Slender Columns and Two-way Slabs

1. Slender Columns and Two-way Slabs

2. Lecture Goals • Slender Column Design • One-way and two-way slab • Slab thickness, h

3. Design of Long Columns- Example A rectangular braced column of a multistory frame building has floor height lu =25 ft. It is subjected to service dead-load moments M2= 3500 k-in. on top and M1=2500 k-in. at the bottom. The service live load moments are 80% of the dead-load moments. The column carries a service axial dead-load PD = 200 k and a service axial live-load PL = 350 k. Design the cross section size and reinforcement for this column. Given YA = 1.3 and YB = 0.9. Use a d’=2.5 in. cover with an sustain load = 50 % and fc = 7 ksi and fy = 60 ksi.

4. Design of Long Columns- Example Compute the factored loads and moments are 80% of the dead loads

5. Design of Long Columns- Example Compute the k value for the braced compression members Therefore, use k = 0.81

6. Design of Long Columns- Example Check to see if slenderness is going to matter. An initial estimate of the size of the column will be an inch for every foot of height. So h = 25 in. We need to be concerned with slender columns

7. Design of Long Columns- Example So slenderness must be considered. Since frame has no side sway, M2 = M2ns, ds = 0 Calculate the minimum M2 for the ratio computations.

8. Design of Long Columns- Example Compute components of concrete The moment of inertia of the column is

9. Design of Long Columns- Example Compute the stiffness, EI

10. Design of Long Columns- Example The critical load (buckling), Pcr, is

11. Design of Long Columns- Example Compute the coefficient, Cm, for the magnification d coefficient

12. Design of Long Columns- Example The magnification factor

13. Design of Long Columns- Example The design moment is Therefore, the design conditions are

14. Design of Long Columns- Example Assume that the r = 2.0 % or 0.020 Use 14 # 9 bars or 14 in2

15. Design of Long Columns- Example The column is compression controlled so c/d > 0.6. Check the values for c/d = 0.6

16. Design of Long Columns- Example Check the strain in the tension steel and compression steel.

17. Design of Long Columns- Example The tension steel is

18. Design of Long Columns- Example Combined forces are

19. Design of Long Columns- Example Combined force is

20. Design of Long Columns- Example Moment is

21. Design of Long Columns- Example The eccentricity is Since the e = 11.28 in. < 13.62 in. The section is in the compression controlled region f = 0.65. You will want to match up the eccentricity with the design.

22. Design of Long Columns- Example We need to match up the eccentricity of the problem. This done varying the c/d ratio to get the eccentricity to match. Check the values for c/d = 0.66

23. Design of Long Columns- Example Check the strain in the tension steel and compression steel.

24. Design of Long Columns- Example The tension steel is

25. Design of Long Columns- Example Combined forces are

26. Design of Long Columns- Example Combined force is

27. Design of Long Columns- Example Moment is

28. Design of Long Columns- Example The eccentricity is Since the e 11.28 in. The reduction factor is equal to f = 0.65. Compute the design load and moment.

29. Design of Long Columns- Example The design conditions are The problem matches the selection of the column.

30. Design of Long Columns- Example Design the ties for the column Provide #3 ties, spacing will be the minimum of: Therefore, provide #3 ties @ 18 in. spacing.

31. Determine eccentricity. Estimate column size required base on axial load. Determine e/h and required fPn/Ag, fMn/(Agh) Determine which chart to use from fc, fy and g. Determine r from the chart. Select steel sizes. Check values. Design ties by ACI code Design sketch Using Interaction Diagrams

32. Two-way Slabs

33. Comparison of One-way and Two-way slab behavior One-way slabs carry load in one direction. Two-way slabs carry load in two directions.

34. Comparison of One-way and Two-way slab behavior One-way and two-way slab action carry load in two directions. One-way slabs: Generally, long side/short side > 1.5

35. Comparison of One-way and Two-way slab behavior Two-way slab with beams Flat slab

36. Comparison of One-way and Two-way slab behavior For flat plates and slabs the column connections can vary between:

37. Comparison of One-way and Two-way slab behavior Flat Plate Waffle slab

38. Comparison of One-way and Two-way slab behavior The two-way ribbed slab and waffled slab system: General thickness of the slab is 2 to 4 in.

39. Comparison of One-way and Two-way slab behavior Economic Choices • Flat Plate suitable span 20 to 25 ft with LL= 60 -100 psf Advantages • Low cost formwork • Exposed flat ceilings • Fast Disadvantages • Low shear capacity • Low Stiffness (notable deflection)

40. Comparison of One-way and Two-way slab behavior Economic Choices • Flat Slab suitable span 20 to 30 ft with LL= 80 -150 psf Advantages • Low cost formwork • Exposed flat ceilings • Fast Disadvantages • Need more formwork for capital and panels

41. Comparison of One-way and Two-way slab behavior Economic Choices • Waffle Slab suitable span 30 to 48 ft with LL= 80 -150 psf Advantages • Carries heavy loads • Attractive exposed ceilings • Fast Disadvantages • Formwork with panels is expensive

42. Comparison of One-way and Two-way slab behavior Economic Choices • One-way Slab on beams suitable span 10 to 20 ft with LL= 60-100 psf • Can be used for larger spans with relatively higher cost and higher deflections • One-way joist floor system is suitable span 20 to 30 ft with LL= 80-120 psf • Deep ribs, the concrete and steel quantities are relative low • Expensive formwork expected.

43. Comparison of One-way and Two-way slab behavior ws =load taken by short direction wl = load taken by long direction dA = dB Rule of Thumb: For B/A > 2, design as one-way slab

44. Two-Way Slab Design Static Equilibrium of Two-Way Slabs Analogy of two-way slab to plank and beam floor Section A-A: Moment per ft width in planks Total Moment

45. Two-Way Slab Design Static Equilibrium of Two-Way Slabs Analogy of two-way slab to plank and beam floor Uniform load on each beam Moment in one beam (Sec: B-B)

46. Two-Way Slab Design Static Equilibrium of Two-Way Slabs Total Moment in both beams Full load was transferred east-west by the planks and then was transferred north-south by the beams; The same is true for a two-way slab or any other floor system.

47. General Design Concepts (1) Direct Design Method (DDM) Limited to slab systems to uniformly distributed loads and supported on equally spaced columns. Method uses a set of coefficients to determine the design moment at critical sections. Two-way slab system that do not meet the limitations of the ACI Code 13.6.1 must be analyzed more accurate procedures

48. General Design Concepts (2) Equivalent Frame Method (EFM) A three-dimensional building is divided into a series of two-dimensional equivalent frames by cutting the building along lines midway between columns. The resulting frames are considered separately in the longitudinal and transverse directions of the building and treated floor by floor.

49. Equivalent Frame Method (EFM) Transverse equivalent frame Longitudinal equivalent frame

50. Equivalent Frame Method (EFM) Perspective view Elevation of the frame