Showing papers on "Tangent stiffness matrix published in 2008"
TL;DR: In this paper, the effect of Rayleigh proportional damping in the analysis of inelastic structural systems is investigated, and it is shown that when the stiffness portion of the system damping matrix is based on the original system stiffness, artificial damping is generated when the structure yields.
Abstract: This paper investigates the consequence of using Rayleigh proportional damping in the analysis of inelastic structural systems. The discussion is presented theoretically, as well as by example through the analysis of a simple five-story structure. It is shown that when the stiffness portion of the system damping matrix is based on the original system stiffness, artificial damping is generated when the structure yields. When the damping matrix is based on the tangent stiffness but the Rayleigh proportionality constants are based on the initial stiffness, a significant but reduced amplification of damping occurs. When the damping is based on the tangent stiffness and on updated frequencies based on this stiffness, virtually no artificial damping occurs. The paper also investigates the influence on effective damping when localized yielding occurs in areas of concentrated inelasticity. In such cases, it is possible to develop artificial viscous damping forces that are extremely high, but that are not easy to detect. Such artificial damping forces may lead to completely invalid analysis. The paper ends with recommendations for performing analysis where the artificial damping is eliminated, or at least controlled.
279 citations
TL;DR: This work presents an approach, based on work by Miehe, for an efficient numerical approximation of the tangent moduli that can be easily implemented within commercial FE codes and will facilitate the incorporation of novel hyperelastic material models for a soft tissue behavior into commercial FE software.
Abstract: Finite element (FE) implementations of nearly incompressible material models often employ decoupled numerical treatments of the dilatational and deviatoric parts of the deformation gradient. This treatment allows the dilatational stiffness to be handled separately to alleviate ill conditioning of the tangent stiffness matrix. However, this can lead to complex formulations of the material tangent moduli that can be difficult to implement or may require custom FE codes, thus limiting their general use. Here we present an approach, based on work by Miehe (Miehe, 1996, "Numerical Computation of Algorithmic (Consistent) Tangent Moduli in Large Strain Computational Inelasticity," Comput. Methods Appl. Mech. Eng., 134, pp. 223-240), for an efficient numerical approximation of the tangent moduli that can be easily implemented within commercial FE codes. By perturbing the deformation gradient, the material tangent moduli from the Jaumann rate of the Kirchhoff stress are accurately approximated by a forward difference of the associated Kirchhoff stresses. The merit of this approach is that it produces a concise mathematical formulation that is not dependent on any particular material model. Consequently, once the approximation method is coded in a subroutine, it can be used for other hyperelastic material models with no modification. The implementation and accuracy of this approach is first demonstrated with a simple neo-Hookean material. Subsequently, a fiber-reinforced structural model is applied to analyze the pressure-diameter curve during blood vessel inflation. Implementation of this approach will facilitate the incorporation of novel hyperelastic material models for a soft tissue behavior into commercial FE software.
96 citations
01 Jan 2008
TL;DR: In this article, the Cartesian stiffness matrix of parallel compliant mechanisms is presented, which is more general than any other stiffness matrix found in the literature since it takes into account the stiffness of the passive joints and remains valid for large displacements.
Abstract: Starting from the definition of a stiffness matrix, the authors present the Cartesian stiffness matrix of parallel compliant mechanisms. The proposed formulation is more general than any other stiffness matrix found in the literature since it can take into account the stiffness of the passive joints and remains valid for large displacements. Then, the conservative property, the validity,and the positive definiteness of this matrix are discussed.
64 citations
TL;DR: In this article, a numerical algorithm for nonlinear thermo-mechanical viscoelastic analyses of orthotropic composite materials and structures that follow thermo rheologically complex behaviors is presented.
59 citations
TL;DR: In this paper, the authors investigated the behavior of thin-walled beams with open section in the presence of large torsion and found that the bifurcation points are in accordance with nonlinear stability solutions.
54 citations
TL;DR: In this article, the effect of volume fraction of the constituent materials in the mechanical behavior of FGM plates and shells is investigated, where the material properties are assumed to be varied in the thickness direction according to a sigmoid function in terms of the volume fraction.
