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Due: Lecture 25
HOME ASSIGNMENT
Warm-Up Exercises
Let’s consider a steel bar that is rigidly clamped at one end and free at the other end with a gap of 0.002L between that end and a wall. The bar is of length L, thickness h, and width b. Bending and instability issues are not of concern. The bar is subjected to a uniform temperature change of ∆T from room temperature of 70°F. Properties of steel are modulus of 30 Msi, Poisson’s ratio of 0.3, and coefficient of thermal expansion of 6 μstrain/°F.
h 0.002L
L
steel
z
x
NOTE: For the following, determine the total strain and the two components of thermal and mechanical.
- Determine the stress and strain state in the x-direction for temperatures where the bar end does not contact the wall.
- Determine the temperature at which the bar first contacts the wall and the corresponding stress and strain state in the x-direction.
- Determine the stress and strain state in the x-direction for temperatures where the bar end does contact the wall.
Fall, 2002
Practice Problems
- A bar is 60” long, 2.5” thick, and 2.5” wide and is held between two rigid walls. The first 20” of the bar is made of aluminum while the second part is made of titanium. The bar is subject to a constant temperature change, ∆T, of 100°F. Buckling is not a concern.
Aluminum Titanium E 10.0 Msi 15.5 Msi
α 12.0 μstrain/°F 5.0 μstrain/°F
Al Ti
z
x
(a) Determine the axial stress, σx, and the total strain, εx, distribution in the
bar as a function of x.
(b) Plot the three components of the strain (total, thermal, and mechanical) as a function of x.
(c) What is the displacement of the material junction point?
(d) Where would you need to apply St. Venant’s principle? Describe what would happen at these points.
(e) Assuming the three-dimensional effects do not change the answer to part (c), determine the full three-dimensional state of stress and strain (total, thermal, mechanical) at the junction point.
Fall, 2002
(a) Determine the axial stress, σx, and the total strain, εx, distributions in the
bar as a function of x (and the bar length, constant A, and the material parameters).
(b) Plot the three components of the strain (total, thermal, and mechanical) in the x-direction as a function of x for a 6-foot long structure. The value of A is 0.2°F/ft^3.
(c) Neglecting end effects, what would be the general state of strain for a bar of this configuration (not just of steel, but of any isotropic material)?
(d) For the case of part (b), there is now a constant thermal differential of 35°F between the top surface and bottom surface that is linear in z. Describe what will change about the behavior of the structure. Determine, as best as possible, the stress and strain states in the x-direction.
