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

1. Lecture 10: Beams By: Prof Dr. Akhtar Naeem Khan chairciv@nwfpuet.edu.pk

2. Beam • A beam is generally considered to be any member subjected principally to transverse gravity or vertical loading.

3. Beam

4. Beam

5. Types of Beams • Girders usually the most important beams. • Stringers Longitudinal bridge beams spanning between floor beams. • Floor Beams In buildings, a major beam usually supporting joists; a transverse beam in bridge floors.

6. Types of Beams

7. Types of Beams

8. Types of Beams • Joists A beam supporting floor construction but not major beams.

9. Types of Beams

10. Types of Beams

11. Types of Beams • Purlins Roof beam spanning between trusses.

12. Types of Beams • Girts Horizontal wall beams serving principally to resist bending due to wind on the side of an industrial building. • Lintels Member supporting a wall over a window or door opening.

13. Sections used for Beams • Among the steel shapes that are used as beam include: • W shapes, which normally prove to be the most economical beam sections and they have largely replaced channels and S sections for beam usage. • Channels are sometimes used for beams subjected to light loads, such as purlins and at places where clearances available require narrow flanges

14. Sections used for Beams

15. Design Approaches Elastic Design • For many years the elastic theory has been the bases for the design and analysis of steel structures. This theory is based on the yield stress of a steel structural element. • However, nowadays, it has been replaced by a more rational & realistic theory the ultimate stress design that is based on plastic capacity of a steel structure.

16. Design Approaches Elastic Design • In the elastic theory the maximum load that a structure could support is assumed to equal the load that cause a stress somewhere in the structure equal the yield stress of the Fy of the material. • The members were designed so that computed bending loads for service loads did not exceed the yield stress divided a factor of safety (e.g. 1.5 to 2)

17. Design Approaches Elastic Design Versus Ultimate Design • According to ASD, one FOS is used that accounts for the entire uncertainty in loads & strength. • According to LRFD(probability-based) different partial safety factors for different load and strength types are used.

18. Design Approach Elastic Design Versus Ultimate Design • Engineering structures have been designed for many years by the allowable stress design(ASD), or elastic design with satisfactory results. • However, engineers have long been aware that ductile members(e.g. steel) do not fail until a great deal of yielding occurs after yield stress is first reached. • This means that such members have great margin of safety against collapse than the elastic theory would seem to suggest.

19. Bending Behavior of Beams • Assumptions & Conditions • Strains are proportional to the distance from the neutral axis. • The stress-strain relationship is idealized to consist of two straight lines. • Deformations are sufficiently small so that • ø = tanø

20. Bending Behavior of Beams

21. Bending Behavior of Beams Rectangular Beam: Elastic Bending

22. Bending Behavior of Beams Bending Stresses • If the beam is subjected to some bending moment the stress at any point may be computed by usual flexural formula • It is important to remember that this expression is only applicable when the maximum computed stress in the beam is below elastic limit.

23. Bending Behavior of Beams Bending Stresses • The value of I/c is a constant for a particular section and is known as section modulus. • The flexural formula may then be written as

24. Bending Behavior of Beams Bending Stresses

25. Bending Behavior of Beams Internal Couple Method

26. Bending Behavior of Beams Internal Couple Method

27. Bending Behavior of Beams Plastic Moment • Stress varies linearly from neutral axis to extreme fibers. • When moment increases there will also be linear increase in moment and stress until yield. • When moment increases beyond yield moment the outer fiber will have the same stress but will yield. • The process will continue with more and more parts of the beam x-section stressed to yield point until finally a fully plastic distribution is approached.

28. Bending Behavior of Beams Plastic Moment

29. Bending Behavior of Beams Plastic Moment

30. Bending Behavior of Beams Plastic Moment

31. Bending Behavior of Beams Plastic Moment

32. Bending Behavior of Beams Plastic Moment

33. Bending Behavior of Beams Plastic Moment

34. Bending Behavior of Beams Plastic Moment Progression of Yield Zone Leading to Fully Plastic Hinge and Collapse • Stresses reach Yield Magnitude at extreme fibers • Yield Zones spreads towards Neutral axis • Yield Zones join, are now spread through entire x-section

35. Bending Behavior of Beams Plastic Hinges • The effect of plastic hinge is assumed to be concentrated at one section for analysis purpose. • However, it should be noted that this effect may extend for some distance along the beam.

36. Bending Behavior of Beams Plastic Moment

37. Bending Behavior of Beams Plastic Modulus • The resisting moment at full plasticity can be determined in a similar manner. • The result is the so-called plastic moment Mp.

38. Bending Behavior of Beams Plastic Modulus b d d/2 Fy

39. Bending Behavior of Beams Plastic Modulus • The plastic moment is equal to the yield stress Fy times the Plastic modulus Z. • From the foregoing expression for a rectangular section, the plastic modulus Z can be seen equal to bd2/4.

40. Bending Behavior of Beams Shape Factor • The shape factor which is equal to • So, for rectangular section the shape factor is equal to 1.5

41. Bending Behavior of Beams Shape Factor

42. Bending Behavior of Beams Shape Factor

43. Bending Behavior of Beams Neutral Axis for Plastic Condition • The neutral axis for plastic condition is different than its counterpart for elastic condition. • Unless the section is symmetrical, the neutral axis for the plastic condition will not be in the same location as for the elastic condition. • The total internal compression must equal the total internal tension.

44. Bending Behavior of Beams Neutral Axis for Plastic Condition • As all the fibers are considered to have the same stress Fy in the plastic condition, the area above and below the plastic neutral axis must be equal.

45. Bending Behavior of Beams Plastic Modulus

46. Bending Behavior of Beams Plastic Modulus: Unsymmetrical Shape • The areas above and below the neutral axis must be equal for Plastic analysis

47. Bending Behavior of Beams Plastic Modulus: Assignment • Determine the yield moment My, the Plastic Mp and the plastic modulus Z for the simply supported beam having the x-section as given. • Also calculate the shape factor. • Calculate nominal load Pn acting transversely through the mid span of the beam. Assume the Fy=36 ksi

48. Bending Behavior of Beams Advantages of Plastic Design

49. Bending Behavior of Beams Advantages of Plastic Design There is 50% increase in strength above the computed elastic limit (stage !) due to plasticization of the x-section

50. Bending Behavior of Beams Advantages of Plastic Design: Wide Flange Section My = Fy S Mp = Fy Z