Reinforced ColumnsReinforced Columns
Concrete Construction
Lecture GoalsLecture Goals
• Columns– Short Column Design– Long Column Design
Behavior under Combined Bending and Behavior under Combined Bending and Axial LoadsAxial Loads
Usually moment is represented by axial load times eccentricity, i.e.
Behavior under Combined Bending and Behavior under Combined Bending and Axial LoadsAxial Loads
Interaction Diagram Between Axial Load and Moment ( Failure Envelope )
Concrete crushes before steel yields
Steel yields before concrete crushes
Any combination of P and M outside the envelope will cause failure.
Note:
Behavior under Combined Bending and Behavior under Combined Bending and Axial LoadsAxial Loads
Axial Load and Moment Interaction Diagram -General
Behavior under Combined Bending and Behavior under Combined Bending and Axial LoadsAxial Loads
Resultant Forces action at Centroid
( h/2 in this case )s2
positive is ncompressio
cs1n TCCP
Moment about geometric center
2*
22*
2* 2s2c1s1n
hdT
ahCd
hCM
Columns in Pure TensionColumns in Pure Tension
Section is completely cracked (no concrete axial capacity)
Uniform Strain y
N
1iisytensionn AfP
ColumnsColumnsStrength Reduction Factor, (ACI Code 9.3.2)
Axial tension, and axial tension with flexure. = 0.9
Axial compression and axial compression with flexure.
Members with spiral reinforcement confirming to 10.9.3
Other reinforced members
(a)
(b)
ColumnsColumnsExcept for low values of axial compression, may be increased as follows:
when and reinforcement is symmetric
and
ds = distance from extreme tension fiber to centroid of tension reinforcement.
Then may be increased linearly to 0.9 as Pn decreases from 0.10fc Ag to zero.
psi 000,60y f
70.0s
h
ddh
ColumnColumn
ColumnsColumns
Commentary:
Other sections:
may be increased linearly to 0.9 as the strain s increase in the tension steel. Pb
Design for Combined Bending and Design for Combined Bending and Axial Load (short column)Axial Load (short column)
Design - select cross-section and reinforcement to resist axial load and moment.
Design for Combined Bending and Design for Combined Bending and Axial Load (short column)Axial Load (short column)
Column Types
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
1)
2)
General ProcedureGeneral 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.
General Procedure for construction General Procedure for construction of an interaction diagramof an interaction diagram
– Compute P0 and determine maximum Pn in compression
– Select a c value.• 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.
General Procedure for construction General Procedure for construction of an interaction diagramof an interaction diagram
– Repeat with series of c values (10) to obtain a series of values.
– Obtain the maximum tension value.– Plot Pn verse Mn.– Determine Pn and Mn.
• Find the maximum compression level.• Find the will vary linearly from 0.65 to 0.9 for the
strain values • The tension component will be = 0.9
Example: Axial Load Vs. Moment Example: Axial Load Vs. Moment Interaction DiagramInteraction Diagram
Consider an square column (20 in x 20 in.) with 8 #10 ( = 0.0254) and fc = 4 ksi and fy = 60 ksi. Draw the interaction diagram.
Example: Axial Load Vs. Moment Example: Axial Load Vs. Moment Interaction DiagramInteraction Diagram
Given 8 # 10 (1.27 in2) and fc = 4 ksi and fy = 60 ksi
2 2st
2 2g
8 1.27 in 10.16 in
20 in. 400 in
A
A
Example: Axial Load Vs. Moment Example: Axial Load Vs. Moment Interaction DiagramInteraction Diagram
Given 8 # 10 (1.27 in2) and fc = 4 ksi and fy = 60 ksi
0 c g st y st
2 2 2
0.85
0.85 4 ksi 400 in 10.16 in 60 ksi 10.16 in
1935 k
P f A A f A
n 0
0.8 1935 k 1548 k
P rP
Example: Axial Load Vs. Moment Example: Axial Load Vs. Moment Interaction DiagramInteraction Diagram
Determine where the balance point, cb.
n 0
0.8 1935 k 1548 k
P rP
Example: Axial Load Vs. Moment Example: Axial Load Vs. Moment Interaction DiagramInteraction Diagram
Determine where the balance point, cb. Using similar triangles you can find cb
bb
b
17.5 in. 0.00317.5 in.
