Inexact Inference for Rule-Based Damage Assessment of Existing Structures

INEXACT INFERENCE FOR RULE-BASED DAMAGE ASSESSMENT OF EXISTING STRUCTURESt

Mitsuru Ishizuka, K. S. Fu and James T. P. Yao

School of Electrical Engineering Purdue University West Lafayette, Indiana 47907

School of Civil Engineering Purdue University West Lafayette, Indiana 47907

ABSTRACT

The knowledge organization of a rule-based dam­ age assessment system of existing structures sub­ jected to earthquake excitation is outlined in this paper. A principle of inexact inference to obtain a rational solution is presented. The fuz­ zy set theory and the production system with cer­ tainty factor are employed jointly in the inexact inference to deal with the continuous nature of the damage state and to attain the modularity of uncertain knowledge, respectively.

I. INTRODUCTION

The role of damage assessment of existing structures has received increasing attention re­ cently [3~,33,34]. Existing structures refer to those already built and in existence. Frequently, there exists a need to evaluate the safety and re­ liability of a particular structure or a number of existing structures either as a part of periodic inspection program or immediately following a given hazardous event [5,31]. As an example, con­ sider the aftermath of a strong-motion earthquake in a metropolitan area.

Prior to construction, each structure is analyzed and designed on the basis of some mathematical formulations, which are the results of idealization and generalizations from available knowledge and past experience. Once a structure is built, it has its own characteristics, which can no longer be precisely ·described by the same initial mathematical models used in the design phase [8,17,30]. More realistic behavior of ex­ isting structures can be obtained during earth­ quakes. For this purpose, accelerometers and oth­ er instruments have been installed to record the dynamic behavior of certain building structures [37]. System identification techniques [4,6,~4] and damage assessment can be employed jointly to examine the real behavior and to assess the safety state of these existing structures so that a correct decision may be reached i n terms of any immediate alarm, the need of repairing, the pred­ iction of future damage and the improvement of technologies for aseismic structures.

The state-of-the-art in damage assessment of existing structures is that only very small number of experienced engineers are well qualified to practice it. Moreover, the transfer of this complex decision-making practice to younger engineers depends primarily on many years of close working relationship with these experienced and qualified engineers [33]. To-date, several methods of structural damage assessment have been proposed [32], and some related works on the failure resis­ tance evaluation or estimation of existing build­ ings have been reported [~,5,2~,29]. However, a complete and rational solution of the damage as­ sessment problem is not yet available.

tThis work was supported by NSF under Grant P FR 79-0!Y. 96 ·

Fu and Yao [11] suggested that the problem of the damage assessment can be considered in terms of the theory of pattern recognition. In pattern recognition [12], when using decision-theoretic [14] or syntactic approaches [10], it requires a description of the patterns under study in terms of a certain mathematical model, which requires a fairly clear or statistical knowledge about the patterns. Such complete knowledge is frequently unavailable in complex or ill-defined problems, or problems involving subjective human factor, such as damage assessment and medical diagnosis [21,~7]. Accordingly, a recent damage assessment study [18] suggested the use of a rule-based pro­ duction system to realize a highly effective knowledge utilization of the structural experts and an inexact inference procedure. Relation between pattern recognition and some AI approaches has been ·discussed in [20].

This paper describes a rule-based damage as­ sessment system of the existing structures sub­ jected to earthquake excitation, called SPERIL <structural Peril). After a brief description of the relevant knowledge organization, the principle of an inexact inference employed in SPERIL is described.

ll· KNOWLEDGE OR�~NIZATION

structures are commonly classified according to their structural materials into following types [8,17,30]: (a) wooden, (b) masonry, (c) reinforced concrete, and (d) steel. During construction, certain parts of the structure can be pre­ fabricated for economical reasons. In particular, reinforced concrete can be further classified into <c-1> poured-in-place (or in-situ> reinforced­ concrete and <c-~ > precast <or prestressed) reinforced-concrete. As a structure with a mixed property of reinforced concrete and steel frame, (e) steel-framed reinforced-concrete structures are built in Japan [17]. Among these types, be­ cause wooden and masonry construction are fre­ quently limited to low-rise buildings, we will concentrate our attention on reinforced concrete and steel structures herein.

Generally speaking, for high-rise structure, steel frame is usually preferable because of its high strength, high ductility and uniform quality. The construction cost of steel structure, however, is frequently higher than that of reinforced con­ crete.

As the first step to the system design, define the grade of the damage state of existing struc­ tures in terms of a numerical quantity between 0 and 10, where 0 and 10 correspond to "no damage" and "total collapse", respectively. In addition, define the damage state in terms of verbal in­ terpretation as shown in Fig. 1. This classifica­ tion is not strict. However, each class is as­ sumed to be associated with a suitable recommenda­ tion and the cost for proper repair action.

Now the problem is one to construct a rational decision-making system for confirming the hy­ pothesis that the structure in question is severe­ ly damaged, to be true or false, or to be more reasonable than other hypotheses from possible ob­ servations. The observations may come from (i) visual inspection at various portions of the structure, (ii) reading of accelerometer records during the earthquake, (iii) nondestructive test­ ing, and (iv) loading tests before and after the earthquake. Although we will primarily consider the observations (i) and (ii) in this paper, ac­ ceptability of other observations should be con­ sidered in the design.

