Buckling and post-buckling of long pressurized elastic thin-walled tubes under in-plane bending

The present paper focuses on the structural stability of long uniformly pressurized thin elastic tubular shells subjected to in-plane bending. Using a special-purpose non-linear finite element technique, bifurcation on the pre-buckling ovalization equilibrium path is detected, and the post-buckling path is traced. Furthermore, the influence of pressure (internal and/or external) as well as the effects of radius-to-thickness ratio, initial curvature and initial ovality on the bifurcation moment, curvature and the corresponding wavelength, are examined. The local character of buckling in the circumferential direction is also demonstrated, especially for thin-walled tubes. This observation motivates the development of a simplified analytical formulation for tube bifurcation, which considers the presence of pressure, initial curvature and ovality, and results in closed-form expressions of very good accuracy, for tubes with relatively small initial curvature. Finally, aspects of tube bifurcation are illustrated using a simple mechanical model, which considers the ovalized pre-buckling state and the effects of pressure.     

Kinematic hardening of trusses

Summary Polycrystalline materials are modelled using plastic trusses. The rods of the truss simulate the slip systems of a single crystal while the geometry of the truss controls the interaction between the different slip systems. For calculations of the evolution of the yield locus in the space of the nodal forces, i.e. in the generalized stress space, due to the geometric variation of the truss, (kinematic hardening) a direct formulation is looked for, in which only quantities of the stress space are involved. It is found that for an adequate approach, besides the nodal forces and the nodal velocities also moments and rotations of the rods have to be introduced. If this result would be transferred to a polycrystal then it would mean that kinematic hardening could be described, in a sensible way by means of a Cosserat continuum only. Comparing for a truss the yield loci of the direct formulation with yield loci calculated by an indirect method a good agreement is achieved.

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Kinematische Verfestigung von Stabwerken

Übersicht Vielkristalline Werkstoffe werden durch plastische Stabwerke modelliert, deren Einzelstäbe die Gleitsysteme im Einzelkorn simulieren, während die Geometrie des Stabwerkes die wechselseitige Beeinflussung der Gleitsysteme regiert. Für die Änderung des Fließortes im Zustandsraum der Stabkräfte als Folge der sich ändernden Geometrie des Stabwerkes (kinematische Verfestigung) wird ein direkter Ansatz gesucht, der nur Größen aus dem betrachteten Zustandsraum enthalten soll. Es zeigt sich, daß hierzu neben den Knotenkräften und den Knotenverschiebungen auch Stabmomente und Drehgeschwindigkeiten eingeführt werden müssen. Wenn man dieses Ergebnis auf den Vielkristall überträgt, so bedeutet es, daß eine sinnvolle Beschreibung der kinematischen Verfestigung nur im Cosserat-Kontinuum gelingt. Zur Kontrolle der Ansätze werden Fließorte aus der direkten und einer indirekten Rechnung verglichen. Es ergibt sich eine gute Übereinstimmung.

     

可展桁架运动过程动力学模拟

针对可展桁架运动过程分析的算法已经被提出。本文在已有算法的基础上,着重对违约的修正、约束雅可比矩阵的生成、雅可比矩阵的Moore-Penrose广义逆计算进行推导和修正,进一步完善用广义逆方法求解可展桁架运动过程的动力学仿真算法。算例说明了算法的有效性。     

Non linear analysis of space trusses

Summary A combined nonlinear analysis of spatial truss systems is per formed. Local buckling of elements is accounted for.The approach given herewith is able to describe the post buckling behavior of the system.

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Sommario Si effettua lo studio di sistemi reticolari spaziali considerando la non linearità di comportamento dovuta. sia ai grandi spostamenti sia ad una legge costitutiva non lineare.L'analisi si estende alla fase post-critica e tiene conto dell'instabilità locale di singole aste.

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This research has been carried out with the financial support of ENEL (Electricity Board of Italy).     

