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Robert Levy - One of the best experts on this subject based on the ideXlab platform.

  • Geometrically nonlinear analysis of shell structures using a flat triangular shell finite element
    Archives of Computational Methods in Engineering, 2006
    Co-Authors: Robert Levy
    Abstract:

    This paper presents a state of the art review on Geometrically nonlinear analysis of shell structures that is limited to the co-rotational approach and to flat triangular shell finite elements. These shell elements are built up from flat triangular membranes and plates. We propose an element comprised of the constant strain triangle (CST) membrane element and the discrete Kirchhoff (DKT) plate element and describe its formulation while stressing two main issues: the derivation of the Geometric Stiffness matrix and the isolation of the rigid body motion from the total deformations. We further use it to solve a broad class of problems from the literature to validate its use.

  • The Geometric Stiffness of thick shell triangular finite elements for large rotations
    International Journal for Numerical Methods in Engineering, 2006
    Co-Authors: Robert Levy, Erez Gal
    Abstract:

    This paper is concerned with the development of the Geometric Stiffness matrix of thick shell finite elements for Geometrically nonlinear analysis of the Newton type. A linear shell element that is comprised of the constant stress triangular membrane element and the triangular discrete Kirchhoff Mindlin theory (DKMT) plate element is ‘upgraded’ to become a Geometrically nonlinear thick shell finite element. Perturbation methods are used to derive the Geometric Stiffness matrix from the gradient, in global coordinates, of the nodal force vector when stresses are kept fixed. The present approach follows earlier works associated with trusses, space frames and thin shells. It has the advantage of explicitness and clear physical insight. A special procedure, tailored to triangular elements is used to isolate pure rotations to enable stress recovery via linear elastic constitutive relations. Several examples are solved. The results compare well with those available in the literature. Copyright © 2005 John Wiley & Sons, Ltd.

  • The Geometric Stiffness of triangular composite-materials shell elements
    Computers & Structures, 2005
    Co-Authors: Erez Gal, Robert Levy
    Abstract:

    This paper is concerned with the development of the Geometric Stiffness matrix for Newton type large rotation analysis of composite thin shell structures. The Geometric Stiffness matrix is derived from load perturbation of the discrete equilibrium equations of a given linear finite element formulation. The Geometric Stiffness matrix is extracted from the gradient, in global coordinates, of the element nodal force vector when stresses are kept fixed. In order to overcome the difficulties in taking derivatives of the rotation matrix with respect to the nodal coordinates, gradient evaluations are performed in the local coordinate system to result in an in-plane Geometric Stiffness matrix. An out-of-plane Geometric Stiffness matrix is then introduced to account for the effect of rigid body rotations on member forces. A unique procedure is used for the removal of rigid body displacements and rotations that enables stress recovery via linear, kinematic and constitutive, relationships. The Geometric Stiffness matrix derived was used to study several examples whose results compare well with the literature.

  • Geometric Stiffness of membranes using symbolic algebra
    Engineering Structures, 2004
    Co-Authors: Robert Levy, Chuin-shan Chen, Cheng-wei Lin, Yeong-bin Yang
    Abstract:

    This paper is concerned with derivation of the Geometric Stiffness matrix for membrane shells which are represented by constant stress triangular finite elements. Symbolic algebra is used to calculate the gradient of the member nodal force vector of each element when the stresses are kept fixed. This gradient defines the Geometric Stiffness matrix of the element in global coordinates. The present approach follows the earlier works associated with trusses, plane frames and space frames. It has the advantage of explicitness in derivation, while showing clear physical insight. For the case of small rotations, all the mathematical manipulations can be handled by hand. However, for the case of finite rotations, one must have recourse to symbolic algebra programs. The Geometric Stiffness matrices derived were implanted into an existing nonlinear membrane analysis program that was used to study two examples from the available literature.

