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.

Second-order analysis of planar steel frames considering the effect of spread of plasticity

Abstract: This paper presents a method of elastic-plastic analysis for planar steel frames that provides the accuracy of distributed plasticity methods with the computational efficiency that is greater than that of distributed plasticity methods but less than that of plastic-hinge based methods. This method accounts for the effect of spread of plasticity accurately without discretization through the cross-section of a beam-column element, which is achieved by the following procedures. First, nonlinear equations describing the relationships between generalized stresses and strains of the cross-section are derived analytically. Next, nonlinear force-deformation relationships for the beam-column element are obtained through lengthwise integration of the generalized strains. Elastic-plastic flexibility coefficients are then calculated by differentiating the above element force-deformation relationships. Finally, an elastic-plastic stiffness matrix is obtained by making use of the flexibility-stiffness transformation. Adding the conventional geometric stiffness matrix to the elastic-plastic stiffness matrix results in the tangent stiffness matrix, which can readily be used to evaluate the load carrying capacity of steel frames following standard nonlinear analysis procedures. The accuracy of the proposed method is verified by several examples that are sensitive to the effect of spread of plasticity.

Force deformation equations for initially curved laterally loaded beam columns

Abstract: The analytical inclusion of the effects of initial curvature and lateral load in the force deformation equations of a two-noded beam-column element is considered. Current position-(Eulerian) based equations are developed that give both the end-shortening and nodal forces for a deformed element under the action of a significant axial compression. Consideration is restricted to an initial curvature in the form of a single sinusoidal half-wave, and to a sinusoidal approximation of a symmetric triangular lateral load. The equations developed are compatible with the existing Eulerian force deformation equations presented by Oran for straight axially loaded beam columns. A tangent stiffness matrix compatible with the new force deformation equations is not presented. The correctness of the new formulation is demonstrated by various analytical comparisons with the results of some standard beam-column problems, and by a number of numerical comparisons between results obtained using the new formulation and those predicted by conventional nonlinear analyses, which employ a number of standard beam-column elements to model each structural member. The use of the new equations can lead to a significant savings in the core storage required for the analysis of full-size lattice domes.

PWL approximation of hyperbolic tangent and the first derivative for VLSI implementation

Abstract: Hyperbolic tangent function is approximated using piecewise linear approximation. This approximation can be used in any embedded hardware architecture where occupied chip space is a challenging factor. The presented recursive algorithm makes a trade-off between circuit delay and accuracy, where low memory consumption is required. In the presented centered linear approximation, hyperbolic tangent and its first derivative is approximated and optimized using maximum error and mean square error of the approximation. Hyperbolic tangent approximation using maximum error shows better results while the first derivative of hyperbolic tangent is better approximated using mean square error. It is demonstrated that a mean square error of 0.02 can be achieved after specific number of iterations in the approximation of hyperbolic tangent.

Determination of branches of limit points by an Asymptotic Numerical Method

Abstract: This paper deals with parameter dependence in nonlinear structural stability problems. The main purpose is the study of the influence of imperfections on a structure. This analysis implies the calculation of the so called fold curve connecting the critical points of the equilibrium path when a structural defect varies. This is traditionally achieved by adding a well-chosen constraint equation demanding the criticality of the equilibrium. The crucial feature of the paper lies in the use of the Asymptotic Numerical Method (A.N.M.) for the numerical resolution of the obtained augmented system. The theoretical framework upon which the A.N.M. is based as well as its advantages over incremental-iterative strategies are presented. The numerical isolation of an initial starting limit point is described. The extended system and its resolution with the A.N.M. are discussed. From a numerical point of view, it leads to an efficient treatment which takes the singularity of the tangent stiffness matrix into account. Emphasis is given on a geometrical shape imperfection.