Linear static isogeometric analysis of an arbitrarily curved spatial Bernoulli–Euler beam

Abstract

The present research is focused on the linear analysis of a spatial Bernoulli–Euler beam. Metrics of the reference and deformed configurations are rigorously defined with respect to the convective coordinate frame of reference. No higher order terms are neglected which makes the formulation ideally suited for analysis of arbitrarily curved spatial beams in the frame of finite (but small) strain theory. The well-known issue of nonorthogonality of local coordinate system at an arbitrary point of a spatial beam is solved by the introduction of a new coordinate line that is orthogonal to the normal plane of the beam axis at each point. Generalized coordinates of the present model are translations of the beam axis and the angle of twist of a cross section. Two different parameterizations of this angle are discussed and implemented. Both geometry and kinematics are described with the same set of NURBS functions, in line with the isogeometric approach. Numerical analysis proved that the theoretical considerations are correct and some limits of applicability are defined. An in-depth analysis of convergence properties has confirmed the fact that models with the highest interelement continuity have an improved accuracy per DOF, for problems that result in a smooth structural response.