“Reliability Analysis of Structures with Interval...

Giuseppe Muscolino1 , Roberta Santoro1 and Alba Sofi2

1 Department of Civil Engineering, University of Messina, Messina, Italy

2 Department of Mechanics and Materials, University “Mediterranea” of Reggio Calabria (Italy)

““Reliability Analysis of Structures with Interval Reliability Analysis of Structures with Interval Uncertainties under Stochastic ExcitationsUncertainties under Stochastic Excitations””

The credibility of probabilistic reliability methodsprobabilistic reliability methods relies on the availability of sufficient data to describe accurately the probabilistic distribution of the uncertain variables, especially in the tails. Indeed, reliability estimates are very sensitive to small variations of the assumed probabilistic models.

IntroductionIntroduction

2

Uncertainties affecting both structural parameters and external loads need to be included in structural reliability assessment to obtain credible estimates of failure probability.

If available information is fragmentary or incomplete, nonnon--probabilistic probabilistic approachesapproaches, such as convex models, fuzzy set theory or interval models (Ben-Haim ,Elishakoff, 1995; Elishakoff, Ohsaki, 2010), can be alternatively applied for handling structural uncertainties.

Non-probabilistic methods are complementary rather than competitive to probabilistic methods (Moens, Vandepitte, 2005).

However, while the numerous available data permit to model with good accuracy the excitations as stochastic processes, unfortunately the data about the structural parameters are frequently quite limited.

Reliability for stochastic excitationsReliability for stochastic excitations

3

Studies on reliability analysisreliability analysis of randomly excited structures have been carried out mainly introducing the extreme value processextreme value process.

{ }max 0( )= max ( )

t TX T X t

≤ ≤

( ) [ ]max s max, ( , ) ( ) ; ( )XL b T T B X t B 0 t T X T B≡ = ≤ ≤ ≤ = ≤⎡ ⎤⎣ ⎦P P P

The probability of failureprobability of failure coincides with the first passage probability, i.e. the probability that the extreme value random process firstly exceeds the safety bounds within the time interval [0,T].

t

B

-B

X (k)(t)

T

The reliability functionreliability function represents the probability that the extreme value process is equal to or less than the barrier level B within the time interval [0,T ]

Reliability for stochastic excitationsReliability for stochastic excitations

4

Studies on reliability analysisreliability analysis of randomly excited structures with deterministic properties have shown that the reliability function, for zero-mean Gaussian exciting process can be expressed as (Vanmarke,1975):

⎮X (r)(t)⎮

X max(r) (T )

T t pXmax(b,T ) FXmax

(b,T )

b b

1-p

p XT,p

1-pp

σX

( ) [ ]max max 0,, ( ) ( ) exp ( )X X XL b T X T b P b T bη⎡ ⎤= ≤ = −⎣ ⎦P

0, max( ) [ (0) ]XP b X b= ≤Pdenotes the initial probability, that is the probability of not exceeding the deterministic level b at time t=0

2 22,

0, 0,

1 1( ) exp exp2 π 2 2 π 2

XXX

X X X X

b bbλσ

ησ σ λ λ

⎛ ⎞⎛ ⎞⎜ ⎟= − = −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

is the so-called hazard function (or limiting decay rate).spectral momentsspectral moments

,0

=2 ( )dX Xλ ω ω ω∞

∫ G

(Vanmarke, 1972)

The present contribution deals with the reliability evaluation of linear reliability evaluation of linear structural systems with uncertainstructural systems with uncertain--butbut--bounded parameters subjected to bounded parameters subjected to stationary Gaussian random excitationstationary Gaussian random excitation.

Interval ReliabilityInterval Reliability

5

Interval reliability evaluation involves zerozero-- and secondand second--order spectral order spectral momentsmoments of a selected stationary response process: the underlying idea is to derive the interval spectral moments of the stochastic response process and the corresponding interval reliability function in approximate closed-form.

The proposed approachproposed approach relies on:ù the use of Interval Rational Series ExpansionInterval Rational Series Expansion (IRSEIRSE) in conjunction with the improved interval analysisimproved interval analysis to obtain an analytical approximation of the interval reliability function.ù the derivation of interval reliability sensitivitiesinterval reliability sensitivities with respect to the uncertain parameters by direct differentiation.

ù the use of firstfirst--order interval Taylor series expansionorder interval Taylor series expansion to obtain estimates of the upper and lower bounds of the interval reliabilityupper and lower bounds of the interval reliability.

