CE 72.52 Lecture 4 - Ductility of Cross-sections

1
CE 72.52 Advanced Concrete
Lecture 4:
Ductility of
Cross-sections and
Members
Naveed Anwar
Executive Director, AIT Consulting
Director, ACECOMS
Affiliate Faculty, Structural Engineering, AIT
August - 2015

Key-1
Ductility
is the Key to good
(seismic) performance of
Structures
Performance Based Design Relies on Ductility
2

Typical Force-Displacement
or Stress-Strain
3

• Ductility can be defined
as the “ratio of
deformation and a given
stage to the maximum
deformation capacity”
• Normally ductility is
measured from the
deformation at design
strength to the maximum
deformation at failure
Ductility – Definition and Usage
Yield/
Design
Strength
Load
Deformation
Dy Du
Ductility = Du / Dy

Ductility
• The ability to sustain deformation without
fracture/failure
• Ductility Ratio
• Ductility Levels
• Material Level
• Cross-section Level
• Member Level
• Structure Level

Structure Stiffness and Response
Deformation
Force
Curvature
Moment
Section Stiffness
Member Stiffness
Structure Stiffness
Material Stiffness
Structure Geometry
Member Geometry
Cross-section Geometry
Rotation
Moment
Strain
Stress

Action – Deformation Curves
• Relationship between action and corresponding deformation
• These relationships can be obtained at several levels
• The Structural Level: Load - Deflection
• The Member Level: Moment - Rotation
• The Cross-section Level: Moment - Curvature
• The Material Level : Stress-Strain
• The Action-Deformation curves show the entire response of the
structure, member, cross-section or material

Action Deformation Curve
• The entire response of structure or a member can be
determined, in an integrated manner from the Action-
Deformation Curve
DEFORMATION
LOAD
P
A
B C
D
O-A - Serviceability Range
A - Cracking Limit
B - Strength Limit
C-D - Failure Range
O

Limiting Points on Load Deformation Curve
• A - The point up to which the relationship
between load and deformation can be
considered nearly linear and the
deformations are relatively small
• B - The point at which the deformation
starts to increase suddenly, at more or less
constant load value or with relatively small
increase in the load
• C - The point at which the load value
starts to drop with increasing
deformations
• D - The point where load value become
nearly zero and member loses all capacity
to carry any loads and collapses or fails
completely

Design Stages
• Region OA corresponds to the
serviceability design considerations
and working strength or allowable
strength design concepts related to
linear, small deformation state
• Point ‘A’ roughly corresponds to the
ultimate strength considerations or
the design capacity consideration
based on the material strength or
material yielding criterion

Design Stages
• Point ‘B’ roughly corresponds to the
maximum load carrying capacity of the
member and is a measure of the maximum
load based performance level. This point is
often called as a measure of ductility in
many cases.
• Point ‘C’ is a clear indication of
deformation based performance of the
member, but is rarely used in actual design
considerations due to a certain level of
uncertainty near that point
• Region A-C represents the ductility of the
system

What Effects Ductility!
• The most important factor effecting ductility of
reinforced concrete cross-section is the confinement of
concrete
• Amount of confinement steel
• Shape of confinement steel
• Other factors include:
• Presence of Axial Load
• Stress-strain curve of rebars
• Amount of rebars in tension
• Amount of rebars in compression
• The shape of cross-section

How to Get Action-Deformation Curves
• By actual measurements
• Apply load, measure deflection
• Apply load, measure stress and strain
• By computations
• Use material models, cross-section dimensions to get
Moment-Curvature Curves
• By combination of measurement and computations
• Calibrate computation models with actual measurements
• Some parameters obtained by measurement and some by
computations

Material Level Ductility
• This is measured from the material stress-strain curve
• The stress-strain curve is obtained from testing of
material, in tension or in compression
• Materials with low ductility will generally produce
cross-sections of low ductility

Material Ductility - Steel
• Various Stress-Strain Curves for Steel reinforcement and steel sections


y h su
syf
suf
Parabola


y h su
syf
suf
Parabola


syf
y su

syf
y su


y h su
suf


y h su
suf


y h su
syf
suf
Parabola


y h su
syf
suf
Parabola
Various Stress-Strain Curves for Steel reinforcement and steel sections.

