Stability conditions for tensegrity structures

Stability conditions for tensegrity structures are derived based on positive definiteness of the tangent stiffness matrix, which is the sum of the linear and geometrical stiffness matrices. A necessary stability condition is presented by considering the affine motions that lie in the null-space of the geometrical stiffness matrix. The condition is demonstrated to be equivalent to that derived from the mathematical rigidity theory so as to resolve the discrepancy between the stability theories in the fields of engineering and mathematics. Furthermore, it is shown that the structure is guaranteed to be stable, if the structure satisfies the necessary stability condition and the geometrical stiffness matrix is positive semidefinite with the minimum rank deficiency for non-degeneracy.     

Axially symmetric thermoviscous elastoplastic deformation of bifurcated shell structures

Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Prikladnaya Mekhanika, Vol. 25, No. 2, pp. 35–44, February, 1989.     

Stability conditions of prestressed pin-jointed structures

Pin-jointed structures are first classified to trusses, tensile structures, and tensegrity structures in view of their respective stability properties. A sufficient condition for stability of an equilibrium state is derived for tensegrity structures. The condition is based on the bilinear forms of the linear and geometrical stiffness matrices considering the flexibility of members. The stability is defined by the positive definiteness of the tangent stiffness matrix, whereas the definition of prestress-stability is based on the geometrical stiffness matrix and the infinitesimal mechanisms. Numerical examples verify that the so-called super-stability condition might not be satisfied by a stable tensegrity structure, and that a prestress-stable structure can be unstable if the prestresses are moderately large.     

Conforming radial point interpolation method for spatial shell structures on the stress-resultant shell theory

The implementation of the conforming radial point interpolation method (CRPIM) for spatial thick shell structures is presented in this paper. The formulation of the discrete system equations is derived from a stress-resultant geometrically exact theory of shear flexible shells based on the Cosserat surface. A discrete singularity-free mapping between the five degrees of freedom of the Cosserat surface and the normal formulation with six degrees of freedom is constructed by exploiting the geometry connection between the orthogonal group and the unit sphere. A radial basis function is used in both the construction of shape functions based on arbitrarily distributed nodes as well as in the surface approximation of general spatial shell geometries. The major advantage of the CRPIM is that the shape functions possess a delta function property and the interpolation function obtained passes through all the scattered points in the influence domain. Thus, essential boundary conditions can be easily imposed, as in finite element method. A range of shape parameters is studied to examine the performance of CRPIM for shells, and optimal values are proposed. The phenomena of shear locking and membrane locking are illustrated by presenting the membrane and shear energies as fractions of the total energy. Several benchmark problems for shells are analyzed to demonstrate the validity and efficiency of the present CRPIM. The convergence rate of the results using a Gaussian (EXP) radial basis is relatively high compared to those using a multi-quadric (MQ) radial basis for the shell problems.     

he analytical solutions of thick-walled cylinder of softening material and its stability

将弹塑性材料的应力应变全过程曲线简化为三线性模型(弹性-线性软化-残余理想塑性), 并 假设材料服从Tresca屈服准则和关联流动法则,推导出受内压厚壁筒的解析解. 在这个解析解 的基础上,讨论了厚壁筒的平衡稳定性问题,内压达到临界载荷时,厚壁筒丧失稳定性,其 临界载荷就是软化塑性材料厚壁筒的承载能力.     

THEORY OF THICK-WALLED SHELLS AND ITS APPLICATION IN CYLINDRICAL SHELL

In this paper, a theory of thick-walled shells is established by means of Hellinger-Reissner's variational principle, with displacement and stress assumptions. The displacements are expanded into power series of the thickness coordinate. Only the first four and the first three terms are used for the displacements parallel and normal to the middle surface respectively. The normal extruding and transverse shear stresses are assumed to bt, cubic polynomials and to satLyfy the boundary stress conditions on the outer and inner surfaces of the shell. The governing equations and boundary conditions are derived by means of variational principle. As an example, a thick-walled cylindrical.shell is disscussed with the theory proposed. Furthermore, a photoelastic experiment has been carried out, and the results are in fair agreement with the computations.     

循环对称结构的多尺度拓扑优化方法

研究了循环对称结构多尺度拓扑优化问题,阐明了均匀化等效性能的基本性质,提出了循环对称单胞的特征参数概念,建立了均匀化映射计算方法,构造了等效性能的三元参数插值模型,有效简化了不同特征参数、微结构构型与体分比下的单胞均匀化等效过程.通过微结构的特征驱动建模与B样条参数化,克服了微结构拓扑优化变量多和变量离散难以保证其光滑...     

Problems of fatigue strength definition for shell structures

Stresses caused by oscillations of shell structures may vary significantly even for the same oscillation form. Therefore, during fatigue tests it is very important to solve the loading problem for these structures and further evaluate their fatigue strength statistical parameters. A new method for solution of these difficult problems is proposed in this paper. The basic idea is to take into account relative load values of fatigue tests instead of often used absolute load values. The fatigue loads are reproduced on a test bed by the same means by which the dynamic loads were determined. Levels of fatigue loads used in the tests are based on the individual dynamic loads for each of the structures investigated. The basic principles are described for axial fans used in railway transport. Axial fan blades are a characteristic example of shell structures.     

Bloch theorem with revised boundary conditions applied to glide and screw symmetric,quasi-one-dimensional structures

Bloch theorem is useful for analyzing wave propagation in periodic systems. It has been widely used to determine the energy bands of various translationally-periodic crystals and with the advent of nanoscale structures like nanotubes, it has been extended to account for additional symmetries using group theory. However, this extension is restricted to Hamiltonian systems with analytical potentials. For complex problems, as for engineering structures, the periodic unit cells are often discretized and the Bloch method is restricted to translational periodicity.The goal of this paper is to generalize the direct and transfer-matrix propagation Bloch method to structures with glide and screw symmetries by deriving appropriate boundary conditions. Dispersion relations for a set of reduced problems are compared to results from the classical method, when available. It is found that (i) the dispersion curves are easier to interpret, (ii) the computational cost and error are reduced, and (iii) revisited Bloch method is applicable to structures as the Boerdijk–Coxeter helix that do not possess purely-translational symmetries for which the classical method is not applicable.     

