Identification of equivalent structural schemes for coupled systems

EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICS, VOL. 20,317-333 (1991)

IDENTIFICATION OF EQUIVALENT STRUCTURAL SCHEMES FOR COUPLED SYSTEMS

D. BENEDETTI AND G. M. BENZONI Department of Structural Engineering. Politecnico di Milano. Piazza L. ah Vinci 32,20133 Milano, Italy

SUMMARY Some arrays of buildings in historical nuclei of Central Italy are currently monitored by steady-state accelerometers. The aim is to identify the structural scheme of a single building representing the effect on it of the other systems of the array. A method for interpreting the recorded responses is presented. It is based on the a priori assumption of a scheme for the building of interest, thought of as being isolated, and on the identification of two sets of masses and of stiffnesses to add in order to account for the array effect. The method is limited to linear behaviour and to planar systems.

1. INTRODUCTION

In historical European urban nuclei most buildings are connected to one another, giving rise to complex assemblies whose seismic behaviour is difficult to predict. This creates a major problem when structural analysis must be performed on a single building or part of the array (e.g. to design its seismic strengthening). It is basically a matter of determining the appropriate structural scheme to use in the analysis in order to account for actions exerted on the given building by the neighbouring ones. To describe the problem and the procedure presented here more clearly, let us make reference to Figure 1, where the building of interest is denoted by A, and all the other buildings, directly or indirectly connected to it and presumably exerting an influence on its response, are denoted by B. When reference is made to system A (or to its structural scheme) and it is thought of as being isolated, the notation AA will be used. This is the same scheme that would be used for the analysis of the existing building A were it standing alone. The choice of AA is subject to several uncertainties, such as the nature of the scheme to be used and the material properties involved, which in old existing buildings are difficult to assess. However, after establishing the structural scheme AA, based on engineering judgement and research results, as for instance in References 1 and 2, we must determine appropriate distributions of masses and stiffnesses of a system BA which, added to AA, produces the same response recorded at a given location k of A. Figure 2 depicts this new situation: the system AA + BB is equivalent to the configuration of Figure 1, as far as the response of A is concerned, provided BA is correctly identified.

Two things must be clearly noted. First, is that BA is not intended to represent the real structural properties of buildings B, but aims to account for their effects on AA. The role of BA is to ‘force’ the seismic response of AA to be equal to the recorded response of A. The purpose of this study is not to identify the real physical properties of A and B, but to determine a structural scheme which can reproduce responses occurring in A under static and dynamic loading. Second, different systems BA may be identified depending on the choice of AA performed by the analyst. However, a common feature of all possible schemes must be their capacity to reproduce the recorded responses. The method cannot provide a criterion of choice among all the possible AA-dependent equivalent schemes which may be obtained, since these are all equally effective in providing equivalent responses, owing to the action of the different BA systems identified.

The added system BA has a tri-diagonal stiffness matrix and a diagonal mass matrix. This is to say that BA is assumed as a shear type system. Such an assumption has been made in order to simplify analytical derivations. It does not imply that the real behaviour of buildings B is shear type, although this is often the

oO98-8847/9 1 / W 3 17-17$08.5O 0 1991 by John Wiley & Sons, Ltd.

Received 18 November 1989 Revbed 8 October 1990

318 D. BENEDETTI AND G. M. BENZONI

Figure 1. Typical array; original system A + B

Figure 2. Equivalent scheme AA + BA

case for the class of buildings considered here. It is meant that the modes selected for the identification of BA are interpreted as if due to a storey mechanism. BA properties determined by means of this assumption also proved effective in recovering the actual behaviour of A, in cases, as in example W1, discussed in Section 5, where the behaviour of B-buildings was definitely not shear type. The added system BA is connected to AA by means of rigid bars, hinged in AA, so that BA masses undergo the same lateral displacements as the corresponding AA connected points. This prevents the description of relative lateral movements among A and B buildings and, conseqeuntly, the equivalent scheme cannot account for the pounding between adjacent buildings which may occur during strong earthquakes. However, these effects were seldom observed in the arrays of old urban nuclei considered here. The linear coupling between AA and BA also influences the procedure discussed here below. In fact, the selection of modes used in the identification excludes those modes, derived from the analysis of the recorded signals, associated with relative oscillations among A and B. A specific step has been introduced to perform this selection in the identification procedure (3rd phase). The application of this step is described with reference to example W1 of Section 5. In order to determine stiffnesses and masses of BA, it is assumed that both building A and buildings B are subjected to the same base input a, and that response acceleration records are available at some locations k of A. The proposed procedure is based on the spectral analysis of signals, and assumes that the response in direction x at each storey of the building of interest A (Figure 1) is described by a single record, that is, no plan torsional effects are considered.

