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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




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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 and warping of the cross sections.

– When torsional rigidity (GJ) is very large compared with its warping rigidity (EΓ), 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.




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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




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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, then the member would develop only pure torsion.

The total angle of twist (φ ) over a length of z is given by

JG

zTq ⋅=φ

When a member is in non-uniform torsion, the rate of change of angle of twist will vary along the length of the member




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Pure Torsion and Warping - 2

• The warping shear stress (τ w) at a point is given by ,

t

SE wmsw

φτ′′′

−=

Swms = Warping statical moment

• The warping normal stress (σw) due to bending moment

in-plane of flanges (bi-moment) is given by

σw = E .Wnwfs . φ ''

where Wnwfs = 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 φ caused by torsion would be amplified 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 /γm)

• The "capacity check" for major axis bending

σ bx + σ byt +σ w ≤ fy /γm.




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• Buckling Check

whenever lateral torsional buckling governs the

design (i.e. when pb is less than fy) the values of σ w

and σ byt will be amplified.

( )( ) 1

M

M0.51

/fM

M

b

x

my

wbyt

b

x ≤

+

++

γσσ

, equivalent uniform moment = mx Mx

Mb , the buckling resistance moment =

xM

( ) 21pE2BB

pE

MM

MM

−+ φφ




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• 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

σ bx + σ byt +σ w + σ by ≤ fy/γm




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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

=

=

≤

+

+++

σ

γσσ

γwhere




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• Torsional Shear Stress

Torsional shear stresses and warping shear stresses

should also be amplified in a similar manner

( )

++=

b

xwtvt M

M0.51τττ

This shear stress should be added to the shear stresses due to bending in checking the adequacy of 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 ME are

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 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) = η LT. ME Mb

where

( ) 21pE2BB

pEb

MM

MMM

−+=

φφ



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Mp = fy . Zp / γm

( )2

M1M ELTpB

++=

ηφ

ηLT = Perry coefficient, similar to column buckling 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 of the beam is expressed as a function of its slenderness (λ 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.



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EM

M pLT =λ

LTy

2LT f

E λπλ ⋅=

The beam slenderness (λLT) is given by,

where,



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300

200

100

050 100 150 200 250

pb

N/mm2

λLT Fig 1. Bending strength for rolled sections of design

strength 275 N/mm2 according to BS 5950

Beam fails by yield

Beam buckling



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EM

PMLT =λ

Fig.2 Comparison of test data with theoretical elastic critical moments

0.4 0.8 1.20

0.4

1.0

0.8

stocky

intermediate

slender

ME / MP

Plastic 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 <

< 1.2

LTλ

LTλ

LTλ



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- 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



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•In the absence of instability, eqn. 11 may be adopted for the full plastic moment capacity pb for λLT < 0.4 .

•This corresponds to λLT values of around 37 (for

steels having fy= 275 N/mm2) below which the lateral

instability is NOT of concern.



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For more slender beams, pb is a function of λLT which is

given by ,

yLT r

uv=λ

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.



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• Unequal flanged sections

For unequal flanged sections, the following equation is used for finding the buckling moment of resistance.

Mb= pb .Zp



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• 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 αL), the values of φ and its differentials are given by

Tq

α (1-α)



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For 0 ≤ z ≤ α ,

( )

−+−=a

zsinh

acosh

atanh

asinh

a

z1

GJ

aTq

α

α

αφ.

( )

−+−=′a

zcosh

acosh

atanh

asinh

1GJ

Tq

α

α

αφ



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For 0 ≤ z ≤ α ,

a

zsinh

acosh

atanh

asinh

aJG

Tq

−=′′

α

α

φ

a

zcosh

acosh

atanh

asinh

aJG

Tq

−=′′′

α

α

φ2

Similar equations are available for different loading cases and for different values of α.



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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.



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