L7 26 the maximum limit on the web slenderness is to

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L7-26 The maximum limit on the web slenderness 𝑠 𝑡𝑡 𝑤𝑤 is to prevent the compression flange from buckling vertically Some flexural stiffness is needed from the web along the flange-to- web connection to preclude torsional buckling of the flange Assume the flange is a compression member independent of the rest of the girder When the girder is bent, the curvature gives rise to flange force components that causes compression on the edge of the web adjacent to the flange. If the web remains stable, the flange cannot buckle vertically. Assume the flange has zero stiffness to resist vertical buckling Vertical Buckling of the Compression Flange 𝜖𝜖 𝑒𝑒 𝑎𝑎𝑥𝑥 = 𝑎𝑎𝜃𝜃 ⁄ 𝑠 2 (37) 𝑎𝑎𝜃𝜃 = 2𝜖𝜖 𝑓𝑓 𝑠 𝑎𝑎𝑥𝑥 (38) the vertical force component in the web: 𝜎𝜎 𝑒𝑒 𝐴𝐴 𝑒𝑒 𝑎𝑎𝜃𝜃 (39) Stress in the web: 𝑓𝑓 𝑐𝑐 = 𝜎𝜎 𝑓𝑓 𝐴𝐴 𝑓𝑓 𝑑𝑑𝜃𝜃 𝑡𝑡 𝑤𝑤 𝑑𝑑𝑥𝑥 = 2𝜎𝜎 𝑓𝑓 𝐴𝐴 𝑓𝑓 𝜖𝜖 𝑓𝑓 𝑡𝑡 𝑤𝑤 𝑠 (40) Elastic buckling stress for a plate 𝐹𝐹 𝑐𝑐𝑐𝑐 = 𝑘𝑘𝜋𝜋 2 𝐸𝐸 12 1−𝜇𝜇 2 𝑏𝑏 𝑡𝑡 2 (41) Assume web plate hinged at the top and bottom and free along edges parallel to the load: 𝑘𝑘 = 1 , b= 𝜕𝜕 , 𝑡𝑡 = 𝑡𝑡 𝜕𝜕 𝐹𝐹 𝑐𝑐𝑐𝑐 = 𝜋𝜋 2 𝐸𝐸 12 1−𝜇𝜇 2 𝑠 𝑡𝑡 𝑤𝑤 2 (42)
Equating 𝑓𝑓 𝑐𝑐 = 𝐹𝐹 𝑐𝑐𝑐𝑐 2𝜎𝜎 𝑓𝑓 𝐴𝐴 𝑓𝑓 𝜖𝜖 𝑓𝑓 𝑡𝑡 𝑤𝑤 𝑠 = 𝜋𝜋 2 𝐸𝐸 12 1−𝜇𝜇 2 𝑠 𝑡𝑡 𝑤𝑤 2 (43) Let 𝐴𝐴 𝜕𝜕 = 𝑡𝑡 𝜕𝜕 𝑠 𝑡𝑡 𝑤𝑤 = 𝜋𝜋 2 𝐸𝐸 24 1−𝜇𝜇 2 𝐴𝐴 𝑤𝑤 𝐴𝐴 𝑓𝑓 1 𝜎𝜎 𝑓𝑓 𝜖𝜖 𝑓𝑓 (44) Considering the residual stress in the flange 𝐹𝐹 𝑐𝑐 and conservatively expressing the total flange strain as the strain due to the sum of the residual stress and the yield stress: 𝜖𝜖 𝑒𝑒 = ( 𝐹𝐹 𝑐𝑐 +𝐹𝐹 𝑦𝑦𝑓𝑓 ) 𝐸𝐸 (45) Conservatively assuming that 𝜎𝜎 𝑒𝑒 must reach 𝐹𝐹 𝑦𝑦 to achieve the strength of the flange and substituting 𝐸𝐸 = 29,000 and 𝜇𝜇 = 0.3: 𝑠 𝑡𝑡 𝑤𝑤 = 0 . 672 𝐸𝐸 𝐴𝐴 𝑤𝑤 𝐴𝐴 𝑓𝑓 𝐹𝐹 𝑦𝑦𝑓𝑓 𝐹𝐹 𝑦𝑦𝑓𝑓 +𝐹𝐹 𝑐𝑐 (46) L7-27 Vertical Buckling of the Compression Flange
