Stability Analysis of Box-Girder Cable-Stayed Bridges

The objective of this study is to investigate the stability characteristics of box-girder cable-stayed bridges by three-dimensional finite-element methods. Cable-stayed bridges have many design parameters, because they have a lot of redundancies, especially for long-span bridges. Cable-stayed bridges exhibit several nonlinear behaviors concurrently under normal design loads because of large displacements; the interaction among the pylons, the stayed cables, and the bridge deck; the strong axial and lateral forces acting on the bridge deck and pylons; and cable nonlinearity. A typical two-lane, three-span, steel box-girder cable-stayed bridge superstructure was selected for this paper. The numerical results indicate that, if the ratio of the main span length with respect to the total span length, L1∕L, is small, the structure usually has a higher critical load. If the ratio Ip∕Ib increases, the critical load of the bridge decreases, in which Ip is the moment of inertia of the pylon and Ib is the moment of inertia of the bridge deck. When the ratio Ip∕Ib is greater than 10.0, the decrement becomes insignificant. For cable arrangements, bridges supported by a harp-type cable arrangement are the better design than bridges supported by a fan-type cable arrangement on buckling analysis. The numerical results also indicate that use of either A-type or H-type pylons does not significantly affect the critical load of this type of structure. In order to make the numerical results useful, the buckling loads have been nondimensionalized and presented in both tabular and graphical forms.     

Number of Cable Effects on Buckling Analysis of Cable-Stayed Bridges

Cables instead of interval piers support cable-stayed bridges, and the bridge deck is subjected to strong axial forces due to the horizontal components of cable reactions. The structural behavior of a bridge deck becomes nonlinear because of the axial forces, large deflection, and nonlinear behavior of the cables and the large deformation of the pylons as well as their interactions. The locations and amplitude of axial forces acting on the bridge deck may depend on the number of cables. Agrawal indicated that the maximum cable tension decreases rapidly with the increase in the number of cables. This paper investigates the stability analysis of cable-stayed bridges and considers cable-stayed bridges with geometry similar to those proposed in Agrawal's paper. A digital computer and numerical analysis are used to examine 2D finite element models of these bridges. The eigen buckling analysis has been applied to find the minimum critical loads of the cable-stayed bridges. The numerical results indicate that the total cumulative axial forces acting on the bridge girder increase as the number of cables increases, yet because the bridge deck is subjected to strong axial forces, the critical load of the bridges decreases. Increasing the number of cables may not increase the critical load on buckling analysis of this type of bridge. The fundamental critical loads increase if the ratio of Ip∕Ib increases until the ratio reaches the optimum ratio. If the ratio of Ip∕Ib is greater than the optimum ratio, depending on the geometry of an individual bridge, the fundamental critical load decreases for all the types of bridges considered in this paper. In order to make the results useful, they have been normalized and represented in graphical form.     

Estimating Fatigue Life of Bridge Components Using Measured Strains

The design fatigue life of a bridge component is based on the stress spectrum the component experiences and the fatigue durability. Changes in traffic patterns, volume, and any degradation of structural components can influence the fatigue life of the bridge. A fatigue life evaluation reflecting the actual conditions has value to bridge owners. Procedures are outlined in the AASHTO Guide Specifications for Fatigue Evaluation of Existing Steel Bridges to estimate the remaining fatigue life of bridges using the measured strain data under actual vehicular traffic. This paper presents the methodology with an actual case study of Patroon Island Bridge. The Patroon Island Bridge consists of ten spans. Spans 3 through 9 are considered the main spans and consist of steel trusses and concrete decks. Spans 1, 2, and 10 are considered approach spans and consist of plate girders. The overall bridge length is 1,795 feet. Strain data from critical structural members were used to estimate the remaining fatigue life of selected bridge components. The results indicate that most of the identified critical details have an infinite remaining safe fatigue life and others have a substantial fatigue life. Cracked floor beams were not addressed in this analysis, but have been recommended for retrofitting or replacement.     

Control of Suspension Bridge Nonlinear Vibrations due to Moving Loads

The flexibility and low damping of the long-span suspended cables in the suspension bridges make them prone to vibrations due to wind and moving loads, which affect the dynamic response of the suspended cables and the bridge deck. This paper shows the design of two control schemes to control the nonlinear vibrations in the suspended cable and the bridge deck due to a vertical load moving on the bridge deck with a constant speed. The first control scheme is an optimal state feedback controller. The second control scheme is a robust state feedback controller, whose design is based on the design of optimal controllers. The proposed controllers, whose design is based on Lyapunov theory, guarantee the asymptotic stability of the system. A vertical cable between the bridge deck and the suspended cable is used to install a hydraulic actuator able to generate the active control force on the bridge deck. The MATLAB software is used to simulate the performance of the system with the designed controllers. The simulation results indicate that the proposed controllers are capable of significantly reducing the nonlinear oscillations of the system. In addition, the performance of the system with the proposed controllers is compared to the performance of the system controlled with a velocity feedback controller. It is found that the system with the proposed controllers can provide better performance than the system with the velocity feedback controller.     

