Nonlinear analysis of high-speed trains moving on bridges during earthquakes

This paper investigates the derailment of high-speed trains moving on multispan simply supported bridges. A?moving wheel finite element containing contact and separation modes was developed to simulate vehicle?Cbridge interaction problems under seismic loads. Rail irregularities and bridge?Crail?Ctrain interactions were appropriately considered in the nonlinear finite element analysis, which indicates that the derailment coefficients are enlarged with the increase of train speeds for high-speed trains moving on multispan simply supported bridges. The accelerations of elevated bridge girders may be significantly magnified during the seismic load; moreover, gaps between simply supported girders during seismic loads will produce large derailment coefficients.     

Vibration suppression for high-speed railway bridges using tuned mass dampers

This paper deals with the applicability of passive tuned mass dampers (PTMDs) to suppress train-induced vibration on bridges. A railway bridge is modeled as an Euler–Bernouli beam and a train is simulated as series of moving forces, moving masses or moving suspension masses to investigate the influence of various vehicle models on the bridge features with or without PTMD. According to the train load frequency analysis, the resonant effects will occur as the modal frequencies of the bridges are close to the multiple of the impact frequency of the train load to the bridge. A single PTMD system is then designed to alter the bridge dynamic characteristics to avoid excessive vibrations. Numerical results from simply supported bridges of Taiwan High-Speed Railway (THSR) under German I.C.E., Japanese S.K.S. and French T.G.V. trains show that the proposed PTMD is a useful vibration control device in reducing bridge vertical displacements, absolute accelerations, end rotations and train accelerations during resonant speeds, as the train axle arrangement is regular. It is also found that the inner space of bridge box girder of THSR is wide and deep enough for the installation and movement of PTMD.     

Fluid–structural interaction with application to rocket engines

The objective of this work is to develop a finite element model for studying fluid–structure interaction. The geometrically non‐linear structural behaviour is considered and based on large rotations and large displacements. An arbitrary Lagrangian–Eulerian (ALE) formulation is used to represent the compressible inviscid flow with moving boundaries. The structural response is obtained using Newmark‐type time integration and fluid response employs the Lax–Wendroff scheme. A number of numerical examples are presented to validate the structural model, moving mesh implantation of the ALE model and complete fluid–structure interaction. Copyright © 1999 John Wiley & Sons, Ltd.     

Static analysis of rectangular thick plates resting on two-parameter elastic boundary strips

In this paper, the Generalized Differential Quadrature (GDQ) method is used to obtain bending solution of moderately thick rectangular plates. The plate is resting on two-parameter elastic (Pasternak) foundation or strips with a finite width. Various combinations of clamped, simply supported and free boundary conditions are considered. According to the first-order shear deformation theory, the governing equations of the problem consist of three second-order partial differential equations (PDEs) in terms of displacement and rotations of the plate. The governing equations and solution domain is discretized based on the GDQ method. It is demonstrated that the method converges rapidly while providing accurate results with relatively small number of grid points. Accuracy of the results is examined using available data in the literature for Pasternak foundation. Furthermore, due to lack of data for Pasternak strips, all predictions are verified by finite element analysis which can be used as benchmark in future studies.     

Non-linear buckling and postbuckling behavior of thin-walled beams considering shear deformation

The static stability of thin-walled composite beams, considering shear deformation and geometrical non-linear coupling, subjected to transverse external force has been investigated in this paper. The theory is formulated in the context of large displacements and rotations, through the adoption of a shear deformable displacement field (accounting for bending and warping shear) considering moderate bending rotations and large twist. This non-linear formulation is used for analyzing the prebuckling and postbuckling behavior of simply supported, cantilever and fixed-end beams subjected to different load condition. Ritz's method is applied in order to discretize the non-linear differential system and the resultant algebraic equations are solved by means of an incremental Newton-Rapshon method. The numerical results show that the beam loses its stability through a stable symmetric bifurcation point and the postbuckling strength is in relation with the buckling load value. Classical predictions of lateral buckling are conservative when the prebuckling displacements are not negligible and the non-linear buckling analysis is required for reliable solutions. The analysis is supplemented by investigating the effects of the variation of load height parameter. In addition, the critical load values and postbuckling response obtained with the present beam model are compared with the results obtained with a shell finite element model (Abaqus).     

