Shear and Diagonal Tension

1. Shear and Diagonal Tension

2. Acknowledgement This Powerpoint presentation was prepared by Dr. Terry Weigel, University of Louisville. This work and other contributions to the text by Dr. Weigel are gratefully acknowledged.

3. Shear Stresses in Concrete Beams • Flexural stress • Shear stress • Principal angle • Diagonal tension – Mohr’s circle • Principal stresses

4. Shear Strength of Concrete • ACI Code Equation 11-3 – conservative but easy to use • ACI Code Equation 11-5 – less conservative but “difficult” to use

5. Shear Strength of Concrete Equation 11-3 Equation 11-5

6. Shear Cracking of Reinforced Beams • Flexural shear crack – initiate from top of flexural crack • For flexural shear cracks to occur, moment must be larger than the cracking moment and shear must be relatively large • Flexural shear cracks oriented at angle of approximately 45 degrees to the longitudinal beam axis

7. Flexural-shear Cracks

8. Shear Cracking of Reinforced Beams • Web shear crack – form independently • Typically occur at points of small moment and large shear • Occur at ends of beams at simple supports and at inflection points at continuous beam

9. Web-shear Cracks

10. Web Reinforcement • Stirrups • Hangers • At a location, the width of a diagonal crack is related to the strain in the stirrup – larger strain = wider crack • To reduce crack width, stirrup yield stress is limited to 60 ksi

11. Web Reinforcement • Small crack widths promote aggregate interlock • Limiting stirrup yield stress also reduces anchorage problems

12. Types of Stirrups

13. Types of Stirrups

14. Types of Stirrups

15. Types of Stirrups

16. Types of Stirrups

17. Types of Stirrups

18. Types of Stirrups

19. Types of Stirrups

20. Types of Stirrups

21. Behavior of Beams with Web Reinforcement • Truss analogy • Concrete in compression is top chord • Longitudinal tension steel is bottom chord • Stirrups form truss verticals • Concrete between diagonal cracks form the truss diagonals

22. Truss Analogy

23. Required Web Reinforcement – ACI Code • Required for all flexural members except: • Footings and solid slabs • Certain hollow core units • Concrete floor joists • Shallow beams with h not larger than 10”

24. Required Web Reinforcement – ACI Code • Required for all flexural members except: • Beams built integrally with slabs and h less than 24 in. and h not greater than the larger of 2.5 times the flange thickness or one-half the web width • Beams constructed with steel fiber- reinforced concrete with strength not exceeding 6,000 psi and

25. Stirrups • Diagonally inclined stirrups more efficient than vertical stirrups • Not practical • Bent-up flexural bars can be used instead

26. Bent-up Bar Web Reinforcement

27. Shear Cracking • The presence of stirrups does not materially effect the onset of shear cracking • Stirrups resist shear only after cracks have occurred • After cracks occurs, the beam must have sufficient shear reinforcement to resist the load not resisted by the concrete in shear

28. Benefits of Stirrups • Stirrups carry shear across the crack directly • Promote aggregate interlock • Confine the core of the concrete in the beam thereby increasing strength and ductility • Confine the longitudinal bars and prevent cover from prying off the beam • Hold the pieces on concrete on either side of the crack together and prevent the crack from propagating into the compression region

29. Design for Shear • Stirrups crossing a crack are assumed to have yielded • Shear crack forms at a 45 degree angle

30. Design for Shear • ACI Code Equation 11-15 • ACI Code • Equation 11-16

31. Design for Shear

32. ACI Code Requirements for Shear • ACI Code Section 11.4.6.1 – if Vu exceeds one-half fVc, stirrups are required • When shear reinforcement is required, ACI Code Section 11.4.6.3 specifies a minimum amount:

33. ACI Code Requirements for Shear • To insure that every diagonal crack is crossed by at least one stirrup, the maximum spacing of stirrups is the smaller of d/2 or 24 in. • If • maximum • spacings • are halved. See ACI Code Section 11.4.5.3

34. ACI Code Requirements for Shear • See ACI Code Section 11.4.7.9 • ACI Code Section 11.1.2 -> • Stirrups should extend as close as cover requirements permit to the tension and compression faces of the member - anchorage

35. ACI Code Requirements for Shear • Stirrup hook requirements shown on the next slide. See ACI Code Section 8.1 and 12.13 • In deep beams ( l /d < 4), large shear may affect flexural capacity • In most cases, beam can be designed for shear at a distance d from the face of the support . See next three slides for exceptions.

36. End Shear Reduction Not Permitted

37. End Shear Reduction Not Permitted

38. Corbels

39. ACI Code Requirements Stirrup Hooks

40. ACI Code Requirements Stirrup Hooks

41. ACI Code Requirements Stirrup Hooks

42. Shear Design Examples

43. Example 8.1 • Determine the minimum cross section required for a rectangular beam so that no shear reinforcement is required. Follow ACI Code requirements and use a concrete strength of 4,000 psi. Vu = 38 k.

44. Example 8.1 • Shear strength provided by concrete

45. Example 8.1 • ACI Code Section 11.4.6.1 requires: • Use a 24 in x 36 in beam (d = 33.5 in)

46. Example 8.2 • The beam shown in the figure was designed using a concrete strength of 3,000 psi and Grade 60 steel. Determine the theoretical spacing for No 3 U-shaped stirrups for the following values for shear: • (a) Vu = 12,000 lb • (b) Vu = 40,000 lb • (c) Vu = 60,000 lb • (d) Vu = 150,000 lb

47. Example 8.2

48. Example 8.2 • (a) Vu = 12,000 lb (use l = 1)

49. Example 8.2 • (b) Vu = 40,000 lb • Stirrups are needed since

50. Example 8.2 • (b) (con’t) Maximum spacing to provide minimum Av