A sailing enthusiast is seeking materials for lightweight panels to use in a sea-going yacht. The panels are of rectangular cross-section and will be loaded in bending, as shown in Figure 1. The span L and width b of the panels are fixed, but the thickness d may vary (up to a maximum specified value). The required stiffness is specified as a maximum allowable deflection? under a given central load W, for a simply supported span. The designer is interested in two scenarios: (i) minimum mass; (ii) minimum material cost.
NB: The central deflection of a simply supported span under a point load is given by:
?= (WL^3)/48 EI where I= bd^3/12
a) Show that the stiffness and geometric constraints lead to the relationship Ed^3 = constant, where E is Young’s modulus. Explain why the design specification leads to a minimum allowable value for E, and find an expression for this minimum value.
b) Write down an expression for the first objective to be minimised (mass) and use the stiffness constraint to eliminate the free variable (thickness). Hence define the material performance index to maximise for a minimum mass design.
c) How does the short-list of materials in (c) change if the limit on depth leads to a requirement for a minimum required Young's modulus of 5 GPa?