Development of Static Response Based Objective ...

Development of Static Response Based Objective Functions for Finite Element Modeling of Bridges Xiaoyi Wang1, James A. Swanson2, Arthur J. Helmicki3, and Victor J. Hunt4

Abstract:

The basic mechanisms and procedures of finite element (FE) modeling and

calibration are briefly presented in the context of bridge condition assessment.

Different

physical parameters of FE models are adjusted to simulate experimental measurements. To quantify the calibration process, static response based objective functions are carefully developed based on two powerful condition indices: Bridge Girder Condition Indicators (BGCIs) and Unit Influence Lines (UILs). Critical issues related to the indices are discussed in detail. Using an existing calibration strategy, a nominal FE bridge model is optimized by minimizing this global static-response-based objective function. The value of the objective function is reduced from 12.98% to 4.45%, which indicates convergence of the calibration process. It is shown that the automated calibration becomes practical due to the formulation of the static response based objective function.

1

Graduate Research Assistant, ASCE Associate Member, Department of Civil & Environmental Engineering, University of Cincinnati, OH 45221-0071. 2 Associate Professor, ASCE Associate Member, University of Cincinnati Infrastructure Institute, Department of Civil & Environmental Engineering, University of Cincinnati, Cincinnati, OH 45221-0071. 3 Professor, ASCE Member, University of Cincinnati Infrastructure Institute, Department of Electrical and Computer Engineering and Computer Science, University of Cincinnati, Cincinnati, OH 45221-0030. 4 Assistant Research Professor, University of Cincinnati Infrastructure Institute, Department of Electrical and Computer Engineering and Computer Science, University of Cincinnati, Cincinnati, OH 45221-0030.

1

Introduction The civil engineering community has been long aware of the limitations of the current practice of condition assessment (CA) based on visual inspections. Visual inspections do find signs of deterioration when they become visible. By this time, however, the structural condition may be so severely deteriorated that options for retrofit may be limited. During the last several decades, a number of additional techniques for bridge CA have become available to compliment visual inspection.

Although much research on CA has been carried out, there are still fundamental issues to be resolved.

The University of Cincinnati Infrastructure Institute (UCII) has explored the

experimental and analytical aspects of a structural-identification method for a comprehensive CA procedure, which has been successfully applied to numerous types of bridges (Aktan 1997; Aktan 1998a; Helmicki 2000; Hunt 1998; Lenett 1999; Wang 2004). Recent examples include the evaluation of a 6-span steel-stringer bridge for a 900kip truckload (Hunt 2002), the assessment of an 80-year old cantilever through-truss bridge over the Ohio River for strength and fatigue rating (Helmicki 2002), and laboratory and field identification of a new high strength steel material used in the construction of a composite 4-span steel-stringer bridge (Choo et. al. 2005; Kayser et. al., 2006a; Kayser et. al. 2006b). In the CA procedure, a nominal finite element model (based on plans and/or visual inspection) is constructed and then calibrated to better simulate measured responses (Turer 2000). To quantify the comparison between experimental and analytical results, objective functions for model calibration are developed based on structural response indices.

2

Structural response indices that are selected for CA should be able to accurately and completely represent the structural characteristics of the system. A series of indices was proposed for damage identification and health monitoring of structural and mechanical systems in a literature review (Doebling et al. 1996). The sensitivity of vibration-based indices to various levels of damage was evaluated by Farrar and Jauregui (1998) based on data recovered from a steel-girder bridge. A method to evaluate structural deterioration by dynamic response was developed by Chen et al. (1995) based on the study of two simple girders designed to respond similar to fullsize bridges. The CA procedure developed at UCII incorporates both the traditional truckload responses of deflections and strains induced by a truck loading and the more recently accepted modal response data. These two sets of responses are conveniently grouped into static and dynamic responses and are treated in separate discussions. In this paper, objective functions that were developed for finite element (FE) model calibration based on static response indices are discussed. Objective functions for dynamic response indices are discussed in the companion paper (Wang et. al. 2006).

The development of well conceived and defined objective functions is an important step forward in the area of FE model calibration and condition assessment. In its current form, model calibration typically consists of an engineer making successive modifications to an FE model based on subjective assessments of analytical and experimental responses. By providing an objective way of quantifying the differences in the analytical and experimental responses, the calibration process can be reduced to a general optimization problem that is well suited for computerized automation.

