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STRUCTURAL MODEL UPDATING USING EXPERIMENTAL STATIC MEASUREMENTS By Masoud Sanayei, l Member, ASCE, Gregory R. Imbaro 1 Associate Member: ASCE Jennifer A. S. McClain,3 Associate Member, ASCE, and Linfi~ld C. Brown,4 MeU:ber, ASCE ~e results of experiments on a small scale steel frame model are presented to support the "displacement equation error function,". "di.splace~ent o~tput error function," and "strain output error function" met.h0ds ~f structural parameter estimatIOn usmg static nondestructive test data. Both static displacement and static stram mea~urements are used to successfully evaluate the unknown stiffness parameters of the structural components. ~elght factors calculated from the variance of the measured data are applied to reduce error in the parameter estimates.

ABSTRACT:

INTRODUCTION There exists an increasing need for a reliable method to monitor the performance of structures from satellite dishes to highway bridges, to space stations. In all cases, damage to the structure ca.n affect its safety or render it useless; therefore, early de~ectlOn ~f damage is par~ount. There are many nondestructive testmg (NDT) techmques available, including X-ray, RADAR, and acoustical analysis. While many of these can detect and locate damage, they cannot quantify the stiffness properties of a structure for each component. Parameter estimation and model updating can satisfy this need. Structural parameter estimation is a powerful tool that uses NDT data to update the parameters of an a priori model. When compared with earlier versions, the updated model indicates changes in parameters that may affect a structure's load-carrying capacity. Estimated parameters can be any appropriate cross-sectional property (i.e., area or moment of inertia) of the structural components. The response of a structure to nondestructive excitations provides the input for parameter estimation. Using a finite-element model (FEM) of the structure, measured and analytical responses are compared. Parameters that define the model at the elemental level are then updated to minimize the difference between the measured and analytical responses. The revised model can serve as a baseline for model updating and condition assessment. The excitations can be either static or dynamic, and the responses can be displacements, rotations, or strains with static excitation and modal or time history response with dynamic excitation. This paper uses static loading with displacement and strain response measurements. The theory behind parameter estimation is not new, but improvements in computational capability have resulted in significant progress in algorithm development and testing. Static parameter estimation is based on measured deformations induced by static loads such as a slowly moving truck on a bridge, a slowly moving mass on a building, or actuator induced static loading on a space station. While little research I Assoc. Prof.• Dept. of Civ. and Envir. Engrg.• Tufts Univ., Medford, MA 02155. 2S truct . Engr.• Fay. Spofford & Thorndike. Inc., Burlington, MA 01803; formerly. Grad. Student. Dept. of Civ. and Envir. Engrg., Tufts Univ.• Medford, MA. 'Struct. Engr.• Weidlinger Assoc.• Inc.• Cambridge, MA 02142; formerly, Grad. Student. Dept. of Civ. and Envir. Engrg.• Tufts Univ., Medford, MA. 'Prof.. Dept. of Civ. and Envir. Engrg.• Tufts Univ., Medford. MA. Note. Associate Editor: Kevin Z. Truman. Discussion open until November I, 1997. To extend the closing date one month, a written request m~st be filed with t~e ASCE Manager of Journals. The manuscript for thIS paper was submltted for review and possible publication on February 20, 1996. This paper is part of the Journal of Structural Engineering. Vol. 123. No.6. June. 1997. ©ASCE, ISSN 0733-9445/97/0006-07920798/$4.00 + $.50 per page. Paper No. 12655.

