一端固支一端简支变厚度梁的弹性力学解

Elasticity Solution of Clamped-Simply Supported Beams With Variable Thickness

研究了一端固支另一端简支连续变厚度梁在静力荷载作用下的应力和位移分布.通过引入单位脉冲函数和Dirac函数,将固支边等效为简支边与未知水平反力的叠加,利用平面应力问题的基本方程,导出满足控制微分方程及左右两端边界条件的位移函数的一般解,对上下表面的边界方程作Fourier级数展开,结合固支边位移为0的条件确定待定系数,得到的解是高精度的.数值结果与商业有限元软件ANSYS进行了比较,显示出很好的精度.

The stress and displacement distributions of continuously varyingg thickness beams with one end clamped and the other end simply supported under static loads are ,studied. By introducing the unit pulse functions and Dirac functions, the clamped edge can be made equivalent to the simply supported one by adding the unknown horizontal reactions. According to the governing equations of plane stress problem, the general expressions of displacements, which satisfy the governing differential equations and the boundary conditions at two ends of the beam, can be deduced. The unknown coeffidents in the general expressions were then determined by using the Fourier sinusoidal series expansion along the upper and lower boundaries of the beams and using the condition of zero displacements at the clamped edge. The solution obtained has excellent convergence property. The numerical results being compared with those obtained from the commercial software ANSYS, excellent accuracy of the present method is demonstrated.