Demolition of structures considering uncertaintyrec.ce.gatech.edu/resources/6 Demolition of structures considering... · Demolition of structures considering uncertainty ... Institute

Institute for Statics and Dynamics of Structures

Demolition of structuresconsidering uncertainty

Bernd Möller

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Program

1 Introduction, Motivation

4 Fuzzy Multibody Dynamics

5 Fuzzy Probabilistic Multibody Dynamics

6 Conclusions

2 Data Models Fuzziness and Fuzzy Randomness

3 Numerical Model for Blasting

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KW Thierbach,October 2002

Introduction (1)

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Introduction (1)

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structural parametersstructural parameters

geometry

failure zones

material

friction

contact parameterscontact parameters

damping

blasting parametersblasting parameters

grade of local structural damage

detonation impacts

time of detonation

Introduction (2)

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uncertain geometrical parametersuncertain geometrical parameters

h

~

12

s1~

23

22

21

h~

uncertain position of reinforcement

Introduction (3)

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fuzzy random variablefuzzy random variable1

0.5

0 2 4 6 8 10 x

fuzzy variablefuzzy variable

1

0.5

0 2 4 6 8 10 x

deterministic variabledeterministic variable

0 2 4 6 8 10 x

x0

∈sup µ(x) x R =1:(x)

f(x)

Data Models (1)

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a fuzzy number possesses exactly one x with the membership value µX(x) = 1

x

1,0

0,0

membershipfunction µX(x)

membership

value µX(x)

xuxo

fuzzy variable Xfuzzy variable X~

( ) µX X∈ ≥ ∀ ∈X XX = x; µ (x) x ; (x) 0 x

support

Data Models (2) - Fuzziness

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α-level-setα-level-set α-discretizationα-discretization

µX(x)

1,0

0,0

ak

x

xαkl xαkr

Xα k

Xαk= x X µx(x) ≥ αk X = Xα; µ(Xα)( )~

Xαi Xαk αi, αkαi ≥ αk

αi; αk (0,1]

"-level-set

support

Data Models (3) - Fuzziness

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fuzzy function: x(t): T F(X)~

parameter space set of fuzzy variables

t = (θ, τ, n)

x(t) = xt = x(t) t t T ~ ~ ~

x

time τ

µ(x)

τ1 τ2 τ3

fuzzy process: x(τ) = xτ = x(τ) œ τ τ T ~ ~ ~

X~

τ1

Data Models (4) – Fuzzy functions

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fuzzy random variable X

X : Ω 6 F(X )

x(ω5)~

x(ω3)~x(ω2)~

x(ω4)~

x(ω1)~

x(T5) , F(X ) set of allfuzzy numbers in ún

space of the one-dimensional realizations

x 0 X = ú1

ω2

ω3

ω4

ω5

ω 0 Ω

space of the random elementary events

fuzzy number

~

~ ~

ω1

µ(x)1,0

0,0

Data Models (6) – Fuzzy Randomness

~

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x(ω5)~

x(ω3)~x(ω2)~x(ω1)~

x 0 X = ú1

ω2

ω3

ω4

ω5

ω 0 Ω

x(ω4)~

µ(x)1.0

0.0

ω1

an original Xj has the porpertyof a real random variable X

x(ω1)

x(ω1): realization of the realrandom variable Xx(ω1) 0 x(ω1)

~

X := fuzzy set of all originals Xj~

~if for all x(ωi) holds: x(ωi) 0 x(ωi),then: x constitute one

original Xj of X


˜

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( )( )1 2F(x) exp -exp - (x- )s s= ⋅

1 2π

= = − ⋅ σσ ⋅ X X

X

s s; m 0,456

( )F(x) F , xs=

F(x)

1,0

F(x)~

0,0x

fuzzy probability distribution function

e.g. Gumbel

1,0 0,0µ(F(xi))

α-levelFα(xi)

xi


original Fj(x)

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( ) tX(t) X =X t t t = ∀ ∈T

X(t) : T V Ω 6 F(X )

