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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  Â 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.Â
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.Â
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.Â
Compare the predicted deflection with the exact post-buckling solution
given in Section 10.4.3.
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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.Â
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.Â
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.Â
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. Â
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.
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