Simplified Identification Scheme for Structures on a Flexible Base

Transcription

1 Simplified Identification Scheme for Strctres on a Flexible Base L.M. Star California State University, Long Beach G. Mylonais University of Patras, Greece J.P. Stewart University of California, Los Angeles SUMMARY: The objective of system identification is to evalate nnown properties of a dynamic system by developing a mathematical model for the relationship between measred responses (otpts) and measred external demands (inpts) to the system. We describe a new freqency domain procedre which tilizes a graphical bisector scheme to identify freqency and damping for the ible base, psedo-ible base and fixed base behavior of a strctre, from a single set of ible base data. The procedre is applied for analysis of both earthqae data and forced vibration test data. We can validate the bisector procedre by generating simlated data from comptational models of soil-fondation-strctre systems. Responses from these modelled strctres can also be analyzed sing traditional methods to identify ible and fixed base modal parameters. Keywords: soil-strctre interaction, field testing, system identification, shallow fondations 1. INTRODUCTION The objective of system identification is to evalate the nnown properties of a dynamic system by developing a mathematical model for the relationship between measred responses of the system (otpts) and measred external demands (inpts) to the system. For applications to bildings, system identification can be sed to estimate modal freqencies, damping ratios, mode shapes and participation coefficients based on recordings of system vibrations (e.g., Safa, 1991). Depending on the selected inpt-otpt and motions, modal vibration parameters can be identified that describe the behavior of the sperstrctre alone, or the soil-fondation-strctre system. Fndamental-mode freqencies and damping ratios are distinct for the two base fixity conditions. The differences, sch as the ratio of ible/fixed base fndamental mode periods, are a sefl qantification of the significance of Soil-Strctre Interaction (SSI) effects. In this paper, we first describe a new freqency domain system identification procedre to evalate fixed- and ible-base modal properties. The model contains nnown modal freqencies and damping ratios that are estimated sing resonance considerations throgh a graphical scheme referred to as bisector method. The procedre can be applied for analysis of either earthqae data or forced vibration test data. We validate the reslts of the simplified bisector method for synthetic data for which both earthqae and forced vibration test data are available.. THEORY OF EVALUATION OF SSI EFFECTS USING BISECTOR METHOD We first examine a linear, ndamped, mlti degree-of-freedom (MDOF) system with an external shaer force applied at the roof of the strctre. The concept can be easily extended to incorporate fondation compliance, as shown sbseqently. The relative displacements or rotations of each of the n degrees-of-freedom (DOFs) in the strctre relative to the base are described by an n 1 vector U, with corresponding velocity and acceleration vectors U and U, respectively. In matrix form, the wellnown eqation of motion of the sperstrctre can be written as:

2 m K 1b 1 Fsh 0 Mb Ub Kb1 Kbb Ub 0 (1) We have partitioned the matrices to isolate the top DOF on the strctre, so 11 is a 1x1 matrix, and sqare matrix K bb is of ran n-1 x n-1. We can apply the Forier transform to the EOM, casing and its derivatives to be freqency dependent, which is noted with an overbar. By converting the terms from accelerations to displacements, we can move the elements of M b from the mass matrix to the stiffness matrix. This yields the eqation: U b m K 1b 1 Fsh b b1 bb 0 0 U K K Mb Ub 0 () that leaves s with jst a single nodal mass in the mass matrix and allows s to perform algebraic ( static ) condensation to redce the matrices to a single eqivalent degree-of-freedom (SDOF) system. After trivial algebraic operations the condensed eqation of motion is: 1 m b bb b1 1 Fsh K K M K (3) In light of the elementary relation 0 / m (4) the fndamental freqency of the system at hand can be determined from Eqation (3) as: 1 / 0 11 K1b Kbb Mb K b1 m1 (5) It shold be noted that 0 is dependent on. By definition the resonant condition of the ndamped SSI system is reached when the freqency of response is eqal to the freqency of excitation If we loo for soltions for which the two freqencies coincide, there are n distinct vales that will satisfy this condition for a mass matrix of ran n, which correspond to the n fndamental freqencies. As shown in Figre 1, we can graphically solve for the resonant freqencies of the system by plotting the resonant freqency from Eqation (5) as a fnction of the freqency of demand (red line). We plot a bisector line (blac line) that indicates points at which both freqencies are identical, which are interpreted as the resonant freqencies.

