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1. Lecture - Design of Columns
CEN 347

3. Introduction

16. Types of Column

19. Types of Column

20. Types of Column

22. Strength of short or Slender Columns

23. Strength of short or Slender Columns

24. Strength of short or Slender Columns

51. 0.70
0.65

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

57. Analysis and Design of “Short” Columns
General Information
Column:

58. Analysis and Design of “Short” Columns
Tie spacing h (except for seismic)
tie support long bars (reduce buckling)
ties provide negligible restraint to
lateral expose of core
Tie Columns - 95% of all columns in buildings are tied


59. Analysis and Design of “Short” Columns
Pitch = 1.375 in. to 3.375 in.
spiral restrains lateral (Poisson’s effect)
axial load delays failure (ductile)
Spiral Columns

60. 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.

61. Analysis and Design of “Short” Columns
Elastic Behavior
An elastic analysis using the transformed section
method would be:
st
c
c
nA
A
P
f


For concentrated load, P
uniform stress over
section
c
s nf
f 
area
steel
area
concrete
/
s
c
c
s



A
A
E
E
n

62. 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:

63. 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

64. 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.
Creep and shrinkage do not affect the strength
of the member.
Note:

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

66. Behavior, Nominal Capacity and Design
under concentric Axial loads
  st
y
st
g
c
0 *
85
.
0 A
f
A
A
f
P 


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.
Let
Ag = Gross Area = b*h Ast
= area of long steel fc
=concrete compressive strength fy =
steel yield strength

67. Behavior, Nominal Capacity and Design
under concentric Axial loads
Maximum Nominal Capacity for Design Pn (max)
2. 
  0
max
n rP
P 
r = Reduction factor to account for accidents/bending
r = 0.80 ( tied )
r = 0.85 ( spiral )

68. Behavior, Nominal Capacity and Design
under concentric Axial loads
Reinforcement Requirements (Longitudinal Steel Ast)
3.
g
st
g
A
A


- ACI Code 10.9.1 requires
Let
08
.
0
01
.
0 g 
 

69. Behavior, Nominal Capacity and Design
under concentric Axial loads
3.
- Minimum # of Bars ACI Code 10.9.2
min. of 6 bars in circular arrangement
w/min. spiral reinforcement.
min. of 4 bars in rectangular
arrangement
Reinforcement Requirements (Longitudinal Steel Ast)

70. Behavior, Nominal Capacity and Design
under concentric Axial loads
3.
ACI Code 7.10.5



Reinforcement Requirements (Lateral Ties)
# 3 bar if longitudinal bar # 10 bar
# 4 bar if longitudinal bar # 11 bar
# 4 bar if longitudinal bars are bundled


size

71. Behavior, Nominal Capacity and Design
under concentric Axial loads
3. Reinforcement Requirements (Lateral Ties)
16 db ( db for longitudinal bars )
48 db ( db for tie bar )
least lateral dimension of column

Vertical spacing:


s
s
s

72. Behavior, Nominal Capacity and Design
under concentric Axial loads
3. Reinforcement Requirements (Lateral Ties)
Vertical spacing: Arrangement,
At least every other longitudinal bar shall have
lateral support from the corner of a tie with an
included angle 135o.
No longitudinal bar shall be more than 6 in. clear on
either side from “support” bar.
1.)
2.)


73. Behavior, Nominal Capacity and Design
under concentric Axial loads
Examples of
lateral ties.

74. Behavior, Nominal Capacity and Design
under concentric Axial loads
ACI Code 7.10.4

Reinforcement Requirements (Spirals )
3/8 “ f (3/8” f smooth bar, #3 bar dll or
wll wire)
- size
- clear spacing:  1 in.
3 in.


75. Behavior, Nominal Capacity and Design
under concentric Axial loads
Reinforcement Requirements (Spiral)
s
D
A
c
sp
s
4
Core
of
Volume
Spiral
of
Volume



Spiral Reinforcement Ratio, s









s
D
D
A
4
1
:
from 2
c
c
sp
s




76. Behavior, Nominal Capacity and Design
under concentric Axial loads
Reinforcement Requirements (Spiral)







 










y
c
c
g
s *
1
f
f
A
A
 ACI Eqn. 10-6
 
psi
60,000
steel
spiral
of
strength
yield
center)
(center to
steel
spiral
of
tch
spacing(pi
spiral
of
edge
outside
to
edge
outside
:
diameter
core
4
area
core
ent
reinforcem
spiral
of
area
sectional
-
cross
y
c
2
c
c
sp







f
s
D
D
A
A

where

77. Behavior, Nominal Capacity and Design
under concentric Axial loads
4. Design for Concentric Axial Loads
(a) Load Combination
 
w
DL
u
w
LL
DL
u
LL
DL
u
3
.
1
9
.
0
7
.
1
7
.
1
4
.
1
*
75
.
0
7
.
1
4
.
1
P
P
P
P
P
P
P
P
P
P







Gravity:
Gravity + Wind:
and
Etc.

78. Behavior, Nominal Capacity and Design
under concentric Axial loads
4. Design for Concentric Axial Loads
(b) General Strength Requirement
u
n P
P 
f
f = 0.7 for tied columns
f = 0.75 for spiral columns
where,

79. Behavior, Nominal Capacity and Design
under concentric Axial loads
4. Design for Concentric Axial Loads
(c) Expression for Design
 
08
.
0
0.01
Code
ACI g
g
st
g 

 

A
A
defined:

80. Behavior, Nominal Capacity and Design
under concentric Axial loads
    u
c
y
st
c
g
n
steel
85
.
0
concrete
85
.
0 P
f
f
A
f
A
r
P 














 

 






f
f
or
 
  u
c
y
g
c
g
n 85
.
0
85
.
0 P
f
f
f
A
r
P 


 
f
f

81. Behavior, Nominal Capacity and Design
under concentric Axial loads
 
 
85
.
0
85
.
0 c
y
g
c
u
g
f
f
f
r
P
A




f
* when g is known or assumed:
 
 










 c
g
u
c
y
st 85
.
0
85
.
0
1
f
A
r
P
f
f
A
f
* when Ag is known or assumed:

82. Example: Design tied Column for
concentric Axial Load
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 g =0.03.
Select longitudinal transverse reinforcement.

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

86. Behavior under Combined Bending and
Axial Loads
Resultant Forces action at Centroid
( h/2 in this case )
s2
positive
is
n
compressio
c
s1
n T
C
C
P 







Moment about geometric center
























2
*
2
2
*
2
* 2
s2
c
1
s1
n
h
d
T
a
h
C
d
h
C
M

90. 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.

91. Behavior under Combined Bending and
Axial Loads
Axial Load and Moment Interaction Diagram -General

92. Example: Axial Load vs. Moment
Interaction Diagram