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"�21 |x]@=Q� OR u}tND � T} QD=t |=yx}=QO `tH =@

Di�erence =mXi=1

nXj=1j�(i; j)j (21)

|rN=OD |=ysQD p} w� Qvo} w `} RwD "CU= xO=OV}=tv =Q|i}a[ Q=}U@ OQmrta w xDi=}=@ xOWx@U=Lt Ot=U@� u=tR |=yuqB =@ j@=]t "CU= xOQm O}rwD x=@DW= x@ =Q |v=w=Qi|=yOt=U@ QO 'K}LY |=yOt=U@ Q@xwqa p=vo}U |SQv= p} w� Qvo} w `} RwD R= xO=iDU=QO p=vo}U '|SQv= |i}] |r=oJ T=U=Q@ xm 'CU= xOW xO=O V}=tv R}v |Qo}O`} RwD \UwD xOWx@U=Lt Ot=U@� u=tR uqB "CU}v |SQv= RmQtD |=Q=O =yOt=U@ u;'Qo}O CQ=@ax@ "CU= u=tR QwLt pw] QO xS} wx@ |oOWVNB |=Q=O xQ=ov h}]xU}=kt Ot=U@ |xRwL =} u=tR |xRwL =@ xOt;CUOx@ Ot=U@� u=tR uqB xm|t=ovy=} u=tR |xRwL QO p=vo}U |xvt=O uDiQoCOW R= V}B xm OwW|t xOy=Wt 'OwW|wJ `} RwD "CU= V}=Ri= x@ wQ Ot=U@� u=tR |xRwL QO p=vo}U |xvt=O 'Ot=U@|QDtm |oOWVNB |=Q=O xQ=ov h}] `} RwD =@ xU}=kt QO xmv}= OwHw =@ Rt=}r} w�'xDi=}Vy=m |rN=OD `} RwD w xOWQ=wty p} w� Qvo} w x@W `} RwD =@ xU}=kt QO 'CU=`} RwD R= xOt;CUOx@ Ot=U@� u=tR uqB "CU= xOQm O}rwD |QDW}@ |rN=OD |=ysQDxQ=ov h}] `} RwD R= pY=L Ot=U@� u=tR uqB =@ x@=Wt xOWQ=wty p} w� Qvo} w x@W|wJ `} RwD =@ xU}=kt QO "CU= xDi=}Vy=m |oOWVNB Q=Okt xm Cw=iD u}= =@ 'CU=|=ysQD xOWQ=wty p} w� Qvo} w x@W `} RwD =@ xOWx@U=Lt Ot=U@� u=tR uqB 'Rt=}r} w�xOt;CUOx@ Ot=U@� u=tR |=yuqB QO CkO =@ "CU= xOQm O}rwD =Q |QDtm |rN=ODXNWt 'u; |x} Qwi p}O@D w pB |x}=B MU=B p=vo}U =@ |rY= |=ysQD j@=]D wp=vo}U |}x_Lr C}Y=N V}=tv QO xS} wx@ xDi=}Vy=m |rN=OD `} RwD xm OwW|t|=Q@ `} RwD u}= R= j}kLD |xt=O= QO =Pr 'CU= xOQm x�=Q= OwN R= |QDy@ OQmrta

"OW Oy=wN xO=iDU= Qw]k pB |vD@ |=yx}=B MU=B |=yp=vo}U VR=OQB

?}U; X}NWD "5w Ot=U@ |xRwL QO =Q p=vo}U |SQv= u=wD|t Ot=U@� u=tR |=yp}rLD ltm x@QO |SQv= uR=wD =@ s}kDUt |x]@=Q '?}U; "OQm xOy=Wt u=tRsy Qw]x@ u=tR |xRwL|xRwL w Ot=U@ |xRwL QO p=vo}U l} |SQv= `} RwD xJv=vJ u}=Q@=v@ "OQ=O xR=Uu=tR w 'pLt 'COW =@ =Q `} RwD u}= u=wD|t 'OW=@ QwOkt u=tRsy CQwYx@ u=tR

