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Design code: AISC 13ed.

LRFD (Load and Resistance Factor) design:

LRFD design:

- 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:

- 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.

LRFD

- Calculate factored load based on strength design provision of building code
- Calculate factored axial compressive
force, P
_{u}. - Select a trial column size and calculate
normal axial compressive strength, P
_{n}from Chapter E of AISC 13th edition. - Multiply P
_{n}by f = 0.9. and compare fP_{n }with P_{u}.

ASD design:

- Calculate apply load based on allowable stress design provision of building code
- Calculate apply axial compressive force, P.
- Select a trial column size and ccalculate
normal axial compressive strength, P
_{n}from Chapter E of AISC 13th edition - Divide P
_{n}by W = 1.67 and compare P_{n}/W with P.

__Load combination factor:__

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

- 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

- 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.

- If KL/r £ l ,
or Fe ³
0.44 F
_{y }then F_{cr}= (0.685^{Fy/Fe}) F_{y} - If KL/r > l ,
or Fe <
0.44 F
_{y }then F_{cr}= 0.877 F_{e} - LRFD - The axial compressive strength, fP
_{n}= 0.9A_{g}F_{cr}, where Ag is gross section area of member. - ASD - The allowable axial compression, P
_{a}= A_{g}F_{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 in^{2}, r_{x} =
2.56 in, r_{y} =
1.46 in

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

Slenderness ratio, K_{x}L_{ux}/r_{x} =
70.3, K_{y}L_{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 in^{2}, 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_{x}L_{ux}/r_{x} =
72.8, K_{y}L_{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

__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.