fuzzy probabilistic optimisation of building performance

Ž .Automation in Construction 8 1999 437–443

Fuzzy probabilistic optimisation of building performance

Milan HolickyˇKlokner Institute, Czech Technical UniÕersity, SolınoÕa 7, 166 08 Prague 6, Czech Republic´

Abstract

Fundamental performance criteria between action effects and relevant performance requirements for serviceability, safety,comfort and functionality are analyzed, assuming randomness of the effect action and both randomness and fuzziness ofperformance requirements. Randomness due to natural variability of basic variables is handled by commonly usedprobability theory methods, fuzziness due to vague or imprecise definitions of performance requirements is described bybasic tools of the recently developed theory of fuzzy sets. Both types of uncertainties are combined in newly defined fuzzyprobabilistic measures of building performance, damage function and fuzzy probability of performance failure, which arethen analysed and applied similarly as conventional probabilistic quantities. An illustrative example of optimization ofvibration constraints for building structures due to occupancy comfort indicates that commonly considered limiting valuesfor acceleration may be uneconomical. However, theoretical models used to describe fuzzy probabilistic properties ofperformance requirements need to be reexamined using adequate experimental data. q 1999 Elsevier Science B.V. All rightsreserved.

Keywords: Performance requirements; Fuzziness; Randomness; Optimisation

1. Introduction

Performance requirements for serviceability, safe-ty, security, comfort and functionality within thespaces of a building may often be affected by vari-ous uncertainties which can hardly be entirely de-scribed by traditional probabilistic models. As a rule,translation of human needs, particularly those con-

Ž .cerning occupancy comfort, to performance userrequirements often results in a vague and imprecisedefinition of the technical requirements for relevant

Žperformance indicators deflection, velocity, acceler-.ation, etc. .

Thus, in addition to natural randomness of basicvariables, performance requirements may be consid-erably affected by fuzziness in the definition ofcritical conditions. Two types of uncertainty of per-formance requirements are therefore identified here:

randomness, handled by commonly used methods oftheory of probability and, fuzziness, described bybasic tools of the recently developed theory of fuzzy

w xsets 2,7 . Similarly as in the case of structuralw xserviceability 3,4 , the fundamental condition of

building performance, SFR, between an action ef-fect S and a relevant performance requirement R, isanalysed assuming randomness of S and both ran-domness and fuzziness of R.

An illustrative example of continuous vibration inoffices is used throughout the paper to clarify thepresented concepts. In this example, it is shownw x1,5,6 that it is impossible to identify a distinct value

Žof an appropriate indicator root mean square value.of acceleration that would separate satisfactory from

unsatisfactory performance. Typically, a broad tran-sition region is observed, where the building is grad-ually losing its ability to perform adequately and

0926-5805r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved.Ž .PII: S0926-5805 98 00090-9

( )M. HolickyrAutomation in Construction 8 1999 437–443´438

Žwhere the degree of damage inadequate perfor-.mance or malfunction gradually increases.

2. Theoretical model for performance require-ments

Fuzziness due to vagueness and imprecision in thedefinition of performance requirement R is de-

Ž .scribed by membership function Õ x indicating theR

degree of membership of a structure in a fuzzy set ofŽ . w xdamaged unserviceable structures 3,4 ; here x de-

notes a generic point of a relevant performanceŽindicator in the illustrative example the root mean

.square value of acceleration used to assess both Sand R. A simple piece wise linear membership

Ž .function Õ x , shown in Fig. 1, is considered in theR

following analysis. This function describes the non-Ž .random deterministic part of uncertainty in require-

ment R. The randomness of R at each damage levelŽ .ÕsÕ x is described by the normal probability den-R

Ž < . Ž .sity function w x Õ see Fig. 1 , for which theR

normal distribution is considered here.² :The transition region r ,r , where a building is1 2

gradually losing its ability to perform adequately andits damage increases, may be rather broad dependingon the nature of the performance requirement. In theillustrative example of continuous vibration in of-

fices, the upper bound r may be a multiple of the2Ž w xlower bound r see Ref. 1 and the International1

w x.Standards 5,6 . An assessment of the lower boundr can be derived from the root mean square value of1

acceleration limits which are expected to be approxi-mately equal to r . The acceleration constraints for1

continuous vibration in offices suggested in variousŽ w x.countries see critical review in Ref. 1 are within a

range from 0.02 to 0.06 msy2 . As discussed in Ref.w x1 , the vibration threshold may be even lower, withinvalues from 0.01 to 0.02 msy2 . In the case ofcontinuous vibration in offices there is, however, alow probability of adverse comment for accelerationsbelow 0.02 msy2 . Therefore, this value may beconsidered as the lower limit r below which an1

office is assumed to be fully serviceable and performadequately without any damage.

