A novel method of determining the sole configuration of tensegrity structures

A novel analysis method is presented for form-finding of tensegrity structures. The spectral decomposition of the force density matrix and the singular value decomposition of the equilibrium matrix are performed iteratively to find the feasible sets of nodal coordinates and force densities. An algorithm of determining the sole configuration of free-form tensegrities is provided by specifying an independent set of nodal coordinates, which indicates the geometrical and mechanical properties of the structures can be at least partly controlled by the proposed method. Several illustrative examples are presented to demonstrate the efficiency and robustness in finding self-equilibrium configurations of tensegrity structures.     

A Monte Carlo form-finding method for large scale regular and irregular tensegrity structures

We propose a Monte Carlo form-finding method that employs a stochastic procedure to determine equilibrium configurations of a tensegrity structure. This method does not involve complicated matrix operations or symmetry analysis, works for arbitrary initial configurations, and can handle large scale regular or irregular tensegrity structures with or without material/geometrical constraints.     

Adaptive force density method for form-finding problem of tensegrity structures

A numerical method is presented for form-finding of tensegrity structures. Eigenvalue analysis and spectral decomposition are carried out iteratively to find the feasible set of force densities that satisfies the requirement on rank deficiency of the equilibrium matrix with respect to the nodal coordinates. The equilibrium matrix is shown to correspond to the geometrical stiffness matrix in the conventional finite element formulation. A unique and non-degenerate configuration of the structure can then be obtained by specifying an independent set of nodal coordinates. A simple explanation is given for the required rank deficiency of the equilibrium matrix that leads to a non-degenerate structure. Several numerical examples are presented to illustrate the robustness as well as the strong ability of searching new configurations of the proposed method.     

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.     

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.     

A numerical algorithm for tensegrity dynamics with non-minimal coordinates

This paper provides a numerical correction algorithm for implementation of the dynamics of tensegrity systems described by non-minimal coordinates. This correction algorithm corrects any numerical error that would violate the fixed-length bar constraints. A recursive form of the correction algorithm is proposed, and simulation results support the validity of the proposed scheme.     

Geometrical nonlinear elasto-plastic analysis of tensegrity systems via the co-rotational method

An efficient finite element formulation is presented for geometrical nonlinear elasto-plastic analyses of tensegrity systems based on the co-rotational method. Large displacement of a space rod element is decomposed into a rigid body motion in the global coordinate system and a pure small deformation in the local coordinate system. A new form of tangent stiffness matrix, including elastic and elasto-plastic stages is derived based on the proposed approach. An incremental-iterative solution strategy in conjunction with the Newton-Raphson method is employed to obtain the geometrical nonlinear elasto-plastic behavior of tensegrities. Several numerical examples are given to illustrate the validity and efficiency of the proposed algorithm for geometrical nonlinear elasto-plastic analyses of tensegrity structures.     

On the elastica solution of a T3 tensegrity structure

Stability studies of a T3 tensegrity structure are performed. This structure is composed of three slender struts interconnected by six nonlinear elastic tendons and is prestressed. The struts are governed by linear constitutive laws and are allowed to buckle. Since tensegrity is used for modeling structures with quite large deformations, for example the cytoskeleton, and bifurcation theory—valid for small solutions of the nonlinear equations—does not directly apply, a general procedure for studying the stability behavior of the particular tensegrity model based upon the elastica theory is presented. The reference placement is defined by the prestress, and the equilibrium placements are defined by the applied force and moment.     

