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Behaviour of Complex Reinforced
Concrete Structures under Three-
Dimensional Loads
Ashraf Ayoub, PhD, PE, FACI
RAE Chair of Nuclear Infrastructure
Dept. of Civil Eng.
City University London
Research Centre for Civil Engineering Structures
January 13, 2016
OUTLINE
• Experimental Program and Results of Panels
subjected to 3D Loads
• Conclusion and On-Going Work
• Background and Objective
• FE Model of RC Elements subjected to 3D
Loads and Correlation Studies
x
y
z
Fig. 1 Three-Dimensional Panel Elements under Various Types of Stresses
Structures subjected to tri-directional state of stresses
1-Columns subjected to multi-directional loads
2-Beams subjected to bi-directional loads
3-Shells and plates subjected to combination of
membrane and transverse shear
BEHAVIOR OF RC COLUMNS
Shear Failure Bending Failure
Failure by
Combined Loadings
(Axial + Shear + Bending )
Kobe
Earthquake
Northridge
Earthquake in
California
Shinkansen bridge
Japan
P
R
T (http://www.wsdot.wa.gov/projects/viaduct/deis/chapter1_1.htm)
Seattle, WA, US
TORSION IN BRIDGE MEMBERS
Shear Stresses in Containment Wall
• Tangential Shear (in-plane)
• Peripheral Shear (punching)
• Radial Shear (out-of –plane)
Available Experimental Data for Concrete
Elements subjected to 3D Loads
• Tests conducted on cubes or cylinders using
hydraulic triaxial cells (e.g. K. Gurstle, and
Van Mier)
• Specimens are small-scale and unreinforced
• Specimens subjected to compressive loads
only
Experimental Panel Tests
(Adebar, University of Toronto)
0
1
2
3
4
5
6
7
8
9
10
-30 -20 -10 0 10 20 30
Toronto
SP7: (-10.48,2.62)
SP9: (-16.72,2.09)
SP4: (14.32,3.58) SP3: (0,2.93)
SP8: (-27.82,0) SP8: (27.82,0)
τxy / √fc
τ 23
/√f c
Normalized
No
rmal
ized
3D Universal Panel Tester
(University of Houston)
Experimental Program
Reinforcement elevation Reinforcement Sideview
8.0
4.5
30.0
4.5
8.0
16
Group B panels
2#4
2#4
2#4
2#4
A
B
C
D
7
16.0
12.5
11.5 0.5 10.0
0.5
55
55
0
200
400
600
800
1000
1200
0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016 0.018 0.020
In-Plane Shear Capacity of Panels
OP6 (72%)
OP5 (68%)
OP4 (46%)
OP2 (22%)
OP1 (13%)
OP3 (28%)
OP0
In-p
lan
e sh
ear
stre
ss (
psi
)
In-plane shear strain (in./in.)
• Concrete crushing at the south
bottom side
• Severe cracks and spalling of concrete
at the north side
Group B: OP4
South Side North Side
46% of out-of-plan shear capacity
• In-plane shear load/Jack=26 kip
0
1
2
3
4
5
6
7
8
9
10
0 5 10 15 20 25 30
Interaction Diagram between Out-of-Plane and In-Plane
Shear Strength
Normalized in-plane shear strength
No
rmal
ized
ou
t-o
f-p
lane
shea
r st
reng
th
τxy / √fc
τ 23
/√f c
OP0 (14.15,0)
OP2 (13.91,1.23)
OP1 (12.18,0.73)
OP3 (11.32,1.59)
OP4 (8.18,2.91)
OP5 (8.56,4.06)
OP6 (4.17,4.62)
OPR (5.65,0)
OPR
OP3
OP1
OP0
1.0
0.68
0.48
0.29
0.13
0.58 0.60
0.80 0.86
OP5
OP4
1.0 0.68 0.48 0.29
1.0
τxy
τxz
τyz
Interaction Diagram between Tri-directional Shear Stresses
SL
NB
South side
NT
North side
NR
SR
ST
SB
NL
1
2
3
45◦
x y
z
τxy
τxy
τyz
τxz
τxz τyz =
Concrete Behavior in Compression
Concrete in Compression
Softening Coefficient z
'ζ
p
cf
p
'
cf
is compressive strength in test
is compressive strength for cylinder test
15
9.08.5
'
'
c
cf
ff
1
14001
1
f
241f
z ffff c 1
Constitutive Relationships of Concrete
Concrete in Tension
1
1
1
1
when
c
crc cr crf
1 1 when c c crE
crf
'47,000 ( )psic cE f'3.7 (psi)5 ccr ff
0.00008cr
Concrete in Compression
2
2
'
cfz
0z
2
' 2 22 2 c
c
o o
fz
z z
2
2'
2
/ 11
4 / 1
c ocfz
z
z
c
c
16
12υ 0.2 850ε
sf ε ε
sf y
12υ 1.9 ε ε
sf y
Poisson Ratio for RC (Zhu and Hsu, 2002)
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05
Pang's series
Belarbi's series
OP series
Soft
enin
g co
effi
cien
t
Lateral horz. strain (in./in.)
