optymalizacja niezawodnościowa konstrukcji - teoria i...

Optymalizacja niezawodnościowa Optymalizacja niezawodnościowa konstrukcji konstrukcji -- teoria i zastosowaniateoria i zastosowania

Stefan JENDO, Rafał STOCKIStefan JENDO, Rafał STOCKI

Pracownia Metod NumerycznychPracownia Metod Numerycznych

Niezawodności i Optymalizacji

Instytut Podstawowych Problemów TechnikiInstytut Podstawowych Problemów Techniki

Polska Akademia NaukPolska Akademia Nauk Niezawodności i Optymalizacji

Optymalizacja niezawodnościowa konstrukcji – teoria i zastosowania, Warszawa 16 grudnia 2005

Plan prezentacjiPlan prezentacji

Wprowadzenie

Metody analizy niezawodności konstrukcji

Optymalizacja niezawodnościowa konstrukcji

Optymalizacja odporna (robust optimization)

Przykłady numeryczne

Optymalizacja niezawodnościowa konstrukcji kratowej narażonej na utratę stateczności lokalnej i globalnej

Mieszana dyskretno-ciągła optymalizacja niezawodnościowakonstrukcji kratowych

Wielokryterialna niezawodnościowa dyskretna optymalizacja w projektowaniu przekrycia hali sportowej.


Stochastic Stochastic aanalysis and nalysis and ooptimizationptimization -- research centersresearch centers

Fundamental works: A.M. Freudenthal, A.R. Rżanicyn, W. Wierzbicki

Selected scientific centers:

Stanford University, California , USA – Armen Der-KiureghianIowa University, Iowa, USA – K.K. ChoiColorado University, Boulder, Colorado, USA – Dan M. FrangopolTokio University, Japan – Jun KandaOsaka University, Japan – Y. MurotsuKansai University, Japan – Hitoshi FurutaNewcastle University, New South Wales, Australia – Robert E. Melchers TU Munich , Germany – Rüdiger RackwitzTU Denmark, Lyngby, Denmark – Ove Ditlevsen, S. Krenk, N. LindAalborg University, Denmark – Palle Thoft Christensen, John Sorensen

Poland:Politechnika Krakowska – Janusz MurzewskiPolitechnika Wrocławska – Paweł ŚniadyPolitechnia Szczecińska – Witold M. PaczkowskiIPPT, Warszawa – Krzysztof Doliński, Stefan Jendo, Michał Kleiber, Kazimierz Sobczyk


Stochastic Stochastic aanalysis and nalysis and ooptimizationptimization -- research centersresearch centers

Thematic conferences:

ICOSSAR-9 (Structural Safety and Reliability) Congresses organized by IASSAR (Inter. Assoc. for Structural Safety and Reliability)

ICASP-9 (Applications of Statistics and Probability in Civil Engineering) - Congresses organized by CERRA (Civil Engineering Risk and Reliability Association)

ESREL (European Safety and Reliability) Annual Conferences organized by ESRA -European Safety and Reliability Association (Carlos Guedes Soares)

IFIP: TC7: System Modelling and Optimization WG 7.5 – Reliability and Optimization of Structural Systems (Palle. Thoft-Christensen, R. Rackwitz)

WG 7.7: Stochastic Optimization (K. Marti)

WCSMO-6 Congresses organized by ISSMO (Martin Bendsoe)

CSM-4 Computational Stochastic Mechanics ( Pol D. Spanos)

RBO’02 and RBO’03 (Workshop and Course on Reliability-Based Optimization) organized by AMAS IPPT PAN (K. Doliński, S. Jendo, M. Kleiber)


IPPT participation in EU projects dedicated to reliability IPPT participation in EU projects dedicated to reliability and reliabilityand reliability--based optimization problemsbased optimization problems

Thematic Networks: Safety and Reliability of Industrial Products, Systems and Structures

Integrated Project on Advanced Protection Systems

Integrated Project on New Design and ManufacturingProcesses for High Pressure Fluid Power Products


Formulation of structural reliability problemFormulation of structural reliability problem

Vector of basic random variablesrepresents basic uncertain quantities that define the state of the structure, e.g., loads, material property constants, member sizes

x1

x2

0

Ωs

Ωf

g ( x ) = 0

f X ( x ) = const.

