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4/6/2009 IFMA Seminar, France 1
The Stochastic Finite Element Method: Theory and Applications
George Stefanou
Institute of Structural Analysis and Seismic ResearchNational Technical University of Athens, Greece
Clermont-FerrandJune 2009
4/6/2009 IFMA Seminar, France 2
Outline of the presentation
• Stochastic processes and fields
• Simulation of Gaussian stochastic processes/fields
• Simulation of non-Gaussian stochastic processes/fields
• The stochastic finite element method (SFEM)
• SFE (static) analysis of shells with random material and geometric properties -response variability calculation
• SFE-based stability analysis of shells with random imperfections
• Nonlinear dynamic analysis of frames with random material properties under seismic loading
4/6/2009 IFMA Seminar, France 3
Simulation of Gaussian stochastic fields
The spectral representation method (Rice 1954, Shinozuka & Deodatis 1991)
Fundamental theorem:
Spectral representation of field :
is asymptotically a Gaussian stochastic field as due to the central limit theorem
[ ]0
( ) cos( ) ( ) sin( ) ( )f x x du x dvκ κ κ κ∞
= +∫
( )f x1
0
ˆ ( ) cos( )N
n n nn
f x A xκ−
=
= +Φ∑
2 ( )n ff nA S κ κ= Δ n nκ κ= Δ
0,1, 2,..., 1n N= −
u
N
κκΔ =
ˆ ( )f x N →∞
Truncated form (finite number of terms)
4/6/2009 IFMA Seminar, France 4
Its mean value and autocorrelation function are identical to the corresponding targets as
is periodic with period Τ0:
The Fast Fourier Transform (FFT):
M is the number of points on which
f is generated:
FFT-based sample functions are always generated on a domain equal to one period Τ0
The reduction of step Δx (stochastic mesh refinement) leads to a larger number of terms M in the FFT series
N →∞
0
2T
πκ
=Δ
( )21
( )
0
ˆ RenpM i
i Mn
n
f p x B eπ⎛ ⎞− ⎜ ⎟
⎝ ⎠
=
⎡ ⎤Δ = ⎢ ⎥
⎢ ⎥⎣ ⎦∑
( )
2i
nin nB A e φ=
0,1,..., 1n M= −
1,...,1,0 −= Mpμ2=M
ˆ ( )f x
4/6/2009 IFMA Seminar, France 5
The Karhunen-Loève expansion method (Loève 1977, Ghanem & Spanos1991)
Exact form (infinite terms) Truncated form (finite number of terms)
eigenpairs of the covariance function obtained from the solution of the following eigenvalue problem:
(Fredholm integral equation of the second kind)
Exact solution feasible only for simple geometries and special forms of
Numerical solution with the conventional Galerkin approach Dense matrices very costly to compute and invert
∑∞
=
+=1
)()()(),(n
nnn xxxf θξφλμθ ∑=
+=N
nnnn xxxf
1
)()()(),(ˆ θξφλμθ
, ( )n n xλ φ 1 2( , )ffC x x
∫ =D
nnnff xdxxxxC )()(),( 21121 φλφ
1 2( , )ffC x x
4/6/2009 IFMA Seminar, France 6
• Some remarks on K-L expansion
Few K-L terms are needed for the simulation of strongly correlated (narrow-banded) stochastic fields
Few K-L terms are needed for the simulation of stochastic fields with smooth (differentiable) target covariance function
More efficient simulation when the exact (analytical) solution of the eigenvalueproblem is feasible (Huang et al. 2001)
The variance of the stochastic field is underestimated and the simulated field is in general not homogeneous since its variance is a function of x (Field & Grigoriu2004):
)],([])([])([)],(ˆ[ 2
1
2
1
θφλφλθ xfVarxxxfVarn
nn
N
nnn =≤= ∑∑
∞
==
ˆ ( , )f x θ
4/6/2009 IFMA Seminar, France 7
• Non-ergodic characteristics of K-L sample functions (Stefanou & Papadra-kakis 2007)
- Ergodicity in the mean:
Proof:
Without any loss of generality, we consider that
[ ]{ }Pr lim ( ) ( ) 1LL
f x E f x→∞
= =
[ ]( ) ( ) 0E f x f x= =
( ) ( ) ( ) ( )
0 0 01 1
1 1 1ˆ ˆ( ) ( ) ( ) ( )N NL L Li i i i
n n n n n nL
n n
f x f x dx x dx x dxL L L
λ ξ φ λ ξ φ= =
= = =∑ ∑∫ ∫ ∫
( ) ( )
01
1ˆlim ( ) lim ( )N Li i
n n nL LLn
f x x dxL
λ ξ φ→∞ →∞
=
⎡ ⎤= ⎢ ⎥⎣ ⎦∑ ∫
( )
01
?
