Showing papers on "Tangent stiffness matrix published in 2000"

The eigenscrew decomposition of spatial stiffness matrices

Abstract: A manipulator system is modeled as a kinematically unconstrained rigid body suspended by elastic devices. The structure of spatial stiffness is investigated by evaluating the stiffness matrix "primitives"-the rank-1 matrices that compose a spatial stiffness matrix. Although the decomposition of a rank-2 or higher stiffness matrix into the sum of rank-1 matrices is not unique, one property of the set of matrices is conserved. This property, defined as the stiffness-coupling index, identifies how the translational and rotational components of the stiffness are related. Here, we investigate the stiffness-coupling index of the rank-1 matrices that compose a spatial stiffness matrix. We develop a matrix decomposition that yields a set of rank-1 stiffness matrices that identifies the bounds on the stiffness-coupling index for any decomposition. This decomposition, referred to as the eigenscrew decomposition, is shown to be invariant in coordinate transformation. With this decomposition, we provide some physical insight into the behavior associated with a general spatial stiffness matrix.

The spatial stiffness matrix from simple stretched springs

Abstract: Looks at the stiffness matrix of some simple but very general systems of springs supporting a rigid body. The stiffness matrix is found by symbolically differentiating the potential function. After a short example attention turns to the general structure of the stiffness matrix and in particular the principal screws introduced by Ball (1900).

Geometrical method for modeling of asymmetric 6/spl times/6 Cartesian stiffness matrix

Abstract: In this paper, we study the 6/spl times/6 Cartesian stiffness matrices of conservative systems using the method of changing basis in differential geometry of the motion of the rigid body. We show that the stiffness matrix is symmetric at the unloaded equilibrium configuration. When the system is subjected to external loads, the 6/spl times/6 Cartesian stiffness matrix becomes asymmetric. The skew-symmetric part of the stiffness matrix is equal to the negative one-half of the cross-product matrix formed by the externally applied load, referenced to the inertial frame. This method presented in this paper provides a systematic way of constructing 6/spl times/6 stiffness matrix in robotic grasping/manipulation and stiffness control.

Accelerated initial stiffness schemes for elastoplasticity

Abstract: Iterative methods for the solution of non-linear finite element equations are generally based on variants of the Newton–Raphson method. When they are stable, full Newton–Raphson schemes usually converge rapidly but may be expensive for some types of problems (for example, when the tangent stiffness matrix is unsymmetric). Initial stiffness schemes, on the other hand, are extremely robust but may require large numbers of iterations for cases where the plastic zone is extensive. In most geomechanics applications it is generally preferable to use a tangent stiffness scheme, but there are situations in which initial stiffness schemes are very useful. These situations include problems where a nonassociated flow rule is used or where the zone of plastic yielding is highly localized. This paper surveys the performance of several single-parameter techniques for accelerating the convergence of the initial stiffness scheme. Some simple but effective modifications to these procedures are also proposed. In particular, a modified version of Thomas' acceleration scheme is developed which has a good rate of convergence. Previously published results on the performance of various acceleration algorithms for initial stiffness iteration are rare and have been restricted to relatively simple yield criteria and simple problems. In this study, detailed numerical results are presented for the expansion of a thick cylinder, the collapse of a rigid strip footing, and the failure of a vertical cut. These analyses use the Mohr–Coulomb and Tresca yield criteria which are popular in soil mechanics. Copyright © 2000 John Wiley & Sons, Ltd.

Scaled corrector and branch-switching in necking problems

Abstract: A new branch-switching procedure for necking problems by finite element method is proposed. The scaled corrector method adopted in this paper for computing the critical eigenvector of the tangent stiffness matrix, which is indispensable to bifurcation analysis, is quite simple and effective for large scale problems because eigenvalue analysis is not required. The numerical background of the approximate critical eigenvector calculated during equilibrium iterations and the branch-switching procedure in elasto-plastic bifurcation problems are described in detail. As numerical examples, necking bifurcation problems of plane strain/stress states in finite strains are demonstrated to validate the proposed branch-switching procedure.

Modeling of Cable Stayed Bridges for Analysis of Traffic Induced Vibrations

Abstract: This paper presents a method for modeling and analysis of cable-stayed bridges under the action of moving vehicles. Accurate and efficient finite elements are used for modeling the bridge structure. A beam element is adopted for modeling the girder and the pylons. Whereas, a two-node catenary cable element derived using exact analytical expressions for the elastic catenary, is adopted for modeling the cables. The vehicle model used in this study is a so-called suspension model that includes both primary and secondary vehicle suspension systems. Bridge damping, bridge-vehicle interaction and all sources of geometric nonlinearity are considered. An iterative scheme is utilized to include the dynamic interaction between the bridge and the moving vehicles. The dynamic response is evaluated using the mode superposition technique and utilizing the deformed dead load tangent stiffness matrix. To illustrate the efficiency of the solution methodology and to highlight the dynamic effects, a numerical example of a simple cable-stayed bridge model is presented.