Limit state function

Safe domain

Failure domain

Limit state surface


Formulation of structural reliability problemFormulation of structural reliability problem

Probability of failure

probability density function (PDF) of basic variables X

Strict reliability index


Methods for reliability computationMethods for reliability computation

Numerical computation of the integral in definition for large number of random variables (n > 5) is extremely difficult or even impossible. In practice, for the probability of failure assessment the following methods are employed:

First Order Reliability Method (FORM)Second Order Reliability Method (SORM)Simulation methods: Monte Carlo, Importance Sampling,Directional Simulation


FORM FORM –– First Order Reliability MethodFirst Order Reliability Method

u 2

G ( u ) = 0

∆s

∆ f

l ( u ) = 0

δ *

u*

0 u 1

α

region of mostcontribution toprobability integral

ϕn ( u,0,I ) = const


Transformation to the standard normal space Transformation to the standard normal space

Transformation of the basic random variables to the standard normal space,

must guarantee the equivalence of the reliability problem formulation.



When only the mean vector and covariance matrix are known

is diagonal matrix with standard deviations of random variables, is a lower triangular matrix obtainedfrom Choleski decomposition of the correlation matrix

such that

For non-normal and statistically independent variables



Rosenblatt transformation – for statistically dependent random variables

Marginal probability density function


RackwitzRackwitz--Fiessler Fiessler algorithmalgorithm

findsubject to

Rackwitz-Fiessler iteration formula

Gradient vector in the standard space:


FirstFirst--order reliability sensitivity measuresorder reliability sensitivity measures

Sensitivities of the index w.r.t parameters of the distribution function

Sensitivities of the index w.r.t parameters of the limit state function


SORM SORM –– Second Order Reliability MethodSecond Order Reliability Method

u 2

Gv(v) = f v ( v ) – v n = 0

∆ s

∆ f

0 u 1

v n ≡ v n

v ~

~

v n = sv ( v )

v*

′

~


Monte Carlo methodMonte Carlo method

x1

x2

0

Ωs

Ωf

g ( x ) = 0

f X ( x ) = const.

Failure set indicator function

ifif

Probability of failure estimator


u 2

G ( u ) = 0

∆s

∆ f

0 u 1

ϕn ( u,0,I ) = const

u*

ImportanceImportance samplingsampling


Directional samplingDirectional sampling

n

n

G

G

constant density of A

Ωn – unit (hyper-)spherein U space

N vectors of direction cosines are randomly generated

Along the ouward direction r is determined such that

It can be shown that


Response surface techniqueResponse surface technique

Method based on Rackwitz – Fiessler recursive formula where on each step, instead of unknown function G(u) (and ∇G(u)), the following least square linear approximation is used

∑=

+=n

iiiubbG

10)(ˆ u [ ]nbbbG ,,,)(ˆ

21 K=∇ uand then

based on points planed with distance ∆u around the central point and points from previous steps from spherical vicinity with radius

)(ˆ uG

ε+∆= ukR where k and ε are assumed


Response surface techniqueResponse surface technique

u 2

∆s

∆ f

u 1

∆f

1 2

3R

5

4G ( u ) = 0

0ˆ =)(G u

Plans of points

grad ∆u2

∆u1

u2

u1

∆u2

∆u1

u2

u1

axial


Relability analysis methods Relability analysis methods -- summarysummary

FORM – cheap compared to other methods, not very accurate for the problems with highly nonlinear LSF, an efficient implementation requires gradient-based algorithms.Good for small failure probabilities.

SORM – more accurate than FORM but more computationally expensive, the curvatures of the LSF must be computed.For large number of random variables the simulation methods become competitive.

