lecture chp-9&10 – columns
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Lecture Chp-9&10 – Columns. Lecture Goals. Definitions for short columns Columns. Analysis and Design of “Short” Columns. General Information. Column:. Vertical Structural members Transmits axial compressive loads with or without moment - PowerPoint PPT PresentationTRANSCRIPT
Lecture Chp-9&10 – Lecture Chp-9&10 – ColumnsColumns
Lecture GoalsLecture Goals
Definitions for short columnsColumns
Analysis and Design of Analysis and Design of “Short” Columns“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:
Analysis and Design of Analysis and Design of “Short” Columns“Short” Columns
General Information
Column Types:
1. Tied
2. Spiral
3. Composite
4. Combination
5. Steel pipe
Analysis and Design of Analysis and Design of “Short” Columns“Short” Columns
Tie spacing h (except for seismic)
tie support long bars (reduce buckling)
ties provide negligible restraint to lateral expose of core
Tied Columns - 95% of all columns in buildings are tied
Analysis and Design of Analysis and Design of “Short” Columns“Short” Columns
Pitch = 1.375 in. to 3.375 in.
spiral restrains lateral (Poisson’s effect)
axial load delays failure (ductile)
Spiral Columns
Analysis and Design of Analysis and Design of “Short” Columns“Short” Columns
Elastic Behavior
An elastic analysis using the transformed section method would be:
stcc nAA
Pf
For concentrated load, P
uniform stress over section
n = Es / Ec
Ac = concrete area
As = steel areacs nff
Analysis and Design of Analysis and Design of “Short” Columns“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
Analysis and Design of Analysis and Design of “Short” Columns“Short” Columns
Elastic Behavior
An elastic analysis does not work, because creep and shrinkage affect the acting concrete compression strain as follows:
Analysis and Design of Analysis and Design of “Short” Columns“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.
Analysis and Design of Analysis and Design of “Short” Columns“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:
Behavior, Nominal Capacity and Behavior, Nominal Capacity and Design under Concentric Axial Design under Concentric Axial
loadsloads
Initial Behavior up to Nominal Load - Tied and spiral columns.
1.
Behavior, Nominal Capacity and Behavior, Nominal Capacity and Design under Concentric Axial Design under Concentric Axial
loadsloads
Behavior, Nominal Capacity and Behavior, Nominal Capacity and Design under Concentric Axial Design under Concentric Axial
loadsloads
stystgc0 *85.0 AfAAfP
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
Behavior, Nominal Capacity and Behavior, Nominal Capacity and Design under Concentric Axial Design under Concentric Axial
loadsloads
Maximum Nominal Capacity for Design Pn (max) 2.
0maxn rPP
r = Reduction factor to account for accidents/bending
r = 0.80 ( tied )
r = 0.85 ( spiral )ACI 10.3.6.3
Behavior, Nominal Capacity and Behavior, Nominal Capacity and Design under Concentric Axial Design under Concentric Axial
loadsloads
Reinforcement Requirements (Longitudinal Steel Ast)
3.
g
stg A
A
- ACI Code 10.9.1 requires
Let
08.001.0 g
Behavior, Nominal Capacity and Behavior, Nominal Capacity and Design under Concentric Axial Design under Concentric Axial
loadsloads
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
min. of 3 bars in triangular ties
Reinforcement Requirements (Longitudinal Steel Ast)
Behavior, Nominal Capacity and Behavior, Nominal Capacity and Design under Concentric Axial Design under Concentric Axial
loadsloads
3.
ACI Code 7.10.5.1
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
Behavior, Nominal Capacity and Behavior, Nominal Capacity and Design under Concentric Axial Design under Concentric Axial
loadsloads
3. Reinforcement Requirements (Lateral Ties)
Vertical spacing: (ACI 7.10.5.2)
16 db ( db for longitudinal bars ) 48 db ( db for tie bar ) least lateral dimension of column
s s s
Behavior, Nominal Capacity and Behavior, Nominal Capacity and Design under Concentric Axial Design under Concentric Axial
loadsloads
3. Reinforcement Requirements (Lateral Ties)
Arrangement Vertical spacing: (ACI 7.10.5.3)
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.)
