Non-Linear Modeling of Reinforced Concrete Structures for Seismic Applications
02/18/2010
Luis A. MontejoAssistant Professor
Department of Engineering Science and MaterialsUniversity of Puerto Rico at Mayaguez
OutlineJustificationExtended moment curvature analysisFinite Element Modeling:
Lumped plasticityDistributed plasticity / fiber basedDistributed plasticity / fiber basedFiber based lumped plasticity)
Non-linear static analysis (pushover)Non-linear time history analysis:
Seismic InputDamping
Application ExampleConclusion
When do we need non-linear modeling?Design verification of very important and/or unusual structures.
When do we need non-linear modeling?
Have you ever use a force-reduction factor (R) in your design?
WTn
0 2
0.3
0.4
0.5
0.6
0.7
cele
ratio
n [g
]
A [g]
Fel=A [g]*W
0 1 2 3 4 50
0.1
0.2
period [s]ac
cTn
Fel/R
Δ1
Δ1*Sd <=Δlimit
When do we need non-linear modeling?Have you ever use a force-reduction factor (R) in your design?
20
25
e
Fn (Mn) μ1
0 1 2 3 4 5 6 7 80
5
10
15
displacement
late
ral f
orce
first yield
φεc
εy
ΔFΔ1*Sd ~Δlimit
OutlineJustificationExtended moment curvature analysisFinite Element Modeling:
Lumped plasticityDistributed plasticity / fiber basedDistributed plasticity / fiber basedFiber based lumped plasticity
Non-linear static analysis (pushover)Non-linear time history analysis:
Seismic InputDamping
Application ExampleConclusion
Moment-Curvature Analysis
φεc
εy
Stre
ss
Strain0 0.01 0.02 0.03
0
10
20
30
40
50
60
: Confined Concrete: Unconfined Concrete
Stre
ss [M
Pa]
Strain-0.1 -0.05 0 0.05 0.1
-600
-400
-200
0
200
400
600concrete steel
Moment-Curvature Analysis
4000
5000
6000kN
-m)
0 0.05 0.1 0.150
1000
2000
3000
Curvature(1/m)
Mom
ent (
k
From M-Φ to F-Δ
(M) L
Δy ΔpP
actualidealized
φp φy
LpLsp
θp
structure and moment distribution curvature profile displacements
yppy
y
LL
L
φφφ
φφφ
>+Δ=Δ
≤=Δ 3/2
(Park and Priestley, 1988)
From M-Φ to F-Δ
2500
3000
3500
4000kN
)
0 0.05 0.1 0.15 0.20
500
1000
1500
2000
Displacement(m)
Forc
e (k
Shear CapacityV = Vp + Vs + Vc
revised UCSD shear model (Kowalsky and Priestley, 2000), illustration by Pablo Robalino
02
>−
= PL
cDPVp
Shear CapacityV = Vp + Vs + Vc
picture by Pablo Robalino
( )θcots
cdclbHfAV hyhsxs
−+−=
Shear CapacityV = Vp + Vs + Vc
ecs AfV 'αβγ=
α : aspect ratioβ : longitudinal reinforcementγ : aggregate interlock
Shear Capacity
2500
3000
3500
4000
kN)
2500
3000
3500
4000
kN)
2500
3000
3500
4000
kN)
0 0.05 0.1 0.150
500
1000
1500
2000
Displacement(m)
Forc
e (k
0 0.05 0.1 0.150
500
1000
1500
2000
Displacement(m)
Forc
e (k
0 0.05 0.1 0.150
500
1000
1500
2000
Displacement(m)
Forc
e (k
shear failure
Shear Capacity
45
90
-2.4 -1.6 -0.8 0 0.8 1.6 2.4[in]
200
400
e [k
N]
μ8
μ6μ4
-90
-45
0
[kip
s]
-60 -40 -20 0 20 40 60
-400
-200
0
displacement [mm]
late
ral f
orce
μ8
μ6 μ4
Onset of bucklingst
rain
characteristic capacity
flexural tension strain b kli
curvature ductility
stee
l ten
sion
growth straincolumn strain-ductilit
y
behavior
n buckling
Moyer and Kowalsky 2003
Onset of buckling
1020
[kip
s]
-5.9 -3.9 -2.0 0 2.0 3.9 5.9displacement [in]
50
100
e [k
N]
μ8μ6
μ5μ4
-20-10
0
late
ral f
orce
-150 -100 -50 0 50 100 150
-100
-50
0
displacement [mm]
late
ral f
orce
: bar buckl.: theor. buckl.
