功能梯度梁的静动态力学行为分析
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摘要
梁是工程应用中最为常见的结构形式。在航空航天、机械、土木工程等领域中,大多数承受荷载的构件可以抽象为梁或柱。功能梯度梁由于微观成分和性能呈连续性梯度变化,消除了两种材料之间的界面,从而具有高强度、高韧性、减少应力集中、耐高温等优良的力学性能,广泛应用于各个领域。本文研究了材料非线性功能梯度悬臂梁在载荷作用下的大变形问题,及双向功能梯度梁和指数梯度梁的振动自然频率。此外,还计算了弹性地基上非经典剪切梁的自然频率。
本文共分为七章。第一章是引言部分,对课题研究的发展趋势进行概述。第二章至第六章为主体部分,给出了一些重要的结果和结论。最后部分是结论与对本课题的展望。本文取得的主要成果如下：
(1)分析了材料非线性功能梯度悬臂梁自由端受到力偶作用时的大变形。对幂律硬化的功能梯度梁,确定了中性轴的位置。给出了非线性材料的功能梯度梁挠度和转角的显式表达式。分析了梯度变化的弹性模量和材料非线性参数对功能梯度梁变形的影响。结果表明随高度变化的弹性模量和材料非线性对梁的变形有重大的影响,且功能梯度梁比均质梁能承受更大载荷。为确定最佳的梯度分布,给出了具有较轻重量和较大刚度的优化设计。
(2)利用大变形和小变形理论研究了非线性功能梯度悬臂梁自由端受集中力作用时的力学行为。假设弹性模量沿梁高度方向变化,分析了高度相关的弹性模量和非线性参数对功能梯度梁挠度和转角的影响。结果表明,不同梯度指数能改变梁的弯曲刚度,因此选择适当的梯度,功能梯度梁能比均质梁承受较大载荷。且功能梯度梁弯曲应力的分布与均质梁的应力分布完全不同。最大拉应力在梁内部位置取得而不是在表面。这些结论对线性和非线性梁的安全设计是非常有用的。
(3)研究了双向功能梯度梁的自由振动。对材料性质沿轴向和高度方向变化的细长梁,用二维弹性理论推导出其控制方程。然后将它们转化为对应的Fredholm积分方程,得到了材料性质对称分布的简支梁弯曲振动的自然频率。讨论了梯度变化和转动惯量对自然频率的影响。结果表明自然频率对梯度变化敏感。转动惯量在确定自然频率上有重要作用。
(4)分析了轴向非均匀梁的自由振动。对指数梯度梁在不同约束条件下,推导出封闭特征方程。当梯度指数为零时,这些特征方程或频率方程能退化为Euler-Bernoulli梁的经典形式。梯度对频谱有很强的影响,自然频率明显依赖梯度参数的变化和约束条件。对某一指数梯度梁,存在一个依赖梯度参数的临界频率。频率超过临界频率的传播波才能激起振动,否则振动不会产生。对某一梯度指数以小的改变,当自然频率超过临界频率,它会有突变。上述结论可作为分析轴向功能梯度梁自由振动的其它数值程序的一个参照。对某一梯度指数可寻求到最低自然频率,这能帮助工程人员对非均匀梁结构振动进行优化设计
(5)计算了弹性地基上非经典剪切梁的自然频率。高层建筑的弯曲变形主要是平行楼板产生相对滑移引起的,因而可用剪切梁来模拟高层建筑。经典剪切梁不考虑弯矩的存在。由于建筑物墙-墙和墙-板间强相互作用的共同影响引起拉-压力耦合,这种力的耦合使建筑物产生原始弯矩,所以需要修正剪切梁理论。给出了修正剪切梁模型的理论分析。重点计算了在弹性基如泥土上和带有一集中质量的剪切梁的自然频率。取得了修正剪切梁自然频率的特征方程。阐明了平移和转动弹簧刚度、轴向压力和附带的质量对自然频率的影响。
Beams are one of the most common engineering structures. In the field of aerospace, mechanical and civil engineering, many components subjected to applied loading can be understood as beams or columns. Functionally graded materials (FGMs) possess continuously varying material properties. This feature can effectively remove abrupt stress jump when across the interface between two bonded dissimilar materials. Therefore, due to this feature they have been widely used in a variety of engineering fields. In this thesis, the mechanical behaviors of a non-linear functionally graded material cantilever beam subjected to an end force or moment are investigated by using large and small deformation theories. Free vibration of bidirectionally functionally graded beams and axially inhomogeneous beams is analyzed. In addition, the natural frequencies of nonclassical shear beams standing on an elastic foundation and carrying a mass is studied.
