ductility and robustness of concrete structures
TRANSCRIPT
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Ductility and Robustness of
Concrete Structures
Professor Ian GilbertProfessor Ian Gilbert
Centre for Infrastructure Engineering and SafetyCentre for Infrastructure Engineering and Safety
The University of New South WalesThe University of New South Wales
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� Ductility is the ability of a structure or structural member to undergo
large plastic deformations without significant loss of load carrying
capacity.
Introduction
Curve A - Slab S8 (As /bd = 0.0038 Class N bars) - Ductile
25
20
App
lied
Loa
d, (
kN)
DUCTILE BEHAVIOUR
Curve B - Slab S2 (As /bd = 0.0029 Class L mesh) - Brittle
0 40 80 120 160 200
Mid-span Deflection (mm)
15
10
5
0
App
lied
Loa
d, (
kN)
NON-DUCTILE BEHAVIOUR
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Why is Ductility Required in Structures ?
1) to give warning of impending collapse by the development of large
deformations prior to collapse;
2) to enable actions in indeterminate structures to redistribute themselves
near ultimate so that intentional and unintentional deviations from the
‘true’ distribution can be accommodated;
3) to ensure that many of the assumptions routinely made in the analysis
and design of concrete structures are reasonable and appropriate;and design of concrete structures are reasonable and appropriate;
4) in seismic regions, to enable major distortions to be
accommodated and energy to be absorbed without collapse during an
earthquake; and
5) to assist in providing ‘robustness’ (the ability to withstand
unforeseen local accidents without collapse).
Points 2), 3) and 4) have been extensively researched, while the
requirements for 1) and 5) are more difficult to quantify.
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Ductility levels in Reinforced Concrete
� Ductility may be considered at three levels:
(1) Material ductility which depends on the properties and strength
of concrete and steel (i.e. the stress-strain laws)
- Concrete is generally regarded as non-ductile in both
tension and compression
100
120
- Steel reinforcement is generally regarded as ductile
(but this is not necessarily the case for wire mesh)
0 0.001 0.002 0.003 0.004 0.005 0.006Strain
0
20
40
60
80
100
Str
ess
(MP
a)
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800
600
400
StressStress(MPa)(MPa)
Typical StressTypical Stress--Strain Curves for ReinforcementStrain Curves for Reinforcement
εεεεsu
Class L welded wire fabric – non-ductile
fsu
fsy
Hot rolled bar - ductile
0.00 0.05 0.10 0.15 0.20
400
200
0
εεsusu = 0.015 = 0.015 εεsusu = 0.05= 0.05(Minimum ultimate elongations in AS3600-2009
for Class L and Class N, respectively)
StrainStrain
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Ductility levels in Reinforced Concrete
� Ductility may be considered at three levels:
(1) Material ductility which depends on the properties and strength
of concrete and steel (i.e. the stress-strain laws)
(2) Local ductility (or hinge ductility) which depends on the cross-
sectional properties, reinforcement quantities and position, and
the bond-slip relationship at the steel-concrete interface.
This refers to the ductility of the critical cross-section at the locationThis refers to the ductility of the critical cross-section at the location
of maximum moment and this depends on the magnitude of the
ultimate curvature and the length of the plastic hinge that can
develop as the peak moment is approached.
Local ductility is essential for the rotation at the peak moment
region necessary for moment redistribution, for finding
alternative load paths and for the development of the full
plastic capacity.
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Moment
Increasing Ast
Moment vs curvature for slabs with ductile reinforcement
Ast
d
(Mu1, κu1)
(Mu2, κu2)
A more heavily reinforced section, Ast2 > Ast1
Curvature, κ
Hinge length, ℓh ≈ dMulti-crackhinges are typical
Rotation of hinge = ℓh x κu
A ductile under-reinforced section – Ast1
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Ast
dMoment
Increasing Ast
Moment vs curvature for slabs with non-ductile reinforcement
Hinge length, ℓh ≈ 2db to 5db(Mu1, κu1)
(Mu2, κu2)
db
Ast2
Curvature, κ
Hinge length, ℓh ≈ 2db to 5dbSingle-crack hinges are typical in slabs containing WWF
Rotation of hinge = ℓh x κ
u1 κu1
Ast1
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If the hinge ductility is sufficiently high (i.e. if the rotation at the plastic hinges is sufficient), a mechanism can develop and the fullplastic capacity of the system can be realized.