μ8μ6
μ5 μ4
OutlineJustificationExtended moment curvature analysisFinite Element Modeling:
Lumped plasticityDistributed plasticity / fiber basedDistributed plasticity / fiber basedFiber based lumped plasticity
Non-linear static analysis (pushover)Non-linear time history analysis:
Seismic InputDamping
Application ExampleConclusion
Finite element modeling of RC structures
Spring moment - rotationnon-linear spring
Lumped Plasticity Model
Lumped plasticity modelsDisadvantages:
-Axial force-moment interaction and axial-force stiffness interaction are separate from the element behavior
-Need to use M-C analysis to find:Elastic and post-yield stiffnessp yNon-linear axial force/moment interaction envelope
force-moment interaction and axial-force stiffness interaction
-5%
0%+5%
+10%+20%
+40%
T C C T
OutlineJustificationExtended moment curvature analysisFinite Element Modeling:
Lumped plasticityDistributed plasticity / fiber basedDistributed plasticity / fiber basedFiber based lumped plasticity
Non-linear static analysis (pushover)Non-linear time history analysis:
Seismic InputDamping
Application ExampleConclusion
Finite element modeling of RC structures
Longitudinal steel fibers
Unconfined concreteA A
Section A-A
Unconfined concrete cover fibers
Confined concrete core fibers
Material stress - strain
Distributed Plasticity Model
Material constitutive relationships
1.5
2.9
4.4
[ksi
]
10
20
30
Stre
ss [M
Pa]
Mander monotonic envelope
Concrete02 Unconfined concrete
0
0 0.005 0.01 0.015 0.02
0
Strain
0
1.5
2.9
4.4
[ksi
]
0 0.005 0.01 0.015 0.02
0
10
20
30
Strain
Stre
ss [M
Pa]
Mandermonotonicenvelope
Concrete02
Confined concrete
Material constitutive relationshipsReinforcingSteel material (Mohle and Kunnath, 2006), account for degrading strength and stiffness due to cyclic reversals.
73
109
500
750 Raynor monotonic envelope
-109
-73
-36
0
36
73
[ksi
]
-0.02 0 0.02 0.04 0.06-750
-500
-250
0
250
500
Strain
Stre
ss [M
Pa]
ReinforcingSteelmaterial
Distributed plasticity modelsAdvantages:
-No prior M-C analysis required-No need to define hysteretic response (it’s defined by the material models)-The influence of axial load is directly modeledPost peak strength reduction factor resulting from material-Post-peak strength reduction factor resulting from material
strain-softening or failure can be directly modeled.
Disadvantages:
-Shear strength and shear deformations still under development-Time consuming-Strain localization
Strain localization problemas
e sh
ear
3 IP
5 IP 3 E
5 E20 E
Force based Displacement based
Ba
Bas
e cu
rvat
ure
Displacement at top Displacement at top
8 IP
5 IP
3 IP
8 IP
3 E
5 E
20 E
OutlineJustificationExtended moment curvature analysisFinite Element Modeling:
Lumped plasticityDistributed plasticity / fiber basedDistributed plasticity / fiber basedFiber based lumped plasticity
Non-linear static analysis (pushover)Non-linear time history analysis:
Seismic InputDamping
Application ExampleConclusion
Fiber based lumped plasticity model
Force-based-fiber sectionsE, A, I
Linear Elastic
Lpi Lpj
L
node i node j
BeamWithHinges element (Scott and Fenves, 2006)
Fiber based lumped plasticity model
-1
-0.5
0
0.5
1
Nor
mal
ized
For
ce
0
0.5
1
Nor
mal
ized
For
ce
-0.05 0 0.05Drift
N 0 0.02 0.04 0.06 0.080
Drift
N
0 10 20 300
10
20
30
40
cycle #
AB ξ
[%]
0 0.02 0.04 0.06 0.080
10
20
30
Drift
Cur
vatu
re d
uctil
ity: Experimental: Simulation
OutlineJustificationExtended moment curvature analysisFinite Element Modeling:
Lumped plasticityDistributed plasticity / fiber basedDistributed plasticity / fiber basedFiber based lumped plasticity
