多层功能梯度直梁和曲梁的理论分析及其应用研究系研究
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摘要
本文对各向同性及正交各向异性多层功能梯度直梁和曲梁的弹性弯曲问题进行了研究,推导出了相应的悬臂梁、简支梁、简支-固支梁和固支梁等四种梁的弹性力学解和近似解,在此基础上进行了有关应用研究。主要工作和结论如下：
(1)用应力函数法推导出了含功能梯度过渡层的各向同性双材料直梁和多层正交各向异性功能梯度直梁上表面受均布载荷作用时的弹性力学解,其中,各向同性梯度层的泊松比为常数,其弹性模量和正交各向异性梯度层的柔度系数皆为厚度坐标的任意函数。
(2)确定了多层梯度圆弧曲梁外表面受均布载荷作用时的应力函数形式,给出了弹性力学解的求解过程；具体求解出了弹性模量分别为Ei0(r+βi)和Ei0rmeλr、泊松比为常数时的多层各向同性功能梯度曲梁和柔度系数分别为Sklirm(r+βkli)和Sklirmeλr时的多层正交各向异性功能梯度曲梁的弹性力学解。
(3)用有限元数值分析结果和已知解检验了上述直梁及曲梁弹性力学解的正确性和有效性。上述弹性力学解可进行多种形式的退化和应用。
(4)多层功能梯度梁中,不论是梯度层的材料性能还是厚度的变化,对挤压应力的分布和大小基本上没有影响,对最大弯曲正应力和最大弯曲切应力有一定影响；梯度性能的变化对梯度层中的弯曲正应力的影响较大；当梁的跨高比达到10时,梁的横截面在变形后仍基本保持为平面,但对于夹芯梁,由于层间的柔度系数比较大,横截面在变形后呈Z字型。
(5)固支端的约束形式对应力和位移有一定程度的影响；弹性力学解中的固支约束条件BC1(使固支端附近的轴向微分线段固定)比实际约束强,而固支约束条件BC2(使固支端附近的横向微分线段固定)则比实际约束弱,两者的平均值与实际约束比较接近。
(6)分别用Euler-Bernoulli梁理论和Timoshenko梁理论推导出了多层功能梯度直梁和曲梁的基本微分方程组,具体给出了受均布载荷作用时的近似解,并与弹性力学解和有限元解作了对比分析,发现：用近似解计算出的弯曲正应力的精度基本能满足工程需要,直梁中,Timoshenko梁理论解在固支约束条件BC2下的精度最高,曲梁中,两种近似解的区别不大；由近似解计算出的挠度有一定误差,但直梁中的Timoshenko梁理论在固支约束条件BC2下的计算精度较高,与有限元解非常接近。
In this dissertation, the elastic bending of multilayer functionally graded straight and curved beams is studied. Elasticity and approximate solutions are derived for cantilever, simply supported, simply-fixed, and fixed-end beams, respectively. Based on the solutions, some applications are discussed. The main works and some conclusions are listed as follows:
(1) Using the Airy stress function, elasticity solutions are derived for both the bi-material isotropic straight beam with a graded intermediate layer and multilayer orthotropic functionally graded straight beam. The beams are subjected to a uniform load on upper surfaces, in which Poisson's ratio of the isotropic layer is kept a constant, and its Young's modulus and the elastic compliance parameters of the orthotropic layer are both assumed to be arbitrary functions of the thickness coordinate.
(2) The form of the Airy stress function is selected for the multilayer functionally graded circular curved beam subjected to a uniform load on its outer surface, and the process to obtain the elasticity solution is given. The solutions are derived for the multilayer isotropic functionally graded curved beam with Young's modulus respectively being Ei0(r+βi) and Eiormeλr and Poisson's ratio being a constant and the multilayer orthotropic functionally graded curved beam with the elastic compliance parameters being Sklirm (r+βkli,) and Sklirmeλr, respectively.
(3) The above elasticity solutions are demonstrated to be correct and effective by the FEM (finite element method) and the known solutions. The obtained solutions can be degenerated into different forms and also have many applictions.
(4) In multilayer functionally graded beam, neither the thickness nor the material property of the graded layer influences the bearing stress, but they influence the maximum bending and shear stresses a little. The material property of the graded layer obviously affects the bending stress in the graded layer. When the ratio of span to thickness is not less than ten, the cross section of the beam will remain plane. However, the section will become zigzag in a sandwich beam for the compliance ratio between the face-sheet and core is too big.
(5) The type of description for the fixed end partly affects the stresses and displacements. In the elasticity solutions, the constraint condition BC1(an element of the axis of the beam being fixed at the fixed end) is stronger than the real case while the condition BC2(a vertical element of the cross section being fixed at the fixed end) weaker than the real one, but the averages of the results for BC1and BC2are close to the FEM ones.
(6) Based on the Euler-Bernoulli and Timoshenko beam theories, basic differential equations are deduced for multilayer functionally graded straight and curved beams, and the approximate solutions are obtained for the beam subjected to a uniform load on the upper/outer surface. The precision of the bending stress of these solutions can meet the engineering needs; here, the precision of the Timoshenko beam theory with BC2is higher than the others in the straight beam while the precisions of the Euler-Bernoulli and Timoshenko beam theories are similar in the curved beam. For the bending deflection, the only one of the straight beam, obtained by the Timoshenko beam theory with BC2, is close to the FEM result.
