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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. Pu £ fPn.

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

ASD (Allowable Stress ) design:

1. Pa £ Pn/W.

Where Pa is axial compressive force calculated based on allowable stress design provision of building code, Pn 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, Pu.
3. Select a trial column size and calculate normal axial compressive strength, Pn from Chapter E of AISC 13th edition.
4. Multiply Pn by f = 0.9. and compare fPn with Pu.

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, Pn from Chapter E of AISC 13th edition
4. Divide Pn by W = 1.67 and compare Pn/W with P.

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

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

### Determine axial compressive strength,Pn

1. Determine slenderness ratio in both axes.

KxLx/rx and KyLy/ry

where

Lx ,Ly are laterally unsupported lengths in X and Y direction,

rx ,rx are radius of gyrations in X and Y direction

Ky ,Ky are slenderness factors in X and Y direction and can be determined as 1. Calculate, l = 4.71ÖE/Fy or Fe = p E/(KL/r)2 where E is elastic modulus, Fy is yield strength of steel.
1. If KL/r £  l , or Fe ³ 0.44 Fy then Fcr = (0.685Fy/Fe ) Fy
2. If KL/r >  l , or Fe < 0.44 Fy then Fcr = 0.877 Fe
3. LRFD - The axial compressive strength, fPn = 0.9AgFcr, where Ag is gross section area of member.
4. ASD - The allowable axial compression, Pa = AgFcr/1.67.

Example 1:

Situation: A structural column is supporting roof

Design Code: AISC ASD 13th edition

Roof live load: WL = 20 psf

Unsupported Length of column: Lux = 15 ft, Luy = 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, rx = 2.56 in, ry = 1.46 in

Slenderness factor, Kx = 1, Ky = 1

Slenderness ratio, KxLux/rx = 70.3, KyLuy/ry = 123.3

Elastic modulus, E = 29000 ksi

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

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

Fcr = (0.877)(18.83)= 16.5 ksi

LRFD solution

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

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

ASD solution

Allowable column load: Pa = (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, rx = 2.47 in, ry = 0.905 in.

Lateral unsupported length, Lux = 15 ft, Luy = 7.5 ft

Slenderness ratio, KxLux/rx = 72.8, KyLuy/ry = 99.4

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

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

Fcr = [(0.658)(50/28.94)](50) = 24.26 ksi

LRFD solution

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

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

ASD solution

Allowable column load: Pa = (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.