1.1. Improved Interval AnalysisImproved Interval Analysis

2.2. Interval Stochastic AnalysisInterval Stochastic Analysis

3.3. Explicit Interval Reliability FunctionExplicit Interval Reliability Function

4.4. Numerical ApplicationsNumerical Applications

5.5. Concluding RemarksConcluding Remarks

OutlineOutline

Basic interval operationsBasic interval operations: let , and be interval numbers (Moore, 1966)

7

[ ] { }[ , ] , I x xx xx x x x x= ≤ ≤ ∈= R

( )0 , ( ) / 2

2:: ./

x xx x

= +

∆ = −

xx

midpointdeviationx

x∆

x0x xx∆

0

An interval number interval number represents a range of possible values within a closed set:

Ix Iy Iz

( ) ( )[ ]

, ; min , , , ,max , , , ;

, ; / , 1 , 1 if 0 .

I I I I

I I I I I

x y x y xy xy xy xy xy xy xy xy

x y x y

x y x y

x y x yy x yx y

⎡ ⎤⎡ ⎤+ = + + × =⎣ ⎦ ⎣ ⎦⎡ ⎤ ⎡ ⎤− = − − = × ∉⎣ ⎦ ⎣ ⎦

0; 1I I I Ix x x x− ≠ ≠

Properties of interval arithmeticProperties of interval arithmetic:; I I I I I I I Ix y y x x y y x+ = + × = ×

( ) ( ) ( ) ( );I I I I I I I I I I I Ix y z x y z x y z x y z+ ± = + ± × × = × ×

Commutative law Commutative law ::

Associative law Associative law ::

SubdistributiveSubdistributive law law :: ( )I I I I I I Ix y z x y x z× + ⊆ +

Furthermore:

Classical Interval Analysis (CIA)Classical Interval Analysis (CIA)

8

In the classical interval analysis, the accuracy of the results is affected by the so-called dependency phenomenondependency phenomenon arising when different occurrences of a single interval variable in an expression are treated as independent variablesindependent variables.

To limit effects of the dependency phenomenonTo limit effects of the dependency phenomenon

““ReducingReducing the the overstimationoverstimation isis a a crucialcrucial issueissue toto a a successfulsuccessful intervalinterval analysisanalysis””(Muhanna & Mullen, 2001)

Affine Affine ArithmeticArithmetic ((AAAA)) (Comba & Stolfi,1993; Stolfi & De Figueiredo, 2003)

ImprovedImproved IntervalInterval AnalysisAnalysis ((IIAIIA)) ( Muscolino & Sofi, 2012)

ParameterizedParameterized IntervalInterval AnalysisAnalysis ((PIAPIA)) ( Elishakoff & Miglis, 2012)

The dependency phenomenon often leads to an overestimationoverestimation of the interval width.

( ) ; [1,2]I I I If x x x x= − =•• For example : ( ) [1 2,2 1] [ 1,1] 0If x = − − = − ≠

( ) (1 1)I If x x= − ( ) 0If x =

Dependency phenomenonDependency phenomenon

9

0 1 1 2 2 3 3ˆ I I I Ix x x x xε ε ε= + + + +

The ImprovedImproved IntervalInterval AnalysisAnalysis ((IIAIIA)) is based on the definition of the Extra Extra Unitary IntervalUnitary Interval (EUIEUI) î

Ie

0 central valuepartial devia

::

[ 1,1] :

tions

noise symbolsiIi

xx

ε = −Each intermediate result is represented by a linear function with a small remainder interval.

In Affine Affine ArithmeticArithmetic ((AAAA) ) an interval variable is represented by an affineaffine formform , which is a first-degree polynomial:

Ix ˆ Ix

0 Î Ixeα α α= + ∆

EUIEUI enablesenables toto treat variables with multiple occurrence as dependent onestreat variables with multiple occurrence as dependent onesIx

Improved Interval analysis (IIA)Improved Interval analysis (IIA)

( ) ( ) [ ]2ˆ ˆ ˆ ˆ ˆ; ˆ 1,1i i i i i

I I I I I Ii i i i ii i i i i ix ye e e e e ex y x y x y x y± = ± × = =

“Classical”interval analysis ( ) ( )

, ; , ;

min , , , , max , , , ;

I I I Ii i i i i i i i i i i i

I Ii i i i i i i i i i i i i i i i i i

x y x y x y x y x y x y

x y x y x y x y x y x y x y x y x y

⎡ ⎤ ⎡ ⎤+ = + + − = − −⎣ ⎦ ⎣ ⎦⎡ ⎤× = ⎣ ⎦

““Improved Improved Interval AnalysisInterval Analysis””

[ ] [ ]ˆ ˆ ˆ ˆ ˆ ˆ ˆ 1,1 0 1,1 ; ; / 1I I I I I I Ii i i i i i ie e e e e e e− ⇒ − = × = =

The accuracy of the results obtained by the Classical Interval Analysis Classical Interval Analysis (CIA) , the Affine Arithmetic Affine Arithmetic (AA) and the Improved Interval AnalysisImproved Interval Analysis (IIA) is demonstrated through appropriate comparisons with the Exact solution. Exact solution.

