Fuzzy Structural Analysis

FuzzyFuzzyStructural AnalysisStructural Analysis

Dresden University of TechnologyDresden University of TechnologyInstitute of Structural AnalysisInstitute of Structural Analysis

Michael BeerMichael Beer

Introduction

Basic problems

Fuzzy Structural AnalysisFuzzy Structural AnalysisFuzzy Structural AnalysisFuzzy Structural Analysis

Examples

Conclusions

Solution technique - -level optimization


IntroductionIntroductionIntroductionIntroduction

damping dr and dt

dt

h

Pwind

krdr

p(x)

x

P

Uncertain structural parameters

wind load Pwind

~

model height h~

spring stiffness kr

~

~~

earth pressure p(x)(value and shape)

~


Examples of fuzzificationExamples of fuzzificationExamples of fuzzificationExamples of fuzzification

Fuzzification of the foundation as linear elastic elementembedded in the foundation soil

clay,semi-solid

rock

HEB 240

2,00 m

1,00

1,50

m

detail of the foundation model

~ ( )F k triangular fuzzy number

Examples of fuzzificationExamples of fuzzificationExamples of fuzzificationExamples of fuzzification

Fuzzification of end-plate shear connections at the corner of aplane frame

~ ( )F k triangular fuzzy number

detail of the plane frame model

HEB 240

IPE 330

Fuzzy model for theFuzzy model for thematerial behaviour of concretematerial behaviour of concrete

Fuzzy model for theFuzzy model for thematerial behaviour of concretematerial behaviour of concrete

(c)

1.0

0.0c

c

fc0

~

c0~

Cartesian productCartesian productCartesian productCartesian product

ni21 A~

...A~

...A~

A~

K~

n1;...;i ; )(xmin )x( ;x

)x;...;x;(x)x( );x;...;x;(xx

iAKii

n21KKn21

iX

K~

1.0

0.0x1

1.0

0.0x2

K(x)

1.0

0.0

x1

x2

fuzzy input value x1 = A1

~~ fuzzy input value x2 = A2

~~

K = A1 A2

~~~

(x1) (x2)

A1

~

A2

~


Extension principleExtension principleExtension principleExtension principle

...)x;...;(x

z );x;...;f(xz (z)z

n1n1

n1B XX

Z;B~

fuzzy input values,cartesian product

K(x)

1.0

0.0

x1

x2

otherwise 0

)x;...;f(xz if )],(x);...;(x[ min sup(z)

n1n1)x;...;f(xzB n1

ZXX B~;...K~B~K~ n1

B(z)1.0

0.0z

A1

~

A2

~

fuzzy result value z = B~~

K = A1 A2

~~~


Interaction of fuzzy values - exampleInteraction of fuzzy values - exampleInteraction of fuzzy values - exampleInteraction of fuzzy values - example

1.0

0.0

1.0

0.0x2x1

(x1) (x2)

7 8 10 1 5 6

fuzzy input values x1 and x2~~

constraint a priori interaction

4x~2x~ 21

cartesian product with a priori interaction

x1

x2

K(x)

1.0

0.0

K = x1 x2~~~

x2~

x1~


fuzzy intermediate results y1 and y2 with additionalinteraction inside the mapping

~~

y1

y2

(y)

1.0

0.0

y2~

y1~

mapping operator

2122112121 xxy and xxy :with yy2xf(xz );

1.0

0.0

1.0

0.0y2y1

(y1) (y2)

8 13 1 7 9168.5 3

0.1430.4


fuzzy result z~

1.0

0.0

0.143

0.4

z

(z)

withoutinteraction

withinteraction

17 22 23 29 36 39 41

18 22.5 36.2

Extension principle - exampleExtension principle - exampleExtension principle - exampleExtension principle - example

fuzzy input values x1 und x2~~

(x1) (x2)1.0 1.0

-13 -5 0 2 x1 -12 -4 0 4 x2

mapping operator

65 x3 x13x x48 x17x)x;f(xz 222

321

21

3121

z

4

2

0

-12

2 4x2

x1


Extension principle - exampleExtension principle - exampleExtension principle - exampleExtension principle - example

numerical procedure to compute the fuzzy result

- discretization of the support of all fuzzy input values- generation of all combinations of discretized elements- determination of the membership values using the min operator- computation of the results from all element combinations

using the mapping operator- determination of the membership values of the result

elements by applying the max operator- generation of the membership function for the fuzzy result

fuzzy result value z~

0.274 2.570 5.859 z

(z)

exact solutionapproximationby smoothing

numerical result(1023 combinations)

