弹性矩形板动静力分析的有限积分变换法
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摘要
本文以弹性矩形板为研究对象,采用有限积分变换法研究了Kirchhoff薄板、Mindlin中厚板以及从三维弹性力学角度建立的强厚板和层合厚板。在分析过程中,首先对弹性矩形板的基本控制方程进行有限积分变换,将高阶偏微分方程转化为线性代数方程,通过线性方程的求解,并进行相应的积分逆变换,就可以得到实际问题的精确解。
与与传统的叠加法和傅里叶半逆解法相比,有限积分变换法使问题的求解得到了简化,而且在求解过程中出现的待定常数均具有明确的物理意义。在求解过程中,由于不需要人为选取挠度函数,而是直接从板的基本控制方程出发,计算得到了各种边界条件下的精确解,因此求解过程史加合理。
在四边固支板、弹性地基上四边自由板求解的基础上,本文总结了任意边界支承Kirchhoff薄板、Mindlin中厚板的位移函数统一公式,运用该公式,可直接得到矩形板的位移函数。该公式的提出,极大地简化了有限积分变换法的求解过程,克服了基础理论推导比较繁琐的缺点,为编程计算提供了方便。
实际上,矩形板的求解属于三维空间问题,为了便于求解才将其简化为二维平面问题。随着近年来工程领域中矩形板厚度的逐渐增大和复合材料层合板的广泛应用,二维平面解误差逐步增大,已不能满足工程要求。另外,无论是一阶理论还是高阶理论,都会由于采用人为假定的应力或位移函数而导致弹性力学基本方程的不相容,无法包含全部的弹性常数,这意味着某些弹性常数的变化对计算结果毫无影响,这显然与实际情况是不相符的。因此,若要得到矩形板问题真正意义上的精确解,必须从三维弹性力学角度出发进行计算。
文中最后摒弃了Kirchhoff薄板和Mindlin中厚板理论中关于位移、应力的一切人为假定,完全从三维弹性力学基本方程出发,采用空间状态向量与有限积分变换相结合的方法,求得了四边固支强厚板和层合厚板的精确解。与传统弹性力学六阶微分矩阵方法不同的足,预先将关于应力和位移分量的基本方程化为两个彼此独立的四阶、二阶矩阵微分方程后再分别进行求解。由于预先将求解方程进行了降阶处理,因此极大地提高了求解效率
The present paper focuses on the problems of elastic rectangular plates. The finite integral transform method is adopted to solve the static and dynamic problems of Kirchhoff thin plates, Mindlin plates and three-dimensional thick plates. Firstly, the finite integral transform is applied to the governing equations of elastic rectangular plates. High order partial differential equations are then transformed into a system of linear algebraic equations, by which the exact solutions of the problems are obtained via the corresponding inverse transform.
Compared with the traditional superposition method and semi-inverse method, the finite integral transform method simplifies the solution process of the problems while the undetermined constants in the solution procedure have the obvious physical significance. The method eliminates the need to pre-determine the deflection functions and derives the exact solutions of plates under various boundary conditions theoretically because it starts from the governing equations; hence it is more reasonable than other available methods.
Based on the solution of fully clamped plates and plates on elastic foundation, the generalized displacement functions for Kirchhoff thin plates and Mindlin plates with arbitrary boundary conditions are proposed, by which the generalized displacement functions can be directly obtained. The proposed functions greatly simplify the calculations and overcome the shortcomings of the complexity in theoretical derivation, providing obvious convenience in programming.
Solution of elastic plates belongs to the three-dimensional problem but two-dimensional simplification is usually adopted for convenience. With the increased structure thickness in practical engineering and the vast application of composite laminated plates, the error of two-dimensional solution becomes increasingly significant. In addition, both the first-order and the high-order theories pre-determine the stress or deflection function, which induces the incompatibility of basic equations of elasticity as well as the loss of certain elastic constants. This indicates that the changes of some constants do not influence the calculation results, which is obviously not practical. Consequently, the real exact solutions of rectangular plates must be obtained from three-dimensional calculation.
All assumptions about the stresses or the deflections in problems of Kirchhoff thin plates and Mindlin plates are eliminated in the last part of this paper. Based on three-dimensional fundamental equations of elasticity, the theory of state vector space is used in combination with the finite integral transform method for analytic solutions of fully clamped thick plates and laminated plates. In contrast with the traditional method with respect to the sixth-order matrix in elasticity, the basic equations about the stresses and the deflections are separated as two matrix differential equations, one second order and another fourth order. Because of the order reduction, the solution efficiency improves significantly.