For instance, let us consider the multiplication of two interval functions:

( )10 (10 )II I II I Iaz x y b a c= × = + + × − +

Interval Analysis: an exampleInterval Analysis: an example

10

[ ]2, 2 ;Ia = − + [ ]1, 1 ;Ib = − + [ ]1, 1Ic = − +

121

77

[77,121]Exact

[71,121]IIA

[71,129]AA

71

129

[49,169]CIA

( , , )I Iaz b c

( , , )II bz ca

2π− 2π

169

49

( , , )I Ib cz a

Truss structuresThe rank-r change in the stiffness matrix expressed as the superposition of r rank-one matrices:

11

( ) 01

ˆr

I Ti i i i

i

eα=

= + ∆∑K K v vα

Interval Rational Series Expansion (IRSE)Interval Rational Series Expansion (IRSE)

Solution of interval equilibrium equations

( )1

1

01

ˆ ˆ( ) ( ) ; ( )r

I I I I I I Ti i i i i i i

i

e eα α α−

−

=

⎡ ⎤= = ∆ ⇒ = + ∆⎢ ⎥⎣ ⎦∑α α αu K f u K v v f

Neumann Series Expansion

( ) ( )1

1 1 1 10 0 0 0

1 1 1

ˆ ˆ1sr r

sI T I Ti i i i i i i i

i s i

e eα α− ∞

− − − −

= = =

⎡ ⎤ ⎡ ⎤= + ∆ = + − ∆⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦∑ ∑ ∑K K v v K K v v Kα

Since this expansion converges very slowly to the exact solution, the so-called Interval Rational Series Expansion (IRSE), which gives an approximate explicit expression of the inverse of the interval stiffness matexplicit expression of the inverse of the interval stiffness matrix,rix, has been recently proposed ((MuscolinoMuscolino, Santoro & , Santoro & SofiSofi 2012)2012)

Interval equilibrium equations in Statics: ( )( ) ( )I I =K u fα αÎ I

xeα α= ∆

IRSE formula to solve interval equilibrium equations

The IRSE formula can be used to evaluate the inverse of the interval stiffness matrix. In the case of truss structurestruss structures, the IRSE yields:

12

The previous formula allows to obtain explicit solutions for theThe previous formula allows to obtain explicit solutions for the responseresponse

Interval Rational Series Expansion (IRSE)Interval Rational Series Expansion (IRSE)

( )( )1

1 10 0

1

1

11 1

1 1

ˆˆ

ˆˆˆ

ˆˆ1+

ˆ1++

ˆ ˆ

1

Irj

Ik k

I

r rI I T

i

IIri ii i

Ii

jI

ji i i i i j i j

i i

r r r

i j j

j ji i i

kk i k

ki j

jj i

Ij j

Ii i

jkk

k

e

e d

d

eee d

e

ee d

ee

dd

ααα

α αα

αα

α

α =≠

−− −

= =

= = =≠

=

⎡ ⎤= + ∆ = − +⎢ ⎥

∆∆

⎣ ⎦

∆

∆

∆∆

− +

∆∆

∆

∑ ∑

∑∑∑

∑ ∑K K v v K D D

D

α

If , the approximate inverse of the interval stiffness matrix can be evaluated by retaining only the first two terms only the first two terms (Impollonia-Muscolino, 2011):

1iα∆

( )( )1

1 10 0

1 1

ˆˆ1+

Îr

i iI

i i i

rI I T

i ii

i i ii

ed

ee

α αα

−

= =

− −⎡ ⎤= + ∆ ≈ −⎢ ⎥⎣

∆∆⎦

∑ ∑K K v v K Dα

10

1 10 0

; , ,Tis i s

Tis i s

d s j k−

− −

= =

=

v K v

D K v v K

It holds if and only if the following conditions are satisfied : <1; , , ,s sd s i j kα =






OutlineOutline

Problem formulationProblem formulationEquations of motion of a n-DOF linear structure with uncertain-but-bounded

parameters subjected to a stationary multi-correlated Gaussian stochastic process:

[ ]0 ( ) ( ) ( ) ( ) ( ) ( ), ,It t t t, + , + , = ∈ =M U C U K U F α α α αα α α α α

14

0 0 1( ) ( ) c c=C α M + K α is the Rayleigh model of the interval damping matrix.