has been done in this area, there are many instances in which stati7lo~ing is IJ.l0re economical than dynamic loading. Many applicatIOns reqUire only element stiffnesses for condition ass~ssment. In these cases static testing and analysis can prove SImpler and more cost effective. Displacements and rotations can be used easily in FEMbased parameter estimations. Hajela and Soeiro (I 990b) compared the use of incomplete static and dynamic data using an output error method. Static displacements were determined to have an advantage over dynamic measurements in lower computational cost and greater insight into damage assessment. They presented the idea of dominant displacements for both static and vibrational testing, meaning that certain forces and measurements are more representative of the structural system. They also showed that errors are more prevalent when loading does not result in an equal stress distribution in each of the members. In a related paper, Hajela and Soeiro (1 990a) showed through experimental testing that a uniform stress loading produces excellent results. Banan et al. (l994a,b) estimated element stiffnesses using complete and incomplete sets of applied static forces and measured displacements. Bruno (1994) formulated a parameter identification technique to 10ca~e and characterize loose joints of a deployable space truss usmg actuator-induced static loading and unloading. Sanayei and Onipede (1991) extended the method developed by Sanayei and Nelson (1986) to use static displacement test data at different subsets of force and displacement degrees of freedom (DOF). In this method, loads were applied at one subset of the DOF, and displacements were measured at another subset. In both studies, element structural stiffness parameters were accurately estimated. Sanayei et al. (1992) used the preceding method to determine the effects of measurement errors. They also presented a heuristic method to select a small subset of error tolerant force and displacement measurement locations. Measuring translational displacements on full-scale structures can be difficult, but rotations are not because they need not frame of reference. Strain gauges also offer an easier and less costly alternative. Strains on a structural surface are caused by both bending and axial deformations of the member; therefore, strain measurements can capture element behavior well. Sanayei and Saletnik (1996a) developed a method for parameter estimation of linear-elastic structures using static strain measurements and preserving structural connectivity. Numerical simulations on truss and frame structures demonstrated this system's ability to identify all or a portion of the structural cross-sectional properties, including element failures. Weight factors have been successfully incorporated into parameter estimation using dynamic test data, but have not been widely applied to static parameter estimation. Recent research includes Alcoe and Hjelmstad (1992), who considered the use

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of weighting factors to increase the accuracy of eigenvalue or eigenvector estimates. Lim (1991) assessed the performance of a parameter estimation method when measured mode shapes were inaccurate. He used averaged data and found a significant improvement in damage detection when the averaged stiffness reduction factors were weighted using the inverse of their standard deviations. PARAMETER ESTIMATION

This paper presents NDT results to support previously published methods of static parameter identification. To estimate element stiffness parameters (e.g., area, moment of inertia), both static displacement and strain responses are measured. The algorithms used are those of Sanayei and Onipede (1991), Babu (1992), and Sanayei and Saletnik (1996a). The methods are briefly summarized for completeness and as foundation for the development of weight factors. Displacement Equation Error Function

This method of parameter estimation assumes the structure is linear-elastic and experiences small deformations (Sanayei and Onipede 1991). This is reproduced in the laboratory experiments. Static forces are applied at one subset of force DOF (FDOF) of FEM, and displacement measurements are taken at another subset of response DOF (RLOC). These subsets may overlap completely, partially, or not at all. The parameters of FEM are systematically iterated to match the simulated response with the measured displacements. The basic FEM force-displacement relationship is {F}

= [K]{U}

(I)

where {F} = vector of applied forces; {V} = vector of measured displacements; and [K] = stiffness matrix that is a function of the known and unknown parameters {p}. To measure a structure's response a number of sets of forces (NSF) are applied one at a time to FDOF, and NSF sets of displacements are taken at RLOC, which contains a number of measured responses (NMR) per set. NSF may contain single or multiple forces in each load set. However, no set should be a linear combination of any others. These individual force and displacement vectors become corresponding columns of the force and displacement matrices [F]

= [K][U]

(2)

Since the displacements are taken only at a subset of DOF, matrix [V] may be partitioned into [Va] and [Vb] which are the measured and unmeasured displacements, respectively. Giving Fa] [ Fb

=

[KaaIKab] [Ua] KbalKbb Ub

(3)

Using static condensation [Fa]

= ([Kaa]

- [KabHKbbrl[Kba])[Va]

+

[KabHKbbrl[Fb]

(4)

Using applied forces and measured responses, the "displacement equation error function" [e(p)] (Sanayei and Onipede 1991) is defined as [e(p)] = ([K..] -

-

[Kab][Kbbrl[Kba])[Ual"'

+

[Fa]"'

[Kab][Kbbr1[Fbl"' (5)

The superscript m indicates experimentally measured forces and displacements. Displacement Output Error Function

Similar to the displacement equation error function, the displacement output error function is derived from the basic FEM

force-displacement relationship in (2). Rearranging (4) to find an analytical value for [Va] and subtracting [va]m results in the "displacement output error function" (Babu 1992) [e(p)] -

= ([K..]

