微分求积法在计算功能梯度Timoshenko梁临界荷载中的应用研究  

Calculation of critical load for functionally graded Timoshenko beam using differential quadrature method

作　　者：葛仁余 张佳宸 刘凡 陈哲 熊海超 Ge Renyu;Zhang Jiacheng;Liu Fan;Chen Zhe;Xiong Haichao(School of Architecture and Civil Engineering,Anhui Polytechnic University,241000,Wuhu,China)

机构地区：[1]安徽工程大学建筑工程学院,芜湖241000

出　　处：《应用力学学报》2020年第6期2634-2641,I0022,共9页Chinese Journal of Applied Mechanics

基　　金：国家级大学生创新创业训练计划项目(201810363088);安徽省自然科学基金(1808085ME147)。

摘　　要：提出了一种改进型等比数列布点方式研究轴向功能梯度Timoshenko变截面梁的屈曲临界荷载。首先基于Timoshenko梁理论,建立了求解功能梯度材料Timoshenko变截面梁屈曲临界荷载的变系数常微分方程,由微分求积法理论将其转化为标准型的广义代数特征值问题,再采用QR法求解该代数特征方程组,可一次性地计算出轴向功能梯度Timoshenko变截面梁的屈曲临界荷载。数值计算结果表明:当梁上区间单元划分段数N取28时,采用改进型等比数列布点方式和切比雪夫多项式根布点方式时,由微分求积法(DQM)获得的屈曲临界荷载数值解计算精度等价且与实际值完全吻合,证明了本文方法的可行性和计算精度;当N取8和12时,采用切比雪夫多项式根作为布点方式的计算值与实际值误差较大甚至失真,而采用等比数列变步长布点方式时,公比q为控制算法精度的控制参数,通过调整公比q可获得精确值,相对于切比雪夫多项式根作为布点方式,这一优势十分明显。Based on the theory of differential quadrature method(DQM),an improved method of arranging points in equal ratio sequence is proposed to study the buckling critical load of a Timoshenko beam.Firstly,based on Timoshenko beam theory,the calculation of the critical load of the axial functionally gradient material beam with variable cross section is transformed into the eigenvalue problem of variable coefficient ordinary differential equation.From the theory of differential quadrature method,the eigenvalue problem of ordinary differential equation with variable coefficients is transformed into the generalized algebraic eigenvalue problem of standard type,then QR method is used to solve the algebraic characteristic equations.The critical buckling load of an axial functionally gradient material Timoshenko beam with variable cross section can be calculated at one time.The numerical results show that,when the number of section elements on the beam N is 28,using two variable step methods of equal ratio sequence and the roots of Chebyshev polynomials,the accuracy of the numerical solution of critical load obtained by DQM is equivalent to that of the actual value,which proves the feasibility and accuracy of the proposed method.When N is 8 or 12,the error between the calculated value and the actual value using Chebyshev polynomial root as the distribution method is large or even distorted.In the case of constant ratio series with variable step size,the common ratio q is the control parameter of the control algorithm precision,and the precise value can be obtained by adjusting the common ratio q,compared with the Chebyshev polynomial root as the way of distribution,this advantage is very obvious.

关 键 词：TIMOSHENKO梁 切比雪夫多项式 等比数列 屈曲临界荷载 微分求积法 

分 类 号：O326[理学—一般力学与力学基础]