Showing papers on "Tangent stiffness matrix published in 1999"
TL;DR: In this article, a two-node catenary cable element, derived using exact analytical expressions for the elastic catenary, is proposed for the modeling of cables, and the cable element tangent stiffness matrix and internal force vector are evaluated accurately and efficiently using an iterative procedure.
196 citations
TL;DR: In this paper, a method for estimating the tangent stiffness matrix and the internal stress vector of a four-strut tense grity system with large deformation and displacements is presented.
131 citations
TL;DR: In this article, a thermodynamic framework is proposed as a basis of a nonlocal associative plasticity theory in which the plastic yielding laws comply with a (nonlocal) maximum intrinsic dissipation theorem.
Abstract: Elastic-plastic rate-independent materials with isotropic hardening/softening of nonlocal nature are considered in the context of small displacements and strains. A suitable thermodynamic framework is envisaged as a basis of a nonlocal associative plasticity theory in which the plastic yielding laws comply with a (nonlocal) maximum intrinsic dissipation theorem. Additionally, the rate response problem for a (continuous) set of (macroscopic) material particles, subjected to a given total strain rate field, is discussed and shown to be characterized by a minimum principle in terms of plastic coefficient. This coefficient and the relevant continuum tangent stiffness matrix are shown to admit, in the region of active plastic yielding, some specific series representations. Finally, the structural rate response problem for assigned load rates is studied in relation to the solution uniqueness, and two variational principles are provided for this boundary value problem.
63 citations
TL;DR: This work uses balycentric coordinates in order to efficiently track cell-to-cell movement of the tangent curves and produces exact values based upon piecewise linear variation over a tetrahedrization of the domain of interest.
Abstract: We present some very efficient and accurate methods for computing tangent curves for three-dimensional flows. Our methods work directly in physical coordinates, eliminating the usual need to switch back and forth with computational coordinates. Unlike conventional methods, such as Runge-Kutta, for computing tangent curves which give only approximations, our methods produce exact values based upon piecewise linear variation over a tetrahedrization of the domain of interest. We use balycentric coordinates in order to efficiently track cell-to-cell movement of the tangent curves.
57 citations
Journal Article•
TL;DR: In this article, the stabilizing process of unstable structures by using dynamic relaxation method is presented, which is the energy minimization technique that searches the static equilibrium state by simple vector iteration method.
Abstract: In this paper the stabilizing process of unstable structures by using dynamic relaxation method is presented. The process of installing the unstable structures by introducing the prestress is called stabilizing process. The unstable structures such as cable, pneumatic structures or cable domes initially behaves to be unstable state because of having no initial bending stiffness. The dynamic relaxation method is the energy minimization technique that searches the static equilibrium state by simple vector iteration method. In the dynamic relaxation method the tangent stiffness matrix of structure does not need to be assembled to analyze the stabilizing process during each iteration. Thus, computational effort and time can be reduced. The finite difference integration technique is used to integrate the dynamic equilibrium equation for static equilibrium state. Several numerical examples to confirm the efficiency and applicability of dynamic relaxation method.
50 citations
TL;DR: In this article, a quadratically convergent algorithm for the computation of stability points (limit and bifurcation points) in the finite element formulation of the nonlinear structural mechanics is presented.
50 citations
TL;DR: In this article, an incremental elasto-plastic finite element method based on an updated Lagrangian formulation in which a sliding-sticking friction mode is specially considered was proposed to predict the buckling process and load-deflection curves for energy dissipation capacity during the design stage, before trials.
45 citations
TL;DR: In this article, the symmetry of the element tangent stiffness matrix of a spatial elastic beam has been a subject of debate, and it is shown that the stiffness matrix is invariably asymmetric.
Abstract: In the literature, the symmetry of the element tangent stiffness matrix of a spatial elastic beam has been a subject of debate. The symmetry of the tangent stiffness matrices derived by some researchers are tenuously attributed to the use of Lagrangian formulations, while the asymmetry of corotational tangent stiffness matrices is commonly attributed to the noncommutativity of spatial rotations. In this paper, the inconsistency regarding the symmetry of element tangent stiffness matrices formulated in the Lagrangian and the corotational frameworks is resolved. It is shown that, irrespective of the formulation framework, the element tangent stiffness matrix is invariably asymmetric. A “correction matrix” that enforces the proper rotational behavior of nodal moments into the conventional geometric stiffness matrix of an Updated Lagrangian spatial beam element is presented. It is demonstrated through a numerical example that adoption of this correction matrix is necessary for the detection of the lowest buck...
31 citations
TL;DR: In this article, a methodology for computationally efficient formulation of the tangent stiffness matrix consistent with incrementally objective algorithms for integrating finite deformation kinematics and with closest point projection algorithms for integration material response is developed.