Following the interval formalism, the stiffness and damping matrices can be expressed as linear functions of the uncertain physical properties:

( )0

0 0 ; ( ) ;iiα

∂= =

∂α=α

α αK K K K 0 0 0 1 0 c c+C = M K

r∈α R is the vector collecting the symmetric fluctuations of the uncertain parameters , ( ) [ ]ˆ ˆ, 1,..., ; 1,1I I I

i i i ie i r eα α= ∆ = = −

( ) 01

ˆ , r

I Ii i i

ieα

=

= + ∆ ∈∑α α αK K K 0 11

ˆ( ) , r

I Ii i i

ic eα

=

= + ∆ ∈∑α α αC C K

( ) ( )t t= +F FF Xµ Fµis fully characterized by the mean-value and PSD ( )ωF FX XG

The interval response meaninterval response mean--valuevalue, where the input has mean value , can be determined once the inverse of the interval stiffness matrixinverse of the interval stiffness matrix is evaluated, that is:

15

Frequency domain stochastic analysisFrequency domain stochastic analysis

[ ]1( ) E ( , ) ( ) , , .It −= = ∈ =U Fµ α U α K α α α α αµ

( ) [ ]* , ( , ) ( ) ( , ), ,T Iω ω ω ω= ∈ =F FX XUUG α H α G H α α α α α

requiring the evaluation of the interval frequency response function interval frequency response function (FRFFRF) matrix given by:

[ ] [ ]1 110 0 0( , ) ( ) ( , ) ( ) ( , ) ( ), , .I

nω ω ω ω ω ω− −−⎡ ⎤= + = + ∈ =⎣ ⎦H α H P α I H P α H α α α α

120 0 0 0( ) jω ω ω

−⎡ ⎤= − + +⎣ ⎦H M C K ( )1

1

ˆ( , ) 1 jr

Ii i i

ic eω ω α

=

= + ∆∑P α K

E ( )t=fµ f

The interval response PSDinterval response PSD function matrix can be determined by

Due to the linearity of the system, the stationary Gaussian stochastic response process is characterized from a probabilistic point of view by the definition of the meanmean--value vectorvalue vector and the Power Spectral Density (PSD)Power Spectral Density (PSD) function matrix , following a frequency domain analysis approach.

( )Uµ α( , )tU α

( ),ωUUG α

The starting point to derive the IRSEIRSE is the decomposition of the matrix as sum of rank-one matrices:

16

Explicit interval meanExplicit interval mean--value response vectorvalue response vector

Retaining only first-order terms, the IRSEIRSE yields the following approximate approximate explicit expressionexplicit expression of the inverse of the interval stiffness matrixinverse of the interval stiffness matrix :

n n× iK

0 01 1

ˆ ˆr r

I I T Ii i i i i i i

i i

e eα α= =

= + ∆ = + ∆∑ ∑K K K K v v

( ) 1 10

1

ˆˆ1

IrI i i

iIi i i i

ee d

αα

− −

=

∆≈ −

+ ∆∑K K D1

0

1 10 0

;

.

Ti i i

Ti i i

d −

− −

=

=

v K v

D K v v K

( ) { } { }1( ) mid ( ) +dev ( )I I I I I−

≡ = =U fU U Uµ µ α K µ µ α µ α

By applying the IRSEIRSE in conjunction with the improved interval analysis via improved interval analysis via EUIEUI, the approximate interval meanapproximate interval mean--value response vector value response vector can be expressed as:

with LowerLower and UpperUpper Bounds:

{ }{ }

( ) mid ( )

( ) mid ( )

I

I

= −∆

= + ∆

U U

U U

U

U

µ α µ µ α

µ α µ µ α

{ } 10 0,

1

mid ;r

Ii i

i

a−

=

⎡ ⎤= +⎢ ⎥⎣ ⎦∑U fµ K D µ

1

( ) ;r

i ii

a=

∆ = ∆∑U fDµ α µ

( )

( )

2

0, 2

2

;1

.1

i ii

i i

ii

i i

dad

ad

αααα

∆=

− ∆

∆∆ =

− ∆

17

Explicit interval Explicit interval FRFFRF matrixmatrix

11

01

ˆ( ) ( ) ( )r

I T Ii i i i

i

p eω ω ω α−

−

=

⎡ ⎤= + ∆⎢ ⎥⎣ ⎦∑H H v v

Upon substitution of the decomposition , the interval FRF matrixinterval FRF matrix of the structural system with interval parameters takes the following form:

Ti i i=K v v

Then, by applying the IRSEIRSE truncated to first-order terms, an approximate explicit approximate explicit expression of the interval FRF matrix expression of the interval FRF matrix is obtained as:

01

ˆ( )( ) ( ) ( )ˆ1+ ( ) ( )

IrI i i

iIi i i i

p ep e b

ω αω ω ωω α ω=

∆≈ −

∆∑H H B 0

0 0

( ) ( ) ;

( ) ( ) ( ).

