MAterial- Ch17
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Transcript of MAterial- Ch17
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8/13/2019 MAterial- Ch17
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Teaching Resource in Design of Steel Structures
IIT Madras, SERC Madras, Anna Univ., INSDAG
1
TORSION
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Teaching Resource in Design of Steel Structures
IIT Madras, SERC Madras, Anna Univ., INSDAG
3
INTRODUCTION
In a transversely loaded beam if the resultant force
passes through the longitudinal shear centre axis, thebeam only bends and no torsion occurs.
When the resultant acts away from the shear centre axis,
then the beam will not only bend but also twist.
If a beam is subjected to a twisting moment, the
assumption of planarity is simply incorrect except for
solid circular sections and for hollow circular sectionswith constant thickness.
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Teaching Resource in Design of Steel Structures
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UNIFORM AND NON-UNIFORM
TORSION
Shear Centre and Warping
Shear Centre is defined as the point in the cross-
section through which the lateral (or transverse)loads must pass to produce bending without
twisting
Warping of the section does not allow an initiallyplane section to remain as plane after twisting
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Teaching Resource in Design of Steel Structures
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UNIFORM AND NON-UNIFORM TORSION - 2
Classification of Torsion
1. Uniform or Pure Torsion (called St. Venant's
torsion) - Tsv
2. Non-Uniform Torsion, consisting of St.Venant's
torsion (Tsv
) and warping torsion (Tw).
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UNIFORM AND NON-UNIFORM TORSION - 3
Uniform Torsion in a Circular Cross Section
Z
T
T
Twisting of circular section
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UNIFORM AND NON-UNIFORM TORSION - 4
In this case, plane cross sections normal to the
axis of the member remain plane after twisting,i.e. there is no warping.
For a circular section, the St. Venant's torsion is
given by
dz
dGIT psv
where, - angle of twist
G - modulus of rigidity
Tsv
- St. Venant's torsion.
Ip
- the polar moment of inertia
z - direction along axis of the member
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Teaching Resource in Design of Steel Structures
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UNIFORM AND NON-UNIFORM TORSION - 5
Uniform Torsion in Non-Circular Sections
When a torque is applied to a non-circular cross
section , the transverse sections which are plane
prior to twisting, warp in the axial direction
dz
dJGTsv
where J = C. bt3
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Teaching Resource in Design of Steel Structures
IIT Madras, SERC Madras, Anna Univ., INSDAG
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UNIFORM AND NON-UNIFORM TORSION - 7
T
T
Uniform Torsion(Constant Torque :
Ends are free to warp)
bi
ti
Thin walled open section
made of rectangular elements
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UNIFORM AND NON-UNIFORM TORSION - 8
t in flange
tin web
Stress pattern due to pure torsio n
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NON-UNIFORM TORSION
Non uniform Torsio n:Twist ing o f Non-Circu lar Sect ion restrainedagainst free warping (Cons tant Torqu e : End warping is prevented )
h
u
Vf
VfTa
Ta
Z
X
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NON-UNIFORM TORSION - 2
Torsion in sim ply
supp or ted beam
with free end warping
Bending Moment
Shear Force
Pure Torsion
(TP)
Warping
Torsion (Tw)
Total Torsion
(Tn)
e
e
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NON-UNIFORM TORSION - 3
Tors ion in
Canti levers
Bending moment(M)
Shear force
(V)
Pure Torsion
(Tp)
Warping Torsion
(Tw)
Total Torsion
(Tn)
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An Approximate Method of Torsion Analysis
H
Load PYacting eccent r ically w .r.t . y axis and causing tors ion
=h
+X
Y PY
eXPY
+
+
H
PY.eX
Warping stresses due
to bimoment
= PY.eX/h
Load PXact ing eccentr ical ly
and causing tors ion.
Rotation of the crosssection
=
+
Y
X
eY
PX
PX+
H
PX.eY
H
+
= PX.eY/h
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An Approximate Method of Torsion Analysis - 2
Y
Rotation of
cross section
Warping Stresses in Open Cross Sect ion
H
HWarping
normal
stress(w)
In-plane
bending
moment in
the flange
Warping
shear
stress
(Tw)
Flange
shears
Z
X
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An Approximate Method of Torsion Analysis -3
The magnitude of the warping normal stress at any
particular point (w) in the cross section is given by
w = - EW
nwfs
Wnwfs
= normalised warping function at a particular point Sin the
cross section
The magnitude of the warping shear stress at any given
point is given by
t
ESwms
w
Swms
= Warping statical moment of area at a particular point S
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The Effect of Torsional Rigidity (GJ)
and Warping Rigidity(E)When the torsional rigidity (GJ) is very large
compared to the warping rigidity, E ,then thesection will effectively be in "uniform torsion".
If GJis very small compared with E ,the memberwill effectively be subjected to warping torsion.
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The Effect of Torsional Rigidity (GJ)and Warping Rigidity(E) - 2
End Conditions
The end support conditions of the member
Influence the torsional behaviour significantly
Torsion fixed, Warping fixed :
Torsion fixed, Warping free :
Torsion free, Warping free :
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The Effect of Torsional Rigidity (GJ)and Warping Rigidity(E) - 3
Procedures for checking adequacy in Flexure (fill)
G
A
y
t
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The Effect of Torsional Rigidity (GJ)and Warping Rigidity(E) - 4
Cross Sectional Properties for Symmetrical IandH Sections
ww
fff
y
wms
nwfs
yA
Q
yAQ
hI
TBhS
BhW
tTDBTJ
2
.
4
16
4
223
1
2
2
33
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BT
t
Dh
XX X X
Af
yfy
w
X XHalf the area= 0.5A
The Effect of Torsional Rigidity (GJ)and Warping Rigidity(E) - 5
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Uniform torsion applied to the beam would cause atwist
Non-uniform torsion will cause both twisting and
warping of the cross section
Analysis of a beam subjected to torsional moment isoutlined
Simple methods of evaluating the torsional effects are
discussed
CONCLUSIONS
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THANK YOU