Problem Diagram

Recall Dave, a college student, decided to construct a hanging shelf from the ceiling. He plans to place 12, 1-foot square boxes on the shelf. The left boxes will hold heavy books (45 lb each) and the right boxes will hold clothes (20 lb each). The shelf itself weighs 35 lb.

Now, Dave needs to determine what size wood rods are needed to support the shelf and boxes. The rods come in increments of 1/16" and the wood will be Douglas Fir.

First the actual load in each rod needs to be determined. A free body diagram will help analyze the shelf and solve for the rod load. Remember, there are two roads at each end.

Free Body Diagram of Shelf

     ΣF_y = 0
     2 A_y + 2 B_y - 45(6) - 20(6) - 35 = 0
     A_y + B_y = 212.5 lb

Remember, the 2 before A_y and B_y is due to each end having two rods.

     ΣM_A = 0
     -45(6)(3) - 20(6)(9) + 2 B_y (12) - 35 (6) = 0
     B_y = 87.5 lb

     A_y = 212.5 - 87.5 = 125 lb

Thus, the two rods at A will each carry 125 lbs and the two rods at B will carry 87.5 lbs. Using the worse case, 125 lbs, the design load will be 4 times the actual load for the required factor of safety,

     P = 4(125) = 500 lb

Using the appendix, it is determined that Douglas Fir wood can withstand a stress of 7,500 psi before it will fail. Using this failure stress, gives

     σ = P/A
     (7,500 psi) = (500 lb)/A
     A = 0.0667 in²

The diameter of a cicular bar for 0.0667 in² is

     π (D/2)² = A = 0.0667 in²
     D = 0.2913 in

The smallest rod to be at least 0.2913 inches is 5/16"