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Fuzzy

PATTERN RECOGNITION MODEL FOR DIAGNOSING CRACKS IN

RC STRUCTURES

By

Ching-Ju Chao

l

and Fu-Ping Cheng

2

ABSTRACT: This paper examines a diagnostic model based on the concept of cause-and-effect diagramming and fuzzy pattern recognition, which contributes a new methodology for diagnosing engineering problems. Three examples are presented to demonstrate the feasibility of the model in diagnosing crack formations in reinforced concrete structures. Two levels of parameters representing the causes of cracks in concrete are used to form fuzzy sets. The parameters represent the materials used, fabrication of structural elements, loading, and envi-ronmental conditions. An expert system that links the parameters by means of fuzzy set theory is constructed using finite universal sets consisting of membership functions and fuzzy vectors. Pattern recognition is used to identify a fuzzy vector that represents the most likely causes of the crack.

INTRODUCTION

Concrete is a versatile building material that has been used widely in construction since the invention of cement in 1824. Major concrete structures include dams, bridges, highways, buildings, and pipe systems. Because concrete is strong in compression but weak in tension, concrete structures often de-velop cracks that ultimately affect the performance of the structure. The assessment of cracks in concrete members of existing buildings is a complex process that requires infor-mation on the aggregate used, mixing and curing, the prop-erties of the concrete, and loading conditions. In addition, shrinkage and creep often compound the degree of complexity to the extent that engineers are unable to pinpoint the precise causes of cracks or accurately predict the behavior of the cracked concrete structural elements.

In general, concrete develops cracks because of one or more of the following reasons: abnormal setting of cement paste, heat of hydration and expansion of cement paste (Lea 1971; Soroka 1979); alkali aggregate reactions, poor gradation of aggregate (ACI Committee 221 1961); inadequate mixing, concrete construct defects (Powers 1968; ACI Committee 302 1971; ACI Committee 304 1972; ACI Committee 308 1971); overloading or abnormal loading, and other factors. Tradition-ally, assessment of cracks in concrete structures was done by experienced engineers only; however, expert systems have been developed to conserve time, make expertise more widely available, and simplify decision making. Similar expert sys-tems have been successfully used in medicine (Shortliffe 1976; Adlassnig 1982; Binaghi 1990), mineral exploration (Duda and Reboh 1984), structural analysis (Bennet and Engelmore 1979; Adeli 1988; Adeli and Balasubramanyam 1988), construction material selection (Clifton and Oltikar 1987), and building re-pair technology (Kalyanasundaram et al. 1990; Wang et al. 1991).

The causes of cracking in concrete are complicated and in-terrelated, and the characteristics of cracks are difficult to de-scribe precisely. These characteristics include how long after casting cracks develop, the depth of cracks, whether cracks are regular or irregular, the types of concrete members that de-IphD Candidate, Dept. of Civ. Engrg., Nat. Chiao Tung Univ., Lect., Dept. of Civ. Engrg., Ming Hsin Institute of Technology, I Hisn-hsing Rd., Hsin-Fong, Hsinchu, Taiwan R.O.C.

'Assoc. Prof., Dept. of Civ. Engrg., Nat. Chiao Tung Univ., 1001 Ta Hsueh Rd., Hsinchu, Taiwan R.O.C.

Note. Discussion open until September 1, 1998. To extend the closing date one month, a written request must be filed with the ASCE Manager of Journals. The manuscript for this paper was submitted for review and possible publication on November 27, 1995. This paper is part of the

Journal of Computing in Civil Engineering, Vol. 12, No.2, April, 1998.

©ASCE, ISSN 0887-3801/98/0002-0111-0119/$4.00 +$.50 per page. Paper No. 12102.

velop cracks, crack patterns, and crack locations. Fuzzy set theory (Zadeh 1965) has been used to describe similar char-acteristics successfully in the field of medical diagnostic sys-tems and in pile type selection (Mishido et al. 1990). Pattern recognition has been applied in linguistic pattern search tech-niques (Sanchez et al. 1982), character recognition (Chatterjii 1982), texture classification (Hajnal and Koczy 1982), and earthquake engineering (Fu et al. 1982; Ishizuka et al. 1982; Watada et al. 1984).

Cause-and-effect diagrams have been employed in construc-tion management to classify the relaconstruc-tionships between defects and their causes. These diagrams and fuzzy pattern recognition can be combined to identify fuzzy relationships between the cause of cracking and the characteristics cracks exhibit. In this paper we propose a two-level system for compiling data on the causes of cracks; the method can be extended to more levels if necessary.

MATHEMATICAL MODELING

An attempt was made to explore the feasibility of using fuzzy pattern recognition in the investigation of concrete struc-tures exhibiting cracks. The concept of fuzzy set theory and pattern recognition may be new to readers in the field of con-crete, so this section briefly describes some important termi-nology used in this paper.

1. Membership function. In fuzzy sets, an object's mem-bership in a set may be whole, partial, or nonexistent. The degree of membership is expressed as its member-ship function, which is defined as follows: If X is a col-lection of objects denoted by x, then a fuzzy set

A

in X is a set of ordered pairs

A

=

{(x, J.lA(x»lx EX}. J.lA(X) is called the membership function of objectx in

A.

The membership function is a real number 0 :s; J.l..i(x):s; 1. 2. Fuzzy vector. A fuzzy set defined by a finite universal

set X

=

{Xl> X2, ... , xn} can be represented by a vector

A

= (al> a2, ... , an), whereai = J.lA(Xj ) , i = 1, 2, ... , n.

