MECHANICS  CASE STUDY SOLUTION

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^{o}F and it
was perfectly level. Later in the summer, when the temperature is at 125^{o}F,
the two packages are placed on the rack as shown. Thus the change in temperature is 105^{o}F.
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 Thermal Expansion

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_{Tal} =
L_{al} α_{al} (ΔT)
= (5 ft) (12
in/ft) (13×10^{6}/^{o}F) (105^{o} F)
= 0.08190 in
δ_{Tst} =
L_{st} α_{st} (ΔT)
= (3
ft) (12 in/ft) (6.5×10^{6}/^{o}F) (105^{o} F)
= 0.02457 in






Pipe Compression Load and Deflection

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




Pipe CrossSection 

Before finding the deflection, the crosssection area of the pipes needs to
be known.
= 0.08590 in^{2}




Pipe Compression Deflection
Due to Loads 

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






Rack Rotation

Rotation Angle, θ


The angle or rotation can now be determined from both the thermal
and compression deflections.
δ_{al} = δ_{Tal} 
δ_{Fal} = 0.8190  0.00931
= 0.08097
in
δ_{st} = δ_{Tst}  δ_{Fst} =
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.



