Finite element analysis of infinitely long beam resting on continuous viscoelastic foundation subjected to moving loads
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摘要: 把无限长梁、连续粘弹性基础和移动荷载视为一个系统, 并将该系统进行有限单元离散, 梁单元的弯曲形函数采用Hermitian三次方插值函数, 利用弹性系统动力学总势能不变值原理, 得到单元的刚度矩阵、质量矩阵、阻尼矩阵和节点荷载列阵, 建立该系统的振动方程组; 再用Wilsonθ法求解该振动方程组, 得到梁中点的位移时程曲线。举例分析了基础的粘弹性特性和梁的抗弯刚度对梁动力响应的影响。计算结果表明: 增大基础的弹性系数、阻尼系数和梁的抗弯刚度都有利于减小梁的动力响应。Abstract: The beam, foundation and moving loads were considered as a system, and the system was separated into a number of finite elements.Hermitian cubic interpolation function were utilized as the bending shape functions of the two-node beam element.The element stiffness matrix, mass matrix, damping matrix, and vector of element nodal forces could be obtained by the principle of total potential energy with stationary value in elastic system dynamics. The vibration equations of the system were established. The equations were solved by Wilson θ-method, and the displacement time histories of the beam at mid-point were found. Several numerical examples were presented, and the influences of the viscoelastic characteristic of foundation and the bending stiffness of beam on dynamic responses of beam were analyzed.Calculation results show that the increase either of spring stiffness, or of damping coefficient of foundation or of the bending stiffness of beam each leads to the decrease of dynamic responses of beam.
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图 1 连续粘弹性基础支承无限长梁模型
Figure 1. Model of infinitely long beam resting on continuous viscoelastic foundation
图 3 t时刻集中荷载P在梁单元上的位置
Figure 3. Position of concentrated load P on beam element at t time
图 4 用Wilson θ法求解振动方程组(1) 的流程
Figure 4. Flowchart for solving vibration equations (1) using Wilson θ method
图 5 单个移动荷载作用下长梁中点的位移时程曲线(无阻尼)
Figure 5. Displacement time history of the long beam at mid-point subjected to a moving load (without damping)
图 6 单个移动荷载作用下长梁中点的位移时程曲线(有阻尼)
Figure 6. Displacement time history of the long beam at mid-point subjected to a moving load (with damping)
图 7 三个移动荷载作用下长梁中点的位移时程曲线(无阻尼)
Figure 7. Displacement time history of the long beam at mid-point subjected to three moving loads (without damping)
图 8 三个移动荷载作用下长梁中点的位移时程曲线(有阻尼)
Figure 8. Displacement time history of the long beam at mid-point subjected to three moving loads (with damping)
表 1 长梁中点在移动荷载作用下的最大竖向位移
Table 1. Maximum displacement of long beam at mid-point subjected to moving loads
梁及基础参数 单个荷载P=112800 N作用下长梁中点最大竖向位移/m 三个荷载P1与P2、P2与P3之间距离为1.8 m, P1=P2=P3=112800N作用下长梁中点最大竖向位移/m 50型钢轨I=2.037×10-5 m4 ks=8×107 N/m2 c=0 1.1215×10-3 1.2279×10-3 c=1.3×105 Ns/m2 1.0223×10-3 1.0074×10-3 ks=1.6×108 N/m2 c=0 6.8052×10-4 7.3234×10-4 c=1.3×105 Ns/m2 6.0310×10-4 5.7984×10-4 60型钢轨I=3.217×10-5 m4 ks=8×107N/m2 c=0 1.1625×10-3 1.2019×10-3 c=1.3×105 Ns/m2 9.1647×10-4 9.2879×10-4 ks=1.6×108 N/m2 c=0 6.2875×10-4 6.4796×10-4 c=1.3×105 Ns/m2 5.4165×10-4 5.2476×10-4 75型钢轨I=4.49×10-5 m4 ks=8×107 N/m2 c=0 1.1151×10-3 1.038 3×10-3 c=1.3×105 Ns/m2 8.4569×10-4 8.927 7×10-4 ks=1.6×108 N/m2 c=0 6.6603×10-4 5.9460×10-4 c=1.3×105 Ns/m2 5.0035×10-4 4.911 8×10-4 
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