torsion
TRANSCRIPT
Torsion in Structural Design 1. Introduction 1.1. Problems in Torsion
The role of torsion in structural design is subtle, and complex. Some torsional phenomena include
(a) Twist of beams under loads not passing through the shear center
(b) Torsion of shafts
(c) Torsional buckling of columns
(d) Lateral torsional buckling of beams
• Two main types of situation involve consideration of torsion in design
(1) Member's main function is the transmission of a primary torque, or a primary torque combined with bending or axial load (Cases (a) and (b) above.)
(2) Members in which torsion is a secondary undesirable side effect tending to cause excessive deformation or premature failure. (Cases (c) and (d) above.)
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1.2. Development of Torsional analysis- A few key contributors
• 1853 - French engineer Adhemar Jean Barre de Saint-Venant presented the classical torsion theory to the French Academy of Science
• 1899 - A. Michell and L. Prandtl presented results on flexural-torsional buckling
• 1905 - S. P. Timoshenko presented a paper on the effects of warping torsion in I beams
• 1909 - C. Bach noted the existence of warping stresses not predicted by classical torsion theory when the shear center and centroid do not coincide.
• 1929 - H. Wagner began to develop a general theory of flexural torsional buckling
• V. Z. Vlasov (1906-1958) developed the theory of general bending and twisting of thin walled beams
• 1944 - von Karman and Christensen developed a theory for closed sections (approximate theory)
• 1954 - Benscoter developed a more accurate theory for closed sections.
Numerous other contributors, these are just a few highlights.
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2. Uniform Torsion of Prismatic Sections
Consider a prismatic shaft under constant twisting moment along its length.
Classical theory due to St Venant. Assume
• Cross-sections do not distort in plane during twisting, so every point in the section rotates (in plane) through angle )(xφ about the center of twist.
• Out of plane warping is not constrained • Out of plane warping does not vary along the bar
The resulting displacement field is
dxdyxxyw
dxdzxxzv
zydxdu
φ=φ=
φ−=φ−=
ωφ=
)(
)(
),(
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In plane displacements v and w are seen from the figure below…
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Out of plane distortion (warping) of the section is assumed to vary with the rate of twist
dx
xd )(φ=θ =constant θ=φ→ xx)(
and to be a function of the position (y,z) on the cross-section only. Several models may be constructed
• Warping function model.
• Conjugate Harmonic function model
• St Venant's stress function model
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Warping Function Model
Substituting the displ. fields into the diff. eq. of equilibrium from elasticity, we obtain
022
2
2
2
=ω∇=∂
ω∂+∂
ω∂zy
(Laplace's equation)
with b.c.
nzny yazan
−=∂ω∂
where n is the normal direction to the boundary, and ),( nzny aa are the components of the unit normal vector n on
the boundary. It can be shown that the St Venant torsional stiffness of the section is given by
dAz
zy
yzyJA ∂
ω∂−∂ω∂++= ∫∫ 22
and that the angle of twist is related to the torque by
L
JGT φ= or dxdJGT φ=
a result that reduces to the usual polar moment of inertia when the section becomes circular, and the warping function vanishes. Problem: The eqn. is hard to solve with the b.c given.
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St Venant's Stress Function Model
Assume that the non-zero stresses xzxy ττ , are related to a stress function ),( zyΨ by
yz xzxy ∂Ψ∂−=τ
∂Ψ∂=τ
The function ),( zyΨ automatically satisfies equilibrium. In order for the resulting displacements to be compatible (i.e. satisfy continuity) the d.e.
θ−=φ−=∂
Ψ∂+∂
Ψ∂ GdxdG
yy222
2
2
2
be satisfied, where G=the shear modulus. The boundary conditions for this model are Γ=Ψ onconstant ),( yx where Γ is the boundary of the section. In many cases, it is convenient to simply take Ψ=0 on the boundary. Given the stress function, it can be shown that
∫∫Ψ=A
dydzzyJ ),(
over the section.
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This d.e. is somewhat easier to solve because of the simpler b.c. We are particularly interested in rectangular sections. Consider such a section, of dimension 2α x 2β, as shown.
It can be shown (using a Levy type solution) that, for this section
∑∞
=
−
απ
απβαπ−−
πφα=Ψ
,...5,3,1
2/)1(33
2
2cos
)2/cosh()2/cosh(1)1(132
n
n ynn
znn
G
The corresponding stresses are
∑∞
=
−
απ
απβαπ−−
απφα=τ
,...5,3,1
2/)1(22
2
2sin
)2/cosh()2/cosh(1)1(116
n
nxz
ynn
znn
G
∑∞
=
−
απ
απβαπ−−
απφα=τ
,...5,3,1
2/)1(22
2
2cos
)2/cosh()2/sinh()1(116
n
nxy
ynn
znn
G
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Ψ is shown below for two sections
Assume overall dimensions b=2α, t=2β. Assume that b>t. In general, it is useful to write
dxdktG φ=τmax
btkJ 31=
Then, since T∝τmax , we may write
btk
T2
2max =τ
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Values of the coefficients are tabulated below for different aspect ratios
Table 1 Coefficients for Torsion of a Rectangular Section
b/t k k1 k2 1.0 0.675 0.1406 0.208 1.2 0.759 0.166 0.219 1.5 0.848 0.196 0.231 2.0 0.930 0.229 0.246 2.5 0.968 0.249 0.258 3 0.985 0.263 0.267 4 0.997 0.281 0.282 5 0.999 0.291 0.291 10 1.000 0.312 0.312 ∞ 1.000 0.333 0.333
It is seen that for the thin section, the response away from the ends is almost independent of y. Hence, a simplified model takes the form
0)( subject to 22
2
=β±Ψθ−=Ψ Gdzd
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This leads to the simple solution
β
−θβ=Ψ 2
22 1)( zGz with resulting shear stress
( )βθβ−=τ /2 zGxy Hence, at z=t/2,
dxdGtGt φ=θ=τmax
Integrating this approximate Ψ function over the area,
3
31 btJ =
0
0.2
0.4
0.6
0.8
1
1.2
1 1.2 1.5 2 2.5 3 4 5 10 *
kk1k2
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consistent with the limit of the 2-D solution as the width to thickness ratio approaches ∞. The shear stresses in a narrow rectangular beam are as shown below
Thin-walled open section beams Some thin walled open sections: angles, channels, W and S sections, T sections, etc. • Key result 1. The section torsional stiffness J is
approximately equal to the sum of the J's for the constituent thin walled plates. Thus,
∑=
=n
iiitbJ
1
3
31
• Key result 2. The maximum shear stress is estimated as
J
Ttimax
max =τ
except that larger stresses may occur at the corners . • Key result 3. The shear flow around the section, caused
by St Venant torsion, is as shown below.
