基于哈密顿辛对偶体系的若干梁/板结构问题研究
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摘要
工程领域中大量问题可归结为梁/板结构模型,人们对该课题的研究一直没有间断过。因此进一步对其进行研究有着重要科学意义和工程应用价值。以往研究主要是在Lagrange体系下欧几里德空间中一类变量范围内进行,这样不可避免地带来高阶偏微分方程并导致求解受偏微分算子和边界条件限制等问题。而辛对偶理论基于Hamilton体系辛空间,使得有效的数学物理方法如分离变量法、共轭辛正交和辛本征函数向量展开等得以实施,避免了传统方法存在的不足,提供了一种新的求解方法。
本文将辛对偶理论扩展应用到梁/板结构问题中,从基本方程出发,基于能量变分原理,由勒让德变换引入混合型对偶变量并建立正则(对偶)方程组。在辛空间中,问题的求解归结为哈密顿算子矩阵的本征值与本征解问题,从而建立了一套梁/板结构辛体系求解方法。基于此方法重点研究了复合材料叠层结构、地基梁板、薄壁结构剪力滞问题和功能梯度压电材料力电耦合问题及二维弹性平面奇异性等问题,并分析预应力梁在纵横耦合力作用下的动力问题。主要研究工作和结论如下:
(1)从各向异性弹性力学基本方程出发,研究复合材料叠层梁在各种不同铺层形式下弯曲问题,首次得到了适用于任意跨厚比和边界条件的解析解,有效避免了简化理论对剪切变形以及横向正应力等对结构特性的影响估计不确切的缺点,为今后各种简化理论和数值分析方法提供了较好的检验标准。分析了正交铺设和斜角铺设及双参数地基上各向异性梁的弯曲性能,讨论了跨厚比、铺层数和各向异性程度及端部支承条件等参数对力学性能的影响,得出一些有益的结论。
(2)系统地给出多种不同边界条件组合和几何形状情形时对称铺设叠层板的解,突破了传统方法无法得到任意边界条件和各类几何形状的板解析解的瓶颈,结果显示取前几项本征值就可达到较高的精度,并进一步推广应用于分析建筑筏板基础和弹性地基上钢筋混凝土板等实际工程问题。讨论了碳纤维环氧复合材料正交铺设和斜交铺设的弯曲特性及铺设层数、铺设角、材料各向异性程度等对板的力学特性的影响。
(3)建立薄壁结构剪力滞效应的弹性力学辛求解方法,推导出简支箱梁和悬臂箱梁在满跨均布荷载作用下翼板部分的圣维南解,给出了剪力滞系数和有效宽度系数的闭合多项式形式,将结果与有机玻璃模型试验梁实测值、国际规范及数值解进行比较。结果表明,辛求解方法是分析箱形截面剪力滞效应是一种有效而实用的方法,得到的公式表达简单,可快速计算简支和悬臂箱梁桥的有效宽度。
(4)将辛方法推广应用于功能梯度压电材料力电耦合问题中,首次引入材料非均匀性沿纵向分布的假设,突破以往研究仅限于沿厚度方向变化的局限性,构造和材料系数梯度相关的应力分量,提出偏移哈密顿算子矩阵的概念,分析并重新建立了本征解之间的辛正交共轭关系,得到了耦合场问题的解析解,讨论了材料梯度指数对结构宏观性能的影响。为解决非均匀材料多场耦合问题提出一个新思路,也推进了辛体系在智能材料中的应用。
(5)利用辛空间级数自动收敛的特性,研究二维弹性平面奇异性问题,并将问题归结为求解算子矩阵的零本征值本征解和非零本征值本征解。尤其是引入具有局部效应衰减特性的非零本征值本征解,充分体现了对偶体系的特点和优势,给出悬臂梁完整的应力分布情况,固定端附近的位移和应力分布也可得到更精确的分析结果,揭示了边界效应产生的局部现象,为局部效应和边界现象的研究提供一种有效途径。
(6)提出Newmark-β精细耦合Pade级数法。这种改进的时域求解方法避免传统时域逐步积分法存在的不足,克服了精细积分法降阶时遇到的困难并保持较高的精度,并结合采用位移解析解作为试函数建立的剪切梁单元研究双参数地基上预应力混凝土梁在耦合力作用下的动力响应。讨论预应力对固有模态的影响,分析偏心距、荷载速度、激励频率及地基刚度等参数对梁动态响应的影响,得出了一些规律,对路面高架桥、铁路和桥梁等预应力结构特性的理解和设计奠定了基础。
以上研究结果表明,辛体系在对偶的二类变量(位移、应力)范围内研究问题,有收敛快精度高、操作简单、通用性好等优点,具有Lagrange体系无法比拟的优越性,是一种简单、直接、高效的求解方法,突破了传统方法的局限性,具有一般性及较高的理论推广价值,在工程结构分析方面有广阔的应用前景。
There are many problems that can be reduced to beam and plate structural system.The research on this subject has been ongoing for many years. The traditional method solves this problem in Euclidean space under Lagrange system, which involves solving higher orders of partial differential equations and faces the difficulty of handing the boundary conditions.So it is necessary to investigate a new and efficient solving method. Symplectic dual system is introduced to theory of elasticity mechanics by Professor Zhong Wanxie.And then symplectic system of elasticity is established and developed so that many effective mathematics and physics methods such as separation of variables, conjugated symplectic orthogonal and sumplectic eigenfunction expansion method etc. are made to be implemented. So the symplectic solution system of elastic mechanics is introduced to the beam and plate structures in the paper.The main study works and conclusions are as follows:
