Material indices for elastic beams with differing constraints (Figure E4). Start each of the four parts of this problem by listing the function, the objective and the constraints. You will need the equations for the deflection of a cantilever beam with a square cross-section t x t, given in UASSP, 3. The two that matter are that for the deflection of a beam of length L under an end load F:

F L3 =

3EI and that for the deflection of a beam under a distributed load f per unit length:

1 f L⁴ =

8 E I

where I =t⁴ / 12 . For a self-loaded beam f =A g where is the density of the material of the beam, A its cross-sectional area and g the acceleration due to gravity.

Show that the best material for a cantilever beam of given length L and given (i.e. fixed) square cross-section (t x t) that will deflect least under a given end load F is that with the largest value of the index M = E, where E is Young's modulus (neglect self-weight). (Figure E4a.)

Show that the best material choice for a cantilever beam of given length L and with a given section (t x t) that will deflect least under its own self-weight is that with the largest value of M =

E/, where is the density. (Figure E4b.)

Show that the material index for the lightest cantilever beam of length L and square section (not given, i.e., the area is a free variable) that will not deflect by more than under its own weight is

= E / ² . (Figure E4c.)

Show that the lightest cantilever beam of length L and square section (area free) that will not deflect by more than under an end load F is that made of the material with the largest value of

= E^{1/ 2} / (neglect self weight).