Post-buckling analysis of shell and membrane structures by dynamic relaxation method

The family of arc-length method has been used for the post-buckling analysis of structures. However, it often becomes intractable for complex nonlinear problems. For instance, the incremental solution hangs about and never reaches the desired deformation when wrinkling appears after the buckling of membrane structures. In order to cope with this problem, alternative methods have been proposed, e.g., the dynamic relaxation (DR) method and the DR arc-length method. The applicability of these methods to the post-buckling analysis of shell and membrane structure is investigated in the present research, where the mixed interpolation of tensorial components (MITC) shell element is used for the FE modeling. It is shown that the DR and the DR arc-length methods are much more robust than the arc-length method, but for complicated nonlinear problems, a continuous deformation path can not be obtained by any of the three methods.     

Post-buckling analysis of structures by dynamic relaxation

The paper presents a displacement incrementation procedure to handle multiple loadings in post-buckling analysis of structures by Dynamic Relaxation (DR). This procedure is generalized and a ‘variable-arc-length’ method is proposed to automate the tracing of load-deflection path. The resulting algorithm exactly traces limit points and can handle ‘snap-through’ or ‘snap-back’ problems. The efficiency of the proposed method is demonstrated by typical examples of truss, beam and shell structures.     

区间随机桁架结构动力特性分析方法研究

摘 要：针对区间随机桁架结构的动力特性分析,提出了一种区间随机有限元方法。当结构的物理参数和几何尺寸同时具有区间随机性时,利用区间因子法和随机因子法建立了结构的刚度矩阵和质量矩阵;从结构振动的瑞利商表达式出发,利用区间运算推导了结构动力特性区间随机变量的计算式;进而利用随机变量的矩法和代数综合法,推导出了结构特征值的数字特征的计算式。最后通过算例分析了区间随机桁架结构参数的区间随机性对其动力特性的影响,计算结果表明该方法是可行和有效的。
     

A method for space truss analysis in the post-buckling range

A dual load method of truss analysis is presented that allows highly nonlinear member behaviour to be followed. In particular, a brittle type of strut behaviour is considered. The method assumes that the inelastic strut response is known and that piecewise linearization of the behaviour is acceptable. An example is given to illustrate the type of problem that can be handled. The example involves negative stiffness of the members and consequently attention must be given to the equation solving procedure for the structure. A comparison is made with some observed test results.     

大型空间柔性桁架结构等效建模与动力学分析

大型空间柔性桁架结构具有周期性、大柔度、构型复杂等特点,其等效建模是进行振动控制器设计的关键性技术之一。基于能量等效原理和经典Timoshenko梁理论,对刚性连接的大型空间柔性正三棱柱桁架结构进行了等效建模与动力学分析,采用Taylor展开方法推导了等效梁模型的刚度和质量表达式,对比分析桁架结构与等效梁模型的固有振动特性,二者吻合较好。数值结果表明了等效方法的有效性且等效梁模型具有较高的精度。     

Sensitivity analysis of truss structures (virtual distortion method approach)

A new approach to structural sensitivity analysis based on the so-called virtual distortion method is presented. The proposed methodology enables the calculation of derivatives for elastic as well as elasto-plastic structures on the basis of knowledge of current strains, permanent plastic deformations and influence matrix, describing interactions between a chosen member and the entire structure. The analytical basis as well as numerical verification of the concept is demonstrated. Advantages of the proposed approach, in the sense of numerical cost, are summarized in conclusions. © 1998 John Wiley & Sons, Ltd.     

Non-linear structural analysis by dynamic relaxation

Dynamic relaxation, an iterative method for use with digital computers, is described and is shown to be suitable for the solution of a system of linear equations and in particular for such problems derived from structural frame analysis. It is further shown that the method may be modified to include non-linear equations relating to these problems. Some specific examples of linear and non-linear solutions are given and comparisons are made with another computer method which performs the same tasks.     

