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MECHANICS - CASE STUDY SOLUTION


Building Load

 

This case requires the center column for a basic building be designed to carry the roof load. The roof load has a dead and live roof load of, 300 and 250 kg/m2, respectively. Generally, a dead load is the static, non-changing load such as roof weight and equipment on the roof. On the other hand, a live load is changing loads, such as snow or wind. The total (worst case) should be considered when designing the column.

The other major condition is that the selection of the center column needs to be specified from a group of I-beams that are in storage. The best column will be the lightest column that can withstand the roof load. Both buckling and compression failure should be checked.

   
    Center Column Load


Roof Load Area
Carried by Center Column

 

The center column will need to carry the roof load that is half way to each of the other columns. The total roof area carried by the center column is 10 m by 8 m as shown in the diagram at the left. The total roof load over this area is

     F = (10 m)(8 m)(300 + 250 kg/m2)(9.81 m/s2)

        = 431.6 kN

The last term is the standard gravitational constant.

The column requires a factor of safety of 2.5, so the design load needs to be increased by a factor of 2.5, giving

     Pcr = 2.5 (431.6 kN) = 1.079 MN

     
    Required Moment of Inertia


Center Column Load

 


Both Directions must be
Considered for Buckling

 

The minimum moment of inertia is needed so that a suitable wide-flange I-beam can be chosen. Since both ends are assumed to be fixed, the Euler buckling equation is

     

Substituting known values give,

     

Solving for the moment of inertia,

     I = 6.696 × 10-6 m4 = 6.696 × 106 mm4

The I-beam must have this moment of inertia (or greater) in both direction. Generally, I-beams have a higher inertia around the x-axis, but buckling can occur about either axis.

     
    Column Selection

   

There are currently 18 different wide-flange I-beams available for the construction of the building. They are list below.

     
Section
Number
Weight Area
mm2
Ix
106 mm4
Iy
106 mm4
W310 x 67 67 8,530 145 20.7
  x 39 39 4,930 84.8 7.23
  x 33 33 4,180 65.0 1.92
  x 24 24 3,040 42.8 1.16
  x 21 21 2,680 37.0 0.986
W250 x 58 58 7,400 87.3 18.8
  x 45 45 5,700 71.1 7.03
  x 28 28 3,620 39.9 1.78
  x 22 22 2,850 28.8 1.22
  x 18 18 2,280 22.5 0.919
W200 x 59 59 7,580 61.2 20.4
  x 46 46 5,890 45.5 15.3
  x 36 36 4,570 34.4 7.64
  x 22 22 2,860 20.0 1.42
W150 x 37 37 4,730 22.2 7.07
  x 30 30 3,790 17.1 5.54
  x 22 22 2,860 12.1 3.87
  x 24 24 3,060 13.4 1.83

Wide-Flange Beams Available
     
   

Both Ix and Iy must be at least 6.696 × 106 mm4 to satisfy the buckling requirements. The critical moment of inertia is Iy. There are several beams that have moment of inertia's greater than 6.696 in both directions. However, the lightest one is

     W200 x 36

     
    Compression Stress Check

   

Even though the column was designed assuming buckling, the compression stress should be checked to make sure it does not exceed the yield stress of the material. The compression stress is

     σ = P/A = (1.079 MN)/(4,570 mm2)

        = 236.1 MPa

This is less than the yield stress of 250 MPa for structural steel and will not yield in compression.

     
   
 
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