"OQm xDU@sy ?}U; O=O} wQh}] w jx(t)j2 |}x_Lr |SQv= 'P (t; !) Ot=U@� u=tR |SQv= `} RwD l} |=Q@

"OvwW|t h} QaD 16 w 15 |=yx]@=Q R= xO=iDU= =@ jS(!)j2 |SQv= |r=oJ

jx(t)j2 =+1Z�1

P (t; !)d! (15)

jS(!)j2 =+1Z�1

P (t; !)dt (16)

CUOx@ 17 |x]@=Q CQwYx@ p=vo}U pm |SQv= 'OwW =[Q= |RQt \}=QW xJv=vJ%O};|t

E =+1Z�1

P (t; !)d!dt (17)

xt=O= QO xm CU= xOW xO=iDU= ?}U; X}NWD |=Q@ VwQ xU R= j}kLD u}= QO"CU= xOW xO=O u=Wv R}v ?}U; pLt u}}aD QO l} Qy G}=Dv w |iQat

44

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i j@=]D =@ VwQ u}= QO "CU= xOW xO=iDU= CWov= QF= w =OY |}=U=vW QO xOQDUoQO "OwW|t |}=U=vW Q_vOQwt |wor= 'xOWxQ}NP |=ywor= =@ O}OH C=aq]= u}@xO=iDU= ?}U; X}NWD |=Q@ |k@=]D |=yVwQ QO KQ]t sD} Qwor= wO R= j}kLD u}=

"CU= xOW16|oDU@ty "1"2"5

Q=Okt "OwW|t x@U=Lt (Cij) wor= wO u}@ |oDU@ty?} Q[ '|oDU@ty p}rLD QOCQwY QO "OvDUy u=Um} sy =@ ,qt=m wor= wO xm CU= |vat u}O@ 1 |oDU@ty�1 OOa xJv=vJ w CU= wor= wO u}@ pt=m Cw=iD R= |m =L 'QiY OOa |x@U=LtQDl}ORv 1 x@ xH}Dv xJ Qy "OvDUy Qo}Om} Twmat ,=kr]t wor= wO 'OwW pY=LKQW x@ xO=iDU=OQwt |x]@=Q "CU= wor= wO u}@ QDl}ORv j@=]D |xOvyOu=Wv 'OW=@

[17w10]%CU= 22 |x]@=Q

Cij =

nPk=1

(Si(k)� �Si)(Sj(k)� �Sj)snPk=1

(Si(k)� �Si)2s

nPk=1

(Sj(k)� �Sj)2(22)

T=U=Q@ |oDU@ty VwQ R= xO=iDU= =@ xOWO=H}= ?}U; X}NWD Q=Owtv "26 pmW"|vD@ |=yx}=B |q=@ QO xOWC@F MU=B

T=U=Q@ |oDU@ty VwQ R= xO=iDU= =@ xOWO=H}= ?}U; X}NWD Q=Owtv "27 pmW"|vD@ |=yx}=B |xv=}t QO xOWC@F MU=B

|U} QD=t p[=iD VwQ R= xO=iDU= =@ xOWO=H}= ?}U; X}NWD Q=Owtv "24 pmW"|vD@ |=yx}=B |q=@ QO xOWC@F MU=B T=U=Q@

|U} QD=t p[=iD VwQ R= xO=iDU= =@ xOWO=H}= ?}U; X}NWD Q=Owtv "25 pmW"|vD@ |=yx}=B |xv=}t QO xOWC@F MU=B T=U=Q@

xv=}t R=QD wO QO |rm |=yQ}}eD Q=Okt |x@U=Lt j} Q] R= ?}U; OwHw p=tDL= OYQO"O};|t CUOx@ QDnQR@ OOa ?ULQ@ R=QD Qy QO =yu; uOQmp=tQv w =yx}=B |q=@ w

