An exact small deformation model for a curved planar beam incorporating the bending warping

A planar curved beam is modeled by incorporating transverse shear and bending-warping deformation within the curvilinear coordinates under small deformations. Based on the constrained planar postulate that the deformation of a beam is split into the part of center line and the other part of cross section, the kinematics is orthogonally expanded in terms of the generalized displacements for a curved beam. The resultant stresses and generalized strains are defined and the principle of virtual work is then restated. The lower-order theory is eventually proposed including the equilibrium equations and the boundary conditions. The ordinary differential equations with variable coefficients with variable coefficients are formulated. Bending examples are presented that illustrate the influence of the initial curvature on the deflection and the bending warping.