Non-linear static analysis (pushover)Non-linear time history analysis:
Seismic InputDamping
Application ExampleConclusion
Non-linear Static Analysis (Pushover)
1348
1798
6000
8000
kN]
449
899 [kip
s]
0 0.02 0.04 0.06 0.08 0.10
2000
4000
Lateral drift
Forc
e [k
: +20°C: -40°C : serviceability : dam. control
Non-linear Static Analysis (Pushover)Disadvantages:
•Higher mode effects are missed
•With an unidirectional push the hysteretic characteristics f th t t t b l t dof the structure can not be evaluated
•If force controlled: tends to become unstable after the peak force is reached
•If displacement controlled: how do you specified the displacement vector in a multistory building… potential soft-storey building mechanisms can be inhibited
OutlineJustificationExtended moment curvature analysisFinite Element Modeling:
Lumped plasticityDistributed plasticity / fiber basedDistributed plasticity / fiber basedFiber based lumped plasticity
Non-linear static analysis (pushover)Non-linear time history analysis:
Seismic InputDamping
Application ExampleConclusion
Non-linear Time History Analysis: Seismic inputSeismic input:
0.4
0.6
0.8
1
PSA
[%g]
•Real records (Historic)•Artificial records:
•Full artificial (Simqke – Gasparini and Vanmarke, 1976)•Modified historic records:
•Using Fourier: Wes Rascal (Silva and Lee, 1987)•Using CWT: ArtifQuakeLet (Suarez and Montejo, 2003)•Using Wavelets: rspmatch (Hancock et al, 2005)
0 1 2 3 40
0.2
T (s)
Non-linear Time History Analysis: Seismic Input
0.2
0.4
0.6
acce
lera
tion
[g]
T1 T2
0 0.5 1 1.5 2 2.5 30
Period [s]
a
0 0.5 1 1.5 2 2.5 30
0.005
0.01
0.015
0.02
0.025
Period [s]
disp
lace
men
t/g
period shift
error
OutlineJustificationExtended moment curvature analysisFinite Element Modeling:
Lumped plasticityDistributed plasticity / fiber basedDistributed plasticity / fiber basedFiber based lumped plasticity
Non-linear static analysis (pushover)Non-linear time history analysis:
Seismic InputDamping
Application ExampleConclusion
Non-linear Time History Analysis: Damping
Damping = hysteretic + elastic (viscous) elhyst ξξξ +=
Hysteretic damping:
Non-linear Time History Analysis: DampingElastic damping: represents damping not captured by the hysteretic model:
•Hysteretic damping on the elastic range•Foundation compliance and non-linearity•Radiation dampingI t ti b t t t l t t l•Interaction between structural an non-structural
members
Non-linear Time History Analysis: DampingElastic damping:
mkmc
xmkxxcxm
ξωξ 22 ==
−=++ &&&&&
What values of k and ξ are appropriate?
•Traditional lumped plasticity model: 5% concrete, 2% steel •Fiber model: very low 0-2%
•Initial stiffness based viscous damping may result in inelastic damping forces that are unrealistically high. Use tangent-stiffness viscous damping.
Non-linear Time History Analysis: Damping
Petrini et al. 2008
Non-linear Time History Analysis: Damping
Petrini et al. 2008
OutlineJustificationExtended moment curvature analysisFinite Element Modeling:
Lumped plasticityDistributed plasticity / fiber basedDistributed plasticity / fiber basedFiber based lumped plasticity
Non-linear static analysis (pushover)Non-linear time history analysis:
Seismic InputDamping
Application ExampleConclusion
Application: Alaska DOT bridges
Cap Beam
Super Structure
Columnn/Pile
Application: Alaska DOT bridges
8#7 or8#9
#3@60mm
linear potentiometers
457mm OD steel tubeplane stub
post-tensioning
base plate
supportblock
string potentiometer
(2) hydr.jacks
Crossbeam
Crossbeam
1651 mm
(F, Δ)
(2) loadcells
strong floor
pin pin
pin
specimen
pinfixed
dL
API-5L x52OD: 24 inThick.: 12 in
12#7 ASTM A706string pot.
