A geometrically exact higher order beam model and its weak form quadrature element formulation

As an extension of the benchmark higher order beam theory to the realm of geometrical nonlinearity, a geometrically exact higher order beam model is proposed for beam structures undergoing large displacements and rotations. The shear-modification terms are established to be introduced into the kinematic description of geometrically exact spatial beams with rectangular or circular cross-sections for the higher order deformation modelling. The shear-stress-free conditions on the beams’ surfaces are naturally satisfied and a parabolic distribution of transverse shear stress can be obtained on the cross-sections, thus circumventing the dependence of shear coefficients. A weak form quadrature element method is employed to formulate the corresponding beam element that meets the C¹ continuity requirement brought about by the shear-modification terms. This quadrature element formulation owns the advantage of retaining strain objectivity and avoiding shear locking phenomena. Typical numerical tests are presented to demonstrate the accuracy and reliability of the present scheme in nonlinear static and dynamic analysis of beams.