MAterial- Ch17

8/13/2019 MAterial- Ch17

1/23

Teaching Resource in Design of Steel Structures

IIT Madras, SERC Madras, Anna Univ., INSDAG

1

TORSION


2/23


3/23



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.


4/23



4

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


5/23



5

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).


6/23



6


Uniform Torsion in a Circular Cross Section

Z

T

T

Twisting of circular section


7/23



7


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


8/23



8


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


9/23



9


T

T

Uniform Torsion(Constant Torque :

Ends are free to warp)

bi

ti

Thin walled open section

made of rectangular elements


10/23



10


t in flange

tin web

Stress pattern due to pure torsio n


11/23



11

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


12/23



12

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


13/23



13

NON-UNIFORM TORSION - 3

Tors ion in

Canti levers

Bending moment(M)

Shear force

(V)

Pure Torsion

(Tp)

Warping Torsion

(Tw)

Total Torsion

(Tn)


14/23



14

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


15/23



15

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


16/23



16

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


17/23



17

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.


18/23



18

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 :


19/23



19


Procedures for checking adequacy in Flexure (fill)

G

A

y

t


20/23



20


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


21/23



21

BT

t

Dh

XX X X

Af

yfy

w

X XHalf the area= 0.5A



22/23



22

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


23/23



23

THANK YOU

Documents

MAterial- Ch17