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CE 72.52 - Lecture 5 - Column Design
1. 1
CE 72.52 Advanced Concrete
Lecture 5:
Design of Columns
Naveed Anwar
Executive Director, AIT Consulting
Director, ACECOMS
Affiliate Faculty, Structural Engineering, AIT
August - 2015
2. CE 72.52 – Advanced Concrete Structures - August 2014, Dr. Naveed Anwar 2
• Member Strength: Column Slenderness
• Member Capacity Vs. Section Capacity
• The Moment Magnifier Method
• The Importance of Moment Magnification
• Some Issues Regarding Slenderness Effects
• The P-Delta Analysis
• Special Considerations
• Effective Design of Columns
3. CE 72.52 – Advanced Concrete Structures - August 2014, Dr. Naveed Anwar
Column BeamColumn or Beam?
4. CE 72.52 – Advanced Concrete Structures - August 2014, Dr. Naveed Anwar
Mx, ≠ 0, Vy ≠ 0, Tz ≠ 0
or
My, ≠ 0, Vx ≠ 0, Tz ≠ 0
General 3D Beam Section
Nz, Vx, Vy, Mx, My, Tz
Nz, > 0.1 Ncap
Mx ≠ 0, My ≠ 0
Vx, Vy, Tz can be ignored
Design as column section Design as beam sections
Also, Generally no load between supports and are cast vertically
5. CE 72.52 – Advanced Concrete Structures - August 2014, Dr. Naveed Anwar
• 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
6. CE 72.52 – Advanced Concrete Structures - August 2014, Dr. Naveed Anwar
• a) Ideal Situation
• b) Practical Situation
Loads
Material
Shape & Size
Reinforcement
Solution
Loads
Trial Material
Trial Shape & Size
Trial Reinforcement
Design
Acceptable
No
Yes
Repeat
8. CE 72.52 – Advanced Concrete Structures - August 2014, Dr. Naveed Anwar
Load-Shape-Slenderness
Shape
Loading
Length
V.Long
Long
Short
P
P Mx
P Mx My
Most Simple
Problem
Shape
Complexity
Load Complexity
Slenderness
9. CE 72.52 – Advanced Concrete Structures - August 2014, Dr. Naveed Anwar
Load-Bracing-Length
Bracing
Length
Loading
P
10. CE 72.52 – Advanced Concrete Structures - August 2014, Dr. Naveed Anwar
Load-Section-Material
Material
Section
Loading
P
11.
12. CE 72.52 – Advanced Concrete Structures - August 2014, Dr. Naveed Anwar
• 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
13. CE 72.52 – Advanced Concrete Structures - August 2014, Dr. Naveed Anwar
• a) Ideal Situation
• b) Practical Situation
Loads
Material
Shape & Size
Reinforcement
Solution
Loads
Trial Material
Trial Shape & Size
Trial Reinforcement
Design
Acceptable
No
Yes
Repeat
15. CE 72.52 – Advanced Concrete Structures - August 2014, Dr. Naveed Anwar
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
ReviseSection/Material
Not OK
16. CE 72.52 – Advanced Concrete Structures - August 2014, Dr. Naveed Anwar
• 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
18. CE 72.52 – Advanced Concrete Structures - August 2014, Dr. Naveed Anwar
• The curve is generated by
varying the neutral axis
depth
Safe
Un-safe
zi
N
i
si
z A
cny
N
i
si
A
cnx
dAfdzdafM
AfdafN
si
b
si
b
1
1
.)(
)(
19. CE 72.52 – Advanced Concrete Structures - August 2014, Dr. Naveed Anwar 19
20. CE 72.52 – Advanced Concrete Structures - August 2014, Dr. Naveed Anwar
• 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
21. CE 72.52 – Advanced Concrete Structures - August 2014, Dr. Naveed Anwar
• Mx-My curve is plotted at applied axial load, Pu
• Sum of the Ratios of Moment is each direction gives the
Capacity Ratio
22. CE 72.52 – Advanced Concrete Structures - August 2014, Dr. Naveed Anwar
• Mx-My curve is plotted at applied axial load
• Ratio of Muxy vector to Mnxy vector gives the Capacity Ratio
23. CE 72.52 – Advanced Concrete Structures - August 2014, Dr. Naveed Anwar
• P-M Curve is plotted in the direction of the resultant moment
• Ratio of PuMuxy vector to PnMuxy vector gives the Capacity Ratio
24.
