Post on 27-Jun-2022
CE 412 Design of Concrete Structures
Lecture 5:
Cross-sections
Behavior in Combined Axial Load
and Flexure – Columns
November – 2020
Fawad A. NajamDepartment of Structural Engineering
NUST Institute of Civil Engineering (NICE)
National University of Sciences and Technology (NUST)
H-12 Islamabad, Pakistan
Cell: 92-334-5192533, Email: fawad@nice.nust.edu.pk
Naveed Anwar
Vice President, Executive Director, AIT Consulting
Director, ACECOMS
Affiliate Faculty, Structural Engineering,
Asian Institute of Technology (AIT), Thailand
2CE – 412: Design of Concrete Structures – Semester: Fall 2020
What is a Column?
3CE – 412: Design of Concrete Structures – Semester: Fall 2020
What is a Column
• Generally no load between supports and are cast vertically.
4CE – 412: Design of Concrete Structures – Semester: Fall 2020
Classification of Columns
• Material
• Geometry
• Stability
5CE – 412: Design of Concrete Structures – Semester: Fall 2020
The Column Design Problem: Given
• Loads:
• Moments Mz, My, Px at two ends
• Geometry:
• Length, X-Section
• Adjoining Members
• Material:
• Concrete strength
• Rebar Strength
x
y
z
Upper End
Lower End
Column
Connecting
Beams in Z-Axis
Connecting
Beams in Y-Axis
Lower Column
Upper Column
z
y
x
Mz2
My2
Px
Mz1
My1
Px
a) Basic Model a) Column Loads
6CE – 412: Design of Concrete Structures – Semester: Fall 2020
Column Design Problem
a) Ideal Situation
b) Practical Situation
Loads
Material
Shape & Size
Reinforcement
Solution
Loads
Trial Material
Trial Shape & Size
Trial Reinforcement
DesignAcceptable
No
Yes
Repeat
7CE – 412: Design of Concrete Structures – Semester: Fall 2020
Complexity in the Column Design
• Loading
• +P, -P, Mx, My
• Slenderness
• Length (Short, Long, Very Long)
• Bracing (Sway, Non-Sway, Braced, Unbraced)
• Framing (Pin, Fixed, Free, Intermediate..)
• Section
• Geometry (Rectangular, Circular, Complex..)
• Materials (Steel, Concrete, Composite…)
8CE – 412: Design of Concrete Structures – Semester: Fall 2020
Load-Shape-Slenderness
Shape
Loading
Length
V. Lon
g
Long
Short
P
P Mx
P Mx M
y
Most Simple
Problem
ShapeComplexity
Load Complexity
Slenderness
9CE – 412: Design of Concrete Structures – Semester: Fall 2020
Load-Shape-Material
10CE – 412: Design of Concrete Structures – Semester: Fall 2020
Load-Bracing-Length
Bracing
Length
Loading
11CE – 412: Design of Concrete Structures – Semester: Fall 2020
What is Uni-axial Bending
• Uni-axial bending is induced when column bending results in only one moment stress resultants about any of
the mutually orthogonal axis.
No Bending
Mx = 0, My = 0Strain StressSection
x
y𝜺𝒄 𝒇𝒄
P
fs1
fc
fs2
P
x
y
P
ey
P
eyfs1
fs2
fc
Uni-axial Bending
Mx <> 0, My = 0
12CE – 412: Design of Concrete Structures – Semester: Fall 2020
What is Bi-axial Bending
• Biaxial bending is induced when column bending results in two moment stress resultants about two mutually
orthogonal axis.
y
x
eye
ex P
x
P
ey ey
P
x
y y
Overall Design Process - ACI
Estimate Cross-section
based on “Thumb Rules”
Compute
Section Capacity Mn
Determine the
Layout of Rebars
Compute
Transverse Bars
Mn > Mu
Design
Completed
Y
Given P, Mux, Muy, fc, fy, L
Check
Slenderness
Ratio
Compute Mu
Not Slender
Compute
Design Moment
Slender
Rev
ise
Sec
tio
n/ M
ater
ialNot
OK
No
14CE – 412: Design of Concrete Structures – Semester: Fall 2020
Main Design Steps
• Assume section dimensions
• Compute Design Actions
• Elastic analysis results and magnification of moments due to slenderness, minimum eccentricities etc.