"OwW|t |}=U=vW xO}O?}U; |x}=B 'G}=Dv |xU}=kt =@ TBU|@=} RQ= VwQ u}= OQmrta 'xOWx@U=Lt Ot=U@� u=tR |=yuqB R= xO=iDU= =@ p=Lj@=]t "CU= xOW xO=O u=Wv 25 w 24 |=ypmW QO C=@U=Lt G}=Dv "CU= xOWxO=iDU= =@ G} QEQwv w =D} QB=twr 'wOv=vQiuU |xrRrR OQwmQ 3 Q}F-=D CLD '=ypmW u}=|x}=B QO ?}U; OwHw p=tDL= '|O=yvW}B |xOWKqY= |U} QD=t p[=iD VwQ R=?}U; OwHw p=tDL= xm CU= |r=L QO u}= "CU= xOt; CUOx@ �100 =@ Q@=Q@ swUOwW|t xOy=Wt =Pr "CU= xOW x@U=Lt QiY =} w lJwm Q=Okt =yx}=B Q}=U QO'Ow@ xOW O=H}= swU |x}=B QO xm |@}U; X}NWD QO '|O=yvW}B sD} Qwor= u}= xm

"CU= xOQm pta jiwt ,qt=m CQwYx@

|k@=]D |=yVwQ "2"5j@=]D '?}U; X}NWD w |U=vWwor= |=Q@ xOWxDN=vW Q=}U@ |=yVwQ R= |m}=y|vLvt '\=kv� Cr=L wO u}@ Cy=@W u}}aD |=Q@ j@=]D |rm Cr=L QO "CU=Qw]x@ wor= j@=]D VwQ "OwQ|t Q=mx@ 'OvDUy |v=Um} wv |=Q=O xm �=ypmW w

45

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|}=U=vW|xrY=i VwQ R= xO=iDU= =@ |O=yvW}B sD} Qwor= '29 w 28 |=ypmW j@=]t

x�=Q= u; pLt |}=U=vW w ?}U; u}}aD QO =Q |@ wr]t Q=}U@ OQmrta xv}tm |a@ Qt"CU= xOQm

|U} QD=t p[=iD ?}U; X}NWD VwQ wO Qy 'xOt;CUOx@ G}=Dv x@ xHwD =@"OvDUy VN@C}=[Q OQmrta |=Q=O `@ Qt |xv}tm |xrY=i w Ov=xOW KqY=

|Q}oxH}Dv "6Ot=U@� u=tR `@=wD R= xO=iDU= =@ |U} QD=t p[=iD |xOWKqY= VwQ j}kLD u}= QO|=Q@ "CU= xOW O=yvW}B =ypB |x}=B QO u; pLt w ?}U; |}=U=vW |=Q@ |a@ Qt|a@ Qt Ot=U@� u=tR `@=wD w QDt 448 pw] x@ Qw]k pB pOt R= |O=yvW}B VwQ O}}-=D'p} w� Qvo} w |=y`} RwD 'lHwt p}O@D h}] 'x=Dwm u=tR |x} Qwi p}O@D h}] pt=WVR=OQB |=Q@ 'xDi=}Vy=m |rN=OD w xOWQ=wty p} w� Qvo} w x@W 'Rt=}r} w� |wJxDi=}Vy=m |rN=OD Ot=U@� u=tR `@=D w xOW xO=iDU= =ypB |}xRQr MU=B |=yp=vo}UxOW |}=U=vW =ypB |}xRQr MU=B |=yp=vo}U VR=OQB |=Q@ ?wr]t `@=D |xrRvtx@