Abstract: The four-node quasi-conforming shell element was extended in the present article to the case of geometrically non-linear behavior of the FGM plates and shells. The high stress occurring in the FGM structures will affect its integrity and the structures is susceptible to failure. Therefore, we focus on the effect of volume fraction of the constituent materials in the mechanical behavior of FGM plates and shells. The material properties are assumed to be varied in the thickness direction according to a sigmoid function in terms of the volume fraction of the constituents. The series solutions of sigmoid FGM (S-FGM) plates, based on the first-order shear deformation theory and Fourier series expansion are provided as the reference solution for the numerical results. In quasi-conforming formulation, the tangent stiffness matrix is explicitly integrated. This makes the element computationally efficient in the non-linear analysis. Several selected examples of non-linear analysis of FGM shells are included in the...
41 citations
TL;DR: In this paper, a nonlinear viscoelastic constitutive model is expressed with an integral form of a creep function, whose initial, long-term, and transient properties change with stresses and temperatures.
40 citations
TL;DR: In this article, a 9-node co-rotational curved quadrilateral shell element formulation is presented, where the internal force vector and the element tangent stiffness matrix are respectively the first derivative and the second derivative of the element strain energy with respect to the nodal variables.
Abstract: A new 9-node co-rotational curved quadrilateral shell element formulation is presented in this paper. Different from other existing co-rotational element formulations: (1) Additive rotational nodal variables are utilized in the present formulation, they are two well-chosen components of the mid-surface normal vector at each node, and are additive in an incremental solution procedure; (2) the internal force vector and the element tangent stiffness matrix are respectively the first derivative and the second derivative of the element strain energy with respect to the nodal variables, furthermore, all nodal variables are commutative in calculating the second derivatives, resulting in symmetric element tangent stiffness matrices in the local and global coordinate systems; (3) the element tangent stiffness matrix is updated using the total values of the nodal variables in an incremental solution procedure, making it advantageous for solving dynamic problems. Finally, several examples are solved to verify the reliability and computational efficiency of the proposed element formulation.
27 citations
TL;DR: In this paper, a theoretical and numerical model for thin-walled beams with open cross-section in the presence of large torsion is presented, which takes into account for shortening effects, pre-buckling deformation and flexural-torsional coupling.
Abstract: The aim of the present paper is to investigate a theoretical and numerical model which is able to study the behaviour of thin-walled beams with open cross section in presence of large torsion. The presented model takes into account for large torsion, linear and non-linear warping currently named shortening effects, pre-buckling deformation and flexural–torsional coupling. In numerical analysis, a 3D beam with two nodes and seven degrees of freedom per node is adopted. The equilibrium equations and the material behaviour are derived in discrete form without assumption on torsion angle amplitude. Due to large torsion context, all the equilibrium equations are non-linear and highly coupled. The linear behaviour is made possible by disregarding non-linear terms. For non-linear behaviour and stability, the tangent stiffness matrix is carried out. Due to large torsion context, new matrices are present. The element is incorporated in a homemade finite element code. Newton–Raphson iterative methods are used with different control parameters. In order to prove the efficiency of the model many examples are presented in linear and non-linear behaviour with presence of bifurcations.
25 citations
TL;DR: In this article, an exact dynamic stiffness matrix is established for an elastically connected three-beam system, which is composed of three parallel beams of uniform properties with uniformly distributed-connecting springs among them.
24 citations
TL;DR: In this paper, the applicability of a mesh-free approximation method, namely the EFG method, on fully geometrically exact analysis of plates is investigated, based on a unified nonlinear theory of plates, which allows for arbitrarily large rotations and displacements, a Galerkin approximation via MLS functions is settled.