Performance and Repair Design of Stay Cables with Rain/Wind-Induced Vibrations

The results of a multifaceted investigation into large-amplitude vibrations of bridge stay cables are presented. The vibration of the stay cables occurred under light rain and wind conditions, and resulted in low-frequency, large-amplitude oscillations having recorded displacements of over 24?in. (0.6?m). Due to concerns about the implications of the vibrations on the integrity and durability of the stay cables and observed damage to stay-cable anchorage components, an investigation and repair program was developed. The final repair program arose from an investigation that combined the talents of practicing engineers, the Texas Department of Transportation, and representatives from four universities. Ultimately, a combination of supplemental hydraulic dampers and cable restrainers were used to mitigate the vibrations. This paper describes the results of the investigation and presents the practice-based methodology used to develop the implemented repair program.     

Experimental Verification of Smart Cable Damping

Stay cables, such as are used in cable-stayed bridges, are prone to vibration due to their low inherent damping characteristics. Transversely attached passive viscous dampers have been implemented in many bridges to dampen such vibration. However, only minimal damping can be added if the attachment point is close to the bridge deck. For longer bridge cables, the relative attachment point becomes increasingly smaller, and passive damping may become insufficient. A recent analytical study by the authors demonstrated that “smart” semiactive damping can provide increased supplemental damping. This paper experimentally verifies a smart damping control strategy employing H2/linear quadratic Gaussian (LQG) clipped optimal control using only force and displacement measurements at the damper for an inclined flat-sag cable. A shear mode magnetorheological fluid damper is attached to a 12.65?m inclined flat-sag steel cable to reduce cable vibration. Cable response is seen to be substantially reduced by the smart damper.     

Probabilistic Strength Estimates and Reliability of Damaged Parallel Wire Cables

Cable reliability analysis involves the combined evaluation of cable capacity and cable load in a probabilistic manner. Assessment of cable capacity is only possible through visual inspections of the wires, field sampling, laboratory analysis of the degraded wire populations, and analytical techniques. In addition to a brief presentation of cable mechanics and deterministic models that approximate cable strength, this paper discusses inspection methodologies and statistical methods of estimation of the sizes of the degraded wire populations, and wire properties, leading to cable capacities. These capacities are described by probability distributions. The paper also discusses fundamentals of reliability analysis as they apply to bridge cables. Load criteria of present standard specifications (such as AASHTO or other international codes) are not applicable to long-span suspension bridges. The paper discusses criteria of bridge loading and reliability indices for bridge cables. More work is needed in the evaluation of loading for long-span bridges.     

Three-Dimensional Elastic Catenary Cable Element Considering Sliding Effect

The nonlinear behavior of cable-supported bridges is governed by the geometric nonlinearity of cables, which is attributable to sag and sliding effects at the saddle. In a cable-stayed bridge with a midspan saddle, and in all suspension bridges, cable sliding can occur at the saddle under extreme forces, such as those caused by an earthquake or typhoon. However, the conventional method of analysis of cable-supported bridges does not consider the effect of cable sliding at the saddle; instead it regards those cables as fixed. This assumption might lead to a misunderstanding of the global structure system. The goal of this study is to develop a three-dimensional (3D) elastic cable finite element that considers the sliding effect and uses a geometric nonlinear cable finite element based on elastic catenary theory. In this study, two types of sliding were considered: the roller sliding condition without friction and the frictional sliding condition. These were formulated to derive the nodal force vectors and tangential stiffness matrices. To validate the proposed 3D cable sliding element, experiments were conducted for both sliding conditions, and results were compared with calculations of the amount of sliding and displacement at the loading point. In addition, a cable-supported structural system was analyzed to investigate the characteristics of a realistic structure with cable sliding. Overall calculations using the 3D cable sliding model were in very good agreement with the measured values.     

