Unbraced Column Verification Example. AISC Design Examples AISC 13 th Edition. ASDIP Steel is available for purchase online at

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1 Unbraced Column Verification Example AISC Design Examples AISC 3 th Edition IP Steel is available for purchase onle at

2 H-9 Example H.4 W-Shape Subject to Combed Axial Compression and Flexure Given: Select an ASTM A992 W-shape with a 0. nomal depth to carry nomal axial compression forces of 5 s from dead load and 5 s from live load. The unbraced length is 4 ft and the ends are pned. The member also has the followg nomal required moment strengths, not cludg second-order effects: M xd = 5 -ft M yd = 2 -ft M xl = 45 -ft M yl = 6 -ft The member is not subject to sidesway. Solution: Material Properties: ASTM A992 F y = 50 ksi F u = 65 ksi Calculate the required strength, not considerg second-order effects P u =.2(5.00 s) +.6(5.0 s) = 3 s M ux =.2(5.0 -ft) +.6(45.0 -ft) = 9 -ft M uy =.2(2.00 -ft) +.6(6.00 -ft) = 2.0 -ft P a = 5.00 s s = s M ax = 5.0 -ft ft = 6 -ft M ay = ft ft = ft Try a W0 33 Geometric Properties: W0 33 A = S x = Z x = I x = 7. 4 S y = Z y = I y = L p = 6.85 ft L r = 2.8 ft Calculate the available axial strength For a pned-pned condition, K =.0. Sce KL x = KL y = 4.0 ft and r x > r y, the y-y axis will govern. Manual Table - Table 3- Commentary Table C-C2.2 Manual P c = φ c P n = 253 s P c = P n /Ω c = 68 s Table 4-

3 H-0 Calculate the required flexural strengths cludg second order amplification Use Amplified First-Order Elastic Analysis procedure from Section C2.b. Sce the member is not subject to sidesway, only P-δ amplifiers need to be added. Cm B = αp / P r e Eqn. C2-2 C m =.0 X-X axis flexural magnifier ( KL) 2 4 π ( 29,000 ksi)( 7. ) ((.0)( 4.0 ft)( 2./ft) ) π EI = = = 730 s 2 x e 2 2 P x Eqn. C2-5 α =.0.0 B = = s /730 s ( ) M ux =.02(9 -ft) = 9.8 -ft α =.6.0 B = =.02.6 s /730 s ( ) M ax =.02(6 -ft) = 6.2 -ft Eqn. C2-2 Y-Y axis flexural magnifier ( KL y ) 2 4 π ( 29, 000 ksi)( ) ((.0)( 4.0 ft)( 2./ft) ) π EI = = = 37 s Eqn. C2-5 2 y e 2 2 P α =.0.0 B = = s / 37s ( ) M uy =.09(2.0 -ft) = 3. -ft α =.6.0 B = =.09.6 s / 37s ( ) M ay =.09 (8.00 -ft) = ft Eqn. C2-2 Calculate the nomal bendg strength about the x-x axis Yieldg limit state M nx = M p = F y Z x = 50 ksi( ) = or 62 -ft Eqn. F2- Lateral-torsional bucklg limit state Sce L p < L b < L r, Equation F2-2 applies From Manual Table 3-, C b =.4 Manual Table 3-

4 H- M nx = Cb M p-( M p-0.7fysx) Lb Lr - L p - L p M M p M nx = ( ( 50 ksi)( )) = M , therefore use: 3 4.0ft ft 2.8 ft ft Eqn. F2-2 M nx = or 52 -ft controls Local bucklg limit state Per Manual Table -, the member is compact for F y = 50 ksi, so the local bucklg limit state does not apply Calculate the nomal bendg strength about the y-y axis Manual Table - Section F6.2 Sce a W0 33 has compact flanges, only the yieldg limit state applies. M ny = M p = F y Z y M.6F y S y = 50 ksi( ) M.6(50 ksi)( ) = < , therefore Eqn. F6- Use M ny = or ft φ b = 0.90 M cx = φ b M nx = 0.90(52 -ft) = 37 -ft M cy = φ b M ny = 0.90(58.3 -ft) = ft Ω b =.67 M cx = M nx /Ω b = 52 -ft/.67 = 9.0 -ft M cy = M ny /Ω b = ft/.67 = ft Section F2 Check limit for Equation H-a Pr Pu 3 s = = = 0.9, therefore, Pc φ cpn 253 s use Specification Equation H.b P r M M rx ry + + M.0 2P c Mcx M cy 3 s 9.8 -ft 3. -ft + + 2(253 s) 37 -ft ft = M.0 o.k. Pr Pa s = = = 0.9, therefore Pc Pn / Ω c 68 s use Specification Equation H.b P r M M rx ry + + M.0 2P c Mcx M cy s 6.2 -ft ft + + 2(68 s) 9.0 -ft ft = M.0 o.k. Section H. Eqn. H.b

