Rack Storage Structure

A storage rack is set up over a 2 foot rise in the floor in a warehouse. The rack spans a 12 foot space and is supported by two pipes, one at each end. The left pipe is an aluminum pipe (E_al = 10,000 ksi) and right pipe is a steel pipe (E_st = 29,000 ksi). Both pipes have an outside diameter of 1/2" and a wall thickness of 1/16".

The rack was constructed in winter at a temperature of 20^oF and it was perfectly level. Later in the summer, when the temperature is at 125^oF, the two packages are placed on the rack as shown. Thus the change in temperature is 105^oF.

Since it is critical that the shelf remains nearly level, the angle of rotation of the shelf needs to be calculated. This can be done by determining both the thermal expansion and the compression deflection due to the package loads of both pipes.

Pipe Expansion After
Temperature Change

Since each pipe is a different length and made of different materials, their overall length changed due to the temperature change will be different.

     δ_T-al = L_al α_al (ΔT)
            = (5 ft) (12 in/ft) (13×10^-6/^oF) (105^o F)
            = 0.08190 in

     δ_T-st = L_st α_st (ΔT)
            = (3 ft) (12 in/ft) (6.5×10^-6/^oF) (105^o F)
            = 0.02457 in

Reactions at Pipe Supports

Due to the packages on the rack, there will also be compression loads in the pipes which will shorten the pipes slightly. First, the compression loads in each pipe needs to be determined from basic static equilibrium equations.

     ΣM_st = 0
     -R_al (12 ft) + (200 lb)(5 ft) + (300 lb)(2 ft) = 0
     R_al = 133.33 lb

     ΣF_y = 0
     -133.33 lb + 200 lb + 300 lb - R_st = 0
     R_st = 366.67 lb

Before finding the deflection, the cross-section area of the pipes needs to be known.

     
           = 0.08590 in²

Recall, axial deflection is a function of the material length, stiffness, area and load. Now that the load is known, the deflection can be calculated for each pipe as,

     
        = 0.009313 in

     
        = 0.005299 in

Rotation Angle, θ

The angle or rotation can now be determined from both the thermal and compression deflections.

     δ_al = δ_T-al - δ_F-al = 0.8190 - 0.00931

            = 0.08097 in

     δ_st = δ_T-st - δ_F-st = 0.02457 - 0.005299

            = 0.01927 in

     s = δ_al - δ_st = 0.08097 in - 0.01927 in

        = 0.06170 in

The angle or rotation, θ, is

     tanθ = s/144 = 0.06170/144 = 0.0004285

or

     θ = 0.02455^o

This is a small angle, so the rack could be considered level.