25. CE 72.52 – Advanced Concrete Structures - August 2014, Dr. Naveed Anwar
• The member capacity is based on the capacity of cross-section
at various locations along the member length
• The member capacity is almost always less than cross-section
capacity at critical location
• The reduction in member capacity is due to the stability
considerations, P-Delta effects and non-linearity in member
behavior, effect of boundary conditions and interaction with
other load configuration etc.
26. CE 72.52 – Advanced Concrete Structures - August 2014, Dr. Naveed Anwar
Column Capacity (P-M)
M
P
I
II
Moment
Amplification
Capacity
Reduction
II : Mc = P(e + D)
Long Column
P
e
D = f(Mc)C
I. Mc = P.e
Short Column
P
e
C
27. CE 72.52 – Advanced Concrete Structures - August 2014, Dr. Naveed Anwar
• Overall Objective
– To estimate magnification of the “elastic actions” due to geometric and
material in- elasticity or non- linearity.
• Real Situation
– Geometric Effect Alone M = M0 + PD
– Material Effect Alone
• Δo based on E0Ig Cracking Ig Ief
• Δ based on (EI) modified (Nonlinear Ec)
• Correct Approach
– Non linear analysis that includes effect of geometric and material non
linearity of “entire” structure
• Approximate Approach
– Moment magnification factor M = δ Mo
P
D
P
28. CE 72.52 – Advanced Concrete Structures - August 2014, Dr. Naveed Anwar
• The moment due to axial load
multiplied by deflection at
each point along the length
• The deflection is an integration
of total moment diagram,
divided by stiffness at each
point along the length
EI
dxM
L
x
L
D 0
D
L
xx
x
L
IE
dxM
0
LLLt PMM D 0
29. CE 72.52 – Advanced Concrete Structures - August 2014, Dr. Naveed Anwar
• “Effective” Length
– Length used for moment integration
– End Framing and Boundary Conditions
– Lateral Bracing Conditions
• “Effective” Stiffness
– Cross-sections Dimensions and Proportions
– Reinforcement amount and Distribution
– Modulus of Elasticity of Concrete and Steel
– Creep and Sustained Loads
• Loads
– Axial Load
– End Moments and Moments along the Length
30. CE 72.52 – Advanced Concrete Structures - August 2014, Dr. Naveed Anwar
• What is Slenderness ?
– When the Buckling Load controls Ultimate Capacity
– or Secondary Moments become Significant
• ACI Definition of Slenderness
– Braced Frames
• Kl/r > 32-12 (M1b/M2b)
– Unbraced Frames
• Kl/r > 22
M’c = Mc + P.Dc
c Dc
P
31. CE 72.52 – Advanced Concrete Structures - August 2014, Dr. Naveed Anwar
• The Moment and Stress
Amplification Factors are
derived on the basis of pin-
ended columns with single
moment curvature.
(Cm = 1.0)
• For other Moment
Distribution, the correction
factor Cm needs to be
computed to modify the
stress amplification.
Cm = 0.4 to 1.0
M1 is the smaller End
Moment
M2 is the larger End Moment
M1/M2
Positive
M1/M2
Negative
M
1
M
2
M
2
M
1
4.0
2
1
4.06.0
M
M
Cm
32. CE 72.52 – Advanced Concrete Structures - August 2014, Dr. Naveed Anwar
M1= -M M1 = 0 M1 =M M1 =0
M2 = M M2 = M M2 = M
1
2
1
M
M
0
2
1
M
M
1
2
1
M
M 0
2
1
M
M
M1
M2 M1 M1
M1
M2M2
M2
Cm = 1.0 Cm = 0.6 Cm = 0.4 Cm = 0.6
33. CE 72.52 – Advanced Concrete Structures - August 2014, Dr. Naveed Anwar
• Attempt to include,
– Cracking, Variable E, Creep effect
– Geometric and material non linearity
• Ig = Gross Moment of Inertia
• Ise = Moment of Inertia of rebars
• bd = Effect of creep for sustained loads.
