两端固支叠层圆柱厚壳轴对称问题的精确解析解

Exact analytical solutions for axisymmetric problem of laminated cylindrical thick shells with two clamped ends

基于柱坐标系下的三维弹性力学基本方程,采用状态空间法得到两端固支单层与叠层圆柱厚壳轴对称问题的精确解析解。为严格满足固支端的边界条件,将固支端的边界位移函数作为状态变量引入状态方程,采用增维方法把非齐次状态方程变为齐次状态方程,并通过层合渐近技术将变系数状态矩阵转为常系数矩阵进行求解。所得到的解不仅严格满足三维弹性力学基本方程,而且严格满足固支边界条件,是真正意义上的三维精确解。算例表明,本研究解与有限元解吻合,具有很高的精度,且关于级数项数和分层数具有很好的收敛性。另外,通过圆柱厚壳各力学量沿径向和轴向的精确分布规律分析了厚径比和跨径比变化对位移和应力分布的影响。

Based on the three-dimensional basic equations of elasticity in cylindrical coordinates,the state space method was used to solve axisymmetric problem of thick single and laminated cylindrical shells with two clamped ends.To satisfy strictly the boundary conditions of the clamped ends,boundary displacement functions introduced into the state equations as state variables,the nonhomogeneous state equations were transformed into the homogeneous state equations by employing the dimensional expanding method,and variable coefficient state matrixes were changed into constant coefficient matrixes with the approximate laminate technique by dividing the layer into sub-layers.The three-dimensional basic equations of elasticity and all the boundary conditions are satisfied strictly,the exact three-dimensional analytical solutions are obtained.The results of two examples show that the present solutions obtained have high precision by comparing with FEM(Finite Element Method)solutions,and have good constringency with the number of series terms and sub-layers.In addition,the influences of thickness-radius ratio and span-radius ratio on the distributions of the displacements and stresses of cylindrical thick shells along the radial and axial directions are analyzed.