5 Engeer: Javier Encas, PE 7/20/204 IP Steel STEEL COLUMN DESIGN GEOMETRY Column Designation... Steel Yield Strength Fy... Modulus of Elasticity Es... Member Length L... Effective Length Kx-factor Effective Length Ky-factor Unbraced Length Lb... W0X ksi ksi ft ft OK Area.. Depth bf... tw... tf... k des. Ix PROPERTIES ² ⁴ Sx... Zx... rx... Iy... Sy... Zy... ry ³ ³ ⁴ ³ ³ SERVICE LOADS () Loads from a st-order Analysis to be Amplified (, ) Axial Moment X Moment Y X-X Y-Y Total Axial the Story... No Lateral Translation... Lateral Translation Only.. Amplified Loads Story Shear... Interstory Drift.. M/M2 Ratio... Cm-factor COMPRESSION Slenderness Ratio Kx L / rx... Slenderness Ratio Ky L / ry... Max. Slenderness Ratio OK Limit States Nomal Pn Flexural Bucklg Torsional Bucklg Flexural-Torsional Bucklg Nomal Strength Pn... Safety Factor Ω... Allowable Strength Pn/Ω... P / Pn/Ω Design Ratio OK BENDING ABOUT Y-Y Limit States Nomal Mn Yieldg Lateral-Torsional Bucklg Flange Local Bucklg Web Local Bucklg Nomal Strength Mn... Safety Factor Ω... Allowable Strength Mn/Ω... M / Mn/Ω Design Ratio OK

6 Engeer: Javier Encas, PE 7/20/204 IP Steel STEEL COLUMN DESIGN BENDING ABOUT X-X Moment at /4 pot of Lb... Moment at /2 pot of Lb... Moment at 3/4 pot of Lb... L. T. Bucklg Cb-factor... Limit States Yieldg Lateral-Torsional Bucklg Flange Local Bucklg Web Local Bucklg Nomal Strength Mn... Safety Factor Ω... Allowable Strength Mn/Ω... M / Mn/Ω Design Ratio Nomal Mn OK COMBINED FORCES AISC Equation {H-a)... AISC Equation {H-b) OK LOCAL BUCKLING Flanges Flexure... Flanges Compression... Web Flexure... Web Compression... Compact Non-slender Compact Non-slender 2

7 Engeer: Javier Encas, PE 7/20/204 IP Steel STEEL COLUMN DESIGN Column Designation... W0X33 Area ² Sx ³ Steel Yield Strength Fy... 5 ksi Depth 9.7 Zx ³ Modulus of Elasticity Es ksi bf rx Member Length L ft tw Iy ⁴ Effective Length Kx-factor...00 tf Sy ³ Effective Length Ky-factor...00 k des Zy ³ Unbraced Length Lb ft OK Ix ⁴ ry Axial Moment X Moment Y X-X Y-Y Total Axial the Story Story Shear No Lateral Translation Interstory Drift Lateral Translation Only... M/M2 Ratio Amplified Loads Cm-factor Slender unstiffened Qs =.00, Slender stiffened Qa =.00, Q = Qs * Qa =.00 AISC E7 Slenderness ratio.00 * 4.00 / 4.9 = 40. Slenderness ratio.00 * 4.00 /.94 = 86.6 Maximum slenderness ratio = Max (40., 86.6) = 86.6 AISC E3 Elastic bucklg 3.4² * ² = 38.2 ksi AISC Eq. E3-4 (38.2 >= 0.44 *.00 * 5 = 22.0) Flexural bucklg stress AISC Eq. E3-2 Fcr =.00 * 5 * (0.658) = 28.9 ksi AISC Eq. E4-4 = [ π² * * 79.0 (.00 * 4.0 * 2)² * 0.6 ] = 70. ksi (70. >= 0.44 *.00 * 5 = 22.0) Torsional bucklg AISC Eq. E3-2 Fcr =.00 * 5 * (0.658) = 37. ksi Compressive strength 28.9 * 9.7 = AISC Eq. E3- Controllg limit state: Flexural Bucklg Compressive design ratio = = /.67 = 0.2 <.0 OK AISC E

8 Engeer: Javier Encas, PE 7/20/204 IP Steel STEEL COLUMN DESIGN = 2.5 * 6. *.0 / (2.5 * * * * 46.0 =.4 AISC Eq F- Plastic moment Nomal strength 5 * 38.8 = 94 k- 94 / 2 = 6.7 AISC Eq. F2- = Lp = 82.2 AISC Eq. F2-5 = rts = 2.2 AISC Eq. F = 9.3 c =.0 (doubly symmetric I-shape) AISC Eq. F2-8a AISC Eq. F2-6 = Lr = 26.9 AISC Eq. F2-4 =.4 * π² * 2900 Fcr = 75.8 ksi Nomal strength Mnx =.4 * [94 - ( * 5 * 35.0)(4.0 * ) / ( )] / 2 = 52.2 AISC Eq. F2-2 X - Controllg limit state: Lateral-Torsional Bucklg X - Flexural design ratio = = /.67 = 0.67 <.0 OK AISC F Nomal strength M (5 * 4.0,.6 * 5 * 9.2) = 70 k- 70 / 2 = 58.3 AISC Eq. F6- Nomal strength (compact flanges) Y - Controllg limit state: Yieldg Y - Flexural design ratio = = /.67 = 0.25 <.0 OK AISC F 2

9 Engeer: Javier Encas, PE 7/20/204 IP Steel STEEL COLUMN DESIGN Allowable axial strength = = 68.0 AISC E Allowable flexural strength = Allowable flexural strength = = 9. AISC F = 34.9 AISC F Combed forces ratio = + [ + ] AISC Eq. H-b = 2 * [ ] = 0.98 <.0 OK 3