3

Description of Condition Assessment Condition Assessment Procedure: A CA procedure has been developed based on a complete system of field-testing, FE modeling, and load rating. Experimental techniques, including both modal testing and truckload testing, are used to collect data from constructed systems. Threedimensional (3D) FE modeling is used in a process that typically consists of generating an FE model, calibrating that model to match experimental data, and using the results from the calibrated model to rate the condition of the bridge or investigate unique loadings or retrofit schemes (Wang 2004).

3D FE Modeling: A 3D FE model is constructed using various analytical elements at different locations. The pier caps and abutments are defined by four node shell elements. The girders are defined by using four node shell elements for the web and two node beam elements for the flange. The shell and beam elements have six degrees of freedom at each node. The pier caps and abutments are connected to the girders by beam elements hereafter referred to as connecting springs. The deck is modeled using four node shell elements that are connected to the girders by rigid links. The stiffness of the link elements determines the level of composite action present in the model. Pinned supports are defined to simulate the boundary conditions at the bottom of piers and abutments. Cross frames are represented by beam elements and guardrails or parapets are modeled by connecting the edge nodes of the exterior deck elements with beam elements. The properties of these beam elements are computed to approximate the stiffness of the actual guardrail or parapet. Sidewalks, if present, are modeled by including a second layer of shell elements above the first layer of elements that represent the bridge deck. These sidewalk elements are connected by rigid links similar to those connect the deck to the girders. More

4

detailed information regarding the FE modeling approach used in the CA process is presented in dissertations (Turer 2000; Wang 2005) and in other journal articles (Wang et. al. 2004; Padur et. al. 2002).

3D FE Model Calibration: After the nominal FE model is generated, it is typically modified to better simulate the actual conditions of the bridge. Adjustments are made to improve the simulations of boundary conditions, continuity conditions, structural geometry, and material properties at both global and local levels using response indices as measures of accuracy. The selection of critical parameters closely follows earlier work by others (Turer 1999; Turer and Aktan 2000; Wang 2005). The combinations and sequences of those parameters are determined by sensitivity studies (Li 2003) and the evaluation of calibration results. The adjustment of critical parameters is typically separated into those that will most directly affect the static responses and those that will most directly affect the dynamic responses. The static responses of the bridge model that are used for comparisons include girder deflections via Bridge Girder Condition Indicator (BGCI) (Lenett 1998) and strain responses via Unit Influence Lines (UILs) (Turer 1999). To quantify the differences between experimental data and analysis results, a global objective function is defined based on the response indices:

O.F .static = f ( BGCI _ Error , UIL _ Error )

(1)

where O.F.static is an objective function based on static response indices, BGCI_Error is an objective function based on deflections responses, and UIL_Error is an objective function based on measured strains. The relationship between BGCI_Error and UIL_Error may be linear or nonlinear depending on the properties of indices and the optimization process. Two factors are

5

introduced to combine BGCI_Error and UIL_Error and adjust the relative weighting of each term. As a result, a specific static-response-based objective function can be generated as O.F .static = w1 BGCI _ Error + w2UIL _ Error

(2)

where w1 and w2 are weighting factors.

The sum of the weighting factors should be 1 and the deflection and strain objective functions should be defined such that the value of global objective function is between 0 and 1. A value 0 indicates a minimized error and 1 indicates a large error.

The calibration of FE models is accomplished by minimizing the static response based objective function along with a similar objective function based on dynamic response indices that is presented in a companion paper. A thorough discussion of the BGCI_Error and UIL_Error is presented in the following sections.

Condition Assessment Indices Deflection Responses under Static Loads: Any condition variation of a structure, global or local, would result in the change of its flexibility or stiffness. In the UCII CA procedure, the modal flexibility matrix is used to generate the experimental BGCI response from measured modal data. The transformation of the measured natural frequencies and mode shapes to an approximation of the modal flexibility matrix [f] is achieved by n

[f ]= ∑

{ψ } r {ψ }Tr

ω r2

r =1

6

(3)

where [f] is the modal flexibility matrix, n is number of measured mode shapes, {ψ}r is the rth mode shape, and ωr is the angular frequency (rad/s) of the rth mode. The number of mode shapes to be included in the computation of the experimental flexibility is determined by using a modal truncation study (Lenett 1998).