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That is, the stresses vary linearly through the thickness on any of the constituent plates, achieving a maximum on each edge.
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3. Shear Stresses due to bending - thin walled open sections
• Provides important background needed for analysis of torsion
• Three items are needed… • shear stresses and shear flow concept • shear center • sectorial moments
These items can be obtained from a generalization of the analysis studied in undergraduate Mechanics of Materials. We summarize the results of the analysis below.
Normal Stresses and Resultants
From generalized analysis of beams,
zIII
MIMIy
IIIMIMI
yzzzyy
zyzyzz
yzzzyy
yyzzyyxx 22
−−
−−
−=σ
where
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∫∫ ==A
zzA
yy dAyIdAzI 22 = the moments of inertia.
inertia ofproduct == ∫A
yz dAyzI
From equilibrium of the slice of the section cut above,
zyzzzyy
zyzyzzy
yzzzyy
yyzzyy VIII
sQIsQIV
IIIsQIsQI
tq 22
)()()()(−
−+
−−
=τ=
where
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∫∫∫ ∫ ====s
sAz
sA
s
y sytdydAQsztdzdAQ0)(*)(* 0
,
q is the shear flow on the section. If the shear resultants act through the shear center, there is no twist. Summing moments about the centroid, it can be shown that the coordinates of the shear center are
2
2
2
2
yzzzyy
yyzzyyz
yzzzyy
zyzyzzy
IIIIIII
e
IIIIIII
e
−−
=
−−
−=
ωω
ωω
In the above,
∫∫
∫∫
ω==
ω==
ω
ω
S
z
S
zz
S
y
S
yy
sdsQrdssQI
sdsQrdssQI
00
00
)()()(21
)()()(21
are the sectorial products of inertia of the section, and
)(2)( sddssr ω= define the sectorial area (the area shaded in the figure above.) r(s) is the perpendicular distance from the origin to the force q ds needed in calculating moments.
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4. Bending and torsion of open sections For combined bending and torsion, a number of additional sectorial area properties are needed for the section.
Let point A undergo displacements v, w, and rotation φ. A different point s , located at coordinates (y,z) displaces by
φ−+=
φ−−=)(
)(
ys
zS
aywwazvv
This can be resolved into a normal, and a tangential displacement component. The tangential component of displacement is
φ++=η rdsdzw
dsdyv
The shear strain in the tangential plane is given by
0≈∂η∂+
∂∂=γ
xsu
xs (Wagner's assumption)
From which the out of plane displacement is approximately
)()(2)()( 0 xudxds
dxdwsz
dxdvsyu +φω−−−=
In the above eqn.
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∫∫ ω==ωss
sdsrds00
)(2)(2
is the sectorial area at point s. The resulting normal stress is then
)2( 02
2
2
2
2
2
dxdu
dxd
dxwdz
dxvdyE
xuExx +φω−−−=
∂∂=σ
Integrating the normal stress over the area leads to the axial and bending resultants. These may be written as
2
2
2
2
2
2
2
2
2
2
2
2
2
20
2
2
dxdEI
dxwdEI
dxvdEIM
dxdEI
dxwdEI
dxvdEIM
dxdES
dxduEAN
zyzzzz
yyyyzy
x
φ+−−=
φ+−−=
φ−=
ω
ω
ω
In the above, the first sectorial moment
∫∫ ω=ωA
dAsS )(2
has been used. The sectorial products of inertia zy II ωω , were defined in connection with the shear center. Solving for the displacements,
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2
2
222
2
2
2
222
2
2
20
2)(
2)(
dxd
IIIIIII
IIIEIMIM
dxwd
dxd
IIIIIII
IIIEIMIM
dxvd
dxd
AS
AEN
dxdu
yzzzyy
yzzzzy
yzzzyy
yzzzzy
yzzzyy
yzyyyz
yzzzyy
yzyyyz
x
φ−
−+
−−
−=
φ−
−+
−−
−=
φ+=
ωω
ωω
ω
The coefficients multiplying φ" are the distances from A to the shear center, so if point A is taken at the shear center, these terms disappear, and the equations for v and w reduce to
)(
)(
22
2
22
2
yzzzyy
yzzzzy
yzzzyy
yzyyyz
IIIEIMIM
dxwd
IIIEIMIM
dxvd
−−
−=
−−
−=
The equation for "0u can be written in alternative form as
2
2
00 2
dxd
AEN
dxdu x φω+=
where =ω0 the sectorial area between the outside edge of the section and the sectorial centroid, a point for which ωS vanishes.If the s origin were taken at a sectorial centroid, instead of at an outside edge, then this term would disappear.
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Shear Stresses and Resultants
For a section under bending and twisting, the total shear stress at any point is
tq
SVxsxs +τ=τ )(
where
=τ SVxs )( St Venant torsion stresses
=tq shear stresses caused by shear flow on the
section.
Integrating the twisting moments caused by the shearing stresses,
ω+φ= TdxdGJT
where ωT is the torque caused by constrained warping induced shear flow. It is sometimes convenient to express this in terms of applied torque per unit length )(xt . Here,
dx
dTdxdGJ
dxdTxt ω−φ−=−= 2
2
)(
If the free edge is taken as the origin of the s coordinate, the shear flow is given by
∫σ∂−=
sxx dA
dxsq
0
)(
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Hence,
ωω
ω−
−−
+−
−−= V
CQV
IIIQIQI
VIII
QIQIsq z
yzzzyy
zyzyzzy
yzzzyy
yyzzyy22)(
where ∫∫ =ω−ωσ=ω
Axx dAW )(2 0 the bimoment
∫∫ ω−ωσ== ωωA
xx dAdxdW
dxdV )(2 0
∫ ω−ω=ω
s
dAQ0
0 )(2 =the first sectorial moment
∫∫ ω−ω=ωA
dAC 20 )(4 =the warping constant
Integrating q over the section, and recognizing that the bending related shear flow vanishes if the load passes through the shear center (which we may always impose) we obtain, after some manipulation
3
3
dxdECT w
φ−=ω
from which the equation for twist of a thin-walled section
)(2
2
4
4
xtdxdGJ
dxdECw =φ−φ
is obtained. Example Consider an I beam, simply supported at each end, free to warp at the ends, loaded by a twisting moment in the middle.
