A D A M A G E A N A L Y S I S O F S T E E L B E A M S

C E S A R E D A V I N I \O G A T T I ^ and A N T O N I N O M O R A S S I ' ^Università di Udine, Istituto di Meccanica Teorica ed Applicata,

33100 Udine, Italy ^ISMES Spa, 24100 Bergamo. Italy

(Received: 3 December 1991; accepted i n revised f o r n i : 18 June 1992)

A B S T R A C T . M o d a l analysis as a d i agnos t i c t o o l i n s t r u c t u r a l mechanics is g a i n i n g p o p u l a r i t y , bu t its use requires

some c a u t i o n . Here we discuss t w o app l i c a t i ons o f an identifìcation technique , based o n the measurement o f the

eigenfrequencies on l y , to steel beams unde r var i ous damage c o n d i t i o n s .

S O M M A R I O . L ' ana l is i moda l e come s t r u m e n t o d iagnos t i co i n meccanica delle s t r u t t u r e ha suscitato crescente

interesse in a n n i recent i , m a i l suo imp i e go p r a t i c o presenta anco ra aspet t i de l i ca t i . Q u i si d i s cu tono due app l i caz i on i

su t r a v i i n acc ia io so t to var ie c o n d i z i o n i d i dannegg i amen to i n cu i V anal is i e' fa t ta sul la base delle sole frequenze

propr i e .

K E Y W O R D S . D iagnos t i cs , S t r u c t u r a l identifìcation, Beam damages, S t r u c t u r a l mechanics.

1. I N T R O D U C T I O N

M o d a l analysis is widely used, among various app l i ­cations, as a diagnostic too l to detect the s tructura l modif ict ions that may have occurred in structures dur ing service. The final scope is to identify possible damages predict ing their locat ion and degree of severeness. I f the a im is clear, and the evaluat ion of the moda l parameters is easy nowadays, the diagnostic prob lem presents in fact intr insic difficulties that are only part ly understood and managed successfully.

Here we evaluate the results of measurements made on the flexural and axial v ibrat ions o f steel beams. The w o r k is part o f a research program, carried out by I S M E S and ENEL-CR IS , a iming to tune mon i t o r ing and diagnostics o f complex structures [ 1 ] by dynamic methods. A t this stage tests have been performed on simple physical sys­tems under contro l ied damage condit ions. Namely , beams free or s imply supported by elastic hinges at the ends, where the damage consists of an increasing removal o f mater ia l at some cross section. The focus is on local iz ing the posi t ion and quant i fy ing the amount of damage, e.g. the reduct ion of bending stifTness, th rough the use of a suitable identifìcation technique.

When locat ion and geometry are known , the numerical model ing of damages does not exhibit peculiar novelties and has been addressed by various methods i n recent years. The prob lem of reconstruct ing the damage f rom the moda l properties, on the contrary, runs in to a series of difhculties which are typical of the inverse problems in mathematics. So, it either requires the so lut ion of the damaged prob lem to be available i n analytic f o rm [ 2 ] ; or reveals extreme sensitivity to the accuracy of the measure­ments [ 3 ] ; or has to face in various fo rm the effects of non-uniqueness [ 4 ] , [ 5 ] .

These drawbacks are not avoided by hav ing enough or unreahstically accurate in fo rmat ion at band. I n fact, not

Meccanica 28: 2 7 - 3 7 , 1993.

© 1993 Kluwer Academic Publishers. Printed in the Netherlands.

even the knowledge of the full spectrum suffices to identify the physical properties of v ibrat ing strings and rods [ 6 ] and, on the other band, the avai labi l i ty of part ia l i n ­f o rmat i on on the modes gives rise to the difficulties discussed by Ewins in describing the error matr ix tech­nique [ 7 ] . The direct use of the measured modes, e.g. in reconstruct ing str ing and rod stiffness [ 8 ] , is also to be avoided because o f the extreme sensitivity o f the technique to errors and because of the relative coarseness w i th which modes can be measured in practice. Thus one has always to deal w i t h problems in the presence o f insufficient data and is, to say i t w i t h Ewins ' words [ 9 ] , i n that 'grey area' where the mathemat ic ian is probably hopeless and the engineer is to make sensible use o f technical resorts.

We adopt the var iat ional approach to structural ident­ifìcation [ 10 ] . The want of data reflects then into non-convexity of the cost function, w i th the appearance of non-unique locai and global min ima (which may be the explanat ion o f the results reported in [ 11 ] , for instance) and of possible instabilit ies of the min imiza t i on algorithms w i t h respect to slight variations of the data. The numerical study of K o h n and McKenney [ 12 ] , i n a different context, indicates that such difficulties are not peculiar of the identifìcation via moda l techniques. A consequence is that one can hard ly obta in anything good w i thou t a relatively simple physical model. I t also follows that operating in the large is unl ike ly to give useful answers.

I n our analysis we substitute the beam under study by a chain of springs (to be identifìed) and masses (supposed known) , and choose the distance between a certain number of proper frequencies of the model and of the real structure as the cost funct ion to be minimized. TheTdecision to use the frequencies only reflects the fact that they are easier to measure and less affected by experimental errors, but i t lets in some ambiguities that one has to dispose of in some way.