Probabilistic analysis of truss structures with uncertain parameters (virtual distortion method approach)

A new approach for probabilistic characterization of linear elastic redundant trusses with uncertainty on the various members subjected to deterministic loads acting on the nodes of the structure is presented. The method is based on the simple observation that variations of structural parameters are equivalent to superimposed strains on a reference structure depending on the axial forces on the elastic modulus of the original structure as well as on the uncertainty (virtual distortion method approach). Superposition principle may be applied to separate contribution to mechanical response due to external loads and parameter variations. Statically determinate trusses dealt with the proposed method yields explicit analytic solution in terms of displacements while redundant trusses have been studied by means of an asymptotic expansion exhibiting explicit dependence on parameter fluctuations. Probabilistic characterization of the response may then be obtained both for statically determinate and statically indeterminate stochastic trusses.     

Comparative analysis of three-dimensional frames by dynamic relaxation methods

Analysis of three-dimensional frames is a complex process. Each node of these structures has six degrees of freedom, which results in a large set of governing equations. Investigators have utilized the computer power to extend the application of numerical approaches. One of these methods is named dynamic relaxation technique. This strategy explicitly solves the simultaneous system of equations. In this scheme, the static structural equilibrium equations are converted into a dynamic one by adding fictitious mass and damping. To perform nonlinear geometric analysis of 3D frames, 12 classical DR methods are exploited. Previously, the abilities of these approaches in analyzing these structures have not been compared. In each technique, time step, mass, and damping matrices are obtained based on other researchers' works. The structural behavior is assessed with and without considering the shear deformations. For both cases, the load–displacement curves are depicted. In this article, 11 different frames are analyzed. Since a few nonlinear solutions of 3D frames are available, these structures can be used as a benchmark in the future studies. Finally, the used algorithms are graded based on their required number of iterations and analysis time. Findings prove that the Qiang, Rezaiee-Pajand and Sarafrzi techniques perform more successfully in comparison to the other schemes.     

Analysis of partly wrinkled membranes by the method of dynamic relaxation

A version of the method of dynamic relaxation is developed to analyze equilibrium configurations of partly wrinkled membranes. In this method equilibria are regarded as long time limits of a damped dynamical problem. The membrane theory considered is based on the concept of a relaxed strain energy function that automatically incorporates the effects of wrinkling. For neo-Hookean materials, existence theorems of nonlinear elasticity are used to show that the relaxed potential energy possesses minimizers in a certain function space. Moreover, solutions of the equilibrium equations furnish global minima of the energy, for certain classes of boundary data. Such deformations are automatically stable according to the minimum-energy criterion. This result motivates the search for solutions of the equilibrium equations, although the existence theory does not guarantee that energy minimizers possess the degree of regularity required by these equations. Several examples of two-and three-dimensional deformations are presented.     

Inelastic finite strain analysis of structures subjected to nonproportional loading

The inelastic behaviour of elasto-plastic materials is nonlinear, path-dependent, and is a function of the total plastic strain. For finite strain problems, the total inelastic strain in Lagrangian co-ordinates cannot be decomposed additively. A generalized logarithmic strain which is formulated in ‘updated’ Lagrangian coordinates and obtained by numerical integration of the Lagrangian strain rate is therefore introduced in this paper. By the use of this strain measure, which is additively decomposable, the plasticity model proposed by the authors can be extended to the finite strain range. It is shown that by correlating the generalized plastic modulus in the constitutive relations with the experimental uniaxial true stress-logarithmic strain diagrams, the inelastic behaviour of steel structures subjected to nonproportional loading can be analyzed numerically by using the finite element method.     

波谱单元法在空间桁架地震响应分析中的应用

摘 要：针对传统波谱单元法（SEM）只能用于求解节点集中荷载作用下结构动力响应问题的不足,提出了一种通过计算地震等效波谱节点荷载求解桁架结构地震响应的方法。基于虚功原理,利用波谱形函数积分得到地震等效波谱节点荷载的显式表达式,通过修改波谱单元法中单元刚度矩阵的波数,考虑了阻尼对结构动力特征的影响,采用数值拉普拉斯（Laplace）变换替换快速傅立叶（FFT）变换,回避了传统波谱单元法中FFT的周期性问题。利用地震荷载等效后的波谱节点荷载对三维空间桁架结构进行地震响应分析,结果表明,采用本文的方法能方便的计算桁架结构的地震等效波谱节点荷载,精确求解结构的地震响应,与传统有限元法（FEM）相比,大大减少计算单元数量,提高计算精度,且便于编程计算。     

