CE 72.52 - Lecture 5 - Column Design

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

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

CE 72.52 – Advanced Concrete Structures - August 2014, Dr. Naveed Anwar
Column BeamColumn or Beam?

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

• 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

• 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

• 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…)

Load-Shape-Slenderness
Shape
Loading
Length
V.Long
Long
Short
P
P Mx
P Mx My
Most Simple
Problem
Shape
Complexity
Load Complexity
Slenderness

Load-Bracing-Length
Bracing
Length
Loading
P

Load-Section-Material
Material
Section
Loading
P

CE 72.52 - Lecture 5 - Column Design

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

• 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

• 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
.)(
)(



• 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

• Mx-My curve is plotted at applied axial load, Pu
• Sum of the Ratios of Moment is each direction gives the
Capacity Ratio

• Mx-My curve is plotted at applied axial load
• Ratio of Muxy vector to Mnxy vector gives the Capacity Ratio

• P-M Curve is plotted in the direction of the resultant moment
• Ratio of PuMuxy vector to PnMuxy vector gives the Capacity Ratio

• 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.

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

• 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

• 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

• “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

• 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

• 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

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

• 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

• 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 - 

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




• 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

• 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

• Appreciable relative moment of two ends of column
• Sway Limits
c
BT
l
Sway
DD
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”

• 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

• 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

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

Basic Equation
Magnification Factor for Moments
that do not cause sway

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

Basic Equation
Magnification Factor for Moments
that cause sway

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

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

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

• 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

• 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

Load
Range
Length

• 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
– Column Above
• Length = 3m
• Fixed at Base
• The column is part of a non-sway structure

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

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

– Beam on Right:
• Length = 5 m
– Beam on Left:
• Length = 3 m
– Column Above
• Length = 3m
• Fixed at Base

B40 - Variation in kl/r
kl/r=29
kl/r=43.4kl/r=36.2
kl/r=22kl/r=11

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

– Beam on Right:
• Length = 5 m
– Beam on Left:
• Length = 3 m
– Column Above
• Length = 3m
• Fixed at Base

C80 - Variation in kl/r
kl/r=11.2kl/r=5.5 kl/r=14.9
kl/r=22.4kl/r=18.6

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

• 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

• 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.)

• 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

• Step 4. For each load combination, Check sway conditions
CU
U
lV
P
Q 0D

 averageheightstoreyClearl
VVVV
PPPP
C
UUUU
BT
UUUU


DDD

.......
......
321
0
321
PU1
PU2 PU3
PU4
VU1VU1VU1VU1
DT
DB
lC
CaseSwayQ
caseswayNonQIf
:05.0
:05.0


Equation 10-10
ACI 318-11

• 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

• 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

CE 72.52 - Lecture 5 - Column Design

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CE 72.52 - Lecture 5 - Column Design