Design sensitivity analysis and optimization of nonlinear transient dynamics

Abstract: A shape design sensitivity analysis (DSA) and optimization of structural transient dynamics are proposed for the finite deformation elastoplastic materials under impact with a rigid surface. A shape variation of the structure is considered using the material derivative approach in continuum mechanics. Hyperelasticitybased multiplicatively decomposed elastoplasticity is used for the constitutive model. The implicit Newmark time integration scheme is used for the structural dynamics. The design sensitivity equation is solved at each converged time step with the same tangent stiffness matrix as response analysis without iteration. The cost of sensitivity computation is more efficient than the cost of response analysis for the implicit time integration method. The efficiency and the accuracy of the proposed method are shown through the design optimization of a vehicle bumper.

Implicit integration of hypoplastic models

Abstract: The theory of hypoplasticity is reviewed. Next, an algorithm for the integration of the rate equations resulting from hypoplasticity is presented. Herein, the concept of numerical differentiation has been utilized. The integration algorithm is applied to the hypoplastic model proposed by [13]. To prevent unadmissible stress states, an automatic substepping algorithm has been implemented. Results of a finite element simulation of the direct shear box test are presented. Keywords: hypoplasticity, implicit integration, consistent tangent stiffness matrix, quadratic convergence, direct shear box test, softening, localization.

Regularized tangents in computer graphics

Abstract: A method, apparatus and article of manufacture for generating regularized tangents of curves. The method comprises the steps of bounding a length of the arc, computing a chord vector, where the chord vector corresponds to the bounded length of the arc, generating a tangent vector, where the tangent vector is substantially normal to the chord vector, and regularizing the tangent vector, where the regularized tangent vector approximates a true tangent vector to the arc.

Determination of branches of limit points by an Asymptotic Numerical Method

Abstract: This paper deals with parameter dependence in nonlinear structural stability problems. The main purpose is the study of the influence of imperfections on a structure. This analysis implies the calculation of the so called fold curve connecting the critical points of the equilibrium path when a structural defect varies. This is traditionally achieved by adding a well-chosen constraint equation demanding the criticality of the equilibrium. The crucial feature of the paper lies in the use of the Asymptotic Numerical Method (A.N.M.) for the numerical resolution of the obtained augmented system. The theoretical framework upon which the A.N.M. is based as well as its advantages over incremental-iterative strategies are presented. The numerical isolation of an initial starting limit point is described. The extended system and its resolution with the A.N.M. are discussed. From a numerical point of view, it leads to an efficient treatment which takes the singularity of the tangent stiffness matrix into account. Emphasis is given on a geometrical shape imperfection.

Direct computation of paths of limit points using the Asymptotic Numerical Method

Abstract: This paper is concerned with parameter dependent problems for structural instability. The aim is the direct determination of the so called fold curve connecting the limit points of the equilibrium path for a structure subjected to a variable imperfection. This is traditionally achieved by adding a well-chosen constraint equation requiring the criticality of the equilibrium. The crucial feature of the paper lies in the numerical resolution of the obtained augmented system. Indeed, it is solved using the Asymptotic Numerical Method (A.N.M.) which is well-known for its robustness. The theoretical framework upon which the A.N.M. and the extended system are based are presented. From a numerical point of view, it leads to an efficient treatment which takes the singularity of the tangent stiffness matrix into account. Emphasis is given on two specific types of geometrical imperfections. Eventually, the numerical isolation of an initial starting limit point is discussed.

Shape Optimization under Shakedown Constraints

Abstract: Based on Melan’s static shakedown theorem for linear unlimited kinematic hardening material behaviour, we formulate an integrated approach for all necessary variations within direct analysis, shakedown analysis and variational design sensitivity analysis based on convected coordinates. Using a special formulation of the optimization problem of shakedown analysis, we easily derive the necessary variations of residuals, objectives and constraints. Subsequent discretizations w.r.t. displacements and geometry using e.g. an isoparametric finite element method yield the well known tangent stiffness matrix and tangent sensitivity matrix, as well as the corresponding matrices for the variation of the Lagrange-functional. Thus, all expressions on the element level are dependent only on the nodal values of the displacements and the coordinates but not on a single design variable or the corresponding design velocity field. Remarks on the computer implementation and a numerical example show the efficiency of the proposed formulation.

