1. © Teaching Resource in Design of Steel Structures – IIT Madras, SERC Madras, Anna Univ., INSDAG 1 BENDING AND TORSION

2. © Teaching Resource in Design of Steel Structures – IIT Madras, SERC Madras, Anna Univ., INSDAG 2 BENDING AND TORSION Introduction Designing for torsion in practice Pure torsion and warping Combined bending and torsion Design method for lateral torsional buckling Conclusion

3. © Teaching Resource in Design of Steel Structures – IIT Madras, SERC Madras, Anna Univ., INSDAG 3 INTRODUCTION –Torsional moments cause twisting and warping of the cross sections. –When torsional rigidity (GJ) is very large compared with its warping rigidity (E  ), 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.

4. © Teaching Resource in Design of Steel Structures – IIT Madras, SERC Madras, Anna Univ., INSDAG 4 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

5. © Teaching Resource in Design of Steel Structures – IIT Madras, SERC Madras, Anna Univ., INSDAG 5 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 (  ) over a length of z is given by When a member is in non-uniform torsion, the rate of change of angle of twist will vary along the length of the member

6. © Teaching Resource in Design of Steel Structures – IIT Madras, SERC Madras, Anna Univ., INSDAG 6 Pure Torsion and Warping - 2 The warping shear stress (  w ) at a point is given by, S wms = 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.  '' where W nwfs = Normalised warping function

7. © Teaching Resource in Design of Steel Structures – IIT Madras, SERC Madras, Anna Univ., INSDAG 7 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 be amplified by bending moment, inducing additional warping moments and torsional shears.

8. © Teaching Resource in Design of Steel Structures – IIT Madras, SERC Madras, Anna Univ., INSDAG 8 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 (f y /  m ) The "capacity check" for major axis bending  bx +  byt +  w  f y /  m.

9. © Teaching Resource in Design of Steel Structures – IIT Madras, SERC Madras, Anna Univ., INSDAG 9 Combined Bending and Torsion - 3 Buckling Check whenever lateral torsional buckling governs the design (i.e. when p b is less than f y ) the values of  w and  byt will be amplified., equivalent uniform moment = m x M x M b, the buckling resistance moment =

10. © Teaching Resource in Design of Steel Structures – IIT Madras, SERC Madras, Anna Univ., INSDAG 10 Combined Bending and Torsion - 4 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  bx +  byt +  w +  by  f y /  m

11. © Teaching Resource in Design of Steel Structures – IIT Madras, SERC Madras, Anna Univ., INSDAG 11 Combined Bending and Torsion - 5 Buckling Check

12. © Teaching Resource in Design of Steel Structures – IIT Madras, SERC Madras, Anna Univ., INSDAG 12 Combined Bending and Torsion - 6 Torsional Shear Stress Torsional shear stresses and warping shear stresses should also be amplified in a similar manner This shear stress should be added to the shear stresses due to bending in checking the adequacy of the section.

13. © Teaching Resource in Design of Steel Structures – IIT Madras, SERC Madras, Anna Univ., INSDAG 13 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 M E are too complex for routine use -there are limitations to their extension in the ultimate range

14. © Teaching Resource in Design of Steel Structures – IIT Madras, SERC Madras, Anna Univ., INSDAG 14 Design method for lateral torsional buckling - 2 A simple method of computing the buckling resistance of beams is as follows:- - the buckling resistance moment, M b, is obtained as the smaller root of the equation, (M E - M b ) (M p - M b ) =  LT. M E M b where

15. © Teaching Resource in Design of Steel Structures – IIT Madras, SERC Madras, Anna Univ., INSDAG 15 Design method for lateral torsional buckling - 3 M p = f y. Z p /  m  LT = Perry coefficient, similar to column buckling coefficient Z p = Plastic section modulus

16. © Teaching Resource in Design of Steel Structures – IIT Madras, SERC Madras, Anna Univ., INSDAG 16 Design method for lateral torsional buckling - 4 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 ( LT ) - the buckling resistance moment M b is given by M b = p b.Z p where p b = bending strength allowing for susceptibility to lateral torsional buckling. Z p = plastic section modulus.

17. © Teaching Resource in Design of Steel Structures – IIT Madras, SERC Madras, Anna Univ., INSDAG 17 Design method for lateral torsional buckling - 5 The beam slenderness ( LT ) is given by, where,

18. © Teaching Resource in Design of Steel Structures – IIT Madras, SERC Madras, Anna Univ., INSDAG 18 Design method for lateral torsional buckling - 6 300 200 100 0 50 100150 200 250 p b N/mm 2 LT Fig 1. Bending strength for rolled sections of design strength 275 N/mm 2 according to BS 5950 Beam fails by yield Beam buckling

19. © Teaching Resource in Design of Steel Structures – IIT Madras, SERC Madras, Anna Univ., INSDAG 19 Design method for lateral torsional buckling - 7 Fig.2 Comparison of test data with theoretical elastic critical moments 0.4 0.8 1.2 0 0.4 1.0 0.8 sto cky interm ediate slender M E / M P Plastic yieldM / M p

20. © Teaching Resource in Design of Steel Structures – IIT Madras, SERC Madras, Anna Univ., INSDAG 20 Design method for lateral torsional buckling - 8 In Fig. 2 three distinct regions of behaviour can be observed :- - stocky beams which are able to attain the plastic moment M p, for values of below about 0.4. - slender beams which fail at moments close to M E, for values of above about 1.2 - beams of intermediate slenderness which fail to reach either M p or M E. In this case 0.4 < < 1.2

21. © Teaching Resource in Design of Steel Structures – IIT Madras, SERC Madras, Anna Univ., INSDAG 21 Design method for lateral torsional buckling - 9 - 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

22. © Teaching Resource in Design of Steel Structures – IIT Madras, SERC Madras, Anna Univ., INSDAG 22 Design method for lateral torsional buckling - 10 In the absence of instability, eqn. 11 may be adopted for the full plastic moment capacity p b for LT < 0.4. This corresponds to LT values of around 37 (for steels having f y = 275 N/mm 2 ) below which the lateral instability is NOT of concern.

23. © Teaching Resource in Design of Steel Structures – IIT Madras, SERC Madras, Anna Univ., INSDAG 23 Design method for lateral torsional buckling - 11 For more slender beams, p b is a function of LT which is given by, 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.

24. © Teaching Resource in Design of Steel Structures – IIT Madras, SERC Madras, Anna Univ., INSDAG 24 Design method for lateral torsional buckling - 12 Unequal flanged sections For unequal flanged sections, the following equation is used for finding the buckling moment of resistance. M b = p b.Z p

25. © Teaching Resource in Design of Steel Structures – IIT Madras, SERC Madras, Anna Univ., INSDAG 25 Design method for lateral torsional buckling - 13 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  L), the values of  and its differentials are given by TqTq  (1-  )

26. © Teaching Resource in Design of Steel Structures – IIT Madras, SERC Madras, Anna Univ., INSDAG 26 Design method for lateral torsional buckling - 14 For 0  z  ,

27. © Teaching Resource in Design of Steel Structures – IIT Madras, SERC Madras, Anna Univ., INSDAG 27 Design method for lateral torsional buckling - 15 For 0  z  , Similar equations are available for different loading cases and for different values of .

28. © Teaching Resource in Design of Steel Structures – IIT Madras, SERC Madras, Anna Univ., INSDAG 28 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.

29. © Teaching Resource in Design of Steel Structures – IIT Madras, SERC Madras, Anna Univ., INSDAG 29 THANKYOU