2. IntroductionStructural engineers occasionally need to determine the section properties of steel shapes notfound in the current edition of the Handbook of Steel Construction (CISC 2000). The followingpages provide the formulas for calculating the torsional section properties of structural steelshapes.The section properties considered are the St. Venant torsional constant, J, the warpingtorsional constant, Cw, the shear centre location, yO , and the monosymmetry constant, βx .Also included are the torsional constant, C, and the shear constant, CRT, for hollow structuralsections (HSS). Although not a torsional section property, the shear constant is includedbecause it is not easily found in the literature.Some of the formulas given herein are less complex than those used in developing theHandbook and the Structural Section Tables (CISC 1997). Effects such as flange-to-web filletradii, fillet welds, and sloped (tapered) flanges have not been taken into account. Likewise,some of the formulas for monosymmetric shapes are approximations which are only validwithin a certain range of parameters. If needed, more accurate expressions can be found inthe references cited in the text.Simple example calculations are provided for each type of cross section to illustrate theformulas. A complete description of torsional theory or a detailed derivation of the formulas fortorsional section properties is beyond the scope of this discussion; only the final expressionsare given. The references can be consulted for further information.Although no effort has been spared in an attempt to ensure that all data contained herein isfactual and that the numerical values are accurate to a degree consistent with currentstructural design practice, the Canadian Institute of Steel Construction does not assumeresponsibility for errors or oversights resulting from the use of the information containedherein. Anyone making use of this information assumes all liability arising from such use. Allsuggestions for improvement will receive full consideration. 2 CISC 2002
3. St. Venant Torsional ConstantThe St. Venant torsional constant, J, measures the resistance of a structural member to pureor uniform torsion. It is used in calculating the buckling moment resistance of laterallyunsupported beams and torsional-flexural buckling of compression members in accordancewith CSA Standard S16.1-94 (CSA 1994).For open cross sections, the general formula is given by Galambos (1968):  b′ t 3 J = ∑  3   [1]  where b are the plate lengths between points of intersection on their axes, and t are the platethicknesses. Summation includes all component plates. It is noted that the tabulated values inthe Handbook of Steel Construction (CISC 2000) are based on net plate lengths instead oflengths between intersection points, a mostly conservative approach.The expressions for J given herein do not take into account the flange-to-web fillets. Formulaswhich account for this effect are given by El Darwish and Johnston (1965).For thin-walled closed sections, the general formula is given by Salmon and Johnson (1980): 2 4 AOJ= [2] ∫ S ds twhere AO is the enclosed area by the walls, t is the wall thickness, ds is a length elementalong the perimeter. Integration is performed over the entire perimeter S. 3 CISC 2002
4. Warping Torsional ConstantThe warping torsional constant, Cw, measures the resistance of a structural member tononuniform or warping torsion. It is used in calculating the buckling moment resistance oflaterally unsupported beams and torsional-flexural buckling of compression members inaccordance with CSA Standard S16.1-94 (CSA 1994).For open sections, a general calculation method is given by Galambos (1968). For sections inwhich all component plates meet at a single point, such as angles and T-sections, acalculation method is given by Bleich (1952). For hollow structural sections (HSS), warpingdeformations are small, and the warping torsional constant is generally taken as zero.Shear CentreThe shear centre, or torsion centre, is the point in the plane of the cross section about whichtwisting takes place. The shear centre location is required for calculating the warping torsionalconstant