The problem of the uniqueness of the solution of the identification proper has not been treated, since it is here of minor importance, given the premises and purpose of this work. Systems BA with different distributions of masses and stifhesses, but such as to give rise to the same spectral parameters, can, in fact, determine efficient equivalent schemes AA + BA, as will be seen. However, the identification algorithm employed here, consisting in a sequence of monodimensional minimizations, can provide global minima. These may correspond to different values of the parameters, depending on scanning fineness. The corresponding BA masses and stifbesses vary, in such instances, according to common ratios and thus result in the same spectral parameters and functions which are the basis of the equivalence.

In describing the procedure it is first assumed that responses are available at each level k of A; it will then be shown that one record suffices to identify the structural properties of BA. Physical quantities in the time domain will be denoted by small letters and the corresponding Fourier transforms by capital letters. Accordingly, the acceleration at level k of A reads 2i( t ) and its transform Xi(f). Moreover, quantities pertaining to the original array A + B are referred to as 'measured' (accelerations, frequencies, etc.), while the

EQUIVALENT STRUCTURAL SCHEMES FOR COUPLED SYSTEMS 319

corresponding quantities evaluated through the equivalent scheme AA + BA will be said to be 'calculated' (accelerations, frequencies, etc.) and represented by $t), J , etc.

2. THE PROCEDURE

Suppose that the equivalent system of Figure 2 is subjected to ground acceleration a,,. The response acceleration at a given location k can be seen as resulting from the sum of two signals, owing to the hypothesis of rigid connection assumed between AA and BA. The first signal, PM(t), is the response to ground acceleration a&) as determined by AA properties alone; the second, AkA(t), represents the response of BA to the same input. Note that this assumption considers only translational degrees of freedom at k in the analytical derivations which follow. Calling ~ i A ( j ) and f?",(j) the transfer functions at k in AA and BA, we may write in the frequency domain (Figure 3):

3A = i i A + i i A = &A A, + @A A , (1)

The effectively recorded signal n:(t) at the same level k of A (Figure 1) can be similarly expressed in the frequency domain in terms of transfer function H i ( f ) as H i A , . The equivalence between reality and AA + BA at k is imposed by equating the latter product and equation (1). This results in

G k A ( f ) = H i ( f ) - A ! i A ( f ) (2) Of the three complex functions appearing in the above equation only @A is unknown, since Hi is obtained from the recorded signals, and is analytically constructed from the (known) properties of AA. Equation (2) then allows us to determine the BA transfer functions at locations k where recordings are available.

The properties of f ikA are the input values for the identification of the masses and stiffnesses of BA. The equivalence between (A + B) and (AA + BA) is checked by comparing accelerations x i@) at various levels k of A for the two configurations (Figures 1 and 2), and by comparing lateral displacements and storey shears in the building of interest under the action of lateral forces proportional to masses in both situations. While the philosophy of the method is general, the applications which follow are restricted to cases where BA is a three mass system. The same system has given good results (see Section 5 ) in representing the action of B buildings with 5 storeys.

The identification of the structural properties of BA (mass, stiffness, damping) is performed in three steps. We begin by estimating some basic modal parameters from the transfer functions f i & ( f ) . These are: frequencies, damping factors and shape vectors for the selected modes. Selection of modes takes into account both the accuracy desired (the larger the number of modes, the greater the accuracy of the final result) and the readability of the modes themselves. As is known, modes display peaks in moduli of the transfer functions; however, the jaggedness, due to the effects of time variation and amplitude non-linearity of the system, to the finite length of the records and to measurement noise, makes selection critical. In fact, the selected peaks may not correspond to dominant modes or these, in turn, may not be consistent with the hypothesis of rigid links between AA and BA in the equivalent scheme (see Sections 4 and 5). The selection of modes is checked in the third phase of the procedure.

The second step concerns the identification of the structural properties of BA. A mathematical model of BA is built, consistent with the hypothesis of storey mechanism, having masses and storey stiffnesses

Figure 3. Interaction in the frequency domain


unknown. These quantities are identified by minimizing a function z of the squares of the (calculated and measured) circular frequencies of the modes selected in step 1. The calculated frequencies (derived by a standard eigenvalue procedure referred to the assumed mathematical model for BA) are functions of the structural parameters to be identified. When the value of the function z reaches the pre-set tolerance, all the structural and modal properties of BA are assumed to be known. As to modal quantities, a high accuracy is achieved only for the selected modes which are controlled by z, while the values obtained for the other modes are only approximate. The minimization algorithm ensures that a global minimum of z is reached within the assumed range of variation of the parameters.

The third step consists of the comparison between measured accelerations and those calculated using the equivalent model AA + BA so far determined. A measure of fit E is defined whenever its numerical value is higher than a preset tolerance, new values (all or some) of the modal parameters are re-evaluated, and the procedure is rerun from step 1.

