Identificación de propiedades dinámicas y cuantificación ...€¦ · Priors P( ): CV of 10% of...

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

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

Model updating

Bayes Theorem

P(Θ|D,Mj)︸︷︷︸Posterior

Likelihood︷︸︸︷P(D|Θ,Mj)

Prior︷︸︸︷P(Θ|Mj)

P(D)︸︷︷︸Evidence

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

Model updating

Likelihood

P(D|Θ) =1

2π× exp

(X − X (Θ))2

Sampling method: Importance sampling (some samples are moreimportant)Sampling of the distribution

Procedure for probabilistic model updating

Probabilistic model updating flow chart

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

Application in Structural Dynamics

Application 1 / Structural Dynamics

Problem description

Full scale 5 floor building at UCSD (Astroza et al)

Problem description

Modes shapes ID (Astroza et al)

1.905 Hz2.6559 Hz

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

Results

Posterior vs Prior distributions (Modulus of Elasticity)

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

Application in Human-Structure Interaction

Application 2 / HSI

Problem description

Laboratory tests

Empty structure Sand bags (∼72kg) Person (∼72kg)

Application 2 / HSI

Mass-Damper-Spring (MDS) models

Model mimics the human behavior

mh mh1

Human as a mechanical system

Applications in:

AutomotivePhysical therapySpatial program

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

MDS model for groups

Application 2 / HSI

Closed-Loop control system - single person

R(s) E (s) C (s)

Block diagram of a closed-loopcontrol system

TF (s) =G (s)

1 + G (s)H(s)

Application 2 / HSI

Closed-Loop control system - groups of people

R(s) E (S) C (s)

TF (s) =G (s)

1 + G (s)[H1(s) + H2(s) + ...+ Hn(s)]

Application 2 / HSI

Closed-Loop control system

Structure

G (s) =s2

s2 + cm s + k

Controllers:

H(s) = Kp(1 + td s +1

H(s) = Kp(1 +1

PDH(s) = Kp(1 + Td s)

Application 2 / HSI

Closed-Loop control system - Stability

Human-structure interaction system

TF (s) =G (s)

1 + G (s)H(s)

TF (s) =ti s

kptd ti s3 + (kpti + mti )s2 + (kp + tic)s + tik

TF (s) =ti s

ti (kp + m)s2 + (kp + tic)s + tik

TF (s) =s2

kptds3 + (kp + m)s2 + cs + k

Application 2 / HSI

Structure description

177,5"

171,5"

12 Variable distance

Mobile support

Modificable mass

Variable cantilver length

Variable distance

Variable cantilver length

Constant mass at

support

Modificable

Steel Tube 5x4x

Block of 72 x 121/4 x 51/4

Concrete Blocks

72 x 12

Fixed support

PLAN VIEW

SIDE VIEW

Human-Structure Interaction Project

- structure for tests -

Concrete: f'c: 4000psi

Block of 72 x 12

Steel: A36

Rectangular Tube 5x4x

PLAN VIEW

Measure units: inches

SIDE VIEW

Measure units: inches

Plant and side views of lab specimen

Application 2 / HSI

Tests description

Lab structure Excitation

Application 2 / HSI

Procedure for updating HSI models

Flow chart for updating HSI models

θs : parameters of the structure

θh: parameters of the human

Application 2 / HSI

Model updating - Configuration A

Model: G(s) =s2

ms2+ c

mExperimental Data Optimization

0 20 40 60 80 100Frequency (rad/s)

28 30 32 34 36 38Frequency (rad/s)

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

Application 2 / HSI

Model updating - Configuration A

c s[Ns/m

1e−2

ks[N/m

ms[kg]

2 3 4cs[Ns/m]1e1

3.5 4.0ks[N/m]1e5

3.5ms[kg] 1e2

1 2 3 4 5σ

30 32 34 36 38Frequency (rad/s)

Exp. data95% HPD interval

Application 2 / HSI

Occupied structure

0 10 20 30 40 50 60 70 80Frequency (rad/s)

Occupied

Model Structural model Human model1 CKM PID controller2 CKM PI controller3 CKM PD controller4 CKM SDOF5 CKM 2DOF

