Showing papers on "Tangent stiffness matrix published in 1991"

An investigation of a finite rotation four node assumed strain shell element

Abstract: The paper presents a shell formulation based on the ‘degenerated solid approach’. The theory employs covariant strains and performs explicit integration through the shell thickness. The rigid body motion is exactly represented. The consistent tangent stiffness matrix is evaluated for the four node quadrilateral. It is shown, in the final part, that this type of element, which distinguishes itself by a very simple and easily understandable theory, gives good answers for linear as well as non-linear applications.

3‐D Nonlinear Seismic Behavior of Cable‐Stayed Bridges

Abstract: The dynamic behavior of three-dimensional (3-D) long-span cable-stayed bridges under seismic loadings is studied. The cases of synchronous and nonsynchronous support motions due to seismic excitations of these flexible structures are considered; furthermore, effects of the nondispersive traveling seismic wave on the bridge response are studied. Different sources of nonlinearity for such bridges are included in the analysis. Nonlinearities can be due to: (1) Changes of geometry of the whole bridge due to its large deformations including changes in the geometry of the cables due to tension changes (known as the sag effect); and (2) axial force and bending moment interaction in the bridge tower as well as the girder elements. A tangent stiffness iterative procedure is used in the analysis to capture the nonlinear seismic response. Numerical examples are presented in which a comparison is made between a linear earthquake-response analysis (based on the utilization of the tangent stiffness matrix of the bridge at the dead-load deformed state) and a nonlinear earthquake-response analysis using the step-by-step integration procedure. In these examples, two models having center (or effective) spans of 1,100 ft (335.5 m) and of 2,200 ft (671 m) are studied; this range covers both present and future designs. The study sheds some light on the salient features of the seismic analysis and design of these long contemporary bridges.

Computational use of group theory in bifurcation analysis of symmetric structures

Abstract: An efficient computational procedure is proposed for identifying singular points in global bifurcation analysis of the static behavior of symmetric discrete structures such as symmetric truss domes. Assuming group equivariance of the system of equations describing the steady state, and making use of group representation theory, the proposed method decomposes the Jacobian matrix (or tangent stiffness matrix) into block-diagonal form, with possibly repeated occurrences of identical blocks, by means of a suitable “local” coordinate transformation. The “local” transformation is computationally favorable in that it requires a small amount of computation and preserves the sparsity of the original Jacobian matrix fairly well. A concrete procedure is described for symmetric truss structures which are equivariant to dihedral groups; an explicit formula is given to the number of diagonal blocks into which the Jacobian matrix splits, and an estimate of the required number of computations shows the efficiency of the proposed method.

Computational coupled non-associative thermo-plasticity

Abstract: Computationally oriented formulation of the isothermal, rate-independent theory of non-associative elasto-plasticity is extended in this paper to describe coupled thermo-elastic-plastic and thermo-elastic-visco-plastic behaviour of materials. This is done by additionally considering thermal strains, assumming all material properties to be temperature dependent and accounting for the mechanical coupling terms in the non-stationary heat conduction equation. The finite deformation effects are included in the analysis. The theory is employed for the analysis of thermo-mechanical response of ductile metals with damage effects modelled by a generalization of the so-called Gurson approach. This constitutive model is known to generate equations typical of non-associative plasticity and hence it can be consistently incorporated into the present more general considerations. The finite element assessment of combined thermal and damage effects on the axisymmetric necking process illustrates the paper. Numerical aspects such as a ‘tangent’ stiffness for rate-dependent thermo-plasticity and the algorithmic (or consistent) tangent stiffness matrix for non-associative plasticity are discussed as well.

The dynamic stiffness matrix method in the analysis of rotating systems

Abstract: The dynamic stiffness method has been applied to the evaluation of the natural frequencies of rotating systems. To this purpose, a rotating “Rayleigh beam,” defined by adding the effect of the rotary inertia and the gyroscopic effects to the Bernoulli-Eider beam, has been formulated and its dynamic stiffness matrix is presented in this paper. The effects due to the presence of concentrated disks, as well as of elastic, isotropic supports, have been included in the formulation. The usual matrix assembly procedure is used in order to obtain the global dynamic stiffness matrix of the system. The natural frequencies of the system are determined by utilizing an iterative root searching technique. Numerical results, obtained for a rotor system taken from the literature on this subject, are presented. Presented at the 45th Annual Meeting in Denver, Colorado May 7–10, 1990

Analytical Modeling of Bonded Bars under Cyclic Loads

Abstract: A new modeling technique for predicting the behavior of deformed bars anchored in concrete is developed. In the model, the bond between steel and concrete is simulated with discrete springs that connect the bar to concrete along the anchorage length. The simulation technique is based on the displacement method of analysis. Unlike other approaches, the new model does not require iterative solution of the governing nonlinear equations. This technique involves the construction of the tangent stiffness matrix of the anchorage for use in an incremental solution algorithm. The model is thus very time‐efficient for computer analysis. The proposed model is used for simulating the behavior of bonded bars with common types of boundary conditions. The analytical results are compared with results of monotonic and cyclic tests. The model can be used for efficient idealization of anchorages in analytical studies of structural subassemblies and complete structural systems.

