 and 4.00E minus 07 at 10 hertz with a high of 5.00E minus 07 at 19 hertz and between 2.00E minus 07 and 3.00E minus 07 for the 30hertz endpoint. To convert flexibility transfer functions to metric units, inches per poundforce can be multiplied by 0.005708 to give meters per newtons.
Figure 60. Graph. Bent 12 flexibility TFs at node 4 for intact (I), excavated (E), and broken (N) piles.
This graph shows the flexibility transfer functions at node 4 of bent 12 for intact (I), excavated (E) and broken (N) piles. The horizontal axis is frequency and ranges from 10 to 30 hertz. The vertical axis shows flexibility (inches per poundforce) and ranges from 0.00E plus 00 to 8.00E minus 07. All three curves plot similar configurations with intersecting values, and the intact curve shows the lowest values. The values at 10 hertz range between 2.00E minus 07 and 4.00E minus 07. At the 30hertz endpoint, the values taper to 1.00E. To convert flexibility transfer functions to metric units, inches per poundforce can be multiplied by 0.005708 to give meters per newtons.
Figure 61. Graph. Bent 12 flexibility TFs at nodes 6, 10, and 11 for the intact pile.
This graph shows the flexibility transfer functions at nodes 6, 10, and 11 for the intact pile. The loading force is applied at node 6, and the responses are measured by vertical accelerometers at nodes 6, 10, and 11. The horizontal axis ranges in frequency from 10 to 30 hertz. The vertical axis shows flexibility (inches per poundforce) and ranges from 0.00E plus 00 to 8.00E minus 07. All three curves plot similar configurations with the node 10 curve showing the lowest values. The curves begin between 2.00E minus 07 and 4.00E minus 07 at 10 hertz and taper to between 1.00E minus 07 and 3.00E minus 07 at the 30hertz endpoint. To convert flexibility transfer functions to metric units, inches per poundforce can be multiplied by 0.005708 to give meters per newtons.
Figure 62. Graph. Bent 12 flexibility transfer functions for node 8/node 2 for intact (I), excavated (E), and broken (N) piles.
This graph shows the bent 12 flexibility transfer functions for the Vibroseis force at node 2 and the vertical accelerometer response reading at node 8 for intact (I), excavated (E), and broken (N) piles. The horizontal axis ranges in frequency from 10 to 30 hertz. The vertical axis shows flexibility (inches per poundforce) and ranges from 0.00E plus 00 to 8.00E minus 07. All three curves plot similar configurations with intersecting values and the intact curve shows the lowest values. The range of the curves is between 2.00E minus 07 and 5.00E minus 07 and tapers to about 3.00E minus 07 at the 30hertz endpoint. To convert flexibility transfer functions to metric units, inches per poundforce can be multiplied by 0.005708 to give meters per newtons.
Figure 63. Graph. Bent 12 flexibility TFs for node 9/node 4 for intact (I), excavated (E), and broken (N) piles.
This graph shows the bent 12 flexibility transfer functions for the Vibroseis force at node 4 and the vertical accelerometer response reading at node 9 for intact (I), excavated (E), and broken (N) piles. The horizontal axis ranges in frequency from 10 to 30 hertz. The vertical axis shows flexibility (inches per poundforce) and ranges from 0.00E plus 00 to 8.00E minus 07. All three curves plot similar configurations with intersecting values and the intact curve shows the lowest values. The range of the curves is between 1.00E minus 07 and 3.00E minus 07 and descends to about 1.00E minus 07 at the 30hertz endpoint. To convert flexibility transfer functions to metric units, inches per poundforce can be multiplied by 0.005708 to give meters per newtons.
Figure 64. Graph. Bent 12 flexibility TFs for node 10/node 4 for intact (I), excavated (E), and broken (N) piles.
This graph shows the bent 12 flexibility transfer functions for the Vibroseis force at node 4 and the vertical accelerometer response reading at node 10 for intact (I), excavated (E), and broken (N) piles. The horizontal axis ranges in frequency from 10 to 30 hertz. The vertical axis shows flexibility (inches per poundforce) and ranges from 0.00E plus 00 to 8.00E minus 07. All three curves plot similar configurations with some intersecting values and the intact curve shows the lowest values. The range of the curves is between 1.00E minus 07 and almost 4.00E minus 07 and descends to about 1.00E minus 07 at the 30hertz endpoint. To convert flexibility transfer functions to metric units, inches per poundforce can be multiplied by 0.005708 to give meters per newtons.
Figure 65. Graph. Bent 12 flexibility TFs for node 11/node 6 for intact (I) and excavated (E) piles.
This graph shows the bent 12 flexibility transfer functions for the Vibroseis force at node 6 and the vertical accelerometer response reading at node 11 for intact (I) and excavated (E) piles. The horizontal axis ranges in frequency from 10 to 30 hertz. The vertical axis shows flexibility (inches per poundforce) and ranges from 0.00E plus 00 to 8.00E minus 07. Both curves plot similar configurations with one intersecting value of 3.00E minus 07 at 23 hertz, and the intact curve shows the lowest values. The range of the curves is between 2.00E minus 07 to almost 6.00E minus 07 and descends to about 2.00E minus 07 at the 30hertz endpoint. To convert flexibility transfer functions to metric units, inches per poundforce can be multiplied by 0.005708 to give meters per newtons.
Figure 66. Graph. Bent 2 flexibility TFs at node 2 for intact (I), excavated (E), and broken (N) piles.
This graph shows the bent 2 flexibility transfer functions for the Vibroseis force and the vertical accelerometer response reading, both at node 2, for intact (I), excavated (E), and broken (N) piles. The horizontal axis ranges in frequency from 10 to 30 hertz. The vertical axis shows flexibility (inches per poundforce) and ranges from 0.00E plus 00 to 8.00E minus 07. All curves plot similar configurations with a range from about 2.00E minus 07 to more than 6.00E minus 07 (the intact pile peak value at 14 hertz). The values are about 2.00E minus 07 at the 30hertz endpoint. To convert flexibility transfer functions to metric units, inches per poundforce can be multiplied by 0.005708 to give meters per newtons.
Figure 67. Graph. Bent 2 flexibility TFs at node 4 for intact (I), excavated (E), and broken (N) piles.
This graph shows the bent 2 flexibility transfer functions for the Vibroseis force and the vertical accelerometer response reading, both at node 4, for intact (I), excavated (E), and broken (N) piles. The horizontal axis ranges in frequency from 10 to 30 hertz. The vertical axis shows flexibility (inches per poundforce) and ranges from 0.00E plus 00 to 8.00E minus 07. All curves plot similar configurations with the widest range at 14 hertz (between 3.00 minus 07 and 6.00 minus 07); the intact curve shows the lowest values. The range of the curves is between 1.00E minus 07 to 6.00E minus 07 and descends to about 1.00E minus 07 at the 30hertz endpoint. To convert flexibility transfer functions to metric units, inches per poundforce can be multiplied by 0.005708 to give meters per newtons.
Figure 68. Graph. Bent 2 flexibility TFs at node 6 for intact (I) and excavated (E) piles.
This graph shows the bent 2 flexibility transfer functions for the Vibroseis force and the vertical accelerometer response reading, both at node 6, for intact (I) and excavated (E) piles. The horizontal axis ranges in frequency from 10 to 30 hertz. The vertical axis shows flexibility (inches per poundforce) and ranges from 0.00E plus 00 to 8.00E minus 07. Both curves plot similar configurations with one intersecting value of about 2.00E minus 07 at 28 hertz; the intact curve shows the lowest values. The excavated curve has a peak at 14 hertz of 7.00E minus 07. The curves descend to about 1.50E minus 07 at the 30hertz endpoint. To convert flexibility transfer functions to metric units, inches per poundforce can be multiplied by 0.005708 to give meters per newtons.
Figure 69. Graph. Bent 2 flexibility TFs for node 8/node 2 for intact (I), excavated (E), and broken (N) piles.
This graph shows the bent 2 flexibility transfer functions for the Vibroseis force at node 2 and the vertical accelerometer response reading at node 8 for intact (I), excavated (E), and broken (N) piles. The horizontal axis ranges in frequency from 10 hertz to 30 hertz. The vertical axis shows flexibility (inches per poundforce) and ranges from 0.00E plus 00 to 8.00E minus 07. All curves plot similar configurations except where the intact pile curve rises to a peak of almost 7.00E minus 07 at 15 hertz. The range of the curves is between 2.00E minus 07 and 7.00E minus 07 and descends to about 2.00E minus 07 at the 30hertz endpoint. To convert flexibility transfer functions to metric units, inches per poundforce can be multiplied by 0.005708 to give meters per newtons.
Figure 70. Graph. Bent 2 flexibility TFs for node 9/node 2 for intact (I), excavated (E), and broken (N) piles.
This graph shows the bent 2 flexibility transfer functions for the Vibroseis force at node 2 and the vertical accelerometer response reading at node 9 for intact (I), excavated (E), and broken (N) piles. The horizontal axis ranges in frequency from 10 hertz to 30 hertz. The vertical axis shows flexibility (inches per poundforce) and ranges from 0.00E plus 00 to 8.00E minus 07. All curves plot similar configurations except where the intact pile curve rises to a peak of almost 5.00E minus 07 at 15 hertz. The range of the curves is between 1.00E minus 07 and 5.00E minus 07 and descends to about 1.00E minus 07 at the 30hertz endpoint. To convert flexibility transfer functions to metric units, inches per poundforce can be multiplied by 0.005708 to give meters per newtons.
Figure 71. Graph. Bent 2 flexibility TFs for node 10/node 6 for intact (I) and excavated (E) piles.
This graph shows the bent 2 flexibility transfer functions for the Vibroseis force at node 6 and the vertical accelerometer response reading at node 10 for intact (I) and excavated (E) piles. The horizontal axis ranges in frequency from 10 to 30 hertz. The vertical axis shows flexibility (inches per poundforce) and ranges from 0.00E plus 00 to 8.00E minus 07. Both curves plot similar configurations with one intersecting value of 1.00E minus 07 at the 30hertz endpoint. The intact curve shows the highest values with the peak at 6.00E minus 07 at 14 hertz. The range of the curves is between 1.00E minus 07 and almost 6.00E minus 07 and descends to about 1.00E minus 07 at the 30hertz endpoint. To convert flexibility transfer functions to metric units, inches per poundforce can be multiplied by 0.005708 to give meters per newtons.
Figure 72. Graph. Bent 2 flexibility TFs for node 11/node 6 for intact (I) and excavated (E) piles.
This graph shows the bent 2 flexibility transfer functions for the Vibroseis force at node 6 and the vertical accelerometer response reading at node 11 for intact (I) and excavated (E) piles. The horizontal axis ranges in frequency from 10 hertz to 30 hertz. The vertical axis shows flexibility (inches per poundforce) and ranges from 0.00E plus 00 to 8.00E minus 07. The excavated curve is steeper with a peak of almost 8.00E minus 07 at about 14 hertz. Both curves descend to approximately 2.00E minus 07 at the 30hertz endpoint. To convert flexibility transfer functions to metric units, inches per poundforce can be multiplied by 0.005708 to give meters per newtons.
Figure 73. Diagram. Bent 12, Mode 1 Vertical, node 4 (center) loading, frequency, and damping.