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The conditions at the left and right end are similar to those at the end of a simply supported beam in bending. Hence
0)(" 0)0(" 0)( 0)0(
=φ=φ=φ=φ
LL
Rather than use the fourth order differential equation which has discontinuous load at midspan, use the third order differential equations
lxLTGJEC
LxTGJEC≤<−=φ−φ
≤≤=φ−φ
ω
ω
2/ 2/''''2/0 2/''''
The homogeneous eqn.
0'''' =φ−φω GJEC
has the solution
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)cosh()sinh()( 210 xCxCCxh λ+λ+=φ
where ω
=λECGJ . Since the r.h.s. of the non-
homogeneous equations are constant, and the lowest order derivative of φ in the o.d.e. is one, assume a particular solution
xp α=φ
Substituting into the o.d.e.'s,
LxLGJTTGJ
LxGJTTGJ≤<−=α→−=α
≤≤=α→=α2/ 2/2/
2/0 2/2/
Hence, the general solution is given by
LxLGJTxxCxCCLxGJTxxCxCC≤<−λ+λ+=φ
≤≤+λ+λ+=φ2/ 2/)cosh()sinh(
2/0 2/)cosh()sinh(
210
210
The two solutions must be identical at midspan, and by symmetry, 0)2/(' =φ L . Consider the left half of the beam first. For this portion of the beam, the b.c. at the origin are
00)0("
00)0(
22
20
=λ→=φ
=+→=φ
CCC
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Hence, 020 == CC , and the solution reduces to
GJTxxC 2/)sinh(1 +λ=φ
Then, the symmetry condition at midspan
GJTLCL 2/)2/cosh(0)2/(' 1 +λλ==φ
yields )2/cosh(
121 LGJ
TCλλ
−= .
Hence, for the left half of the beam,
λλ−λ
λ=φ
)2/cosh()sinh(
2 Lxx
JGT
For the right hand side of the beam, we may place the origin at the right end, running to the left. Then, it can be shown that the resulting solution (after converting to the original coordinate system) is
λ
−λ−−λλ
=φ)2/cosh())(sinh()(
2 LxLxL
JGT
Next, consider the relative magnitudes of the two components of torsion over the span. We have, for the left half of the span,
λ
λ−=φ=)2/cosh(
)cosh(12
'L
xTGJTSV
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)2/cosh(
)cosh(2
'''L
xTECTW λλ=φ−= ω
For a typical W section, the λ values range from roughly .04 for a heavy column section to less than .01 for some very slender beams.
Contributions of the warping torsion are shown below for a 20' long beam.
When combined bending and twist of a beam occurs, it is necessary to include all three components of shear flow Shear Flow due to bending St Venant Torsion Warping Torsion to evaluate the total shear stresses.
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The shear flow from bending is calculated in the usual way, as are the St Venant torsion shear stresses. The shear flow from torsion may be calculated as
ωω
ωω −= V
CQq
However, it is more direct to calculate the shear flow from the fact that
3
3
2
2
dxdECV
dxdECW φ−=→φ−= ωωωω
So, 3
3
dxdEQq φ= ωω provides the easiest calculation, given φ.
How do we calculate ωQ ? By definition,
∫ ω−ω=ω
s
dAQ0
0 )(2
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In the integral, ASω=ω0 , where ∫∫ ω=ω
A
dAS 2 was defined
earlier. In these integrals, ω is a continuous function of position on the section, starting from 0 at the sectorial origin. In general, calculating these integrals is messy, but we may often use a simpler approach.
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Example: Suppose the above problem involves a W section with properties shown.
Since the beam is doubly symmetrical, the shear center coincides with the centroid. Taking this point as the origin of sectorial areas, and as the sectorial center, the sectorial areas (taken positive when rotated counterclockwise) are as shown below.
The negative and positive ω contributions cancel, so
00 0 =ω→=ωS . Integrating the sectorial areas over the flanges, (from the outside edges), the ωQ functions are as shown. Within the web, the + and - areas from the flanges cancel, so the warping torsion induced shear flow vanishes.
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Then,
)2/cosh()cosh(
23
3
Lx
CTQ
dxdEQq
λλ=φ=
ω
ωωω
In the above,
)2/cosh()cosh(
43
)(24 max
32
Lx
bhT
xqtbh
C
f
ff
λλ=
→= ωω
The same result can be achieved in a much more intuitive manner. The sketch below illustrates the procedure
The lateral translation v introduced into the flange by the rotation φ causes a bending moment unless φ' is constant.
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''vEIM f=
'''212
'''2
'''
3
φ=
φ=
=→
htbE
hEI
vEIV
ff
f
f
This shear produces a torque
'''2122
23)1( φ
−== htb
EhVT fff
A similar torque develops on the bottom flange, so
'''24
23)2()1( φ−=+=
htbETTT ff
ff
But the warping torsion is given as
24
'''23 htb
CECT ff=→φ−= ωωω
in agreement with the previous result. Within the top flange, bending introduces a shear flow
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f
f
IVQ
q =ω
From the above sketch,
12
22
32
2ff
fff
f
tbIy
btQ =
−
=
So
'''222
12
22'''
24
22
3
223
φ
−
=
−
φ
=ω
ybthE
tb
ybt
htb
E
q
ff
ff
ffff
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For the beam analyzed previously,
)2/cosh(
)cosh(2
'''L
xECT
λλ−=φ
ω
so
)2/cosh()cosh(3
2
)2/cosh()cosh(
224
222
)2/cosh()cosh(
2222
32
2
232
2
22
Lx
hbTy
b
LxT
htby
bth
Lx
CTy
bthq
f
f
ff
ff
ff
λλ
−
=
λλ
−
=
λλ
−
=ω
ω
The peak value of warping shear flow occurs at the middle of the flange (y=0) at the midspan of the beam (x=L/2), where
hb
Tqf4
3=ω
The third part of the sketch on p. TO.30 re-establishes compatibility by rotating the flanges, and the web about an angle φ about their respective centroids. This introduces the St Venant torsion, as well as a secondary warping torsion caused by the plate bending of the individual flange and web plates. We usually ignore the secondary warping except for certain torsionally weak sections.