Improved Quasi‐Newton Methods for Large Nonlinear Problems

Abstract: In this work, schemes based on limited‐memory quasi‐Newton methods are investigated, as applied to solving large systems of nonlinear equations with sparse symmetric Jacobian matrices. Problems in mechanics typically give rise to such systems when the method of finite elements is employed to solve them. An attempt is made to develop algorithms that take advantage of sparsity and can effectively use a variable amount of storage according to the availability. The use of preconditioning matrices as initial approximations to the tangent stiffness matrix is suggested in order to accelerate convergence when the available high‐speed storage exceeds the needs of purely vectorial methods but is not sufficient to house a full factorization of the tangent stiffness. The limited‐memory quasi‐Newton methods are also combined with the concept of truncation, based on a preconditioned conjugate gradient iterative solver of the linearized equations, to produce quite efficient algorithms.

Modern Structural Analysis: The Matrix Method Approach

Abstract: Preliminary Considerations Nodal Variables - Stiffness and Flexibility Matrices of a Structure Element Variables - Total Stiffness Matrix of an Element Direct Stiffness Method Supplementary Procedures to the Stiffness Method Basic Element Variables The Equations of Equilibrium and the Equations of Compatibility of a Structure Computation of the Displacements of Statically determinate Framed Structures The Flexibility Method The Displacement or Stiffness Method Appendix A. The International System of Units Appendix B. End Actions in Fixed-At-Both-Ends Elements Appendix C. Vector Quantities Appendix D. Computation of the Total Stiffness Matrix of an Element and the Equivalent Actions to be Placed at the Nodes of a Structure Using Shape Functions.

On special points on load‐displacement paths in the prebuckling domain of thin shells

Abstract: Points on load-displacement paths of thin shells subjected to proportional loading, at which the second and/or the third derivative of the displacement components with respect to the load parameter vanishes, are termed ‘special points’. Such points represent global characteristics of the state of deformation of the shell in the sense that for the respective values of the load parameter the mentioned rate(s) of all displacement components at all points of the shell must vanish. It will be shown that special points on load-displacement paths correspond to special points of one order lower on the Det KT–λ diagram, where Det KT is the determinant of the tangent stiffness matrix within the framework of the finite element method and λ is a dimensionless load parameter. Points of inflection on load-displacement diagrams, for example, correspond to extreme values on the Det KT–λ diagram. The main reason for the occupation with special points on load-displacement paths is that points of inflection and flat points on these paths correspond to special points on eigenvalue curves in the context of accompanying linear stability analyses of geometrically non-linear prebuckling analyses of thin shells by the finite element method, investigated in a companion paper.

Stiffness Formulations of Planar Kinematics

Abstract: A simple theory using the stiffness concept is presented for linkage-motion analysis. The nonlinear incremental mechanism equations are established and their solution procedure is proposed. The kinematic system is first regarded as an instable elastic structure. The tangent stiffness equations are updated for current mechanism configurations. The obtained singular stiffness matrix is then used to predict incremental nodal displacements. The stress-associated deformation is designated as numerical error. The proposed stiffness method follows successively the mechanism motion in an incremental-iterative manner. Computed numerical examples show that the elastic deformation can easily be removed with a few iterations. The proposed stiffness approach to kinematics may be incorporated into existing finite-element computer program systems and can be applied to general three-dimensional mechanisms.

Some Recent Research on the Non-Linear Analysis of Shells

Abstract: The paper describes some recent research on the non-linear analysis of shells using a co-rotational formulation with the constant strain/ constant moment facet-triangular shell element.

Inelastic Analysis of Structures Based on Stress Redistribution

Abstract: An important characteristic of structures that exhibit material nonlinearity is that the internal stresses are redistributed when the stiffness properties of the members change. Based on this physical characteristic of structures, an efficient analytical method for the nonlinear analysis of structures is presented here. In his analytical method, the overall response of the structure is determined by considering: (1) The elastic response of the structure to external loads; and (2) the stress redistribution that results from the member yield excursions. One advantage of this method is that the stress redistribution may be represented in terms of the elastic stiffness matrix of the structure and not the tangent stiffness matrix. Another advantage of this method is that the only parameters that have to be evaluated and updated throughout the analysis are the member stresses and the member deformations. Hence, there is no direct evaluation of the structure displacements or the tangent stiffness matrices.