提出了一种截面插值梁模型，利用该模型求解梁在非均匀热载荷作用下的动静态响应，解决了传统梁理论无法处理受不均匀温度场的梁的问题。首先利用拉格朗日插值函数对梁单元的截面和轴向分别插值，构造梁的位移场。接着，将位移场代入热弹性动力学方程，得到单元应变和应力，再依据虚功原理推导出单元刚度矩阵、质量矩阵以及等效节点载荷列阵，求解得到热应力。最后利用热应力的横向剪切力更新单元刚度矩阵，计算梁在热载荷作用下的振动特性。计算结果表明，本文方法得到的结果与实体单元模型结果吻合，并且更易于处理受非均匀热载荷作用的细长结构，同时能很好地反映截面的形状、受载及响应结果。
Abstract
A cross-section interpolation beam model was proposed to solve the dynamic and static response of the beam under nonuniform thermal load. It solved the problem that the traditional beam theory cannot deal with the beam subjected to nonuniform temperature field. Firstly, the Lagrange interpolation function was used to interpolate the section and axial of the beam element to construct the displacement field of the beam. Then, the displacement field was substituted into the thermoelastic dynamic equation, and the element strain and stress were obtained. According to the principle of virtual work, the element stiffness matrix, mass matrix and equivalent node load array were derived, and the thermal stress was solved. Finally, the element stiffness matrix was updated by the transverse shear force of thermal stress to calculate the vibration characteristics of the beam under thermal load. The calculation results showed that the results obtained by the method in this paper are consistent with the results of the solid element model, and it is easier to deal with the slender structure under non-uniform thermal load, and can well reflect the shape, load and response results of the cross section.
关键词
空间梁模型 /
截面插值 /
截面变形 /
静力计算 /
热模态分析
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Key words
spatial beam model /
cross-section interpolations /
cross-section deformation /
static calculation /
thermal modal analysis
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