Post on 10-Jun-2020
Identificacion de propiedades dinamicas y cuantificacionde incertidumbre en ingenierıa estructural
Albert R. Ortiz
Departamento de Ingenierıa Civil y AmbientalUniversidad del Norte
oalbert@uninorte.edu.edu
September 20, 2018
Universidad del Norte (UN) UQ en ingenierıa estructural September 20, 2018 1 / 60
Probabilistic thinking and Model Updating
Probabilistic thinking and Model Updating
Model (M): representation of a system using general rules andconcepts,
but the model is not the actual system, then...
Uncertainty in parameters (Θ)Uncertainty in the model (M)
Discrepancies between experimentally measured data (D) andcomputational predictions are unavoidable for structural dynamicsystems,
Model updating methods have been developed over the past threedecades to reduce this gap
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Probabilistic thinking and Model Updating
Model updating
Bayes Theorem
P(Θ|D,Mj)︸ ︷︷ ︸Posterior
=
Likelihood︷ ︸︸ ︷P(D|Θ,Mj)
Prior︷ ︸︸ ︷P(Θ|Mj)
P(D)︸ ︷︷ ︸Evidence
where
Θ: model parameters
D: experimental data
Mj: Model j
Prior: prior knowledge about the parameters.
Likelihood: probability of the data given a set of parameters
Posterior: probability of the parameters after considering theexperimental data
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Probabilistic thinking and Model Updating
Model updating
Likelihood
P(D|Θ) =1
σ√
2π× exp
[−1
2
(X − X (Θ))2
σ2
]
Sampling method: Importance sampling (some samples are moreimportant)Sampling of the distribution
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Probabilistic thinking and Model Updating
Procedure for probabilistic model updating
Probabilistic model updating flow chart
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Probabilistic thinking and Model Updating
Model selection
From the Bayes Theorem we can obtain the ratio of probabilities of twomodels M1, M2:
P(M1|D,Θ1)
P(M2|D,Θ2)=
∫P(Θ1|M1)P(D|Θ1,M1)dΘ∫P(Θ2|M2)P(D|Θ2,M2)dΘ
Compare different models
Selection of the model with the highest probability
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Application in Structural Dynamics
Application 1 / Structural Dynamics
Problem description
Full scale 5 floor building at UCSD (Astroza et al)
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Application 1 / Structural Dynamics
Problem description
Modes shapes ID (Astroza et al)
1.905 Hz2.6559 Hz
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Application 1 / Structural Dynamics
Models and Data
Model (M):
OpenSees model (linear elastic) for Bare structure (Modes 1-3)
Data (D):
FrequenciesD1 = {f1, f2, f3}
Modes ShapesD2 = {Φ1,Φ2,Φ3}
Parameters (Θ):
Modulus of Elasticity: Columns, Beams, Walls, Slabs
Priors P(Θ): CV of 10% of reported value
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Application 1 / Structural Dynamics
Results
Posterior vs Prior distributions (Modulus of Elasticity)
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Application 1 / Structural Dynamics
Questions to solve
Is the model correct?
How many modes for running the update?
Variability in the modulus of elasticity?