Simple examples show that even the prescription of ali

28 C E S A R E D A V I N I E T A L .

the frequencies of a discrete system does not single out the relevant parameters uniquefy, so that the cost funct ion is to have several m in ima . In fact, it can be proved that these min ima are isolated points in the space of parameters. Thus, one should select the appropriate one by min imiz ing the cost function in the ' r ight ' well. Things however become worse i f on ly some frequencies of the discrete model are accounted for, because then the min ima are no longer isolated and operat ing in the right well is no longer a guarantee of success.

In order to cope w i t h such difficulties we adopt two guidelines. First, we choose the analyt ical model in a family of discrete systems that contains arb i t ra r i l y good approximations to the cont inuous model under study [13 ] . This is done by relat ing the spring stifTnesses to that of the real beam in a suitable way. In part icular, we use the discrete model corresponding to the undamaged structure as the start ing po int of the m in im i za t i on procedure, assuming that the damage cause slight changes in the global stiffness of the beam and that the in i t i a l po int chosen in this way be located in the r ight well . Eventually, the relat ion between the cont inuous and the discrete system is again used for the local izat ion and the estimate of the damage.

Second, since only a few frequencies are used and the min ima in any wel l are expected to f o rm a cont inuous

set, in m in im i z ing the cost function we supplement the gradient method w i th the requirement that the spring stiffnesses cannot but diminish, in accordance wi th the relat ion between the discrete and the continuous system. We notice incidental ly that this monotonic i ty would be less transparent, or less easily accounted for, if one used more elaborated physical models.

The general aspects of the technique are not discussed here. Its appl icat ion however is encouraging. We treat the flexural v ibrat ions of a beam elastically supported at the ends in Section 4 and the axial v ibrations of a free beam in Section 5. I n bo th cases different levels of damage are successively explored. The identifìcation technique pro-vides a satisfactory description of posit ion and severeness of the damage, w i th low computat ional burden and requir ing the knowledge of few proper frequencies. Results seem to be stable w i th respect to the number of used frequencies.

2. D E S C R I P T I O N O F T H E E X P E R I M E N T S A N D R E S U L T S

The damage analysis has been performed for the two beams represented in the fìgure below. The first model, in Figure l(a), is a steel beam of the series H E 100 B clamped

1220

DAMAGED ZONE HE lOOB BEAM

UNI 5 3 9 7 - 6 4

3 0 0 500 400 5 3 0 3 3 0 3 5 0 3 3 0 400 500 500

5C 00

ACCELEROMETER

60 60 30 I

40 30

C LENGTHS IN MM ]

DAMAGE A

DAMAGE B

60 60

Fig . l (a)

A D A M A G E A N A L Y S I S O F S T E E L B E A M S 29

'/// \

6 7 2 1 6 5 6

WIRE ROPE D - 4 MM

6 7 2

NOTCH ( 1 . 5 MM)

/ V

1 1 2 5

3 0 0 0

ACCELEROMETER [ LENGTHS IN MM ]

1 UNDAMAGED

I

HE 1 0 0 B BEAM

UNI 5 3 9 7 - 6 4

DAMAGE A DAMAGE B

F i g . l ( b )

F i g . L E x p e r i m e n t a l mode l s .

at bo th ends and supported by a frame mounted on the floor of the laboratory . We have studied the lowest three flexural modes and frequencies b o t h i n the undamaged conf igurat ion and i n a sequence o f damaged states where the cross section had been reduced by progressive removal of mater ia l i n some part o f the beam. Here we comment o n the two levels o f damage indicated i n figure and refer to the technical report [ 14 ] for a complete account of the experiment.

The tests have been r u n according to the stepped sine technique, After hav ing chosen nine measurement points, the beam is set i n forced v ib ra t i on by app ly ing a harmonic load to one po in t of the measurement set under contro l i ed ampl i tude and frequency condi t ions. The accelerations were then measured at a l i points simultaneously by means of an automat ic data acquis i t ion system that checked the achievement of the stat ionary condi t ions and prov ided the calculat ion of the components of the response ma t r i x

Pre l iminary tests on the free beam have prov ided an average value o f 1.971 x 10^^ N/m^ for the Young's modulus E. F o r a l i s t ructura l conf igurat ions and for various posit ions o f the shaker we have performed a narrow band analysis i n the interval (/ — 0.5 Hz, /, + 0.5 Hz) a round each proper frequency w i t h a resolut ion A/ = 0.005 Hz. The occurrence of wel l -

separated v ib ra t i on modes and the very small damping al lowed us to identify the moda l parameters by means of the so-called single mode technique. Table I and Figure 2 compare eigenfrequencies and modes for al i test conf igurations.

