Dr. Seshu Adluri. Structural Steel Design Compression Members

Transcription

1 Dr. Seshu Adluri Structural Steel Design Compression Members

2 Columns in Buildings

3 Columns in Buildings

4 Column supports

5 Compression members in trusses

6 Compression members in trusses

7 Compression members in OWSJ

8 Compression members in bridges Howrah bridge, Kolkata, India

9 Compression members in towers Eiffel Tower ( ) The new Tokyo Tower is set to be completed in It will stand 610m high.

10 Compression in equipment

11 Introduction Steel Compression members Building columns Frame Bracing Truss members (chords and bracing) Useful in pure compression as well as in beamcolumns Design Clauses: CAN/CSA-S16 Over-all strength as per Clause 13.3 Local buckling check: Clause 11 (Table 1) Built-up members: Clause 19

12 Column erection

13 Different column c/s shapes

14 Different column c/s shapes

15 Instability and bifurcation Stable, neutral and unstable equilibriums

16 Buckling

17 Instability and bifurcation Instability effect To compress or not to compress? Energy considerations Long column

18 Compression terminology -review Moment of inertia I x Parallel axis theorem Radius of gyration Effective length = A da kl Slenderness ratio kl/r Principal axes (major and minor) Critical Load P cr Factored compressive strength, C r y 2 I x = I x + r = I A 2 Ax h b Symmetric (major) axis Unsymmetric (minor) axis

19 Compression members Bucking Elastic (Euler) buckling Inelastic buckling Buckling modes Overall buckling Flexural buckling Torsional buckling Torsional-flexural buckling Local buckling

20 Elastic Buckling Equilibrium equation Internal moment + applied moment = 0 2 d w EI + Pw = 0; w = y = 0; w = 2 dx πx Solution : w = Asin satisfies the b. c. L Substituting int o the differential equation, y = L EI 2 π A L πx x sin π + P Asin L L = 0 P 2 π L cr EI + P 2 π EI = 2 L = 0

21 Inelastic Buckling

22 Compression members Moment of inertia Radius of gyration Effective length Slenderness ratio σ pl = σ y - κλ/ρ es σ pl = (0.5~1.0)σ y

23 Effective length factors Different end conditions give different lengths for equivalent half-sine wave

24 Theoretical Effective length factors

25 Theoretical Effective length factors

26 Effective length factors US practice

27 Effective lengths in different directions

28 Effective length factors Canadian practice k =.65 k =.8 k = 1.2 k = 1.0 k = 2.0 k = 2.0

29 US recommended values Free-Free Hinged-Free Guided-Free Boundary Conditions Hinged-Hinged (Simply-Supported) Guided-Hinged Guided-Guided Clamped-Free (Cantilever) Clamped-Hinged Clamped-Guided Theoretical Eff. Length, L eff T L L L 2 L 2 L L 2 L 0.7 L L Engrg. Eff. Length L eff E (1.2 L) (1.2 L) L (2.1 L) 2 L 1.2 L 2.1 L 0.8 L 1.2 L Clamped-Clamped 0.5 L 0.65 L

30 Canadian recommended values Appendix F CAN/CSA/S16-01 Free-Free Hinged-Free Guided-Free Boundary Conditions Hinged-Hinged (Simply-Supported) Guided-Hinged Guided-Guided Clamped-Free (Cantilever) Clamped-Hinged Clamped-Guided Theoretical Eff. Length, L eff T L L L 2 L 2 L L 2 L 0.7 L L Engrg. Eff. Length L eff E (1.2 L) (1.2 L) L (2.0 L) 2 L 1.2 L 2.0 L 0.8 L 1.2 L Clamped-Clamped 0.5 L 0.65 L

