Dynamic Response of Suspension Bridge to High Wind and Running Train

A framework is presented for predicting the dynamic response of long suspension bridges to high winds and running trains. A three-dimensional finite-element model is used to represent a suspension bridge. Wind forces acting on the bridge, including both buffeting and self-excited forces, are generated in the time domain using a fast spectral representation method and measured aerodynamic coefficients and flutter derivatives. Each 4-axle vehicle in a train is modeled by a 27-degrees-of-freedom dynamic system. The dynamic interaction between the bridge and train is realized through the contact forces between the wheels and track. By applying a mode superposition technique to the bridge only and taking the measured track irregularities as known quantities, the number of degrees of freedom of the bridge-train system is significantly reduced and the coupled equations of motion are efficiently solved. The proposed formulation is then applied to a real wind-excited long suspension bridge carrying a railway inside the bridge deck of a closed cross section. The results show that the formulation presented in this paper can predict the dynamic response of the coupled bridge-train systems under fluctuating winds. The extent of interaction between the bridge and train depends on wind speed and train speed.     

Safety Analysis of Moving Road Vehicles on a Long Bridge under Crosswind

This paper presents a safety analysis of high-sided road vehicles running on a long span cable-stayed bridge when the road vehicle enters a sharp-edged crosswind gust while the bridge is oscillating under fluctuating winds. Road vehicle accidents, including overturning, excessive sideslip, and exaggerated rotation, are defined first. The mathematical model and the equation of motion of coupled road vehicle–bridge systems under crosswind are then established, which include road surface roughness, vehicle suspension, and the sideslip of the vehicle tire relative to the bridge deck in the lateral direction. A case study using a real long cable-stayed bridge and a high-sided road vehicle is finally conducted, and an extensive computational work is performed to obtain a series of accident vehicle speed against mean crosswind speed, by which the decision on the threshold of mean wind speed above which the bridge should be closed to the road vehicle can be made. The obtained accident vehicle speeds are also compared with those for the same vehicles running on the ground. It is shown that the oscillation of the cable-stayed bridge will lower the accident vehicle speed when wind speed reaches a certain level.     

Dynamic Performance Simulation of Long-Span Bridge under Combined Loads of Stochastic Traffic and Wind

Slender long-span bridges exhibit unique features which are not present in short and medium-span bridges such as higher traffic volume, simultaneous presence of multiple vehicles, and sensitivity to wind load. For typical buffeting studies of long-span bridges under wind turbulence, no traffic load was typically considered simultaneously with wind. Recent bridge/vehicle/wind interaction studies highlighted the importance of predicting the bridge dynamic behavior by considering the bridge, the actual traffic load, and wind as a whole coupled system. Existent studies of bridge/vehicle/wind interaction analysis, however, considered only one or several vehicles distributed in an assumed (usually uniform) pattern on the bridge. For long-span bridges which have a high probability of the presence of multiple vehicles including several heavy trucks at a time, such an assumption differs significantly from reality. A new “semideterministic” bridge dynamic analytical model is proposed which considers dynamic interactions between the bridge, wind, and stochastic “real” traffic by integrating the equivalent dynamic wheel load (EDWL) approach and the cellular automaton (CA) traffic flow simulation. As a result of adopting the new analytical model, the long-span bridge dynamic behavior can be statistically predicted with a more realistic and adaptive consideration of combined loads of traffic and wind. A prototype slender cable-stayed bridge is numerically studied with the proposed model. In addition to slender long-span bridges which are sensitive to wind, the proposed model also offers a general approach for other conventional long-span bridges as well as roadway pavements to achieve a more realistic understanding of the structural performance under probabilistic traffic and dynamic interactions.     

Feasibility of a 1,400 m Span Steel Cable-Stayed Bridge

This paper describes the feasibility of 1,400 m steel cable-stayed bridges from both structural and economic viewpoints. Because the weight of a steel girder strongly affects the total cost of the bridge, the writers present a procedure to obtain a minimum weight for a girder that ensures safety against static and dynamic instabilities. For static instability, elastoplastic, finite-displacement analysis under in-plane load and elastic, finite-displacement analysis under displacement-dependent wind load are conducted; for dynamic instability, multimodal flutter analysis is carried out. It is shown that static critical wind velocity of lateral torsional buckling governs the dimension of the girder. Finally, the writers briefly compare a cable-stayed bridge with suspension bridge alternatives.     