(1) The symplectic method was applied to analyze the bending of composite laminated beam. Based on the equations of elasticity, the analytic solutions were obtained for all kinds of span to depth ratio and boundary conditions, which avoid the lacking of imprecision effect on the structures coming from shearing deformation and transverse normal stress based on simplified theory. The parameters’influence on mechanical property are discussed, such as span to depth ratio, ply angle, Arrangement of plies and anisotropic degree. Some laws are derived.
(2) For symmetric angle-ply laminated plates, the solutions are not limited to simple support boundary condition and geometrical shape, which are further applied to analyze the raft foundation.The results of cross-ply and angle-ply laminated graphite–epoxy composite plate are also shown. The parameters’influence on mechanical property are discussed, such as ply angle, Arrangement of plies and anisotropic degree.
(3) The closed polynomial expressions for shear lag coefficient and effective width are given at the condition of Saint-Venant analytical solutions to full-span uniform load on flange slab of box girder. Availability of formula in this paper is demonstrated by comparing results with international norm and measured values.
(4) The paper proposes the application of the symplectic elasticity solution system in solving of electromechanical problem exists in functionally graded piezoelectric material for the first time.After introducing the material nonhomogeneous along axial direction and constructing the new stress, the electromechanical problem are formulated in the form of symplectic system. The shift Hamiltonian matrix is raised and the conjugated symplectic orthogonal between the eigenvectors are re-established, which provide a new method for solving piezoelectric problems and promote the use of symplectic elasticity solution in solving intellectual material problems at the same time.
(5) Based on the convergence being guaranteed by the completeness of symplectic expansion, the stress singularities problem of anisotropic beam basing on two-dimensional elasticity is investigated, which is reduced to solve the eigenvalue problem of the operator matrix. Especially, the nonzero eigen-solutions that have fading characteristic usually ignored by Saint Venant’s principle can explain the local phenomenon. It is especially good in describing the localized effects and their reduction, which are seldom mentioned in former researches.
(6) A new method of structural dynamic equation is put forward by introducing Pade series expansion.The algorithm is based on the coupled precise time integration and Newmark method,which avoid the deficiency of traditional method and overcome the difficulties for time integration reducing orders and maintain the fine precision.The dynamic response of prestressed Timoshenko beams fully and partially resting on a two-parameter elastic foundation under moving harmonic load is investaged.A beam element with shear deformation is formulated.The effectd of prestress,foundation support ,moving velocity and excitation frequency on dynamic characteristics of the beams are studied and described in detail.
