Analysis of Linearly Elastic Inextensible Frames Undergoing Large Displacement and Rotation

This paper presents an efficient semi-analytical technique for modeling two-dimensional, linearly elastic, inextensible frames undergoing large displacement and rotation. A system of ordinary differential equations governing an element is first converted into a system of nonlinear algebraic equations via appropriate enforcement of boundary conditions. Taylor's series expansion is then employed along with the co-rotational approach to derive the best linear approximation of such system and the corresponding exact element tangent stiffness matrix. A standard assembly procedure is applied, next, to obtain the best linear approximation of governing nonlinear equations for the structure. This final system is exploited in the solution search by Newton-Ralphson iteration. Key features of the proposed technique include that (i) exact load residuals are evaluated from governing nonlinear algebraic equations, (ii) an exact form of the tangent stiffness matrix is utilized, and (iii) all elements are treated in a systematic way via direct stiffness strategy. The first two features enhance the performance of the technique to yield results comparable to analytical solutions and independent of mesh refinement whereas the last feature allows structures of general geometries and loading conditions be modeled in a straightforward fashion. The implemented algorithm is tested for various structures not only to verify its underlying formulation but also to demonstrate its capability and robustness.