Showing papers on "Tangent stiffness matrix published in 1996"
TL;DR: In this article, the authors extended the dynamic stiffness technique for a certain class of plate structure to include in-plane vibrations and derived the stiffness matrix of each component using the equations of motion and edge tractions.
63 citations
TL;DR: In this paper, the spectral properties of tangent operators determine failure conditions at the material level in terms of loss of uniqueness and discontinuous bifurcation of the incremental response.
57 citations
TL;DR: In this paper, a three-dimensional elastoplastic beam element being capable of incorporating large displacement and large rotation is developed and examined, which is used for the analysis of prebuckling and large deflection collapse of spatial framed structures.
Abstract: A three-dimensional elastoplastic beam element being capable of incorporating large displacement and large rotation is developed and examined. Elastoplastic constitutive equations are applied to the beam element based upon the assumption of small deformational strain leading to a material formulation which is completely objective for the application of stress update procedures. The continuum-type equations of plastic model of J2 mixed hardening are transformed into the beam equations by satisfying beam hypotheses. An effective stress update algorithm is proposed to integrate elastoplastic rate equations by means of the so-called multistep method which is a method of successive control of residuals on yield surfaces. It avoids severe divergence when the displacement increments become large which is usual for the continuation methods. Material tangent stiffness matrix is derived by using consistent elastoplastic modulus resulting from the integration algorithm and is combined with geometric tangent stiffness matrix. Different from other elements, the present element is shear flexible and can satisfy the plasticity condition in a pointwise fashion. A great number of numerical examples are analysed and compared with the literature. The proposed beam element is verified to be not only quite accurate but also very effective for the analyses of pre-buckling and large deflection collapse of spatial framed structures.
50 citations
TL;DR: In this paper, a constitutive equation for masonry-like materials with bounded compressive strength was proposed and the general properties of this equation were proved and its solution was explicitly calculated.
44 citations
TL;DR: In this article, the authors derived the explicit expression for the integrated stress along with a flow rule that represents volumetric non-associativity for yield criteria of the Mohr-Coulomb type for elastic and plastic isotropy.
Abstract: In this paper, we discuss the efficient treatment of yield criteria that are of the Mohr–Coulomb type for elastic and plastic isotropy. On the basis of the fully implicit method, we derive the explicit expression for the integrated stress along with a flow rule that represents volumetric non-associativity. The integration algorithm covers all the possible cases of regular, corner and apex solutions including the suitable indicator for each case. We also establish the consequent consistently linearized tangent stiffness modulus tensor, which is shown to appear in the form of an additive modification of the continuum tangent stiffness tensor. The convergence properties of the consistent tangent stiffness tensor are compared with its feasible approximations. The results indicate the strongly sensitivity to the proper treatment of the corner conditions at the establishment of the ATS-tensor.
42 citations
TL;DR: In this paper, a matrix formulation capable of representing the elastic behavior of the composite material is established, which yields a much clearer implementation of linear elastic homogenisation algorithms and a relatively straightforward extension to inelastic behaviour.
Abstract: In this article attention is given to the homogenisation of periodic layered materials. Based on the assumption of a homogeneous state of stress and strain in each layer, a novel matrix formulation capable of representing the elastic behaviour of the composite material is established. The matrix formulation yields a much clearer implementation of linear elastic homogenisation algorithms and a relatively straightforward extension to inelastic behaviour. The theory of plasticity, which is adopted to describe the inelastic behaviour, follows modern concepts, including a unconditionally stable implicit Euler backward return mapping, a local Newton–Raphson method and a consistent tangent stiffness matrix. A comparison between the homogenised continuum and the standard continuum with an exact discretisa tion of the geometry of the composite shows excellent agreement, both in the presence of elastic and inelastic material behaviour.
21 citations
TL;DR: In this article, three methods are presented for sensitivity analysis of bifurcation load factor of finite-dimensional conservative symmetric systems subjected to a set of symmetric proportional loads.