1lim ( )
N Lin n nL
n
x dxL
λ ξ φ→∞
=
=
⎡ ⎤= ⎢ ⎥⎣ ⎦∑ ∫
Existence of limitRight hand side of the expression:
random variableK-L sample functions: in general
not ergodic in the mean
4/6/2009 IFMA Seminar, France 8
- Ergodicity in autocorrelation:
Proof:
( )ˆ ( ) ( )i Tff ffR Rξ ξ= ( )ˆ ( )if x∀
( ) ( ) ( ) ( ) ( )
0
1ˆ ˆ ˆ ˆˆ ( ) ( ) ( ) ( ) ( )Li i i i i
ffL
R f x f x f x f x dxL
ξ ξ ξ= + = +∫( ) ( ) ( ) ( )
0 01 1 1 1
1 1( ) ( ) ( ) ( )
N N N NL Li i i in m n m n m n m n m n m
n m n m
x x dx x x dxL L
λ λ ξ ξ φ ξ φ λ λ ξ ξ φ ξ φ= = = =
= + = +∑∑ ∑∑∫ ∫
( ) ( ) ( ) ( )
01 1
1ˆ ˆlim ( ) ( ) lim ( ) ( )N N Li i i i
n m n m n mL LL
n m
f x f x x x dxL
ξ λ λ ξ ξ φ ξ φ→∞ →∞
= =
⎡ ⎤+ = +⎢ ⎥⎣ ⎦∑∑ ∫
( ) ( )
01 1
1lim ( ) ( )
N N Li in m n m n m
Ln m
x x dxL
λ λ ξ ξ φ ξ φ→∞
= =
⎡ ⎤= +⎢ ⎥⎣ ⎦∑∑ ∫
1
( ) ( ) ( )N
Tff n n n
n
R x xξ λ φ ξ φ=
≠ = +∑ even in the case n=m
K-L sample functions: in general not ergodic in autocorrelation
4/6/2009 IFMA Seminar, France 9
• Solution of Fredholm integral equation
Exact solution feasible only for simple geometries (line, square, circle) and special forms of
Two main categories of numerical methods are available: integration formulae-based methods (e.g. quadrature method) and expansion methods (e.g. Galerkin)
Numerical solution with the conventional Galerkin approach Densematrices very costly to compute and invert
An efficient numerical solution of the Fredholm integral equation is indispensable especially when higher order eigenpairs are needed for an accurate representation of the stochastic field
1 2( , )ffC x x
4/6/2009 IFMA Seminar, France 10
The wavelet-Galerkin method
Wavelet basis functions enhance the performance of the Galerkin method in the solution of integral equations (Phoon et al. 2002)
If the kernel of the integral equation is a rapidly decreasing function, the application of the wavelet-Galerkin method leads to sparse matrices (Beylkin, Coifman & Rokhlin 1991)
Basic steps of the wavelet-Galerkin approach:
1. Selection of a set of M wavelet basis functions
Usually Haar wavelets are used (the simplest form of Daubechies wavelets)
Haar mother wavelet function:1
( ) 1
0
xψ⎧⎪= −⎨⎪⎩
0 0.5
0.5 1
x
x
otherwise
≤ <≤ <
1 2( ), ( ),..., ( )Mx x xψ ψ ψ
1 2( , )ffC x x
4/6/2009 IFMA Seminar, France 11
2. Approximation of each eigenfunction of the covariance kernel by a linear combi-nation of Haar wavelet basis functions:
di(n): wavelet coefficients
M = 2m, m: wavelet level
Remark 1. Number of terms in the truncated K-L expansion:
1( ) ( )
0
( ) ( ) ( )M
n nn i i
i
x d x x Dφ ψ−
Τ
=
= =Ψ∑
3. Solution of the generalized eigenvalue problem: ( ) ( )x D x AHDΤ ΤΨ Λ = Ψ
Remark 2. A crucial difference of this approach with the conventional Galerkin method: the computation of matrices A and H does not require numerical integration
Matrix A is the 2D wavelet transform of and H a diagonal MxM matrix
Remark 3. Matrix A is sparse by nature and can be made further sparse byignoring elements below a threshold value This may lead to computational instabilities in the numerical procedure
1 2( , )ffC x x
N M≤
4/6/2009 IFMA Seminar, France 12
• Numerical examples (Stefanou & Papadrakakis 2007)
1. First order Markov stochastic field with exponential covariance function
2/2
2 2Case 1: ( ) ( ) ( )
(1 )b
ff ff ff
bC R e S
bξ σξ ξ σ κ
π κ−= = ⇒ =
+- zero mean, unit variance- selected threshold values in step 3: 10-3, 10-7
2. Stochastic field with square exponential covariance function
- zero mean, unit variance- selected threshold value in step 3: 10-12
A smaller threshold value is selected for case 2 because the square exponential kernel is sparser by its nature
Parameters: Wavelet level m=7 (M=27=128 eigenpairs can be computed at most), number of K-L terms-points in the discretization of the wave number domain N=16, number of terms in the FFT series M=128
2 22 2 2
2 /Case 2: ( ) ( ) ( ) exp42
bff ff ff
b bC R e Sξ σ κξ ξ σ κ
π− ⎛ ⎞
= = ⇒ = −⎜ ⎟⎝ ⎠
4/6/2009 IFMA Seminar, France 13
Ensemble variance of truncated K-L expansion:
- Wavelet level m = 7- Threshold values in matrix A: 10-3, 10-7
Mean variance: 0.912 Good approximation for 5 K-L terms
Threshold value 10-3: substantial oscillations observed throughout the range [-1,1] due to the numerical instabilities in the calculation of eigenpairs
Threshold value 10-7: a stable solution is obtained
0.8
0.85
0.9
0.95
1
0 32 64 96 128
x [-1,1]
Va
ria
nc
e
thres.10^-3
thres.10^-7
2
1
ˆ[ ( )] ( )N
n nn
Var f x xλ φ=
=∑
The variance is fluctuating w.r.t. xThe error at the boundaries is larger compared
to the middle region
4/6/2009 IFMA Seminar, France 14
• Eigenvalue decay for θ=0.2, 0.4, 1.0 και 2.0 (θ: Vanmarcke’s scale of fluctuation)
The most rapid eigenvalue decay is observed in case 2 for θ=2.0 (strongly correlated stochastic field with smooth autocovariance function)
Few K-L terms are needed for an accurate simulation of random fields of this kind
Case 2: bθ π=Case 1: 2bθ =
0
0.2
0.4
0.6
0.8
1
1.2
0 8 16 24 32
index i
eig
enva
lue
θ=0.2
θ=0.4
θ=1.0
θ=2.0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 8 16 24 32
index i
eig
enva
lue
θ=0.2
θ=0.4
θ=1.0
θ=2.0
4/6/2009 IFMA Seminar, France 15
Case 1: θ=0.2
0
0.2
0.4
0.6
0.8
1
1.2
0 4 8 12 16 20 24
Number of terms N
En
se
mb
le v
ari
an
ce
K-L
Spectral
0
0.2
0.4
0.6
0.8
1
1.2
0 4 8 12 16 20 24
Number of terms N
En
se
mb
le v
ari
an
ce
K-L
Spectral
Case 1: θ=2.0
0
0.2
0.4
0.6
0.8
1
1.2
0 4 8 12 16 20 24
Number of terms N
En
sem
ble
var
ian
ce
K-L
Spectral
0
0.2
0.4
0.6
0.8
1
1.2
0 4 8 12 16 20 24
Number of terms N
En
se
mb
le v
ari
an
ce
K-L
Spectral
Case 2: θ=0.2
Case 2: θ=2.0Convergence of each approach to the target
variance