Vibration Control of Stay Cables of the Shandong Binzhou Yellow River Highway Bridge Using Magnetorheological Fluid Dampers

The Shandong Binzhou Yellow River Highway Bridge is a three-tower, cable-stayed bridge in Shandong Province, China. Because the stay cables are prone to vibration, 40 magnetorheological (MR) fluid dampers were attached to the 20 longest cables of this bridge to suppress possible vibration. An innovative control algorithm for active and semiactive control of mass-distributed dynamic systems, e.g., stay cables, was proposed. The frequencies and modal damping ratios of the unimpeded tested cable were identified through an ambient vibration test and free vibration tests, respectively. Subsequently, a series of field tests were carried out to investigate the control efficacy of the free cable vibrations achieved by semiactive MR dampers, “Passive-off” MR dampers and “Passive-on” MR dampers. The first three modal damping ratios of the cable incorporated with the MR dampers were also identified from the in situ experiments. The field experiment results indicated that the semiactive MR dampers can provide significantly greater supplemental damping for the cable than either the Passive-off or the Passive-on MR dampers because of the pseudonegative stiffness generated by the semiactive MR dampers.     

Cable Erection Test at Splay Band for Spatial Suspension Bridge

The Yongjong Grand Bridge includes a self-anchored suspension bridge with inclined cable planes. The bridge uses splay bands (cable collars) to flare the main cables at the anchorage, which is located at the end of a stiffening truss. During cable erection, some of the wires at the splay band were expected to experience lateral displacement and/or lift phenomena because of the large flare angles at the splay band. Mockup cable erection tests at the anchorage were carried out to find the degree of displacement of the wires and to determine appropriate measures to deal with these problems. Through these tests, methods to arrange wires at the splay bands were devised and tried, and the selected method was successfully used for the actual bridge.     

Roebling Suspension Bridge. II: Ambient Testing and Live-Load Response

This is the second part of a two-part paper on the evaluation of the historic Roebling suspension bridge using dynamic-analysis techniques. Dynamic properties are determined using ambient field testing under natural excitation. The finite-element (FE) model described in the first part of this two-part paper is modified to more accurately represent current bridge properties. Modifications of the model are based on correlating the FE model frequencies with ambient test frequencies by adjusting the FE model stiffness parameters. The updated 3D FE model is subsequently subjected to an extreme live-load condition to evaluate static safety margins. In addition, cable areas are reduced by 10 to 40% to simulate further deterioration and corrosion. The safety margin of the main cables is demonstrated to be good even when assuming a very conservative 40% cable area reduction, and truss member forces remain within the maximum load-carrying capacity even when the cable areas are reduced by 40%.     

Estimation of Tension in Cables with Intermediate Elastic Supports Using Finite-Element Method

Cables with intermediate elastic supports are widely used in bridge engineering, but there is a lack of effective methods for estimating the tensions in these cables. In this paper, an analytical form of the equilibrium equation for a tension bar has been used to provide a shape function to develop a suitable finite-element model for a cable. A program for estimating the tension in cables with intermediate elastic supports has been developed in which the bending stiffness of the cable, the stiffness of the end restraints, and the stiffness of the intermediate supports are taken into account by using this element and MATLAB. The program has been verified by comparison of the results from laboratory experiments. The method has been applied to compute the tension in the tie bars of a concrete-filled steel tube arch bridge. This demonstrates that the cable tension estimated by the present method is suitably accurate.     

Field Static Load Test on Kao-Ping-Hsi Cable-Stayed Bridge

Field load testing is an effective method for understanding the behavior and fundamental characteristics of a cable-stayed bridge. This paper presents the results of field static load tests on the Kao-Ping-Hsi cable-stayed bridge, the longest cable-stayed bridge in Taiwan, before it was open to traffic. A total of 40 loading cases, including the unit and distributed bending and torsion loading effects, were conducted to investigate the bridge behavior. The atmospheric temperature effect on the variations of the main girder deflections was also monitored. The results of static load testing include the main girder deflections, the flexural strains of the prestressed concrete girder, and the variations of the cable forces. A three-dimensional finite-element model was developed. The results show that the bridge under the planned load test conditions has linear superposition characteristics and the analytical model shows a very good agreement with the bridge responses. Further discussion of deflection and cable forces of the design specifications for a cable-stayed bridge is also presented.     