The second model that has been tested is a beam of the series H E 100 B suspended by means of two steel wire ropes fixed at 0.672 m f rom the ends, cf. Figure l(b). I n this case the object o f the experiment was to study the effects o f a no tch on the axial v ibrat ions of the beam. Since the measurement of the moda l shapes was here more dif f icult and less accurate, we just measured the eigenfreq­uencies. T o do so we adopted an impulsive excitation at one end and measured the acceleration at the other. The signals were acquired w i t h a sampling frequency of 20 k H z and were then worked out i n the frequency domain in the usuai way. Aga in we have studied the response of the undamaged beam and the two damaged configurations described i n Figure l (b). The results of the measurements are shown i n Table I I . I t 's only w o r t h not ic ing the appearance o f spurious resonances i n correspondence w i th the f ou r th and fifth eigenfrequencies ( in parentheses in the table) whose o r i g in has not been clarified. The choice to consider them spurious, however, is corroborated by the compar ison w i t h the frequencies of an ideal rod , i n the case

T A B L E L M e a n values o f the e xpe r imen ta l eigenfrequencies

(*)A/;% = (/;(damage)-y;(undam.))/y;.(undam.) x 100

U n d a m a g e d D a m a g e A D a m a g e B

y;.(Hz) (exp. 1 6 - 3 2 ) (exp. 6 7 - 6 8 ) ^fX{*) (exp. 8 0 - 8 2 ) A/%(* )

A 16.85 16.54 - 1 . 8 4 15.23 - 9 . 6 1

/2 53.14 52.36 - 1 . 4 7 43.18 - 1 8 . 7 4

/3 120.78 119.28 - 1 . 2 4 111.06 - 8 . 0 5

30 C E S A R E D A V I N I E T A L .

0 . 2 0

F I R S T MODE

0 . 2 0

- 0 . 1 0 -

- 0 . 2 0

0 . 2 0

0 . 1 0 -

u n d a m a g e d d a m a g e A d a m a g e B

S E C O N D MODE

THIRD MODE

F i g . 2. E x p e r i m e n t a l modes.

of the undamaged conf igurat ion, and by a study done on the basis o f a simple mathemat ica l model of the notched beam i n the remaining cases.

3, T H E I D E N T I F I C A T I O N T E C H N I Q U E

I n our analysis we refer to special classes o f lumped parameter systems. Namely , i n the damage identifìcation f rom axial v ibrat ions we imagine the r od axis to be div ided into N subintervals Af = [ x f _ i , x f ] and model the actual structural element by a chain o f springs and mass points w i t h mass

p{x) dx,

and stiffness

k^(i) = m i n

óv{i) = 1

EA{xy\x)dx,

(3.1)

(3.2)

where p{x\ are the mass density and the axial stiffness of the rod , respectively, and òv{i) = v{x^)-v{xt,).

Likewise, i n the study of the flexural vibrations, we use a discrete model consisting o f a chain of r ig id, massless segments Aj^ jo ined to one another by elastic hinges of mass

m^(0 =

and stiffness

_ p ( x )dx

m i n

Sv'ii) = 1

EJ(x)i;"^(x)dx. A f

(3.3)

(3.4)

I n (3.3), (3.4) is the interval [ x f , xf^ J = [(xf^ + xf_ 0/2, ( x f - f x f+ i )/2] centered at the point x f ; 5v'{i) = ^'(^?+i) — ' (^f); and EJ{x) is the bending stiffness of the beam.

The so lut ion of the m i n i m u m problems (3.2) and (3.4) respectively yields the we l l -known formulae

k^ii) =

1

A f EA{x)

• 1

d x

Ar EJ{x) dx

(3.5),

(3.5)2

that relate the spring stiffness o f the discrete models to that o f the cont inuous system under study. Here the usuai theory o f rods and beams is assumed to describe the damaged structure adequately and the damage is expected to reflect i n some way in t o a modi f icat ion of the effective stiffnesses EA(x) or EJ{x),

Formulae (3.5), 2 enable us to calculate the correspond­ing k^(i) when EA{x) or EJ{x) are k n o w n and, conversely, they give in fo rmat ion on these quantities when k^(i) are evaluated in some way, Since i t can be proved [ 13 ] that the

T A B L E I L E x p e r i m e n t a l eigenfrequencies (*) A y ;% = (/ (damage) -/ (undam. ) )/y ; (undam. ) x 100

U n d a m a g e d D a m a g e A Damage B

/ ( H z ) (Test 1) (Test 2) {Test 3)

/ i 860.4 805.7 -6 Al 737.6 - 1 4 . 3 7

fi 1722.2 1664.5 - 3 . 3 5 1600.0 - 7 . 1 0

h 2582.9 2541.9 - 1 . 5 9 2505.3 - 3 . 0 0

h 3434.2 3162.2 - 7 . 9 2 3016.0 - 1 2 . 1 8

(3293.0) - 4 . 1 1

h (4115.8) (4107.4) - 0 . 2 0 (4093.3) - 0 . 5 5 4353.6 4332.2 - 0 . 4 9 4310.2 - 1 . 0 0

fe 5174.4 4961.1 - 4 . 1 2 4812.6 - 6 . 9 9

A D A M A G E A N A L Y S I S O F S T E E L B E A M S 31

above families of discrete models, w i th the choice (3.5) i 2 , provide an arb i t rar i l y good approx imat ion of the eigenpairs of the cont inuous system, it seems reasonable to found an identifìcation procedure on this basis.