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Secondary Warping: For certain sections, the only resistance to warping is the secondary warping resistance. Several such sections are shown below…
The secondary warping torsional resistance can often be computed for such sections using an analogy to the approach used for the flange above. Consider a plate, rotated about its base through an angle φ which varies with x. At y,
''')(''' φ= yyv
'''12
''')()(3
φ=φ= ωω ydytEyEIydV ss
'''12
)()( 23
φ== ωω ydytEyydVydT ss
Integrating over the height of the plate,
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36
'''36
''' 12
33
33
0
23
ps
ph
s
htC
htEdyytET
p
=→
φ=φ=
ω
ω ∫
All of the cases shown above can be constructed by variations of the above. For example, a plate of height h rotated about its centroid is equivalent to two plates of h/2 rotated about their respective bases. Hence,
14436
)2/(23333 hthtC s ==ω
Likewise, the T section has warping stiffness
36144
32
331 htbtC s +=ω
and the cruciform shown has warping stiffness
72144
23333 hthtC s ==ω
(You should calculate the corresponding warping stiffness for an angle with unequal legs.) If the plates are thin, the warping stiffness may be relatively small. This can have an important effect, as we'll see.
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Design Approximations for Torsion Analysis using the differential equation for torsion is complicated, even for simple loads, so various approximations have been introduced. Idea: In warping torsion, the flanges act like beams bending laterally. ∴ A (conservative) approximate analysis might be 1. convert the torque into couples acting on the flanges by dividing by the beam height. 2. Apply the "equivalent" lateral loads as forces on the flanges. 3. Analyze the flanges as beams under these loads.
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Example:
For the beam analyzed before, the torque converts into two lateral loads
hTP /=
applied at the top and bottom flanges.
The resulting shear is half of this, on each end…
hTV 2/ =
The shear flow is thus
3
22
23
/f
f
ff bh
Tyb
IVQq
−
=≈ω
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The max. shear flow at y=0 is
fbh
Tq43
max=ω
which clearly is an upper bound to the actual shear flow. This approach is often overly conservative, because
• The normal stresses caused by the bimoment are most important quantitatively, and may be as large as the bending stresses for a beam in combined bending and torsion
• The flange shear flow predicted by the bending analogy will over-estimate the actual warping torsion shear flow over much of the beam. Since, in the same way that
∫ ξξ+=→=x
dVMxMdx
dMV0
)()0()(
We also have
∫ ξξ+=→= ωωωω
ω
x
dVWxWdx
dWV0
)()0()(
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For example, by the beam analogy,
The beam analogy yields
22L
hTM f =→
ω from which
f
fxx I
bLh
T 2/22
=σω
However, the actual shear is
)2/cosh(
)cosh(2
L
xh
TVf λλ=
Hence, the integrated lateral flange moment is
2222)2/cosh(
)cosh(2
2/
0
Lh
TLh
TdxL
xh
TML
f <β=λλ= ∫ω
where β is a reduction factor, dependent upon • λL • The end conditions • The specifics of the applied torque S & J give values of β for some common load cases.
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Are the stresses caused by torsion large enough to be a problem? Let's see. Example: A W18x71 beam spanning 24 ft. is loaded with a concentrated load of 20 kips at midspan. The load acts 2'' away from the Z axis. The beam is fixed at both ends.
(Variation on S&J example.) For the beam, the properties are
623
23
in 685,424
)"81."47.18)("810(.)"635.7(24
=−=
=ω
htbC ff
[ ]4
333
in 39.3
)"495)(.62.147.18()"810)(."635.7(231
3=
−+==∑btJ
6.2)3.01(2)1(2/ =+=ν+=GE
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∴
in.01668.0
6.2)685,4(39.3 ===λ
ωECJG
For the fixed-fixed case, we can show that
[
( )
λ−λ
λ−+
λ−λλ
=φ
))cosh(1()2/sinh(
)2/cosh(1
)sinh(2
xL
L
xxJGT
so ( )
λλ
λ−−λ−=φ )sinh()2/sinh(
)2/cosh(1)cosh(12
' xL
LxJGT
( )
λλ
λ−−λ−λ=φ )cosh()2/sinh(
)2/cosh(1)sinh(2
" xL
LxJG
T
(a) Saint-Venant Torsion
[)]2/sinh(/)sinh())2/cosh(1(
)cosh(12
'
LxL
xJ
TtGt fSV
λλλ−+
λ−=
φ=τ
Here, 477.5)2/sinh( =λL , 567.4)2/cosh(1 −=λ− L , so
Torsion in Structural Design - Notes 11/30/01
41
)]01668sinh(.8338.0)01668cosh(.1[)in 39.3(2
)"81.0)("40(4 xxk
SV +−=τ
For the fixed-fixed case, the Saint-Venant torsion is zero at both support and centerline, and is maximum at 4/L , where ksi 141.21 =τSV (b) Warping Torsion Shear: For the W section, following the above reasoning,
'''16
2
maxφ=τω
hbE f
From the solution,
( )
λλ
λ−−λ−λ=φ )sinh()2/sinh(
)2/cosh(1)cosh(2
'''2
xL
LxJG
T
so
[ ])01668sinh(.8338.0)01668cosh(.2556.0
)sinh()2/sinh(
)2/cosh(1)cosh(32
2
max
xx
xL
LxC
Thbf
+−=
λλ
λ−−λ−=τω
ω
maxωτ reaches its max. amplitude at the end, and at the
centerline, where the warping restraint is greatest. There
2556.0)2/()0( =τ=τ ωω L
Torsion in Structural Design - Notes 11/30/01
42
It is smallest at the quarter points where warping restraint is least.
(c) Lateral bending stresses introduced by warping torsion The bending stress caused by warping is given by
2
2
dxdExx
φω−=σω
From the above analysis, the largest magnitude stress occurs at the flange tips, where 4/hbf=ω . Thus,
( ) ]2/sinh(/)cosh()2/cosh(1)sinh([24
"4
LxLxJT
GEhb
Ehb
f
fxx
λλλ−−λ−λ=
φ=σω
at the flange tips Substituting the numerical values leads to
Torsion in Structural Design - Notes 11/30/01
43
[ ])]01668cosh(.8338.0)01668sinh(.[625.8
)01668cosh(.8338.0)01668sinh(. )in 39.3(2
)k" 40)(in./01668.0()6.2(4
)"66.17)("635.7(4
xxxx
xx
+−=+−
=σω
The maximum values of
ωσxx occur at the ends, and at
midspan, where ksi 192.7=σωxx (compressive on one side
of the flange, tensile on the other.