Abstract: The applicability of a meshfree approximation method, namely the EFG method, on fully geometrically exact analysis of plates is investigated. Based on a unified nonlinear theory of plates, which allows for arbitrarily large rotations and displacements, a Galerkin approximation via MLS functions is settled. A hybrid method of analysis is proposed, where the solution is obtained by the independent approximation of the generalized internal displacement fields and the generalized boundary tractions. A consistent linearization procedure is performed, resulting in a semi-definite generalized tangent stiffness matrix which, for hyperelastic materials and conservative loadings, is always symmetric (even for configurations far from the generalized equilibrium trajectory). Besides the total Lagrangian formulation, an updated version is also presented, which enables the treatment of rotations beyond the parameterization limit. An extension of the arc-length method that includes the generalized domain displacement fields, the generalized boundary tractions and the load parameter in the constraint equation of the hyper-ellipsis is proposed to solve the resulting nonlinear problem. Extending the hybrid-displacement formulation, a multi-region decomposition is proposed to handle complex geometries. A criterium for the classification of the equilibrium's stability, based on the Bordered–Hessian matrix analysis, is suggested. Several numerical examples are presented, illustrating the effectiveness of the method. Differently from the standard finite element methods (FEM), the resulting solutions are (arbitrary) smooth generalized displacement and stress fields.
TL;DR: In this paper, the effect of inclination on the static behaviors of inclined variable-arc-length (VAL) beams has been developed via the variational approach, and the critical values of uniform self-weight of the inclined VAL beams are obtained by equating the determinant of the tangent stiffness to zero.
TL;DR: In this article, the authors investigated the application of topology optimization to electromechanical microdevices for the purpose of delaying this unstable behavior by maximizing their pull-in voltage by using a monolithic finite element formulation combined with a normal flow algorithm.
TL;DR: In this paper, a method for online estimation of experimental tangent stiffness is proposed, which is tailored for fast online applications by transforming the measurements into a coordinate system, which reduces the number of unknown stiffness coefficients that need to be updated during the simulation and is used in a modified operator splitting integration scheme to improve the accuracy of hybrid simulations with highly nonlinear experimental substructures.
Abstract: Improved numerical integration procedures are essential for the extension of hybrid numerical and experimental simulation to large and complex structural systems While implicit integration algorithms are widely used in pure numerical simulations for their superior stability and accuracy, their direct application to hybrid simulation has been partially limited by difficulties in estimating the tangent stiffness matrix of multi-degree-of-freedom experimental substructures Current applications of hybrid simulation using integrators with improved stability have mostly resorted to methods that are noniterative or utilize the initial stiffness matrix for iterative corrections To improve the accuracy of integration procedures for hybrid simulation, a new method for online estimation of experimental tangent stiffness is proposed The stiffness estimation procedure is tailored for fast online applications by transforming the measurements into a coordinate system, which reduces the number of unknown stiffness coefficients that need to be updated during the simulation The updated experimental stiffness matrix is used in a modified operator-splitting integration scheme to improve the accuracy of hybrid simulations with highly nonlinear experimental substructures The application and effectiveness of the proposed approach is demonstrated through hybrid simulations with multi-degree-of-freedom experimental substructures
15 Oct 2008
TL;DR: The formula of general stiffness model at tool tip is derived from virtual-work principle and point transformation matrix method, which indicates the transformation relationship of elastic displacement on both ends of flexible axes as well as the transformed relationship of force.
Abstract: In this paper, a new method for computing general stiffness model at tool tip for multi-axis machine tool is presented. The formula of general stiffness model at tool tip is derived from virtual-work principle and point transformation matrix method. Point transformation matrix method indicates the transformation relationship of elastic displacement on both ends of flexible axes as well as the transformation relationship of force. By multiplying the point transformation matrix, the final stiffness matrix at tool tip will be obtained efficiently. In this modeling method, the final stiffness matrix is composed of local compliance matrixes according to their contribution to the final stiffness matrix.
TL;DR: In this article, a generalized plane strain implicit formulation of the cross-sectional expansion of an extruded aluminum tube with pressurized fluid to fill a hydroforming die is presented, and the axial feed is implemented by imposing either a compressive force or strain in the tube length direction.
01 Jan 2008
TL;DR: In this paper, a finite element method has been employed to obtain a three-dimensional mathematical model for the bridge system, where the vehicle is modeled as a threeaxle six-wheel system and the dynamic response is evaluated using the modal method and the step-by-step integration method.