Nonlinear analysis of shear deformable beam-columns partially supported on tensionless three-parameter foundation

In this paper, a boundary element method is developed for the nonlinear analysis of shear deformable beam-columns of arbitrary doubly symmetric simply or multiply connected constant cross-section, partially supported on tensionless three-parameter foundation, undergoing moderate large deflections under general boundary conditions. The beam-column is subjected to the combined action of arbitrarily distributed or concentrated transverse loading and bending moments in both directions as well as to axial loading. To account for shear deformations, the concept of shear deformation coefficients is used. Five boundary value problems are formulated with respect to the transverse displacements, to the axial displacement and to two stress functions and solved using the Analog Equation Method, a BEM-based method. Application of the boundary element technique yields a system of nonlinear equations from which the transverse and axial displacements are computed by an iterative process. The evaluation of the shear deformation coefficients is accomplished from the aforementioned stress functions using only boundary integration. The proposed model takes into account the coupling effects of bending and shear deformations along the member as well as the shear forces along the span induced by the applied axial loading. Numerical examples are worked out to illustrate the efficiency, wherever possible, the accuracy and the range of applications of the developed method.     

Numerical Analysis of Axisymmetric Buckling of a Conical Shell under Radial Compression

The nonlinear boundary-value problem of the axisymmetric buckling of a simply supported conical shell (dome) under a radial compressive load applied to the supported edge is formulated for a system of six first-order ordinary differential equations for independent fields of finite displacements and rotations. Multivalued solutions are obtained using the shooting method with specified accuracy. For various values of the loading parameter, bifurcation of the solutions of the problem is studied and a parametric branching diagram is constructed. The buckling modes are obtained for three branches of the solution. Curves of the buckling modes corresponding to three isolated branches of the solution are given.     

地基变形计算的修正Flamant解

针对地表分布载荷作用下地基变形的半平面问题,利用对称化方法对Flamant位移解答进行修正,得到有限深度底部光滑刚性支撑的地基位移的精确解.该解与相同条件下的有限元数值计算结果基本一致.修正解与原解的对比表明,两者差异较大;但当基点距离载荷较近时,两者所得到的地表附近的相对沉陷分布形式和大小差别不大.     

Shear buckling of infinite plates resting on tensionless elastic foundations

The buckling problem of an infinite thin plate resting on a tensionless Winkler foundation and subjected to shearing loads is investigated. The infinite plate is simplified to a one-dimensional mechanical model by assuming a lateral buckling mode function and a borderline function between contact and non-contact regions. After the governing differential equations for the plate sections in the contact and non-contact regions have been solved, the problem reduces to two nonlinear algebraic equations. Buckling coefficients for plates with simply supported edges and clamped edges are determined for a range of relative foundation stiffness factors. Comparison of the results with existing theory and finite element analyses shows good agreement.     

Non-linear flexural–torsional dynamic analysis of beams of arbitrary cross section by BEM

In this paper, a boundary element method is developed for the non-linear flexural–torsional dynamic analysis of beams of arbitrary, simply or multiply connected, constant cross section, undergoing moderately large deflections and twisting rotations under general boundary conditions, taking into account the effects of rotary and warping inertia. The beam is subjected to the combined action of arbitrarily distributed or concentrated transverse loading in both directions as well as to twisting and/or axial loading. Four boundary value problems are formulated with respect to the transverse displacements, to the axial displacement and to the angle of twist and solved using the Analog Equation Method, a BEM based method. Application of the boundary element technique leads to a system of non-linear coupled Differential–Algebraic Equations (DAE) of motion, which is solved iteratively using the Petzold–Gear Backward Differentiation Formula (BDF), a linear multistep method for differential equations coupled to algebraic equations. The geometric, inertia, torsion and warping constants are evaluated employing the Boundary Element Method. The proposed model takes into account, both the Wagner's coefficients and the shortening effect. Numerical examples are worked out to illustrate the efficiency, wherever possible the accuracy, the range of applications of the developed method as well as the influence of the non-linear effects to the response of the beam.     