For the first vertical mode and loading at node 4, the center node, this figure shows frequencies in hertz and damping percentage at nodes 2, 4, 6, and 8 through 15 of bent 12. The frequencies and damping percentages at each node are for the sound pile case, the excavated pile case, and the broken pile case. The nodes are located as indicated in figures 20 and 58: nodes 1 through 7 extend horizontally from left to right at the top; nodes 8 and 12 extend vertically downward from node 1; nodes 9 and 13 extend vertically downward from node 3; nodes 10 and 14 extend vertically downward from node 5; and nodes 11 and 15 extend vertically downward from node 7. Following are the data by node. Node 2: sound pile case, 14.814 hertz; sound pile case damping, 8.16 percent; excavated pile case, 13.913 hertz; excavated pile case damping, 9.22; broken pile case, 14.202 hertz; broken pile case damping, 8.27 percent. Node 4: sound pile case, 14.729 hertz; sound pile case damping, 9.28 percent; excavated pile case, 13.911 hertz; excavated pile case damping, 9.94; broken pile case, 13.941 hertz; broken pile case damping, 9.46 percent. Node 6: sound pile case, 14.828 hertz; sound pile case damping, 11.90 percent; excavated pile case, 13.903 hertz; excavated pile case damping, 10.10; broken pile case, 13.739 hertz; broken pile case damping, 9.94 percent. Node 8: sound pile case, 14.812 hertz; sound pile case damping, 7.57 percent; excavated pile case, 13.855 hertz; excavated pile case damping, 8.29; broken pile case, 14.224 hertz; broken pile case damping, 7.45 percent. Node 9: sound pile case, 14.916 hertz; sound pile case damping, 8.59 percent; excavated pile case, 13.509 hertz; excavated pile case damping, 9.804; broken pile case, 14,110 hertz; broken pile case damping, 9.06 percent. Node 10: sound pile case, 14.876 hertz; sound pile case damping, 10.23 percent; excavated pile case, 13.912 hertz; excavated pile case damping, 10.16; broken pile case, 13.851 hertz; broken pile case damping, 9.86 percent. Node 11: sound pile case, 14.411 hertz; sound pile case damping, 10.18 percent; excavated pile case, 13.866 hertz; excavated pile case damping, 10.00; broken pile case, 13.654 hertz; broken pile case damping, 10.11 percent. Node 12: sound pile case, 14.814 hertz; sound pile case damping, 7.44 percent; excavated pile case, 13.844 hertz; excavated pile case damping, 7.99; broken pile case, 14,199 hertz; broken pile case damping, 7.66 percent. Node 13: sound pile case, 14.807 hertz; sound pile case damping, 7.84 percent; excavated pile case, 13.938 hertz; excavated pile case damping, 9.68; broken pile case, 14,053 hertz; broken pile case damping, 8.82 percent. Node 14: sound pile case, 14.857 hertz; sound pile case damping, 9.39 percent; excavated pile case, 13.891 hertz; excavated pile case damping, 9.75; broken pile case, 13.879 hertz; broken pile case damping, 9.56 percent. Node 15: sound pile case, 14.647 hertz; sound pile case damping, 11.14 percent; excavated pile case, 13.939 hertz; excavated pile case damping, 9.91; broken pile case, 13.684 hertz; broken pile case damping, 9.98 percent.
Figure 74. Diagram. Bent 12, Mode 1 Vertical, node 4 (center) loading, magnitude and phase.
For the first vertical mode and loading at node 4, the center node, this figure shows the absolute magnitude of the mode shape (normalized to node 4) and the phase (plus or minus in degrees relative to node 4) at nodes 2, 4, 6, and 8 through 15 of bent 12. The configuration of the nodes was given in the description of figure 73. The magnitude and phase at each node are for the sound pile case, the excavated pile case, and the broken pile case. Following are the data by node. Node 2: sound pile case magnitude, 1.04; sound pile case phase, minus 3; excavated pile case magnitude, 0.7888; excavated pile case phase, 2; broken pile case magnitude, 0.863; broken pile case phase, minus 24. Node 4: sound pile case magnitude, 1.00; sound pile case phase, 0.0; excavated pile case magnitude, 1.00; excavated pile case phase, 0.0; broken pile case magnitude, 1.00; broken pile case phase, 0.0. Node 6: sound pile case magnitude, 1.09; sound pile case phase, minus 10; excavated pile case magnitude, 1.20; excavated pile case phase, minus 4; broken pile case magnitude, 1.18; broken pile case phase, 15. Node 8: sound pile case magnitude, 1.00; sound pile case phase, minus 2; excavated pile case magnitude, 0.601; excavated pile case phase, 4; broken pile case magnitude, 0.674; broken pile case phase, minus 32. Node 9: sound pile case magnitude, 1.06; sound pile case phase, minus 13; excavated pile case magnitude, 0.886; excavated pile case phase, 2; broken pile case magnitude, 0.954; broken pile case phase, minus 13. Node 10: sound pile case magnitude, 1.07; sound pile case phase, minus 15; excavated pile case magnitude, 1.12; excavated pile case phase, minus 4; broken pile case magnitude, 1.09; broken pile case phase, 5. Node 11: sound pile case magnitude, 0.775; sound pile case phase, minus 3; excavated pile case magnitude, 1.26; excavated pile case phase, minus 6; broken pile case magnitude, 1.28; broken pile case phase, 20. Node 12: sound pile case magnitude, 0.948; sound pile case phase, minus 4; excavated pile case magnitude, 0.584; excavated pile case phase, 5; broken pile case magnitude, 0.655; broken pile case phase, minus 30. Node 13: sound pile case magnitude, 1.00; sound pile case phase, minus 5; excavated pile case magnitude, 0.911; excavated pile case phase, minus 1; broken pile case magnitude, 0.898; broken pile case phase, minus 10. Node 14: sound pile case magnitude, 1.04; sound pile case phase, minus 17; excavated pile case magnitude, 1.06; excavated pile case phase, minus 2; broken pile case magnitude, 1.05; broken pile case phase, 3. Node 15: sound pile case magnitude, 0.894; sound pile case phase, minus 9; excavated pile case magnitude, 1.26; excavated pile case phase, minus 12; broken pile case magnitude, 1.24; broken pile case phase, 18.
Figure 75. Diagram. Bent 12, Mode 1 Vertical, loading at nodes 2 and 4, frequency and damping.
For the first vertical mode and the sound pile case, this figure gives frequency in hertz and damping in percent at nodes 2, 4, 6, and 8 through15 for node 4 loading and node 2 loading. The configuration of the nodes was given in the description of figure 73. Data for each node include loading at node 4 and at node 2. Over all nodes, the averages for node 4, or center, loading are 14.779 hertz and 9.46 percent. The averages for node 2 loading are 14.782 hertz and 8.38 percent. Following are the data for each node. Node 2: frequency for node 4 loading, 14.814 hertz; damping for node 4 loading, 8.16 percent; frequency for node 2 loading, 14.498 hertz; damping for node 2 loading, 8.98 percent. Node 4: frequency for node 4 loading, 14.729 hertz; damping for node 4 loading, 9.28 percent; frequency for node 2 loading, 14.797 hertz; damping for node 2 loading, 7.76 percent. Node 6: frequency for node 4 loading, 14.828 hertz; damping for node 4 loading, 11.90 percent; frequency for node 2 loading, 15.113 hertz; damping for node 2 loading, 7.08 percent. Node 8: frequency for node 4 loading, 14.812 hertz; damping for node 4 loading, 7.57 percent; frequency for node 2 loading, 14.376 hertz; damping for node 2 loading, 10.24 percent. Node 9: frequency for node 4 loading, 14.916 hertz; damping for node 4 loading, 8.59 percent; frequency for node 2 loading, 14.713 hertz; damping for node 2 loading, 8.22 percent. Node 10: frequency for node 4 loading, 14.876 hertz; damping for node 4 loading, 10.23 percent; frequency for node 2 loading, 14.955 hertz; damping for node 2 loading, 6.26 percent. Node 11: frequency for node 4 loading, 14.411 hertz; damping for node 4 loading, 10.18 percent; frequency for node 2 loading, 15.295 hertz; damping for node 2 loading, 6.86 percent. Node 12: frequency for node 4 loading, 14.814 hertz; damping for node 4 loading, 7.44 percent; frequency for node 2 loading, 14.382 hertz; damping for node 2 loading, 10.20 percent. Node 13: frequency for node 4 loading, 14.807 hertz; damping for node 4 loading, 7.84 percent; frequency for node 2 loading, 14.734 hertz; damping for node 2 loading, 8.37 percent. Node 14: frequency for node 4 loading, 14.857 hertz; damping for node 4 loading, 9.39 percent; frequency for node 2 loading, 14.967 hertz; damping for node 2 loading, 6.32 percent. Node 15: frequency for node 4 loading, 14.647 hertz; damping for node 4 loading, 11.14 percent; frequency for node 2 loading, 15.265 hertz; damping for node 2 loading, 8.18 percent.
Figure 76. Diagram. Bent 12, Mode 1 Vertical, loading at nodes 2 and 4, magnitude and phase.
For the first vertical mode and the sound pile case, this figure gives magnitude (normalized to node 4) and phase (plus or minus in degrees relative to node 4) at nodes 2, 4, 6, and 8 through 15 for node 4 loading and node 2 loading. The configuration of the nodes was given in the description of figure 74. Following are the data for each node. Node 2: magnitude for node 4 loading, 1.04; phase for node 4 loading, minus 3; magnitude for node 2 loading, 1.10; phase for node 2 loading, 51. Node 4: magnitude for node 4 loading, 1.00; phase for node 4 loading, 0; magnitude for node 2 loading, 1.00; phase for node 2 loading, 0. Node 6: magnitude for node 4 loading, 1.09; phase for node 4 loading, minus 10; magnitude for node 2 loading, 1.07; phase for node 2 loading, minus 57. Node 8: magnitude for node 4 loading, 1.00; phase for node 4 loading, minus 2; magnitude for node 2 loading, 1.29; phase for node 2 loading, 69. Node 9: magnitude for node 4 loading, 1.06; phase for node 4 loading, minus 13; magnitude for node 2 loading, 1.06; phase for node 2 loading, 20. Node 10: magnitude for node 4 loading, 1.07; phase for node 4 loading, minus 15; magnitude for node 2 loading, 0.888; phase for node 2 loading, minus 32. Node 11: magnitude for node 4 loading, 0.775; phase for node 4 loading, minus 3; magnitude for node 2 loading, 1.22; phase for node 2 loading, minus 89. Node 12: magnitude for node 4 loading, 0.948; phase for node 4 loading, minus 4; magnitude for node 2 loading, 1.30; phase for node 2 loading, 67. Node 13: magnitude for node 4 loading, 1.00; phase for node 4 loading, minus 5; magnitude for node 2 loading, 1.09; phase for node 2 loading, 16. Node 14: magnitude for node 4 loading, 1.04; phase for node 4 loading, minus 17; magnitude for node 2 loading, 0.90; phase for node 2 loading, minus 34. Node 15: magnitude for node 4 loading, 0.894; phase for node 4 loading, minus 9; magnitude for node 2 loading, 1.40; phase for node 2 loading, minus 75.
