For a bolted assembly, the stiffness of each bolt is kb = 4 Mlbf/in and the stiffness of the members is km = 12 Mlbf/in per bolt. The joint is subject to occasional disassembly for maintenance and should be preloaded accordingly. A fluctuating external load is applied to the entire joint with Pmax = 80 kips and Pmin = 20 kips. Assume the load is equally distributed to all the bolts. It has been determined to use 1/2 in-13 UNC grade 8 bolts with rolled threads.

(a) Determine the minimum number of bolts necessary to avoid yielding of the bolts.

(b) Determine the minimum number of bolts necessary to avoid joint separation.