Strength of Parallel Wire Cables for Suspension Bridges

8th ASCE Specialty Conference on Probabilistic Mechanics and Structural Reliability

PMC2000-222

STRENGTH OF PARALLEL WIRE CABLES FOR SUSPENSION BRIDGES Gongkang Fu, M.ASCE Wayne State University, Detroit, MI 48202 [email protected] Fred Moses, M.ASCE University of Pittsburgh, Pittsburgh PA PA15261 Dyab A. Khazem, M.ASCE Parsons Transportation Group, New York, NY 10038 [email protected]

Abstract This paper focuses on modeling the strength of suspension bridge cables made of parallel high-strength wires. Previously proposed models are briefly reviewed first. It is pointed out that ductility limits and their uncertainty have not been adequately covered in these models. A new probabilistic model is then proposed, using the Monte Carlo simulation method to cover wires’ ductility limits and associated uncertainty. Application of the model is illustrated by an example. The results agree with physical test observations well. This also shows that it is important to acquire stress-strain relation in wire sample testing and use it in estimating cable strength. Further work is underway to evaluate the proposed model for application to cables with a larger number of wires.

Introduction There are approximately 50 suspension bridges in this country, representing a relatively older group of the entire suspension bridge population in the world. Most of these US bridges are reaching an age when their structural safety has become or will soon become of concern. The main cables in these structures are primary load-carrying members which are of special importance in safety evaluation. It is thus urgent to develop rational guidelines for the practice of cable inspection and safety evaluation. Although a large amount of experience has been cumulated for cable inspection in recent years, cable safety evaluation has not been consistently practiced. This paper focuses on modeling the strength of suspension bridge cables consisting of parallel high-strength wires. Such modeling is critical to evaluating the safety of suspension bridges, which represent significant investments in the transportation infrastructure system. Review of Previously Proposed Models For cable evaluation, the most extensive test data reported to date are from the Williamsburg Bridge in New York City (Steinman 1988). Using this set of data, Matteo et at (1994) developed a method to estimate the cables’ strength for evaluating their safety. It perhaps is the first systematic attempt reported in this area. For load sharing among wires in a cable, Matteo et al (1994) used two models to describe the cable

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behavior at an ultimate load: ductile and ductile-brittle models. The ductile model implicitly assumes that each and every unbroken wire in the cable cross-section possesses perfect ductility, so that they ultimately fail simultaneously. The ductile-brittle model also assumes perfect ductility for the wires, but with brittle wires excluded from the cable coss-seciotn. Brittle wires were defined as those that have an elongation limit less than 0.6 percent. Actually, test results show that all wires are brittle to some degree, as shown by a typical wire stress stain relation in Fig.1 (Bieniek 1990). This model ignores the interaction among wires in load- and deformation-sharing, as well as the uncertainty in this process. This interaction became impossible to model when thousands of wires were averaged as done. Perry (1999) proposed another set of models for cable strength estimation. The Type III extreme distribution was used to model wire strength uncertainty and the same concept was used for averaging wire properties over the cable cross section. Namely, no distinction was made among deteriorated and intact wires in modeling wire strength. Further, the same ductile model as in (Matteo et al 1994) was used as one model. The second model was for brittle wires and it used the statistics for the so-called Daniels Model (1945). That model is advantageous in that the threads (or wires) are modeled as brittle material. Namely, there is a strength limit for a thread (or wire) to break. On the other hand, the Daniels Model was developed with an assumption that all threads in the system have an identical probability distribution and there exists no deformation limit for the threads to elongate. This assumption may be reasonable for textile fibers for which the model was originally developed, but is questionable for wires in a suspension bridge cable. Betti et al. (1998, 1999) proposed a model that plausibly addressed the role of sequential wire breakage in the process of cable failure. They also provided test results showing that cable strength (by testing) is indeed lower than that predicted by the ductile model discussed above - by more than 10 percent. Although the cable force was first described as a function of strain in the proposed model, the subsequent simplification in the analysis treated the cable force as a function of only the wire forces with the variable of wire strain excluded. This simplification was based on an assumption that the wire stress-strain curves are essentially identical. In reality, this may hardly be the case, as shown in test data (Steinman 1988). The above models focused on a section of the cable, believed to be the weakest section. This cable section is assumed to be clamped at its two ends. Clamping is provided by bands connecting the cable and the suspensions. Based on field testing results, the length of a clamped section is about 1 to 3 intervals between bands (Matteo et at. 1994, Steinman 1988). The cable actually has a large number of these sections. Being a nonredundant system, the cable may fail when any one of these sections fails. When these sections are highly statistically correlated, the cable’s safety (which is a systemreliability) is about equal to the smallest value of any section’s safety (which are the component reliabilities). When the correlation between these sections is reduced (for example, due to degradation), the system reliability decreases considerably if a large number of sections are involved. This will decrease the system reliability possibly below