This is just a full-scale test, on a bare structure
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Application in Human-Structure Interaction
Application 2 / HSI
Problem description
Laboratory tests
Empty structure Sand bags (∼72kg) Person (∼72kg)
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Application 2 / HSI
Mass-Damper-Spring (MDS) models
Model mimics the human behavior
mh mh1
mh2
Human as a mechanical system
Applications in:
AutomotivePhysical therapySpatial program
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Application 2 / HSI
Mass-Damper-Spring (MDS) models
! Developed for single humans
! Model could be used for groups
! Single and multiple degrees offreedom
% Large number of parameters
% Overtfitting
% Reduced feedback
mstructure
Mh11
Mh12
Mh21
Mh22
Mh31
Mh32
MDS model for groups
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Application 2 / HSI
Closed-Loop control system - single person
G (s)
H(s)
R(s) E (s) C (s)
−
B(s)
Block diagram of a closed-loopcontrol system
TF (s) =G (s)
1 + G (s)H(s)
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Application 2 / HSI
Closed-Loop control system - groups of people
G (s)
H1(s)
H2(s)
Hn(s)
R(s) E (S) C (s)
B(S)
TF (s) =G (s)
1 + G (s)[H1(s) + H2(s) + ...+ Hn(s)]
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Application 2 / HSI
Closed-Loop control system
Structure
G (s) =s2
m
s2 + cm s + k
m
Controllers:
PID
H(s) = Kp(1 + td s +1
ti s)
PI
H(s) = Kp(1 +1
ti s)
PDH(s) = Kp(1 + Td s)
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Application 2 / HSI
Closed-Loop control system - Stability
Human-structure interaction system
TF (s) =G (s)
1 + G (s)H(s)
PID
TF (s) =ti s
2
kptd ti s3 + (kpti + mti )s2 + (kp + tic)s + tik
PI
TF (s) =ti s
2
ti (kp + m)s2 + (kp + tic)s + tik
PD
TF (s) =s2
kptds3 + (kp + m)s2 + cs + k
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Application 2 / HSI
Structure description
177,5"
72"
19,5"
19,5"
17"
171,5"
12 Variable distance
5"
4"
4"
4"
Mobile support
Modificable mass
Variable cantilver length
Variable distance
Variable distance
12"
Variable cantilver length
4"
Constant mass at
support
Modificable
mass
Steel Tube 5x4x
3
16
Block of 72 x 121/4 x 51/4
Concrete Blocks
72 x 12
1
4
x 5
1
4
Fixed support
PLAN VIEW
SIDE VIEW
Human-Structure Interaction Project
- structure for tests -
Concrete: f'c: 4000psi
Block of 72 x 12
1
4
x 5
1
4
Steel: A36
Rectangular Tube 5x4x
3
16
PLAN VIEW
Measure units: inches
SIDE VIEW
Measure units: inches
Plant and side views of lab specimen
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Application 2 / HSI
Tests description
Lab structure Excitation
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Application 2 / HSI
Procedure for updating HSI models
Flow chart for updating HSI models
θs : parameters of the structure
θh: parameters of the human
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Application 2 / HSI
Model updating - Configuration A
Model: G(s) =s2
ms2+ c
ms+ k
mExperimental Data Optimization
0 20 40 60 80 100Frequency (rad/s)
−70
−60
−50
−40
−30
−20
−10
Mag
nitu
de (d
B)
28 30 32 34 36 38Frequency (rad/s)
−40
−35
−30
−25
−20
−15
−10
−5
Mag
nitu
de (d
B)
Exp. dataOptimization
Priors
Parameter units PDF Parameterscs Ns/m Normal µ = 31.0, σ = 3.10ks kN/m Uniform lower = 300.0, upper = 500.00ms kg Normal µ = 377.0, σ = 16.00σs Inverse Gamma α = 10.0, β = 3.07
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Application 2 / HSI
Model updating - Configuration A
−5
0
5
c s[Ns/m
]
1e−2
3.0
3.5
4.0
4.5
ks[N/m
]
1e5
3.0
3.5
4.0
ms[kg]
1e2
2 3 4cs[Ns/m]1e1
1
2
3
4
5
σ
3.5 4.0ks[N/m]1e5
3.5ms[kg] 1e2
1 2 3 4 5σ
30 32 34 36 38Frequency (rad/s)
−50
−40
−30
−20
−10
0
Mag
nitu
de (d
B)
Exp. data95% HPD interval
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Application 2 / HSI
