system identification of civil engineering structures

12
System identification of civil engineering structures J.-G. B~LIVEAU' Dtparftnet~f of Civil Etlgitleerit~g~ l d Mechcrt~iccrl Etrgit~eeritrg, Ut~iversify oJ' Verttlotlr, Bio.lit~grot~. VT 05405, U.S.A Received August 15. 1985 Revised manuscript accepted September 2, 1986 The comparison of measured dynamic characteristics or response of large structurcs with that of an appropriate finite clement model with all its underlying assumptions often reveals discrepancies. This may be due to inlproperly determined parameters, such as interstory stiffness, mass of different stories, and the modulus of elasticity of the concrete, as well as the inadequacies of the model. The measured dynamic response generally occurs in one of three forms: timc response, frequency response, and modal data. For time response data, either in free vibration or for a known input. parameters are estimated by proper adjustments to match more closely the measured motion. For steady-state frequency response, a sinusoidal load (or synchronized loads) is input mechanically and the response, both in amplitude and in phase, is measured for different frequencies of excitation. Damped resonant frequencies, the associated modal damping ratios, and the corresponding mode shapes are the measured quantities for modal data. The finite element models used for civil engineering structures often incorporate a large number of degrees of freedom. Measured response is sparse and usually limited to the lower frequency range. A procedure for estimating these parameters must be able to allow for the small amount of data and must utilize cfficient numerical algorithms to determine the best parameters. Nonlinear least squares, within a Bayesian framework, is such a method. It can be applied to time-history data, steady-state response, and modal characteristics. This method is used to determine aerodynamic coefficients of a scale model of a suspension bridge deck from free response data in a wind tunnel, stiffness parameters from frequency measurenlents of a 5-story steel building frame loaded by mechanical exciters on the roof, and stiffness parameters from modal data of a 12-story reinforced concrete frame, as obtained from transient wind observation of lateral accelerations. Le comportement dynamique mesure de structures a plusieurs degres de libertt ne coincide pas avec celui calculC 8 partir de modttles d'CICments finis qui comprennent des hypothkses de simplification. Plusieurs raisons pour cette diffkrence existent. Citons, par exemple, les valeurs de paramktres utilises, tel que la rigidit6 entre les etages, la massc calculCe dcs divers plachers, le module du bCton, qui peuvent ktre pas tout 8 fait justes et, bien sCr, les hypotheses possiblement inadequates du modttle numCriaue. Les mesures du comportement dynamique sont gdnkralement repres6ntCes sous une de trois formcs: mesures de sdries de temps, mesures de reponse en rCgime permanent. et mesures modales. L'adjustement des paramktres, pour approcher les mesures de series de temps par moindre carre. par exemple. se fait traditionnellcment sur les mesures temporelles, soit en regime transitoire, c'est-8-dire vibration libre, ou dans le cas ou I'excitation est egalement mesurCe. En rCgime permanent, on mesure la rCponse en amplitude et phase pour un ou plusieurs excitation toutes a la mCme frCquencc, frCquence qui est variCe a I'intCrieur d'une gamme predeterminke. L'analyse modalc consiste en mesures de frkquences de resonance, rapports d'amortissement critiques, et les modes de vibration, souvent determines 8 partir d'excitations alCatoires. Les modklcs d'ClCments finis pour des structurcs de genic civil contiennent plusieurs degrCs de IibertC. Les mesures du comportement dynamique sont IimitCes a quclqucs degrds de IibertC, ou une gamme de frCquences restreintes. Une mcthode d'estimation des parametres doit tenir compte de ccs points et doit utiliser des algorithmes en consCquence. L'approche d'estimation bayesienne est une mCthode ayant les caractCristiques desirees et s'applique aussi bien sur des mesures tempo- relles, fr6quentielle.s ou bien modales. Dans cct article, elle est utilisCe pour determiner dcs coefficients atrodynamiques d'un modkle rCduit de pont suspcndu test6 en soufflerie a partir de mesures de siries de temps et les paramktres de rigidit6 de deux bbtiments, le premier a 5 etages, soumis 8 des charges sinuso'idales appliqukes par des excitatcurs mdcaniques sur le toit, et un 12 de douze Ctages, avec des mesures modales d'accil6rateurs horizontaux dans des conditions ambiantes. Can. J. Civ. Eng. 14. 7-18 (1987) Introduction The area of system identification has long been associated with the forecasting and control of dynamic processes in elec- trical and chemical engineering applications (Sage and Melsa 197 1; Graupe 1972) with some mechanical engineering appli- cations (Pilkey and Cohen 1972). The application to structural dynamics is relatively recent, but is actually undergoing rapid expansion (Hart and Yao 1977; Shinozuka et al. 1982; Beliveau 1974b). The structural engineer is often asked to evaluate the strength, stiffness, and stability aspects of existing structures. NOTE: Written discussion of the papcr is welcomed and will be received by the Editor until May 3 1, 1987 (address inside front cover). ' Previous address: Department of Civil Engineering, Faculty of Applied Sciences, Universitd de Sherbrooke, Sherbrooke, Que., Can- ada JIK 2RI. Often, these are the object of extensive retrofitting or mod- ifications, and their structural integrity is critical in the eco- nomic decisions taken with regards to the structure. In this paper, these two disciplines are brought together. Measured data of scale models and of actual structures can be used in estimating parameters of a mathematical model, often based on finite elements. The data may be in the form of observed response in time to measured loads and to random excitation. It may also be in the form of frequency response amplitude and phase within a given frequency range, or in modal format such as resonant frequencies, modal damping ratios, and their respective vibration mode shapes. Civil engineering structures, rather than mechanical ele- ments, are investigated. In one example, a section model of a suspension bridge deck tested in a wind tunnel with time- history data is studied (Beliveau 1974n). A 5-story steel-frame building and 12-story reinforced concrete shear-wall building Can. J. Civ. Eng. Downloaded from www.nrcresearchpress.com by University of Queensland on 11/23/14 For personal use only.

