基于波谱单元法的结构动力分析
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摘要
土木工程中广泛应用的数值仿真软件都是基于各种离散的数值计算方法：有限元法、有限差分法、无网格法等等,离散的数值计算方法都具有一个共同的特点,就是依赖于结构划分的单元或节点,划分单元或节点不仅使得以上各种离散算法在应用时受到了一定程度的限制,也降低了计算精度和效率,在结构动力分析领域尤其如此。而基于连续介质力学的波谱单元法对结构进行动力分析时,对于连续均匀构件,无需划分单元,是一种精确的动力学计算方法。传统的波谱单元法主要应用于求解集中动荷载作用下结构的动力响应问题,而对于分布动荷载和其他形式动荷载的分析并不多见。本文在现有波谱单元法的基础上,通过理论公式推导和数值计算分析等手段在以下几个方面展开了研究,取得了一些有价值的研究成果：
(1)通过在结构平衡微分方程中引进阻尼项,推导了考虑结构外部粘滞阻尼和内部粘弹性阻尼的轴向振动杆、扭转振动杆以及弯曲振动梁的波谱单元刚度矩阵。推导结果表明,与不考虑阻尼的波谱单元法相比,只需要通过修改波谱刚度矩阵中的波数就可以方便的计算阻尼对结构动力响应的影响。将波谱单元法中的中阻尼系数与有限元法Rayleigh阻尼系数的推导过程进行对比,得到了波谱单元法阻尼系数与有限元法Rayleigh阻尼系数的关系。对考虑阻尼的桁架结构,多跨连续梁和框架结构进行分析,数值分析结果表明,波谱单元法能十分简便的计算结构考虑了外部粘滞阻尼和内部粘弹性阻尼后的动力响应,并大大减少计算单元数量,提高计算效率。
(2)分别利用基于线性叠加原理的等效方法和基于虚功原理利用波谱形函数积分的等效方法得到分布动荷载的等效节点荷载的计算公式,利用等效节点荷载对分布动荷载作用下的结构进行动力分析。推导了均布动荷载、三角形分布动荷载以及梯形分布动荷载三类特殊荷载作用的伯努利-欧拉梁的等效节点荷载的表达式以及承受轴向均布动荷载的轴向振动杆和承受均布扭转动荷载的扭转振动杆的等效节点荷载的表达式。利用分布动荷载等效为节点集中动荷载的方式,采用得到的等效节点荷载对分布动荷载作用下的结构进行动力分析并与有限元法的结果对比,数值分析结果表明,利用等效节点荷载能十分方便有效的计算分布动荷载作用下结构的动力响应。
(3)地震荷载是施加于结构上的惯性力,利用均布动荷载等效的方式得到地震荷载的等效节点荷载。在相同位置的节点上对不同单元的等效节点荷载进行叠加,得到每个节点总的地震等效节点荷载,把总的地震等效节点荷载看成集中荷载施加于结构上用来计算结构受地震荷载作用的动力响应。为证明这种地震等效节点荷载计算方法的有效性,利用波谱单元法对空间桁架和框架结构进行地震响应分析,结果表明,在波谱单元法中把地震荷载看成均布惯性力再进行节点荷载等效的方式能精确有效的计算结构的地震响应。
(4)现有的波谱单元法都是应用于荷载作用位置固定不变的问题,本文将波谱单元法推广到移动荷载作用下桥梁的动态响应问题。通过引进单位脉冲函数描述移动荷载的空间位置变化,再利用数值Laplace变换将时域内的移动荷载转换频域内,基于能量原理,在频域内通过波谱形函数将作用在桥梁上的移动荷载等效为桥梁端部位置固定不变的节点集中荷载,利用波谱单元法对移动荷载作用下的桥梁结构进行分析。不论移动荷载的个数多少和方向如何,波谱单元法都能通过简单叠加求得结构的动力响应,从而有效回避了有限元法中由于移动荷载位置变化带来的单元划分问题。将波谱单元法的计算结果分别与解析解和高精度精细积分法比较,结果表明,波谱单元法不但精度高,而且具有很高的计算效率。
Numerical simulation software, which has been widely used in Civil Engineering, is based on all kinds of discrete numerical methods, such as the finite element method (FEM), the finite difference method (FDM), the meshless Method, etc. All these numerical methods have a common characteristic. That is the structure must be meshed before analysis. Elements or nodes meshed form structures are necessary in the analysis process of these numerical methods. The calculation accuracy and efficiency will decrease greatly because of the discrete mesh in these numerical methods, especially for structural dynamic analysis. Sometimes, these discrete numerical methods may not converge or may be failure because mesh generation is not simple under certain circumstances. The spectral element method (SEM), which is based on the theory of continuum mechanics, is a high precise method in structural dynamics. Only one spectral element need be placed between any two joints for structures with uniform area. Conventional SEM can be applied only to structures subjected to concentrated loads applied on nodes. Furthemore, there are few literatures about dynamic analysis for structures under distributed loads and other complex loads. In this dissertation, the following aspects are studied by theoretical and numerical analysis, some important results and conclusions have been achieved as follows:
(1) The spectral stiffness matrices of axial vibration rod, torsional vibration rod and flexural vibration Bernoulli-Euler beam were derived from the vibration differential equations of these members. Both internal viscoelastic damping and external viscous damping were considered in the vibration equations. The derived results indicate that the damping effect in SEM can be easily considered by modifying the wave number in the spectral stiffness matrix. The relationship of damping coefficients in SEM and classic Rayleigh damping coefficients in FEM was obtained in the derivation process. The space truss, multi-span continuous beam and space frame structures were analyzed. The SEM has been proved to be an efficient method to analyze the dynamic responses of structures while the number of elements can greatly decrease.