Figure 77. Diagram. Bent 12, Mode 2 Vertical, node 4 (center) loading, frequency and damping.
For the second vertical mode and loading at node 4, the center node, this figure shows frequencies in hertz and damping in percent at the nodes 2, 4, 6, and 8 through 15 of bent 12. The configuration of the nodes was given in the description of figure 73. Following are the summary results for each node. Node 2: sound pile case, 21.982 hertz; damping, 8.12 percent; excavated pile case, 21.563 hertz; damping, 7.39 percent; broken pile case, 21.119 hertz; damping, 5.77percent. Node 4: sound pile case, 22.019 hertz; damping, 8.30 percent; excavated pile case, 21.608 hertz; damping, 6.71 percent; broken pile case, 20.853 hertz; damping, 6.30 percent. Node 6: sound pile case, 22.087 hertz; damping, 8.30 percent; excavated pile case, 21.284 hertz; damping, 4.52 percent; broken pile case, 20.062 hertz; damping, 4.81 percent. Node 8: sound pile case, 21.865 hertz; damping, 7.87 percent; excavated pile case, 21.439 hertz; damping, 7.82 percent; broken pile case, 21.002 hertz; damping, 6.05 percent. Node 9: sound pile case, 21.897 hertz; damping, 8.16 percent; excavated pile case, 21.511 hertz; damping, 7.61 percent; broken pile case, 21.189 hertz; damping, 6.43 percent. Node 10: sound pile case, 21.848 hertz; damping, 8.26 percent; excavated pile case, 21.299 hertz; damping, 6.07 percent; broken pile case, 20.151 hertz; damping, 5.68 percent. Node 11: sound pile case, 21.926 hertz; damping, 8.27 percent; excavated pile case, 20.934 hertz; damping, 4.00 percent; broken pile case, 19.934 hertz; damping, 2.71 percent. Node 2: sound pile case, hertz; damping, percent; excavated pile case, hertz; damping, percent; broken pile case, hertz; damping, percent. Node 12: sound pile case, 21.850 hertz; damping, 7.55 percent; excavated pile case, 21.346 hertz; damping, 8.08 percent; broken pile case, 20.936 hertz; damping, 6.07 percent. Node 13: sound pile case, 21.795 hertz; damping, 7.47 percent; excavated pile case, 21.373 hertz; damping, 8.33 percent; broken pile case, 21.191 hertz; damping, 6.99 percent. Node 14: sound pile case, 21.719 hertz; damping, 8.19 percent; excavated pile case, hertz 21.225; damping, 6.73 percent; broken pile case, 20.538 hertz; damping, 6.59 percent. Node 15: sound pile case, 21.781 hertz; damping, 8.09 percent; excavated pile case, 20.871 hertz; damping, 4.23 percent; broken pile case, 20.200 hertz; damping, 4.55 percent.
Figure 78. Diagram. Bent 12, Mode 2 Vertical, node 4 (center) loading, magnitude and phase.
For the second vertical mode and loading at node 4, the center node, this figure shows the absolute magnitude of the mode shape (normalized to node 4) and the phase, plus or minus in degrees relative to node 4, at nodes 2, 4, 6, and 8 through 15 of bent 12. The configuration of the nodes was given in the description of figure 73. Following are the data for each node. Node 2: sound pile case magnitude, 1.380;sound pile case phase, 7; excavated pile case magnitude, 1.505; excavated pile case phase, 15; broken pile case magnitude, 1.29; broken pile case phase, 6. Node 4: sound pile case magnitude, 1.000; sound pile case phase, 0; excavated pile case magnitude, 1.000; excavated pile case phase, 0; broken pile case magnitude, 1.00; broken pile case phase, 0. Node 6: sound pile case magnitude, 0.772; sound pile case phase, minus 12; excavated pile case magnitude, 0.614; excavated pile case phase, 11; broken pile case magnitude, 1.04; broken pile case phase, 60. Node 8: sound pile case magnitude, 1.460; sound pile case phase, 17; excavated pile case magnitude, 1.806; excavated pile case phase, 27; broken pile case magnitude, 1.68; broken pile case phase, 18. Node 9: sound pile case magnitude, 1.040; sound pile case phase, 9; excavated pile case magnitude, 1.109; excavated pile case phase, 12; broken pile case magnitude, 1.15; broken pile case phase, minus 20. Node 10: sound pile case magnitude, 0.766; sound pile case phase, 8; excavated pile case magnitude, 0.740; excavated pile case phase, 20; broken pile case magnitude, 1.32; broken pile case phase, 51. Node 11: sound pile case magnitude, 0.688; sound pile case phase, minus 5; excavated pile case magnitude, 0.616; excavated pile case phase, 34; broken pile case magnitude, 0.863; broken pile case phase, 37. Node 12: sound pile case magnitude, 1.310; sound pile case phase, 17; excavated pile case magnitude, 1.906; excavated pile case phase, 34; broken pile case magnitude, 1.71; broken pile case phase, 23. Node 13: sound pile case magnitude, 0.912; sound pile case phase, 21; excavated pile case magnitude, 1.308; excavated pile case phase, 22; broken pile case magnitude, 1.41; broken pile case phase, minus 25. Node 14: sound pile case magnitude, 0.738; sound pile case phase, 19; excavated pile case magnitude, 0.880; excavated pile case phase, 25; broken pile case magnitude, 1.48; broken pile case phase, 3. Node 15: sound pile case magnitude, 0.678; sound pile case phase, 7; excavated pile case magnitude, 0.652; excavated pile case phase, 48; broken pile case magnitude, 1.50; broken pile case phase, 17.
Figure 79. Diagram. Bent 12, Mode 2 Vertical, loading at nodes 2 and 4, frequency and damping.
For the second vertical mode and the sound pile case, this figure gives frequency in hertz and damping in percentage at nodes 2, 4, 6, and 8 through 15 for node 4 loading and node 2 loading. The configuration of the nodes was given in the description of figure 73. Following are the data for each node. Node 2: frequency for node 4 loading, 21.982 hertz; damping for node 4 loading, 8.12 percent; frequency for node 2 loading, 21.626 hertz; damping for node 2 loading, 5.40 percent. Node 4: frequency for node 4 loading, 22.019 hertz; damping for node 4 loading, 8.30 percent; frequency for node 2 loading, 21.786 hertz; damping for node 2 loading, 7.06 percent. Node 6: frequency for node 4 loading, 22.087 hertz; damping for node 4 loading, 8.30 percent; frequency for node 2 loading, 22.034 hertz; damping for node 2 loading, 7.48 percent. Node 8: frequency for node 4 loading, 21.865 hertz; damping for node 4 loading, 7.87 percent; frequency for node 2 loading, 21.422 hertz; damping for node 2 loading, 4.51 percent. Node 9: frequency for node 4 loading, 21.897 hertz; damping for node 4 loading, 8.16 percent; frequency for node 2 loading, 21.563 hertz; damping for node 2 loading, 6.36 percent. Node 10: frequency for node 4 loading, 21.848 hertz; damping for node 4 loading, 8.26 percent; frequency for node 2 loading, 21.780 hertz; damping for node 2 loading, 7.47 percent. Node 11: frequency for node 4 loading, 21.926 hertz; damping for node 4 loading, 8.27 percent; frequency for node 2 loading, 22.032 hertz; damping for node 2 loading, 7.34 percent. Node 12: frequency for node 4 loading, 21.850 hertz; damping for node 4 loading, 7.55 percent; frequency for node 2 loading, 21.371 hertz; damping for node 2 loading, 4.64 percent. Node 13: frequency for node 4 loading, 21.795 hertz; damping for node 4 loading, 7.47 percent; frequency for node 2 loading, 21.390 hertz; damping for node 2 loading, 6.16 percent. Node 14: frequency for node 4 loading, 21.719 hertz; damping for node 4 loading, 8.19 percent; frequency for node 2 loading, 21.672 hertz; damping for node 2 loading, 7.26 percent. Node 15: frequency for node 4 loading, 22.781 hertz; damping for node 4 loading, 8.09 percent; frequency for node 2 loading, 22.123 hertz; damping for node 2 loading, 7.50 percent.
Figure 80. Diagram. Bent 12, Mode 2 Vertical, loading at nodes 2 and 4, magnitude and phase.
For the second vertical mode and the sound pile case, this figure gives magnitude, normalized to node 4, and phase, plus or minus in degrees relative to node 4, at nodes 2, 4, 6, and 8 through 15 for node 4 loading and node 2 loading. The configuration of the nodes was given in the description of figure 73. Following are the data for each node. Node 2: magnitude for node 4 loading, 1.38; phase for node 4 loading, 7; magnitude for node 2 loading, 0.930; phase for node 2 loading, 48. Node 4: magnitude for node 4 loading, 1.00; phase for node 4 loading, 0; magnitude for node 2 loading, 1.00; phase for node 2 loading, 0. Node 6: magnitude for node 4 loading, 0.772; phase for node 4 loading, minus 12; magnitude for node 2 loading, 1.30; phase for node 2 loading, minus 50. Node 8: magnitude for node 4 loading, 1.46; phase for node 4 loading, 17; magnitude for node 2 loading, 0.889; phase for node 2 loading, 87. Node 9: magnitude for node 4 loading, 1.04; phase for node 4 loading, 9; magnitude for node 2 loading, 0.917; phase for node 2 loading, 34. Node 10: magnitude for node 4 loading, 0.766; phase for node 4 loading, 8; magnitude for node 2 loading, 1.02; phase for node 2 loading, minus 18. Node 11: magnitude for node 4 loading, 0.688; phase for node 4 loading, minus 5; magnitude for node 2 loading, 1.50; phase for node 2 loading, minus 58. Node 12: magnitude for node 4 loading, 1.31; phase for node 4 loading, 17; magnitude for node 2 loading, 0.915; phase for node 2 loading, 95. Node 13: magnitude for node 4 loading, 0.912; phase for node 4 loading, 21; magnitude for node 2 loading, 0.869; phase for node 2 loading, 52. Node 14: magnitude for node 4 loading, 0.738; phase for node 4 loading, 19; magnitude for node 2 loading, 0.956; phase for node 2 loading, minus 10. Node 15: magnitude for node 4 loading, 0.678; phase for node 4 loading, 7; magnitude for node 2 loading, 1.43; phase for node 2 loading, minus 52.
Figure 81. Diagram. Bent 12, Mode 2 Horizontal, node 4 (center) loading, frequency and damping.
