Dynamic Response of Curved Beam under Moving Load

In this paper an integral transform method is used to analyze the dynamic response of simple supported curved beam under single moving load with constant speed, and some parameters are defined. These parameters, such as radius of curvature, ratio of stiffness, velocity, warping stiffness, which may influence the response, are also discussed.

Dynamic response of a beam on a Pasternak foundation and under a moving load

The dynamic response of an infinite beam placed on a Pasternak foundation when the system was subjected to a moving load was investigated.We used the double Fourier transform and its inversion to solve the formulations of the problem.A closed form analytic solution of the beam was obtained by the theorem of residues.We selected a numerical example to illustrate the dynamic response of the beam on Pasternak and Winkler foundations,respectively.We discuss the effect of the moving load velocity on the dynamic displacement response of the beam.The maximum deflection of the beam increases slightly with increased load velocity but increases significantly with reduced shear modulus of subgrade at a given velocity.The maximum deflection of a beam resting on a Pasternak foundation is much smaller than that of a beam on a Winkler foundation.

Dynamic response of multi-scale structure in flexible pavement to moving load

In order to study the dynamic responses in the microstructures of the pavement structure, the multi-scale modeling subjected to moving load is analyzed using the discrete element method （DEM）. The macro-scale discrete element model of the flexible pavement structure is established. The stress and strain at the bottom of the asphalt concrete layer under moving load are calculated. The DEM model is validated through comparison between DEM predictions and the results from the classical program. Based on the validated macro-scale DEM model, the distribution and the volumetric fraction of coarse aggregate, mastics and air voids at the bottom of the asphalt layer are modeled, and then the multi-scale model is constructed. The dynamic response in the microstructures of the multi-scale model are calculated and compared with the results from the macro model. The influence of mastic stiffness on the distribution of dynamic response in the microstructures is also analyzed. Results show that the average values and the variation coefficient of the tensile stress at the aggregate-mastic interface are far more than those within the mastics. The dynamic response including stress and strain distributes non-uniformly in both mastics and the interface. An increase in mastic stiffness tends to a uniform distribution of tensile stress in asphalt concrete.

STEADY-STATE RESPONSE OF A TIMOSHENKO BEAM ON AN ELASTIC HALF-SPACE UNDER A MOVING LOAD

By introducing the equivalent stiffness of an elastic half-space interacting with a Timoshenko beam, the displacement solution of the beam resting on an elastic half-space subjected to a moving load is presented. Based on the relative relation of wave velocities of the half-space and the beam, four cases with the combination of different parameters of the half-space and the beam, the system of soft beam and hard half-space, the system of sub-soft beam and hard half- space, the system of sub-hard beam and soft half-space, and the system of hard beam and soft half-space are considered. The critical velocities of the moving load are studied using dispersion curves. It is found that critical velocities of the moving load on the Timoshenko beam depend on the relative relation of wave velocities of the half-space and the beam. The Rayleigh wave velocity in the half-space is always a critical velocity and the response of the system will be infinite when the load velocity reaches it. For the system of soft beam and hard half-space, wave velocities of the beam are also critical velocities. Besides the shear wave velocity of the beam, there is an additional minimum critical velocity for the system of sub-soft beam and hard half-space. While for systems of （sub-） hard beams and soft half-space, wave velocities of the beam are no longer critical ones. Comparison with the Euler-Bernoulli beam shows that the critical velocities and response of the two types of beams are much different for the system of （sub-） soft beam and hard half-space but are similar to each other for the system of （sub-） hard beam and soft half space. The largest displacement of the beam is almost at the location of the load and the displacement along the beam is almost symmetrical if the load velocity is smaller than the minimum critical velocity （the shear wave velocity of the beam for the system of soft beam and hard half-space）. The largest displacement of the beam shifts behind the load and the asymmetry of the displacement along the beam increases with the increase of the load velocity due to the damping and wave racliation. The displacement of the beam at the front of the load is very small if the load velocity is larger than the largest wave velocity of the beam and the half space. The results of the present study provide attractive theoretical and practical references for the analysis of ground vibration induced by the high-speed train.

