Abstract
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.
Highlights
Problems of finite spatial deformations of elastic rods with coupled torsion and bending are mainly treated using finite element or other variational formulations; examples of solutions based on differential equations are rare to find in the literature
It can be analytically shown that a straight rod with a symmetric cross section assumes a helical form under the action of a dead-end moment, see Eliseev [1]
Further we demonstrate the asymptotic character of the solution for flat rods, whose out-of-plane bending stiffness is high, by deriving and solving the equations for the leading order terms
Summary
Problems of finite spatial deformations of elastic rods with coupled torsion and bending are mainly treated using finite element or other variational formulations; examples of solutions based on differential equations are rare to find in the literature. This increases the value of such semi-analytic or even closed-form solutions, in particular for studies of parameter sensitivity, asymptotic treatment, etc. Attributing the discrepancies to the shell-specific deformation forms, which cannot be described by the kinematics of a rod, we determine the little known boundary of the range of applicability of the Kirchhoff theory to strips with a thin cross section at coupled bending and torsion. We conclude the paper by a practically relevant study of the effect of axial pre-stretch on the lateral and torsional deformations of a horizontally clamped strip owing to its natural curvature in the horizontal plane
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