The free vibrations of the discrete models above are governed by the differential equat ion

d 'u (0/dr ' -h C^u(r) = 0, (3.6)

w i th the global mass and stifTness matrices M ^ , suitably bui l t up w i th the m^{i) and k^{i).

Let Ài = cof, / = 1 , . . . , A , be the eigenvalues o f the free v ibra t ion problem (3.6), and I , the corresponding values measured in the experiments. We choose the Euclidean distance between the M-vectors /.^ = (À, (^) , . . . , Àj^(P)) and ;T _

Fili) = £ (3, - W))' (3.7)

as the cost funct ion to be minimized. P^ = (/c^(/i), /c (/2),..., /c%)), ì^i,<Ì2<'"<ÌQ = Q^^ is the e-vector of s tructural stiffness to be identifìed (structural vector). Then, i f we accept that the damage is expected not to increase the effective stiffness EA{x) or EJ{x) i n (3.5), we reduce the damage identifìcation to the constrained varia­t ional problem:

r e ff^^: nr) = m i n £ ( I , - A,(/^))\ (3.8)

w i t h the vector of the springs stiffnesses corresponding to the undamaged rod or beam. We observe that a problem of type (3.8) can be treated on the basis of the QA-updat ing technique worked out at the Kathol ieke Univer-siteit of Leuven and used successfully to study several practical applications [ 15 ] , [ 16 ] .

Since the dependence o f the frequencies upon the struc­tura l vector is not available in explicit form, recourse to a numerical approach is i n order. So we have worked out an i terat ion a lgor i thm based on the gradient method in order to solve the prob lem (3.8)-(3.9). Accordingly, at each step the structural vector p is updated in the d irect ion of the max imal descent

where

WF(P)= - 2 . SU

(3.9)

(3.10)

and is chosen so as to minimize F{P) in the d irect ion of VF(P(^j^). I n order to case computat ion , however, it is expedient to use the value of tf^j^ that minimizes the quadratic approx imat ion of F at = j ^ ) in the d irect ion of VF(p^j^y Observe that

V'F(P) = 2 'di' T

L J (7. - X(p) (3.11)

reduces to the first term i n the prox imi ty of A = /, so the procedure is further simpli f ied by tak ing just this term in

(3.11) and by using the approximate opt imal value

|VF(/?,,,)i^ (3.12)

2x|[a//a^,,JVF(/i,,,),

in (3.9). The program checks that, w i th this choice, does not violate the constraints and picks up the point on the boundary of the al lowed domain along the direction of VFiP^j^) otherwise, Also, when P^j^ lies on the boundary, and VF{P^j^) does not po int inside the domain 0 < P ^ p^, the program takes the projection of the gradient on the boundary when updat ing the structural vector. Once P^j+1) is known , the updated eigenfrequencies / = ÀiP^j^^^) are calculated by solving a new eigenvalue problem. The i terat ions goes on unt i l the relative var iat ion of F(P) and p satisfy a chosen cr i ter ion o f smallness.

Since the var iat ional problem is not convex, the success of the technique crucially depends on the choice of a good in i t i a l estimate P^q^ of the structural parameters to be identifìed. O u r experience suggests that tak ing the struc­tura l vector p^ corresponding to the undamaged con-fìguration serves the scope even for rather severe levels of damage.

I t is w o r t h not ing that the sensitivity matr ix [dÀ/dp'] in (3.10) takes the explicit f o rm

dPn. (3.13)

where u ' is the a h eigenvector evaluated at the yth step and normal ized w i t h respect to M ^ . So, the sensitivity ma t r i x can be easily calculated once we known how is assembled f rom the k^(i).

4. D A M A G E I D E N T I F I C A T I O N F R O M F L E X U R A L V I B R A T I O N S

I n the case of Figure l(a), the mass and stiffness matrices M ^ and corresponding to the discrete model described above take the form, cf. [ 6 ] ,

M ^ = diag(m^( l ) , . . . ,m^(N) ) ,

(C^),, = ( E L - ' E ^ K E L - ' E ^ ) , , -

X ( ^ . Ì V ^ J N - I + ^ . N - I ^ J N ) +

/ / \) f/c^yV) + 4

(4.1)

where k]^ and A: are the stiffness of the elastic jo ints at the

left and r ight end, respectively;

1 i f M = m w i t h M, m = l , 2,

- 1 i f n = m + l w i th m = 1 , 2 , . . . , N - 1 (4.2)

0 otherwise,

is the assemblage matr ix ; L = d i a g ( / o , . . . J N - Ì ) ^^^^

32 C E S A R E D A V I N I E T A L .

T A B L E I I I . Expe r imen ta l a n d numer i c a l frequencies (*) Ay ;% = (y;(numer.)-/(exper.))/y;(exper.) x 100 V _ _ _ _. _

M o d e l

E x p e r i m e n t a l n u m e r i c a l

16.85 ^ 16.85 0.00

53.14

56.25 5.85

A/3% A A/4%

120.78 — 120.21 - 0 . 4 7

212.13 — 207.48 - 2 . 1 9

= x^^,-x^\; and ó^j is the

Kronecker symbol.

To start w i th , we assume N = 20 and a par t i t i on o f the

beam in pieces of equal length / . Then

m(%(/)^4.86kg,

E J f<(0){i) = Tv" = 3.72 X 10^ N m/rad.