Available features for damage classification from the visual inspection may include the detec­ tion of deformations and cracks in columns, beams, joints, floors, ceilings, external !· internal walls, doors, windows, stairs, nonstructural partitions, utilities, elevators, etc. Features to be derived from the accelerometer records by using system identification techniques may include the change of natural frequency of the building vibra­ tion, the change of damping factor, the maximum interstory drift and the total energy absorption and dissipation during the earthquake. (Time his­ tories of above changes are sometimes also good information for experts.) In addition, when we try to infer the damage state from above-mentioned features, we should consider many other conditions regarding the structures, such as structural ma­ terial, height or number of stories, areas of floors, shapes, soil condition and foundation, the year that the building was built, building use, design parameters if available, existence of walls, experience of human inspector, etc. which are stored as reference data apart from inspection data and utilized for the inference.

The approach of production system [6,~6,~8] al­ lows us to decompose a complex problem into a num­ ber of simpler sub-problems, the relations among which are hierarchical (parent and son> or paral­ lel (brothers>. In addition, in order to accommo­ date knowledge efficiently from human experts, these sub-problems are fit into knowledge units of the experts. With this in mind, the framework for knowledge representation is determined and is shown in Fig. 2, where several intermediate diag­ nostic states are introduced. Each numbered node corresponds to a set of rules in the production system for inference. Each double circled node denotes the data analysis process for obtaining the feature from the accelerometer records.

ill· PRODUCTION SYSTEM~ CERTAINTY FACTOR

Before going into the description of SPERIL's inexact inference mechanism, let us see why a direct application of existing methods is incon­ venient.

The certainty factor was first introduced into

Table 1 - An example of rules

the production system ·of MYCIN [25,26] to deal with the uncertainty of knowledge expressed by various experts. The combining function of cer­ tainty factor plays an imoortant role in the pro­ duction system to keep knowledge modularity even in uncertain situations.

Production rules with uncertainty are ex­ pressed, for example, as, Rule 1 IF: X

Rule 2 IF: Y

THEN: there is indication (Ch ) that H, ,y where ch ,x and Ch are certainty factors of ,y the hypothesis t i sfi ed. H when the premises are absolutely sa-

The definition of this relation to probabilities certainty factor and its is as follows,

where P(H) and P<HIX> are a priori and conditional probabilities, respectively.

When a hypothesis H is confirmed from two dif­ ferent rules, say Rule 1 and Rule 2, the resultant certainty factor Ch ( ) can be calculated by the , x,y combining function. Unlike the heuristic combin­ ing functions employed in MYCIN or EMYCIN which is a problem-domain-independent version of MYCIN, a consistent combining function is derived in [19] theoretically based on Bayes' theorem providing that the two evidences are independent in the con­ firmation of H.

RULEII

~01

IF: · 1) STI is severe, and

2> MAT is reinforced concrete,

THEN: there is considerable indication <0.6) that GLO is severe.

IF: 1) STI is severe, and

2) MAT is steel,

THEN: there is strong indication (0.7) that GLO is severe.

203 IF: STI is moderate or destructive

THEN: there is weak indication <0.3> that GLO is severe.

204 IF: STI is no,

THEN: there is weak negative indication (-0.3) that GLO is severe.

• 205 IF: FRG is severe, THEN: there is considerable indication (0.4) that GLO is severe.

207 IF: 1) WAG is severe, and

~) MAT is reinforced concrete,

THEN: there is considerable indication (0.5) that GLO is severe.

"AbiirevTations------------------------------­

GLO damage of global nature

STI diagnosis of stiffness

FRG diagnosis of global nature of frames from field inspections

WAG diagnosis of global nature of structural walls from field inspections

MAT structural material

where

and a is a parameter defined by

Because X and Y are interchangeable, the case of ch < 0 and ch > 0 is omitted in Eq. <~>. It ,x ,y - is worth noting that Eq. (2) reduces to one ex­ pression in the particular case of a = 1.

Suppose that the same grade expressions as the final damage state of Fig. 1 are used for the in­ termediate diagnostic states. For the purpose of illustration, consider node No.2 in Fig.~. Ac­ cording to the approach of production system with certainty factor, a set of rules for confirming that structural damage of global nature (GLO) is severe may be listed like RULE 201-207 of Table 1.

The combining function of the certainty factor can work well only in the case that the rules to be combined are mutually independent in confirming a sub-goal. Thus, a problem arises. Although the combining function may work well, for example, among RULE 201, 205, and 207, it does not work well and sometimes leads into incorrect results such as an overestimation in the confirmation, among RULE 201, 203, and 204. The reasons are: 1> the decision is preserved until the final goal in the production system and therefore there exists several possibilities of different hypotheses at one time in an intermediate state, and 2) the in­ ferred sub-goal is continuous in nature in the damage assessment of existing structures. Some minor changes to solve the above problem are pos­ sible, but they tend to lose the consistency and knowledge modularity of the production system.

