Problem Sheet 3: material indices for strength-limited design

The numbers in square brackets refer to the number of the problem in your textbook.

1. What is the material index for a light, strong tie-rod, given that the length of the rod is specified (L) and the cross-sectional area is the free parameter? What is this index called? Compare this with the material index for a light, stiff tie rod. What do you observe?
2. What is the material index for a light, strong panel, given that the width "b" and the span "L" are fixed and that the thickness "t" is a free variable?
3. In the problems (1) and (2) above, if the constraint is the minimize cost rather than weight, derive the material indices.
4. [E7.6] A centrifuge has a rotor that can be idealized as a uniform disk. It has a diameter of 200 mm and is made of a material of strength 450 MPa and density 7900 kg/m3. How fast can it be spun (in radians per second) before the stresses it carries exceed its yield strength?
5. [E7.7] A material is sought for a high-speed centrifuge. The objective is to achieve as high an angular velocity $\omega$ of the centrifuge disk as possible. The constraint is that the stress created by the centrifugal force must not exceed the yield strength of the material of which it is made. Derive an index to guide the choice of material to allow the maximum $\omega$.
6. [E7.8] The engine of a car is mounted on four shear bolts designed to fail in shear in a front-end collision, detaching the engine from the car, if the deceleration exceeds 10 g. Assume that all four bolts carry the same load. The mass of the engine is 80 kg. Because of space limitations, the maximum diameter of each shear bolt cannot exceed 5 mm. Calculate the constraint on the yield strength $\sigma_y$ of the material of the shear bolt, assuming that the shear strength is $\sigma_y/2$ and that the mounting is done in such a way that the shear can take place without friction. IF the bolts are to be as light as possible, what will be your choice metal (Use figure 7.8).
7. [E7.9] A plate with rectangular section 500 mm by 15 mm carries a tensile load of 50 kN. It is made of a ductile metal with a yield strength of 50 MPa. The plate contains an elliptical hole of length 100 mm and a minimum radius of 1 mm, oriented as shown. What are the nominal and maximum stresses in the plate? Will the plate start to yield? Will it collapse completely?
8. [E7.10] Valve springs for high-performance automobile engines must be light to minimize inertial loads since part of their mass moves with the valves; at high engine speeds the valves, if heavy, bounce out of contact with the valve itself ('valve bounce'), impeding the flow of gas into and out of the combustion chamber. Derive the index for light springs to guide choice for these, remembering that engine temperatures can reach 200 degree C.
page revision: 10, last edited: 04 Feb 2009 09:32