Ti i i

Ti i i

b ω ω

ω ω ω

=

=

v H v

B H v v H

1( ) (1 ).p j cω ω= +with

Alternatively, by applying the improved interval analysis via EUI:improved interval analysis via EUI:

{ } { }( , ) mid ( ) +dev ( )I Iω ω ω=H α H H

mid 0 0,1

( , ) ( ) ( ) ( )r

i ii

aω ω ω ω=

= + ∑H α H B dev1

ˆ( ) ( ) ( )r

Ii i i

i

a eω ω ω=

= ∆∑H B

2

0, 2 2

( ) ( ) ( )( ) ; ( ) .1 ( ) ( ) 1 ( ) ( )

i i ii i

i i i i

p b pa ap b p b

ω α ω ω αω ωω α ω ω α ω

⎡ ⎤∆ ∆⎣ ⎦= ∆ =⎡ ⎤ ⎡ ⎤− ∆ − ∆⎣ ⎦ ⎣ ⎦

with

18

Explicit Interval Explicit Interval PSD PSD matrixmatrixSubstituting the interval transfer function matrix, the interval PSD function matrixinterval PSD function matrix

of the structural response can be expressed as :

where{ } { }

* ( , ) ( ) ( , ) ( ) ( , )

ˆ mid ( , ) dev ( , )

I Tω ω ω ω ω

ω ω

≡ =

= +

f fU U X XUU

UU UU

G α G H G H α

G α G α

α

{ }{ } ( ) ( )

*mid mid

**mid dev dev mid

mid ( ) ( , ) ( ) ( , );

ˆdev ( ) ( , ) ( ) ( ) + ( ) ( ) ( , )

I T

TI I I T

ω ω ω ω

ω ω ω ω ω ω ω

=

=

f f

f f f f

UU X X

UU X X X X

G H α G H α

G H α G H H G H α

Finally, approximate analytical expressions of the interval spectral momentsinterval spectral moments of order of the random response, useful for structural reliabilityuseful for structural reliability, can be computed as:

{ } { } [ ], , ,( ) mid ( , ) dev ( , ) , , ; 0,1, 2Iω ω= + ∈ = =UU UU UUλ α λ α λ α α α α α

where { } { } { } { }, ,0 0

ˆmid =2 mid ( ) d ; dev =2 dev ( ) d .I Iω ω ω ω ω ω∞ ∞

∫ ∫UU UU UU UUλ G λ G

It is worth to emphasize that explicit relationships between the interval statistics of the displacement vector and the radius of input parameters, useful for the analytical evaluation of the correspondinganalytical evaluation of the corresponding interval sensitivities, interval sensitivities, have been provided.

( , )tU α iα∆






OutlineOutline

20

Interval Reliability FunctionInterval Reliability FunctionFor a structure with uncertain-but-bounded parameters, the extreme value random extreme value random

processprocess, over a specified time interval [0,T], is mathematically defined as:

[ ]max 0( , ) max ( , ) , ,I

t TU T U t

≤ ≤= ∈ =α α α α α α

The cumulative distribution function (CDF)cumulative distribution function (CDF) of the extreme value of the extreme value random processrandom process is called the reliability functionreliability function . It represents the probability that I is equal to or less than the barrier within the time interval [0,T] and is commonly expressed as (Vanmarcke, 1975):

max( , ( ), )UL b Tα α

max ( , )U Tα ( ) ( )Ub b= −α αµ

( ) [ ]max max 0,, ( ), ( , ) ( ) ( , ( )) exp ( , ( )) , ,I

U U UL b T U T b P b T bη⎡ ⎤ ⎡ ⎤= ≤ = − ∈ =⎣ ⎦ ⎣ ⎦α α α α α α α α α α α αP

Initial probabilityInitial probability

Hazard functionHazard function0, max( , ( )) [ ( ,0) ( )]UP b U b= ≤α α α αP

2

0,

( )( , ( )) ( )exp2 ( )U U

U

bbη νλ

+ ⎛ ⎞= −⎜ ⎟

⎝ ⎠

αα α ααBy applying the classical Rice’s formula:

21

By applying the classical Rice’s formula and denoting with the largest dimensionless extreme process, the interval reliabilityinterval reliability can be expressed as:

max ( , )IX Tα

( ) 00, ,( ) ( ) / ( )I IU

IUUbβ λµ= −α αα

Zero and second-order interval spectral moments

dimensionlessdimensionless interval interval peak factor processpeak factor process

( )

[ ]

max 0, max 0,

20,

, ( ), ( , ) ( )

( )exp ( ) exp , , .