For the second horizontal mode and loading at node 4, the center node, this figure shows frequencies in hertz and damping in percentage at nodes 2, 4, 6, and 8 through 15 of bent 12. The configuration of the nodes was given in the description of figure 73. Following is a summary of results for each node. Node 2: sound pile case, 21.598 hertz; sound pile case damping, 4.80 percent; excavated pile case, 20.528 hertz; excavated pile case damping, 6.82 percent; broken pile case, 20.765 hertz; broken pile case damping, 10.05 percent. Node 4: sound pile case, 21.427 hertz; sound pile case damping, 5.38 percent; excavated pile case, 20.413 hertz; excavated pile case damping, 7.44 percent; broken pile case, 20.898 hertz; broken pile case damping, 9.36 percent. Node 6: sound pile case, hertz; sound pile case damping, 5.02 percent; excavated pile case, 20.687 hertz; excavated pile case damping, 6.91 percent; broken pile case, 20.488 hertz; broken pile case damping, 8.70 percent. Node 8: sound pile case, 21.944 hertz; sound pile case damping, 4.62 percent; excavated pile case, 21.041 hertz; excavated pile case damping, 7.71 percent; broken pile case, 21.013 hertz; broken pile case damping, 8.58 percent. Node 9: sound pile case, 21.611 hertz; sound pile case damping, 6.27 percent; excavated pile case, 20.791 hertz; excavated pile case damping, 7.32 percent; broken pile case, 20.710 hertz; broken pile case damping, 10.26 percent. Node 10: sound pile case, 21.953 hertz; sound pile case damping, 3.21 percent; excavated pile case, 20.860 hertz; excavated pile case damping, 5.40 percent; broken pile case, 20.812 hertz; broken pile case damping, 8.42 percent. Node 11: sound pile case, 21.742 hertz; sound pile case damping, 4.38 percent; excavated pile case, 20.527 hertz; excavated pile case damping, 7.66 percent; broken pile case, no data; broken pile case damping, no data. Node 12: sound pile case, 21.844 hertz; sound pile case damping, 3.94 percent; excavated pile case, 20.231 hertz; excavated pile case damping, 5.25 percent; broken pile case, 20.963 hertz; broken pile case damping, 8.41 percent. Node 13 sound pile case, 21.801 hertz; sound pile case damping, 6.89 percent; excavated pile case, 20.945 hertz; excavated pile case damping, 8.00 percent; broken pile case, 19.880 hertz; broken pile case damping, 4.48 percent. Node 14: sound pile case, 22.098 hertz; sound pile case damping, 1.70 percent; excavated pile case, 20.611 hertz; excavated pile case damping, 3.44 percent; broken pile case, 20.131 hertz; broken pile case damping, 4.11 percent. Node 15: sound pile case, 22.723 hertz; sound pile case damping, 3.54 percent; excavated pile case, 20.621 hertz; excavated pile case damping, 8.88 percent; broken pile case, no data; broken pile case damping, no data.
Figure 82. Diagram. Bent 12, Mode 2 Horizontal, node 4 (center) loading, magnitude and phase.
For the second horizontal mode and loading at node 4, the center node, this figure shows the absolute magnitude of the mode shape, normalized to node 4, and the phase, plus or minus in degrees relative to node 4, at nodes 2, 4, 6, and 8 through 15 of bent 12. The configuration of the nodes was given in the description of figure 73. Following are the data for each node. Node 2: sound pile case magnitude, 0.146; sound pile case phase, 74; excavated pile case magnitude, 0.310; excavated pile case phase, 138; broken pile case magnitude, 0.829; broken pile case phase, 65.Node 4: sound pile case magnitude, 0.144; sound pile case phase, 96; excavated pile case magnitude, 0.289; excavated pile case phase, 157; broken pile case magnitude, 0.666; broken pile case phase, 63. Node 6: sound pile case magnitude, 0.138; sound pile case phase, 86; excavated pile case magnitude, 0.237; excavated pile case phase, 136; broken pile case magnitude, 0.671; broken pile case phase, minus 79. Node 8: sound pile case magnitude, 0.221; sound pile case phase, 20; excavated pile case magnitude, 0.614; excavated pile case phase, 74; broken pile case magnitude, 1.11; broken pile case phase, 30. Node 9: sound pile case magnitude, 0.277; sound pile case phase, 85; excavated pile case magnitude, 0.531; excavated pile case phase, 117; broken pile case magnitude, 1.09; broken pile case phase, 59. Node 10: sound pile case magnitude, 0.131; sound pile case phase, 34; excavated pile case magnitude, 0.269; excavated pile case phase, 155; broken pile case magnitude, 0.917; broken pile case phase, 90. Node 11: sound pile case magnitude, 0.123; sound pile case phase, 58; excavated pile case magnitude, 0.555; excavated pile case phase, 103; broken pile case magnitude, no data; broken pile case phase, no data. Node 12: sound pile case magnitude, 0.235; sound pile case phase, 18; excavated pile case magnitude, 0.516; excavated pile case phase, 151; broken pile case magnitude, 1.28; broken pile case phase, 1. Node 13: sound pile case magnitude, 0.361; sound pile case phase, 82; excavated pile case magnitude, 0.500; excavated pile case phase, 111; broken pile case magnitude, 0.292; broken pile case phase, 71. Node 14: sound pile case magnitude, 0.184; sound pile case phase, minus 7; excavated pile case magnitude, 0.265; excavated pile case phase, minus 135; broken pile case magnitude, 0.499; broken pile case phase, minus 151. Node 15: sound pile case magnitude, 0.103; sound pile case phase, minus 64; excavated pile case magnitude, 0.614; excavated pile case phase, 157; broken pile case magnitude, no data; broken pile case phase, no data.
Figure 83. Diagram. Bent 12, Mode 2 Horizontal, loading at nodes 2 and 4, frequency and damping.
For the second horizontal mode and the sound pile case, this figure gives frequency in hertz and damping in percentage at nodes 2, 4, 6, and 8 through 15 for node 4 loading and node 2 loading. The configuration of the nodes was given in the description of figure 73. Following are the data at each node. Node 2: frequency for node 4 loading, 21.598 hertz; damping for node 4 loading, 4.80 percent; frequency for node 2 loading, 21.408 hertz; damping for node 2 loading, 10.56 percent. Node 4: frequency for node 4 loading, 21.427 hertz; damping for node 4 loading, 5.38 percent; frequency for node 2 loading, 21.846 hertz; damping for node 2 loading, 9.98 percent. Node 6: frequency for node 4 loading, 21.482 hertz; damping for node 4 loading, 5.02 percent; frequency for node 2 loading, 21.714 hertz; damping for node 2 loading, 9.96 percent. Node 8: frequency for node 4 loading, 21.944 hertz; damping for node 4 loading, 4.62 percent; frequency for node 2 loading, 20.774 hertz; damping for node 2 loading, 12.04 percent. Node 9: frequency for node 4 loading, 21.611 hertz; damping for node 4 loading, 6.27 percent; frequency for node 2 loading, 21.446 hertz; damping for node 2 loading, 9.13 percent. Node 10: frequency for node 4 loading, 21.953 hertz; damping for node 4 loading, 3.21 percent; frequency for node 2 loading, 21.264 hertz; damping for node 2 loading, 11.52 percent. Node 11: frequency for node 4 loading, 21.742 hertz; damping for node 4 loading, 4.38 percent; frequency for node 2 loading, 21.671 hertz; damping for node 2 loading, 11.91 percent. Node 12: frequency for node 4 loading, 21.844 hertz; damping for node 4 loading, 3.94 percent; frequency for node 2 loading, 20.008 hertz; damping for node 2 loading, 11.37 percent. Node 13: frequency for node 4 loading, 21.801 hertz; damping for node 4 loading, 6.89 percent; frequency for node 2 loading, 21.405 hertz; damping for node 2 loading, 7.30 percent. Node 14: frequency for node 4 loading, 22.098 hertz; damping for node 4 loading, 1.70 percent; frequency for node 2 loading, 20.241 hertz; damping for node 2 loading, 9.45 percent. Node 15: frequency for node 4 loading, 22.723 hertz; damping for node 4 loading, 3.54 percent; frequency for node 2 loading, 20.376 hertz; damping for node 2 loading, 8.50 percent.
Figure 84. Diagram. Bent 12, Mode 2 Horizontal, loading at nodes 2 and 4, magnitude and phase.
For the second horizontal mode and the sound pile case, this figure gives magnitude, normalized to node 4, and phase, plus or minus in degrees relative to node 4, at nodes 2, 4, 6, and 8 through 15 for node 4 loading and node 2 loading. The configuration of the nodes was given in the description of figure 73. Following are the data at each node. Node 2: magnitude for node 4 loading, 0.146; phase for node 4 loading, 74; magnitude for node 2 loading, 0.787; phase for node 2 loading, 97. Node 4: magnitude for node 4 loading, 0.144; phase for node 4 loading, 96; magnitude for node 2 loading, 0.622; phase for node 2 loading, 96. Node 6: magnitude for node 4 loading, 0.138; phase for node 4 loading, 86; magnitude for node 2 loading, 0.522; phase for node 2 loading, 88. Node 8: magnitude for node 4 loading, 0.221; phase for node 4 loading, 20; magnitude for node 2 loading, 1.22; phase for node 2 loading, 117. Node 9: magnitude for node 4 loading, 0.277; phase for node 4 loading, 85; magnitude for node 2 loading, 0.991; phase for node 2 loading, 102. Node 10: magnitude for node 4 loading, 0.131; phase for node 4 loading, 34; magnitude for node 2 loading, 1.01; phase for node 2 loading, 110. Node 11: magnitude for node 4 loading, 0.123; phase for node 4 loading, 58; magnitude for node 2 loading, 1.14; phase for node 2 loading, 111. Node 12: magnitude for node 4 loading, 0.235; phase for node 4 loading, 18; magnitude for node 2 loading, 1.66; phase for node 2 loading, 125. Node 13: magnitude for node 4 loading, 0.361; phase for node 4 loading, 82; magnitude for node 2 loading, 0.877; phase for node 2 loading, 89. Node 14: magnitude for node 4 loading, 0.184; phase for node 4 loading, minus 7; magnitude for node 2 loading, 0.663; phase for node 2 loading, 133. Node 15: magnitude for node 4 loading, 0.103; phase for node 4 loading, minus 64; magnitude for node 2 loading, 0.430; phase for node 2 loading, 161.
Figure 85. Graph. Woodville and Old Reliance Bridges, flexibility TFs, node 9/node 17.
This graph shows the flexibility transfer functions for the Woodville and Old Reliance Bridges with the Vibroseis force applied at node 17 and the response reading taken from the vertical accelerometer at node 9. The horizontal axis ranges in frequency from 0 to 50 hertz. The vertical axis shows flexibility (inches per poundforce) and ranges from 0.00E plus 00 to 4.00E minus 07. The transfer function for the Old Reliance Bridge shows a greater range from 0 to 4 hertz (0.00E plus 00 to 4.00E minus 07). Between 8 hertz and 24 hertz, the transfer function values for both bridges vary from less than 1.00E minus 07 to almost 3.00E minus 07 until the curves roughly mirror each other at less than 1.00E minus 07 from 28 hertz to the 50hertz endpoint. To convert flexibility transfer functions to metric units, inches per poundforce can be multiplied by 0.005708 to give meters per newtons.
Figure 86. Graph. Woodville and Old Reliance Bridges, flexibility TFs, node 9/node 18.
This graph shows the flexibility transfer functions for the Woodville and Old Reliance Bridges with the Vibroseis force applied at node 18 and the response reading taken from the vertical accelerometer at node 10. The horizontal axis ranges in frequency from 0 to 50 hertz. The vertical axis shows flexibility (inches per poundforce) and ranges from 0.00E plus 00 to 4.00E minus 07. The transfer function for the Old Reliance Bridge shows a peak of about 3.00E minus 07 at 20 hertz; the Woodville Bridge peak is less than 3.00E minus 07 at about 15 hertz. Starting at about 24 hertz, the curves roughly coincide with ranges from almost 2.00E minus 07 to nearly 00.E plus 00 at the 50hertz endpoint. To convert flexibility transfer functions to metric units, inches per poundforce can be multiplied by 0.005708 to give meters per newtons.