  • Geometric Stiffness of Space Frames Using Symbolic Algebra
    International Journal of Structural Stability and Dynamics, 2003
    Co-Authors: Robert Levy, Cheng-wei Lin, Erez Gal, Yeong-bin Yang
    Abstract:

    This paper is concerned with the derivation of the Geometric Stiffness matrix for space frames. Symbolic algebra is used to calculate the gradient of the member force vector that defines, in global coordinates, the Geometric Stiffness matrix. Members of solid cross-sections with no warping are considered. The independent nodal variables are the common position coordinates of the member ends. An additional independent variable, which cannot be related to the position coordinates, is the angle of twist at one end of each member. Since the angle of twist is defined locally, its effect on the Stiffness matrix is found separately and then transformed to the global coordinates. The advantages of the present approach are its explicitness in derivation and clear physical insight. Finally, the Geometric Stiffness matrix is implanted into an existing 3D nonlinear frame analysis program. For the analytical, numerical and experimental benchmark examples studied, the present results appear to be in good agreement with the results available in the literature.

Erez Gal - One of the best experts on this subject based on the ideXlab platform.

  • The Geometric Stiffness of thick shell triangular finite elements for large rotations
    International Journal for Numerical Methods in Engineering, 2006
    Co-Authors: Robert Levy, Erez Gal
    Abstract:

    This paper is concerned with the development of the Geometric Stiffness matrix of thick shell finite elements for Geometrically nonlinear analysis of the Newton type. A linear shell element that is comprised of the constant stress triangular membrane element and the triangular discrete Kirchhoff Mindlin theory (DKMT) plate element is ‘upgraded’ to become a Geometrically nonlinear thick shell finite element. Perturbation methods are used to derive the Geometric Stiffness matrix from the gradient, in global coordinates, of the nodal force vector when stresses are kept fixed. The present approach follows earlier works associated with trusses, space frames and thin shells. It has the advantage of explicitness and clear physical insight. A special procedure, tailored to triangular elements is used to isolate pure rotations to enable stress recovery via linear elastic constitutive relations. Several examples are solved. The results compare well with those available in the literature. Copyright © 2005 John Wiley & Sons, Ltd.

  • The Geometric Stiffness of triangular composite-materials shell elements
    Computers & Structures, 2005
    Co-Authors: Erez Gal, Robert Levy
    Abstract:

    This paper is concerned with the development of the Geometric Stiffness matrix for Newton type large rotation analysis of composite thin shell structures. The Geometric Stiffness matrix is derived from load perturbation of the discrete equilibrium equations of a given linear finite element formulation. The Geometric Stiffness matrix is extracted from the gradient, in global coordinates, of the element nodal force vector when stresses are kept fixed. In order to overcome the difficulties in taking derivatives of the rotation matrix with respect to the nodal coordinates, gradient evaluations are performed in the local coordinate system to result in an in-plane Geometric Stiffness matrix. An out-of-plane Geometric Stiffness matrix is then introduced to account for the effect of rigid body rotations on member forces. A unique procedure is used for the removal of rigid body displacements and rotations that enables stress recovery via linear, kinematic and constitutive, relationships. The Geometric Stiffness matrix derived was used to study several examples whose results compare well with the literature.

  • Geometric Stiffness of Space Frames Using Symbolic Algebra
    International Journal of Structural Stability and Dynamics, 2003
    Co-Authors: Robert Levy, Cheng-wei Lin, Erez Gal, Yeong-bin Yang
    Abstract:

    This paper is concerned with the derivation of the Geometric Stiffness matrix for space frames. Symbolic algebra is used to calculate the gradient of the member force vector that defines, in global coordinates, the Geometric Stiffness matrix. Members of solid cross-sections with no warping are considered. The independent nodal variables are the common position coordinates of the member ends. An additional independent variable, which cannot be related to the position coordinates, is the angle of twist at one end of each member. Since the angle of twist is defined locally, its effect on the Stiffness matrix is found separately and then transformed to the global coordinates. The advantages of the present approach are its explicitness in derivation and clear physical insight. Finally, the Geometric Stiffness matrix is implanted into an existing 3D nonlinear frame analysis program. For the analytical, numerical and experimental benchmark examples studied, the present results appear to be in good agreement with the results available in the literature.

Yeong-bin Yang - One of the best experts on this subject based on the ideXlab platform.