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Page 1: System identification of civil engineering structures

System identification of civil engineering structures

J.-G. B ~ L I V E A U ' Dtparf tne t~f o f Civil Etlgitleerit~g ~ l d Mechcrt~iccrl Etrgit~eeritrg, Ut~iversify oJ' Verttlotlr, Bio.lit~grot~. VT 05405, U.S.A

Received August 15. 1985 Revised manuscript accepted September 2, 1986

The comparison of measured dynamic characteristics or response of large structurcs with that of an appropriate finite clement model with all its underlying assumptions often reveals discrepancies. This may be due to inlproperly determined parameters, such as interstory stiffness, mass of different stories, and the modulus of elasticity of the concrete, as well as the inadequacies of the model.

The measured dynamic response generally occurs in one of three forms: timc response, frequency response, and modal data. For time response data, either in free vibration or for a known input. parameters are estimated by proper adjustments to match more closely the measured motion. For steady-state frequency response, a sinusoidal load (or synchronized loads) is input mechanically and the response, both in amplitude and in phase, is measured for different frequencies of excitation. Damped resonant frequencies, the associated modal damping ratios, and the corresponding mode shapes are the measured quantities for modal data.

The finite element models used for civil engineering structures often incorporate a large number of degrees of freedom. Measured response is sparse and usually limited to the lower frequency range. A procedure for estimating these parameters must be able to allow for the small amount of data and must utilize cfficient numerical algorithms to determine the best parameters. Nonlinear least squares, within a Bayesian framework, is such a method. I t can be applied to time-history data, steady-state response, and modal characteristics. This method is used to determine aerodynamic coefficients of a scale model of a suspension bridge deck from free response data in a wind tunnel, stiffness parameters from frequency measurenlents of a 5-story steel building frame loaded by mechanical exciters on the roof, and stiffness parameters from modal data of a 12-story reinforced concrete frame, as obtained from transient wind observation of lateral accelerations.

Le comportement dynamique mesure de structures a plusieurs degres de libertt ne coincide pas avec celui calculC 8 partir de modttles d'CICments finis qui comprennent des hypothkses de simplification. Plusieurs raisons pour cette diffkrence existent. Citons, par exemple, les valeurs de paramktres utilises, tel que la rigidit6 entre les etages, la massc calculCe dcs divers plachers, le module du bCton, qui peuvent ktre pas tout 8 fait justes et, bien sCr, les hypotheses possiblement inadequates du modttle numCriaue.

Les mesures du comportement dynamique sont gdnkralement repres6ntCes sous une de trois formcs: mesures de sdries de temps, mesures de reponse en rCgime permanent. et mesures modales. L'adjustement des paramktres, pour approcher les mesures de series de temps par moindre carre. par exemple. se fait traditionnellcment sur les mesures temporelles, soit en regime transitoire, c'est-8-dire vibration libre, ou dans le cas ou I'excitation est egalement mesurCe. En rCgime permanent, on mesure la rCponse en amplitude et phase pour un ou plusieurs excitation toutes a la mCme frCquencc, frCquence qui est variCe a I'intCrieur d'une gamme predeterminke. L'analyse modalc consiste en mesures de frkquences de resonance, rapports d'amortissement critiques, et les modes de vibration, souvent determines 8 partir d'excitations alCatoires.

Les modklcs d'ClCments finis pour des structurcs de genic civil contiennent plusieurs degrCs de IibertC. Les mesures du comportement dynamique sont IimitCes a quclqucs degrds de IibertC, ou une gamme de frCquences restreintes. Une mcthode d'estimation des parametres doit tenir compte de ccs points et doit utiliser des algorithmes en consCquence. L'approche d'estimation bayesienne est une mCthode ayant les caractCristiques desirees et s'applique aussi bien sur des mesures tempo- relles, fr6quentielle.s ou bien modales. Dans cct article, elle est utilisCe pour determiner dcs coefficients atrodynamiques d'un modkle rCduit de pont suspcndu test6 en soufflerie a partir de mesures de siries de temps et les paramktres de rigidit6 de deux bbtiments, le premier a 5 etages, soumis 8 des charges sinuso'idales appliqukes par des excitatcurs mdcaniques sur le toit, et un 12 de douze Ctages, avec des mesures modales d'accil6rateurs horizontaux dans des conditions ambiantes.

Can. J. Civ. Eng. 14. 7-18 (1987)

Introduction The area of system identification has long been associated

with the forecasting and control of dynamic processes in elec- trical and chemical engineering applications (Sage and Melsa 197 1; Graupe 1972) with some mechanical engineering appli- cations (Pilkey and Cohen 1972). The application to structural dynamics is relatively recent, but is actually undergoing rapid expansion (Hart and Yao 1977; Shinozuka et al. 1982; Beliveau 1974b).

The structural engineer is often asked to evaluate the strength, stiffness, and stability aspects of existing structures.

NOTE: Written discussion of the papcr is welcomed and will be received by the Editor until May 3 1 , 1987 (address inside front cover).

' Previous address: Department of Civil Engineering, Faculty of Applied Sciences, Universitd de Sherbrooke, Sherbrooke, Que., Can- ada J I K 2RI .

Often, these are the object of extensive retrofitting or mod- ifications, and their structural integrity is critical in the eco- nomic decisions taken with regards to the structure.