= Pud/Pu
d
gC
d
sesgC
IE
or
IEIE
EI
b
b
1
4.0
1
2.0
12
3
bh
I g
2
. bbse yAI
h
b
Ab
yb
34. CE 72.52 – Advanced Concrete Structures - August 2014, Dr. Naveed Anwar
• To account for “Axial-Flexural Buckling”
• Indicates the “total bent” length of column between
inflection points
• Can vary from 0.5 to Infinity
• Most common range 0.75 to 2.0
0.5 1.0
2.0
0.5 - 1.0 1.0 -
35. CE 72.52 – Advanced Concrete Structures - August 2014, Dr. Naveed Anwar
Members Part of Framed Structure
IncreasesKIncreaseGGK
BeamsLEI
ColumnsLEI
G C
,
)/(
)/(
21
20
20
mm
m
GforG
G
K
2)1(9.0 mm GforGK
0.105.085.0
0.1)(05.07.0
m
BT
GK
GGK
Unbraced
Frames
Braced
Frames
(smaller of)
BTm
B
T
GandGofMinimumG
EndBottomG
EndTopG
36. CE 72.52 – Advanced Concrete Structures - August 2014, Dr. Naveed Anwar
• How about “I” Gross? Cracked? Effective?
• ACI Rules Beams I = 0.35 Ig, Column I = 0.7Ig
• E for column and beams may be different
IncreasesKIncreaseK
BeamslEI
ColumnslEI C
,
)/(
)/(
)(
)(
21
21
BB
CC
T
IIE
IIE
Example
C2
C3
C1
B1 B2
B4B3
Lc
37. CE 72.52 – Advanced Concrete Structures - August 2014, Dr. Naveed Anwar
• Sway is dependent upon the structural configuration
as well as type of loading
For Non-sway Frames (Very rigid or braced)
For Sway Frames (Open frames, not braced,
depends on loads also)
0.1
0.1
ns
s
0.1
0.1
ns
s
Non Sway Sway May be Sway
38. CE 72.52 – Advanced Concrete Structures - August 2014, Dr. Naveed Anwar
• Appreciable relative moment of two ends of column
• Sway Limits
c
BT
l
Sway
DD
D0
05.1)
05.0)
6)
0
D
M
M
c
lV
P
b
EIEIa
m
CU
U
ColumnswallsBracing
DT
DB
lc
Frame considered
as “Non-Sway”
39. CE 72.52 – Advanced Concrete Structures - August 2014, Dr. Naveed Anwar
• Braced Column (Non-Sway)
• Unbraced Column (Sway)
• Most building columns may
be considered “Non-Sway”
for gravity loads
• More than 40% of columns
in buildings are “Non-Sway”
for lateral loads
• Moment Magnification for
“Sway” case is more
significant, more
complicated and more
important
40.
41. CE 72.52 – Advanced Concrete Structures - August 2014, Dr. Naveed Anwar
• The Moment Magnifier Method
– An Approximate Method to account for Slenderness Effects
– May be used instead of P-D Analysis
– Not to be used when Kl/r > 100
– Separate Magnification for Sway and Non-Sway Load Cases
– Separate Magnification Factors for moment about each axis
– Moment magnification generally 1.2 to 2.5 times
– Mostly suitable for building columns
42. CE 72.52 – Advanced Concrete Structures - August 2014, Dr. Naveed Anwar
ssnsnsm MMM
Larger Sway Moment
Larger Non- Sway Moment
Final
Design
Moment
Magnification of
moment that do not
cause sway
Magnification of
moment that
cause sway
43. CE 72.52 – Advanced Concrete Structures - August 2014, Dr. Naveed Anwar
Basic Equation
ssnsnsm MMM
Magnification Factor for Moments
that do not cause sway
44. CE 72.52 – Advanced Concrete Structures - August 2014, Dr. Naveed Anwar
Calculation of ns (Non-Sway)
C
u
m
ns
P
P
C
75.0
1
Moment curvature
Coefficient
Applied column load
2
2
)(
)(
U
C
Kl
EI
P
Critical buckling load
Effective Length Factor
Flexural Stiffness
Equation 10-12
ACI 318-11
Equation 10-13
ACI 318-11
45. CE 72.52 – Advanced Concrete Structures - August 2014, Dr. Naveed Anwar
Basic Equation
ssnsnsm MMM
Magnification Factor for Moments
that cause sway
46. CE 72.52 – Advanced Concrete Structures - August 2014, Dr. Naveed Anwar
Determination of ds (Sway)
