Pointwise error estimate for a consistent beam theory

This paper studies the planar deformations of a beam composed of a linearly elastic material. Starting from the field equations for the plane-stress problem and adopting a series expansion for the displacement vector about the bottom surface, we deduce the beam equations with two unknowns in a consistent manner. The success relies on using the field equations together with the bottom traction conditions to establish the exact recursion relations, such that all quantities can be represented in terms of the two leading expansion coefficients of the displacements. Another feature is that the remainders of the series can be carried over to the beam equations. Then, based on the general solutions and the error terms of the beam equations, pointwise error estimates for displacement and stress fields are rigorously established. Three benchmark problems are considered, for which the two-dimensional exact solutions are available. It is shown that this new beam theory recovers the exact solutions for these problems. Two cases with boundary layer effects are also discussed in the appendix.