"CU=VwQ wO =@ x=Qty |O=yvW}B |U} QD=t p[=iD VwQ '?}U; |}=U=vW |=Q@'|O=yvW}B sD} Qwor= |@=} RQ= Qw_vtx@ "Ov=xOW xU}=kt Qo}Om} =@ w xO=iDU= |k@=]D|xvwtv |=Q@ 'CU= |}xS} w |=y|oO}J}B w xHwDp@=k O=a@= |=Q=O xm Qw]k pB'pB OwHwt `[w =@ j@=]t |r}rLD pOt CN=U CyH "CU= xOW ?=NDv= |}xR=UxOt; CUOx@ hrDNt |=yV}=tR; j} Q] R= <=[a= X=wN w Kr=Yt C=YNWt

"CU=w q=@ QO |}=yxU=UL ?Yv =@ 'xO=iDU= p=LQO w xOWxDN=U pB l} |=Q@xm |]=kv QO =Q xR=U |}xRQr MU=B |HwQN |=yp=vo}U u=wD|t uwDU Qy \Uww =yxRQrV}B wkw u=tR QO |v=tR |xJN} Q=D CQwYx@ 'CU= xOW ?Yv xU=ULxDi=}Vy=m |rN=OD `} RwD R= xO=iDU= =@ TBU w OQm CW=OQ@ w C@F |rY= |xrRrR|HwQN |}xRQr |=yp=vo}U u}= '|O=yvW}B |xOWKqY= T} QD=t p[=iD VwQ w\ki |O=yvW}B VwQ xmv}= x@ xHwD =@ =Pr "OQm |}=U=vW =Q ?}U; pLt w VR=OQB =QpOt h} QaD x@ |R=}v ptaQO 'CU= |HwQN |}xRQr |=yp=vo}U VR=OQB T=U=Q@

"CU}v |r}rLD|}xRQr |=yp=vo}U VR=OQB |x}=B Q@ VwQ u}= T=U= xm =Hv; R= u}vJtyOvv=t� Qo}O |=yVwQ R= |Q=}U@ hqNQ@ |O=yvW}B VwQ QO 'CU= |HwQN|Q}oxR=Ov= x@ |R=}v '�"""w =tQ; 'Cr=L |=[i pOt p}@k R= l} QDt=Q=B |=yVwQ

"CU}v R}v |OwQw |wQ}v=Q xR=U ,=OOHt xrRrR wkw R= TB '?}U; X}NWD |=Q@ =yVwQ R= |NQ@ QOxOWC@F |=yp=vo}U VR=OQB =@ w OvyO|t Q=Qk l}vwtQ=y |Q=@H= l} QLD CLDQo}O |m} =t= "OwW|t s=Ok= ?}U; pLt X}NWD x@ C@Uv l}vwtQ=y l} QLD R=xrRrR wkw u=tR QO xOWC@F |=yp=vo}U OwN R= xO=iDU= VwQ u}= |=yC} Rt R=|Q=@H= V=aDQ= QwDwt |=yx=oDUO R= xO=iDU= =@ pB |Q=@H= l} QLD x@ |R=}v|@ w

"CU= |}xRQr ?}U; X}NWD |=Q@p[=iD |=yVwQ '=yx}=B |q=@ w xv=}t R=QD wO QO xOWs=Hv= C=@U=Lt T=U=Q@Q}F-=DCLD swU|x}=B QO =Q?}U; OwHw 'xv}tm|a@ Qt |xrY=i w xOWKqY=|U} QD=tp=tDL= |x@U=Lt QO =]N Q=Okt u} QDW}@ "Ov=xOQm |}=U=vW |@ wNx@ xrRrR OQwmQ 3|=yp=vo}U x@ \w@ Qt Ot=U@� u=tR |=yT} QD=t OQ@ Q=m =@ =yx}=B QO ?}U; OwHw

|=ywor= |Q=OQ@ Q}O=kt Sj(k) w Si(k) 'wor= QO |Q=OQ@ \=kv O=OaD n 'x]@=Q u}= QOC=@U=Lt "OvDUy j w i|=ywor= u}ov=}t �Sj w �Si Q}O=kt wCU= k |x]kv QO j w iQO u; pLt u}}aD w ?}U; X}NWD |=Q@ |oDU@ty VwQ R= xO=iDU= =@ xOWs=Hv=