As can be seen, the minimization algorithm is not applied here to identify directly modal parameters from the measured records. This is done, for instance, by Beck4, ' and McVerry,6 in time and frequency domains respectively, achieving an excellent match between the recorded accekrations and those calculated through superposition of the identified modes. In Reference 5 Beck shows that, if the structural scheme and masses are known, the other structural parameters (stiffness and damping matrices) can be uniquely determined, provided acceleration records are available at half or more of the degrees of freedom of the structure.

In the present approach optimization is performed with respect to structural parameters of BA (both masses and stiffnesses), while modal parameters are evaluated through transfer functions (step 1) and subsequently checked (with possible re-evaluation) in step 3. The accuracy of the modal parameters obtained depends on the number of loops of the procedure, that is, on the assumed tolerance for the measure of fit E. If this is kept rather high, as is the case for the applications of Section 5, the match between computed and measured accelerations is less satisfactory than that obtained by the approach of References 4-6. Neverthe- less, static checks proved to be fully satisfactory, as is shown in Section 5.

3. DETERMINATION OF Z&A ( f )

Once AA is known, modal analysis gives all the quantities of interest here: modal masses m A A , i , modal stiffnesses k A A , i and modal shapes &A,i. The choice of the modal damping factors [AA,i is somewhat arbitrary, although hints may come from knowledge of damping values determined experimentally for systems of the same type of buildings (see for example References 1 and 2). It is noted, however, that uncertainties in the assessment of both the scheme of AA and its structural properties do not play an important role in the determination of the equivalent scheme AA + BA. In fact, once system BA has been identified, it forces the responses at any level k of AA to be equal to those actually recorded, correcting any initially poor estimate of AA properties.

In the following, suffix i refers to the ith mode and k to the storey. The lateral acceleration relative to the ground at level k is given by

n

i = 1 = 4 i A , i j A A , i ( t ) (3)

y A A , i ( t ) being the ith modal acceleration. Dynamic equilibrium for the ith mode is expressed by

j A A , i ( l ) + 2cAA, i(2nfAA, i) i A A , i(t) + 'AA, i (2nfAA, i)' = - PAA, i (4)

where fAA,i is the modal frequency for the considered mode. By taking Fourier transforms of both sides of equation (4), and using the following relationships between

transforms of displacements, velocities and accelerations under the hypothesis of zero initial conditions (which is consistent with the possibility of modern equipment of pre-event recording),

EQUIVALENT STRUCTURAL SCHEMES FOR COUPLED SYSTEMS 32 1

where j is the imaginary unit, one obtains

The transfer function (PAA,i $'A, i(f)) represents the correlation between the base input and the ith modal response. Using equation (3) it is easily seen that

(8) - @ i A ( f ) = ( d i A , i PAA, i *LA, i (f )) A g ( f ) i = l

where the term multiplying A g ( f ) is the transfer function of the relative acceleration at level k. The transfer function of the absolute acceleration at k is then

4. IDENTIFICATION OF BA PROPERTIES

Once HBA has been determined, it is possible to estimate the characteristics of the selected normal modes. Let 1 be the number of such modes (I 6 N owing to the assumption of the storey mechanism for BA, N being the total number of degrees of freedom of BA). The estimated modal characteristics are: resonant frequenciesf,, damping factors ci and modal shapes dik. Let us suppose that all the 1 selected modes can be read at all N levels of A, meaning that N recordings are available. This hypothesis will be removed later on.

The damping factor Ci is initially evaluated in terms of the half-power-point bandwidth, and modal shapes are estimated through the one-sided autospectral density function of the response acceleration at the ith normal mode and at the kth l ~ a t i o n . ~ Such evaluations are limited to small values of damping, which may not be the case for masonry buildings. Moreover, peaks in the output autospectrum refer to resonant frequencies and not to undamped natural frequencies which are assumed in the identification scheme and in the relevant equations that follow. These factors as well make it necessary to perform iterations of the identification procedure, with new values for damping and modal shapes, until good agreement between the recorded and the model response is achieved (step 3 of the procedure). The following classical modal equations, for the 1 considered modes, are used

with +,,A,i (known) the ith modal shape vector,f;- (known) the modal frequencies, M B A and K B A respectively the mass and the stiffness matrices of BA (unknown).