4/6/2009 IFMA Seminar, France 16
Case 1 Case 2
Spectral representation produces ergodic sample functions in a sample-by-sample sense i.e. every sample function has the target mean and SDF (see below SDF plots)
0.0000.0711-0.01031.6556Skewness
1.0000.78480.78480.7848Variance
0.0000.00000.00000.0000Mean
2.02261.86843.2233Max. value
-1.9070-1.9034-1.3331Min. value
TargetAverageBestWorst
Samples generated with the spectral representation (N=16, θ=0.2)
0.000-0.07610.0153-1.1962Skewness
1.0000.67910.70320.8954Variance
0.000-0.0115-0.1370-0.3434Mean
1.59811.77061.1902Max. value
-1.7178-2.0820-3.3076Min. value
TargetAverageBestWorst
Samples generated with the K-L expansion(N=16, θ=0.2)
0.0000.0678-0.00041.7822Skewness
1.0000.90690.90690.9069Variance
0.0000.00000.00000.0000Mean
2.03282.11473.4019Max. value
-1.9791-2.3388-1.2252Min. value
TargetAverageBestWorst
Samples generated with the spectral representation (N=16, θ=0.2)
0.0000.05950.0166-0.5614Skewness
1.0000.75920.88030.5237Variance
0.0000.08720.08130.5846Mean
1.84551.89321.9598Max. value
-1.5988-1.7148-1.5411Min. value
TargetAverageBestWorst
Samples generated with the K-L expansion(N=16, θ=0.2)
4/6/2009 IFMA Seminar, France 17
0
0.2
0.4
0.6
0.8
1
1.2
-3 -2 -1 0 1 2 3
f
CD
F(f
)
Sample
Target
0
0.2
0.4
0.6
0.8
1
1.2
-3 -2 -1 0 1 2 3
f
CD
F(f
)
Sample
Target
K-L expansion: ensemble average
Spectral representation: ensemble average
Case 2θ=0.2
K-L expansion: Case 1, θ=0.2, m=9
Spectral representation: Case 1, θ=0.2, m=9
0
0.2
0.4
0.6
0.8
1
1.2
-3 -2 -1 0 1 2 3
f
En
se
mb
le C
DF
(f)
Sample
Target
0
0.2
0.4
0.6
0.8
1
1.2
-3 -2 -1 0 1 2 3
f
En
se
mb
le C
DF
(f)
Sample
Target
4/6/2009 IFMA Seminar, France 18
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0 10 20 30 40 50 60
wave number
SD
F
Target
Sample
10-33x10-4Spectral representation (1 sample)
60.02.5K-L expansion (1 sample)
97Wavelet level, power in FFT series (m)
Computational performance of K-L expansion and spectral representation
Ergodicity of spectral representation-based samples with regard to SDF: a perfect matching of the target SDF is observed (cases 1-2, θ=0.2)
Case 1
Case 2
02
( ) ( )
0 0
1 ˆ( ) ( ) exp( )2
Ti i
ffS f x i x dxT
κ κπ
= −∫
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0 10 20 30 40 50 60
wave number
SD
F
Target
Sample
4/6/2009 IFMA Seminar, France 19
Simulation of non-Gaussian stochastic fields
Theoretically all the joint multi-dimensional density functions are needed to fully characterize a non-Gaussian stochastic field
Much of the existing research has focused on a more realistic way of defining a non-Gaussian sample function e.g. as a simple transformation of some under-lying Gaussian field with known second-order statistics:
Translation field theory (Grigoriu 1984, 1998):
- Spectral distortion:
- Compatibility of F - :
)]([)( 1 xgFxf Φ⋅= −
( ) ( )Tff ffS Sκ κ≠
1 11 2 1 2 1 2( ) [ ( )] [ ( )] [ , ; ( )]T
ff ggR F g F g g g R dg dgξ φ ξ∞ ∞
− −
−∞ −∞
= Φ Φ ⋅∫ ∫
)(ξTffR [ ]
1 1
( ) ( ) ( ) [ ( )] [ ( )]F F
TffR E f x f x E h g x h g xξ ξ ξ
− −⋅Φ ⋅Φ↓ ↓
⎧ ⎫⎪ ⎪= + = + ⇒⎨ ⎬⎪ ⎪⎩ ⎭
4/6/2009 IFMA Seminar, France 20
and the joint density of
• Characteristics of translation fields
The relationship between the two autocorrelation functions can have a closed form only in few cases
Strictly speaking, if the target F και are proven to be incompatible, there is no translation field with the prescribed characteristics.
The problem of incompatibility becomes even greater for highly skewed narrow-banded stochastic fields (Grigoriu 1998)
Analytical expressions of crossing rates are available for translation fields
1 ( ),g g x= )(2 ξ+= xgg
)](;,[ 21 ξφ ggRgg },{ 21 gg
)(ξTffR
where
4/6/2009 IFMA Seminar, France 21
- In order to address the problem of spectral distortion Iterative methods (e.g. Yamazaki-Shinozuka 1988, Deodatis-Micaletti 2001)
Repeated updating of in every iteration:
Unwanted correlations between the terms of the spectral representation series
After the first iteration, the underlying Gaussian field is no more Gaussian and homogeneous. Finally, the generated non-Gaussian sample functions will not have the prescribed marginal PDF
Use of extended non-Gaussian to non-Gaussian mapping:
- To address the issue of possible incompatibility between F and : Spectralpreconditioning Generated non-Gaussian field: approximately translationfield
)(κggS )()(
)()( )(
)(
)1( κκ
κκ
α
jggj
ff
Tffj
gg SS
SS
⎥⎥⎦
⎤
⎢⎢⎣
⎡=+
)]([)( 1 xgFFxf ∗− ⋅=
)(ξTffR
4/6/2009 IFMA Seminar, France 22
- Correlations between the terms of the spectral representation series (proof):
1( 1) ( 1)0
0
( ) 2 ( ) cos( )N
j jgg n n n
n
g x S xκ κ κ φ−
+ +
=
= Δ +∑2 2
( ) ( ) 1 ( )0 0
0 0
1 1( ) ( ) exp( ) [ ( )]exp( )
2 2
T Tj j j
ffS f x i x dx F g x i x dxT T
κ κ κπ π
−= − = Φ −∫ ∫
21
1 ( )
00
12 ( ) cos( ) exp( )
2
T Nj
gg n n nn
F S x i x dxT
κ κ κ φ κπ
−−
=
⎡ ⎤= Φ Δ + −⎢ ⎥⎣ ⎦∑∫
in iteration j+1
depends on the random phase angles φn
)()(
)()( )(
)(
)1( κκ
κκ
α
jggj
ff
Tffj
gg SS
SS
⎥⎥⎦
⎤
⎢⎢⎣
⎡=+
( ) ( )jffS κ
also depends on the random phase angles φn alteration of themarginal PDF of due to c.l.t.
( 1) ( )jggS κ+
( 1)0 ( )jg x+
4/6/2009 IFMA Seminar, France 23
Proposed enhanced hybrid method (Lagaros, Stefanou & Papadrakakis 2005)
Basic idea: replace the updating scheme of the Gaussian spectrum (source of all important difficulties in the simulation) by a NN-based regression model
The unwanted correlations between the terms of the spectral representation series become negligible
Use of the extended mapping:
Spectral preconditioning is not required an algorithm covering a widerrange of non-Gaussian fields is obtained (not only translation fields).