Figure 87. Equation. K subscript SSS.
Bridge foundation stiffness, K subscript SSS, equals a 6 by 6 matrix. Following are the rows from left to right. First row: K subscript XX, 0, 0, 0, 0, K subscript phi, which also has a subscript Z, times subscript X; second row: 0, K subscript YY, 0, 0, 0, 0; third row: 0, 0, K subscript ZZ, K subscript phi, which also has a subscript X, times subscript Z, 0, 0; fourth row: 0, 0, K subscript Z times subscript phi subscript X, K subscript phi, which also has a subscript X, times subscript phi subscript X, 0, 0; fifth row: 0, 0, 0, 0, K subscript phi, which also has a subscript Y, times subscript phi subscript Y, 0; sixth row: K subscript X times subscript phi subscript Z, 0, 0, 0, 0, K subscript phi, which also has a subscript Z, times subscript phi subscript Z.
Figure 88. Equation. M subscript SSS.
Bridge foundation mass, M subscript SSS, equals a 6 by 6 matrix. Following are the rows from left to right. First row: M subscript XX, 0, 0, 0, 0, 0; second row: 0, M subscript YY, 0, 0, 0, 0; third row: 0, 0, M subscript ZZ, 0, 0, 0; fourth row: 0, 0, 0, M subscript phi, which also has a subscript X, times subscript phi subscript X, 0, 0; fifth row: 0, 0, 0, 0, M subscript phi, which also has a subscript Y, times subscript phi subscript Y, 0; sixth row: 0, 0, 0, 0, 0, M subscript phi, which also has a subscript Z, times subscript phi subscript Z.
Figure 89. Equation. K subscript SSS (3 by 3 matrix).
Bridge foundation stiffness, K subscript SSS, equals a 3 by 3 matrix. Following are the rows from left to right. First row: K subscript XX, 0, K subscript phi X; second row: 0, K subscript YY, 0; third row: K subscript X phi, 0, K subscript phi phi.
Figure 90. Equation. M subscript SSS (3 by 3 matrix).
Bridge foundation mass, M subscript SSS, equals a 3 by 3 matrix. Following are the rows from left to right. First row: M subscript XX, 0, 0; second row: 0, M subscript YY, 0; third row: 0, 0, M subscript phi phi.
Figure 91. Equation. KE subscript max (distributed properties).
KE subscript max, which is the maximum kinetic energy of a structure, equals onehalf times the squared term omega subscript N, which is the resonant frequency, times the integral from 0 to L of the product of M(X), which is the distributed mass, and the square of phi subscript N, which is the mode shape associated with the resonant frequency.
Figure 92. Equation. SE subscript max (distributed properties).
SE subscript max, which is the maximum potential or strain energy of a structure, equals onehalf times the integral from 0 to L of the product of EI(X), which is the distributed stiffness, and the square of phi with a subscript N and a superscript of double prime, phi being the mode shape associated with the resonant frequency.
Figure 93. Equation. KE subscript max (lumped properties).
KE subscript max, which is the maximum kinetic energy of a structure, equals the product of onehalf times the squared term omega subscript N, which is the resonant frequency, times the sum of the set of I equals 1 to M of the product of the lumped mass M subscript I, times the square of phi subscript I, which is the mode shape associated with the resonant frequency.
Figure 94. Equation. SE subscript max (lumped properties).
SE subscript max, which is the maximum potential or strain energy of a structure, equals onehalf times the integral from 0 to L of the product of EI(X), which is the distributed stiffness, times the square of phi with a subscript N and a superscript of double prime, phi being the mode shape associated with the resonant frequency.
Figure 95. Equation. Alpha subscript N.
Alpha subscript N equals the resonant frequency, which is omega subscript N, in two dimensions divided by the resonant frequency, which is omega subscript N, in three dimensions, all squared.
Figure 96. Equation. Matrix K times vector phi.
The stiffness matrix K of a structure times the Ith mode shape, which is the vector phi, equals the product of the corresponding resonant frequency, which is omega, squared, times the mass matrix M of the structure, times the Ith mode shape, which is the vector phi.
Figure 97. Equation. Matrix K times vector phi (partitioned).
The stiffness matrix K consisting of K subscript AA, K subscript AB, K subscript BA, and K subscript BB times the mode shape, which is the vertical vector phi subscript A and phi subscript B, equals the product of the corresponding resonant frequency, which is omega, squared, times the mass matrix M consisting of M subscript AA, M subscript AB, M subscript BA, and M subscript BB, times the vertical vector phi subscript A and phi subscript B.
Figure 98. Equation. Stiffnessbased residual modal error function.
The residual modal error function of the Ith iteration of E subscript S, times P, equals a multicomponent term. The first component is K subscript AA minus the product of the Ith iteration of omega squared times M subscript AA. From this component is subtracted the product of components two, three, and four. The second component is K subscript AB minus the product of the Ith iteration of omega squared times M subscript AB. The third component is the inverse of K subscript BB minus the product of the Ith iteration of omega squared times M subscript BB. The fourth component is K subscript BA minus the product of the Ith iteration of omega squared times M subscript BA. The result of all of the preceding is multiplied by the Ith iteration of phi subscript A, which is the mode shape.
Figure 99. Equation. Flexibilitybased residual modal error function.
The residual modal error function of the Ith iteration of E subscript F, times P, equals a multicomponent term. The first component is the product of the Ith iteration of omega squared times the sum of the product of F subscript AA times M subscript AA plus the product of F subscript AB times M subscript BA. To this component is added the product of the second component, which is the Ith iteration of omega to the fourth power, times components three, four, and five. The third component is the sum of the product of F subscript AA times M subscript AB plus the product of F subscript AB times M subscript BB. The fourth component is the inverse of the difference between the identity matrix I and the product of the Ith iteration of omega squared times the sum of the product of F subscript BA times M subscript AB plus the product of F subscript BB times M subscript BB. The fifth component is the difference between the sum of the product of F subscript BA times M subscript AA plus the product of F subscript BB times M subscript BA, and the identity matrix I. The result of all of the preceding is multiplied by the Ith iteration of phi subscript A, which is the mode shape.
Figure 100. Equation. Error function approximation.
The stiffness or flexibilitybased error function of the parameter vector plus the change in parameter vector is approximately equal to the stiffness or flexibilitybased error function plus the product of the sensitivity matrix times the change in parameter vector.
Figure 101. Equation. Objective function J.
The objective function of the parameter vector plus the change in parameter vector equals the product of the stiffness or flexibilitybased error function of the parameter vector plus the change in parameter vector, raised to the T power, times the stiffness or flexibilitybased error function of the parameter vector plus the change in parameter vector.
Figure 102. Equation. Change in parameter vector.
The change in each unknown parameter equals the negative of the product of: first, the inverse of the product of the sensitivity matrix raised to the T power times the sensitivity matrix; second, the sensitivity matrix raised to the T power; and third, the stiffness or flexibilitybased error function.
Figure 103. Equation. Parameter iteration.
The unknown parameter of subscript K plus 1 equals the unknown parameter of subscript K plus the change in unknown parameter of subscript K.
Figure 104. Equation. Resonant frequency with artificial errors.
The vector of resonant frequency, which is omega subscript M, equals the vector of computersimulated resonant frequency, which is omega subscript C, plus the product of the matrix of measurement error, which is E subscript omega, times the random numbers in the vector R subscript omega.
Figure 105. Equation. Mode shape with artificial errors.
The vector of mode shape, which is phi subscript M, equals the vector of computersimulated mode shape, which is phi subscript C, plus the product of the matrix of measurement error, which is E subscript phi, times the random numbers in vector R subscript phi.
Figure 106. Equation. Grand mean percentage error.
The grand mean percentage error, GPE, equals the product of four terms. The first term is 1 divided by the product of the number of unknown parameters (NUP) times the number of observations (NOBS). The second term is the sum of the set from I equals 1 to NOBS. The third term is the sum of the set from J equals 1 to NUP. The fourth term is PE, which is the percentage error and has superscript I, subscript J.
Figure 107. Equation. Grand standard deviation percentage error.
The grand standard deviation percentage error, GSD, equals the square root of the product of four terms. The first term is 1 divided by the product of the number of unknown parameters (NUP) times the number of observations (NOBS). The second term is the sum of the set from I equals 1 to NOBS. The third term is the sum of the set from J equals 1 to NUP. The fourth term is the square of the difference of PE, which is the percentage error and has superscript I and subscript J, and the GPE, which is the grand mean percentage error from figure 106.
Figure 108. Equation. Percentage error.
Percentage error, which is PE superscript I, subscript J, equals the difference of the estimated value, which is P superscript I subscript J, and the true value, which is P superscript T subscript J, divided by the true value, and all multiplied by 100.
Figure 109. Diagram. Vibration tests of bent 12 of the Trinity River Relief Bridge.
This cross section shows nodes 13 and 15 with the excitation point in the middle of the concrete slab having a downloading arrow. The top deck is 7.68 meters (25 feet 2 inches) wide. The asphalt curb is 0.19 meters (7.5 inches) thick. The thickness of the concrete slab at its edge to the underside of its lip is 0.49 meters (1 foot 7.5 inches). The lip of the concrete slab is 0.20 meters (6.5 inches) thick. The lip of the beam cap has a thickness of 0.41 meters (1 foot 4 inches). The drawing also shows the 1.93 meters (6 feet) of exposed column with the 2.75 meters (9 feet) excavation between columns 3 and 4. The space between columns is 1.94 meters (6 feet 4 inches), and the width of a column is 0.35 meters (1 foot 2 inches).
Figure 110. Graphs. A. Vibroseis chirp forcing function in poundforce at center of bent 12. B. Accelerometer 15 response in inches per second squared.
These graphs, which appear to be screen dumps or captured computer screens, show the time histories of the vertical excitation or forcing function and the vertical accelerationvibration response along the column at accelerometer 15. The horizontal axis in both charts is time in seconds from 0 to 6. The forcing function vertical axis in graph A is amplitude in poundforce from negative 3 times 10 to the fourth power to 3 times 10 to the fourth power. The vibration vertical axis in graph B is amplitude in inches per second squared ranging from minus 200 to 200. The frequency of both the forcing function and the response increased almost linearly from 5 hertz at 0.5 second to 75 hertz at 5.5 seconds. These exact frequencies are not discernable on the graphs, but the increase in frequency is evident from the increase in the number of up and down cycles from the left side of the graphs to the right side. The first peak in both graphs occurs at approximately 1.6 seconds.
Figure 111. Graphs. Fourier spectra (F.S.) of figure 110 data.
