Showing papers on "Tangent stiffness matrix published in 1997"

Affine connections for the Cartesian stiffness matrix

Abstract: We study the 6/spl times/6 Cartesian stiffness matrix. We show that the stiffness of a rigid body subjected to conservative forces and moments is described by a (0,2) tensor which is the Hessian of the potential function. The key observation of the paper is that since the Hessian depends on the choice of an affine connection in the task space, so will the Cartesian stiffness matrix. Further, the symmetry of the Hessian and thus of the stiffness matrix depends on the symmetry of the connection. The connection that is implicit in the definition of the Cartesian stiffness matrix through the joint stiffness matrix (Salisbury, 1980) is made explicit and shown to be symmetric. In contrast, the direct definition of the Cartesian stiffness matrix in Griffis (1993), Ciblak and Lipkin (1994) and Howard et al. (1996) is shown to be derived from an asymmetric connection. A numerical example is provided to illustrate the main ideas of the paper.

Effective Elastoplastic Algorithms for Ductile Matrix Composites

Abstract: A three-dimensional computational algorithm is proposed to implement the micromechanical framework derived by the writers. The proposed formulation is capable of predicting effective elastoplastic behavior of two-phase particle reinforced ductile matrix composites (PRDMCs) containing may randomly dispersed interacting elastic spheres. A strain-driven algorithm is presented to determine the stress for a given strain by using the two-step operator splitting methodology applicable to any arbitrary loading and unloading histories. Furthermore, continuum and consistent tangent moduli are derived in closed form to facilitate the assembly of the global tangent stiffness. Extension is also made to predict effective rate-dependent elasto-viscoplastic behavior of particle-reinforced ductile matrix composites. Finally, finite-element implementation and numerical examples are presented.

The role of softening in the numerical analysis of R.C. framed structures

Abstract: Reinforced Concrete beams with tension and compression softening material constitutive laws are studied. Energy-based and non-local regularisation techniques are presented and applied to a R.C. element. The element characteristics (sectional tangent stiffness matrix, element tangent stiffness matrix restoring forces) are directly derived from their symbolic expressions through numerical integration. In this way the same spatial grid allows us to obtain a non-local strain estimate and also to sample the contributions to the element stiffness matrix. Three examples show the spurious behaviors due to the strain localization and the stabilization effects given by the regularisation techniques, both in the case of tension and compression softening. The possibility to overestimate the ultimate load level when the non-local strain measure is applied to a non softening material is shown.

Finite element method for thermomechanical response of near-incompressible elastomers

Abstract: The present study addresses finite element analysis of the coupled thermomechanical response of near-incompressible elastomers such as natural rubber. Of interest are applications such as seals, which often involve large deformations, nonlinear material behavior, confinement, and thermal gradients. Most published finite element analyses of elastomeric components have been limited to isothermal conditions. A basic quantity appearing in the finite element equation for elastomers is thetangent stiffness matrix. A compact expression for theisothermal tangent stiffness matrix has recently been reported by the first author, including compressible, incompressible, and near-incompressible elastomers. In the present study a compact expression is reported for the tangent stiffness matrix under coupled thermal and mechanical behavior, including pressure interpolation to accommodate near-incompressibility. The matrix is seen to have a computationally convenient structure and to serve as a Jacobian matrix in a Newton iteration scheme. The formulation makes use of a thermoelastic constitutive model recently introduced by the authors for near-incompressible elastomers. The resulting relations are illustrated using a near-incompressible thermohyperelastic counterpart of the conventional Mooney-Rivlin model. As an application, an element is formulated to model the response of a rubber rod subjected to force and heat.

Automated solution procedures for negotiating abrupt non‐linearities and branch points

Abstract: In the inelastic stability analysis of plated structures, incremental‐iterative finite element methods sometimes encounter prohibitive solution difficulties in the vicinity of sharp limit points, branch points and other regions of abrupt non‐linearity. Presents an analysis system that attempts to trace the non‐linear response associated with these types of problems at minor computational cost. Proposes a semi‐heuristic method for automatic load incrementation, termed the adaptive arc‐length procedure. This procedure is capable of detecting abrupt non‐linearities and reducing the increment size prior to encountering iterative convergence difficulties. The adaptive arc‐length method is also capable of increasing the increment size rapidly in regions of near linear response. This strategy, combined with consistent linearization to obtain the updated tangent stiffness matrix in all iterative steps, and with the use of a “minimum residual displacement” constraint on the iterations, is found to be effective in avoiding solution difficulties in many types of severe non‐linear problems. However, additional procedures are necessary to negotiate branch points within the solution path, as well as to ameliorate convergence difficulties in certain situations. Presents a special algorithm, termed the bifurcation processor, which is effective for solving many of these types of problems. Discusses several example solutions to illustrate the performance of the resulting analysis system.