The maximum normal stress on one edge of the flange is shown over a half beam length. Maximum warping restraint exists at the support, and at midspan. (d) Bending analogy… For this case
Torsion in Structural Design - Notes 11/30/01
44
kips 265.266.17
40k"40 ,"66.17 ==→== fPTh
For a fixed-fixed beam, max moments are
k"54.818
)"288)(kips 265.2(8
===LP
M f
Then, if 322
in 8696.76
)"635.7)("81.0(6
=== fffl
btS
ksi 361.10in 8696.7k"54.81
3.==σ
equivxx
As noted above, this is overly conservative. From table 8.6.2, the interpolated β value is 696.0)76.0)(2.0()68.0)(8.0( =+=β Then, ksi 211.7)ksi 361.10(696.0 =≈σ
ωxx which is pretty close to the maximum value obtained from the closed form solution. Observations:
Torsion in Structural Design - Notes 11/30/01
45
• The Saint-Venant torsion shear stresses are somewhat larger than the warping torsion shear stresses, but both are relatively small.
• The warping normal stresses may be significant. (e) To see how significant, let's compute the bending stresses…
k"7208
)"288)(kips 20(8
=== PLM
3in 127=xS ksi 67.5in 127/k"720 3 ==σ
bendingxx In this case, the warping normal stresses are larger than the bending stresses!
Torsion in Structural Design - Notes 11/30/01
46
5. Other Approaches: Calculating φ may be simplified by tabulated solutions for certain idealized boundary conditions, or a matrix stiffness equation
φφφφ
−−−−
−−
∆=
ω
ω
'
'
2
2
1
1
3242
2121
4232
2121
2
1
2
1
aaaaaaaa
aaaaaaaa
JG
WTWT
where λ+λλ−λλ=∆ 2)cosh(2)sinh(2 LLL
LLaLLLa
LaLaλ−λ=λ−λλ=−λλ=λλ=
)sinh( )sinh()cosh(]1)[cosh( )sinh(
43
22
1
may be useful. This stiffness matrix may be used to replace the 2x2 torsional stiffness submatrix
φφ
−
−=
2
1
2
1
1111
LJG
TT
commonly used in the beam (or frame) equations, and incorporates the end conditions commonly encountered.
Torsion in Structural Design - Notes 11/30/01
47
This approach has the advantage that it can produce reasonable torsional stiffnesses for applications involving torsional moments caused by eccentrically connected beams (See S&J, section 8.7) The associated solution ')()(')()()( 24231211 φ+φ+φ+φ=φ xfxfxfxfx can then be used to obtain all derivatives of φ needed to calculate the shear flow, where
[ ][ ][ ][ ] ∆λλ−−λ−=
∆λ−−λλ−=∆λλ−+λ−+−=
∆λλ+λ+−−=
//)sinh(()cosh(1(/))cosh(1()/)sinh((
//)sinh()())cosh(1(//)sinh())cosh(1()(
244
213
21322
1211
xxaxafxaxxaf
xaLaxaxafxaxaxLaf
are the "shape functions" for the problem. Distributed torques can also be accomodated.
Torsion in Structural Design - Notes 11/30/01
48
S&J suggest that the best approach for determining the applied torque is one that establishes compatibility between the twisting rotation φ of the beam supporting a torque producing member, and the end bending rotation of the attached element. They note that • Applying the shear at the face of the web
underestimates the torsion • Applying the shear at the centerline of the connection
overestimates the torsion • Applying the shear so as to achieve rotational continuity
with the end of the beam framing in tends to be about right.
Torsion in Structural Design - Notes 11/30/01
49
6. LRFD for Torsional Moments
Assumption: Torsion stresses add to bending stresses, but lateral torsional instability is prevented.
i.e. lateral restraint exists, but laterally restraining members may induce self-limiting torsion before restraint becomes effective. Other problems exist as well, for which torsional rotation may not be self limited. AISC Philosophy: (consistent with ASD) • Assume the limit state occurs when yielding is reached.
∴ Elastic analysis is applicable. • Compute factored stresses using LD ww , , etc, e.g.
LDu www 6.12.1 += , … • Once the factored design moment and factored design
lateral flange moments are known, the stresses are superpositioned with the limit state. Formally,
ybz
b FSM
SM
z φ≤+ω
ω
or, in terms of the equivalent lateral flange moment
yby
f
z
b FSM
SM
z φ≤+2
In the above, 9.0=φb =resistance factor.
Torsion in Structural Design - Notes 11/30/01
50
• Since λ is a section property that influences the ωM term, through the β factor, or through the d.e. solution, iterative solution is necessary. • Guess a value of λ in the
right approximate range. • Use the corresponding value of Lλ , to determine the
value of ωM to be used with the design bending moment.
• Use the bounds to estimate a section size. • Use the section size to update the estimated value of
λ., etc.
