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TORSIONAL SECTION PROPERTIES OF STEEL SHAPES Canadian Insiue of Seel Consrucion, 00 Conens Page Inroducion S. Venan Torsional Consan
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TORSIONAL SECTION PROPERTIES OF STEEL SHAPES Canadian Insiue of Seel Consrucion, 00 Conens Page Inroducion S. Venan Torsional Consan Warping Torsional Consan Shear Cenre Monosymmery Consan Shear Consan A) Open Cross Secions: 1. Doubly-Symmeric Wide-Flange Shapes (W-Shapes and I-Beams) Channels Angles T-Secions Monosymmeric Wide-Flange Shapes Wide-Flange Shapes wih Channel Cap B) Closed Cross Secions: 7. Hollow Srucural Secions, Round Hollow Srucural Secions, Square and Recangular References CISC 00 Inroducion Srucural engineers occasionally need o deermine he secion properies of seel shapes no found in he curren ediion of he Handbook of Seel Consrucion (CISC 000). The following pages provide he formulas for calculaing he orsional secion properies of srucural seel shapes. The secion properies considered are he S. Venan orsional consan, J, he warping orsional consan, C w, he shear cenre locaion, y O, and he monosymmery consan, β x. Alhough no a orsional propery, he shear consan, C RT, is also included for hollow srucural secions (HSS), as i is no easily found in he lieraure. Some of he formulas given herein are less complex han hose used in developing he Handbook and he Srucural Secion Tables (CISC 1997). Effecs such as flange-o-web fille radii, fille welds, and sloped (apered) flanges have no been aken ino accoun. Likewise, some of he formulas for monosymmeric shapes are approximaions which are only valid wihin a cerain range of parameers. If needed, more accurae expressions can be found in he references cied in he ex. Simple example calculaions are provided for each ype of cross secion o illusrae he formulas. A complee descripion of orsional heory or a deailed derivaion of he formulas for orsional secion properies is beyond he scope of his discussion; only he final expressions are given. The references can be consuled for furher informaion. Alhough no effor has been spared in an aemp o ensure ha all daa conained herein is facual and ha he numerical values are accurae o a degree consisen wih curren srucural design pracice, he Canadian Insiue of Seel Consrucion does no assume responsibiliy for errors or oversighs resuling from he use of he informaion conained herein. Anyone making use of his informaion assumes all liabiliy arising from such use. All suggesions for improvemen will receive full consideraion. CISC 00 S. Venan Torsional Consan The S. Venan orsional consan, J, measures he resisance of a srucural member o pure or uniform orsion. I is used in calculaing he buckling momen resisance of laerally unsuppored beams and orsional-flexural buckling of compression members in accordance wih CSA Sandard S (CSA 1994). For open cross secions, he general formula is given by Galambos (1968): b J = [1] where b' are he plae lenghs beween poins of inersecion on heir axes, and are he plae hicknesses. Summaion includes all componen plaes. I is noed ha he abulaed values in he Handbook of Seel Consrucion (CISC 000) are based on ne plae lenghs insead of lenghs beween inersecion poins, a mosly conservaive approach. The expressions for J given herein do no ake ino accoun he flange-o-web filles. Formulas which accoun for his effec are given by El Darwish and Johnson (1965). For hin-walled closed secions, he general formula is given by Salmon and Johnson (1980): J = 4 A S ds O [] where A O is he enclosed area by he walls, is he wall hickness, ds is a lengh elemen along he perimeer. Inegraion is performed over he enire perimeer S. CISC 00 Warping Torsional Consan The warping orsional consan, C w, measures he resisance of a srucural member o nonuniform or warping orsion. I is used in calculaing he