2

X X X

X IX

L T X T

T

β β

βν +

= ≤⎡ ⎤⎣ ⎦⎡ ⎤⎛ ⎞

≈ − − ∈ =⎢ ⎥⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

α α α α

αα α α α α

P

2,

0,

1( )2

((π

))

IUI

IU

Uνλλ

+ =αα

α

maxma

0,

x( , )( )

),

(

I

IU

I U TX Tλ

=αα

α

mean upmean up--crossing rate crossing rate

at levelat level ( ) 0IUµ >α

2 2 2

0 00, 2,( ) 2 ( ,( )d ; ( ) =2 ( , )d) ( )I I I I I

U U U UUI

U U UG Gλ λσ ω ω σ ω ω ω∞ ∞

≡ = ≡ =∫ ∫α α α αα α

dimensionless interval barrier

Explicit Interval Spectral MomentsExplicit Interval Spectral Moments

22

( )( ) ( )

( ) ( ) ( )( )

max

max

, 0, 2 ,

0,0, 0,

(0)20 0 0(0) 0,0, 0, 0, 0, , ,(0)

2,

, ( ),,

( , ) 2 1

X

U U U

IX UI

L i Yi

I I U IU U U i U i i

U

L Ts T

C T s s sµ λ λ

ββ

α

λβ β λ β

λ

∆ =

∂=

∂∆

⎧ ⎫⎪ ⎪⎡ ⎤= + − +⎨ ⎬⎢ ⎥⎣ ⎦⎪ ⎪⎩ ⎭

α 0

α α

Interval sensitivity Interval sensitivity of the meanof the mean--valuevalue

Interval sensitivities of the zero Interval sensitivities of the zero --and and second second --order spectral momentsorder spectral moments

The interval reliability sensitivityinterval reliability sensitivity can be derived analytically by differentiating with respect to the deviation amplitude of the i-thuncertain parameter:

in terms of interval sensitivities of the meaninterval sensitivities of the mean--valuevalue , zerozero--orderorder and second second --order spectral momentsorder spectral moments.

,U

Iisµ 0, ,U

Iisλ

2, ,U

Iisλ

iα∆max

IXL

Since explicit relationships between interval statistics of the response and interval parameters have been determined, interval sensitivitiesinterval sensitivities can be evaluated analytically by direct differentiation with respect to the uncertain parameters.

Explicit Interval Reliability SensitivityExplicit Interval Reliability Sensitivity

To this aim, can be approximated by applying the firstfirst--order interval order interval Taylor series expansionTaylor series expansion:

23

Bounds of the interval reliability functionBounds of the interval reliability function

By applying the improved interval analysis via EUIimproved interval analysis via EUI, the Lower BoundLower Bound and the Upper BoundUpper Bound of can be evaluated as follows:( )

max 0, ,IX UL Tβ

( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )( )

max max max

max max max

0 0 0(0)0, 0, 0,

1

0 0 0(0)0, 0, 0,

1

, , , , ;

, , , , .

X

X

r

X U X U L ,i U ii

r

X U X U L ,i U ii

L T L T s T

L T L T s T

β β β α

β β β α

=

=

= − ∆ ∆

= + ∆ ∆

∑

∑

α

α

( )max 0, ,I

X UL Tβ

( )( ) ( )( ) ( )( )( )( ) ( )( )

max max max

max max

0 0 0(0)0, 0, , 0,

1

0 0(0)0, , 0,

1

, , ,

ˆ , ,

X

X

rI IX U X U L i U i

ir

IX U L i U i i

i

L T L T s T

L T s T e

β β β α

β β α

=

=

= ∆

= ∆ ∆

∑

∑

+

+

Evaluation of an analytical approximation of the interval reliability function of the peak factor, along with its sensitivities, allows to estimate its bounds.

( )( ) ( )( ), maxmax

0 00, , 0, ˆ, ,

L i XX

I IU L i U is T s T eβ β= ∆

nominal or midpoint CDFnominal or midpoint CDF with

ii--thth peak factor reliability sensitivitypeak factor reliability sensitivity

deviation deviation amplitudeamplitude

24

Bounds of the interval reliability functionBounds of the interval reliability functionThe knowledge of the bounds of the interval reliability function of the peak

factor allows to evaluate the parameters of the interval response process needed to guarantee the desired safety level.

denotes the interval interval fractilefractile of the peak factor of the peak factor of order p and can be evaluated as solution of the following nonlinear interval equation:

( )max

, ,X p Tρ α

( )( )max max, , , ,X Xp L p T Tρ= α α

Sketch of: a) UB and LB of the peak factor CDF and b) PDFs of the UB and LB of the peak factor CDF.

