By: Prof Dr. Akhtar Naeem Khan chairciv@nwfpuet.pk

1. Lecture 09: Compression Members By: Prof Dr. Akhtar Naeem Khan chairciv@nwfpuet.edu.pk

2. Effective length of columns in frames • Rotation of the ends of the columns in building frames is usually limited by the beams connecting to them, while compression members in trusses may have restricted end rotations because of other members connecting at the joints.

3. Effective length of columns in frames • KL is called effective length of column and K effective length factor.

4. Effective length of columns in frames

5. Effective length of columns in frames • So far, we have looked at the buckling strength of individual columns. These columns had various boundary conditions at the ends, but they were not connected to other members with moment (fix) connections. • The effective length factor K for the buckling of an individual column can be obtained for the appropriate end conditions from Table C-C2.1 of the AISC Manual

6. Effective length of columns in frames • However, when these individual columns are part of a frame, their ends are connected to other members (beams etc.). • These frames are sometimes braced and sometimes un braced.

7. Effective length of columns in frames • A Braced frame is one in which a sideway (joint translation) is prevented by means of bracing, shear walls, or lateral support from adjoining structure.

8. Effective length of columns in frames • A Un Braced does not have any bracing and must depend on stiffness of its own members and rotational rigidity of joints between frame members to prevent lateral buckling.

9. Effective length of columns in frames Conclusions • Effective length coefficient increases with decreasing stiffness of the beam and becomes unity with zero stiffness. • Critical loads for a column depends on: • Its stiffness relative to that of beams framing into it and • Presence or absence of restraint to lateral displacement of its ends.

10. Effective length of columns in frames Braced and Un-braced Frames

11. Effective length of columns in frames Braced and Un-braced Frames • Similarly you can analyze multi bay, multistory frames. • Assumptions • Subjected to vertical loads only • All columns become unstable simultaneously • All joint rotations at floor are equal • Restraining moment distributed in proportion to stiffness.

12. Effective length of columns in frames Method of Analysis • First, you have to determine whether the column is part of a braced frame or an unbraced (moment resisting) frame. • Then, you have to determine the relative rigidity factor G for both ends of the column

13. Effective length of columns in frames Method of Analysis • G is defined as the ratio of the summation of the rigidity (EI/L) of all columns coming together at an end to the summation of the rigidity (EI/L) of all beams coming together at the same end. • G = • It must be calculated for both ends of the column.

14. Effective length of columns in frames Method of Analysis • Then, you can determine the effective length factor K for the column using the calculated value of G at both ends, i.e., GA and GB and the appropriate alignment chart. • There are two alignment charts provided by the AISC manual.

15. Effective length of columns in frames Method of Analysis • One is for columns in braced (side sway inhibited) frames. See Figure C-C2.2a on page 16.1-191 of the AISC manual. 0 < K ≤ 1 • The second is for columns in un-braced (side sway uninhibited) frames. See Figure C-C2.2b on page 16.1-192 of the AISC manual. 1 < K ≤ ∞

16. Effective length of columns in frames Alignment Chart

17. Effective length of columns in frames Method of Analysis: Inelastic Case • G is a measure of the relative flexural rigidity of the columns (EIc/Lc) with respect to the beams (EIb/Lb) • However, if column buckling were to occur in the inelastic range (lc < 1.5), then the flexural rigidity of the column will be reduced because Ic will be the moment of inertia of only the elastic core of the entire cross-section.

18. Effective length of columns in frames Method of Analysis: Inelastic Case

19. Effective length of columns in frames Method of Analysis: Inelastic Case • The beams will have greater flexural rigidity when compared with the reduced rigidity (EI­c) of the inelastic columns. As a result, the beams will be able to restrain the columns better, which is good for column design.

20. Effective length of columns in frames Method of Analysis: Inelastic Case The ratio Fcr/ Fe is called Stiffness reduction factor

21. Effective length of columns in frames Method of Analysis: Inelastic Case The ratio Fcr/ Fe is called Stiffness reduction factor

22. Procedure for Column Design • Design Load • Assume Fcr • ØPn = ØAg Fcr = Pu • Find Ag • Select a section • Find • Find lc =

23. Procedure for Column Design For lc ≤ 1.5 Fcr = Fy For lc > 1.5 Fcr = Fy • Fcr Calculated > Fcr Assumed • ØP > Pu…………………. Check

24. Procedure for Column Design Using Design Aids • LRFD mannual contains variety of Design aids, helpful in making original trial section. • Design load • Find section for Corresponding P & KL using table 4-21 • Calculate an equivalent (KL)eq =

25. Procedure for Column Design Using Design Aids • Use the calculated (KL)eq value to find (ØcPn) the column strength. • Select section and its properties • Find lc • Find Fcr • Find ØP

26. Problem 4-11-1

27. Problem 4-11-1

28. Problem 4-11-1

29. Example Problem 01 ASD

30. Example Problem 01 ASD

31. Example Problem 01 ASD

32. Example Problem 01 LRFD

33. Example Problem 01 LRFD

34. Example Problem 01 LRFD

35. Example Problem 01 LRFD

36. Example Problem 02 LRFD

37. Example Problem 02 LRFD

38. Example Problem 02 LRFD

39. Example Problem 02 LRFD

40. Example Problem 02 LRFD

41. Example Problem 02 LRFD

42. Column Bases

43. Column Bases

44. Example ASD

45. Example ASD

46. Example ASD

47. Example ASD

48. Example LRFD

49. Example LRFD

50. Example LRFD