变厚度梁板结构的弹性静力学分析
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摘要
本文基于小变形的精确线弹性理论,不引入任何人为假设,研究了变厚度梁板结构的弹性静力学特性。首先,对于简支边界条件,利用傅立叶级数展开法,分别给出了任意载荷作用下两端简支变厚度梁的二维弹性力学解和四边简支变厚度矩形板的三维弹性力学解。其次,对于非简支边界条件,以固支边为例,通过引入单位脉冲函数,采用边界松弛法,给出了一端固支一端简支变厚度梁的弹性力学解。最后,将这种方法推广应用,研究了在热荷载与机械荷载共同作用下各向同性材料、正交各向异性材料、功能梯度材料、压电材料变厚度梁和变厚度矩形板的弯曲问题。
具体的说,本文的主要内容包括：
(1)对于两端简支变厚度梁,从二维平面弹性力学的基本方程出发,导出满足控制微分方程和两端简支边界条件的位移函数一般解,对梁上下表面的边界方程作傅立叶级数展开确定待定系数。
(2)对于四边简支变厚度矩形板,从三维弹性力学的基本方程出发,导出满足控制微分方程和四边简支边界条件的位移函数一般解,对板上下表面的边界方程作双重傅立叶正弦级数展开确定待定系数。
(3)对于一端固支一端简支变厚度梁,引入单位脉冲函数和Dirac函数,将固支边等价为简支边加上水平方向的未知边界力,求得其精确解析解,对梁上下表面的边界方程作傅立叶级数展开并与固支边位移为零的条件共同确定未知系数。
(4)对于功能梯度材料,假设弹性模量沿厚度方向指数变化,泊松比为常数,首先求得简支边界条件下控制微分方程的解析解,然后对结构上下表面的边界方程作傅立叶级数展开,分析了两端简支变厚度功能梯度梁和四边简支变厚度功能梯度矩形板的弯曲问题。
(5)对于压电材料,首先求解简支边界条件下精确满足控制微分方程的结构位移场和压电场,然后对结构上下表面的边界方程作傅立叶级数展开,分析了两端简支变厚度压电梁和横观各向同性四边简支变厚度压电矩形板的弯曲问题。
(6)对于受温度作用的两端简支变厚度梁和四边简支变厚度板,首先根据温度的边界条件,采用傅立叶正弦级数展开求解梁和板内的温度分布,然后再将温度荷载施加于梁和板上,给出了机械荷载与热荷载共同作用下变厚度梁的二维热弹性力学解和变厚度矩形板的三维热弹性力学解。
(7)对于多跨的板结构,首先求得满足控制微分方程和四边简支边界条件的矩形板位移函数的一般解,将支承反力看作是作用于板上的待求反力,利用板上下表面的边界方程确定待定系数。给出了点支、线支和弹性地基上简支矩形板以及面内受线支作用的功能梯度矩形板的三维弹性力学解。
This thesis studies the static properties of beams and plates with variable thickness, based on the small-strain linear elasticity theory which does not rely on any artificial hypotheses. Firstly, for the simple-supported boundary conditions, the two-dimensional elasticity solution of simple-supported beams with variable thickness and the three-dimensional elasticity solution of simple-supported rectangular plates with variable thickness under arbitrary loads are presented by using the Fourier series expansion method. Then, for the non-simply-supported boundary conditions, the clamped boundary condition is taken as an example. The elasticity solution of varying thickness beams with one end clamped and the other end simply supported under static loads are presented by introducing the unit pulse functions and using the boundary relaxation method. Finally, the Fourier series expansion method is extended to study the bending problem of beams with variable thickness and rectangular plates with variable thickness, subjected to thermo-mechanical load where the beams and the plates are, respectively, made of isotropic materials, orthotropic materials, functionally graded materials and piezoelectric materials.
The detailed contents of the thesis are given as follows:
(1) On the basis of the two-dimensional plane elasticity theory, the general expressions of displacements, which exactly satisfy the governing differential equations and the simply-supported boundary conditions at two ends of the beam, have been deduced. The unknown coefficients in the solution are then determined by using the Fourier sinusoidal series expansion to the boundary equations on the upper and lower surfaces of the beams.