Postbifurcation equilibrium paths modified by nonlinear configuration-dependent conservative loading using nonsmooth analysis

Abstract: This paper discusses the effect of deformation-sensitive loading devices because the nature of loading is generally not perfectly dead, being independent of the deflections that occur. This paper presents the effect of nonlinear variable load. Postbifurcation equilibrium paths and structural tangent stiffness are modified on the basis of a polygonal approximation and nonsmooth analysis. The effects of dead and variable loads are compared. Configuration-dependent loading devices can be characterized by some load-deflection functions, much like the nature of material behavior can be characterized by stress-strain functions. The effect of a deformation-sensitive load is similar to that of the material. Consequently, in the stability analysis of structures, a configuration-dependent loading program can be handled like material behavior. Thus, in the tangent stiffness of the structure, much like the tangent modulus of the material, the tangent modulus of the load appears. Previous research has shown t...

The Stiffness Matrix

Abstract: This introduction to stiffness matrices will show how to derive the appropriate matrix connecting ‘force’ to ‘displacement’ for (i) a bar under uniaxial stressing, (ii) a beam in bending and (iii) a shaft under torsion, using the work of previous chapters. The approach used is similar to the finite element method (FEM) where we shall refer to the uniform bar as an element and to its ends as the nodes. Using the stiffness or displacement method of analysis the displacements at the ends of the bar are the unknowns. These nodal-point displacements can be found from their relation to the nodal forces. The stiffness matrix connects nodal forces to displacements and has a unique form depending upon the number of degrees of freedom for the element in question. Once the displacements are known, the strains follow from the strain-displacement relations and, finally, the stresses are found from Hooke’s law.

Synthesis of Multi-Body Systems for Desired Eigenfrequencies

Abstract: The eigenvalues in the modal co-ordinate frame are varied and the corresponding changes in the stiffness matrix is investigated. To represent the modified eigenvalues as a function of the stiffness matrix is the main focus of this paper. It is shown that the change in eigenvalue is proportional to the change in the stiffness matrix. This approach may be applied to shift certain natural frequencies of a structure away from the critical operating frequency by structural modification.Copyright © 1997 by ASME

Non-linear analysis with an axisymmetric thick shell element

Abstract: The paper highlights the results of numerical experimentation with a geometrically non-linear formulation of a thick axisymmetric shell element involving non-axisymmetric deformation. The formulation itself has earlier been presented in the context of a non-linear local–global analysis of shells of revolution, where very little attention was paid to an independent evaluation of the element's performance. A Fourier decomposition of the loads and the displacements has been used in the circumferential co-ordinate, in order to describe the non-axisymmetric behavior. Due to the interaction between different harmonic terms in the non-linear analysis, the tangential stiffness matrix is no longer block-diagonal. A pseudoload method and a conjugate gradient like iterative scheme have been used to overcome the problem of a large tangent stiffness matrix, and thus most of the advantages of the semi-analytical method have been retained in the non-linear analysis. The accuracy of the predictions in the study has been benchmarked by analysing the same examples using the quadrilateral shear deformable shell element available in the commercial code NISA II. A comparison with other results available in the literature hints that the effect of transverse shear deformation should not be neglected in the geometrically non-linear analysis of shells which are traditionally considered thin. © 1997 by John Wiley & Sons, Ltd.

Tangent Stiffness Equations for Laterally Distributed Loaded Member

Abstract: Synopsis Tangent stiffness equations for a beam-column which is subjected to either uniformly or sinusoidally distributed lateral load are presented. The equations have been derived by differentiating the slope-deflection equations under axial forces for a member. Then, the tangent stiffness equations take into account axial forces, a bowing effect and laterally distributed loads. Elastic buckling behavior of parallel chord latticed beams with laterally distributed loads is investigated, to compare the results of the present method with a conventional method in which the distributed loads are considered as concentrated loads at additional nodes of a member. Furthermore, buckling tests were carried out to confirm the derived equations and to make clear the buckling behavior of space frame structures. As a result, the new equations can lead to a good efficiency of estimating equilibrium paths and a significant savings in the core storage and computing time required for the analysis of space frame structures.

特殊直交群so(3)を含んだ有限要素法定式化 : 第2報, 梁要素の剛性要素の導出

Abstract: In the previous paper, the relationship between the stress rate and the tangent stiffness was derived using the special orthogonal group SO (3) in the finite element formulation. The stress rates used are the Truesdell stress rate, the Jaumann stress rate, the Neo-Green stress rate and the Ishihara stress rate. In this work, the stiffness elements of the tangent stiffnesses of beam elements with SO (3) are given. These elements include material stiffness ΔδПm, geometric stiffness of rigid rotation ΔδП10, geometric stiffness of stretch to stress direction ΔδПsg, stress, geometric stiffness of stretch perpendicular to area ΔδПsg, area and geometric stiffness of stretch to deformation rate ΔδПsg, deform. In a future paper, the tangent stiffness of these stress rates of a beam element will be given.