1.0

0.0

0.4

0.8

0.6

0.2

problems: tremendous high numerical effortexactness of the result


A(x)1.0

0.0

k

xlkx rk

xkA

Fuzzy structural analysisFuzzy structural analysisFuzzy structural analysisFuzzy structural analysis

-level set

-discretization

(x) x AXA

A(x)

1.0

0.0

k

xS(A)~

kiki ;ki

AA


fuzzy input values

n-dimensionaluncertaininput subspace

-level-optimization-level-optimization-level-optimization-level-optimization

Generation of an uncertain input subspace

(x1)

1.0

= i

0.0x1 l x1 r x1

(x2)

1.0

= i

0.0x2 l x2 r x2

(x3)

1.0

= i

0.0x3 l x3 r x3

x1 l x1 r x1

x2 l

x2 r

x2

x3 l

x3 r

x3


mapping operator

x1 x1 x2 x2

0.0

1.0

0.0

1.0

x1x1x1 l x1 r x2

x2x2 l x2 r

z z

(x1) (x2)

0.0

1.0

zzz l z r

(z)


fuzzy input values x1 und x2~ ~

fuzzy result value z~


-level optimization

uncertain input subspace

uncertain result subspace

at t = t1

Mapping of subspacesMapping of subspacesMapping of subspacesMapping of subspaces

at t = t2

- optimum point at t = t2

x1 l x1 r x1

x2 l

x2 r

x2

x3 l

x3 r

x3

z1 l z1 r z1

z2

z1 l z1 r z1

z2



fuzzy input values, uncertain input subspace

search for the optimum points xopt

and computation of the assigned results zj

fuzzy result values

optimization problem


objective functions zj = fj(x) (discrete set)are described by the mapping operator

- no special properties, generally only implicit -

standard optimization methods are only partly suitable to solve the problem

modified evolution strategy with elements of theMonte-Carlo method and the gradient method

uncertain input subspace for two fuzzy input values

lower bound for offspring pointsupper bound for offspring points

starting point

optimum point xopt,

improvement of zj

no improvement of zj

1

0

2

345

67

x1 l x1 r x1

x2 l

x2 r

x2

Modified evolution strategyModified evolution strategyModified evolution strategyModified evolution strategy


permissible domains for 3

permissible domains for = 3

not permissible points for = 3

permissible points for = 3, points to be checkedoptimum point xopt for = 3 from -level optimization"better" point = new starting point new optimum point for = 3

Post-computation

Modified evolution strategyModified evolution strategyModified evolution strategyModified evolution strategy

uncertain input subspace for two fuzzy input valuesand six -levels, post-computation for = 3

x1 l x1 r x1

x2 l

x2 r

x2


Dynamic fuzzy structural analysisof a multistory frame

Example 1Example 1Example 1Example 1

fuzzy differential equation system

v3

v2

v1

Cauchy problemwith fuzzy initial conditions and fuzzy coefficients

EI2

EI2

EI2

EI1EI1

EI1EI1

EI1EI12.7

m2.

7 m

2.5

m

EI2 ; EA


Investigation IInvestigation IInvestigation IInvestigation I

fuzzy dampingparameters

dii

(dii)

EI1 = 4103 kNm2

fuzzy initialconditions

mappingoperator

deterministic parameters


fuzzy displacement-time dependencyof the lowest story

0.000 0.075 0.150 0.225 0.300 t [s]-6.0

-4.0

-2.0

0.0

2.0

4.0

6.0

8.0

10.0

v1 [mm] = 0.0 = 1.0

(v1)1.0

0.50.0

v1

Investigation IInvestigation IInvestigation IInvestigation I


6.62

mapping operator

Investigation IIInvestigation IIInvestigation IIInvestigation II

I1 = 1.33310-4 m4deterministic parameters

fuzzy elastic modulus

(E)

107 107 107 E [kNm-2]

deterministicinitial conditions

deterministicdamping parameters

dii = 0.05 kNsm-1; i = 1; ...; 3



0.000 0.075 0.150 0.225 0.300 t [s]

v1 [mm] = 0.0 = 1.0

fuzzy displacement-time dependencyof the lowest story

-6.0

-4.0

-2.0

0.0

2.0

4.0

6.0

8.0

(v1)1.0

0.50.0

v1

(v1)1.0

0.50.0

v1


-2.29

-4.06

input subspace for = 0

result subspace for = 0

0


107 107 E [kNm-2]

107

107

107

v1

v3

v2

v1 l v1 r

v1 l = -4.04 mmv1 r = -3.84 mm

107

107

107

v10

v2

v3

v1 l

v1 r

v1 l = -4.06 mmv1 r = -3.92 mm

at t = t1 = 0,099 s

at t = t2 = 0,100 s



Dynamic fuzzy structural analysis of aprestressed reinforced concrete frame

8.00 m6.