Abstract: Three methods are presented for sensitivity analysis of bifurcation load factor of finite-dimensional conservative symmetric systems subjected to a set of symmetric proportional loads. In the first method, a conventional method with diagonalization is utilized to derive an explicit formula of sensitivity coefficients corresponding to a minor imperfection. Next, a new concept is introduced to find the sensitivity coefficients of the load factor, displacements and the eigenmodes under fixed lowest eigenvalue of the tangent stiffness matrix. Based on this concept, a method is presented for finding approximate sensitivity coefficients of the buckling load factor. Finally, a direct method is presented to find the accurate sensitivity coefficients of the bifurcation load factor, displacements at buckling and the buckling mode of a symmetric system. Note that different formula should be used for sensitivity analysis of a limit point load factor. In the examples, the proposed three methods are compared in view of accuracy of the results and simplicity in coding.
17 citations
TL;DR: In this paper, a general formulation for static and dynamic analysis of spatial elastic beams capable of undergoing finite rotations and small strains is presented, where tangent maps associated to the finite rotation vector are used to compute the tangent iteration matrices used to integrate the equations of motion in descriptor form.
Abstract: A general formulation is presented for static and dynamic analysis of spatial elastic beams capable of undergoing finite rotations and small strains. The tangent maps associated to the finite rotation vector are used to compute the tangent iteration matrices used to integrate implicitly the equations of motion in descriptor form. A total Lagrangian primal corotational method and an updated Lagrangian mixed variational method are proposed to compute the tangent stiffness matrix. The tangent inertia matrices, including the gyroscopic and centrifugal terms, are also obtained by using the tangent maps of rotation. The numerical examples analyzed in this paper include static (pre- and postbuckling) and dynamic analysis of flexible beams structures. The new finite elements show a very good performance, in terms of fewer number of elements used and accuracy during the simulation, both for static and dynamic problems.
11 citations
01 Jan 1996
TL;DR: In this article, the stiffness of a conservative mechanical system is described by a ( 2 0 ) tensor and components of the Cartesian stiffness matrix are given by evaluating this tensor on a pair of basis twists.
Abstract: In the paper we study the Cartesian stiffness matrix using methods of differential geometry. We show that the stiffness of a conservative mechanical system is described by a ( 2 0 ) tensor and that components of the Cartesian stiffness matrix are given by evaluating this tensor on a pair of basis twists. Our formulation leads to three important results: (a) The stiffness matrix does not depend on the parameterization of the manifold; (b) The stiffness matrix depends on the choice of a connection on the manifold; and (c) The standard definition of the Cartesian stiffness matrix assumes an asymmetric connection and this is the reason that the matrix is, in general, asymmetric.
8 citations
TL;DR: The tangent stiffness matrices developed in this paper represent the final conditions of the material, such as the most recent deformed geometry, the current total stress level, the amount of plastic strains present, as well as the current material constants in the form of either the tangent or secant modulus of elasticity.
8 citations
TL;DR: In this paper, the authors discuss solution schemes for the incremental elastic-plastic structural problem, discretized by means of the Finite Element method, focused on their formulation and implementation in a parallel computing environment.
01 Jan 1996
TL;DR: In this paper, a non-linear finite element method is used for the analysis of flexible rotating mechanical systems. But it is shown that certain terms can be neglected in the derivation of the tangent stiffness matrix.
Abstract: The paper discusses some recent advances in the non-linear finite element method for the analysis of structures. In particular, the paper concentrates on beams and shells. It starts with statics and describes a new co-rotational formulation which relies on a matrix defining the relationship between the pseudo-vector associated with the spin of the local element frame and the changes in the global displacement variables. It is shown that certain terms can be neglected in the derivation of the tangent stiffness matrix. The paper moves on to consider implicit non-linear dynamics and describes various energy conserving or approximately energy conserving techniques. It shows, that in contrast to the widely used trapezoidal Newmark method (and also the α method), these techniques are very stable in the non-linear regime. The dynamic solution procedures are used in conjunction with a set of non-linear master-slave relationships which allow the modelling of joints. Using these various techniques, the finite element method becomes an attractive tool for the analysis of flexible rotating mechanical systems.