Three graphs show the corresponding Fourier spectra for the time domain traces of figure 110. All three graphs (A, B, and C) have frequency in hertz as the horizontal axes: for graphs A and B, the frequency is from 0 to 180 hertz; for graph C, the frequency is from 0 to 40 hertz. The vertical axes shows the amplitudes, but the types of units are not identified. Graph A is the Fourier spectrum of the forcing function, and the vertical axis is labeled from 0 to 15 times 10 to the fifth power. The graph has peaks of approximately 10 times 10 to the fifth power at approximately 20 hertz and 60 hertz. Graph B is the Fourier spectrum of the vibration, and the vertical axis is from 0 to 10,000. The graph has intermediate peaks at approximately 20 hertz, 28 hertz, and 43 hertz, and a major peak of approximately 7,000 at approximately 70 hertz. Graph C is the normalized Fourier spectrum, which is also the accelerance transfer function and the ratio of the middle graph’s Fourier spectrum of vibration to the top graph’s Fourier spectrum of the forcing function. The vertical axis is labeled from 0 to 4 times 10 to the negative three power. The first peak is at approximately 14.8 hertz.
Figure 112. Graphs. Morlet (graph B) and Db5 (graph C) wavelet spectra of the forcing function (graph A).
These graphs are screen dumps, or captured computer screens. Graph A is approximately the same as figure 110A, which is the time domain graph of the forcing function. The vertical axis is from negative 4 times 10 to the fourth power to 4 times 10 to the fourth power. Both the Morlet (graph B), and the Db5 (graph C), wavelet spectra have horizontal axes of 0 to 2,000, which is time in seconds multiplied by .00305. The horizontal axes are labeled “Scale (1:100).” The Morlet wavelet spectrum, graph B, shows one light color band between roughly 0 and 800 on the horizontal axis, and a smattering of lighter color between 1,400 and 1,800 on the horizontal axis. The Db5 wavelet spectrum, graph C, shows three light color bands between roughly 0 and 800 on the horizontal axis.
Figure 113. Graphs. Accelerance TFs for intact, excavated, and broken pile states.
Three graphs show accelerance transfer plots at sensor 15 for bent 12. The horizontal scale is frequency in hertz from 0 to 40, and the vertical scale is amplitude ranging from 0 to 8 times 10 to the negative three power. The plots show similar configurations. They all begin near the top of the vertical axis and fall to the bottom by 5 hertz. Then there are gentle rises to between 4 to 6 times 10 to the negative 3 in the vicinity of 27 hertz, followed by taperings to between 2 to 4 times 10 to the negative 3 at the 40hertz endpoint.
Figure 114. Equation. X(T).
The data X as a function of T equals the sum of the set from J equals 1 to N of the sum of the term C subscript J as a function of T plus R subscript N as a function of T, T being time and R being the residue.
Figure 115. Equation. Z(T).
Analytical signal Z as a function of T, T being time, equals the data C as a function of T plus I times the Hilbert transform Y as a function of T, all of which in turn equals small letter A as a function of T times Euler’s formula in which the real number portion of the exponent is theta as a function of T. Euler’s formula is the natural logarithm E raised to the exponential I times a real number where I is the imaginary unit. Two subequations are also given in the equation. First, small letter A as a function of T equals the square root of the sum of C squared as a function of T plus Y squared as a function of T. Second, the exponential I times theta as a function of T can also be expressed as the arctan of the quotient of Y as a function of T divided by C as a function of T.
Figure 116. Equation. X(T) in Hilbert transform form.
The data X as a function of T, T being time, equals the real part of the sum of the set from J equals 1 to N of the product of small letter a subscript J, the combination being a function of T, times Euler’s formula in which the real number portion of the exponent is the integral of small omega subscript J, the combination being a function of T. Euler’s formula is the natural logarithm E raised to the exponential I times a real number where I is the imaginary unit. In a subequation, small omega subscript J, the combination being a function of T, equals the derivative of theta subscript J, the combination being a function of T, the derivative being taken with respect to T.
Figure 117. Equation. X(T) in Fourier series representation.
The data X as a function of T, T being time, equals the real part of the sum of the set from J equals 1 to N of the product of A subscript J times Euler’s formula in which the real number portion of the exponent is the product of large omega subscript J times T. Euler’s formula is the natural logarithm E raised to the exponential I times a real number where I is the imaginary unit.
Figure 118. Graph. HHT spectrum of forcing function, bent 12, intact state.
This graph is a screen dump, or captured computer screen. The horizontal scale is time in seconds from 0 to 6, and the vertical scale is frequency in hertz from 0 to 100. A color spectrum key is on the right side of the graph and begins at the bottom with dark blue. Going up the spectrum, the color gradually lightens. The color is green about halfway to the top. Over the last half of the spectrum, the color changes to yellow, orange, red, and finally, at the top, dark amber. The scale of the color spectrum goes from 0.2 times 10 to the fourth power at the bottom to 2 times 10 to the fourth power at the top. The scale’s units are not identified. The plot itself is predominantly a dark line rising from 5 hertz at 0.5 seconds on the left to 75 hertz at 5.5 seconds on the right.
Figure 119. Graph. HHT spectrum of vibration, accelerometer 15, bent 12, intact state.
This graph is a screen dump, or captured computer screen. The horizontal scale is time in seconds from 0 to 6, and the vertical scale is frequency in hertz from 0 to 100. A color spectrum key is on the right side of the graph and begins at the bottom with dark blue. Going up the spectrum, the color gradually lightens. The color is green about halfway to the top. Over the last half of the spectrum, the color changes to yellow, orange, red, and finally, at the top, dark amber. The scale of the color spectrum goes from 20 near the bottom to 140 at the top. The scale’s units are not identified. The central feature of the plot is a dark line rising from 5 hertz at 0.5 seconds on the left to 75 hertz at 5.5 seconds on the right. The other main grouping of colors is between 10 hertz and 20 hertz between 1 second and 2 seconds. Various lesser color groupings are scattered about the plot.
Figure 120. Graphs. Empirical mode decomposition (EMD) with the eight IMF components of vibration, accelerometer 15, bent 12, intact state.
Eight graphs separate and plot the 8 intrinsic mode function (IMF) components from the vibration response depicted in figure 119. The horizontal scales are time from 0 to 6 seconds and the vertical scales are amplitude of unspecified units ranging from minus 100 to 100 for the top graph, which probably depicts the vibration component resulting from the Vibroseis excitation at the DIDF, to minus 2 to 2 for the seventh graph, which probably depicts one of the less pronounced natural frequencies. In the top graph, the frequency of the plot increases linearly with time. The plots in the other graphs have no such linear increase. The amplitudes of the second and third plots are significantly higher than the amplitudes in the remaining 5 plots.
Figure 121. Graphs. Fourier spectra of IMF components in figure 120.
Eight graphs plot the Fournier spectra of the 8 IMF components plotted in figure 120. The horizontal scales are frequency in hertz ranging from 10 to the 0 to 10 squared. The vertical scales are unspecified amplitude labeled from C1 to C8. The dominant frequencies decrease from approximately 70 hertz in the top graph to approximately 16 hertz, 10 hertz, and 6 hertz in graphs 2, 3, and 4 respectively. The dominant frequencies in the remaining graphs are less clear, but they appear to be near or below 1 hertz.
Figure 122. Graph. HHT spectrum of first IMF component of vibration, accelerometer 15, bent 12, intact state.
This graph is a screen dump, or captured computer screen. The horizontal scale is time in seconds from 0 to 6, and the vertical scale is frequency in hertz from 0 to 100. A color spectrum key is on the right side of the graph and begins at the bottom with dark blue. Going up the spectrum, the color gradually lightens. The color is green about halfway to the top. Over the last half of the spectrum, the color changes to yellow, orange, red, and finally, at the top, dark amber. The scale of the color spectrum goes from 20 near the bottom to 140 at the top. The scale’s units are not identified. The central feature of the plot is a dark line rising from approximately 20 hertz at 2 seconds on the left to approximately 70 hertz at 5.5 seconds on the right. A grouping of colors also is found between 20 hertz and 70 hertz between 1 second and 2 seconds. Several lesser color groupings are scattered about the plot.
Figure 123. Graph. HHT spectrum of second IMF component of vibration, accelerometer 15, bent 12, intact state.
This graph is a screen dump, or captured computer screen. The horizontal scale is time in seconds from 0 to 6, and the vertical scale is frequency in hertz from 0 to 100. A color spectrum key is on the right side of the graph and begins at the bottom with dark blue. Going up the spectrum, the color gradually lightens. The color is green about halfway to the top. Over the last half of the spectrum, the color changes to yellow, orange, red, and finally, at the top, dark amber. The scale of the color spectrum goes from 5 near the bottom to 35 near the top. The scale’s units are not identified. The plot consists of various groupings of dark colors between 10 and 50 hertz, with only the slight appearance of a rising trend from left to right.
Figure 124. Graph. HHT spectrum of vibration, accelerometer 15, bent 12, intact state.
This graph is a screen dump, or captured computer screen. The horizontal scale is time in seconds from 0 to 3, and the vertical scale is frequency in hertz from 0 to 40. A color spectrum key is on the right side of the graph and begins at the bottom with dark blue. Going up the spectrum, the color gradually lightens. The color is green about halfway to the top. Over the last half of the spectrum, the color changes to yellow, orange, red, and finally, at the top, dark amber. The scale of the color spectrum goes from 20 near the bottom to 140 at the top. The scale’s units are not identified. The plot consists primarily of an irregular dark band rising from approximately 10 hertz at 1 second to 40 hertz at 3 seconds. A pronounced dark area is also found at 14 hertz to 15 hertz at approximately 1.5 seconds. In addition, groupings of light lines are found from approximately 7 to 35 hertz at 0.4 to 0.9 seconds, and from approximately 0 to 7 hertz at 1 to 1.7 seconds.
Figure 125. Graph. HHT spectrum of vibration, accelerometer 15, bent 12, excavated state.
This graph is a screen dump, or captured computer screen. The horizontal scale is time in seconds from 0 to 3, and the vertical scale is frequency in hertz from 0 to 40. A color spectrum key is on the right side of the graph and begins at the bottom with dark blue. Going up the spectrum, the color gradually lightens. The color is green about halfway to the top. Over the last half of the spectrum, the color changes to yellow, orange, red, and finally, at the top, dark amber. The scale of the color spectrum goes from 20 near the bottom to 150 at the top. The scale’s units are not identified. The plot consists primarily of a very irregular band rising from approximately 10 hertz at 1 second to 40 hertz at 3 seconds. A pronounced dark area is also found at approximately 7 hertz at approximately 1.4 seconds. In addition, the plot includes other less pronounced concentrations of colors.
Figure 126. Graph. HHT spectrum of vibration, accelerometer 15, bent 12, broken state.
This graph is a screen dump, or captured computer screen. The horizontal scale is time in seconds from 0 to 3, and the vertical scale is frequency in hertz from 0 to 40. A color spectrum key is on the right side of the graph and begins at the bottom with dark blue. Going up the spectrum, the color gradually lightens. The color is green about halfway to the top. Over the last half of the spectrum, the color changes to yellow, orange, red, and finally, at the top, dark amber. The scale of the color spectrum goes from 10 near the bottom to 100 near the top. The scale’s units are not identified. The plot consists primarily of an extremely irregular band rising from approximately 10 hertz at 1 second to 40 hertz at 3 seconds. A pronounced dark area is also found at approximately 3 hertz at approximately 1.7 seconds. In addition, the plot includes other concentrations of colors.