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that of the worst section. This effect needs to be taken into account in cable strength modeling. Proposed Cable Strength Model The failure of a cable under increasing load is a complex process, because the composing wires interact with each other in sharing load and deformation. It is also a process with uncertainty. For example, the strength and ductility-limit are not constant for all the wires. A model for describing cable failure under load is proposed here. This model is intended to provide a rational method for determining a nominal cable capacity for safety evaluation. This model should also be used for understanding the cable capacity’s expected value and uncertainty. Accordingly a probabilistic model is used here. A wire segment may fail under a load due to either excessive force or excessive deformation (possibly occurring in a local area). When any one of these two conditions is realized, the segment fails. Ideally perfect ductility that permits a wire to yield without strain limit does not exist. With this view, it is not difficult to state that all wires are brittle with various levels of brittleness. The ductility limit is emphasized here, which was not adequately covered in the previous models. Furthermore, associated uncertainties are also addressed in this model. A section of the cable is modeled here as a system of N wires in parallel, and each wire consists of M segments. The mechanical properties of a segment can be described by its stress-strain curve as shown in Fig.1 as an example for a wire segment extracted from the Williamsburg Bridge (Bieniekw 1990). Under a load S, these N wires (if not broken) will be subject to the same amount of deformation. Note that these wires may not carry the same load (i.e., S is not equally shared by the wires) depending on the wires’ cross sections, yield strengths, etc. Further, strain and thus deformation may not be uniformly distributed along a wire, because the ductility of each segment may not be identical. As a matter of fact, deformation may “concentrate” in certain segments where severe deterioration has taken place, making them break earlier than other segments and wires. Thus, when the load increases, a segment of wire will fail first due to excessive stress or strain, then the wire containing this segment fails. The load carried by the broken wire then has to be taken over by other wires in the cross-section. This process will continue until all wires fail. The maximum load reached in this process is defined as the section’s cable capacity. Note that there may be significant uncertainties in the capacities of these segments and wires. This model can be realized by computer simulation to include the interaction among wires. For applying the proposed model to a cable with a large number (thousands) of wires, over the cross-section of the cable, the wires need to be grouped according to their behavior in strength and ductility, as well as the type and severity of deterioration. Then, a number of wire samples need to be taken and physically tested, and statistics describing the wire’s strength and ductility limits should be established for each group using these in-situ samples. This cable section’s strength can be modeled using the Monte Carlo simulation. Mechanical properties of the wires are treated as random variables for each Fu, Moses, and Khazem

