Miran Saje

A kinematically exact space finite strain beam model - finite element formulation by generalized virtual work principle

The present paper presents a novel finite element formulation for static analysis of linear elastic spatial frame structures extending the formulation given by Simo and Vu-Quoc [A geometrically-exact rod model incorporating shear and torsion-warping deformation, Int. J. Solids Structures 27 (3) (1991) 371-3931, along the lines of the work on the planar beam theory presented by Saje [A variational principle for finite planar deformation of straight slender elastic beams, Internat. J. Solids and Structures 26 (1990) 887-9001. We apply exact non-linear kinematic relationships of the space finite-strain beam theory, assuming the Bernoulli hypothesis and neglecting the warping deformations of the cross-section. Finite displacements and rotations as well as finite extensional, shear, torsional and bending strains are accounted for in the formulation. A deformed configuration of the beam is described by the displacement vector of the deformed centroid axis and an orthonormal moving frame, rigidly attached to the cross-section of the beam. The position of the moving frame relative to a fixed reference frame is specified by an orthogonal matrix, parametrized by the rotational vector which rotates the moving frame from an arbitrary position into the deformed configuration in one step. Also, the incremental rotational vector is introduced, which rotates the moving frame from the configuration obtained at the previous iteration step into the current configuration of the beam. Its components relative to the fixed global coordinate system are taken to be the rotational degrees of freedom at nodal points. Because in 3-D space both the axial and the follower moments are non-conservative, not the variational principle but the principle of virtual work has been introduced as a basis for the finite element discretization. Here we have proposed the generalized form of the principle of virtual work by including exact kinematic equations by means of a procedure, similar to that of Lagrangian multipliers. This makes possible the elimination of the displacement vector field from the principle, so that the three components of the incremental rotational vector field remain the only functions to be approximated in the finite element implementation of the principle. Other researchers, on the other hand, employ the three components of the incremental rotational vector field and the three components of the incremental displacement vector field. As a result, more accurate and efficient family of beam finite elements for the non-linear analysis of space frames has been obtained. A one-field formulation results in the fact that in the present finite elements the locking never occurs. Any combination of deformation states is described equally precisely. This is in contrast with the elements developed in literature, where, in order to avoid the locking, a reduced numerical integration has to be applied, which unfortunately, diminishes the accuracy of the solution. Polynomials have been chosen for the approximation of the components of the rotational vector. In this case the order of the numerical integration can rationally be estimated and the computer program can be coded in such a way that the degree of polynomials need not be limited to a particular value. The Newton method is used for the iterative solution of the non-linear equilibrium equations. In an non-equilibrium configuration, the tangent stiffness matrix, obtained by the linearization of governing equations using the directional derivative, is non-symmetric even for conservative loadings. Only upon achieving an equilibrium state, the tangent stiffness matrix becomes symmetric. Thus, obtained tangent stiffness matrix can be symmetrized without affecting the rate of convergence of the Newton method. For non-conservative loadings, however, the tangent stiffness matrix is always non-symmetric. The numerical examples demonstrate capability of the present formulation to determine accurately the non-linear behaviour of space frames. In numerical examples the out-of-plane buckling loads are determined and the whole pre-and post-critical load-displacement paths of a cantilever and a right-angle frame are traced. These, in the analysis of space beams standard verification example problems, show excellent accuracy of the solution even when employing only one element to describe the displacements of the size of the structure itself, the rotations of 2n, and extensional strains much beyond the realistic values of linear elastic material.

A quadratically convergent algorithm for the computation of stability points: The application of the determinant of the tangent stiffness matrix