Abstract: A methodology for computationally efficient formulation of the tangent stiffness matrix consistent with incrementally objective algorithms for integrating finite deformation kinematics and with closest point projection algorithms for integrating material response is developed in the context of finite deformation plasticity. Numerical experiments illustrate an excellent performance of the proposed formulation in comparison with other algorithms. Copyright © 1999 John Wiley & Sons, Ltd.
27 citations
Journal Article•
TL;DR: In this paper, a geometrically nonlinear nine-node assumed strain shell element with six DOF per node was used to estimate the torsional buckling loads of composite cylinders with angle plies.
Abstract: Buckling of composite cylinders under torsion has been analyzed using the geometrically nonlinear nine-node assumed strain shell element with six DOF per node. Buckling loads are pinpointed by finding the lowest torsion load that makes the determinant of the global tangent stiffness matrix zero in the course of the geometrically nonlinear analysis. Torsional buckling loads calculated from the present method agree well with the linear buckling loads obtained from various shell theories for most composite cylinders. For cylinders with [30/30/-60]s ply orientation, however, large discrepancy between the two buckling loads has been observed. Comparing the numerical test results with experimental data, we conclude that the present method can estimate the torsional buckling loads of composite cylinders with angle plies. The present paper may provide new reference solutions for torsional buckling loads of composite cylinders
20 citations
TL;DR: In this paper, a triangular flat finite element for the analysis of thin shells which undergo large displacements is proposed, based upon the geometrically nonlinear theory of von Karman for thin plates and the total Lagrangian approach.
Abstract: A triangular flat finite element for the analysis of thin shells which undergo large displacements is proposed. It is based upon the geometrically nonlinear theory of von Karman for thin plates and the total Lagrangian approach. It has a total of only twelve degrees of freedom, namely, three translations at each vertex and one rotation at each mid-side. The stiffness matrix and the tangent stiffness matrix are derived explicitly. The element is tested against nonlinear patch test solutions and its performance is evaluated by solving several standard problems. The directional derivatives of the potential energy function required for the stability analysis are also provided.
TL;DR: In this paper, the Generalized Method of Cells (GMC) is extended to deal with plasticity problems. But the authors use a tangent plasticity matrix approach as opposed to an initial stiffness matrix approach.
Abstract: A recently developed computationally efe cient implementation of the Generalized Method of Cells is extended here to deal with plasticity problems. The new formulation can be used to perform linear elastic and plastic micromechanical analysis of composites with complex microstructures. It makes use of a tangent plasticity matrix approach as opposed to an initial stiffness matrix approach employed in the original Generalized Method of Cells. This tangent stiffness matrix approach makes it possible to perform plastic analyses of composites with microstructures that require a high-resolution, cell ‐subcell model. To demonstrate the capabilities of the new formulation, it is applied to the problem of determining the ine uence of e ber shape and e ber architecture on the plastic behavior of two-phase unidirectional composites. Some results from previous studies on this subject are cone rmed. New results that could not be obtained with previous implementations are also reported. Problems modeled with up to 10,000 subcells are presented.
TL;DR: Based on analytical equations, a catenary element is presented for the finite element analysis of cable structures and a program with the proposed element is of less computer time and better accuracy.
Abstract: Based on analytical equations, a catenary element is presented for the finite element analysis of cable structures. Compared with usually used element (3-node element, 5-node element), a program with the proposed element is of less computer time and better accuracy.
TL;DR: The non-linear Q1-P0 method as mentioned in this paper is a modified version of the three-field mean dilatation method, which has been shown to be over-stiff in comparison to the original procedure.
Abstract: The mean dilatation method effectively involves bi-linear displacements and a constant pressure and is often known as the Q1-P0 formulation. Its non-linear implementation was originally derived as a three-field formulation which included the volume ratio via the Jacobian, J, of the deformation gradient as an additional separate variable. However, the latter term was not directly required in the numerical implementation once J was assumed constant along with the pressure. This formulation will here be termed the non-linear Q1-P0 method. It is known to give good solutions for many practical large-strain elasto-plastic problems. However, for some problems, it has been shown to be prone to severe ‘hour-glassing’. With a view to remedying this situation, we here re-visit the three-field formulation and derive a modified form, which although variationally valid, is over-stiff in comparison to the original procedure (here simply called the Q1-P0 method). However, the concepts lead to a natural method for stabilising the Q1-P0 technique. The associated tangent stiffness matrix is symmetric. Copyright © 1999 John Wiley & Sons, Ltd.