Occupied structure
0 10 20 30 40 50 60 70 80Frequency (rad/s)
−80
−70
−60
−50
−40
−30
−20
−10
0
Am
plit
ud (
dB
)
Empty
Occupied
Model Structural model Human model1 CKM PID controller2 CKM PI controller3 CKM PD controller4 CKM SDOF5 CKM 2DOF
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Application 2 / HSI
Controller models
Prior PDFs of the structure’s parameters
Parameter units PDF Parameterscs Ns/m Normal µ = 28.9, σ = 4.9ks kN/m Normal µ = 383245.2, σ = 13438.2ms kg Normal µ = 350.9, σ = 12.3σs Normal µ = 2.3, σ = 0.3
Prior definition for parameters of the PID, PI, and PD controller
Parameter Model Distribution ValuesKp 1, 2, 3 Uniform min = −10E3, max = 10E3Td 1, 3 Uniform min = −10E3, max = 10E3Ti 1, 2 Uniform min = −10E3, max = 10E3
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Application 2 / HSI
Controller models
Parameters of the controllers
−5
0
5
kp
1e−2
−0.5
0.0
0.5
1.0
Td
1e−1
3 4kp 1e1
0
2
4
Ti
1e−2
0.0 0.5Td 1e−1
0 2 4Ti 1e−2
−5
0
5
kp
1e−2
3.0 3.5 4.0kp 1e1
2
3
4
5
Ti
1e−2
2 3 4 5Ti 1e−2
−5
0
5
kp
1e−2
0.0 0.5kp 1e2
−2
0
2
4
6
Td
1e−3
−2 0 2 4 6Td 1e−3
Model 1 (PID Controller) Model 2 (PI controller) Model 3 (PD controller)
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Application 2 / HSI
Controller models
Controller vs structure parameters
2.8
3.0
c s[Ns/m
]
1e1
3.80
3.85
ks[N/m
]
1e5
3 4kp 1e1
3.50
3.55
ms[kg]
1e2
0.0 0.5Td 1e−1
1 2 3 4Ti 1e−2
2.8
3.0
c s[Ns/m
]
1e1
3.80
3.85
ks[N/m
]
1e5
3.0 3.5 4.0kp 1e1
3.50
ms[kg]
1e2
3 4Ti 1e−2
2.6
2.8
c s[Ns/m
]
1e1
3.70
3.75
ks[N/m
]
1e5
0.0 0.5kp 1e2
3.40
3.42
3.44
ms[kg]
1e2
0 2 4Td 1e−3
Model 1 (PID Controller) Model 2 (PI controller) Model 3 (PD controller)
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Application 2 / HSI
Controller models
Posterior predictive check
26 28 30 32 34 36Frequency (rad/s)
−45
−40
−35
−30
Mag
nitu
de (d
B)
Exp. data95% HPD interval
26 28 30 32 34 36Frequency (rad/s)
−45
−40
−35
−30M
agni
tude
(dB
)
Exp. data95% HPD interval
26 28 30 32 34 36Frequency (rad/s)
−60
−50
−40
−30
−20
−10
0
10
Mag
nitu
de (d
B)
Exp. data95% HPD interval
Model 1 (PID Controller) Model 2 (PI controller) Model 3 (PD controller)
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HSI Model Updating
Controller models single individuals
Age and weight of people during tests (9 people)
ID Age (years) BMI (kg/m2)
P1 30 24.5P2 17 24.0P3 17 35.1P4 16 22.8P5 23 21.8P7 34 21.9P8 19 20.2P9 30 30.0
P10 28 23.4
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HSI Model Updating
Controller models single individuals
Moments of variables describing the PI controller for the people: P1:P9
ID Param. Kp ti
P1
Mean 121.1 1.230STD 3.8 1.710
95%HPD(113.6, (0.328,128.5) 5.532)
ID Param. Kp ti
P2
Mean 129.8 0.218STD 7.75 0.073
95%HPD(114.9, (0.112,145.6) 0.359)
ID Param. Kp ti
P3
Mean 123.7 0.576STD 4.3 0.344
95%HPD(115.5, (0.235,132.7) 1.063)
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HSI Model Updating
Controller models single individuals
Age vs controller parameters (9 people)
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HSI Model Updating
Controller models single individuals
Weight vs controller parameters (9 people)
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HSI Model Updating
Controller models single individuals
Body mass index vs controller parameters (9 people)
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HSI Model Updating
Partial conclusions
Three different controllers were updated using a Bayesian probabilisticapproach
The PID and the PI controllers were able to model the phenomenonThe PD did not show good fit to the human-structure interaction
The PI is the best controller model among the others evaluated formodeling the human-structure interaction