Softening coefficient
100000178.01
241
4001
19.0
8.5
1
'
xx
fc
z
100ou
ox
OP2
OP3
OP5
OP4
OP1
OP6
Out-of-plane shear
Reduction factor
Macro Models
– Based on Frame-Type
Elements
– Suitable for System
Evaluation (e.g.
Buildings/Bridges)
Micro Models
– Based on 2D
Membranne/3D Brick
Elements
– Suitable for Component
Evaluation (e.g. Joints)
3D FEM Analysis of RC Structures
Nonlinear Frame Elements
Plastic Hinges
– Lumped Plasticity
– Computationally
Efficient
– Suitable for Evaluation
of Global Parameters
(e.g. Drifts)
– Not suitable for
Evaluation of Local
Details (e.g.
Reinforcement and
Concrete Strains)
– Input Parameters not
Easily Defined
Fiber Beam Elements
– Distributed Plasticity
– Suitable for Evaluation of
both Global and Local
Parameters
– Input Parameters based
on Basic Material
Properties
– Computationally more
Expensive
X
Y
Z
Y
Z
yj
zj
FEM Analysis - Fiber Element for RC Columns
22
Moment Mu Curvature κ
N
P
Mixed frame models assume “exact” force
Interpolation functions
Element result is exact for the beam theory
Same structural DOF’s as plastic hinge
Accurate in representation of curvature
localization at plastic zone
1 0 0
1 1( ) 0
0 1
N xL L
x x
L L
Force Interpolation
function
Section Force = 1
2
( )
N N
V N x M
M M
0 ( )
( ) ( )
( )
x N
x f x V
x M
( )f x = Flexibility Matrix
Strains
x
L
THEORETICAL MODEL : Mixed Formulation
FIBER ELEMENT - Accounting for Shear Effects
• Adopt a Timoshenko Beam Formulation with Shear
Deformations
• Impose Equilibrium in Transverse Direction to Determine
Concrete Lateral Strain
• Rotate Stresses to Principal (Crack) Directions
• Use 3D Concrete Constitutive Models
FIBER ELEMENT -Timoshenko Formulation
ys yc
Concrete Beam Stirrups
x
b
y
x
y
STIRRUP - Equilibrium in Transverse Direction
Equilibrium in Vertical Direction:
y
, ,, , 0y c y s
i i i i
y c y sA A , ,, , 0
z c z s
i i i i
z c z sA A
z
Ay,s Stirrup Area
Longitudinal steel Area
i
Ax,s
t
Az,s Stirrup Area
y
z
y
z
y’
z’ '
Stirrup
Fiberi
y
z
y’
z’ '
Stirrup
Fiberi
RC Circular Section (No Warping)
2 ' ' ' 2 '
y' yy yz zzε = cos (θ )ε - 2cos(θ )sin(θ ) sin (θ )ε
Fz xy Fy
xz
y' y'
2 '
y'
2 '
y'
= f(ε )
= cos (θ )
= sin (θ )
yy
zz
Fiber Element – Softened Membrane Model
Concrete Element in
Cartesian Coordinate System Concrete Element in Principal
1-2 Coordinate
of Applied Stress System
cracks
P Stress/Strain Curve in Principal
1-2 Coordinate of Applied Stress
Non Softened
Curve
Softened Curve
1
212
1
2
l
t
12
lt
lt
1 2
1
c
12
c 1
c
2
c
2
c
12
c
12 12
c G
ς
2D BIAXIAL STRENGTH ENVELOPE - Constitutive Model
Biaxial Compression region Compression/Tension region
Biaxial Tension region
Tensile strength is constant
ip i cf z
2
1 0.92 0.76ip ip
i
c cf f
z
2 p cf z
1 1 2 3cf f f fz
2
5.80.9c
c
f ff
3 1
24f
1 1
1
1
1 400f
FE Model I: In-Plane
Analysis under the
Effect of Out-of-Plane
Loads
Select
Assume
Assume
Calculate , and , Eqs. and
Calculate , , , , and Eqs. , , , and
Calculate and Eq. to
Calculate and from and
Calculate and Eqs. and
Calculate and Eqs. and
End
No (assume )
No (assume )
No Yes
Yes
Yes
2
12
1
t 11 12
zc
2 c
1 12c 15 14 13 16 17
f tf 18a 18e
1)( tt ff 1)( tt ff f tf
2)( tt ff 2)( tt ff 19 20
?0)()( 12 tttt ffff
?0)()( 12 tttt ffff
t t 3 6a
? Is 02
modified
Verified with the
experimental data
Prediction of Model for In-Plane Shear Strength of OP Series:
0
200
400
600
800
1000
1200
0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016 0.018 0.020
Experimental
Analytical
Shea
r st
ress
(p
si)
Shear strain (in./in.)