0.003 0.003 0.00207 0.003 0.00207
10.36 in.
cc
c
Example: Axial Load Vs. Moment Example: Axial Load Vs. Moment Interaction DiagramInteraction Diagram
Determine the strain of the steel
bs1 cu
b
bs2 cu
b
2.5 in. 10.36 in. 2.5 in.0.003 0.00228
10.36 in.
10 in. 10.36 in. 10 in.0.003 0.000104
10.36 in.
c
c
c
c
Example: Axial Load Vs. Moment Example: Axial Load Vs. Moment Interaction DiagramInteraction Diagram
Determine the stress in the steel
s1 s s1
s2 s s1
29000 ksi 0.00228
66 ksi 60 ksi compression
29000 ksi 0.000104
3.02 ksi compression
f E
f E
Example: Axial Load Vs. Moment Example: Axial Load Vs. Moment Interaction DiagramInteraction Diagram
Compute the forces in the column
c c 1
2s1 s1 s1 c
2s2
0.85 0.85 4 ksi 20 in. 0.85 10.36 in.
598.8 k
0.85 3 1.27 in 60 ksi 0.85 4 ksi
215.6 k
2 1.27 in 3.02 ksi 0.85 4 ksi
0.97 k neglect
C f b c
C A f f
C
Example: Axial Load Vs. Moment Example: Axial Load Vs. Moment Interaction DiagramInteraction Diagram
Compute the forces in the column
2s s s
n
3 1.27 in 60 ksi
228.6 k
599.8 k 215.6 k 228.6 k
585.8 k
T A f
P
Example: Axial Load Vs. Moment Example: Axial Load Vs. Moment Interaction DiagramInteraction Diagram
Compute the moment about the center
c s1 1 s 32 2 2 2
0.85 10.85 in.20 in.599.8 k
2 2
20 in. 215.6 k 2.5 in.
2
20 in. 228.6 k 17.5 in.
2
6682.2 k-in 556.9 k-ft
h a h hM C C d T d
Example: Axial Load Vs. Moment Example: Axial Load Vs. Moment Interaction DiagramInteraction Diagram
A single point from interaction diagram, (585.6 k, 556.9 k-ft). The eccentricity of the point is defined as
Now select a series of additional points by selecting values of c. Select c = 17.5 in.
6682.2 k-in11.41 in.
585.8 k
Me
P
Example: Axial Load Vs. Moment Example: Axial Load Vs. Moment Interaction DiagramInteraction Diagram
Determine the strain of the steel, c =17.5 in.
s1 cu
s1
s2 cu
s2
2.5 in. 17.5 in. 2.5 in.0.003 0.00257
17.5 in.
74.5 ksi 60 ksi (compression)
10 in. 17.5 in. 10 in.0.003 0.00129
17.5 in.
37.3 ksi (compression)
c
c
f
c
c
f
Example: Axial Load Vs. Moment Example: Axial Load Vs. Moment Interaction DiagramInteraction Diagram
Compute the forces in the column
c c 1
2s1 s1 s1 c
2s2
0.85 0.85 4 ksi 20 in. 0.85 17.5 in.
1012 k
0.85 3 1.27 in 60 ksi 0.85 4 ksi
216 k
2 1.27 in 37.3 ksi 0.85 4 ksi
86 k
C f b c
C A f f
C
Example: Axial Load Vs. Moment Example: Axial Load Vs. Moment Interaction DiagramInteraction Diagram
Compute the forces in the column
2s s s
n
3 1.27 in 0 ksi
0 k
1012 k 216 k 86 k
1314 k
T A f
P
Example: Axial Load Vs. Moment Example: Axial Load Vs. Moment Interaction DiagramInteraction Diagram
Compute the moment about the center
c s1 12 2 2
0.85 17.5 in.20 in.1012 k
2 2
20 in. 216 k 2.5 in.