Example: An A36 beam with torsionally simply supported ends must carry two 20 kip loads acting at an eccentricity of 6" (5 Kips DL, 15 Kips LL) (Variation on S&J example)
kips 30)kips 15(6.1)kips 5(2.1 =+=uP
Guess plf 180)150(2.1plf 150 ==→= ubeam ww Bending moments:
Torsion in Structural Design - Notes 11/30/01
51
Torsion:
k."180)kips 30)("6( ==uT
Using beam analogy
hhTP uf /k "180/ ==
h
hLPM ff
/kin 920,25
/)"144)(k"180(3/2=
==
The β reduction factor depends upon λL. Guess 015.0=λ , (for a majority of W sections, 02.01. <λ< ) Thus, 48.6)"432(015. ==λL 36.0≈β→
Torsion in Structural Design - Notes 11/30/01
52
The β value for the torques in the middle of the beam aren't exactly known, but the estimates tend to be somewhat conservative. Guessing a 14" beam, the reduced flange moment is
kM f "667"14/)k-in 920,25(36.0 2 ==
For a midsized W14,
6.2/ ≈zy SS , taking both flanges into account, so
ksi) 36(9.06.2/
2k"667k"4670 ≤×+yy SS
3in 251≥→ yS Try W14x159. For this section, 64.2/ =zy SS , so the guess is o.k. for now. Also, for this section
2.681 ==λ
ω
GJEC (p. 1-148 of AISC-LRFD)
so 33.62.68/"432 ==λL This implies a revised 37.0=β Now,
Torsion in Structural Design - Notes 11/30/01
53
k"695"79.13/)920,25(37.0 ==fM
So
3in 25764.2/4.32
)2)(k"695(ksi 4.32
k" 670,4 =+≥yS
The W14x159 is not quite big enough, since 3in 254=yS . (actually, it's probably o.k., in view of the approximations in calculating β, but without a more precise calculation, we can't be sure.) Try a W14x176 Revise the moment upward to 4,720"k to account for the extra dead load. From the tables, 86.69.62/4329.62/1 ==λ→=λ L =yS 63.2/ =zy SS Revised 35.0=β "91.13=h k"65291.13/)920,25(35. ==fM
ksi 432ksi 2963.2/2812652
2814720 .<=×+
o.k. A W14x176 is more than adequate here
Torsion in Structural Design - Notes 11/30/01
54
7. Presence of Axial Loads (Torsional Instability) In presence of axial loads, or under certain lateral load conditions, instability can occur. In this case, formulation in the displaced state is necessary. The relevant displacements are φ and, wv . So, the first equation (for 0u ) is not changed. Formulation is complicated, (as is the result) so, we'll just present the resulting equations for φ and , wv . The complete equations are nonlinear. A linear version which omits the non-linear terms is presented here.
yzzyyyyzzyy
yzyyyzxlzyzlyyy
xyzxyyyzzzyy
IqIqdxdVIVI
dxdIeIeNMIMI
dxwdNI
dxvdNI
dxvdIIIE
−=φ++
φ+−++
+−−
)(
)]([
)(
2
2
2
2
2
2
4
42
Torsion in Structural Design - Notes 11/30/01
55
yzyzzzzyzyzz
yzzzzyxlyyzlzzz
xyzxzzyzzzyy
IqIqdxdVIVI
dxdIeIeNMIMI
dxvdNI
dxwdNI
dxwdIIIE
−=φ−−
φ++−−+
+−−
)(
)]([
)(
2
2
2
2
2
2
4
42
)(
)()( 2
2
2
2
2
2
4
4
xtdxdV
CHVCVC
dxdwV
dxdvV
dxwdNeM
dxvdNeM
dxdW
CHMCMCN
AIGJ
dxdEC
zyyz
yzxylzxzly
lyylzzxE
=φ
++−
−+−−−+
φ
++++−φ
ωω
ω
ωω
ωω
In these equations, a number of additional quantities are to be defined.
=lzly MM , Moments caused by transverse loads only =+++= AeeIII zyzzyyE )( 22 polar moment of inertia about
shear center Three new geometrical quantities are
Torsion in Structural Design - Notes 11/30/01
56
( )
∫∫
∫∫
∫∫
+ω−ω=
+=
+=
ωA
Az
Ay
dAzyH
dAzyyH
dAzyzH
))((2
)(
220
22
22
From these,
yyzzzyy
yyzzyyz
zyzzzyy
zyzyzzy
eIII
HIHIC
eIII
HIHIC
2
2
2
2
−−
−=
−−
−=
The equations are three simultaneous equations in three unknowns. Although the geometrical properties are constant, the load coefficients ωω VWVVMM zylzly ,,,,, all vary with x. ∴ The d.e. coefficients are variable. Often, the d.e.s can be simplified, by taking advantage of special properties of different classes of problems. We will want to do this in practice, since the original set of equations is unwieldy.
Torsion in Structural Design - Notes 11/30/01
57
First, assume that the y-z coordinates are principal coordinate axes. Then 0=yzI , and the eqns reduce to
yyzzxlyxzz qdxdV
dxdeNM
dxvdN
dxvdEI =φ+φ−+− 2
2
2
2
4
4
)(
zyyxlzxyy qdxdV
dxdeNM
dxwdN
dxwdEI =φ−φ+−+− 2
2
2
2
4
4
)(
)(
)()( 2
2
2
2
2
2
4
4
xtdxdV
CHVCVC
dxdwV
dxdvV
dxwdNeM
dxvdNeM
dxdW
CHMCMCN
AIGJ
dxdEC
zyyz
yzxylzxzly
lyylzzxE
=φ
++−
−+−−−+
φ
++++−φ
ωω
ω
ωω
ωω
If, in addition, the transverse loads zy qq , are zero, and the transverse moments are constant, then the shears zy VV , are also zero, and the additional simplification
0)( 2
2
2
2
4
4
=φ−+−dxdeNM
dxvdN
dxvdEI zxlyxzz
0)( 2
2
2
2
4
4
=φ+−+−dxdeNM
dxwdN
dxwdEI yxlzxyy
Torsion in Structural Design - Notes 11/30/01
58
)()()( 2
2
2
2
2
2
4
4
xtdx
wdNeMdx
vdNeM
dxdW
dxd
CH
dxdMCMCN
AIGJ
dxdEC
xylzxzly
lyylzzxE
=−−−+
φ−φ
+++−φ
ωω
ωω
is possible. Now, assuming that the bar is subjected only to an axial compressive load through the centroid, the moments due to transverse loads are zero, the applied torque vanishes, and if the loading is applied so that the bimoment is zero, the equations become
02
2
2
2
4
4
=φ++dxdeN
dxvdN
dxvdEI zxxzz
02
2
2
2
4
4
=φ−+dxdeN
dxwdN
dxwdEI yxxyy
02
2
2
2
2
2
4
4
=−+φ
−−φ
ω dxwdNe
dxvdNe
dxdN
AIGJ
dxdEC xyxzx
E
These equations represent a generalization of the elastic column buckling problem. (Note sign change in xN because of compression.) Assume a pinned end column for which the appropriate boundary conditions are
0" 0" 0"0 0 0
=φ===φ==
wvwv
Torsion in Structural Design - Notes 11/30/01
59
Then a solution of the form
)/sin()/sin(
)/sin(
0
0
0
LxLxww
Lxvv
πφ=φπ=
π=
where 000 ,, φwv are the values of φ,,wv at midspan, will satisfy the resulting equations provided
0)( 002
2
=φ−−πzxxzz eNvN
LEI
0)( 002
2
=φ+−πyxxyy eNwN
LEI
002
2
00 =φ
−+π++− ω xE
xyxz NAIGJ
LECwNevNe
or, in matrix form
=
φ
−+π−
−π
−−π
ω
000
0
0
0
0
0
2
2
2
2
2
2
wv
NAIGJ
LECeNeN
eNNL
EI
eNNL
EI
xE
yxzx
yxxyy
zxxzz
Non-trivial solutions exist iff the determinant of coefficients is zero. Rather than solve for the general case, let's consider two special cases.