Crude MC – computationally efficient only for problems with the explicit or very “cheap” LSF, can be arbitrarily accurate, insensitive to the shape of LSF, no sensitivity analysis is needed


Relability analysis methods Relability analysis methods -- summarysummary

Importance sampling – requires much less sample points than crude MC provided that the sampling density is properly chosen. Usually applied to improve accuracy of FORM.

Adaptive MC – sampling density is “moved” each time a point close to the design point is sampled. More expensive than Importance Sampling but does not require the prior knowledge of the design point.

Directional Simulation – simulation in the polar coordinate standard normal space. Efficient for not-to-large problems (say, up to 5 random variables) and particularly useful for limit state surfaces which are nearly spherical (in the standard normal space).


System reliabilitySystem reliability

series

mixed

parallel

FORMFORM--based series system reliabilitybased series system reliability1 2 3 m

Ditlevsen bounds


Failure modes correlationG1

G2

G3

System reliability index



Time variant failure probabilityTime variant failure probabilityThe probability of entering the failure domain for the first time given that the component was in the safe state at t = 0 in the time interval [0, t]

[ ] ( ) ( ) 1 ( ) 1 ( 0, : ( , ) 0 )fP t P T t P T t P t gτ τ= ≤ = − > = − ∀ ∈ > x

T – first passage time

The rate of outcrossings into the failure domain F = g(X(t), t) ≤ 0

0

1( , ) lim ( ( ( ), ) 0 ( ( ), ) 0 )F P g gν τ τ τ τ τ+

∆→⎡ ⎤= > ∩ + ∆ + ∆ ≤⎣ ⎦∆

X X

0

( ) (0) ( , )t

f fP t P F dν τ τ+≤ + ∫

The upper bound on failure probability (Bolotin 1981)


Time variant failure probabilityTime variant failure probability

Stationary rectangular wave renewal vector process (Breitung, Rackwitz 1982)Limit state surface given by a hyperplane δF = αTu + β = 0

22

1 1( , ) ( ) ( , ;1 ) ( )

n n

i i ii i

Fν τ λ β β β α λ β+

= =

⎡ ⎤≈ Φ − − Φ − − − ≤ Φ −⎣ ⎦∑ ∑

Stationary differentiable Gaussian process (Veneziano et al. 1977)

200( ) ( )

2TF ων ϕ β ω

π+ = = − &&with α αR

– the matrix of second derivatives of the matrix of correlation functions &&αR

1 2

2

1 2 |1 2

( , ) ; , 1, ,ij i j nτ τ τρ τ ττ τ = =

⎧ ⎫∂= =⎨ ⎬∂ ∂⎩ ⎭

&& KR


ReliabilityReliability--based based designdesign optimization optimization (RBDO)(RBDO)

( )( )( )

min

min max

minimize ( )

subject to 1, ,0 1, ,0 1, ,

1, ,

i i R

j R I

I Ek

l l l

C

β i mh j m mh k m mp p p l n

β≥ =≥ = += = +

≤ ≤ =

pp

ppp

K

K

K

K

( )1

( ) ( ) ( )R

i

m

I f ii

C C C β=

= + Φ −∑p p p

d

α µ

σ

⎧ −⎪= = −⎨⎪ −⎩

px

x xx

deterministic parameters

mean values

standard deviations

Cost function

Design variables


RRBDO problem BDO problem -- ilustration ilustration

G2(u(x µ, x σ), x d) = 0

u1

u2

βmin

u1

u2

βmin

βmin0 0

g1(x, x d) = 0

x1

x2

0g3(x, x d) = 0

g2(x, x d) = 0

G1(u(x µ, x σ), x d) = 0

G3(u(x µ, x σ), x d) = 0

x1

x2

0

fX ( x) = const.

g1(x, x d) = 0

g3(x, x d) = 0

g2(x, x d) = 0

fX ( x) = const.

G2(u(x µ, x σ), x d) = 0

G1(u(x µ, x σ), x d) = 0

G3(u(x µ, x σ), x d) = 0

X – space, initial design X – space, optimal design

U – space, initial design U – space, optimal design


converged?

realizations xof stochastic var. X

FEA/DSA

REL

N

T

FEA/DSA

currentdesign

acceptable?

