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The Dam age Asse ssm ent o f Concre te S t ruc turesBy Tim e f req ue ncy Distr ibut ionsb y R S O l iv it o a n d L S u r a c eA B S T R A C T - - T h i s p a p e r d i s c u s s e s th e u s e o f u l tr a s o n icmethodology w h e n s t u d y i n g t h e d a m a g e c a u s e d t o c u b e -shap ed conc re t e spe c i m en s by un i ax i a l com press i ve load i ng .It uses t he t i m e - f requency d i s t ri bu t ion o f t h e c ross -am b i gu i t yf unc t i on C AF) . An a l go r i t hm capab l e o f com pu t i ng t he c o e ff ic ien ts o f the Z a k t r a n sf o r m needed to d iscret ize t he c ross -am b i gu i t y function i s a l so g i ven , toge t he r w i th t he definit ionof a coe f f i c i en t t ha t pe rm i t s t he study of different d a m a g em e chan i sm s and t he i r evo l u t ion . The se p rov ide a m eansof e f fect ive ly measur ing the extent o f the resu l t ing damage.The resu l t s o f t he expe r i m en t show t ha t t h i s m e t hodo l ogy i se x t r e m e ly sens i t i ve in de t ec t i ng and m on i to r i ng dam ge p ro -c e s s e s i n concrete.ntroduction

Nondes t ruc t ive u l t ra son ic t echn iques have evoked grea tin te re s t in the f i e ld o f c iv i l eng inee r ing dur ing recen t yea rsbecause they do no t dam age the s t rnc ture they a re app l ied to .The fo l low ing expenm enta l equ ipm e nt i s needed to ca r ry ou tan analy s is of ul t rasonic s ignals : an ul t rasonic uni t , a t rans-duce r s e t , a un ive rsa l wav eform ana lyze r , a pe rsona l com -pute r and a p lo t te r . 1 Me thodo logy i s ve ry s im ple and con-s i st s o f de tec t ing the va r ia t ions underg one by an u l t ra son ics igna l em i t t ed by m a te r ia l s unde r t e s t ing . In fac t , u l tra son icwaves t rave ling th rough non hom ogen eous m a te r ia l s such a sconcre te a re m o dula ted , re f l ec ted and a t t enua ted bo th by in -te rna l de fec t s and by aggrega te m a t r ix in te r faces , and thesephenom ena inc rease in the p resence o f dam age due to ex-te rna l loads . Com plex m ic roc rack ing proces ses in conc re tea re o f ten caused by e x t rem e s t res ses, v io len t im pac t , fa t iguecondi t ions and so fo r th . T he m ic roc racks spread , b ranch andjo in each o the r , fo rm ing w ide c racks tha t l ead , even tua l ly , tothe loca l co l l apse o f the m a te r ia l .The deve lopm ent o f m ic roc racks in conc re te was re -cen t ly inves t iga ted us ing ex pe r im ents tha t in t roduced dam -age coe f f i c ien t s re la ted to the m e asurem en t o f the quan t i tyem ploy ed; 1-2-3 i .e . , veloci ty, ampli tu de an d frequ ency. Inth i s pape r, the da m ag e coe f f i c ien t i s ob ta ined b y m e a n s o fthe t im e-f requency d i s t r ibu t ion adopted and by a s sum ing are fe rence s igna l ob ta ined by p lac ing the two p iezoe lec t r i ct ransduce rs used in co n tac t wi th the conc re te cube .T im e-f req uency d i s t r ibu t ions a re wide ly app l ied in the de -s ign and eva lua t ion o f aud io sys tem s , and were in t roducedto so lve p rob lem s a r i s ing f rom the F our ie r t rans form F T) ,R. S . OHvi to i s Assoc ia t e Professor and L . Surace i s a Doctora lS tuden t De-par tm en t o f S t ruc tura l Eng ineer ing Un iver s i t y o f Ca labr ia 87036 ReudeItaly.F ina l man uscr ip t r ece i ved : Ma rch 21 1997 .