(4.3)

W i t h this value the identifìcation technique based on the

fìrst three frequencies and w i t h = (k^, k'j)) yields

/c{) ^ 3.3 X 10 ' N m/rad, k'^ ^ 3.3 x 10 ' N m/rad. (4.4)

Table I I I and Figure 3 conta in a compar ison between the

frequencies and the modes, respectively, o f the real beam

and of the discrete model so identifìed. Apar t f rom the

second frequency, the agreement is very good. The

estimate (4.4) is further val idated by the comparison w i th

0 . 2 0

0 . 1 0 -

0 . 0 0

0 . 2 0

F I R S T MODE

0 . 1 0 -

0 . 0 0

- 0 . 1 0 -

- 0 . 2 0

0 . 2 0

e x p e r i m e n t a l n u m e r i c a l

S E C O N D MODE

0 . 1 0 -

0 . 0 0 -

- 0 . 1 0 -

- 0 . 2 0

THIRD MODE

Fig . 3. Expe r imen ta l a n d n u m e r i c a l undamaged m o d e shapes.

the frequencies obtained from the finite element method,

w i t h the same exception for the second frequency. Which

suggests that the disagreement is probably due to a lack of

accuracy in mode l ing the jo ints .

Let US consider now the effect of the damage. Figure 2

gives an account of the modes measured in different

damage configurations. I t is seen that modes change very

sl ightly even for severe levels of damage. Tak ing the

measurement uncertainty into account, i t follows that it is

hard to extract f rom them more than the indicat ion that

the damage is probably localized in the left half of the

beam. I f we make this assumption, in fact the assumption

that the damage has a localized character, and choose

P^ = {k^(\\...^k^(\{S)) as the structural vector, w i th

Ao)(0 = ^(O)(0 for / = 1 , . . . , 10, the identifìcation technique

yields results that are in fairly good agreement w i th the

real local izat ion of the damage both for level A and B, cf.

Figure 4.

Figure 5 accounts for an attempt to refìne the de­

scr ipt ion of the stiffness i n the damaged zone by using a

fìner mesh {N = 39 and = 0.125m, Constant) and con-

fìning the var iab i l i ty of the spring stiffness to the elastic

hinges wh i ch are dose to the cross section that had

5 E + 0 0 6

O

O E + 0 0 0

5 E + 0 0 6

o

E z:

O E + 0 0 0 -1 I — I — I — I — I — I — I — r 6 11

^ d a m a g e a x i s

Fig . 4. O p t i m a l d i s t r i b u t i o n o f stiffness for the damaged mode l w i t h N - 20 a n d Q = IO.

A D A M A G E A N A L Y S I S O F S T E E L B E A M S 33

1 E - f 0 0 7

O

O E + 0 0 0 -

\\\

1 /

l\

V 1

lì lì

/ d a m a g e B

V 'ì ' Q = 5 Tpar. 8 - 1 2 Ì

\'j Q = 8 ( p a r . 7 - 1 4 )

1 E + 0 0 7

J 1 21 31

d a m a g e a x i s

Fig . 5. O p t i m a l d i s t r i b u t i o n o f stiffness for the damaged m o d e l w i t h

N = 39 and Q = 5,8.

appeared to be the most weakened f rom the previous analysis. The results refer to the choice o f Q = 5 and g = 8 structural parameters to be identifìed and seem fairly stable. I t is w o r t h not ing that i f a l i stiffnesses k^(i), i = 1,2,..., 20 of the left hal f o f the beam were al lowed to vary, then one obtains well-localized results for the most severe level of damage by use o f the fìrst three frequencies only, cf. Figure 6.

Final ly , f r om the knowledge o f the op t ima l k^{i) i t is possible to estimate EJ{x) o f the real beam by using (3.5)2. Namely, i f we approximate EJ{x) by a Constant i n each A f and use the values o f k^{i) o f Figures 4 and 5, the moment of inert ia i n the weakest cross section is

«/li ìn = ^ = 1 -9x10 -^ m^ (Ar = 20)

4 T . = ^ = 2 . 8 5 x l O - ^ m ^ (iV = 39)

(4.5)

for levels A and B, respectively. The value o f k i n the two cases is the mean between the k^{i) o f the nearest hinges. These values are fair ly dose to those o f the real beam:

[ ildam = 2 . 0 6 x l 0 - ^ m ^

4?l,dan. = 3 . 0 x l 0 - ^ m ^ . (4.6)

We note that , extending the eva luat ion to the other A^(0 one obtains a descript ion o f the region influenced by the damage. By this crude procedure the length of the affected zone is estimated to be 25 cm, wh i ch can be regarded as a decay length o f the per turba t i on caused by the locai reduct ion i n the cross section. M o r e accurate use o f the identifìcation results above cou ld probab ly be obtained by combin ing them w i t h classical studies on the locai effects of cracks and notches i n elastostatics i n the way Shen and Pierre [ 17 ] seem to do.