Torsion in Structural Design - Notes 11/30/01
60
(a) The section is doubly symmetric. Then 0== zy ee , and the equations reduce to
=
φ
−+π
−π
−π
ω
000
00
00
00
0
0
0
2
2
2
2
2
2
wv
NAIGJ
LEC
NL
EI
NL
EI
xE
xyy
xzz
The corresponding determinantal equation
02
2
2
2
2
2
=
−+π
−π
−πω x
Exyyxzz N
AIGJ
LECN
LEIN
LEI
is already factored, and has the three solutions
zzEzzx PL
EIN =π= 2
2
(buckling about z axis)
yyEyyx PL
EIN =π= 2
2
(buckling about y axis)
TE
x PGJL
ECIAN =
+π= ω 2
2
(Torsional, or twist buckling)
The 3rd solution corresponds to a new mode of buckling, shown below, for a cruciform section.
Torsion in Structural Design - Notes 11/30/01
61
Dividing by A and noting that pE II = for a doubly symmetrical section yields the critical stress
+π−=σ ω GJL
ECI p
xxcr 2
21
If this critical stress is less than the minor axis Euler stress, then torsional buckling will preceed Euler buckling of the section, and must be accounted for in design.
Torsion in Structural Design - Notes 11/30/01
62
Example: Consider an equal leg cruciform section with h=6",t=1/4". Section properties:
42222
in 508.412
])"25(.)"6)[("25)(."6(12
)( =+=+== ththII zzyy
4in 016.9=+= zzyyp III
"239.1in 938.2)"25(.)"25)(."6(22 222 ==→=−=−= zy rrtthA
63333
in 04688.072
)"6()"25(.72
===ωhtC
4
333
in 0625.03
)"25)(."6(23
23
)2/(4 ==== htthJ
For Euler buckling, both axes have the same I, so
22
2
2
2
2)( 530,454)"239.1()000,30(
)/( LLrLEEuler
xxcr=π=π=σ
For torsional buckling,
Torsion in Structural Design - Notes 11/30/01
63
80540,1
)]0625.0)(540,11()04688.0)(000,30[(016.91
)(1
2
2
2
2
2)(
+=
+π=
+π=σ ω
L
L
GJL
ECI p
Torsionxxcr
(The GJ term dominates the torsional buckling equation unless the column is very short.) Torsion controls if )()( Euler
xxTorsion
xx crcrσ<σ , so
"2.75530,45480154022 <→<+ L
LL
• In a symmetrical section, torsional buckling tends to
control only for short columns. • The thinner the walls of the column, relative to the plate
widths (Thin plates) the more likely it is that torsional buckling will control. (Increasing the wall thickness to 3/8" in the above calculations decreases the length column for which torsion controls to 50".)
• It may be feasible to ignore the warping torsion term in the above in the practical range of interest.
• For the torsion buckling, it is appropriate to replace E with )1/( 2ν−E consistent with dominant plate action.
(b) The section is singly symmetrical. If the z axis is an axis of symmetry, then 0=ye , and
Torsion in Structural Design - Notes 11/30/01
64
=
φ
−+π−
−π
−−π
ω
000
0
00
0
0
0
0
2
2
2
2
2
2
wv
NAIGJ
LECeN
NL
EI
eNNL
EI
xE
zx
xyy
zxxzz
From which, the determinantal equation is
0222
2
2
2
2
2
=
−
−+π
−π
−πω zxx
Exzzxyy eNN
AIGJ
LECN
LEIN
LEI
Before solving this problem, it is useful to substitute the relations
2
2
LEIP zzEzz
π= 2
2
LEIP yyE yy
π= GJL
ECPAI
TE +π= ω 2
2
Then, the determinantal eqn. takes the form
( ) ( )( ) 022 =
−−−− zxxTxE
ExE eNNPNP
AINP
zzyy
This is an important case, as T beams, one section commonly used for compression elements (e.g. truss members) is singly symmetrical. The leading factor is just Euler column buckling in the plane of symmetry. The other factor leads to a coupled flexural-torsional buckling mode. The formal solution is
Torsion in Structural Design - Notes 11/30/01
65
−
−−+−+
=
E
z
TEE
zTETE
x
IAe
PPI
AePPPPN
zzzzzz
2
22
12
14)()(
Typically, this critical load will be smaller than either the twist buckling load, or the flexural buckling mode.
Torsion in Structural Design - Notes 11/30/01
66
Dividing the equation by the area permits the relationship to be written in terms of the critical stresses for the individual buckling modes.
−
σσ
−−σ+σ−σ+σ
=σ
E
z
Txx
Exx
E
zTxx
Exx
Txx
Exx
FTxx
IAe
IAe
crcrcrcrcrcr
cr 2
)()(2
2)()()()(
)(
12
14)()(
where
2
2
)/( y
Exx rL
Ecr
π=σ
ν+
+π=σ ω )1(22
2)( J
LC
IE
E
Txxcr
One can use the latter equation to define an equivalent radius of gyration for torsional buckling. Equating
ν+
+π=πω )1(2)/( 2
2
2
2 JL
CIE
rLE
EE
yields the resulting equivalent r
ν+π
+= ω )1(21
2
2JLCI
rE
E
Torsion in Structural Design - Notes 11/30/01
67
Example: Consider a WT12x27.5: 2in 10.8=A
4in 117=yyI 4in 5.14=zzI "8.3=zr "34.1=yr
"248.3=ze 4in 588.0=J 6in 764.2=ωC
42244 in 217)"248.3)(in 1.8(in 5.14in 117 =++=EI
22
2
2
2
2 700,531)"34.1(000,30)/( LLrL
Ey
Exxcr
=π=π=σ
26.31771,36.2
588.0764.2217
000,30)1(2
2
2
2
2
2)(
+=
+π=
ν+
+π=σ ω
L
LJ
LC
IE
E
Txxcr
Substituting into the critical stress eqn yields
213.1
3.31771,3300,289,1)3.31500,535()3.31500,535( 222
22)(
+−+−+
=σ LLLLFTxxcr
This result is shown below in two plots. • First plot shows the interaction of the critical stresses
over a large theoretical range • Second plot shows the interaction of the critical stresses
over a practical range
Torsion in Structural Design - Notes 11/30/01
68
Flexural-Torsional interaction significantly reduces the buckling load for this column, extending well into the elastic range!