Ast1
A
Ast2
P
Ast3
Bending Moment Diagram
Mu.1
Mu.3
Mu.2
Rotation is required at left support for load to increase from P1 to P2
Rotation is required at both left and right supports for load to increase from P2 to P3
P = P1P = P2P = P3(collapse load)
As P increases, the first hingeforms at the left support when P = P1
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Load
Load vs deflection for under-reinforced continuous slab with ductile reinforcement
PSecond hinge formation
P2
Formation of mechanism
P3
Moment redistribution
DeflectionFirst cracking
Further cracking at service loads
First hinge formation (at overload)P1
Second hinge formation
This deflection is often relatively small
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Ductility levels in Reinforced Concrete
� Ductility may be considered at three levels:
1) Material ductility which depends on the properties and strength
of concrete and steel (i.e. the stress-strain laws)
2) Local ductility (or hinge ductility) which depends on the cross-
sectional properties, reinforcement quantities and position, and
the bond-slip relationship at the steel-concrete interface.
3) Structural or system ductility .3) Structural or system ductility .
Structural ductility is related to the plastic deformation of the
member or structure after the plastic mechanism has formed
(i.e. after all the plastic hinges have developed).
Structural ductility affects the robustness of the structural system
and its ability to absorb and dissipate inelastic energy without
substantial loss of load resistance and without jeopardising the
integrity or stability of the overall structural system.
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Load
Load vs deflection for under-reinforced continuous slab with ductile reinforcement
P
P2
P3
The hinge rotation required to facilitate moment redistribution and produce a mechanism is usually small compared to the hinge rotation required to produce the plastic deformation associated with adequate structural ductility
Pmax
Deflection
P1
∆1 ∆2 ∆u
Structural ductility
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Ductile slab after ultimate load test
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Load Pmax, ∆u
Ductility and warning of overstress
� In order to provide warning of failure, it is necessary for a member to
have a strain-hardening response, not merely for it to be ductile.
� The visible damage or deformation that is going to provide warning of
failure must occur at loads slightly below the collapse load.
Deflection
Pmax, ∆uP3
i.e. Pmax must be greater than P3 and ∆u must be much greater than ∆2
∆2
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Ductility and warning of overstress
� Codes of practice generally provide little design guidance (although it is
often argued that the limit on the neutral axis depth for beams and slabs
at the ultimate limit state provides an indirect means of providing
warning of failure).
� If warning of failure is required, it could be logically introduced into
codes by requiring that beams and slabs be able to sustain their design codes by requiring that beams and slabs be able to sustain their design
loads while deflecting to some specified amount – say span/50 – a highly
visible deflection.
A span/50 mid-span deflection corresponds to a support rotation of 0.04
rad or a rotation capacity of the mid-span section of 0.08 rad.
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Ductility and Robustness
� Robustness is the requirement that structures should be able to
withstand damage without total collapse.
“A structure should be detailed so that it can withstand an event
without being damaged to an extent disproportionate to that event.”
� The requirement that surrounding members should not rupture and
collapse requires that the members and materials have adequate
ductility.
� In particular, reinforcement, which is assumed in the design to constitute
the ties in rc structures, must obviously be highly ductile.
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Slabs containing low ductility reinforcement
� This rest of this presentation summarises a research project conducted
at the University of New South Wales on the strength and ductility of
reinforced concrete slabs containing low ductility reinforcement in the
form of Class L welded wire fabric (WWF).
� Over 50 slabs containing Class L reinforcement were tested.
� The project was funded by the Australian Research Council from 2003 to
2010.
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Simply-supported one-way slabs
Continuous one-way slabs
Corner-supported two-way slabs
Edge-supported two-way slabs
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Typical Behaviour of a Simply-Supported One-Way Slab
containing Class L Reinforcement
- this slab has the minimum reinforcement permitted by
AS3600-2009 (and L/D=22)
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A typical Continuous One-Way Slab containing
Class L Reinforcement
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Experimental Program – Two-way Slabs
Full range load tests on eleven two-way, corner-supported
reinforced concrete slab panels containing either Class L WWF
or Class N deformed bars were tested.
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� The slabs were subjected to transverse loads applied by a deformation
controlled actuator in a stiff testing frame.