5. DAMAGE I D E N T I F I C A T I O N F R O M AXIAL VIBRATIONS

Let US consider now the axial v ibrat ions o f the r od described i n Figure l (b). The corresponding discrete para-

o

O E + 0 0 0

1 E + 0 0 7

o

I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I

11 21 31 d a m a g e a x i s

O E + 0 0 0 I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I 1 11 21 31

d a m a g e a x i s

F i g . 6. O p t i m a l d i s t r i b u t i o n o f stiffness for the damaged mode l w i t h

N = 39 a n d e = 20.

meter system has then structural matrices given by

(M^)^„ = m » ^ ^ „ ,

k\m-\)-^k\m) i f n = m w i t h

n , m = l , 2 , . . . , N

-fc^(m) i f { n = m - l w i t h

m = 2, , . . , iV }

a n d i f { n = mH- l w i t h

m = l , . . . , N - l }

fc^(O) = k^{N) = 0 otherwise.

(C^).n= <

I n the whole analysis below we assume N = Al. Accord­ingly, = 0.075 m and the quantities m|^)(0, k^oi^ relative to the undamaged confìguration have been evaluated assuming a mass density p = 20.4 kg/m, a cross section ^ = 2.6 X 10~^m^, and Young's modulus (obtained f rom dynamica l measurements) E ^ 2.094 x 10^^ N/m^. There is no uncerta inty here due to the model ing of the end supports, since free vibrat ions tests were performed, and the discrete model fìts very wel l w i t h the real rod , cf. Table IV .

I n the experiments no measurement o f modes was made. So, ambigu i ty between symmetric confìgurations is un-avoidable, but again the assumption that the damage is confìned to one hal f of the beam is a reasonable start. Accord ing ly we have assumed the structural vector to be )S = (/c^(l),..., F (20) ) , imagined to refer to the haff on the left side and impose the bound jS^ = k^oiJ) = 7.26 X lO^N/m.

Figures 7 and 8 give the opt ima l d is t r ibut ion of stiffness

34 C E S A R E D A V I N I E T A L .

T A B L E I V . E x p e r i m e n t a l a n d n u m e r i c a l eigenfrequencies for the u n ­

damaged beam (*) Ay;.% = (y;(num.fy;.(exp.))/y;.(exp.) X 100

1 E 4 - 0 1 0

/ [ H z ] E x p e r i m e n t a l N u m e r i c a l A/;%(*)

/, 861.4 860.9 - 0 . 0 6

fi 1722.2 1721.3 - 0 . 0 5

h 2582.9 2577.3 - 0 . 2 2

h 3434.2 3430.1 - 0 . 1 2

h 4353.6 4280.9 - 1 . 6 7

h 5174.4 5119.7 - 1 . 0 6

1 E + 0 1 0 -

O E + 0 0 0 -

M = 5 M = 6

M = 4 r 3 0 t h it. 4 0 t h it.) 4 0 t h it.)

— I— I— I 1 — I — I — I — I 1 — r -11 s16 d a m a g e a x i s

1 E + 0 1 0

E

O E + 0 0 0

-

- V» 7 : 5 t h it. Y

1 0 t h it. ( M = 5 ) it. - 1 5 t h it. ( M = 5 ) it.

6 11 J 6 d a m a g e a x i s

1 E + 0 1 0

OE+OOO

2 0 t h it. 2 5 t h it. ( M = 5 ) 4 0 t h it.

- i — I — I — I — I — I — I — I — I — I — r 11 16

d a m a g e a x i s

F ig . 7. O p t i m a l d i s t r i b u t i o n o f stiffness for the damage A .

for the damage levels A and B respectively, for different number M of experimental frequencies accounted for i n the cost funct ion. I n bo th cases we see that considering four frequencies on ly yields two wells, though o f different size, i n the op t ima l d i s t r ibu t i on o f stiffness. The smaller well reduces progressively when increasing the number of frequencies accounted for i n the cost funct ion, un t i l i t almost disappears for M = 6. I n this case the m i n i m u m of stiffness occurs between nodes 15 and 16, which agrees

O E + 0 0 0

M = 4 ( 2 5 t h it.) M = 5 ( 4 0 t h it.) M = 6 ( 4 0 t h it.)

-T 1 1 1 1 1 1 1 1 1 r 1 1 1 1 1 r-1 1 .16

d a m a g e a x i s

1 E + 0 1 0 - r

O E + 0 0 0

5 th it. l O t h it. (M = 5 ) 1 5 t h it.

-I 1 \ 1 1 1 1 1 1—-1 r" 6 1 1 16

d a m a g e a x i s

1 E + 0 1 0

O E + O O O

2 0 t h it. 2 5 t h it. ( M = 5 ) 4 0 t h it.