A total number of 31 equations is thus available, while unknown quantities are 2(N + I ) , i.e. the 1 modal masses mi, the 1 modal stitlhesses ki, the N masses of the system BA and the N stiffnesses of BA. Let the remaining (2N - I) parameters be grouped into a vector a:

a = ( m , . . . k, . . .) (13) These are the variables used by the minimization algorithm. For any a, equations (10) and (1 1) express the remaining 31 parameters as a function of the (2N - 1 ) variables of minimization (using the measured modal shapes). The corresponding eigenvalue problem can be solved, obtaining a set of undamped calculated modal

322 D. BENEDETTI AND G. M. BENZONI -

frequencies x. These are compared with those measured and a residual error is computed: 1

z(a) = C (w? - d?) i = 1

with wi = 2.nJ and di = 2zJ respectively the ith measured and the calculated circular frequencies. Note that frequencies are obtained by modifiying the undamped frequencies as follows:

BA structural properties are determined by finding the values of a which minimize z(a):

min, z(a) (14) Minimization is performed by means of a sequence of (2N - I) one-dimensional linear optimizations,'.* one for each of the (2N - I) parameters, keeping the others constant. The scheme does not require evaluation of the gradient of function z(a). The crucial phase is the choice of the range of variation for each parameter and the number of steps within it. In the applications discussed in the next section, a wide range of variations of parameters with broad intervals was first selected to find one approximate minimum which was then refined by means of finer scanning. In spite of its conceptual simplicity, the described approach proves capable of identifying the absolute minimum of z(a), although convergence is rather lengthy.

After finding M,A and K,A and all the relevant modal quantities, it is possible to build the corresponding transfer function &A at location k in the same way shown in the previous section for diA(f>. Here too one can consider all or part of the N identified modes of BA. The transfer function of the assembled equivalent system (AA + BA) is then evaluated:

(17)

This function is used to calculate the response acceleration at levels k by applying the inverse Fourier transform operator:

The comparison between the calculated and the measured accelerations at k, gi( t ) and Xi( t ) , is performed by evaluation of the following quantity E, as the lower E, the more satisfactory the match:

fi",.f) = & A ( f ) + @ A ( . f >

Kk,(t) = IFT(A:(~)A,(~)) (18)

7 (19)

r.m.s. [XA,p - ~ , p ] E =

1XA,maxI

where ZP is the sequence of peaks in acceleration x( t ) and IX,,,l the maximum absolute value of the measured record. The quantity E is computed at all the degrees of freedom of A where accelerations have been recorded, and the highest value is retained as the control parameter in this third step of the procedure.

High values of E may be due to poor initial estimation of modal damping factors ci, or to an inappropriate choice of BA modes for the identification procedure. The selected modes of BA may be inconsistent with the basic hypothesis (rigid link between AA and BA) assumed to derive the equivalent structural scheme. This is the case (see next section) when one of the chosen modes is associated with relative oscillations between different parts of the overall original system A + B. If this occurs, the calculated complex functions &(f> do not contain the dominant frequencies of the effectively measured responses P(t) which can not therefore be recovered from equation (18). Consequently a new selection of the significant modes to use in the procedure is necessary. When this selection is appropriate (low E values) further refinement is achieved by minimizing &as a function of the modal damping factors. Minimization halts when E < 5, I being the preset tolerance.

The accuracy of the results, determined by the loops governed by E, affects not only the match between calculated and measured accelerations, but the structural parameters of BA as well, whenever the third phase involves any modification of BA shape vectors. However, at least in the realm of the applications discussed in the next section, satisfactory static responses (compared to the real ones) have been obtained for rather high values of E, that is for a limited number of loops of the procedure, although the match between accelerations is not perfect.


Up to now it has been assumed that the number of records available sulTices for the determination of d,, at each level k of BA. However, this may not be the case, owing to instrumental malfunction or to the limited number of accelerometers installed on the structure. The above procedure may nevertheless be applied, with some minor changes, to identify BA properties even in the limit case in which only one recording (plus the base input) is available, provided the first mode can be clearly read.

In this case it is not possible to evaluate BA modal shapes, but the significant modes, and their frequencies, can again be recovered. An arbitrary shape vector is assumed for the first mode (e.g. linear variation of d, along the height of BA). The procedure is activated for 1 = 1 and an arbitrary shape is used in the first iteration of the minimization to express the 31 BA properties which depend on the (2N - I ) variables of the optimization. During subsequent iterations the shape of the first mode resulting from the solution of the relevant eigenproblem is used, as determined by the current set of parameters a. This differs from the procedure described above: there the measured modal shapes are constantly used. However, in this case as well, modal shapes are assumed to be known, since in any step of the minimization the shape vectors obtained in the previous step are used. Function (14) contains all the frequencies which may be read in the only fiBA which is available. In the applications of Section 5, responses, both static and dynamic, very close to those obtained through the availability of N recordings, have also been obtained for very bad initial estimates of the first shape vector. Note that a single recording proved sufficient for Udwadia and Sharma9 to identify the structural properties of a linear chain system (as BA is assumed to be). There, however, the masses were assumed to be known, while here the first mode shape is considered as known and the masses are unknown.