Absolute versus relative formulations of the moving oscillator problem

In the general framework of the bridge-vehicle dynamic interaction, the so-called “moving oscillator” problem is revisited in order to provide a deeper insight into some analytical and physical aspects not specifically analyzed in previous investigations. Without lack of generality, the case of a stream of moving oscillators crossing a simply supported beam with arbitrary time law is considered. The formulations in terms of both absolute and relative displacements of the moving oscillators are critically reviewed and compared, and alternative sets of differential equations with time-dependent coefficients are derived. The study enlightens, both theoretically and numerically, that impulsive contributions to the dynamic response appear in the relative displacement formulation at the time instants in which each vehicle enters or exits the bridge. It is demonstrated that such contributions, somehow “hidden” in the absolute displacement formulation, may have a significant influence on the vibration of the moving oscillators, and thus cannot be a priori neglected in the analysis. It is also shown that the analytical and computational difficulties associated with these additional impulses make preferable the use of the absolute displacement formulation. Far from being restricted to the case of simply supported beams, these findings are valid for any type of bridge structure which induces a discontinuity in the slope of the road profile experienced by the vehicles.     

Non-linear stability analysis of imperfect thin-walled composite beams

The static non-linear behavior of thin-walled composite beams is analyzed considering the effect of initial imperfections. A simple approach is used for determining the influence of imperfection on the buckling, prebuckling and postbuckling behavior of thin-walled composite beams. The fundamental and secondary equilibrium paths of perfect and imperfect systems corresponding to a major imperfection are analyzed for the case where the perfect system has a stable symmetric bifurcation point. A geometrically non-linear theory is formulated in the context of large displacements and rotations, through the adoption of a shear deformable displacement field. An initial displacement, either in vertical or horizontal plane, is considered in presence of initial geometric imperfection. Ritz's method is applied in order to discretize the non-linear differential system and the resultant algebraic equations are solved by means of an incremental Newton-Rapshon method. The numerical results are presented for a simply supported beam subjected to axial or lateral load. It is shown in the examples that a major imperfection reduces the load-carrying capacity of thin-walled beams. The influence of this effect is analyzed for different fiber orientation angle of a symmetric balanced lamination. In addition, the postbuckling response obtained with the present beam model is compared with the results obtained with a shell finite element model (Abaqus).     

Effects of the dynamic wheel–rail interaction on the ground vibration generated by a moving train

Based on Biot’s fully dynamic poroelastic theory, the dynamic responses of the poroelastic half-space soil medium due to quasi-static and dynamic loads from a moving train are investigated semi-analytically. The dynamic loads are assumed to be generated from the rail surface irregularities. The vehicle dynamics model is used to simulate the axle loads (quasi-static loads) and the dynamic loads from a moving train. The compatibility of the displacements at wheel–rail contact points couple the vehicle and the track–ground subsystem, and yield equations for the dynamic wheel–rail loads. A linearized Hertzian contact spring between the wheel and rail is introduced to calculate the dynamic loads. Using the Fourier transform, the governing equations for the poroelastic half-space are then solved in the frequency–wavenumber domain. The time domain responses are evaluated by the fast inverse Fourier transform. Numerical results show that the dynamic loads can make important contribution to dynamic response of the poroelastic half-space for different train speed, and the dynamically induced responses lie in a higher frequency range. The ground vibrations caused by the moving train can be intensified as the primary suspension stiffness of the vehicle increases.     

