technical note torsional analysis of steel sections[1]

7/30/2019 Technical Note Torsional Analysis of Steel Sections[1]

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182 / ENGINEERING JOURNAL / FOURTH QUARTER / 2002

Torsional analysis of rolled steel sections is generallyaccomplished with torsional function curves whichhave been published by the American Institute of Steel Con-

struction in Design Guide No. 9, Torsional Analysis of

Structural Steel Members (Seaburg and Carter, 1997),

which is an update to an earlier Bethlehem Steel publication

(Heins and Seaburg, 1963). Such problems are more easily

solved with a personal computer than with charts, and this

paper presents the equations in a form suitable for pro-

gramming.

The equations for the angle of twist, , for twelve casesof torsional loading and end conditions are given in AISCs

Design Guide No. 9, but the successive derivatives of (', '', and ''') are not given. These derivatives are requiredto determine the torsional stresses in the following equa-

tions:

Pure Torsional Shear Stress

Warping Normal Stress

Warping Shear Stress

where

G = shear modulus of elasticity; 11,200 ksi for steel

t = thickness of the element, in.

E = modulus of elasticity; 29,000 ksi for steel

Wns = normalized warping function at a point s on the

cross section, in.2

Sws = warping statical moment at a point s on the

cross section, in.4

The reader is referred to the AISC Design Guide No. 9

(Seaburg and Carter, 1997) for a general discussion of tor-

Technical Note:Torsional Analysis of Steel Sections

sional stresses, end conditions and combining 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 , ', ' ', and ''' for the twelve com-monly 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 dimen-

sional factors and may be used with any consistent set of

units. Each set of equations has been tested against a solu-

tion using the curves from the AISC Design Guide (Seaburg

and Carter, 1997). Although the additional precision

afforded by using the equations (instead of the curves) is of

doubtful value, the ease and speed of computation is

immensely helpful.

REFERENCES

Seaburg, P.A. and Carter, C.J. (1997), Torsional Analysis of

Structural Steel Members, Steel Design Guide Series No.9, AISC, Chicago, IL.

Heins, C.P. and Seaburg, P.A. (1963), Torsion Analysis of

Rolled Steel Sections, Bethlehem Steel Company Steel

Design File, 1963-B.

WILLIAM E. MOORE II and KEITH M. MUELLER

William E. Moore, II, is consultant, Ferro Products Corpora-

tion, Charleston, WV.

Keith M. Mueller is senior engineer, American Institute of

Steel Construction, Inc., Chicago, IL.

'Gt=

''ns

E W=

'''wsES

t=

(1)

(2)

(3)


2/7

The following variables and constants are used throughout

the appendix:

a =

G = shear modulus of elasticity; 11,200 ksi for steel

E = modulus of elasticity; 29,000 ksi for steel

Cw = warping constant, in.2

J = torsional constant, in.4

T = concentrated torque, kip-in.

t = running torque load, kip-in./ft

L = span length, in.

z = length from left support to cross-section ana-

lyzed, in.

ENGINEERING JOURNAL / FOURTH QUARTER / 2002 / 183

APPENDIX

wEC

GJ

T

L

T

'

'' ''' 0

Tz

GJ

T

GJ

=

=

= =

T

L

T

1 2 2

1

2

1

22

1

23

1

2

cosh sinh

' 1.0 cosh sinh

'' cosh sinh

''' cosh sinh

where

tanh2

z z zc c c

a a a

c z zc

a a a

c z zc

a aa

c z zc

a aa

Tac

GJ

Lc

a

= +

= +

=

= +

=

=

Case 1Concentrated Torques with Free Ends

Case 2Concentrated Torques with Fixed Ends


3/7


Case 4Uniformly Distributed Torque with Pinned Ends

t

L

2

2

1 2

1

1

1

1

cosh tanh sinh 1.022

2' 1.0 sinh tanh cosh

22

'' cosh tanh sinh 1.02

''' sinh tanh cosh2

where

L z z L zc a z

a a aLa

L z z L zc a

a a aa L

z L zc

a a a

c z L z

a a a a

tc

GJ

= +

= + =

=

=

Case 5Linearly Varying Torque with Pinned Ends

t

L

32 2

1 2 2

22

1 2 2

1

1

1

sinh

6 6sinh

cosh1'6 2sinh

sinh''

sinh

cosh 1'''

sinh

where

zzza a az

LL LL L

a

zza a a

c LL LL L

a

zza

cL L

a

z

ac

L Laa

tc

GJ

c

+

= +

= =

=

=

Case 3Concentrated Torque with Pinned Ends

T

L L(1 )

( )

( )

1 2

1 2

1 2

1 2

2

1

1

0

1.0 sinh

' 1.0 cosh

'' sinh

''' cosh

sinh( ) sinh sinh cosh

tanh

sinh' cosh sinh sinh

tanh

z L

zc z c a

a

zc c

a

c c z

a a

c c z

aa

L z L

L

a Lz zc L z a

L a a a

a

L

a Lz zc

L a a a

a

= + = +

=

=

<

= +

=

1

1

2

1

2

sinh'' sinh sinh cosh

tanh

sinh''' cosh sinh sinh

tanh

where

sinh coshtanh

Lc a Lz z

a L a a a

a

Lc a Lz z

L a a aaa

Tc

GJ

L

a LcL a

a

=

=

=

=


4/7


Case 6Concentrated Torque with Fixed Ends

1 2

1 2

1

2

1

22

1 3 4 5

1 4 5

0

cosh 1.0 sinh

' sinh cosh 1.0

'' cosh sinh

''' sinh cosh

cosh sinh

' sinh cosh 1.

