圆截面梁的高阶理论和梯度梁的动静态与稳定性分析
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摘要
由金属和陶瓷组成的功能梯度材料相比于金属或陶瓷的单一材料具有更好的陶瓷热阻能力和更强的机械性能,被广泛应用于航空航天,汽车,国防工业等领域。本论文建立了圆截面柱形梁杆的高阶理论,基于该理论模型讨论了径向功能梯度圆截面梁杆的弯曲、自由振动和屈曲问题,与Euler-Bernoulli梁和Timoshenko梁理论等经典模型不同,我们的解可以满足梁的圆周边界自由条件。另外,提出了一种解析的积分方程方法来处理轴向功能梯度Euler-Bernoulli梁的自由振动和屈曲问题。前一部分从弹性力学基本方程出发我们得到了均匀和径向功能梯度圆截面梁的弯曲、自由振动和屈曲时的耦合控制方程,建立了圆截面梁的高阶理论；后三章主要研究材料性质轴向任意连续变化的功能梯度材料和变截面Euler-Bernoulli梁的自由振动和屈曲问题。每一章都给出了利用新方法讨论计算相关问题的数值结果,以及与精确解和其他文献的数值结果对比,这些都表明本文方法研究圆截面功能梯度梁力学性质的有效性。
本文取得主要成果如下：
(1)建立了圆截面梁的高阶理论。基于圆周上应力自由的边界条件,我们构造出了轴向位移的表达式,并进一步建立了关于挠度和转角的两个耦合方程,且分析了均匀圆截面梁弯曲和自由振动问题。与Euler-Bernoulli梁,Timoshenko梁理论不同的是,构建中我们既考虑了剪切变形和惯性矩的影响,又不需要做出截面上剪力恒定的假设和引入类似于Timoshenko梁中的修正剪切系数,而且满足了圆周上的边界自由条件。将著名的Levinsion梁对矩形截面的高阶理论扩展到了圆截面梁。利用高阶理论讨论圆截面梁的动静态问题与三维弹性理论得到的数值结果比较吻合。
(2)利用建立的高阶理论模型分析径向功能梯度圆截面梁的弯曲、自由振动,这种径向变化可以是任意的连续变化。给出了各种常见边界条件下功能梯度梁弯曲时的应力和位移分布与自由振动的自然频率及模态特征方程,研究了幂函数梯度参数对挠度、应力分布、自然频率等影响,并将高阶理论模型应用于求解圆柱壳的自由振动问题。
(3)利用建立的高阶理论研究了径向功能梯度圆截面梁的稳定性问题。给出了利用高阶理论与Euler-Bernoulli梁和Timoshenko梁计算功能梯度梁屈曲时的临界荷载之间的关系,研究了剪切模量和材料梯度性质对临界荷载的影响,并讨论了双壁碳纳米管的屈曲问题。
(4)分析了材料性质沿轴线方向任意连续变化的功能梯度梁和变截面梁的自由振动性质。结合梁两端的边界条件利用积分方法,提出了将微分控制方程转化为对应的Fredholm积分方程,然后通过模态的级数展开使得自由振动低阶和高阶的自然频率都由特征多项式方程确定。
(5)研究材料性质沿轴线方向任意连续变化的功能梯度梁和变截面梁的稳定性,以及在弹性约束下的屈曲问题。利用新方法使得临界荷载由特征多项式方程确定,而且基于这种方法我们给出了简支圆柱形变截面梁的次优化设计分析,发现存在一个最优的设计使得梁的承载能力最强。
Due to the advantages of better thermal resistance of the ceramic phase and exhibiting stronger mechanical performance, functionally graded materials of metals and ceramics have been increasingly used in aerospace, automobile and defence industries. In this thesis, we have established a higher-order theory of circular cylindrical beams. Based on the theory, we have discussed the bending,vibration and buckling behaviors of homogeneous beams and circular cylindrical radially functionally graded beams. Different from other classical beam theories, such as Euler-Bernoulli beam and Timoshenko beam theories, the obtained solutions have satisfied the stress-free condition at the circumferential surface. Moreover, we present an analytical integral method for dealing with the free vibration and buckling of axially functionally graded Euler-Bernoulli beams. In the first part, we have obtained the two coupled governing equations for analyzing the bending,vibration and buckling behaviors of the circular cylindrical radially functionally graded beams starting from the elastic theory. Next we have mainly discussed the free vibration and buckling of functionally graded beams with varying materials properties along the axial direction. Based on the introduced methods, we have given the calculation of related issues in each chapter and compared the results with closed-form solutions as well as the existing numerial results, which show the effectiveness of the method for dealing with circular cylindrical functionally graded beams.