Material Ductility - Concrete
• Stress- Strain Relation as given in British code










cc cu
ccf  cuf 


cc cu
ccf  cuf 
Stress-Strain Relation for
Confined Concrete
Stress-Strain Relation for
Concrete after Whitney
cf 
uf 
Stress-Strain Relation as given in British code
General Stress Strain curve


Stress-Strain Relation for Un
Confined Concrete
cc
cf 


Stress-Strain Relation for Un
Confined Concrete
cc
cf 


0.0035
m
cuf

4
104.2 

m
cuf

67.0


0.0035
m
cuf

4
104.2 

m
cuf

67.0
cf 85.0

Various Concrete Confinement Models
18

Key - 2
Confinement
is the Key for Ductility in
Reinforced Concrete Members
19

Some proposed stress-strain curves for
concrete confined by rectilinear ties
20

Concrete Behavior and Confinement
• Unconfined Concrete Stress-Strain Behavior

• Idealized Stress-Strain Behavior of Unconfined Concrete

Due to spiral reinforcement, triaxial compression increase the strength of concrete. From
experiments, it is found
21 1.4 fff c 
Role of spiral reinforcement

• Confined Concrete Stress-Strain Behavior

• Idealized Stress-Strain Behavior of Confined Concrete

Comparison of Confine and Un-Confined
Concrete
• Unconfined Concrete Stress-Strain
Behavior
• Confined Concrete Stress-Strain Behavior

Behavior of spirally reinforced & tied
columns

Role of Reinforcement Splices in
confinement

Moment Curvature
Relationships

Cross-Section Ductility
• Cross-section ductility is governed by the materials
used, their distribution, cross-section shape and
dimensions as well as loads
• Axial-Flexural cross-section ductility is often
determined from Moment Curvature Curve

Key-3
Moment Curvature Relationship
is the Key for computing
Cross-section and Member Ductility
32

Moment Curvature Relationships
• Curvature:
• In geometry, it is rate of change of rotation
• In structural behavior, Curvature is related to Moment
• For a cross-section undergoing flexural deformation, it can
computed as the ratio of the strain to the depth of neutral axis

R
dP
P
CrackSteel
Neutral axis
M M
Steel
Es
Ec
kd
dx

dkdkdR
kdkdR
kd
dx
kd
dx
R
dx
scsc
sc
sc











)1(
1
)1(
1
)1(

• Curvature:
• The curvature will actually vary along the length of the
member because of the fluctuation of neutral axis and the
strains between the cracks.
• If the element length is small and over a crack, the curvature is
given by
• The relationship between moment M and curvature Ø is given
by the classical equation
• Significant information can be obtained from Moment
Curvature Curve to compute: Yield Point, Failure Point,
Ductility, Stiffness, Crack Width, Rotation, Deflection, Strain
dkdkdR
Curvature scsc 





)1(
1
)(

M
MREI 

First CrackFirst Crack
First yield of steel
reinforcement (Unconfined Concrete)
Crushing of concrete
commences before steel
yields
Moment MMoment M
Curvature Curvature
M M
Under-Reinforced Section Over-Reinforced Section

• With increase in moment, cracking of the concrete
reduces the flexural rigidity (EI) of the section, the
reduction of rigidity is higher for under-reinforced
section than over reinforced section
• For under-reinforced concrete section, the M-φ
relationship can be idealized by tri-linear relationship.
The first stage cracking, second yielding and third to
the limit of useful strain of the concrete.
• Over reinforced section shows the brittle failure unless
confined by closed stirrup.

• In many cases, the M- φ relationships can be idealized
as bilinear relationship which give progressive degree
of approximation.
• Once cracks have developed , as would be the case in
most beams under service loading, M- φ relationship is
nearly linear from zero to the onset of yield. Therefore,
the bilinear M- φ relationship can be approximated for
initially cracked beams.

First Crack
First yield of steel
reinforcement
Moment M
Curvature
Moment M
Curvature
Mu
Tri-linear M- φ
Relationship
Idealized bilinear M- φ
Relationship

Determination of M-Fi Curve
• The main idea behind generation of the moment
curvature curve is to obtain the neutral axis depth and
the corresponding strain at the compression extreme
for a given set of axial load and moment.
• There is no direct solution possible and an iterative
approach needs to be used.
• It is often easier to fix the strain first and iterate on the
depth of neutral axis until equilibrium with the axial
load is achieved. The corresponding moment capacity
at that depth of neutral axis and strain level is then
used, along with the curvature at that point.