Vehicle Induced Dynamic Behavior of Short-Span Slab Bridges Considering Effect of Approach Slab Condition

In this paper the vehicle induced dynamic bridge responses are calculated by modeling the bridge and vehicle as one coupled system. The dynamic behavior of short slab bridges with different span lengths induced by the AASHTO HS20 truck is investigated. A parametric study is conducted to analyze the effects of different truck speeds and different road surface conditions. Critical truck speeds that result in peaks of dynamic response are found to follow the rule that describes the resonant vibration of bridges due to train loading. The approach slab condition that consists of faulting at the ends and deformation along the span is considered in the analysis. Although the effect of the along-span deformation on the dynamic response of bridges is trivial, the faulting condition of the approach slab is found to cause significantly large dynamic responses in short-span slab bridges. Impact factors obtained from numerical analyses are compared with those values specified in the AASHTO codes.     

Vibration Control of Railway Bridges under High-Speed Trains Using Multiple Tuned Mass Dampers

In this paper, multiple tuned mass dampers (MTMDs) are considered for suppressing the vibration of railway bridges under high-speed trains. The interaction equations of motion between the vehicle and the bridge with MTMDs have been developed. The effectiveness of MTMDs on suppressing resonant vibration of railway bridges is examined and the optimum parameters of MTMDs for suppressing the resonant vibration are proposed. The results indicate that the use of the MTMD with the optimum parameters reduces the displacement and acceleration responses of railway bridges significantly.     

Dynamic and Impact Behavior of Half-Through Arch Bridges

The objective of this paper is to present the results of an investigation of the dynamic and impact characteristics of half-through arch bridges with rough decks caused by vehicles moving across them. Seven arch bridges modeled as three-dimensional structures with overall span lengths ranging from 20?to?200?m (65.5?to?656.2?ft) are analyzed. The American Association of State Highway and Transportation Officials Specifications HS20-44 truck is the applied vehicle loading used in the analysis and is simulated as a three-dimensional, nonlinear vehicle model with 11 degrees of freedom. Truck components include the body, suspension, and tires. The bridge deck surface is assumed to have a “good” surface roughness and is simulated using a stochastic process (power spectral density function). The effect on impact factors of span length, rise-to-span ratio, and vehicle speed is discussed. The results of the analyses show that the impact factors of bending moment and axial force will not exceed 0.4 and 0.25, respectively. The proposed impact equations are simple and conservative and can be used in the design of half-through arch bridges.     

Fatigue Damage of Steel Bridges Due to Dynamic Vehicle Loads

This paper focuses on the fatigue damage caused in steel bridge girders by the dynamic tire forces that occur during the crossing of heavy transport vehicles. This work quantifies the difference in fatigue life of a short-span and a medium-span bridge due to successive passages of either a steel-sprung or an air-sprung vehicle. The bridges are modeled as beams to obtain their modal properties, and air-sprung and nonlinear steel-sprung vehicle models are used. Bridge responses are predicted using a convolution method by combining bridge modal properties with vehicle wheel forces. A linear elastic fracture mechanics model is employed to predict crack growth. For the short-span bridge, the steel-sprung vehicle caused fatigue failure up to 6.5 times faster than the air-sprung vehicle. For the medium-span bridge, the steel-sprung vehicle caused fatigue failure up to 277 times faster than the air-sprung vehicle.     