A very small number (<50) of iterations is required until convergence drastic reduction of the computational effort required for simulation
)(κggS
)]([)( 1 xgFFxf ∗− ⋅=
4/6/2009 IFMA Seminar, France 24
0
0.2
0.4
0.6
0.8
-4 -2 0 2 4
f
Ga
us
sia
n P
DF
(f) Target PDF
Sample PDF
0
0.2
0.4
0.6
-4 -2 0 2
f
PD
F(f
)
Target PDF
Sample PDF
Exact and sample Gaussian PDF at first iteration
Exact and sample beta PDF at final iteration (translation field case)
2
1
1(w) [ ( ) ( )]
2
NT
ff j ff jj
S κ S κ=
= −∑E
4/6/2009 IFMA Seminar, France 25
• Numerical example: a highly skewed narrow-banded stochastic field
Characteristics of target non-Gaussian field:- Lognormal distribution defined in the range [-1.30, 10.0]skewness γ = 2.763, kurtosis δ = 19.085
- Target correlation structure: σ = 1 και b = 5]exp[4
1)( 232 κκσκ bbS T
ff −=
-2
0
2
4
6
8
10
0 64 128 192 256
x (m)
Sa
mp
le f
un
cti
on
-2
0
2
4
6
8
10
0 64 128 192 256
x (m)
Sa
mp
le f
un
cti
on
Sample function generated using the D-M algorithm: [-1.23, 7.87]
Sample function generated using the proposed EHM: [-1.25, 8.89]
4/6/2009 IFMA Seminar, France 26
0
0.2
0.4
0.6
0.8
-2 0 2 4 6 8 10
f
PD
F(f)
Target PDF
Sample PDF
0
0.2
0.4
0.6
0.8
-2 0 2 4 6 8 10
f
PD
F(f)
Target PDF
Sample PDF
Marginal PDF of sample function generated using the D-M algorithm
versus target lognormal PDF
Marginal PDF of sample function generated using the proposed EHM
versus target lognormal PDF
0
0.2
0.4
0.6
0.8
0 2 4 6 8
wave number (rad/m)
Sp
ec
tra
l de
ns
ity
Starget
Sng
SDF of sample function generated using the D-M algorithm versus target SDF
0
0.2
0.4
0.6
0.8
0 2 4 6 8
wave number (rad/m)
Sp
ec
tra
l de
ns
ity Starget
Sng
SDF of sample function generated using the proposed EHM versus target SDF
4/6/2009 IFMA Seminar, France 27
Perfect matching of the target PDF due to the (exact) extended mappingPerfect matching of the target SDF achieved by the proposed ΕΗΜ (use of Rprop
training, Riedmiller & Brown 1993)
0.832EHM-Rprop
1.245EHM-Quickprop
0.625EHM-CG
2.082EHM-SD
14629279D-M*
Time (sec)IterationsMethod
* : without spectral preconditioning
Computational performance of D-M and proposed algorithm (EHM)
Substantially smaller cost of proposed EHM (~ 2 orders ofmagnitude)
Reduction of cost for the stochastic analysis of realistic structures with uncertain non-Gaussian properties
4/6/2009 IFMA Seminar, France 28
Statistical comparison of D-M and EHM algorithms
2951582
2861618
2910628
3353295
3594494
315935
2924335
284414
3846843
3229279
EHM-Rprop (Iterations)D-M* (Iterations)
* : without spectral preconditioning
D-M*
Mean value: 38242St. dev.: 28701Max: 94494Min: 4414
EHM-RpropMean value: 31.2St. dev.: 3.5Max: 38Min: 28
4/6/2009 IFMA Seminar, France 29
Identification of random shapes from images based on polynomial chaos expansion (Stefanou, Nouy & Clement 2009)
Physical problems with random geometry: disordered systems and random media, fluctuating domain boundaries, shells with cut-outs, heterogeneous materials with random distribution of inclusions…
Usually a limited number of samples of the geometry is available in practice: its complete probabilistic characterization (in terms of joint densities) is infeasible
Shape recovery from simple images allows obtaining many samples at a low cost + it is relatively precise
Material layout of a quadratic microstructure
Cylindrical shell with rectangular cut-out
4/6/2009 IFMA Seminar, France 30
• Short description of the procedure
- Shape recovery with the level-set technique: construction of a collection of discreti-zed level-set functions corresponding to a collection of images - Reduction of information through empirical Karhunen-Loève expansion:
( ) ( )
1
mk k
i ii
Xφφ μ=
≈ +∑U
- Probabilistic identification of random vector from samplesdecomposition on a polynomial chaos basis of degree p in dimension m(Wiener 1938):
- Identification of chaos coefficients:1. Without independence hypothesis: maximum likelihood estimation for random vector X2. With independence hypothesis: maximum likelihood estimation for each random variable or projection method based on empirical CDF
1( ,..., )mX X=X ( )kX
,
( ( ))m p
Hα αα
θ∈ℑ
≈ ∑X X ξ
4/6/2009 IFMA Seminar, France 31
The stochastic finite element method
Solution of stochastic elliptic boundary value problems
• Well posed problem (existence and uniqueness of solution):
και
• Weak form:
• Finite element approximation selection of suitable function spacese.g. and
[ ]( , ) ( , ) ( , ),
( , ) 0,
u f
u
κ θ θ θθ
−∇ ⋅ ∇ =
=
x x x
x
D∈x
D∈∂x
1( )C Dκ ∈
min max0 κ κ< <min maxPr ( , ) [ , ], 1x x Dκ θ κ κ⎡ ⎤∈ ∀ ∈ =⎣ ⎦
:u D×Θ→
( , , )PΘ ℑ
( , ) ( )A u v B v= 10 ( )v X H D∀ ∈ =
( , ) ,D
A u v u vdκ= ∇ ⋅∇∫ x ( )D
B v fvd= ∫ x ,u v X∈
deterministic
( , ) ( )A u v B v= 2 ( )Pv X L∀ ∈ ⊗ Θ stochastic
1 2{ , ,..., }x
hNX span Xφ φ φ= ⊂
1 2{ ( ), ( ),..., ( )}hNW span Wξ
ψ ψ ψ= ⊂ξ ξ ξ
1( ,..., )Mξ ξ=ξ
4/6/2009 IFMA Seminar, France 32
Formulation of the stochastic stiffness matrix
- Case of material randomness:
- Discretization of the stochastic fields
• Midpoint method
• Local average method:
• Weighted integral method:
[ ]),,(1),,( 0 zyxfDzyxD +=
( ) ( ) ( ) ( ) ( )
( )
( ) ( ) ( ) ( ) ( ) ( )
( )0 0 , ,
e e
e e e e e e e e e eT T
V V
k B D B dV B D B f x y z dV= +∫ ∫ ( ) ( ) ( )eee kkk Δ+= 0
( ) ( ) ( )0 (1 )e e ek k a= +
( ) ( ) ( ) ( )0
1
WNe e e e
l ll
k k X k=
= + Δ∑
3D elasticity problem:
4/6/2009 IFMA Seminar, France 33
Response variability calculation
• Direct Monte Carlo Simulation (e.g. Rubinstein 1981)
Solution of ΝSIM deterministic problems substantial computational cost especially for high dimensional problems and for large number of simulations
It must be combined with efficient discretization methods (e.g. local average method)
Particularly suitable for parallel computing environment (“embarrassingly parallel”)
Assisted by the spectacular growth of computing power, the only available method for the solution of large-scale realistic problems
∑=
=NSIM
jii ju
NSIMuE
1
)(1
)( ⎥⎦
⎤⎢⎣
⎡⋅−
−= ∑
=
NSIM
jiii uENSIMju
NSIMu
1
222 )()(1
1)(σ
4/6/2009 IFMA Seminar, France 34
• The perturbation method (Kleiber & Hien 1992)
Taylor series expansion of the stochastic finite element matrix and of the resulting response vector:
Calculation of the stochastic displacement vector:
Satisfactory results only for small coefficients of variation of the uncertain input parameters
Improvement in accuracy obtained using higher order approximations: small compared to the disproportional increase of computational effort
I II0
1 1 1
1...