应用动力单元法进行空间索网结构自由振动分析

索网结构具有成型跨度大而结构重量轻的优点,因此在空间航天领域得到了广泛应用。目前对索网结构的动力学分析主要是采用集中质量法或有限单元法进行低阶模态的分析,这些方法得到的较高阶频率和模态,其精度和实用性都不理想。动力单元法采用含有固有频率的动态形函数作为动力学分析离散单元中的插值函数,该形函数由单元动力学控制微分方程导出,因此能给出比有限元法更高的求解精度。在总结研究动力单元法理论的基础上,推导了张紧索单元的动力单元矩阵。采用动力单元法对几个典型的空间张紧索网结构进行动力学特性的分析,并将分析结果与传统有限元分析结果进行比较。     

Explicit finite difference method in the dynamic analysis of composite structures

An approach for numerical analysis of composite thin-walled structural elements is presented. The analysis is based on the applied theory of composite shells and the grid-characteristic method of finite differences. The explicit and stable homogeneous method for numerical integration of dynamic equations of thin-walled composite structures is discussed. The splitting procedure is used to extract strongly oscillating components from the solution. Implementation of this procedure provides the stability of explicit finite differences schemes. The method considered makes it possible to carry out computations for a wide range of variation of structural parametres and could be used for the analysis of composite structures under dynamic loading.     

Inelastic transient dynamic analysis of Reissner–Mindlin plates by the D/BEM

A direct domain/boundary element method (D/BEM) for dynamic analysis of elastoplastic Reissner–Mindlin plates in bending is developed. Thus, effects of shear deformation and rotatory inertia are included in the formulation. The method employs the elastostatic fundamental solution of the problem resulting in both boundary and domain integrals due to inertia and inelasticity. Thus, a boundary as well as a domain space discretization by means of quadratic boundary and interior elements is utilized. By using an explicit time‐integration scheme employed on the incremental form of the matrix equation of motion, the history of the plate dynamic response can be obtained. Numerical results for the forced vibration of elastoplastic Reissner–Mindlin plates with smooth boundaries subjected to impulsive loading are presented for illustrating the proposed method and demonstrating its merits. Copyright © 2000 John Wiley & Sons, Ltd.     

A generalized complex eigenvector method for dynamic analysis of heterogeneous viscoelastic structures

A generalized complex eigenvector method which can be used to a linear dynamic analysis of viscoelastic structures is described. Here dynamic analysis is understood as transient analysis and frequency response analysis. The generalized complex eigenvector method is based on finite element discretization of structure, approximation of viscoelastic properties by differential operators and mode superposition technique. Coefficients of differential operator are defined from the condition of best coincidence of complex characteristic of viscoelastic material and complex characteristic of differential operator in preset frequency range. Advantage of this method is that it allows to take into account the real changes of the viscoelastic property in frequency range. Also, the generalized complex eigenvector method permit to describe a viscoelastic properties by two functions (complex Young's modulus, complex Poisson's ratio). The method is verified with the help of comparing with solutions obtained by complex modulus method. An influence of viscoelastic Poisson's ratio on transient and frequency responses of structure is demonstrated. Copyright © 2001 John Wiley & Sons, Ltd.     

复合材料加筋结构后屈曲的自开发程序

针对复合材料层合加筋结构的后屈曲问题,开发出自编Fortran程序,并嵌入Patran进行前后处理。采用该程序计算出线性屈曲模态,将其作为后屈曲非线性计算的初始挠度;采用增量迭代技术、弧长控制及能量收敛准则求解非线性方程;最后引入复合材料Tsai-Hill失效准则,对单元的损伤进行判断及刚度折减。采用复合材料加筋板的压缩试验对该系统进行初步验证。针对不同加载方式、选材及结构型式的机翼盒段,利用该系统进行后屈曲计算和分析,数值结果与试验数据吻合较好。