Curve Veering of Eigenvalue Loci of Bridges with Aeroelastic Effects

The eigenvalues of bridges with aeroelastic effects are commonly portrayed in terms of a family of frequency and damping loci as a function of mean wind velocity. Depending on the structural dynamic and aerodynamic characteristics of the bridge, when two frequencies approach one another over a range of wind velocities, their loci tend to repel, thus avoiding an intersection, whereas the mode shapes associated with these two frequencies are exchanged in a rapid but continuous way as if the curves had intersected. This behavior is referred to as the curve veering phenomenon. In this paper, the curve veering of cable-stayed and suspension bridge frequency loci is studied. A perturbation series solution is utilized to estimate the variations of the complex eigenvalues due to small changes in the system parameters and establish the condition under which frequency loci veer, quantified in terms of the difference between adjacent eigenvalues and the level of mode interaction. Prior to the discussion of bridge frequency loci, the curve veering of a two-degree-of-freedom system comprised of a primary structure and tuned mass damper is discussed, which not only provides new insight into the dynamics of this system, but also helps in understanding the veering of bridge frequency loci. To study this more complicated dynamic system, a closed-form solution of a two-degree-of-freedom coupled flutter is obtained, and the underlying physics associated with the heaving branch flutter is discussed in light of the veering of frequency loci. It is demonstrated that the concept of curve veering in bridge frequency loci provides a correct explanation of multimode coupled flutter analysis results for long span bridges and helps to improve understanding of the underlying physics of their aeroelastic behavior.     

Equivalent Wheel Load Approach for Slender Cable-Stayed Bridge Fatigue Assessment under Traffic and Wind: Feasibility Study

In the current AASHTO LRFD specifications, the fatigue design considers only one design truck per bridge with 15% dynamic allowance. While this empirical approach may be practical for regular short and medium span bridges, it may not be rational for long-span bridges (e.g., span length >152.4?m or 500?ft) that may carry many heavy trucks simultaneously. Some existent studies suggested that fatigue may not control the design for many small and medium bridges. However, little research on the fatigue performance of long-span bridges subjected to both wind and traffic has been reported and if fatigue could become a dominant issue for such a long-span bridge design is still not clear. Regardless if the current fatigue design specifications are sufficient or not, a real understanding of the traffic effects on bridge performance including fatigue is desirable since the one truck per bridge for fatigue design does not represent the actual traffic condition. As the first step toward the study of fatigue performance of long-span cable-stayed bridges under both busy traffic and wind, the equivalent dynamic wheel load approach is proposed in the current study to simplify the analysis procedure. Based on full interaction analyses of a single-vehicle–bridge–wind system, the dynamic wheel load of the vehicle acting on the bridge can be obtained for a given vehicle type, wind, and driving condition. As a result, the dimension of the coupled equations is independent of the number of vehicles, through which the analyses can be significantly simplified. Such simplification is the key step toward the future fatigue analysis of long-span bridges under a combined action of wind and actual traffic conditions.     

Impact Factors for Curved Continuous Composite Multiple-Box Girder Bridges

The results from a parametric study on the impact factors for 180 curved continuous composite multiple-box girder bridges are presented. Expressions for the impact factors for tangential flexural stresses, deflection, shear forces and reactions are deduced for AASHTO truck loading. The finite-element method was utilized to model the bridges as three-dimensional structures. The vehicle axle used in the analysis was simulated as a pair of concentrated forces moving along the concrete deck in a circumferential path with a constant speed. The effects of bridge configurations, loading positions, and vehicle speed on the impact factors were examined. Bridge configurations included span length, span-to-radius of curvature ratio, number of lanes, and number of boxes. The effect of the mass of the vehicle on the dynamic response of the bridges is also investigated. The data generated from the parametric study and the deduced expressions for the impact factors would enable bridge engineers to design curved continuous composite multiple-box girder bridges more reliably and economically.     

Train-Induced Vibration Control of High-Speed Railway Bridges Equipped with Multiple Tuned Mass Dampers

This paper deals with the applicability of multiple tuned mass dampers (MTMDs) to suppress train-induced vibration on bridges. A railway bridge is modeled as an Euler-Bernoulli beam and a train is simulated as a series of moving forces, moving masses, or moving suspension masses. According to the train load frequency analysis, resonant effects will occur as the modal frequencies of a bridge are close to the multiple of the impact frequency of the train load to the bridge. An MTMD system is then designed to alter the bridge dynamic characteristics to avoid excessive vibrations. Numerical results from simply supported bridges of the Taiwan High-Speed Railway (THSR) under real trains show that the proposed MTMD is more effective and reliable than a single TMD in reducing dynamic responses during resonant speeds, as the train axle arrangement is regular. It is also found that the inner space of a bridge box-girder of the THSR is wide and deep enough for installation and movement of MTMDs.     