Buckling of long thick-walled circular cylindrical shells of isotropic incompressible hyperelastic materials under uniform external pressure

For isotropic incompressible hyperelastic materials, the problem of determining the critical external pressure at which a long thick-walled circular cylindrical shell will buckle involves solving a fourth-order system of highly non-homogeneous, ordinary differential equations. Closed-form solutions of this system are derived here for plane-strain conditions and for the particular case of the Varga material. These solutions are used to derive the buckling criterion and numerical values are obtained for the resulting critical pressures. They are found to be in good agreement with existing theoretical and experimental results for the neo-Hookean material.     

Large displacement analysis of elastic pyramidal trusses

The paper examines the equilibrium stability problem for a simple class of elastic space trusses in the shape of a regular pyramid. Joints located at the vertices of the base polygon are fixed while the joint at the apex is subjected to a proportionally increasing load acting in either the vertical direction, in the horizontal plane, or along a generic oblique direction. Exact closed-form solutions are derived for each load condition under the common hypotheses of linear material law, small or moderate axial deformation in bars and large nodal displacements. Despite their seeming simplicity, these mechanical systems exhibit a wide variety of post-critical responses, not exhausted by the classical snapping and bifurcation phenomena. In addition to regular primary and secondary branches, the equilibrium paths may include neutral branches, namely branches entirely composed of bifurcation or limit points. Besides their immediate theoretical interest, these branches are particularly difficult to handle by the standard numerical procedures of non-linear analysis, so the given solutions may represent severe benchmark tests.     

Complete equilibrium paths for Mises trusses

This paper presents determination of equilibrium paths for Mises trusses with different ratio of height to span. Unsymmetrical deformation modes are considered and the structure is treated as a two DOF system. First, a few special equilibrium configurations are resolved from considerations of free body diagrams. Complete equilibrium paths are determined by solving numerically the governing non-linear equilibrium equations. The stability of possible equilibrium configurations is checked using the second partial derivative test for the total potential energy. The positive definiteness of the appropriate Hessian matrices is checked numerically using the Sylvester criterion.     

Duality between plane trusses and grillages

In the paper, projective plane duality, that is, a point-to-line, line-to-point, incidence-to-incidence correspondence between plane trusses and grillages of simple connection is treated. By means of linear algebra it is proved that the rank of the equilibrium matrix of plane trusses and grillages does not change under projective transformations and polarities: consequently the number of infinitesimal inextensional mechanisms and the number of independent states of self-stress are preserved under these transformations. The results obtained are also applied to structures with unilateral constraints, and by using several examples it is shown that plane tensegrity trusses have projective dual counterparts among grillages which can be physically modelled with popsicle sticks by weaving.     

Optimal design of trusses for alternative loads

Summary The paper treats optimal design of a plane elastic truss (Fig. 1) subject to two alternative loads L and L, when an upper bound is given for the absolute value of the stress in any bar. While the load L is fixed, the load L is varied subject to the restriction to nonnegative values of P and Q (Fig. 1). The possibility that the optimal truss, that is the truss with the smallest total volume of bars, is statically determinate is not excluded. Statically indeterminate trusses, and particularly fully stressed trusses of this kind are found to play a less important role than has generally been believed.

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Übersicht Es wird der optimale Entwurf von ebenen elastischen Fachwerken nach Abb. 1 behandelt. Das Fachwerk wird durch zwei alternativ wirkende Belastungen L und L beansprucht und es wird angenommen, daß eine obere Grenze für den Absolutwert der Spannung für jeden der Stäbe vorhanden ist. Bei fest vorgegebener Last L wird L so verändert, daß die Komponenten P und Q nicht negativ werden. Als optimales Fachwerk wird ein solches mit kleinstem Gesamtvolumen der Stäbe definiert; es kann statisch bestimmt sein. Es wird gezeigt, daß statisch unbestimmte sowie voll ausgelastete Fachwerke eine weniger bedeutende Rolle spielen, als allgemein angenommen wird.

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Dedicated to Professor Udo Wegner on his 70th birthday.     