基于余能原理的有限变形问题有限元列式

利用基面力概念,推导了一种基于余能原理的有限变形问题显式有限元列式,可应用于结构的大位移、大转动问题。以基面力为状态变量来表达单元的余能,将有限变形情况下的单元余能分解为变形余能部分和转动余能部分,利用Lagrange乘子法推导出余能原理有限元的控制方程,编制了相应的非线性有限元程序。通过算例分析,说明该列式和程序的可靠性和精确性。     

基于共旋列式的空间梁的稳定性分析

基于独立于单元的共旋列式(EICR),将一种几何线性的无剪切锁死的Timoshenko梁单元扩展用于空间梁结构的几何非线性分析。考虑到三维分析中发生大转动时转动顺序的不可交换性,也即转动自由度不能作为向量采用加法规则更新,采用了四元变量来存储和更新转动自由度,使得共旋列式适用于位移任意大和转动任意大但应变很小的几何非线性分析。同时改进了Riks弧长法使之适用于带有大转动的三维几何非线性分析。给出了几个数值算例,结果表明本文方法具有较高的计算精度和效率。     

Phenomenological invariants and their application to geometrically nonlinear formulation of triangular finite elements of shear deformable shells

A phenomenological definition of classical invariants of strain and stress tensors is considered. Based on this definition, the strain and stress invariants of a shell obeying the assumptions of the Reissner–Mindlin plate theory are determined using only three normal components of the corresponding tensors associated with three independent directions at the shell middle surface. The relations obtained for the invariants are employed to formulate a 15-dof curved triangular finite element for geometrically nonlinear analysis of thin and moderately thick elastic transversely isotropic shells undergoing arbitrarily large displacements and rotations. The question of improving nonlinear capabilities of the finite element without increasing the number of degrees of freedom is solved by assuming that the element sides are extensible planar nearly circular arcs. The shear locking is eliminated by approximating the curvature changes and transverse shear strains based on the solution of the Timoshenko beam equations. The performance of the finite element is studied using geometrically linear and nonlinear benchmark problems of plates and shells.     

Symplectic system based analytical solution for bending of rectangular orthotropic plates on Winkler elastic foundation

This paper analyses the bending of rectangular orthotropic plates on a Winkler elastic foundation.Appropriate definition of symplectic inner product and symplectic space formed by generalized displacements establish dual variables and dual equations in the symplectic space.The operator matrix of the equation set is proven to be a Hamilton operator matrix.Separation of variables and eigenfunction expansion creates a basis for analyzing the bending of rectangular orthotropic plates on Winkler elastic foundation and obtaining solutions for plates having any boundary condition.There is discussion of symplectic eigenvalue problems of orthotropic plates under two typical boundary conditions,with opposite sides simply supported and opposite sides clamped.Transcendental equations of eigenvalues and symplectic eigenvectors in analytical form given.Analytical solutions using two examples are presented to show the use of the new methods described in this paper.To verify the accuracy and convergence,a fully simply supported plate that is fully and simply supported under uniformly distributed load is used to compare the classical Navier method,the Levy method and the new method.Results show that the new technique has good accuracy and better convergence speed than other methods,especially in relation to internal forces.A fully clamped rectangular plate on Winkler foundation is solved to validate application of the new methods,with solutions compared to those produced by the Galerkin method.     

高速铁路连续箱型梁桥的动力响应与减振分析

根据列车具体的轴距和轴重,建立了和谐号动车组CRH380AL型列车简化模型;对高速铁路两跨连续梁桥采用多自由度欧拉伯努利梁单元进行主梁的模拟,并将液体黏滞阻尼器模拟为有限元阻尼单元;采用Newmark直接积分方法求解了高速列车作用下的连续梁桥运动方程,数值分析了列车车速以及液体黏滞阻尼器的阻尼系数对于高速铁路连续梁桥振动响应的影响。结果表明:黏滞阻尼器对于桥梁具有明显的减振效果,阻尼力不仅与阻尼系数有关还与列车时速有关;同一黏滞阻尼器条件下,桥梁的最大加速度并不随列车速度的增加而单调增加,而是在某些特定列车车速下桥梁的最大加速度出现了峰值,且随着黏滞阻尼器的阻尼系数增大,桥梁振动响应峰值处的最大加速度减幅不同;同一列车时速的条件下,桥梁的减振效果并不是随着阻尼系数的递增呈正比递增,而是随着阻尼系数的增大,阻尼器的减振效果增幅在减小。     