OPT

N

User interaction- visualization- modification of the

design model

Final project

initial design)

T

RBRBDDO system O system data flowdata flow

0

*p

0 =p 0 0

design point search loop


Streicher & Rackwitz (2002) ( ) ( ) ( ) ( )Z B C D= − −p p p p

Structure is given up after failure Systematic reconstruction

Failure due to time invariant loads:

* *( ) ( ) ( )fB C B H P− − +p p ( )( ) ( ( ) )

1 ( )f

f

Pb C C HPγ

− − +−

pp p

p

Poissonian disturbances (Hasofer 1974): ( )

( )( ) ( )

f

f f

Pb C HP P

λγ λ γ λ

− −+ +

pp

p p ( )

( ) ( ( ) ) fPb C C Hλ

γ γ− − +

pp p

Poissonian failures (Rosenblueth 1976, Rackwitz 2000) ( )( )

( ) ( )b C H λ

γ λ γ λ− −

+ +pp

p p ( )( ) ( ( ) )b C C H λ

γ γ− − +

pp p

General asymptotic result (Rackwitz 2000)

- 1( ) ( ( ) )[ ]

b C C HE Tγ γ

− − +p p

General result (Rackwitz 2000) * *(1 ( , )) ( ) ( , )b f C H fγ γ

γ− − −p p p *( ) ( ( ) ) ( , )b C C H h γ

γ− − +p p p

Cost functions Z(p) for various replacement strategies and failure models

Benefit Constructioncost

Damagecost

RBDO RBDO –– cost functionscost functions

maximize


RBDO RBDO –– oneone--level formulationlevel formulation

TI

TV

1

1

( ): ( ) ( ) ( ( ) )

1 ( )

subject to: ( , ) 0

( ) ( , ) ( , ) 0;

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

k k

s

f kk

s

f kk

k k

k i k k k k i k

f k

PbZ C p C p H

P

g

u g g

i n k sP P

γ=

=

− = − + + +−

=

∇ + ∇ =

= − =

−

∑

∑

u u

pminimize p

p

u p

u p u p u

p max ) 0f k ≤

Kuschel & Rackwitz 2000

Streicher/Rackwitz 2002

1

admissib

( , ): ( ) ( ) ( ( ) )

subject to: ( , ) 0

( ) ( , ) ( , ) 0;

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

k k

s

kk

k k

k i k k k k i k

k

FbZ C p C p H

g

u g g

i n k sF

ν

γ γ

ν ν

+

=

+

− = − + + +

=

∇ + ∇ =

= − =

−

∑

u u

pminimize p

u p

u p u p u

p le ) 0k ≤


Robust optimizationRobust optimization

( )21, 2, 2

1 1, 2,

:

: ( ( ), ( )) ( ) ( )l

i iYi i Yi

i i i

w wF M

s sµ σ

=

⎡ ⎤= − +⎢ ⎥

⎢ ⎥⎣ ⎦∑Y Y

p, X p, X p, X p, X

pFind the set of design variables that

minimizes µ σ

mean on target variance reduction

6 sigma type probabilistic constraints

: ( ) ( )) Lower specification limit, 1,...,Yi Yin i lµ σ

− ≥ =p, X p, Xsubject to( ) ( )) Upper specification limit, 1,...,Yi Yin i l µ σ+ ≤ =p, X p, X


Robust optimizationRobust optimizationMethods for estimating the mean and variance of the performance vector Y

Monte Carlo Simulation

Sensitivity-based variability estimation. First order Taylor’s expansion2

2 2

1j

ni

Yi Xj j

Yx

σ σ= =

⎛ ⎞∂⎜ ⎟=⎜ ∂ ⎟⎝ ⎠

∑xx µ

( )Yi iYµ = Xµ


RBDO of the geometrically nonlinear space trussRBDO of the geometrically nonlinear space truss



The structural volume is minimized with 26 reliability constraints corresponding to the following limit states:

• admissible displacement of the central node

• admissible stress / local buckling

• global loss of stability



• cross-sectional areas of 3 groups of elements (lognormal) • Young modulus (lognormal)• yield stress (lognormal)• load factor (Gumbel)• coordinates of nodes 1-7 (normal)

Stochastic description: (27 random variables)

• mean values of the cross-sections of the groups of elements • mean values of x and y coordinates of nodes 2-7

Design parameters:

The minimal admissible β values for displacement and stress type failure functions are equal to 3.7 (Pf = 1.0·10-4) while for global stability type failure condition it is 4.7 (Pf = 1.3·10-6)



RBDO results:• The optimal point was found after 6 iterations of the NLPQL algorithm• The local and global stability type reliability constraints were active• The volume of the structure was reduced by 17% • The gradients of failure functions had to be calculated more than 1000times



initial designoptimal design


Mixed discrete continuous RBDO problem formulationMixed discrete continuous RBDO problem formulation

minimize

subject to:


Transformation to continuous optimization problemTransformation to continuous optimization problem

Discrete variables:

are replaced with the new discrete variables:

by the transformation:

where:

and



The variables can be made continuous by taking values in [0,1].

It was proved (Wang et al. 1998) that at the optimal point theytake values for only one and for all otherif the additional constraints are imposed, where:



minimize

subject to:


Controlled enumeration methodControlled enumeration method

(N,1)

(N-1, 1)

(1, 1)

(2, 1)

(1, 2)

(0, 0)

2

(N-1, J )N-1

(N, J )N

N

N -1

(2, J )

(1, J )1

2

1

Truss structure volume :

Selection of the first design variable

- volume of the solution from continuous optimization


Selection of the second design variable

Condition of limited variation of

- the optimal continuous design

Controlled enumeration methodControlled enumeration method

(N,1)

(N-1, 1)

(1, 1)

(2, 1)

(1, 2)

(0, 0)

2

(N-1, J )N-1

(N, J )N

N

N -1

(2, J )

(1, J )1

2

1


Mixed RBDO Mixed RBDO eexamplexample: 10 element plane truss: 10 element plane truss



Random and design variables

Rand. var. Description Distr. type Mean value Std. dev. Design var.

X1,...,X10 Cross-sec. area–elem. No.1÷10Constant c.o.v.: 5% log-normal 20.0 cm2 1.0 cm2 p 1...,p10

X 11 Young modulus of material for all elements log-normal 21000 kN/cm2 1050 kN/cm2

X 12 Yield stress of material for all elements log-normal 21 kN/cm2 1 kN/cm2

X 13 x coor. – node No. 1 normal 720 cm 2 cm X 14 y coor. – node No. 1 normal 360 cm 2 cm p11 X 15 x coor. – node No. 2 normal 720 cm 2 cm X 16 y coor. – node No. 2 normal 0 cm 2 cm p12 X 17 x coor. – node No. 3 normal 360 cm 2 cm X 18 y coor. – node No. 3 normal 360 cm 2 cm p13 X 19 x coor. – node No. 4 normal 360 cm 2 cm X 20 y coor. – node No. 4 normal 0 cm 2 cm p14 X 21 First load case factor Gumbel 1.0 0.2 X 22 Second load case factor log-normal 1.0 0.05



Failure functions

Exceeding admissible displacement of the node No. 2

Yield stress or local buckling of the elements

6 4

5 3 1

2



Continuous RBDO problem

minimize

subject to:


Mixed RBDO eMixed RBDO examplexample: 10 element plane truss: 10 element plane truss

Mixed RBDO problem

minimize

subject to:

Catalogues of equal-sided angle cross-sections



Equivalent continuous RBDO problem

minimize

subject to:



Optimal designs

Continuous problem Mixed discrete problem


Design variable Description Initial

value Continuous optimization

Mixed discrete optimization

p1 Cross-sec. area–elem. No.1 20.0 cm2 28.85 cm2 29.90 cm2 /L130x130x12 p2 Cross-sec. area–elem. No.2 20.0 cm2 18.57 cm2 18.70 cm2/L90x90x11 p3 Cross-sec. area–elem. No.3 20.0 cm2 40.21cm2 43.00 cm2/L150x150x15 p4 Cross-sec. area–elem. No.4 20.0 cm2 16.40cm2 18.70 cm2/L90x90x11 p5 Cross-sec. area–elem. No.5 20.0 cm2 5.09cm2 5.69 cm2/L50x50x6 p6 Cross-sec. area–elem. No.6 20.0 cm2 5.09cm2 5.69 cm2/L50x50x6 p7 Cross-sec. area–elem. No.7 20.0 cm2 6.03cm2 6.91 cm2/L60x60x6 p8 Cross-sec. area–elem. No.8 20.0 cm2 24.60cm2 27.50 cm2/L120x120x12 p9 Cross-sec. area–elem. No.9 20.0 cm2 5.75cm2 5.82 cm2/L60x60x5 p10 Cross-sec. area–elem. No.10 20.0 cm2 35.53cm2 37.2 cm2/L160x160x12 p11 y coor. – node No. 1 360 cm 150.0 cm 150.0 cm p12 y coor. – node No. 2 0 cm 10.1 cm 100.0 cm p13 y coor. – node No. 3 360 cm 256.9 cm 276.4 cm p14 y coor. – node No. 4 0 cm 35.3 cm 38.46 cm

Optimization results


Initial value 83929.4 cm3

Continuous optimization 69878.4 cm3Structural volumeMixed optimization 74391.5 cm3



Continuous RBDO: reliability constraints

-9-8

-7-6

-5-4-3

-2-1

012

34

56

789

1011

1213

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33

iterationrelia

bilit

y co

nstr

aint

s

1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11

a a a a aa a


Mixed RBDO optimization: reliability constraints

-1

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

iteration

relia

bilit

y co

nstr

aint

s

1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11

Mixed RBDO example: 10 element plane trussMixed RBDO example: 10 element plane truss


Mixed discrete optimization: Q type constraints and shape design variables

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

iteration

Q ty

pe c

onst

rain

ts

0

10

20

30

40

50

60

70

80

90

100

shap

e pa

ram

eter

s

Q - 1; Q - 2; Q - 3; Q - 4; Q - 5; Q - 6; Q - 7; Q - 8; Q - 9; Q - 10; x12; x14

[cm]

shape parameters

Q type constraints

Mixed RBDO example: 10 element plane trussMixed RBDO example: 10 element plane truss


Discrete Discrete multicriteriamulticriteria rreliabilityeliability--based optimizationbased optimization of of spatial trussesspatial trusses

Problem statementProblem statementThere are given:

Numerical model of an objectVector of decision variables and parameter set

x = xn, n = 1,...,NVector of objective functions

f (x) = fj (x) , j = 1,...,JVectors of constraints

g (x) = gk (x) ≤ 0, k = 1,...,Kh (x) = hm (x) = 0, m = 1,...,M



Problem statementProblem statement contcont..

Vector of random variablesX = Xm, m = 1,...,MXm : (probability distribution, parameters)

Vector of limit state functionsG (X) = Gt(X), t = 1,...,T


0)(:)()( <== ∫ XXXx X G,dfPf ffj

f

ΩΩ

Find the sets of so called nondominated solutions and nondominatedevaluations



Transformation of the space of solutions Transformation of the space of solutions AAto theto the space of evaluations space of evaluations BB

y1ND=

x1 f2(x)

f1(x)

Y⊂ B

y3ND

y4ND=

y2

y1

Λ0 = B+

B

x2

AX⊂ A

y3

y4

x1ND=

x2ND

x3ND

x4ND=

x1x2

x3

x4

y=f (x)

XND YND

2x(

1x(

2y(

1y(

y2NDyp

yid

xp



The set of nondominated evaluations

The set of nondominated solutions

X j J JX f ( ) f ( ) f ( )< f ( )

i

k k kND ND j i j ND t i t NDx t∈ ∈ ∈

⎡ ⎤ ⎡ ⎤∈ ⇔ ¬ ∃ ∀ ≤ ∧ ∃⎢ ⎥ ⎣ ⎦⎣ ⎦x x x x x

Problem Problem solutionsolution

Λ+∈∧≠∃¬∈=∈ i

kND

kNDiYy

kNDND yyyy:YyY

i



Geometrical interpretation ofGeometrical interpretation of the evaluationsthe evaluations

f2(x)

f1(x)