2
N N N
i i ij i ji i j
K K K a K a a= = =
= + + +∑ ∑∑ I ,0
ii
KK
a a
∂∂
==
2II
0ij
i j
KK
a a a
∂∂ ∂
==
I II0
1 1 1
1...
2
N N N
i i ij i ji i j
u u u a u a a= = =
= + + +∑ ∑∑ 10 0 0u K P−=
( )I 1 I I0 0i i iu K P K u−= −
( )II 1 II I I I I II0 0ij ij i j j i iju K P K u K u K u−= − − −
4/6/2009 IFMA Seminar, France 35
• The spectral stochastic finite element method (Ghanem & Spanos 1991)
Karhunen-Loève expansion of the stiffness matrix + polynomial chaos expansion (PCE) of the displacement vector:
Application (mainly) to linear problems with small variability + Gaussian assumption for the uncertain input parameters (Sudret & der Kiureghian 2002)
Prohibitive computational cost for the solution of problems with large stochasticdimension (increase of the order of PCE)
1
0
( ) ( ),P
j jj
θ θ−
=
= Ψ∑U U
( ) ( ) ( )0
1
( ) ( ),M
e e el i
l
θ ξ θ=
= +∑k k ke
Tii
ei d
e
Ω= ∫Ω
BDBxk 0)( )(φλ
( )!1
! !
M pP
M p
++ =
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
⋅
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−−−−−
−
−
1
1
0
1
1
0
1,10,1
1,110
1,000
PPPPP
P
P
F
F
F
U
U
U
KK
KK
KK
…
……
NPNP ×
4/6/2009 IFMA Seminar, France 36
- Spectral SFEM: recent advances and extensions
Cases of non-smooth solutions (non-linearities, discontinuities): use of other basis functions e.g. wavelets (Le Maitre et al. 2004) or adaptive sparse generalized PCE (Wan & Karniadakis 2005, Blatman & Sudret 2008).
Stochastic reduced basis methods (Nair et al. 2002-2008): limited to random linear systems
Non-intrusive, stochastic response surface approaches (Baroth et al. 2006, Berveilleret al. 2006): can take advantage of powerful deterministic FE codes
Spectral SFEM in a multi-scale setting (Xu 2007): stochastic variational approach + scale-bridging multi-scale shape functions
X-SFEM (Nouy et al. 2007-8): implicit representation of complex geometries using random level-set functions
4/6/2009 IFMA Seminar, France 37
Stochastic finite element analysis of shells
The multi-layered triangular shell element TRIC (Argyris et al. 1997)
TRIC is a triangular, shear-deformable facet shell element suitable for the analysis of thin and moderately thick isotropic as well as composite plate and shell structures
The element formulation is based on the natural mode finite element method (Argyris et al. 1979)
The treatment of the element kinematics eliminates automatically shear locking phenomena (Argyris et al. 2000)
No need to perform numerical integration for the computation of the deterministic stiffness matrix which is carried out in closed form derivation of a cost-effective stochastic stiffness matrix
Computational efficiency in large-scale stochastic finite element computations
4/6/2009 IFMA Seminar, France 38
kqc : axial and symmetric bending stiffness terms
kqh : anti-symmetric bending and shear stiffness terms
kaz : stiffness terms due to in-plane rotations (azimuth stiffness terms)
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
ΟΟ
ΟΟ
ΟΟ
=
×
×
×
×Ν
)33(
)33(
)66(
)1212(
az
qh
qc
k
k
k
k
The natural stiffness of the TRIC element is based only on the 12 independent natural straining modes:
4/6/2009 IFMA Seminar, France 39
• Derivation of the stochastic stiffness matrix
- Random variation of Young modulus (Argyris, Papadrakakis & Stefanou 2002)
( ) [ ]),(1, 0 yxfEyxE +=
∫=× V
tcctTtcqc dVBBk κ
)66(
( ) 0)66(
)(1 qcqc kak +=×
( )a f x y d= ∫1
ΩΩ
Ω
,
It is proved that the axial and symmetric bending stiffness terms have a local average form
( )[ ] [ ]{ } 111
33
−−−
×
+= sqh
bqhqh kkk
∫=V
thctTth
bqh dVBBk κ ( )
6
01
b bqh qh i i
i
k k X k=
= + Δ∑
dVBXBkV
shsTsh
sqh ∫= ( ) 0)(1 s
qhsqh kak +=
6 weighted integrals Χi
Local average form
1.
2.
2a.
2b.