In this paper, these two disciplines are brought together. Measured data of scale models and of actual structures can be used in estimating parameters of a mathematical model, often based on finite elements. The data may be in the form of observed response in time to measured loads and to random excitation. It may also be in the form of frequency response amplitude and phase within a given frequency range, or in modal format such as resonant frequencies, modal damping ratios, and their respective vibration mode shapes.

Civil engineering structures, rather than mechanical ele- ments, are investigated. In one example, a section model of a suspension bridge deck tested in a wind tunnel with time- history data is studied (Beliveau 1974n). A 5-story steel-frame building and 12-story reinforced concrete shear-wall building

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8 CAN. J . CIV. ENG.

are then identified using frequency (Favillier 1979) and modal data (Chater 198 I) , respectively. Unknown parameters are considered to be aerodynamic coefficients for the suspension bridge example and interstory stiffness coefficients for the buildings, though mass parameters as well as member stiffness and geometric properties may be considered as the unknown parameters (Bkliveai~ et (11. 1984).

Only two degrees of freedom are investigated for the sus- pension bridge and each of the stories in the two buildings is assumed to have three degrees of freedom only, two lateral displacen~ents and one torsional rotation about the vertical axis. These correspond to a rigicl-slab assumption. Furthermore, the lateral and torsional motions are assumed to be uncoupled. The two degrees of freedom of the s c ~ ~ l e model of the bridge deck are the vertical displacement and rotation about the center line. These are assumed to be structurally uncoupled, all coupling is due to the assumed quasi-steady aerodynamic loads.

Thus, the identification is reduced to a problem of parameter estimation rather than a general system identification adapta- tion. Off-line con~plete data is assumed rather than on-line sequential estimation procedures. There is no discussion of the control and forecasting aspects of the dynamic behavior, though the sensitivity relations presented are useful in these areas. Finally, parametric rather than nonparametric statistical modes are assumed, the parameters to be estimated being phys- ical rather than modal.

The numerical algorithms use the special nature of the matrix system in structural dynamics and in particular real symmetric matrices are assumed with banded and sparse nature of both the mass and the stiffness matrices, one of which is assumed to be positive definite. The damping matrix is linlited to the viscous type. The last two examples of structures presented (the two buildings) are, however, assumed to have normal modes. The eigenvalues of the undamped system and the first-order repre- sentation of the damped systems are assumed to be non- repeated. In addition, these eigenvalues and their correspond- ing eigenvectors are assumed to occur in complex conjugate pairs for the damped structural representation.

The paper is divided into four sections. The first section presents the linear ordinary differential equations with constant coefficients used in structural dynamics, in both second- and first-order state variable form. Expressions for modal informa- tion, frequency response, and time-history response are given. Parameter estimation is next presented as a linear relation be- tween the n~easured quantities and the parameters. It is presented as an optimization problem, on a statistical basis, and finally as a linear algebraic problem. It is then generalized to the case of nonlinear least squares.

The third section presents numerically efficient methods for calculating the sensitivity coefficients for modal data, time series analysis, and frequency response. In the last section, the three examples mentioned earlier are presented.

Structural dynamics The dynamic behavior of structures about an equilibrium

position is often given by a system of second-order linear ordinary differential equations

where [MI, [ C ] , and [K] are the mass, damping, and stiffness matrices, respectively. {x), {i), and {:;) are the vectors of displacements from equilibrium, their velocities, and ac- clerations, respectively. Often both the mass and the stiffness

matrices are symmetric, with both or at least one of them, i~sually the stiffness matrix, being positive definite. { f ' ) is the vector of forces of dimension 11, where 11 is the number of independent degrees of freedom characterizing the motion. All matrices are square and of dimension 11 X 11 and are assumed to have constant coefficients. Viscous damping is assumed.

First-order Jorrn~~latio~z For purposes of numerical sin~ulation and integration, [ 1 ] is

often written as a system of first-order equations

121 { i } = [Ql {Y> + {g}

where

101 I 111 -------- + ------- -[MI-' [K] I -[MI-' [ C ] 1

with - I representing the inverse, and [ I ] is the identity matrix.

Naturcll fieyuet1cie.s The undamped natural frequencies are obtained by setting

{ f ) equal to zero in [ I ] , neglecting damping, and assuming a sinusoidal solution

161 {s} = {+) sin O t

This yields the generalized eignevalue problem

where 0, are the undamped natural frequencies and (4,) are the corresponding normal modes of vibration.

181 [+I' [MI [+I = 111

[+I' [MI [+I = [Rl'

where [O]' is a diagonal matrix.

Eigenchnracteristics ,for rlatnped systems The quadratic eigenvalue problem corresponding to the

damped system ([ I ] ) is obtained by assuming a solution of the form

[ lo] {x) = { u ) ehr

for the unforced system with both the eigenvalues, A and the eigenvector {u) being complex. This yields the quadratic eigen- value problem

[I I] [h2[M] + A[C] + [K]] { u ) = (0)

The eigenvalues A and eigenvectors [V] of the matrix [Q] ([3]) are related to A and { u ) by the following relations (Bkliveau 1977):

[ U ] [u:!:] ----- + ------ [U] [A] I [U*] [A'%]

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Page 3: System identification of civil engineering structures

written in the context of a system of linear algebraic ecluations

1241 I A I { P I = {(I} L . -

where the elements of [ A ] are the known cosine and sine terms, Freqlretlcy re.spoi~.se { / I ) are unknown coefficients, and the vector {cl) is the dis-

The steady-state sinusoidal response [or the linear system of cretized series, i.e., the measured data. equations of structural dynamics to sinusoidal input at a radial Premultiplication of 1241 by the transpose of [A] yieltls the frequency w is given by symmetric system of equations