Y⊂By1ND

y2ND

y3ND y4

ND

y2

y1∈ y2+Λ

y2+Λ

Λ0 = B+

B



Solution Solution algorithmalgorithm

Probability of failure Pf

Problem statement

Set of nondominated evaluations YNDand nondominated solutions XND

Numerical model

Realization of the vector of decision variables xl

(2) for the selected solution

Determination of internal forces and optimum analysis of the cross section selection

Reliability analysis of the variant of the structureValues of deterministic

objective functions

Evaluations of the solution f(x)

Stop conditions fulfilled?

Y

Selection of preferred evaluation yPand preferred solution xP

N

INN

ER L

OO

P



Numerical example: RBDO of the cover of sports hall

x

y

40 m

80 m



Numerical example: RBDO of the cover of sports hall

Problem statement

Vector of decision variables

x = xn, n = 1,...,4x1 – number of the shape of the coverx2 – number of divisions in the upper layer, described by

x2,1 and x2,2, where x2,1 stands for divisions in x directionand x2,2 – in y direction

x3 – depth of the coverx4 – rise of the cover



one-sloped cover x1 = 1

x4 x3

40 m

number of division in the upper layer x2,1

two-sloped cover x1 = 2

x4

x3 40 m

number of division in the upper layer

cylindrical coverx1 = 3

x4x3

number of division in the upper layer x

2,1

40 m



VARIABLE PROBABILITY DISTRIBUTION FUNCTION

Description Notation Type of distribution Expected value Standard deviation

Diameter of the bar no. 1 X1 log – normal 30.0 mm 0.300 mm





Thickness of the bar no. 1 X6 log – normal 2.9 mm 0.029 mm





Yield stresses of steel X11 log – normal 360 MPa 36 MPa

Young modulus X12 log – normal 225 GPa 11.25 GPa

Wind load multiplier X13 Gumbel for max 1 0.2

Snow load multiplier X14 Frechet 1 0.2



Limit state functions (LSF)

0ilt , if 0

,, if 0

ii cr

i i

σ σσ

σ σ

⎧ ≥⎪= ⎨<⎪⎩

lt1)(σσ

−= i1G X

lt1)(qq

G i2 −=X

Stress type LSF

Displacement type function

min(min , )qiβ β β=

Reliability index



Vector of objective functions

( ) ( ), j 1,...,3jf= =f x x

∑ ⋅⋅=i

ii1 lAf ρ)(x

∑ ⋅⋅⋅

=i i

ii2 AE

lNNf )(x

3( ) ( )ff P β= = Φ −x



Constraint functions

26

66)(

2

31 <<

xxg :x

31:)( ≤≤ 12 xg x

Ixh 11 ∈:)(x

IiixIiixh

2,2

1,22

∈⋅=

∈⋅=

4, 2,:)(x

Iiixh ∈⋅= 0.6;)( 33 :x

Iiixh 44 ∈⋅= 2,:)(x2820 1410:)(

≤≤

≤≤

2,2

1,23

xxg x

4.23.0:)( 34 ≤≤ xg xI – set of integer numbers

6.02.0:)( 45 ≤≤ xg x

26 f f f( ): , 5 10lt ltg P P Pβ −= Φ(− ) ≤ = ⋅x

max7 ( ): , 0.16 mlt ltg q q q≤ =x



Preferred solution

xp = 3; (10x20); 4.2; 4.0

Preferred evaluation

yp = 25.3 kg/m2; 6.35 cm; 3.975

cylindrical cover x1 = 3

x4 = 4.0 m x3 = 4.2 m

number of divisions 10 x 20

optymalizacja niezawodnościowa konstrukcji - teoria i...

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