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8/13/2019 Material -Ch18
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Teaching Resource in Design of Steel Structures IIT Madras, SERC Madras, Anna Univ., INSDAG
1
BENDING AND TORSION
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Teaching Resource in Design of Steel Structures IIT Madras, SERC Madras, Anna Univ., INSDAG
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BENDING AND TORSION
Introduction
Designing for torsion in practice
Pure torsion and warping
Combined bending and torsion
Design method for lateral torsional
buckling
Conclusion
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INTRODUCTION
Torsional moments cause twisting andwarping of the cross sections.
When torsional rigidity (GJ) is very large
compared with its warping rigidity (E), thesection would effectively be in uniform torsionand warping moment would be unlikely to be
significant.
The warping moment is developed only if
warping deformation is restrained.
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Designing for Torsion in Practice
" Avo id Tors ion - i f you can "
The loads are usually applied in such a manner that their
resultant passes through the centroid in the case of
symmetrical sections and shear centre in the case of
unsymmetrical sections. Arrange connections suitably.
Where significant eccentricity of loading (which would
cause torsion) is unavoidable, alternative methods of
resisting torsion like design using box, tubular sectionsor lattice box girders should be investigated
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Pure Torsion and Warping
When a torque is applied only at the ends of a
member such that the ends are free to warp, thenthe member would develop only pure torsion.
The total angle of twist ( ) over a length of z isgiven by
JG
zTq
When a member is in non-uniform torsion, the rate
of change of angle of twist will vary along the lengthof the member
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Pure Torsion and Warping - 2
The warping shear stress (w) at a point is given by ,
t
SE wmsw
Swms = Warping statical moment
The warping normal stress (w)due to bending moment
in-plane of flanges (bi-moment) is given by
w= E .W
nwfs. ' '
whereW
nwfs = Normalised warping function
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Combined Bending and Torsion
There is interaction between the torsional and
flexural effects, when a load produces both
bending and torsion
The angle of twist caused by torsion would beamplified by bending moment, inducing
additional warping moments and torsional
shears.
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Combined Bending and Torsion - 2
Maximum Stress Check or "Capacity check"
The maximum stress at the most highly stressed
cross section is limited to the design strength
(fy/m)
The "capacity check" for major axis bending
bx+ byt+w fy/m.
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Combined Bending and Torsion - 3
Buckling Check
whenever lateral torsional buckling governs the
design (i.e. when pbis less than fy) the values of w
and bytwill be amplified.
1MM
0.51/fM
M
b
x
my
wbyt
b
x
, equivalent uniform moment = mxMx
Mb , the buckling resistance moment=
xM
21
pE2
BB
pE
MM
MM
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Combined Bending and Torsion - 4
Applied loading having both Major axis and Minor
axis momentsWhen the applied loading produces both major
axis and minor axis moments, the "capacity
checks" and the "buckling checks" are modified.
Capacity Check
bx+ byt+w+ by fy/m
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Combined Bending and Torsion - 5
Buckling Check
yybyt
yyy
b
x
my
wbyt
myy
y
b
x
Z/M
MmM
1M
M0.51
/f/Zf
M
M
M
where
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Combined Bending and Torsion - 6
Torsional Shear Stress
Torsional shear stresses and warping shear stresses
should also be amplified in a similar manner
b
xwtvt
MM0.51
This shear stress should be added to the shear
stresses due to bending in checking the adequacyof the section.
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Design method for lateral torsional
buckling
the basic theory of elastic lateral stability cannot
be directly used for the design purpose because
-the formulae for elastic critical moment MEare
too complex for routine use
-there are limitations to their extension in the
ultimate range
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Design method for lateral torsional buckling - 2
A simple method of computing the bucklingresistance of beams is as follows:-
- the buckling resistance moment, Mb, is obtained
as the smaller root of the equation,
(ME- M
b) (M
p- M
b) =
LT. M
EM
b
where
21pE2BBpE
bMM
MMM
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Design method for lateral torsional buckling - 3
Mp = fy. Zp/ m
2
M1M ELTpB
LT = Perry coefficient, similar to columnbuckling coefficient
Zp= Plastic section modulus
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Design method for lateral torsional buckling - 4
In order to simplify the analysis, BS5950: Part 1
uses a curve, in which the bending strength ofthe beam is expressed as a function of its
slenderness (LT)
- the buckling resistance moment Mbis given byM
b= p
b.Z
p
where
pb= bending strength allowing for susc ept ib i l i ty to
lateral torsio nal buckl ing .
Zp= plast ic sect ion modulus.
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Design method for lateral torsional buckling - 5
EM
Mp
LT
LT
y
2LT
f
E
The beam slenderness (LT) is given by,
where,
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Design method for lateral torsional buckling - 6
300
200
100
050 100 150 200 250
pb
N/mm2
LT
Fig 1. Bending strength for ro l led sect ions of des ignstrength 275 N/mm2accord ing to BS 5950
Beam failsby y ield
Beam buc kl ing
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Design method for lateral torsional buckling - 7
EM
PM
LT
Fig.2 Comparison o f test data with th eoret ical elast ic cr i t ical
moments
0.4 0.8 1.20
0.4
1.0
0.8
s to
cky
interm
ediate
slender
ME/ MP
Plast ic yield
M / Mp
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Design method for lateral torsional buckling - 8
In Fig. 2 three distinct regions of behaviour can
be observed:-- stocky beams which are able to attain the
plastic moment Mp, for values of below
about 0.4.
- slender beams which fail at moments close to
ME, for values of above about 1.2
- beams of intermediate slenderness which fail
to reach either Mp or ME . In this case 0.4
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Design method for lateral torsional buckling - 9
- Beams having short spans usually fail byyielding
- Beams having long spans would fail by lateral
buckling
- Beams which are in the intermediate range
without lateral restraint, design must be based
on considerations of inelastic buckling
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Design method for lateral torsional buckling - 10
In the absence of instability, eqn. 11 may be
adopted for the full plastic moment capacity pbforLT< 0.4.
This corresponds to LT values of around 37 (forsteels having fy= 275 N/mm
2) below which the lateral
instability is NOT of concern.
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Design method for lateral torsional buckling - 11
For more slender beams, pb
is a function of LT
whichis
given by ,
y
LTr
uv
uis called the buckling parameter and x,the torsionalindex.
Please refer paper for the expressions for buckl ing
parameter and the tors ional index corresponding to
f langed sect ions symmetr ical about the mino r axis andf langed sect ions symmetr ical about th e major axis .
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Design method for lateral torsional buckling - 12
Unequal flanged sections
For unequal flanged sections, the following
equation is used for finding the buckling moment
of resistance.
Mb= p
b.Z
p
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Design method for lateral torsional buckling - 13
Evaluation of differential equations
For a member subjected to concentrated torquewith torsion fixed and warping free condition at
the ends ( torque applied at varying values of L ),
the values of and its differentials are given byTq
(1-)
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Design method for lateral torsional buckling - 14
For 0 z ,
a
zsinh
acosh
atanh
asinh
a
z1
GJ
aTq
.
a
zcosh
acosh
atanh
asinh
1GJ
Tq
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Design method for lateral torsional buckling - 15
For 0 z ,
a
zsinh
acosh
atanh
asinh
aJG
Tq
a
zcosh
acosh
atanh
asinh
aJG
Tq
2
Similar equations are available for different loading casesand for different values of .
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THANKYOU