(1)用应力函数法推导出了含功能梯度过渡层的各向同性双材料直梁和多层正交各向异性功能梯度直梁上表面受均布载荷作用时的弹性力学解,其中,各向同性梯度层的泊松比为常数,其弹性模量和正交各向异性梯度层的柔度系数皆为厚度坐标的任意函数。
(2)确定了多层梯度圆弧曲梁外表面受均布载荷作用时的应力函数形式,给出了弹性力学解的求解过程；具体求解出了弹性模量分别为Ei0(r+βi)和Ei0rmeλr、泊松比为常数时的多层各向同性功能梯度曲梁和柔度系数分别为Sklirm(r+βkli)和Sklirmeλr时的多层正交各向异性功能梯度曲梁的弹性力学解。
(3)用有限元数值分析结果和已知解检验了上述直梁及曲梁弹性力学解的正确性和有效性。上述弹性力学解可进行多种形式的退化和应用。
(4)多层功能梯度梁中,不论是梯度层的材料性能还是厚度的变化,对挤压应力的分布和大小基本上没有影响,对最大弯曲正应力和最大弯曲切应力有一定影响；梯度性能的变化对梯度层中的弯曲正应力的影响较大；当梁的跨高比达到10时,梁的横截面在变形后仍基本保持为平面,但对于夹芯梁,由于层间的柔度系数比较大,横截面在变形后呈Z字型。
(5)固支端的约束形式对应力和位移有一定程度的影响；弹性力学解中的固支约束条件BC1(使固支端附近的轴向微分线段固定)比实际约束强,而固支约束条件BC2(使固支端附近的横向微分线段固定)则比实际约束弱,两者的平均值与实际约束比较接近。
(6)分别用Euler-Bernoulli梁理论和Timoshenko梁理论推导出了多层功能梯度直梁和曲梁的基本微分方程组,具体给出了受均布载荷作用时的近似解,并与弹性力学解和有限元解作了对比分析,发现：用近似解计算出的弯曲正应力的精度基本能满足工程需要,直梁中,Timoshenko梁理论解在固支约束条件BC2下的精度最高,曲梁中,两种近似解的区别不大；由近似解计算出的挠度有一定误差,但直梁中的Timoshenko梁理论在固支约束条件BC2下的计算精度较高,与有限元解非常接近。
In this dissertation, the elastic bending of multilayer functionally graded straight and curved beams is studied. Elasticity and approximate solutions are derived for cantilever, simply supported, simply-fixed, and fixed-end beams, respectively. Based on the solutions, some applications are discussed. The main works and some conclusions are listed as follows:
(1) Using the Airy stress function, elasticity solutions are derived for both the bi-material isotropic straight beam with a graded intermediate layer and multilayer orthotropic functionally graded straight beam. The beams are subjected to a uniform load on upper surfaces, in which Poisson's ratio of the isotropic layer is kept a constant, and its Young's modulus and the elastic compliance parameters of the orthotropic layer are both assumed to be arbitrary functions of the thickness coordinate.
(2) The form of the Airy stress function is selected for the multilayer functionally graded circular curved beam subjected to a uniform load on its outer surface, and the process to obtain the elasticity solution is given. The solutions are derived for the multilayer isotropic functionally graded curved beam with Young's modulus respectively being Ei0(r+βi) and Eiormeλr and Poisson's ratio being a constant and the multilayer orthotropic functionally graded curved beam with the elastic compliance parameters being Sklirm (r+βkli,) and Sklirmeλr, respectively.
(3) The above elasticity solutions are demonstrated to be correct and effective by the FEM (finite element method) and the known solutions. The obtained solutions can be degenerated into different forms and also have many applictions.
(4) In multilayer functionally graded beam, neither the thickness nor the material property of the graded layer influences the bearing stress, but they influence the maximum bending and shear stresses a little. The material property of the graded layer obviously affects the bending stress in the graded layer. When the ratio of span to thickness is not less than ten, the cross section of the beam will remain plane. However, the section will become zigzag in a sandwich beam for the compliance ratio between the face-sheet and core is too big.
(5) The type of description for the fixed end partly affects the stresses and displacements. In the elasticity solutions, the constraint condition BC1(an element of the axis of the beam being fixed at the fixed end) is stronger than the real case while the condition BC2(a vertical element of the cross section being fixed at the fixed end) weaker than the real one, but the averages of the results for BC1and BC2are close to the FEM ones.
(6) Based on the Euler-Bernoulli and Timoshenko beam theories, basic differential equations are deduced for multilayer functionally graded straight and curved beams, and the approximate solutions are obtained for the beam subjected to a uniform load on the upper/outer surface. The precision of the bending stress of these solutions can meet the engineering needs; here, the precision of the Timoshenko beam theory with BC2is higher than the others in the straight beam while the precisions of the Euler-Bernoulli and Timoshenko beam theories are similar in the curved beam. For the bending deflection, the only one of the straight beam, obtained by the Timoshenko beam theory with BC2, is close to the FEM result.
引文
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