- 1 — I — I — I — I — I — I — I — I — f — 6 11

-1 1 1 1—r-16

d a m a g e a x i s

F i g . 8. O p t i m a l d i s t r i b u t i o n o f stiffness for the damage B.

very wel l w i t h the real locat ion o f the notch in th real rod. Figures 7(b) and (c) and 8(b) and (c) show the result of

the analysis for the two levels of damage for different i terations i n apply ing the gradient method. A sharp local izat ion o f the damage is apparent even w i th five frequencies i n the cost function. As to the value of the structura l vector there is some f luctuat ion that is sensitive to the number o f frequencies used. Figure 9(a) shows the approx imat i on o f the m i n i m u m of F obtained by different iterations. Fo r M = 5, F oscillates near the m i n i m u m (cf. Figure 9(a)), and this seems to be related to a certain instabi l i ty i n the value of /c^(16) (cf. Figures 9(b) and (c)), probably coming f r om the use of the gradient method (cf. [18, p. 71]). Calculat ions made by Newton ' smethod , w i th M = 4,5,6, to identify the three u n k n o w n structural para­meters /c^(0, / = 15, 16, 17, for the two damage levels confìrm the above estimates and el iminate the oscillations.

W i t h the op t ima l structural vector so identifìed one can calculate frequencies and modes of the approximat ing

A D A M A G E A N A L Y S I S O F S T E E L B E A M S 35

o d a m a g e A (M = 5 )

-A7VyvAA7VAA_A ^ , 1 , , , P -

1 6 11 16 21 2 6 31 3 6 ITERAT IONS

0 . 2 5

0 . 0 0

- 0 . 2 5

S E C O N D MODE

1 E + 0 1 0

CD

O E + O O O i \ 1 1 \ r

0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0 4 5 5 0 I T E R A T I O N S

1 E + 0 1 0

O E + O O O

I T E R A T I O N S

Fig . 9. Osc i l l a t o r y character o f F near i ts m i n i m u m .

discrete model . I n Figures 10(a) and (b) the dot ted lines refer to the modes of the undamaged beam, whereas the solid lines describe those o f the damaged one as calculated f rom the discrete model. Table V (relative to the results for M = 5, cf. Figures 7(b),(c) and 8(b),(c)) shows that the approx imat ion of the first five frequencies is very good.

0 . 0 0

- 0 . 2 5

THIRD MODE

- 0 . 2 5

F IFTH MODE

(a)

0 . 2 5

0 . 0 0

- 0 . 2 5

u n d a m a g e d d a m a g e A ( M = 5 )

0 . 2 5

0 . 0 0

- 0 . 2 5

u n d a m a g e d d a m a g e B ( M = 5 )

F I R S T M O D E F I R S T MODE

36 C E S A R E D A V I N I E T A L .

0 . 2 5

0 . 0 0

- 0 . 2 5

/ 7

\\

\\

\\

\

0 . 2 5

0 . 0 0

- 0 . 2 5

- 0 . 2 5

0 . 2 5

0 . 0 0

- 0 . 2 5

S E C O N D MODE

\\

\\

\

/ ~

/ f

\

\

N. \ \

/ / / /

/ / / / \/ /

-

1 T •• —T 1 1

THIRD MODE

F O U R T H MODE

f \ /

1 \

// \

F I F T H MODE

(b)

F ig . 10. F i r s t identifìed modes for (a) damage A , and (b) damage B.

T A B L E V. Dev ia t i ons between numer i ca l and exper i ­

m e n t a l eigenfrequencies for the identifìed damaged beam

(•)A/;% = (y;(num.)-y;(exp.))/y;.(exp.) X 100

/ [ H z ] Damage A Damage B

/ i - 0 . 0 5 - 0 . 2 4

h 0.34 0.78

h - 0 . 6 6 - 1 . 6 4

k 0.07 0.80

h - 1 . 5 3 0.41

[ 16 ] , [19 ] ) . The results above indicate that the discrete models that we have used are quite accurate in spile of their crudeness, showing that more elaborate ones, based for instance on the t rad i t iona l finite element approach, may be redundant for beams. I n particular, the approach is l ikely to w o r k for more general structures, such as frames and struts where the one-dimensional character is main-tained. This remark is relevant because, in undetermined problems l ike those posed by diagnostics and moni tor ing , i t may be cruciai to keep the physical model as simple as possible i n order to get useful in format ion.

Also, measurement of a few frequencies, when supplied w i t h reasonable assumptions, seems sufficient for the ident i f i cat ion of the structure. Here a number of 3 to 6 is enough, bu t the levels of damage were rather severe in b o t h experiments. I t wou ld be interesting to study the corre la t ion between detectable damages and the number o f frequencies to be measured.

F ina l ly , since the cross section of the beam under testing was Constant, the peculiarity of the choice (3.5) has been only par t l y exploited. I n fact, g iv ing stiffnesses the values found f r om (3.5) defines a discrete model that, for any assignment o f nodal displacements (in the case of axial v ibrat ions, for instance), exhibits the min ima l strain energy. Thus, w i t h a caveat for the part played i n Ray-leigh's quot ient by the kinet ic term (which would require a specifìc evaluation), the model is somewhat opt imal in approx imat ing the eigenfrequencies of the real beam. This can expla in the accuracy of the results and suggests that the choice o f (3.5) may be impor tan t i n dealing w i th beams where the structural properties are changing more dramat ica l ly .

6. C O N C L U D I N G R E M A R K S

We have studied two simple problems in the identi f icat ion of damages in elastic beams. I t must be said that similar cases have been considered by several authors in recent years in the context of specific analyses or as simpler applications of general ident i f icat ion techniques ( [15 ] ,

A C K N O W L E D G E M E N T S

This research has been financially supported by E N E L ; the par t i c ipa t ion of C. Dav in i has been funded by M U R S T . The authors acknowledge the active interest in this w o r k o f Ing. G. Giuseppett i .