4/6/2009 IFMA Seminar, France 40
the natural (triangular) coordinates
e.g. weighted integrals Χ1, Χ4 : ( )X f da a12= ∫ζ ζ ζ ζβ γ, , Ω
Ω
( )X f da a4 = ∫ζ ζ ζ ζ ζβ β γ, , ΩΩ
ζ ζ ζβ γa , ,
Matrix Δk1 of the fluctuating part of the anti-symmetric bending stiffness terms:
⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
++
−−−−++=Δ
γγγ
γγ
βγγβ
γγ
ββ
β
γγβββ
ββ
β
kzl
kzl
kzl
lsymm
kzll
kzll
lkz
ll
lkz
l
llkz
lkz
lkz
l
lk
a
a
aa
a
a
a
a
a
aa
a
a
a
aa
a
2
2
2
2
2
4
2
22
2
2
2
2
4
2
2
2
2
2
4
2
1
96.
93396000
( )/ 2
2 2 3 31
1/2
1
3
h Nk
ij ij ij k kkh
z k z dz z zκ κ +=−
= = −∑∫ γβα ,,, =ji
- κij the entries of the constitutivematrix κct
- k = 1,2,…,N the number of layers
4/6/2009 IFMA Seminar, France 41
( ) ( ) ( ) ( ) ( ) ( ){ }k a k a k a kz za
z z= + + +max , ,1 1 1β γ
The azimuth stiffness terms have also an inherent local average form
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−−−−−−
=⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
=×
15.05.0
5.015.0
5.05.01
.)33(
z
z
zz
az
az
aaz
az k
ksymm
kk
kkk
kγγ
βγββ
γβ
3.
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
⋅Ω= ∫∫∫ −−−
2/
2/
22
2/
2/
22
2/
2/
22
1,
1,
1max
h
h
h
h
h
ha
z dzzl
dzzl
dzzl
k γγγ
βββ
αα κκκ
kz the maximum of the three edge bending stiffness values:
( ) ∫−Ω
=2/
2/
22
h
h iii
iz dzz
lk κ
( )a f x y d= ∫1
ΩΩ
Ω
,
4/6/2009 IFMA Seminar, France 42
- Combined random variation of Young modulus and Poisson ratio (Stefanou & Papadrakakis 2004)
The entries of the elasticity matrix are nonlinear functions of Poisson ratio consideration of random variation of Lamé constants λ and μ
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡+
+=
μμλλ
λμλκ
200
02
02
1221 ν
νλ−
=E
( )νμ+
==12
EG
( ) [ ]),(1, 0 yxfyx λλλ += ( ) [ ]),(1, 0 yxfyx μμμ +=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−−
=ν
νν
νκ
100
01
01
1 212
E
+
Axial and symmetric bending stiffness terms: local average formAnti-symmetric bending stiffness terms: 12 weighted integralsAnti-symmetric shear stiffness terms: local average formAzimuth stiffness terms: local average form
4/6/2009 IFMA Seminar, France 43
- Random variation of the thickness
The thickness h appears into the integrals expressing the stiffness matrix of the shell element
The application of the weighted integral method is not straightforward The local average approach is applied leading to greater computational efficiency
( ) )1(, 0 hahyxh += ∫Ω ΩΩ
= dyxfa hh ),(1
4/6/2009 IFMA Seminar, France 44
2. Scordelis-Lo shell
C
ν = 0.3
Α
rigid diaphragm
rigid diaphragm R = 300 mm L = 600 mm h = 3.0 mm E = 3000 N/mm2 ν = 0.3
L
P
P
C
h
x
y
z
R
1. Pinched cylinder
• Numerical examples(Stefanou & Papadrakakis 2004)
4/6/2009 IFMA Seminar, France 45
3. Hyperboloid shell
R1 = 4800 mm
R2 = 8000 mm
L = 20000 mm
h = 40 mm
E = 21000 N/mm2
ν = 0.25
L
y
xz
R1
R2
A
C
4/6/2009 IFMA Seminar, France 46
Use of 2D homogeneous Gaussian stochastic fields for the representation of the uncertain material and geometric properties
Sample functions of the stochastic fields generated by the spectral representation method
Selection of two different correlation structures:
[ ])(exp)(),( 22
22
21
21
22
22
21
2121
2
21 κκκκπσ
κκ bbbbbbS fff +−+=
( )⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎠⎞
⎜⎝⎛−⎟
⎠⎞
⎜⎝⎛−=
2
22
2
1121221 22
exp4
,κκ
πσκκ
bbbbS fff
Calculation of COV of displacement at the characteristic node C:
Investigation of the effect of various parameters of the stochastic fields (σf, Sff, b1, b2) on the response variability
)(
)()(
i
ii uE
uuCOV
σ=
1
2
4/6/2009 IFMA Seminar, France 47
1
2
σf = 0.2 b1 = b2 = 2.6
4/6/2009 IFMA Seminar, France 48
0
0,05
0,1
0,15
0,2
0,25
0,3
0,2 2 4 8 80 200
Correlation length parameter b1
CO
V o
f w
at
no
de
C E
h
E,h
0
0.05
0.1
0.15
0.2
0.25
0.065 0.65 1.3 2.6 26 65 130
Correlation length parameter b
CO
V o
f w
at
no
de
C
Eh
E,h
• Scordelis-Lo shell (σf = 0.1):
Random variation of Young modulus and thickness (local average method)
Important effect of thickness variation
• Hyperboloid shell (σf = 0.1):
Random variation of Young modulus and thickness (local average method)
Important effect of thickness variation
4/6/2009 IFMA Seminar, France 49
0
0.05
0.1
0.15
0.065 0.65 1.3 2.6 26 65 130
Correlation length parameter b
CO
V o
f w
at
no
de
C
E E,v (Local average) E,v (Weighted integral)
0
0,05
0,1
0,2 2 4 8 80 200
Correlation length parameter b1
CO
V o
f w
at
no
de
C
E E (Weighted integral)
E,v (Local average) E,v (Weighted integral)
• Scordelis-Lo shell (σf = 0.1):
Random variation of Young modulus and Poisson ratio
Negligible effect of Poisson ratio variation
• Hyperboloid shell (σf = 0.1):
Random variation of Young modulus and Poisson ratio
Negligible effect of Poisson ratio variation
4/6/2009 IFMA Seminar, France 50
0
0,05
0,1
0,15
0,2
0,25
0,3
0,2 2 4 8 80 200
Correlation length parameter b1
CO
V o
f w
at
no
de
C
hh(P.s.2)
0
0,05
0,1
0,15
0,2
0,25
0,3
0,35
0,2 2 4 8 80 200
Correlation length parameter b1
CO
V o
f w
at
no
de
C E,h
E,h(P.s.2)
• Hyperboloid shell (σf = 0.1):
Random variation of Young modulus and thickness (local average method)
Investigation of the effect of correlation structure (power spectrum)
( ) ( )C ffVar w S κ∼
4/6/2009 IFMA Seminar, France 51
• Conclusions
The stochastic stiffness matrix of the TRIC shell element is formulated in terms of a minimum number of random variables
Good agreement between weighted integral and local average methods
Significant effect of correlation length parameter b on the response variability
Slight effect of correlation structure (expressed via the form of the power spectrum) on the response variability
Random variation in the shell thickness has significant effect on the response variability compared to the effect of random Young modulus