[Kab][Kbbrl[Kba])-I([Fal"' -

[Kab][Kbbrl[Fb]"')

(6)

[Ua ]"'

Due to the inversion of the submatrices of the stiffness matrix, both displacement error functions, (5) and (6), will be an algebraic nonlinear function of the stiffness parameters {p}. The force-displacement relationship remains linear. To identify the parameters the norm of [e(p)] is minimized with respect to the parameters using the Gauss-Newton or gradient methods. Strain Output Error Function

As seen in (1), forces are related to displacements through a stiffness matrix. To accommodate strains a mapping matrix is developed to relate displacements to strains. Mapping matrices for both two- and three-dimensional frame elements are developed by Sanayei and Saletnik (1996a). Only two-dimensional frames will be examined. Assumptions made include that strains are measured only in the local longitudinal or x-direction and that torsional effects are negligible. Bending and axial deformations are developed separately and combined using the principle of superposition. The mapping matrix is defined as {e}.

= [B]{ V}

(7)

where {E}. = vector of strains for element n; and [B] = corresponding mapping matrix. An expression for {V} is found from (1) and substituted into (7). Similarly to (2), multiple load sets and corresponding strains are then appended horizontally to produce

[~] =[::] [K]-'[F]

(8)

Strains in (8) are partitioned into [Ea ] and [Eb ] as measured and unmeasured strains [e(p)]

= [ea]a -

[eal"'

(9)

Therefore [e(p)] = [BaHK(p)r'[F)"' - [Eal"'

(10)

Superscript a indicates the analytical strains, and superscript

m indicates experimentally measured forces and strains. Again, due to the inversion of [K(p)], this error function will be an algebraic nonlinear function of the stiffness parameters {p}. As with displacement measurements, the error function is minimized to solve for the parameters using either Gauss-Newton's method or the gradient method. WEIGHT FACTORS

Errors in the parameter estimates may arise from many sources, the most significant of which are measurement errors and model errors. Displacement measurements using dial indicators may include errors due to mounting deformation, leveling, alignment, or the accuracy of the measurements. Strain measurements using strain gauges can be contaminated by the size of the gauge, its application, its location and alignment, thermal expansion, reading technique, or the accuracy of the measurements. In addition to these sources of error, the mathematical model does not capture variations in cross-sectional properties, existing deformations, residual stresses, stress concentrations, or variations in connection stiffness. Other differences between the physical and analytical models used arise JOURNAL OF STRUCTURAL ENGINEERING / JUNE 1997/793


from assumptions such as homogeneous linear-elastic behavior, rigid connections, and no shear or axial deformations. In some cases, data points may be of questionable accuracy due to errors or testing conditions. To blindly discard data that mayor may not be correct may leave too little information to successfully estimate parameters. The amount of information available for estimating should be maximized, but if data is questionable, its influence on the error function can be reduced by weight factors. Output error functions using displacement [(6)] or strain [(10)] measurements can be treated similarly. For these two error functions, all of the entries in a given row of the error matrix are the result of measurements at a single sensor. If a strain or dial gauge is faulty, the corresponding row of the error matrix is contaminated. Similarly, each column of the error matrix is associated with one load case. If there is some question of the validity of a given loading, doubt is cast on the associated column of the error matrix. The error functions are differences between analytical and measured quantities. To account for imprecise data associated with a given response degree of freedom, these error functions can be premultiplied by a diagonal weighting matrix [Wu]. The values along the diagonal represent the confidence in measurements recorded by individual sensors. If confidence is high, the measurement will influence the error function more. If it is low, the measurement will contribute very little. Similarly, confidence in individual load cases can be introduced by postmultiplying the error function with a different diagonal weighting matrix [WF ]. Pre- and postmultiplying [e] by weighting matrices produces the weighted error function [ew].