(2) The dynamic analysis of structures under distributed loads using Laplace-based SEM was proposed. The distributed loads applied on structures were equivalent to concentrated nodal forces using two different methods. The first method is based on the principle of linear superposition and the other is based on the principle of virtual work. The computational expressions were derived in both two methods. While the equivalent forces were obtained, the dynamic responses of structures can be calculated without any difficulty by use of traditional SEM which solved the vibration problem of structures under concentrated forces. The analytical expressions of equivalent nodal forces under uniform distributed dynamic loads, triangle distributed dynamic loads and trapezium distributed dynamic loads were derived. The equivalent nodal forces of rod subjected to uniform axial dynamic loads and torsional dynamic loads were also presented. By comparison of the numerical results obtained from SEM and those from FEM, the ability and effectiveness of SEM under distributed dynamic loads have been proved.
(3) The seismic load applied on structures could be considered as inertia force of structures. Therefore, for homogeneous structures, the seismic loads were uniform distributed loads acted on structures. In other words, the seismic loads can also be equivalent to concentrated forces applied on nodes based on the principle of virtual work, which is the same way as the uniform distributed loads. The equivalent nodal forces got form adjacent elements were superimposed at the same node. The total seismic equivalent nodal forces were the sum of the equivalent nodal forces obtained form adjacent elements. The calculated total seismic equivalent forces are then used to analyze seismic responses of structures. To evaluate the accuracy of SEM, the dynamic responses of space truss and frame structures under seismic load are analyzed. It has been found that the SEM provides good dynamic results under seismic load by using the equivalent forces.
(4) The traditional SEM is limited to structures subjected to loads with fixed positions. An extended SEM for dynamic analysis of bridges subjected to moving loads was proposed here. The Dirac function was introduced to simulate the moving loads applied on beam bridge. The moving loads in time domain were transformed into frequency domain by use of Laplcace tranform. The moving loads in frequency were then equivalent to fixed concentrated nodal forces by integrating the shape function in SEM based on the principle of virtual work. These equivalent nodal forces were used to calculate the dynamic responses of bridges in SEM. No matter how many moving loads and their directions acted on the continuous beam bridge, their equivalent nodal forces could be simply superimposed using linear superposition principle at the end of bridges. Only one spectral element need to be placed between any two joints in SEM and this avoid the element meshing in other discrete numerical methods. The numerical results calculated by SEM were compared with those obtained from FEM and the precise time integration method. It has been shown that the present extended SEM provides very high precision results while high efficiency could be obtained.
