Download - 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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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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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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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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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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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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Mp = fy. Zp/ m

2

M1M ELTpB

LT = Perry coefficient, similar to columnbuckling coefficient

Zp= Plastic section modulus


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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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EM

Mp

LT

LT

y

2LT

f

E

The beam slenderness (LT) is given by,

where,


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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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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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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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- 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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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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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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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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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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For 0 z ,

a

zsinh

acosh

atanh

asinh

a

z1

GJ

aTq

.

a

zcosh

acosh

atanh

asinh

1GJ

Tq


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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

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