Ch 9. Columns Multimedia Engineering Mechanics BasicColumns FixedColumns EccentricLoads
 Chapter 1. Stress/Strain 2. Torsion 3. Beam Shr/Moment 4. Beam Stresses 5. Beam Deflections 6. Beam-Advanced 7. Stress Analysis 8. Strain Analysis 9. Columns Appendix Basic Math Units Basic Equations Sections Material Properties Structural Shapes Beam Equations Search eBooks Dynamics Fluids Math Mechanics Statics Thermodynamics Author(s): Kurt Gramoll ©Kurt Gramoll

MECHANICS - CASE STUDY SOLUTION

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.

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

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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