Experimental Evaluation of Dynamic Effects for a Selected Highway Bridge

The paper presents an experimental study of the actual dynamic effects for a preselected typical highway bridge. Knowledge of the dynamic impact factors is important for accurate determination of the ultimate load capacity and performance assessment of constructed bridges. Static and dynamic field tests were performed on a two-lane concrete highway bridge built in 1999 on U.S. 90 in northwest Florida. During the tests, one or two fully loaded trucks crossed over the bridge, which was instrumented with strain gauges, accelerometers, and displacement transducers. A wooden plank was placed across the lanes for some runs to trigger extensive dynamic vibration and to simulate poor road surface conditions. Data collected from the tests were used for comprehensive assessment of the bridge under dynamic loading. Impact factors obtained from the tests with higher speeds were found larger than corresponding values recommended by bridge codes. Analysis revealed that stiff vehicle suspension, road surface imperfection, and “bouncing” of the truck loading contributed to the high impact factors. Experimental data were also used for validation of the finite-element models developed for the vehicle–bridge system.     

Impact Factors for Horizontally Curved Composite Box Girder Bridges

This paper presents a method for determining the dynamic impact factors for horizontally curved composite single- or multicell box girder bridges under AASHTO truck loading. The bridges are modeled as three-dimensional structures using commercially available software. The vehicle is idealized as a pair of concentrated forces, with no mass, traveling in two circumferential paths parallel to the curved centerline of bridges. An extensive parametric study is conducted, in which over 215 curved composite box girder bridge prototypes are analyzed. The key parameters considered in this study are: Number of cells, number of lanes, degree of curvature, arc span length, slope of the outer steel webs, number and area of bracing and top chord systems, and truck(s) speed and truck(s) positioning. Based on the data generated from the parametric study, expressions for dynamic impact factors for longitudinal moment, reaction, and deflection are proposed as function of the ratio of the arc span length to the radius of curvature. The results from this study would enable bridge engineers to design horizontally curved composite box girder bridges more reliably and economically. Furthermore, the results can be used to potentially increase the live-load capacity of existing bridges to prevent posting or closing of the bridge.     

Application of Suspension Bridges Stiffened by Prestressed Concrete Slabs in China

Four suspension bridges stiffened by prestressed concrete slabs were designed and constructed on highways in southwestern mountainous areas of China. These bridges are the first applications of its kind in China. This paper discusses the site condition, adaptability, and design and construction features of these bridges. These bridges have single suspension spans between 278 and 388?m and deck width between 14.4 and 15.0?m. The longitudinal distance between hangers is only 5?m, which is relatively small for this bridge type, and there are only two lanes. The dual direction prestressed concrete slabs are 0.6?m deep, and its wind blocking area is relatively small. Dynamic analysis and wind tunnel tests verify that the wind resistance requirements are easily satisfied.     

Reliability Analysis of Suspension Bridges against Fatigue Failure from the Gusting of Wind

A fatigue reliability analysis of suspension bridges due to the gustiness of the wind velocity is presented by combining overall concepts of bridge aerodynamics, fatigue analysis, and reliability analysis. For this purpose, the fluctuating response of the bridge deck is obtained for buffeting force using a finite-element method and a spectral analysis in frequency domain. Annual cumulative fatigue damage is calculated using Palmgren–Miner’s rule, stress-fatigue curve approach and different forms of distribution for stress range. In order to evaluate the reliability, both first-order second-moment (FOSM) method and full distribution procedure (assuming Weibull distribution for fatigue life) are used to evaluate the fatigue reliability. Probabilities of fatigue failure of the Thomas Bridge and the Golden Gate Bridge for a number of important parametric variations are obtained in order to make some general observations on the fatigue reliability of suspension bridges. The results of the study show that the FOSM method predicts a higher value of the probability of fatigue failure as compared to the full distribution method. Further, the distribution of stress range used in the analysis has a significant effect on the calculated probability of fatigue failure in suspension bridges.     

