4 fuzzy stochastic finite element methodrec.ce.gatech.edu/resources/4 fuzzy stochastic finite...
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Institute for Statics und Dynamics of Structures
Fuzzy Stochastic Finite Element Method
-
FSFEM
Bernd Möller
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Slide 2
Institute for Static and Dynamics of Structures
Fuzzy Stochastic Analysis
from fuzzy stochastic sampling to fuzzy stochastic analysis
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Slide 3
Institute for Static and Dynamics of Structures
Fuzzyrandom variables X
Fuzzy Random functions X(s, t)
~
~
Fuzzy Stochastic Analysis
Uncertain input data Mapping Uncertain result data
α-discretization yields:original Xj of X with assigned probability
distribution function Fj (x)
~
X Z
~ ~~
Stochastic analysis withthe Xj and MCS plus
algorithm for structural analysis
Combining the results Fj(z)to the fuzzy probability distribution function F(σ,z)of the fuzzy random result Z
~
~
(empirical) probabilitydistribution function Fj(z)for structural responseparameter Zj(at regarded α-level )
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Slide 4
Institute for Static and Dynamics of Structures
kσm1,α
kσ1,α
elementsof s~
elementsof σ~
s1,α
sn1,α
sk,α
k
k
k
^
^
^
^
^
Fuzzy Stochastic Analysis: Mapping Operator
kF1,α (x)
x
1
Input space of the F1,α (x), ... , Fp ,α (x)1 kk
kFp ,α (x)
1
x
1
kF1,α (x)
rF1,α (x)
F1,α (x)l
Fp ,α (x)k1
Fp ,α (x)l1
Fp ,α (x)r1
Stochastic analysis MCS plusMapping model
kF1,α (z)
z
1
kFq ,α (z)
1
z
1
Output space of the F1,α (z), ... , Fq ,α (z)1 kk
kF1,α (z)
F1, α (x)l
F1, α (x)r
kFq ,α (z)
1l
Fq ,α (z)1
rFq ,α (z)
1
( )k 1 k k 1 k1, q , 1, p ,ˆ ˆ ˆ ˆF (z), ...,F (z) g F (x), ...,F (x)α α α α=
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Slide 5
Institute for Static and Dynamics of StructuresFuzzy Stochastic Analysis
~F(x1)
x1
1µ(s1)
s1
s1,αα
fuzzy randomness
1µ(s3)
s3
s3,α
fuzziness
α
simultaneous consideration of
F(x2)
x2
1µ(s2)
s2
randomness
s2,αα
s1,α min
s1,α max
s1,α
s3,α max
s3,α min
s3,α
input space of the fuzzybunch parameters
s1
s2
s3
s2,α
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Slide 6
Institute for Static and Dynamics of Structures
Fuzzy Stochastic Finite Element Method
FSFEM
from fuzzy stochastic sampling tofuzzy stochastic finite element method
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Slide 7
Institute for Static and Dynamics of Structures
Fuzzy randominput variables
Fuzzy randomresult variables
FSFEM:
→F : X( t ) Z( t )
⋅ + ⋅ + ⋅ = ∀ ∈t t t tm (.) v d (.) v k (.) v f (.) t T
Discretization
Differentional equation of motion
Fuzzy Stochastic Finite Element Method (FSFEM)
M(.) v D(.) v K(.) v F(.)⋅ + ⋅ + ⋅ =
( )= θ τ ϕt , ,
K(.) v F(.)⋅ =
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Slide 8
Institute for Static and Dynamics of Structures
Fuzzy random
Fuzzy random
structural analysis
input variables
result variables
Fuzzy stochastic
1 Fuzzy random fields
2 Discretization offuzzy random fields