Figure 127. Equation. Theoretical frequency in the intact state (in table 15).
F subscript I equals 1 divided by the product of 2 times L, all times the square root of the quotient of E divided by P.
Figure 128. Equation. Theoretical frequency in the minor damage state (in table 15).
F subscript M equals 1 divided by the product of 2 times L, all times the square root of the quotient of E divided by P.
Figure 129. Equation. Theoretical frequency in the severe damage state (in table 15).
F subscript S equals 1 divided by the product of 2 times L, all times the square root of the quotient of E divided by P.
Figure 130. Graph. HHT spectrum of vibration, accelerometer 13, bent 12, intact state.
This graph is a screen dump, or captured computer screen. The horizontal scale is time in seconds from 0 to 3, and the vertical scale is frequency in hertz from 0 to 40. The plot consists primarily of an extremely irregular band rising from approximately 15 hertz at 1 second to 40 hertz at 3 seconds. Other scattered concentrations of colors are also present.
Figure 131. Graph. HHT spectrum of vibration, accelerometer 13, bent 12, excavated state.
This graph is a screen dump, or captured computer screen. The horizontal scale is time in seconds from 0 to 3, and the vertical scale is frequency in hertz from 0 to 40. The plot consists primarily of an extremely irregular band rising from approximately 15 hertz at 1 second to 40 hertz at 3 seconds. A pronounced dark area is also found at approximately 10 hertz at approximately 2 seconds. In addition, the plot includes other concentrations of colors.
Figure 132. Graph. HHT spectrum of vibration, accelerometer 13, bent 12, broken state.
This graph is a screen dump, or captured computer screen. The horizontal scale is time in seconds from 0 to 3, and the vertical scale is frequency in hertz from 0 to 40. The plot consists primarily of an extremely irregular band rising from approximately 13 hertz at 1.25 seconds to 40 hertz at 3 seconds. A pronounced dark area is also found at approximately 15 hertz at approximately 2.25 seconds. In addition, the plot includes other concentrations of colors.
Figure 133. Graph. HHT spectrum, second component of vibration, accelerometer 15, bent 12, intact state.
This graph is a screen dump, or captured computer screen. The horizontal scale is time in seconds from 0 to 3, and the vertical scale is frequency in hertz from 0 to 40. The plot displays a very irregular rise from left to right, with several groupings of colors extending toward the top of the graph or in a downward direction. The low point at approximately 1 second is approximately 10 hertz.
Figure 134. Graph. HHT spectrum, second component of vibration, accelerometer 13, bent 12, intact state.
This graph is a screen dump, or captured computer screen. The horizontal scale is time in seconds from 0 to 3, and the vertical scale is frequency in hertz from 0 to 40. The plot displays a fairly discernable rise from left to right, with several groupings of colors extending above or below the trend. The low point at approximately 1 second is approximately 10 hertz.
Figure 135. Graph. HHT spectrum, second component of vibration, accelerometer 15, bent 12, excavated state.
This graph is a screen dump, or captured computer screen. The horizontal scale is time in seconds from 0 to 3, and the vertical scale is frequency in hertz from 0 to 40. The plot displays a fairly discernable rise from left to right, with several groupings of colors extending above or below the trend. The low point at approximately 1.4 seconds is approximately 7 hertz.
Figure 136. Graph. HHT spectrum, second component of vibration, accelerometer 13, bent 12, excavated state.
This graph is a screen dump, or captured computer screen. The horizontal scale is time in seconds from 0 to 3, and the vertical scale is frequency in hertz from 0 to 40. The plot displays a fairly discernable rise from left to right, with several groupings of colors extending above or below the trend. The low point at approximately 1 second is approximately 10 hertz.
Figure 137. Graph. HHT spectrum, second component of vibration, accelerometer 15, bent 12, broken state.
This graph is a screen dump, or captured computer screen. The horizontal scale is time in seconds from 0 to 3, and the vertical scale is frequency in hertz from 0 to 40. The plot displays a fairly discernable but very irregular rise from left to right, with several groupings of colors extending above or below the trend. The low point at approximately 0.9 seconds is approximately 5 hertz.
Figure 138. Graph. HHT spectrum, second component of vibration, accelerometer 13, bent 12, broken state.
This graph is a screen dump, or captured computer screen. The horizontal scale is time in seconds from 0 to 3, and the vertical scale is frequency in hertz from 0 to 40. The plot displays a fairly discernable rise from left to right, with several groupings of colors, predominantly at the left side of the graph, extending above or below the trend. The low point at approximately 1.2 seconds is approximately 10 hertz.
Figure 139. Diagram. Locations of accelerometers 9 and 11 on bent 2.
This cross section shows locations of accelerometers 9 and 11 on piles of bent 2 of the Trinity River Relief Bridge, and the bent’s dimensions. Accelerometer 9 is located near the top of the second pile from the left. Accelerometer 11 is located near the top of the fourth pile from the left. The top deck is 7.68 meters (25 feet 2 inches) wide. The asphalt curb is 0.191 meters (7.5 inches) high. The concrete slab is 0.39 meters (1 foot 3.5 inches) high. The beam cap is 7.17 meters (23 feet 6 inches) long and 0.71 meters (2 feet 5 inches) high. The four piles are 1.93 meters (6 feet 4 inches) apart. The buried length of each pile is 4.8 meters (15 feet 9 inches). The footing is 0.61 meters (2 feet) thick and 2.14 meters (7 feet), which is the length of the exposed portions of the piles or columns, below the beam cap.
Figure 140. Graph. HHT spectrum of vibration, accelerometer 11, bent 2, intact state.
This graph is a screen dump, or captured computer screen. The horizontal scale is time in seconds from 0 to 3, and the vertical scale is frequency in hertz from 0 to 40. A color spectrum key is on the right side of the graph and begins at the bottom with dark blue. Going up the spectrum, the color gradually lightens. The color is green about halfway to the top. Over the last half of the spectrum, the color changes to yellow, orange, red, and finally, at the top, dark amber. The scale of the color spectrum goes from 10 near the bottom to 90 at the top. The scale’s units are not identified. The plot consists primarily of an irregular dark band rising from approximately 10 hertz at 1 second to 40 hertz at 3 seconds. Several groupings of colors extend above and below the trend. One significant dark cluster is at approximately 10 hertz at approximately 1.5 seconds.
Figure 141. Graph. HHT spectrum of vibration, accelerometer 11, bent 2, excavated state.
This graph is a screen dump, or captured computer screen. The horizontal scale is time in seconds from 0 to 3, and the vertical scale is frequency in hertz from 0 to 40. A color spectrum key is on the right side of the graph and begins at the bottom with dark blue. Going up the spectrum, the color gradually lightens. The color is green about halfway to the top. Over the last half of the spectrum, the color changes to yellow, orange, red, and finally, at the top, dark amber. The scale of the color spectrum goes from 10 near the bottom to 90 near the top. The scale’s units are not identified. The plot consists primarily of an irregular dark band rising from approximately 10 hertz at 1 second to near 40 hertz at 3 seconds. Several groupings of colors extend above and below the trend. One significant dark cluster is at approximately 7 hertz at approximately 1.5 seconds.
Figure 142. Graph. HHT spectrum of vibration, accelerometer 11, bent 2, broken state.
This graph is a screen dump, or captured computer screen. The horizontal scale is time in seconds from 0 to 3, and the vertical scale is frequency in hertz from 0 to 40. A color spectrum key is on the right side of the graph and begins at the bottom with dark blue. Going up the spectrum, the color gradually lightens. The color is green about halfway to the top. Over the last half of the spectrum, the color changes to yellow, orange, red, and finally, at the top, dark amber. The scale of the color spectrum goes from 50 near the bottom to 400 near the top. The scale’s units are not identified. The plot consists primarily of an irregular band rising from approximately 10 hertz at 1 second to 40 hertz at 3 seconds. Several groupings of colors extend above and below the trend. One significant cluster is at approximately 5 hertz at 1.5 seconds.
Figure 143. Graph. HHT spectrum of vibration, accelerometer 9, bent 2, intact state.
This graph is a screen dump, or captured computer screen. The horizontal scale is time in seconds from 0 to 3, and the vertical scale is frequency in hertz from 0 to 40. A color spectrum key is on the right side of the graph and begins at the bottom with dark blue. Going up the spectrum, the color gradually lightens. The color is green about halfway to the top. Over the last half of the spectrum, the color changes to yellow, orange, red, and finally, at the top, dark amber. The scale of the color spectrum goes from 20 near the bottom to 180 at the top. The scale’s units are not identified. The plot consists primarily of an irregular dark band rising from approximately 10 hertz at 1 second to 40 hertz at 3 seconds. Several groupings of colors extend above and below the trend.
Figure 144. Graph. HHT spectrum of vibration, accelerometer 9, bent 2, excavated state.
This graph is a screen dump, or captured computer screen. The horizontal scale is time in seconds from 0 to 3, and the vertical scale is frequency in hertz from 0 to 40. A color spectrum key is on the right side of the graph and begins at the bottom with dark blue. Going up the spectrum, the color gradually lightens. The color is green about halfway to the top. Over the last half of the spectrum, the color changes to yellow, orange, red, and finally, at the top, dark amber. The scale of the color spectrum goes from 20 near the bottom to 160 near the top. The scale’s units are not identified. The plot consists primarily of an irregular dark band rising from approximately 10 hertz at 1 second to 40 hertz at 3 seconds. Several groupings of colors extend above and below the trend.
Figure 145. Graph. HHT spectrum of vibration, accelerometer 9, bent 2, broken state.
This graph is a screen dump, or captured computer screen. The horizontal scale is time in seconds from 0 to 3, and the vertical scale is frequency in hertz from 0 to 40. A color spectrum key is on the right side of the graph and begins at the bottom with dark blue. Going up the spectrum, the color gradually lightens. The color is green about halfway to the top. Over the last half of the spectrum, the color changes to yellow, orange, red, and finally, at the top, dark amber. The scale of the color spectrum goes from 20 near the bottom to 140 near the top. The scale’s units are not identified. The plot consists primarily of an irregular dark band rising from approximately 10 hertz at 1 second to 40 hertz at 3 seconds. Several groupings of colors extend above and below the trend.
Figure 146. Graph. Normalized HHT spectrum of vibration, accelerometer 15, bent 12, intact state.
This graph is a screen dump, or captured computer screen. The horizontal scale is time in seconds from 0 to 6, and the vertical scale is frequency in hertz from 0 to 100. A color spectrum key is on the right side of the graph and begins at the bottom with dark blue. Going up the spectrum, the color gradually lightens. The color is green about halfway to the top. Over the last half of the spectrum, the color changes to yellow, orange, red, and finally, at the top, dark amber. The scale of the color spectrum goes from 1 times 10 to the negative 3 power near the bottom to 9 times 10 to the negative 3 power at the top. The scale’s units are not identified. The plot consists primarily of an irregular dark band rising from approximately 10 hertz at 1 second to 70 hertz at 5.5 seconds. A number of color groupings extend above and below the trend, including a significant grouping between 20 hertz and 35 hertz between 0 and 1 second.
Figure 147. Graph. Amplitude of force applied to bent 12, intact state.