Abstract: Moving vehicle loads, associated with roadway traffic can induce significant dynamic effects in the structural behavior of bridges, especially for long-span bridges. The main objective of this research is to study traffic induced dynamic responses of long-span bridges, particular for box-girder bridges and cable-stayed bridges.
The finite element method has been employed in this study to obtain a three-dimensional mathematical model for the bridge system. For box-girder bridges, the deck, the box girders and the diaphragms are modeled as 8-node shell element. Linear elastic structural responses are considered. For cable-stayed bridges, the box-girder bridge deck and diaphragms are modeled as 8-node shell element, while the pylons are modeled as beam element. Stability functions are applied to both shell element and beam element to account for geometric nonlinearity. A two-node catenary cable element is adopted using exact analytical expressions for modeling stayed cables. Non-linear structural responses are considered.
For the present analytical study of the bridge-vehicle system, the vehicle is modeled as a three-axle six-wheel system. The equation derivations are described in details. The dynamic response is evaluated using the modal method and the step-by-step integration method. The natural frequencies and mode shapes of the bridge are obtained based on the deformed dead load tangent stiffness matrix. For cable-stayed bridge, an iterative scheme is utilized to obtain the deformed dead load tangent stiffness matrix due to its nonlinear characteristics.
The dynamic responses of bridges subjected to traffic loads are complicated because the dynamic effect induced by moving vehicles is basically influenced by the interaction between the bridge and the moving vehicle. The bridge-vehicle interaction is affected by many factors, such as vehicle speed, road roughness, damping of bridge and vehicle, dynamic characteristics of bridge, and etc. Parametric study has been carried out to investigate those factors that influence the bridge-vehicle interaction.
The finite element computer program GLBA in FORTRAN 90 and MATLAB is developed to carry out the proposed research. GLBA includes four main modules: static linear analysis, static non-linear analysis, free vibration analysis and dynamic analysis due to traffic load.
TL;DR: In this paper, an optimization approach for force design of tensegrity structures by enumeration of the vertices of the feasible region of the prestresses was presented, which is defined as the linear combinations of the coefficients of the self-equilibrium force vectors.
Abstract: An optimization approach is presented for force design of tensegrity structures by enumeration of the vertices of the feasible region of the prestresses, which is defined as the linear combinations of the coefficients of the self-equilibrium force vectors. The unilateral properties of the stresses in cables and struts are taken into consideration. In order to design the stiffest structure against uncertain external loads as well as specific external loads, a multiobjective optimization problem is formulated for simultaneous maximization of the lowest eigenvalue of the tangent stiffness matrix and minimization of the compliance against a specified set of external loads. In the numerical example, Pareto optimal solutions are found by enumerating the vertices of the feasible region of prestresses of a tensegrity grid, and the monotonicity properties of the objective functions are investigated.
Posted Content•
TL;DR: In this article, a variational multiscale formulation of the Navier-Stokes equations in 3D is presented. But the main contributions of this work are a systematic study of the variational multi-scale method for three-dimensional problems, and an implementation of a consistent formulation suitable for large problems with high nonlinearity, unstructured meshes, and non-symmetric matrices.
Abstract: In the following paper, we present a consistent Newton-Schur solution approach for variational multiscale formulations of the time-dependent Navier-Stokes equations in three dimensions. The main contributions of this work are a systematic study of the variational multiscale method for three-dimensional problems, and an implementation of a consistent formulation suitable for large problems with high nonlinearity, unstructured meshes, and non-symmetric matrices. In addition to the quadratic convergence characteristics of a Newton-Raphson based scheme, the Newton-Schur approach increases computational efficiency and parallel scalability by implementing the tangent stiffness matrix in Schur complement form. As a result, more computations are performed at the element level. Using a variational multiscale framework, we construct a two-level approach to stabilizing the incompressible Navier-Stokes equations based on a coarse and fine-scale subproblem. We then derive the Schur complement form of the consistent tangent matrix. We demonstrate the performance of the method for a number of three-dimensional problems for Reynolds number up to 1000 including steady and time-dependent flows.