3. Fuzzy relation. Fuzzy relationship is an important con-cept in fuzzy set theory. A relationship is an association between elements, which is also called a mapping be-causeitassociates elements from one domain with those in another domain. Let X, Y ~ Ube universal sets. Then,

R

= {«x, y), f.'-..(x, y»!(x, y) ~X

x

Y} is a fuzzy relation of X x Y. A fuzzy relation is a fuzzy subset in the

Car-tesian product universe. A convenient way of represent-ing a relationship is by means of a matrix. The Cartesian product of two crisp sets X and Y, denoted by X X Y, is

the crisp set of all ordered pairs such that the first ele-ment in each pair is a member of X and the second el-ement is a member of Y. Such a set can be represented

as X X Y= {(x, y)lx E X, y E V}. Zadeh and other

JOURNAL OF COMPUTING IN CIVIL ENGINEERING / APRIL 1998/111

(2)

FIG. 2. Membership Functions of Linguistic Variables Sets A andB

authors have suggested additional definitions for fuzzy set operations, such as algebraic product and weighted Hamming distance.

4. Algebraic product. The algebraic product of two fuzzy sets

A

and

R

is defined as

C

=

A'R;

then,

C

=

{(x, f.Lc(x))Ix EX}, where f.Lc(x)= f.LA:(x)· f.Ls(x). 5. Weighted Hamming distance. Let

A

and

R

be two fuzzy

vectors on X

=

{x" X2, ... :..xn]. The weighted Hamming

distance is defined as dw(A, B) = ~7=1 W(Xj)' [f.LA:(Xi) -f.Ls(x;)], i = I, 2, ... ,n, where W(xi) is the value of the weight for Xi' If f.LA:(Xi) - f.Ls(xi) ;;:::: 0, it is a positive distance from

A

to

R.

If f.L..i(Xi) - f.Ls(Xj) < 0, it is a negative distance from

A

to

B.

(5) (8) (6) (7) f.Lmedium(X)= SeX; 0, 0.5, 1.0)

=

0.5 f.Llow(X) = sex;0, 0.5, 1.0)=0.25 wherex

=

0.35. wherex= 0.2. f.Lvery_low(X)=Sex; 0, 0.5, 1.0)=0.1 wherex= 0.5. f.Lhigh(X)

=

sex; 0, 0.5, 1.0)=0.75 wherex= 0.65.

Fuzzy sets and linguistic variables can be used to quantify concepts used in natural language, which can then be manip-ulated. A linguistic variable must have a valid syntax and se-mantics, which can be specified by fuzzy sets or rules. A syn-tactic rule defines the well-formed expressions in T(L). The term setT(L) of a linguistic variable, L, is the set of values it may take. For example, T(Age) = {very_young, young, me-dium, old, very _old}, where each of these values may itself be a linguistic variable that can take on values that are fuzzy sets. The membership function could be defined as the S func-tion f.Lol,,(X)

=

Sex;60, 70, 80).

A cause-and-effect diagram is shown in Fig. I; the cause parameters can be divided into several groups according to their properties. We call these groups the primary-level cause parameters and denote them by V = {V., V2, ... , Vn }. Each

primary-level cause parameter contains several subparameters, which are known as secondary-level cause parameters and ex-pressed as Vi = {Vi., Vi2, ... , Vim} (the number of levels can be extended if necessary). Q

=

{q., q2, ... , qr} is the crisp universal set of all characteristics.

Linguistic variables are used to describe the degree of re-lationship between a causeVi and a characteristicqb which is defined as a set A = {very_low, low, medium, high, very_ high}. Then, the fuzzy set is defined asA(x), x= {O, 0.1, 0.2, ... , 1.0}, and the membership function is defined as f.L..i(x) = Sex; 0, 0.5, 1.0), x E X, where X is the relation space. Lin-guistic variables are also used to describe the degree to which it is confirmed that a characteristic qk is exhibited, which is defined as a set B = {very_low, low, medium, high, very_ high}. The fuzzy set is then R(x), x

=

{O, 0.1, 0.2, ... , 1.0}, and the membership function is defined asf.Ls(x) =Sex;0, 0.5, 1.0), x E Y, where Yis the confirmation space.

In this paper, the membership functions of the element sets A and B can be chosen from among the following equations:

y

.5 .65 .8

f3

.2 .35

low

FIG. 1. Cause-and-Effect Diagram

medium

o

a

---..:l_---...

.,..---I.Q {ql. q1• ... }

_ _ _ _ '::ly-high

_

hi h

----~---L..-_ _" " ' - _..._ ..._ ..._ ..._ ..._ _

"'--_x

1.0 0.9 0.5 0.1

o

0.75 0.25

Zadeh defines standard piecewise quadratic functions (Za-deh 1981) as

where a

=

0;

13

=

0.5; and 'Y

=

1.0.

ft(x; a, ~,-y)

=

Sex; a, ~, -y) (1)

(.l. {fl(X;

~

- a,

~

- al2,

~);

x::5

~

(2)

j2(x; a, 1-')

=

1 - ft(x; ~, ~

+

a/2, ~

+

a); x>

13

j,(x; a, ~,-y)

=

1 - ft(x; a, ~,-y) (3) whereSex; a,

13,

'Y)is an S-function that is often used in fuzzy sets as a membership function and in this paper is defined as follows: (9) VI V2 V3 Vn q,

[~U

f.L12 f.L13

~:'.]

q2 f.L21 f.L22 -(I) (10) R = ~3 f.L.31 qr f.Lrl f.Lm f.Lvery_high(X) =SeX; 0, 0.5, 1.0)=0.9 wherex = 0.8.

