Technical Note: Torsional Analysis of Steel Sections WILLIAM E. MOORE II and KEITH M. MUELLER
T
orsional analysis of rolled steel sections is generally accomplished with torsional function curves which have been published by the American Institute of Steel Construction in Design Guide No. 9, Torsional Analysis of Structural Steel Members (Se (Seabu aburg rg and Carter Carter,, 199 1997), 7), which is an update to an earlier Bethlehem Steel publication (Heins and Seaburg, Seaburg, 1963). Such problems are are more easily solved with a personal computer than than with charts, and this paper presents the equations in a form suitable for programming. The equations for the angle of twist, θ, for twelv twelvee cases cases of torsional loading and end conditions are given in AISC’s Design Guide No. No. 9, but the successive successive derivativ derivatives es of θ (θ', θ'''',, an and d θ''') are not given. These derivatives derivatives are required to determine the torsional stresses in the following equations: (1) Pure Torsional Torsional Shear Stress = Gt θ ' Warping Normal Stress Warping Shear Stress where G t E W ns
= E W θ '' ns
=−
ES ws t
θ ''''''
sional stresses, sional stresses, end conditions conditions and combining combining torsional torsional stresses with bending and shear stresses. The user is encouraged to program the pinned and fixed end conditions on the same output in order to compare the trade-off between rotational stiffness and warping normal stress. The equations for θ, θ', θ'''',, an and d θ''' for the twelve commonly encountered loading and end conditions presented in the AISC Design Guide No. 9 are given in the Appendix of this paper. The case numbers are consistent with those given in the design guide. These equations contain no dimensional factors and may be used with any consistent set of units. Each set of equations has been tested against a solution using the curves from the AISC Design Guide (Seaburg and Carter, Carter, 1997). Altho Although ugh the additional additional precisio precision n afforded by using the equations (instead of the curves) is of doubtful doubt ful value, value, the ease ease and speed speed of computati computation on is immensely helpful.
(2) REFERENCES (3)
= = = =
shear modulus of elasticity; elasticity; 11,200 ksi for steel steel thickn thi ckness ess of the the elemen element, t, in. modulus modul us of elasticity; elasticity; 29,000 29,000 ksi for steel steel normalize norm alized d warping warping function function at a point point s on the 2 cross sect section, ion, in. S ws = warp warping ing static statical al moment moment at a point point s on the 4 cross sect section, ion, in. The reader is referred to the AISC Design Guide No. 9 (Seaburg and Carter, Carter, 1997) for a general discussion of tortor-
William E. Moore, II, is consultant, Ferro Ferro Products Corporation, Charleston, WV. Keith M. Mueller is senior engineer, engineer, American American Institute of Steel Construction, Inc., Chicago, IL.
182 / ENGINEERING JOURNAL JOURNAL / FOURTH QUARTER QUARTER / 2002
Seaburg, P.A. and Carter, Carter, C.J. (1997), Torsional Analysis of Structural Steel Members , Steel Design Guide Series No. 9, AI AISC SC,, Ch Chic icag ago, o, IL IL.. Heins, C.P Heins, C.P.. and Seaburg, Seaburg, P.A. (1963), (1963), Torsion Analysis of Rolled Steel Sections, Beth Bethlehem lehem Steel Steel Company Company Steel Steel Design Desi gn File, 19631963-B. B.
APPENDIX The following variables and constants are used throughout the appendix: a
=
Case 2—Concentrated Torques with Fixed Ends T
EC w
T
GJ
G E C w J T t L z
= = = = = = = =
shear modulus of elasticity; 11,200 ksi for steel modulus of elasticity; 29,000 ksi for steel warping constant, in. 2 torsional constant, in.4 concentrated torque, kip-in. running torque load, kip-in./ft span length, in. length from left support to cross-section analyzed, in.