Data Models – Fuzzy Random Functions

given: set of fuzzy random variables X at points T~

with t = τ, θ, τ timeθ = θ1, θ2, θ3 spatial coordinates

definition: A fuzzy random function X(t) is theset of fuzzy random variables Xt in T

~~

special cases:~

1 no randomness: X(t) 6 x(t) fuzzy function~

~2 no fuzziness: X(t) 6 X(w) random function

~3 for fixed t : X(t) 6 X(q) fuzzy random field

~

t ∈

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Numerical Model (1)

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problem of multibody dynamics

with data uncertainties

flexiblebodies =failure regions

rigidbodies

Numerical Model (2)

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Numerical Model (3)

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FE-Model

considering the nonlinear behavior

of reinforcement concrete

2 1

Numerical Model (4)

rigid body

flexiblebody

rigidbody

rigidbody

flexiblebody

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nonlinear relations between forces and displacements

1

11 11 12 122

3

1

2

11 11 12 123

121 21

2

3

1

221

3

F (1)k (1,1) k (1,6) k (1,1) k (1,6)F (1)

F (1)

M (1)

M (1)k (6,1) k (6,6) k (6,1) k (6,6)

M (1)

F (2)k (1,1) k (1,6)

F (2)

F (2)

M (2)

M (2) k (6M (2)

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

=⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

1

2

3

1

2

3

122 22

2

3

1

221 22 22

3

v (1)v (1)v (1)

(1)

(1)

(1)

v (2)k (1,1) k (1,6)

v (2)

v (2)

(2)

(2),1) k (6,6) k (6,1) k (6,6)(2)

⎡ ⎤⎢ ⎥⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥ϕ⎢ ⎥⎢ ⎥

⎢ ⎥⎢ ⎥ ϕ⎢ ⎥⎢ ⎥⎢ ⎥ϕ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥ ϕ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥ ϕ

⎢ ⎥ ⎢ ⎥⎣ ⎦ϕ⎢ ⎥⎣ ⎦

i

Numerical Model (5)

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A-priori-computation of force-deformation relationship

Springelement:

Hypothesis:

Objective:

diagonal element >> off diagonal element

Decoupledsystem of equations:

i i i iF k (v ) v= ⋅ i ii i

v Fk (v )

= ⋅1

bzw.

Numerical Model (7)

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3

2

3

1

2

1

3

1

2

3

2

1

F (i) . . . . . . . . . . . .F (i) . . . . . . . . . . . .

. . . . . . . .M (i) . . . . . . . . . . . .

. . . . . . . .M (i) . . . . . . . . . . . .F (k) . . . . . . . . . . . .F (k) . . . .

F (i) x x x x

M (i) x x x x

F (k) x

M (k

. . . . . . . .. .

M (k)

M (k))

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥ =⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

3

2

1

2

1

3

3

1

2

2

1

3

v (i)v (i)

(i)

(i)v (k)v (k)

. . . . . .(k). . . . . . . . . . . .

. . . . . . . .(k). .

v (i)

(i)

v (k)

. . . . . . . . . .

x x x

(k)x x x x

ϕϕ

ϕ

ϕ

ϕ

ϕ

⎡ ⎤⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥ • ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

3

2

3

1

2

1

3

1

2

3

2

1

F (i) . . . . . . . . . . . .F (i) . . . . . . . . . . . .

. . . . . . . .M (i) . . . . . . . . . . . .

. . . . . . . .M (i) . . . . . . . . . . . .F (k) . . . . . . . . . . . .F (k) . . . .

F (i) x x x x

M (i) x x x x

F (k) x

M (k

. . . . . . . .. .

M (k)

M (k))

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥ =⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

3

2

1

2

1

3

3

1

2

2

1

3

v (i)v (i)

(i)

(i)v (k)v (k)

. . . . . .(k). . . . . . . . . . . .

