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Design of steel columns (ASD, Allowable Stress design)

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

Design requirements

1. Maximum axial compressive stress, f_a must not exceed allowable stress, F_a.

Design procedure:

1. Calculate axial compression load and
2. Select a trial beam size and calculate axial compressive stress, f_a.
3. Calculate allowable axial stress, F_a (see below)

Determine axial compression stress

Axial compression stress shall be determined as

f_a= P/A            where P is axial load, A is cross section area.

Determine allowable axial stress F_a (ASD)

1. Determine slenderness ratio in both axes.

K_xL_x/r_x and K_yL_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

2. Compare the larger value of  K_xL_x/r_x and K_yL_y/r_y with

C_c = Ö2p²E/F_y  

where E is elastic modulus, F_y is yield strength of steel.

3. If KL/r £ C_c, then

4. 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 in², 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_xL_ux/r_x = 70.3, K_yL_uy/r_y = 123.8

Elastic modulus, E = 29000 ksi

Factor, C_c = Ö2p²E/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 in², 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_yL_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*106.9³)]} = 14.8 ksi O.K.