Problems for Chapter 10
Approximate Theories for Solids with Special Shapes:
Rods, Beams, Membranes, Plates and Shells
10.4. Exact Solutions to Problems Involving Slender Rods
10.4.1. A slender, linear elastic rod has shear modulus and an elliptical cross-section, as illustrated in the figure. It is subjected to equal and opposite axial couples with magnitude Q on its ends. Using the general theory of slender rods, and assuming that the rod remains straight:
10.4.1.1. Write down the internal force and moment distribution in the rod
10.4.1.2. Calculate the twist per unit length of the shaft
10.4.1.3. Find an expression for the displacement field in the shaft
10.4.1.4. Find an expression for the stress distribution in the shaft
10.4.1.5. Find an expression for the critical couple Q that will cause the shaft to yield
10.4.2. A slender, linear elastic rod has shear modulus and an equilateral triangular cross-section, as illustrated in the figure. It is subjected to equal and opposite axial couples with magnitude Q on its ends. Using the general theory of slender rods, and assuming that the rod remains straight:
10.4.2.1. Write down the internal force and moment distribution in the rod
10.4.2.2. Calculate the twist per unit length of the shaft
10.4.2.3. Find an expression for the displacement field in the shaft
10.4.2.4. Find an expression for the stress distribution in the shaft
10.4.2.5. Find an expression for the critical couple Q that will cause the shaft to yield
10.4.3. The figure shows a flexible cable, subjected to a transverse force per unit length and forces and acting at its ends. In a flexible cable, the area moments of inertia can be neglected, so that the internal moments . In addition, the axial tension
is the only nonzero internal force. Show that under these conditions the virtual work equation reduces to
where is the virtual velocity of the cable, and is the corresponding rate of change of arc-length along the cable. Show that if the virtual work equation is satisfied for all and compatible , the internal force and curvature of the cable must satisfy
10.4.4. The figure shows a flexible string, which is supported at both ends and subjected to a tensile force . The string is subjected to a uniform transverse for p per unit length. Calculate the deflection of the string, assuming small deflections.
10.4.5. The figure shows a flexible string with length L, which is pinned at both ends. The string is subjected to a uniform transverse for p per unit length. Calculate the deflection of the string, assuming small deflections.
10.4.6. The figure shows a flexible cable with length L and weight m per unit length hanging between two supports under uniform vertical gravitational loading. In a flexible cable, the area moments of inertia can be neglected, so that the internal moments
10.4.6.1. Write down the curvature vector of the cable in terms of the angle shown in the figure
10.4.6.2. Hence, show that the equations of equilibrium for the cable reduce to
10.4.6.3. Hence, show that , where H is a constant. Interpret the equation physically.
10.4.6.4. Deduce that
10.4.6.5. Hence, deduce that and calculate as a function of
10.4.6.6. Finally, calculate the internal forces in the cable.
10.4.6.7. Show that, as the full solution approaches the small deflection solution calculated in Problem 5. Find the value of for which the discrepancy between the full solution and the small deflection solution is 10%.
10.4.7. The figure shows an inextensible cable with weight per unit length m, and length 2L that is pinned at both ends. The cable is supported by a frictionless pulley midway between the two ends. Find all the possible equilibrium values of the sags of the cable. Display your results by plotting a graph showing the equilibrium values of as a function of . You will need to solve problem 10.4.6 before attempting this one.
10.4.8. The figure shows a flexible string, which is supported at both ends and subjected to a tensile force . The string has mass per unit length m and can be approximated as inextensible. Calculate the natural frequencies of vibration and the corresponding mode shapes, assuming small transverse deflections.
10.4.9. Estimate the fundamental frequency of vibration for the stretched string described in the preceding problem, using the Rayleigh-Ritz method. Use the approximation for the mode shape. Compare the estimate with the exact solution derived in problem 8.2.11.
10.4.10. The figure shows an Euler-Bernoulli beam with Young’s modulus E, area moments of inertia and length L, which is clamped at and pinned at . It is subjected to a uniform load p per unit length. Calculate the internal moment and shear force in the beam, and calculate the transverse deflection.
10.4.11. The figure shows an initially straight, inextensible elastic rod, with Young’s modulus E, length L and principal in-plane moments of area , which is subjected to end thrust. The ends of the rod are constrained to travel along a line that is parallel to the undeformed rod, but the ends are free to rotate. Use the small-deflection solution for beams subjected to significant axial force given in Section 10.3.3 to calculate the value of P required to hold the rod in equilibrium with a small nonzero deflection, and find an expression for the deflected shape. Compare the predicted deflection with the exact post-buckling solution given in Section 10.4.3.
10.4.12. The theory describing small-deflections of beams subjected to significant axial force given in Section 10.3.3 can be extended to obtain an approximate large deflection solution. Consider the beam shown in the figure. The beam has Young’s modulus E, cross-sectional area A, and principal transverse moments of inertia . The bar is subjected to load per unit length , and axial forces at its two ends. Assume that the displacement field can be described as . Deformation measures are to be expanded up to second order in transverse deflection, so that
· The axial stretch can be approximated as
· The curvature can be approximated as
10.4.12.1. 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.
10.4.12.2. Solve the governing equations for the beam problem described in Problem 10.4.11. Compare the predicted deflection with the exact post-buckling solution given in Section 10.4.3, for the limiting case of an inextensible beam.
10.4.13. An initially straight tent-pole with Young’s modulus E and hollow circular cross-section with external radius a, moment of inertia is to be bent into an arc with height and base as shown in the figure. Calculate expressions for
10.4.13.1. The force P required to bend the pole into shape
10.4.13.2. The total length of the pole
10.4.13.3. The maximum stress in the pole
10.4.14. An initially straight, elastic rod with Young’s modulus E, area moments of inertia and axial effective inertia is subjected to an axial couple , which remains fixed in direction as the rod deforms.
10.4.14.1. Show that the straight rod, with an appropriate twist is a possible equilibrium configuration for all values of Q, and calculate the value of twist
10.4.14.2. Show that, for a critical value of Q, the rod may adopt a helical shape, with one complete turn and arbitrary height h and radius r. Calculate the critical value of Q
10.4.14.3. What can you infer about the stability of a straight rod subjected to end couples?
10.4.15. The figure shows a rod, which is a circular arc with radius R in its stress free configuration, and is subjected to load per unit length and forces , on its ends that cause a small change in its shape. In this problem, we shall neglect out-of-plane deformation and twisting of the rod, for simplicity. Let denote the arc length measured along the undeformed rod, and let the displacement of the rod’s centerline.
10.4.15.1. Note that approximate expressions for the resulting (small) change in arc length and curvature of the rod can be calculated using the time derivatives given in Section 10.2.3. Hence, show that
· The derivative of the change in arc-length of the deformed rod is
· The change in curvature vector is
10.4.15.2. The geometric terms in the equilibrium equations listed in Section 10.2.9 can be approximated using the geometry of the undeformed rod. Show that internal forces and internal moment must satisfy the following static equilibrium equations
10.4.15.3. Assume that the rod is elastic, with Young’s modulus E and area moment of inertia , and can be idealized as inextensible. Show that under these conditions the axial displacement must satisfy
and write down expressions for the boundary conditions at the ends of the rod.
10.4.15.4. As a particular example, consider a rod which is a semicircular arc between , subjected to equal and opposite forces acting on its ends, as shown in the figure. Assume that the displacement and rotation of the rod vanish at , for simplicity. Calculate and for the rod.
(c) A.F. Bower, 2008