Showing papers on "Tangent stiffness matrix published in 2001"

Remarks on tension instability of Eulerian and Lagrangian corrected smooth particle hydrodynamics (CSPH) methods

Abstract: The paper discusses the problem of tension instability of particle-based methods such as smooth particle hydrodynamics (SPH) or corrected SPH (CSPH). It is shown that tension instability is a property of a continuum where the stress tensor is isotropic and the value of the pressure is a function of the density or volume ratio. The paper will show that, for this material model, the non-linear continuum equations fail to satisfy the stability condition in the presence of tension. Consequently, any discretization of this continuum will result in negative eigenvalues in the tangent stiffness matrix that will lead to instabilities in the time integration process. An important exception is the 1-D case where the continuum becomes stable but SPH or CSPH can still exhibit negative eigenvalues. The paper will show that these negative eigenvalues can be eliminated if a Lagrangian formulation is used whereby all derivatives are referred to a fixed reference configuration. The resulting formulation maintains the momentum preservation properties of its Eulerian equivalent. Finally a simple 1-D wave propagation example will be used to demonstrate that a stable solution can be obtained using Lagrangian CSPH without the need for any artificial viscosity.

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

Non-linear spatial Timoshenko beam element with curvature interpolation

Abstract: The paper presents a spatial Timoshenko beam element with a total Lagrangian formulation. The element is based on curvature interpolation that is independent of the rigid-body motion of the beam element and simplifies the formulation. The section response is derived from plane section kinematics. A two-node beam element with constant curvature is relatively simple to formulate and exhibits excellent numerical convergence. The formulation is extended to N-node elements with polynomial curvature interpolation. Models with moderate discretization yield results of sufficient accuracy with a small number of iterations at each load step. Generalized second-order stress resultants are identified and the section response takes into account non-linear material behaviour. Green–Lagrange strains are expressed in terms of section curvature and shear distortion, whose first and second variations are functions of node displacements and rotations. A symmetric tangent stiffness matrix is derived by consistent linearization and an iterative acceleration method is used to improve numerical convergence for hyperelastic materials. The comparison of analytical results with numerical simulations in the literature demonstrates the consistency, accuracy and superior numerical performance of the proposed element. Copyright © 2001 John Wiley & Sons, Ltd.

Die shape design optimization of sheet metal stamping process using meshfree method

Abstract: A die shape design sensitivity analysis (DSA) and optimization for a sheet metal stamping process is proposed based on a Lagrangian formulation. A hyperelasticity-based elastoplastic material model is used for the constitutive relation that includes a large deformation effect. The contact condition between a workpiece and a rigid die is imposed through the penalty method with a modified Coulomb friction model. The domain of the workpiece is discretized by a meshfree method. A continuum-based DSA with respect to the rigid die shape parameter is formulated using a design velocity concept. The die shape perturbation has an effect on structural performance through the contact variational form. The effect of the deformation-dependent pressure load to the design sensitivity is discussed. It is shown that the design sensitivity equation uses the same tangent stiffness matrix as the response analysis. The linear design sensitivity equation is solved at each converged load step without the need of iteration, which is quite efficient in computation. The accuracy of sensitivity information is compared to that of the finite difference method with an excellent agreement. A die shape design optimization problem is solved to obtain the desired shape of the workpiece to minimize spring-back effect and to show the feasibility of the proposed method. Copyright © 2001 John Wiley & Sons, Ltd.

Numerical aspects of finite elastoplasticity with isotropic ductile damage for metal forming

Abstract: This work is devoted to the study of an efficient numerical algorithm for evaluating damaged-plastic response of a material submitted to large plastic deformations. Fully coupled constitutive equations accounting for both combined isotropic and kinematic hardening as well as the ductile damage are formulated in the framework of Continuum Damage Mechanics (CDM). The associated numerical aspects concerning both the local integration of the coupled constitutive equations and the (global) equilibrium integration schemes are presented and implemented into a general purpose Finite Element code (ABAQUS). For the local integration of the fully coupled constitutive equations an efficient implicit and asymptotic scheme is used. Special care is given to the consistent tangent stiffness matrix derivation as well as to the reduction of the number of constitutive equations. Some numerical results are presented to show the numerical performance of the proposed stress calculation algorithm and the capability of t...

