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Design code: AISC Allowable Stress Design 9^{th} edition,
1989.

- Maximum axial compressive stress, f
_{a}must not exceed allowable stress, F_{a}.

Design procedure:

- Calculate axial compression load and
- Select a trial beam size and calculate
axial compressive stress, f
_{a}. - Calculate allowable axial stress, F
_{a}(see below)

Axial compression stress shall
be determined as

f_{a}= P/A where
P is axial load, A is cross section area.

- Determine slenderness ratio in both axes.

K_{x}L_{x}/r_{x} and
K_{y}L_{y}/r_{y}

where

L_{x} ,L_{y} are
laterally unsupported lengths in X and Y direction,

r_{x} ,r_{x} are
radius of gyrations in X and Y direction

K_{y} ,K_{y} are
slenderness factors in X and Y direction and can be determined as

- Compare the larger value of K
_{x}L_{x}/r_{x}and K_{y}L_{y}/r_{y}with

C_{c} = Ö2p^{2}E/F_{y}

where E is elastic modulus, F_{y} is
yield strength of steel.

- If KL/r £ C
_{c}, then

- If KL/r > C
_{c}, then

**Example 1:**

__Situation__: A structural column is
supporting roof

Design Code: AISC ASD 9th edition_{}

Roof live load: W_{L} =
20 psf

Roof dead load: W_{D} =
20 psf

Unsupported Length of column: L_{ux} =
15 ft, L_{uy} =
15 ft

Top and bottom of column is pinned

Tributary area 30 ft x 30 ft

Material: ASTM A36, yield strength, Fy = 50 ksi

__Requirements__: Select
a W8 beam

Solution:

Try a W6x15 column, A = 4.43 in^{2},
r_{x} = 2.56 in,
r_{y} = 1.46 in

Total column load: P = [(20+20)*30*30+15*15]/1000 = 36.2 kip

Compressive stress, fa = 36.2/4.43 = 8.2 ksi

Slenderness ratio factor: K_{x} =
1, K_{y} = 1

Slenderness ratio: K_{x}L_{ux}/r_{x} =
70.3, K_{y}L_{uy}/r_{y} =
123.8

Elastic modulus, E = 29000 ksi

Factor, C_{c} = Ö2p^{2}E/F_{y} =
106.9 < 123.8

Allowable compressive stress,

Fa = [(12)(3.14)^{2}(29000)]/[(23)(123.8)^{2}]
= 9.8 ksi O.K.

**Example 2**:

Situation: In example 1 assume that the column is supported
laterally at mid-height of column in minor axis,

Requirement: Select a more economical column

Solution:

Try W6x9, A = 2.68 in^{2}, r_{x} =
2.47 in, r_{y} =
0.905 in.

Compressive stress, fa = 26.2/2.68 = 13.5 ksi

Unsupported Length of column: L_{ux} =
15 ft, L_{uy} =
7.5 ft

Slenderness ratio, K_{y}L_{uy}/r_{y} =
99.4 < Cc = 106.9

Allowable compressive stress

F_{a} =
{[1-(99.6)^{2}]/[2*(106.9)^{2}]}/{5/3+[3*99.4/(8*106.9)]-[(99.4)^{2}/(8*106.9^{3})]}
= 14.8 ksi O.K.