Improved numerical integration for locking treatment in isogeometric structural elements, Part I: Beams

A general mathematical framework is proposed, in this paper, to define new quadrature rules in the context of B-spline/NURBS-based isogeometric analysis. High order continuity across the elements within a patch turned out to have higher accuracy than C0 finite elements, as well as a better time efficiency. Unfortunately, a maximum regularity accentuates the shear and membrane locking in thick structural elements. The improved selective reduced integration schemes are given for uni-dimensional beam problems, with basis functions of order two and three, and can be easily extended to higher orders. The resulting B-spline/NURBS finite elements are free from membrane and transverse shear locking. Moreover, no zero energy modes are generated. The performance of the approach is evaluated on the classical test of a cantilever beam subjected to a distributed moment, and compared to Lagrange under-integrated finite elements. •We model Timoshenko isogeometric curved beams.•We examine membrane and shear locking in pure bending problems.•Increasing the continuity across the elements accentuates the numerical locking.•Higher continuity elements exhibit superior accuracy when no locking occurs.•We propose a general mathematical framework to define improved integration rules for locking treatment.