轴向功能梯度Timoshenko变截面梁的屈曲分析

Buckling analysis of axially functionally graded Timoshenko beams with variable cross-section

运用插值矩阵法研究了不同边界条件下轴向功能梯度材料变截面Timoshenko梁的屈曲性能问题。基于Timoshenko梁基本理论,将轴向功能梯度变截面Timoshenko梁临界荷载的计算转化为一组变系数常微分方程特征值问题,然后运用插值矩阵法可一次性地计算出轴向功能梯度变截面梁在不同边界条件下的屈曲临界荷载。当区间划分点数n为80时,在不同的边界条件下均质材料等截面Timoshenko梁量纲为一的临界荷载的本文计算值与解析解有7位有效数字相同,轴向功能梯度Timoshenko锥形梁量纲为一的临界荷载的本文计算值与已有文献计算结果有3~5位有效数字相同,数值计算结果表明了本文方法的有效性和较高的计算精度。同时,本文方法可获取相应的挠度模态函数,而且对于材料梯度函数和截面几何轮廓的具体形式无任何限制条件。

In this paper, the buckling behaviors of axially functionally graded and variable cross-section Timoshenko beams with different boundary conditions are investigated using interpolating matrix method(IMM). Based on the Timoshenko beam theory, the governing equations of free transverse bending analysis of axially functionally graded Timoshenko beams under the buckling critical load are transformed into a set of nonlinear characteristic ordinary differential equations. Then the interpolating matrix method is employed to solve numerically the nonlinear governing equation of an axially functionally graded Timoshenko beam with different boundary conditions, all the buckling critical load of axially functionally graded beam with variable cross-section are calculated. Under different boundary conditions, the dimensionless critical load of an uniform Timoshenko beam obtained by using IMM converge to the analytic solution up to the seventh significant figure when the interval division numbers is 80, and the dimensionless critical load of an axially FG Timoshenko beam with different taper coefficient obtained by using IMM converges to the exist solution up to the third or fifth significant figure when the interval division numbers is 80. Therefore the validity and accuracy of the proposed method are illustrated. Simultaneously, the critical load companying with the corresponding deflection mode functions of an axially functionally graded beam are calculated at a time. The present methods do not pose any restrictions on both the type of material gradation and the variation of the cross section profile.