3 Figre 1. Schematic of bisector soltion for identifying resonant freqency of a two DOF ndamped system Let s now consider a damped MDOF system with a compliant fondation as shown in Figre the strctre consists of n strctral masses (m s,i ), pls two fondation degrees of freedom. The strctre is sbjected to an external harmonic force F sh (e.g., imposed by a shaer) acting at the roof node having mass m s,n. The compliance of the soil is modeled throgh freqency-dependent springs, x, and yy that enable fondation translation ( f ) and rotation (θ f ), respectively. The soil-strctre interaction is modeled by complex-valed springs of nnown modls i,j, which can be written in the form: ( 1i ) (6) where is the spring stiffness and the dimensionless damping ratio. The displacement at the nth degree of freedom, the top of the strctre, is denoted by st,n. The displacement is measred relative to an nmoving point on the grond. Figre. A simple SSI System sbjected to forced vibration at roof node The eqations of motion can be written in matrix form:

4 ms, n st, n 0 m s,1 0 0 st,1 0 0 If 0 f m f f 1,1 1, n 1, n1 1, n st, n Fsh n n n n n,1,, 1 n, n st,1 0 n1,1 n1, n n1, n1 n1, n f 0 n,1 n, n n, n1 n, n f 0 (7) By converting to the freqency domain, rearranging terms, and performing static condensation as illstrated above, the system can be redced to an eqivalent SDOF one described by the eqation of motion m F (8) s, n st, n st, n sh Here is the complex-valed stiffness of the ible base SSI system. Assming mass m s,n is nown, we can solve Eqation (8) for the stiffness of the ible base soilstrctre system: F m st, n sh s, n st (9) By writing in the complex form as in Eqation (6) one can solve for spring modls Re( ) (10) where Re( ) denotes real part. Sbstitting into Eqation (4) and dividing by (π) gives the freqency-dependent n-damped natral freqency of the system: f 0 1 m sn, 1/ (11) Note that a single mass, m s,n, is sed in the above expression becase the algebraic condensation has moved all other masses into the stiffness matrix. In the same spirit, writing in the complex form as in Eqation (6) and rearranging term, yields the effective freqency dependant damping ratio of the SSI system in the form: 1 s Imag( ) (1)

5 In this formlation the resonant freqency of the strctre is dependent on the freqency of the excitation becase is dependent on the excitation (shaer) freqency. As previosly shown in Figre 1, we can graphically solve for the resonant freqencies of the system by plotting the resonant freqency from Eqation (11) as a fnction of the freqency of demand (red line). We plot a bisector line (blac line) that indicates points at which both freqencies are identical. By identifying the point(s) at which the freqency fnction and the bisector intersect, we can identify freqencies of resonance. Once the resonant freqencies are identified the appropriate damping can then be recognized by selecting beta at the identified resonant freqency. The freqencies identified throgh this process are n-damped freqencies of vibration, which are related to damped freqencies throgh the well-nown relation (Chopra, 007): d 0 1 (13) The above derivation provides a method for evalating the stiffness and damping for a ible base SSI system. New inpts and otpts are needed to derive the stiffness and damping of the strctre nder fixed base conditions. To illstrate how the appropriate inpt-otpts can be selected, we trn or attention to a simple system consisting of two springs in series, with stiffnesses of 1 and, sbjected to force, F. Springs 1 and experience deformations, 1 and respectively. We can show that: total total (14) Where total is the sm of the two spring displacements, and total is the stiffness of the system. Eqation (14) provides a direct relation between the stiffness of a single component spring and the stiffness of the total system. This indicates that as long as we measre the total displacement of the system and the relative deformation of one spring of interest, we can solve for the stiffness of the other component spring from the stiffness of the total system. Retrning to the MDOF SSI system, in order to determine the fixed base modal parameters, we frame the eqations of motion so that the stiffness of the strctre alone appears. It is convenient to formlate the eqations of motion in terms of the translation of the strctre relative to the fondation translations and rocing. H (15) sr,n st,n f f The ible base stiffness can be thoght of as the total stiffness of a system consisting of soil springs and a strctre spring. If we tae to be analogos to total, and the combined effects of fondation rotation and translation to be analogos to 1, then the stiffness of the strctre alone,, is analogos to. By applying Eqation (14), we can then solve for : st,n fixed (16) sr,n If we sbstitte Eqation (16) into Eqation (8) we find: m F (17) s, n st, n fixed sr, n sh We can solve this eqation for the stiffness of the fixed base soil-strctre system as:

6 fixed Fsh ms, n st, n (18) sr, n The stiffness of the fixed base strctre can be sed to solve for the natral freqency and damping in the same manner as for the ible base case. It is sometimes of se to investigate the strctre as if it were allowed to roc, bt was fixed against translation. Following prior convention (Stewart and Fenves, 1998), we call this the psedo-ible base case. More information abot the derivation of stiffness eqations for this case can be fond in Star, (011)..1 Fixed and Flexible Base Parameters for Earthqae Loading of a System To extend the method developed above for earthqae loading, we consider an MDOF SSI system with a compliant fondation similar to the one illstrated in Figre Error! Reference sorce not fond.. The eqation of motion of a strctre sbjected to earthqae loading is given by (Chopra, 007) as: MU( t) KU( t) M1 ( t) (19) g where 1 is a vector with ones for each of the strctre translational degrees of freedom along the direction of shaing and zeros for all other degrees of freedom, and ü g is the appropriate grond acceleration, which is discssed in more detail below. The strctre consists of n strctral masses (m s,i ), pls two fondation degrees of freedom. The compliance of the soil is modeled throgh springs, x, and yy, that enable fondation translation ( f ) and rotation (θ f ) respectively. The soil springs are complex-valed. The displacements and accelerations on the left side of the eqations are given relative to the grond. The earthqae forces can be transferred to the left side of the eqations and we can rewrite the acceleration vector in terms of absolte displacements where: (0) T s, n st, n g By converting to the freqency domain, rearranging terms, and performing static condensation as illstrated in the previos sections, the system can be redced to an eqivalent SDOF system, with an eqation of motion eqal to: m T T s, n s, n s, n 0 (1) where is now a fictitios stiffness for degree of freedom n that acconts for all of the deformations within the strctral system as well as grond displacements g. As in the forced vibration case we are interested in solving for the fixed, ible and psedo-ible base system stiffness terms. Recalling Eqation (14), we can find as a fnction of, the absolte displacement at the top of the strctre, and the total displacement at the top of the strctre relative to the grond: T s,n () st,n

7 If we sbstitte Eqation () into Eqation (1) we find: T s, n st, n st, n m 0 (3) We can solve this eqation for the stiffness of the ible base soil-strctre system as: m st, n T s, n s, n (4) Here is the complex-valed stiffness term for the ible base SSI system. We can also develop eqations similar to (4) for the fixed base cases: fixed ms,n sr T s (5) As was shown for the forced vibration case in Eqations (11) and (1), by writing, and fixed in the complex form we can separate the real from the imaginary portion of the complex stiffness and we can solve for the freqency and damping of the SSI system. Unlie in the forced vibration loading case, the strctre mass cancels ot of the eqations for freqency and damping, maing the calclations more robst. One potential pitfall is that the calclations are dependent on the inematically appropriate grond motion. Grond motions are altered by the presence of the fondation, so that the grond motions that the strctre experiences are not identical to the free-field grond motions (Kim and Stewart, 003). It is difficlt to determine the tre grond motion experienced by the fondation experimentally. There will be some error introdced to the reslts if the measred free-field motion is sbstitted. 3. VERIFICATION OF IDENTIFICATION METHODS USING MODELED DATA We can validate the methods of system identification described above by generating simlated data from comptational models of soil-fondation-strctre systems. Responses from these modeled strctres are ideal for testing the method becase the strctres have nown properties and can be analyzed sing traditional methods to identify ible and fixed base modal parameters. The first model developed is a simple SDOF strctre on top of a fondation with translational and rotational soil springs. The soil stiffness and damping are modeled sing freqency dependant theoretical soil springs as developed by Pais and Kasel, (1988), with an addition hysteretic soil material damping. The strctre is assigned a nown stiffness, and hysteretic strctral damping. Table 1 gives the soil and strctre properties sed in the model. The system is excited by a broadband excitation fnction at the top strctre node. Translation and rotation accelerations at the fondation were compted as well as roof translation. Table 1. SSI system modeling parameters (B = footing width, L= footing length, D=embedment) Soil Parameters Strctre Parameters Fondation Parameters V s = 110 m/s = 1.73 Mg/m 3 = 0.45 soil = % G = 1.98 x 10 4 KN/m K x, yy, x, yy from Pais and Kasel, 1988 m s = 6.98 Mg H =.9 m EI = 3.8 x 10 7 KN m strctre = % m f = Mg B =.13 m L = 4.6 m H f = 0.6 m D = 0