The membership functions are illustrated in Fig. 2.

For the primary level, we can define a fuzzy relation R(1)

on the set QX V in which membership function f.LA<ll(qj, "}), (qi' E Q, Vj E V)indicates the degree of relationship between

characteristic qi and cause

"l.

This relation can be expressed in matrix form:

where f.Lij

=

f.L..i(qi,

"l);

i

=

I, 2, ... ,r;j

=

I, 2, ... ,n. For the secondary level, using the cause parameters for each

"l,

define a fuzzy relation

Ry)

on the setQX \ojin which the membership function f.LiiJ2)(qn 11,), (qs E Q, Vj, E

"l)

indicates (4) x::5a 1; 0; ( X -

a)2

2 - - ; "'I-a I - 2

(!...::...:i)

2 ; 'Y- a f.L..i(x)

=

Sex; a, ~,'Y)

=

(3)

the degree of relationship between characteristicqsand cause

Vj"

ay)

can also be written in matrix form:

'Ujl Vj2 'Uj3 'Ujm

q'[",

'Y12 'Y13

'~.

]

q2 'Y21 'Y22 - (2) (11) RI = ~J "'I:' qh "'Ihl "'Ihm

where "'1st

=

J.LA:(q.. vI'); S

=

1, 2, ... ,h; t

=

I, 2, ... ,m; and j = 1,2, ... ,n.

An observational fuzzy vector

P

can be defined on a set Q

= {ql> q2, ... , qr} to indicate the degree to which it is_con-firmed that a particular crack characteristic is exhibited. P can be expressed as

P

=: (PI' P2' ... , P,), wherePi

=

J.Ljj(x), i

=

1, 2, ... ,r. Let pO) indicate the degree of confirmation for the primary level, and let p(2)indicate the degree of confirmation for the secondary-level.

We define a matrix H = (hi)rx" whose component hi) is called the importance factor. Ifhi)= k, this means crack char-acteristicqiisktimes more important than crack characteristic

ql for the cause of cracks, and hji =: 11k. Then the weighting

vector W is defined as W =(L}..I hlj , Lj=1 h2j, ... ,Lj•• hrj)= (WI>W2, • . . ,w,)and the sum of the components of this vector is unity,L~=l Wi= 1.Wi>

°

are weights that express the relative

importance of the crack characteristics setQ.LetW(I)indicate

the primary-level weighting vector and W(2) indicate the

sec-ondary-level weighting vector.

For the primary level, cause parameters define a fuzzy vec-tor

Vi

on Q(lJ

=

{ql> q2, ... , qs} as a fuzzy pattern, which is represented by a vector

Vi

= (al> a2, ... , as), i = 1, 2, ... ,

n, where aj

=

J.Lvlqj), j =: 1, 2, ... ,s. We then pelform

pa!-tern comparison for each pair of fuzzy patpa!-terns VI and VI using the weighted Ham,!l1in,g distance dw(Vj, Vj) = ~ ~_Wk' [J.Lv,(qk) - J.LV;(q,JJ. !fdw(Vi , VI) > 0, the fuzzy_pattern_Vi is

selected. Ifaw(Yi, Yj) = 0, the fuzzy pattern Yi and VI are selected. Ifdl'o.(Vi , V) < 0, the fuzzy pattern VI is selected.

In each step of the process one pattern is screened out. Even-tually, only one fuzzy pattern is left; this pattern is identified as the cause on the primary level.

From the observational fuzzy vector

p(1),

the last selected fuzzy pattern

Vi,

and the weighted vector WO

), the degree of confirmation C;lJfor the fuzzy pattern

Vi

can be computed by the following formula: C?) =: L;"I (WI).pO» .

Vi'

i = 1, 2,

"', n.

After

Vi

is selected, a fuzzy vector"ik on Q(2) =: {ql> q2,

... , qp} is defined as a fuzzy pattern on the secondary level. This vector can be expressed as follows:

Vik

=

(''I" "12•... , 'Yp),i

=

1, 2, ...•n,k

=

1, 2, ... ,m. where 'Yj

=

J.L,;,,(ql)'j

=

I, 2, ... ,p. _

For the secondary level, using "II' l\b p(2), W(2), and C~;) instead ofaj,

Vi'

p(I),

W(l), andC}lJrespectively, we duplicate the procedures performed for the primary level to obtain the fuzzy pattern Viiand the degree of confirmation. If necessary, the procedure can be repeated for more levels.

CRACK MODELS

According to actual investigations and suggestions reported in engineering publications (Lerch 1957;ACI Committee224 1972; Beaufait and Hoadley 1973; Price 1974;ACI 1974)the primary causes of cracks in reinforced concrete structural el-ements can be classified into four primary-level parameters. These can be expressed as V = {Vir V2, V3 , V4 }, where VI

represents causes related to the quality of the concrete mate-rial, V2represents causes related to the procedure used to

con-struct the concrete, V3 represents causes related to

environ-mental factors, and V4 represents causes related to the applied

loads. Each primary-level cause parameter and its subcauses are shown in Fig. 3 and in Tables 1-5.

Cracks can be described on the basis of seven characteris-tics: how soon after casting they develop, their depth, their regularity, whether they appear only in a concrete member or throughout the overall structure, the type of member in which they appear, their patterns, and their locations. The first four of these characteristics will be referred to as primary-level characteristics and the other as secondary-level characteristics. The primary-level characteristics, QI = {ql' q2' q3, q4}, are shown in Table 6, and the secondary-level characteristics,Q2

=:{qs, q6, Q7},are shown in Table 7. The fuzzy relation matrix

R

for each level is shown in Tables 8-12. All the data and characteristics of cracks can be extended properly by experts or experienced engineering.

