Direct stiffness method

About: Direct stiffness method is a research topic. Over the lifetime, 2584 publications have been published within this topic receiving 53131 citations.

The stiffness matrix in elastically articulated rigid-body systems

Abstract: Discussed in this paper is the Cartesian stiffness matrix, which recently has received special attention within the robotics research community. Stiffness is a fundamental concept in mechanics; its representation in mechanical systems whose potential energy is describable by a finite set of generalized coordinates takes the form of a square matrix that is known to be, moreover, symmetric and positive-definite or, at least, semi-definite. We attempt to elucidate in this paper the notion of “asymmetric stiffness matrices”. In doing so, we show that to qualify for a stiffness matrix, the matrix should be symmetric and either positive semi-definite or positive-definite. We derive the conditions under which a matrix mapping small-amplitude displacement screws into elastic wrenches fails to be symmetric. From the discussion, it should be apparent that the asymmetric matrix thus derived cannot be, properly speaking, a stiffness matrix. The concept is illustrated with an example.

Finite element analysis of inelastic structures.

Abstract: Differential stress-strain relationships are used to generate a system of simultaneous firstorder differential force-displacement equations which are integrated numerically to obtain the stresses, strains, and displacements in inelastic structures. For the biaxially stressed element, the concept of isotropic hardening and a generalized stress are used to evaluate an effective modulus and Poisson's ratio, which vary continuously from their initial values during elastic straining action to their asymptotic values during intense plastic straining action. The surface of plasticity for this element closely approximates the von Mises surface when the generalized stress is set equal to the von Mises stress and the strain distribution is essentially identical to that obtained by the Prandtl-Reuss incremental flow theory. The analysis of the MIT shear lag structure is presented to demonstrate the applicability of the method to systems of practical size and interest. Nomenclature A = equilibrium matrix B = compatibility matrix C = stress-strain matrix C = differential stress matrix E = Young's modulus Et = tangent modulus Es = secant modulus K — stiffness matrix K = differential stiffness matrix P = applied load parameter u = element nodal displacements X = element nodal forces X = load constant n = Poisson's ratio fjLt = tangent Poisson's. ratio Us = secant Poisson's ratio e = strain a- = normal stress

Structural Analysis of Solids

Abstract: No new pitfalls are revealed in this use of the stiffness method for analyzing three-dimensional solids. Numerical experiments show the importance of using the monotonic convergence criteria. They demonstrate the desirability of using energy relations to define loading. They verify the superior accuracy of the prism geometry over the tetrahedron. The relative accuracy of various matrices correlates well with the relative values of the stiffness matrix trace and latent roots. The bases for development of four stiffness matrices are presented and coefficients cited for the matrices of special interest. Six loadings on a rectangular prism are analyzed. Exact solutions are compared with numerical results for one, eight, and sixty-four subprisms. All subprisms have the same shape. Using this shape, the trace and latent roots of each matrix are obtained. It is shown that the smaller the trace and roots, the more accurate the matrix is in solving problems.