Problems for Chapter 10
Approximate Theories for Solids with Special Shapes:
Rods, Beams, Membranes, Plates and Shells
10.3. Simplified Versions of the General Theory of Deformable Rods
10.3.1. Consider a flexible, inextensible cable subjected to transverse loading (e.g. due to gravity) as illustrated in the figure.
10.3.1.1. Express the basis vector in terms of and (derivatives of) .
10.3.1.2. Find an expression for the curvature vector for the cable in terms of (derivatives of)
10.3.1.3. Find the two equilibrium equations relating the axial tension to the external loading and geometry of the cable, by substituting into the general equations of motion for a rod.
10.3.2. Consider a long, straight rod, with axis parallel to , which is subjected to pure twisting moments acting at its ends. The rod may be idealized as a linear elastic solid with shear modulus . The deformation of the rod can be characterized by the twist and the transverse displacement of the cross-section . Assume that the only nonzero internal moment component is , and the nonzero internal stress components are . Simplify the general governing equations for a deformable rod to obtain:
10.3.2.1. A simplified expression for the curvature tensor for the deformed rod, in terms of
10.3.2.2. Equations of equilibrium and boundary conditions for and
10.3.2.3. Expressions relating to and the warping function w. Show that the equilibrium equation for reduces to the governing equation for the warping function given in Section 10.2.10.
10.3.2.4. Expressions relating to .
10.3.3. An initially straight beam, with axis parallel to the direction and principal axes of inertia parallel to is subjected to a force per unit length . The beam has Young’s modulus and mass density , and its cross-section has area A and principal moments of inertia . Assume that a large axial internal force is developed in the beam, either by a horizontal force per unit length or horizontal forces acting at the ends of the beam. Suppose that the beam experiences a finite transverse displacement , so that the stretch of the beam and its curvature must be approximated by
Show that the static equilibrium equations for the displacement components can be reduced to
and list the boundary conditions on the ends of the beam.
10.3.4. The goal of this problem is to derive the equation of motion for an inextensible stretched string subjected to small displacements by a direct application of the principle of virtual work. Assume that at some instant the string has transverse deflection and velocity as indicated in the figure.
10.3.4.1. Write down an equation for the curvature of the string, accurate to first order in
10.3.4.2. Write down an expression for the relative velocity of the end B of the string relative to A, in terms of and .
10.3.4.3. Write down the rate of virtual work done by the transverse forces in terms of a virtual velocity
10.3.4.4. Write down the rate of virtual done by the applied tension .
10.3.4.5. Hence use the principle of virtual work to derive the equation of motion for the string.
(c) A.F. Bower, 2008