OP4 (46%)
OP2 (22%)
OP1 (13%)
OP3 (28%)
FE Analysis and Prediction Yoshimura 1996
Tested short RC columns under bi-directional shear loads
1- Lateral shear load equal to 0, 33, 55, 77 kip was applied and kept constant in
direction-2.
2- Shear load in direction-1 was applied and increased till failure.
FE Analysis and Prediction Kobayashi et. al. 1986
0
25
50
75
100
125
0 0.05 0.1 0.15 0.2 0.25 0.3
0
100
200
300
400
500
600
0 1 2 3 4 5 6 7 8
Lo
ad (
kip
)
Deflection (in.)
Lo
ad (
kN
)
Deflection (mm)
D00 experimental
D00 analytical
0
25
50
75
100
125
0 0.05 0.1 0.15 0.2 0.25 0.3
0
100
200
300
400
500
600
0 1 2 3 4 5 6 7 8
Lo
ad (
kip
)
Deflection (in.)
Lo
ad (
kN
)
Deflection (mm)
S35 experimental
S35 analytical
Fig. (7) 3D Concrete failure Surface
3D Concrete Constitutive Model
Menetrey-Willam (1995) 3-Parameter
Model
2
( , , ) ( , ) 0f f ff A B r e C cz z
2 2 2
2 1/ 2
4(1 )cos (2 1)( , )
2(1 )cos (2 1)
e er e
e e D
1.5 1 1; ;
6 3f f f
c c c
A B Cf f f
FE Model II: Stress De/Re-Composition Approach
Maekawa model (2003)
Vecchio and Selby Model (1991)
3D strain field is decomposed into
three in-plane sub-spaces
In each sub-space, the component stresses are
computed using the 2D crack model and
constitutive laws
3 3 1( , )f
1 1( )f
2 2 1( , )f
100000178.01
241
4001
19.0
8.5
1
'
xx
fc
z
xx
zz yy
xy
xz
36
Support Blocks
Strong Floor
Hydralic JackLoad Cell
Steel Strands
(Inside Column)
Strong Wall
Test Unit
Two Hydraulic
Actuators
Load Stub
Axial Force using Prestressing
(approximately 7 % of predicted axial capacity)
12 ft
2 ft
Belarbi et al.(2007)
'
cf = 4.85 ksi
Longitudinal yield stress = 66.4 ksi
Transverse yield stress = 65.2 ksi
MS&T Column
MS&T Column (Combined Bending/Torsion) – T/M =0.2
NUMERICAL RESULTS – Static Load Displacement
0
10
20
30
40
50
60
0 2 4 6 8 10
Displacement (in)
Lo
ad (
kip
)
Experiment T/M 0.2Analysis T/M 0Analysis T/M 0.2
Load - Displacement response matched well with the experiment
Increase of the torque reduced the load resisting capacity
-50
-40
-30
-20
-10
0
10
20
30
40
50
-10 -8 -6 -4 -2 0 2 4 6 8 10
Displacement (in)
Lo
ad
(k
ip)
Analysis T/M 0.4
Experiment T/M 0.4
MS&T Column (Combined Bending/Torsion) – T/M =0.4
NUMERICAL RESULTS – Cyclic Load Displacement
3D Element predicted well the cyclic response
NUMERICAL RESULTS – Earthquake Analysis
Axial Load = 356 kN
Concrete Strength = 36.7 MPa ksi
Spiral Strength = 397.4 MPa
Long. Bar Strength =458.2 MPa
Long. Steel Ratio = 3.5% (16 #6)
Spiral Ratio = 0.92% ( 0.25”, 1.5” C/C)
UNR Column - Laplace et al.