TL;DR: In this article, an error estimator of the flux projection type, based on the mechanical deformation power, is developed to facilitate the study of these results, which is also used as a refinement criterion for the FE mesh.
Abstract: The present paper proposes a procedure for the resolution of non‐linear structural problems. It includes a study of the reliability of the results and the adaptive meshing. The iterative phase of the solution of the equilibrium equations entails an adaptive strategy for updating the tangent stiffness matrix, with a control of the load step. This results in a higher rate of convergence for the iterative process. The mechanical deformation processes here considered may give rise to considerable geometric distortion in the finite elements of the mesh. If they do, the consequence will be not only that the FE analysis fails to yield precise results, but also, owing to problems deriving from the numerical ill‐conditioning, that continuation may be impractical. To facilitate the study of these results, we developed an error estimator of the flux projection type, which is based on the mechanical deformation power. It is also used as a refinement criterion for the FE mesh. Distorted meshes can be fully or partiall...
TL;DR: In this article, the analysis of the tangent stiffness matrix is extended to nonconvex strain energy functionals, namely to material softening, and convex and nonlinear external potential due to deformation-sensitive loading devices.
Abstract: This paper surveys the tangent stiffness matrix as the basis for nonlinear structural analysis. Systematic derivation of the family of tangent stiffness matrices and the possible linearization and approximation aspects are discussed. The analysis of the tangent stiffness matrix is extended to nonconvex strain energy functionals, namely to material softening. Moreover it is extended to convex and nonconvex external potential due to deformation-sensitive loading devices.
TL;DR: In this paper, a class of rational curves with an ordinary singular point satisfying the condition required in [GO] and whose tangent cone is always reduced was constructed, and it was shown that the tangent cones are not always reduced.
Abstract: We study a class of rational curves with an ordinary singular point, which was introduced in [GO]. We find some conditions under which the tangent cone is reduced and we show that the tangent cone is not always reduced. We construct another class of rational curves with an ordinary singular point satisfying the condition required in [GO] and whose tangent cone is always reduced.
TL;DR: In this article, a continuum-based shape design sensitivity analysis (DSA) method for 2D solid components with rate-independent elasto-plastic material is presented, where the design sensitivity equation is derived using the incremental form of the equilibrium equation and increments of the static response with respect to shape design variables.
Abstract: A continuum-based shape design sensitivity analysis (DSA) method is presented for 2-D solid components with rate-independent elasto-plastic material. The material derivative of continuum mechanics is utilized to develop a continuum-based shape DSA method. The design sensitivity equation is derived using the incremental form of the equilibrium equation and increments of the static response with respect to shape design variables. The direct differentiation method is utilized to obtain the first-order variation of the performance measure explicitly in terms of variations of shape design variables. With the consistent tangent stiffness matrix employed at the end of each load step to compute the design sensitivity, the method does not require iterations to compute the design sensitivity. Numerical results are presented for a hollow cylinder model and a membrane with a hole model to validate the proposed DSA method.
TL;DR: Tangent stiffness equations for a beam-column, which is subjected to either uniformly or sinusoidally distributed lateral loads, are presented in this article, where the equations have been derived by differentiating the slop-deflection equations under axial forces for a member.
Abstract: Tangent stiffness equations for a beam-column, which is subjected to either uniformly or sinusoidally distributed lateral loads, are presented. The equations have been derived by differentiating the slop-deflection equations under axial forces for a member. Thus, the tangent stiffness equations take into consideration axial forces, bowing effect, and laterally distributed loads. As a numerical example, elastic buckling behavior of parallel chord latticed beams with laterally distributed loads is investigated to compare the results obtained from the present method with those from the conventional matrix method in which the distributed loads are considered as a series of concentrated loads at additional intermediate nodes of a member. Furthermore, buckling tests were carried out to confirm the equations derived as well as to clarify the buckling behavior of space frame structures. In conclusion, it can be said that the new equations can provide a good efficient way of estimating the equilibrium paths and buckling loads. They can also lead to a significant savings in core storage and computing time required for the analysis of space frame structures.
01 Jan 1999
TL;DR: In this article, an imperfect beam-column element for second-order analysis of two-and three-dimensional frames is presented, where the initial imperfection of element is restricted to a curvature in the form of a single sinusoidal half-wave.