In the fuzzy set theory [9,15,35,36J, a mem­ bership function and its operations play a major role in the expression of ambiguous facts and inferences. Maximum and/or minimum are fundamen­ tal operations for the manipulation of the mem­ bership function. The theory of fuzzy set could provide a convenient tool to the inference of dam­ age assessment which is continuous in nature. Moreover, recent applications of fuzzy set theory to civil or structural engineering problems [1,3,7,13,33) have shown that fuzzy membership functions provide a good measure for the interpre­ tation of low-level features in damage assessment.

Consider the inference example in Section III in terms of fuzzy set theory under the framework of production system. First of all, we define the damage grades as shown in Fig. tic variables. For example, severe damage state. Then the can be specified as follows: 1 as fuzzy Linguis­ let F denote the membership function

where d denotes the numerical grade of the damage state.

Suppose that we have a rule

Rule "3 IF: Sx is F

G, where F and G are fuzzy Dx and Dh respectively, subsets of and ch > ,x universe 0. Let sets

Then we can use the following composition [22J to generate a fuzzy relation R from Rule 3, because it satisfies the inference of modus ponens;

where 11 denotes min. operation. If the evidence that Sx is F' which is somewhat different from F is found, then the fuzzy set G'' of Sh can be cal­ culated through a fundamental fuzzy inference as,

where V denotes max. operation.

Several fuzzy sets of {GLO}, for example, can be inferred in this way from Rule 201, 20"3, 205, 207. According to the elementary operations of the fuzzy set theory, the final {GLO} is eventual­ ly obtained by taking the maximum membership func­ tion of these {GLO}s at each d.

The advantages of using fuzzy set theory in this application are that, 1) the range covered by a rule is broad, 2) redundant rules are allowed because only one effective element is selected through the max. and/or min. operations, and "3) the inference is realized for the continuous vari­ ables by a smart mapping calculations of fuzzy re­ lations. Hence, the problem described in the previous section would not occur.

However, because a strong rule acts to mask the other rules, the property of accumulating several confirmations from different evidences cannot oe expected. To attain the accumulation effect, we have to combine the rules to create a new rule. This implies the loss of knowledge modularity.

V. CONFIRMATION BY FUZZY INFORMATION

we have seen that while the production system with certainty factor and the fuzzy set theory in production system have interesting properties in some respects, their direct applications are not necessarily appropriate in damage assessment be­ cause of their critical drawbacks.

The idea of inference mechanis~ employed in SPERIL is simple but very important. That is, the premise and/or conclusion of the rule are regarded as fuzzy sets along the degree of damage state d. Individual inference with a rule is conducted in terms of fuzzy information, which will be described subsequently. After this individual inference, several resultant certainty factors of the damage state from different rules are combined by using the consistent combining function of Eq. (~).

Suppose that we have the abovementioned Rule~. The probability of fuzzy state that sh is G is defined by,

where p(dh) is the probability density at dh. A posterior probability of fuzzy state Shafter the acquisition of fuzzy state that sx is F can be calculated by,

using these definitions, Rule 3 can be interpreted as that if the premise js completely satisfied, P<GIF> is associated with P(G) and ch,x as fol-

lows,

A posterior probability density p(dhiF> is then calculated to satisfy Eqs. (8) and (9); however, this solution is not unique. Therefore we impose a heuristic restriction such that p(dhl F) is increased over p(dh) in proportion to uG(dh) if ch,x > 0. · Then we can have

This implies that the certainty factor at dh can be inferred by

I Ch in Eq. (11) will be if replaced by C as h,x ,x described subsequently premise is not perfect. the machining of the

The certainty factor calculated as follows, of the premise part can be

where cx(dx) is the inferred density of certainty factor of sx. consequently, ch,x in Eq. (11) is replaced by,

where the lower limitation is introduced to elim­ inate the effect of disconfirmed states. Similar­ ly, equations for Ch < 0 can be derived; they ,x are omitted in this paper.

Both the modularity of the knowledge and the capability of expressing continuous nature can be achieved by taking advantages of the certainty factor and fuzzy membership function, respective­ ly.

.'!l· CONCLUSIONS

The knowledge organization of a rule-based dam­ age assessment system of existing structures sub­ jected to earthquake excitation is outlined. A principle of inexact inference to reach a rational solution has been described. Fuzzy set theory and production system with certainty factor are em­ ployed jointly in the inexact inference to deal with the continuous nature of the damage state and to attain the modularity of uncertain knowledge, respectively.

No special strategy to speed up the inexact inference process is adopted at present. It is important for the system to build up gradually by accepting new knowledge. Particularly, recent full-scale dynamic tests of buildings are expected to provide useful information to this problem.

ACKNOWLEDGEMENTS

The authors are grateful to Prof. M. A. Sozen of univ. of Illinois for his comment and suoplying many earthquake experimental data, and our co­ worker Mr. s. Toussi for his assistance in system identification. Comments of Prof. K. Tanaka of osaka univ. are also acknowledged.