EL Centro (1940) record
Squat Column
Longitudinal
Reinforcement 3.5%
Shell Region
Core Region
406.4 mm Dia.
Transverse Reinforcement
0.92%
2.5 x EL Centro NS
NUMERICAL RESULTS – Time History
UNR Column- 9S1
-40
-30
-20
-10
0
10
20
30
40
0 5 10 15 20 25 30
Time (s)
Dis
pla
cem
ent
(mm
)Experiment
Flexural Element
Flexure Element is not able to predict the displacement
NUMERICAL RESULTS – Time History
UNR Column 9S1
-50
-40
-30
-20
-10
0
10
20
30
40
50
0 5 10 15 20 25 30
Time (s)
Dis
pla
cem
ent
(mm
)
Experiment
Shear Element
Shear Element is able to predict the displacements rather well
NUMERICAL RESULTS- Shear Element Load Displacement
UNR Column- 9S1
-500
-400
-300
-200
-100
0
100
200
300
400
500
-50 -45 -40 -35 -30 -25 -20 -15 -10 -5 0 5 10 15 20 25 30 35 40 45 50
Displacement (mm)
Bas
e sh
ear
(kN
)
ExperimentShear Element
Shear Element response is rather well
Arias-Acosta and Sanders (2010)
UNR Column- C1
'
cf = 32 MPa
0.6x Petrolia at Mendocino (1992)
Earthquake
X
Y
III. EXPERIMENT : Earthquake Analysis
UNR Column- C1
X C o m p o n e n t
-1 5 0
-1 0 0
-5 0
0
5 0
1 0 0
1 5 0
-4 0 -2 0 0 2 0 4 0 6 0
D ix p la c e m e n t (m m )
Ba
se
Sh
ea
r (
kN
)
E x p e r im e n t
A n a ly s is
Y C o m p o n e n t
-6 0
-4 0
-2 0
0
2 0
4 0
6 0
-3 0 -2 0 -1 0 0 1 0 2 0 3 0
D ix p la c e m e n t (m m )
Ba
se
Sh
ea
r (
kN
)
E x p e r im e n t
A n a ly s is
3D element prediction is fairly accurate compared to experimental values
ANALYTICAL STUDY : Load - Displacement
Ongoing Research
Evaluation of the behavior of concrete nuclear structures
under multi-hazards:
Long term effects: Temperature, Radiation,
Shrinkage, Creep, Corrosion
Natural and man-made hazards: Earthquake,
Impact/Blast, Loss of Coolant (LOCA)
Multi-Scale Analysis of Concrete Structures
Meso-Modeling for long term deterioration effects
Macro-Modeling for hazard loads
Improving the design and behavior of concrete structures
Experimental evaluation of new design systems
Enhancement of Concrete Properties
• Fibre Reinforced Concrete
• Steel Fibre Reinforced Concrete (SFRC)
• Carbon Nano-Fibre Reinforced Concrete
(CNFRC)
• Reactive Powder Concrete
• Engineering Cementitious Composites (ECC)
New Construction Techniques for Nuclear Structures:
Steel-Concrete (SC) Walls
FRCM/TRM Composites
Resin is made of inorganic materials
Suitable for use in high temperatures
Summary and Conclusion
• New test data for full scale reinforced concrete panels subjected to
tri-directional loads were obtained
• The softening coefficient for concrete subjected to tri-directional
load was established
• Based on the test results, a new finite element model for concrete
structures subjected to 3D loads was developed
• The model was implemented in FE fiber beam elements, and the
behavior of structures subjected to multi-directional loads was
accurately correlated with experimental results
Future Work
• Experimental and analytical evaluation of the constitutive law to
model FRC (using steel & CNF fibers)
• Evaluation and Improvement of the behavior of SC panels
• Simulation of Impact/Blast loads (SHPB tests, large deformation,
explicit algorithm)
• Modeling of bond-slip (RC & SC panels) and simulation of Steel
Buckling
• Behavior under elevated temperature – Multi-hazard analysis
• Evaluation of strengthening techniques using FRCM/TRM
composites