This thesis is composed of seven chapters. Chapter one is the introduction, in which the development of relative subjects is outlined. Chapter two to six are the main chapters, and some novel results and conclusions are drawn. The final chapter gives a simple summary and expectation of future researches. Here, main results are emphasized as follows:
(1) The analysis of the large deformation of a non-linear cantilever functionally graded beam is made. For an FGM beam of power-law hardening, the location of the neutral axis is determined. When subjected to an end moment, explicit expressions for deflection and rotation are derived for an FGB. The effects of the gradient distribution of Young's modulus and the material non-linearity parameter on the deflections of the FGB are analyzed. Our results show that depth-dependent Young's modulus and material non-linearity have a significant influence on the deflections of the beam, and an FGB can bear larger applied load than a homogeneous beam. Moreover, to determine a optimal gradient distribution, an optimum approach for choosing a beam with a lighter weight and larger stiffness is given.
(2) The mechanical behaviors of a non-linear functionally-graded-material cantilever beam subjected to an end force are investigated by using large and small deformation theories. The Young's modulus is assumed to be depth-dependent. The effects of the depth-dependent Young's modulus and nonlinearity parameter on the deflections and rotations of the FGM beam are analyzed. Our results show that different gradient indexes may change bending stiffness of the beam so that the FGM beam may bear larger applied load than a homogeneous beam when choosing appropriate gradients. Moreover, the bending stress distribution in an FGM beam is completely different from that in a homogeneous beam. The bending stress arrives at the maximum tensile stress at an internal position rather than at the surface. Obtained results are useful in safety design of linear and non-linear beams.
(3) The free vibration of bidirectionally functionally graded beams is studied. For a slender beam with axially varying and depth-dependent material properties, a governing equation is derived from two-dimensional elasticity theory. By converting it to a Fredholm integral equation, numerical results of the natural frequencies for free vibration are obtained for simply-supported beams with symmetrically-distributed material properties. The effects of gradient variation and rotary moment of inertia on the natural frequencies are discussed. It is found that the natural frequencies are sensitive to the gradient variation. Rotary inertia plays an important role in determining the natural frequencies.
(4) Free vibration of axially inhomogeneous beams is analyzed. For exponentially graded beams with various end conditions, characteristic equations are derived in closed form. These characteristic or frequency equations can analytically reduce to the classical forms of Euler-Bernoulli beams if the gradient index disappears. The gradient has a strong influence on the frequency spectrum, and the natural frequencies noticeably depend on the variation of the gradient parameter and end support conditions. For certain beams with exponential gradients, there exists a critical frequency depending on the gradient parameter. Vibration can be only excited by propagating waves with frequencies in excess of the critical frequency, and otherwise vibration is prohibited. For some gradient index, the natural frequencies have an abrupt jump when across its critical frequencies. Obtained results can serve as a benchmark for other numerical procedures for analyzing transverse vibration of axially functionally graded beams. The minimal natural frequency can be sought for certain gradient index, and this helps engineers to optimally design vibrating inhomogeneous beam structures.
(5) The natural frequencies of a shear beam standing on an elastic foundation and carrying a mass are obtained. Tall buildings may be modelled as shear beams since the corresponding bending is mainly induced by relative sliding of parallel floor slabs. The classical shear beams do not consider the presence of bending moment. Since the bending moment of buildings can be originated from tension-compression force couple due to combined effects of strong wall-to-wall and wall-to-slab interactions, a modified shear beam theory with consideration of bending moment is needed. The final chapter gives a theoretical analysis of a refined shear beam model. Emphasis is placed on the determination of the natural frequencies of a shear beam on an elastic base such as soil and carrying a mass. The characteristic equation for free vibration of a modified shear beam is obtained. The influences of translational and rotational spring stiffnesses, axially compressive loads and attached mass on the natural frequencies are illuminated.
