Large curvature analysis of linear elastic, inextensible structures by a direct stiffness method

Abstract

This study proposes a simple technique that has a capability of analysis of linearly elastic, inextensible beam and frame structure undergoing large curvature and rotation. The technique utilizes a standard direct stiffness strategy along with a standard nonlinear solver based on Newtow-Ralphson iteration. The element tangent stiffness matrix is derived directly from the governing nonlinear differential equations; in particular, a standard procedure analogous to that used to construct solutions for classical elastica problems is employed as a basis of the development. The length constraint equation arising form the beam inextensibility is additionally incorporated to the element tangent stiffness matrix in a straightforward fashion and, as a result, increasing the dimension of the matrix by one. The derived element tangent stiffness matrix is exact and essentially symmetric, and its final expression is given in a concise and explicit form in terms of elliptic integrals and other integrals of the same kind. A key advantage of exploiting exact element tangent stiffness matrices in the direct stiffness method is that an exact solution can directly be obtained without any refinement on the size of elements. In this sense, it additionally provides a means for generating results that can be used as benchmark solutions for a comparison purpose. To demonstrate the capability and robustness of the current technique, extensive analysis of beam and frame structures are preformed and results are then reported and discussed.