Sizing optimization of truss structures by method of centers and force formulation

This paper describes a new approach to optimum weight design of truss structures. The force method is incorporated in an optimization algorithm based on the method of center points. Design variables are the member cross-sectional areas and the redundant forces evaluated for each independent loading condition acting on the structure. The optimization method utilizes the largest hyperspheres inscribed within the feasible space. The method of hyperspheres has been enhanced here to handle the compatibility equality constraints as well. By including the analysis step in the optimization cycle there is no longer the need to perform separate structural analyses thus saving computation time. The viability and efficiency of the proposed method are demonstrated for truss structures subject to multiple loading conditions and constraints on member stresses, nodal displacement and minimum gage. Numerical results are compared with those reported in the literature.     

宽翼缘T梁温差应力的弹性力学解法

为了充分反映宽翼缘梁的温差自应力分布特点,克服按梁理论计算温差自应力时的不足,从弹性力学的分析方法入手,按平面应力状态分析翼缘板和腹板的温差应力.根据翼缘板与腹板连接处的变形协调及平衡条件建立补充方程,求解艾瑞应力函数中的积分常数,导出翼缘板与腹板的温差应力及位移的解析式.对一宽翼缘T梁的计算表明,当翼缘板相对于腹板发生温差变化时,沿其宽度的纵向温度自应力分布很不均匀,在翼缘板根部自应力较大而在悬臂端则显著减小.     

Numerical approach for detecting bifurcation points of the compatibility paths of symmetric deployable structures

New internal mechanisms of a deployable structure could be generated, when the structure undergoes significant transformations along its compatibility path. Because of such kind of kinematic bifurcation, the structure might not transform into the desired configuration. To design novel deployable structures, it is necessary to detect all possible bifurcation points of the compatibility paths and study the bifurcation behavior. Here, on the basis of the nonlinear prediction–correction algorithm with variable increment size, we will propose an efficient approach to detect all the possible bifurcation points of the compatibility path for a symmetric deployable structure. Null space of the Jacobian matrix is studied iteratively, to follow the complete compatibility path. The variable increment size at each step is determined by evaluating whether the configuration is close to the singular configuration. Numerical examples of several 2D and 3D symmetric deployable structures are presented, to verify the feasibility and computational complexity of the proposed approach. The results show that the proposed method is computationally efficient, and could detect different bifurcation points of the compatibility path. Further, it turns out that all the analyzed symmetric structures experience kinematic bifurcation on certain conditions.     

基于仿射算法的深部隧道衬砌力学状态区间分析

根据深部地层隧道结构物理、力学参数信息资料缺乏及各类不确定性模型的特征,揭示了区间模型于深部地层隧道工程不确定性分析的优越性。将限制条件少、适用范围广的链杆模式的矩阵位移理论与响应面思路结合,提出了衬砌结构力学状态参量解析式的搭建方法,并进一步推导出了基于仿射运算的响应面函数矩阵表达形式及其区间上、下界计算公式,构筑起基于矩阵表达的响应面函数区间仿射运算模型。将区间离散逐步逼近规则引入,建立了该模型的优化求解方法。利用该模型及方法分析了某衬砌墙脚轴力及拱顶弯矩分布区间,与蒙特卡洛法计算结果比较,显示本文模型及方法有效地克服了经典区间方法在复杂状态函数中的扩张和溢出这一致命缺陷,提高了准确性和计算效率,分析结果精确度满足工程使用要求。     

宽翼缘梁非线性温差自应力级数解

为了客观反映宽翼缘梁在非线性温差作用下的自应力分布特点,克服按梁理论计算温度自应力时的不足,本文从弹性力学的分析方法入手,按平面应力状态分析翼缘板和腹板的温度应力。通过翼缘板与腹板连接处的变形协调条件及平衡条件建立补充方程,求解艾瑞应力函数中的积分常数,推导了翼缘板与腹板的纵、横向正应力、剪应力及位移分量解析式。导出的翼缘板纵向正应力公式可自动考虑宽翼缘梁沿横截面宽度自应力不均匀分布的特点。对一宽翼缘T梁的计算表明,当翼缘板相对于腹板发生温差变化时,沿其宽度的纵向温度自应力分布很不均匀,在翼缘板根部自应力较大而在悬臂端则显著减小。按通常基于梁理论的温度自应力计算方法,无法反映这种应力分布规律。     