z L

z z zc a c

a aa

z zc ca a

c z zc

a a a

c z zc

a aa

L z L

z z zHc a c c c

a a a

z zHc c ca a

= +

= +

=

=

<

= + +

= +

1

4 5

1

4 52

0

'' cosh sinh

''' sinh cosh

where

1.0 cosh cosh 1.0

sinhtanh sinh

cosh cosh cosh cosh 1.0

( 1sinh

Hc z zc c

a a a

Hc z zc c

a aa

L L

a a L L

L L a a

a a

HL L L L

a a a a L

L a

a

= +

= +

+ +

= + +

1

2

3

4

.0) sinh

( 1)

cosh cosh1 1sinh sinh

sinh tanh tanh tanh

cosh 1.0 cosh cosh sinh

sinh sinh

1.0 cosh 1.0

tanh

L

a

Tc

H GJ

L L

L a L ac H

L a L a L L

a a a a

L L L L L

a a a a ac

L LH

a a

L

ac

LH

a

=+

= + + +

+ = +

= +

5

cosh cosh

sinh

cosh 1.0

cosh

L L

a a

L

a

L

a Lc

aH

= +

T

L L(1 )


5/7


Case 7Uniformly Distributed Torque with Fixed Ends Case 8Linearly Varying Torque with Fixed Ends

t

L

1 2

1 2

1

2

1

22

1

2

(cosh 1.0) (1 ) sinh

2' sinh 1.0 cosh

2'' cosh sinh

''' sinh cosh

where

2

1 cosh

sinh

z z z zc a c

a a L a

zz zc c

a aL

c az zc

a a aL

c z zc

a aa

tLc

GJ

L

ac

L

a

= + = +

=

=

=

+=

t

L

3

1 2 3

2

1

2 3

2

1

22 3

3

123 3

2

1

2

cosh 1.0 sinh6

' sinh cosh2

'' cosh sinh

'' sinh cosh

where

tanh22 sinh

2

z z z zc c S

a a a L

c z az zc S S

a aa L

c zaz zc S

a aa L

c az zc Sa aa L

tLc

GJ

a Lc S

L aL

a

a

LS

= +

= +

= +

= +

=

=

=

sinh

cosh 1.0

6.0sinh 2.0 2cosh

L

L a

a

L L L

a a a

+


6/7


Case 9Concentrated Torque with Fixed and Free End Case 10Partially Uniformly Distributed Torque with

Fixed and Free End

T

L L(1 )

1 2

1 2

1

2

1

22

1 3 4 4

1 4

0

cosh 1.0 sinh

' sinh cosh 1.0

'' cosh sinh

''' sinh cosh

tanh cosh sinh

' tanh sinh

z L

z z zc a c

a aa

z zc c

a a

c z zca a a

c z zc

a aa

L z L

L z z Lc a c c c

a a a a

L zc c

a a

= + = +

=

=

<

= + +

=

1 4

1 4

2

1

2

3

4

cosh

'' tanh cosh sinh

''' tanh sinh cosh

where

sinh tanh cosh tanh

tanh cosh tanh sinh

cosh 1.0

z

a

c cL z z

a a a a

c c L z z

a a aa

Tc

GJ

L LL Lc

a a a a

L LL Lc

a a a a

Lc

a

+

= +

= +

=

= +

=

=

t

L L(1-)

1 2

1 2

12

1

22

2

1 3 4 5

0

cosh 1.0 sinh2

' sinh cosh

'' cosh sinh

''' sinh cosh

cosh sinh

'

z L

z L zzc a c a L z

aa a a

z zc c a L L z

a a

c z zc a L aa a a

c z zc a L

a aa

L z L

z zc a c c c

a a

= + = +

=

=

<

= +

1 4 5

1 4 5

1

4 5

1

2

2 2

3 2

4

sinh cosh

'' cosh sinh

''' sinh cosh

where

tanh sinh cosh

tanh sinh cosh tanh 1.0 2

sinh

z zc a c c

a a

z zc c c

a a

c z zc c

a a a

tc

GJ

L L LLc

a aa a

L L L LL Lc

a a a a a a

L Lc

a a

= +

= +

= +

=

= +

= + +

=

5

tanh

sinh

L

a

L Lc

a a

=


7/7


Case 11Linearly Varying Torque with Free and

Fixed End

Case 12Uniformly Distributed Torque with Fixed

and Pinned Ends

t

L

2

2 2

1 3

2

2

1

1

2

5 11.0 tanh

26

sinh

2 6cosh

cosh'

2 2cosh

sinh''

2 cosh

L a L LLz

aL a La a

c za L za

LL a a La

zc a L a L z a

a LL a L a aL

a

zc a L a

LL aaa

+ + = +

= + +

= +

1

3

2

1

cosh'''

2 cosh

where

z

L

zc a L aa

LL a Laa

tac

GJ

=

=

t

L

1 2

2

1

1

2

tanh tanh cosh sinh

cosh1

2cosh cosh

sinh

' 1.0 tanh sinh cosh

cosh

cosh

'' tanh cosh sinh 1

cosh

L z L z zH

a a a a a

zcza

L L a

a a

z

c L z z z aH

a La a a a

a

z

c L z z aH

La a aa

a

+ = +

= + +

= + +

1

3

2

1

2

2

.0

sinh

''' tanh sinh cosh

cosh

where

1 11.0

2 cosh tanh

z

c L z z aH

La a aa

a

tac

GJ

LH

L L La

a a a

= + +

=

= +

technical note torsional analysis of steel sections[1]

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