On the local stability condition in the planar beam finite element

Abstract: In standard finite element algorithms, the local stability conditions are not accounted for in the formulation of the tangent stiffness matrix. As a result, the loss of the local stability is not adequately related to the onset of the global instability. The phenomenon typically arises with material-type localizations, such as shear bands and plastic hinges. This paper addresses the problem in the context of the planar, finite-strain, rate-independent, materially non-linear beam theory, although the proposed technology is in principle not limited to beam structures. A weak formulation of Reissner`s finite-strain beam theory is first presented, where the pseudocurvature of the deformed axis is the only unknown function. We further derive the local stability conditions for the large deformation case, and suggest various possible combinations of the interpolation and numerical integration schemes that trigger the simultaneous loss of the local and global instabilities of a statically determined beam. For practical applications, we advice on a procedure that uses a special numerical integration rule, where interpolation nodes and integration points are equal in number, but not in locations, except for the point of the local instability, where the interpolation node and the integration point coalesce. Provided that the point of instability is an end-point of the beam-a condition often met in engineering practice-the procedure simplifies substantially; one of such algorithms uses the combination of the Lagrangian interpolation and Lobatto`s integration. The present paper uses the Galerkin finite element discretization, but a conceptually similar technology could be extended to other discretization methods.

Note on the normal form of a spatial stiffness matrix

Abstract: There has been some recent interest in the problem of designing compliance mechanisms with a given spatial stiffness matrix. A key result that has proven useful in the design of such mechanisms is Loncaric's normal form. When a spatial stiffness matrix is described in an appropriate coordinate frame, it will have a particularly simple structure. In this form the 3/spl times/3 off-diagonal blocks of the stiffness matrix are diagonal. It has been shown that generically, a spatial stiffness matrix can be written in normal form. For example, it is fairly well known that this is possible for any positive definite spatial stiffness matrix. In this article, it is shown that any symmetric positive semi-definite matrix can be written in normal form. As an application this result is used to design a compact parallel compliance mechanism with a prescribed positive semi-definite spatial stiffness matrix.

Consistent tangent matrices for density-dependent finite plasticity models

Abstract: SUMMARY The consistent tangent matrix for density-dependent plastic models within the theory of isotropic multiplicative hyperelastoplasticity is presented here. Plastic equations expressed as general functions of the Kirchho! stresses and density are considered. They include the Cauchy-based plastic models as a particular case. The standard exponential return-mapping algorithm is applied, with the density playing the role of a "xed parameter during the nonlinear plastic corrector problem. The consistent tangent matrix has the same structure as in the usual density-independent plastic models. A simple additional term takes into account the in#uence of the density on the plastic corrector problem. Quadratic convergence results are shown for several representative examples involving geomaterial and powder constitutive models. Copyright 2001 John Wiley & Sons, Ltd.

Shock spectra and damage boundary curves for hyperbolic tangent cushioning system and their important features

Abstract: By applying the method suggested in the author's previous paper, shock spectra and damage boundary curves are investigated for a hyperbolic tangent cushioning system under the action of rectangular, half-sine, terminal-peak saw-tooth and initial-peak saw-tooth acceleration pulses, respectively. The shock spectrum is affected not only by the damping parameter but also by the dimensionless pulse peak, and both the damping parameter and the dimensionless fragility influence the damage boundary curve for this cushioning system. Some important features of a hyperbolic tangent cushioning system that differs from a tangent cushioning system are discussed in detail. Copyright © 2001 John Wiley & Sons, Ltd.

A Classification of Spatial Stiffness Based on the Degree of Translational–Rotational Coupling

Abstract: Previously, we have shown that, to realize an arbitrary spatial stiffness matrix, spring components that couple the translational and rotational behavior along/about an axis are required. We showed that, three such coupled components and three uncoupled components are sufficient to realize any full-rank spatial stiffness matrix and that, for some spatial stiffness matrices, three coupled components are necessary. In this paper, we show how to identify the minimum number of components that provide the translational-rotational coupling required to realize an arbitrarily specified spatial stiffness matrix. We establish a classification of spatial stiffness matrices based on this number which we refer to as the degree of translational-rotational coupling (DTRC). We show that the DTRC of a stiffness matrix is uniquely determined by the spatial stiffness mapping and is obtained by evaluating the eigenstiffnesses of the spatial stiffness matrix. The topological properties of each class are identified. In addition, the relationships between the DTRC and other properties identified in previous investigations of spatial stiffness behavior are discussed.

Double Nonlinear Analysis of The Loading Capacity of Drilling Derrick Steel Structures Containing Defects and Defacements

Abstract: This paper presents a method of analyzing the loading capacity of drilling derrick steel structures containing defects and defacements in keeping with the current demand in the oil industry for evaluating safety and the loading capacity of derricks during their service lives. Adopting the three dimensional beam element as the basic element for FEM analysis, the problem of large deformation analysis is treated using the incremental tangent stiffness matrix method incorporating both the geometric and deformation stiffness matrices of the element, and the material nonlinearity is simplified as a concentrated inelasticity approach. Thus allowing for here two types of non-linearity analysis of the loading capacity of a perfect derrick structure has been studied. Applying a new Referential Uniform Reduction Method, special defects and defacements could then be introduced into the mathematical model of the structure, and a group of load-displacement curves representing different states of imperfections are obtai...