2
4213 k-in 351.1 k-ft
h a hM C C d
Example: Axial Load Vs. Moment Example: Axial Load Vs. Moment Interaction DiagramInteraction Diagram
A single point from interaction diagram, (1314 k, 351.1 k-ft). The eccentricity of the point is defined as
Now select a series of additional points by selecting values of c. Select c = 6 in.
4213 k-in3.2 in.
1314 k
Me
P
Example: Axial Load Vs. Moment Example: Axial Load Vs. Moment Interaction DiagramInteraction Diagram
Determine the strain of the steel, c =6 in.
s1 cu
s1
s2 cu
s2
s2 cu
2.5 in. 6 in. 2.5 in.0.003 0.00175
6 in.
50.75 ksi (compression)
10 in. 6 in. 10 in.0.003 0.002
6 in.
58 ksi (tension)
17.5 in. 6 in.
c
c
f
c
c
f
c
c
s2
17.5 in.0.003 0.00575
6 in.
60 ksi (tension)f
Example: Axial Load Vs. Moment Example: Axial Load Vs. Moment Interaction DiagramInteraction Diagram
Compute the forces in the column
c c 1
2s1 s1 s1 c
2s2
0.85 0.85 4 ksi 20 in. 0.85 6 in.
346.8 k
0.85 3 1.27 in 50.75 ksi 0.85 4 ksi
180.4 k C
2 1.27 in 58 ksi
147.3 k T
C f b c
C A f f
C
Example: Axial Load Vs. Moment Example: Axial Load Vs. Moment Interaction DiagramInteraction Diagram
Compute the forces in the column
2s s s
n
3 1.27 in 60 ksi
228.6 k
346.8 k 180.4 k 147.3 k 228.6 k
151.3 k
T A f
P
Example: Axial Load Vs. Moment Example: Axial Load Vs. Moment Interaction DiagramInteraction Diagram
Compute the moment about the center
c s1 1 s 32 2 2 2
0.85 6 in.346.8 k 10 in.
2
180.4 k 10 in. 2.5 in.
228.6 k 17.5 in. 10 in.
5651 k-in 470.9 k-ft
h a h hM C C d T d
Example: Axial Load Vs. Moment Example: Axial Load Vs. Moment Interaction DiagramInteraction Diagram
A single point from interaction diagram, (151 k, 471 k-ft). The eccentricity of the point is defined as
Select point of straight tension
5651.2 k-in37.35 in.
151.3 k
Me
P
Example: Axial Load Vs. Moment Example: Axial Load Vs. Moment Interaction DiagramInteraction Diagram
The maximum tension in the column is
2n s y 8 1.27 in 60 ksi
610 k
P A f
ExampleExample
Point c (in) Pn Mn e
1 - 1548 k 0 0
2 20 1515 k 253 k-ft 2 in
3 17.5 1314 k 351 k-ft 3.2 in
4 12.5 841 k 500 k-ft 7.13 in
5 10.36 585 k 556 k-ft 11.42 in
6 8.0 393 k 531 k-ft 16.20 in
7 6.0 151 k 471 k-ft 37.35 in
8 ~4.5 0 k 395 k-ft infinity
9 0 -610 k 0 k-ft
ExampleExampleColumn Analysis
-1000
-500
0
500
1000
1500
2000
0 100 200 300 400 500 600
M (k-ft)
P (
k)
Use a series of c values to obtain the Pn verses Mn.
ExampleExample
Column Analysis
-800
-600
-400
-200
0
200
400
600
800
1000
1200
0 100 200 300 400 500
Mn (k-ft)
Pn
(k
)
Max. compression
Max. tension
Cb
Location of the linearly varying