and the monosymmetry constant. It is also required to determine the stabilizing ordestabilizing effect of gravity loading applied below or above the shear centre, respectively(SSRC 1998). The coordinates of the shear centre location (xO, yO) are calculated withrespect to the centroid. A calculation method is given by Galambos (1968).Monosymmetry ConstantThe monosymmetry constant, βX, is used in calculating the buckling moment resistance oflaterally unsupported monosymmetric beams loaded in the plane of symmetry (CSA 2000). Inthe case of a monosymmetric section that is symmetric about the vertical axis, the generalformula is given by SSRC (1998): ∫ y (x ) 1βX = 2 + y 2 dA − 2 y O [3] IX Awhere IX is moment of inertia about the horizontal centroidal axis, dA is an area element andyO is the vertical location of the shear centre with respect to the centroid. Integration isperformed over the entire cross section. The value of βX is zero for doubly-symmetricsections. 4 CISC 2002
5. HSS Torsional ConstantThe torsional constant, C, is used for calculating the shear stress due to an applied torque. Itis expressed as the ratio of the applied torque, T, to the shear stress in the cross section, τ : TC= [4] τHSS Shear ConstantThe shear constant, CRT, is used for calculating the maximum shear stress due to an appliedshear force.For hollow structural sections, the maximum shear stress in the cross section is given by: VQτ max = [5a] 2t Iwhere V is the applied shear force, Q is the statical moment of the portion of the section lyingoutside the neutral axis taken about the neutral axis, I is the moment of inertia, and t is thewall thickness.The shear constant is expressed as the ratio of the applied shear force to the maximum shearstress (Stelco 1981): V 2t IC RT = = [5b] τ max Q 5 CISC 2002
6. A) Open Cross Sections1. Doubly-Symmetric Wide-Flange Shapes (W-Shapes and I-Beams) Fig. 1a Fig. 1bTorsional section properties (flange-to-web fillets neglected): 2 b t 3 + d ′w 3J= (Galambos 1968) [6] 3Cw = (d ′)2 b 3 t (Galambos 1968, Picard and Beaulieu 1991) [7] 24d′ = d − t [8]Example calculation: W610x125d = 612 mm, b = 229 mm, t = 19.6 mm, w = 11.9 mmd = 592 mmJ = 1480 x 103 mm4Cw = 3440 x 109 mm6 6 CISC 2002
7. 2. Channels Fig. 2a Fig. 2bTorsional section properties (flange slope and flange-to-web fillets neglected): 2 b′ t 3 + d ′ w 3J= (SSRC 1998) [9] 3 3 1 − 3 α α2  d ′ w C w = (d ′) (b ′) t  2 + 1 +   (Galambos 1968, SSRC 1998) [10]  6 2  6 b ′ t   1α= [11] d ′w 2+ 3 b′ td′ = d − t , b′ = b − w 2 [12]Shear centre location: wxo = x + b′ α − (Galambos 1968, Seaburg and Carter 1997) [13] 2Example calculation: C310x31d = 305 mm, b = 74 mm, t = 12.7 mm, w = 7.2 mm(Actual flange slope = 1/6; zero slope assumed here for simplicity)d = 292 mm, b = 70.4 mmJ = 132 x 103 mm4α = 0.359, Cw = 29.0 x 109 mm6x = 17.5 mm (formula not shown)xO = 39.2 mm 7 CISC 2002
8. 3. Angles Fig. 3a Fig. 3bTorsional section properties (fillets neglected):J= (d ′ + b ′)t 3 [14] 3Cw = t3 36 [ ] (d ′)3 + (b′)3 (Bleich 1952, Picard and Beaulieu 1991) [15] t td′ = d − , b′ = b − [16] 2 2The warping constant of angles is small and often neglected. For double angles, the values ofJ and Cw can be taken equal to twice the value for single angles.The shear centre (xO, yO) is located at the intersection of the angle leg axes.Example calculation: L203x102x13d = 203 mm, b = 102 mm, t = 12.7 mmd = 197 mm, b = 95.7 mmJ = 200 x 103 mm4Cw = 0.485 x 109 mm6 8 CISC 2002
9. 4. T-Sections Fig. 4a Fig. 4bTorsional section properties (flange-to-web fillets neglected): b t 3 + d ′w 3J= [17] 3 b 3 t 3 (d ′) w 3 3Cw = + (Bleich 1952, Picard and Beaulieu 1991) [18] 144 36 td′ = d − [19] 2The warping constant of T-sections is small and often neglected.The shear centre is located at the intersection of the flange and stem plate axes.Example calculation: WT180x67d = 178 mm, b = 369 mm, t = 18.0 mm, w = 11.2 mmd = 169 mmJ = 796 x 103 mm4Cw = 2.22 x 109 mm6 9 CISC 2002