High-precision solution to the moving load problem using an improved spectral element method

In this paper, the spectral element method(SEM)is improved to solve the moving load problem. In this method, a structure with uniform geometry and material properties is considered as a spectral element, which means that the element number and the degree of freedom can be reduced significantly. Based on the variational method and the Laplace transform theory, the spectral stiffness matrix and the equivalent nodal force of the beam-column element are established. The static Green function is employed to deduce the improved function. The proposed method is applied to two typical engineering practices—the one-span bridge and the horizontal jib of the tower crane. The results have revealed the following. First, the new method can yield extremely high-precision results of the dynamic deflection, the bending moment and the shear force in the moving load problem.In most cases, the relative errors are smaller than 1%. Second, by comparing with the finite element method, one can obtain the highly accurate results using the improved SEM with smaller element numbers. Moreover, the method can be widely used for statically determinate as well as statically indeterminate structures. Third, the dynamic deflection of the twin-lift jib decreases with the increase in the moving load speed, whereas the curvature of the deflection increases.Finally, the dynamic deflection, the bending moment and the shear force of the jib will all increase as the magnitude of the moving load increases.

Analytical study on the dynamic displacement response of a curved track subjected to moving loads

A closed-form out-of-plane dynamic displacement response of a curved track subjected to moving loads was proposed.The track structure was modeled as a planar curved Timoshenko beam periodically supported by the double-layer spring-damping elements.The general dynamic displacement response induced by the moving loads along the curve on the elastic semi-infinite space was firstly obtained in the frequency domain,according to the Duhamel integral and the dynamic reciprocity theorem.In the case of the periodic curved track structure subjected to moving loads,the dynamic displacement equation was simplified into a form of summation within the basic track cell instead of the integral.The transfer function for the curved track was expressed in the form of a transfer matrix.Single and series moving loads were involved in the calculation program.For the verification of the analytical model,the mid-span vertical deflection of a simply support curved beam subjected to moving load was recalculated and compared with the same case in the reference.The research results indicate that:under the same moving loads,the displacement response of the curved track decreases slightly with the increasing track radius,and the displacement response of the curved track with the radius greater than or equal to 600 m is almost equivalent to the displacement response of the straight track;the frequency spectrum of the curved track is more abundant than that of the straight track,which may result in more wheel-rail resonance and rail corrugation in the curved lines.

DYNAMIC ANALYSIS TO INFINITE BEAM UNDER A MOVING LINE LOAD WITH UNIFORM VELOCITY

Based on the principle of linear superposition, this paper proves generalized Duhamel's integral which reserves moving dynamical load problem to fixed dynamical load problem. Laplace transform and Fourier transform are used to solve patial differential equation of infinite beam. The generalized Duhamel's integral and deflection impulse response function of the beam make it easy for us to obtain final solution of moving line load problem. Deep analyses indicate that the extreme value of dynamic response always lies in the center of the line load and travels with moving load at the same speed. Additionally, the authors also present definition of moving dynamic coefficient which reflects moving effect.

DYNAMIC RESPONSE OF A DOUBLE-LAYERED SUBGRADE WITH ROCK SUBSTRATUM TO A MOVING POINT LOAD

In this paper, an analytical solution for the dynamic response of a double-layered subgrade with rock substratum to a moving point load is derived. The subgrade profile is divided into two layers. The upper layer is modeled by an elastic medium and the lower layer by a fully saturated poroelastic medium governed by Biot’s theory. In the meanwhile, the subgrade is resting on the rock substratum. The analytical solutions for stress, displacement and pore pressure are derived by using the Fourier transform. Numerical results obtained by using the inverse fast Fourier transform (IFFT) are used to analyze the influence of the moving load velocity, the thickness of an elastic medium layer and a fully saturated poroelastic medium layer on the dynamic response.