8 Figre 3 shows the graphical otpt of the bisector system identification procedre for ible-base conditions [Eqations (11) and (1)], and the analogos eqations for the psedo-ible and fixed base conditions. The intersection of the natral freqency crve and the bisector for the ible base case indicates the first mode freqency. Table smmarizes the reslting freqencies and damping ratios. The fndamental freqency reslts of the system identification procedre can be checed against the reslts of a classical eigenvale analysis (neglecting damping matrix) of the modeled strctre with a ible base and the modeled strctre nder fixed base, and psedo-ible conditions. The identification procedre had a discrepancy of abot 1% or less compared to the eigenvale analysis. The fixed base system identification procedre provides a damping of %, which is consistent with the strctre damping employed in the model. Figre 3. Bisector identification of freqency and damping for forced vibration (FV) loading of a SDOF modeled strctre for (a) ible base, (b) psedo-ible base and (c) fixed base conditions.

9 Table. Freqency and damping from the bisector system identification procedre and traditional eigenvale analysis of 1 DOF modeled strctre: Flexible Psedo- Flexible Fixed T / T f Freqency FV (Hz) Bisector System ID FV (%) Freqency EQ (Hz) EQ (%) Eigen Freqency (Hz) The same simple SSI strctre can also be excited with a broadband earthqae excitation. The procedre for system identification of an earthqae excited strctre can be applied. The reslts of the graphical otpt for the bisector method are shown in Figre 4. As this is the same strctre as examined previosly, we expect to find approximately the same freqency and damping for each base fixity condition. The reslts for the Eigen analysis, and the graphical identification methods are tablated in Table 3. The system identification method for the earthqae loading prodces the same damping vale for the fixed base case as the forced vibration cases and marginally smaller damping levels for the other base fixity conditions. Figre 4. Bisector identification of freqency and damping for earthqae (EQ) loading of a SDOF modeled strctre for: (a) ible base, (b) psedo-ible base and (c) fixed base conditions.

10 Table 3. Freqency and damping from the bisector system identification procedre and traditional eigenvale analysis of three DOF modeled strctre (assming i =%) Flexible Psedo- Fixed T / T f Bisector System ID Flexible Freqency FV (Hz) FV Eigen Analysis Freqency EQ (Hz) EQ Freqency (Hz) DISCUSSION AND CONCLUSIONS A simple, physically-motivated procedre was presented to identify freqency-dependent effective stiffness and damping of strctral systems for SSI applications. The method tilizes a graphical bisector scheme to identify fndamental freqency and damping for the ible base, psedo-ible base and fixed base behavior of a strctre, withot the formidable mathematics associated with system identification theory. Implementation of the procedre reqires measrements of horizontal displacements at the roof and fondation level of the strctre, vertical fondation displacements (from which rotations are derived), shaer forces, and limited system masses. The procedre is applied for compter-simlated strctres. The strctre was loaded by forced vibration and earthqae excitation, allowing comparisons of the reslts from both methods. The system identification procedre provides reasonable estimates of system freqency and damping for strctres with relatively low levels of damping. The error in the first mode freqency for most strctre system will be very small. The error may limit possibility of identifying higher modes in some cases. In order to identify the ible base freqency and damping for the earthqae loading case it is necessary to now the fondation inpt motion, however this motion is difficlt to measre experimentally. There will be some error introdced to the reslts becase the measred free-field motion will sally be sbstitted. REFERENCES Chopra, A. K. (007). Dynamics of Strctres: Prentice Hall International Series in Civil Engineering and Engineering Mechanics. Kim, S., and Stewart, J. P. (003). Kinematic Soil-Strctre Interaction from Strong Grond Motion Recordings. Jornal of Geotechnical and Geoenvironmental Engineering, 19:4, doi: //asce/ /003/19:4/33. Pais, A., and Kasel, E. (1988). Approximate Formlas for Dynamic Stiffness of Rigid Fondations. Soil Dynamics and Earthqae Engineering, 7 (4), Safa, E. (1991). Identification of Linear Strctres Using Discrete-Time Filters. Jonal of Strctral Engineering, 117:10, Star, L. M. (011). Seismic Vlnerability of Strctres: Demand Characteristics and Field Testing to Evalated Soi-Strctre Interaction Effects, Department Civil and Environmental Engineering, University of California, Los Angeles. Stewart, J. P., and Fenves, G. L. (1998). System Identification for Evalating Soil-Strctre Interaction Effects in Bildings from Strong Grond Motion Recordings. Earthqae Engineering and Strctral Dynamics, 7,