3 Numerical techniques of fuzzystochastic sampling
4 Result evaluation
FSFEM:
OUTLET:
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Slide 9
Institute for Static and Dynamics of Structures
B f ú2θ1 θ2
x(θ1, θ2)
FSFEM: Fuzzy random fields
fuzzy random field
X( ) X =X( ) θθ = θ θ∈ ⊆B ún
time invariant fuzzy random variables
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Slide 10
Institute for Static and Dynamics of Structures
B f ú2θ1 θ2
x(θ1, θ2)
Pi
FSFEM: Fuzzy random fields
fuzzy random field
time invariant fuzzy random variables
fuzzy random variable of Pi(θ1, θ2)
X(θ1, θ2): Ω 6 F( X ) =~ x x ∈~~ ú1~ X X( ) X =X( ) θθ = θ θ∈ ⊆B ún
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Slide 11
Institute for Static and Dynamics of Structures
µ(xk)1.00.0
x
xk~
ωk 6 xk~~
~realization of X(θ1, θ2)
B f ú2θ1 θ2
x(θ1, θ2)
Pi
FSFEM: Fuzzy random fields
fuzzy random field
time invariant fuzzy random variables
fuzzy random variable of Pi(θ1, θ2)
X(θ1, θ2): Ω 6 F( X ) =~ x x ∈~~ ú1~ X X( ) X =X( ) θθ = θ θ∈ ⊆B ún
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Slide 12
Institute for Static and Dynamics of Structures
µ(xk)1.00.0
x
xk~
ωk 6 xk~~
~realization of X(θ1, θ2)
B f ú2θ1 θ2
x(θ1, θ2)
Pi
FSFEM: Fuzzy random fields
fuzzy random field
time invariant fuzzy random variables
realization of thefuzzy random field
fuzzy random variable of Pi(θ1, θ2)
X(θ1, θ2): Ω 6 F( X ) =~ x x ∈~~ ú1~ X X( ) X =X( ) θθ = θ θ∈ ⊆B ún
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Slide 13
Institute for Static and Dynamics of Structures
1.0µ(s)
s
s~
0.0
F(x(θ1))1.0
x0.0
F(x(θ2))1.0
x
α
Sα Xα(θ1) Xα(θ2)
θ1 θ2
x(θι1, θι2)
P1(θ1) P2(θ2)
realization of one original function of the fuzzy random field
FSFEM: Fuzzy random fields
representation with fuzzy bunch parameters s: X(θ) = X(s, θ)~ ~ ~
sj 0 Xj(θ1) 0 Xj(θ2) 0
x(θ2)x(θ1)
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Slide 14
Institute for Static and Dynamics of Structures
1.0µ(s)
s
s~
0.0
F(x(θ1))1.0
x0.0
F(x(θ2))1.0
x
α
Sα Xα(θ1) Xα(θ2)
θ1 θ2
x(θι1, θι2)
P1(θ1) P2(θ2)realization ofone original of the fuzzy random field
FSFEM: Fuzzy random fields
representation with fuzzy bunch parameters s: X(θ) = X(s, θ)~ ~ ~
numerical evaluation with α-discretization
( ) ))((X );(X ) ,sX( θµθ=θ αα~
( ) ( ] α αθ = θ ∀ ∈ α ∈X ( ) X s , s s ; 0,1S1] (0; ))((X ∈α∀α=θµ α
sj 0 Xj(θ1) 0 Xj(θ2) 0
x(θ2)x(θ1)
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Slide 15
Institute for Static and Dynamics of Structures
FSFEM: Fuzzy random fields
15
point discretization
midpoint method
method of local averaging
methods of weightedintegrals
Representation of continuous fuzzy random fields by afinite number of discrete fuzzy random variables
Discretization
nodal point method
series extension Karhunen-Loeve extension
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Slide 16
Institute for Static and Dynamics of Structures
FSFEM: Fuzzy random fields
Special case: stationary isotropic fuzzy random field
Fuzzy covariance function for( ) ( ) ( )1 2 1 2K COV X Xθ θ = θ θ⎡ ⎤⎣ ⎦, ,
