Analytical beam theory for the in-plane deformation of a composite strip with in-plane curvature

The variational-asymptotic-method (VAM) provides a mathematically rigorous way to reduce a three-dimensional elasticity formulation to a one-dimensional beam theory without ad hoc assumptions. In this work, the VAM is employed to develop a beam theory to analyze the in-plane deformation of a laminated strip-beam with initial in-plane curvature. The cross-sectional stiffness constants and recovery relations for stress and strain are presented as analytical expressions. For the case of zero initial curvature, consistency of the expressions with those of plate theory is demonstrated. For strip-beams with initial curvature in the in-plane direction, results obtained show explicit dependence on the curvature. Results are verified by comparison with those obtained from VABS, the accuracy and consistency of which with three-dimensional finite elements has been reported in several published works. In addition to the internal consistency check this work provides and its utility in helping to validate VABS (which is based on the principles of VAM), it is hoped that the results obtained herein, since they are all analytical expressions, will help researchers and engineers validate the effect of initial curvature in their beam theories, whether existing or new.