2. Reinforced Concrete Design-I Design of Axial members
By Dr. Attaullah Shah Swedish College of Engineering and Technology Wah Cantt.

7. Analysis and Design of “Short” Columns
General Information
Column:
Vertical Structural members
Transmits axial compressive loads with or without moment
transmit loads from the floor & roof to the foundation

8. Analysis and Design of “Short” Columns
General Information
Column Types:
Tied
Spiral
Composite
Combination
Steel pipe

9. Analysis and Design of “Short” Columns
Tied Columns % of all columns in buildings are tied
Tie spacing h (except for seismic)
tie support long bars (reduce buckling)
ties provide negligible restraint to lateral expose of core

10. Analysis and Design of “Short” Columns
Spiral Columns
Pitch = in. to in.
spiral restrains lateral (Poisson’s effect)
axial load delays failure (ductile)

11. Analysis and Design of “Short” Columns
Elastic Behavior
An elastic analysis using the transformed section method would be:
For concentrated load, P
uniform stress over section
n = Es / Ec
Ac = concrete area
As = steel area

12. Analysis and Design of “Short” Columns
Elastic Behavior
The change in concrete strain with respect to time will effect the concrete and steel stresses as follows:
Concrete stress
Steel stress

13. Analysis and Design of “Short” Columns
Elastic Behavior
An elastic analysis does not work, because creep and shrinkage affect the acting concrete compression strain as follows:

14. Analysis and Design of “Short” Columns
Elastic Behavior
Concrete creeps and shrinks, therefore we can not calculate the stresses in the steel and concrete due to “acting” loads using an elastic analysis.

15. Analysis and Design of “Short” Columns
Elastic Behavior
Therefore, we are not able to calculate the real stresses in the reinforced concrete column under acting loads over time. As a result, an “allowable stress” design procedure using an elastic analysis was found to be unacceptable. Reinforced concrete columns have been designed by a “strength” method since the 1940’s.
Note:
Creep and shrinkage do not affect the strength of the member.

16. Behavior, Nominal Capacity and Design under Concentric Axial loads
Initial Behavior up to Nominal Load - Tied and spiral columns. 1.

17. Behavior, Nominal Capacity and Design under Concentric Axial loads

18. Behavior, Nominal Capacity and Design under Concentric Axial loads
Let
Ag = Gross Area = b*h Ast = area of long steel fc = concrete compressive strength fy = steel yield strength
Factor due to less than ideal consolidation and curing conditions for column as compared to a cylinder. It is not related to Whitney’s stress block.

19. Behavior, Nominal Capacity and Design under Concentric Axial loads2.
Maximum Nominal Capacity for Design Pn (max)
r = Reduction factor to account for accidents/bending
r = ( tied )
r = ( spiral )
ACI

20. Behavior, Nominal Capacity and Design under Concentric Axial loads3.
Reinforcement Requirements (Longitudinal Steel Ast)
Let
- ACI Code requires

21. Behavior, Nominal Capacity and Design under Concentric Axial loads3.
Reinforcement Requirements (Longitudinal Steel Ast)
- Minimum # of Bars ACI Code
min. of 6 bars in circular arrangement w/min. spiral reinforcement.
min. of 4 bars in rectangular arrangement
min. of 3 bars in triangular ties

22. Behavior, Nominal Capacity and Design under Concentric Axial loads3.
Reinforcement Requirements (Lateral Ties)
ACI Code
size
# 3 bar if longitudinal bar # 10 bar # 4 bar if longitudinal bar # 11 bar # 4 bar if longitudinal bars are bundled

23. Behavior, Nominal Capacity and Design under Concentric Axial loads3.
Reinforcement Requirements (Lateral Ties)
Vertical spacing: (ACI )
16 db ( db for longitudinal bars ) db ( db for tie bar ) least lateral dimension of column
s s s

24. Behavior, Nominal Capacity and Design under Concentric Axial loads3.
Reinforcement Requirements (Lateral Ties)
Arrangement Vertical spacing: (ACI )
At least every other longitudinal bar shall have lateral support from the corner of a tie with an included angle o.
No longitudinal bar shall be more than 6 in. clear on either side from “support” bar.
1.)
2.)