• The curvature is simply the ratio between the strain and
the depth of neutral axis. This curvature is measured in
the units of radians/length units used to define the
neutral axis depth.
• Once one moment-curvature set is obtained, the
extreme fiber strain is changed and another solution is
attempted to obtain yet another pair of moment and
curvature.
• Several points are computed, using a small strain
increment to plot a smooth curve.

• The generation of moment curvature curve can be
terminated based on any number of specific conditions
such as,
• The maximum specified strain is reached
• The first rebar reaches yield stress a any other strain level
• The concrete reaches a certain strain level.
• Also, during the generation of the moment curvature curve
the failure or key response points can be recorded and
displayed on the curve.

Determination of Axial Load-Shortening Curve
• The axial load-shortening curve can be generated in a
manner similar to that described for the generation of
moment curvature curve.
• However, in this case the iteration to determine the
depth of neutral axis is not needed, as the neutral axis
is assumed to be horizontal, in the absence at any
moment.
• The strain is incremented and at each increment of
strain, the corresponding axial load is determined using
the appropriate material models.

Outputs from M-Phi Curve
y
u
Ductility


3 -
1 -Yield Point
2 -Failure Point

46
• 4 - Stiffness of the Section at given M and Phi
• 5 - Slope of the section at given Moment


M
EI
EI
M


dx
EI
M
b
a


47
• 6 - Deflection of the section at given Moment
• 7 - Strain at given Moment
dxx
EI
M
b
a
 






c 
c = distance from the
the point where strain
required

48
• 8 - Crack Width at given crack spacing
• 9 - Crack Spacing at given crack width
XW
XW
y
s




y
s
W
X
W
X




Specified Crack Spacing = X
y
s

Rebar Centroid
NA
W

CE 72.52 – Advanced Concrete Structures - August 2012, Dr. Naveed Anwar
Plot M-Phi Curve
Determine curvature
at known moment
Determine Flexural
Stiffness (EI)
Determine Slope
Determine Deflection
Determine Strain
Determine Crack
Spacing/Width

M
EI 
dx
EI
M
b
a

dxx
EI
M
b
a
 






c 
XW s
s
W
X


Outputs from M-Phi Curve - Summary

Outputs from M-Phi Curve - Example
50
• For M=600 Phi = 0.00006
• From M-Phi Diagram
• EI=600x12/0.00006
• EI=1.2E8 k-in^2
• Slope at Mid Span
• =600x7.5x144/1.2E8
• =0.0054 rad

M
EI 
15 ft
P=160 K
M=600 k-ft
L/2
36 in
24 in
dx
EI
M
b
a


51
• Deflection at Mid Span
• From M-Phi Diagram
• =600x7.5x144x15x12/(6x1.2E8)
• =0.162 in
• Strain in Steel
• M = 600 k-ft, y=16
• =0.00006x16
• =0.00096
y
s

Rebar Centroid
NA
W
dxx
EI
M
b
a
 






c 

52
• Crack Width
• Assuming crack spacing of 18 in
• =0.00096 x 18
• =0.01728 in
• Crack Spacing
• Assuming crack width of 0.02 in
• =0.02/0.00096
• =20.8 in
XW s
s
W
X


y
s

Rebar Centroid
NA
W

Ductility of Unconfined
Beam Sections

Reinforced Concrete Behavior

Ductility of Unconfined Beam Sections
• The ductility of a member is usually expressed as the
ratio of the ultimate deformation to the deformation at
first yield.
• Taking Doubly Reinforced Concrete Beam Section
u
y
Ductility




• Doubly reinforced beam section with flexure (a) at first yield (b) at ultimate
(a) (b)

• At first yield
• If concrete stress at extreme compression fiber does not
exceed 0.7fc’ when the steel reaches yield strength, the depth
to the neutral axis may be calculated using elastic theory
formula, and hence the M-φ value for first yield can be
calculated.
)1(
/
)'(
''
2)'(
2/1
22
kd
Ef
jdfAM
nn
d
d
nk
sy
y
ysy





















• At first yield
• If concrete stress exceed 0.7fc’, then the neutral axis depth at
first yield of the tension steel should be calculated using
actual curved stress-strain curve of concrete (parabola),
however for approximation can be obtained from straight line
formula.