Deflected shapes of a bridge obtained by virtually loading the modal flexibility by different load patterns have been shown to be sensitive to deterioration or damage (Lenett 1999). The BGCI is the deflected shape of a girder under a series of unit loads. It is not the deflected shape of the bridge when the bridge is uniformly loaded. Instead, it is the deflected shape of each girder when only the test nodal points over a reference girder are loaded with vertical unit loads. The flexibility deflection profile, or BGCI, is computed by multiplying the flexibility matrix by a load vector comprised of downward loads of unit magnitude positioned at each accelerometer location along the reference girder line and is defined mathematically as

{∆}BGCI ,i = [ f ]{F }BGCI ,i

(4)

where {∆}BGCI,i is the deflected shape of all girders in the bridge and {F}BGCI is unit force vector applied to the accelerometer locations over girder i.

The BGCIs can be used in either vector or matrix form or as scalar values by summing the terms of the vectors corresponding to one or more girders. It is a powerful condition index, which can be used for assessing many possible impacts on structural reliability.

7

When direct deflection measurements from displacement transducers are available, they are not directly used in the CA process. Instead, they are used to independently verify the modal flexibility measurements and calculations.

Strain Responses under Static and Moving Truck Load: Goble et al. (1991) demonstrated the concept of conducting a crawl-speed loading test and used the results for rating bridges by analyzing possible changes in strains/stresses due to different trucks. The Unit Influence Line (UIL) of a critical strain response can be extracted from a crawl test as a condition index. As a truck with a known weight and axle spacing crosses over a bridge, the response of the bridge at critical locations can be measured using strain and deflection gages. Assuming linear behavior, the measured response can be decomposed into a normalized influence line for the measured response as if a unit concentrated load had traveled over the bridge.

The measured response

(strain or deflection) is the superposition of a number of responses that may be obtained from bridge UILs after accounting for the locations of axles and scaling for the different axle weights. Therefore, if the measured response, axle weights, and distances between the axles are known, it is possible to solve the problem in reverse and compute the UIL (Turer and Aktan 1999; Hunt 2000). The UIL obtained in this way can further be used for predicting the response of a super load with various axle load and patterns, for load rating, and health monitoring purposes as long as the moving load does not cause dynamic amplification and linearity is satisfied (Hunt 2000; Helmicki 2002; Hunt 2002; Kambli 2002).

8

Static Response Based Objective Functions An objective function based on static response indices is shown in Equation (2). This equation consists of a deflection component and a strain component. In the calibration of an FE model to experimental data, this static objective function and a similar dynamic objective function (Wang et. al 2006) are simultaneously minimized using either an automated or manual updating process.

Deflection Response Based Objective Function: In the modal tests, accelerometers are placed on the road surface above some or all of the girders. After the modal frequencies and mode shapes are extracted from measurements, the approximate flexibility matrix can be computed using Equation (3). Then experimental BGCIs can then be computed by Equation (4). The analytical BGCIs can be computed applying unit concentrated loads to the top the girders of FE bridge models. A separate load case is used for each loaded girder line.

To quantify the difference between experimental and analytical BGCIs, two requirements should be satisfied. First, the difference between experimental and analytical nodal displacements corresponding to each accelerometer location should be included. This is the foundation of the criterion, which assures the comparison is correct from a mathematical view. Second, a standard domain should be applied to each term of the global static response based objective function, which makes the determination of weighting factors practical. In the current study, the domain is set between 0 and 1. Zero indicates a minimized error and 1 indicates an infinite error.

To satisfy the two requirements described above, the hyperbolic tangent function is introduced,

9

e2x − 1 Tanh( x) = 2 x e +1

(5)

where Tanh(x) is the hyperbolic tangent of a variable x.

Taking the absolute value of each side of Equation (5) gives, Tanh( x) =

e2x − 1 e2x + 1

(6)

The curve of the function with -∝ < x < +∝ is plotted in Figure 1.

It can be proven that the

function equals zero when x equals zero and that it approaches unity as x approaches infinity. Note that the slope of the function at a given point is an indication of the proximity to the optimum solution. As a result, the derivative of Tanh(x) can be used to determine step sizes in an iterative optimization procedure.

Thus, if x =

∆ anal ,ij − ∆ exp,ij ∆ exp − max

is substituted into Equation (6), the following results.