Numerical techniques advantageous for symmetric matrices can then be used to determine {p). An alternate approach would be to rewrite (241 in the following manner by adding experi- mental errors {E}:

where the complex vectors {X} and {F) are related by the frequency response matrix [HI. 1261 {El = ((1) - [ A 1 {PI

[I71 {XI = [HI {FJ

[HI is the inverse of the impedance matrix [Z]

A least-squares optimization approach to solving this would be to minimize the sum of the squares of the errors (Jacoby e t a / . 1969)

where with respect to the parameter p yielding also 1251. Alterna- tively, if the errors {E} are assumed to be statistical quantities,

1191 [Z] = [K] - w2[M] + iw[C] then one should maximize the likelihood function, which, for a normal distribution about a zero mean, is

In the case of real normal modes, it can be shown that (Beliveau 1977)

I201 [HI = [+I 1DI-l [+I'

where the diagonal elements of [Dl are given by where N is the number of measured data. The niaximization of 1281 is obviously the same as minimization of 1271. Thus the

1211 D, = 0: - w2 + 2il ; ,R,w results of a statistical analysis, an optinlization procedure in the - ~ ,

forni of least squares, and an ordinary algebraic manipulation More general expressions are available in the case of arbitrary

,f the equations a l l lead to the system of equations [25] viscous damping yielding real eigenvalues (Lancaster 1966) for the parameters, F~~ different weights of experimental and for pairs of conlplex eigenvalues (Beliveau 1979). observations and a priori information on the parameters, this Tiine history information may be incorporated into a ~ a i e s i a n statistical

The time-history response to arbitrary excitation {g) for lin- mdysis, optimization of a weighted l e a s t - S ~ U ~ ~ C S criterion, or ear ordinary differentia] equations having constant coefficients a generalized pseudo-inverse of the algebraic equations and arbitrary initial conditions {y,,} is readily given for [2] by (Lawson and Hanson 1974). the following Duhamel integral kxpression having a homoge- neous (owing to initial conditions) and a particular solution Nori,~nl er/ucrtiotls oJ' rveigl~retl least .sc/unre.s

In amlications for which the measured auantities are linear L L

combinations of the unknown parameters, with perhaps differ- ent weights attached to each measurement, [24] becomes

where the weighting matrix [W,,] is normally diagonal, for where [V] and [A] are the eigenvectors and eigenvalues of the uncorrelated measurements, with the diagonal elements equal matrix [ e l ( [ 121) (Massoud et 01. 1985). to the inverse of the square of the standard deviation of the Ranrlotn re.sponse measurement.

For random-response situations, it is practical from a param- eter estimation point of view to consider the impulse response j + i matrix [Iz(t)] obtained by taking the inverse Fourier transform of the measured frequency response function /H(w)] (Lin 1967; Beliveau et a/. 1986): A priori information of the parameters may also be weighted:

1 " 1231 [h ( t)] = [H (w)] el"" dw

- 7.

13 1 I lW,l {PI = lW,I {P) where {p} are initial estimates and [W,] is also diagonal for statistically uncorrelated and independent parameters. These

Parameter estimation may then be combined accordingly.

One long-standing time series analysis technique is the deter- mination of coefficients in Fourier series expansion of mea-

1Wdl l 101

sured random signals (Jenkins and Watts 1969). This may be

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Page 4: System identification of civil engineering structures

10 C A N . J . CIV. ENG. VOL. 14. lLJX7

which, when symmetry is imposed by a premultiplication of late sensitivity by changing each of the parameters individu- [A]' [ I ] , yields the normal equations of weighted least squares ally, calculating the time history, modal properties, and fre- incorporating a priori information on the parameters quency response for each variation, and then calculating the - .

appropriati sensitivity. In this section, efficient sensitivity re- [331 [[A]' [W,l[A1 + [Wp]l {p} = [Wdl{dl + [wpl{p) lations are presented for the three types of data available in - - For weights of the data relatively large with respect to the initial structural d h m i c s . parameters, the parameters are

Real normal modes [341 {P) = [[A]' [Wd [Al l - ' [A]' [W,l{d) Since the matrices of structural dynamics are often large and

whereas for relatively small weights of the data, the parameters are equal to the initial estimates {p ) (BCliveau 1976).

The variance-covariance matrix of the estimated parame- ters may be determined through statistical arguments (Jenkins and Watts 1969).

Notilinear least squares in a Bayesian framework In practical situations, the measured quantities, i.e., time

series, modal information, and frequency response, are not linear combinations of the parameters of interest, i.e., masses, stiffness elements, etc. A first-order Taylor series approxi- mation to the parameters yields

[361 [A1 (6') = {dl - id')

[371 (6') = {P) - {PI)

where {p') and {dl) are the parameters and the corresponding calculated quantities at the lth iteration, 6' are corrections to the parameters, and the elements of the matrix [A] are calculated with parameters at the lth iteration

These are the sensitivity coefficients. The co~~esponding normal equations, with appropriate

weighting matrices, are then

[391 [[A]' [WdI [A1 + [Wpll {S') = [A]' [Wdl {d - dl )

+ [Wpl {P - p i )

Equation [39] is solved iteratively until convergence. For the data to influence the parameters, not only should [W,] be rela- tively large when compared with [W,], but the measured quan- tities must be sensitive with respect to the parameters as is evident from [38] and [39]. Again, an approximate variance- covariance matrix of the estimates upon convergence is given by [35]. To implement this procedure, efficient algorithms are required to evaluate the sensitivity coefficients. The objective function corresponding to [39] is

Sensitivity calculations It is obvious that system identification, as envisaged in this

paper, requires fairly sophisticated computational capabilities. The procedure, as outlined so far, is fairly straightforward, provided a large enough computer is available. Much of the numerical effort is in determining the sensitivity expressions required in [38]. As this is done iteratively, normally on large matrix systems, and on modal, time series, and frequency response information, it is essential that the algorithms used be efficient and require a minimum in computer memory and transfer. It would be foolish, for instance, to numerically calcu-

have certain useful properties (i.;., they are real and-often symmetric, have a band structure, have many zero entries, and have either a positive definite mass or a positive definite stiff- ness matrix), algorithms that utilize these properties effectively are required. With regards to the eigenvalue problem, for in- stance, the subspace iteration technique has proved to be useful (Clough and Penzien 1975). It determines in sequence only the lower natural frequencies and associated mode shapes. Thus, sensitivity formulas should use only those particular character- istics and not all the eigenvectors and eigenvalues as in other methods (Nelson 1976).