Benchmark case studies in structural design optimization using the force method

A structural optimization algorithm is developed for truss and beam structures under stress–displacement or frequency constraints. The algorithm combines the mathematical programming based on the Sequential Quadratic Programming (SQP) technique and the finite element technique based on the Integrated Force Method. A new approach based on the single value decomposition technique has been developed to derive the compatibility matrix required in the force method. Benchmark case studies illustrate the procedure and allow the results obtained to be compared with those reported in the literature. It is shown that the computational effort required by the force method is significantly lower than that of the displacement method and in some cases such as structural optimization problems with multiple frequency constraints, the analysis procedure (force or displacement method) significantly affects the final optimum design and the structural optimization based on the force method may result in a lighter design.     

Analysis of prestressed mechanisms

A new theory is presented for the matrix analysis of prestressed structural mechanisms made from pin-jointed bars. The response of a prestressed mechanism to any external action is decomposed into two almost separate parts, which correspond to extensional and inextensional modes. A matrix algorithm which treats these two modes separately is developed and tested. It is shown that the equilibrium requirements for the assembly, in its initial configuration as well as in deformed configurations which are obtained through infinitesimal inextensional displacements, can be fully described by a square equilibrium matrix. It is also shown that any set of extensional nodal displacements has to satisfy some equilibrium conditions as well as standard compatibility equations, and that the resulting system of linear equations defines a square kinematic matrix. Theoretical as well as experimental evidence supporting this approach is given in the paper ; two simple experiments which were of crucial importance in arriving at the equilibrium conditions on the extensional displacements are described.The interaction between the two modes of action of a prestressed mechanism is discussed, together with a rapidly converging iterative procedure to handle it. A study of the non-linear effect by which the self-stress level in a statically indeterminate assembly rapidly increases if an inextensional mode is excited, supported by further experimental results, concludes the paper. This work is relevant to the analysis of most cable systems, pneumatic domes, fabric roofs, and “Tensegrity” frameworks.     

杆件去除与内力变更时空间网架结构的简捷计算法

本文采用矩阵位移法,引入初内力的概念,提出并建立了初内力准方程式,再运用性叠加原理来简化分析计算杆件去除与内力变更时的空间时的空间网架结构。这种简捷计算法的所讨论的工况下只要一次性的建立、分解和应用网架结构的总刚度矩阵,无须重复多次建立和分解总刚度矩阵,从而使会计工作量成倍地甚至几十倍地大幅度减少。特别是当网架结构多根杆、依次先后断杆、预应力及分期分批施加预应力时,采用本文方法可更方便地求得各工况     

基于构件层次的铰接杆系结构几何稳定性讨论

考察构件刚度和构件撤除对杆件系统几何稳定性的影响.从常规结构稳定理论的角度审视铰接杆件系统几何稳定性问题.基于结构稳定的能量准则和刚度矩阵的构成分析,重新考察了Maxwell准则和平衡矩阵准则的充分必要性.解释了构件零刚度和构件撤除对体系几何稳定性影响的一致性.利用自应力矩阵的特性,提出并证明了一种快速识别杆系结构中“必需杆”的方法.一种多根构件撤除后体系几何稳定性的判别准则进而被发展.该判别准则的数值效率体现在仅利用原结构平衡矩阵一次分解后的信息,杆件撤除后体系平衡矩阵的秩可通过两小规模矩阵秩之间的关系来表示.     

Non-linear dynamic analysis of tensegrity structures using a co-rotational method

A new formulation is presented for the non-linear dynamic analysis of space truss structures. The formulation is based on the dynamics of 3D co-rotational rods. In the co-rotation method, the rigid body modes are assumed to be separated from the total deformations at the local element level. In this paper a new co-rotational formulation is proposed based on the direct derivation of the inertia force vector and the tangent dynamic matrix. A closed-form equation is derived for the calculation of the inertia force, the tangent dynamic matrix, the mass matrix and the gyroscopic matrix. The new formulation is used to perform dynamic analysis of example tensegrity structures. The developed formulation is applicable to tensegrity structures with non-linear effects due to internal mechanisms or geometric non-linearities, and is applied to two numerical examples. The efficiency of the proposed approach is compared to the conventional Lagrangian method, and savings in computation of about 55%, 54% and 37% were achieved.     

