Dynamic Behavior of Orthotropic Rectangular Plates under Moving Loads

The dynamic behavior of an orthotropic plate simply supported on a pair of parallel edges and under a system of moving loads is analyzed based on Lagrange equation and modal superposition. Thin plate theory is assumed for the plate model and no restriction is placed on the type of loading. Parameters of the plate affecting its dynamic behavior are discussed, and a new classification of the plates for computing the mode shapes and natural frequencies is proposed. The impact factors and the dynamic responses of a typical bridge deck are studied using the proposed method. Preliminary results indicate that the effect of eccentric loads on the impact factor depends on the proportion ratio between the flexural and torsional rigidities of the bridge deck, and the multilane loading case is less critical than a single-lane loading case.     

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.     

Vibration Characteristics of K?mürhan Highway Bridge Constructed with Balanced Cantilever Method

K?mürhan Highway Bridge is a reinforced concrete box girder bridge located on the 51st km of Elaz??–Malatya Highway over the F?rat River. Because of the fact that the K?mürhan Bridge is the only bridge in this part of F?rat, it has major logistical importance. So, this paper aims to determine dynamic characteristics such as natural frequencies, mode shapes, and damping ratios of the bridge using experimental measurements and finite-element analyses to evaluate current behavior. The experimental measurements are carried out by ambient vibration tests under traffic loads. Due to the expansion joint in the middle of the bridge, special measurement points are selected and experimental test setups are constituted. Vibration data are gathered from the both box girder and bridge deck. Measurement time, frequency span, and effective mode number are determined by considering similar studies and literature. The peak picking method in the frequency domain is used for the output-only modal identification. An analytical modal analysis is performed on the developed two- and three-dimensional finite-element model of the bridge using SAP2000 software to provide the analytical frequencies and mode shapes. At the end of the study, dynamic characteristics of the Elaz?? and Malatya parts of the bridge obtained from the experimental measurements are compared with each other and transverse effects on the bridge are determined. Also, experimental and analytical dynamic characteristics are compared. Good agreement is found between dynamic characteristics in the all measurement test setups performed on the box girder and bridge deck and analytical modal analyses.     

Study of Passive Deck-Flaps Flutter Control System on Full Bridge Model. I: Theory

A passive aerodynamic control method for suppression of the wind-induced instabilities of a very long span bridge is presented in this paper. The control system consists of additional control flaps attached to the edges of the bridge deck. Control flap rotations are governed by prestressed springs and additional cables spanned between the control flaps and an auxiliary transverse beam supported by the main cables of the bridge. The rotational movement of the flaps is used to modify the aerodynamic forces acting on the deck and provides aerodynamic forces on the flaps used to stabilize the bridge. A time-domain formulation of self-excited forces for the whole three-dimensional suspension bridge model is obtained through a rational function approximation of the generalized Theodorsen function and implemented in the FEM formulation. This paper lays the theoretical groundwork for the one that follows.     

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.     

Lateral-Torsional Buckling of Stepped Beams with Continuous Bracing

Continuous span multibeam steel bridges are common along the state and interstate highways. The top flange of the beams is typically braced against lateral movement by the deck slab, and in many bridges the cross section is stepped at discrete points along the span. Design equations for lateral–torsional buckling (LTB) resistance in the American Association of State Highway and Transportation Officials “Load and resistance factor design bridge design specifications” are for prismatic beams and ignore the lateral restraint provided by the bridge deck. A new design equation is proposed that can be applied to I-shaped stepped beams with continuous top flange lateral bracing. By including the effects of the change in cross section size and the continuous top flange bracing, the calculated LTB resistance is significantly increased. Critical bending moment values from the proposed equation are compared to values from finite element method buckling analyses. The new equation is sufficiently accurate for use in design and in the evaluation of existing bridges.     