( ) ( )1 2X and Xθ θ
( ) ( )2
1 2 X X 12K k Lθ θ = σ ⋅,
16
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Slide 17
Institute for Static and Dynamics of Structures
µ(Lx)
Lx
fuzzy correlation lenght Lx~
1
L12
0
1kx(L12) fuzzy correlation function kx(L12)
1212 x
xx 12
L1 if L 0 LLk L
0 otherwise
⎧ − ∈⎪= ⎨⎪⎩
[ , ]( )~ ~
~
FSFEM: Fuzzy random fields
Special case: stationary isotropic fuzzy random field
Fuzzy covariance function for( ) ( ) ( )1 2 1 2K COV X Xθ θ = θ θ⎡ ⎤⎣ ⎦, ,
( ) ( )1 2X and Xθ θ
( ) ( )2
1 2 X X 12K k Lθ θ = σ ⋅,
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Slide 18
Institute for Static and Dynamics of Structures
µ(Lx)
Lx
fuzzy correlation lenght Lx~
1
L12
0
1kx(L12) fuzzy correlation function kx(L12)
1212 x
xx 12
L1 if L 0 LLk L
0 otherwise
⎧ − ∈⎪= ⎨⎪⎩
[ , ]( )~ ~
~
µ[kx(L12)] 1 0
kx(L12)
FSFEM: Fuzzy random fields
Special case: stationary isotropic fuzzy random field
Fuzzy covariance function for( ) ( ) ( )1 2 1 2K COV X Xθ θ = θ θ⎡ ⎤⎣ ⎦, ,
( ) ( )1 2X and Xθ θ
( ) ( )2
1 2 X X 12K k Lθ θ = σ ⋅,
18
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Slide 19
Institute for Static and Dynamics of Structures
Fuzzy random
Fuzzy random
structural analysis
input variables
result variables
Fuzzy stochastic
1 Fuzzy random fields
2 Representation offuzzy random fields
3 Numerical techniquesof FSFEM
4 Result evaluation
FSFEM:
OUTLET:
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Slide 20
Institute for Static and Dynamics of Structures
FSFEM: Numerical techniques
Fuzzy random function
i i i i
1
F (x) F(x, ) F(s , x, )
i 1, ...,pθ = θ = θ
=1
1
p
s
s
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦ 1
1
1 2
1 2
1 2 3
1 2 3
1 2 3 4
1
n
n 1
n n
n n 1
n n n
n n n 1
n n n n
s
s
s
ss
s
s
s
s
+
+
+ +
+ +
+ + +
+ + +
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
Construction of the bunch parameter space
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Slide 21
Institute for Static and Dynamics of Structures
1
1
1 2
1 2
1 2 3
1 2 3
1 2 3 4
1
n
n 1
n n
n n 1
n n n
n n n 1
n n n n
s
s
s
ss
s
s
s
s
+
+
+ +
+ +
+ + +
+ + +
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
Fuzzy random functions
Fuzzy functions
i i i i
1
F (x) F(x, ) F(s , x, )
i 1, ...,pθ = θ = θ
=
i i i
2
x( ) x(s , )i 1, ...,p
θ = θ=
2
1
p
s
s
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
1
1
p
s
s
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
FSFEM: Numerical techniques
Construction of the bunch parameter space
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Slide 22
Institute for Static and Dynamics of Structures
1
1
1 2
1 2
1 2 3
1 2 3
1 2 3 4
1
n
n 1
n n
n n 1
n n n
n n n 1
n n n n
s
s
s
ss
s
s
s
s
+
+
+ +
+ +
+ + +
+ + +
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
FSFEM: Numerical techniques
Fuzzy random functions
Fuzzy functions
Random functions
i i i i
1
F (x) F(x, ) F(s , x, )
i 1, ...,pθ = θ = θ
=
i i i
2
x( ) x(s , )i 1, ...,p
θ = θ=
i i i i
3
F (x) F(x, ) F(s , x, )
i 1, ...,pθ = θ = θ
=
1
1
p
s
s
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