⎛ ∆ anal ,ij − ∆ exp,ij Tanh⎜ ⎜ ∆ exp − max ⎝

⎛ ∆ anal , ij − ∆ exp, ij 2⎜ ⎜ ∆ exp − max ⎝

⎞ ⎟ ⎟ ⎠

⎞ e −1 ⎟= ⎛ ⎞ ∆ anal , ij − ∆ exp, ij ⎟ ⎟ 2⎜ ⎠ ⎜ ∆ exp − max ⎟ ⎝ ⎠ e +1

(7)

where ∆anal,ij is the analytical vertical deflection at the jth degree of freedom in the ith load case,

∆exp,ij is the experimental vertical deflection at the jth degree of freedom in the ith load case, and ∆exp-max is the experimental vertical deflection with the maximum magnitude for all load cases. Taking all the degrees of freedom and all the loads cases into account, the deflection response based objective function is defined as,

10

BGCI _ Error =

⎛ ∆ anal ,ij − ∆ exp,ij 1 m n Tanh⎜ ∑∑ ⎜ ∆ mn i =1 j =1 exp − max ⎝

⎞ ⎟ ⎟ ⎠

(8)

where m is the number of load cases and n is the number of degrees of freedom.

The numerator (∆anal,ij - ∆exp,ij) expresses the difference between experimental and analytical deflections at each accelerometer location. The denominator ∆exp-max is used to generate a relative difference with respect to the experimental vertical deflection with the maximum magnitude. BGCI deflections are expected to be different for different bridges due to their different physical conditions. A deflection with a smaller magnitude may result in an error with a smaller magnitude, which does not necessarily indicate a better calibration result. A relative value is more objective in evaluating the BGCI_Error. As one of the most important properties of the BGCI curve, the experimental vertical deflection with the maximum magnitude for load cases is selected as the denominator. Since both requirements described above are satisfied, the deflection response based objective function can be combined with the strain response based objective function to form the global one.

Strain/Stress Response Based Objective Function: In the truckload test, strain gages are placed on the top and bottom flanges of girders located at mid-span and over piers. Trucks are driven across the bridge as strain responses are recorded. One response is used for each lane on the bridge. In the FE model, a series of static concentrated loads are applied on each lane of the bridge to simulate the crawl speed truckloads.

11

The maximum response recorded by a strain gage is considered to be one of the most important properties of the UILs. Therefore, the normalized absolute differences of maximum stress responses can be used to quantify the comparisons between experimental and analytical UILs.

The objective function with respect to maximum stress response can be generated based on Equation (6), UIL _ Error =

1 m

n

∑∑ F i =1 j =1

⎛ u anal ,ij −max − uexp,ij −max ⎜ F Tanh ∑∑ conf −i , j ⎜ u exp,ij −max i =1 j =1 ⎝ m

conf −i , j

n

⎞ ⎟ ⎟ ⎠

(9)

where m is the number of strain gages, n is the number of loaded lanes for each bridge, Fconf-i,j is the confidence coefficient for the ith strain gage of the jth test lane, uexp,ij-max is the strain value with the maximum magnitude in the experimental UIL vector for the ith strain gage of the jth test lane, and uanal,ij-max is the analytical strain value at the same location as uexp,ij.

Typically, a set of UIL data is obtained from truck-load responses that are recorded by strain gages when a truck moves along a lane of the bridge. A confidence coefficient is used to account for the uncertainty of the UIL data recorded by a strain gage for one test lane. The value of each confidence coefficient is set between 0 and 1. A value of 1 is assigned when there is a high degree of confidence in the measured data. A value of 0 is assigned when there is a low degree of confidence in the measured data. After scaling, the value of the objective function will be between 0 and 1.

Note that the form of the UIL_Error shown in Equation (9) makes use of the maximum values of strain from the experimental and analytical UILs. While these maximum values are probably the

12

most critical indicators for the UILs, it should be noted that there is often more than a single local maxima for a given UIL. For continuous girders, there are local maxima and minima for each span of the structure. To accommodate multiple maxima in the UIL_Error, Equation (9) is modified and shown as Equation (10).

UIL _ Error =

1 m

n

∑∑ F i =1 j =1

⎛ Fconf −i , j ⎜ ∑ ∑ ⎜ l i =1 j =1 ⎝ m

conf −i , j

n

⎛ u anal ,ij −k − uexp,ij −k Tanh⎜ ∑ ⎜ uexp,ij −k k =1 ⎝ l

⎞ ⎞⎟ ⎟ ⎟⎟ ⎠⎠

(10)

where l is the number of peak values per strain gage per lane, uexp,ij-k is the kth peak strain value in the experimental UIL vector for the ith strain gage of the jth test lane, and uanal,ij-k is the analytical strain value at the same location as uexp,ij-k.

