Tangent stiffness matrix

About: Tangent stiffness matrix is a research topic. Over the lifetime, 1031 publications have been published within this topic receiving 21140 citations.

Efficient formulation of a two‐noded geometrically exact curved beam element

Abstract: The article extends the formulation of a 2D geometrically exact beam element proposed by Jirásek et al. (2021) to curved elastic beams. This formulation is based on equilibrium equations in their integrated form, combined with the kinematic relations and sectional equations that link the internal forces to sectional deformation variables. The resulting first-order differential equations are approximated by the finite difference scheme and the boundary value problem is converted to an initial value problem using the shooting method. The article develops the theoretical framework based on the Navier–Bernoulli hypothesis, with a possible extension to shear-flexible beams. Numerical procedures for the evaluation of equivalent nodal forces and of the element tangent stiffness are presented in detail. Unlike standard finite element formulations, the present approach can increase accuracy by refining the integration scheme on the element level while the number of global degrees of freedom is kept constant. The efficiency and accuracy of the developed scheme are documented by seven examples that cover circular and parabolic arches, a spiral-shaped beam, and a spring-like beam with a zig-zag centerline. The proposed formulation does not exhibit any locking. No excessive stiffness is observed for coarse computational grids and the distribution of internal forces is not polluted by any oscillations. It is also shown that a cross effect in the relations between internal forces and deformation variables arises, that is, the bending moment affects axial stretching and the normal force affects the curvature. This coupling is theoretically explained in the Appendix.

Design of non-diagonal stiffness matrix for assembly task

Abstract: Compliance control is an increasingly employed technique used in the robotic field. It is known that various mechanical properties can be reproduced depending on the design of the stiffness matrix, but the design theory that takes advantage of this high degree of design freedom has not been elucidated. This paper, therefore, discusses the non-diagonal elements of the stiffness matrix. We proposed a design method according to the conditions required for achieving stable motion. Additionally, we analyzed the displacement induced by the non-diagonal elements in response to an external force and found that to obtain stable contact with a symmetric matrix, the matrix should be positive definite, i.e., all eigenvalues must be positive, however its parameter design is complicated. In this study, we focused on the use of asymmetric matrices in compliance control and showed that the design of eigenvalues can be simplified by using a triangular matrix. This approach expands the range of the stiffness design and enhances the ability of the compliance control to induce motion. We conducted experiments using the stiffness matrix and confirmed that assembly could be achieved without complicated trajectory planning.

Structural optimization of elastoplastic structures under shakedown conditions

Abstract: First, it is outlined that common structural shape optimization methods (e.g. weight minimization for static loads) are problematic in case of elasto-plastic deformations combined with load spaces. This is because the growth of plastic deformations for a given n-dimensional load space with an arbitrary number of load cycles can only be modeled by an adequate shakedown theorem (or more refined theorem), in this paper by Melan’s static shakedown theorem for linear unlimited kinematic hardening material behavior. Herein, no aging by micro- or meso-damage is considered, allowing for an infinite number of load cycles within the given load space [1], see also [2, 3]. An adequate formulation of the structural optimization problem, including shakedown analysis, requires variations of residuals, objectives, and constraints, which is presented briefly [1]. 2D problems are discretized with isoparametric finite elements, yielding the ‘tangent stiffness matrix’ and the ‘tangent sensitivity matrix’ as well as the corresponding matrices for the variation of the Langrangian functional. The computer implementation is discussed, and the efficiency of the proposed algorithm is shown by examples. Important effects of shakedown conditions in shape optimization with elasto-plastic deformations are highlighted, comparing the results for optimized systems with elastic and elasto-plastic material behavior, from which the necessity of shape optimization under shakedown conditions for elasto-plastic material deformations and n-dimensional load spaces is deduced, i.e. for guaranteeing structural safety.

A numerical model for unilateral materials in finite deformations, part ii: finite element implementation and applications

Abstract: The implementation of the model for unilateral no-Tension materials described in part one in a standard isoparametric nonlinear finite elements code is described and discussed. A formulation based on the tensorial components of the Cauchy-Green deformation tensor has been used. Explicit expressions for the components of the material and geometric tangent stiffness matrices and for the equivalent internal nodal forces are furnished. Special attention has been focused on the material tangent stiffness matrix, in order to obtain satisfactory convergence properties. Some examples complete the paper.