2
1
p
s
s
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
3
1
p
s
s
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
Construction of the bunch parameter space
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Slide 23
Institute for Static and Dynamics of Structures
1
1
1 2
1 2
1 2 3
1 2 3
1 2 3 4
1
n
n 1
n n
n n 1
n n n
n n n 1
n n n n
s
s
s
ss
s
s
s
s
+
+
+ +
+ +
+ + +
+ + +
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
FSFEM: Numerical techniques
Fuzzy random functions
Fuzzy functions
Random functions
Dependencies
i i i i
1
F (x) F(x, ) F(s , x, )
i 1, ...,pθ = θ = θ
=
i i i
2
x( ) x(s , )i 1, ...,p
θ = θ=
i i i i
3
F (x) F(x, ) F(s , x, )
i 1, ...,pθ = θ = θ
=
ix 12 x i 12
4
k (L ) k (s ,L )
i 1, ...,p
=
=
1
1
p
s
s
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
2
1
p
s
s
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
3
1
p
s
s
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
4
1
p
s
s
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
Construction of the bunch parameter space
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Slide 24
Institute for Static and Dynamics of Structures
~F(x1)
x1
1µ(s1)
s1
s1,αα
fuzzy randomness
s1,α min
s1,α max
s1,α
1µ(s3)
s3
s3,α
fuzziness
α
s3,α max
s3,α min
s3,α
FSFEM: Numerical techniques
simultaneous consideration of
input space of the fuzzybunch parameters
s1
s2
s3
F(x2)
x2
1µ(s2)
s2
randomness
s2,ααs2,α
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Slide 25
Institute for Static and Dynamics of Structures
1
E1,α
µ(σ1)
αk
µ(σm1)
αk
1
σ1 σm1
kEm1,αk
ks1,α
1
sn1,α
sk,α
αk
µ(s1)
kS1,α
αk
1
1
Sn1,αk
k
Sk,αk
µ(sn1)
µ(sk)
αk
^
^
^
F1,α 0 Σ1,α
s1,α 0 S1,αk k
sn1,α 0 Sn1,αk k
sk,α 0 Sk,α kk
k k
Fm1,α 0 Σm1,αk k
^
^
^
^
^
mapping operator
(σ1, ..., σm1) = f(s1, ..., sn1)
α-level-optimization
FSFEM: Numerical techniques
25
a-level optimization
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Slide 26
Institute for Static and Dynamics of Structures
kσm1,α
kσ1,α
elementsof s~
elementsof σ~
s1,α
sn1,α
sk,α
k
k
k
^
^
^
^
^
FSFEM: Numerical techniques
kF1,α (x)
x
1
Input space of the F1,α (x), ... , Fp ,α (x)1 kk
kFp ,α (x)
1
x
1
kF1,α (x)
rF1,α (x)
F1,α (x)l
Fp ,α (x)k1
Fp ,α (x)l1
Fp ,α (x)r1
Stochastic finite element method withdeterministic basic solution
kF1,α (z)
z
1
kFq ,α (z)
1
z
1
Output space of the F1,α (z), ... , Fq ,α (z)1 kk
kF1,α (z)
F1, α (x)l
F1, α (x)r
kFq ,α (z)
1l
Fq ,α (z)1
rFq ,α (z)
1
( )k 1 k k 1 k1, q , 1, p ,ˆ ˆ ˆ ˆF (z), ...,F (z) g F (x), ...,F (x)α α α α=
26
Mapping model
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Slide 27
Institute for Static and Dynamics of Structures
FSFEM: Numerical techniques
Stochastic finite element method withdeterministic FEM algorithm
( )k 1 k k 1 k1, q , 1, p ,ˆ ˆ ˆ ˆF (z), ...,F (z) g F (x), ...,F (x)α α α α=
27
Perturbationmethods
PolynomialChaos
Monte-CarloSimulation
x2
x1
f (s, x1, x2)generation of n crisprealizations for allinput variables
computing of nresult variables zk
checking of the variance of zk!