Innovative Bridge Condition Assessment from Dynamic Response of a Passing Vehicle

An innovative approach for damage assessment of a bridge deck is proposed with the measured dynamic response of a vehicle moving on top of a structure. The simply supported bridge deck is modeled as a Euler–Bernoulli beam. The moving vehicle serves as a smart sensor and force transducer in the structural system. The damage is defined as the flexural stiffness reduction in the beam finite element. The identification algorithm is based on dynamic response sensitivity analysis, and it is realized with a regularization technique from the measured vehicle acceleration measurement. Measurement noise, road surface roughness, and model errors are included in the simulations, and the results indicate that the proposed algorithm is computationally stable and efficient, and the identified results are acceptable and not sensitive to the different parameters studied.     

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.     

Wavelet-Galerkin Method for Free Vibrations of Elastic Cable

The wavelet-Galerkin method is applied to study the free vibration of a horizontally suspended catenary cable. Antiderivatives of the Daubechies compactly supported wavelets have been used with multilevel representations. Comparison between wavelet and Fourier methods is presented for natural frequencies, mode shapes, and dynamic tension of the cable. Both methods showed that they are converging fast in obtaining the natural frequencies and mode shapes. However, as the dynamic tension is obtained, the Fourier solution shows many oscillations and the existence of the Gibbs phenomenon at the cable supports, whereas, these oscillations do not appear in the wavelet solutions. Comparison is also made with the linear theory of cable vibration. Due to the inclusion of the inertia term of the longitudinal component in our solution, new modes have been found. Those modes are reverting but swapping modes in which the longitudinal displacement component is larger than the transverse displacement component.     

Free Vibration of Three-Dimensional Orthotropic Uniform Shear Beam-Columns with Generalized End Conditions

The free vibration analysis of asymmetrical three-dimensional (3D) uniform shear beam-columns with generalized boundary conditions (semirigid flexural and torsional restraints, lateral bracings, and lumped masses at both ends) subjected to an eccentric end axial load in addition to a linearly distributed eccentric axial load along its span is presented in a classic manner. The five coupled governing equations of dynamic equilibrium (i.e., two shear equations, two bending moment equations, and the pure torsion moment equation) are sufficient to determine the natural frequencies and modal shapes. The proposed model which is an extension of a 2D model presented previously by the writer includes the simultaneous 3D coupling effects among the lateral deflections, deformations of the cross section along the member (shear, torsional and rotational), the translational, rotational and torsional inertias of all masses considered, an eccentric end axial load in addition to a linearly distributed axial load along its span, and the end restraints. Deformations caused by shear forces and pure torsion are considered. The effects of axial deformations, warping torsion and torsional stability are not included. The proposed model shows that the dynamic behavior of 3D shear beam-columns is highly sensitive to the coupling effects just mentioned, particularly in members with both ends free to rotate. Analytical results indicate that except for doubly symmetric members with concentric axial loads and with perfectly clamped ends, the natural frequencies and modal shapes of 3D shear beam-columns are determined from the eigenvalues of a full 8×8 matrix, rather than from the uncoupled equations of transverse (or shear-wave equations) and torsional moment equilibrium. Two comprehensive examples are presented that show the effectiveness of the proposed method.     

Dynamic Analysis of Curved Continuous Multiple-Box Girder Bridges

The use of horizontally curved composite multiple-box girder bridges in modern highway systems is quite suitable in resisting torsional and warping effects induced by highway curvatures. Bridge users react adversely to vibrations of a bridge and especially where torsional modes dominate. In this paper, continuous curved composite multiple-box girder bridges are analyzed, using the finite-element method, to evaluate their natural frequencies and mode shapes. Experimental tests are conducted on two continuous twin-box girder bridge models of different curvatures to verify and substantiate the finite-element model. Empirical expressions are deduced from these results to evaluate the fundamental frequency for such bridges. The parameters considered herein are the span length, number of lanes, number of boxes, span-to-radius of curvature ratio, span-to-depth ratio, end-diaphragm thickness, number of cross bracings, and number of spans.     