Folding analysis of reversal arch by the tangent stiffness method

Abstract: This paper presents the tangent stiffness method for 3-D geometrically nonlinear folding analysis of a reversal arch. Experimental tests are conducted to verify the numerical analysis. The tangent stiffness method can accurately evaluate the geometrical nonlinearity due to the element translating as a rigid body, and the method can exactly handle the large rotation of the element in space. The arch in the experiment is made from a thin flat bar, and it is found that the folding process of the arch may be captured exactly by the numerical analysis with a model consisting of only 18 elements with the same properties.

Second-order analysis of planar steel frames considering the effect of spread of plasticity

Abstract: This paper presents a method of elastic-plastic analysis for planar steel frames that provides the accuracy of distributed plasticity methods with the computational efficiency that is greater than that of distributed plasticity methods but less than that of plastic-hinge based methods. This method accounts for the effect of spread of plasticity accurately without discretization through the cross-section of a beam-column element, which is achieved by the following procedures. First, nonlinear equations describing the relationships between generalized stresses and strains of the cross-section are derived analytically. Next, nonlinear force-deformation relationships for the beam-column element are obtained through lengthwise integration of the generalized strains. Elastic-plastic flexibility coefficients are then calculated by differentiating the above element force-deformation relationships. Finally, an elastic-plastic stiffness matrix is obtained by making use of the flexibility-stiffness transformation. Adding the conventional geometric stiffness matrix to the elastic-plastic stiffness matrix results in the tangent stiffness matrix, which can readily be used to evaluate the load carrying capacity of steel frames following standard nonlinear analysis procedures. The accuracy of the proposed method is verified by several examples that are sensitive to the effect of spread of plasticity.

Buckling of Imperfect Elastic Shells using the Asymptotic Numerical Method

Abstract: This paper is concerned with stability behaviour and imperfection sensitivity of elastic shells. The aim is to determine the reduction of the critical buckling load as a function of the imperfection amplitude. For this purpose, the direct calculation of the so-called fold line connecting all the limit points of the equilibrium branches of the imperfect structures is performed. An augmented system demanding the criticality of the equilibrium is used. In order to solve the augmented system, the Asymptotic Numerical Method is used as an alternative to Newton-like incremental-iterative procedures. It results in a very robust and efficient path-following algorithm that takes the singularity of the tangent stiffness matrix into account. Two specific types of imperfections are detailed and several numerical examples are discussed.

Geometrically nonlinear performance of portal frame structures with tapered members

Abstract: In order to discuss the stability of the lightweight steel portal frames with tapered mem- bers, its geometrically nonlinear analysis is made by adopting the theory of the finite element, the tangent stiffness matrix considering shear deformation is presented for tapered beam element. The pro- gram for the geometrically nonlinear analysis of the structures is made employing the arc -Iength method. The instability behavior of portal frame subjected to roof uniform loading is investigated some results are gained by comparing the load-deflection curves and buckling mode of the different portal frames, the instability of portal frame is closely related with the rigidity ratio of column to rafter. The buckling behavior of the portal frame is symmetrical when the columns are strong. The buckling be- havior of portal frame is anti-symmetric when the columns are weak.

Development of the Tangent Modulus from the Euler-Problem to Nonsmooth Materials

Abstract: History and development of the tangent modulus from the origins to the recent nonsmooth damaging versions are presented. Load history and stability analyses of structures of nonlinear reversible or irreversible materials are based on the concept of tangent modulus. Generally, instantaneously changing tangent modulus is needed and the solution yields iteration process. In the case of inelastic problems, the switch from loading to unloading of the material behaviour results in nonsmooth material functions. Nonsmooth, generally saw-tooth like behaviour happens in composite, laminated or rock type materials, or in the interaction of concrete and the reinforcement, too. Recently, damage and localization are in the focus of structural analyses, extending the tangent modulus to the negative cases, as well. Consequently, an overview of the history and development of the tangent modulus containing the recent modifications seems to be necessary. On the other hand, the more than a century long history of the tangent modulus is a marvellous study of the parallel development of mechanics and mathematics, by following the mutual inspiring effect of them through the activity of such pioneers like P.D. Panagiotopoulos in creating Nonsmooth Mechanics.

A lateral torsion buckling analysis of elastic beam under axial force and bending moment

Abstract: The buckling moment of doubly symmetric spatial beams under different types of end bending moment and compressive axial force is investigated using finite element method. A co-rotational total Lagrangian finite element formulation is employed here. An incremental-iterative method based on the Newton-Raphson method combined with constant arc length of incremental displacement vector is employed for the solution of nonlinear equilibrium equations. The zero value of the tangent stiffness matrix determinant of the structure is used as the criterion of the buckling state. Numerical examples are presented to investigate the effect of compressive force on the buckling moment of spatial beams under different types of bending moment.