1. Provide a general description (1-2 sentences) of the Raleigh-Ritz method.
What is minimized?
Which feature of the solution is approximated?
Which feature of the solution is exactly satisfied?
How are the coefficients determined?
e. Write an appropriate polynomial trial function (generic term of the series, let it be
different as the number of the series term changes) for a beam of length L with zero
displacements at the ends x=0, x=L.
For fixed domain and boundary conditions, what kind of solution are we trying to find?
Over what part of the domain? Is the solution exact? Why use this method?
What is a variation? If we have a variation in a displacement function with displacement
conditions at the ends (essential or rigid boundary conditions), what is the variation at
Is it possible in general, for a problem in the calculus of variations (i.e. not just the
Raleigh-Ritz method), to find the boundary conditions as part of the solution?
i. If the variation in a displacement solution is 8(x), the trial solution is u(x), , and the
exact solution is y(x), write an equation relating 8(x), u(x) and y(x). .
2. What are the 3 key characteristics of an exact solution to an elasticity boundary value problem?
3. Plane problems.
a. Write the defining assumptions for plane stress (stress components) and plane strain
(displacement components). Show that the same simplified equilibrium equations
result. Show that the z-direction equilibrium equation is identically satisfied for both
plane strain and plane stress.
b. Substitute the expressions defining the Airy stress function Q(x,) y) (use Cartesian
coordinate expressions) into the remaining 2 equilibrium equations and show that
equilibrium is satisfied. Is there an important assumption about the specific form chosen
for the function that is needed to assure that equilibrium is satisfied?
4. Consider the following state of stress in a cylindrical body with z-axis normal to its cross-
[o] = 0 0
0 0 O2 (x, y)
Use the 6 compatibility equations to show that, in the absence of body forces, the most general form of
O E (x,y) is given by
o z (x,y) = OCX + By + Y,
where O, Band yare constants.
It can be shown that the displacements, corresponding to a stress field which is axisymmetric
about the z-axis, are given as (with the rigid-body constants set to zero):
u, r ]
- Inr - + +
where E is the Young's modulus and A, B, Care undetermined constants. The corresponding stresses
Tro =0 =
Discuss why, for a thick-walled cylinder (Fig 8-8), or a pressurized hole (Fig 8-10) a3 = 0, while for a
problem involving a jump discontinuity in us (e.g. a mode III crack) we would want a3 0.
What are the resulting stresses and displacements for a thick-walled cylinder? (Do not evaluate
6. Consider the stress function - where a is the radius of a solid circular
torsion bar and C is a constant. Show that this stress function solves the torsion problem for a
circular cylinder of radius a (work it out, don't try to change the stress function, solve for C). Use
Eq. (9.3.18) (hint: this is easily accomplished with polar coordinates r2 =x2 + y2 2 ) to determine
the expression for the applied torque in terms of a and the angle of twist per unit length a. Then
assuming the applied torque is known, give the expression for the angle of twist per unit length
a in terms of the torque and the radius a.
7. Consider the Airy stress function 0 = ox'y with zero body forces. (a) Verify that it satisfies the
biharmonic equation. (b) Determine the in-plane stresses o , oy, I (c) Determine the
tractions on the four rectangular boundaries X = 0, x = b, Y = 0, V = C. . (d) As a plane strain
solution, determine Tx2,T,22,00 and all of the strain components. (e) As a plane stress solution,
determine Tx2,Ty2,02 and all of the strain components.
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