In the case of a symmetric system with normal modes, the sensitivity expressions for the undamped natural frequencies are readily obtained by chain-rule differentiation of [7] and a premultiplication by {+,}' yielding

I- 1

and the mode sensitivity is given by the relation

One of the lines of this singular relation may be replaced by a normalization condition (appropriately weighted) such as a fixed value for one of the elements of {+), say, +, (BCliveau and Chater 1984).

The corresponding column in 1421 is then set to zero except for the diagonal entry. This would then maintain the convenient nature of the system matrices in solving for the sensitivity.

Complex modes For the more general case ( [ l I]), the sensitivity relations are

given by (indices are again omitted for simplicity)

where {w)' are the left eigenvectors and satisfy

[45] [ h ' [ ~ ] ' + h[CI1 + [K]' {IV) = (0) 1

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Page 5: System identification of civil engineering structures

FIG. I. Scale model of suspension bridge deck of example 1.

The complex eigenvector has a sensitivity given by

[46] [h2[M] + A [C] + [K] ] {e] = - /A2[%]

1

which must also be augmented by a normalization condition [431.

In these expressions, both the eigenvalue sensitivity and the eigenvector sensitivity are functions of only the eigen- characteristics about which sensitivity is calculated and the sensitivity of the mass, damping, and stiffness matrices. The sensitivity of the amplitude and phase of a complex quantity are functions of the sensitivity of the real and imaginary parts of the complex number (BCliveau 1976).

Frequency response sensitivity The frequency response matrix [HI is the inverse of the

impedance matrix. Thus its sensitivity may also be expressed as a function of the sensitivity of the mass, damping, and stiffness matrices (BCliveau and Favillier 1981)

Depending on whether the loads have unknown parameters or not, the sensitivity of {X) of [17] is then

Time-history sensitivity Chain-rule differentiation of [ l ] readily yields the sensitivity

of data in the form of time history (BCliveau 1 9 7 4 ~ )

which is of the same form as [ I ] and involves sensitivity of the force vector, should it contain unknown parameters. Normally, inititial conditions, both displacements and velocities, are also

FIG. 2. Five-story steel-frame building of example 2.

FIG. 3. Twelve-story reinforced concrete building of example 3

treated as unknowns in this approch and the right-hand side of [49] is then zero for these parameters which are assumed to be independent and thus have a unit initial sensitivity followed by a free vibration. The equations are then integrated numerically in time at each iteration in the parameters by a Runge-Kutta scheme, for example.

Examples The dynamic behavior of structures is, of course, a function

of the materials used, the type of structural system utilized, and

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V A N . J . CIV. ENC;. V O L . 14. 1987

FIG. 4. 'I'wo degrees o f freedom of bridge deck of example I

the loads imposed. For instance, the wind effect on a bridge deck may not only impose loads but may, in fact, influence the stiffness properties and, more importantly, the damping char- acteristics.

Mechanical exciters operating in sinusoidal mode are useful in studying the dynamic behavior of civil engineering struc- tures in the low-frequency regime. Though the equipment re- quired is somewhat cumbersome, a dynamic test of this sort is much less demanding than a static test on a full-scale structure.

The response to ambient conditions is the simplest experi- mental procedure. With a portable spectral analyzer, field-data is readily obtained and analyzed on site, without recourse to exciters. Simplifying assumptions reduce the data to modal form.

In this section, three exanlples of system identification of civil engineering structures are presented. The aerodynamic behavior of a scale model of a sectional niodel of a suspension bridge deck is first studied. Only the first modes in vertical and torsional motions are simulated by the four springs shown in Fig. 1. At increasing values of horizontal wind velocity in an open-ended wind tunnel, arbitrary initial conditions were im- posed on the motlel, and the subsequent time-history response in free vibration was recorded. A least-squares fit to this data was then performed utilizing the ideas presented earlier with the aerodynamic coefficients as the unknown parameters (Bkliveau 1 9 7 4 ~ ) .

The 5-story steel-frame office building having two syn- chronized mechanical exciters at the roof is shown in Fig. 2. By measuring the amplitude and phase of the acceleration at thc different stories' relative to the exciters, interstory stiffness pa- rameters are determined for both lateral directions and in tor- sion (Favillier 1979; BCIiveau and Favillier 1981).

The third example is a 12-story reinforced concrete apart- ment building with combined frame-wall lateral stiffness. In- terstory stiffnesses are also the unknown parameters with the corresponding shear stiffness matrix added to a finite element model. The response to ambient wind, both before and after the exterior walls and interior separations were completed, is con- verted to modal data which is then used in the parameter esti- mation procedure. An elevation view of the building is shown in Fig. 3.

Ex-ample I . Time-history ~.esporzse of CI section model of n suspe?zsiorz bridge

As mentioned previously, the spring mounts were scaled to model the first modes in vertical and torsional motion of the bridge deck, as shown in Fig. 4. Further information on the model and testing is available (Beliveau 1974a). Bccause of the symmetry of the deck about the center line, the center of

' 0.00 1 lo 2 :o 1 3.0

T IME

FIG. 5. Time-history response of vertical (+) and torsional ( x ) motion and the best-fit curves (solid lines) at the wind velocity of 3.8 m/s (12.6 ft/s).