MODEL DESIGN

Physical Model The model used for testing is a two-story, one-bay scale steel frame shown in Fig. 1. The model is designed as a twodimensional frame for in-plane testing and analysis only. With the two-story frame mounted in a rigid steel testing frame, loads are applied and measurements are recorded at several points on the scale structure. The frame model is mounted at the center of the test frame to retain overall symmetry. The heights of the first and second floors are 350 mm for a total height of 700 mm. The length of the bay is 600 mm. The beams and columns each have an in-plane thickness of 8 mm, and an out of plane width of 25 mm. This ratio of in-plane to out-of-plane dimensions is designed to provide sufficient outof-plane rigidity to limit the response to in-plane. The in-plane moment of inertia is 1,066.7 mm 4 , and the area is 200 mm 2 • The true cross-sectional properties are specified by the supplier, High-Tech Scientific Limited, England, and verified by several precision measurements. Although it is possible that some variation exists along the length of each member, these "true" values provide the best available benchmark for assessing estimates. Vertical and lateral loads are applied by hanging weights directly on the frame or with pulleys mounted to the test frame. There is no provision to allow the application of moments to the structure. Displacements are measured using dial indicators with a resolution of 0.01 mm and a total range of 0-10 mm. To measure strains, strain gauges with a range of

(11)

Either weighting matrix can be set equal to the identity matrix, should no weighting be justified. With the displacement equation error function (5), the correlation between rows of the error function and individual sensors is lost. Examination of (5) shows that the measured displacements are premultiplied by a fully populated matrix. Each entry in the error function includes contributions from multiple rows, but only one column. Thus, it is meaningful to weight columns of the equation error function, but not rows. [Wu] is set equal to the identity matrix, and weighting matrix [WF ] is used to represent the confidence in each given load case. Weight factors can be calculated from the variances of the response data. Measured responses are either displacement or strain measurements. If NSF is the number of sets of forces per experiment and NMR is the number of measured responses per set, then each response matrix is of size NMR X NSF. The number of experiments (NE) is the number of times the entire NDT is repeated. Each row of a response matrix corresponds to a sensor location and each column corresponds to a load case. Using these NE experiments, variances of the measurements are calculated and placed into a variance matrix of size NMR X NSF. Weight factors are derived from this matrix. Pooled variances are calculated row-wise, and their inverses are the diagonal entries of [Wu]. The reciprocal of the columnwise pooled variances are the diagonal entries of [WF ]. The pooled variances are calculated from (12)

where Vk = number of statistical degrees of freedom used to calculate the kth variance s i. The number of statistical degrees of freedom is equal to nk - 1, where nk = number of measurements used to calculate each variance in the NMR X NSF variance matrix.

FIG. 1.

Physical Model and Test Frame

1

350

7

6¥--

8

5

MM

4.c£1

If: 2 350

10

9£6

7

6&

1-

4

3

MM

J

2

Y"

% % %

1---300

FIG. 2.

MM

-I,

300

MM-4

Finite-Element Model



1

TABLE 1

Maximum Stre...., DI.placement., and Strain. for Final Load Sets Load Case

Maximum responses

1 FDOF 1: 117.7 N FDOF 6: 117.7 N (2)

(3)

101.4 N/mrn 2

100.6 N/mrn2

40.1 N/mrn2

110.3 N/mm 2

0.41 7.45 mrn -198 x 10- 6

0.41 -2.29 mrn 56 X 10- 6

0.16 3.09 mrn 90 X 10- 6

0.44 2.81 mrn -101 X 10- 6

Load set 4 FDOF 8: 313.6 N (5)

-0.0000239 0.0000240 0.0000237 -0.0001006

(1 ) IT.... IT...,.lITylold

Am..

e....

3 FDOF 1: -117.7 N FDOF 6: 117.7 N (4)

2

FDOF 3: -313.6 N

TABLE 2

4 FDOF 8: 313.6 N (5)

Averaged Strain Data Strains (mm/mm)

RLOC

(1)

Load set 1 FDOF 1: -117.7 N FDOF 6: -117.7 N (2)

Load set 2 FDOF 3: -313.6 N

(3)

Load set 3 FDOF 1: 117.7 N FDOF 6: -117.7 N (4)