5. APPLICATIONS

The above procedure has been applied to two different classes of systems. The first consists of three storey structures (two frames and a wall) for which equivalence is sought by adding an equal (3) number of masses to AA. The second class deals with a frame system where both A and B are five storeys high and BA is still a three mass system, having its masses connected only to some AA levels. For each structural assembly the subsystems A and B were chosen and the response accelerations numerically evaluated by finite element simulation, at all the levels k of A (in the overall structure). These responses constitute the input measured quantities for the procedure.

(a) Three storey buildings The two frame systems (Figure 4) (denoted in the following as F1 and F2) present the same geometry and

masses, but different stiffness distributions: in F2 the subsystem B is much stiffer than in F1, while subsystems A are the same for F1 and F2. Both for A and B columns at each level have the same lateral stiffness. Moreover, A has two spans and B three spans. Table I shows the main characteristics of F1 and F2.

M BA

Figure 4. Original configuration and equivalent scheme for F1 and F2

324 D. BENEDETTI A N D G. M. BENZONI

Masses rn, are expressed in N s2/cm and refer to each storey level, stiffnesses ki are in kN/cm and represent the lateral stiffness of a single column at the given level. Dynamic analyses of F1 and F2 were performed by assuming a damping factor C; = 0.05 for all the modes.

Figure 5 shows the wall system, denoted by W1, to which the procedure was applied. This wall represents a real building in Gubbio which is monitored through Teledyne A-700 accelerometers. The subsystems A and B are also shown in the figure together with geometrical sizes. A rigid slab is simulated at each storey level of A. The assumed weight per volume unit of the wall is 16 kN/m3, while the elastic properties used in the modelling are:

E = 360.0 MPa

G = 40.0 MPa

These numerical values simulate masonry. For the dynamic analysis the damping factor assumed was ri = a/4.lrf,; a = 4,fi being the modal frequencies of the finite elements model. W1 is a mass distributed case, while in F1 and F2 masses are concentrated at each storey level. In order to reduce the original systems to the equivalent structures of Figures 4 and 5, the schemes AA for subsystems were derived by assuming frames with rigid beams and with lateral stiffnesses equal to those of A for F1 and F2. For W1 this was done by considering masonry piers at each storey as columns of equivalent lateral stiffness determined as illustrated in References 1 and 2.

The frequencies of the three natural modes of AA are respectively 2445.46 and 8.5 Hz for F1 and F2, and 1.95,5-47 and 8.4 Hz for W1. As an example, Figure 6 shows the amplitude of the transfer function of W1 (at the third storey), determined in the manner previously described.

In order to apply equation (2) to evaluate function a:,,, the transfer functions H:(f) were obtained by submitting F1 and F2 to the base excitation of Figure 7(a), and W1 to that of Figure 7@). The first input is the

Table I. Mass and stiffness distributions for F1 and F2

A

m1 m2 m3 k l k2 k3 F1 398.0 306.0 184-0 105.0 105-0 60.0 F2 398.0 306.0 184.0 105.0 105.0 60.0

B

m1 m2 m3 k , k2 k3 F1 490.0 392,*0 294.0 80.0 46.0 10.0 F2 490.0 392.0 294.0 4800 277.0 60.0

aa.9 9.5 11.40 - - >I

AA BA

I I

Figure 5. Original configuration ( s h in metres) and equivalent scheme for W1


Figbre 6. Modulus of AA transfer function for W1 (third storey)

n I

-100.0 ' 1 I , I 1 I I . 0 . b 2.00 4.00 6. a.b 10.00 1 :

Figure 7(a). Base input for F1, F2 and F3

-5.0 ' 0.60 1.b 2.b 3.W 4 . b 5 . b 6.

time (s) Figure 7(b). Base input for W 1

Jo

0

Sturno record (28/11/80) scaled to aM = 95 c m / s 2 , while the second input was actually recorded in Gubbio in 1984 (aM = 3-6 cm/s2). It is now assumed that output recordings are available at all the levels k of A. Functions H,(f) have been evaluated through a Kaiser windowing of the waveforms. Figure 8 shows, by way of example, the amplitude of H i (f) for W 1.

The transfer functions of systems BA can now be evaluated as an example Figure 9 shows such a function for the third storey of W1.


Figure 8. Modulus of the measured transfer function for W1 (scheme A + B, third storey)

Figure 9. Modulus of BA transfer function for W1 (third storey)

Table 11. Measured BA frequencies

F1 1.95 4.35 F2 3.51 6.83 w1 2.34 6.00

The final selection of the modes for the identification was performed by using the third phase of the procedure described in the previous section, i.e. keeping those modes showing good agreement between model and measured responses. In order to assess the efficiency of the procedure, only two modes have been used to construct the l?;,,(f), needed to apply equation (17). The frequencies of such modes are shown in Table 11.