TL;DR: In this article, it was shown that the tangent stiffness used for determining loss of stability has a specific, simple form, and that these two terms are not mutually conditional, whereas the term "linear prebuckling paths" refers to the shape of the load-displacement diagrams before stability loss.
Abstract: Stability problems are termed linear if the tangent stiffness used for determining loss of stability has a specific, simple form. The term “linear prebuckling paths” refers to the shape of the load–displacement diagrams before loss of stability. In this note, it will be shown that these two terms are not mutually conditional.
Journal Article•
TL;DR: In this paper, the stiffness of a hybrid machine tool using a redundantly actuated parallel mechanism was analyzed by considering the deformation of the straight rolling slide-way pair and bearings.
Abstract: The stiffness of a hybrid machine tool using a redundantly actuated parallel mechanism was analyzed by considering the deformation of the straight rolling slide-way pair and bearings. Stiffness assembly theory was used to develop the stiffness matrix for the machine tool. The impact of the straight rolling guide-way pair on the whole machine was also analyzed to optimizate design of the weakest part of the machine tool. The analysis results show that the position stiffness is symmetric along the x, y and z directions within a specified workspace, with the best position stiffness along the z direction and the worst along the x direction. The results also show that the straight rolling guide-way pair significantly affects the whole machine stiffness.
01 Mar 2008
TL;DR: In this article, a new model for simulating cracks in cementitious composites using embedded planes with local plasticity is presented, where the embedded planes of degradation can not only undergo separation but can also regain contact, with the state of contact being controlled by a local contact model.
Abstract: A new model for simulating cracks in cementitious composites using embedded planes with local plasticity is presented. In the model it is assumed that the embedded planes of degradation (PODs) can not only undergo separation but can also regain contact, with the state of contact being controlled by a local contact model. A local plasticity model is included to capture permanent relative displacements associated with rough contact between the surfaces of embedded cracks. This model is a significant development of the Craft model developed by the second author, which allows shear contact, or aggregate interlock, behaviour, as well as crack opening/closing behaviour, to be simulated more accurately. A fully consistent tangent stiffness matrix and stress recovery algorithm is derived. The model has been implemented in a constitutive driver program and also in the finite–element program Lusas. The model is assessed against a series of single-point stress–strain paths, and against data from normal-shear and cyc...
Journal Article•
TL;DR: In this article, the tangent stiffness matrix of space beam element under general linear elastic relationship of stress and strain was derived based upon general balance equation of nonlinear questions or nonlinear geometric equation of Space beam element, this stiffness matrix contains initial stress stiffness matrix generated from initial stress and initial strain, which established the foundation for compiling space finite element programs.
Abstract: When carrying out geometric nonlinear analysis or ultimate bearing capacity calculation for stem structures such as beam,roof,reticulated shell structures or grid structures,tangent stiffness matrix of space beam element is requestedThe explicit tangent stiffness matrix of space beam element under general linear elastic relationship of stress and strain shall be derived based upon general balance equation of nonlinear questions or nonlinear geometric equation of space beam element,this stiffness matrix contains initial stress stiffness matrix generated from initial stress and initial strain,which establishes the foundation for compiling space finite element programs
TL;DR: In this article, a numerical method is presented for stability analysis of cable-bar structures, where the minimum value of the incremental total potential energy and the associated displacement increments can be found with good accuracy in about 10 steps of iteration.
Journal Article•
TL;DR: In this article, a continuous stiffness nonlinear mapping general model of spatial parallel mechanism is established, including the elastic deformation of active and passive hinge and gravities factor, and the change of the Jacobian matrix is not neglected under the outside force.