Responses of HFR-LWC beams under close-range blast loadings accompanying membrane action

The load-carrying capacities and failure patterns of reinforced concrete components can be significantly changed by membrane effects.However,limited work has been carried out to investigate the blast resistance of Hybrid Fiber Reinforced Lightweight Aggregate Concrete(HFR-LWC)members accompanying membrane action.This paper presents a theoretical approach to quantitatively depicting the membrane behavior and its contribution on the behavior of HFR-LWC beams under close-range blast loadings,and the suitability of the proposed model is validated by a series of field tests.An improved Single-Degree-of-Freedom(SDOF)model was employed to describe the dynamic responses of beam-like members under blast loadings accompanying membrane action,where the mass-load coefficient is determined according to the nonuniformly distributed load induced by close-range explosion,and the membrane action is characterized by an in-plane(longitudinal)force and a resisting moment.The elastoplastic and recovery responses of HFR-LWC beams under the combined action of blast load and membrane force were analyzed by the promoted model.A specially built end-constrain clamp was developed to provide membrane action for the beam member when they are subjected to blast load simultaneously.It is demonstrated that the analytical displacement-time histories are in good agreement with experimental results before peak deflections and that the improved SDOF model is an acceptable tool for predicting the behavior of HFR-LWC beams under blast loadings accompanying membrane action.

Numerical Simulation-Based Analysis of the Impact of Overloading on Segmentally Assembled Bridges

Segmentally assembled bridges are increasinglyfinding engineering applications in recent years due to their unique advantages,especially as urban viaducts.Vehicle loads are one of the most important variable loads acting on bridge structures.Accordingly,the influence of overloaded vehicles on existing assembled bridge structures is an urgent concern at present.This paper establishes thefinite element model of the segmentally assembled bridge based on ABAQUS software and analyzes the influence of vehicle overload on an assembled girder bridge struc-ture.First,afinite element model corresponding to the target bridge is established based on ABAQUS software,and the load is controlled to simulate vehicle movement in each area of the traveling zone at different times.Sec-ond,the key cross-sections of segmental girder bridges are monitored in real time based on the force character-istics of continuous girder bridges,and they are compared with the simulation results.Finally,a material damage ontology model is introduced,and the structural damage caused by different overloading rates is compared and analyzed.Results show that thefinite element modeling method is accurate by comparing with on-site measured data,and it is suitable for the numerical simulation of segmental girder bridges;Dynamic sensors installed at 1/4L,1/2L,and 3/4L of the segmental girder main beams could be used to identify the dynamic response of segmental girder bridges;The bottom plate of the segmental girder bridge is mostly damaged at the position where the length of the precast beam section changes and the midspan position.With the increase in load,damage in the direction of the bridge develops faster than that in the direction of the transverse bridge.Thefindings of this study can guide maintenance departments in the management and maintenance of bridges and vehicles.

AN ANALYSIS oF THE SHEAR FAILURE OF RIGID-LINEAR HARDENING BEAMS UNDER IMPULSIVE LOADING

A theoretical rigid-plastic analysis for the dynamic shear failure of beams under impulsive loading is presented when using a travelling plastic shear hinge model which tabes into account material strain hardening. The maximum dynamic shear strain and shear strain-rate can be predicted in addition to the permanent transverse deflections and other parameters. The conditions for the three modes of shear failure, i.e., excess deflection failure, excess shear strain failure and adiabatic shear failure are analyzed. The special case of an infinitesimally small plastic zone is discussed and compared with Nonaka's solution for a rigid, perfectly plastic material. The results can also be generalized to examine the dynamic response of fibre-reinforced beams.

Application of semi-analytical finite element method to analyze the bearing capacity of asphalt pavements under moving loads

To facilitate long term infrastructure asset management systems, it is necessary to determine the bearing capacity of pavements. Currently it is common to conduct such measurements in a stationary manner, however the evaluation with stationary loading does not correspond to reality a tendency towards continuous and high speed measurements in recent years can be observed. The computational program SAFEM was developed with the objective of evaluating the dynamic response of asphalt under moving loads and is based on a semi-analytic element method. In this research project SAFEM is compared to commercial finite element software ABAQUS and field measurements to verify the computational accuracy. The computational accuracy of SAFEM was found to be high enough to be viable whilst boasting a computational time far shorter than ABAQUS. Thus, SAFEM appears to be a feasible approach to determine the dynamic response of pavements under dynamic loads and is a useful tool for infrastructure administrations to analyze the pavement bearing capacity.