FIG. 6. Torsional damping coefficient A2 as a function of dinien- sionless velocity V.

mass is at the center of rigidity (a = 0 in Fig. 4) and the coupled vertical and torsional equations of motion, for quasi-steady aerodynamic loads, are

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FIG. 7. ( a ) Typical floor plan, (b) roof plan, and (c) column schcdule of cxamplc 2 ( I ft = 0.3048 m, I in. = 25.4 mm).

TABLE 1. Mass and mass moment of inertia - example 2

Mass Mass moment of inertia Story (kip. s2/ft) (kip. ft . s')

where R,, and R, are the undamped natural frequencies for the uncoupled vertical and torsional matrices and 5,, and 5, are the respective modal damping ratios, both for zero wind velocity. h is at the center of the section model, a is the angle of attack of the deck with the oncoming horizontal wind having a wind velocity U , and B is the width of the scale model. H , , H 2 , H I , H,, A , , A?, A,, and A, are the unknown aerodynamic coeffi- cients, generally functions of k. rn and 1 are the mass and mass moment of inertia of the deck per unit length. The dynamic pressure q and nondimensional reduced frequency k are given by

TABLE 2. Measured and estimated natural fre- quencies (rad/s) - example 2

Mode:': Measured Initial estimate Best-fit

* EW = east-west, NS = north-south, and T = tor- sional.

ing in the angular motion when it is negative. This coefficient is plotted in Fig. 6 as a function of the reduced velocity V. Also shown in a cross-hatched band are earlier experimental results (Scanlan and Tomko 197 1).

At high wind speed, this coefficient becomes positive, thus causing divergence once its effect is larger than the mechanical damping 5, in the structure, even in uncoupled rotational mo- tion about the center line.

1 Ex~~n lp l e 2 . Forced rcsporzse of 5-story steel-Jinme builclirzg [52] q = 2 p ~ ? The typical floor and roof plans for the 5-story structure

BR studied are given in Fig. 7. The location of the eccentric mass [53] k = -

U exciter is also given on the roof plan. The first story did not have a slab. Also shown is the column schedule. Further infor-

A typical discretized time-history free-vibration response, mation on the building, the exciters, and the testing procedure for 3 s at a wind velocity of 3.8 m/s (12.6 ft/s), is shown in is given elsewhere (Favillier 1979). Though the structure has an Fig. 5, together with the best-fit curve (the solid lines). As can obvious unsymmetric location of lateral resistance, no coupling be seen, the torsional motion is divergent at this wind speed, of the motions is considered in the study presented. owing in large part to the A2 coefficient associated with damp- Owing to the nature of the inertial excitation, and the limits

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CAN. J . CIV. ENG. VOL. I?. 19x7

TABLE 3. Results of parameter estimation - example 2

East-west (kip/ft) North-south (kip/ft) Torsion ( x lo5 kip. ft)

Pararnctcrs PO (70 1) I PI) (70 17 I /711 (71) 1) I

(T;,,,,,,1

~ , > l l . l \ C

No. of iterations S I / s l ,

2 x 10 " g I x I0 'rad

5 0.41

2 x 10--'g I x 10.' rad

8 0.36

5 x lo-" g/ft I x lo-' rad

9 0.18

on the mechanical exciters, motion about the two lowest modes was excited in both lateral directions and in torsion. The weights in the rotating baskets had to be reduced at the higher frequencies, not to exceed the design base shear of the build- ing. For a given frequency, accelerometers were placed at different levels in the building to obtain both lateral and tor- sional motions. A least-squares fit to the digitized data gave the amplitude and phase angle relationship with the roof motion. This was then plotted as a function of frequency. The measured acceleration as a percent of gravity and the phase angle, for the roof response in the east-west direction, are plotted in Fig. 8.

A simple shear building model, based on assuming infinite stiffness of the beams, gave the curve of initial estimate. Once the frequency response at each floor was used in the system identification technique presented in this paper, the best fit is somewhat closer to the measured response, at least near the resonant frequency. A simple lumped-mass model with values given in Table 1 was used. and the measured, initially calcu- lated, and best-fit values of the six resonant frequencies are given in Table 2. Initial p,, ant1 final p, values of the stiffness parameters and modal damping ratios estimated are given in Table 3, along with the number of iterations and the ratio of the final ob.jective function ([40]) to the intitial objective function for the standard deviations u. The mass and stiffness matrices are assumed to have the following form:

Exarnple 3. Eslit?l~liorz of rl 12-slory reitzforcecl cor?crete frame from rnodnl rlrrta

The lumped-mass properties for the unfinished and finished building used in this study are given in Table 4. As in the previous example, the mass moment of inertia is approximated by equally distributing the mass of the story in its rectangular floor plan. Plan and elevation views of the reinforced concrete

TABLE 4. Mass and mass moment of inertia - example 3

Mass Mass moment of inertia (kip. s2/ft) ( X 10.' kip-ft.s2)

Story Finished Unfinished Finished Unfinished

NOTE: I kip.s'/ft = 14.593 kN.s2/m, I kip.ft.s' = 1.356 kN . m . s'.

+ D 4 T 4

--- l N l T l 4 L ESTlMdiTE

- BEST- F IT

FIG. 8. ( a ) Amplitude response and (0) phase angle response for the 5th story in the east-west direction (example 2).

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Page 9: System identification of civil engineering structures

FIG. 9. ( a ) Elevation X, (b ) elevation Y, and ( c ) plan vicw of cxarnplc 3 ( I ft = 0.3048 rn, 1 in. = 25.4 nirn).

