
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/m^{2}, respectively. Generally,
a dead load is the static, nonchanging 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 Ibeams 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/m^{2})(9.81
m/s^{2})
= 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
P_{cr} = 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 wideflange Ibeam
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} m^{4} =
6.696 × 10^{6} mm^{4}
The Ibeam must have this moment of inertia (or greater) in both direction.
Generally, Ibeams have a higher inertia around the xaxis, but buckling can
occur about either axis.






Column Selection



There are currently 18 different wideflange Ibeams available for the construction
of the building. They are list below.




Section
Number 
Weight 
Area
mm^{2} 
I_{x}
10^{6} mm^{4} 
I_{y}
10^{6} mm^{4} 
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 
WideFlange Beams Available 





Both I_{x} and I_{y} must be at least 6.696 × 10^{6} mm^{4} to
satisfy the buckling requirements. The critical moment of inertia is I_{y}.
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 mm^{2})
= 236.1 MPa
This is less than the yield stress of 250 MPa for structural steel and will
not yield in compression.