Self-stress design of tensegrity grid structures with exostresses

A numerical method is presented for initial self-stress design of tensegrity grid structures with exostresses, which is defined as a linear combination of the coefficients of independent self-stress modes. A discussion on proper division of the number of member groups for the purpose of existence of a single integral feasible self-stress mode has been explicitly given. Dummy elements to transform the tensegrity grid structure with statically indeterminate supports into self-stressed pin-jointed system without supports are employed. The unilateral properties of the stresses in cables and struts are taken into account. Evaluation of the stability for the structure is also considered. Several numerical examples are presented to demonstrate the efficiency and robustness in searching initial single integral feasible self-stress mode for tensegrity grid structures.     

Experimental and numerical study on the self-stress design of tensegrity systems

Tensegrity systems as kinematically and statically indeterminate pin-jointed systems are characterized by mechanisms and self-stress states. Unlike the other reticulated systems, in tensegrity systems, unilateral behavior of cables causes some problems in determining the basis of compatible self-stress states. At the present study, self-stress design of tensegrity systems is presented. Experimental study on two 3×3×0.7 m tensegrity grids was conducted to verify the accuracy and validity of the numerical method. Using supporting constraints, an effective method for the implementation of self-stress states in a much reduced number of stages is proposed and calibrated. Considering the results of the present study, the self-stress design of these systems can be improved to obtain specific desired behavior.     

Numerical form-finding of tensegrity structures

A novel and versatile numerical form-finding procedure that requires only a minimal knowledge of the structure is presented. The procedure only needs the type of each member, i.e. either compression or tension, and the connectivity of the nodes to be known. Both equilibrium geometry and force densities are iteratively calculated. A condition of a maximal rank of the force density matrix and minimal member length, were included in the form-finding procedure to guide the search of a state of self-stress with minimal elastic potential energy. It is indeed able to calculate novel configurations, with no assumptions on cable lengths or cable-to-strut ratios. Moreover, the proposed approach compares favourably with all the leading techniques in the field. This is clearly exemplified through a series of examples.     

A Numerical Method to Material and Geometric Nonlinear Analysis of Cable Structures

In this article, a numerical method is presented for computing the stiffness matrix and compatibility relations of cables under uniform distributed loads on any direction. Both the geometrical and material nonlinearities, including softening and yielding of material, are taken into account and the catenary cable element with gauss integration scheme is employed. The proposed formulation includes the effect of uniform distributed loads on any direction and 3D nodal forces to assess the geometric and possible material nonlinearity of cables. The derived equations are then applied to the analysis of cable structures. The accuracy of the responses obtained by the proposed method is evaluated by several benchmark solutions available in the literature. Results of numerical examples indicate the capability of the proposed stiffness matrix in prediction of the elastic and inelastic responses of cables. The proposed formulation is therefore recommended for cable elements to be used in the analysis of cable structures.     

基于改进残余力向量法的桁架结构损伤诊断

提出一种基于改进残余力向量法的桁架结构损伤诊断方法. 先由灵敏度分析, 求出结构刚度联系矩阵,再由刚度联系矩阵将损伤后的刚度摄动矩阵展开成对角矩阵,代入 残余力向量方程,得到由刚度联系矩阵表示的新的残余力向量方程,此方程可以直接求解, 即可诊断出桁架结构的损伤杆件及其损伤程度. 对于实测中难以获得完备振型的情况,采用 模态扩阶的方法来获得完备的测试振型. 最后以一桁架结构进行数值仿真分析,证实了该方 法的有效性.     