TABLE 5. Frcqucncies (Hz) - exarnplc 3

X Y 0

1 2 3 I 2 3 1 2 3

Unfinished Initial 1.40 5.44 10.99 1.95 7.38 15.69 2.14 8.15 14.10 Measured 1.36 5.17 9.84 1.60 5.91 12.70 1.81 6.90 17.12 Best-fit 1.35 5.17 9.84 1.60 5.91 12.90 1.82 6.90 14.10

Finished Initial 1.17 4.60 - - - - 1.84 7.01 -

Measured 1.56 5.39 - - - - 1.91 6.76 -

Best-fit 1.54 5.39 - - - - 1.91 6.75 -

TABLE 6. Standard deviations - exarnplc 3

PI-PI, 105 10" 10' PI3 lo-' lo-' lo-' Frequencies I I 1 Modes 0 lo-' lo-'

shear-wall and frame building are given in Fig. 9. Again no coupling is considered in the three degrees of freedom per story. Further information appears in a recent M.Sc. thesis (Chater 198 1).

The stiffness matrix was assumed to be a linear combination of one calculated from a finite element analysis of the building and a shear model with 12 interstory stiffnesses ([56j) which are 12 of the unknown parameters. The 13th parameter consid- ered was a scalar multiplier of the finite element stiffness, with its initial estimate adjusted to match the first measured natural frequency in each of the three motions. No damping parameters were considered in this example, although the method has been applied to that case in another application (BCliveau 1976).

Not all stories were instrumented for modal data. Neverthe- less, the technique presented here for normal modes was able to more closely match the measured mode shapes (Fig. 10) as well as the natural frequencies (Table 5), for both the finished

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CAN. J . CIV. ENC. VOL. 14. 1987

TABLE 7. Results of paramctcr estimation - cxamplc 3

Finishcd building Unfinished building

Elements X (kip/ft) 0 (kip ft) X (kip/ft) Y (kip/ft) 0 (k ip- ft)

No. of iterations 7 7 I I 9 9 sf /so 0.008 0.055 0.001 0.0001 0.007

(0) ( b ) HDOE 1 r l o a ~ 2 rOOE 3

MOOE I M O D E 2 LATERAL ( x )

I MOOE 2

TORSIONAL 8

FIG. 10. Mode shapes for ((1) the finished building and (b) the unfinished building of example 3.

and the unfinished buildings. Table 6 gives the values of the standard deviations of the initial estimates and measured fre- quencies and mode shapes. The second mode shape in the Y direction for the finished building and, similarly, the third mode shape for the unfinished building were not measured. Table 7 gives the initial and final values of the parameters, the number of iterations, and the ratio of initial and final objective functions ([40]).

Conclusion Today's structural engineer is often consulted to evaluate the

structural performance of existing structures or structures still

at the design stage. This could be, for instance, in determining the dynamic stability behavior of a particular bridge and in deciding whether a given building is sufficiently sound to merit retrofitting or important modifications to its structural ele- ments. System identification is a tool which can be useful at arriving at these decisions with quantitative as well as qual- itative information. Combined with model tests, either in a laboratory or as calculated by a computer, parameters may be estimated, which hopefully bring the calculated results closer to those observed on the real structure.

The methods presented here, though requiring sophisticated numerical and computational capabilities, are useful to this end. Data in the form of difficult-to-obtain time data, in fre- quency response form, or in the easier modal format may be utilized. The algorithms used take into consideration the spe- cial nature of the matrices found in structural dynamics. Pro- vided initial estimates to the parameters are available, a limited amount of data can be incorporated in obtaining better cor- relation between the model and the actual structure.

Acknowledgements Thanks go to R. H. Scanlan, G. C. Hart, and J . H. Rainer,

who were instrumental in obtaining the data used here, and to M. Favillier and S. Chater, who analyzed it. The support of the Natural Sciences and Engineering Research Council of Canada and of the Ministere de 1'Cducation du QuCbec is gratefully acknowledged.

BELIVEAU, J.-G. 1974a. Suspension bridge aeroelasticity - nonlincar least squares techniques for system identification. Ph.D. thesis, Departmcnt of Civil and Geological Engineering, Princeton Univer- sity, Princeton, NJ.

1974b. The role of system idcntification in design. 111

Proceedings, basic questions of design theory. Edifed by W. R. Spillers. Elsevier North-Holland Publishing Co. Inc., New York, NY, pp. 21-37.

1976. Identification of viscous damping in structures from modal information. Journal of Applied Mechanics, 98, pp. 335-339.

1977. Eigenrelations in structural dynamics. Journal of the American Institute of Aeronautics and Astronautics, 15, pp. 1039- 1041.

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Page 11: System identification of civil engineering structures

1979. First order formulation of resonance testing. Journal of Sound and Vibration. 65. pp. 319-327.

BELIVEAU, J.-G., and CHATER, S. 1984. System idcntification of structures from ambient wind measurcmcnts. Proceedings, Eighth World Conference on Earthquake Engineering, vol. IV. Response of Structures, San Francisco, CA, pp. 307-3 14.

BELIVEAU. J.-G.. and FAVILLIER, M. 1981. Parameter estimation from full-scale cyclic testing. Proceedings, Dynamic Rcsponse of Structures: Experimentation, Observation, Prediction and Control, Atlanta, GA, editcrl /J? G. C. Hart, pp. 775-793.

BELIVEAU, J.-G., MASSOUD, M., BOURASSA, P., LAUZIER. C., VIG- NERON, F. R., and SOUCY, Y. 1984. Statistical identification of the dynamic parameters of an ASTROMAST from finite elemcnt and test results using Bayesian inference and sensitivity analysis. Proceedings, Sccond International Modal Analysis Confcrcnce, Orlando, FL, pp. 89-95.