Dynamic coupling of fluid and structural mechanics for simulating particle motion and interaction in high speed compressible gas particle laden flow

In this work, structural finite element analyses of particles moving and interacting within high speed compressible flow are directly coupled to computational fluid dynamics and heat transfer analyses to provide more detailed and improved simulations of particle laden flow under these operating conditions. For a given solid material model, stresses and displacements throughout the solid body are determined with the particle–particle contact following an element to element local spring force model and local fluid induced forces directly calculated from the finite volume flow solution. Plasticity and particle deformation common in such a flow regime can be incorporated in a more rigorous manner than typical discrete element models where structural conditions are not directly modeled. Using the developed techniques, simulations of normal collisions between two 1 mm radius particles with initial particle velocities of 50–150 m/s are conducted with different levels of pressure driven gas flow moving normal to the initial particle motion for elastic and elastic–plastic with strain hardening based solid material models. In this manner, the relationships between the collision velocity, the material behavior models, and the fluid flow and the particle motion and deformation can be investigated. The elastic–plastic material behavior results in post collision velocities 16–50% of their pre-collision values while the elastic-based particle collisions nearly regained their initial velocity upon rebound. The elastic–plastic material models produce contact forces less than half of those for elastic collisions, longer contact times, and greater particle deformation. Fluid flow forces affect the particle motion even at high collision speeds regardless of the solid material behavior model. With the elastic models, the collision force varied little with the strength of the gas flow driver. For the elastic–plastic models, the larger particle deformation and the resulting increasingly asymmetric loading lead to growing differences in the collision force magnitudes and directions as the gas flow strength increased. The coupled finite volume flow and finite element structural analyses provide a capability to capture the interdependencies between the interaction of the particles, the particle deformation, the fluid flow and the particle motion.     

Reliability analysis for the uncertainties in vehicle and high-speed railway bridge system based on an improved response surface method for nonlinear limit states

The paper deals with the reliability analysis for the high-speed railway bridge systems. Although the bridge–vehicle interactive system has much more uncertainties in the resistance and loads of trains moving at very high speed compared with static structural analysis, little concern has been engaged to identify which random variable has to be considered in the probabilistic analysis, or what criteria should be selected to determine the probabilistic safety or serviceability. The considered design parameters thus involve uncertainties in stiffness, moment of inertia, damping ratio of primary suspension in terms of load, geometry of girders and slabs, and the mechanical properties of girders in terms of resistance. The considered limit states embrace the safety of trains and comfort of passengers, and the acceptability criteria are based on UIC code. For evaluating the reliability of the time-dependent nonlinear behavior of complex structures, an improved Response Surface Method (RSM) is developed. An adaptive technique and a weight matrix are utilized as an optimizing technique that accelerates the convergence in the reliability analysis. The results of improved RSM, compared with the basic and adaptive RSM, are verified with the improved convergence to the exact solution. The bridge response is analyzed using a new three-dimensional finite element model of high-speed train–bridge interactions. The track structures are idealized using beam elements with the offset of beam nodes and beams on a two-parameter elastic foundation. The vehicle model developed for a 300 km/h train is employed. The calculated reliabilities for performance of the considered bridges and the passenger comfort on board of high-speed trains are compared to the conventional safety indices. The results of this study allow identifying the quantification of uncertainties that can control quality of the high-speed train service.