Abstract: Publisher Summary
This chapter presents the imperfect beam-column element for second-order analysis of two- and three-dimensional frames. The initial imperfection of element is restricted to a curvature in the form of a single sinusoidal half-wave. The simplest and most typical stiffness matrix method of analysis is to extend the cubic Hermite element to the nonlinear case by inclusion of the geometric stiffness to the linear stiffness matrix, to form the tangent stiffness matrix. The method of stability function is employed as an exact solution of the beam-column. The method develops the element matrix by solving the differential equilibrium equation of a beam-column under the action of axial load. The accuracy of the analysis using stability function is affected only by the numerical truncating error. The exact stiffness matrix of an imperfect member under a large axial force is derived and incorporated into a second-order analysis computer program, Nonlinear Integrated Design and Analysis (NIDA), for analysis of skeletal structures. The element is accurate even when the axial force is four times the Euler's buckling load, which refers to the extreme case of buckling in a column with both ends fixed in direction and in rotation.
01 Jan 1999
TL;DR: A dynamic model for gear transmissions acting around a static working point, and taking into account the complete mechanical components is described, providing global dynamic behaviour of automobile gearboxes and consequences of technological choices could be evaluated.
Abstract: This paper describes a dynamic model for gear transmissions acting around a static working point, and taking into account the complete mechanical components. Gearbox casing behavior is introduced by using substructure analysis, and a tangent stiffness matrix could be defined for each roller body bearing element. This model is used to study the dynamic behaviour of an automobile gearbox. The first studies highlight the influence of the roller bearings on the dynamic behaviour of the kinematic chain and show that the bearings have to be modelled accurately. In a second step, the flexible casing of the gearbox is taken into account and dynamic couplings between deformations of the casing and the kinematic chain are evaluated. As a result of these works, global dynamic behaviour of automobile gearboxes can be provided and consequences of technological choices could be evaluated.
01 Jan 1999
TL;DR: This chapter reviews the formulation and logic for some of the more commonly accepted and used non-linear solution methods for displacement control.
Abstract: Publisher Summary
This chapter reviews the formulation and logic for some of the more commonly accepted and used non-linear solution methods. The relative slope of the tangent along the equilibrium path is always used as a yardstick to check the state of stiffness of a structure. In some methods, when this relative stiffness is less than a certain value of its original stiffness, the method is converted to a pure incremental method, and when the stiffness is large, the iterative procedure can be activated since divergence is unlikely. The displacement control method possesses the capacity of traversing the limit point without destroying the symmetrical property of the tangent stiffness matrix. The arc-length method is based on using the complete displacement vector for the control of the advance of the equilibrium path. The arc distance, taken as the dot product of the displacement vectors, is fixed in a particular load cycle. The objective of an iterative process is to eliminate the residual displacement or the unbalanced forces, instead of meeting the constant work done or the constant arc-distance.
01 Jan 1999
TL;DR: In this paper, the elastic-plastic tangent stiffness matrix of three-dimensional degenerated shell element is derived by the use of updated Lagrangian formulation and the postbuckling equilibrium path of the web plates of girders is traced by cylindrical arc-length Method.
Abstract: Publisher Summary
This chapter investigates the postbuckling analysis of web plates of girders by three-dimensional degenerated shell element method. The elastic-plastic tangent stiffness matrix of three-dimensional degenerated shell element is derived by the use of updated Lagrangian formulation and the postbuckling equilibrium path of the web plates of girders is traced by cylindrical arc-length Method. The Nonlinear Postbuckling Analysis for Web Plates of Girders (NPAWPG) is designed using FORTRAN 90 on the desktop of Powerstation 4.0 under Windows 95 and some typical examples are analyzed to fully prove that the program is reliable and in common use. The experimental results of twelve plate girders and three box girders have been compared with their theoretical results respectively, and the errors range from one percent to fourteen percents. The theoretical results have also been compared with the results of European Code, International Standard Organization, and American Code. It is found that the results of American Code are higher than the theoretical results, but the other two results are lower than the corresponding theoretical results.
01 Aug 1999
01 Jan 1999
TL;DR: In this article, the second-order elastic analysis of framed structures using the Newton-Raphson method is introduced, which is based on a mathematical idealization that the structure deforms only in the loaded directions until the bifurcation load is reached.
Abstract: Publisher Summary
This chapter introduces the second-order elastic analysis of framed structures using the Newton-Raphson method. Bifurcation analysis is based on a mathematical idealization that the structure deforms only in the loaded directions until the bifurcation load is reached. As a result, the element stiffness is only a function of force in the structural members and other basic material and geometrical properties before loaded. All structures contain a certain degree of initial imperfection and the loads are not perfectly concentric so that deflections, however small, occur in directions other than the loading plane. The increase or decrease of load can be monitored by the sign of the determinant of the tangent stiffness matrix. In the iterative process, a deflection increment due to the unbalanced force is added to the nodal coordinates of the structure to correct the structural geometry in response to the unbalanced force and the value for total deflection. The Newton–Raphson method gives the response of a structure allowing for various second-order effects at a specified design load level, normally taken as the design load.