Problems for Chapter 4
Solution to Simple Problems
4.1. Axially and
Spherically Symmetric Solutions for Elastic Solids
4.1.1. A solid cylindrical bar with radius a and length L is subjected to a uniform pressure on its ends.
The bar is made from a linear elastic solid with Young’s modulus and Poisson’s ratio .
Write down the
components of the stress in the bar.
Show that the stress satisfies the equation of static equilibrium, and
the boundary conditions on all its surfaces. Express your answer as components in a
Cartesian basis with parallel to the axis of the cylinder.
Find the strain
in the bar (neglect temperature changes)
displacement field in the bar
formula for the change in length of the bar
Find a formula
for the stiffness of the bar (stiffness = force/extension)
Find the change
in volume of the bar
total strain energy in the bar.
4.1.2. Elementary calculations predict that the stresses in a
internally pressurized thin-walled sphere with radius R and wall thickness t<<R
are . Compare this estimate with the exact solution
in Section 4.1.4. To do this, set and expand the formulas for the stresses as a Taylor series in t/R.
Suggest an appropriate range of t/R
for the thin-walled approximation to be
4.1.3. A baseball can be idealized as a small rubber core
with radius a, surrounded by a shell
of yarn with outer radius b. As a first approximation, assume that the
yarn can be idealized as a linear elastic solid with Young’s modulus and Poisson’s ration ,
while the core can be idealized as an incompressible material. Suppose that ball is subjected to a uniform
pressure p on its outer surface. Note that, if the core is incompressible, its
outer radius cannot change, and therefore the radial displacement at . Calculate the full displacement and stress
fields in the yarn in terms of p and
relevant geometric variables and material properties.
4.1.4. Reconsider problem 3, but this time assume that the
core is to be idealized as a linear elastic solid with Young’s modulus and Poisson’s ration . Give expressions for the displacement and
stress fields in both the core and the outer shell.
4.1.5. Suppose that an elastic sphere, with outer radius ,
and with Young’s modulus and Poisson’s ratio is inserted into a spherical shell with
identical elastic properties, but with inner radius and outer radius . Assume that so that the deformation can be analyzed using
linear elasticity theory. Calculate the
stress and displacement fields in both the core and the outer shell.
planet with outer radius a has a
radial variation in its density that can be described as
As a result, the interior of the solid is subjected to
a radial body force field
where g is
the acceleration due to gravity at the surface of the sphere. Assume that the planet can be idealized as a
linear elastic solid with Young’s modulus and Poisson’s ratio . Calculate the displacement and stress fields
in the solid.
4.1.7. A solid, spherical nuclear fuel pellet with outer
radius is subjected to a uniform internal
distribution of heat due to a nuclear reaction. The heating induces a
steady-state temperature field
where and are the temperatures at the center and outer
surface of the pellet, respectively.
Assume that the pellet can be idealized as a linear elastic solid with
Young’s modulus ,
Poisson’s ratio and thermal expansion coefficient . Calculate the distribution of stress in the
4.1.8. A long cylindrical pipe with inner radius a and outer radius b has hot fluid with temperature flowing through it. The outer surface of the pipe has temperature
. The inner and outer surfaces of the pipe are
traction free. Assume plane strain
deformation, with . In addition, assume that the temperature
distribution in the pipe is given by
stress components in the pipe.
Find a formula
for the variation of Von-Mises stress
the tube. Where does the maximum value
The tube will
yield if the von Mises stress reaches the yield stress of the material. Calculate the critical temperature difference
that will cause yield in a mild steel pipe.