"OvDUy xOy=Wt p@=k 27 w 26 |=ypmWhqDN= |xOvyOu=Wv QDlJwm OOa |oDU@ty VwQ QO 'OW xDio xm Qw]u=typLt |xOvyOu=Wv QDlJwm |=yQ=Owtv 27 w 26 |=ypmW QO u}=Q@=v@ "CU= QDW}@X}NWD |=Q@ |oDU@ty VwQ xm OwW|t XNWt G}=Dv |UQQ@ =@ "OvDUy ?}U;

"CU= xDW=Ov |@ wr]t OQmrta ?}U; pLt17�LSD� xv}tm |a@ Qt |xrY=i "2"2"5

CWov= QF= w =OY X}NWD |=Q@ xOQDUo Qw]x@ �LSD� xv}tm |a@ Qt |xrY=i[17w10]%OwW|t h} QaD 23 |x]@=Q CQwYx@ VwQ u}= "CU= xOW xO=iDU=

dij = (nXk=1

(Si(k)� Sj(k))2)0 5 (23)

QDW}@ |oDU@ty |xOvyOu=Wv 'OW=@ QDtm xv}tm |a@ Qt |xrY=i Q=Okt xJ QyQ}O=kt Sj(k) w Si(k) 'wor= QO |Q=OQ@ \=kv O=OaD n 'x]@=Q u}= QO "CU=wO u}@ xv}tm |a@ Qt |xrY=i dij "OvDUy k |x]kv QO j w i |=ywor= |Q=OQ@Cy=@W |xOvyOu=Wv 'OW=@ QDtm xv}tm |a@ Qt |xrY=i Q=Okt xJ Qy "CU= wor=xOW xO=O V}=tv 29 w 28 |=ypmW QO C=@U=Lt G}=Dv "CU= wor= wO QDW}@

"CU=

|xv}tm |xrY=i VwQ R= xO=iDU= =@ xOWO=H}= ?}U; X}NWD Q=Owtv "28 pmW"|vD@ |=yx}=B |q=@ QO xOWC@F MU=B T=U=Q@ `@ Qt

|xv}tm |xrY=i VwQ R= xO=iDU= =@ xOWO=H}= ?}U; X}NWD Q=Owtv "29 pmW"|vD@ |=yx}=B |xv=}t QO xOWC@F MU=B T=U=Q@ `@ Qt

46

"""w|Ot

L==[Q

O}tL'wHW

v=OO=yQ

i =@ =Pr "CU= xOt; CUOx@ �14 'xv}tm |a@ Qt |xrY=i w xOWKqY= |U} QD=t=ypB |HvUCtqU |=Q@ Ov=wD|t wor= u}= '|O=yvW}B sD} Qwor= |oO=U x@ C}=va

"OwW xO=iDU=

w �6 'xOWKqY= |U} QD=t p[=iD VwQ R= xO=iDU= =@ '=yx}=B |q=@ QO xOWC@FQ=Okt u; Q@xwqa "Ov=xOW x@U=Lt �7 'xv}tm |a@ Qt |xrY=i VwQ R= xO=iDU= =@p[=iD VwQ wO Qy QO =yx}=B |xv=}t QO xOWC@F |=yp=vo}U R= xO=iDU= =@ =]N

=yCWwv=B1. auto-regressive moving average2. time-frequency representation(TFR)3. spectrogram4. scalogram5. Wigner-Ville distribution(WVD)6. Choi-Williams distribution(CWD)7. reduced interference distribution(RID)8. smoothed pseudo Wigner-Ville distribution(SPWVD)9. short time fourier transform (STFT)10. spectrogram11. wavelet transform12. mother wavelet13. Morlet14. Hilbert15. Cauchy principal value16. correlation17. least square distance (LSD)

�References� `@=vt1. Doebling, S., Farrar, C., Prime, M. and Shevitz, D.