For W1 the mode corresponding tof= 3.7 Hz (see Figure 9) was neglected, since it gave rise to responses with a frequency content that differed from that of the measured ones. A more detailed analysis, performed on the basis of outputs of the finite element model of W1 showed that this frequency is associated with relative oscillations of the two main portions of the wall. The results of the identification of BA properties are summarized in Table 111. Effwtive participation factors, storey masses and stiffnesses are shown for each system BA. The table also shows the minimum values of z(a) obtained. With respect to these it should be noted that z(a) is affected by the squares of circular frequencies. Hence the reported values for zmi, denote a good agreement between the measured frequencies (shown in parentheses) and the evaluated ones, which are given by the last solution of the eigenvalue problem.

Tabl

e 11

1. R

esul

ts o

f the

iden

tific

atio

n (a

ll re

cord

s av

aila

ble)

F1

F2

w1 k

=l

k=

2

k=

3

k=

l k

=2

k

=3

k

=l

k=

2

k=

3

0.27

8 0.

685

1.50

5 04

24

0.60

5 -0

568

0.29

7 -0

.290

0.

063

755.

17

4905

0 51

208

494.

42

403.

29

207.

10

0.25

1-

949

(1-9

5)

4.35

0 (4

.35)

7.

009

(6-8

3)

0.43

8 Q

853

1.35

6

0.14

2 -0

.199

00

52

0.41

9 0.

347

- 0.409

2119

2 13

298

9378

3 50

33.8

7 27

46.8

82

840

3.46

3.49

1 (3

.51)

6.

797

(6.8

3)

11.1

83

(11.

32)

0.50

9 0.

882

1.24

9 0.

310

0.26

8 - 0.

286

0.17

9 -0

150

0036

14

09-9

8 15

2604

16

35.0

3 17

37.7

2 19

62.0

11

99.0

1.

17

2.33

7 (2

.34)

6.

001

(6.0)

9.8

(10.

35)

8 7 C

w

N 4


The frequencies of the third modes were not controlled by z(a) in the present cases (for which it was assumed I = 2). Consequently their shift with respect to the measured ones is slightly greater than those occurring for the other lower frequencies.

Figures lqa), lqb), lqc) show the model absolute accelerations [computed by equation (lS)] compared with the measured accelerations for the three systems F1, F2 and W1 (respectively at the 3rd, 1st and 3rd storey). Dashed lines refer to calculated responses and solid lines to the measured ones.

300.0 -- u) 2 0 . 0 1 1

- measured _....____ calculated

-300.0 ' I 1 - 0.60 2.60 4.60 6.60 8.dO 10.00 1;

time (s) 10

Figure lqa). Measured and calculated response accelerations for F1 (third storey)

150.0 I 4 u1 100.0

- measured 1 _ _ _ _ _ _ _ _ _ calculated

0 1

- 150.0 , ' , ' I * 0.b 2.60 4.60 6.00 8.00 10.00 12.00

time (s)

Figure lqb). Measured and calculated response accelerations for F2 (first storey)

5.0

- measured ....____. calculated

-5.0 ' I 0 . b 1.60 2.60 3.60 4.60 5.00 6.

time (s) 0

Figure lqc). Measured and calculated response accelerations for W1 (third storey)


Table IV shows the errors resulting for these three responses, and the initial and final estimates of [ I . It will be noted that E values are relatively high. This is mainly due to the fact that the length of the records used in the procedure was limited to 1024 points, thus conditioning the accuracy of the results. This reduced length was required by the use of a personal computer for all the computations, with a limited data handling capability. This choice was deliberate; it was in fact meant to propose a method of analysis which would give reliable results even with the limited computational tools available in most practical applications. For the same reason, minimization was performed in only ten steps within each range of variation of the variables, progressively restricting the ranges, as described in Section 4, and a rather large tolerance ( E = 0.20) is employed. A slightly better match occurs if all BA modes are used in identification. The comparison between signals is not shown here owing to space limitations.

Of particular interest is the comparison between the static responses of (A + B) and of (AA + BA). Since the masses differ in the two situations, comparison cannot be made by applying a constant set of lateral forces. It is instead performed by applying, to both the original systems and the models, lateral storey forces proportional to the actual storey masses (g times the masses). These forces are applied to the left hand side of the system, simulating an equivalent static analysis.

Table V shows the results of such analysis. Storey displacements and total base shears in A and AA respectively are reported for the three systems F1, F2, W1. As can be seen, shifts of calculated displacements with respect to the measured ones range, for both F1 and F2, from 1 to 8 per cent, while base shears differ by a maximum of 5 per cent. The shifts are larger for W1, reaching 20 per cent for displacements at the first storey, while for the base shear the shift between calculated and real values is limited to 6 per cent. These larger variations are due to the enforcement of a simpler framed scheme for AA instead of its original 2-D configuration. Moreover, the equivalent model is obtained by the addition of a simple 3 DOF system. For F2, where B is much stiffer than A, the calculated base shear for AA is considerably lower than the one pertaining to A masses. The opposite is true for F1, where B is less stiff than A. Similar shifts between calculated and measured responses hold for all the locations of AA not referenced in figures and tables shown here.