L
z z z θ = c c cosh − c + − sinh a a a c z z θ ' = 1.0 − cosh + c sinh a a a c z z θ '' = c cosh − sinh a a a c z z θ ''' = − cosh + c sinh a a a 1
2
2
1
2
1
Case 1—Concentrated Torques with Free Ends
2
2
1
T
T
2
3
where L
θ= θ' =
Tz GJ
Ta
c1
=
c2
= tanh
GJ L
2a
T GJ
θ '' = θ ''' = 0
ENGINEERING JOURNAL / FOURTH QUARTER / 2002 / 183
Case 3—Concentrated Torque with Pinned Ends
Case 4—Uniformly Distributed Torque with Pinned Ends t
T
L
(1− α )
α L
L
L z z L z θ=ca − + − − z cosh tanh sinh 1.0 2a a a 2a L L 2 z θ ' = c a 1.0 − + sinh z − tanh L cosh z a a 2a L 2a z L z θ '' = c cosh − tanh sinh −1.0 a a 2a c z L z θ ''' = sinh − tanh cosh a a a 2a 2
2
1
0 ≤ z ≤ αL
θ = c (1.0 − α ) z + c a sinh z a θ ' = c (1.0 − α )+ c cosh z a 1
θ '' = θ ''' =
1
2
1
1
2
c1c2 a c1c2 a
2
2
sinh z
1
a
cosh z a
where
=
c1
α L < z ≤ L sinh α L a sinh z − sinh α L cosh θ = c ( L − z) α + a L a a tanh a sinh α L α a L z z θ' = c cosh − sinh sinh − α a a a tanh L a sinh α L c α a L z z θ '' = sinh − sinh cosh a L a a a tanh a sinh α L c a θ ''' = cosh z − sinh α L sinh z L a a a a tanh a 1
z a
t GJ
Case 5—Linearly Varying Torque with Pinned Ends t
1
L
1
1
2
sinh z z a a z za + − θ = c L 6 − L L L 6 L sinh a cosh z z a a a 1 θ' = c L − + − 6 L L sinh L 2 L a sinh z z a θ '' = c − sinh L L a cosh z 1 a θ ''' = c − a sinh L L a 2
1
2
2
1
where c1
c2
= =
T GJ
sinh α L a
tanh L a
2
1
− cosh αL a
1
where c1
184 / ENGINEERING JOURNAL / FOURTH QUARTER / 2002
3
2
=
t GJ
2
2
2
Case 6—Concentrated Torque with Fixed Ends 0 ≤ z ≤ αL
z θ = c a c cosh − 1.0 − sinh z + z a a a z z θ ' = c c sinh − cosh + 1.0 a a c z z θ '' = c cosh − sinh a a a c z z θ ''' = c sinh − cosh a a a 1
2
1
T
2
1
2
(1− α )
α L
L
1
2
2
α L < z ≤ L z z z θ = Hc a c + c cosh + c sinh − a a a z z θ ' = Hc c sinh + c cosh − 1. 0 a a Hc z z c cosh θ '' = + c sinh a a a Hc z z c sinh θ ''' = + c cosh a a a 1
3
1
4
4
5
5
1
4
5
4
5
1
2
where
α L αL 1.0 − cosh a cosh a −1.0 + + sinh αL − αL a a tanh L sinh L a a H = L L αL αL cosh a + cosh a cosh a − cosh a −1.0 L α L + (α − 1.0) − sinh a a sinh L a c1
c2
c3
=
T
+ 1)GJ cosh α L cosh α L 1 1 L a L a α α + sinh − = H + sinh − + sinh L a a tanh L tanh L tanh a a a cosh α L − 1.0 cosh α L − cosh L + Lsinh L a a a a a = + ( H
H sinh
c4
sinh
L a
1.0 − cosh α L 1.0 − cosh α L cosh L a a a = + H tanh
c5
L a
L a
=
L a
cosh α L − 1.0 a H
sinh L a
+ cosh αL a
ENGINEERING JOURNAL / FOURTH QUARTER / 2002 / 185
Case 7—Uniformly Distributed Torque with Fixed Ends
Case 8—Linearly Varying Torque with Fixed Ends
t
t
L L
θ = c a c (cosh z − 1.0) + z (1 − z ) − sinh z a a L a 2 z θ ' = c c sinh z + 1.0 − − cosh z a a L c 2a θ '' = c cosh z − − sinh z a a a L c z z θ ''' = c sinh − cosh a a a 1