which proces ses t rans ien t s igna l s wi th a d i f fe ren t f requencyconten t a t d i f fe ren t t im es , a s in the case o f u l t ra son ics ignals . 5-6-7-8-9 The nons ta t ionari ty usua l ly resul ts e i therf rom va r ia t ions in the p ropaga t ion m edia , f rom channe l cha r -ac te r va r ia t ions o r f rom nonl inea r e f fec t s in a sys tem . Un-t i l recent ly, these s ignal propert ies were described mainlyin the f requency dom ain us ing va r ious quas i - s t a t iona ry a s-sum pt ions . The inadeq uacy of such desc r ip t ions has l ed re -s ea rchers to sugg es t adopt ing s igna l represen ta t ions tha t usebo th t im e and f requency as indepen dent va r iab le s .In the p resen t pape r , the t im e f requency used i s the c ros s-am bigu i ty func t ion CAF ) . An a lgor i thm based on the com -puta t ion o f the d i s c re te Zak t rans form of the u l t ra son ic s igna lat the receiver-t ransducer output is a lso given. T his a lgori th mperm i t s the com puta t ion o f the coe f f i c ien t s o f double sumexpans ion o f the t im e-va ry ing t rans ien t s igna ls em i t t ed bya conc re te spec im en sub jec ted to com pres s ive load ing , andtoge the r wi th the CA F a l lows the de f in i t ion o f a coe f f i c ien tto s tudy dam age to conc re te spec im ens under t e s t. The t rendof the dam age coe f f i c ien t adopted show s the d i f fe ren t phasesof the dam age proces s and , in pa r t i cu lar , d i s t ingu i shes theevolu t ion and deve lopm ent o f d i f fe ren t f ractu re m echanism ssuch as opening and s l iding.Theory

The t im e-f requency d i s t r ibu t ions m ap a s igna l f rom thet im e dom ain in to the two-d im ens iona l t im e-f requency do-m ain , and were in t roduced to so lve p rob lem s of s igna l s tha thave a d i f fe ren t f requency con ten t5-6-7 at diff ere nt times.The F o ur ie r t rans form , de f ined by

O0X f ) = x t ) e - i2 ~ X f t d t , 1 )

gives in form a t ion on the f requenc ies p resen t in the s igna l bu tdoes no t g ive in form a t ion on the evo lu t ion o f the f requenc iesor the cor re sponding am pl i tudes . F o r exam ple , cons ide r F ig .1 a ), w hich i s cha rac te r i zed by th ree d i f fe ren t f requency com -ponents , wi th am pl i tudes va ry ing in t im e . I t can be s een howthe F our ie r trans form of F ig . l b ) ind ica te s a ll the f requenc iespresen t , and how the t im e-f requen cy dom ain , ob ta ined f romthe c ros s -am bigu i ty func t ion in F igs . l c ) and l d ) whichwi l t be de f ined l a te r in th i s pape r) , shows the f requenc iestaken f rom F T and the i r occur rence and in tens i ty in t im e .Le t ano the r f requency com p onent be added . The va r ia -t ions in the spec t rum and in the t im e-f requency represen ta t ion

E x p e r i m e n t a l M e c h a n i c $ 9 355

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I , 9 , [

0 ~ ~

o li~

0 20 40 60 80 100[See]

I . . . .

0 I 2 3 4 5[rlzl

' ' iIS ~l

Fig. 1--(a) Time domain signal with three frequency compo-nents, (b) signal spectrum, (c) mesh of the cross-ambiguityfunction, (d) contour of the cross-ambiguity function

I , , , , 1.5 , , ,

9 o~ ", ' ~;'~'~'~, ~ .....l0 2 4 0 6 0 g O 100 0 I J 3lSed [rM

5

2 4..0K ~ . . . .20 4 0 6 0 gO

[S~c]

Fig. 2--(a) time domain signal with four frequency compo-nents, (b) signal spectrum, (c) mesh of the cross-ambiguityfunction, (d) contour of the cross-ambiguity function

can be ob served in Fig. 2. It ma y be noted how signal vari-a t ion is much more evident in the cross-ambigui ty funct ion[Fig. 2(d)] , which gives a new frequency component , than inthe original Fourier t ransform, in which the ampli tude of thethi rd frequency com ponen t is only s l ight ly increased. There-fore, where the re are nonstatio nary signals, i t is necessary touse t ime-frequency representat ions .A previous paper demonstra ted that the cross-ambiguityfunct ion furnishes more sat is factory resul ts when s tudy-ing damage to concrete than other distributions, such asthe short - t ime Fourier t ransform (STFT) and the Gaborrepresentations. 4 The cross-am biguity function o f a signalf t ) relative to a referenc e function g is defined by

A f , g ) u , v ) = f f t ) g * t - v ) ex p 2 7 z iu t ) d t u , v ~ R ,2)

where g* ( t - v) i s the complex conjugate of the funct iong. The space of funct ions f and g forms the Hi lbert spaceL2( R). It is assumed that the signals unde r consideration

have finite energy, with I g[ 12 > 0. If the fun ctionguy t) = g t - v) exp 2~xiut) (3)

is the set of base fu nctions that represe nt the signal f t ) , eq(2) can be written in the following form:A f , g) u, v) = f , guo), (4)

where ( , ) denotes the inner product of the functions . Bymeans of eq (3) , the CAF in the t ime-frequency plane canbe calculated by the internal product between the funct ion fand the function guy. The CAF can therefore be seen as at ime-frequency general izat ion o f the Fourier t ransform.