张拉膜结构力密度法混合找形分析

论述了张拉膜结构力密度法混合找形的基本理论;所谓力密度法混合找形,即部分单元力密度控制,部分单元弹性控制;力密度控制采用线性求解,弹性控制采用非线性求解,通过迭代计算混合找形求出各结点的坐标值。据此编制了相应的计算软件;对工程实例进行了验算,结果表明,本文给出的计算结果与德国著名软件easy的计算结果相吻合。     

A necessary condition for stability of kinematically indeterminate pin-jointed structures with symmetry

Stability conditions are the key to transform kinematically indeterminate structures into prestressed structures or deployable structures. From the viewpoint of symmetry, a necessary condition is presented for the stability of symmetric pin-jointed structures with kinematic indeterminacy. The condition is derived from the positive definiteness of the quadratic form of the tangent stiffness matrix. Numerical examples verify that the proposed necessary stability condition is in accord with the conventional theory of structural rigidity, and is considered to be more comprehensible. It is robust and easy to implement. Results show that a symmetric prestressed structure is guaranteed to possess integral prestress modes, if the necessary condition is satisfied. Further, a pin-jointed structure with fully symmetric mechanism modes is proved to be unstable, if it does not satisfy the condition.     

Stiffness formulations and necessary and sufficient conditions for exponential stability of prestressable structures

Prestressable structures composed of elastic tendons and rigid bars are investigated for their stiffness and stability properties. Two analytical formulations of the tangent stiffness matrix are analyzed, the first using tendon lengths and forces, and the second using tendon lengths squared and force densities. Several necessary and sufficient conditions for the exponential stability of prestressable configurations are proved using both formulations. Advantages of these conditions, primarily with respect to the traditional test which uses the matrix of the first order system, are discussed and illustrated via examples. Advantages of the second formulation of the tangent stiffness matrix in analytical manipulations and computations are also revealed.     

Prismatic tensegrity structures with additional cables: Integral symmetric states of self-stress and cable-controlled reconfiguration procedure

A simple energy method is put forward to determine the prestress distribution for symmetric tensegrity structures with multiple states of self-stress and a class of prismatic tensegrities with additional cables are introduced to show the accuracy of presented method. For the purpose of modifying structural shape as well as mechanical properties, a cable-controlled reconfiguration procedure is subsequently proposed for these structures. By defining the length adjustments as the control parameters, the reconfiguration procedure is regarded as a quasi-static process, consisting of a sequence of equilibrium configurations with varying control parameters. Then the nonlinear iterative algorithm based on the tangent stiffness of the structure is presented to simulate and follow this reconfiguration process. By way of example, a particular class of reconfigurations coined symmetrical reconfigurations are investigated carefully and the key features as well as the potential applications are given.     

A direct approach to design of geometry and forces of tensegrity systems

In the process of designing a tensegrity system, some constraints are usually introduced for geometry and/or forces to ensure uniqueness of the solution, because the tensegrity systems are underdetermined in most cases. In this paper, a new approach is presented to enable designers to specify independent sets of axial forces and nodal coordinates consecutively, under the equilibrium conditions and the given constraints, to satisfy the distinctly different requirements of architects and structural engineers. The proposed method can be used very efficiently for practical applications because only linear algebraic equations are to be solved, and no equation of kinematics or material property is needed. Some numerical examples are given to show not only efficiency of the proposed method but also its ability of searching new configurations.     

A computational strategy for the random response of assemblies of structures

This paper presents a dedicated approach to the calculation of the random response of assemblies with uncertain interface characteristics. The random response is constructed using a polynomial chaos expansion (PCE). A decomposition of the assemblies into substructures and interfaces is defined and associated with a dedicated computational strategy which leads to a local/global algorithm enabling the treatments of the substructure and of the interface problems to be uncoupled. Since the only uncertain parameters are those which appear in the interface equations, this approach results in a drastic reduction of the computational costs. This paper first presents the classical stochastic finite element strategy for this kind of problem, then details the proposed dedicated approach. The applications concern structures assembled with uncertain elastic bonded joints. The proposed approach is compared to the Monte Carlo method and to the stochastic finite element method.     

