Showing papers on "Tangent stiffness matrix published in 1990"

A study of incremental-iterative strategies for non-linear analyses

Abstract: The paper describes a study of incremental-iterative solution techniques for geometrically non-linear analyses. The solution methods documented are based on a modified Newton-Raphson approach, meaning that the tangent stiffness matrix is computed at the commencement of each load step but is then held constant throughout the equilibrium iterations. A consistent mathematical notation is employed in the description of the iterative and load incrementation strategies, enabling the simple inclusion of several solution options in a computer program. The iterative strategies investigated are iteration at constant load, iteration at constant displacement, iteration at constant ‘arc-length’, iteration at constant external work, iteration at minimum unbalanced displacement norm, iteration at minimum unbalanced force norm and iteration at constant ‘weighted response’. The load incrementation schemes investigated include strategies based on the number of iterations required to achieve convergence in the previous load step, strategies based on the ‘current stiffness parameter’ and a strategy based on a parabolic approximation to the load-deflection response. Criteria for detecting when the applied external load increment should reverse sign are described. A challenging example of a circular arch exhibiting snap-through (load limit point) behaviour and snap-back (displacement limit point) behaviour is solved using several different iterative and load incrementation strategies. The performance of the solution schemes is evaluated and conclusions are drawn.

Consistent Tangent Moduli for a Class of Viscoplasticity

Abstract: Consistent (algorithmic) tangent moduli for the generalized Duvaut-Lions viscoplasticity model are derived in this work. The derivations are based on consistent linearization of the residual functions associated with two alternative unconditionally stable constitutive integration algorithms; namely, the implicit backward Euler and the “full integration” algorithms. This “consistent linearization” procedure is equally applicable to the Perzyna-type viscoplasticity formulations. In particular, the von Mises isotropic/kinematic hardening viscoplasticity model is chosen as a model problem for demonstration. Consistent viscoplastic tangent moduli for other choices of (single or multiple) loading surfaces can be derived in a similar fashion provided that consistent elastoplastic (inviscid) tangent moduli are available. It is noted that since continuum tangent moduli do not exist at all for viscoplasticity, use of the proposed consistent tangent modul is not only desirable but necessary in the Newton-type finite-element computations. In addition, due to the difference in the two constitutive integration algorithms used, the corresponding consistent tangent moduli are not the same even when time steps are small. Numerical examples are also presented to illustrate the remarkable quadratic performance of the proposed consistent tangent moduli for the generalized Duvaut-Lions viscoplasticity model.

A full tangent stiffness field-boundary-element formulation for geometric and material non-linear problems of solid mechanics

Abstract: The field-boundary-element method naturally admits the solution algorithm in the incompressible regimes of fully developed plastic flow. This is not the case with the generally popular finite-element method, without further modifications to the method such as reduced integration or a mixed method for treating the dilatational deformation. The analyses by the field-boundary-element method for geometric and material non-linear problems are generally carried out by an incremental algorithm, where the velocities (or displacement increments) on the boundary are treated as the primary variables and an initial strain iteration method is commonly used to obtain the state of equilibrium. For problems such as buckling and diffused tensile necking, involving very large strains, such a solution scheme may not be able to capture the bifurcation phenomena, or the convergence will be unacceptably slow when the post-bifurcation behaviour needs to be analysed. To avoid this predicament, a full tangent stiffness field-boundary-element formulation which takes the initial stress–velocity gradient (displacement gradient) coupling terms accurately into account is presented in this paper. Here, the velocity field both inside and on the boundary are treated as primary variables. The large strain plasticity constitutive equation employed is based on an endochronic model of combined isotropic/kinematic hardening plasticity using the concepts of material director triad and the associated plastic spin. A generalized mid-point radial return algorithm is presented for determining the objective increments of stress from the computed velocity gradients. Numerical results are presented for problems of diffuse necking, involving very large strains and plastic instability, in initially perfect elastic–plastic plates under tension. These results demonstrate the clear superiority of the full tangent stiffness algorithm over the initial strain algorithm, in the context of the integral equation formulations for large strain plasticity.

Non‐linear earthquake‐response analysis of long‐span cable‐stayed bridges: Applications

Abstract: The dynamic non-linear behaviour of three-dimensional long-span cable-stayed bridges under seismic loadings is studied. The cases of multiple-support as well as uniform seismic excitations of these long and flexible structures are considered. Different sources of non-linearity for such bridges are included in the analysis, as outlined in the companion paper. In this accompanying analysis a tangent stiffness iterative procedure is utilized to estimate the non-linear seismic response. Numerical examples are presented in which a comparison between a linear earthquake-response analysis (based on the utilization of the tangent stiffness matrix of the bridge at the dead-load deformed state which is obtained from the geometry of the bridge under gravity load conditions) and a non-linear earthquake response analysis using the step-by-integration procedure is made. In these examples two three-dimensional bridge models representing recent and future trends in cable-stayed bridge design are utilized. The study sheds some light on the salient features of the seismic analysis and design of these long contemporary bridges. In addition, parameters affecting the seismic response of these bridges are discussed: other factors considered are non-linearity, uniformity and spatial variation of ground motion inputs and structural configuration.

Co-Rotational Beam Elements for Two- and Three-Dimensional Non-Linear Analysis

Abstract: The paper describes the development of both Kirchhoff and Timoshenko beam elements which are embedded in a continuously rotating frame. Emphasis is placed on the consistent derivation of both the internal force vector and the tangent stiffness matrix.

Nonlinear Analysis of Steel Space Structures

Abstract: A second‐order nonlinear analysis of steel space structures has been presented. Of the two types of nonlinearities, material and geometric, only geometric nonlinearity has been considered. The material of the structure steel has been assumed to be linearly elastic. In geometric nonlinearity, the effects of instability produced by axial forces, the bowing of the deformed members, and finite deflections have all been included. For this purpose, the secant stiffness matrix in the deformed state and the modified kinematic matrices along with the geometric matrix necessary for formulating the tangent stiffness matrix, have been developed. These matrices are used in the analysis, which is carried out by the displacement method through an iterative‐incremental procedure based on Newton‐Raphson technique. The iterations that take into account the latest geometry are repeated until the unbalanced loads become negligible and equilibrium is obtained. The equilibrium equations are solved by Cholesky's method. Results...

On the optention of the tangent matrix for geometrically nonlinear analysis using continuum based beam/shell finite elements

Abstract: The incremental finite element equations for geometrically non linear problems are obtained via the full incremental form of the principle of virtual displacements using a Generalized Lagrangian approach. This leads to the expression of the tangent matrix in a straight forward manner and an example of application for 2D elasticity is presented. For large displacements/large rotations beam/shell problems the incremental equations are derived using a quadratic approximation for the increment of the reference vectors in terms of the nodal rotation increments. It is shown how this approach leads to a complete tangent matrix which in the examples analyzed seems to be competitive with respect to the simplified form obtained by linearizing the changes in the reference vectors.