Problems for Chapter 5
Analytical Techniques and Solutions for
Linear Elastic Solids
5.7.1. A shaft with length L and square cross section is fixed at
one end, and subjected to a twisting moment T
at the other. The shaft is made from a
linear elastic solid with Young’s modulus E
and Poisson’s ratio . The torque causes the top end of the shaft to
rotate through an angle .
the following displacement field
Show that this is a kinematically admissible
displacement field for the twisted shaft.
the strains associated with this kinematically admissible displacement field
show that the potential energy of the shaft is
You may assume that the potential energy of the
torsional load is
the value of that minimizes the potential energy, and hence
estimate the torsional stiffness of the shaft.
5.7.2. In this problem you will use the principle of minimum
potential energy to find an approximate solution to the displacement in a
pressurized cylinder. Assume that the
cylinder is an isotropic, linear elastic solid with Young’s modulus and Poisson’s ratio ,
and subjected to internal pressure p
radial displacement field as ,
where are constants to be determined. Assume all other components of displacement
are zero. Calculate the strains in the
an expression for the total strain energy of the cylinder per unit length, in
terms of and relevant geometric and material parameters
write down the potential energy (per unit length) of the cylinder.
the values of that minimize the potential energy
a graph showing the normalized radial displacement field as a function of the normalized position in the cylinder, for ,
and and . On the same graph, plot the exact solution,
given in Section 4.1.9.
5.7.3. A bi-metallic strip is made by welding together two
materials with identical Young’s modulus and Poisson’s ratio ,
but with different thermal expansion coefficients ,
as shown in the picture. At some
arbitrary temperature the strip is straight and free of stress. The temperature is then increased to a new
causing the strip to bend. Assume that,
after heating, the displacement field in the strip can be approximated as ,
where are constants to be determined.
describe the physical significance of the shape changes associated with .
distribution of (infinitesimal) strain associated with the kinematically
admissible displacement field
the strain energy density distribution in the solid. Don’t forget to account for the effects of
potential energy to determine values for in terms of relevant geometric and material
5.7.4. By guessing the deflected shape, estimate the
stiffness of a clampedclamped beam
subjected to a point force at mid-span.
Note that your guess for the deflected shape must satisfy ,
so you can’t assume that it bends into a circular shape as done in class. Instead, try a deflection of the form ,
or a similar function of your choice (you could try a suitable polynomial, for
example). If you try more than one guess
and want to know which one gives the best result, remember that energy
minimization always overestimates stiffness.
The best guess is the one that gives the lowest stiffness.
5.7.5. A slender rod with length L and cross sectional area A is subjected to an axial body force . Our objective is to determine an approximate
solution to the displacement field in the rod.
that the displacement field has the form
where the function w
is to be determined. Find an expression
for the strains in terms of w and hence deduce the strain energy density.
Show that the
potential energy of the rod is
To minimize the
potential energy, suppose that w is
perturbed from the value the minimizes V to a value . Assume that is kinematically admissible, which requires
that at any point on the bar where the value of w is prescribed. Calculate the potential energy and show that it can be expressed in the form
is a function of w only, is a function of w and ,
and is a function of only.
As discussed in
Section 8 of the online notes (or in class), if is stationary at ,
then . Show that, to satisfy ,
we must choose w to satisfy
first term by parts to deduce that, to minimize, V, w must satisfy
that this is equivalent to the equilibrium condition
deduce that if is not prescribed at either or both, then the boundary conditions on the
end(s) of the rod must be
that this corresponds to the condition that at a free end.
Use your results
in (2.5) to estimate the displacement field in a bar with mass density ,
which is attached to a rigid wall at ,
is free at ,
and subjected to the force of gravity (acting vertically downwards…)