(2) On the basis of three-dimensional elasticity theory, the general expressions for displacements and stresses of the rectangular plate under static loads, which exactly satisfy the governing differential equations and the simply-supported boundary conditions at four edges of the plate, are analytically derived. The unknown coefficients in the stress solutions are approximately determined by using the double Fourier sinusoidal series expansions to the boundary conditions on the upper and lower surfaces of the plates.
(3) For the varying thickness beams with one end clamped and the other end simply-supported, 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. Then the exact analytical solution is obtained. The unknown coefficients can be 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.
(4) For the functionally graded materials, the Young's modulus is graded through the thickness following the exponential-law and the Poisson's ratio keeps constant. Firstly, the analytical solution of the governing differential equations can be obtained. Then using the Fourier sinusoidal series expansions to the boundary conditions on the upper and lower surfaces of the structures, the bending problem of simply-supported functionally graded beams with variable thickness and simply-supported functionally graded rectangular plates with variable thickness are studied.
(5) For the piezoelectric materials, the general expressions of displacement fields and piezoelectric field, which exactly satisfy the governing differential equations and the simply-supported boundary conditions, are derived firstly. The unknown coefficients in the solution are then determined by using the Fourier sinusoidal series expansion to the boundary equations on the upper and lower surfaces of the structures. The bending problem of simply-supported piezoelectric beams with variable thickness and simply-supported transversely isotropic piezoelectric rectangular plates with variable thickness are studied.
(6) For the simple-supported beam with variable thickness and simple-supported rectangular plate with variable thickness under the temperature field, we need to solve the temperature distributions on beam and plate by using Fourier sinusoidal series expansions according to the temperature boundary condition at first. Then the temperature load is exerted to the beam and the plate. The two-dimensional thermoelastic analysis of beams with variable thickness subjected to thermo-mechanical loads and the three-dimensional thermoelastic analysis of rectangular plates with variable thickness subjected to thermo-mechanical loads are presented.
(7) For the multi-span plates, the exact expressions of the displacements, which satisfy the governing differential equations and the simply supported boundary conditions at four edges of the plate, are analytically derived firstly. The reaction forces of the intermediate supports are regarded as the unknown external forces acting on the lower surface of the plate. The unknown coefficients are then determined by the boundary conditions on the upper and lower surfaces of the plate. Three-dimensional elasticity solution of simple-supported rectangular plate on point supports, line supports and elastic foundation are studied and the simply-supported functionally graded rectangular plates with internal elastic line supports are also presented.
具体的说,本文的主要内容包括：
(1)对于两端简支变厚度梁,从二维平面弹性力学的基本方程出发,导出满足控制微分方程和两端简支边界条件的位移函数一般解,对梁上下表面的边界方程作傅立叶级数展开确定待定系数。
(2)对于四边简支变厚度矩形板,从三维弹性力学的基本方程出发,导出满足控制微分方程和四边简支边界条件的位移函数一般解,对板上下表面的边界方程作双重傅立叶正弦级数展开确定待定系数。
(3)对于一端固支一端简支变厚度梁,引入单位脉冲函数和Dirac函数,将固支边等价为简支边加上水平方向的未知边界力,求得其精确解析解,对梁上下表面的边界方程作傅立叶级数展开并与固支边位移为零的条件共同确定未知系数。
(4)对于功能梯度材料,假设弹性模量沿厚度方向指数变化,泊松比为常数,首先求得简支边界条件下控制微分方程的解析解,然后对结构上下表面的边界方程作傅立叶级数展开,分析了两端简支变厚度功能梯度梁和四边简支变厚度功能梯度矩形板的弯曲问题。
(5)对于压电材料,首先求解简支边界条件下精确满足控制微分方程的结构位移场和压电场,然后对结构上下表面的边界方程作傅立叶级数展开,分析了两端简支变厚度压电梁和横观各向同性四边简支变厚度压电矩形板的弯曲问题。
(6)对于受温度作用的两端简支变厚度梁和四边简支变厚度板,首先根据温度的边界条件,采用傅立叶正弦级数展开求解梁和板内的温度分布,然后再将温度荷载施加于梁和板上,给出了机械荷载与热荷载共同作用下变厚度梁的二维热弹性力学解和变厚度矩形板的三维热弹性力学解。
(7)对于多跨的板结构,首先求得满足控制微分方程和四边简支边界条件的矩形板位移函数的一般解,将支承反力看作是作用于板上的待求反力,利用板上下表面的边界方程确定待定系数。给出了点支、线支和弹性地基上简支矩形板以及面内受线支作用的功能梯度矩形板的三维弹性力学解。