Specific contents are as follows:
(1) We have established the higher-order theory of a cicular beam. Based on the traction-free condition at the circumferential surface of the beam, expression for the axial displacements is constructed, then two coupled equations for the deflection and rotation are deduced, and they are used to investigate the bending and vibration of the homogeneous circular cylindrical beam. Different from the Euler-Bernoulli beam and Timoshenko beams, we not only take into account the effects of the rotary inertia and shear deformation, but also need not assume the shear stress uniform or introduce the revised shear correction factor. Moreover, the shear stress-free surface condition is satisfied at the circumferential surface. This higher-order theory is an extension to circular cylindrical beams of the Levinson higher-order beam theory suitable for rectangular beams. The results obtained by the proposed higher-order theory show very good agreement with the results of the three dimensional elastic analysis.
(2) Based on the higher-order theory, we have studied the bending and free vibration of the circular cylindrical functionally graded beam, where the graded material properties are assumed to vary arbitrarily in the radial direction. For different boundary conditions, we have obtained the bending solutions, such as deflection and stress distribution, and charateristc equations of natural frequencies as well as the characteristic mode equations. The effects of gradient variation on deflection, stress distribution and natural frequencies are analyzed. Moreover, we apply this model to analyze the free vibration of an isotropic circular cylindrical shell.
(3) We have exploited the higher-order theory to discuss the buckling of radial circular cylindrical functionally graded beam. The relationship of critical buckling loads, which is calculated through different beam theories, has been given. The effects of shear deformation and gradient variation critical buckling loads are analyzed. Furthermore, we have investigated the buckling behaviors of double-walled carbon nanotubes based on the higher-order theory.
(4) We have analyzed the dynamic behaviors of graded non-uniform beams, where the material properties are assumed to vary arbitrarily in the axial direction. Combining with the boundary condition as well as the integral method, we present a novel approach for transforming the governing equation with varying coefficients to Fredholm integral equations. Then by expanding the mode shapes as power series, the natural frequencies can be determined by the polynomial characteristic equation.
(5)We have investigated the buckling behaviors of axially graded non-uniform beams, and the buckling of axially non-uniform columns with elastic restrained has also been studied. Through the new method, critical buckling loads can be determined by the polynomial characteristic equation. Based on this method, an suboptimal design of varying cross-section simply supported beam is illustrated for a cylindrical bar. It has been found that there is an optimal design such that the bar achieves its maximum load-carrying capacity.
本文取得主要成果如下：
(1)建立了圆截面梁的高阶理论。基于圆周上应力自由的边界条件,我们构造出了轴向位移的表达式,并进一步建立了关于挠度和转角的两个耦合方程,且分析了均匀圆截面梁弯曲和自由振动问题。与Euler-Bernoulli梁,Timoshenko梁理论不同的是,构建中我们既考虑了剪切变形和惯性矩的影响,又不需要做出截面上剪力恒定的假设和引入类似于Timoshenko梁中的修正剪切系数,而且满足了圆周上的边界自由条件。将著名的Levinsion梁对矩形截面的高阶理论扩展到了圆截面梁。利用高阶理论讨论圆截面梁的动静态问题与三维弹性理论得到的数值结果比较吻合。
(2)利用建立的高阶理论模型分析径向功能梯度圆截面梁的弯曲、自由振动,这种径向变化可以是任意的连续变化。给出了各种常见边界条件下功能梯度梁弯曲时的应力和位移分布与自由振动的自然频率及模态特征方程,研究了幂函数梯度参数对挠度、应力分布、自然频率等影响,并将高阶理论模型应用于求解圆柱壳的自由振动问题。
(3)利用建立的高阶理论研究了径向功能梯度圆截面梁的稳定性问题。给出了利用高阶理论与Euler-Bernoulli梁和Timoshenko梁计算功能梯度梁屈曲时的临界荷载之间的关系,研究了剪切模量和材料梯度性质对临界荷载的影响,并讨论了双壁碳纳米管的屈曲问题。
(4)分析了材料性质沿轴线方向任意连续变化的功能梯度梁和变截面梁的自由振动性质。结合梁两端的边界条件利用积分方法,提出了将微分控制方程转化为对应的Fredholm积分方程,然后通过模态的级数展开使得自由振动低阶和高阶的自然频率都由特征多项式方程确定。
(5)研究材料性质沿轴线方向任意连续变化的功能梯度梁和变截面梁的稳定性,以及在弹性约束下的屈曲问题。利用新方法使得临界荷载由特征多项式方程确定,而且基于这种方法我们给出了简支圆柱形变截面梁的次优化设计分析,发现存在一个最优的设计使得梁的承载能力最强。
