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In this paper we demonstrate the use of B\'{e}zier projection to alleviatelocking phenomena in structural mechanics applications of isogeometricanalysis. Interpreting the well-known $\bar{B}$ projection in two differentways we develop two formulations for locking problems in beams and nearlyincompressible elastic solids. One formulation leads to a sparse symmetricsymmetric system and the other leads to a sparse non-symmetric system. Todemonstrate the utility of B\'{e}zier projection for both geometry and materiallocking phenomena we focus on transverse shear locking in Timoshenko beams andvolumetric locking in nearly compressible linear elasticity although theapproach can be applied generally to other types of locking phenemona as well.B\'{e}zier projection is a local projection technique with optimalapproximation properties, which in many cases produces solutions that arecomparable to global $L^2$ projection. In the context of $\bar{B}$ methods, theuse of B\'ezier projection produces sparse stiffness matrices with only aslight increase in bandwidth when compared to standard displacement-basedmethods. Of particular importance is that the approach is applicable to anyspline representation that can be written in B\'ezier form like NURBS,T-splines, LR-splines, etc. We discuss in detail how to integrate this approachinto an existing finite element framework with minimal disruption through theuse of B\'ezier extraction operators and a newly introduced dual basis for theB\'{e}zierprojection operator. We then demonstrate the behavior of the twoproposed formulations through several challenging benchmark problems.

B\'{e}zier $\bar{B}$ Projection