• Direct determination of design actions using P-Delta or full nonlinear analysis
• Check Capacity for Design Actions
• Assume failure criteria
• Assume material layout and material models
• Compute capacity and check against actions
15CE – 412: Design of Concrete Structures – Semester: Fall 2020
Reinforced Concrete Short Columns
Behaviour of Spiral Column and Tied Columns
Axially loaded column Eccentrically loaded column
Reference: James G. MacGregor (1997) Reinforced Concrete Mechsnics and Design, Third Edition.
Spiral Column Tied Column
Reference: James G. MacGregor (1997) Reinforced Concrete Mechsnics and Design, Third Edition.
Spiral Column Tied Column
Damage by 1971 San Fernando EarthquakeReference: James G. MacGregor (1997) Reinforced Concrete Mechsnics and Design, Third Edition.
19
Axially-Loaded Short
Columns
20CE – 412: Design of Concrete Structures – Semester: Fall 2020
Minimum Spiral Reinforcement Ratio, ACI Section 10.9.3
0.45 1g c
s
c ys
A fP
A f
• ACI Eq. 10-5 can be solved for max spacing.
• For spirals to bind the core, they must be spaced close together. ACI sec 7.10.4.3 limits the clear spacing to not
more than 3 in.
• For concrete placing, spirals should be as far apart as possible. Clear spacing > 1-⅓ times size of aggregate
(ACI sec 3.3.3) and never less than 1 in (ACI sec 7.10.4.3) [also see ACI secs 7.9.1 & 7.10.4.6 to 7.10.4.8].
(Eq. 10-5)𝜌𝑠
21
RC Columns: Basic
Reinforcement
Configurations and Tie
Arrangements
22CE – 412: Design of Concrete Structures – Semester: Fall 2020
1. Ties restrain the longitudinal bars from buckling out through the surface of the column.
• ACI sec 7.10.5.1
# 3 bars for longitudinal # 10 bars
# 4 bars for # 11, # 14, # 18 & bundled
• ACI sec 7.10.5.2
s < 16 db longitudinal bars
< 48 db tie bars
< least dimension (bmin)
• ACI sec 7.10.5.3
2. Ties hold the reinforcement cage together during the construction process.
3. Properly detailed ties confine the concrete core, providing increased ductility.
4. Ties serve as shear reinforcement for columns
• s < d/2 for effective shear reinforcement (ACI sec 11.5.4.1)
• ACI eq 11-2 and 11-14, if shear governs.
Look for ACI sec 7.10.5.4, 7.10.5.5 and 7.9.1 for further requirements.
Spacing and Construction Requirements for Ties
23CE – 412: Design of Concrete Structures – Semester: Fall 2020
P-M Interaction Curve
(Capacity Curve)
Uniaxial Bending - Short Columns
Safe
Un-safe
24
Column Capacity –
Concentrically Loaded
Columns
Column Capacity
Capacity is property of the cross-
section and does not depend on the
applied actions or loads.
25
Column Capacity???
Column Capacity –
Eccentrically Loaded
Columns
26
Column Capacity –
Eccentrically Loaded
Columns
27CE – 412: Design of Concrete Structures – Semester: Fall 2020
J. G. Macgregor (1997): Reinforced
Concrete Mechanics and Design, 3rd Edition.
Interaction diagram for an elastic column,
𝒇𝒄𝒖 = 𝒇𝒕𝒖
28
J. G. Macgregor (1997): Reinforced Concrete
Mechanics and Design, 3rd Edition.
Effect of Compressive or
Tensile Strength on
Interaction curve
29
Effect of Axial Load on
Moment-Curvature Curve
30CE – 412: Design of Concrete Structures – Semester: Fall 2020
• The curve is generated by varying the
neutral axis depth.
P-M Interaction Curve
Safe
Un-safe
...),(1
....,1
...),(1
...,1
121
2
121
1
i
n
i
ii
x y
x
x y
n
i
iiz
yyxAydydxyxM
yxAdydxyxN
Uniaxial Bending
31
An Example P-M
Interaction Diagram used
for the Design of Short
RC Columns
32CE – 412: Design of Concrete Structures – Semester: Fall 2020
What is Capacity?
• The axial-flexural capacity of the cross-section
is represented by three stress resultants.
• Capacity is property of the cross-section and
does not depend on the applied actions or
loads.
Mx
P
MyP-M Interaction Surface
Biaxial Bending
33CE – 412: Design of Concrete Structures – Semester: Fall 2020
What is Capacity?
• Capacity is dependent on failure criteria, cross-section geometry and material properties.
• Maximum strain
• Stress-strain curve
• Section shape and Rebar arrangement etc.
