Corotational stiffness matrices for 3D truss elements

Revista de Estrutura do Aço

With the increasing use of high strength steels, steel structures are become more resilient and also slender than before. Therefore, the phenomena of structural instability must be considered and prevented. The goal of this paper is to help structural engineer to understand, in detail, analytically and numerically, the nonlinearities of slender structural systems. Moreover, it explains the detection and classification of critical points present in the primary equilibrium path of slender structural systems. This work uses the Total Lagrangian Formulation to describe the kinematics of a truss 3D bar element. Through this formulation the authors obtain the internal forces vector and tangent stiffness matrix that take into account the effects of geometric non-linearity. An elastic linear constitutive model for the uniaxial stress-strain state is assumed. The Green-Lagrange deformation and the axial stress of the second Piola-Kirchhoff tensor, energetically conjugated, are adopted. As a case study, a simple physical system with three degrees of freedom is presented. Such system is composed of two truss 3D bars and a linear spring. Finally, the geometric and physical conditions for the coalescence between the limit and bifurcation points are determined.

2008

In the nonlinear analysis of elastic structures, the displacement increments generated at each incremental step can be decomposed into two components as the rigid displacements and natural deformations. Based on the updated Lagrangian (UL) formulation, the geometric stiffness matrix [kg] is derived for a 3D rigid beam element from the virtual work equation using a rigid displacement field. Further, by treating the three-node triangular plate element (TPE) as the composition of three rigid beams lying along the three sides, the [kg] matrix for the TPE can be assembled from those of the rigid beams. The idea for the UL-type incremental-iterative nonlinear analysis is that if the rigid rotation effects are fully taken into account at each stage of analysis, then the remaining effects of natural deformations can be treated using the small-deformation linearized theory. The present approach is featured by the fact that the formulation is simple, the expressions are explicit, and all memb...

In this work, an alternative rod element formulation is proposed for the nonlinear dynamic analysis of trusses. The classical, geometrically nonlinear elastic rod element formulation is extended by implicitly defining new, hysteretic, degrees of freedom, subjected to an evolution equation of the Bouc Wen type with kinematic hardening. An interpolation field is proposed for the new degrees of freedom, which are regarded as hysteretic strains. By means of the principle of virtual work a geometrically nonlinear elastoplastic stiffness matrix is derived. This stiffness matrix together with the hysteretic evolution equations fully describes the constitutive behavior of the element. Solutions are obtained by simultaneously solving the three sets of governing equations of the structure, namely the global equilibrium equations, global compatibility equations and local constitutive equations. A Livermore solver for stiff differential equations is implemented. Following this approach, the linearization of the constitutive relations is avoided, contrary to the usual step-by-step solution approaches. Furthermore, stability problems can be studied as a dynamic phenomenon. The efficiency of the proposed method is demonstrated with a characteristic example.

Meccanica, 2014

This paper deals with the numerical simulation of the dynamic response of frame structures undergoing large displacements and 3D rotations.The corotational and the geometrically exact approaches are combined in the formulation of a 3D beam element. Inertia forces are evaluated starting from an exact expression of the element kinetic energy. This last is defined only in terms of nodal variables in the global inertial frame of the mechanical problem, so to avoid cumbersome calculations coming from the corotational decomposition of the motion. The proposed procedure is formally independent from the definition of the corotated frame. The performance of the proposed formulation is discussed with reference to some classic benchmarks involving slender frame structures undergoing large 3D rotations and displacements.

Computer Methods in Applied Mechanics and Engineering, 2005

This article presents a unified theoretical framework for the corotational (CR) formulation of finite elements in geometrically nonlinear structural analysis. The key assumptions behind CR are: (i) strains from a corotated configuration are small while (ii) the magnitude of rotations from a base configuration is not restricted. Following a historical outline the basic steps of the Element Independent CR formulation are presented. The element internal force and consistent tangent stiffness matrix are derived by taking variations of the internal energy with respect to nodal freedoms. It is shown that this framework permits the derivation of a set of CR variants through selective simplifications. This set includes some previously used by other investigators. The different variants are compared with respect to a set of desirable qualities, including self-equilibrium in the deformed configuration, tangent stiffness consistency, invariance, symmetrizability, and element independence. We discuss the main benefits of the CR formulation as well as its modeling limitations.

