A FULLY NONLINEAR MULTI-PARAMETER ROD MODEL INCORPORATING GENERAL CROSS-SECTIONAL IN-PLANE CHANGES AND OUT-OF-PLANE WARPING

Abstract

THIS WORK PRESENTS A FULLY NONLINEAR GEOMETRICALLY-EXACT MULTI-PARAMETER ROD MODEL THAT INCORPORATES GENERAL IN-PLANE CROSS-SECTIONAL CHANGES AS WELL AS GENERAL OUT-OF-PLANE CROSSSECTIONAL WARPING. THE FORMULATION CONSTITUTES AN EXTENSION OF THE EARLIER WORKS, IN THE SENSE THAT THE RESTRICTIONS TO A RIGID CROSS-SECTION AND TO A SAINT-VENANT-LIKE ELASTIC WARPING ARE NOW REMOVED FROM THE THEORY. THE DE¯NITION OF ENERGETICALLY CONJUGATED CROSS-SECTIONAL RESULTANTS IN TERMS OF GENERALIZED STRESSES AND STRAINS, BASED ON THE CONCEPT OF A CROSS-SECTION IRECTOR, SIMPLI¯ES THE DERIVATION OF EQUILIBRIUM EQUATIONS AND THE ENFORCEMENT OF BOUNDARY CONDITIONS, IN EITHER WEAK OR STRONG SENSES. IN ADDITION, THE CORRESPONDING TANGENT BILINEAR WEAK FORM IS OBTAINED IN A MORE EXPEDIENT WAY, RENDERING ALWAYS SYMMETRIC FOR HYPER-ELASTIC MATERIALS AND CONSERVATIVE LOADINGS. THE DE¯NITION OF A CROSS-SECTION DIRECTOR ALLOWS ALSO THE INTRODUCTION OF INDEPENDENT DEGREES-OF-FREEDOM TO DESCRIBE BOTH THE IN-PLANE CROSS-SECTIONAL CHANGES AND THE OUT-OF-PLANE WARPING. FULLY THREE-DIMENSIONAL ¯NITE STRAIN CONSTITUTIVE EQUATIONS CAN THEREFORE BE EMPLOYED WITH NO SPURIOUS STI®ENING. FINITE ROTATIONS ARE TREATED CONSISTENTLY BY THE EULER-RODRIGUES FORMULA IN A PURE LAGRANGEAN FRAMEWORK. ALTOGETHER, THE PRESENT ASSUMPTIONS ALLOW A CONSISTENT BASIS FOR THE PROPER REPRESENTATION OF PRO¯LE (DISTORTIONAL) DEFORMATIONS, WHICH ARE TYPICAL OF COLD-FORMED THIN-WALLED ROD STRUCTURES. THIS IS ONE OF THE MAIN FEATURES OF THE FORMULATION, AS THE USE OF MORE COMPLEX SHELL MODELS IN ORDER TO CAPTURE SUCH PHENOMENA CAN BE NEEDLESS.