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 > 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 = in. < in. The section is in the compression controlled region f = 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 in. The reduction factor is equal to f = 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 18 in. spacing.

31. Using Interaction Diagrams
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

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= 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= 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= 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= 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= 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 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

51. Method of Analysis (1) Elastic Analysis
Concrete slab may be treated as an elastic plate. Use Timoshenko’s method of analyzing the structure. Finite element analysis

52. Method of Analysis (2) Plastic Analysis
The yield method used to determine the limit state of slab by considering the yield lines that occur in the slab as a collapse mechanism.
The strip method, where slab is divided into strips and the load on the slab is distributed in two orthogonal directions and the strips are analyzed as beams.
The optimal analysis presents methods for minimizing the reinforcement based on plastic analysis

53. Method of Analysis (3) Nonlinear analysis
Simulates the true load-deformation characteristics of a reinforced concrete slab with finite-element method takes into consideration of nonlinearities of the stress-strain relationship of the individual members.

54. Column and Middle Strips
The slab is broken up into column and middle strips for analysis

55. Minimum Slab Thickness for Two-way Construction
The ACI Code specifies a minimum slab thickness to control deflection. There are three empirical limitations for calculating the slab thickness (h), which are based on experimental research. If these limitations are not met, it will be necessary to compute deflection.

56. Minimum Slab Thickness for Two-way Construction
(a) For
fy in psi. But not less than 5 in.

57. Minimum Slab Thickness for Two-way Construction
(b) For
fy in psi. But not less than 3.5 in.

58. Minimum Slab Thickness for Two-way Construction
(c) For
Use the following table 9.5(c)

59. Minimum Slab Thickness for Two-way Construction
Slabs without interior beams spanning between supports and ratio of long span to short span < 2
See section For slabs with beams spanning between supports on all sides.

60. Minimum Slab Thickness for two-way construction
The definitions of the terms are:
h = Minimum slab thickness without interior beams
ln =
b =
am=
Clear span in the long direction measured face to face of column
the ratio of the long to short clear span
The average value of a for all beams on the sides of the panel.

61. Definition of Beam-to-Slab Stiffness Ratio, a
Accounts for stiffness effect of beams located along slab edge reduces deflections of panel adjacent to beams.

62. Definition of Beam-to-Slab Stiffness Ratio, a
With width bounded laterally by centerline of adjacent panels on each side of the beam.

63. Beam and Slab Sections for calculation of a

64. Beam and Slab Sections for calculation of a

65. Beam and Slab Sections for calculation of a
Definition of beam cross-section
Charts may be used to calculate a

66. Minimum Slab Thickness for Two-way Construction
Slabs without drop panels meeting and ,
tmin = 5 in
Slabs with drop panels meeting and ,
tmin = 4 in

67. Example - Slab
A flat plate floor system with panels 24 by 20 ft is supported on 20 in. square columns. Determine the minimum slab thickness required for the interior and corner panels. Use fc = 4 ksi and fy = 60 ksi

68. Example - Slab
Slab thickness, from table 9.5(c) for fy = 60 ksi and no edge beams

69. Example - Slab
Slab thickness, from table 9.5(c) for fy = 60 ksi and no edge beams for a = am = 0 (no beams)

70. Example – a Calculations
The floor system consists of solid slabs and beams in two directions supported on 20-in. square columns. Determine the minimum slab thickness, h, required for the floor system. Use fc = 4 ksi and fy = 60 ksi

71. Example – a Calculations
The cross-sections are:

72. Example – a Calculations
To find h, we need to find am therefore Ib, Islab and a for each beam and slab in long short direction. Assume slab thickness h = 7 in. so that x = y < 4 tf

73. Example – a Calculations
Compute the moment of inertia and centroid

74. Example – a Calculations
Compute the a coefficient for the long direction
Short side of the moment of inertia

75. Example – a Calculations
Compute the a coefficient for short direction
The average am for an interior panel is

76. Example – a Calculations
Compute the b coefficient
Compute the thickness for am > 2
Use slab thickness, 6.5 in. or 7 in.

77. Example – a Calculations
Compute the moment of inertia and centroid for the L-beam

78. Example – a Calculations
Compute the am coefficient for long direction
Short side of the moment of inertia

79. Example – a Calculations
Compute the am coefficient for the short direction

80. Example – a Calculations
Compute the am coefficient for the edges and corner

81. Example – a Calculations
Compute the am coefficient for the edges and corner

82. Example – a Calculations
Compute the largest length ln of the slab/beam, edge to first interior column.

83. Example – a Calculations
Compute the thickness of the slab with am > 2
The overall depth of the slab is 7 in.
Use slab thickness, 6.5 in. or 7 in.