25. Behavior, Nominal Capacity and Design under Concentric Axial loads
Examples of lateral ties.

26. Behavior, Nominal Capacity and Design under Concentric Axial loads
Reinforcement Requirements (Spirals )
ACI Code
size
3/8 “ dia. (3/8” f smooth bar, #3 bar dll or wll wire)
clear spacing between spirals
1 in.
3 in.
ACI

27. Behavior, Nominal Capacity and Design under Concentric Axial loads
Reinforcement Requirements (Spiral)
Spiral Reinforcement Ratio, rs

28. Behavior, Nominal Capacity and Design under Concentric Axial loads
Reinforcement Requirements (Spiral)
ACI Eqn. 10-5
where

29. Behavior, Nominal Capacity and Design under Concentric Axial loads4.
Design for Concentric Axial Loads
(a) Load Combination
Gravity:
Gravity + Wind:
and
etc.
Check for tension

30. Behavior, Nominal Capacity and Design under Concentric Axial loads4.
Design for Concentric Axial Loads
(b) General Strength Requirement
where,
f = 0.65 for tied columns
f = 0.7 for spiral columns

31. Behavior, Nominal Capacity and Design under Concentric Axial loads4.
Design for Concentric Axial Loads
(c) Expression for Design
defined:

32. Behavior, Nominal Capacity and Design under Concentric Axial loads

33. Behavior, Nominal Capacity and Design under Concentric Axial loads
* when rg is known or assumed:
* when Ag is known or assumed:

34. Example: Design Tied Column for Concentric Axial Load
Pdl = 150 k; Pll = 300 k; Pw = 50 k
fc = 4500 psi fy = 60 ksi
Design a square column aim for rg = Select longitudinal transverse reinforcement.

35. Example: Design Tied Column for Concentric Axial Load
Determine the loading
Check the compression or tension in the column

36. Example: Design Tied Column for Concentric Axial Load
For a square column r = 0.80 and f = 0.65 and r = 0.03

37. Example: Design Tied Column for Concentric Axial Load
For a square column, As=rAg= 0.03(15.2 in.)2 =6.93 in2
Use 8 #8 bars Ast = 8(0.79 in2) = 6.32 in2

38. Example: Design Tied Column for Concentric Axial Load
Check P0

39. Example: Design Tied Column for Concentric Axial Load
Use #3 ties compute the spacing
< 6 in. No cross-ties needed

40. Example: Design Tied Column for Concentric Axial Load
Stirrup design
Use #3 stirrups with 16 in. spacing in the column

41. Behavior under Combined Bending and Axial Loads
Usually moment is represented by axial load times eccentricity, i.e.

42. Behavior under Combined Bending and Axial Loads
Interaction Diagram Between Axial Load and Moment ( Failure Envelope )
Concrete crushes before steel yields
Steel yields before concrete crushes
Note:
Any combination of P and M outside the envelope will cause failure.

43. Behavior under Combined Bending and Axial Loads
Axial Load and Moment Interaction Diagram – General

44. Behavior under Combined Bending and Axial Loads
Resultant Forces action at Centroid
( h/2 in this case )
Moment about geometric center

45. Columns in Pure Tension
Section is completely cracked (no concrete axial capacity)
Uniform Strain

46. Columns Strength Reduction Factor, f (ACI Code 9.3.2) (a)
Axial tension, and axial tension with flexure. f = 0.9
Axial compression and axial compression with flexure.
(b)
Members with spiral reinforcement confirming to f = 0.70
Other reinforced members f = 0.65

47. Columns
Except for low values of axial compression, f may be increased as follows:
when and reinforcement is symmetric
and
ds = distance from extreme tension fiber to centroid of tension reinforcement.
Then f may be increased linearly to 0.9 as fPn decreases from 0.10fc Ag to zero.

48. Column

49. Columns Commentary: Other sections:
f may be increased linearly to 0.9 as the strain es increase in the tension steel. fPb

50. Design for Combined Bending and Axial Load (Short Column)
Design - select cross-section and reinforcement to resist axial load and moment.

51. Design for Combined Bending and Axial Load (Short Column)
Column Types 1)
Spiral Column - more efficient for e/h < 0.1, but forming and spiral expensive
Tied Column - Bars in four faces used when e/h < 0.2 and for biaxial bending 2)

52. General Procedure
The interaction diagram for a column is constructed using a series of values for Pn and Mn. The plot shows the outside envelope of the problem.