(1)通过在结构平衡微分方程中引进阻尼项,推导了考虑结构外部粘滞阻尼和内部粘弹性阻尼的轴向振动杆、扭转振动杆以及弯曲振动梁的波谱单元刚度矩阵。推导结果表明,与不考虑阻尼的波谱单元法相比,只需要通过修改波谱刚度矩阵中的波数就可以方便的计算阻尼对结构动力响应的影响。将波谱单元法中的中阻尼系数与有限元法Rayleigh阻尼系数的推导过程进行对比,得到了波谱单元法阻尼系数与有限元法Rayleigh阻尼系数的关系。对考虑阻尼的桁架结构,多跨连续梁和框架结构进行分析,数值分析结果表明,波谱单元法能十分简便的计算结构考虑了外部粘滞阻尼和内部粘弹性阻尼后的动力响应,并大大减少计算单元数量,提高计算效率。
(2)分别利用基于线性叠加原理的等效方法和基于虚功原理利用波谱形函数积分的等效方法得到分布动荷载的等效节点荷载的计算公式,利用等效节点荷载对分布动荷载作用下的结构进行动力分析。推导了均布动荷载、三角形分布动荷载以及梯形分布动荷载三类特殊荷载作用的伯努利-欧拉梁的等效节点荷载的表达式以及承受轴向均布动荷载的轴向振动杆和承受均布扭转动荷载的扭转振动杆的等效节点荷载的表达式。利用分布动荷载等效为节点集中动荷载的方式,采用得到的等效节点荷载对分布动荷载作用下的结构进行动力分析并与有限元法的结果对比,数值分析结果表明,利用等效节点荷载能十分方便有效的计算分布动荷载作用下结构的动力响应。
(3)地震荷载是施加于结构上的惯性力,利用均布动荷载等效的方式得到地震荷载的等效节点荷载。在相同位置的节点上对不同单元的等效节点荷载进行叠加,得到每个节点总的地震等效节点荷载,把总的地震等效节点荷载看成集中荷载施加于结构上用来计算结构受地震荷载作用的动力响应。为证明这种地震等效节点荷载计算方法的有效性,利用波谱单元法对空间桁架和框架结构进行地震响应分析,结果表明,在波谱单元法中把地震荷载看成均布惯性力再进行节点荷载等效的方式能精确有效的计算结构的地震响应。
(4)现有的波谱单元法都是应用于荷载作用位置固定不变的问题,本文将波谱单元法推广到移动荷载作用下桥梁的动态响应问题。通过引进单位脉冲函数描述移动荷载的空间位置变化,再利用数值Laplace变换将时域内的移动荷载转换频域内,基于能量原理,在频域内通过波谱形函数将作用在桥梁上的移动荷载等效为桥梁端部位置固定不变的节点集中荷载,利用波谱单元法对移动荷载作用下的桥梁结构进行分析。不论移动荷载的个数多少和方向如何,波谱单元法都能通过简单叠加求得结构的动力响应,从而有效回避了有限元法中由于移动荷载位置变化带来的单元划分问题。将波谱单元法的计算结果分别与解析解和高精度精细积分法比较,结果表明,波谱单元法不但精度高,而且具有很高的计算效率。
Numerical simulation software, which has been widely used in Civil Engineering, is based on all kinds of discrete numerical methods, such as the finite element method (FEM), the finite difference method (FDM), the meshless Method, etc. All these numerical methods have a common characteristic. That is the structure must be meshed before analysis. Elements or nodes meshed form structures are necessary in the analysis process of these numerical methods. The calculation accuracy and efficiency will decrease greatly because of the discrete mesh in these numerical methods, especially for structural dynamic analysis. Sometimes, these discrete numerical methods may not converge or may be failure because mesh generation is not simple under certain circumstances. The spectral element method (SEM), which is based on the theory of continuum mechanics, is a high precise method in structural dynamics. Only one spectral element need be placed between any two joints for structures with uniform area. Conventional SEM can be applied only to structures subjected to concentrated loads applied on nodes. Furthemore, there are few literatures about dynamic analysis for structures under distributed loads and other complex loads. In this dissertation, the following aspects are studied by theoretical and numerical analysis, some important results and conclusions have been achieved as follows:
(1) The spectral stiffness matrices of axial vibration rod, torsional vibration rod and flexural vibration Bernoulli-Euler beam were derived from the vibration differential equations of these members. Both internal viscoelastic damping and external viscous damping were considered in the vibration equations. The derived results indicate that the damping effect in SEM can be easily considered by modifying the wave number in the spectral stiffness matrix. The relationship of damping coefficients in SEM and classic Rayleigh damping coefficients in FEM was obtained in the derivation process. The space truss, multi-span continuous beam and space frame structures were analyzed. The SEM has been proved to be an efficient method to analyze the dynamic responses of structures while the number of elements can greatly decrease.