buckling momen resisance of laerally unsuppored beams and orsional-flexural buckling of compression members in accordance wih CSA Sandard S (CSA 1994). For open secions, a general calculaion mehod is given by Galambos (1968). For secions in which all componen plaes mee a a single poin, such as angles and T-secions, a calculaion mehod is given by Bleich (195). For hollow srucural secions (HSS), warping deformaions are small, and he warping orsional consan is generally aken as zero. Shear Cenre The shear cenre, or orsion cenre, is he poin in he plane of he cross secion abou which wising akes place. The shear cenre locaion is required for calculaing he warping orsional consan and he monosymmery consan. I is also required o deermine he sabilizing or desabilizing effec of graviy loading applied below or above he shear cenre, respecively (SSRC 1998). The coordinaes of he shear cenre locaion (x O, y O ) are calculaed wih respec o he cenroid. A calculaion mehod is given by Galambos (1968). Monosymmery Consan The monosymmery consan, β X, is used in calculaing he buckling momen resisance of laerally unsuppored monosymmeric beams loaded in he plane of symmery (CSA 000). In he case of a monosymmeric secion ha is symmeric abou he verical axis, he general formula is given by SSRC (1998): X ( x + y ) da yo 1 β X = y I [] A where I X is momen of ineria abou he horizonal cenroidal axis, da is an area elemen and y O is he verical locaion of he shear cenre wih respec o he cenroid. Inegraion is performed over he enire cross secion. The value of β X is zero for doubly-symmeric secions. 4 CISC 00 Shear Consan The shear consan, C RT, is used for deermining he maximum shear sress in he cross secion due o an applied shear force. For hollow srucural secions, he maximum shear sress in he cross secion is given by: V Q τ max = [4] I where V is he applied shear force, Q is he saical momen of he porion of he secion lying ouside he neural axis aken abou he neural axis, I is he momen of ineria, and is he wall hickness. The shear consan is expressed as he raio of he applied shear force o he maximum shear sress (Selco 1981): V I C RT = = τ Q max [5] 5 CISC 00 A) Open Cross Secions 1. Doubly-Symmeric Wide-Flange Shapes (W-Shapes and I-Beams) Fig. 1a Fig. 1b Torsional secion properies (flange-o-web filles negleced): J b + d w = (Galambos 1968) [6] ( d ) b C w = (Galambos 1968, Picard and Beaulieu 1991) [7] 4 d = d [8] Example calculaion: W610x15 d = 61 mm, b = 9 mm, = 19.6 mm, w = 11.9 mm d' = 59 mm J = 1480 x 10 mm 4 C w = 440 x 10 9 mm 6 6 CISC 00 . Channels Fig. a Fig. b Torsional secion properies (flange slope and flange-o-web filles negleced): b + d w J = (SSRC 1998) [9] 1 α α d w ( d ) ( b ) + 1+ C = w 6 6 b (Galambos 1968, SSRC 1998) [10] 1 α = [11] d w + b d = d, b = b w [1] Shear cenre locaion: x o w = x + b α (Galambos 1968, Seaburg and Carer 1997) [1] Example calculaion: C10x1 d = 05 mm, b = 74 mm, = 1.7 mm, w = 7. mm (Acual flange slope = 1/6; zero slope assumed here for simpliciy) d' = 9 mm, b' = 70.4 mm J = 1 x 10 mm 4 α = 0.59, C w = 9.0 x 10 9 mm 6 x = 17.5 mm (formula no shown) x O = 9. mm 7 CISC 00 . Angles Fig. a Fig. b Torsional secion properies (filles negleced): J ( d + b ) = [14] [( d ) + ( b ) ] C w = (Bleich 195, Picard and Beaulieu 1991) [15] 6 d = d, b = b [16] The warping consan of angles is small and ofen negleced. For double angles, he values of J and C w can be aken equal o wice he value for single angles. The shear cenre (x O, y O ) is locaed a he inersecion of he angle leg axes. Example calculaion: L0x10x1 d = 0 mm, b = 10 mm, = 1.7 mm d' = 197 mm, b' = 95.7 mm J = 00 x 10 mm 4 C w = x 10 9 mm 6 8 CISC 00 4. T-Secions Fig. 4a Fig. 4b Torsional secion properies (flange-o-web filles negleced): b + d w J = [17] ( d ) b w C w = + (Bleich 195, Picard and Beaulieu 1991) [18] d = d [19] The warping consan of T-secions is