Geometrically Exact Analysis of Spatial Frames

Department of Structural and Foundation Engineering, Escola Politecnica Universidade de Sao Paulo, CP 61548, 05424-970 Sao Paulo, SP, Brazil A fully nonlinear, geometrically exact, finite strain rod model is derived from basic kinematical assumptions. The model incorporates shear distortion in bending and can take account of torsion warping. Rotation in 3D space is handled with the aid of the Euler-Rodrigues formula. The accomplished parametrization is simple and does not require update algorithms based on quaternions parameters. Weak and strong forms of the equilibrium equations are derived in terms of cross section strains and stresses, which are objective and suitable for constitutive description. As an example, an invariant linear elastic constitutive equation based on the small strain theory is presented. The attained formulation is very convenient for numerical procedures employing Galerkin projection like the finite element method and can be readily implemented in a finite element code. A mixed formulation of Hu-Washizu type is also derived, allowing for independent interpolation of the displacement, strain and stress fields within a finite element. An exact expression for the Frechet derivative of the weak form of equilibrium is obtained in closed form, which is always symmetric for conservative loading, even far from an equilibrium state and is very helpful for numerical procedures like the Newton method as well as for stability and bifurcation analysis. Several numerical examples illustrate the usefulness of the formulation in the lateral stability analysis of spatial frames. These examples were computed with the code FENOMENA, which is under development at the Computational Mechanics Laboratory of the Escola Politecnica. INTRODUCTION The interest on geometrically nonlinear analysis of struc­tures has increased in the recent few years. Besides the practical importance of nonlinear static and dynamic analysis of flexible rod and shell assemblages, the de­velopment of convenient geometrically exact models has contributed to this fact. These models show many ben­efits, which have been emphasized by many authors, as one can verify in a non-exhaustive list reproduced in the references. This work derives a geometrically exact rod model from the kinematic assumption that cross sections, which are initially orthogonal to the axis, remain plane and undistorted during the deformation. The theory accom­modates finite strains, large displacements and rotations, and accounts for shear distortion in bending. Torsion warping can be effortlessly acquainted for, provided elas­tic behavior is assumed. On the other hand, the intro­duction of elastic-plastic, visco-plastic and visco-elastic constitutive equations in terms of cross section general­ized strains and stresses is straightforward. The consid­eration of cross section inertia is direct as well. The accomplished formulation can be readily applied to the nonlinear analysis of spatial frames through the finite element method and presents the following advan­tages: (a) rotations in 3D space are treated in a consistent but convenient way through the Euler-Rodrigues for­mula: update algorithms based on quaternion pa­rameters are not required; (b) there is no need of approximate strain-displacement relationships or additional assumptions like moder­ate rotations, small curvatures and small cross sec­tion dimensions; (c) generalized cross section strains and stresses, which are energetically conjugate, can be consistently de­fined; (d) generalized cross section displacements and external loadings, which are energetically conjugate, can be consistently defined; (e) equilibrium and motion equations are consistently derived in weak form as well as in strong form; (f) boundary conditions are obtained by variational ar­guments;