APPLICATIONS

Three examples were used to verify the applicability of the model. The data used in these three examples are listed in Tables 8-12.

Example 1

In a reinforced concrete structure, fine cracks occurred on the slab surface three days after casting. The cracks are random in nature and have no regularity.

The degree of confirmation for each of the crack character-istics was as follows:

Crack time (ql): very _high :. PI

=

J.LV.ry_high(X)

=

0.90 Crack depth(q2): high ...P2

=

J.Lhigh(X)

=

0.75

Crack regularity(q3): very_high ...P3

=

J.LV.ry_high(X)

=

0.90 Crack range(q4): very_high ... P4

=

!J..v.ry_high(X)

=

0.90 Crack member(q5): very_high P5

=

J.Lv.ry_higlx)

=

0.90 Crack pattern (q6): very _high P6= J.Lv.ry_high(X) =:0.90 Crack location(q7): high :. P7=: J.Lhigh(X)=0.75.

According to these crack characteristics(QJ

=

q12' q2=: Q2l' q3

=:q32, q4= Q41), four fuzzy patterns for the primary level were found in Table 8, and the primary-level observational fuzzy vectorsp(llwere expressed as follows:

Vi

=:(q12 q21 q32 q41) i

=

1, 2, 3, 4.

VI

=

(.50 .90 .90 .90), V2

=

(.75 .50 .50 .90), V3 = (.25 .50 .50 .90),

V

4

=

(.25 .10 .10 .90), and p(l)

=

(PI, P2 P3' P4) (.90 .75 .90 .90).

For the primary-level characteristics, the importance factors assigned were qt Q2 q3 q4 Ql

p,

I 1.5

15]

H =:Q2 1 1.5 1.5 qJ 0.67 1 1 q4 0.67 0.67 1 I

Therefore, W(I)= (5 5 3.34 3.34),the weighting vector was

normalized as W(I)= (.3 .3 .2 .2).

The weighted Hamming distance was computed by the fol-lowing process:

JOURNAL OF COMPUTING IN CIVIL ENGINEERING / APRIL 1998/113

(4)

Q

Acids and sulfate attack (V34 )

Fire or e,,-posed to high temperature (V33 )

New building constmcted nearby(V44)

Freezingandthawing interaction (VJ2)

Different material bonding(V4$ ) Temperature and moisture change(V3/)

riod(V26)

Inadequate mixing of concrete(V2/)

Drying and sluinkage of concrete(V)7)

Segregation occurs during placement(V23)

Undervibrated or oyervibrated during Concrete bleeding, segregationand settlement (V16 )

concrete placing(Vu)

Improper curing procedure (V2$)

Alkali-aggregate reaction (V1$)

Abnonnal setting of cement paste(V) } )

QUALrfYO)i' CONCRETE

Vi

MATERIAL

CONCRETE CONSTRUCTION

(Vu)Overly long mixing time

(V28)Inadequate rebar layout (V29)Insufficient concrete coyer (V12)Heat of hydration

(Vu)Aggregate contains impurities (V13 )Expansion of cement paste

(V212)Support settlement

(V210)Defonnation offormwork

FIG. 3. Cauae-and-Effect Diagram of Cracks

TABLE 1. Causes of CrackofPrimary Level TABLE 3. Concrete Construction Procedure (V2 )

Cause (1 ) Characteristic (2) Cause (1) Characteristic (2) 4

e\l)

=

2:

(W(I)·p(l»·V

I

=

0.662 ]-1 dw(VI ,

V

2 ) = .3'(.50-.75)

+

.3'(.9-.5)

+

.2'(.9-.5)

+

.2'(.9-.9)= 0.125 dw(VI>

V

2) = 0.125>0;

V

Iis selected. dw(VI> V3) = .3, (.50-.25)

+

.3, (.9-.5)

+

.2, (.9-.5)

+

.2, (.9-.9) = 0.275 dw(VI> V3) = 0.275> 0; VI is selected. dw(VI> V4 ) = .3'(.50-.25)

+

.3'(.9-.1)

+

.2'(.9-.1)

+

.2, (.9-.9) = 0.475 dw(VI> V4) = 0.475>0; VI is selected. Characteristic (2)

Temperature and moisture change Freezing and thawing interaction Fire or exposed to high temperature Acids and sulfate attack

Corrosion of rebar Inadequate mixing of concrete Overly long mixing time

Segregation occurs during placement

Undervibrated or overvibrated during concrete placing Improper curing procedure

Freezing and thawing in early period Inappropriate construction joint treatment Inadequate rebar layout

Insufficient concrete cover Deformation of formwork Form removed too early Support settlement Inadequate surface finishing

TABLE 4. Environmental Factors ( V3 )

Cause (1 )

The primary-level crack cause is related to the quality of the concrete material (fuzzy pattern

V

I), and the degree of confir-mation is 66.2%.

For the secondary level there are seven fuzzy patterns re-lated to the quality of the concrete material. The fuzzy vector Quality of concrete material

Concrete construction procedure Environmental factors

Applied loads

Characteristic (2)

Abnormal setting of cement paste Heat of hydration

Expansion of cement paste Aggregate contains impurities Alkali-aggregate reaction

Concrete bleeding, segregation, and settlement Drying and shrinkage of concrete

TABLE 2. Quality of Concrete Material (V,) Cause

(1 )

(5)

TABLE 6. Primary-Level Characteristic Data

a.