34CE – 412: Design of Concrete Structures – Semester: Fall 2020
Development of P-M Interaction Diagrams for Uniaxially Loaded RC
Short Columns – Solved Example
Courtesy: Dr. Khaliq ur Rashid Kayani (Architectural & Civil Engineering Services (ACES), Pakistan).
35CE – 412: Design of Concrete Structures – Semester: Fall 2020
General Points on P-M
Interaction Diagram
Axial Load (Compressive)
1 2 2 3 3
3
1
2
0.85
0.85 .
o c si s s c
si si y c
c c
o c si
i
hP X C d C d C d C
C A f f
C f b h
P C C
36CE – 412: Design of Concrete Structures – Semester: Fall 2020
Axial Load (Tensile)
3
1
3
1
t si
i
si si y
t t si i
i
P T
T A f
P X T d
37CE – 412: Design of Concrete Structures – Semester: Fall 2020
38CE – 412: Design of Concrete Structures – Semester: Fall 2020
39CE – 412: Design of Concrete Structures – Semester: Fall 2020
1
should always be +ve
c c
si si si ci
C K f bkd
C A f f
Points below Balanced - Yield
40CE – 412: Design of Concrete Structures – Semester: Fall 2020
Solved Example
Symmetric Section
112
c t
hX X
60
0.00207
4000
y
y
c
f ksi
f psi
41CE – 412: Design of Concrete Structures – Semester: Fall 2020
Section dimensions
Width = 22 Height =
Steel areas and depths
Layer no. 1 area = 4 depth = 3
Layer no. 2 area = 2 depth = 8
Layer no. 3 area = 2 depth = 14
Layer no. 4 area = 4 depth = 19
Concrete properties
Comp. strength (psi) beta1 Concrete Crushing Strain 0.0034000 0.85
Conc. strain at max comp. stress (Hognestad’s Model)= 0.002
Slope of stress vs. strain curve beyond peak stress (Hognestad’s Model)= z = 150
Steel properties
Yield stress (Ksi) Yield strain Elastic Modulus (ksi)
60 2.068 E-03 29000
42CE – 412: Design of Concrete Structures – Semester: Fall 2020
AXIAL COMPRESSIVE LOAD ( Po)
Cc = 0.85 x 4 x 22 x 22
= 1645.6 K
Cs1 = 4 x (60 - .85 x 4) = 226.4 K
Cs2 = 2 x (60 - .85 x 4) = 113.2 K
Cs3 = 113.2, Cs4 = 226.4
Po = 2324.8 K
AXIAL TENSILE LOAD ( Pt )
Ts1 = 4 x 60 = 240 K
Ts2 = 2 x 60 = 120 K
Ts3 = 2 x 60 = 120 K
Ts4 = 4 x 60 = 240 K
Pt = 720 K
43CE – 412: Design of Concrete Structures – Semester: Fall 2020
Solved Example: Complete P-M Interaction Diagram
1: Balanced Point
60
0.00207
4000
y
y
c
f ksi
f psi
11
622
633
0.00319 11.2 in.
0.003
0.003 0.00220
0.003 865 10
0.003 736 10
bal
y
s y
s
s
C
c d
c
c d
c
c d
c
1 2 360 , 25.1 , 21.3 s s sf ksi f ksi f ksi
44CE – 412: Design of Concrete Structures – Semester: Fall 2020
1
1 1 1
2 2 2
3 3 3
4 4 4
0.85 715
0.85 226
0.85 43.4
42.6
240
c c
s s s c
s s s c
s s s
s s s
C f b c k
C A f f k
C A f f k
T A f k
T A f k
702 balP k
1
6
2 2 2 2
8430 k-in
0.003 268 10 rad/in
bal c si i si ii i
bal
balbal
ch h hM C C d T d
M
C
Express as Positive
45CE – 412: Design of Concrete Structures – Semester: Fall 2020
2: A Point Above Balanced
4 4Pick C 19 in 0sd
1
2
2
6
3
3
19 30.003 0.00253
19
19 80.003 0.00154
19
50.4 ksi
19 140.003 790 10
19
22.9 ksi
s y
s
s
s
s
f
f
1
1 1 1
2 2 2
3 3 3
0.85 1208
0.85 226
0.85 94
0.85 39
c c
s s s c
s s s c
s s s c
C f b c k
C A f f k
C A f f k
C A f f k
1570 P C k
46CE – 412: Design of Concrete Structures – Semester: Fall 2020
1
6
2 2 2
3533 1808 282 117
5510 k-in
0.003 158 10 rad/in19 in
c si ii
ch hM C C d
M
47CE – 412: Design of Concrete Structures – Semester: Fall 2020
3: A Point Below Balance, Nominal Diagram
Set 2s y
1 1
6
2 2
3 3
4
0.0037.98
0.003 2
0.00187 , 54.3 ksi
6 10 , 0
0.00226 , 60 ksi
2 , 60 ksi
y
s s
s s
s s
sy y s
C d
f
f
f
f
=
48CE – 412: Design of Concrete Structures – Semester: Fall 2020
1
1 1 1
2
3 3
4 4
0.85 508
0.85 204
0
120
240
c c
s s s c
s
s s y
s s y
C f b c k
C A f f k
T
T A f k
T A f k
352 P k=
49CE – 412: Design of Concrete Structures – Semester: Fall 2020
1
6
2 2 2 2
7780 k-in.