Ambient Vibration of Long-Span Cable-Stayed Bridge

The investigation of dynamic response for long-span cable-stayed bridges largely depends on a detailed understanding of their dynamic characteristics, such as the natural frequencies, mode shapes, and modal damping ratios. In this paper, the dynamic characteristics of a fairly long cable-stayed bridge in Hong Kong are studied using finite-element analysis and ambient vibration measurements. A three-dimensional finite-element model is first established for the bridge based on design drawings. The dynamic characteristics are then analyzed from the statically deformed configuration. Ambient vibration measurements are also conducted to obtain the dynamic characteristics of the bridge. Comparison between these two results shows that, for the most part, a total of 31 modes can be correlated with a reasonable agreement. However, the frequency differences of the higher modes can range between 15 and 30%. This implies that, if the measurement is more reliable, a finite-element model updating is necessary in order to achieve better correlation between these two results.     

Finite-Element Model Updating for the Kap Shui Mun Cable-Stayed Bridge

This paper presents the implementation of the finite-element model updating for the Kap Shui Mun Bridge, a 430 m main span double-deck cable-stayed bridge in Hong Kong. The dynamic characteristics of the bridge have been studied through both three-dimensional finite-element prediction and field vibration measurement previously. In this paper, the developed finite-element model is updated based on the field measured dynamic properties. A comprehensive sensitivity study to demonstrate the effects of various structural parameters (including the connections and boundary conditions) on the modes of concern is first performed, according to which a set of structural parameters are then selected for adjustment. The finite-element model is updated in an iterative fashion so as to minimize the differences between the predicted and the measured natural frequencies. The final updated finite-element model for the Kap Shui Mun Bridge is able to produce natural frequencies in good agreement with the measured ones, and can be helpful for a more precise dynamic response prediction.     

Monitoring and Analysis of a Bridge with Partially Restrained Bearings

This paper reports on the monitoring and analysis of a two-span bridge in which the bearings were partially restrained. In an earlier experimental study, it was shown that the natural frequencies changed in colder weather, and it appeared that this was due to restraints in the end bearings. This research was conducted to verify this initial conclusion and to develop an analytical approach based on the finite-element method to model this change. Additional field measurements were made. The nonlinear dynamic finite-element analysis is based on a planar model that includes the influence of both the deck cracking and the eccentric axial forces, which develop when the bearings are restrained. Both the flexural and the torsional modes are evaluated. Although the changes in the bearings and the overall structural behavior were relatively small, the results show that it was nevertheless possible to verify the changes with a nonlinear dynamic finite-element analysis calibrated with field measurements.     

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.     

Output-Only System Identification of Posttensioned Segmental Concrete Highway Bridges

This paper discusses the application of system identification of a highway bridge using finite-element method and ambient-vibration testing. The posttensioned Gülburnu Highway Bridge located on the Giresun-Espiye state highway was selected as a case study. A finite-element model of the bridge was developed using SAP2000 software, and dynamic characteristics were obtained analytically. During the test, sources of ambient excitations were provided by the traffic effects over the bridge. Ambient-vibration tests were applied to the bridge to identify dynamic characteristics. The selection of measurement time, frequency span, and effective mode number was considered from similar studies in the literature. Two output-only system identification methods, enhanced frequency domain decomposition and stochastic subspace identification, were used to estimate the dynamic characteristics of the bridge experimentally. The accuracy and efficiency of both methods were investigated and compared with finite-element results. Results suggest that ambient-vibration measurements are sufficient to identify structural modes with a low range of natural frequencies. In addition, the dynamic characteristics obtained from the finite-element model of the bridge have a good correlation with experimental frequencies and mode shapes.     

