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Abstract Continuous‐assumed‐strain (CAS) elements were recently introduced (Casquero and Golestanian. Comput Methods Appl Mech Eng. 2022; 399:115354.) to remove the membrane locking present in quadratic ‐continuous NURBS‐based discretizations of linear plane curved Kirchhoff rods. In this work, we generalize CAS elements to remove shear and membrane locking from quadratic NURBS‐based discretizations of linear plane curved Timoshenko rods. CAS elements are an assumed strain treatment that interpolates the shear and membrane strains at the knots using linear Lagrange polynomials. Consequently, the inter‐element continuity of the shear and membrane strains is maintained. The numerical experiments considered in this work show that CAS elements excise the spurious oscillations in shear and membrane forces caused by shear and membrane locking. Furthermore, when using CAS elements with either full or reduced integration, the convergence of displacements, rotations, and stress resultants is independent of the slenderness ratio up to while the convergence is highly dependent on the slenderness ratio when using NURBS elements. We apply the locking treatment of CAS elements to quadratic ‐continuous NURBS and the resulting element type is named discontinuous‐assumed‐strain (DAS) elements. Comparisons among CAS and DAS elements show that once locking is properly removed, continuity across element boundaries results in higher accuracy than continuity across element boundaries. Lastly, CAS elements result in a simple numerical scheme that does not add any significant computational burden in comparison with the locking‐prone NURBS‐based discretization of the Galerkin method.