53. General Procedure for Construction of ID
Compute P0 and determine maximum Pn in compression
Select a “c” value (multiple values)
Calculate the stress in the steel components.
Calculate the forces in the steel and concrete,Cc, Cs1 and Ts.
Determine Pn value.
Compute the Mn about the center.
Compute moment arm,e = Mn / Pn.

54. General Procedure for Construction of ID
Repeat with series of c values (10) to obtain a series of values.
Obtain the maximum tension value.
Plot Pn verse Mn.
Determine fPn and fMn.
Find the maximum compression level.
Find the f will vary linearly from 0.65 to 0.9 for the strain values
The tension component will be f = 0.9

55. Example: Axial Load vs. Moment Interaction Diagram
Consider an square column (20 in x 20 in.) with 8 #10 (r = ) and fc = 4 ksi and fy = 60 ksi. Draw the interaction diagram.

56. Example: Axial Load vs. Moment Interaction Diagram
Given 8 # 10 (1.27 in2) and fc = 4 ksi and fy = 60 ksi

57. Example: Axial Load vs. Moment Interaction Diagram
Given 8 # 10 (1.27 in2) and fc = 4 ksi and fy = 60 ksi
[ Point 1 ]

58. Example: Axial Load vs. Moment Interaction Diagram
Determine where the balance point, cb.

59. Example: Axial Load vs. Moment Interaction Diagram
Determine where the balance point, cb. Using similar triangles, where d = 20 in. – 2.5 in. = 17.5 in., one can find cb

60. Example: Axial Load vs. Moment Interaction Diagram
Determine the strain of the steel

61. Example: Axial Load vs. Moment Interaction Diagram
Determine the stress in the steel

62. Example: Axial Load vs. Moment Interaction Diagram
Compute the forces in the column

63. Example: Axial Load vs. Moment Interaction Diagram
Compute the forces in the column

64. Example: Axial Load vs. Moment Interaction Diagram
Compute the moment about the center

65. Example: Axial Load vs. Moment Interaction Diagram
A single point from interaction diagram, (585.6 k, k-ft). The eccentricity of the point is defined as
[ Point 2 ]

66. Example: Axial Load vs. Moment Interaction Diagram
Now select a series of additional points by selecting values of c. Select c = 17.5 in. Determine the strain of the steel. (c is at the location of the tension steel)

67. Example: Axial Load vs. Moment Interaction Diagram
Compute the forces in the column

68. Example: Axial Load vs. Moment Interaction Diagram
Compute the forces in the column

69. Example: Axial Load vs. Moment Interaction Diagram
Compute the moment about the center

70. Example: Axial Load vs. Moment Interaction Diagram
A single point from interaction diagram, (1314 k, k-ft). The eccentricity of the point is defined as
[ Point 3 ]

71. Example: Axial Load vs. Moment Interaction Diagram
Select c = 6 in. Determine the strain of the steel, c =6 in.

72. Example: Axial Load vs. Moment Interaction Diagram
Compute the forces in the column

73. Example: Axial Load vs. Moment Interaction Diagram
Compute the forces in the column

74. Example: Axial Load vs. Moment Interaction Diagram
Compute the moment about the center

75. Example: Axial Load Vs. Moment Interaction Diagram
A single point from interaction diagram, (151 k, 471 k-ft). The eccentricity of the point is defined as
[ Point 4 ]

76. Example: Axial Load vs. Moment Interaction Diagram
Select point of straight tension. The maximum tension in the column is
[ Point 5 ]

77. Example: Axial Load vs. Moment Interaction Diagram
Point c (in) Pn Mn e k
k 253 k-ft 2 in
k 351 k-ft in
k 500 k-ft in
k 556 k-ft in
k 531 k-ft in
k 471 k-ft in
8 ~ k 395 k-ft infinity
k k-ft

78. Example: Axial Load vs. Moment Interaction Diagram
Use a series of c values to obtain the Pn verses Mn.

79. Example: Axial Load vs. Moment Interaction Diagram
Max. compression
Location of the linearly varying f. Cb
Max. tension