. . . . . . . .(k). .

v (i)

(i)

v (k)

. . . . . . . . . .

x x x

(k)x x x x

ϕϕ

ϕ

ϕ

ϕ

ϕ

⎡ ⎤⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥ • ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

v3 (i)

M2(k)

φ2 (k)

M2(k)

φ2 (i)

M2(i)

φ2 (k)

M3(k)

force deformation realation v(F)force deformation realation v(F)

Numerical Model (6)

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fuzzy function: v = v(F)fuzzy function: v = v(F)

~ ~

dis

pla

cem

ent

v

v(F)1 0

Fi

:(v

)

v

force F

Numerical Model (6) – Fuzzy Function

~ ~

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F(v)

disp

lace

men

t v

force F

~

v

Numerical Model (7) – Fuzzy Random Function

fuzzy random function v = v(F)fuzzy random function v = v(F)~ ~

Slide 24 of 45


orientation of the rigid body i in R3orientation of the rigid body i in R3

ii i ir R A u= +

TTTir

iiq R⎡ ⎤= ⎢ ⎥⎣ ⎦θ

Numerical Model (8) – Algorithmic Procedure

A transformation matrix

body i

Slide 25 of 45


kinematical constraintskinematical constraints

nb rigid bodies:

holonomic constraints none holonomic constraints

qr ncr r r rC C(q , t) C (q , t),C (q , t), ,C (q , t) =0⎡ ⎤= = ⎣ ⎦1 2 …

Tnbr r r r

q q ,q , ,q⎡ ⎤= ⎣ ⎦1 2 …


Slide 26 of 45


Lagrangian equation of motion

rigid bodies only:

rigid and flexible bodies:

qr

T

e vrMq C Q Q+ λ = +

Te vqrr r r rrr rf

Tfr ff ff e vqf f ff f

q q (Q ) (Q )Cm m 0 0m m 0 Kq q (Q ) (Q )C

⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤⎢ ⎥+ + λ = +⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦ ⎣ ⎦


Slide 27 of 45


fuzzy valuesfuzzy values

deterministicvalues

deterministicvalues

X(t)~ ~

fuzzyvaluesfuzzyvalues

Z(t)~~

Fuzzy multibodydynamics

fuzzy analysis

deterministicfundamental

solution

Fuzzy multibodydynamics

fuzzy analysis

deterministicfundamental

solution

Fuzzy Multibody Dynamics (1)

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mapping z = f(s1, s2)s1

: (s1,s2)

0

1

"k

:(z)

z

1

0

space offuzzy input values s1 and s2

"kS"k

sj

(s1, s2) z~~ ~mapping:

~ ~

"-level-Optimierung

s2

Fuzzy Multibody Dynamics (2)

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fuzzy randomvalues

fuzzy randomvalues

deterministicvalues

deterministicvalues

X(t)~ ~ Z(t)~~

uncertain

results

uncertain

results

fuzzy probabilisticmultibody dynamic

fuzzy stochasticanalysis

deterministicfundamental solution

fuzzy probabilisticmultibody dynamic

fuzzy stochasticanalysis

deterministicfundamental solution

Fuzzy Probabilistic Multibody Dynamics (1)

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s 1

: (s1,s2)

0

1

space of fuzzy input parameterss1 und s2

"k S"ksj,"k

~ ~s2

s1,"k^

k

s2,"k^

elem

ents

of

s~

1 1 1F (x) F (x,s )=fuzzy randominput parameters: =2 2 2F (x) F (x,s )and

sj (s1, s2)


fuzzy probability distribution functions

Slide 31 of 45


s1,"k^

k

s2,"k^

Stochsatische Mehrkörperdynamik

( )k k k k, q , , p ,ˆ ˆ ˆ ˆF (z),...,F (z) g F (x),...,F (x)α α α α=1 11 1

stochastic analysis

( )k k k k, q , , p ,ˆ ˆ ˆ ˆF (z),...,F (z) g F (x),...,F (x)α α α α=1 11 1

x

1F

2 , " (x)k

input space F

11k

F2 ," (x)F (x)F (x)