The integrative term, ti , is significant to the controllerThe ti parameter deals with the velocity component of thehuman-structure systemThe importance of the velocity for the balance and stability of thehuman body has been reported
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HSI Model Updating
Partial conclusions
Parameters may be related to physical properties of the person
Kp and Ti look related to the massKp and Ti look independent of the age (*)
(Looking) Indirect identification of human body using vibrationsHealth problems
Stability/PhysicalStability/Disorders
Body skills
SportsOther activities
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Application in Material Modelling
HSI Model Updating Application 3 / Material modeling
Problem description
Old railroad tiesDamaged wood ties
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HSI Model Updating Application 3 / Material modeling
Problem description
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HSI Model Updating Application 3 / Material modeling
Problem description
Working on the development of a HSRM-HPC for railroad ties(Collaboration with Dimitris Rizos, USC)
Similar StrengthReduced Modulus of Elasticity
The reduced Modulus of Elasticity is attributed to the use ofaggregates
Influence of the coarse aggregates properties in the concrete
Concrete model that captures this behavior
Model relates properties of the raw material to the properties of theproduced concrete
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HSI Model Updating Application 3 / Material modeling
Objectives
Compare the Modulus of Elasticity of FE models and Experiments ofHSRM-HPC
Influence of the coarse aggregates Modulus of Elasticity in the propertiesof a concrete
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HSI Model Updating Application 3 / Material modeling
Methodology
Properties of the coarse aggregates
ID Density Modulus of Elasticity
(kg/m3) (MPa)CA-1 2589.4 56945CA-2 2667.1 27800CA-3 2643.1 22845
Compression test (Eyy)Core extraction
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HSI Model Updating Application 3 / Material modeling
Methodology
Properties of the coarse aggregates
Sieve Analysis
Sieve analysis for aggregates modeledUniversidad del Norte (UN) UQ en ingenierıa estructural September 20, 2018 44 / 60
HSI Model Updating Application 3 / Material modeling
Methodology
Concrete mix
Concrete Mix
Coarse Agg. Fine Agg. Cement Water(kg) (kg) (kg) (kg)
1m3 1091 791 356 1386”x12” cylinder 6.07 4.40 1.98 0.774”x8” cylinder 1.80 1.30 0.59 0.23
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HSI Model Updating Application 3 / Material modeling
Aggregate modeling
Sieve fit for a cylinder
0.0 5.0 10.0 15.0 20.0 25.0
Grain Diameter (mm)
0.0
500.0
1000.0
1500.0
2000.0
Weight Passing (gr
)
ModeledExperimental
CA-1
0.0 5.0 10.0 15.0 20.0 25.0
Grain Diameter (mm)
0.0
500.0
1000.0
1500.0
2000.0
Weight Passing (gr
)
ModeledExperimental
CA-2
0.0 5.0 10.0 15.0 20.0 25.0
Grain Diameter (mm)
0.0
500.0
1000.0
1500.0
2000.0
Weight Passing (gr
)
ModeledExperimental
CA-3
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HSI Model Updating Application 3 / Material modeling
Aggregates distribution in cylinder
Spatial distribution in cylinder
Polar coordinates
r = triangular
(0, rc −
φe2, rc −
φe2
)
θ = uniform
(0, 2π
)Height
h = uniform
(0 +
φe2, hc −
φe2
)
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HSI Model Updating Application 3 / Material modeling
FE model
Mortar Part Aggregates PartsAssembled Model
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HSI Model Updating Application 3 / Material modeling
FE model
Regular 6x12 and small 4x8 Cylinder Use of the symmetry
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HSI Model Updating Application 3 / Material modeling
