A COMPUTER ANALYSIS OF THE VINCENT THOMAS SUSPENSION BRIDGE 81 Raymond W. Wolfe Hany J. Farran A COMPUTER ANALYSIS OF THE VINCENT THOMAS SUSPENSION BRIDGE Civil Engineering Civil Engineering Given the increasing assimilation of high density population centers, particularly in earthquake prone regions, engineering emphasis is shifting towards the evaluation of existing structures ability to withstand seismic loadings. With the advent of the computer revolution, scientists and engineers have increasingly more powerful computational tools with which to analyze structural system performance. This study centers on a three-dimensional analysis of the Vincent Thomas Suspension Bridge suspended spans. Figure 1. The Vincent Thomas Suspension Bridge Introduction The Port of Los Angeles lays claim to being one of the world s busiest commerce ports. Yet Los Angeles is located within the geographical region known as the Pacific Rim of Fire, encompassing all of the nations bordering the Pacific Ocean, a region plagued by innumerable earthquakes. The faults underlying the Southern California area store enormous amounts of strain energy, which when released can lead to catastrophic loss of life and property. Thus, in a precautionary effort, today s engineers are predominantly charged with the responsibility of developing and implementing retrofit schemes to protect major regional structures and safeguard the public and property against the inexorable power of nature. The Vincent Thomas Suspension Bridge spans the Los Angeles harbor from San Pedro on the mainland to the expanded port facilities on Terminal Island. The bridge was originally completed in 1963 under the supervision of engineers from the State of California. The total length of the bridge is 6060 feet, including the numerous approach spans. The main suspended span is 1500 feet, making the Vincent Thomas Bridge the third largest suspended structure in the State of California. The clear height of the main span at the high water mark is 185 feet, with the towers reaching 365 feet in the air. The main and side suspended spans are comprised of steel trusses supporting a 6.5 inch thick lightweight concrete deck.
82 WOLFE, FARRAN Fall 1996 Suspension cables were comprised of 19 strands of 212 wires each, spun from No. 6 galvanized wire. The bridge remains today as one of the only vital links from the mainland to the expanded port facilities on Terminal Island. In an effort to thoroughly analyze the structural behavior of the Vincent Thomas Suspension Bridge, a three-dimensional computer model was created. After verifying the integrity of the model under various static load cases, a dynamic forcing function derived from data retrieved from the 1987 Whittier Earthquake was introduced. The results of the model further substantiated recent findings by various experts in the field of structural engineering. P.I. @ Splay Saddle P.I. El.163.20 Tower # 1 Tower # 2 P.I. El. 365.26 C L Cable Cable Bent # 1 Top of Pier Elev. = 25.00 MHHW Elev. = +5.4 MLLW Elev. = 0.00 Cable Bent # 2 151-6 506-6 1500 506-6 151-6 Figure 2. Elevation of Vincent Thomas Bridge The Model The structural analysis of the computer model herein described was run on a 486-DX2 IBM compatible machine with 8 megabytes of RAM and a 426 megabyte hard drive. To perform the analysis of the bridge, a finite element program entitled SAP90 (Habibullah & Wilson, 1989) was utilized. As the design of a suspension bridge naturally lends itself to a frame analysis, the SAP90 finite element specification, FRAME, was utilized in this study. The completed input file contained 2421 joints, 4743 members, and 28 member section properties, excluding the cable elements. For comparative purposes, both linear and nonlinear analyses were investigated. Only four cable element properties were necessitated for the linear analysis, one for the main cable between the anchorage and the cable bent, another for the remaining length of the main cable, and two defining the suspender ropes. To model the nonlinear cable effects, the Ernst equation (Nazmy, 1987) was utilized. Symmetry relations allowed the number of modified cable modulus calculations to reduce to 39 suspender ropes and 41 cable elements. Thus, for the nonlinear case, the number of separate cable elements necessary to define the structure increased from 4 to 80, still substantially less than the total number of cable elements, 318.