Small effect of random variation of Poisson ratio on the results
4/6/2009 IFMA Seminar, France 52
Stability analysis of shells with random imperfections
Large scatter in the measured buckling loads
Big discrepancy between the deterministic predictions of buckling loads and the corresponding experimental results
Presence of initial imperfections which occur during the manufacturing and construction stages
Realistic description of imperfections in the framework of a robust stochastic finite element formulation
Importance of available data banks for the realistic simulation of imperfections (estimation of probability distribution, correlation structure), e.g. Elishakoff & Arbocz (1982, 1985)
4/6/2009 IFMA Seminar, France 53
Methods based on random variables
Use of 2D Fourier series with random Fourier coefficients (e.g. Chryssanthopoulos& Poggi 1995, Noirfalise et al. 2007) for the representation of geometric imperfect-ions
This idea is related to the fact that an analytical buckling analysis of cylindrical shells yields a 2D Fourier series representation of the critical modes
Methods based on stochastic fields
Simulation of the imperfections using stochastic fields combined with the finite element method for the solution of the stability problem
Use of direct Monte Carlo simulation for the computation of buckling load variability
4/6/2009 IFMA Seminar, France 54
Stability analysis of cylindrical shells with random geometric imperfections(Schenk & Schueller 2003)
Use of the data bank by Arbocz & Abramovich (1979) for the estimation of the probability distribution and correlation structure of the imperfections
Selection of 2D non-homogeneous Gaussian stochastic fields for the description of the imperfections
250 direct Monte Carlo simulations for the computation of buckling load variability (finite element model with geometric non-linearity)
The experimentally determined scatter in the limit load can be predicted numerically
A more accurate prediction requires the incorporation of other kinds of imperfections e.g. material, thickness, boundary conditions
4/6/2009 IFMA Seminar, France 55
Stability analysis of cylindrical shells with random geometric, material, thickness and boundary imperfections (Papadopoulos, Stefanou & Papadrakakis 2009)
Use of TRIC shell element with geometric and material non-linearitySelection of 2D non-homogeneous Gaussian stochastic fields for the description
of geometric imperfections (data bank by Arbocz & Abramovich, 1979)Effect of non-Gaussian assumption for E, t on buckling load variability
0 1( , ) [1 ( , )]E x y E f x y= +
0 2( , ) [1 ( , )]t x y t f x y= +
Randomly varying quantities:
X
Z
Y
Loading P
θ
L
R
E=104410N/mm2
ν=0.3 L=202.3mm R=101.6mm t=0.11597mm
f1, f2: homogeneous non-Gaussian translation fields
( ) 0 1r , R a ( , ) g ( , )x y x y x y= + +
( ) [ ]0 2P P 1 g ( )x x= +
4/6/2009 IFMA Seminar, France 56
• Initial geometric imperfections: ( ) 0 1r , R a ( , ) g ( , )x y x y x y= + +
3442.9108940196.850.1110101.6A-14
3108.8104110196.850.1128101.6A-13
3853.0104800209.550.1204101.6A-12
3196.9102730203.200.1204101.6A-10
3724.8101350203.200.1153101.6A-9
3673.8104800203.200.1179101.6A-8
3036.4104110203.200.1140101.6A-7
P (N)E (N/mm2)L (mm)t (mm)R(mm)Shell
Geometry, material properties and experimental buckling loads of A-shells (Arbocz & Abramovich, 1979)
Measured initial unfolded shape of shell A7
a0(x,y)
The ensemble average a0(x,y) as well as the evolutionary power spectrum of stochastic field g1(x,y) are derived from a statistical analysis of experimentally measured imperfections of A-shells
4/6/2009 IFMA Seminar, France 57
• Material and thickness imperfections: use of non-Gaussian translation fields
q=1.6p=0.80.26-0.13L-beta - Case 3
q=0.8p=0.80.16-0.16U-beta - Case 2
q=12p=120.5-0.5Beta - Case 1
--+-1Lognormal
Shape parametersUpper boundLower bound
∞
( )2 2
2 1 2 1 1 2 21 2, exp
4 2 2gg f
b b b bS
κ κκ κ σπ
⎡ ⎤⎛ ⎞ ⎛ ⎞= − −⎢ ⎥⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦
Correlation structure of the imperfections
• In-plane edge (boundary) imperfections: non-uniform axial loading
It is assumed that boundary imperfections are produced by a non-uniform random axial load distribution on the upper edge of the cylinder, modeled as a 1D homoge-neous Gaussian stochastic field
0 1( , ) [1 ( , )]E x y E f x y= +
0 2( , ) [1 ( , )]t x y t f x y= +
( ) [ ]0 2P P 1 g ( )x x= +
22 21
( ) exp ( )4 4
ggg g gS b b
σκ κ
π⎡ ⎤= −⎢ ⎥⎣ ⎦
loading surface
cylinder
4/6/2009 IFMA Seminar, France 58
• Numerical results
bg = 100 mm with σg = 5% are selected for the description of g2
b1 = b2 = 50 mm with σf = 10% are selected for the description of Young modulus and thickness these values are responsible for the minimum mean(Pu) and it is likely to lead to “worst case” scenarios w.r.t. lowest Pu
The choice of PDF has a significant effect on first and second order properties of Pudistribution
Max Cov(=0.16) for a Gaussian PDF, Min Cov(=0.06) for an L-beta PDF
NSIM=100
4/6/2009 IFMA Seminar, France 59
Gaussian E, t Lognormal E, t
Beta E, t U-beta E, t
(perfect)uP 5350N=
4/6/2009 IFMA Seminar, France 60
L-beta E, t
Experimental
• Conclusions
The choice of marginal PDF of E, t is crucial: it affects significantly the shape as well as the extreme values of Pu distribution
A large magnification of uncertainty has been observed in the Gaussian case: Cov(Pu) ~ 1.6σf
The lognormal and beta non-Gaussian assumptions led to estimates of the scatter of Pucloser to the experimental measurements
The tri-modal shape of buckling loads observed in the experiments has been reproduced by the corresponding numerical simulations
The lowest Pu has been found to represent only the 28-60% of the Pu of the perfect shell
