  In this paper we demonstrate the use of B\'{e}zier projection to alleviate
locking phenomena in structural mechanics applications of isogeometric
analysis. Interpreting the well-known $\bar{B}$ projection in two different
ways we develop two formulations for locking problems in beams and nearly
incompressible elastic solids. One formulation leads to a sparse symmetric
symmetric system and the other leads to a sparse non-symmetric system. To
demonstrate the utility of B\'{e}zier projection for both geometry and material
locking phenomena we focus on transverse shear locking in Timoshenko beams and
volumetric locking in nearly compressible linear elasticity although the
approach can be applied generally to other types of locking phenemona as well.
B\'{e}zier projection is a local projection technique with optimal
approximation properties, which in many cases produces solutions that are
comparable to global $L^2$ projection. In the context of $\bar{B}$ methods, the
use of B\'ezier projection produces sparse stiffness matrices with only a
slight increase in bandwidth when compared to standard displacement-based
methods. Of particular importance is that the approach is applicable to any
spline representation that can be written in B\'ezier form like NURBS,
T-splines, LR-splines, etc. We discuss in detail how to integrate this approach
into an existing finite element framework with minimal disruption through the
use of B\'ezier extraction operators and a newly introduced dual basis for the
B\'{e}zierprojection operator. We then demonstrate the behavior of the two
proposed formulations through several challenging benchmark problems.