This graph is of the average amplitude of the HHTprocessed input force—the dominant linear chirp force—used to normalize the corresponding HHTprocessed vibration response for the intact state of bent 12. The horizontal axis is time from 0 to 6 seconds. The vertical axis is numbered from 0 to 2.5 times 10 to the fourth power. The units of the vertical axis are not identified. The graph has peaks of approximately 1.7 times 10 to the fourth power at approximately 1.8 seconds and approximately 2.2 times 10 to the fourth power at approximately 4.75 seconds.
Figure 148. Graph. Normalized HHT spectrum of vibration, accelerometer 15, bent 12, excavated state.
This graph is a screen dump, or captured computer screen. The horizontal scale is time in seconds from 0 to 6, and the vertical scale is frequency in hertz from 0 to 100. A color spectrum key is on the right side of the graph and begins at the bottom with dark blue. Going up the spectrum, the color gradually lightens. The color is green about halfway to the top. Over the last half of the spectrum, the color changes to yellow, orange, red, and finally, at the top, dark amber. The scale of the color spectrum goes from 0.005 near the bottom to 0.025 near the top. The scale’s units are not identified. The plot consists primarily of an irregular dark band rising from approximately 10 hertz at 1 second to 70 hertz at 5.5 seconds. Several small color groupings are found above and below the trend.
Figure 149. Graph. Amplitude of force applied to bent 12, excavated state.
This graph is of the average amplitude of the HHTprocessed input force—the dominant linear chirp force—used to normalize the corresponding HHTprocessed vibration response for the intact state of bent 12. The horizontal axis is time from 0 to 6 seconds. The vertical axis is numbered from 0 to 2.5 times 10 to the fourth power. The units of the vertical axis are not identified. The graph has peaks of approximately 2.3 times 10 to the fourth power at approximately 1.2 seconds, approximately 2.4 times 10 to the fourth power at approximately 2 seconds, and approximately 1.8 times 10 to the fourth power at approximately 5.1 seconds.
Figure 150. Graph. Normalized HHT spectrum of vibration, accelerometer 15, bent 12, broken state.
This graph is a screen dump, or captured computer screen. The horizontal scale is time in seconds from 0 to 6, and the vertical scale is frequency in hertz from 0 to 100. A color spectrum key is on the right side of the graph and begins at the bottom with dark blue. Going up the spectrum, the color gradually lightens. The color is green about halfway to the top. Over the last half of the spectrum, the color changes to yellow, orange, red, and finally, at the top, dark amber. The scale of the color spectrum goes from 0.002 near the bottom to 0.02 at the top. The scale’s units are not identified. The plot consists primarily of an irregular dark band rising from approximately 10 hertz at 1 second to 70 hertz at 5.5 seconds. A number of color groupings extend above and below the trend, including a significant grouping between 20 hertz and 35 hertz between 0 and 1 second.
Figure 151. Graph. Amplitude of force applied to bent 12, broken state.
This graph is of the average amplitude of the HHTprocessed input force—the dominant linear chirp force—used to normalize the corresponding HHTprocessed vibration response for the intact state of bent 12. The horizontal axis is time from 0 to 6 seconds. The vertical axis is numbered from 0 to 2.5 times 10 to the fourth power. The units of the vertical axis are not identified. The graph has a major peak of approximately 2.3 times 10 to the fourth power at approximately 2 seconds and several lesser peaks.
Figure 152. Equation. Damping matrix [C].
The damping matrix C equals the sum of the constant coefficient a subscript 0 times the matrix of mass M plus the constant coefficient a subscript 1 times the matrix of stiffness K.
Figure 153. Diagram. ANSYS Model for bent 12.
The diagram is a twodimensional model of bent 12. The model consists of one horizontal line and four vertical lines extending downward from the horizontal line. The horizontal line represents the bent’s beam and the vertical lines represent the bent’s four piles. The second vertical line from the left is labeled “Pile 13,” and the fourth vertical line from the left is labeled “Pile 15.” Node 37 is located in the midpoint of pile 13, and node 53 is located in the midpoint of pile 15. A vertical arrow labeled “Forcing Function” points from above into the midpoint of the horizontal line.
Figure 154. Graph. HHT spectra of vibration of 2D FEM model, node 37, bent 12, intact state, D equals 0.
This graph is a screen dump, or captured computer screen. The graph shows the HHT spectra of vibrations for the 2D FEM model at node 37 of bent 12 in the intact state. The damping, D, equals 0. To the right of the graph is a color spectra key that begins at the bottom with dark blue. Going up each spectrum, the color gradually lightens. The color is green about halfway to the top. Over the last half of the spectrum, the color changes to yellow, orange, red, and finally, at the top, dark amber. The scale of the spectrum goes from 100 near the bottom to 700 near the top. The scale’s units are not identified. The horizontal axis is time in seconds. The horizontal axis is labeled from 0 to 6. The vertical axis is frequency in hertz, labeled from 0 to 100. The graph has an extremely irregular broad band of dark colors rising from left to right and two broad bands of horizontal colors.
Figure 155. Graph. HHT spectra of vibration of 2D FEM model, node 37, bent 12, intact state, D equals 0.00198.
This graph is a screen dump, or captured computer screen. The graph shows the HHT spectra of vibrations for the 2D FEM model at node 37 of bent 12 in the intact state. The damping, D, equals 0.00198. To the right of the graph is a color spectra key that begins at the bottom with dark blue. Going up the spectrum, the color gradually lightens. The color is green about halfway to the top. Over the last half of the spectrum, the color changes to yellow, orange, red, and finally, at the top, dark amber. The scale of the spectrum goes from 20 near the bottom to 120 near the top. The scale’s units are not identified. The horizontal axis is time in seconds labeled from 0 to 2. The vertical axis is frequency in hertz labeled from 0 to 30. The graph has an extremely irregular broad band of dark colors rising from left to right.
Figure 156. Graph. HHT spectra of vibration of 2D FEM model, node 37, bent 12, intact state, D equals 0.05305.
This graph is a screen dump, or captured computer screen. The graph shows the HHT spectra of vibrations for the 2D FEM model at node 37 of bent 12 in the intact state. The damping, D, equals 0.05305. To the right of the graph is a color spectra key that begins at the bottom with dark blue. Going up the spectrum, the color gradually lightens. The color is green about halfway to the top. Over the last half of each spectrum, the color changes to yellow, orange, red, and finally, at the top, dark amber. The scale of the spectrum goes from 5 near the bottom to 35 near the top. The scale’s units are not identified. The horizontal axis of is time in seconds labeled from 0 to 2. The vertical axis is frequency in hertz labeled from 0 to 30. The graph has an extremely irregular broad band of dark colors rising from left to right.
Figure 157. Graph. Spectra of marginal amplitude, node 37, bent 12, intact state, D equals 0.
The graph shows the spectra of the marginal amplitude at node 37 of bent 12 in the intact state. The graph has a horizontal axis of 0 to 100 hertz and a vertical axis identified only as amplitude and labeled from 0 to 2.5 times 10 to the fifth power. The plotted line has peaks of 2.9 times 10 to the fifth power, 2.5 times 10 to the fifth power, and 2.3 times 10 to the fifth power at, respectively, 12 hertz, 20 hertz, and 70 hertz.
Figure 158. Graph. Spectra of marginal amplitude, node 37, bent 12, intact state, D equals 0.00198.
The graph has a horizontal axis of 0 to 30 hertz and a vertical axis identified only as amplitude and labeled from 0 to 3500. The plotted line begins above 3500 at 0 hertz and drops almost immediately, remaining in the range of 0 to 1,000, with numerous small peaks, from 0.5 to 30 hertz.
Figure 159. Graph. Spectra of marginal amplitude, node 37, bent 12, intact state, D equals 0.05305.
The graph has a horizontal axis of 0 to 30 hertz and a vertical axis identified only as amplitude and labeled from 0 to 1000. The plotted line begins above 1,000 at 0 hertz and drops almost immediately, remaining in the range of 0 to 300 hertz, with numerous small peaks, from 0.5 to 30 hertz.
Figure 160. Graph. HHT spectra of vibration of 2D FEM model, D equals 0.003937, bent 12, broken state, node 37.
This graph is a screen dump, or captured computer screen. The graph shows the HHT spectra of vibrations for the 2D FEM model at node 37 of bent 12 in the broken state. The damping, D, is 0.003979. To the right of the graph is a color spectra key that begins at the bottom with dark blue. Going up each spectrum, the color gradually lightens. The color is green about halfway to the top. Over the last half of the spectrum, the color changes to yellow, orange, red, and finally, at the top, dark amber. The scale of the spectrum goes from 10 near the bottom to 100 near the top. The scale’s units are not identified. The horizontal axis of graph is time in seconds from 0 to 2. The vertical axis is frequency in hertz from 0 to 30. The graph has a broad band of dark colors rising from left to right and an additional pronounced area of color, indicating a concentration of vibration energy. The additional area is at approximately 12 hertz between 0.6 to 1 second.
Figure 161. Graph. HHT spectra of vibration of 2D FEM model, D equals 0.003937, bent 12, broken state, node 53.
This graph is a screen dump, or captured computer screen. The graph shows the HHT spectra of vibrations for the 2D FEM model at node 53 of bent 12 in the broken state. The damping, D, is 0.003979. To the right of the graph is a color spectra key that begins at the bottom with dark blue. Going up each spectrum, the color gradually lightens. The color is green about halfway to the top. Over the last half of the spectrum, the color changes to yellow, orange, red, and finally, at the top, dark amber. The scale of the spectrum goes from 5 near the bottom to 35 near the top. The scale’s units are not identified. The horizontal axis of graph is time in seconds from 0 to 2. The vertical axis is frequency in hertz from 0 to 30. The graph has a broad band of dark colors rising from left to right and an additional pronounced area of color, indicating a concentration of vibration energy. The additional area is at approximately 7 hertz between 0.5 to 1.2 seconds.
Figure 162. Graph. Spectra of marginal amplitude, d equals 0.003937, bent 12, broken state, node 37.
The graph shows the spectra of the marginal amplitude at node 37 of bent 12 in the broken state. The horizontal axis is frequency in hertz from 0 to 30. The vertical axis is identified only as amplitude and labeled from 0 to 1,500. The plotted line drops sharply from 1,500 at 0 hertz to nearly 0 at 1.5 hertz and then remains between 0 and 1,000.
Figure 163. Graph. Spectra of marginal amplitude, D equals 0.003937, bent 12, broken state, node 53.
The graph shows the spectra of the marginal amplitude at node 53 of bent 12 in the broken state. The horizontal axis is frequency in hertz from 0 to 30. The vertical axis is identified only as amplitude and labeled from 0 to 1,500. The plotted line drops sharply from approximately 200 at 0 hertz to almost 0 at 2 hertz and then remains between 0 and approximately 400.
Figure 164. Graph. Hypothetical probability density functions for percent change in vertical stiffness.
The graph shows two bell curves. The horizontal axis is percentage of change in vertical stiffness. It is labeled from 0 to 80. The units of the vertical axis are poundforce, and the axis is labeled from 0 to 0.09. One bell curve is for an undamaged substructure. The curve peaks at 0.04 poundforce at just over 20 percent. The second bell curve is for a damaged substructure. It is a much narrower curve and peaks at 0.08 poundforce at 50 percent.