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## Design of steel columns (ASD, Allowable Stress design)

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

### Design requirements

1. Maximum axial compressive stress, fa must not exceed allowable stress, Fa.

Design procedure:

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

### Determine axial compression stress

Axial compression stress shall be determined as

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

### Determine allowable axial stress Fa(ASD)

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. Compare the larger value of  KxLx/rx and KyLy/ry with

Cc = Ö2p2E/Fy

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

1. If KL/r £ Cc, then 1. If KL/r > Cc, then Example 1:

Situation: A structural column is supporting roof

Design Code: AISC ASD 9th 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 W8 beam

Solution:

Try a W6x15 column, A = 4.43 in2, rx = 2.56 in, ry = 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: Kx = 1, Ky = 1

Slenderness ratio: KxLux/rx = 70.3, KyLuy/ry = 123.8

Elastic modulus, E = 29000 ksi

Factor, Cc = Ö2p2E/Fy = 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, rx = 2.47 in, ry = 0.905 in.

Compressive stress, fa = 26.2/2.68 = 13.5 ksi

Unsupported Length of column: Lux = 15 ft, Luy = 7.5 ft

Slenderness ratio, KyLuy/ry = 99.4 < Cc = 106.9

Allowable compressive stress

Fa = {[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.