Via the Planchere l theorem, 5 we are able to write the fol-lowing:

= f f [a f, g) u, v)l 2 du dr, (5)Igll2Zllfll 2 i / t /R Rwhich, up to the positive scale factor [[gll 2 > 0, equates theenergy of the s ignal wi th the energy of the s ignal wi th theenergy o f the cross-ambigui ty funct ion:

A f, g) u, v) = 0 ==~ I lf lh = 0 u, v ~ g . (6)Consequent ly, the CAF of the ul t rasonic s ignal uniquely de-fines the signal. T he discrete for m of eq (4)is

A f, g) m, n) = f, gmn) m, n ~ Z, (7)where Z is the field of integer numb ers. Equa tion (7) can becalculated by us ing the Z ak t ransform defined by

Z f ) x , y ) = E f y + n) exp -2rt inx) 8)n~---OO- c~ < x, y < oo.

By us ing the propert ies of the Zak t ransform, the productZ f ) x , y ) . Z* g) x , y ) period ic in x and y is obtained andcan be dev eloped in the Fourier series.

+oo +c~Z f ) . Z* g) = E E bmn exp -27 [i nx Jc- my).

l l~ O Q m ~ O Q 9)The coefficients bran are obtained by inverting eq (7). Inparticular, applying the Zak transform to base functions gmngives

bran = Z f) x, y), Z gmn) X, y)). (10)In Ref. 8, i t was shown ho w the Zak transform Z is an isom-etry from the Hi lbert space LZ(R) onto the Hi lbert spaceL2o C), and we can ident i fy the bran coeff ic ient from eq (10) .Equation 9 shows that the Zak transform is an isometry,and it is possible to determin e the coefficients bmn as

bran = f, gmn) m, n ~ Z. (11)For digi ta l computat ion, we need a f inite number of coeff i -cients bran. Then, eq (9) becomes

N - 1 M- IZ f ) . Z* g) = ~ ~ f , gmn) exp --2rti nx + my).

n=0 m=0 (12)

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TABLE 1--TY PICA L PROPERTIES OF MATERIALSUnit Weight

Serie s (kN/m 3) Water-Cement RatioiiCemen t Failure Stress Numb er of

(kg/m 3) (M Pa) Cub es TestedConcrete A 24.0 0.5 360 44.75 25Concrete B 23.5 0.4 400 69.40 28Mortar C 21.0 0.5 360 8.70 22

Fig. 4--U ltraso nic measuring apparatus (scheme)

SPECIMEN Concrete ype"C")MESH 0 [MPa]J

r

SPECIMEN Concrete ype "C")MESH 8 70 MPa]

SPFX]~AISN Conc~te ype A ES H 0 [MPa]

f [l~,] ~ [ [ ~ e ]

x1 CON' I 'OUP. 4 CONT OUR

1 ~ ~ 4 5 i 2 3 ~ 5[SEe] xl@ [SEq xlO q

Fig. 5 Series A specimen mesh and contour of the CAF atzero and maximum stress

I SPECIMEN (Co~re le type B ) IVlESH 69.40 [MPa]

x10 C O N T O U R xt0 C 0 i ~ D R

[SEG] xl04 [SECI xl04F ig . 6 - -S e r i e s B spe c im en- -m e sh and con tour of the CAF a tze ro and m axim um s t re s s

J LA--I s~c]

Fig. 7--Serie s C --me sh comparison of the CAF at zero andmaximum stress

i ii . Neglig ible diss ipat ion for X > > d; i . e ., the harmo nicdoes not show the presen ce of any defect .

Bearing in mind that energy losses due to dispers ion arenegligible in defect s izes smaller than 1/100 of the wavelengthbut beco me re levant for s izes greater than 1/ t 0 , and that som eexper im ents have shown tha t the m axim um s ize reached bymicrocr acks is about 4 mm , a reference wav elength (Lr) of40 m m was s et . Th i s is the m axim um wave leng th capab leof unde rgo ing re levan t d i s s ipa t ion when a m axim um -s izedcrack is encountered. In consequence, only energy associ-a ted with harmonics of wavelengths smaller than or equalto the reference ones wil l be taken into account . This canbe done becau se i t is poss ible to ident ify these harm onics inthe t ime-frequen cy plane. Indeed, i f the fol lowing re la t ion isconsidered,

the p rogres s ive d i s appea rance o f s lower ha rm onics was ob-served. The A and B series specim ens had a natural frequencyof 10 kHz, and those o f C series had a natural frequenc y of 8kHz .The energy diss ipat ion is s t r ic t ly connected to the wave-length ~ . and to the defect s ize d according to the fol lowingconsiderat ions :

i . Com plete diss ipat ion for X < < di i. Consid erable diss ipat ion for X = d

l constf = - - , (18)X 9 twhere f is the frequency of the s ignal , l is the specimenlength and t is the arr ival t ime, then eq (19) represents anequila tera l hyperbole that a l lows us to obta in two domainsin the t ime-frequ ency plane characterized by k < Lr and)~ > Zr. By considering the domain L < ~r for the presentanalys is , we observe a progress ive diminution of the s lowerha rm onics . Th i s phenom enon occurs because the s lower ha r -monics have waveleng ths smaller than the crack dimensions ,