� The results of the tests are presented and evaluated, with particular
emphasis on the strength, ductility and failure mode of the slabs.
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Experimental Program
Free edges
x
y
A ALy
Pinsupport
Rollersupport
d
D
dx
y
Section A-A
Ly = 2080 mmor 3280 mm
Rollersupport
Rollersupport
x
Free edges
Ly
L
PlanRoller support detail
Lx = 2080 mm
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Reinforcement ratio Slab ID
Ly (mm)
Lx (mm)
D (mm)
Steel Class & Type b
Bar dia.
(mm) px
(%) py
(%)
εsu
(%)
S2S-1 2080 2080 103.1 L - SL62 6.0 0.18 0.17 2.47 S2S-2 2080 2080 101.4 L - SL82 7.6 0.30 0.27 2.11 S2S-3 2080 2080 100.1 L - SL102 9.5 0.47 0.41 3.73 S2S-4 2080 2080 100.0 N - N12 12.0 0.51
Experimental Program
S2S-4 2080 2080 100.0 N - N12 12.0 0.51 0.44 7.69 S2S-5 2080 2080 106.1 N - N10 10.0 0.52 0.46 9.65 S2S-6 2080 2080 100.0 N - N12 12.0 0.48 0.44 14.11 S2R-1 3280 2080 103.7 L - SL62 6.0 0.17 0.16 2.46 S2R-2 3280 2080 95.9 L - SL82 7.6 0.30 0.27 2.30 S2R-3 3280 2080 106.7 L - SL102 9.5 0.42 0.38 3.09 S2R-4 3280 2080 100.0 N - N12 12.0 0.51 0.44 7.69 S2R-5 3280 2080 101.6 N - N10 10.0 0.55 0.48 9.65
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Load versus deflection
50
60
70
80
90
100
S2R-5
S2R-4
S2R-3
Tot
al L
oad
(kN
)
0 20 40 60 80 100 120 140 160 180 2000
10
20
30
40
50
S2R-2
Mid-panel deflection (mm)
Tot
al L
oad
(kN
)
S2R-1
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80
100
120
140
S2S-6
S2S-5
S2S-4
Tot
al L
oad
(kN
)
Load versus deflection
0 20 40 60 80 100 120 140 160 180 2000
20
40
60
S2S-4
S2S-3S2S-2
Mid-panel deflection (mm)
Tot
al L
oad
(kN
)
S2S-1
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40
50
60
70
80
Tot
al L
oad
(kN
)
Load versus deflection
S2S-2
S2R-2
0 10 20 30 40 50 60 70 80 90 100 110 120 130 1400
10
20
30
Mid-panel deflection (mm)
Tot
al L
oad
(kN
)
Slabs containing Class L meshpy = 0.30%
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SQUARE SLAB S2S-2 CONTAINING CLASS L MESH
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60
70
80
90
100
110
120T
otal
Loa
d (k
N)
Load versus deflection
S2S-4
S2R-4
punching shearfailure
0 10 20 30 40 50 60 70 80 90 100 110 120 130 1400
10
20
30
40
50
60
Mid-panel deflection (mm)
Tot
al L
oad
(kN
)
Slabs containing Class N barpy =0.44%
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Load versus deflection
0.78
1.04
1.30
1.56
1.82
2.08
0.00 0.26 0.52 0.78 1.04 1.30 1.56 1.82 2.08
0.78
1.04
1.30
1.56
1.82
2.08
1.1
Cle
ar s
pan
alon
g y-
axis
(m
)
1.3
0.00 0.26 0.52 0.78 1.04 1.30 1.56 1.82 2.08
0.78
1.04
1.30
1.56
1.82
2.08
Cle
ar s
pan
alon
g y-
axis
(m
)
33
0.00 0.26 0.52 0.78 1.04 1.30 1.56 1.82 2.08
0.00
0.26
0.52
0.00
0.26
0.52
Cle
ar s
pan
alon
g y-
axis
(m
)
Clear span along x-axis (m)
0.2
0.8
.06
(a) At a load of 35 kN before cracking.
0.00 0.26 0.52 0.78 1.04 1.30 1.56 1.82 2.08
0.00
0.26
0.52
Cle
ar s
pan
alon
g y-
axis
(m
)
Clear span along x-axis (m)
9
17
25
(b) At a load of 64 kN in the post-peak stage
Deflection contours of slab S2S-2.