Abstract This paper presents a quadratically convergent algorithm for the computation of stability points (limit and bifurcation points) in the finite element formulation of the problems of the nonlinear structural mechanics. One such approach was given by Wriggers and coworkers [Comput. Methods Appl. Mech. Engrg. 70 (1988) 329–347 and Int. J. Numer. Methods Engrg. 30 (1990) 155–176]. Their approach employs the eigenvector equation K T ψ = 0 as the characterization of the stability point. In the present paper, an alternative approach is proposed, which uses the equation det K T = 0 as the stability point condition. In combination with Newtons iteration scheme, this condition has so far been considered unsuitable for the implementation in the finite element analysis because it has been believed that it requires the full assembly of the derivative of the global tangent stiffness matrix, which is a very costly operation. Objectives of our paper are: (i) to derive a new quadratically convergent algorithm for the computation of stability points, which uses the condition det K T = 0 for the characterization of the stability point; (ii) to show that the full assembly of the global tangent stiffness matrix derivative is not required, and that the linearization of the determinant of the global tangent stiffness matrix can be done in an element-by-element fashion; and (iii) to indicate that, in terms of the number of algebraic operations, memory requirements and the computer programming effort, our algorithm practically equals the one based on an eigenvector equation, but has the advantage of a larger radius of convergence. An additional benefit of the proposed algorithm is that it can also be used without any modification as a comprehensive path-following procedure to calculate regular and singular (stability) points. The effectiveness of our algorithm is illustrated by numerical examples. Its convergence characteristics are compared with those of the algorithm based on the eigenvector equation. We show stability analyses of elastic-plastic planar frames. The algorithm is, of course, valid generally and is not limited to this particular kind of structures.

Finite element formulation of finite planar deformation of curved elastic beams

Abstract The objective of this paper is to present a finite element formulation of finite deformation analysis of an arbitrarily curved, extensible, shear-flexible, elastic planar beam. The formulation is based on a modified Hu-Washizu variational principle in which exact non-linear kinematic equations of Reissner [J. appl. Math. Phys. (ZAMP)23, 795–804 (1972)] are taken into account. In such a case a new variational principle can be derived which is expressed in terms of only one function, the rotation of the cross-section of the beam. Thus only the rotation in the interior of an element needs to be approximated in its finite element implementation. The Euler-Lagrange equations of this principle are, among others, exact kinematic and equilibrium conditions for the beam. The solution capabilities are illustrated with numerical examples. Several finite elements of different order are examined. Excellent convergence of the solution of non-linear equilibrium equations by the Newton method and high accuracy of the solution for all elements considered is demonstrated. The results indicate that the accuracy of the present elements is not notably influenced by the length of the element and the order of numerical integration. Relatively large load increments are allowed. In some cases the results are insensitive to the number of load steps. These finite elements do not exhibit any kind of locking and describe equally precise extensible and inextensible beams as well as shear-stiff or shear-flexible ones. A beam subjected to a variety of loads and extremely deformed, may be modeled with only one element, but still with a very high precision.

A consistent equilibrium in a cross-section of an elastic-plastic beam

Abstract A phenomenon of inequality of equilibrium and constitutive internal forces in a cross-section of elastic–plastic beams is common to many finite element formulations. It is here discussed in a rate-independent, elastic–plastic beam context, and a possible treatment is presented. The starting point of our discussion is Reissners finite-strain beam theory, and its finite element implementation. The questions of the consistency of interpolations for displacements and rotations, and the related locking phenomena are fully avoided by considering the rotation function of the centroid axis of a beam as the only unknown function of the problem. Approximate equilibrium equations are derived by the use of the distribution theory in conjunction with the collocation method. The novelty of our formulation is an inclusion of a balance function that measures the error between the equilibrium and constitutive bending moments in a cross-section. An advantage of the present approach is that the locations, where the balance of equilibrium and constitutive moments should be satisfied, can be prescribed in advance. In order to minimize the error, explicit analytical expressions are used for the constitutive forces; for a rectangular cross-section and bilinear constitutive law, they are given in Appendix A . The comparison between the results of the two finite element formulations, the one using consistent, and the other inconsistent equilibrium in a cross-section, is shown for a cantilever beam subjected to a point load. The problem of high curvature gradients in a plastified region is also discussed and solved by using an adapted collocation method, in which the coordinate system is transformed such to follow high gradients of curvature.

Buckling of layered wood columns

Slip between layers, material properties of layers and geometric non-linearities largely dictate both the bearing capacity and the ductility of a composite beam. That is why the accuracy of their modelling in the numerical analysis of the composite beams is of utmost importance. In the present paper we present a new strain-based finite element model which considers these issues in a highly effective manner. Each layer of the beam is modelled by geometrically exact Reissners beam model. The layers are assumed to stay in the contact during deformation but the relative tangential displacement (slip) is possible. The non-linear load-slip law of the interface is considered. The formulation is found to be accurate, reliable and computationally time-effective. The further objective of the paper is an analysis of the buckling force of axially compressed layered wood columns, being simply supported, fixed-pinned or continuous. We compare the present numerical results with the analytical values of [Girhammar UA, Gopu VKA. Composite beam-columns with interlayer slip-exact analysis, J Struct Engng ASCE 1993;199(4):1265-82] and with the values, recommended by the European code for timber structures [Eurocode 5, Design of timber structures, Part 1-1: General rules and rules for buildings, 1993; ENV 1995-1-1]. The comparisons indicate that the European code for timber structures provides very conservative estimates for the buckling load.

