Showing papers on "Tangent stiffness matrix published in 1993"

A three‐dimensional non‐linear Timoshenko beam based on the core‐congruential formulation

Abstract: A three-dimensional, geometrically nonlinear two-node Timoshenkoo beam element based on the total Larangrian description is derived. The element behavior is assumed to be linear elastic, but no restrictions are placed on magnitude of finite rotations. The resulting element has twelve degrees of freedom: six translational components and six rotational-vector components. The formulation uses the Green-Lagrange strains and second Piola-Kirchhoff stresses as energy-conjugate variables and accounts for the bending-stretching and bending-torsional coupling effects without special provisions. The core-congruential formulation (CCF) is used to derived the discrete equations in a staged manner. Core equations involving the internal force vector and tangent stiffness matrix are developed at the particle level. A sequence of matrix transformations carries these equations to beam cross-sections and finally to the element nodal degrees of freedom. The choice of finite rotation measure is made in the next-to-last transformation stage, and the choice of over-the-element interpolation in the last one. The tangent stiffness matrix is found to retain symmetry if the rotational vector is chosen to measure finite rotations. An extensive set of numerical examples is presented to test and validate the present element.

Fracture and Slip of Interfaces in Cementitious Composites. II: Implementation

Abstract: This paper focuses on the numerical treatment of an interface model that was presented in the companion paper. A generalized trapezoidal rule is proposed for integrating the slip rule. The new model is used in finite element predictions; whereby Newton iterations to achieve equilibrium require the algo- rithmic tangent stiffness matrix that is pertinent to the incremental problem. In order to ensure convergence to a physically acceptable (stable) solution, one should employ a modified tangent stiffness that is always positive definite. The analysis of a simple composite microstructure confirms that the model is capable of pre- dicting interface failure. Application of the interface model as a generalized ficti- tious crack model concludes the paper. Preliminary results show that it is possible to obtain a crack pattern that extends from the aggregate-mortar boundary into the mortar itself along predefined interelement boundaries.

Coincident collocation of displacement and tangent derivative boundary integral equations in elasticity

Abstract: The regular boundary integral equations of elastostatics are combined with regularized versions of the tangent derivative equations and collocated at the same points to formulate the elasticity problem in terms of displacements, tractions and the tangential displacement gradients. Hermitian cubic polynomials are used for functional interpolation on certain elements to formulate the boundary element method in terms of displacements, tractions and their tangent derivatives. Commensurate accuracy of nodal values of these functions and the tangent derivatives is obtained and makes possible the accurate and immediate recovery of all stress components. An example problem demonstrates the accuracy and atility of the approach.

Method and apparatus for predicting post-buckling deformation of sheet metal

Abstract: Method for predicting post-buckling deformation of a sheet of metal during a draw forming process, for use with a computer including memory, by introducing a set of springs to stabilize the sheet metal close to the onset of buckling, thereby enhancing convergence of numerical solutions. The method is for use with sheet metal forming tools including a draw die, a punch and binder having surfaces designed to form the sheet metal into a part, the sheet metal being represented as a mesh including a plurality of nodes. The method includes applying a displacement increment to the sheet metal nodes and identifying a singularity in a tangent stiffness matrix associated with the sheet metal close to the onset of buckling. The method also includes introducing a plurality of springs at the sheet metal nodes so as to eliminate the singularity and enhance convergence of the numerical solution of the displacement increment.

A non‐linear numerical method for accurate determination of limit and bifurcation points

Abstract: This paper presents a numerical procedure for accurate determination of a limit or a bifurcation point. The method minimizes simultaneously the first and the second variations of an admissible functional or iterates to satisfy the equilibrium and the semi definite condition for the tangent stiffness matrix. It can be readily incorporated into a computer program for non-linear finite element analysis to improve its accuracy in the location of critical points.

Vibration of thermally buckled composite plates with initial deflections using triangular elements

Abstract: A consistent finite element formulation is presented for the analysis of thermal postbuckling and free vibration of thermally buckled thin, laminated composite plates subjected to large temperature change. The influence of moderately large initial imperfections in deflection on the thermal postbuckling deflection and the vibration characteristics of the buckled plate is also investigated. The finite element equations of motion are derived from the principle of virtual work. These equations can be mathematically separated into two sets and solved in sequence. The first set of equations yields the particular solution of static thermal postbuckling deflection, and the second set of equations gives the homogeneous solution of vibration characteristics on the buckled plate. The first set of static equations is solved by using Newton-Raphson iteration method. The tangent stiffness matrix in the final iteration is equal to the total stiffness matrix of the second set of dynamic equations. This feature saves tremendous computation time in comparing with using the conventional approach. The influence of lamination angle, temperature distribution, plate planform of arbitrary shape, and boundary support conditions on postbuckling and vibration behavior are investigated.