Towards a robust design of imperfect shell structures
4/6/2009 IFMA Seminar, France 61
Nonlinear stochastic dynamic analysis of frames (Stefanou & Fragiadakis 2009)
• A 3-storey steel moment-resisting frame designed for a Los Angeles site (SAC/FEMA program)
• Fundamental mode period: T1=1.02 sec
• 5 integration sections defined in every beam-column element
• Geometrical nonlinearities not considered in the analysis
• Material law: bilinear with pure kinematic hardening• Loading: 3 sets of 5 strong ground motion records corresponding to 3 levels of
increasing hazard: low, medium and high + spectrum-compatible artificial accelerograms
4/6/2009 IFMA Seminar, France 62
0.638–,D16.96.9090WAHOLoma Prieta, 198915 (5/3)
0.370–,D16.96.9000WAHOLoma Prieta, 198914 (4/3)
0.371–,D28.86.9000Hollister South & PineLoma Prieta, 198913 (3/3)
0.358C,D25.56.7360LA, Hollywood Storage FFNorthridge, 199412 (2/3)
0.2C,D24.46.7360Wildlife Liquefaction ArraySuperstition Hills, 198711 (1/3)
0.209C,D28.86.9360Sunnyvale Colton AveLoma Prieta, 198910 (5/2)
0.18C,D24.46.7090Wildlife Liquefaction ArraySuperstition Hills, 19879 (4/2)
0.207C,D28.86.9270Sunnyvale Colton AveLoma Prieta, 19898 (3/2)
0.057C,D31.76.5090Plaster CityImperial Valley, 19797 (2/2)
0.239B,B31.36.7090LA, Baldwin HillsNorthridge, 19946 (1/2)
0.147C,D32.66.5285CompuertasImperial Valley, 19795 (5/1)
0.074C,D15.16.5090Westmoreland Fire StationImperial Valley, 19794 (4/1)
0.11C,D15.16.5180Westmoreland Fire StationImperial Valley, 19793 (3/1)
0.057C,D31.76.5135Plaster CityImperial Valley, 19792 (2/1)
0.042C,D31.76.5045Plaster CityImperial Valley, 19791 (1/1)
PGASoil4R3Mw2φο1StationEarthquakeID
( level)
The ground motion records used
1 Component 2 Moment magnitude 3 Closest distance to fault rupture 4 USGS, Geomatrix soil class
4/6/2009 IFMA Seminar, France 63
• 1D stochastic variation of Young modulus and yield stress described by zero-mean lognormal and L-betatranslation fields with σf =10%
• The variability of the maximum inter-storey drift θmax is examined using 1000 Monte Carlo simulations:
Lower bound Upper bound Shape parameters
Lognormal -1 +∞ - - L-beta -0.13 0.26 p=0.8 q=1.6
maxmax
max
( )( )
( )
σCOV
E
θθ
θ=
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
-1 -0.5 0 0.5 1
normalised length (m)
Sam
ple
func
tion
b=0.2
b=1
b=2
b=10
b=20
b=100
• Investigation of the sensitivity of θmax w.r.t. the correlation length parameter b
A set of sample functions of lognormal stochastic fields characterizing the spatial variation of Young modulus E in a beam
4/6/2009 IFMA Seminar, France 64
Mean value, COV of θmax for different values of correlation length parameter b and the ground motion records of slide 61 (lognormal distribution of E, σy)
The different characteristics of the seismic records are transferred to the response statistics significant record-to-record variability
The effect of b on COV is important in many cases
A large magnification of uncertainty is observed in some cases max COV=18% (~1.8σf) for record 4/2
The mean value is practically not affected by b
1/1 2/1 3/1 4/1) 5/1 1/2 2/2 3/2 4/2 5/2 1/3 2/3 3/3 4/3 5/30
0.005
0.01
0.015
0.02
0.025
record
mea
n of
θm
ax
b=0.2b=1.0b=2.0b=10b=20b=100
1/1 2/1 3/1 4/1) 5/1 1/2 2/2 3/2 4/2 5/2 1/3 2/3 3/3 4/3 5/30
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
record
CO
V( θ
max
) =
σ /
μ
b=0.2b=1.0b=2.0b=10b=20b=100
4/6/2009 IFMA Seminar, France 65
Mean value, COV of θmax for different values of correlation length parameter b and artificial accelerograms compatible with records 1/1, 4/2, 1/3 (lognormal distribution of E, σy)
The different characteristics of the seismic records are transferred to the response statistics significant record-to-record variability
The effect of b on COV is important in some cases
The magnification of uncertainty is even more pronounced in this case: max COV=23% (~2.3σf) for record 1/1
The mean value is practically not affected by b
1/1 4/2 1/30
0.005
0.01
0.015
record
mea
n of
θm
ax
b=0.2b=1.0b=2.0b=10b=20b=100
1/1 4/2 1/30
0.05
0.1
0.15
0.2
0.25
record
CO
V( θ
max
) =
σ /
μ
b=0.2b=1.0b=2.0b=10b=20b=100
4/6/2009 IFMA Seminar, France 66
Skewness of θmax for different values of correlation length parameter b and the ground motion records of slide 61 (lognormal, L-beta distribution of E, σy)
Important record-to-record variability
Significant influence of b in many cases
Values of skewness substantially different from those of the system properties due to the strong non-linearity of the problem
1/1 2/1 3/1 4/1) 5/1 1/2 2/2 3/2 4/2 5/2 1/3 2/3 3/3 4/3 5/3−3.5
−3
−2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
record
Ske
wne
ss
b=0.2b=1.0b=2.0b=10b=20b=100
1/1 5/1 1/2 2/2 2/3 5/3−3.5
−3
−2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
record
Ske
wne
ss
b=0.2b=2.0b=100
Input skewness values:- Lognormal distribution: 0.3- L-beta distribution: 0.59
4/6/2009 IFMA Seminar, France 67
• Conclusions
A stochastic response history analysis of a steel frame having uncertain non-Gaussian material properties and subjected to seismic loading has been performed.
The Young modulus and the yield stress were described by uncorrelated homogeneous non-Gaussian translation fields.
The effect of the probability distribution of the input parameters on the response variability was negligible due to the small input COV(=10%).
The significant influence of the scale of correlation of the stochastic fields and of the different seismic records on the response variability has been revealed.
A large magnification of uncertainty has been observed in some cases.
Importance of a realistic uncertainty quantification and propagation in nonlinear dynamic analysis of engineering systems.
4/6/2009 IFMA Seminar, France 68
Thank you for your attention!