00 m

cross bar columns

reinforcement steeltendonsanchors

**

*

h = 600 mmb = 400 mmprefabricated

structure, cross sections, materials

concrete:fc0,m = 43 Nmm-2

c0,m = 2.9 ‰fct,m = 3.8 Nmm-2

Em = 29 800 Nmm-2

reinforcement steel:fy,m = 551 Nmm-2

fu,m = 667 Nmm-2

y,m = 2.62 ‰u,m = 7.0 %

tendon parameters:A = 4 × 3 cm2; F = 4 × 292 kN; = 0.18; = 0.4° m-1

E = 195 000 Nmm-2 (remain in the linear range)conduits: d = 40 mm; wedge slip: d = 0 mm

8 16 10 16



structural model and loading

Mt

Pv

Ph(t)

vh(t)

p1 h(t)

p2 h(t)

p2 v

m2

m1

8.00 m6.

00 m

Mt = 5.0 Mgm1 = 0.587 Mgm-1

m2 = 2.373 Mgm-1

m1

Mt

Pv

horizontal acceleration of the foundation soil a(t)

p1 h(t)

statical loadsPv = 49.05 kNp2 v = 17.66 kNm-1

dynamic loadsPh(t) = a(t)Mt

p2 h(t) = a(t)m2

p1 h(t) = a(t)m1

t [s]

1.0

0.0

-1.0

0.2 0.7 0.9

1.1 1.3

0.1 0.8

1.2

a(t)a(t)k k

a(t)a0


Specification of the fuzzy analysisSpecification of the fuzzy analysisSpecification of the fuzzy analysisSpecification of the fuzzy analysis

mapping operator

fuzzy input values

fuzzy result value

fuzzy rotationalspring stiffness

fuzzy acceleration

fuzzy displacement (horizontal)of the left hand frame corner

(k)

k [MNmrad-1]

(a0)

a0 [g]


Deterministic fundamental solutionDeterministic fundamental solutionDeterministic fundamental solutionDeterministic fundamental solution

plane structural model withimperfect straight bars and layered cross sections

numerical integration of the differential equationsystem 1st order in the bars

interaction of internal forces

incremental-iterative solution techniqueto consider complex loading processes

consideration of all essentialgeometrical and physical nonlinearities

large displacements and moderate rotations

realistic material description of reinforcedconcrete including cyclic and damage effects


Numerical simulationNumerical simulationNumerical simulationNumerical simulation

constructing and loading process

prestressing of all tendons andgrouting of the conduits, no dead weight

dead weight of the columns, hinged connectionof the columns and the cross bar,

dead weight of the cross bar

transformation of the hinged jointsinto rigid connections (frame corners)

additional translational mass at the frame cornersand along the cross bar (statical loads)

dynamic loads due to the time dependentuncertain horizontal acceleration a(t)

dynamic analysis

numerical time step integrationusing a modified Newmark operator, t = 0.025 s


Fuzzy resultsFuzzy resultsFuzzy resultsFuzzy results


horizontal displacement vh(t)of the left hand frame corner

~

0.0 t [s]

0

vh [mm] = 0.0 = 1.0

60

-60

-120

-180

120

0.5 1.0 1.5 2.0 2.5

largest fuzzy bending moment Mb

at the right hand column base

~

0.0

1.0(Mb)

Mb [kNm]222.6182.8141.80.0 81.855.9 109.0

linear nonlinear analysis

= 1.0 (linear analysis)

Consideration of theConsideration of thenonlinear material behaviornonlinear material behavior

Consideration of theConsideration of thenonlinear material behaviornonlinear material behavior

cross sectional points to observe thenonlinear behavior of the material


1concrete

layer

reinforcement steeltendonsanchors

**

*

2reinforcement

layer

stress-strain dependencies forconcrete 1 and reinforcement steel 2

c( 1 ) [Nmm-2]50

-25

-45-4.0 0.0 4.0 [‰]

s( 2 ) [Nmm-2]600360

0

-360-600

0.0 8.0 16.0 [‰]

k = 7.0 MNmrad-1

a0 = 0.45 g


ConclusionsConclusionsConclusionsConclusions

„There is nothing so wrong with the analysis as believing the answer!“

Richard P. Feynman

suitably matched computational models

reliable input and model parameters,uncertainty has to be accounted forin its natural form

Conditions forrealistic structural analysis andsafety assessment

Documents

Fuzzy Structural Analysis