Problem Sheet 1

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

  1. For a panel (which is loaded in bending by a central load), the minimum stiffness S*, the length L and the width b are specified. The objective is to minimize the mass. The panel thickness h is the free variable. Derive the material index for this optimization problem.
  2. For a beam of square cross-section, the minimum stiffness S* and the length L are specified. The objective is to minimize the mass. The area of the cross-section A is the free variable. Derive the material index for this optimization problem. Is the result dependent on the shape of the beam? Explain.
  3. In the problems (1) and (2) above, if the constraint is the minimize cost rather than weight, derive the material indices.
  4. [E5.4] The objective in selecting a material for a panel of given in-plane dimensions for the casing of a portable computer is that of minimizing the panel thickness h while meeting a constraint on bending thickness, S*. What is the appropriate material index?
  5. [E5.5] Derive the material index for a torsion bar with a solid circular section. The length L and the (torsional) stiffness S* are specified, and the torsion bar is to be as light as possible. Given the moment $K = \frac{\pi}{2} r^{4}$ m4.
  6. [E5.7] A material is required for a cheap column with a solid circular cross-section that must support a load Fcrit without buckling. It is to have a height L. Write down an equation for the material cost of the column in terms of its dimensions, the price per kg of the material, Cm, and the material density$\rho$. The cross-section A is a free variable—eliminate it by using the constraint that the buckling load must not be less than Fcrit ($= \frac{n^{2} \pi^{2} E I}{L^{2}}$). What is the material index for finding the cheapest tie?
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