Order reduction method for locking free isogeometric analysis of Timoshenko beams

The order reduction (OR) method is proposed to provide an effective and efficient locking-free formulation in the isogeometric analysis of plane curved Timoshenko beams. The shear and membrane strains are represented by the reduced order B-spline basis functions, and the locking phenomenon can be prevented. The modified strains are equivalent to the original ones in the sense of the least square approximation. Different order reduction strategies are discussed, and it is suggested to use multiple sets of lower order B-spline basis functions to improve the accuracy. The theoretical analysis of the formulation and the computational cost is carried out to assess the efficiency of the OR method. The performance of the proposed method is tested by the straight beam, curved beams with constant and varied curvatures problems, and compared with other methods such as the selective and reduced integration method, the local discrete shear gap method, and the local B̄ method (without strain smoothing). The numerical results show that the proposed method is competitive, which leads to an efficient locking free formulation.