TL;DR: In this paper, Ishihara et al. showed that the geometric stiffness of large strain in terms of the variation of the deformation gradient tensor before rigid-body rotation becomes non-symmetric away from equilibrium.
Abstract: Finite-element analysis of a 3-dimensional large-rotation problem of a beam has been made possible by means of the nonlinear beam model developed by Simo and Vu-Quoc. Their beam model is regarded as geometrically exact, and is described by a configuration manifold which involves the rotation group. As a result, the tangent stiffness matrix becomes non-symmetric away from equilibrium. However, it have been shown in Ishihara that the tangent stiffness of semitangential rotation became symmetric, and by analyzing of fundamental problem, it was proven that symmetric tangent stiffness was valid. The beam model was assumed to have large rotation and small strain. Next, it was shown in Ishihara that the geometric stiffness of large strain in terms of the variation of the deformation gradient tensor before rigid-body rotation. However, that formulation was a total lagrangian formulation parametrized by the coordinate of the centerline of the reforence configuration. It was also shown in Ishihara that the large-strain beam model of an updated lagrangian formulation pararetrized by the coordinate of the centerline of the current configuration. Here, it will be shown that numerical examples of large-rotation and largestrain analysis of a beam.
TL;DR: In this paper, a finite element anlaysis is performed for large deformations of a felxible beam, which involves finite rotations from the Eulerian point of view.
Abstract: A finite element anlaysis is performed for large deformations of a felxible beam. The total Lagrangian formulation for a general large deformation, which involves finite rotations, is chosen and the exponential map is used to treat finite rotations from the Eulerian point of view. The finite elements results are confirmed for several cases of deformations through comparison to a first order elasticity solution obtained by numerical integration, and the agreement between the two is found to be excellent. For lateral buckling, the point of vanishing determinant of the resulting unsymmetric tangent stiffness is traced to examine its relationship to bifurcation points. It is found that the points of vanishing determinant is not corresponding to bifurcation points for large deformation in general, which suggests that the present unsymmetric tangent stiffness is not an exact first derivative of internal forces with respect to displacement.
TL;DR: In this paper, the relationship between the stress rate and the tangent stiffness of a beam element was shown, and it was shown that all tangent stiffnesses have the second variation of velocity gradient tensor because of SO(3.
Abstract: Three-dimensional large-rotation analysis of a beam by the finite-element method is possible using a beam element with the special orthogonal group SO(3). It is known that the stress rate used in the constitutive equation governs the tangent stiffness. Therefore, in this paper, the relationship between the stress rate and the tangent stiffness will be shown. The stress rates used are Truesdell stress rate, Jaumann stress rate, Neo-Green stress rate and Ishihara stress rate. It will also be shown that all tangent stiffnesses have the second variation of velocity gradient tensor because of S0(3), and the tangent stiffness of Ishihara stress rate has only material stiffness and geometric stiffness of rigid rotation. In the upcoming paper, tangent stiffnesses of those stress rates of a beam element will be shown.
01 Jan 1996
TL;DR: In this paper, the effects of both transverse shear deformation and Saint-Venant's torsional deformation are taken into account in addition to that of bending and axial deformations.
Abstract: Publisher Summary
This chapter presents a versatile numerical method for the geometric and material nonlinear analysis of space frames. The geometric nonlinearity is precisely considered by using a corotational technique, where no restrictions are needed on the magnitude of finite rotations in space. In the beam element, the effects of both transverse shear deformation and Saint-Venant's torsional deformation are taken into account in addition to that of bending and axial deformations. With the help of the theorem of virtual work, a symmetric incremental tangent stiffness matrix is derived for the beam element to improve the computational efficiency of the nonlinear analysis. A numerical example is described to demonstrate the accuracy of the proposed method.