Non-linear closed-form computational model of cable trusses

In this paper the non-linear closed-form static computational model of the pre-stressed suspended biconvex and biconcave cable trusses with unmovable, movable, or elastic yielding supports subjected to vertical distributed load applied over the entire span and over a part (over the half) of the span is presented. The paper is an extension of the previously published work of authors [S. Kmet, Z. Kokorudova, Non-linear analytical solution for cable trusses, Journal of Engineering Mechanics ASCE 132 (1) (2006) 119-123]. Irvine's linearized forms of the deflection and the cable equations are modified because the effects of the non-linear truss behaviour needed to be incorporated in them. The concrete forms of the system of two non-linear cubic cable equations due to the load type are derived and presented. From a solution of a non-linear vertical equilibrium equation for a loaded cable truss, the additional vertical deflection is determined. The computational analytical model serves to determine the response, i.e. horizontal components of cable forces and deflection of the geometrically non-linear biconvex or biconcave cable truss to the applied loading, considering effects of elastic deformations, temperature changes and elastic supports. The application of the derived non-linear analytical model is illustrated by numerical examples. Resulting responses of the symmetric and asymmetric cable trusses with various geometries (shallow and deep profiles) obtained by the present non-linear closed-form solution are compared with those obtained by Irvine's linear solution and those by the non-linear finite element method. The conditions for the use of the linear and non-linear approach are briefly specified.     

可展桁架展开过程模拟的违约校正

在 Yoon等学者的基础上 ,利用泰勒级数和 Moore-Penrose广义逆对主动校正法进行了深入、清晰的阐述 ,并针对可展桁架结构展开模拟所遇到的完整定常约束 ,发展了一种简单实用且精度较高的能量和速度违约校正方法。文中给出的算例说明了本文的违约校正算法的有效性     

可视化杆系结构分析程序设计

利用可视化编程语言将杆系优化计算封装为动态连接库 (DLL) ,采用面向对象的方法描述了杆系结构分析所需的类 ,并用数据库技术和三维图形模式对杆系结构分析进行格式化输入输出。通过对杆系结构的特点分析 ,将程序分为三个主要的设计模块 :显示模块、数据管理模块和结构分析模块 ,实现了程序的模块化设计。通过对实例的测试 ,说明该程序已达到对一般多工况杆系可视化设计的目的     

Stochastic finite element analysis of non-linear plane trusses

—This study considers the responses of geometrically and materially non-linear plane trusses under random excitations. The stress-strain law in the inelastic range is based on an explicit differential equation model. After a total Lagrangian finite element discretization, the nodal displacements satisfy a system of stochastic non-linear ordinary differential equations with right-hand-sides given by random functions of time. The exact solution of the above stochastic differential equation is generally difficult to obtain. To seek an approximate solution with good accuracy and reasonable computational effort, the stochastic linearization method is used to find the first and second statistical moments (i.e. the mean vector and the one-time covariance matrix) of the nodal displacements. Results of simple structures under Gaussian white-noise excitation indicate that the proposed method has good accuracy (generally underestimates the r.m.s. stationary response by 5–14%) and requires only a small fraction of the computation time of the time-history Monte-Carlo method.     

The propagation of waves in impact-loaded spatial trusses

Summary The propagation of elastic and plastic waves in arbitrary spatial trusses under impact loading is considered in this paper. Particular attention is payed to the derivation of the equations which are needed to determine the distribution of the waves at the pins. Subsequently, the equations are written compactly as a matrix equation which is solved numerically. Various examples illustrate the method.

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Ausbreitung von Wellen in räumlichen Stabwerken unter Stoßbelastung

Übersicht In dieser Arbeit wird die Ausbreitung von elastischen und plastischen Wellen in beliebigen räumlichen Stabwerken unter Stoßbelastung behandelt. Dabei wird der Herleitung der Gleichungen, die zur Ermittlung der Wellenaufteilung an einem Knoten benötigt werden, besondere Aufmerksamkeit gewidmet. Die Gleichungen werden anschließend zu einem Matrizensystem zusammengefaßt, das numerisch gelöst wird. Beispiele illustrieren die Methode.