Problems for Chapter 10
Approximate Theories for Solids with Special Shapes:
Rods, Beams, Membranes, Plates and Shells
10.2. Motion and Deformation of Slender Rods
10.2.1. The figure shows an inextensible rod that is bent into a helical shape. The shape of the helix can be characterized by the radius r of the generating cylinder, and the number of turns n in the helix per unit axial length. Consider a point on the axis of the rod specified by the polar coordinates .
10.2.1.1. Write down an expression for in terms of r, n and .
10.2.1.2. Write down the position vector of the point as components in the basis.
10.2.1.3. Calculate an expression for the unit vector that is tangent to the rod, in terms of the basis vectors and appropriate coordinates.
10.2.1.4. Assume that is perpendicular to the axis of the cylinder. Use this and the solution to 1.3 to find expressions for the basis vectors in terms of .
10.2.1.5. Calculate the normal and binormal vectors to the curve and hence deduce an expression for the torsion of the curve.
10.2.1.6. Deduce an expression for the curvature vector of the rod.
10.2.1.7. Suppose that the stress state in the deformed rod is a simple axial distribution . Calculate the stress components in the basis.
10.2.2. An initially straight, inextensible slender bar with length L and circular cross-section with radius a is bent into a circle with radius R by terminal couples, as shown in the figure. Assume that cross-sections of the rod remain circles with radius a and remain transverse to the axis of the rod after deformation.
10.2.2.1. Write down an expression for the position vector of a position vector on the axis of the deformed rod, expressing your answer as components in the basis
10.2.2.2. Find an expression for the basis vectors as a function of arc-length s, expressing each unit vector as components in . Hence find an expression for the orthogonal tensor R that maps onto .
10.2.2.3. Write down the curvature vector for the deformed rod, and verify that
10.2.2.4. Write down expressions for the deformation gradient in the rod, expressing your answer as both components in and
10.2.2.5. Find an expression for the Lagrange strain tensor in the rod, expressing your answer as both components in and . Neglect second-order terms.
10.2.2.6. Hence deduce expressions for the Material stress and Cauchy stress in the rod.
10.2.2.7. Calculate the resultant internal moment and force acting on a generic internal cross-section of the rod.
10.2.2.8. Show that the internal moment satisfies the equations of equilibrium.
10.2.3. Consider a deformable rod, as shown in the figure. Let denote the arc-length of a point on the axis un-deformed rod from some arbitrary origin, and let denote the twist of the rod, as defined in Section 10.2.2. In addition, let and denote the position vector and velocity of this point on the deformed rod, let denote its arc-length after deformation, and let denote the curvature vector of the rod, where are basis vectors aligned with the deformed rod as discussed in Section 10.2.2 Show that
10.2.3.1. The time derivative of a unit vector tangent to the rod can be computed as
10.2.3.2. The time derivative of the the rate of stretching of the rod’s centerline is related to its velocity by
10.2.3.3. The angular velocity of the basis vectors can be calculated as
10.2.3.4. The angular acceleration of the basis vectors can be calculated as
(c) A.F. Bower, 2008