板结构计算模型的新发展

现代复合材料层合板具有高强和轻型的突出优点,从而在军工和民用等诸多领域发挥着重要作用。这种板结构的特点是随着纤维走向的不同,层间材料的物理-力学特性发生剧烈变化。沿板厚方向变形的梯度比较陡峭,并在层间结合面处发生强不连续,呈现zig-zag （锯齿状）现象。这导致横向剪应变在板的静态和动态响应中发生重要作用,不计横向变形的经典组合板计算模型CLPT难以适应现代多层板计算分析的需要。考虑横向剪切变形影响的板的计算模型得到重视和发展。需要指出,现有各种考虑剪切变形影响的计算模型虽然有了很大的发展,但在全面和准确性上仍然存在一定的不足,难以适应现代多层组合板横向力和物理性能多变的情况。模型预测的沿板厚方向位移和应力的变化规律难以通过严格的检验。本文提出的以比例边界有限元为基础的正交各向异性板的数值计算模型,同时可适用于各种薄板与厚板的分析,对现代复合材料层合板的分析具有特殊的优越性。所得到的板的位移、正应力和剪应力沿板厚方向的变化,与三维弹性理论的标准解高度吻合。数值算例进一步表明,随着层间纤维走向的变化,板内位移场和应力场沿板厚方向剧烈变化所呈现的锯齿现象均可以精准地进行模拟。据此,本文建议方法对现代板分析的广泛适应性和高度准确性得到了充分论证。     

A computational model for the limit analysis of three-dimensional masonry structures

This paper extends previous work on the limit analysis of ductile frames and plane masonry arches to the limit analysis of three-dimensional masonry structures. A lower-bound approach is developed which can handle three-dimensional collapse mechanisms involving any combination of sliding, twisting and hingeing at the block interfaces. A computer program for determining the collapse load of such structures is used to study (a) the equilibrium limits of a block with four contact points resting on an inclined plane and (b) the collapse of a semicircular arch of four blocks. The paper also describes experimental and computational work on a radially symmetric model dome of 380 blocks subject to foundation settlement.

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Sommario Il presentre contributo estende al campo delle structture tridimensionali in muratura un precedente lavoro sull'analisi limite di telai duttili ed archi in muratura piani. Si e' sviluppato un approccio statico che analizza meccanismi di collasso tridimensionale ottenuti per combinazione dei meccanismi semplici di scorrimento e rotazione nel piano e fuori dal piano delle superfici di interfaccia tra i blocchi. Si descrivono (a) i limiti di equilibrio di un blocco con 4 punti di contatto su base inclinata, (b) le condizioni di collasso di un arco semicircolare costituito da quattro blocchi, applicando un programma di calcolo redatto per l'analisi e la definizione del carico di collasso di tali strutture. La terza parte dell'articolo presenta il lavoro sperimentale e di calcolo sviluppato su un modello di cupola a simmetria radiale costituita da 380 blocchi soggetta a cedimenti fondali.

     

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.     

索网天线找形分析的参变量变分有限元法

针对柔性空间索网天线的非线性特点,建立了基于参变量变分描述索网拉压非线性和共旋列式描述几何非线性的有限元控制方程,应用Lernke与改进牛顿法进行求解.通过对索网预张力平衡计算,证明改进牛顿法比Newton-Raphson法具有更强的收敛能力.进一步将力密度法迭代原理与有限元法结合应用于索网天线的非线性找形分析中,获得了理想的索网构型.本文的索网找形方法可广泛应用于空间索网天线结构的设计.     

A theoretical and computational framework for anisotropic continuum damage mechanics at large strains

The main objective of this work is the formulation and algorithmic treatment of anisotropic continuum damage mechanics at large strains. Based on the concept of a fictitious, isotropic, undamaged configuration an additional linear tangent map is introduced which allows the interpretation as a damage deformation gradient. Then, the corresponding Finger tensor – denoted as damage metric – constructs a second order, internal variable. Due to the principle of strain energy equivalence with respect to the fictitious, effective space and the standard reference configuration, the free energy function can be computed via push-forward operations within the nominal setting. Referring to the framework of standard dissipative materials, associated evolution equations are constructed which substantially affect the anisotropic nature of the damage formulation. The numerical integration of these ordinary differential equations is highlighted whereby two different schemes and higher order methods are taken into account. Finally, some numerical examples demonstrate the applicability of the proposed framework.