Roebling Suspension Bridge. I: Finite-Element Model and Free Vibration Response

This first part of a two-part paper on the John A. Roebling suspension bridge (1867) across the Ohio River is an analytical investigation, whereas Part II focuses on the experimental investigation of the bridge. The primary objectives of the investigation are to assess the bridge’s load-carrying capacity and compare this capacity with current standards of safety. Dynamics-based evaluation is used, which requires combining finite-element bridge analysis and field testing. A 3D finite-element model is developed to represent the bridge and to establish its deformed equilibrium configuration due to dead loading. Starting from the deformed configuration, a modal analysis is performed to provide the frequencies and mode shapes. Transverse vibration modes dominate the low-frequency response. It is demonstrated that cable stress stiffening plays an important role in both the static and dynamic responses of the bridge. Inclusion of large deflection behavior is shown to have a limited effect on the member forces and bridge deflections. Parametric studies are performed using the developed finite-element model. The outcome of the investigation is to provide structural information that will assist in the preservation of the historic John A. Roebling suspension bridge, though the developed methodology could be applied to a wide range of cable-supported bridges.     

Design for Shear in Curved Composite Multiple Steel Box Girder Bridges

Modern highway bridges are often subject to tight geometric restrictions and, in many cases, must be built in curved alignment. These bridges may have a cross section in the form of a multiple steel box girder composite with a concrete deck slab. This type of cross section is one of the most suitable for resisting the torsional, distortional, and warping effects induced by the bridge’s curvature. Current design practice in North America does not specifically deal with shear distribution in horizontally curved composite multiple steel box girder bridges. In this paper an extensive parametric study, using an experimentally calibrated finite-element model, is presented, in which simply supported straight and curved prototype bridges are analyzed to determine their shear distribution characteristics under dead load and under AASHTO live loadings. The parameters considered in this study are span length, number of steel boxes, number of traffic lanes, bridge aspect ratio, degree of curvature, and number and stiffness of cross bracings and of top-chord systems. Results from tests on five box girder bridge models verify the finite-element model. Based on the results from the parametric study simple empirical formulas for maximum shears (reactions) are developed that are suitable for the design office. A comparison is made with AASHTO and CHBDC formulas for straight bridges. An illustrative example of the design is presented.     

Aeroelastic Analysis of Bridges: Effects of Turbulence and Aerodynamic Nonlinearities

Current linear aeroelastic analysis approaches are not suited for capturing the emerging concerns in bridge aerodynamics introduced by aerodynamic nonlinearities and turbulence effects. These issues may become critical for bridges with increasing spans and/or with aerodynamic characteristics sensitive to the effective angle of incidence. This paper presents a nonlinear aerodynamic force model and associated time domain analysis framework for predicting the aeroelastic response of bridges under turbulent winds. The nonlinear force model separates the aerodynamic force into low- and high-frequency components according to the effective angle of incidence. The low-frequency force component is modeled utilizing quasi-steady theory. The high-frequency force component is based on the frequency dependent unsteady aerodynamic characteristics, which are similar to the traditional force model but vary in space and time following the low-frequency effective angle of incidence. The proposed framework provides an effective analysis tool to study the influence of structural and aerodynamic nonlinearities and turbulence on the bridge aeroelastic response. The effectiveness of this approach is demonstrated by utilizing an example of a long span suspension bridge with aerodynamic characteristics sensitive to the angle of incidence. The influence of mean wind angle of incidence on the aeroelastic modal properties and the associated aeroelastic response and the sensitivity of bridge response to nonlinear aerodynamics and low-frequency turbulence are examined.     

Refined Stick Model for Dynamic Analysis of Skew Highway Bridges

Stick models are widely employed in the dynamic analysis of bridges when only approximate results are desired or when detailed models are difficult or time-consuming to construct. Although the use of stick models for regular bridges has been validated by various researchers, the application of such models to skew highway bridges continues to present challenges. The conventional single-beam stick model used to represent the bridge deck often fails to capture certain predominant vibration modes that are important in obtaining the true dynamic response of the bridge. In this paper, a refined stick model is proposed for the preliminary dynamic analysis of skew bridges. The model utilizes a dual-beam stick representation of the bridge deck. The validity of the model is established by comparing results obtained from the proposed model with numerical solutions obtained for skew plates and a skew bridge. It is shown that this dual-beam stick model is superior to the conventional single-beam model in estimating the natural vibration frequencies and in predicting the predominant vibration modes of the bridge. Because of its simplicity and relative accuracy, this model is recommended for the preliminary dynamic analysis of skew highway bridges.