The research results show that the symplectic method fits the beam/plate problems well and have complete space of eigensolutions, which is simple, straight, effective and have more advantages than the traditional method, which make a breakthrough of higher orders of partial differential equations and the difficulty of handing the boundary conditions.Based on the results of herein, one may conclude that the symplectic method for beam/plate structure is usefule in engineering application and has good development and applied prospects for structural engineering.
本文将辛对偶理论扩展应用到梁/板结构问题中,从基本方程出发,基于能量变分原理,由勒让德变换引入混合型对偶变量并建立正则(对偶)方程组。在辛空间中,问题的求解归结为哈密顿算子矩阵的本征值与本征解问题,从而建立了一套梁/板结构辛体系求解方法。基于此方法重点研究了复合材料叠层结构、地基梁板、薄壁结构剪力滞问题和功能梯度压电材料力电耦合问题及二维弹性平面奇异性等问题,并分析预应力梁在纵横耦合力作用下的动力问题。主要研究工作和结论如下:
(1)从各向异性弹性力学基本方程出发,研究复合材料叠层梁在各种不同铺层形式下弯曲问题,首次得到了适用于任意跨厚比和边界条件的解析解,有效避免了简化理论对剪切变形以及横向正应力等对结构特性的影响估计不确切的缺点,为今后各种简化理论和数值分析方法提供了较好的检验标准。分析了正交铺设和斜角铺设及双参数地基上各向异性梁的弯曲性能,讨论了跨厚比、铺层数和各向异性程度及端部支承条件等参数对力学性能的影响,得出一些有益的结论。
(2)系统地给出多种不同边界条件组合和几何形状情形时对称铺设叠层板的解,突破了传统方法无法得到任意边界条件和各类几何形状的板解析解的瓶颈,结果显示取前几项本征值就可达到较高的精度,并进一步推广应用于分析建筑筏板基础和弹性地基上钢筋混凝土板等实际工程问题。讨论了碳纤维环氧复合材料正交铺设和斜交铺设的弯曲特性及铺设层数、铺设角、材料各向异性程度等对板的力学特性的影响。
(3)建立薄壁结构剪力滞效应的弹性力学辛求解方法,推导出简支箱梁和悬臂箱梁在满跨均布荷载作用下翼板部分的圣维南解,给出了剪力滞系数和有效宽度系数的闭合多项式形式,将结果与有机玻璃模型试验梁实测值、国际规范及数值解进行比较。结果表明,辛求解方法是分析箱形截面剪力滞效应是一种有效而实用的方法,得到的公式表达简单,可快速计算简支和悬臂箱梁桥的有效宽度。
(4)将辛方法推广应用于功能梯度压电材料力电耦合问题中,首次引入材料非均匀性沿纵向分布的假设,突破以往研究仅限于沿厚度方向变化的局限性,构造和材料系数梯度相关的应力分量,提出偏移哈密顿算子矩阵的概念,分析并重新建立了本征解之间的辛正交共轭关系,得到了耦合场问题的解析解,讨论了材料梯度指数对结构宏观性能的影响。为解决非均匀材料多场耦合问题提出一个新思路,也推进了辛体系在智能材料中的应用。
(5)利用辛空间级数自动收敛的特性,研究二维弹性平面奇异性问题,并将问题归结为求解算子矩阵的零本征值本征解和非零本征值本征解。尤其是引入具有局部效应衰减特性的非零本征值本征解,充分体现了对偶体系的特点和优势,给出悬臂梁完整的应力分布情况,固定端附近的位移和应力分布也可得到更精确的分析结果,揭示了边界效应产生的局部现象,为局部效应和边界现象的研究提供一种有效途径。
(6)提出Newmark-β精细耦合Pade级数法。这种改进的时域求解方法避免传统时域逐步积分法存在的不足,克服了精细积分法降阶时遇到的困难并保持较高的精度,并结合采用位移解析解作为试函数建立的剪切梁单元研究双参数地基上预应力混凝土梁在耦合力作用下的动力响应。讨论预应力对固有模态的影响,分析偏心距、荷载速度、激励频率及地基刚度等参数对梁动态响应的影响,得出了一些规律,对路面高架桥、铁路和桥梁等预应力结构特性的理解和设计奠定了基础。
以上研究结果表明,辛体系在对偶的二类变量(位移、应力)范围内研究问题,有收敛快精度高、操作简单、通用性好等优点,具有Lagrange体系无法比拟的优越性,是一种简单、直接、高效的求解方法,突破了传统方法的局限性,具有一般性及较高的理论推广价值,在工程结构分析方面有广阔的应用前景。
There are many problems that can be reduced to beam and plate structural system.The research on this subject has been ongoing for many years. The traditional method solves this problem in Euclidean space under Lagrange system, which involves solving higher orders of partial differential equations and faces the difficulty of handing the boundary conditions.So it is necessary to investigate a new and efficient solving method. Symplectic dual system is introduced to theory of elasticity mechanics by Professor Zhong Wanxie.And then symplectic system of elasticity is established and developed so that many effective mathematics and physics methods such as separation of variables, conjugated symplectic orthogonal and sumplectic eigenfunction expansion method etc. are made to be implemented. So the symplectic solution system of elastic mechanics is introduced to the beam and plate structures in the paper.The main study works and conclusions are as follows:
(1) The symplectic method was applied to analyze the bending of composite laminated beam. Based on the equations of elasticity, the analytic solutions were obtained for all kinds of span to depth ratio and boundary conditions, which avoid the lacking of imprecision effect on the structures coming from shearing deformation and transverse normal stress based on simplified theory. The parameters’influence on mechanical property are discussed, such as span to depth ratio, ply angle, Arrangement of plies and anisotropic degree. Some laws are derived.
