Showing papers on "Tangent stiffness matrix published in 1992"

The (symmetric) Hessian for geometrically nonlinear models in solid mechanics: intrinsic definition and geometric interpretation

Abstract: The tangent stiffness matrix (i.e., the Hessian) for nonlinear structural models in computational solid mechanics is typically computed by linearization of the weak form of the equilibrium equations via the directional derivative formula. Depending on the specific mechanical model, away from equilibrium this procedure will in general yield a nonsymmetric tangent stiffness matrix. By contrast, if the directional (Gateaux) derivative is replaced by the covariant derivative (relative to a certain Riemannian metric) an intrinsic definition of the Hessian is obtained which is always symmetric away from equilibrium. It is shown that by endowing the rotation group with the standard bi-invariant metric, the resulting (symmetric) Hessian is simply the symmetrization of the usual expression obtained via the Gateaux derivative. In a finite element context, this property provides a rigorous justification for the common practice of symmetrizing the ‘seemingly nonsymmetric tangent’ away from equilibrium. This apparently ad hoc symmetrization procedure yields, in fact, the actual Hessian.

A consistent co‐rotational formulation for shells using the constant stress/constant moment triangle

Abstract: The facet-shell formulation involves the combination of the constant-strain membrane triangle with a constant-curvature bending triangle. The paper describes a technique whereby this facet-formulation is extended to handle geometric non-linearity by means of a co-rotational procedure. Emphasis is placed on the derivation of a technique that is increment-independent with both the internal force vector and tangent stiffness matrix being derived from the «total strain measures» in a «consistent manner».

Compression Failure of Quasibrittle Material: Nonlocal Microplane Model

Abstract: The previously presented constitutive model of microplane type for nonlinear triaxial behavior and fracture of concrete is used in nonlocal finite element analysis of compression failure in plane strain rectangular specimens. For specimens with sliding rigid platens there is a bifurcation of the loading path at the beginning of postpeak softening; a symmetric (primary) path exists but the actual (stable) path is the nonsymmetric (secondary) path, involving an inclined shear‐expansion band that consists of axial splitting cracks and is characterized by transverse expansion. The secondary path is indicated by the first eigenvalue of the tangent stiffness matrix but can be more easily obtained if a slight nonsymmetry is introduced into the finite element model. In specimens with bonded rigid platens there is no bifurcation; they fail symmetrically, by two inclined shear‐expansion bands that consist of axial splitting cracks. The transverse expansion produces transverse tension in the adjacent material, which...

Finite-Element Solution

Abstract: Heat transfer problems can be formulated either by a given differential equation with boundary conditions or by a given functional equivalent to the differential equation (Kleiber and Shuzalec 1983a, b, 1984; Stuzalec 1987a, b, 1988a). The weighted residual method is an approximate solution method for differential equations. This method widens the range of problems amenable to solution since it does not require a variational formulation of the problem.

An Efficient Formulation for a Shell Element Considering Finite Rotation Increments and its Assessmest.

Abstract: An efficient formulation for a 4-node shell element which includes the effect of large rotation increments is presented. The formulation is based on the MITC element proposed by Bathe et al. A complete and symmetric tangent stiffness matrix is obtained by considering up to the quadratic term of Taylor expansion of a finite rotation tensor. Several numerical examples are demonstrated to show the superior convergence by the present formulation compared with the conventional MITC formulation which assumes infinitesimal rotation increments. It is also shown in sensitivity analysis that accurate gradients are always obtained by the complete tangent stiffness, although erroneous gradients can be obtained by the conventional ones.

Nonlinear Postbuckling Analysis of Plates and Shells by Four-Noded Strain Element

Abstract: This paper deals with the nonlinear postbuckling analysis of plates and shells by the finite element method. The four-noded C° strain element for nonlinear plate anaysis developed by the authors is first extended to the nonlinear shell analysis and then used for the nonlinear postbuckling analysis of plates and shallow shells. The element tangent stiffness matrix presented here is given explicitly, i.e., without any numerical integration. Consequently, this C° strain element is much more computationally economical and efficient than the widely used elements obtained from numerical integration. The efficiency and accuracy of the present four-noded strain element for postbuckling analysis are demonstrated by five numerical examples of various plates and shells.