Behavior, Nominal Capacity and Behavior, Nominal Capacity and Design under Concentric Axial Design under Concentric Axial
loadsloads
Examples of lateral ties.
Behavior, Nominal Capacity and Behavior, Nominal Capacity and Design under Concentric Axial Design under Concentric Axial
loadsloads
ACI Code 7.10.4
Reinforcement Requirements (Spirals )
3/8 “ dia.(3/8” smooth bar, #3 bar dll or wll wire)
size
clear spacing between spirals
3 in. ACI 7.10.4.31 in.
Behavior, Nominal Capacity and Behavior, Nominal Capacity and Design under Concentric Axial Design under Concentric Axial
loadsloads
Reinforcement Requirements (Spiral)
sD
A
c
sps
4
Core of Volume
Spiral of Volume
Spiral Reinforcement Ratio, s
sD
DA
41
:from
2c
csps
Behavior, Nominal Capacity and Behavior, Nominal Capacity and Design under Concentric Axial Design under Concentric Axial
loadsloads
Reinforcement Requirements (Spiral)
y
c
c
gs *1*45.0
f
f
A
A ACI Eqn. 10-5
psi 60,000 steel spiral ofstrength yield
center) (center to steel spiral ofpitch spacing
spiral of edge outside toedge outside :diameter core
4
area core
entreinforcem spiral of area sectional-cross
y
c
2c
c
sp
f
s
D
DA
A
where
Behavior, Nominal Capacity and Behavior, Nominal Capacity and Design under Concentric Axial Design under Concentric Axial
loadsloads
4. Design for Concentric Axial Loads
(a) Load Combination
u DL LL
u DL LL w
u DL w
1.2 1.6
1.2 1.0 1.6
0.9 1.3
P P P
P P P P
P P P
Gravity:
Gravity + Wind:
and
etc. Check for tension
Behavior, Nominal Capacity and Behavior, Nominal Capacity and Design under Concentric Axial Design under Concentric Axial
loadsloads
4. Design for Concentric Axial Loads
(b) General Strength Requirement
un PP = 0.65 for tied columns
= 0.7 for spiral columns
where,
Behavior, Nominal Capacity and Behavior, Nominal Capacity and Design under Concentric Axial Design under Concentric Axial
loadsloads
4. Design for Concentric Axial Loads
(c) Expression for Design
08.00.01 Code ACI gg
stg
A
A
defined:
Behavior, Nominal Capacity and Behavior, Nominal Capacity and Design under Concentric Axial Design under Concentric Axial
loadsloads
ucystcgn
steel
85.0
concrete
85.0 PffAfArP
or
ucygcgn 85.085.0 PfffArP
Behavior, Nominal Capacity and Behavior, Nominal Capacity and Design under Concentric Axial Design under Concentric Axial
loadsloads
85.085.0 cygc
ug
fffr
PA
* when g is known or assumed:
cg
u
cy
st 85.085.0
1fA
r
P
ffA
* when Ag is known or assumed:
Example: Design Tied Column for Example: Design Tied Column for Concentric Axial Load Concentric Axial Load
Example: Design Tied Column Example: Design Tied Column for for Concentric Axial Load 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.
Example: Design Tied Column Example: Design Tied Column for for Concentric Axial Concentric Axial LoadLoad
Determine the loading
u dl ll
u dl ll w
1.2 1.6
1.2 150 k 1.6 300 k 660 k
1.2 1.0 1.6
1.2 150 k 1.0 300 k 1.6 50 k 560 k
P P P
P P P P
u dl w0.9 1.3
0.9 150 k 1.3 50 k 70 k
P P P
Check the compression or tension in the column
Example: Design Tied Column Example: Design Tied Column for for Concentric Axial Concentric Axial LoadLoad
For a square column r = 0.80 and = 0.65 and = 0.03
ug
c g y c
2
2g
r 0.85 0.85
660 k
0.85 4.5 ksi0.65 0.8
0.03 60 ksi 0.85 4.5 ksi
230.4 in
15.2 in. 16 in.
