Problem Sheet 1

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

- 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. - 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. - In the problems (1) and (2) above, if the constraint is the minimize cost rather than weight, derive the material indices.
- [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? - [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 $K = \frac{\pi}{2} r^{4}$ m^{4}. - [E5.7] A material is required for a cheap column with a solid circular cross-section that must support a load F
_{crit}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, C_{m}, 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 F_{crit}($= \frac{n^{2} \pi^{2} E I}{L^{2}}$). What is the material index for finding the cheapest tie?

page revision: 3, last edited: 04 Jan 2009 09:50