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MECHANICS - CASE STUDY SOLUTION


Free Body Diagram of
Typical Box Beam

 

The roof of a new room addition supports 80 lb/ft2 roof load. The 20 ft long box beams are evenly spaced every 4 ft. Thus the total load on a single beam (not end beams) is

     w = (80 lb/ft2) (4 ft) = 320 lb/ft

The wall reaction forces are equal due to symmetry.

     RA = RB = (320 lb/ft)(20 ft)/2 = 3,200 lb = 3.2 kip

     
    Shear and Moment Diagrams


Moment and Shear Diagrams

 

A shear and moment diagram of the channel will help identify the maximum shear and moment. Both of these values are needed to determine the maximum shear and bending stress.

Only one cut is needed due to the continuous loading.

     ΣF = 0
     V = 3.2 - 3.2x  kip

     ΣM = 0
     M + (0.32x)(x/2) - 3.2x = 0
     M = 3.2x - 0.16x2

The shear load is maximum at each end of the beam (this is one reason old wood beams crack near the center at each end). To design the screw spacing, it is best to take the worst load, 3,200 lb, even though it changes along the beam.

The maximum moment, 16.0 kip-ft, occurs at the beam mid-point.

     
    Shear Stress


Shear Stress on
Horizontal Surfaces (A)


Shear Stress on
Vertical Surfaces (B)

 

There are two possible location to shear the screws. The first location is the horizontal joints and involves the full top section (label Area A in first diagram). The left side is part of the left 2x6 and it is assumed the wood will not shear. The second location is the vertical joints and involves only part of the top section (label Area B in the second diagram). In both cases, there are two surfaces that will carry the shear load.

First determine Q for both cases.

     QA = AAyA = (7.0)(1.5)(3.5 - 0.75) = 28.875 in3

     QB = AByB = (4.0)(1.5)(3.5 - 0.75) = 16.5 in3

Next, the moment of inertia for the full cross sections needs to be known.

     I = 7(7)3/12 - 4(4)3/12 = 178.75 in4

Using the shear stress equation gives,

     

This shear stress needs to be carried by the screws. Each screw can carry 800 lb. The minimum spacing (sA and sB) of the screws can be found using

     FScrew = τ A

For each surface, this gives the spacing as

     800 lb = (172.3 lb/in2) [(1.5 in) sA]
     sA = 3.095 in

     800 lb = (98.46 lb/in2) [(1.5 in) sB]
     sB = 5.417 in

Thus, the screws should be every 3 inches on the horizontal joints and every 5 inches on the vertical surfaces.

     
    Bending Stress

   

The bending stress should also be checked. Unlike shear stress, the maximum bending stress occurs at the point furthest from the neutral axis. This is at the top or bottom surface. Using the bending stress equation gives,

     σ = -My/I
        = -(16 kip-ft)(12 in/ft)(-3.5 in) / (178.75 in4)
        = 3.759 ksi

This is lower than the allowable bending stress for wood so it should be safe.

     
   
 
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