1
75.0
1
1
)
5.1
0.1
1
1
) 0
D
c
u
s
s
cu
u
s
P
P
b
thenIf
lV
P
Qwhere
Q
a
Sum of Critical Buckling Load of
all columns in floor
Sway Quotient
Equation 10-10
ACI 318-11
Equation 10-20
ACI 318-11
Equation 10-21
ACI 318-11
47. CE 72.52 – Advanced Concrete Structures - August 2014, Dr. Naveed Anwar
Sway Quotient Q and Pc
cu
u
lV
P
Q 0D
Sum of column loads in one floor
Relative displacement
Determined from Frame Analysis
Storey shear (sum of
shear in all columns)
Storey height
2
2
)(
)(
U
C
Kl
EI
P
Effective Length Factor
Flexural Stiffness
48. CE 72.52 – Advanced Concrete Structures - August 2014, Dr. Naveed Anwar
ssnsnsm MMM
1
75.0
1
1
)
5.1
0.1
1
1
)
0
D
c
u
s
s
cu
u
s
P
P
b
thenIf
lV
P
a
C
u
m
ns
P
P
C
75.0
1
Larger Sway Moment
Larger Non- Sway Moment
Final Design Moment
2
2
)(
)(
U
C
Kl
EI
P
4.0
2
1
4.06.0
M
M
Cm
Equation 10-16
ACI 318-11
Equation 10-12
ACI 318-11
49.
50. CE 72.52 – Advanced Concrete Structures - August 2014, Dr. Naveed Anwar
• According to ACI 318-11 Code:
– For Braced Frames (Non-sway)
• Kl/r > 34-12(M1b/M2b)
– For Un-braced Frames (Sway)
• Kl/r > 22
– Or When Secondary Moments become Significant
• These provisions do not consider other factors, such as P,
lateral deflection, lateral loads, section material or properties
51. CE 72.52 – Advanced Concrete Structures - August 2014, Dr. Naveed Anwar
• Computation of Slenderness Effects for 3 column sections for
different axial load and lengths
– A = 30x30 cm
– B = 40x40 cm
– C = 80x80 cm
• Braced (Non-Sway) frames assuming shear walls prevent large
lateral displacements
52. CE 72.52 – Advanced Concrete Structures - August 2014, Dr. Naveed Anwar
Load
Range
Length
53. CE 72.52 – Advanced Concrete Structures - August 2014, Dr. Naveed Anwar
• Column Cross-Section = 30cmx30cm reinforced with 6-d20
• Connecting Members
– Beam on Right:
• Length = 5 m
• Cross-section = 30cmx50cm
– Beam on Left:
• Length = 3 m
• Cross-section = 30cmx50cm
– Column Above
• Length = 3m
• Cross-section = 40cmx40cm
• Fixed at Base
• The column is part of a non-sway structure
54. CE 72.52 – Advanced Concrete Structures - August 2014, Dr. Naveed Anwar
A30 - Variation in kl/r
kl/r=14.5 kl/r=28.9 kl/r=38.1
kl/r=47.7 kl/r=57.3
55. CE 72.52 – Advanced Concrete Structures - August 2014, Dr. Naveed Anwar
Variation of Moment Magnification with Axial Load for Various
kl/r ratios
0
0.5
1
1.5
2
2.5
3
0.20 0.30 0.40 0.50 0.60 0.70 0.80
MomentMagnificationFactor
kl/r=28.9
kl/r=38.1
kl/r=47.7
kl/r=57.3
kl/r=14.5
Normalized Axial Load Pu/Pno
A30 – Moment Magnification
30 cm
30 cm
56. CE 72.52 – Advanced Concrete Structures - August 2014, Dr. Naveed Anwar
• Column Cross-Section = 40cmx40cm reinforced with 6-d20
• Connecting Members
– Beam on Right:
• Length = 5 m
• Cross-section = 30cmx50cm
– Beam on Left:
• Length = 3 m
• Cross-section = 30cmx50cm
– Column Above
• Length = 3m
• Cross-section = 40cmx40cm
• Fixed at Base
• The column is part of a non-sway structure
57. CE 72.52 – Advanced Concrete Structures - August 2014, Dr. Naveed Anwar
B40 - Variation in kl/r
kl/r=29
kl/r=43.4kl/r=36.2
kl/r=22kl/r=11
58. CE 72.52 – Advanced Concrete Structures - August 2014, Dr. Naveed Anwar
Variation of Moment Magnification with Axial Load for
Various kl/r ratios
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Normalized Axial Load Pn/Pu
MomentMagnification
Factor
kl/r=11
kl/r=22
kl/r=29
