Analysis Of Suspension Bridge Research Articles

Nonstationary response analysis of suspension bridges for multiple support excitations

A method for the nonstationary response analysis of suspension bridges subjected to earthquake ground excitations at multiple supports is developed. The equation of motion is formulated for the vertical motion of the bridge girder using a continuous co-ordinate system. The nonstationarity of the earthquake excitation is modeled by utilizing a time-varying envelope function. The correlation effects between ground excitations at different supports are investigated, for various cases of traveling seismic waves with different wave propagation velocities. The non-stationary responses are obtained in terms of time-dependent variance functions. Expected peak responses are also evaluated, thereafter. Numerical results from the example analysis indicate that the correlation effects between different support excitations are very significant on the response of the suspension bridge.

GRAVITATIONAL STIFFNESS IN TORSIONAL OSCILLATION OF SUSPENSION BRIDGES AND IT'S EFFECT ON NATURAL FREQUENCY

Gravitational stiffness in torsional oscillation of suspension bridge is discussed, and it's effect on the fundamental frequency of torsional mode is investigated theoretically with a parameter of span length. It is found that the effect of gravitational stiffness becomes larger with the increase of span length and that the gravitational stiffness should be taken into account in order to estimate accurately natural frequencies of extremely long-spanned suspension bridges. The three dimensional frame model with gravitational stiffness is also discussed for the analysis of suspension bridges.

Aerodynamic flutter analysis of suspension bridges by a modal technique

Suspension bridges are long, slender flexible structures which have the potential to be susceptible to a variety of types of wind-induced instabilities, the most serious of which are divergence (due to stationary wind forces) and flutter (due to aerodynamic forces). Flutter occurs at certain wind speeds where aerodynamic forces acting on the deck feed energy into an oscillating structure, so increasing the vibration amplitudes. If this situation is approached the basic safety of the bridge is threatened. This paper describes a computational method for predicting flutter speed based on a modal technique. A selection of the lowest vertical and torsional natural mode shapes is included with the aerodynamic forces in an interaction analysis, which yields an unsymmetric matrix eigenvalue problem. Flutter instability is indicated when, at some wind speed, one of the complex eigenvalue pairs resulting from the solution of the eigenvalue problem has a zero real part and a non-zero imaginary part.

Second order influence line analysis of suspension bridges

An influence line theory for suspension bridge analysis is sketched, based on a second order theory, which permits us to avoid some cumbersome procedures. The method is iterative, but the convergence is quite rapid, so allowing a preliminary check of the most dangerous load placements. Finally, a numerical example is performed, in which a recently designed one-span suspension bridge is studied.

Nonlinear dynamic analysis of suspension bridges under random wind loads by stochastic linearization

For the investigation of the geometric nonlinear effect of suspension bridges under random wind loads, a frequency domain method based on a stochastic linearization technique is presented. The equation of motion is formulated by including the coupling between the vertical and the torsional motions. In the linearization procedure, the nonlinear terms are approximated as a sum of linear and constant terms in order to take into account the nonzero mean components as well as the fluctuating components of structural responses. The verification has been made on a case with four modal degrees of freedom. Example analyses have been carried out on two structures for various wind speeds and wind force parameters. Numerical results indicate that, by including the nonlinearity into the analysis, the dynamic responses of the bridges, particularly in the vertical direction, change considerably.

Three subroutines for the analysis of suspension bridges

A new procedure for analysis of suspension bridges has been recently presented (Franciosi and Franciosi, Comput. Struct. 26, 499–512, 1987), in which the so-called cell method was employed in order to obtain static and dynamic response of a one-span suspension bridge. In the present paper three efficient subroutines are introduced, which allow us to calculate the strain energy matrix, Lagrangian mass matrix and participation factors of each vibration mode. Every matrical operation is avoided, so that the proposed method is very fast and manageable. Both the matrices can be immediately built, and a standard eigenvalue package will furnish eigenvalues and eigenvectors. Then, the third subroutine calculates the participation factors of the modes, for synchronous and asynchronous earthquakes. A numerical example shows that the cell method leads to good approximations, even when the discretization is very rough.

Finite element analysis of suspension bridges

A finite element analysis of static and dynamic response of suspension bridges is presented in this paper. Several finite element models of the three-dimensional bridge structure, with varying degrees of complexity and accuracy, are discussed. The formulation takes into account the geometric nonlinearities of the cables and some elements of the girders-bracingsdeck system as well as the nonlinear material properties of the components. Special attention is given to the effects of steady and unsteady wind forces. Examples of application include calculations of the static and dynamic response of a bridge subjected to wind and moving loads.

Structural Analysis of Suspension Bridges

A practical structural analysis of suspension bridges by the stiffness matrix method is presented. This analysis is based on deflection theory. In this method all live loads are applied at arbitrary points and locations along a stiffening girder or truss because of the inclusion of the load terms derived by means of the Laplace transformation. Therefore, it is possible to reduce the number of division elements. Horizontal cable tension can be determined by integral of Green's function with respect to deflection. Irrespective of the number of division elements, the magnitude of horizontal cable tension can be obtained precisely. The method is applicable to an analysis of suspension bridges with any number of spans and variable cross sections. The validity of the stiffness matrix method proposed herein is examined and checked by numerical calculations. This stiffness matrix method can be developed and extended to a three‐dimensional analysis of suspension bridges.