Cause

(1 )

Characteristic

(1 )

TABLE 5. Applied Loads (V.)

Characteristic (2)

Overloading Earthquake force

Uneven settlement of structure New building constructed nearby Different material bonding

Value (2)

dw(VII, V12)

=

.3' (.9-.1)

+

.45 . (.9-.1)

+

.25' (.9-.25)

=

0.7625>0 Vl1 is selected.

dw(Vlh VI3)

=

0.405; VII is selected. dw(Vl1'VI.)

=

0.293; VII is selected.

dw(Vlh VIS)

=

0.413; Vl1 is selected. dw(Vlh V16)

=

0.405; Vl1 is selected. dw(Vl1'V17)= 0.068; Vl1 is selected.

Fuzzy pattern Vl1 is selected, but fuzzy pattern Vl7 is near V11;therefore,

v

17 is also selected:

3

cw

=

2:

(W(21·p(2»·Vl1

=

0.776 ]-1

TABLE 7. Secondary-Level Characteristic Data~ Time(q,)

Depth(q2)

Regularity(q,)

Range(q.)

qll:One hour-one day

q,2:One day-28 days ql3: More than 28 days

q21:Shallow and fine on surface

q22: Deep and wide

q'l:Regular

q32: Random

q.l: Member (beam, column, slab, wall)

q.,: Overall structure

3

cW

=

2:

(W(21·p(2»·VI7

=

0.716 ]-1

Thus, the cause related to the quality of the concrete material involves two issues:

1. Abnormal setting of cement paste (fuzzy pattern Vl1), with a degree of confirmation at 77.6%.

2. Drying and shrinkage of concrete (fuzzy pattern VI7 ), with a degree of confirmation of 71.6%.

Characteristic (1 ) Pattern(q.) Location(q,) Value (2) q,,:Beam q,,: Column q,,:Slab q,.: Wall q,,: Overall structure q.,:Longitudinal q.2: Transverse q.,: Diagonal q..:X shape q.,:1\shape q..:\Ishape q.,: +shape q..:Turtle-back shape qf>j: Random shape q.lO: Honeycombing q.lI: Spall q.,2: Spiral shape q.l3: Corrosion of rebar q71: End parts qn:Central parts q73: Comer parts q,.: Member surface q"Opening hole

q,.

Joint of members Example 2

In a reinforced concrete structure, cracks were found in wall surfaces one year after casting. The cracks were very deep and formed a regular pattern, like an X shape, near the center of the walls.

The degree of confirmation for each of the crack character-istics was as follows:

Crack time(ql): very _high :. PI

=

!J.v<ry_high(X)

=

0.90

Crack depth(q2): very_high :. PZ

=

!J.v<ry_high(X)

=

0.90 Crack regularity(ql): high :. P3

=

!J.high(X)

=

0.75 Crack range (q.): high :.P.

=

!J.high(X)

=

0.75

Crack member(q,): very_high :. Ps

=

!J.v<ry_high(X)

=

0.90 Crack pattern (q6): very _high :. P6

=

!J.very_high(X)= 0.90 Crack location(q7): high :. P,

=

!J.high(X)

=

0.75.

TABLE 8. Primary-Level: Causes-Characteristics Matrices

V

I

V

2

V

3

V

4

qll

[9

.75

.1

I ]

ql ~ {j12

.5

.75

.25

.25

{j13

.75

.25

.9

.9.

The importance factors assigned were

Therefore, weighted vector W(Zl= (.30 .45 .25).

.1 ]

.9

V

2

V

3

.5

.5

.5

.75

v;

V

2

V

3

V

4

%1

[.1

.5

.5

.9]

%

~ q32

.9

.5

.5

.1

VI

V

2

V,l

V

4

q'f

.9 .9

.9]

q4~

.1

.9

.9

{j42

.1

1.5]

i

,W2)=(3.17 4.52.17) 0.67 1 0.5 VI}= (qs3 q69 q7.) j = 1, 2, ... , 7. VII

=

(.90 .90 .90), V,2

=

(.10 .10 .25), VI3

=

(.75 .10 .90),

V,. =

(.90 .25 .90),

VIS

=

(.50 .25 .90), VI.

=

(.75 .10 .90), VI'

=

(.90 .75 .90),

p(2)

=

(Ps P. p,)

=

(.90 .90 .75).

was found from Table 9 (qs

=

qS3, q6

=

q69' q7

=

q7.), the observational fuzzy vectors

p(2)

were expressed as follows:

JOURNAL OF COMPUTING IN CIVIL ENGINEERING / APRIL 1998/115

(6)

TABLE 9. Secondary-Level:Quality of Concrete Material Ma- dw

(V

3,

V

4)

=

.3'(.90-.90)

+

.3'(.75-.90)

+

.2'(.50-.90)

trices

+

.2·(.90-.90)

=

-0.125

dw

(V

3,

V

4)

=

-0.125<0;

V

4is selected.

VII VI2

"\3

"14 VIS "16

"\7

The degree of confinnation for fuzzy pattern

V

4

is

q'l

.25

.75

.5

.75

.75

.9

.1

4

CJS2

.25

.1

.25

.5

.9

.75

.1

cil)

=

L

(W(1)·tJ<1».

V

4

=

0.756.

)-1

CJs => qS3

.9

.1

.75

.9

.5

.75

.9

Therefore, the 5ause of cracking is related to applied loads

%4 .75

.5

.75

.75

.5

.75

.9

(fuzzy patternV

4),

and the degree of confinnation is 75.6%.