A consistent formulation of a dilatant interface element

Abstract: The paper considers a plane joint or interface element suitable for implementation into a standard non-linear finite element code. The element is intended to model discontinuities with rough contact surfaces, such as rock joints, where dilatant behaviour is present. Of particular concern is the formulation of a constitutive model which fully caters for all possible histories of opening, closing and sliding (accompained by dilation or contraction) in any direction. The non-linear incremental constitutive equations are formulated in a manner appropriate for a back-ward difference discretization in time along the path of loading. The advantage of such an approach is that no essential distinction need be drawn between opening, closing and sliding. Further, a convenient formulation of the constitutive equations is facilitated by representing the different contact conditions in relative displacement space. The state diagram in relative displacement space, however, changes from one time step to the next, and evolution equations for the updating must be formulated. These concepts are illustrated for two rock-joint models: a sawtooth asperity model and a limited dilation model. The models are based on a penalty formulation to enforce the contact constraints, and explicit equations for the tangent stiffness matrix and for the corrector step of the standard Newton–Raphson iterative algorithm are derived. These equations have been implemented as an user element into the finite element code ABAQUS7. Three examples are presented to illustrate the predictions of the formulation.

Force deformation equations for initially curved laterally loaded beam columns

Abstract: The analytical inclusion of the effects of initial curvature and lateral load in the force deformation equations of a two-noded beam-column element is considered. Current position-(Eulerian) based equations are developed that give both the end-shortening and nodal forces for a deformed element under the action of a significant axial compression. Consideration is restricted to an initial curvature in the form of a single sinusoidal half-wave, and to a sinusoidal approximation of a symmetric triangular lateral load. The equations developed are compatible with the existing Eulerian force deformation equations presented by Oran for straight axially loaded beam columns. A tangent stiffness matrix compatible with the new force deformation equations is not presented. The correctness of the new formulation is demonstrated by various analytical comparisons with the results of some standard beam-column problems, and by a number of numerical comparisons between results obtained using the new formulation and those predicted by conventional nonlinear analyses, which employ a number of standard beam-column elements to model each structural member. The use of the new equations can lead to a significant savings in the core storage required for the analysis of full-size lattice domes.

Stiffness Expressions for Element with Central and End Springs

Abstract: The force‐deformation equations of a two‐noded three‐dimensional (3‐D) beam‐column element with both central and end rotational springs, which take account of the effects of the axial load on the lateral element stiffness, are derived Current position‐based equations are developed that give both the nodal forces and end shortening for a deformed element under the action of a significant axial force The tangent stiffness matrix for this element is derived from the force‐deformation relations by following the procedures adopted by Oran for a two‐noded beam‐column prismatic element Where appropriate, a successful analytical comparison with either Oran's or Chilton's expressions is made for the cases of no axial force, infinite end joint stiffness, and finite/infinite central joint stiffness The stiffness expressions of this element can be used to represent several commonly used members in real structures by employing the appropriate stiffnesses of the central and end springs The use of the new equations

Large Rotations in Structural Mechanics — Overview

Abstract: Some basic aspects on the handling of finite rotations in structural mechanics are presented in this overview.

Modified cyclic micromechanical model for concrete

Abstract: A detailed cyclic micromechanical model based on the micromechanics of granular material is proposed for concrete. In the current study, concrete is idealized to have two kinds of contacts; aggregate-aggregate and aggregate- mortar contacts. The behavior of these contacts is examined and distinguished for both cyclic and virgin loadings. Finally, an explicit formula which expresses the tangent stiffness matrix of the material as a summation of the contributions of all contacts inside any representative volume is derived. Moreover, the nonhomogeniety of the microstructure and the nonuniformity of strain distribution are considered. This is in contrast to Bazant's microplane model in which concrete has been treated as a homogenous brittle aggregate material having a single kind of contact with uniform strain distribution.