6 reinforced columns
DESCRIPTION
Reinforced ColumnsTRANSCRIPT
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Reinforced ColumnsReinforced Columns
Concrete Construction
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Lecture GoalsLecture Goals
• Columns– Short Column Design– Long Column Design
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Behavior under Combined Bending and Behavior under Combined Bending and Axial LoadsAxial Loads
Usually moment is represented by axial load times eccentricity, i.e.
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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:
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Behavior under Combined Bending and Behavior under Combined Bending and Axial LoadsAxial Loads
Axial Load and Moment Interaction Diagram -General
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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
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Columns in Pure TensionColumns in Pure Tension
Section is completely cracked (no concrete axial capacity)
Uniform Strain y
N
1iisytensionn AfP
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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)
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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
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ColumnColumn
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ColumnsColumns
Commentary:
Other sections:
may be increased linearly to 0.9 as the strain s increase in the tension steel. Pb
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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.
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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)
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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.
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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.
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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
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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.
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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.
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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