Convergence of size effects
0.00 0.20 0.40 0.60 0.80 1.00
# of Agg. used / # of Total Agg.
0.6
0.7
0.8
0.9
1.0E
c i/E
c
6in x 12 in4in x 8inSymmetry
Modulus of Elasticity as a function of the percentage of the aggregates for threedifferent models
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HSI Model Updating Application 3 / Material modeling
Modulus of Elasticity for different models
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0
Test number
0.990
0.995
1.000
1.005
1.010E
c i/E
cTestMeanMean ± STD
Ec for different models using same propertiesUniversidad del Norte (UN) UQ en ingenierıa estructural September 20, 2018 51 / 60
HSI Model Updating Application 3 / Material modeling
Results of experimental and analytical models
Modulus of Elasticity (Cylinder 4”x8”)
Mix Agg. Wagg Modulus of Elasticity Modulus of Elasticity, Ec
Model Aggregate Mortar FE Model Experimental CV[kg] [MPa] [MPa] [MPa] [MPa] [%]
CM9 CA1 1.8 56945 25790 34746 31159 10.3CM11 CA2 1.8 27800 25790 26670 23427 6.7CM10 CA3 1.8 22845 25790 24219
22056 5.9CM10 CA3(*) 1.8 14500 25790 20432
(* from the literature)
Reduced Modulus of Elasticity for the concrete created using aggregatesCA-2 and CA-3
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HSI Model Updating Application 3 / Material modeling
Probabilistic Model Updating using DIC data
Integration with DIC
Isotropic material, Ea distribution, ν distribution
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HSI Model Updating Application 3 / Material modeling
Current work - Parameters estimation
Parameters Ea and νadistributions
Compliance form of thestiffness matrix:
εxxεyyεzzεyzεzxεxy
=1
E
1 −ν −ν 0 0 0
−ν 1 −ν 0 0 0−ν −ν 1 0 0 0
0 0 0 1 + ν 0 00 0 0 0 1 + ν 00 0 0 0 0 1 + ν
σxxσyyσzzσyzσzxσxy
Isotropic material, Ea distribution, ν distribution
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HSI Model Updating Application 3 / Material modeling
Distributions of the parameters
Ea (MPa)Universidad del Norte (UN) UQ en ingenierıa estructural September 20, 2018 55 / 60
HSI Model Updating Application 3 / Material modeling
Distributions of the parameters
ν (MPa)Universidad del Norte (UN) UQ en ingenierıa estructural September 20, 2018 56 / 60
HSI Model Updating Application 3 / Material modeling
FE model
Mortar Part Aggregates Parts Assembled Model
Em and νm are random from thedistribution (Mortar)
Ea and νa of each aggregate are randomfrom the distribution (Aggregates)
Concrete modulus of elasticity influencedby Aggregates
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HSI Model Updating Application 3 / Material modeling
FE model vs Experiment
21000 22000 23000 24000 25000 26000
Young's Modulus (Mpa)
0
2
4
6
8
10
12
Frequency
µ
µ± σ
Models
Histogram of the modulus of elasticity of concrete cylinders models using 30 samples compared to the mean (µ) and µ± σ ofexperimental test
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HSI Model Updating Application 3 / Material modeling
Conclusions
The performance of Finite Element models of concrete cylinders madewith different types of aggregates was evaluated
Models show that the concrete created with crushed weatheredgranite aggregates have a reduced Modulus of Elasticity whencompared with Limestone aggregates currently being used in tiesfabrication
The smallest 20% of the aggregates does not affect the predictedYoung’s modulus
FE Models show that the Modulus of Elasticity is between 7.4% and12.1% different from the Experimental Data.
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Identificacion de propiedades dinamicas y cuantificacionde incertidumbre en ingenierıa estructural
Albert R. Ortiz
Departamento de Ingenierıa Civil y AmbientalUniversidad del Norte
oalbert@uninorte.edu.edu
September 20, 2018
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