Assessment of an upper limit r is even more2

difficult than appraisal of the lower limit r . The1

upper limit r , above which an office is fully unser-2

viceable, may vary considerably depending on thedefinition of fully unserviceable state of a building.

w xIn accordance with the discussion in Ref. 1 , adversecomment is probable for accelerations above 0.10msy2 . Although this value may not imply full dis-ability of a building space to be used as an office, itis accepted here as an assessment of the upper limitr . To show the effect of the upper limit of the2

Fig. 1. Fuzzy probabilistic model for performance requirement R.

( )M. HolickyrAutomation in Construction 8 1999 437–443´ 439

transition region on optimum constraints, two indica-Ž y2 .tive values r s3r s0.06 ms and r s5r2 1 2 1

Ž y2 .s0.10 ms are considered in the following anal-ysis.

In addition to fuzziness, performance require-ments are also dependent on the natural randomnessof user needs. As already indicated above, this uncer-tainty is described in Fig. 1 by the normal probability

Ž < . Ž < .density function w x Õ . The mean of w x Õ forR R

a given damage level Õ is considered as the value ofŽ .the indicator x for which ÕsÕ x , the standardR

deviation is taken as independent of x and equal toŽ y2 .0.1r 0.002 ms .1

The above described theoretical model of perfor-mance requirements, including fuzziness and ran-domness characteristics, should however be consid-ered only as a conceivable representation of actualuser needs. In order to determine more accurate andmore precise fuzzy probabilistic model of perfor-mance requirements, there is an urgent need forfurther development in definitions of newly intro-duced characteristics of performance uncertaintiesusing appropriate experimental data.

3. Fuzzy probabilistic measures of building per-formance

Ž .The damage function D x is defined as theR

weighted average of damage probabilities reduced bythe corresponding damage level

x1 1 X X<D x s n w x n d x dn 1Ž . Ž . Ž .H HR Rž /N 0 y`

where N denotes a factor normalising the damageŽ . ² :function D x to the conventional interval 0,1R

and xX is a generic point of x. The damage functionŽ . Ž . w xD x defined by Eq. 1 may be used 3,4 toR

Ž .specify the design or characteristic value of perfor-mance requirements guaranteeing an acceptable levelof the total expected damage. Thus, for a given set offuzziness characteristics r and r , and the random-1 2

ness characteristic s the design value of a perfor-R

mance requirement R may be specified in a rationalway using fuzzy probabilistic elements.

ŽThe fuzzy probability of performance serviceabil-. w xity failure p is then defined as 3,4

`

ps w x D x d x 2Ž . Ž . Ž .H S Ry`

Ž .where w x is the probability density function ofS

the action effect S. Similarly the fuzzy probability ofŽ .performance failure p defined by Eq. 2 enables

formulation of various design criteria in terms ofrelevant randomness as well as fuzziness character-istics. Then, however, besides the fuzziness charac-teristics r , r and the randomness characteristic s1 2 R

of the performance requirement R, the character-istics of actions effect S, particularly the mean mS

and standard deviation s are also needed. In theSŽ .following, a symmetric normal Laplace–Gauss dis-

tribution of S is accepted. The general case ofasymmetric three parameter log normal distribution

w xis considered in earlier studies 3,4 .

4. Optimisation procedure

The optimum value of the fuzzy probability ofperformance failure can be estimated using the tech-

w xnique of design optimisation 3,4 . It is assumed thatthe objective function is given by the total cost Cexpressed as a sum

CsC qp C 3Ž .0 D

where C is given as the sum of the construction and0

maintenance cost, p C is the expected malfunctionDŽcost; here C denotes the cost of full damage fullD

.malfunction or serviceability failure . It has beenw xshown 3,4 that this equation can be used if the

malfunction cost due to damage level Õ is given asŽthe multiple ÕC in the illustrative example it repre-D

sents the cost due to disturbance and the lower.efficiency of occupancies in the offices . Further, it

is assumed that both the initial cost C and the fuzzy0

probability of performance failure p are dependentŽon a decision parameter j in the illustrative exam-

.ple it is mass per unit length of a floor componentwhile the cost of full damage C is independent ofD

j .If C is proportional to the decision parameter j ,0

and the load effect S is proportional to a power


yk Ž .j kG1 , then the optimum ratio C rC may beD 0w xexpressed 3,4 as

y1Ep Ep

C rC s k m q kq1 s 4Ž . Ž .D 0 s sž /Em Ess s

where the quantities C , m , s are dependent on0 S S

the decision parameter j . Partial derivatives of theŽ .fuzzy probability of failure p in Eq. 4 are to be

Ž .determined using Eq. 2 and numerical methods.