(2) For symmetric angle-ply laminated plates, the solutions are not limited to simple support boundary condition and geometrical shape, which are further applied to analyze the raft foundation.The results of cross-ply and angle-ply laminated graphite–epoxy composite plate are also shown. The parameters’influence on mechanical property are discussed, such as ply angle, Arrangement of plies and anisotropic degree.
(3) The closed polynomial expressions for shear lag coefficient and effective width are given at the condition of Saint-Venant analytical solutions to full-span uniform load on flange slab of box girder. Availability of formula in this paper is demonstrated by comparing results with international norm and measured values.
(4) The paper proposes the application of the symplectic elasticity solution system in solving of electromechanical problem exists in functionally graded piezoelectric material for the first time.After introducing the material nonhomogeneous along axial direction and constructing the new stress, the electromechanical problem are formulated in the form of symplectic system. The shift Hamiltonian matrix is raised and the conjugated symplectic orthogonal between the eigenvectors are re-established, which provide a new method for solving piezoelectric problems and promote the use of symplectic elasticity solution in solving intellectual material problems at the same time.
(5) Based on the convergence being guaranteed by the completeness of symplectic expansion, the stress singularities problem of anisotropic beam basing on two-dimensional elasticity is investigated, which is reduced to solve the eigenvalue problem of the operator matrix. Especially, the nonzero eigen-solutions that have fading characteristic usually ignored by Saint Venant’s principle can explain the local phenomenon. It is especially good in describing the localized effects and their reduction, which are seldom mentioned in former researches.
(6) A new method of structural dynamic equation is put forward by introducing Pade series expansion.The algorithm is based on the coupled precise time integration and Newmark method,which avoid the deficiency of traditional method and overcome the difficulties for time integration reducing orders and maintain the fine precision.The dynamic response of prestressed Timoshenko beams fully and partially resting on a two-parameter elastic foundation under moving harmonic load is investaged.A beam element with shear deformation is formulated.The effectd of prestress,foundation support ,moving velocity and excitation frequency on dynamic characteristics of the beams are studied and described in detail.
The research results show that the symplectic method fits the beam/plate problems well and have complete space of eigensolutions, which is simple, straight, effective and have more advantages than the traditional method, which make a breakthrough of higher orders of partial differential equations and the difficulty of handing the boundary conditions.Based on the results of herein, one may conclude that the symplectic method for beam/plate structure is usefule in engineering application and has good development and applied prospects for structural engineering.
引文
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