Dynamic Analysis of Structural Columns Subjected to Impulsive Loading

For a building structure subjected to impulsive loading, particularly shock and impact loading , the response of the critical columns is crucial to the behaviour of the entire system during and after the blast loading phase. Therefore, an appropriate evaluation of the column response and damage under short-duration impulsive loading is important in a comprehensive assessment of the performance of a building system. This paper reports a dynamic analysis approach for the response of RC columns subjected to impulsive loading. Considering that the dynamic response of a column in a frame structure can also be affected by the floor movement which relates to the global vibration of the frame system, a generic column-mass model is used, in which a concentrated mass is attached to the column top to simulate the effect of a global vibration. To take into account the high shear effect under impulsive load, the model is formulated using Timoshenko beam theory, and three main nonlinear mechanisms are considered. Two typical scenarios, one under a direct air blast loading, and another under a blast-induced ground excitation, are analyzed and the primary response features are highlighted.

Vibration characteristics of frozen soil under moving track loads

Vibration due to moving traffic loads is an important factor which induces frozen soil damage; this paper analyzed these vibration characteristics of frozen soil foundation under track loads. Firstly, seismic observation array （SOA） technology was applied to monitor the three dimensional dynamic characteristics of frozen soil under movable track load in a per- mafrost region and seasonal frozen soil area. Secondly, a numerical simulation for the response of frozen soil under movable track load was performed based on finite element analysis （FEA）, The results show that dynamic characteristics of frozen soil in perpendicular and parallel direction of the track are obviously different. In the direction perpendicular to the track, the vertical acceleration amplitude had an abrupt increase in the 9-10 m from the track line. In the direction parallel to the track, the acceleration in vertical and horizontal direction had a quick attenuation compared to the other direction. Lastly, various parameters were analyzed for the purpose of controlling the dynamic response of frozen soil and the vibration attenuation in frozen soil layer.

Quasi-static and dynamical analyses of a thermoviscoelastic Timoshenko beam using the differential quadrature method

The quasi-static and dynamic responses of a thermoviscoelastic Timoshenko beam subject to thermal loads are analyzed. First, based on the small geometric deformation assumption and Boltzmann constitutive relation, the governing equations for the beam are presented. Second, an extended differential quadrature method(DQM)in the spatial domain and a differential method in the temporal domain are combined to transform the integro-partial-differential governing equations into the ordinary differential equations. Third, the accuracy of the present discrete method is verified by elastic/viscoelastic examples, and the effects of thermal load parameters, material and geometrical parameters on the quasi-static and dynamic responses of the beam are discussed. Numerical results show that the thermal function parameter has a great effect on quasi-static and dynamic responses of the beam. Compared with the thermal relaxation time, the initial vibrational responses of the beam are more sensitive to the mechanical relaxation time of the thermoviscoelastic material.

结构刚塑性动力解的弹性补偿

近年来,我国学者以膜力因子法和饱和分析方法相结合为理论工具,对梁、板等结构件在脉冲载荷作用下的塑性大变形行为作了全面深入的研究,为脉冲加载下结构的最终挠度提供了优于历史上各种刚塑性近似解的最佳刚塑性预测公式。然而,由于实际工程应用中金属结构弹塑性动力响应的复杂性和数值模拟的局限性,与考虑材料弹性效应的结果相比,刚塑性解对脉冲加载下结构所预测的最终挠度的误差有多大,是一个亟待解决的关键问题。对这个问题的首阶段研究成果厘清了材料弹性对脉冲加载下结构塑性动态大变形的影响,定量评估了由最佳刚塑性理论解与弹塑性数值模拟得到的最终挠度预测结果之间的差异。在此基础上,提出了补偿弹性效应的策略和方法,即:在已有的最佳刚塑性解预测的挠度基础上添加一个补偿项,将补偿项表达为脉冲载荷强度的效应与结构自身刚度的效应分离的变量函数,并尽量减少待定系数/指数的数量,以求表达式的简洁;根据这些原则在金属结构的主要工程应用领域内选定结构刚度和外载参数的变化范围,对固支梁和固支方板的案例实施拟合与补偿,最后得到了对梁和板增添补偿项后的简单而实用的最终挠度预测公式,其相对误差在3%的范围之内,很适合工程设计应用。文末列表给出了符号与公式的一览,并对梁和方板的结果作了综合和比较。