-0.0001047 -0.0001070 0.0001985 0.0001163

0.0000561 -0.0000563 -0.0000561 0.0000252

-0.0000432 -0.0000438 0.0000761 0.0000903

E,

Ez E3 E7

TABLE 3. Estimated Moments of Inertia Using Averaged Strain Data METHOD Gauss-Newton Output Error No Weighting Average Data Element (1 )

Iz (mm 4 ) (2)

% Error (3)

1 2 3 4 5 6 7 8

1,018.9 1,008.6 1,131.5 1,021.1 1,073.9 1,135.7 1,099.7 1,021.1

-4.5 -5.4 6.1 -4.3 0.7 6.5 3.1 -4.3

Minimum error Maximum error RMS error Iterations

0.7% 6.5% 4.7% 4

5% are applied to the frame model on the extreme fibers of an element. The indicator unit has a resolution of IIJ.£ and an accuracy of ±0.05%.

Analytical Model The FEM of the frame shown in Fig. 2 consists of eight elements and 10 DOP. Intermediate nodes along the beams are used for loading and measurement. The model is used with the PARIS program for parameter estimation, and to determine loads that would not overload the structure and would produce measurable displacements or strains. When modeling the frame, it is assumed that members are prismatic and homogeneous with linear-elastic behavior and small deformations. Shear and axial deformations, which are small compared to bending deformation, are taken to be zero. The support conditions are taken as 100% fixed, and the connections between the members are assumed to be rigid. The dimensions of the FEM are taken from centerline to centerline of the members in the physical model. Each frame element has two parameters {P} associated with it: area, and moment of inertia. Because the axial stiffness is

high when compared with the bending stiffness, areas are treated as known parameters while moments of inertia are unknown, estimated parameters. The initial values for all moments of inertia are 1,000 mm" with an expected true value of 1,066.7 mmd , based on cross-sectional dimensions provided by the supplier of the frame. The areas are assumed to remain constant at 200 mm 2• NONDESTRUCTIVE TESTING

Using the heuristic method recommended by Sanayei et al. (1992) and Sanayei and Saletnik (1996b), an error tolerant subset of measurement and loading locations is selected and verified through Monte Carlo analysis (lmbaro 1993). The number and location of loads in each load set are chosen such that the stress distribution in each of the members would be nearly equal (Hajela and Soeiro 1990b). Load magnitudes selected are large enough to produce strains and displacements with a low noise-to-signal ratio, but not so large as to overstress the model. The final load cases are applied at DOF, 1, 3, 6, and 8 as single or multiple force sets as reported in Table 1. Predicted maximum bending stresses, displacements, and strains are O'max, am.", and £max' respectively. Four load sets are used. Four strains and four displacements are measured simultaneously for each load set. Each of the load sets is applied one at a time. The dial indicators and strain gauges are zeroed before the load set is applied, and each measurement is taken at the required response location, RLOC. Each load set is repeated 10 times resulting in 10 sets of data and a total of 40 displacement measurements and 40 strain measurements. Averaged measurements from all four load sets are used as input for parameter estimation. The size of the averaged displacement or strain measurement matrices is 4 X 4 each. TEST DATA AND PARAMETERS ESTIMATES

Forces and Strains The data from the strain experiments is shown in Table 2. Columns represent each of the four load sets. The location, magnitude, and direction of the loads are given. Averaged measured strains from the four elements are separated into rows. The maximum strains almost exactly compare to the predicted maximum strains in Table 1. Estimated moments of JOURNAL OF STRUCTURAL ENGINEERING 1 JUNE 19971795


TABLE 4.

Averaged Displacement Data Displacements (mm)

RLOe (1 )

Load set 1 FDOF 1: 117.7 N FDOF 6: 117.7 N (2)

Load set 2 FDOF 3: -313.6 N (3)

Load set 3 FDOF 1: -117.7 N FDOF 6: 117.7 N (4)

Load set 4 FDOF 8: 313.6 N (5)

3.572 0.004 7.204 0.100

-0.020 -2.288 0.010 0.257

0.683 0.007 3.069 0.050

-0.100 -0.230 0.087 2.848

u, U3 U.

u.

TABLE 5.

Estimated Moments of Inertia Using Displacement Data METHOD

Gauss-Newton Equation Error No Weighting Average Data

Gradient Equation Error No Weighting Average Data

Gradient Equation Error No Weighting Multiple Data

Gradient Equation Error Weighting Average Data

'.