Double Nonlinear Analysis of the Loading Capacity of Drilling Derrick Steel Structures with Defects and Defacements

Abstract: Aiming at the demand in oil industry for evaluating the safe state and the loading capacity of derricks in service, this paper presents a method of analyzing the loading capacity of drilling derrick steel structures with defects and defacements Firstly, by adopting the three dimensional beam element as the basic element for FEM analysis, the problem of large deformation was considered by the method of incremental tangent stiffness matrix including both geometric and deformation stiffness matrix of the element, and the material nonlinearity was simplified using the method of Concentrated Inelasticity Approach with the result that the double nonlinear analysis of the loading capacity of a perfect derrick structure could be proceeded After it, the Referential Uniform Reduction Method is to be applied Special defects and defacements could be introduced into the mathematical model of the structure, and a group of the load-displacement curves representing different states of imperfections were obtained The ultimate strength of the derrick in service could be determined finally through the comparison between the computed curves and the full-scale displacement testing results At the end of this article, two examples of different types of derrick structures were analyzed

Finite-element formulation with special orthogonal group SO (3) * (Part 3, consideration upon formulation with SO (3))

Abstract: The relationship between stress rates and tangent stiffness values with special orthogonal group SO (3) for finite-element formulation was shown in Part1, and stiffness elements of finite beam element using SO (3) were shown in Part2. Those stress rates are the Truesdell stress rate, the Jaumann stress rate, the Neo-Green stress rate and the Ishihara stress rate, a nonsymmetric stress rate that will be defined by this author in this report. In this study, the constitutive equation of the Ishihara stress rate will be considered. It will be shown that although the stress rate and the strain rate are nonsymmetric, the material stiffness matrix is symmetric. Next, it is shown that although the geometric stiffness of a rigid rotation is nonsymmetric, even if only the symmetric part of the stiffness is used, the second-order convergent rate determined by the Newton-Raphson method is conserved. Finally, it will be shown that the tangent stiffness formulated with SO (3) is different from that of the conventional formulation with linear rotation.

Buckling analysis of plates by hierarchical elements

Abstract: 本論文では, 全ラグランジュ表記法によるハイアラーキ平面シェル要素とソリッド要素の接線剛性行列を求め, 線形座屈理論に基づいたハイアラーキ要素による座屈解析法を開発する. 本解析法の精度を調べるために平板の座屈解析を行って, 提案する要素と各非線形項の影響を検討する. 平板の解析では, 一般に Kirchhoff 理論, Mindlin 理論と3次元弾性論が用いられるが, ハイアラーキ要素による解法では, それぞれの理論解に対して高精度の値が得られることを示す. また, 実用的な観点から, 計算効率の良い Mindlin 要素を用いて, 弾性論による解に匹敵する値が得られるせん断補正係数を提案する.

Solution of non-linear problems with full stiffness matrices

Abstract: In this contribution the non-incremental system of FEM equations containing the full non-linear tangent stiffness matrix is derived from the principle of virtual work. This formulation is applied to the derivation of basic FEM formulae for 1D-bar element. Two iterative solution methods of non-linear equations are presented and their effectiveness is verified on several numerical experiments.

Design sensitivity anal ysis and optimiza tion of nonlinear transient dynamics

Abstract: A shape design sensitivity analysis and optimization of structural transient dynamics are proposed for the finite deformation elastoplastic materials under impact with a rigid surface. A shape variation of the structure is considered using the material derivative approach in continuum mechanics. A penalty method, modified Coulomb friction law, and the slave—master concept are used for the impact problem. Hyperelasticity-based multiplicatively decomposed elastoplasticity is used for the constitutive model. The implicit Newmark time integration scheme is used for the structural dynamics. The design sensitivity equation is solved at each converged time step with the same tangent stiffness matrix as response analysis without iteration. The cost of sensitivity computation is more efficient than the cost of response analysis for the implicit time integration method. The efficiency and the accuracy of the proposed method are shown through the design optimization of a vehicle bumper.

Illustration of the Interaction of Strain and Displacement Nonlinearities in the Structural Tangent Stiffness

Abstract: Systematization of complex nonlinearities, the wide-ranging linearization concepts are detailed in [3], [4] related to material, strain, displacement and loading type nonlinearities and their interaction. In this paper, an illustration of the full geometric nonlinearity, the interaction of the strain and displacement nonlinearities are presented, by means of the finite element model of the Timoshenko beam.