2
1
2
1
2
1
2
2
z z z z θ = c c cosh − 1.0 + S sinh − − a a a 6L c z a θ ' = c sinh z + S cosh z − S − a a a 2 L c za θ '' = c cosh z + S sinh z − a a a L c a θ '' = c sinh z + S cosh z − a a a L 3
1
2
3
2
1
2
3
2
1
2
2
3
3
1
where c1
c2
2
3
= =
3
tL
2GJ 1+
cosh L a
sinh L
where c1
=
c2
=
a
2
tL
GJ a
2 L sinh L a
− S tanh L
2a
L sinh a L − a 2 L cosh a − 1.0 6.0 S = L sinh L + 2.0 − 2 cosh L a a a
186 / ENGINEERING JOURNAL / FOURTH QUARTER / 2002
Case 9—Concentrated Torque with Fixed and Free End
Case 10—Partially Uniformly Distributed Torque with Fixed and Free End
T
t
(1− α )
α L
L(1-α )
α L
L
0 ≤ z ≤ αL
0 ≤ z ≤ αL
z θ = c a c cosh − 1.0 − sinh z + z a a a z z θ ' = c c sinh − cosh + 1.0 a a c z z θ '' = c cosh − sinh a a a c z z θ ''' = c sinh − cosh a a a 1
2
1
2
1
2
1
2
2
α L z z θ = c a c a cosh − 1.0 − α Lsinh z + z − a a 2 a a z z θ ' = c c a sinh − αL cosh + α L − z a a c z z θ '' = c a cosh − αLsinh − a a a a c z z θ ''' = c a sinh − αL cosh a a a 1
2
1
2
1
2
1
2
2
α L < z ≤ L
α L < z ≤ L αL L z z θ = c a c − c tanh cosh + c sinh + a a a a L z z θ ' = c c − tanh sinh + cosh a a a cc L z z θ '' = − tanh cosh + sinh a a a a cc L z z θ ''' = − tanh sinh + cosh a a a a 1
3
4
4
1 4
1 4
1 4 2
z z θ = c a c − c cosh + c sinh a a z z θ ' = c a − c sinh + c cosh a a z z θ '' = c − c cosh + c sinh a a c z z θ ''' = − c sinh + c cosh a a a 2
1
3
1
4
5
4
1
5
4
5
1
4
5
where
where T
c1
=
c2
= sinh α L − tanh L cosh α L + tanh
c3
= tanh L cosh α L − tanh
c4
= cosh α L − 1.0
c1
GJ a
a
a
a
a
L
a
a
L
− sinh α L
a
a
c2
c3
c4
c5
=
t GJ
α L αL αL = tanh L − sinh + cosh a a a a = tanh L sinh α L − cosh αL − α L tanh a
a
a
a
L a
+1.0 + α
2
2a 2
α L αL L = sinh − tanh a a a = sinh α L − αL a
2
L
a
ENGINEERING JOURNAL / FOURTH QUARTER / 2002 / 187
Case 11—Linearly Varying Torque with Free and Fixed End
Case 12—Uniformly Distributed Torque with Fixed and Pinned Ends
t
t
L
L
a L L 5 L 1 − − tanh L + z − + 1.0 − a L 2 a L a 6a θ=c z a L sinh a z − + − 6a L L 2a cosh La cosh z c a L a L z a θ ' = − + + − − a L L a L 2 a cosh a 2 aL sinh z c a L a z θ '' = + − L 2a cosh L L a a cosh z c a L a a θ ''' = − − L 2 a cosh L L a a 2
2
1
2
3 2
2
1
1
2
1
3
where c1
=
ta
L z L z z H tanh a − a − tanh a cosh a + sinh a θ = c cosh z z 1 a + − − cosh L cosh L 2a a a z sinh c L z z z a θ ' = H −1.0 − tanh sinh + cosh + − a L a a a a cosh a z cosh c L z z a θ '' = H − tanh cosh + sinh + − 1 .0 L a a a a cosh a z sinh c L z z a θ ''' = H − tanh sinh + cosh + L a a a a cosh a 1
2
2
1
1
2
1
3
2
where
GJ c1
=
ta
2
GJ
1 1 L − 1.0 + H = L L L 2a cosh tanh − a a a 2
2
188 / ENGINEERING JOURNAL / FOURTH QUARTER / 2002
