Showing papers on "Tangent stiffness matrix published in 1973"

Tangent Stiffness in Plane Frames

Abstract: With reference to planar framed structures, a tangent stiffness matrix is derived that does not require any approximation beyond those used in the conventional beam-column theory. The matrix is given in such a form as to clearly separate the contributions of large rigid body displacements from elastic and locally nonlinear effects, and its numerical evaluation appears to be routine. Possible approximations, analogous to those used by previous investigators, are also examined.

Analysis of Biaxially Loaded Steel H-Columns

Abstract: A number of problems on the inelastic stability of biaxially loaded columns were solved by utilizing an incremental approach. The matrix of influence coefficients, referred to herein as the tangent stiffness matrix, varies with the loading and depends upon the penetration of yielding across the section, the magnitude of external loads, and the displacements of the column. The matrix must therefore be revised with each increment of load, and during each cycle of iteration. Successive corrections are made until the condition of equilibrium between the internal and external forces is satisfied. The analysis covered two different types of loading, proportional and nonproportional. The influences of material yielding, residual stress, end warping restraint, end bending restraint, and initial imperfection were included. The problems of centrally loaded columns and lateral-torsional buckling of beam-columns were also examined by using the biaxial bending theory and the concept of initial imperfection.

Viscoelastoplastic Response of Axisymmetric Shells under Impulsive Loadings

Abstract: A derivation of finite element equations of and solution to the viscoelastoplastic response of an isotropic axisymmetric shell are presented herein. The generalized Maxwell model is incorporated into the von Mises isotropic yield function. This permits a derivation of the incremental stress as a function of elastic, viscous, and plastic strains. With this relationship inserted into the incremental equation of motion, a direct numerical integration scheme is then used to solve for incremental responses. The plastic tangent stiffness matrix is updated at each incremental time step. Numerical results are presented for a circular plate to verify correctness of the program and subsequently for a spherical cap subjected to uniformly distributed transverse impulsive load of infinite duration.

Mechanical Interpretations for Invariant Imbedding

Abstract: The Scott method is examined within the context of solid mechanics, with particular emphasis upon interpretations of the various occurring functions. Two alternative approaches arise, as the basic Riccati variable is taken to be: (1) the stiffness matrix at the right end of the member corresponding to zero load on the body and at the left end; or (2) the flexibility matrix at the right end corresponding to zero body load and zero displacements at the left end. In the stiffness matrix approach the other three quantities appearing are a transmission matrix for displacements, a virtual boundary load, and a displacement that would result from this virtual boundary load. Analogous interpretations hold within the flexibility matrix approach. A clarifying analysis is presented of the distinction between the stiffness matrix in approach 1 and that defined as the inverse of the flexibility matrix in approach 2.

Inelastic Response of Column Segments under Biaxial Loads

Abstract: An analytical formulation of the generalized stress-strain relationships for biaxially loaded column segments is presented. The generalized stresses are an axial force, two bending moments about the principal axes, and a bimoment. The corresponding generalized strains are an axial strain at the centroid, two bending curvatures, and a warping curvature. The material characteristics are assumed to be trilinear, i.e., elastic, plastic, and linear strain hardening. Elastic unloading of yielded fibers and “Bauschinger effects” are included in the analysis. Based upon the equations formulated, a computer program was developed to provide numerical results. The element of the tangent stiffness matrix was evaluated numerically by dividing the wide flange section into finite element.