31 Effective lengths in frame columns

32 Effective lengths in frame columns

33 Real columns -Factors for consideration Partially plastic buckling Initial out-ofstraightness (L/2000 to L/1000)

34 Real columns - Factors for consideration Residual stresses in Hot-rolled shapes (idealized)

35 Real columns - Factors for consideration Residual stresses in Hot-rolled shapes (idealized)

36 Perfect column failure

37 Perfect column failure

38 Practical column failure

39 Column curve

40 Material Short Column (Strength Limit) Intermediate Column (Inelastic Stability Limit) Long Column (Elastic Stability Limit) Slenderness Ratio ( kl/r = L eff / r) Structural Steel kl/r < < kl/r < 150 kl/r > 150 Aluminum Alloy AA T6 kl/r < < kl/r < 66 kl/r > 66 Aluminum Alloy AA T6 kl/r < < kl/r < 55 kl/r > 55 Wood kl/r < < kl/r < (18~30) (18~30)<kL/r<50

41 Over-all buckling Flexural Torsional Torsional-flexural

42 Flexural Buckling About minor axis (with higher kl/r) for doubly symmetric shapes About minor axis (the unsymmetric axis) for singly symmetric shapes 1964 Alaska quake, EqIIS collection

43 Flexural Buckling

44 Torsional buckling Short lengths Usually kl/r less than approx. 50 doubly symmetric sections Wide flange sections, cruciform sections, double channels, point symmetric sections,. Not for closed sections such as HSS since they are very strong in torsion

45 Torsion Torque is a moment that causes twisting along the length of a bar. The twist is also the torsional deformation. For a circular shaft, the torque (or torsional moment) rotates each c/s relative to the nearby c/s.

46 Torsional deformation

47 Torsion of non-circular sections Torsion of non-circular sections involves torsional shear and warping. Torsional shear needs the use of torsion constant J. J is similar to the use of polar moment of inertia for circular shafts. J=Σbt 3 /3 Warping calculation needs the use od the constant C w. Both J and C w are listed in the Handbook In addition, we need to use the effective length in torsion (k z L z ). Usually, k z is taken as 1.0

48 Torsional buckling of open sections Buckling in pure torsional mode (not needed for HSS or closed sections): K z is normally taken as 1.0. C w, J, r x, r y are given in the properties tables, x and y are the axes of symmetry of the section. E= MPa (assumed), G= MPa (assumed). F ez λ = 2 1 π EC w = + GJ 2 2 Ar o ( KL z ) F y F r = x + y + r + r o o o x y ( ) 2n C 1 1 e r = φafy + λ n

49 Shear centre Sections always rotate about shear centre Shear centre lies on the axis of symmetry

50 Torsionalflexural buckling For of singly symmetric sections, about the major axis For unsymmetric sections, about any axis Rotation is always about shear centre

51 Torsional-flexural buckling

52 Shear flow

53 Shear flow

54 Shear flow

55 Shear centre

56 Shear flow effect

57 Shear centre

58 Shear centre

59 Local (Plate) buckling

60 Plate buckling

61 Plate buckling Effective width concept

62 Plate buckling Different types of buckling depending on b/t ratio end conditions for plate segments Table 1 for columns Table 2 for beams and beam-columns

63 Web buckling

64 Plate buckling b/t ratio effect

65 Built-up columns Two or more sections Stitch bolts Batten plates Lacing Combined batten & lacing Perforated cover plates

66 Built-up columns Two or more sections Stitch bolts Batten plates Lacing Combined

67 Built-up columns

68 Built-up columns Closely spaced channels

69 Built-up columns Built-up member buckling is somewhat similar to frame buckling Batten acts like beams Battens get shear and moment due to the bending of the frame like built-up member at the time of buckling

70 Battened column

71 Built-up columns Design as per normal procedure Moment of inertia about the axis which shifts due to the presence of gap needs parallel axis theorem Effective slenderness ratio as per Cl. 19.1

72 References AISC Digital Library (2008) ESDEP-the European Steel Design Education Programme - lectures Earthquake Image Information System Hibbeler, R.C., Mechanics of Solids, Prentice-Hall