0.003 376 10 rad/in.
c si i si ii i
ch h hM C C d T d
M
C
Express as Positive
50CE – 412: Design of Concrete Structures – Semester: Fall 2020
4 : A Point Below Balance, Yield Diagram
1 2Set 0.002 0.667 , 0.375cm k k
1 1
1 1
6
2 2
2 2
6
3 3
9.34 in.
0.00136
39.4 ksi , 3.59 ksi
286 10
8.30 ksi , 1.06 ksi
999 10 , 29.0 ksi
cm
cm y
s c
s c
s c
s c
s s
kd d
f f
f f
f
51CE – 412: Design of Concrete Structures – Semester: Fall 2020
352 P k
1
1 1 1 1
2 2 2 2
3 3 3
4 4 4
548
143
14.5
58.0
240
c c
s s s c
s s s c
s s s
s s s
C k f bkd k
C A f f k
C A f f k
T A f k
T A f k
52CE – 412: Design of Concrete Structures – Semester: Fall 2020
2
6
2 2 2
7390 k-in.
0.002 214 10 rad/in.
c si i si ii i
h h hM C k kd C d T d
M
kd
Express as Positive
53CE – 412: Design of Concrete Structures – Semester: Fall 2020
**** RESULTS FOR NOMINAL DIAGRAM ****
AXIAL LOAD (KIPS) NOMINAL MOMENT (K-IN.) CURAVTURE (RAD./IN.)
2324.8 0 0
1819.415 3990.68 1.36364 E-04
1724.501 4630.547 1.43541 E-04
1626.811 5219.912 1.51515 E-04
1539.457 5652.607 1.60428 E-04
1434.626 6149.517 1.70455 E-04
1325.143 6606.003 1.81818 E-04
1210.01 7027.416 1.94805 E-04
1087.924 7420.75 2.0979 E-04
963.9466 7774.937 2.27273 E-04
822.1076 8143.495 2.47934 E-04
702.1297 8436.976 2.66788 E-04
631.755 8374.65 2.854 E-04
557.8183 8286.174 3.06804 E-04
472.0286 8112.867 3.31679 E-04
377.8782 7893 3.60943 E-04
308.777 7572.274 3.95871 E-04
227.7132 7168.089 4.38283 E-04
138.3848 6692.336 4.90873 E-04
37.41104 6127.631 5.57804 E-04
-75.04004 5462.657 6.4587 E-04
**** RESULTS FOR YIELD DIAGRAM ****
AXIAL LOAD (KIPS) MOMENT (K-IN.) CURAVTURE (RAD./IN.)
702.1297 8436.976 2.667877 E-04
543.4567 8063.764 2.330024 E-04
362.9939 7160.652 2.091632 E-04
210.9273 6270.086 1.915271 E-04
95.02707 5507.163 1.780181 E-04
-3.005219 4842.844 1.67392 E-04
END OF PROGRAM
54CE – 412: Design of Concrete Structures – Semester: Fall 2020
55CE – 412: Design of Concrete Structures – Semester: Fall 2020
Properties of P-M Interaction Diagrams of RC Columns
56
Wight and Macgregor (2012): Reinforced
Concrete Mechanics and Design, 6th Edition.
Strain profiles
corresponding to points
at interaction diagram
57CE – 412: Design of Concrete Structures – Semester: Fall 2020
58
J. G. Macgregor (1997): Reinforced Concrete
Mechanics and Design, 3rd Edition.
Contributions of Steel
and Concrete to Column
Capacity
59
J. G. Macgregor (1997): Reinforced Concrete
Mechanics and Design, 3rd Edition.
