Mba admission in india

MBA Admission in India
By:
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© Teaching Resource in Design of Steel Structures –
IIT Madras, SERC Madras, Anna Univ., INSDAG
2
BENDING AND TORSION

3
BENDING AND TORSION
• Introduction
• Designing for torsion in practice
• Pure torsion and warping
• Combined bending and torsion
• Design method for lateral torsional
buckling
• Conclusion

4
INTRODUCTION
– Torsional moments cause twisting and
warping of the cross sections.
– When torsional rigidity (GJ) is very large
compared with its warping rigidity (EG ), the
section would effectively be in uniform torsion
and warping moment would be unlikely to be
significant.
– The warping moment is developed only if
warping deformation is restrained.

5
Designing for Torsion in Practice
• "Avoid Torsion - if 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 sections
or lattice box girders should be investigated

Tq × z
6
Pure Torsion and Warping
• When a torque is applied only at the ends of a
member such that the ends are free to warp, then
the member would develop only pure torsion.
The total angle of twist (f ) over a length of z is
given by
G J
f =
When a member is in non-uniform torsion, the rate
of change of angle of twist will vary along the length
of the member

E Swms
t f
7
Pure Torsion and Warping - 2
• The warping shear stress (t w) at a point is given by ,
t
w
¢¢¢
= -
Swms = Warping statical moment
• The warping normal stress (s w) due to bending moment
in-plane of flanges (bi-moment) is given by
s w = E .Wnwfs . f ''
where
Wnwfs = Normalised warping function

8
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 f caused by torsion would be
amplified by bending moment, inducing
additional warping moments and torsional
shears.

9
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 /gm)
• The "capacity check" for major axis bending
sbx + sbyt +sw £ fy /gm.

é
+
1 0.5 M
s s
úû
x byt w
x
£ © Teaching Resource in Design of Steel Structures –
10
• Buckling Check
whenever lateral torsional buckling governs the
design (i.e. when pb is less than fy) the values of sw
and sbyt will be amplified.
( )
( ) 1
M
M
M f /
b
y m
b
ù
êë
+
+
g
, equivalent uniform moment = mx Mx
Mb , the buckling resistance moment =
Mx
M M
( 2
) 1
2
E p
B B
E p
M M
f + f -

11
• Applied loading having both Major axis and Minor
axis moments
When the applied loading produces both major
axis and minor axis moments, the "capacity
checks" and the "buckling checks" are modified.
Capacity Check
sbx + sbyt +sw + sby £ fy/gm

y
x
M
M
M =
m M
y y y
s s
byt w
é
+
1 0.5 M
12
Buckling Check
( )
( )
byt y y
x
b
y m
y y m
b
M / Z
1
M
f Z / f /
M
=
ù
£ úû
êë
+
+ +
s
g
g
where

ö
æ
x
t t t 1 0.5 M
13
• Torsional Shear Stress
Torsional shear stresses and warping shear stresses
should also be amplified in a similar manner
( ) ÷ ÷ø
ç çè
= + +
b
vt t w M
This shear stress should be added to the shear
stresses due to bending in checking the adequacy
of the section.

14
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 ME are
too complex for routine use
-there are limitations to their extension in the
ultimate range

M M
=
f f
15
Design method for lateral torsional buckling - 2
• A simple method of computing the buckling
resistance of beams is as follows:-
- the buckling resistance moment, Mb, is obtained
as the smaller root of the equation,
(ME - Mb) (Mp - Mb) = h LT. ME Mb
where
( 2
) 1
2
E p
B B
E p
b
M M
M
+ -

16
Mp = fy . Zp / g m
( )
2
Mp LT 1 ME
B
+ +
=
h
f
h LT = Perry coefficient, similar to column
buckling coefficient
Zp = Plastic section modulus

17
• In order to simplify the analysis, BS5950: Part 1
uses a curve, in which the bending strength of
the beam is expressed as a function of its
slenderness (l LT )
- the buckling resistance moment Mb is given by
Mb= pb .Zp
where
pb = bending strength allowing for susceptibility to
lateral torsional buckling.
Zp = plastic section modulus.

The beam slenderness (l LT) is given by,
l = p E × l
Mp
18
E M
lLT =
LT
y
2
LT f
where,

Beam fails by yield
Beam buckling
19
300
200
100
0
50 100 150 200 250
pb
N/mm2
lLT
Fig 1. Bending strength for rolled sections of design
strength 275 N/mm2 according to BS 5950

MP
ME / MP
0.4 0.8 1.2
1.0
0.8
0.4
20
ME
l LT =
Fig.2 Comparison of test data with theoretical elastic critical
moments
0
sto
cky
interm
ediate
slender
Plastic
yield
M / Mp

21
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 <
< 1.2
lLT
lLT
lLT

22
- Beams having short spans usually fail by
yielding
- 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

23
•In the absence of instability, eqn. 11 may be
adopted for the full plastic moment capacity pb for
l LT < 0.4 .
•This corresponds to l LT values of around 37 (for
steels having fy= 275 N/mm2) below which the lateral
instability is NOT of concern.

l = uv 
LT r
24
For more slender beams, pb is a function of l LT which is
given by ,
y
u is called the buckling parameter and x, the torsional
index.
Please refer paper for the expressions for buckling
parameter and the torsional index corresponding to
flanged sections symmetrical about the minor axis and
flanged sections symmetrical about the major axis.

25
• Unequal flanged sections
For unequal flanged sections, the following
equation is used for finding the buckling moment
of resistance.
Mb= pb .Zp

26
• Evaluation of differential equations
For a member subjected to concentrated torque
with torsion fixed and warping free condition at
the ends ( torque applied at varying values of a L),
the values of f and its differentials are given by
Tq
a  (1-a)

ì
ù
é

Tq a 
1 z
.
= - + -
sinh z
ì
ù
é

Tq 
¢ = - + -
cosh z
27
For 0 £ z £ a ,
( )
ü
ï ïþ
ï ïý
ï ïî
ï ïí
ú ú ú ú
û
ê ê ê ê
ë
a
a
cosh
a
tanh
a
sinh
a
GJ

a
a
f a
( )
ü
ï ïþ
ï ïý
ï ïî
ï ïí
ú ú ú ú
û
ê ê ê ê
ë
a
a
cosh
a
tanh
a
sinh
1
GJ

a
a
f a

sinh z
Tq
ù
é

cosh z
Tq
ù
é

¢¢¢ = - 
28
For 0 £ z £ a ,
a
a
cosh
a
tanh
a
sinh
G J a
ú ú ú ú
û
ê ê ê ê ë
¢¢ = - 

a
a
f
a
a
cosh
a
tanh
a
sinh
G J a
ú ú ú
û
ê ê ê
ë

a
a
f 2
Similar equations are available for different loading cases
and for different values of a .

29
CONCLUSION
A simple method of evaluating torsional effects
and to verify the adequacy of a chosen cross
section when subjected to torsional moments has
been discussed.

30
THANKYOU

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