Dynamic Analysis of Tower Cranes

Tower cranes are widely used in the construction of high-rise buildings and the analysis of their dynamic behaviors is of both theoretical and practical significance. A parameterized super element formulation for modeling the multiple-pulley cables in a crane system is proposed based on the friction-free assumption between the cable and pulleys. The cable passages over intermediate pulleys are considered as new degrees-of-freedom in the proposed element model and equal tensions at different segments of a continuous cable are exactly achieved. Numerical results show that these cable passages have significant effect on both static tensions and dynamic properties of tower cranes and the proposed super element model provides more accurate and realistic results in dynamic analyses of the crane system. It is found that some modes are insensitive to the change of jib angle or bracings at crane mast, while others change drastically depending on the mode shapes. Picking up the payload from ground or payload accelerating/decelerating during lift operation will induce finite impulsive excitations and, therefore causes noticeable dynamic responses of the crane system.     

Effect of Hanger Flexibility on Dynamic Response of Suspension Bridges

Linearized continuum models of a suspended span with unloaded backstays and of a symmetric three-span suspension bridge are used to study the effects of the flexibility of the hangers on the vertical vibrations of suspension bridges. The models include elastic parabolic cables, flexible distributed hangers with variable length, and a stiffening girder represented by an elastic beam. It is shown that the free vibrations of a suspended span with unloaded backstays are controlled by five dimensionless parameters, while six dimensionless parameters control the response of a symmetric three-span suspension bridge. The results indicate that the flexibility of the hangers has a significant effect on the natural frequencies of the higher modes only when the relative stiffness of the girder is very high. The effects of hanger flexibility on the response of a suspension bridge to localized impulsive loads are also found to be small. These findings confirm the traditional, albeit previously untested, assumption of inextensible hangers. Finally, the threshold amplitudes of free vibrations that would result in the incipient slackening of the hangers are determined.     

Modal Testing, Finite-Element Model Updating, and Dynamic Analysis of an Arch Type Steel Footbridge

This paper describes an arch type steel footbridge, its analytical modeling, modal testing, finite-element model updating, and dynamic analysis. A modern steel footbridge which has an arch type structural system and is located on the Karadeniz coast road in Trabzon, Turkey is selected as an application. An analytical modal analysis is performed on the developed three-dimensional finite-element model of footbridge to provide analytical frequencies and mode shapes. Field ambient vibration tests on the footbridge deck under natural excitation such as human walking and traffic loads are conducted. The output-only modal parameter identification is carried out by using peak picking of the average normalized power spectral densities in the frequency domain and stochastic subspace identification in the time domain, and dynamic characteristics such as natural frequencies, mode shapes, and damping ratios are determined. The finite-element model of the footbridge is updated to minimize the differences between analytically and experimentally estimated modal properties by changing some uncertain modeling parameters such as material properties. Dynamic analyses of the footbridge before and after finite-element model updating are performed using the 1992 Erzincan earthquake record. At the end of the study, maximum differences in the natural frequencies are reduced from 22 to only 5% and good agreement is found between analytical and experimental dynamic characteristics such as natural frequencies and mode shapes by model updating. Also, maximum displacements and principal stresses before and after model updating are compared with each other.     

Model-Based Damage Identification in a Continuous Bridge Using Vibration Data

Damage often causes changes in the dynamic characteristics of a structure such as frequencies and mode shapes. Vibration-based damage identification techniques utilize the changes in the dynamic characteristics of a structure to determine the location and extent of damage in the structure. Such techniques are applied in this study to the Crowchild Bridge, a steel-free deck continuous bridge located in western Canada. While the numerical models of the bridge are correlated with the measured dynamic characteristics, computer simulation is used to study the identification of a number of different damage patterns, and the effects of measurement errors and incomplete mode shapes on the quality of results are evaluated. The effectiveness of some selected damage identification techniques is examined; the potential difficulties in identifying the damage are outlined; and areas of further research are suggested. A three-dimensional finite-element model and a simple two-dimensional girder model of the bridge have been constructed to study the usefulness of the selected damage identification methods. Another promising damage detection method proposed here is based on the application of neural networks that combines a vibration-based method.