\Damage identi�cation and health monitoring of struc-tural and mechanical systems from changes in their vi-bration characteristics: A literature review", LA-13070-MS, Los Alamos National Laboratory, USA (1996).

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47

"""|vD@|

=yx}=BQO

?}U;X

}NWDw

xYNWt

|}=U=vW23. Track & Technical Structures Department of Iran Rail-

ways, \Preliminary design of seismic retro�t of ghotourrailway bridge", I.R. of Iran, Tehran (In Persain) (2009).

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27. Beskhyroun, S., Oshima, T., Mikami, S. and Tsubota,Y. \Structural damage identi�cation algorithm based onchanges inpower spectral density", J. of Applied Mechan-ics, 8, pp. 73-84 (2005).

28. Beskhyroun, S., Oshimay, T. and Mikami, S. \Wavelet-based technique for structural damage detection", Struc-tural Control and Health Monitoring, 17, pp. 473-494(2010).

48

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h SYSTEM IDENTIFICATION ANDDAMAGE DETECTION OF BRIDGECONCRETE PIERS, USING TIME-FREQUENCY REPRESENTATIONAND A NEW MODIFIED MATRIXSUBTRACTION METHODF. Daneshjoo(corresponding author)danesh [email protected]. [email protected]. of Civil and Environmental EngineeringTarbiat Modares UniversityG. Mobtakerha [email protected] & Technical Structures Department I.R.Iran Railways

Sharif Civil Engineering JournalVolume 29, Issue 2, Page 37-48, Original Article

c Sharif University of Technology

� Received 8 March 2011; received in revised form 3 July 2011;accepted 5 October 2011.

AbstractIn this research, a new modi�ed matrix subtractionmethod is proposed using square time-frequency repre-sentations to detect bridge pier damage and its loca-tion. The new proposed method is con�rmed, using theseismic response of the 448 meter long Ghotour Bridgemodel, and compared with correlation and least squaredistance methods. Time frequency plans of damagedand non-damaged bridge matrix elements are calculatedusing reduced interference distribution. The di�erencematrix is calculated by subtraction of corresponding ma-trix elements. The possibility of an existing damage in-dex is calculated by summation of all elements of di�er-ence matrices and then normalizing them with a maxi-mum value. Linear time history analyses and earthquakeacceleration records of San Fernando, Loma Prieta andNorthridge earthquakes are used for seismic responseanalysis.In previous response analyses of the Ghotour Bridge, asstated in reference [23], the third pier from the left is con-sidered more vulnerable to seismic damage. Therefore,for the purpose of this study, a reduction of thirty per-cent in the sti�ness of that pier is considered, to simulatethe seismic damage in the damaged analytical model. Itis shown that the proposed method could satisfactorilyidentify the damage location. For the seismic responseof the top of the bridge piers, the maximum error in lo-cating the damage is 6 percent, while, for the seismicresponse of the middle of the bridge piers, it is 14 per-cent.

The time-frequency representations used include: theshort time Fourier transform spectrum, wavelet trans-form spectrum, Wigner-Ville Distribution, Choi-Williamsdistribution, smoothed pseudo Wigner-Ville distributionand reduced interference distribution, which are �nallyidenti�ed as optimal performance time-frequency rep-resentation for bridge seismic response signal process-ing. Reduced interference distribution and Wigner-Villedistribution are both in the Cohen class, but reducedinterference distribution methods are more appropriatefor processing seismic bridge transient nonstationary re-sponse signals. Time-frequency planes have been cal-culated and dynamic speci�cations of the system havebeen estimated.The proposed algorithm is a seismic output-only method.Therefore, it has the advantage of not needing to de�nethe bridge analytical model and measuring seismic inputloading, and, also, not needing to use harmonic forced vi-bration analysis after earthquake occurrence for bridgeseismic damage detection, as is general in some othermethods.

Key Words: Bridge system identi�cation, damage detec-tion, time-frequency representation, modi�ed matrix sub-traction.

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