'.

% Error (5)

'.

(6)

'.

% Error

(4)

(8)

(9)

(10)

'.

% Error (11 )

819.5 1,366.5 1,486.9 837.9 875.3 1,425.9 897.7 1,191.9

-23.2 28.1 39.4 -21.4 -17.9 33.7 -15.8 11.7

819.5 1,366.3 1,486.9 837.9 875.3 1,425.7 897.7 1,191.8

993.8 1,098.4 1,105.3 1,052.7 978.4 1,043.7 996.6 1,011.0

-6.8 3.0 3.6 -1.3 -8.3 -2.2 -6.6 -5.2

1,127.6 1,136.1 1,053.2 1,057.7 1,016.9 1,097.4 1,069.8 1,086.5

5.7 6.5 -1.3 -0.8 -4.7 2.9 0.3 1.9

Element (1 )

(2)

1 2 3 4 5 6 7 8

846.2 1,362.2 1,518.6 831.4 867.2 1,277.3 796.4 1,523.4

Minimum error Maximum error RMS Iterations

% Error (3) -20.7 27.7 42.4 -22.1 -18.7 19.7 -25.3 42.8 18.7% 42.8% 28.9% 5

11.7% 39.4% 25.5% 59

inertia from the averaged strain data are shown in Table 3. Using the Gauss-Newton method with the output error function, the solution converged in four iterations. To measure the success of the parameter estimation and to compare these results to those using displacement test data, the minimum, maximum, and root mean square (RMS) of percent error are calculated against the "true" moment of inertia of 1,066.7 mm 2 • With a minimum error of 0.7%, a maximum error of 6.5%, and an RMS error of 4.7%, the parameters are satisfactorily close to the directly measured "true" value. Since strains were measured with a high precision and had a small scatter leading to accurate parameter estimates, weight factors were not applied to the strain measurements. Weight factors will be applied to the less accurate displacement measurements. Forces and Displacements The data for the measured static displacements is shown as Table 4. The arrangement is identical to the strain experiments with the exception that RLOC indicates average measured displacements. The same load sets are used. The maximum displacements from each load set compare well to the predicted values in Table 1. Table 5 summarizes the estimated parameters using these data. As with strains the minimum, maximum, and RMS error are calculated against the true value of 1,066.7 mm 4 • Estimates of moments of inertia produced using the Gauss-Newton method with a displacement equation error function on the averaged data without weight factors are presented in column 2 of Table 5. It is quickly seen that much higher errors exist in the parameter estimates. A minimum and maximum error of 18.7 and 42.8% is clearly unacceptable. In an effort to improve the results, several other methods for estimating the unknown parameters from the displacement test data are also presented in Table 5. Estimates are made using the gradient method, multiple data sets, averaged data

% Error (7) -23.2 28.1 39.4 -21.4 -17.9 33.7 -15.8 11.7 11.7% 39.4% 25.4% 59

Gradient Output Error Weighting Average Data

1.3% 8.3% 5.2% 55

0.3% 6.5% 3.7% 51

sets, weighting, the "displacement equation error function," and the "displacement output error function." The second technique shown uses equation error and averaged data with the gradient method to estimate the parameters. After 59 iterations, the estimation converges. The estimates and percent errors are shown in columns 4 and 5 of Table 5. At 11.7, 39.4, and 25.5%, the minimum percent error, the maximum percent error, and RMS are lower than the previous case but are still quite high. At this point, loss of meaningful information through averaging is considered. If the complete set of data is used, will the estimate improve or remain the same? In the third estimation approach, the equation error function is used with the gradient method again. All 10 sets of raw data are used without averaging. The results of this estimation are shown in column 6 of Table 5. After 59 iterations, the estimates converge to nearly the same values as in the averaged case. The differences can be attributed to rounding of averaged. data. Therefore, we can conclude that averaging the data does not affect the estimate for the better or worse. In an attempt to improve the estimates further, weight factors are calculated using the method recommended previously. Using the 10 data sets, variances are calculated at each degree of freedom under each load case. The variances are shown in Table 6. Some variances have zero values because all of the measurements at some locations are identical under repeated load cases. In these cases one entry in each set of 10 measurements is increased by 0.01 mm, which corresponds to the precision of the displacement measurements. The variances are then recalculated as shown in Table 7. Next, the pooled variances are calculated columnwise. Their inverses are used as the diagonal entries for [WF ). The row-wise pooled variances are also calculated. These variances and the corresponding weight factors are shown in Table 8. Column 1 indicates which rows correspond to the columnwise variances and weight fac-



