Energy-consistent formulation and order deficiency of linear and nonlinear shear-deformable beam theories

This paper presents an energy-consistent beam theory and uses it to show the order deficiency, shear locking, and other problems of shear-deformable beam theories. The order of spatial differentiation of both Timoshenkoʹs first-order shear-deformable beam theory and the Euler–Bernoulli beam theory is four, but Timoshenkoʹs theory can account for shear deformation but the other cannot. However, Timoshenkoʹs beam theory is shown to be order-deficient by two, and this order deficiency causes the shear locking problem in finite-element analysis of thin beams. Moreover, the use of one shear correction factor alone cannot fully account for the influence of shear warping on the total strain energy of composite beams. The Cosserat rod theory with Green–Lagrange strains is often used to model highly flexible beams because it can account for geometric nonlinearity and first-order shear deformation. Unfortunately, the use of first-order shear theory makes it order-deficient and is prone to shear locking problems. Moreover, it requires the use of three Rodrigues parameters or four Euler parameters for modeling large rotations, but these math-based variables may experience soft singularity (due to discontinuous spatial derivatives) and artificial strains. Furthermore, the use of Green–Lagrange strains to account for geometric nonlinearity is well known to be problematic. It is also shown that shearing- and bending-induced displacements are relative quantities and they cannot be used as nodal degrees of freedom in finite-element analysis.