Finite Elements in Analysis and Design, 2006

This paper presents a new geometric nonlinear formulation for static problems involving space trusses. Based on the finite element method (FEM), the proposed formulation uses nodal positions rather than nodal displacements to describe the problem. The strain is determined directly from the proposed position concept, using a Cartesian coordinate system fixed in space. Bilinear constitutive hardening relations are considered here to model the elastoplastic effects, but any other constitutive model can be used. The proposed formulation is simple and yields good results, as shown in the example section. Four examples are presented here to validate the formulation. ᭧

Shock and Vibration

The aim of this study is to compose a corotational finite element formulation for space frames with geometrically nonlinear behavior under dynamic loads. Using a moving frame through three successive rotations similar to Euler angles is one of the oldest techniques; however, there are still some gaps that require attention, mainly due to singularity. Hence, alternative techniques had been developed, sometimes elusive and computationally expensive. In this paper, we went back to the old technique and filled the gaps. Three-coordinate systems are used, i.e., the fixed global coordinate system, the fixed local coordinate system that is attached individually to every element, and the corotational local frame for each element that moves and rotates with the element. The deformation is always small relative to the corotational frame. The successive rotations between different coordinate systems are expressed using Tait–Bryan angles. Lagrange’s equation is used to derive the equation of mo...

SUN MeDIA eBooks, 2009

in Germany. The authors performed coordinated and cooperative research on nonlinear structural analysis and on computer-supported civil engineering over a period of several years. Many of the innovative aspects of this book were invented and developed in the course of the research effort. It is hoped that the results illustrate the potential of global interaction in university research. The increasing size of the world population and the necessity to raise the standard of living in developing areas place new demands on the civil engineering profession. Since the available amount of material, energy and labour is limited, their use must be optimised. One of the strategies to achieve this goal is to reduce the amount of material that is required for structural systems. This strategy reduces the mass of the buildings as well as the energy that is required for their construction and for the production of building materials. 5 Monographs on structural analysis with finite elements usually contain chapters on the theory of elasticity, the structural mechanics of specific types of structural components and the matrix formulation of different kinds of finite elements. In addition, there are chapters on the constitutive laws of materials and the methods of numerical analysis of nonlinear governing equations. Special chapters may be devoted to structural stability, contact problems and nonlinear dynamic analysis. Of particular interest is a publication in the Journal of Aeronautical Sciences by Turner, Clough, Martin and Topp [63] on "Stiffness and deflection analysis of complex structures" which dates back to 1956. This publication sparked a broad and well-financed research programme on the finite element method in the United States of America, which was strongly supported by both government and the aerospace industry. While this research was not dedicated specifically to nonlinear structural analysis, it did lead to significant advances in the area, which were summarised by Oden [42] in his monograph Finite elements for nonlinear continua in 1972. Research on nonlinear structural analysis in Russia started in the 1940s. The first classification of nonlinear problems was presented by V. Novozhilov [40]. The nonlinear theory of elasticity was further developed in the fundamental work of I. Goldenblatt [17] and A. Lurie [36]. A. Ilyushin [22, 23] developed the theory of plasticity for small deformations. Physically nonlinear problems and related questions were first investigated in the works of G. Geniev [16] and A.A. Tolokonnikov [60]. Research in geometrically nonlinear structural behaviour was mostly concentrated on geometrically nonlinear problems of plates and shells. Best known are the works of V. Vlasov [64], A. Volmir [65], Kh. Mushtari and K. Galimov [39], V. Feodosiev [14], M. Kornishin [28], P.M. Ogibalov and M. Koltunov [43] as well as V. Petrov [47, 48]. Nonlinear behaviour of rod systems was investigated in the work of A. Gemmerling [15] and P. Lukash [35]. D. Davidenko [12, 13] introduced the parameter variation method for solving nonlinear equations that is now known as the parameter continuation method. V. Vlasov and V. Petrov [47] proposed a load step method that was applied to the analysis of plates and shells in the subsequent works of V. Petrov and his progeny [48]. This method has been used intensively for solving a wide range of problems in nonlinear mechanics.

2013

ario/Natal-RN/Brazil Abstract. This paper presents the positional nonlinear geometric formulation for plane truss prob- lems in static analysis. The positional formulation presents an alternative approach for non linear problems, since it considers nodal positions as variables of the nonlinear system instead of dis- placements (widely found in literature). The strain determination is done directly from the proposed positional concept. The initial configuration is assumed as the basis of calculation, i.e., Hooke's law relates reference stress and engineering strain. Some simple numerical examples are shown proving the accuracy of the proposed formulation.

Within the large strains regime, manifold theory is used in order to define the truss kinematics in the most fundamental way. The complete deformation gradient of the truss is revealed and consequently, Hill's large strain measures are computed. It is shown that for an isotropic, hyperelastic truss, stress - strain conjugacy depends on the reckoned truss volume. If conjugacy is defined in the current truss volume, then even for the simplest linear hyperelastic constitutive model, the truss's structural behavior depends on its cross sectional kinematics. This shows that the large strain hyperelastic truss is at most a semi - one - dimensional problem. Thus, it is seen that a general formulation of the large strains hyperelastic truss requires precise definitions of its kinematics, kinetics, constraints, constitutive laws and notions of conjugacy and description. These issues are comprehensively dealt with in the present paper.