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
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