Realistic Assessment for Safety and Service Life of Reinforced Concrete Decks in Girder Bridges

The lack of safety of deck slabs in bridges generally causes frequent repair and strengthening. The repair induces great loss of economy, not only due to direct cost by repair, but also due to stopping the public use of such structures during repair. The major reason for this frequent repair is mainly due to the lack of a realistic and accurate assessment system for bridge decks. The purpose of the present paper is therefore to develop a realistic assessment system which can estimate reasonably the safety, as well as the service life of concrete bridge decks, based on the deterioration models that are derived from both the traffic loads and environmental effect. A deterioration model due to chloride ingress is first established. The damage models due to repetitive traffic loads considering the dry and wet conditions of deck slabs are proposed. These models are used to calculate the remaining life of a bridge deck slab. A prediction method for service life of deck slab due to loading and environmental effects is developed based on material, as well as structural evaluation. The proposed method includes the assessment of corrosion in material level, and the analyses of flexure, shear, and fatigue in structural level. Finally, an assessment system for prediction of safety and remaining service life is developed based on the theories established in this study. The developed assessment system will allow the correct diagnosis of damage state and the realistic prediction of service life of concrete decks in girder bridges.     

Statewide Vessel Collision Design for Bridges

Design Method II for vessel collision design is recommended by AASHTO specifications for most bridges. With its required resources more elaborate than Method I, Method II classifies vessel traffic passing under a specific bridge into different categories. To achieve an economical vessel collision design, a strategy is proposed to perform design Method II analysis on a statewide basis. In association with the waterways of Florida, the vessel traffic data are collected from available resources and a synthesizing methodology is developed to statistically process these data by vessel types, sizes, and loading condition. This study provides statewide input data ready for vessel collision analysis adopting Method II. As a numerical example, an illustrative bridge with multiple piers is designed in accordance with Method II. Waterborne traffic data from Indian River in Florida are utilized as input data. Comparison among annual frequencies of collapse due to various vessel categories indicates that barge trains are statistically significant in the collision design. Future traffic growth is considered using two different approaches and the possible difference is studied.     

Design of Mechanical Viscous Dampers for Stay Cables

External dampers have been utilized in a number of cable-stayed bridges to suppress transverse cable vibrations. However, simple and accurate damper design recommendations that concurrently consider all important cable parameters are lacking. Previous efforts have been based on the idealization of cables as taut strings. In this paper, the governing differential equation for vibration of cables containing a viscous damper was first converted to a complex eigenvalue problem containing nondimensional cable parameters. Then, a parametric study was conducted involving repeated solutions of the eigenvalue problem for a wide range of nondimensional parameters. Based on the results of the parametric study, the effects of dampers on first mode vibration frequencies and first mode cable damping ratios were presented in nondimensional format. It is shown that for the range of parameters involved in most stay cables, the influence of cable sag is insignificant, whereas the cable bending stiffness can have a significant influence on the resulting cable damping ratios. Simplified nondimensional relationships are proposed for calculating damper-induced changes in the first mode cable damping ratios. Results of laboratory tests on a scaled model cable are compared with the estimated values using the formulation presented. Finally, example problems are presented for comparison with other relationships, and for the design of mechanical viscous dampers for suppression of cable vibrations including rain-wind induced vibrations.     

Suspension Cable Design of the New San Francisco–Oakland Bay Bridge

Various factors contribute to the difficulty in designing the main suspension cable for the new San Francisco–Oakland Bay Bridge Self-Anchored Suspension Span (or East Bay Bridge Suspension Span). The key factors are bridge design life, cable geometry, cable anchorage layout, cable construction method, and cable corrosion protection system. This paper describes the unique main suspension cable geometry layout for the East Bay Bridge Suspension Span, reviews the available technologies for each of the aforementioned design considerations, and presents the final cable design recommendations.     

Vibration Studies of Tsing Ma Suspension Bridge

A three-dimensional dynamic finite element model is established for the Tsing Ma long suspension Bridge in Hong Kong. The two bridge towers made up of reinforced concrete are modeled by three-dimensional Timoshenko beam elements with rigid arms at the connections between columns and beams. The cables and suspenders are modeled by cable elements accounting for geometric nonlinearity due to cable tension. The hybrid steel deck is represented by a single beam with equivalent cross-sectional properties determined by detailed finite element analyses of sectional models. The modal analysis is then performed to determine natural frequencies and mode shapes of lateral, vertical, torsional, longitudinal, and coupled vibrations of the bridge. The results show that the natural frequencies of the bridge are very closely spaced; the first 40 natural frequencies range from 0.068 to 0.616 Hz only. The computed normal modes indicate interactions between the main span and side span, and between the deck, cables, and towers. Significant coupling between torsional and lateral modes is also observed. The numerical results are in excellent agreement with the measured first 18 natural frequencies and mode shapes. The established dynamic model and computed dynamic characteristics can serve further studies on a long-term monitoring system and aerodynamic analysis of the bridge.