PA
f f f
A d d d
Example: Design Tied Column Example: Design Tied Column for for Concentric Axial Concentric Axial LoadLoad
For a square column, As=Ag= 0.03(15.2 in.)2 =6.93 in2
ust c g
y c
2
2
10.85
r0.85
1
60 ksi 0.85 4.5 ksi
660 k * 0.85 4.5 ksi 16 in
0.65 0.8
5.16 in
PA f A
f f
Use 8 #8 bars Ast = 8(0.79 in2) = 6.32 in2
Example: Design Tied Column Example: Design Tied Column for for Concentric Axial Concentric Axial LoadLoad
Check P0
0 c g st y st
2 2 2
n 0
0.85
0.85 4.5 ksi 256 in 6.32 in 60 ksi 6.32 in
1334 k
0.65 0.8 1334 k 694 k > 660 k OK
P f A A f A
P rP
Example: Design Tied Column Example: Design Tied Column for for Concentric Axial Concentric Axial LoadLoad
Use #3 ties compute the spacing
b stirrup# 2 cover
# bars 1
16 in. 3 1.0 in. 2 1.5 in. 0.375 in.
24.625 in.
b d ds
< 6 in. No cross-ties needed
Example: Design Tied Column Example: Design Tied Column for for Concentric Axial Concentric Axial LoadLoad
Stirrup design
b
stirrup
16 16 1.0 in. 16 in. governs
48 48 0.375 in. 18 in.
smaller or 16 in. governs
d
s d
b d
Use #3 stirrups with 16 in. spacing in the column
Behavior under Combined Behavior under Combined Bending and Axial LoadsBending and Axial Loads
Usually moment is represented by axial load times eccentricity, i.e.
Behavior under Combined Behavior under Combined Bending and Axial LoadsBending and Axial 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 Behavior under Combined Bending and Axial LoadsBending and Axial Loads
Axial Load and Moment Interaction Diagram – General
Behavior under Combined Behavior under Combined Bending and Axial LoadsBending and Axial 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 Columns in Pure TensionTension
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
ColumnsColumnsCommentary:
Other sections:
may be increased linearly to 0.9 as the strain s increase in the tension steel. Pb
Design for Combined Design for Combined Bending and Axial Load Bending and Axial Load
(Short Column)(Short Column)
Design - select cross-section and reinforcement to resist axial load and moment.
Design for Combined Design for Combined Bending and Axial Load Bending and Axial Load
(Short Column)(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 General Procedure for Construction of IDConstruction 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.
General Procedure for General Procedure for Construction of IDConstruction 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 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. Example: Axial Load vs. Moment Moment Interaction Interaction DiagramDiagram
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. Example: Axial Load vs. Moment Moment Interaction Interaction DiagramDiagram
Given 8 # 10 (1.27 in2) and fc = 4 ksi and fy = 60 ksi
2 2st
2 2g
2st
2g
8 1.27 in 10.16 in
20 in. 400 in
10.16 in0.0254
400 in
A
A
A
A
Example: Axial Load vs. Example: Axial Load vs. Moment Moment Interaction Interaction DiagramDiagram
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
[ Point 1 ]
Example: Axial Load vs. Example: Axial Load vs. Moment Moment Interaction Interaction DiagramDiagram
Determine where the balance point, cb.
Example: Axial Load vs. Example: Axial Load vs. Moment Moment Interaction Interaction DiagramDiagram
Determine where the balance point, cb. Using similar triangles, where d = 20 in. – 2.5 in. = 17.5 in., one can find cb
b
b
b
17.5 in.
0.003 0.003 0.002070.003
17.5 in.0.003 0.00207
10.36 in.
c
c
c
Example: Axial Load vs. Example: Axial Load vs. Moment Moment Interaction Interaction DiagramDiagram
Determine the strain of the steel
bs1 cu
b
bs2 cu
b
2.5 in. 10.36 in. 2.5 in.0.003
10.36 in.