与与传统的叠加法和傅里叶半逆解法相比,有限积分变换法使问题的求解得到了简化,而且在求解过程中出现的待定常数均具有明确的物理意义。在求解过程中,由于不需要人为选取挠度函数,而是直接从板的基本控制方程出发,计算得到了各种边界条件下的精确解,因此求解过程史加合理。
在四边固支板、弹性地基上四边自由板求解的基础上,本文总结了任意边界支承Kirchhoff薄板、Mindlin中厚板的位移函数统一公式,运用该公式,可直接得到矩形板的位移函数。该公式的提出,极大地简化了有限积分变换法的求解过程,克服了基础理论推导比较繁琐的缺点,为编程计算提供了方便。
实际上,矩形板的求解属于三维空间问题,为了便于求解才将其简化为二维平面问题。随着近年来工程领域中矩形板厚度的逐渐增大和复合材料层合板的广泛应用,二维平面解误差逐步增大,已不能满足工程要求。另外,无论是一阶理论还是高阶理论,都会由于采用人为假定的应力或位移函数而导致弹性力学基本方程的不相容,无法包含全部的弹性常数,这意味着某些弹性常数的变化对计算结果毫无影响,这显然与实际情况是不相符的。因此,若要得到矩形板问题真正意义上的精确解,必须从三维弹性力学角度出发进行计算。
文中最后摒弃了Kirchhoff薄板和Mindlin中厚板理论中关于位移、应力的一切人为假定,完全从三维弹性力学基本方程出发,采用空间状态向量与有限积分变换相结合的方法,求得了四边固支强厚板和层合厚板的精确解。与传统弹性力学六阶微分矩阵方法不同的足,预先将关于应力和位移分量的基本方程化为两个彼此独立的四阶、二阶矩阵微分方程后再分别进行求解。由于预先将求解方程进行了降阶处理,因此极大地提高了求解效率
The present paper focuses on the problems of elastic rectangular plates. The finite integral transform method is adopted to solve the static and dynamic problems of Kirchhoff thin plates, Mindlin plates and three-dimensional thick plates. Firstly, the finite integral transform is applied to the governing equations of elastic rectangular plates. High order partial differential equations are then transformed into a system of linear algebraic equations, by which the exact solutions of the problems are obtained via the corresponding inverse transform.
Compared with the traditional superposition method and semi-inverse method, the finite integral transform method simplifies the solution process of the problems while the undetermined constants in the solution procedure have the obvious physical significance. The method eliminates the need to pre-determine the deflection functions and derives the exact solutions of plates under various boundary conditions theoretically because it starts from the governing equations; hence it is more reasonable than other available methods.
Based on the solution of fully clamped plates and plates on elastic foundation, the generalized displacement functions for Kirchhoff thin plates and Mindlin plates with arbitrary boundary conditions are proposed, by which the generalized displacement functions can be directly obtained. The proposed functions greatly simplify the calculations and overcome the shortcomings of the complexity in theoretical derivation, providing obvious convenience in programming.
Solution of elastic plates belongs to the three-dimensional problem but two-dimensional simplification is usually adopted for convenience. With the increased structure thickness in practical engineering and the vast application of composite laminated plates, the error of two-dimensional solution becomes increasingly significant. In addition, both the first-order and the high-order theories pre-determine the stress or deflection function, which induces the incompatibility of basic equations of elasticity as well as the loss of certain elastic constants. This indicates that the changes of some constants do not influence the calculation results, which is obviously not practical. Consequently, the real exact solutions of rectangular plates must be obtained from three-dimensional calculation.
All assumptions about the stresses or the deflections in problems of Kirchhoff thin plates and Mindlin plates are eliminated in the last part of this paper. Based on three-dimensional fundamental equations of elasticity, the theory of state vector space is used in combination with the finite integral transform method for analytic solutions of fully clamped thick plates and laminated plates. In contrast with the traditional method with respect to the sixth-order matrix in elasticity, the basic equations about the stresses and the deflections are separated as two matrix differential equations, one second order and another fourth order. Because of the order reduction, the solution efficiency improves significantly.
引文
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