.1

.1

.1

.1

.1

.1

.1

On the secondary level there are five fuzzy patterns related

CJss to the applied loads. The fuzzy vector was fonnulated from

VII VI2 VI3 VI4 VIS VI6 VI7 Table 12, and

p<2l

are as follows:

q61

.1

.5

.1

.1

.1

.75

.9

V4j

=

(qS4 q64 qn) j

=

1, 2, ... , 5. V41

=

(.50 .50 .90),V42

=

(.90 .90 .90),V43

=

(.25 .50 .25) q62

.1

.5

.1

.1

.1

.1

.9

V44

=

(.50 .50 .25), V4S

=

(.90 .10 .10), tJ<2)

=

(.90 .90 .75). CJ63

.1

.1

.1

.1

.1

.1

.75

CJ64

.1

.1

.1

.1

.1

.1

.1

Weighted vector: W(2)

=

(.30 .45 .25). Q6S

.1

.1

.1

.1

.1

.1

.75

dw(V4h V42).90)

=

-0.3

=

.3, (.50-.90)

+

.45· (.50-.90)

+

.25, (.90-CJ66

.1

.1

.1

.1

.1

.1

.75

dw(V4h V42)

=

-0.3 <0;

v

42 is selected. dW(V42,V43)

=

0.538 >0;

v

42is selected. CJ6 => q67

.5

.25

.5

.25

.1

.75

.1

dw(V42,

v

44)

=

0.463>0;

v

42 is selected. dW(V42,V4S)

=

0.56>0; V42 is selected. q(IX

.25

.1

.75

.9

.5

.1

.75

.9

.1

.1

.25 .25

.1

.75

and q69 3 q610

.1

.1

.1

.1

.1

.1

.1

ci~

=

L

(Wi2). P(2». V42

=

0.776. )-1 CJ611

.1

.1

.75

.25

.9

.1

.1

Hence the specific cause of these loading-related cracks was

q611

.1

.1

.1

.1

.1

.1

.1

earthquake force (fuzzy pattern V42), with a degree of

confir-.1

.1

mation of77.6%.

CJ613

.1

.1

.1

.1

.1

VII VI2 VI3 Viol VIS VI6 "17 Example 3

.1

.1

.1

.1

.1

.1

.1

In a reinforced concrete column, honeycombing occurred on

CJ71 the member surface seven days after casting. The cracks were

qn

.1

.1

.1

.1

.1

.1

.1

deep and had no regularity.The degree of confinnation for each of the crack

character-CJ73

.1

.1

.1

.1

.1

.1

.9

istics was as follows:

q7 =>

.9

.25

.9

.9

.9

.9

.9

Crack time (q.): very _high :. PI

=

ILve'Y_high(X)

=

0.90

q74

.1

.1

.1

.1

.1

.1

.9

Crack depth(q2): very_high :. P2

=

ILve'Y_high(X)

=

0.90

q75 Crack regularity (q3): high :.P3

=

ILhigh(X)

=

0.75

'176

.1

.1

.1

.1

.1

.1

.9

Crack range(q4): very_high :. P4

=

ILve'Y_high(X)

=

0.90 Crack member(qs): very_high :. Ps

=

ILve'Y_high(X)

=

0.90 Crack pattern(qo): high :. po

=

ILhigh(X)

=

0.75

Crack location(q7): high :. P7

=

ILhi.h(X)

=

0.75. According to the crack data above, the method and process

used in Example 1, four fuzzy patterns for the primary level were found in Table 8, and the primary-level observational fuzzy vectors

p(1)

were expressed as follows:

V/

=

(q13 q22 q31 q41) i

=

I, 2, 3,4.

VI = (.75 .10 .10 .902,

\'2

= (.25 .50 .50 .90),

"3

=

(.90 .75 .50 .90), V4

=

(.90 .90 .90 .90), tJ<n (.90 .90 .75 .75),andW(l)

=

(.3 .3 .2 .2).

The weighted Hamming distance was computed as follows:

dw(V

h

V

2)

=

.3'(.75-.25)

+

.3'(.10-.50)

+

.2'(.10-.50)

+

.2, (.90-.90)

=

-0.05

dw('V

h

V

2)

=

-0.05 <0;

V

2is selected. dw

(V

2 ,

V

3)

=

.3·(.25-.90)

+

.3'(.50-.75)

+

.2'(.50-.50)

+

.2'(.90-.90)

=

-0.27 dw

(V

2 ,

V

3)

=

-0.27<0;

V

3is selected.

According to the crack data above, the method and process used in Example 1, four fuzzy patterns for the primary level were found in... Table 8, and the primary-level observational fuzzy vectorsp(1) were expressed as follows.

Vi

=

(q12 q22 q32 q41) i = I, 2, 3, 4.

V.

= (.50 .10 .90 .902,

"2

= (.75 .50 .50 .90),

\'3

=

(.25 .75 .50 .90), V4

=

(.25 .90 .10 .90), p(l)

(.90 .90 .75 .90),andWin

=

(.3 .3 .2 .2).

The weighted Hamming distance was computed as follows:

dw('\'t,

V

2 )

=

.3'(.50-.75)

+

.3·(.10-.50)

+

.2'(.90-.50)

+

.2'(.90-.90) = -0.115

dw(V

h

V

2 )

=

-0.115<0;

V

2is selected. dw

(V

2,

V

3)

=

.3, (.75-.25)

+

.3, (.50-.75)

+

.2, (.50-.50)

+

.2, (.90-.90)

=

0.075 dw

(V

2,

V

3)

=

0.075>0;

V

2is selected.