A kinematically exact finite element formulation of elastic-plastic curved beams

Abstract A finite element, large displacement formulation of static elastic–plastic analysis of slender arbitrarily curved planar beams is presented. Non-conservative and dynamic loads are at present not included. The Bernoulli hypothesis of plane cross-sections is assumed and the effect of shear strains is neglected. Exact non-linear kinematic equations of curved beams, derived by Reissner are incorporated into a generalized principle of virtual work through Lagrangian multipliers. The only function that has to be interpolated in the finite element implementation is the rotation of the centroid axis of a beam. This is an important advantage over other classical displacement approaches since the field consistency problem and related locking phenomena do not arise. Numerical examples, comprising elastic and elastic–plastic, curved and straight beams, at large displacements and rotations, show very nice computational and accuracy characteristics of the present family of finite elements. The comparisons with other published results very clearly show the superior performance of the present elements.

Necking of a cylindrical bar in tension

Abstract The paper numerically analyses the necking of a cylindrical, elastic-plastic, strain-hardening bar in axisymmetric tension. Both geometric and material nonlinearities are taken into account. The Lagrangian formulation is used. The solution is obtained by a stepwise integration of the equilibrium equations, using the finite difference method. The results also include stress and strain distributions across the minimum bar section at various deformation stages up to 67% reduction of the bars area. Comparisons with the results of previous investigators (Bridgman, Chen, Needleman, Norris) are also given.

The three-dimensional beam theory: Finite element formulation based on curvature

Abstract The article introduces a new finite element formulation of the three-dimensional ‘geometrically exact finite-strain beam theory’. The formulation employs the generalized virtual work principle with the pseudo-curvature vector as the only unknown function. The solution of the governing equations is obtained by using a combined Galerkin-collocation algorithm. The collocation ensures that the equilibrium and the constitutive internal force and moment vectors are equal at a set of chosen discrete points. In Newton’s iteration special update procedures for the pseudo-curvature and rotational vectors have to be employed because of the non-linearity of the configuration space. The accuracy and the efficiency of the derived numerical algorithm are demonstrated by several examples.

Finite element formulation of hyperelastic plane frames subjected to nonconservative loads

Abstract A finite element formulation for finite deformation static analysis of plane hyperelastic frames subjected to nonconservative loads is presented. A rubber-like material is considered for which the behaviour in tension and in compression differs substantially. A new proposal for the strain energy function of rubber at uniaxial stress state is given, convenient for the present deformation analysis. The finite element formulation is based on a new variational principle of the Hu-Washizu type where exact nonlinear kinematic equations of one-dimensional finite-strain beam theory are taken into account. The contribution of the shear deformations to the total potential energy and the initial curvature of the beam are neglected. The functional of the variational principle is expressed in terms of only one function, the rotation of the cross-section of the beam. Thus only one function needs to be approximated in the functional in the finite element implementation of the variational principle. The outstanding accuracy and high efficiency of the method are illustrated by numerical examples. The application of the present method for the analysis of hyperelastic frames subjected to static nonconservative forces is shown, and some results for critical loads for the dynamic instabilities in the form of flutter are given.

Integrating rotation from angular velocity

The integration of the rotation from a given angular velocity is often required in practice. The present paper explores how the choice of the parametrization of rotation, when employed in conjuction with different numerical time-integration schemes, effects the accuracy and the computational efficiency. Three rotation parametrizations - the rotational vector, the Argyris tangential vector and the rotational quaternion - are combined with three different numerical time-integration schemes, including classical explicit Runge-Kutta method and the novel midpoint rule proposed here. The key result of the study is the assessment of the integration errors of various parametrization-integration method combinations. In order to assess the errors, we choose a time-dependent function corresponding to a rotational vector, and derive the related exact time-dependent angular velocity. This is then employed in the numerical solution as the data. The resulting numerically integrated approximate rotations are compared with the analytical solution. A novel global solution error norm for discrete solutions given by a set of values at chosen time-points is employed. Several characteristic angular velocity functions, resulting in small, finite and fast oscillating rotations are studied.