5. Vibration of a floor member

Vibration of a load bearing horizontal membersupporting the floor structure of a building may be

w xanalysed using the equation of motion 1 for abeam:

E 2n z ,t E 4n z ,tŽ . Ž .j qEJ sF z ,t 5Ž . Ž .2 4E t E z

Ž .where Õ z,t denotes the vertical deflection andŽ .F z,t denotes a load function of a generic point z

and time t, j denotes the mass of the beam per unitlength and EJ the stiffness of the beam. In the case

w xof vibration criteria for building structures 6 ensur-

ing human comfort, the relevant variable used toverify the serviceability conditions of a beam is the

Ž . Ž .acceleration a z,t which follows from Eq. 5 as

E 2n z ,t E 4n z ,tŽ . Ž .y1a z ,t s s F z ,t yEJ jŽ . Ž .2 4ž /E t E z

6Ž .

If the decision parameter is the mass per unit lengthj , then the load effect S, being the root mean squarevalue of acceleration, can be expressed in terms of j

as

SsK z jy1 7Ž . Ž .Ž .where K z is a function expressing the shape of the

deflection curve. Thus, in this case of vibration of afloor member the load effect S is proportional to jy1

Ž .and the parameter k, entering Eq. 4 , is equal to 1.The optimum cost ratio C rC obtained from Eq.D 0

Ž .4 for ks1, s s0.1 r , and for r rr s3 isR 1 2 1

shown in Fig. 2 for selected values of m and s .S S

Similar results are shown in Fig. 3 for r rr s5.2 1

Assuming s s0.1r and r rr s5 it followsR 1 2 1

from Fig. 3 that the optimum values of C rC areD 0

slightly higher than those corresponding to s sR

0.1r and for r rr s3, which are indicated above1 2 1

Fig. 2. The optimum cost ratio C rC for ks1, s s0.1r and r rr s3.D 0 R 1 2 1


Fig. 3. The optimum cost ratio C rC for ks1, s s0.1r and r rr s5.D 0 R 1 2 1

Ž .in Fig. 2. In both cases for r rr s3 and r rr s52 1 2 1

the optimum cost ratio C rC for m )r is veryD 0 S 1Ž .low less than 100 and almost independent of the

standard deviation s .S

It is interesting to note that the optimum probabil-ity ratio p C rC is relatively stable. Fig. 4 showsD 0

this quantity for the input data considered in theillustrative example for ks1, s s0.1r and r rrR 1 2 1

Žs3 the resulting values for ks1, s s0.1r andR 1.r rr s5 are almost exactly the same . It follows2 1

that in the analysed case the optimum fuzzy proba-

bility of performance failure may be assessed usingand approximate relation

pf0.1C rC 8Ž .0 D

Ž .This relationship 8 may be used for a first assess-ment of the optimum fuzzy probability of perfor-mance failure p and required structural character-istics for ks1, s s0.1 r and, for both casesR 1

r rr s3 and r rr s5. For example, if the ex-2 1 2 1

pected cost ratio C rC s100, then the optimumD 0

Fig. 4. The optimum probability ratio p C rC for ks1, s s0.1r and r rr s3.D 0 R 1 2 1


fuzzy probability of performance failure p may beŽ .assessed using Eq. 8 as 0.001.

It should be noted that the acceleration constraintsw xrequired for various building spaces and activities 1

Žmay significantly differ may be characterized by.different lower and upper limits from those consid-

Ž .ered here; thus Eq. 8 should not be applied withoutappropriate analysis.

6. Discussion

Here we have considered the illustrative exampleof acceleration constraints for continuous vibrationin offices. General concepts are, however, applicableto many other common aspects of building perfor-mance, particularly to those affected by significantfuzziness. Assuming r rr s3 and s s0.1r it2 1 R 1

follows from Fig. 2 that for the cost ratio C rCD 0

equal to about 100, that the building should bedesigned in such a way that the characteristics ofaction effects should correspond to the horizontalline at the level C rC s100; for example if s sD 0 S