A COMPUTER ANALYSIS OF THE VINCENT THOMAS SUSPENSION BRIDGE 83 The Ernst equivalent modulus for cable elements is given by the equation (1) E eq = E 1 + [ ( ω L) 2 AE ] 12 T 2 L c T where E eq = Ernst equivalent cable modulus of elasticity E = the cable material effective modulus L = the horizontal projected length of the cable T L ω = weight per unit length of the cable element A = the cross-sectional area of the cable T = the cable tension As this calculation incorporates the effects of geometrical deformations, it should necessarily be revised each time deformations are encountered. Hence, the solution technique should include a means of recomputing the Ernst equivalent modulus as deformations change. In applying the Ernst equation to the classical suspension bridge problem, a cable element would be defined as each individual segment of the main cables between the suspender ropes, and each individual suspender rope. Loads As previously noted, the model was subjected to a variety of load combinations prior to introducing the dynamic loading of the Whittier Earthquake. First, the dead load of the structure was computed by SAP90 to 29, 158 kips. The modified Type 50A.2 concrete barrier railing constructed on the bridge in 1978 was added to the model as an additional dead load. Next, the live loading, computed in accordance with the current AASHTO criteria of 640 pounds per lineal foot per lane, was combined with the dead and additional dead loads, and the model analyzed. No provision was made for pattern loading as it was determined that the dynamic spectrum would govern the structural behavior of the bridge. Finally, the threedimensional dynamic forcing function derived from the Whittier Earthquake was superimposed onto the structure. On October 1, 1987, the Los Angeles area was rocked by a Richter magnitude 6.1 earthquake, hereinafter referred to as the Whittier earthquake. Caltech estimated the location and magnitude of the earthquake (CSMIP, 1987) as Epicenter: 34.058N, 118.075W Depth = 9 km. Origin Time: 14:42:20 GMT (07:42:20 PDT)
84 WOLFE, FARRAN Fall 1996 Due to the recent instrumentation efforts of the California Strong-Motion Instrumentation Program (CSMIP), a total of 128 strong-motion records were recovered from various structures located throughout the Los Angeles area. These records represent 641 channels of data, the largest data recovery from a single seismic incident at the time (CSMIP, 1987). Of particular interest and relevance to this study, data was recovered from the twenty-six (26) instruments located on the Vincent Thomas Suspension Bridge, 40 km from the epicenter. The spatial dynamic forcing function used in this analysis was derived from accelerograms positioned on the structure at locations of structural interest. Base data was not available for each direction at all cable bents. Unfortunately, the SAP90 software limits the dynamic forcing function specification to only one three-dimensional base acceleration. This is generally acceptable for building structures due to their limited footprint, but long span suspension bridges should be analyzed with different base accelerations applied at each tower to investigate the effects of torsion. Thus, it was necessary to extrapolate from one point the spectra used as the dynamic forcing function for the entire structure. For this model, the time history data from tower number 1 was applied throughout the structure as it provided the largest input accelerations. Analysis of Output Data The first computer simulation performed by the SAP90 software was the application of the dead loads resulting from the structure s self-weight and the additional load of the Type 50A.2 barrier rail (California, 1978). The midspan deflection was measured at 5.44 feet. This deflection is due to three basic nonlinear effects: the behavior of cables under large sustained loads, substantial geometric changes due to loading, and the interaction between the bending and axial deformations. In his analysis of cable-stayed bridges (Fleming, 1989), Dr. Fleming concludes, the primary contribution to the nonlinearity is from the cable members. The effects of geometry change in the structure and interaction between the bending and axial deformations in the members have only a very small effect, even for loads well beyond the normal design range. The effects of these two sources of nonlinearity can be safely ignored. As the dead load deflection of the structure was only 5.44 feet, each stiffening truss element rotated from its initial prescribed geometrical location by 0.42 degrees, certainly negligible. Under the application of the additional dead load and the AASHTO live load, this rotation increased to 0.80 degrees, still resulting in negligible nonlinear effects. Dr. Fleming continues, stating that once the dead load displaced position of the structure is reached, the structure tends to behave essentially in a linear manner (Fleming, 1989). The elastic modulus of the cable elements appear to assume a constant value through the application of additional loading. Hence, the overall stiffness of the structure remains constant for loading beyond the structure s dead load weight. This allows further studies to be based on linear analysis techniques as opposed to the more difficult and complex nonlinear techniques. Finally, the dynamic forcing function derived from the Whittier Earthquake (CSMIP, 1987) was imposed. Not surprisingly, the results revealed an overall lack of correlation of the model to the actual data retrieved from the instrumented structure. Given that the data from the nonlinear analysis essentially correlated with that of the linear run, the main contributing factor to the lack of continuity of the data presented, when compared to the actual accelerations measured on the Vincent Thomas Bridge during the 1987 Whittier earthquake, was the effect of nonsynchronous vibrations at the supports, which were not modelled due to the limitations of the SAP90 software.
A COMPUTER ANALYSIS OF THE VINCENT THOMAS SUSPENSION BRIDGE 85 Conclusion The main conclusion drawn from this study was that suspension bridges tend to behave linearly when subjected to loads nearly equivalent to their own dead loads and beyond. This finding has been proposed by several others, namely Dr. John F. Fleming (1989). It is apparent that the complexity of a nonlinear analysis is not necessary to achieve valid results for loading conditions exceeding the self-weight of the structure under investigation. Finally, this study has proven that software capable of analyzing models subjected to nonsynchronous vibrations must be utilized for long span structures. The effects of variable phasing tends to amplify or attenuate accelerations at individual frequencies as nonsynchronous waveforms meet. These effects are further modified at the various points of interest along the structure depending on the geometry changes encountered by the traversing waveforms. Thus, probably the single most important aspect to consider in modelling a long span structure, such as a suspension bridge, is the effect of nonsynchronous vibration at the supports. The SAP90 software, unfortunately, is unable to account for this. Further studies of this structure should incorporate a software routine capable of analyzing the structure given the nonsynchronous vibrations experienced at the supports during the 1987 Whittier Earthquake.
86 WOLFE, FARRAN Fall 1996 References California Department of Transportation, Standard Plans (Sacramento: Caltrans, 1978): 23. CSMIP Strong-Motion Records from the Whittier, California Earthquake of October 1, 1987, California Department of Conservation, Division of Mines and Geology, Office of Strong Motion Studies, Report OSMS 87-05 (1987): 139-47. Habibullah, A., and Wilson, E.L., SAP90-A Series of Computer Programs for Finite Element Analysis of Structures - Users Manual (Berkeley: Computers and Structures, Inc., 1989). Nazmy, Aly S., Seismic Response Analysis of Cable-stayed Bridges Subjected to Uniform and Multiple-support, Unpublished PhD Dissertation, Princeton University (1987). Suspension bridge rivals Golden Gate, Southwest Builder and Contractor (August, 1962): 14-17.