Due to the advantages of better thermal resistance of the ceramic phase and exhibiting stronger mechanical performance, functionally graded materials of metals and ceramics have been increasingly used in aerospace, automobile and defence industries. In this thesis, we have established a higher-order theory of circular cylindrical beams. Based on the theory, we have discussed the bending,vibration and buckling behaviors of homogeneous beams and circular cylindrical radially functionally graded beams. Different from other classical beam theories, such as Euler-Bernoulli beam and Timoshenko beam theories, the obtained solutions have satisfied the stress-free condition at the circumferential surface. Moreover, we present an analytical integral method for dealing with the free vibration and buckling of axially functionally graded Euler-Bernoulli beams. In the first part, we have obtained the two coupled governing equations for analyzing the bending,vibration and buckling behaviors of the circular cylindrical radially functionally graded beams starting from the elastic theory. Next we have mainly discussed the free vibration and buckling of functionally graded beams with varying materials properties along the axial direction. Based on the introduced methods, we have given the calculation of related issues in each chapter and compared the results with closed-form solutions as well as the existing numerial results, which show the effectiveness of the method for dealing with circular cylindrical functionally graded beams.
Specific contents are as follows:
(1) We have established the higher-order theory of a cicular beam. Based on the traction-free condition at the circumferential surface of the beam, expression for the axial displacements is constructed, then two coupled equations for the deflection and rotation are deduced, and they are used to investigate the bending and vibration of the homogeneous circular cylindrical beam. Different from the Euler-Bernoulli beam and Timoshenko beams, we not only take into account the effects of the rotary inertia and shear deformation, but also need not assume the shear stress uniform or introduce the revised shear correction factor. Moreover, the shear stress-free surface condition is satisfied at the circumferential surface. This higher-order theory is an extension to circular cylindrical beams of the Levinson higher-order beam theory suitable for rectangular beams. The results obtained by the proposed higher-order theory show very good agreement with the results of the three dimensional elastic analysis.
(2) Based on the higher-order theory, we have studied the bending and free vibration of the circular cylindrical functionally graded beam, where the graded material properties are assumed to vary arbitrarily in the radial direction. For different boundary conditions, we have obtained the bending solutions, such as deflection and stress distribution, and charateristc equations of natural frequencies as well as the characteristic mode equations. The effects of gradient variation on deflection, stress distribution and natural frequencies are analyzed. Moreover, we apply this model to analyze the free vibration of an isotropic circular cylindrical shell.
(3) We have exploited the higher-order theory to discuss the buckling of radial circular cylindrical functionally graded beam. The relationship of critical buckling loads, which is calculated through different beam theories, has been given. The effects of shear deformation and gradient variation critical buckling loads are analyzed. Furthermore, we have investigated the buckling behaviors of double-walled carbon nanotubes based on the higher-order theory.
(4) We have analyzed the dynamic behaviors of graded non-uniform beams, where the material properties are assumed to vary arbitrarily in the axial direction. Combining with the boundary condition as well as the integral method, we present a novel approach for transforming the governing equation with varying coefficients to Fredholm integral equations. Then by expanding the mode shapes as power series, the natural frequencies can be determined by the polynomial characteristic equation.
(5)We have investigated the buckling behaviors of axially graded non-uniform beams, and the buckling of axially non-uniform columns with elastic restrained has also been studied. Through the new method, critical buckling loads can be determined by the polynomial characteristic equation. Based on this method, an suboptimal design of varying cross-section simply supported beam is illustrated for a cylindrical bar. It has been found that there is an optimal design such that the bar achieves its maximum load-carrying capacity.
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