  • Solution strategy and rigid element for nonlinear analysis of elastically structures based on updated Lagrangian formulation
    Engineering Structures, 2007
    Co-Authors: Yeong-bin Yang, Shihpo Lin, L.j. Leu
    Abstract:

    Abstract Three phases are essential to the incremental–iterative analysis of elastically nonlinear structures: the predictor , corrector and error-checking phases. The predictor relates to solution of the structural displacements for given load increments, which affects only the number of iterations. The corrector is concerned with recovery of the element forces for given element displacements, which governs the accuracy of solution. By choosing a robust incremental–iterative scheme, the use of only the linear Stiffness matrix [ k e ], via the predictor and corrector, is good enough for solving a wide range of moderately nonlinear problems, and this is sufficient for most practical purposes. For highly nonlinear problems, i.e., for those with winding loops in the postbuckling responses, a rigid-body qualified Geometric Stiffness matrix [ k g ] should be added in the predictor to ensure proper directions of iteration. The Geometric Stiffness matrix [ k g ] that is rigid-body qualified is derived from the virtual work equation by assuming the displacement field to be of the rigid type. The above ideas are demonstrated in the solution of several nonlinear problems.

  • Geometric Stiffness of membranes using symbolic algebra
    Engineering Structures, 2004
    Co-Authors: Robert Levy, Chuin-shan Chen, Cheng-wei Lin, Yeong-bin Yang
    Abstract:

    This paper is concerned with derivation of the Geometric Stiffness matrix for membrane shells which are represented by constant stress triangular finite elements. Symbolic algebra is used to calculate the gradient of the member nodal force vector of each element when the stresses are kept fixed. This gradient defines the Geometric Stiffness matrix of the element in global coordinates. The present approach follows the earlier works associated with trusses, plane frames and space frames. It has the advantage of explicitness in derivation, while showing clear physical insight. For the case of small rotations, all the mathematical manipulations can be handled by hand. However, for the case of finite rotations, one must have recourse to symbolic algebra programs. The Geometric Stiffness matrices derived were implanted into an existing nonlinear membrane analysis program that was used to study two examples from the available literature.

  • Geometric Stiffness of Space Frames Using Symbolic Algebra
    International Journal of Structural Stability and Dynamics, 2003
    Co-Authors: Robert Levy, Cheng-wei Lin, Erez Gal, Yeong-bin Yang
    Abstract:

    This paper is concerned with the derivation of the Geometric Stiffness matrix for space frames. Symbolic algebra is used to calculate the gradient of the member force vector that defines, in global coordinates, the Geometric Stiffness matrix. Members of solid cross-sections with no warping are considered. The independent nodal variables are the common position coordinates of the member ends. An additional independent variable, which cannot be related to the position coordinates, is the angle of twist at one end of each member. Since the angle of twist is defined locally, its effect on the Stiffness matrix is found separately and then transformed to the global coordinates. The advantages of the present approach are its explicitness in derivation and clear physical insight. Finally, the Geometric Stiffness matrix is implanted into an existing 3D nonlinear frame analysis program. For the analytical, numerical and experimental benchmark examples studied, the present results appear to be in good agreement with the results available in the literature.

  • A simple nonlinear triangular plate element and strategies of computation for nonlinear analysis
    Computer Methods in Applied Mechanics and Engineering, 1999
    Co-Authors: Yeong-bin Yang, Jiann-tsair Chang, J. D. Yau
    Abstract:

    According to the rigid body rule, for a solid member subjected to rigid body rotations, the initial forces acting on the member that form an equilibrating set must rotate following the rigid body rotations, while remaining unchanged in magnitude. Such a rule is physically intuitive and is employed in this paper to derive an approximate Geometric Stiffness matrix for a three-node triangular plate element (TPE) containing three translational and three rotational degrees of freedom (DOFs) at each node. An element such as this is attractive, since it can be easily used along with the 12-DOF beam element to simulate various plate and shell assemblies. Another advantage with the Geometric Stiffness matrix derived is that it can be explicitly given, which renders numerical integrations unnecessary. Finally, the element and procedure proposed are demonstrated to be robust in that solutions of good accuracy can always be obtained if a practically fine mesh has been used, and that the solutions converge rapidly to the exact one upon mesh refinement.

Steen Krenk - One of the best experts on this subject based on the ideXlab platform.