0.00228
10 in. 10.36 in. 10 in.0.003
10.36 in.
0.000104
c
c
c
c
Example: Axial Load vs. Example: Axial Load vs. Moment Moment Interaction Interaction DiagramDiagram
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. Example: Axial Load vs. Moment Moment Interaction Interaction DiagramDiagramCompute the forces in the column
c c 1
s1 s1 s1 c
2
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. Example: Axial Load vs. Moment Moment Interaction Interaction DiagramDiagramCompute the forces in the column
2s s s
n c s1 s2 s
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 C C C T
Example: Axial Load vs. Example: Axial Load vs. Moment Moment Interaction Interaction DiagramDiagramCompute 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. Example: Axial Load vs. Moment Moment Interaction Interaction DiagramDiagram
A single point from interaction diagram, (585.6 k, 556.9 k-ft). The eccentricity of the point is defined as
6682.2 k-in11.41 in.
585.8 k
Me
P
[ Point 2 ]
Example: Axial Load vs. Example: Axial Load vs. Moment Moment Interaction Interaction DiagramDiagram
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)
s1 cu
s1
s2 cu
s2
2.5 in. 17.5 in. 2.5 in.0.003
17.5 in.
0.00257 74.5 ksi 60 ksi (compression)
10 in. 17.5 in. 10 in.0.003
17.5 in.
0.00129 37.3 ksi (compression)
c
c
f
c
c
f
Example: Axial Load vs. Example: Axial Load vs. Moment Moment Interaction Interaction DiagramDiagram
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. Example: Axial Load vs. Moment Moment Interaction Interaction DiagramDiagram
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. Example: Axial Load vs. Moment Moment Interaction Interaction DiagramDiagram
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. Example: Axial Load vs. Moment Moment Interaction Interaction DiagramDiagram
A single point from interaction diagram, (1314 k, 351.1 k-ft). The eccentricity of the point is defined as
4213 k-in3.2 in.
1314 k
Me
P
[ Point 3 ]
Example: Axial Load vs. Example: Axial Load vs. Moment Moment Interaction Interaction DiagramDiagramSelect c = 6 in. Determine the strain of the steel, c =6 in.
s1 cu
s1
s2 cu
s2
s3 cu
2.5 in. 6 in. 2.5 in.0.003
6 in.
0.00175 50.75 ksi (compression)
10 in. 6 in. 10 in.0.003
6 in.
0.002 58 ksi (tension)
17.5 in. 6 in.
c
c
f
c
c
f
c
c
s3
17.5 in.0.003
6 in.
0.00575 60 ksi (tension)f
Example: Axial Load vs. Example: Axial Load vs. Moment Moment Interaction Interaction DiagramDiagramCompute the forces in the column
c c 1
s1 s1 s1 c
2
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. Example: Axial Load vs. Moment Moment Interaction Interaction DiagramDiagram
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. Example: Axial Load vs. Moment Moment Interaction Interaction DiagramDiagram
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. Example: Axial Load Vs. Moment Interaction DiagramMoment Interaction Diagram
A single point from interaction diagram, (151 k, 471 k-ft). The eccentricity of the point is defined as
5651.2 k-in37.35 in.
151.3 k
Me
P
[ Point 4 ]
Example: Axial Load vs. Example: Axial Load vs. Moment Moment Interaction Interaction DiagramDiagram
Select point of straight tension. The maximum tension in the column is
2n s y 8 1.27 in 60 ksi
610 k
P A f
[ Point 5 ]
Example: Axial Load vs. Example: Axial Load vs. Moment Moment Interaction Interaction DiagramDiagram
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
Example: Axial Load vs. Example: Axial Load vs. Moment Moment Interaction Interaction DiagramDiagram
Column 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.
Example: Axial Load vs. Example: Axial Load vs. Moment Moment Interaction Interaction
DiagramDiagram
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