Suspension Bridge Vibration: Continuum Formulation

The methodology of the free vibration analysis of suspension bridges with horizontal decks is refined and extended, utilizing a continuum approach to include both the coupled vertical-torsional vibration and the effect of cross-sectional distortion. Variational principles are used to obtain the coupled equations of motion in their most general, and nonlinear form. The general equations are linearized, eliminating the coupling effect, and solved for a specific bridge; the resulting natural frequencies and modes of vertical and torsional vibrations are compared with those calculated using the finite element approach and with those estimated from low-amplitude full-scale ambient vibration tests which were conducted on the bridge. In addition, the general, nonlinear equations of motion governing the lateral vibration of suspension bridges are presented and the linearized forms are obtained; a numerical example and a comparison among the analytical, numerical and full-scale test results are also presented.

Discussion of “Vertical Vibration Analysis of Suspension Bridges”

Discussion of “Vertical Vibration Analysis of Suspension Bridges”

Initial-value discrete suspension bridge analysis

A discrete formulation is presented for the analysis of suspension bridges. The resulting equations of equilibrium, which are nonlinear in the displacement terms, are solved by a Newton-Raphson procedure. The linear solution for each cycle of the Newton-Raphson method is treated as a set of linear initial-value problems. Numerical difficulties require that a suppression scheme be employed within each linearization. Numerical problems are presented for both stiffened and unstiffened structures and the results are compared with the findings of other investigators. The effects of hanger spacing on the results, especially in the end regions of the stiffening truss, are studied.

Static Analysis of Suspension Bridges

With the advent of digital computers it has become possible to analyze suspension bridges taking account of hangers as individual members. As the behavior is nonlinear satisfactory convergence is most likely to be obtained in the solution of the governing equations if a Newton-Raphson method is used for correcting the basic variables. Compatibility and equilibrium methods are developed which produce symmetric equations to be solved at each iteration. By suitable choice and ordering of the variables these equations can be made to fit well into a variable bandwidth store thus producing an efficient form of computer solution. The most efficient of these methods is the compatibility method in which the horizontal component of hanger force is ignored. In cases where the stiffening girder is a Warren truss the compatibility method also shows advantage, but in other cases the equilibrium method is likely to be preferred because the equations are more easily constructed.

Analysis of Cables as Equivalent Two-Force Members

In recent years, cable structures have become more widely used because of growing demand for long-span construction for aesthetic, economic, or other practical reasons. Aside from classical suspension bridges, examples of this type of structure are tall guyed towers, cable-stiffened bridges, and suspended roofs. The methods of analysis of suspension bridges have been extensively explored, and will not be discussed herein. Instead, this paper presents a simplified analysis of cable members ordinarily encountered in the other types of structures mentioned above, by treating them as equivalent two-force members. The nonlinear behavior of such cables is accounted for by the use of an equivalent modulus of elasticity as well as equivalent strains. Problems commonly associated with cables, such as stresses (and change of sags) caused by a change of temperature or superimposed loads, or resulting from relative displacement of supports, can be readily solved by this method. It should be noted that the proposed procedure is also applicable to the analysis of transmission lines.

Theoretical Analysis of Suspension Bridges

THE ANALYSIS OF SUSPENSION BRIDGES IS A TYPICAL PROBLEM OF STRUCTURAL FRAMES THAT ARE ASSOCIATED WITH LARGE DEFLECTIONS THAT PLAY AN IMPORTANT PART IN THE BEHAVIOR OF THE FRAME. THEORETICAL AND PRACTICAL TREATISES ON SUSPENSION BRIDGES WERE DEVELOPED BY MANY AUTHORS. THEIR ATTEMPT TO IDEALIZE THE PROBLEM AND MAKE IT SUITABLE FOR HAND CALCULATIONS HAS EXPOSED THESE SOLUTIONS TO CRITICISM. A SYSTEMATIC METHOD IS INTRODUCED FOR THE ANALYSIS OF SUSPENSION BRIDGES BASED ON THE FINITE DEFLECTION THEORY AND THE N-DIMENSIONAL GENERALIZATION OF THE NEWTON-RAPHSON PROCESS FOR SOLVING THE SIMULTANEOUS NON-LINEAR EQUATIONS. THE BASIC ASSUMPTIONS ARE REPORTED AND THE APPLICATION OF THE NEWTON-RAPHSON PROCESS IS EXAMINED. A DERIVATION IS GIVEN FOR THE TANGENT STIFFNESS MATRIX OF A STRUCTURAL FRAME AT A STATE OF DEFORMATION. THE ANALYSIS FOR TWO PROBLEMS, A CABLE AND A TRUSSED GIRDER, IS GIVEN TO INTRODUCE THE APPLICATION OF THE NEW APPROACH. THE CONCEPT OF THE TANGENT STIFFNESS MATRIX IS A NEW TECHNIQUE FOR RAPID CONVERGENCE OF THE ANALYSIS. THE METHOD IS GENERAL AND OF PARTICULAR INTEREST FOR THE ANALYSIS OF SUSPENSION BRIDGES UNDER CONSTRUCTION CONDITIONS. IT IS ALSO SUITABLE FOR ELECTRONIC COMPUTERS. /AUTHOR/

THE LATERAL MOTION OF SUSPENSION BRIDGES

The fundamental equations of small lateral motion of a suspension bridge structure are presented under the proper assumptions. Then, the author derived the formulas to compute the natural frequencies of lateral vibrations, which contain some dimensionless parameters. The coupling oscillations between cables and stiffening frame that is missing in other papers should be taken into account. The effect of the vertical distortion of the structure incidental to its lateral movement is estimated by means of the energy method. Finally, the results are applied to the analysis of suspension bridges subjected to the static lateral forces.