1," (x), ... , Fp ," (x)(x), ... , (x)(x), ... , , (x)1 kk

F

result space F

kF1,α (z)

z

1kF1,α (z)

1kF1,α (z)

1

kF1,α (z)kF1,α (z)kF1,α (z)

1," (z), ... , Fp ," (z)1 kk

k2 ," (z)

z

111

kF ,α (z)

2 kF , (z)

kF , (z)

elem

ents

of

s~

kσ1,α1,α

kσm1,αm1,α^

^

elem

ents

of

F~

kF 1,α (x)

k1,α k1,α

F1,α (x)F1,α (x)F1,αk(x)

1


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n1

n2

fuzzyload-displacement-dependencies

n1(M) = s1 @ n1(M)~ ~

n2(M) = s2 @ n2(M)~ ~

distance r

Fuzzy Multi Body Dynamics (1) – Example 1

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1

01,50,5 1,0

: (s )1

s1

fuzzy bunch parameter s1~

1

01,20,8 1,0

: (s )2

s2

fuzzy bunch parameter s2~


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n

[rad]

M [kNm]0

0,02

0,04

0,06

0,08

0,1

0,12

0,14

0 500 100 150 200 250 300 350 400 450

n2(M) deterministic function

" = 0

" = 0

" = 1

" = 0

" = 0

" = 1n1(M) deterministic

function

n[r

ad]

fuzzy load displacement relation

n1(M) = s1 @ n1(M)~ ~n2(M) = s2 @ n2(M)~ ~


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z

F(z)

"

1

~fuzzy result z

~

( ) ( )( ) ( )

j j 1 n 1 n

j j 1 n 1 n

z f x ; ...; x max x ; ...; x X

z f x ; ...; x min x ; ...; x X

α

α

= ⇒ ∈

= ⇒ ∈

objective function:•

F(s1,s2)

s1

s2α

0

1

1

z = f(x1,x2)

Xα

space offuzzy input values s1 and s2

~ ~

Fuzzy Multi Body Dynamics (2)

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0,50,7

0,91,1

1,31,5 0,8

0,9

1

1,1

1,26,456,5

6,556,6

6,656,7

6,756,8

6,856,9

6,95

S2

S1

r [m]

rmin

rmax

S" = 0

space of fuzzy inputparameters


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distance r distance r

:(r)

r [m]

0,5

1,0

6,96,86,76,60,0

6,5

deterministic result


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m

µ(cr)

0,2 0,4 0,45cr

fuzzy friction parameter

0,0

1,0

0,8 1,0 2,00,0

1,0

s

µ(s)

fuzzy bunch parameterof the load-displacement-dependencies

Fuzzy Multibody Dynamics (6) – Example 1

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r

debris distance radius r


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α=1α=0

α=0


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fuzzy distance r~

3m3m

3m

23 m22 m

15 m


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load displacement relation

n1(M,s1) und n2(M,s2)~ ~

n1

n2

are modeled as fuzzy random function

uncertain lognormal distributionwithfuzzy standard devations s~

~ ~

Achse 1

Achse 2

debris domain

Fuzzy Probabilistic Multibody Dynamics (1) - Example

ni(M,si)~ ~

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F(v)

disp

lace

men

tv

force F

~ ~

v

fuzzy random function

v = v(F)fuzzy random function

v = v(F)~ ~


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Achse 1

Achse 2

fuzzy probability,that parts of debris fall outside the debris domain

0,5

1,0

042,731,318,1

:(Pf)

Pf [10-2]

deterministic probability

debris domain


debris domain is insufficiently designed

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In the case of data uncertainty fuzziness and fuzzy randomness are useful mathematical models

The application on blasting processes demonstratesthe information profit in practicel problems

Conclusions

Slide 47 of 45


Thank you !

Documents

Demolition of structures considering uncertaintyrec.ce.gatech.edu/resources/6 Demolition of structures considering... · Demolition of structures considering uncertainty ... Institute