Nonlinear statics of three-dimensional curved geometrically exact beams by a hierarchal quadrature element method

Nonlinear static analyses of three-dimensional (3D) curved geometrically exact beams are carried out using a hierarchal quadrature element method (HQEM) in this work. The initial value of the rotational quaternions is computed from an initial-value problem with arc-length as the “time” variable, so that the quaternions can be differentiated in subsequent computation. The 3D curved geometrically exact beams are first expressed by NURBS and then transformed to be expressed by the hierarchical quadrature bases. The geometrically exact beams are then analyzed by the same hierarchical quadrature bases, which conforms to the isogeometric concept. This procedure of analysis is new. The weak-form quadrature element method (QEM) is a special case of the hierarchical quadrature element method (HQEM). It is found that the best accuracy may be obtained when the number of integration nodes is one degree less than the nodal points, instead of taking them the same as in the weak-form QEM. Numerical tests through several examples show that the HQEM can present results with high accuracy using only a few degrees of freedom and is not sensitive to shear locking.