Stress-strain distribution for same compressive force in
concrete when steel reaches yield stress.
Curved Concrete
stress distribution
Shaded areas are
equal
εs=fy/E
Triangular Stress
Distribution

• At ultimate state
• Condition (a) : when compression steel is yielding
s
y
ys
c
u
scu
c
ysys
E
f
fAAsfy
bf
dc
SteelnCompressioAt
a
c
c
c
ddfyA
a
dabfM
bf
fAfA
a





























'
'
1
1
''
'
'
85.0
'1
)'(
2
85.0
85.0




• At ultimate state
• Condition (b) : when compression steel is not yielding then to
determine neutral axis a quadratic equation need to be
solved
a
c
c
c
dd
a
da
EAs
a
dabfcM
foundbecanaeauationthisSolving
df
dE
f
fyE
d
a
d
a
u
scu
c
sc
c
sc
1
1
'
1
'
2
)'(
'
2
'85.0
"",
0
7.1
''
7.1
'
2
1




















 







• Ductility:
• At first yield:
• At ultimate state:
• IF
• (a) Compression steel yield
• (b) Compression steel does not yield
1/
)1(
/ 



a
kd
Ef sy
c
y
u 
























2/1
22
2
'
1 ''
2)'()'(1
)'(
85.0
n
d
d
nn
f
fE
y
cs
y
u c






'
2/1
'
1
2
'
2/1
22
1
7.1
'
85.0
''
7.1
'
''
2)'()'(1
c
ycs
c
cs
c
ycs
y
cs
y
u
f
fE
df
dE
f
fE
n
d
d
nn
f
E





















 















• From the above equations:
• 1) An increase in tension steel content decreases the ductility.
• - both k and a increased =>φy increased and φu decreased
• 2) An increase in the compression steel content increases the
ductility
• both k and a decreased=>φy decreased and φu increased
• 3) An increase in the steel yield strength deceases the ductility
• both fy/Es and a increased =>φy increased and φu decreased
• 4) An increase in the concrete strength increases the ductility
• both k and a decreased=>φy decreased and φu increased
• 5)An increase in the extreme fiber concrete strain at ultimate
increases the ductility because φu increased

Ductility of Unconfined
Column Sections

Ductility of Unconfined Column Sections
• The curvature of the section is influenced by the axial
load, hence there is no unique M-φ relationship for a
given column section.
• However, it is possible to plot the combination of axial
load P and Moment M which cause the section to
reach the ultimate capacity.
• It is evident that the ductility of the column section is
significantly reduced by the presence of axial load.
• The axial load levels greater than the balanced failure
load, the ductility decreases, being due only to the
inelastic deformation of the concrete.

• At the levels of load less than the balance load, the
ductility increases as the load level is reduced.
• Because of the brittle failure of the unconfined columns
at moderate axial load, ACI code recommends that the
ends of the columns in ductile frame in earthquake
areas be confined by closely spaced transverse
reinforcement when axial load is greater than 0.4 times
balanced load.

• The curvature of the section is influenced by the axial load
Interaction diagram
(Blume et al., 1961)
P/Po
1.0
0.8
0.6
0..4
0.2
0.0

• The curvature of the section is influenced by the axial
load,
Strength and ductility of section (Blume et al., 1961)
• At the levels of
load less than the
balance load, the
ductility
increases as the
load level is
reduced.

Ductility of Confined
Beam/Column Sections

Ductility of Confined Beam/Column
Sections
• The concrete section may fail in brittle manner if there
is not enough confinement to the concrete
• If the compression zone of a member is confined by
closely spaced transverse reinforcement in the form of
stirrups, ties , hoops or spirals, the ductility of the
concrete may increased significantly.
• When compressive stress approaching the compressive
strength of concrete, the transverse strains in the
concrete increased rapidly and the concrete expands
against the transverse reinforcement .

Ductility of Confined Beam/Column
Sections
• The retaining pressure applied by the reinforcement to
the concrete considerably improves the stress-strain
behavior of the concrete at higher strain. Thus, helps to
improve the ductility of the member.
• Circular spirals confine the concrete more effectively
than rectangular stirrups, ties or hoops because
confining steel in the shape of circle applies a uniform
radial pressure to the concrete, whereas a rectangle
tends to confine the concrete mainly at the corner.