(2) The dynamic analysis of structures under distributed loads using Laplace-based SEM was proposed. The distributed loads applied on structures were equivalent to concentrated nodal forces using two different methods. The first method is based on the principle of linear superposition and the other is based on the principle of virtual work. The computational expressions were derived in both two methods. While the equivalent forces were obtained, the dynamic responses of structures can be calculated without any difficulty by use of traditional SEM which solved the vibration problem of structures under concentrated forces. The analytical expressions of equivalent nodal forces under uniform distributed dynamic loads, triangle distributed dynamic loads and trapezium distributed dynamic loads were derived. The equivalent nodal forces of rod subjected to uniform axial dynamic loads and torsional dynamic loads were also presented. By comparison of the numerical results obtained from SEM and those from FEM, the ability and effectiveness of SEM under distributed dynamic loads have been proved.
(3) The seismic load applied on structures could be considered as inertia force of structures. Therefore, for homogeneous structures, the seismic loads were uniform distributed loads acted on structures. In other words, the seismic loads can also be equivalent to concentrated forces applied on nodes based on the principle of virtual work, which is the same way as the uniform distributed loads. The equivalent nodal forces got form adjacent elements were superimposed at the same node. The total seismic equivalent nodal forces were the sum of the equivalent nodal forces obtained form adjacent elements. The calculated total seismic equivalent forces are then used to analyze seismic responses of structures. To evaluate the accuracy of SEM, the dynamic responses of space truss and frame structures under seismic load are analyzed. It has been found that the SEM provides good dynamic results under seismic load by using the equivalent forces.
(4) The traditional SEM is limited to structures subjected to loads with fixed positions. An extended SEM for dynamic analysis of bridges subjected to moving loads was proposed here. The Dirac function was introduced to simulate the moving loads applied on beam bridge. The moving loads in time domain were transformed into frequency domain by use of Laplcace tranform. The moving loads in frequency were then equivalent to fixed concentrated nodal forces by integrating the shape function in SEM based on the principle of virtual work. These equivalent nodal forces were used to calculate the dynamic responses of bridges in SEM. No matter how many moving loads and their directions acted on the continuous beam bridge, their equivalent nodal forces could be simply superimposed using linear superposition principle at the end of bridges. Only one spectral element need to be placed between any two joints in SEM and this avoid the element meshing in other discrete numerical methods. The numerical results calculated by SEM were compared with those obtained from FEM and the precise time integration method. It has been shown that the present extended SEM provides very high precision results while high efficiency could be obtained.
引文
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