If you want to support a precision optical system laser metr

If you want to support a precision optical system (laser metrology equipment, for instance) on a stable platform, you put it on a granite slab supported on end plinths to bring it to working height. (Granite can be ground to a flat surface and is thermally very stable and hard wearing.) The granite chosen for one such table has a fracture toughness of 0.9 MPa.m^1/2 and is known, from NDT procedures, to contain internal cracks up to 5 mm in length. If the table is 2 m long and 1 m deep, simply supported at its ends and must carry a uniformly distributed load of 2000 N on its upper surface (as in Figure 7.2 of the text), what is the minimum thickness the slab must have? Include the self-weight of the slab in the analysis. Assume that at least one of the cracks will lie in the part of the beam that carries the highest tensile stress - that is, at the lower surface. (The density of granite is 2700 kg/m^3).

Solution

L = 2000*2 = 4000N

Self weight of slab = 27000*2*1*t*9.81 N = 52974*t N

Total load on slab = 4000+ (52974*t)

Total load per area = fracture toughness

4000+ (52974*t)/(2*1) = 0.9*10⁶

Therefore thickness of slab = 3.3903 m