10. 5. Monosymmetric Wide-Flange Shapes Fig. 5a Fig. 5bTorsional section properties (fillet welds neglected): b1 t 1 + b2 t 2 + d ′ w 3 3 3J= (SSRC 1998) [20] 3 (d ′)2 b13 t 1 αCw = (SSRC 1998, Picard and Beaulieu 1991) [21] 12 1α= [22] 1 + (b1 b2 ) (t 1 t 2 ) 3 (t 1 + t 2 )d′ = d − [23] 2Subscripts "1" and "2" refer to the top and bottom flanges, respectively, as shown on Fig. 5b.Shear centre location: t1YO = YT − − α d ′ (Galambos 1968) [24] 2The sign of YO calculated from Eq. 24 indicates whether the shear centre is located above(YO > 0) or below (YO < 0) the centroid. The shear centre is generally located between thecentroid and the wider of the two flanges. For doubly-symmetric sections, YO is equal to zerosince the centroid and shear centre coincide. 10 CISC 2002
11. Monosymmetry constant:   I 2  Iyβ X ≈ δ 0.9 (2 ρ − 1) d ′ 1 −  y   ,   ≤ 0.5 (Kitipornchai and Trahair 1980) [25]   Ix     Ix I y TOPρ= = 1− α [26] IyEq. 25 is an approximate formula and is only valid if IY ≤ 0.5 IX, where IY and IX are themoments of inertia of the section about the vertical and horizontal centroidal axes,respectively. A more accurate expression is given by SSRC (1998).The value of δ depends on which flange is in compression: + 1 If the top flange is in compressionδ = [27] − 1 If the bottom flange is in compressionGenerally, the value of βX obtained from Eq. 25 will be positive when the wider flange is incompression and negative when in tension.Example calculation: WRF1200x244 The top flange is in compression.d = 1200 mm, b1 = 300 mm, b2 = 550 mm, t1 = t2 = 20.0 mm, w = 12.0 mmd = 1180 mmα = 0.860, ρ = 1 - α = 0.140J = 2950 x 103 mm4Cw = 53 900 x 109 mm6YT = 695 mm (formula not shown)YO = -330 mmSince YO is negative, the shear centre is located between the centroid and the bottom flange.IX = 7240 x 106 mm4, IY = 322 x 106 mm4 (formulas not shown)IY / IX = 0.0445 < 0.5 OKδ = +1βX = -763 mm (The top flange is narrower and in compression.) 11 CISC 2002
12. 6. Wide-Flange Shapes with Channel Cap Fig. 6a Fig. 6bTorsional section properties (flange-to-web fillets neglected):A conservative estimate of the St. Venant torsional constant is given by:J ≈ JW + JC [28]The "w" and "c" subscripts refer to the W-shape and channel, respectively. A more refinedexpression for J is proposed by Ellifritt and Lue (1998).Shear centre location: tW + w CYO = YT − − a + e (Kitipornchai and Trahair 1980) [29] 2a = (1 − ρ ) h , b = ρ h [30] I y TOP I y TOPρ= = [31] I y TOP + I y BOT Iywhere Iy TOP, Iy BOT, and Iy are the moments of inertia of the built-up top flange (channel + topflange of the W-shape), the bottom flange, and the entire built-up section about the verticalaxis, respectively. With the channel cap on the top flange, as shown on Fig. 6, the value of YOobtained from Eq. 29 will be positive, indicating that the shear centre is located above thecentroid. 12 CISC 2002
13. The distance between the shear centres of the top and bottom flanges is given by: wCh = dW − tW + +e [32] 2The distance between the shear centre of the built-up top flange and the centre line of thechannel web and W-shape top flange, taken together as a single plate, is given by: 2 2 bC d C t Ce= [33] 4 ρ IyWarping constant of the built-up section:Cw = a 2 I y TOP + b 2 I y BOT (Kitipornchai and Trahair 1980) [34]A simplified formula for Cw is also given by Ellifritt and Lue (1998).Monosymmetry constant:   I 2   b  Iyβ X ≈ δ 0.9 (2 ρ − 1) h 1 −  y   1 + C  , I   2d  ≤ 0.5 (Kitipornchai and Trahair 1980) [35]   x    Ix  where d is the built-up section depth:d = dW + w C [36]Eq. 35 is an approximation which is only valid if IY ≤ 0.5 IX, where Ix is the moment of inertiaof the built-up section about the horizontal centroidal axis. See page 11 for the value of δ andthe sign of βX . A further simplified expression is given by Ellifritt and Lue (1998). 13 CISC 2002
14. Example calculation: W610x125 and C310x31W-shape: W610x125dW = 612 mm, bW = 229 mm, tW = 19.6 mm, wW = 11.9 mmJW = 1480 x 103 mm4 (previously calculated, p. 6)Channel cap: C310x31dC = 305 mm, bC = 74 mm, tC = 12.7 mm, wC = 7.2 mmJC = 132 x 103 mm4 (previously calculated, p. 7)Built-up section, with the top flange in compression:J = 1610 x 103 mm4YT = 255 mm (formula not shown)Iy TOP = 73.1 x 106 mm4 (formula not shown)Iy BOT = 19.6 x 106 mm4 (formula not shown)Iy = 92.7 x 106 mm4ρ = 0.789e = 22.1 mmh = 618 mma = 130 mmb = 488 mmYO = 134 mmSince the calculated value of YO is positive, the shear centre is located above the centroid(see Fig. 6b).CW = 5900 x 109 mm6Ix = 1260 x 106 mm4 (formula not shown)d = 619 mmIy / Ix =0.0736 < 0.5 OKδ = +1 (top flange in compression)βX = 339 mm (wider top flange in compression) 14 CISC 2002