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wire group. They include the controlling points in the stress stain curves, such as ε1, σ1, ε2, σ2, Ε2, and εu in Fig.1. Within each group, the same probability distributions are used to describe these random variables. The statistical parameters of these random variables are determined using the test results. The Monte Carlo simulation will start with generating realizations of these random variables. These realizations are then used to calculate realizations of the ultimate strength of the cable. The loading process is simulated using a deformation control approach. Namely, the deformation of the cable (equal to that of the wires) will be increased by increments. At each increment, the wires’ ductility limits are checked. When this limit is reached for a group of wires, the cable’s load capacity is then computed. Then this wire group is excluded from the cable cross section, and the next deformation increment is imposed for repetitive computations until all the groups of wires have failed. This process establishes a load-deformation curve for the cable. The maximum load reached in the curve is taken as the cable strength. When an adequate number of cable strength realizations is obtained using this simulation method, relevant statistical parameters can be estimated for the cable stength, such as the mean, standard deviation, and possibly the probability distribution. For a cable made of many sections, any one or more sections’ failure will lead to the cable’s failure. This “chain” system’s reliability depends on the sections’ mutual statistical dependence. The system (cable) reliability could be significantly reduced when the sections’ correlation reduces and the number of sections becomes large (Fu 1994). Above discussions for cable failure refer to one single time in the service life of a cable. However, the strength of a cable reduces with time due to cable degradation. The stength estimation for a cable should reflect its time-dependent degradation. For the proposed model it is not difficult to update the wires’ strength and ductility limits at different times when such information becomes available. Cable degradation is possibly influenced by a variety of factors. Examples are atmospheric corrosion, pitting corrosion, fatigue corrosion, stress corrosion, and hydrogen embritlement. An Example For simplicity and advantages of available test data, an example of cable with 7 wires are included here for illustration. These wires were taken from the Williamsburg Bridge in New York (Steinman 1988). Three wire samples were tested to obtain their stress-strain curve. A typical example is shown in Fig.1. These curves consist of three stages. The first stage exhibits a linear elastic behavior from the origin to point (ε1,σ1). The second stage shows a nonlinear behavior (also inelastic in a later portion), starting at (ε1,σ1) and ending at (ε2,σ2). The third one exhibits a linear stress-stain relation, starting at (ε2,σ2) and ending at an ultimate strain εu. This linear relation is characterized by this stage’s constant tangent marked as E2 in Fig.1. The wire test results are used here to obtain the statistical parameters needed for the Monte Carlo simulation. For estimating the ultimate strength of the 7-wire cables using the proposed model, a total of 5 random variables are used in this example, namely, ε2, σ2, Ε2, εu, and A, where A is the cross section of wire. Due to the limited data used here, the correlation of these


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random variables could not be reliably estimated. It is assumed then that these random variables are not correlated. A total of 20 cable samples are used here. Each cable sample consists of 7 wires, each having 5 realizations respectively for the 5 random variables. Two choices of the probability distributions for these 5 random variables are used: normal and lognormal distributions for all the random variables. This is because the limited data did not permit reliable estimation for the probability distribution types. The mean value of the cable strength was found to be 43.06 and 42.01 kips (191.5 and 186.1 kN) respectively for the normal and lognormal distribution assumptions. The test results give an average of 42.17 kips (187.6 kN) based on the three 7-wire cables tested (Bieniek 1990). These results are reasonably close to each other. Conclusions Previously proposed strength estimation models for suspension bridge cables do not adequately cover the ductility limits of the wires. The model proposed herein uses a probabilistic approach to taking this important aspect into account. The example shows that the proposed model can reasonably predict the ultimate strength of cables, based on comparison with physical test results. It highlights the importance for acquiring full information on the stress-strain relation for wires. Further work is needed to evaluate the proposed model for cables with a larger number of wires.

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Figure 1 Typical Stress Strain Relation of Suspension Bridge Cable Wire Acknowledgements It is acknowledged that the work reported here was performed under support from the CULMA of Wayne State University and the US Federal Highway Administration. This suport is gratefully appreciated. References Betti,R. and Bieniek,M.P. (1998) “The Condition of Suspension Bridge Cables”, Technical Report to NYCDOT, NYS Bridge Authority, Port Authority of NY&NJ, Triborough Bridge &Tunnel Authority, Columbia Univ., New York, NY


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Betti,R. and Yanev,B. (1999) “ Conditions of Suspension Bridge Cables: The New York City case Study”, 1999 TRB Annual Meeting Bieniek,M.P. (1990) “Williamsburg Bridge Cable Rehabilitation – Testing Program of Resin-Poured Sockets”, Final Report to Steinman, Aug. 1990 Daniels,H.E. “The Statistical Theory of the Strength of Bundles of Threads. I” Proceedings of the Royal Society of London, Series A, 183, p.405 Fu,G. (1994) "Variance Reduction by Truncated Multimodal Importance Sampling", International Journal of Structural Safety, Vol.13, 1994, p.267 Matteo,J., Deodadis,G., and Billinginton,D.P. (1994) “Safety Analysis of Suspension Bridge Cables: Williamsburg Bridge”, ASCE J. of Structural Engineering, Vol.120, No.11, 1994 Perry,R.J. (1999) “Estimating Suspension Cable Strength”, ASCE Struc.Cong, New Orleans, LA, 4/19/1999, p.442 Steinman, Boynton, Gronquist, and Birdsall (1988) in Association with Columbia University “Williamsburg Bridge Cable Investigation Program”, Final Report to New York State DOT and New York City DOT, 1988


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