本文共分为七章。第一章是引言部分,对课题研究的发展趋势进行概述。第二章至第六章为主体部分,给出了一些重要的结果和结论。最后部分是结论与对本课题的展望。本文取得的主要成果如下：
(1)分析了材料非线性功能梯度悬臂梁自由端受到力偶作用时的大变形。对幂律硬化的功能梯度梁,确定了中性轴的位置。给出了非线性材料的功能梯度梁挠度和转角的显式表达式。分析了梯度变化的弹性模量和材料非线性参数对功能梯度梁变形的影响。结果表明随高度变化的弹性模量和材料非线性对梁的变形有重大的影响,且功能梯度梁比均质梁能承受更大载荷。为确定最佳的梯度分布,给出了具有较轻重量和较大刚度的优化设计。
(2)利用大变形和小变形理论研究了非线性功能梯度悬臂梁自由端受集中力作用时的力学行为。假设弹性模量沿梁高度方向变化,分析了高度相关的弹性模量和非线性参数对功能梯度梁挠度和转角的影响。结果表明,不同梯度指数能改变梁的弯曲刚度,因此选择适当的梯度,功能梯度梁能比均质梁承受较大载荷。且功能梯度梁弯曲应力的分布与均质梁的应力分布完全不同。最大拉应力在梁内部位置取得而不是在表面。这些结论对线性和非线性梁的安全设计是非常有用的。
(3)研究了双向功能梯度梁的自由振动。对材料性质沿轴向和高度方向变化的细长梁,用二维弹性理论推导出其控制方程。然后将它们转化为对应的Fredholm积分方程,得到了材料性质对称分布的简支梁弯曲振动的自然频率。讨论了梯度变化和转动惯量对自然频率的影响。结果表明自然频率对梯度变化敏感。转动惯量在确定自然频率上有重要作用。
(4)分析了轴向非均匀梁的自由振动。对指数梯度梁在不同约束条件下,推导出封闭特征方程。当梯度指数为零时,这些特征方程或频率方程能退化为Euler-Bernoulli梁的经典形式。梯度对频谱有很强的影响,自然频率明显依赖梯度参数的变化和约束条件。对某一指数梯度梁,存在一个依赖梯度参数的临界频率。频率超过临界频率的传播波才能激起振动,否则振动不会产生。对某一梯度指数以小的改变,当自然频率超过临界频率,它会有突变。上述结论可作为分析轴向功能梯度梁自由振动的其它数值程序的一个参照。对某一梯度指数可寻求到最低自然频率,这能帮助工程人员对非均匀梁结构振动进行优化设计
(5)计算了弹性地基上非经典剪切梁的自然频率。高层建筑的弯曲变形主要是平行楼板产生相对滑移引起的,因而可用剪切梁来模拟高层建筑。经典剪切梁不考虑弯矩的存在。由于建筑物墙-墙和墙-板间强相互作用的共同影响引起拉-压力耦合,这种力的耦合使建筑物产生原始弯矩,所以需要修正剪切梁理论。给出了修正剪切梁模型的理论分析。重点计算了在弹性基如泥土上和带有一集中质量的剪切梁的自然频率。取得了修正剪切梁自然频率的特征方程。阐明了平移和转动弹簧刚度、轴向压力和附带的质量对自然频率的影响。
Beams are one of the most common engineering structures. In the field of aerospace, mechanical and civil engineering, many components subjected to applied loading can be understood as beams or columns. Functionally graded materials (FGMs) possess continuously varying material properties. This feature can effectively remove abrupt stress jump when across the interface between two bonded dissimilar materials. Therefore, due to this feature they have been widely used in a variety of engineering fields. In this thesis, the mechanical behaviors of a non-linear functionally graded material cantilever beam subjected to an end force or moment are investigated by using large and small deformation theories. Free vibration of bidirectionally functionally graded beams and axially inhomogeneous beams is analyzed. In addition, the natural frequencies of nonclassical shear beams standing on an elastic foundation and carrying a mass is studied.