If only one response record is available, identification is made as described in Section 4. Suppose that for W1 only the response at the third storey of A is known. The transfer function can still be evaluated (Figure 9). It supplies the significant modal frequencies, but modal shapes cannot now be measured, Two different hypotheses are assumed for the first shape vector to activate the modified procedure. The first,

Table IV. Initial and final damping values and control parameters E

E li(init.) li(final)

F1 (3rd storey) 0.184 005 0.045 F2 (1st storey) 0.167 0.05 0.030 W1 (3rd storey) 0.131 0-05 0.030

Table V. Static displacements and base shears

~

F1 real 3.261 1 5.6166 7.7667 1023.41 F1 model 3.2506 5.3188 7.1009 102043 F2 real 0.93 1 7 1.7638 29017 292.53 F2 model 0.9830 1.8804 2.9875 308.62 W1 real 2-4200 4.2000 5*0800 305591 W1 model 2-9310 4.7780 60140 2870.40

3 30 D. BENEDETTI AND G. M. BENZONI

denoted as (a), assumes a linear variation of c # ~ ~ , ~ ~ along the height, according to the ratios 1 :2: 3. The second hypothesis, denoted as (b), assumes shape variations described by the ratios 1 :2: 1. These values are used in the first iteration of minimization, being the function z(a) expressed in terms of all the known frequencies.

Results of the two identifications are shown in Table VI, which gives effective participation factors, BA masses, stiffnesses and frequencies, static storey displacements, total AA base shear.

As can be seen, the two hypotheses, although quite different from one another, give similar results, rather close to those obtained by assuming that all records are available. In spite of the different initial modal shapes the modified procedure converges to effective participation factors close to those of Table 111. This in turn produces a very good similarity between response accelerations (not shown here). It is of interest to point out the similarity of static responses as well: the modified procedure can be used successfully to give reliable schemes for static analysis, needed for practical purposes of design. This is also the case when the available recording concerns a location other than the firststorey, as assumed in Reference 9.

(b) Five storey buildings It has been supposed so far that the added system BA has the same number of masses as the storeys of

buildings B, whose effects on A it is to represent. Consider now the array of Figure 11, denoted as F3, the properties of which are summarized in Table VII, where storey masses and column lateral stiffnesses, expressed in N s2/cm and kN/cm respectively, are reported.

The equivalent scheme to be identified is also shown in Figure 11. It will be noted that BA masses are connected only with the first, third and fifth level of AA. This implies that recordings are available at these

Table VI. Results of the identification (missing records)

Hypotheses (a) Hypothesis (b) k = l k = 2 k = 3 k = l k = 2 k = 3

0.495 0915 0333 0215 0.171 - 0166

1704.97 1667-7 1962.0 1885-69

2.339 6-0 9-19

2-87 484 28 13.5

1.261 0530 0.9 13 1.255 - 0.312 0.297 0235 - 0.298 0.05 1 0172 - 0.149 0.042

1689-28 1512.7 1634-3 1580-4 1329979 1743.98 1962.0 1253.55

2.339 5.999 9.5 1

2919.45 602 2-90 4.84 6.028

A B

Figure 11. Original configuration and equivalent scheme for 5 storey arrays


locations. Different positions of the instruments would call for different locations of the rigid links between AA and BA. AA has been derived in turn by assuming a two-span frame with rigid beams (while A and B beams are flexible) and with storey lateral stiffnesses equal to those of A. The base input for F3 is the same as that used for F1 and F2. In the first instance three response accelerations are assumed to be known (at k = 1, 3,Sk then only one response is assumed (at k = 1). All the relevant complex functions needed for the analysis are derived from the recorded signals, and identifications are performed as in previous cases, When only one signal is available, the same two hypotheses above concerning modal shapes are used. Table VIII reports the basic results of the procedure in the case of three records available in a format similar to that of Table 111. Initial and final estimates of damping are also reported. The calculated and real relative accelerations at the first and second levels of A are shown respectively in Figures 12 and 13. Both show a satisfactory agreement