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

Application 2 / HSI

Controller models

Parameters of the controllers

1e−2

−0.5

1e−1

3 4kp 1e1

1e−2

0.0 0.5Td 1e−1

0 2 4Ti 1e−2

1e−2

3.0 3.5 4.0kp 1e1

1e−2

2 3 4 5Ti 1e−2

1e−2

0.0 0.5kp 1e2

1e−3

−2 0 2 4 6Td 1e−3

Model 1 (PID Controller) Model 2 (PI controller) Model 3 (PD controller)

Application 2 / HSI

Controller models

Controller vs structure parameters

c s[Ns/m

ks[N/m

3 4kp 1e1

ms[kg]

0.0 0.5Td 1e−1

1 2 3 4Ti 1e−2

c s[Ns/m

ks[N/m

3.0 3.5 4.0kp 1e1

ms[kg]

3 4Ti 1e−2

c s[Ns/m

ks[N/m

0.0 0.5kp 1e2

ms[kg]

0 2 4Td 1e−3

Application 2 / HSI

Controller models

Posterior predictive check

26 28 30 32 34 36Frequency (rad/s)

−30M

26 28 30 32 34 36Frequency (rad/s)

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

HSI Model Updating

Moments of variables describing the PI controller for the people: P1:P9

ID Param. Kp ti

Mean 121.1 1.230STD 3.8 1.710

95%HPD(113.6, (0.328,128.5) 5.532)

ID Param. Kp ti

Mean 129.8 0.218STD 7.75 0.073

95%HPD(114.9, (0.112,145.6) 0.359)

ID Param. Kp ti

Mean 123.7 0.576STD 4.3 0.344

95%HPD(115.5, (0.235,132.7) 1.063)

HSI Model Updating

Age vs controller parameters (9 people)

HSI Model Updating

Weight vs controller parameters (9 people)

HSI Model Updating

Body mass index vs controller parameters (9 people)

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

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

Application in Material Modelling

HSI Model Updating Application 3 / Material modeling

Problem description

Old railroad tiesDamaged wood ties

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

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

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

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

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

Aggregate modeling

Sieve fit for a cylinder

0.0 5.0 10.0 15.0 20.0 25.0

Grain Diameter (mm)

1000.0

1500.0

2000.0

Weight Passing (gr

ModeledExperimental

0.0 5.0 10.0 15.0 20.0 25.0

Grain Diameter (mm)

1000.0

1500.0

2000.0

Weight Passing (gr

ModeledExperimental

0.0 5.0 10.0 15.0 20.0 25.0

Grain Diameter (mm)

1000.0

1500.0

2000.0

Weight Passing (gr

ModeledExperimental

Aggregates distribution in cylinder

Spatial distribution in cylinder

Polar coordinates

r = triangular

(0, rc −

φe2, rc −

θ = uniform

(0, 2π

)Height

h = uniform

φe2, hc −

FE model

Mortar Part Aggregates PartsAssembled Model

FE model

Regular 6x12 and small 4x8 Cylinder Use of the symmetry

Convergence of size effects

0.00 0.20 0.40 0.60 0.80 1.00

# of Agg. used / # of Total Agg.

6in x 12 in4in x 8inSymmetry

Modulus of Elasticity as a function of the percentage of the aggregates for threedifferent models

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

1.010E

cTestMeanMean ± STD

Ec for different models using same propertiesUniversidad del Norte (UN) UQ en ingenierıa estructural September 20, 2018 51 / 60

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

Probabilistic Model Updating using DIC data

Integration with DIC

Isotropic material, Ea distribution, ν distribution

Current work - Parameters estimation

Parameters Ea and νadistributions

Compliance form of thestiffness matrix:

εxxεyyεzzεyzεzxεxy

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

Distributions of the parameters

Ea (MPa)Universidad del Norte (UN) UQ en ingenierıa estructural September 20, 2018 55 / 60

Distributions of the parameters

ν (MPa)Universidad del Norte (UN) UQ en ingenierıa estructural September 20, 2018 56 / 60

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

FE model vs Experiment

21000 22000 23000 24000 25000 26000

Young's Modulus (Mpa)

Frequency

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

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.

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

Identificación de propiedades dinámicas y cuantificación ...€¦ · Priors P( ): CV of 10% of...

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