This thesis studies the static properties of beams and plates with variable thickness, based on the small-strain linear elasticity theory which does not rely on any artificial hypotheses. Firstly, for the simple-supported boundary conditions, the two-dimensional elasticity solution of simple-supported beams with variable thickness and the three-dimensional elasticity solution of simple-supported rectangular plates with variable thickness under arbitrary loads are presented by using the Fourier series expansion method. Then, for the non-simply-supported boundary conditions, the clamped boundary condition is taken as an example. The elasticity solution of varying thickness beams with one end clamped and the other end simply supported under static loads are presented by introducing the unit pulse functions and using the boundary relaxation method. Finally, the Fourier series expansion method is extended to study the bending problem of beams with variable thickness and rectangular plates with variable thickness, subjected to thermo-mechanical load where the beams and the plates are, respectively, made of isotropic materials, orthotropic materials, functionally graded materials and piezoelectric materials.
The detailed contents of the thesis are given as follows:
(1) On the basis of the two-dimensional plane elasticity theory, the general expressions of displacements, which exactly satisfy the governing differential equations and the simply-supported boundary conditions at two ends of the beam, have been deduced. The unknown coefficients in the solution are then determined by using the Fourier sinusoidal series expansion to the boundary equations on the upper and lower surfaces of the beams.
(2) On the basis of three-dimensional elasticity theory, the general expressions for displacements and stresses of the rectangular plate under static loads, which exactly satisfy the governing differential equations and the simply-supported boundary conditions at four edges of the plate, are analytically derived. The unknown coefficients in the stress solutions are approximately determined by using the double Fourier sinusoidal series expansions to the boundary conditions on the upper and lower surfaces of the plates.
(3) For the varying thickness beams with one end clamped and the other end simply-supported, 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. Then the exact analytical solution is obtained. The unknown coefficients can be 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.
(4) For the functionally graded materials, the Young's modulus is graded through the thickness following the exponential-law and the Poisson's ratio keeps constant. Firstly, the analytical solution of the governing differential equations can be obtained. Then using the Fourier sinusoidal series expansions to the boundary conditions on the upper and lower surfaces of the structures, the bending problem of simply-supported functionally graded beams with variable thickness and simply-supported functionally graded rectangular plates with variable thickness are studied.
(5) For the piezoelectric materials, the general expressions of displacement fields and piezoelectric field, which exactly satisfy the governing differential equations and the simply-supported boundary conditions, are derived firstly. The unknown coefficients in the solution are then determined by using the Fourier sinusoidal series expansion to the boundary equations on the upper and lower surfaces of the structures. The bending problem of simply-supported piezoelectric beams with variable thickness and simply-supported transversely isotropic piezoelectric rectangular plates with variable thickness are studied.
(6) For the simple-supported beam with variable thickness and simple-supported rectangular plate with variable thickness under the temperature field, we need to solve the temperature distributions on beam and plate by using Fourier sinusoidal series expansions according to the temperature boundary condition at first. Then the temperature load is exerted to the beam and the plate. The two-dimensional thermoelastic analysis of beams with variable thickness subjected to thermo-mechanical loads and the three-dimensional thermoelastic analysis of rectangular plates with variable thickness subjected to thermo-mechanical loads are presented.
(7) For the multi-span plates, the exact expressions of the displacements, which satisfy the governing differential equations and the simply supported boundary conditions at four edges of the plate, are analytically derived firstly. The reaction forces of the intermediate supports are regarded as the unknown external forces acting on the lower surface of the plate. The unknown coefficients are then determined by the boundary conditions on the upper and lower surfaces of the plate. Three-dimensional elasticity solution of simple-supported rectangular plate on point supports, line supports and elastic foundation are studied and the simply-supported functionally graded rectangular plates with internal elastic line supports are also presented.
引文
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