LRFD - Steel Design

1. LRFD-Steel Design Dr. Ali I. Tayeh First Semester

2. Steel DesignDr.Ali I. Tayeh Chapter 6-A

3. Beam-Columns Definition • Most beams and columns are subjected to some degree of both bending and axial load • Consider the rigid frame in behind Figure. • For the given loading condition, the horizontal member AB must not only support the vertical uniform load but must also assist the vertical members in resisting the concentrated lateral load P1 , Member CD is a more critical case, because it must resist the load P1 + P2without any assistance from the vertical members.

4. Beam-Columns Definition The reasonis that the x-bracing,indicated by dashed lines, prevents side sway in the lower story. For the direction of P2 shown, member ED will be in tension and member CF will be slack, provided that the bracing elements have been designed toresist only tension. Forthis condition to occur, however, member CD must transmit the load P1+ P2 from C to D. The vertical members of this frame must also be treated as beam-columns. In the upper story, members AC and BD will bend under the influence of P1 .

5. Beam-Columns Interaction formulas The inequalityof equationcan be writtenas : If both bendingand axial compressionare acting , the interaction formula would be where For biaxial bending, there will be two bending ratios:

6. Beam-Columns Interaction formulas Two formulas are given in the Specification: one for small axial load and one for large axial load. If the axial load is small, the axial load term is reduced. For large axial load, the bending term is slightly reduced.

7. Beam-Columns Example 6.1:

8. Beam-Columns Example:

9. Beam-Columns Example:

10. Beam-Columns MOMENT AMPLIFICATION The presence of the axial load produces secondary moments, and unless the axial load is relatively small, these additional moments must be accounted for. For an explanation, see the next Figure.. which shows a beam-column with an axial load and a transverse uniform load. At an arbitrary point 0, there is a bending moment caused by the uniform load and an additional moment Py, caused by the axial load acting at an eccentricity from the longitudinal axis of the member.

11. Beam-Columns MOMENT AMPLIFICATION This secondary moment is largest where the deflection is largest - in this case, at the centerline, where the total moment is wL2/8 + PD. Of course, the additional moment causes an additional deflection over and above that resulting from the transverse load. Iterative numerical techniques, called second-order methods, can be used to find the deflections and secondary moments, but these methods are impractical for manual calculations and are usually implemented with a computer program. This method entails computing the maximum bending moment resulting from flexural loading (transverse loads or member end moments) by a first order analysis, then multiplying by a moment amplification factor to account for the secondary moment.

12. Beam-Columns MOMENT AMPLIFICATION The next figure shows a simply supported member with an axial load and an initial out-of-straightness. This initial crookedness can be approximated by where e is the maximum initial displacement, occurring at mid-span. For the coordinate system shown. the moment-curvature relationship can be written as

13. Beam-Columns MOMENT AMPLIFICATION

14. Beam-Columns MOMENT AMPLIFICATION

15. Beam-Columns MOMENT AMPLIFICATION

16. Beam-Columns MOMENT AMPLIFICATION • Example 6.2:

17. Beam-Columns MOMENT AMPLIFICATION

18. Beam-Columns WEB LOCAL BUCKLING IN BEAM-COLUMNS

19. Beam-Columns WEB LOCAL BUCKLING IN BEAM-COLUMNS

20. Beam-Columns WEB LOCAL BUCKLING IN BEAM-COLUMNS Example 6.3 :

21. Beam-Columns WEB LOCAL BUCKLING IN BEAM-COLUMNS Example 6.3 :

22. Beam-Columns • The web will be compact when

23. Beam-Columns BRACED VERSUS UNBRACED FRAMES • Two amplification factors are used in LRFD: • one to account for amplification resulting from the member deflection and one to account for the effect of sway when the member is part of an ubraced frame. • the member is restrained against sideway, and the maximum secondary moment is Po , which is added to the maximum moment within the member. If the frame is actually unbraced, there is an additional component of the secondary moment-as shown in the next figure- that is caused by sideway. This secondary moment has a maximum value of P, which represents an amplification of the end moment.

24. Beam-Columns BRACED VERSUS UNBRACED FRAMES • The amplified moment to be used in design is computed from the factored loads and moments as follows Where:

25. MEMBERS IN BRACED FRAMES • Two amplification factors are used in LRFD: • The amplification factor given by Expression 6.4 was derived for a member braced against sides way- that is, one whose ends cannot translate with respect to cad other. Next figure shows a member of this type subjected to equal end moments producing single-curvature bending(bending that produces tension or compression on one side throughout the length of the member). Maximum moment amplification occurs at the center, where the def1ection is largest. For equal end moments, the moment is constant throughout the length of the member, so the maximum primary moment also occurs at the center. Thus the maximum secondary moment and maximum primary moment are additive.

26. Beam-Columns MEMBERS IN BRACED FRAMES

27. Beam-Columns MEMBERS IN BRACED FRAMES

28. Beam-Columns MEMBERS IN BRACED FRAMES The maximum moment in a beam-column therefore depends on distribution of bending moment within the member. This distribution is accounted for by a factor Cm , • The amplification factor below was derived for the worst case so Cm will never be greater than 1.0 • Amplification factor is : where

29. Beam-Columns Evaluation of Cm • The factor Cmapplies only to the braced condition. There are two categories of members; those with transverse loads applied between the ends and those with no transverse loads. Next Figure illustrate these two cases (member AB is the beam-column under consideration).

30. Beam-Columns Evaluation of Cm

31. Beam-Columns Evaluation of Cm

32. Beam-Columns Evaluation of Cm

33. Beam-Columns Evaluation of Cm Example 6.4: Solution:

34. Beam-Columns

35. Beam-Columns Evaluation of Cm

36. Beam-Columns Evaluation of Cm

37. Beam-Columns Evaluation of Cm

38. Beam-Columns MEMBERS IN UNBRACED FRAMES • In a beam column whose ends are free to translate, the maximum primary moment resulting from the side sway is almost always at one end. As was illustrated in the next Figure the maximum secondary moment from the sides way is always at the end. As a consequence of this condition, the maximum primary and secondary moments are usually additive and there is no need for the factor Cm; in effect, Cm = 1.0. • The amplification factor for the sides way moments, B2, is given by two equations. • Either may be used; the choice is usually one of convenience: OR

39. Beam-Columns Evaluation of Cm • The summations for Pu and Pe2apply to all columns that are in the same story as the column under consideration. The rationale for using the summations is that B2

40. Beam-Columns -Steel Design End