small and ofen negleced. The shear cenre is locaed a he inersecion of he flange and sem plae axes. Example calculaion: WT180x67 d = 178 mm, b = 69 mm, = 18.0 mm, w = 11. mm d' = 169 mm J = 796 x 10 mm 4 C w =. x 10 9 mm 6 9 CISC 00 5. Monosymmeric Wide-Flange Shapes Fig. 5a Fig. 5b Torsional secion properies (fille welds negleced): b1 1 + b + d w J = (SSRC 1998) [0] ( d ) b1 1 α C w = (SSRC 1998, Picard and Beaulieu 1991) [1] 1 1 α = [] 1+ ( b b ) ( ) 1 ( + ) 1 1 d = d [] Subscrips 1 and refer o he op and boom flanges, respecively, as shown on Fig. 5b. Shear cenre locaion: 1 YO = YT α d (Galambos 1968) [4] The sign of Y O calculaed from Eq. 4 indicaes wheher he shear cenre is locaed above (Y O 0) or below (Y O 0) he cenroid. The shear cenre is generally locaed beween he cenroid and he wider of he wo flanges. For doubly-symmeric secions, Y O is equal o zero since he cenroid and shear cenre coincide. 10 CISC 00 Monosymmery consan: I y Iy β 0.9 ( 1) 1 X δ ρ d, Ix Ix 0. 5 (Kiipornchai and Trahair 1980) [5] I y TOP ρ = = 1 α I y [6] Eq. 5 is an approximae formula and is only valid if I Y 0.5 I X, where I Y and I X are he momens of ineria of he secion abou he verical and horizonal cenroidal axes, respecively. A more accurae expression is given by SSRC (1998). The value of δ depends on which flange is in compression: + 1 If he op flange is in compression δ = [7] 1 If he boom flange is in compression Generally, he value of β X obained from Eq. 5 will be posiive when he wider flange is in compression and negaive when in ension. Example calculaion: WRF100x44 The op flange is in compression. d = 100 mm, b 1 = 00 mm, b = 550 mm, 1 = = 0.0 mm, w = 1.0 mm d' = 1180 mm α = 0.860, ρ = 1 - α = J = 950 x 10 mm 4 C w = x 10 9 mm 6 Y T = 695 mm (formula no shown) Y O = -0 mm Since Y O is negaive, he shear cenre is locaed beween he cenroid and he boom flange. I X = 740 x 10 6 mm 4, I Y = x 10 6 mm 4 (formulas no shown) I Y / I X = 0.5 OK δ = +1 β X = -76 mm (The op flange is narrower and in compression.) 11 CISC 00 6. Wide-Flange Shapes wih Channel Cap Fig. 6a Fig. 6b Torsional secion properies (flange-o-web filles negleced): A simple and conservaive esimae of he S. Venan orsional consan is given by: J J W + J C [8] The w and c subscrips refer o he W-shape and channel, respecively. A more refined expression for J is proposed by Ellifri and Lue (1998). Shear cenre locaion: W + w C YO = YT a + e (Kiipornchai and Trahair 1980) [9] a = ( 1 ρ )h, b = ρ h [0] I y TOP I y TOP ρ = = [1] I + I I y TOP y BOT y where I y TOP, I y BOT, and I y are he momens of ineria of he buil-up op flange (channel + op flange of he W-shape), he boom flange, and he enire buil-up secion abou he verical axis, respecively. Wih he channel cap on he op flange, as shown on Fig. 6, he value of Y O obained from Eq. 9 will be posiive, indicaing ha he shear cenre is locaed above he cenroid. 1 CISC 00 The disance beween he shear cenres of he op and boom flanges is given by: w C h = dw W + + e [] The disance beween he shear cenre of he buil-up op flange and he cenre line of he channel web and W-shape op flange, aken ogeher as a single plae, is given by: e b d = C C C [] 4 ρ I y Warping consan of he buil-up secion: C w = a I + b I (Kiipornchai and Trahair 1980) [4] y TOP y BOT A simplified formula for C w is also given by Ellifri and Lue (1998). Monosymmery consan: I y b I 0.9 ( 1) C y β X δ ρ h, Ix d Ix 0. 5 (Kiipornchai and Trahair 1980) [5] where d is he buil-up secion deph: d = d W + w C [6] Eq. 5 is an approximaion which is only valid if I Y 0.5 I X, where I x is he momen of ineria of he buil-up secion abou he horizonal cenroidal axis. See page 11 for he value of δ and he sign of β X. A furher simplified expression is given by Ellifri and Lue (1998). 