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Design of W, I, S, HSS steel columns for axial compression (13 edition)

Design code: AISC 13ed.

Design requirements

LRFD (Load and Resistance Factor) design:

LRFD design:

1. P_u £ fP_n.

Where P_u is factored axial compressive force calculated based on strength design provision of building code.  P_n is  axial compressive strength calculated based on AISC 13th edition, f = 0.9 is resistance factor.

ASD (Allowable Stress ) design:

1. P_a £ P_n/W.

Where P_a is axial compressive force calculated based on allowable stress design provision of building code, P_n is  normal compressive strength calculated based on AISC 13th edition, W = 1.67 is safety factor.

Design procedure:

LRFD

1. Calculate factored load based on strength design provision of building code
2. Calculate factored axial compressive force, P_u.
3. Select a trial column size and calculate normal axial compressive strength, P_n from Chapter E of AISC 13th edition.
4. Multiply P_n by f = 0.9. and compare fP_n with P_u.

ASD design:

1. Calculate apply load based on allowable stress design provision of building code
2. Calculate apply axial compressive force, P.
3. Select a trial column size and ccalculate normal axial compressive strength, P_n from Chapter E of AISC 13th edition
4. Divide P_n by W = 1.67 and compare P_n/W with P.

Determine factored axial force, P_u for strength design & P for allowable stress design

Load combination factor: 

shall be determined according to applicable building code.  Use ASCE 7 when building code is not available.

Determine axial compressive strength,  P_n

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. Calculate, l = 4.71ÖE/F_y or F_e = p E/(KL/r)² where E is elastic modulus, F_y is yield strength of steel.

3. If KL/r £  l , or Fe ³ 0.44 F_y then F_cr = (0.685^Fy/Fe ) F_y  
4. If KL/r >  l , or Fe < 0.44 F_y then F_cr = 0.877 F_e  
5. LRFD - The axial compressive strength, fP_n = 0.9A_gF_cr, where Ag is gross section area of member.
6. ASD - The allowable axial compression, P_a = A_gF_cr/1.67.

Example 1:

Situation: A structural column is supporting roof

Design Code: AISC ASD 13th 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 W6 beam

Solution:

Try W6x15, A = 4.43 in², r_x = 2.56 in, r_y = 1.46 in

Slenderness factor, K_x = 1, K_y = 1

Slenderness ratio, K_xL_ux/r_x = 70.3, K_yL_uy/r_y = 123.3

Elastic modulus, E = 29000 ksi

parameter, l = (4.71)Ö(29000/50) = 113.4   < 123.3

F_e = (3.14)²(29000)/(123.3)² = 18.83 ksi

F_cr = (0.877)(18.83)= 16.5 ksi  

LRFD solution

Compressive strength of column, fP_n = 0.9(4.43)(16.5) = 65.8 kips   O.K.

Factored column load: P_u = {[(1.2)(20)+(1.6)(20)](30)(30)+1.2(15)(15)}/1000 = 50.67 kip

ASD solution

Allowable column load: P_a = (16.5)(4.43)/1.67 = 43.8 kips    

Apply column load: P = {[(20)+(20)](30)(30)+(15)(15)}/1000 = 36.2 kip  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 an economical column

Solution:

Try W6x9, A = 2.68 in², r_x = 2.47 in, r_y = 0.905 in.

Lateral unsupported length, L_ux = 15 ft, L_uy = 7.5 ft

Slenderness ratio, K_xL_ux/r_x = 72.8, K_yL_uy/r_y = 99.4

Slenderness parameter, l = (4.71)Ö(29000/50) = 113.4   >  99.4

F_e = (3.14)²(29000)/(99.4)² = 28.94 ksi

F_cr = [(0.658)^(50/28.94)](50) = 24.26 ksi  

LRFD solution

Compressive strength of column, fP_n = 0.85(2.68)(26) = 59.2 kips   

Factored column load: P_u = {[(1.2)(20)+(1.6)(20)](30)(30)+1.2(9)(15)}/1000 = 50.56 kip  O.K.

ASD solution

Allowable column load: P_a = (16.5)(4.43)/1.67 = 43.8 kips

Apply column load: P = {[(20)+(20)](30)(30)+(9)(15)}/1000 = 36.1 kip  O.K.