Abstract: Based on influence coefficient method and principle of virtual work,continuous stiffness nonlinear mapping general model of spatial parallel mechanism is established,including the elastic deformation of active and passive hinge and gravities factor Different from the traditional model,the stiffness nonlinear general mapping model of spatial parallel mechanism is reverted,not only considering the influence of elastic deformation of active and passive binge and gravities factor on the mechanism stiffness,but also considering the stiffness continuous changeThe change of the Jacobian matrix is not neglected under the outside force,intro- ducing the second-order influence coefficient matrix of mechanismThen,simple restricted equations of parallel mechanism are de- scribed,and several continuous stiffness mapping matrixes are deduced respectivelyOn this foundation,combining Rayleigh quo- tient of the continuous stiffness matrix,the performance index k′used to estimate the mechanism stiffness is defined,a stiffness analysis example of spatial parallel mechanism is presentedThe stiffness matrixes in two models are calculated,and the stiffness performances are contrasted,respectively
TL;DR: In this article, shape functions are proposed for the spectral finite element method aiming to find a nodal spectral stiffness matrix, which obtain a nearly diagonal ĞD stiffness matrix with better conditioning than using the Lagrange and Jacobi bases.
Abstract: In this paper, shape functions are proposed for the spectral finite-element method aiming to finding a nodal spectral stiffness matrix. The proposed shape functions obtain a nearly diagonal ĞD stiffness matrix with better conditioning than using the Lagrange and Jacobi bases.
TL;DR: In this article, a numerical procedure based on the finite element method for materially and geometrically nonlinear analysis of reinforced and prestressed concrete slender columns with arbitrary section subjected to combined biaxial bending and axial load is developed.
Abstract: In this study, a numerical procedure based on the finite element method for materially and geometrically nonlinear analysis of reinforced and prestressed concrete slender columns with arbitrary section subjected to combined biaxial bending and axial load is developed. In order to overcome the low computer efficiency of the conventional section integration method in which the reinforced concrete section is divided into a large number of small areas, an efficient section integration method is used to determine the section tangent stiffness. In this method, the arbitrary shaped cross section is divided into several concrete trapezoids according to boundary vertices, and the contribution of each trapezoid to section stiffness is determined by integrating directly the trapezoid. The space frame flexural theory is utilized to derive the element tangent stiffness matrix. The nonlinear full-range member response is traced by an updated normal plane arc-length solution method. The analytical results agree well with the experimental ones.
01 Jan 2008
TL;DR: In this article, the stiffness analysis of a planar 2-DoF tense-grity mechanism is performed based on an existing stiffness matrix model and several stiffness indices having physical meaning are introduced.
Abstract: This paper presents a general method to perform the stiffness analysis of tensegrity mechanisms. The method is based on an existing stiffness matrix model. Several stiffness indices having physical meaning are introduced. As an example, the method is applied to a planar 2-DoF tensegrity mechanism. Stiffness mappings based on the stiffness indices are generated for the mechanism’s workspace. It is shown for the example mechanism that the effect of the prestress on the stiffness is not significant when linear stiffness models of the components are assumed.Copyright © 2008 by ASME
TL;DR: In this article, a contact search algorithm between the sheet nodes and the interpolated tool surfaces and the consistent tangent stiffness matrix for the sliding nodes were developed, which was introduced into the static-explicit elastoplastic finite-element method code STAMP3D.
Abstract: Treatment of contact between a sheet and tools is one of the most difficult problems to deal with in finite-element simulations of the sheet metal forming processes. In order to obtain more accurate tool models without increasing the number of elements, this paper describes new formulations for contact problems using local interpolation proposed by Nagata for tool surfaces. A contact search algorithm between the sheet nodes and the interpolated tool surfaces and the consistent tangent stiffness matrix for the sliding nodes were developed. The developed algorithms were introduced into the static-explicit elastoplastic finite-element method code STAMP3D. Simulations of a square cup drawing process with a very coarsely discretized punch model were carried out. The simulated results showed that the proposed algorithms gave the proper deformation process, thus demonstrating the validity of the proposed formulations.
TL;DR: In this paper, an example of a C1-smooth real function of two variables whose gradient range is an arc with no tangent at any point is constructed, where the gradient is defined as an arc.
Abstract: We construct an example of a C1-smooth real function of two variables whose gradient range is an arc with no tangent at any point.