1 CISC 00 Example calculaion: W610x15 and C10x1 W-shape: W610x15 d W = 61 mm, b W = 9 mm, W = 19.6 mm, w W = 11.9 mm J W = 1480 x 10 mm 4 (previously calculaed, p. 6) Channel cap: C10x1 d C = 05 mm, b C = 74 mm, C = 1.7 mm, w C = 7. mm J C = 1 x 10 mm 4 (previously calculaed, p. 7) Buil-up secion, wih he op flange in compression: J = 1610 x 10 mm 4 Y T = 55 mm (formula no shown) I y TOP = 7.1 x 10 6 mm 4 (formula no shown) I y BOT = 19.6 x 10 6 mm 4 (formula no shown) I y = 9.7 x 10 6 mm 4 ρ = e =.1 mm h = 618 mm a = 10 mm b = 488 mm Y O = 14 mm Since he calculaed value of Y O is posiive, he shear cenre is locaed above he cenroid (see Fig. 6b). C W = 5900 x 10 9 mm 6 I x = 160 x 10 6 mm 4 (formula no shown) d = 619 mm I y / I x =0.076 0.5 OK δ = +1 (op flange in compression) β X = 9 mm (wider op flange in compression) 14 CISC 00 B) Closed Cross Secions 7. Hollow Srucural Secions (HSS), Round Fig. 7 S. Venan orsional consan: 4 4 [ d ( d ) ] π J = I = (Selco 1981, Seaburg and Carer 1997) [7] where d is he ouer diameer, I is he momen of ineria, and is he wall hickness. The warping consan C w is aken equal o zero. Shear consan: I C RT = (Selco 1981) [8] Q I = π [ d ( d ) ] ( d 6 d 4 ) Q = + (Selco 1981) [40] 6 [9] Example calculaion: HSS610x9.5 d = 610 mm, = 9.5 mm J = x 10 mm 4 I = 810 x 10 6 mm 4 Q = 170 x 10 mm C RT = 8980 mm 15 CISC 00 8. Hollow Srucural Secions (HSS), Square and Recangular Fig. 8 A conservaive esimae of he S. Venan orsional consan is given by: 4 AP J (Salmon and Johnson 1980) [41] p Mid-conour lengh: [( d ) + ( b )] R ( π ) p = 4 [4] C Enclosed area: ( d )( b ) ( π ) A = 4 [4] P R C Mean corner radius: R RO + Ri = 1. [44] C 5 where d and b are he ouside dimensions, and is he wall hickness. R O and R i are he ouer and inner corner radii aken equal o and, respecively. The warping consan C w is usually aken equal o zero. 16 CISC 00 An approximae expression for he shear consan is given by Selco (1981): ( h ) C RT 4 [45] where h is he ouer secion dimension in he direcion of he applied shear force. Example calculaion: HSS0x10x6.4 d = 0 mm, b = 10 mm, = 6.5 mm R O = 1.7 mm R i = 6.5 mm R C = 9.5 mm p = 568 mm A p = mm J = x 10 mm 4 I is assumed ha he shear force acs in a direcion parallel o he longer dimension, d. h = d = 0 mm C RT = 60 mm 17 CISC 00 References Bleich, F Buckling Srengh of Meal Srucures, McGraw-Hill Inc., New York, N.Y. CISC Srucural Secion Tables (SST Elecronic Daabase), Canadian Insiue of Seel Consrucion, Willowdale, On. CISC Handbook of Seel Consrucion, 7 h Ediion, nd Revised Prining. Canadian Insiue of Seel Consrucion, Willowdale, On. CSA Limi Saes Design of Seel Srucures. CSA Sandard S Canadian Sandards Associaion, Rexdale, On. CSA Canadian Highway Bridge Design Code. CSA Sandard S6-00. Canadian Sandards Associaion, Rexdale, On. El Darwish, I.A. and Johnson, B.G Torsion of Srucural Shapes. ASCE Journal of he Srucural Division, Vol. 91, ST1. Erraa: ASCE Journal of he Srucural Division, Vol. 9, ST1, Feb. 1966, p Ellifri, D.S. and Lue, D.-M Design of Crane Runway Beam wih Channel Cap. Engineering Journal, AISC, nd Quarer. Galambos, T.V Srucural Members and Frames. Prenice-Hall Inc., Englewood Cliffs, N.J. Kiipornchai, S. and Trahair, N.S Buckling Properies of Monosymmeric I-Beams. ASCE Journal of he Srucural Division, Vol. 106, ST5. Picard, A. and Beaulieu, D Calcul des charpenes d'acier. Canadian Insiue of Seel Consrucion, Willowdale, On. 18 CISC 00 Salmon, C.G. and Johnson, J.E Seel Srucures, Design and Behavior, nd Ediion. Harper & Row, Publishers. New York, N.Y. Seaburg, P.A. and Carer, C.J Torsional Analysis of Srucural Seel Members. American Insiue of Seel Consrucion, Chicago, Ill. SSRC Guide o Sabiliy Design Crieria for Meal Srucures, 5 h Ediion. Edied by T.V. Galambos, Srucural Sabiliy Research Council, John Wiley & Sons, New York, N.Y. Selco Hollow Srucural Secions - Sizes and Secion Properies, 6 h Ediion. Selco Inc., Hamilon, On. 19 CISC 00
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