Contributions of Steel
and Concrete to Column
Capacity
60
Symmetrical Column
Section
Unsymmetrical Column
Section
Effect of Symmetry of RC
Column Section
61
Effect of Column Type on
Shape of Interaction
Diagram
62CE – 412: Design of Concrete Structures – Semester: Fall 2020
Effect of Reinforcement Ratio Effect of Reinforcement Spacing
63CE – 412: Design of Concrete Structures – Semester: Fall 2020
64CE – 412: Design of Concrete Structures – Semester: Fall 2020
P-M Interaction Surface
(Capacity Surface)
Biaxial Bending - Short Columns Mx
P
My
65CE – 412: Design of Concrete Structures – Semester: Fall 2020
𝑃 −𝑀𝑥 −𝑀𝑦 Interaction Surface
• The surface is generated by changing
Angle and Depth of Neutral Axis.
...),(1
....,1
...),(1
....,1
...),(1
...,1
121
3
121
2
121
1
i
n
i
ii
x y
y
i
n
i
ii
x y
x
x y
n
i
iiz
xyxAxdydxyxM
yyxAydydxyxM
yxAdydxyxN
66
𝑃 −𝑀𝑥 −𝑀𝑦 Interaction
Surface
67CE – 412: Design of Concrete Structures – Semester: Fall 2020
68CE – 412: Design of Concrete Structures – Semester: Fall 2020
69CE – 412: Design of Concrete Structures – Semester: Fall 2020
70CE – 412: Design of Concrete Structures – Semester: Fall 2020
Mx-My Interaction
-Mz
Muy
(-) Mnz
(+) Mnz
+ My
- My
+ Mz
Mx-My Interaction is the basis for many
approximate methods
P-Mx Interaction Curve
Safe
Un-safe
71CE – 412: Design of Concrete Structures – Semester: Fall 2020
How to Check Capacity?
• How do we check capacity when there are three simultaneous actions and three interaction stress
resultants
• Given: Pu, Mux, Muy
• Available: Pn-Mnx-Mny Surface
• We can use the concept of Capacity Ratio, but which ratio
• Pu/Pn or Mux/Mns or Muy/Mny or …
• Three methods for computing Capacity Ratio
• Sum of Moment Ratios at Pu
• Moment Vector Ratio at Pu
• P-M vector Ratio
72CE – 412: Design of Concrete Structures – Semester: Fall 2020
Sum of 𝑀𝑥 and 𝑀𝑦
• 𝑀𝑥 −𝑀𝑦 curve is plotted at applied
axial load, 𝑃𝑢
• Sum of the Ratios of Moment is each
direction gives the Capacity Ratio.
73CE – 412: Design of Concrete Structures – Semester: Fall 2020
• 𝑀𝑥 −𝑀𝑦 curve is plotted at applied axial load.
• Ratio of 𝑀𝑢𝑥𝑦 vector to 𝑀𝑛𝑥𝑦 vector gives the Capacity Ratio.
Vector Moment Capacity
74CE – 412: Design of Concrete Structures – Semester: Fall 2020
True 𝑃 −𝑀 Vector Capacity
• 𝑃 − 𝑀 Curve is plotted in the direction of the
resultant moment.
• Ratio of 𝑃𝑢 −𝑀𝑢𝑥𝑦 vector to 𝑃𝑛 −𝑀𝑢𝑥𝑦vector
gives the Capacity Ratio.
What is Capacity
1- Based on Sum of Moments at Pu 2- Based on Moment Vector at PU
3- Based on True Capacity Vector in 3D
76CE – 412: Design of Concrete Structures – Semester: Fall 2020
Other Forms of Failure Surfaces
77CE – 412: Design of Concrete Structures – Semester: Fall 2020
Design of Rectangular Columns Subjected to Biaxial Loads
1) The strain-compatibility method
2) The equivalent eccentricity method
3) Method based on 45-degree slice through interaction surface
4) Load Contour Method (Bresler and Parme)
5) Bresler’s reciprocal load method
78
Design of Rectangular
Columns Subjected to
Biaxial Loads
79
The equivalent
eccentricity method
80
Bresler’s reciprocal load
method
81CE – 412: Design of Concrete Structures – Semester: Fall 2020
82CE – 412: Design of Concrete Structures – Semester: Fall 2020
84
An Example P-M
Interaction Diagram used
for the Design of Short
RC Columns
87CE – 412: Design of Concrete Structures – Semester: Fall 2020