BELIVEAU, J.-G., VICNERON, F. R., SOUCY, Y., ancl DICAISEY, S. 1986. Modal parameter estimation from base excitation. Journal of Sound and Vibration, 107, pp. 435-449.

CHATEII, S. 198 1. SensibilitC et estimation des parametres dyna- miques d'une batisse a partir de mesures modalcs. M.Sc. thesis, Department of Civil Engineering. UniversitC de Sherbrooke, Shcr- brookc, Que.

CLOUGH, R. W., and PENZIEN, J. 1975. Dynamics of structures. MeGraw-Hill Book Company, New York, NY.

FAVILLIEII, M. 1979. Identification des paramittrcs dynarniques d'un batiment soumis a des charges cycliques. M.Sc. thesis, Department of Civil Engineering, UnivcrsitC de Sherbrookc, Sherbrooke, Que.

GRAUPE, D. 1972. Identification of systems. Van Nostrand Reinhold, New York, NY.

HART, G. C., and YAO, J. T. P. 1977. System identification in structural dynamics. ASCE Journal of the Engineering Mechanics Division, 103(EM6), pp. 1089- 1 101.

JACOBY, S. L. G. . KOWALIK. J . G . , and Plzzo, J. T. 1972. Iterative n~ethods for nonlinear optimization problems. Prentice-Hall. Inc., Englewood Cliffs, NJ.

JENKINS, G. M., and WATTS, D. G. 1969. Spectral analysis and its applications, Holden-Day, London, England.

LANCASTER, P. 1966. Lambda matrices and vibrating systems. Per- gamon Press, Toronto, Ont.

LAWSON, C. S . , and HANSON. R. J. 1974. Solving least squarcs problems. Prentice-Hall, Inc., Englewood Cliffs, NJ.

LIN, Y. K. 1967. Probabilistic theory of structural dynamics. MeGraw-Hill Book Company, New York. NY.

MASSOUD. M. , BELIVEAU, J.-G., and LEFEBVRE. D. 1985. A low- order transition matrix of the state space equation of motion. Pro- ceedings of the Third International Modal Analysis Conference, Orlando, FL, pp. 888-894.

NELSON, R. B. 1976. Simplified calculation of eigenvector deriva- tives. Journal of the American Institute of Aeronautics and Astro- nautics, 14, pp. 1201 - 1205.

PILKEY, W. D., and COHEN, R. editor.^. 1972. System identification of vibrating structures: mathematical nlodels from test data. 1972 Winter Annual Meeting of the American Society of Mechanical Engineers, New York, NY.

SAGE, A. P., and MELSA, J. L. 1971. System identification. Academ- ic Press, New York, NY.

SCANLAN, R. H.. and TOMKO, J. J. 1971. Airfoil and bridge flutter derivatives. ASCE Journal of the Engineering Mechanics Division, 97(EM6), pp. 17 17- 1737.

SHINOZUKA, M., YUN, C. B. , and I M A I , H. 1982. Identification of linear structural dynamic systems. ASCE Journal of the En- gineering Mechanics Division, 108(EM6), pp. 1371 - 1390.

List of symbols a partial derivative { } vector quantity 1 1 matrix quantity

[ h l [ H I H i , H?,

H3, H A I

[I1 I

transpose of quantity matrix inverse inverse of transpose complex conjugate complex conjugate transpose velocity and acceleration sensitivity matrix, matrix for solution or eigenvalue element of [ A ] location of center of gravity of bridge deck

aerodynamic coefficients width of bridge deck element of interstory damping matrix viscous damping matrix measured and calculated data quantities diagonal matrices elements of a diagonal matrix exponential quantity diagonal exponential matrix frequency parameter in Fourier series forcing vector complex representation of forces imaginary matrix Fourier transform load vector downward vertical motion of bridge deck center of gravity impulse response matrix frequency response matrix

aerodynamic coefficients imaginary symbol d- I identify matrix mass moment of suspension bridge per unit distance imaginary portion of complex quantity indices stiffness matrix element of interstory stiffness matrix reduced frequency iteration counter likelihood function mass of suspension bridge per unit distance mass matrix lumped-mass or mass moment of inertia number of terms in discretized time series number of degrees of freedom the zero matrix estimated parameters and parameters at the Lth iteration initial estimates to the parameters dynamic pressure matrix used in first-order formulation real portion of complex quantity symmetric matrices for first-order formulation diagonal decomposition of [ R ] and [ S ] real and imaginary portion of complex quantity objective function time total time in a given series Cholesky decomposition matrix of the mass matrix mean horizontal wind velocity

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18 CAN. J. CIV. ENG. VOL. 14. 1987

complex mode shapes matrix of complex mode shapes matrix of complex mode shapes of first-order formulation columns of [ V ] complex number and its amplitude reduced velocity matrix of variance-covariance matrix of estimated parameters vector of left eigenvector matrix weighting matrices for data and parameters, respective1 y vector of complex response quantitites vector of degrees of freedom and their velocities and accelerations lateral horizontal motion of building state vector and its time derivative used in first- order formulation impendance matrix torsional motion of bridge deck

scalar multiples used in proportional damping scalar multiplier time increment vector of parameter corrections at lth iteration vector of errors diagonal matrix of modal damping ratios and its elements phase angle of complex quantity or torsional motion of building story diagonal matrices of eigenvalues and their elements real and imaginary part of eigenvalue, respectively the numerical value col~esponding to 180' air density standard deviation of statistical variable time variable matrix of real normal modes and one column excitation frequency diagonal matrix of undamped natural frequencies and its elements

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