Table VII. Mass and stiffness distributions for 5 storey buildings

A 398.0 306-0 184.0 184.0 184.0 B 500.0 400.0 3000 300.0 300.0

kl k2 k3 k4 kS

A 105.0 105.0 600 13.0 130 B 64.0 37.0 37.0 13.0 13.0

Table VIII. Results of the identification for 5 storey buildings (three records available)

k = l k = 3 k = 5 Cnit. Cfin.,

Mode 1 0-0576 0.2490 1.1350 0.05 0.08 Mode 2 0.4 126 1.1350 - 0.150 0 0 5 0.06 Mode 3 Q5306 - 0.384 0.0155 0.05 0.04

kLI* orN/f=4 165.0 474 9.21 fl (Hz) 0.977 (0.98) f 2 (Hd 2-710 (2.73) f 3 (Hz) 5-40 1 (5.37)

m, (N s'/cm) 203.0 101.0 195.0

150.0 - m w s u d

0 - 1 sao

0.60 2.60 4 . b 6.00 8.00 10.00 1; time (s)

m

Figure 12. Measured and calculated response accelerations for 5 storey arrays (first storey)

332 D. BENEDE'ITI AND G. M. BENZONI

180.0 I

120.0

-60.0

-120.0

1 __ measured ............. calculated

-180.0 - I , I , 1 I I

0.00 2.00 4.00 6.00 8.00 10.00 12 t ime (s)

0

Figure 13. Measured and calculated response accelerations for 5 storey arrays (second storey)

Table IX. Static displacements and storey shears for 5 storey buildings

Displacement (cm)

AA storey shears (kN)

k Orig. 3R lR.a lR.b Orig. 3R lR.a lR.b

1 4.770 5.050 4.150 5.040 1527.0 1618.0 1330.0 1615-0 2 9040 7.810 6.860 7.820 13660 875.6 866.3 887.2 3 12.96 10.92 9.890 1096 725.2 569.8 5605 581.4 4 25.05 21.24 2012 21.25 483.4 413.4 409.4 411.3 5 31.09 27.20 25.77 26.94 241.8 230.0 226.0 227.9

between model and reality. Note that the latter figure refers to a location where no connection with BA exists. Measures of fit E between calculated and measured accelerations range from 0.2 to 0.29 for all AA storeys: these values are higher than the ones recorded for three storey assemblies owing to the addition to A of a system with a limited number of modes. Static responses, determined by applying lateral forces g-times proportional to masses, are shown in Table IX. Columns denoted by 3R refer to results obtained using three recorded signals, and columns 1R.a, b refer to those obtained from only one signal applying hypotheses (a) and (b) respectively. Original quantities are also shown in the table.

As can be seen, the original quantities and those determined with equivalent schemes show close agreement at the locations where BA is connected to AA. The maximum shift recorded there is 10 per cent lower than the reference data. Larger shifts are seen at other locations, especially at the second storey. The compensation for this minor inaccuracy is, however, the possibility of refemng to a simplified scheme suitable for engineering purposes.

6. CONCLUSIONS

The procedure described allows the transformation of a complex system, with several interacting parts, into a simpler structure suitable for design analysis, thus avoiding the need to consider the overall assembly, which would be impractical in real cases. Results are satisfactory, in spite of the simplicity of the equivalent schemes obtained and the use of a three mass added system. Their accuracy might be improved by greater computer time and by the use of more powerful computing tools. In the present case all computations were performed on a personal computer for the reasons given. Many aspects merit further investigation, which is currently in progress: e.g. the inclusion of torsional effects and the analysis of systems with significant differences in height.


ACKNOWLEDGEMENTS

This research has been carried out with the financial support of Regione Umbria, Italy. The authors wish specially to thank Dr Menichetti for supplying the instrumental network from which this work originated.

REFERENCES

1. M. Tomazevic, ‘Dynamic modelling of masonry buildings: Storey mechanism model as a simple alternative’, Earthquake eng. struct.

2. D. Benedetti, G. Benzoni and P. Pezzoli, ‘Seismic behaviour of a non-symmetric masonry building’, Eur. earthquake eng. 1, 20-30

3. J. S. Bendat and A. G. Piersol, Engineering Applications of CorreZation and Spectral Analysis, Wiley, New York, 1980. 4. J. L. Beck, ‘Structural identification using linear model and earthquake records’, Earthquake eng. struct. dyn. 8, 145-160 (1980). 5. J. L. Beck, ‘Determining models of structures from earthquake records’, Report No. EERL 78-01, California Institute of Technology,

6. G. H. McVerry, ‘Structural identification in the frequency domain from earthquake records’, Earthquake eng. struct. dyn. 8,161-180

7. C. Y. Pen& ‘Generalized modal identification of linear and nonlinear dynamic systems’, Report No. EERL 87-05, California Institute

8. A. 0. Cifuentes and W. D. Iwan, ‘Nonlinear system identification based on modelling of restoring force behaviour’, Soil dyn.

9. F. E. Udwadia and D. K. Sharma, ‘Some uniqueness results related to building structural identification’, SIAM j . appl. math. 34,

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Identification of equivalent structural schemes for coupled systems