含固支边或自由边矩形叠层厚板的状态空间解法研究

发布时间：2018-07-03 04:51

本文选题：矩形叠层厚板 + 固支/自由边　；参考：《合肥工业大学》2017年博士论文

【摘要】：经典薄板理论和各种中厚板理论都是建立在某些人为假设基础上的,若将这些理论用于求解叠层厚板问题,会产生不可忽略的误差。为了获得叠层厚板的三维精确解,一些基于三维弹性力学基本方程的厚板理论逐渐被提出来。状态空间法作为厚板理论中最有效、最流行的解法,能很好地处理层间的连续性问题,且其解的形式简单而统一,便于理解和应用。采用状态空间法求解矩形叠层厚板问题时,通过双傅里叶级数展开进行变量分离,恰好能严格满足四边简支的边界条件。但是对于含非简支边的矩形板,求解依然存在一些难度,常见的方法是通过在非简支边上假定待定的边界位移函数,并采用分层的办法进行求解,所得到的解在非简支边上并不能沿厚度方向严格满足边界条件。本文以含固支边或自由边的矩形叠层厚板作为研究对象,采用状态空间法求解该类矩形叠层厚板静力问题的三维精确解。在求解过程中,为了严格满足固支边或自由边的边界条件,在该边上假定边界位移函数,并将其作为状态变量引入状态方程,建立不同边界条件下的矩形单层与叠层厚板的齐次状态方程,得到相应静力问题的三维精确解。整个求解过程简单清晰,无需处理大量未知量,便于应用。在第三章到第七章中,针对不同边界条件下的矩形单层与叠层厚板,分别建立了齐次状态方程,得到了相应静力问题的三维精确解。算例表明,本文解与有限元解吻合得很好,具有很高的精度和很好的收敛性,而且具有很广的适用性。与经典薄板理论和各种中厚板理论相比,本文解严格满足三维弹性力学基本方程,考虑了所有的弹性参数,是正真意义上的三维精确解,能够给出位移和应力分量沿厚度方向的精确分布规律;而且该解不受板的厚度和材料属性的限制,能很好地处理叠层板的层间连续性问题,充分体现了状态空间法求解叠层厚板问题的优越性。与现有三维精确解相比,本文完全采用解析方法建立了不同边界条件下的矩形单层与叠层厚板的齐次状态方程,使固支边或自由边也能严格满足边界条件,并在这些边界上得到了非常精确的位移和应力结果。这表明,本文解突破了现有三维精确解对于求解含非简支边矩形叠层厚板的限制。此外,对与固支边或自由边的边界位移函数相关联的多项式函数的次数作了比较和研究,结果表明,多项式的次数对本文解的精度和收敛性影响不大。
[Abstract]:The classical thin plate theory and all kinds of plate theories are based on some artificial assumptions. If these theories are used to solve the laminated thick plate problems, the errors can not be ignored. In order to obtain the exact three-dimensional solution of thick laminated plates, some theories of thick plates based on the basic equations of three-dimensional elasticity have been proposed gradually. As the most effective and popular method in thick plate theory, the state space method can deal with the continuity problem between layers well, and the form of the solution is simple and uniform, which is easy to understand and apply. When the state space method is used to solve the problem of rectangular laminated thick plates, the variables are separated by double Fourier series expansion, which can exactly satisfy the boundary condition of simply supported on four sides. However, for rectangular plates with non-simply supported edges, there are still some difficulties in solving the problem. The common method is to assume the undetermined boundary displacement function on the non-simply supported edges, and to solve the problem by stratification. The obtained solution does not satisfy the boundary condition strictly in the direction of thickness on the edge of non-simple support. In this paper, a rectangular laminated thick plate with clamped or free edges is used as the research object. The state space method is used to solve the three-dimensional exact solution of the static problem of the rectangular laminated thick plate. In order to satisfy the boundary conditions of clamped or free edges strictly, the boundary displacement function is assumed to be a state variable and is introduced into the equation of state. The homogeneous state equations of rectangular monolayer and laminated thick plates under different boundary conditions are established and the exact three-dimensional solutions of the corresponding static problems are obtained. The whole solution process is simple and clear, it does not need to deal with a large number of unknown quantities, so it is easy to be applied. In the third to seventh chapters, the homogeneous equation of state is established for the rectangular monolayer and laminated thick plates under different boundary conditions, and the three dimensional exact solution of the corresponding static problem is obtained. The numerical examples show that the proposed solution is in good agreement with the finite element solution, has high accuracy and good convergence, and has a wide range of applicability. Compared with the classical thin plate theory and various plate theories, the solution in this paper satisfies the basic equations of three-dimensional elasticity strictly, and considers all the elastic parameters. It is a exact three-dimensional solution in the sense of positive truth. The displacement and stress components can be accurately distributed along the thickness direction, and the solution is not limited by the thickness of the plate and the properties of the material, so it can deal with the continuity problem between the laminated plates well. The advantages of state space method for solving thick laminated plates are fully demonstrated. Compared with the existing three dimensional exact solutions, the homogeneous state equations of rectangular monolayer and laminated thick plates under different boundary conditions are completely established by using analytical method, so that the clamped or free edges can satisfy the boundary conditions strictly. The results of displacement and stress on these boundaries are very accurate. It is shown that the solution in this paper breaks through the limitations of the existing three-dimensional exact solutions for solving thick rectangular laminated plates with non-simply supported edges. In addition, the degree of polynomial function associated with the boundary displacement function of fixed or free edges is compared and studied. The results show that the degree of polynomial has little effect on the accuracy and convergence of the solution in this paper.
【学位授予单位】：合肥工业大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：TB33

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