2 4 60

0.8

1.6

( )max

00,

X

U

L

β

∂

∂

( )max

(0)

00,

X

U

L

β

∂

∂( )max

00,

X

U

L

β

∂

∂

( )00,Uβ2 4 6

0

0.5

1

maxXLmaxXL

max

(0)XL

( )00,Uβ

maxXρmaxXρ)a )b






OutlineOutline

Numerical ApplicationNumerical Application24-bar truss structure subjected to turbulent wind loads in the x-direction

with uncertain-but-bounded Young’s moduli of the diagonal bars (r=9)

8 20 = 2.1×10 kN/m iE α α∆ = ∆ with and

4 2 -10, 0 0 15 10 m ( 1,2,..., 24); =3 m; =500kg; =3.517897 =0.000547iA A i L M c s c s−= = × = and

( )0 ˆ1 , ( 16,17,...,24)I Ii i iE E e iα= + ∆ =

Wind velocity ( , ) ( ) ( , )sW z t w z W z t= +

( ),10( ) 10s sw z w zγ

= Mean value

( , )W z t Fluctuating component modelled as a zero-mean stationary Gaussian random field

( )2

20 ,10 4/32

( ) 41

sWWG K w χωω χ

=+

(Davenport)

( ), , ,

2

( , ) ( , )1 ( , ) , ( 1,4,7)2

sx i i x i x i i

D i s D i i s

F z t F F z t

C A w C AW z t w iρ ρ

= +

≈ + =

26

L

L

L

LL

1

2

3

4

5

6

7

8

9

14 15

10 11

12 13

2316

20

17

2122

19

24

18

u1 u2 u3

u4 u5 u6

u7 u8 u9

v1 v2 v3

v4 v5 v6

v7 v8 v9

1 2 3

4 5 6

7 8 9

wind

( ),7 7 ,xF z t

( ),4 4 ,xF z t

( ),1 1,xF z t

x

z

Comparison between the proposed and exact reliability functionsreliability functions of the peak factor processes and of the horizontal displacements of nodes1 and 7 of the truss structure (T=1000T0) setting the interval Young moduli of the diagonal bars at their upper bounds.

Numerical Results/1Numerical Results/1

1max ( , )IX Tα 7max ( , )IX Tα

27

( )1max 1

(0)0, ,X UL Tβ

0.15α∆ =

ˆ( 1 )IiExact e i= ∀

ˆ( 1 )IiProposed e i= ∀

Nominal

1

(0)0,Uβ

0.1α∆ =

( )1max 1

(0)0, ,X UL Tβ

0.15α∆ =



Nominal

1

(0)0,Uβ

0.1α∆ =

( )7 max 7

(0)0, ,X UL Tβ

7

(0)0,Uβ

0.15α∆ =

0.1α∆ =



Nominal

( )7 max 7

(0)0, ,X UL Tβ

7

(0)0,Uβ

0.15α∆ =

0.1α∆ =



Nominal

0 (1 ) ( 16,17,...,24)iE E i iα= + ∆ =


( )1max 1

(0)0, ,I

X UL Tβ

UB

LB

0.025α∆ =

Exact

Proposed

Nominal

1

(0)0,Uβ

( )1max 1

(0)0, ,I

X UL Tβ

UB

LB

0.025α∆ =

Exact

Proposed

Nominal

1

(0)0,Uβ

7

(0)0,Uβ

( )7 max 7

(0)0, ,I

X UL Tβ

0.025α∆ =

Exact

Proposed

Nominal

UB

LB

7

(0)0,Uβ

( )7 max 7

(0)0, ,I

X UL Tβ

0.025α∆ =

Exact

Proposed

Nominal

UB

LB

28

Comparison between the exact and proposed UBUB and LBLB of reliability functionsreliability functionsof the peak factor processes and of the horizontal displacements of nodes 1 and 7 of the truss structure with interval Young’s moduli of the diagonal bars.