  • the role of Geometric Stiffness in momentum and energy conserving time integration
    International Journal for Numerical Methods in Engineering, 2007
    Co-Authors: Steen Krenk
    Abstract:

    A momentum and energy conserving time integration algorithm is developed for the motion of elastic bodies described in terms of the quadratic Green strain. Momentum conserving algorithms are formulated from an integral of the equations of motion and energy conservation has traditionally been obtained by evaluating the contribution from the internal forces by use of a combined mean value of stresses and virtual strains on the element level. It is here demonstrated that momentum and energy conservation can be obtained from the classic central difference formulation by including an extra global term in the form of the increment of the Geometric Stiffness matrix over the current time step, usually directly available in global form in existing finite element programmes. The theory is derived by the use of a state-space formulation, where this extra term is located in the same position as the viscous damping matrix, indicating that the effect of the extra incremental Geometric Stiffness term in the non-linear algorithm is equivalent to a variable damping term depending on the change of the state of stress over a time increment. In the actual numerical algorithm, the new value of the velocity vector is eliminated, leaving a non-linear equation for the displacement increment alone, followed by an explicit vector update of the velocity increment. Copyright © 2006 John Wiley & Sons, Ltd.

  • Global Formulation of Conservative Time Integration by the Increment of the Geometric Stiffness
    III European Conference on Computational Mechanics, 1
    Co-Authors: Steen Krenk
    Abstract:

    A momentum and energy conserving time integration algorithm is developed for the motion of elastic bodies described in terms of the quadratic Green strain. Momentum conserving algorithms are formulated from an integral of the equations of motion, and energy conservation has traditionally been obtained by evaluating the internal forces by combining the mean value of stresses and virtual strains at the element level [1]. It is here demonstrated that momentum and energy conservation can be obtained from the classic central difference formulation by including an extra global term in the form of the increment of the Geometric Stiffness matrix. The Geometric Stiffness matrix is usually available in assembled form in existing programs, and thus a global form is attained that avoids the need for modifying the classic element implementation.

Chen Wanji - One of the best experts on this subject based on the ideXlab platform.

  • Buckling Analysis of Angle-ply Composite and Sandwich Plates by Combination of Geometric Stiffness Matrix
    Computational Mechanics, 2006
    Co-Authors: Wu Zhen, Chen Wanji
    Abstract:

    Buckling response of angle-ply laminated composite and sandwich plates are analyzed using the global-local higher order theory with combination of Geometric Stiffness matrix in this paper. This global-local theory completely fulfills the free surface conditions and the displacement and stress continuity conditions at interfaces. Moreover, the number of unknowns in this theory is independent of the number of layers in the laminate. Based on this global-local theory, a three-noded triangular element satisfying C1 continuity conditions has also been proposed. The bending part of this element is constructed from the concept of DKT element. In order to improve the accuracy of the analysis, a method of modified Geometric Stiffness matrix has been introduced. Numerical results show that the present theory not only computes accurately the buckling response of general laminated composite plates but also predicts the critical buckling loads of soft-core sandwiches. However, the global higher-order theories as well as first order theories might encounter some difficulties and overestimate the critical buckling loads for soft-core sandwich plates.

  • The application of a refined non-conforming quadrilateral plate bending element in thin plate vibration and stability analysis
    Finite Elements in Analysis and Design, 2000
    Co-Authors: Y.k. Cheung, Yixia Zhang, Chen Wanji
    Abstract:

    In this paper, the refined non-conforming quadrilateral thin plate element RPQ4 is used to analyze the vibration and stability problems of thin plates. For vibration analysis, a modified mass matrix is formulated to calculate the natural frequencies and in the same way, a modified Geometric Stiffness matrix is formed to calculate the critical load in the stability analysis of thin plates. The numerical results have demonstrated that the modified mass matrix and modified Geometric Stiffness matrix are efficient in improving the accuracy of natural frequency in the free vibration analysis and critical load of buckling in the analysis of stability of thin plates.link_to_subscribed_fulltex

  • Geometric nonlinear analysis by using refined triangular thin plate element and free form membrane locking
    Computers & Structures, 1997
    Co-Authors: Zhu Ju-fen, Chen Wanji
    Abstract:

    Abstract Based on a large deformation variational principle with relaxed interelement continuity requirement in the total Lagranginan description, a refined triangular thin plate element for Geometric nonlinear analysis has been developed. By introducing special element displacement functions into the Geometric Stiffness matrix, the membrane locking phenomenon is relieved effectively. The numerical results are presented to show that the present element possesses higher accuracy and an ability to free form membrane locking.