If 𝐴𝐴 𝑤𝑤 𝐴𝐴 𝑓𝑓 = 0.5 and 𝐹𝐹 𝑐𝑐 = 0.3 𝐹𝐹 𝑦𝑦 𝑠 𝑡𝑡 𝑤𝑤 = 0. 475𝐸𝐸 / 𝐹𝐹 𝑦𝑦𝑒𝑒 ( 𝐹𝐹 𝑦𝑦𝑒𝑒 + 0.3 𝐹𝐹 𝑦𝑦𝑒𝑒 ) (47) Tests on hybrid girders with A516 (100ksi) flanges indicate that 𝑠 𝑡𝑡 𝑤𝑤 can conservatively be accepted as: ℎ 𝑡𝑡 𝜕𝜕 = 250 if 𝑎𝑎 ℎ ≤ 1.0 (48) ℎ 𝑡𝑡 𝜕𝜕 = 200 if 1.0 < 𝑎𝑎 ℎ ≤ 1.5 (49) For other 𝐹𝐹 𝑦𝑦𝑒𝑒 , take ℎ 𝑡𝑡 𝜕𝜕 = 2000 𝐹𝐹 𝑦𝑦𝑓𝑓 200 if 𝑎𝑎 ℎ ≤ 1.5 (50) AISC F 13.2 slenderness limitations a) 𝑎𝑎 ℎ ≤ 1.5 𝑠 𝑡𝑡 𝑤𝑤 11.7 𝐸𝐸 𝐹𝐹 𝑦𝑦𝑓𝑓 = 2000 𝐹𝐹 𝑦𝑦𝑓𝑓 (F13-3) b) 𝑎𝑎 ℎ > 1.5 𝑠 𝑡𝑡 𝑤𝑤 0. 475𝐸𝐸 / 𝐹𝐹 𝑦𝑦𝑒𝑒 ( 𝐹𝐹 𝑦𝑦𝑒𝑒 + 0.3 𝐹𝐹 𝑦𝑦𝑒𝑒 ) = 0 . 42𝐸𝐸 𝐹𝐹 𝑦𝑦𝑓𝑓 (F13-4) L7-28 Vertical Buckling of the Compression Flange
Since the web of plate girder usually has a high ℎ 𝑡𝑡 𝜕𝜕 ratio, buckling may occur as a result of the bending in the plane of the web After this buckling occurs, there is post-buckling strength When the girder is sized most efficiently, the web will buckle before the nominal strength of the girder is reached The elastic buckling stress is 𝐹𝐹 𝑐𝑐𝑐𝑐 = 𝑘𝑘 𝜋𝜋 2 𝐸𝐸 12 1−𝜇𝜇 2 𝑠 𝑡𝑡 𝑤𝑤 2 (26) (5) Bucking coefficient k varies with the aspect ratio 𝑎𝑎 ℎ and with the support conditions at the edges For full fixity parallel to the loading: 𝑘𝑘 𝑚𝑚𝑑𝑑𝑛𝑛 = 39.6 for any 𝑎𝑎 ℎ For no fixity parallel to the loading: 𝑘𝑘 𝑚𝑚𝑑𝑑𝑛𝑛 = 23.9 for any 𝑎𝑎 ℎ For 𝐸𝐸 = 29,000 𝑘𝑘𝑠𝑠𝑠𝑠 and 𝜇𝜇 = 0.3 Equation (26) becomes: 𝐹𝐹 𝑐𝑐𝑐𝑐 = 627 , 000 𝑠 𝑡𝑡 2 𝑘𝑘𝑠𝑠𝑠𝑠 for k=23.9 (51) 𝐹𝐹 𝑐𝑐𝑐𝑐 = 1 , 038 , 000 𝑠 𝑡𝑡 2 𝑘𝑘𝑠𝑠𝑠𝑠 for k=39.6 (52) For welded plate girders the condition is closer to full fixity.

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