( )0 ˆ1 , ( 16,17,...,24)I Ii i iE E e iα= + ∆ =

00.025 ; 1000i T Tα α∆ = ∆ = =


( )1max 1

(0)0, ,I

X UL Tβ

0.05α∆ =

Exact

Proposed

Nominal

1

(0)0,Uβ

UB

LB

( )1max 1

(0)0, ,I

X UL Tβ

0.05α∆ =

Exact

Proposed

Nominal

1

(0)0,Uβ

UB

LB

( )7 max 7

(0)0, ,I

X UL Tβ

0.05α∆ =

Exact

Proposed

Nominal

7

(0)0,Uβ

UB

LB

( )7 max 7

(0)0, ,I

X UL Tβ

0.05α∆ =

Exact

Proposed

Nominal

7

(0)0,Uβ

UB

LB

29

( )0 ˆ1 , ( 16,17,...,24)I Ii i iE E e iα= + ∆ =

00.05 ; 1000i T Tα α∆ = ∆ = =

Comparison between the exact and proposed UBUB and LBLB of reliability functionsreliability functionsof the peak factor processes and of the horizontal displacements of nodes 1 and 7 of the truss structure with interval Young’s moduli of the diagonal bars.


Numerical Results/4: sensitivity analysisNumerical Results/4: sensitivity analysis

30

The proposed approximate closed-form expression of the peak factor interval peak factor interval reliability sensitivityreliability sensitivity is applied to investigate the rate of change in the interval CDF due to changes in the structural parameters. To identify the most influential uncertain parameters, a percentage measure of the influence of the generic interval variable on the CDF of the selected peak factor process can be defined by introducing a function of sensitivityfunction of sensitivity :

16i =

20i =

23i =

( )( )11max

(0), 0, , %

Xi L U Tϕ β

1

(0)0,Uβ

16i =

20i =

23i =

( )( )11max

(0), 0, , %

Xi L U Tϕ β

1

(0)0,Uβ

17i =

20i =

21i =

23i =

( )( )77 max

(0), 0, , %

Xi L U Tϕ β

7

(0)0,Uβ

17i =

20i =

21i =

23i =

( )( )77 max

(0), 0, , %

Xi L U Tϕ β

7

(0)0,Uβ

( )( ),max

00, ,

L iX

IUs Tβ

( )( )( )( )

( )( )max

max

max

0, 0,0

, 0, 0(0)0,

,, (%) 100

,X

X

L i Y

i L Y iX Y

s TT

L T

βϕ β α

β

∆= ∆ ×


31

( )0 ˆ1 , ( 16,20,23)I Ii i iE E e iα= + ∆ =

00.05 ; 1000i T Tα α∆ = ∆ = =

Comparison between the exact and proposed UBUB and LBLB of reliability functionsreliability functionsof the peak factor processes and of the horizontal displacements of nodes 1 and 7 of the truss structure with interval Young’s moduli of the diagonal bars, neglecting terms associated to the least influential parameters in the first-order Taylor series expansion based on the results of the reliability sensitivity analysis.


( )7 max 7

(0)0,

IX UL Tβ

0.05α∆ =

Exact

Proposed

UB

LB

0, 16,18,19, 22, 24i iα∆ = =

7

(0)0,Uβ

( )7 max 7

(0)0,

IX UL Tβ

0.05α∆ =

Exact

Proposed

UB

LB

0, 16,18,19, 22, 24i iα∆ = =

7

(0)0,Uβ

( )1max 1

(0)0, ,I

X UL Tβ

0.05α∆ =

Exact

Proposed

1

(0)0,Uβ

UB

LB

0, 17,18,19, 21, 22, 24i iα∆ = =

( )1max 1

(0)0, ,I

X UL Tβ

0.05α∆ =

Exact

Proposed

1

(0)0,Uβ

UB

LB

0, 17,18,19, 21, 22, 24i iα∆ = =

( )0 ˆ1 , ( 16,17,20,21)I Ii i iE E e iα= + ∆ =






OutlineOutline

The interval reliability function has been evaluated in approximate closed-form by applying the Interval Rational Series ExpansionInterval Rational Series Expansion in conjunction with the improved improved interval analysisinterval analysis, recently developed by the authors.

ConclusionsConclusions

33

An analytical approach to evaluate the reliability function for structures with uncertain-but-bounded parameters subjected to stationary Gaussian random excitation has been proposed.

The Interval Rational Series ExpansionInterval Rational Series Expansion provides an approximate explicit expression of the inverse of an interval matrix with modifications. The improved improved interval analysisinterval analysis allows to limit the overestimation of the interval solution width due to the dependency phenomenon occurring in classical interval analysis.

Remarkable features of the proposed approach are: i) the capability of handling a large number of uncertainties and evaluate analytically the interval reliability function; ii) the possibility of providing very accurate explicit estimates of the bounds of the interval reliability in the framework of the firstfirst--order interval Taylor series order interval Taylor series expansionexpansion.

A wind-excited truss structure with interval axial stiffness of the diagonal bars has been analyzed. Appropriate comparisons with the exact reliability bounds obtained by the vertex method have demonstrated the accuracy of the proposed procedure.

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