15. B) Closed Cross Sections7. Hollow Structural Sections (HSS), Round Fig. 7St. Venant torsional constant (valid for any wall thickness):J = 2I = π 32 [d 4 − (d − 2 t ) 4 ] (Stelco 1981, Seaburg and Carter 1997) [37]where d is the outer diameter, I is the moment of inertia, and t is the wall thickness. Thewarping constant Cw is taken equal to zero.HSS torsional constant (valid for any wall thickness): 2JC= (Stelco 1981, Seaburg and Carter 1997) [38a] dHSS shear constant: 2t IC RT = (Stelco 1981) [38b] QI= π 64 [d 4 − (d − 2 t ) 4 ] [39]Q= t 6 ( 3d 2 − 6d t + 4t 2 ) (Stelco 1981) [40]Example calculation: HSS324x9.5d = 324 mm, t = 9.53 mmJ = 233 000 x 103 mm4, I = 116 x 106 mm4 , Q = 471 x 103 mm3C = 1440 x 103 mm3, CRT = 4710 mm2 15 CISC 2002
16. 8. Hollow Structural Sections (HSS), Square and Rectangular Fig. 8St. Venant torsional constant (valid for thin-walled sections, b / t ≥ 10): 2 4 AP tJ≈ (Salmon and Johnson 1980) [41] pMid-contour length:p = 2 [(d − t ) + (b − t )] − 2 RC (4 − π ) [42]Enclosed area:AP = (d − t )(b − t ) − RC (4 − π ) 2 [43]Mean corner radius: RO + R iRC = ≈ 1 .5 t [44] 2where d and b are the longer and shorter outside dimensions, respectively, and t is the wallthickness. RO and Ri are the outer and inner corner radii taken equal to 2t and t, respectively.The warping constant Cw is usually taken equal to zero. 16 CISC 2002
17. HSS torsional constant (valid for thin-walled sections, b / t ≥ 10):C ≈ 2 t Ap (Salmon and Johnson 1980, Seaburg and Carter 1997) [45a]HSS shear constant (approximate):C RT ≈ 2 t (h − 4 t ) (Stelco 1981) [45b]where h is the outer section dimension in the direction of the applied shear force. A moreaccurate expression is also given by Stelco (1981).Example calculation: HSS203x102x6.4d = 203 mm, b = 102 mm, t = 6.35 mmRO = 12.7 mmRi = 6.35 mmRC = 9.53 mmp = 568 mmAp = 18 700 mm2J = 15 600 x 103 mm4It is assumed that the shear force acts in a direction parallel to the longer dimension, d.h = d = 203 mmC = 238 x 103 mm3CRT = 2260 mm2 17 CISC 2002
18. ReferencesBleich, F. 1952. Buckling Strength of Metal Structures, McGraw-Hill Inc., New York, N.Y.CISC. 1997. Structural Section Tables (SST Electronic Database), Canadian Institute of Steel Construction, Willowdale, Ont.CISC. 2000. Handbook of Steel Construction, 7th Edition, 2nd Revised Printing. Canadian Institute of Steel Construction, Willowdale, Ont.CSA. 1994. Limit States Design of Steel Structures. CSA Standard S16.1-94. Canadian Standards Association, Rexdale, Ont.CSA. 2000. Canadian Highway Bridge Design Code. CSA Standard S6-00. Canadian Standards Association, Rexdale, Ont.El Darwish, I.A. and Johnston, B.G. 1965. Torsion of Structural Shapes. ASCE Journal of the Structural Division, Vol. 91, ST1. Errata: ASCE Journal of the Structural Division, Vol. 92, ST1, Feb. 1966, p. 471.Ellifritt, D.S. and Lue, D.-M. 1998. Design of Crane Runway Beam with Channel Cap. Engineering Journal, AISC, 2nd Quarter.Galambos, T.V. 1968. Structural Members and Frames. Prentice-Hall Inc., Englewood Cliffs, N.J.Kitipornchai, S. and Trahair, N.S. 1980. Buckling Properties of Monosymmetric I-Beams. ASCE Journal of the Structural Division, Vol. 106, ST5.Picard, A. and Beaulieu, D. 1991. Calcul des charpentes dacier. Canadian Institute of Steel Construction, Willowdale, Ont. 18 CISC 2002
19. Salmon, C.G. and Johnson, J.E. 1980. Steel Structures, Design and Behavior, 2nd Edition. Harper & Row, Publishers. New York, N.Y.Seaburg, P.A. and Carter, C.J. 1997. Torsional Analysis of Structural Steel Members. American Institute of Steel Construction, Chicago, Ill.SSRC. 1998. Guide to Stability Design Criteria for Metal Structures, 5th Edition. Edited by T.V. Galambos, Structural Stability Research Council, John Wiley & Sons, New York, N.Y.Stelco. 1981. Hollow Structural Sections - Sizes and Section Properties, 6th Edition. Stelco Inc., Hamilton, Ont. 19 CISC 2002
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