A D A M A G E A N A L Y S I S O F S T E E L B E A M S 37

R E F E R E N C E S

L G a t t i , F. and G iuseppe t t i , G., ' App l i ca z i one del la System Ident i f i ca­

t i o n ad u n m o d e l l o a r ig idezza v inco la re var iab i l e ' , Atti X Congresso Nazionale AIMETA, Pisa 2-5 Octoher 1990.

2. Rizos, P.F., Aspraga thos , N . and D i m a r o g o n a s , A .D . , ' I d en t i f i c a t i on

o f crack l o ca t i on a n d m a g n i t u d e in a cant i l ever beam f r om the

v i b r a t i o n modes ' , J . Sound Vibration, 138(3) (1990) 3 8 1 - 3 8 8 .

3. Lammens , S., Hey l en , W . a n d Sas, P., ' E r r o r l o ca l i z a t i on methods : an

overv iew o f e va lua t i on at K U L ' , Proc. I5th Internai. Seminar on Modal Analysis. Pa r t I ( l A ) , K a t h o l i e k e Un i v e r s i t e i t Leuven , 1990.

4. U d w a d i a , F.E. a n d Sha rma , D .K . , 'Some uniqueness results re lated to

b u i l d i n g s t r u c t u r a l i d en t i f i c a t i on ' , SIAM J . Appi Math., 34 (1978)

104 -118 .

5. U d w a d i a , F.E., Sha rma , D . K . a n d Shah, P.C., 'Uniqueness o f

d u m p i n g and stiffness d i s t r i b u t i o n s i n the i d en t i f i c a t i on o f soi l a n d

s t ruc tu ra l systems', J . Appi. Mech., 45 (1978) 1 8 1 - 1 8 7 .

6. G l a d w e l l G . M . L . , Inverse Problems in Vibration, M a r t i n u s NijhofT

Publ ishers, D o r d r e c h t , 1986.

7. Ewins , D.J. , Modal Testing: Theory and Practice, Research Studies

Press, L o n d o n , 1988.

8. U d w a d i a , F.E. a n d G a r b a , J.A., ' Pa ramete r i d en t i f i c a t i on p rob l ems

i n s t r u c t u r a l a n d geotechnica l eng ineer ing ' , J . Engng. Mech. (ASCE), 110(9) (1984) 1409 -1432 .

9. Ewins , D.J. , ' M o d a l analys is i n an imper fec t w o r l d ' , Proc. 13th Internai. Seminar on Modal Analysis. Pa r t I , (ed. P. Sas), K a t h o l i e k e

Un i ve rs i t e i t Leuven , 1988.

10. N a t k e , H .G . , Identification of Vibrating Structures, Spr inger-Ver lag ,

C I S M , U d i n e , 1982.

11. T o r k a m a n i , A . M . and A h m a d i , A.K. , 'Stiffness ident i f i ca t ion of

frames us ing s imu la t ed g r o u n d exc i ta t ion ' , J . Engng. Mech. (ASCE), 114(5) (1988) 7 5 3 - 7 7 6 .

12. K o h n , R.V. a n d M c K e n n e y , A., ' N u m e r i c a l i m p l e m e n t a t i o n o f a

v a r i a t i o n a l m e t h o d for e lectr ical impedance t o m o g r a p h y ' . Inverse Problems, 6 (1990) 3 8 9 - 4 1 4 .

13. D a v i n i C., ' N o t e o n a paramete r l u m p i n g in the v i b ra t i ons of elastic

beams ' (p r i va te c o m m u n i c a t i o n ) .

14. I S M E S , I n t e r n a i T e chn i ca l Report , Bergamo, 1991.

15. Janter , T . a n d Hey l en , W., ' Q A - M o d e l u p d a t i n g ' , Proc. 13ih Internai. Seminar on Modal Analysis. Part C (C-13), K a t h o l i e k e Univers i te i t

L euven , 1988.

16. L i e fooghe , C., Janter , T . a n d Sas, P., 'Case studies o f the app l i ca t i on

o f the Q A - M o d e l u p d a t i n g a l g o r i t h m ' , Proc. 13lh Internai. Seminar on Modal Analysis. Par t C (C-14), K a t h o l i e k e Un ive rs i t e i t Leuven,

1988.

17. Shen, M . - H . H . a n d Pierre, C , ' N a t u r a i modes o f B e r n o u l l i - E u l e r

beams w i t h s ymme t r i c cracks ' , J . Sound Vibration, 138(1) (1990) 115 -

134.

18. H i m m e l b l a u , D . M . , Applied Non-Linear Programming, M c G r a w -

H i l l , N e w Y o r k , 1972.

19. A l l e m a n g , R., ' C o r r e c t i o n o f finite e lement m o d e l v ia selected

phys i ca l parameters ' , Proc. 13th Internai. Seminar on Modal Analy­sis, Pa r t C (C-12), K a t h o l i e k e Un ivers i t e i t Leuven , 1988.