Explicit thickness integration for three‐dimensional shell elements applied to non‐linear analysis

Abstract: The problem of multilayered degenerated 3-D shell elements for which the numerical integration is performed for each ply is that of the high generation time in non-linear analysis when the number of plies is important. But these elements give accurate results for thin and moderately thick shells, so in order to reduce the generation time explicit thickness integration is investigated. We first write an expansion of the strain-displacement matrix in power series of the thickness variable in order to obtain explicit expressions of the tangent stiffness matrix and internal force vector, appearing in the non-linear formulation. Explicit expressions of non-linear stiffness matrices are presented, using the explicit integration-first approximation. Simple expressions of several matrices, sub-matrices and vectors appearing in the formulation are given here in order to obtain an important computing-time gain. Next, some numerical validation tests comparing the classical element with numerical thickness integration and this one are discussed to prove validity of this formulation.

Parametric Study of Cable-Stayed Bridges Response Due to Traffic-Induced Vibration

Abstract: The dynamic response of long-span cable-stayed bridges due to moving traffic loads is investigated utilizing three-dimensional models. Modal analysis is conducted using the deformed dead-load tangent stiffness matrix. A new concept, presented by discretization of cable into several elements in the finite element modeling, is used to study the effect of cable vibration on bridge dynamics. A computer algorithm is developed to simulate the applied traffic loads in both directions of the bridge deck. The algorithm is flexible in terms of handling different loading capacities, speeds and configurations. Parametric studies are conducted to investigate the effect of cable vibration, damping, vehicle-structure interaction, random roughness of the bridge deck, as well as span length and vehicle-speed. Cases of asymmetric traffic loading clustered in one direction are also considered in order to study the torsional response of the bridge. Results are discussed and summarized.

Numerical Modeling of Transient Creep in Polycrystalline Ice

Abstract: Transient creep, an important deformation mechanism for polycrystalline ice at quasi‐static strain rates, is characterized by rate and temperature sensitivity, by isotropic and kinematic strain hardening, as well as by fabric and deformation‐induced anisotropy A physically based constitutive model, using internal state variables, has been developed by Shyam Sunder and Wu (1989a, b) to describe the multiaxial behavior of ice undergoing transient creep To solve boundary value problems using this constitutive theory requires the numerical time integration of a coupled set of stiff and highly nonlinear first‐order differential equations A closed‐form Newton‐Raphson (tangent) formulation, in conjunction with the α‐method of integration, is developed to solve the constitutive equations The fully consistent constitutive Jacobian matrix that is used to assemble the finite element tangent stiffness matrix is also established in closed form This algorithm is implemented as a subroutine in the finite element pr

Complete Stiffness Matrices for Buckling Analysis of Frames

Abstract: The present paper attempts to explain the significance of the various terms that must be included in an element stiffness matrix of a beam finite element to yield correct results in buckling analyses of spatial frames. The complete stiffness matrices for the buckling and post-buckling analysis of three-dimensional elastic framed structures are derived for the \IC\N\u0-two-noded straight prismatic beam element with double symmetric cross section. The assumed small-strain hypothesis permitted closed-form expressions to be arrived at. Derivations of the elastic and geometric matrices of a generic beam element are first reviewed. All stress resultants are included in the geometric stiffness matrix. The contribution of the conservative external surface loads having moment resultants to the stiffness matrix is then formulated. The so-called energy method of stiffness derivation used commonly by engineers is argued to yield included. The resulting expressions are easier to understand than their counterparts known from the papers advocating semitangential rotations and moments. Finally, the actual way of application of the external moments is included in the system stiffness matrix and it is shown that the tangential operator of the assembled system is symmetric. The present element stiffness matrices can be easily incorporated into finite-element codes by only correcting those already implemented.

Detecting symmetry-breaking bifurcation points of symmetric structures

Abstract: SUMMARY In engineering applications often problems with symmetric system and symmetric loading occur It is well known that these symmetry conditions can be used to reduce the computational effort Thus, only a symmetric reduced system is treated with sufficient boundary and loading conditions Especially for non-linear problems this procedure is very effective Such a strategy allows the computation of solution paths with the constraint that the solution has to be symmetric Consequently in a stability analysis, only limit points and bifurcation points with associated symmetrical eigenvectors can be found Often the stability behaviour is dominated by symmetry-breaking bifurcation points which cannot be detected considering only the tangent stiffness matrix of the reduced system Hence, in case of stability considerations a calculation of the complete system is necessary This paper introduces a special form of stability analysis of the complete system using only certain matrices known from the symmetric reduced system, and some transformations concerning the topology of the total system The proposed methods base on a substructure technique for symmetry under reflections and rotations, and are formulated for the finite element method Numerical examples are given to show the efficiency of the proposed procedures and algorithms

Influence of Different Soil Modeling Criteria in SSI Analysis

Abstract: This paper is in the form of two nearly independent parts. In the first, an overview of the mechanistic aspects of modeling soil structure systems and solution techniques are presented in textbook fashion. The second part deals with a particular application in which blind predictions were made for the measured earthquake response of a 1/4-scale nuclear containment structure built on the SMART site near Lotung, Taiwan.