Δ
actuator 1 actuator 2
TOP HINGE
BOTTOM HINGE
Application: Alaska DOT bridges
TOP HINGE
BOTTOM HINGE
picture by Lennie Gonzales
Application: Alaska DOT bridges
44.9
56.20.8 1.6 2.4 3.1 3.9 4.7
[in]
200
250
N] 202.2
0.0 2.0 3.9 5.9 7.9 9.8 11.813.815.717.719.7[in]
900
1200
N]
MOMENT-CURVATURE ANALYSIS
TOP HINGE
11.2
22.5
33.7
[kip
s]
20 40 60 80 100 1200
50
100
150
displacement [mm]
late
ral f
orce
[kN
: measured: calculated: εs = εy
: εs = 0.015
: εs = 0.06
67.4
134.8
[kip
s]
0 50 100 150 200 250 300 350 400 450 5000
300
600
displacement [mm]
late
ral f
orce
[kN
: measured: calculated: εst = εy
: εst = 0.008
: εst = 0.028
BOTTOM HINGE
Application: Alaska DOT bridges
0
1
mal
ized
forc
e
0.5
1
mal
ized
forc
e
TOP HINGE
FIBER MODEL CALIBRATION
-0.05 0 0.05-1
drift
norm
0 0.02 0.04 0.060
drift
norm
0 10 20 300
20
40
cycle #
AB ξ
[%]
0 0.02 0.04 0.060
20
40
drift
curv
atur
e du
ctili
ty: Experimental: Simulation
Application: Alaska DOT bridges
BOTTOM HINGE
0
1
mal
ized
forc
e
0.5
1
mal
ized
forc
e
Experimental
FIBER MODEL CALIBRATION
-0.05 0 0.05-1
drift
norm
0 0.01 0.02 0.03 0.04 0.050
drift
norm
0 10 20 300
20
40
cycle #
AB
ξ [%
]
0 0.01 0.02 0.03 0.04 0.050
5
10
drift
curv
atur
e du
ctili
ty
pe e aSimulation
Application: Alaska DOT bridges
BeamWithHingesLinearelastic
FINITE ELEMENT MODEL
p-ysprings Distributed
plasticity
Application: Alaska DOT bridges
8847
11796
12000
16000
-m]
: top hinge: bottom hinge: first yied: serviceability: damage control
M-C ANALYSIS
2949
5898 [kip
s-ft]
0 0.02 0.04 0.06 0.08 0.10
4000
8000
φD
Mom
ent [
kN- : damage control
Application: Alaska DOT bridges
1798
2247
8000
10000
12000
kN]
fi t i ld b tt hi
damage control top hingeserviceabilitybottom hinge
PUSHOVER RESULTS
449
899
1348
1798
[kip
s]
0 0.05 0.1 0.150
2000
4000
6000
8000
late
ral f
orce
[
drift
first yield top hinge
serviceability top hinge
first yield bottom hinge
Application: Alaska DOT bridges
1
1.2
atio
n [g
]
0 02
0.03
nt/g
SEISMIC INPUT
0 0.5 1 1.5 2
0.2
0.4
0.6
0.8
Period [s]
Pse
udo
Acc
eler
a
0 0.5 1 1.5 20
0.01
0.02
Period [s]
Dis
plac
emen
Application: Alaska DOT bridgesINCREMENTAL DYNAMIC ANALYSIS
0.08
0.1
0.12t
: Average: Eq. records damage control
0 0.5 1 1.50
0.02
0.04
0.06
peak ground acceleration [g]
late
ral d
rift
0.20g
0.76g
serviceability
first yield
OutlineJustificationExtended moment curvature analysisFinite Element Modeling:
Lumped plasticityDistributed plasticity / fiber basedDistributed plasticity / fiber basedFiber based lumped plasticity
Non-linear static analysis (pushover)Non-linear time history analysis:
Seismic InputDamping
Application ExampleConclusion
Conclusion
It was shown that non-linear analyses provide us with valuable information regarding the seismic behavior of RC structures otherwise impossible to obtain through conventional linear analyses.
It is expected that, with the available computational tools, non-linear analyses become more popular in the design office environment.
Available (FREE) toolsMoment-curvature analysis:
http://www.ecf.utoronto.ca/~bentz/home.shtml http://blogs.uprm.edu/montejo/
Nonlinear FEM (lumped and distributed plasticity):Nonlinear FEM (lumped and distributed plasticity):
http://opensees.berkeley.edu/index.php http://www.seismosoft.com
Generation of spectrum compatible earthquake records:
contact the author http://blogs.uprm.edu/montejo/