重载与中强地震联合作用下的曲线独柱墩桥梁倾覆性分析

为探讨移动车辆和中强地震联合作用对曲线梁桥的抗倾覆影响,建立车-桥-地震分析模型,提出考虑地震作用的曲线梁桥车-桥耦合振动分析方法。以多跨曲线连续梁桥为研究对象,对比分析中强地震和重载车辆偏载组合作用下的支座反力和主梁倾覆时整体转角大小。研究地震动类型、峰值、车辆关键参数等因素对曲线梁桥倾覆稳定性的影响。结果表明:不同车辆和地震荷载组合会改变桥梁的受力情况,纵向间隔2 m的3辆100 t车辆以3 m/s速度行驶时,三跨曲线梁桥的倾覆最明显;综合考虑重载及地震联合作用,对抗倾覆性规范中的稳定系数计算公式进行了修正;分析了五跨曲线梁桥中墩支座偏心距对抗倾覆稳定系数的影响,计算结果表明中墩两相邻支座设置偏心时曲线桥最大抗倾覆系数为1.55。

累计冲击荷载作用下GFRP-混凝土组合梁动态响应分析

为探究玻璃纤维复合材料(glass fiber reinforce plastic,GFRP)-混凝土组合梁在多次落石冲击荷载下的动态响应,采用摇臂式落锤试验机,对1根纯GFRP梁及4根GFRP-混凝土组合梁进行累计冲击试验,得到了不同工况下试验梁位移、冲击力、应变时程曲线及裂缝开展状态,并分阶段讨论了梁体损伤演化过程。试验结果表明:纯GFRP梁具有优异的抗冲击性能,但冲击荷载作用下变形过大;组合梁的破坏特征均为混凝土主裂缝贯通,GFRP-混凝土界面发生剥离,良好的界面黏结性能是确保GFRP-混凝土组合梁抗冲击性能提升的关键;竖向冲击荷载下,混凝土厚度的增大,可明显减小同一冲击高度下梁体跨中的变形响应;试验梁抵抗侧向冲击的能力优于其抵抗竖向冲击的能力。采用显示动力分析软件LS-DYNA建立了GFRP-混凝土组合梁累计冲击模型,并基于该模型分析了冲击锤质量、冲击速度及截面高度比等参数对GFRP混凝土组合梁动态响应的影响规律。

移动荷载下考虑钢筋作用的混凝土裂缝梁动力响应分析

由于钢筋的约束作用,混凝土梁裂缝截面的局部柔度与无筋时不同。建立了移动荷载作用下考虑钢筋影响的混凝土裂缝梁动力方程,基于力平衡和钢筋与混凝土变形协调对裂缝截面的局部柔度进行修正,采用无质量弹簧模拟裂缝,得到计入钢筋作用的裂缝梁自振特性,通过Newmark方法求解控制方程获取钢筋混凝土梁的动力响应。以受移动车辆荷载的简支梁为例,分析不同荷载速度、裂缝深度、裂缝位置、配筋率等条件下的钢筋混凝土裂缝梁响应。研究结果表明,裂缝处钢筋的约束作用可以显著地影响梁的动力响应,计入钢筋提供的刚度后裂缝梁的位移响应总体减小,且随着开裂损伤程度的增加钢筋的影响呈现变大的趋势。移动荷载下裂缝梁的共振现象较无损梁更为不利,考虑钢筋作用后裂缝梁的共振速度有所提高,共振响应则有一定程度的降低。由于钢筋混凝土裂缝梁和无损梁的共振速度存在差异,在某些荷载速度下无损梁的位移响应大于裂缝梁。

基于切比雪夫拟谱法的移动荷载作用下简支梁动力响应计算

采用切比雪夫拟谱法求解移动荷载作用下简支梁的动力响应问题,该方法通过重新选取插值点的方式并借助转换矩阵对边界条件进行施加,对集中荷载采用高斯函数法进行正则化处理。将本文方法计算结果与解析解和有限元解进行比较,结果表明,在划分较少单元数量时,采用本文方法计算结果与解析解吻合良好,求解效率高于有限元方法。采用本文方法对Dirac函数的正则化参数σ_(n)进行敏感性分析,结果表明,当在±3σ_(n)范围内布置配点数6~10个时,计算结果能取得较好的精度。