M-Ø Relationship and Ductility
• Effect of Axial Load
• Effect of Reinforcement ratio
• Effect of Compression Steel
• Effect of Confinement Model
• Effect of Confinement Shape

Confinement Model
75
ACI Whitney, Not Confined Mander, Confined

Axial Load and Ductility
12#8 bars

Reinforcement ratio and ductility -
Reinforcement in tension

Effect of ratio of Tension to Compression
Reinforcement on Moment
79

Effect of ratio of Tension to Compression
Reinforcement on curvature
80

Compression Steel and Ductility
8#8 bars
a)
2#8 bars
8#8 bars
b)
4#8 bars
8#8 bars
c)
8#8 bars
8#8 bars
d)
8#8 bars

Confinement Model and Ductility
Effect of Concrete Confinement Model on Ductility of Cross-Section
0
50
100
150
200
250
300
350
0 0.001 0.002 0.003 0.004 0.005 0.006
Curvature (rad/in)
Moment(kip-ft)
Whitney Rectangle
Mander Circular Confined
Mander Pipe Filled
8#8 bars
Whitney Rectangle
(both)
a) b) c)
8#8 bars
Whitney Rectangle (outside)
Mander Circular Confined (inside)
8#8 bars
Whitney Rectangle (outside)
Mander Pipe Filled (inside)

Confinement Steel and Ductility
Effect of Confinement Steel Spacing on Ductility
-20
0
20
40
60
80
100
120
140
160
-0.0005 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025
Curvature (in/rad)
Moment(kip-ft)
Spacing = 3in
Spacing = 6 in
Spacing = 12 in
8#6 bars
Mander’s Rectangular
Confined
a)

Confinement Shape and Ductility
8#6 bars
Mander’s Rectangular
Confined
8#6 bars
Mander’s Circular
Confined
a) b)
8#6 bars
Whitney Rectangle
a)

Strategies to Improve Ductility
• Use low flexural reinforcement ratio
• Add compression reinforcement
• Add confining reinforcement

Other Functions of Confining Steel
• Acts as shear reinforcement
• Prevents buckling of longitudinal reinforcement
• Prevents bond splitting failures

Moment redistribution and
plastic hinge rotation

Limit Design
• The limit design approach allows any distribution of
bending moments at ultimate load to be used, provide
the following conditions are met.
• The distribution of bending moments is statically
admissible. That is, the bending moment pattern
chosen does not violate the laws of equilibrium for the
structure as a whole or for any member of it.
• The rotation capacity of plastic hinge regions is
sufficient to enable the assumed distribution of
moments to be developed a the ultimate load.

Limit Design
• The cracking and deflection at the service load are not
excessive.
• These requirements can be stated as limit equilibrium,
rotation compatibility ,and serviceability .

Limit Design Methods
• An example of possible limit bending moment diagram
for a continuous beam with ultimate uniformly
distributed load wu per unit length.
• We can have an infinite number of useable positions
for the fixing moment line, because a section can be
reinforced to give ultimate resisting moment as
required.

• For instance, we can set all supports moment wul2/16,
and the required maximum positive moments for the
interior spans and end spans are wul2/16 and
0.0958wul2
wu per unit length
Fixing moment line
Free bending moments
Wul2/8

• The advantage of limit design is patterns of moments
can be chosen to avoid congestion of reinforcement at
the supports of the members.
• Also, substantial economies can be result from
designing to moment obtained by dividing free
bending moments between the negative and positive
moments, rather than designing to the peaks of
bending moments found form the elastic theory
moment envelop for different position of loading.
• The method also gives the designer an appreciation of
the real behavior of the structure.

Design for seismic loading
• Chapter 21 of AC1 318-11 provides necessary guidelines for
design of earthquake resistant buildings.
• Special provisions for longitudinal and transverse
reinforcement
• Provisions about
• Rebar spacing
• Splices
• Hooks and bend angles

Sections After Strengthening

CE 72.52 Lecture 4 - Ductility of Cross-sections

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CE 72.52 Lecture 4 - Ductility of Cross-sections