A rod model for large bending and torsion of an elastic strip with a geometrical imperfection

We consider an initially horizontal curved elastic strip, which bends and twists under the action of the varying length of the span between the clamped ends and of the gravity force. Equations of the theory of rods, linearized in the vicinity of a largely pre-deformed state, allow for semi-analytical (or sometimes closed-form) solutions. A nonlinear boundary value problem determines the vertical bending of a perfect beam, while the small natural curvature additionally leads to torsion and out-of-plane deflections described by the linear equations of the incremental theory. Numerical experiments demonstrate perfect correspondence to the finite element rod model of the strip. Comparisons to the predictions of the shell model allow estimating the range of applicability of the three-dimensional theory of rods. Practically relevant conclusions follow for the case of high pre-tension of the strip.