tors [WF ), and which correspond to the row-wise variances and weight factors fWd. The pooled variances are listed in the second column. The inverses of the pooled variance are the weight factors listed in the third column. In the fourth estimation technique reported in Table 5, weight factors are combined with equation error and the gradient method to identify the unknown parameters from the averaged data. Since equation error is used, only columnwise weighting is meaningful and only [WF ) is used. After 55 iterations, the estimates converged to the values shown in the eighth column of Table 5. The maximum and minimum percent errors drop significantly to 8.3 and 1.3%, respectively, and the RMS decreased substantially to 5.2%. These decreases in percent error between the unweighted and weighted estimations clearly represent the value in using weight factors. In the final method, the moments of inertia are estimated identically to the previous case, except that output error is used instead of equation error. Because output error preserves the correlation between individual measurements and the error function, both weight factor matrices [WF ) and [Wu ) are used: the columnwise weighting used previously, and row-wise weighting. Using this combination of methods, the estimates converged in 51 iterations. As the values in column 11 of Table 5 clearly indicate, the estimates are further improved by using output error instead of equation error. The RMS decreases to 3.7%. The maximum error and minimum error are only 6.5 and 0.3%. Variances in the data can be accounted more completely when both columnwise and row-wise weighting are used. From these five cases, some conclusions about solution methods and estimate quality can be drawn. When weighting is applied, the output error function produces substantially better-quality parameter estimates; despite its higher computational costs, the gradient method works better than the GaussNewton method when real test data are used, and there is no TABLE 6.

Variances of Displacement Data x 10-4 mm 2 Load Cases

ALOC (1 )

1 (2)

2 (3)

3 (4)

4 (5)

1 3 6 8

0.178 0.267 0.267 0

0 0.178 0 0.678

1.789 0.233 7.878 0

0 0 0.456 4.178

TABLE 7.

Variances of Modified Displacement Data x 10-4 mm 2 Load Cases

ALOC (1)

1 (2)

(3)

3 (4)

(5)

1 3 6 8

0.178 0.267 0.267 0.100

0.100 0.178 0.100 0.678

1.789 0.233 7.878 0.100

0.100 0.100 0.456 4.178

TABLE 8. Weight factors (1 ) Wp Wp Wp Wp

1 2 3 4

Wu Wu Wu Wu

1 3 6 8

2

4

Pooled Variances and Weight Factors Pooled variances x 10-4 Weight factors x 10-4 (2) (3) 0.203 4.932 0.264 3.789 2.500 0.400 1.208 0.828 0.542 1.846 0.194 5.143 2.175 0.460 1.264 0.791

real difference between using average test data and using the complete data set. Therefore, if displacement measurements are used, the weighted "displacement output error function" with the gradient method is recommended.

CONCLUSIONS The results from laboratory tests on a one-bay, two-story, two-dimensional, steel frame model verified the previously developed displacement equation error function, displacement output error function, and strain output error function methods of parameter estimation using static displacement and strain test data. Using the experimental data, the parameters of the structural elements (moment of inertia of each element) were successfully updated. Using strain measurements, all parameters were successfully identified with low deviations from the true values. This is the result of having strain measurements with high precision and low noise-to-signal ratios as well as an error function with no bias. Since displacement measurements are less accurate than strain measurements, weight factors were formulated for both of the displacement error functions used. The quality of the parameter estimates were improved when weight factors were used. Based on the results from the experiment presented in this paper, when displacement measurements are taken under static loads, the recommended parameter estimation method is the weighted displacement output error function combined with the gradient method. This method led to fairly accurate parameter estimates signifying the value of using weight factors.

APPENDIX I.

BIBLIOGRAPHY

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APPENDIX II.

REFERENCES

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