Development of the large increment method for elastic perfectly plastic analysis of plane frame structures under monotonic loading

The displacement-based finite element method dominates current practice for material nonlinear analysis of structures. However, there are several characteristics that may limit the effectiveness of this approach. In particular, for elastoplastic analysis, the displacement method relies upon a step-by-step incremental approach stemming from flow theory and also requires significant mesh refinement to resolve behavior in plastic zones. This leads to computational inefficiencies that, in turn, encourage the reconsideration of force-based approaches for elastoplastic problems.One of these force algorithms that has been recently developed is the large increment method. The main advantage of the flexibility-based large increment method (LIM) over the displacement method is that it separates the global equilibrium and compatibility equations from the local constitutive relations. Consequently, LIM can reach the solution in one large increment or in a few large steps, thus, avoiding the development of cumulative errors. This paper discusses the extension of the large increment methodology for the nonlinear analysis of plane frame structures controlled by an elastic, perfectly plastic material model. The discussion focuses on the power of LIM to handle these nonlinear problems, especially when plastic hinges form in the frame and ultimately as the structure approaches the collapse stage. Illustrative planar frame examples are presented and the results are compared with those obtained from a standard displacement method.     

具有任意连接和约束的空间杆系结构静力分析的回传波矩阵法

本文发展了具有任意连接和约束的空间杆系结构静力分析的回传波矩阵法。以杆端位移和转角为基本未知量,通过结构所有节点的平衡方程和位移协调条件,推导出传递分配矩阵和载荷源向量,并进一步利用设定的同一杆件两个局部坐标系下杆端位移之间的关系,最终得到结构的回传矩阵。据此可求出结构所有杆件的杆端位移及杆端内力。对不同的杆件连接形式,如刚接、铰接、半刚接,以及不同的约束情况,如固定支座、铰支座、定向支座等,本文推导出了空间杆系结构的回传波矩阵表达式,可直接用于相应空间杆系结构内力的计算。同时,针对一个具体刚架结构进行了算例分析,并通过与弯矩分配法和有限元结果的比较,验证了本文方法的精确度。     

Green’s functions for bending problem of a thin anisotropic plate with an elliptic hole

The Green’s functions have not been studied in open literatures for the bending problem of an anisotropic plate with an elliptic hole subjected to a normal concentrated force and a concentrated moment. In this paper, the problem is investigated and the Green’s functions are first obtained by using the complex potential approach. The techniques of conformal mapping transformation and analytic continuation are used to derive the closed-form complex stress functions. The Green’s functions obtained have some potential applications in the analysis of composite structures such as the modification of the displacement compatibility model for notched stiffened composite panels and the formulation of a new method for interlaminar stress analysis around holes of laminates.     

Geometric and material nonlinear analysis of tensegrity structures

A numerical method is presented for the large deflection in elastic analysis of tensegrity structures including both geometric and material nonlinearities.The geometric nonlinearity is considered based on both total Lagrangian and updated Lagrangian formulations,while the material nonlinearity is treated through elastoplastic stress-strain relationship.The nonlinear equilibrium equations are solved using an incremental-iterative scheme in conjunction with the modified Newton-Raphson method.A computer program is developed to predict the mechanical responses of tensegrity systems under tensile,compressive and flexural loadings.Numerical results obtained are compared with those reported in the literature to demonstrate the accuracy and efficiency of the proposed program.The flexural behavior of the double layer quadruplex tensegrity grid is sufficiently good for lightweight large-span structural applications.On the other hand,its bending strength capacity is not sensitive to the self-stress level.     

考虑剪力滞后效应的钢-混凝土组合梁力法单元

本文通过在Newmark模型中引入翘曲形函数来描述楼板纵向位移的横向非均匀分布．然后采用虚余功原理建立了考虑剪力滞后效应的钢—混凝土组合梁力法单元。该力法单元严格地满足平衡条件．而仅在积分意义上满足变形协调条件，但具有与通用位移有限元法相一致的刚度矩阵形式。由于受平衡条件的限制结点外力不相互独立，而翘曲位移引起的双弯矩和双剪力使得内力关系变得非常复杂，本文给出了推导力法梁单元内力形函数的通用方法。此外．本文还考虑了梁间荷载的存在对内力形函数的影响。算例分析表明，所提出的力法单元有较高的精度，并发现：在有梁间分布荷载时．采用将梁间分布荷载等效为结点荷载的方法将显著降低应力的精度，而采用高次力法单元对于提高情度没有明显的作用。