Ž0.1r , then theoptimum mean is m s0.9r which1 S 1

is equal to 0,018 msy2 if r s0.02 msy2 and1.r rr s3 . For r rr s5 it follows from Fig. 3 that2 1 2 1

for s ss s0.1r , the optimum mean is m srR S 1 S 1Ž y2 .which is equal to 0.02 ms . Thus, with increasingupper limit r the optimum mean m also increases2 SŽ y2 .from 0.018 to 0.02 ms . Note, that for the stan-dard deviation s s0.2 r the optimum mean m isS 1 S

Žin both cases considerably lower from 0.8 r to1.0.95 r .1

If the cost ratio C rC is 104, then for r rr s3D 0 2 1Ž .Fig. 2 and, as above for s ss s0.1r , the opti-R S 1

mum mean m of the load effect is about 0.63rS 1Ž y2 . Ž .0.0136 ms , for r rr s5 Fig. 3 the optimum2 1

Žmean m of the load effect is about 0.72 r 0.0144S 1y2 .ms . Again, with increasing upper limit r the2

Žoptimum mean m also increases from 0.0136 toSy2 .0.0144 ms . For the standard deviation s s0.2 rS 1

the optimum mean m is in both cases again consid-SŽ .erably lower less than 0.4 r .1

Generally, with increasing cost ratio C rC andD 0

increasing standard deviations s and s the opti-R S

mum mean m and standard deviation s decreasesS S

of the load effect S. For higher values of thesequantities the optimum values for the mean m andS

standard deviation s may be quite severe and mayS

not be achievable without introducing adequatestructural measures. In some cases it may be neces-sary to revise the overall design of the building.Acceleration constraints considered in various inter-

w xnational documents 1,5,6 , which are generallygreater than the lower limit r , correspond to the1

optimum cost ratio C rC in the range from 1 toD 0Ž .100 see Figs. 2 and 3 . In the case of office

buildings such values of cost ratio seem to be ratherlow. Consequently, the values of acceleration con-

w xstraints recommended in 1,5,6 may be uneconomi-cal.

However, the presented results are derived fromhypothetical theoretical models based on availableacceleration constraints indicated in the literature,which may not completely fit real conditions. Tomake a more accurate assessment of the optimumstructural characteristics, appropriate experimentaldata enabling realistic definition of theoretical mod-

Ž .els, particularly for the membership function Õ xR

and for relevant probability distributions, are needed.Especially both limits r and r and the type of the1 2

Ž . Žfunction Õ x which may be a nonlinear functionR.of the indicator x should be defined using adequate

experimental data. Nevertheless, the submitted me-thodical principles and auxiliary computer programsseem to provide effective tools for further develop-ment of the concept of building performance.

7. Conclusions

Ž .1 Functional requirements are generally affectedby two types of uncertainty: randomness and fuzzi-ness.

Ž .2 Newly defined fuzzy probabilistic measures,the damage function and fuzzy probability of perfor-mance failure provide effective tools to analysebuilding performance.

Ž .3 Optimisation analysis indicates that commonlyused acceleration constraints for continuous vibrationin offices may be uneconomical.

Ž .4 Adequate experimental data leading to moreaccurate definition of fuzzy probabilistic models areneeded for further development and practical appli-cations of the proposed technique for optimisation ofbuilding performance.


Acknowledgements

This paper presents a part of the findings ofˇresearch project GACR 103r94r0137 ‘Fuzzy Prob-

abilistic Concept of Time Dependent Structural Reli-ability’ supported by the Grant Agency of the CzechRepublic.

References

w x1 H. Bachmann, W. Ammann, Vibration in Structures Inducedby Man and Machines, IABSE, Zurich, 1987.

w x2 C.B. Brown, J.T.P. Yao, Fuzzy sets and structural engineer-

Ž . Ž .ing, Journal of Structural Engineering 109 5 1983 1211–1225.

¨w x3 M. Holicky, L. Ostlund, Probabilistic design concept, Proc.´International Colloquium IABSE: Structural Serviceability ofBuildings, Goteborg, 1983, pp. 91–98.¨

w x4 M. Holicky, Fuzzy optimisation of structural reliability, Proc.´ICOSSAR’93, A.A. Balkema, Rotterdam, 1994, pp. 1379–1382.

w x5 ISO 2631-2, Evaluation of human exposure to wholerbodyvibration, Part 2: Continuous and shock-induced vibration in

Ž .buildings 1 to 80 Hz , 1989.w x6 ISO 10137, Basis for design of structures—serviceability of

buildings against vibration, 1991.w x7 N. Shiraishi, H. Furuta, Structural design as fuzzy decision

model. Proc. ICASP 4, Pitagora Editrice, Bologna, 1983, pp.741–752.

fuzzy probabilistic optimisation of building performance

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