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Ductility Ratio:
20
30
40S
pan
Load
, P (
kN)
W
Ductility ratio:
∆1 ∆2
0 10 20 30 400
10
Mid-span deflection (mm)
Spa
n Lo
ad, P
(kN
)
W0
W1 µw = (W1+W0)/W0
∆2
If µw < 2.0 – brittle (regarded as undesirable)
If µw > 5.0 – ductile (regarded as desirable)
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Ductility
3
4
5
6
Duc
tility
Rat
io, (
W0+
W1)
/W0
Ductility ratio versus uniform elongation
0
1
2
0 2 4 6 8 10 12
Uniform Elongation, ε su (%)
Duc
tility
Rat
io, (
W
1.5%
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Summary and Conclusions
1. The two-way corner supported slabs reinforced with Class L welded wire fabric fail in a brittle mode by fracture of the tensile reinforcement and, generally, not by crushing of the compressive concrete.
2. Two-way corner-supported slabs containing Class L welded wire fabric are unable to undergo significant welded wire fabric are unable to undergo significant plastic deformation without a significance reduction in the applied load. This is true for both square and rectangular slabs.
3. All slabs containing Class L welded wire fabric had ductility ratios (W1+W0)/W0 less than 2.0.
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Summary and Conclusions
The existing procedures for the design and analysis of reinforced concrete slabs at the ultimate limit state have been developed based on the assumption that the reinforcing steel is elastic-plastic with unlimited strain capacity.
This is not the case when using Class L reinforcing steel.
The brittle nature of the failure of the slabs containing
4.
5.
6. The brittle nature of the failure of the slabs containing Class L reinforcement has resulted in the recent change to the Australian Standard, AS3600-2009, wherein the strength reduction factor for slabs is effectively and appropriately reduced from φ = 0.8 for Class N steel to φ = 0.64 for Class L reinforcement.
Such a reduction in φ is consistent with the code approachfor non-ductile members where the ductility ratio (W1+W0)/W0 is less than 2.0.
6.
7.
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Edge-supported slabs
Plan
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Edge-
supported
slabs
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100
150
200
250
300
Tot
al A
pplie
d Lo
ad (
kN)
Peak load
Extensive yielding
First yield
Tot
al A
ppli
ed L
oad
(kN
)
W0 +W1
0
50
0 50 100 150 200Midspan Deflection (mm)
Mid-panel deflection (LVDT1) (mm)
First cracking Tot
al A
ppli
ed L
oad
(kN
)
Slab containing Class L Slab containing N barsS1R-1 S1R-2
W0
Ductility ratio:
µw = (W1 + W0)/W0 = 4.3
Ductility ratio:
5.7
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Two clamped edges
Clamped area
Clam
ped area
=
143
0
3600
2400
Clampededges
A A
B
Rollersupport
Freeedges
Clam
ped area
L
= 1
430
L = 2630
2400
B
y
x 160
1395 1395
Origin
y
Plan
Large deformation without collapse
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3
4
5
6
Du
ctili
ty R
atio
, (W
0+W
1)/W
0
Clamped on two sides
Supported on all four sides
0
1
2
0 2 4 6 8 10 12
Uniform Elongation, ε su (%)
Du
ctili
ty R
atio
, (W
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Concluding Remarks
The corner supported slabs reinforced with Class L have the same ductility issues as one-way slabs, collapsing much like a one-way slab with the steel in one direction fracturing in a single failure crack across the mid-span region in one direction.
1.
The six edge-supported slabs containing Class L tested at UNSW were surprisingly deformable. It appears that slabs reinforced with Class L perform more satisfactorily, as they become more redundant and there are more possible load paths.
2.
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Concluding Remarks
In the edge-supported slabs tested at UNSW, the failure load exceeded the yield-line load, as loads were carried by membrane action and torsion (as well as in bending).
There were many cracks at close centres in the peak moment regions, in contrast to the single crack hingesthat characterize one-way slabs.
3.
4.
that characterize one-way slabs.
The slabs deformed significantly and continued to carry load even after wires in particular areas fractured.
There was no sudden collapse in these very redundantedge-supported slabs and the ductility ratio is less dependent on the ductility of the reinforcement.
5.
6.
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