358 9 VoI. 37, N 3, September 1997

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xl04

1 2 7[aN]

SPECIMEN a, f to type C9como 8.70

5 -- : L.- ....... .

0 ._Z . . . . . . . : :L.. . . . . . . .

como 0

, ~ - -~ . .. . 7. . . . . . C?~. .~

4 5xlO

1 2 3 4[~l xl0~

Fig. 8--Se ries C--c onto ur comparison of the CAF at zero andmaximum stress

1 00:AiIA till SERIES A /~1 -u- on ~RII~S rl /7 /

0o

0 40

i ~ 0b'~'6.~6 6.;,b' '6.bb lik'0' iNL o a d / L o a d m a x

Fig. 9- -D am ag e coefficient comparison betwee n the threeseries of tested specimens

and for this reason a complete dissipation of the energy ofthe harmonics an d their disappearance in the time-frequency

representation occurs. Based on these considerations, andfrom the experimental investigation carried out, we define asa significant parameter for the evaluation of con crete dama ge(D) the followin g ratio:

Vo V7D = Vo Vmax (19)

where V o and I,~ are the volu me values, as alre ady explaine d,of the cross-ambiguity function for the unloaded and fo r theith loading level, respectively. Th e trend of the da mage co-efficient D is given in Fig. 9 and shows the sensitivity ofthe ultrasonic procedure described in d etecting and assessingmicrocracking processes within concrete material from thestart of the loading process.cknowledgment

This work was funded by the Italian Ministero perl Universit~t e la Ricerca Scien tifica e Tecn olog ica (MU RST40 percent, Com itato 0.8 E.F. 1995).References

1. Daponte, P., Maceri, E and Ofivito, R.S., Frequency-Do main Anal-ysis of Ultrasonic Pulses for the Measure of Damage Growth in StructuralMaterials, Ultrasonic Symposium IEEE, Decembe r, 1113-1118 (1990).2. Suaris, W. and Fernando, V., Detection o f Crack Growth in ConcreteFrom Ultrasonic Measuremen t, Materiaux et Contructions, 20, 214-220(1987).3. Dhir, R.K. and Sanghia, C.M. , Development and Propagation ofMicrocracks in Plain Concrete, Material et Costructions, 7, 17-2 3 (1974).4. Fa zio , G., Molinaro, A. and Olivito, R.S., Le RappresentazioniTempo-Frequenza Applicate Allo Studio D el Danneggiamento Del Calces-truzzo, XXIII Convegno Nazionale AIAS-24, September, Rende-Cosenza(1994).5. Auslander, L., Gartner, L C. and To timeri, R., Th e Discrete ZakTransform Application to Time-Frequen cy Analysis and Synthesis o f Tran-sient Signals, IE EE Trans. Signal Processing, 39 (4), 825 ~3 5 (1991).6. Aaslander, L. and Tolimeri, R., Compu te Decimated Cross-AmbiguityFunction, IEEE Trans. on Acoust., Speech, Signal Processing, 36 (3), 359-364 (1988).7. Fitting, D. W. and Adler, L., Ultrasonic Spectral Analysis for Nonde -structive Evaluation, Plenum Press, New York (1981).8. Janssen, A.J.E.M., The Zak Transform of Some Counter Examples inTime-Frequen cy Analysis, IEE E Trans. Inform. Th eory, 38 (1), 168-171(1992).9. Nassbaume r, H.J., Fast Fourier Transform and Convolution Algo-rithms, Springer-Verlag, Berlin, Heidelberg, New York (1981).10. Hlawatsch, E and Boudreaux-B artels, G. E, Linear and QuadraticTime-Frequency Signal Representation, IEE E SP Magazine, April (1992).1l. Stein, E.M. and Weiss, G., Introduction to Fourier Analysis on Eu-clidean Spaces, Princeton U niversity Press, Princeton, New Jersey (1973).12. Sugiura, M., Unitary Representations and Harmonic Analysis, Ko-danska LTD, Tokyo (1990).

Expe r i r nen ta l Mechan i cs 9 359