This thesis is composed of seven chapters. Chapter one is the introduction, in which the development of relative subjects is outlined. Chapter two to six are the main chapters, and some novel results and conclusions are drawn. The final chapter gives a simple summary and expectation of future researches. Here, main results are emphasized as follows:
(1) The analysis of the large deformation of a non-linear cantilever functionally graded beam is made. For an FGM beam of power-law hardening, the location of the neutral axis is determined. When subjected to an end moment, explicit expressions for deflection and rotation are derived for an FGB. The effects of the gradient distribution of Young's modulus and the material non-linearity parameter on the deflections of the FGB are analyzed. Our results show that depth-dependent Young's modulus and material non-linearity have a significant influence on the deflections of the beam, and an FGB can bear larger applied load than a homogeneous beam. Moreover, to determine a optimal gradient distribution, an optimum approach for choosing a beam with a lighter weight and larger stiffness is given.
(2) The mechanical behaviors of a non-linear functionally-graded-material cantilever beam subjected to an end force are investigated by using large and small deformation theories. The Young's modulus is assumed to be depth-dependent. The effects of the depth-dependent Young's modulus and nonlinearity parameter on the deflections and rotations of the FGM beam are analyzed. Our results show that different gradient indexes may change bending stiffness of the beam so that the FGM beam may bear larger applied load than a homogeneous beam when choosing appropriate gradients. Moreover, the bending stress distribution in an FGM beam is completely different from that in a homogeneous beam. The bending stress arrives at the maximum tensile stress at an internal position rather than at the surface. Obtained results are useful in safety design of linear and non-linear beams.
(3) The free vibration of bidirectionally functionally graded beams is studied. For a slender beam with axially varying and depth-dependent material properties, a governing equation is derived from two-dimensional elasticity theory. By converting it to a Fredholm integral equation, numerical results of the natural frequencies for free vibration are obtained for simply-supported beams with symmetrically-distributed material properties. The effects of gradient variation and rotary moment of inertia on the natural frequencies are discussed. It is found that the natural frequencies are sensitive to the gradient variation. Rotary inertia plays an important role in determining the natural frequencies.
(4) Free vibration of axially inhomogeneous beams is analyzed. For exponentially graded beams with various end conditions, characteristic equations are derived in closed form. These characteristic or frequency equations can analytically reduce to the classical forms of Euler-Bernoulli beams if the gradient index disappears. The gradient has a strong influence on the frequency spectrum, and the natural frequencies noticeably depend on the variation of the gradient parameter and end support conditions. For certain beams with exponential gradients, there exists a critical frequency depending on the gradient parameter. Vibration can be only excited by propagating waves with frequencies in excess of the critical frequency, and otherwise vibration is prohibited. For some gradient index, the natural frequencies have an abrupt jump when across its critical frequencies. Obtained results can serve as a benchmark for other numerical procedures for analyzing transverse vibration of axially functionally graded beams. The minimal natural frequency can be sought for certain gradient index, and this helps engineers to optimally design vibrating inhomogeneous beam structures.
(5) The natural frequencies of a shear beam standing on an elastic foundation and carrying a mass are obtained. Tall buildings may be modelled as shear beams since the corresponding bending is mainly induced by relative sliding of parallel floor slabs. The classical shear beams do not consider the presence of bending moment. Since the bending moment of buildings can be originated from tension-compression force couple due to combined effects of strong wall-to-wall and wall-to-slab interactions, a modified shear beam theory with consideration of bending moment is needed. The final chapter gives a theoretical analysis of a refined shear beam model. Emphasis is placed on the determination of the natural frequencies of a shear beam on an elastic base such as soil and carrying a mass. The characteristic equation for free vibration of a modified shear beam is obtained. The influences of translational and rotational spring stiffnesses, axially compressive loads and attached mass on the natural frequencies are illuminated.
引文
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