kl/r=36.2
kl/r=43.4
B40 – Moment Magnification
40 cm
40 cm
59. CE 72.52 – Advanced Concrete Structures - August 2014, Dr. Naveed Anwar
• Column Cross-Section = 80cmx80cm reinforced with 6-d20
• Connecting Members
– Beam on Right:
• Length = 5 m
• Cross-section = 30cmx50cm
– Beam on Left:
• Length = 3 m
• Cross-section = 30cmx50cm
– Column Above
• Length = 3m
• Cross-section = 40cmx40cm
• Fixed at Base
• The column is part of a non-sway structure
60. CE 72.52 – Advanced Concrete Structures - August 2014, Dr. Naveed Anwar
C80 - Variation in kl/r
kl/r=11.2kl/r=5.5 kl/r=14.9
kl/r=22.4kl/r=18.6
61. CE 72.52 – Advanced Concrete Structures - August 2014, Dr. Naveed Anwar
C80 – Moment Magnification
Variation of Moment Magnification with Axial Load for Various
kl/r ratios
0.99
1
1.01
1.02
1.03
1.04
1.05
1.06
1.07
1.08
1.09
0.20 0.30 0.40 0.50 0.60 0.70 0.80
Normalized Axial Load Pn/Pu
MomentMagnification
Factor
kl/r=5.5
kl/r=11.2
kl/r=14.9
kl/r=18.6
kl/r=22.4
80 cm
80 cm
62.
63. CE 72.52 – Advanced Concrete Structures - August 2014, Dr. Naveed Anwar
• Special Considerations
– Limited size and shape due to architectural and space constraints
– Generally very high axial load, specially in lower floors of high rise
buildings
– Consideration of differential axial shortening
– Consideration of slenderness effects, specially in sway (unbraced)
frames
– Presence of biaxial moments in the corner columns due to gravity loads
and all columns due to diagonal wind or seismic load direction
– Use of high strength concrete and related special considerations
– Requires high ductility in seismic zones
64. CE 72.52 – Advanced Concrete Structures - August 2014, Dr. Naveed Anwar
• Step1. Carry out frame analysis separately for all major load
cases
– Dead loads
– Live loads
– Wind loads
– Seismic loads
• Step 2. Select a “Critical” floor
– (maximum height, maximum loads, maximum deflection etc.)
65. CE 72.52 – Advanced Concrete Structures - August 2014, Dr. Naveed Anwar
• Step 3. Calculate “Factored Load” for various load
combinations
• 1) 1.4D +1.7L
• 2) 1.05D +1.3L +1.3W
• 3) 0.9D +1.3 W
66. CE 72.52 – Advanced Concrete Structures - August 2014, Dr. Naveed Anwar
• Step 4. For each load combination, Check sway conditions
CU
U
lV
P
Q 0D
averageheightstoreyClearl
VVVV
PPPP
C
UUUU
BT
UUUU
DDD
.......
......
321
0
321
PU1
PU2 PU3
PU4
VU1VU1VU1VU1
DT
DB
lC
CaseSwayQ
caseswayNonQIf
:05.0
:05.0
Equation 10-10
ACI 318-11
67. CE 72.52 – Advanced Concrete Structures - August 2014, Dr. Naveed Anwar
• Step 5. Determine Magnified Moment for each load combination
– If combination is non-sway then Mm =M dns
– If combination is “Sway” then Mm =Mns + Ms ds
– usually 1.05D +1.3L+1.3W
Non-sway part of
combination 1.05D + 1.3L
Sway part of
combination 1.3W
68. CE 72.52 – Advanced Concrete Structures - August 2014, Dr. Naveed Anwar 68
69.
70. CE 72.52 – Advanced Concrete Structures - August 2014, Dr. Naveed Anwar
• The program can include the P-Delta effects in almost all Non-
linear analysis types
• Specific P-Delta analysis can also be carried out
• The P-Delta analysis basically considers the geometric
nonlinear effects directly
• The material nonlinear effects can be handled by modification
of cross-section properties
• The Buckling Analysis is not the same as P-Delta Analysis
• No magnification of moments is needed if P-Delta Analysis has
been carried out