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Slide 28
Institute for Static and Dynamics of Structures
gleichverteilteZufallszahlen
Zufallszahlenbeliebiger Verteilungf(y) f(x)
F(yi) = F(xi)
F(y) F(x)
1 1
1 yi xi
0
0
0
0y x
X = FX-1(Y)Method of inverse
distribution function
Monte Carlo Simulation (1)
equally distributedrandom variable
arbitrarily distributedrandom variable
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Slide 29
Institute for Static and Dynamics of Structures
Fuzzy random
Fuzzy random
structural analysis
input variables
result variables
Fuzzy stochastic
1 Fuzzy random fields
2 Representation offuzzy random fields
3 Numerical techniquesof FSFEM
4 Result evaluation
FSFEM:
OUTLET:
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Slide 30
Institute for Static and Dynamics of Structures
FSFEM: Result evaluation
histogram
pk(z) , f(z)
zz
= ⋅∑ k kn
z z p
( )σ = − ⋅∑z k kn
z z p
random sample with n sample elementsfor each result values z
parameter estimation of the sample parameter(mean value, variance)
test of different typs of probability distribution functions –determination of the parameter
estimation of quantiles of the sample
empirical distribution function
bunch parameter σ
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Slide 31
Institute for Static and Dynamics of Structures
200
mm
Finegrade concrete
8
100AR glass filament yarnreinforcement66 x 310 tex
Example: textile reinforced test specimen (1)
test specimen
3 reinforcementlayersEglass=80000N/mm²
1 finegrade concrete layer
FE-Model
assumption of homogeneous material leads to simultaneous cracking wrong
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Slide 32
Institute for Static and Dynamics of Structures
Example: textile reinforced test specimen (2)
uncertainty of material parameters as consquence of:
0 2 4 6 8 10 12 14 16
18
16
14
12
10
8
6
4
2
0
F [N/mm²]
g [‰]
I IIa IIb
Test series B1-004 [1]NEG-ARG310-01Vf = 1.371
different stress-strain curves for different tests
gradually cracking of the finegrade concreteleads to slope
32
function by cripstensile strength
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Slide 33
Institute for Static and Dynamics of Structures
Tensile strength and compressive strength are correlatedcaused by the material law of concrete.
Compressive strength are modeled asstationary fuzzy random field.
Discretization yields 10 fuzzy random variables with equalfuzzy probality distribution function.
200 mm
8 σ
finegrade concrete
NEG-ARG310-0166 parallel laidrovings in 3 layers
realization of the compressive strength α = 0α = 1α = 0
Example: textile reinforced test specimen (3)
( )
σ− +
σ
−∞
⎛ ⎞= σ = −⎜ ⎟⎜ ⎟π ⎝ ⎠
∫
2u
u
lnx ln6.268 22
i u1 uF x,s exp du
22
σu
µ σu( )
6 7 8
fuzzy bunch parameter σu1
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Slide 34
Institute for Static and Dynamics of Structures
( )[ ]⎧ − ∈ ∀ ∈⎪= σ ⋅ ⎨
⎪⎩
12x x x2
x1 2 u
L1 if L 0,L L LLCOV X , X
0 otherwise
Example: textile reinforced test specimen (4)
( ) σ212 uCOV L /
1
L12
kx(L12)
µ(Lx)
Lx [mm]
fuzzy correlation length Lx~1
100 200 300
( ) ( )( ) ( )( )1 2 1 1 2 2COV X , X E X E X X E X⎡ ⎤= − −⎣ ⎦i
Correlation between the 10 fuzzy random variables are describedby the linear fuzzy-covariance function.