Strain gradient plasticity solution for an internally pressurized thick-walled spherical shell of an elastic–plastic material

An analytical solution is presented for an internally pressurized thick-walled spherical shell of an elastic strain-hardening plastic material. A strain gradient plasticity theory is used to describe the constitutive behavior of the material undergoing plastic deformations, whereas the generalized Hooke’s law is invoked to represent the material response in the elastic region. The solution gives explicit expressions for the stress, strain and displacement components. The inner radius of the shell enters these expressions not only in non-dimensional forms but also with its own dimensional identity, unlike classical plasticity-based solutions. As a result, the current solution can capture the size effect. The classical plasticity-based solution of the same problem is shown to be a special case of the present solution. Numerical results for the maximum effective stress in the shell wall are also provided to illustrate applications of the newly derived solution. The new solution can be used to construct improved expanding cavity models in indentation mechanics that incorporate both the strain-hardening and indentation size effects.     

板壳结构选型优化设计

提出了最佳结构型式概念;证明了结构存在最佳结构型式的充要条件;建立了启发式算法;实例计算证明,选型优化不仅可巧妙地融合形状、拓扑、布局优化的优点,而且计算省、结果可靠。     

Stability conditions for elastic-plastic prandtl-reuss solids

Summary The algebraic condition ensuring the validity of Hadamard's criterion of local stability in a Prandtl-Reuss solid is examined. Four fundamental situations depending on the possible transitions of state are considered.

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Sommario Si esaminano le condizioni algebriche assicuranti la validità del criterio di Hadamard di stabilità locale in un solido di tipo Prandtl-Reuss. Quattro situazioni fondamentali dipendenti dalle possibili permanenze o conversioni di stato sono considerate.

     

Stability of a cylindrical shell under programmed variation of an axial compressive force under creep conditions

Journal of Applied Mechanics and Technical Physics - The stability of cylindrical shells with initial deflection is examined under creep with a programmed load. Data are presented from an...     

Stability and Hamiltonian Hopf bifurcation for a nonlinear symmetric bladed rotor

The modal interaction which leads to Hamiltonian Hopf bifurcation is studied for a nonlinear rotating bladed-disk system. The model, which is discussed in the paper, is a Jeffcott rotor carrying a number of planar blades which bend in the plane of the motion. The rigid rotating disk is supported on nonlinear bearings. It is supposed that this dynamical system is a Hamiltonian system which is perturbed by small dissipative and nonlinear forces. Krein’s theorem is employed for obtaining a stability criterion. The nonlinear eigenvalue equations on the stability boundary are turned into ordinary differential equations (ODEs) by differentiating them over the rotating speed. By solving these ODEs, the eigenmodes and the eigenvalues on the stability boundary are obtained. The bifurcation analysis is performed by applying multiple scales method around the boundary. The rotor nonlinear behavior and damping effects are studied for different conditions on the rotating speed and nonlinearity type by the bifurcation equation. It is shown that the damping distribution between the blades and bearings may shift the unstable mode. Depending on the nonlinearity type, subcritical and supercritical Hopf bifurcation are possible.     

A new curved gradient deficient shell element of absolute nodal coordinate formulation for modeling thin shell structures

A curved gradient deficient shell element for the Absolute Nodal Coordinate Formulation (ANCF) is proposed for modeling initially thin curved structures. Unlike the fully parameterized elements of ANCF, a full mapping of the gradient vectors between different configurations is not available for gradient deficient elements, therefore it is cumbersome to work in a rectangular coordinate system for an initially curved element. In this study, a curvilinear coordinate system is adopted as the undeformed Lagrangian coordinates, and the Green–Lagrange strain tensor with respect to the curvilinear frame is utilized to characterize the deformation energy of the shell element. As a result, the strain due to the initially curved element shape is eliminated naturally, and the element formulation is obtained in a concise mathematical form with a clear physical interpretation. For thin structures, the simplified formulations for the evaluation of elastic forces are also given. Moreover, an approach to deal with the on-surface slope discontinuity is also proposed for modeling general curved shell structures. Finally, the developed element of ANCF is validated by several numerical examples.     

A new high-precision stress finite element for analysis of shell structures

A new high-precision finite element for analysis of shell structures is presented. It is derived from a slightly generalized equilibrium principle. Accordingly both stresses and displacements are obtained as primary result of analysis. At the assembly level the element has 45 degrees of freedom, all of them generalized displacements. For the price of some additional computational effort on the elemental level of analysis the proposed element is believed to gain certain advantages over the recently developed high-precision displacement elements. Thin as well as thick shell structures of arbitrary shape and loading can be equally analyzed. Engineering accuracy is attained with only very few elements. A variety of numerical examples demonstrates the applicability of the new element to all kinds of situations occuring in practice. A review of the existing high-precision shell elements is also included.     

Design of reinforcement for concrete co-planar shell structures using optimization techniques

The absence of universally accepted solutions for the design of reinforcement in plates and shells in the structural concrete codes, and the constant development of computers combined with powerful numerical methods, reveal the need for a standard procedure to calculate the required reinforcement in thin elements subject to membrane and flexural forces. In the present study, the amount of reinforcement is optimized locally for each finite element of the mesh that models the geometry of the problem. Some numerical examples are given and compared to the results provided by other authors, achieving significant savings in reinforcement.