(7)

TABLE 10. Secondary-Level: Concrete Construction Procedure Matrices q51

.9

%2

.9

q5 =>q53

.9

q54

.9

q55

.1

.9

.5

.9

.9

.9

.25

.9

.9

.1

.1

.75

.9

.75

.9

.1

.25

.1

.9

.

.1

.1

.5

.1

.9

.1

.1

.75

.1

.1

.9

.1

.75

.5

.9

.25

.1

.5

.75

.9

.75

.1

.9

.75

.9

.25

.1

.9

.75

.9

.5

.1

.9

.25

.9

.5

.1

.75

.25

.9

.9

.1

1121 1122

%1

.5

.9

lJ62

.5

.9

q63

.1

.1

%4

.1

.1

'ltis

.1

.1

q66

.1

.1

q6

=>

q67

.1

.1

q68

.9

.9

q69

.75

.5

lJ610

.5

.5

q6JI

.1

.1

q612

.1

.1

q613

.1

.1

V23 V24

.5

.5

.5

.5

.9

.5

.1

.1

.1

.1

.1

.1

.1

.1

.1

.1

.1

.5

.75 .9

.1

.1

.1

.1

.1

.1

V25 1126 V 27

.5

.9

.9

.5

.9

.9

.25

.1

.9

.1

.1

.1

.1

.1

.1

.1

.1

.1

.1

.1

.1

.9

.9

.1

.9

.5

.1

.1

.1

.1

.1

.1

.1

.1

.1

.1

.75

.1

.1

V28

.1

.75

.5

.1

.1

.1

.1

.5

.5

.1

.1

.1

.1

V29

.9

.9

.1

.1 .1

.1

.1

.75

.75

.1

.75

.1

.9

V210

.9

.9

.9

.1

.1

.1

.1

.1

.1

.1

.1

.1

.1

V2\I

.9

.9

.9

.1

.1

.1

.1

.75

.5

.1

.1

.1

.1

V212 V 213

.1

.1

.9

.1

.9

.1

.1

.1 .1

.1

.1

.1

.1

.1

.1

.9

.1

.75

.1

.1

.1

.1

.1

.1

.1

.1

'hi .5

.5

.5

.5

.5

.5

.5

.75

.5

.5

.5

.5

.75

.9

.1

.5

.5

.9

.9

.1

.5

.9

.5

.9

.1

.5

.5

.75

.9

.9

.75

.75

.5

.9

.1

.5

.75

.5

.9

.1

.75

.9

.5

.5

.75

.1

.9

.9

.1

.5

.75

.25

.9

.9

.75

.5

.1

.5

.5

.25

.5

.5

.75

.75

.1

.1

.1

.1

.1

.1

.1

.1

.1

.1

.1

.9

.1

.1

dw

(V

2,

Y

4) = .3'(.75-.25)

+

.3·(.50-.90)

+

.2'(.50-.10)

+

.2, (.90-.90) = 0.11 dw(Y2,

Y

4) = 0.11 > 0;

Y

2is selected.

The degree of confirmation for fuzzy pattern

Y

2 is

4

C~I)=

2:

(W(1)·p(1»·Y2 = 0.575. ]-1

Therefore, the cause of cracking is related to the concrete con-struction procedure (fuzzy pattern

Y

2), and the degree of con-firmation is57.5%.

On the secondary level there are 13 fuzzy patterns related to the concrete construction procedure. The fuzzy vector was formulated from Table 10 and P<2)as follows:

V~

=

(QS2 q.IO q74) j

=

1, 2, ... , 13. V21

=

(.90 .50 .50), V22 = (.90 .50 .50), V23 = (.90 .75 .75) V24 = (.90 .90 .90), V2S = (.10 .10 .90), V26 = (.10 .10 .75) v27=(.10 .10 .75),V28 = (.50 .10 .1O),v29 = (.75 .10.50) V2\O = (.75 .10 .50), V211 = (.75 .10 .75), V2\2 = (.25 .10 .752 V2\3 = (.25 .10 .90),

p(2)

= (.90 .75 .75), W(2) = (.30 .45 .25). dw(V21' V22) = .3' (.90-.90)

+

.45, (.50- .50)

+

.25' (.50-.50) = 0 dw(V2h V22) = 0; V21 orV22is selected. dw(V2h V23)

=

-0.175<0; V23 is selected. dw(V23'V24)

=

-0.105<0; V24is selected. dw(V24'V2S)

=

0.6>0; V24 is selected. dw(V24'V26)

=

0.6375>0; V24is selected. dw(V24'V27)

=

0.6375 >0; V24is selected. dw(V24'V28)

=

0.68 >0; V24is selected. dw(V24'V29)

=

0.505 >0; V24is selected. dw(V24'V21O) = 0.505 >0; V24 is selected. dw(V24'V2I\) = 0.4425>0; V24is selected.

JOURNAL OF COMPUTING IN CIVIL ENGINEERING / APRIL 1998/117

(8)

dwCV24' V2l2)

=

0.5925 >0; V24is selected. dw(V24' V2I3)

=

0.555>0; V24is selected.

TABLE 12. Secondary-Level: Applied Loads Matrices

and

3

c~2J.

=

2:

(W<2l·jJ(2»·V24

=

0.716.