bunch parameter
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Slide 35
Institute for Static and Dynamics of Structures
0
4
8
12
16
0 5 10 15
F[N/mm²]
ε [‰]
Example: textile reinforced test specimen (5)
Simulation with FSFEM yields fuzzy random function for stress strain dependency.
test series B1-004NEG-ARG310-01
Vf = 1,371 %1
1 F(F)
8
6
F [N/mm²]
empiricalfuzzy probabilisticdistribution function
Functional values are fuzzy random values with fuzzy probability distribution function
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Slide 36
Institute for Static and Dynamics of Structures
Example: reinforced folded plate structure (1)
folded plate plane 2folded plate plane 1
x1
x3 x2 p
1.5 m
1.0
1.2
concrete C 20/25:layer 1-7reinforcement S 500:layer 8 7.85 cm²/mlayer 9 2.52 cm²/mlayer 10 2.52 cm²/mlayer 11 7.85 cm²/m
4.5m0.
1 m
1
4
7
8
1110
9
geometry
12
34
56
7
89
1011
1213
14
1516
1718
1920
21
2223
2425
2627
28
2930
3132
3334
35
3637
3839
4041
42
4344
4546
4748
49
5051
5253
5455
56
5758
5960
6162 63
12
34
56
78
910
1112
1314
1516
1718
1920
2122
2324
2526
2728
2930
3132
3334
3536
3738
3940
4142
4344
4546
4748
finite element model0v 312 =ρ=ρ= ~~~
0v 321 =ρ=ρ= ~~~0v3 =~
36
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Slide 37
Institute for Static and Dynamics of Structures
Example: reinforced folded plate structure (2)
stationary isotropic fuzzy random fieldfor concrete compressive strength
22 26 300.0
1.0
βc [N/mm²]
F(βc)
F(βc)~
∫∞−
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛⎟⎠
⎞⎜⎝
⎛
σπ⋅σ=β
x 2
2c dtm-t0.5-exp2
1)(F~~
24.5 25.5 26.50.0
1.0µ(m)
mfuzzy expected value m~
2.0 4.5 20.00.0
1.0µ(LX)
Lx [m]fuzzy correlation length Lx
~
, σ = 1.5
random field for superficial load
5 6 70.0
1.0
p [kN/m²]
F(p)
∫∞−
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛⎟⎠
⎞⎜⎝
⎛
σπ⋅σ=
x 2
2 dtm-t0.5-exp2
1(p)F
σ = 0.3, m = 6.0
Lx = 4, perfect correlatedrandom field
37
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Slide 38
Institute for Static and Dynamics of Structures
Example: reinforced folded plate structure (3)
realizations of the fuzzy random fieldin dependency of thefuzzy correlation length Lx
2.0 4.5 20.0
1
0Lx [m]
µ(Lx)
~Lx=2.0m
22 29concrete compressive strength [N/mm²]
Lx=4.5m
Lx=20.0m
38
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Slide 39
Institute for Static and Dynamics of Structures
Example: reinforced folded plate structure (4)
features of the deterministic finite element algorithm
• physical nonlinear model on layer-to-layer basis
• consideration of the governing nonlinearities of reinforced concrete- cracking- tension stiffening- nonlinear material laws for concrete and steel
• incremental iterative solution strategy
39
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Slide 40
Institute for Static and Dynamics of Structures
Example: reinforced folded plate structure (5)
empirical fuzzy probability distribution function for displacement v63(63)
14 16 18 20 220
1
displacement v3(63) [mm]
F(v3(63))
α = 0α = 1
40
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Slide 41
Institute for Static and Dynamics of Structures
Example: reinforced folded plate structure (6)
41
fuzzy expected value of the displacement v3(63)
17.58 17.89 18.34 E[v3(63)][mm]
µ(E[v3(63)])
0
1solution withoutuncertainty
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Slide 42
Institute for Static and Dynamics of Structures
Thank you !