L7-26•The maximum limit on the web slenderness �𝑠𝑡𝑡𝑤𝑤is 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 bucklingVertical Buckling of the Compression Flange𝜖𝜖𝑒𝑒𝑎𝑎𝑥𝑥=𝑎𝑎𝜃𝜃 ⁄𝑠2(37)𝑎𝑎𝜃𝜃=2𝜖𝜖𝑓𝑓𝑠𝑎𝑎𝑥𝑥(38)the vertical force component in the web: 𝜎𝜎𝑒𝑒𝐴𝐴𝑒𝑒𝑎𝑎𝜃𝜃(39)Stress in the web:𝑓𝑓𝑐𝑐=𝜎𝜎𝑓𝑓𝐴𝐴𝑓𝑓𝑑𝑑𝜃𝜃𝑡𝑡𝑤𝑤𝑑𝑑𝑥𝑥=2𝜎𝜎𝑓𝑓𝐴𝐴𝑓𝑓𝜖𝜖𝑓𝑓𝑡𝑡𝑤𝑤𝑠(40)Elastic buckling stress for a plate𝐹𝐹𝑐𝑐𝑐𝑐=𝑘𝑘𝜋𝜋2𝐸𝐸121−𝜇𝜇2�𝑏𝑏𝑡𝑡2(41)Assume web plate hinged at the top and bottom and free along edges parallel to the load: 𝑘𝑘= 1, b=ℎ𝜕𝜕, 𝑡𝑡=𝑡𝑡𝜕𝜕𝐹𝐹𝑐𝑐𝑐𝑐=𝜋𝜋2𝐸𝐸121−𝜇𝜇2�𝑠𝑡𝑡𝑤𝑤2(42)
Equating 𝑓𝑓𝑐𝑐=𝐹𝐹𝑐𝑐𝑐𝑐2𝜎𝜎𝑓𝑓𝐴𝐴𝑓𝑓𝜖𝜖𝑓𝑓𝑡𝑡𝑤𝑤𝑠=𝜋𝜋2𝐸𝐸121−𝜇𝜇2�𝑠𝑡𝑡𝑤𝑤2(43)Let 𝐴𝐴𝜕𝜕=𝑡𝑡𝜕𝜕ℎ�𝑠𝑡𝑡𝑤𝑤=𝜋𝜋2𝐸𝐸241−𝜇𝜇2𝐴𝐴𝑤𝑤𝐴𝐴𝑓𝑓1𝜎𝜎𝑓𝑓𝜖𝜖𝑓𝑓(44)Considering the residual stress in the flange 𝐹𝐹𝑐𝑐and conservatively expressing the total flange strain as the strain due to the sum of the residual stress and the yield stress:𝜖𝜖𝑒𝑒=�(𝐹𝐹𝑐𝑐+𝐹𝐹𝑦𝑦𝑓𝑓)𝐸𝐸(45)Conservatively assuming that 𝜎𝜎𝑒𝑒must reach 𝐹𝐹𝑦𝑦to achieve the strength of the flange and substituting 𝐸𝐸=29,000and 𝜇𝜇= 0.3:�𝑠𝑡𝑡𝑤𝑤=0.672 𝐸𝐸⁄𝐴𝐴𝑤𝑤𝐴𝐴𝑓𝑓𝐹𝐹𝑦𝑦𝑓𝑓𝐹𝐹𝑦𝑦𝑓𝑓+𝐹𝐹𝑐𝑐(46)L7-27Vertical Buckling of the Compression Flange
If 𝐴𝐴𝑤𝑤𝐴𝐴𝑓𝑓= 0.5and 𝐹𝐹𝑐𝑐= 0.3𝐹𝐹𝑦𝑦�𝑠𝑡𝑡𝑤𝑤= 0.475𝐸𝐸/𝐹𝐹𝑦𝑦𝑒𝑒(𝐹𝐹𝑦𝑦𝑒𝑒+ 0.3𝐹𝐹𝑦𝑦𝑒𝑒)(47)Tests on hybrid girders with A516 (100ksi) flanges indicate that �𝑠𝑡𝑡𝑤𝑤can conservatively be accepted as:⁄ℎ 𝑡𝑡𝜕𝜕= 250if ⁄𝑎𝑎 ℎ ≤1.0(48)⁄ℎ 𝑡𝑡𝜕𝜕= 200if ⁄1.0 <𝑎𝑎 ℎ ≤1.5(49)For other 𝐹𝐹𝑦𝑦𝑒𝑒, take �ℎ 𝑡𝑡𝜕𝜕=2000𝐹𝐹𝑦𝑦𝑓𝑓≯200if ⁄𝑎𝑎 ℎ ≤1.5(50)AISC F 13.2 slenderness limitationsa) ⁄𝑎𝑎 ℎ ≤1.5�𝑠𝑡𝑡𝑤𝑤≤11.7𝐸𝐸𝐹𝐹𝑦𝑦𝑓𝑓=2000𝐹𝐹𝑦𝑦𝑓𝑓(F13-3)b) ⁄𝑎𝑎 ℎ> 1.5�𝑠𝑡𝑡𝑤𝑤≤0.475𝐸𝐸/𝐹𝐹𝑦𝑦𝑒𝑒(𝐹𝐹𝑦𝑦𝑒𝑒+ 0.3𝐹𝐹𝑦𝑦𝑒𝑒) =0.42𝐸𝐸𝐹𝐹𝑦𝑦𝑓𝑓(F13-4)L7-28Vertical 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 is𝐹𝐹𝑐𝑐𝑐𝑐=𝑘𝑘𝜋𝜋2𝐸𝐸121−𝜇𝜇2�𝑠𝑡𝑡𝑤𝑤2(26)(5)•Bucking coefficient kvaries with the aspect ratio ⁄𝑎𝑎 ℎand with the support conditions at the edges•For full fixity parallel to the loading: 𝑘𝑘𝑚𝑚𝑑𝑑𝑛𝑛= 39.6for any ⁄𝑎𝑎 ℎ•For no fixity parallel to the loading: 𝑘𝑘𝑚𝑚𝑑𝑑𝑛𝑛= 23.9for any ⁄𝑎𝑎 ℎ•For 𝐸𝐸= 29,000𝑘𝑘𝑠𝑠𝑠𝑠and 𝜇𝜇= 0.3Equation (26) becomes:𝐹𝐹𝑐𝑐𝑐𝑐=627,000�𝑠𝑡𝑡2𝑘𝑘𝑠𝑠𝑠𝑠for k=23.9(51)𝐹𝐹𝑐𝑐𝑐𝑐=1,038,000�𝑠𝑡𝑡2𝑘𝑘𝑠𝑠𝑠𝑠for k=39.6(52)•For welded plate girders the condition is closer to full fixity.