]-1

Therefore, the crack cause related to concrete construction was undervibrated or overvibrated during concrete placing (fuzzy pattern V24), with a degree of confirmation of 71.6%.

qs,.9

.5

lJS2

.75

.9

qs=>Q53 .75

.5

lJ54.5

.9

Qss

.5

.75

.75

.75

.1

.25

.25

.J

.5

.5

·.1

.25

.5

.9

.9

.9

.1

CONCLUSION

TABLE 11. Secondary-Level: Environmental Factors Matrices

A fuzzy pattern recognition model based on cause-and-ef-fect diagramming and fuzzy pattern recognition has been de-veloped and applied to diagnose cracks in reinforced concrete structures. The following conclusions can be drawn from this paper:

.75

.75

.1

.25

.25

.1

.75

.75

.1

.75

.75

.1

.5

.5

.1

.5

.5

.9

.1

.1

.1

.9

.1

.1

.1

.5

.75

.75

.5

.1

.9

.5

.5

.1

.1

.9

.9

.1

.1

.9

.9

.J

.1

.1

.1

.1

.1

.1

.1

.1

.1

.1

.1

.1

.J

.J

.J

.1

.J

.J

.1

.5

.5

.25

.25

.J

.J

.J

.1

.J

'171.9

.9

lJn.9

.9

'173

.5

.1

Q7 ::;. Q74

.5

.1

'175

.75

.75

lJ76

.75

.75

q61

.1

Q62

.9

Q63 .9

%4

.5

q6S

.1

%6

.1 q6 ::;. Q67

.1

%8

.1

lJ69

.1

Q610

.1

q611

.1

Q612

.75

q613

.1

1. Cause-and-effect diagrams can be effectively used to es-tablish a diagnostic model, particularly because they clearly depict the relationships among the causes of cracks and the characteristics of the cracks.

2. Fuzzy sets and fuzzy pattern recognition enable us to deal effectively with the ambiguity in diagnosing the causes of cracks. This ambiguity is almost impossible to solve using traditional mathematical models.

3. By combining fuzzy pattern recognition and cause-and-effect diagrams, one can narrow down the possible causes of crack formation .

4. Ifthe data base, the weighted vector, and Hamming dis-tance formula are valid, which depends on the data col-lected and the experience of the base designer, the pro-posed model produces reliable diagnostic results. The model offers an effective tool for diagnosing cracks in concrete structures and may be useful for professionals in the field of concrete engineering.

.5

.9

.25

.9

.9

.9

.9

.9

.25

.25

V34 11 35

.25

.5

.25

.5

.1

.1

.1

.1

.1

.J

.J

.)

.5

.25

.5

.1

.75

.1

.J

.1

.9

.75

.1

.1

.J

.9

V34 V35

.J

.1

.1

.1

.1

.1

.9

.9

.1

.1

.1

.1

V3~ V 33

.1

.1

.1

.1

.25

.1

.9

.9

.1

.1

.25

.1

V31 '171

.25

qn

.25

Q73 .9

'17

=>

lJN

.75

lJ75 .9

'176

.5

V 31 1132 V33

%,.5

.1

.5

lJ62

.5

.1

.5

qOJ.1

.1

.1

C!£,~.9

.1

.1

q6S

.9

.1

.1

C!£,(,.9

.1

.1

q6 => lJ67

.l

.5

.9

q68

.1

.75

.75

lJm

.1

.75

.1

q61o

.l

.1

.1

q611

.1

.75

.5

q

612

.l

.1

.1

Q6l3.1

.1

.1

V31 V32 V33

qs,.5

.5

.9

qS2.5

.5

.9

qs

=>qSJ

.5

.75

.9

qS4.

9

.9

.9

qS5

.9

.75

.25

(9)

APPENDIX II. NOTATION

The following symbols are used in this paper:

=

degree of confirmation of fuzzy pattern VI for pri-mary level;

=

degree of confirmation of fuzzy pattern iiit for

sec-ondary level;

~eighte~ Hamming distance between fuzzy pattern VIand

\'J;

=

observational fuzzy vector for primary level;

=

observational fuzzy vector for secondary level; = ith characteristic of crack;

=

fuzzy relation matrix of characteristic and cause for primary level;

=

fuzzy relation matrix of characteristic and cause for secondary level;

ith fuzzy pattern on primary level; = mth fuzzy pattern on secondary level;

weighting vector of primary level; and

=

weighting vector of secondary level. C(2)

It

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Gupta. M. M., and Sanchez. E. (1982b).Approximate reasoning in de-cision processes. North-Holland Publishing Co., New York, N.Y. Hajnal, M., and Koczy,L. T. (1982). "Classification of textures by

vec-torial fuzzy sets." Fuzzy informationanddecision processes, M. M. Gupta, and E. Sanchez, eds., North-Holland Publishing Co.• New York, N.Y.. 157-164.

Ishizuka, M., Fu, K. S., and Yao, J. T. P. (1982). "A rule-based inference with fuzzy set for structural damage assessment." Approximate rea-soning in decision process,M. M. Gupta and E. Sanchez. eds.• North-Holland Publishing Co., New York, N.Y., 261-275.

JOURNAL OF COMPUTING IN CIVIL ENGINEERING / APRIL 1998/119

數據

FIG. 1. Cause-and-Effect Diagram
TABLE 1. Causes of Crack of Primary Level TABLE 3. Concrete Construction Procedure ( V 2 )
TABLE 8. Primary-Level: Causes-Characteristics Matrices
TABLE 9. Secondary-Level: Quality of Concrete Material Ma- d w (V 3, V 4) = .3'(.90-.90) + .3'(.75-.90) + .2'(.50-.90)
+3

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