Torsion in Structural Design - Notes 11/30/01
69
7. References Bleich, F. (1952), Buckling Strength of Metal Structures, McGraw Hill Book Co., New York, N.Y. Boresi, A. P., Schmidt, R. J., and Sidebottom, O. M., (1993) Advanced Mechanics of Materials, 5th Edition, John Wiley and Sons, Ch. 6 and 8. Chang, F. K., and Johnston, B. G., (1953), "Torsion of Plate Girders," Transactions, ASCE, 118, 337-396. Chen, M.-T., and Jolissaint, D. E. Jr., (1983), "Pure and Warping Torsion Analysis of Rigid Frames," Journal of Structural Engineering, ASCE, 109, 8, 1999-2003. Chu, K-H., and Johnson, R. B., (1974), "Torsion in Beams with Open Sections," Journal of the Structural Division, ASCE, 100, ST7, 1397-1419. Chu, K-H., and Longinow, A., (1967), "Torsion in Sections with Open and Closed Parts," Journal of the Structural Division, ASCE, 93, ST6, 213-227. Dean, D. L., (1994), "Torsion of Regular Multicellular Members," Journal of Structural Engineering, 120, 12, 3675-3678 Driver, R. G., and Kennedy, D. J. L., (1989), "Combined Flexure and Torsion of I-shaped Steel Beams," Canadian Journal of Civil Engineering, 16, 124-139.
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El Darwish, I. A., and Johnston, B. G., (1965), "Torsion of Structural Shapes," Journal of the Structural Division, ASCE, 91, ST1, 203-227, Errata: 92, ST1, 471. Evick, D. R., and Heins, C. P. Jr., (1972), "Torsion of Nonprismatic Beams of Open Section," Journal of the Structural Division, ASCE, 98, ST12, 2769-2784. Felton, L. P., and Dobbs, M. W., (1967), "Optimum Design of Tubes for Bending and Torsion," Journal of the Structural Division, ASCE, 93, ST4, 185-200. Goldberg, J. E., (1953), "Torsion of I-Type and H-Type Beams," Transactions, ASCE, 118, 771-793. Goodier, J. N., and Barton, M. V., (1944), "The Effects of Web Deformation on the Torsion of I Beams," Journal of Applied Mechanics, ASME, p A-35. Heins, C. P. Jr., (1975) Bending and Torsional Design in Structural Members, Lexington Books, Lexington, MA. Heins, C. P., Jr., and Kuo, J. T. C., (1972), "Torsional Properties of Composite-Girders," Engineering Journal, AISC, 9, 2, 79-85. Heins, C. P. Jr., and Potocko, R. A., (1979), "Torsional Stiffening of I-Girder Webs," Journal of the Structural Division, ASCE, 105, ST8, 1689-1698.
Torsion in Structural Design - Notes 11/30/01
71
Hotchkiss, J. G., (1966), "Torsion of Rolled Steel Sections in Building Structures," Engineering Journal , AISC, 3, 1, 19-45. Johnston, B. G., (1982), "Design of W Shapes for Combined Bending and Torsion," Engineering Journal, AISC, 19, 2, 65-85. Kubo, G. G., Johnston, B. G., and Eney, W. J., (1956), "Nonuniform Torsion of Plate Girders," Transactions, ASCE, 121, 759-785. Lin, P. H., (1977), "Simplified Design for Torsional Loading of Rolled Steel Members," Engineering Journal, AISC, 14, 3, 97-107. Lin, P. H., Discussion of "Design of W-Shapes for Combined Bending and Torsion," by B. G. Johnston, Engineering Journal, AISC, 20, 2, 82-87. Lue, T. and Ellifritt, D. S., (1993), "The Warping Constant for the W-Section with a Channel Cap," Engineering Journal, AISC, 30, 1, 31-33. McGuire, W., (1968) Steel Structures, Prentice Hall, Englewood Cliffs N.J., pp. 346-400. Murray, N. W., (1984), Introduction to the Theory of Thin-walled Structures, Oxford Univ. Press, Oxford, Ch. 2 and 3.
Torsion in Structural Design - Notes 11/30/01
72
Oden, J. T., (1967), Mechanics of Elastic Structures, McGraw Hill, N.Y., Ch. 3, 5 and 7. Pi, Y. L., and Trahair, N. S., (1995a) "Inelastic Torsion of Steel I-Beams, Journal of Structural Engineering, ASCE, 121, 4, 609-620. Pi, Y. L., and Trahair, N. S., (1995b), "Plastic Collapse Analysis of Torsion," Journal of Structural Engineering, ASCE, 121, 10, 1389-1395. Pi, Y. L., and Trahair, N. S., (1994a), "Inelastic Bending and Torsion of Steel I-Beams," Journal of Structural Engineering, ASCE, 120, 12, 3397-3417. Pi, Y. L., and Trahair N. S., (1994b), "Nonlinear Inelastic Analysis of Steel Beam-Columns I: Theory," Journal of Structural Engineering, 120, 7, 2041-2061. Pi, Y. L., and Trahair N. S., (1994c), "Nonlinear Inelastic Analysis of Steel Beam-Columns II: Applications," Journal of Structural Engineering, 120, 7, 2062-2085. Salmon, C. G., and Johnson, J. E., (1996) Steel Structures, Design and Behavior, Harper and Row, New York. Shermer, C. L., (1980), "Torsional Strength and Stiffness of Steel Structures," Engineering Journal, AISC, 17, 2, 33-37.
Torsion in Structural Design - Notes 11/30/01
73
Siev, A. (1966), "Torsion in Closed Sections," Engineering Journal, AISC, 3, 1, 46-54. Timoshenko, S., (1941), Strength of Materials, Part II, 2d ed., New York, Van Nostrand Co. Inc., Ch. 6. Vacharajittiphan, P., and Trahair, N. S., (1974), "Warping and Distortion at I-Section Joints," Journal of the Structural Division, ASCE, 100, ST3, 547-564. Vlasov, V. Z., (1961) Thin Walled Elastic Beams, Israel Program for Scientific Translations, Jerusalem.