One final consideration with regard to the denominator inside the Tanh() function is that in the form of Equations (9) and (10), the response from each gage is normalized by the maximum output from that particular gage. This situation leads to an emphasis of the gages with smaller responses. Consider two gages with different responses; say the first has a maximum response of 100 µε and the second has a response of 10 µε. If a difference of 5 µε exists between the experimental and analytical strains at both gage locations, the second gage will dominate the objective function because its relative error is 50% while the relative error for the first gage is only 5%. This situation can lead to unexpected and undesirable results during the calibration process and, as a result, the authors recommend using the maximum strain value of all gage locations for a loading in the jth lane as the denominator in the Tanh() function. This final form of the UIL_Error objective function is shown below as Equation (11),

13

UIL _ Error =

1 m

m

n

∑∑ F i =1 j =1

⎛ uanal ,ij − max − uexp,ij − max ⎞ ⎞⎟ ⎜ ⎟ Tanh conf − i , j ⎜ ⎟⎟ u j =1 exp, j − max ⎝ ⎠⎠ ⎝ n

⎛

∑∑ ⎜⎜ F

conf − i , j

i =1

(11)

where uexp,j-max is the strain value with the maximum magnitude from all gage locations due to a loading in the jth lane.

Although a comparison of strain magnitudes is important to the success of a model calibration, the shape of the UILs is also important. Several approaches have been developed to compare shapes of two vectors. In this paper, the ‘Modal Assurance Criterion’ (MAC) (Allemang and Brown 1982) is selected to compare the shapes of the experimental and analytical UIL vectors. The MAC provides a measure of the least squares deviation or ‘scatter’ of the points from the straight-line correlation. The MAC between two UIL vectors {u}anal,ij and {u}exp,ij is defined as:

{u}Texp,ij {u}anal ,ij MAC ({u}anal ,ij , {u}exp,ij ) = ({u}Texp,ij {u}exp,ij )({u}Tanal ,ij {u}anal ,ij ) 2

(12)

where {u}anal,ij is the analytical UIL vector for the ith strain gage of the jth test lane and {u}exp,ij is the experimental UIL vector for the ith strain gage of the jth test lane.

The MAC is a scalar quantity that ranges from 0 to 1. A MAC value close to 1 indicates that two vectors are well correlated and a value close to 0 is indicative of orthogonal vectors.

Based on Equation (11), the UIL_Error can be defined as: UIL _ Error =

1 m

m

n

∑∑ F i =1 j =1

conf −i , j

n

∑∑ F i =1 j =1

14

conf −i , j

{MAC ({u}

anal ,ij

, {u}exp,ij )}

(13)

It is important to understand, however, that the MAC measures the difference in shape between two vectors but not the difference in magnitude. As a result, Equation 12 should not be used alone. Instead, it should be combined with Equation (11). The final static response based objective function for calibration is a combination of Equations (11) and (13), which is defined as: UIL _ Error =

1 m

n

∑∑ F i =1 j =1

⎡ ⎛ u anal ,ij − max − u exp,ij − max ⎜ F ∑∑ conf − i , j ⎢c1 Tanh⎜ u exp, j − max ⎢⎣ i =1 j =1 ⎝ m

conf − i , j

n

⎤ ⎞ ⎟ + c 2 MAC ({u}anal ,ij , {u}exp,ij )⎥ ⎟ ⎥⎦ ⎠

(14) where c1 and c2 are scaling factors that are used to adjust the relative importance of errors in magnitude and shape between the experimental and analytical UIL vectors. The sum of c1 and c2 should be 1. After scaling, Equation (14) results in a value between 0 and 1. A value 0 indicates a minimized error and 1 indicates an infinite error.

Equation (2) is evaluated by combining Equations (8) and (14). The static response based objective function will be minimized from its initial value when adjustments to critical parameters in the FE model are made to simulate measured data. The maximum limitation of the initial value is set as 15%. If the initial value is larger than the maximum limitation, the nominal model will be re-evaluated and improved before the calibration. Although the ideal value for the static objective function is 0, this goal may not be achieved because critical parameters are adjusted within certain reasonable limitations to ensure the validity of the bridge model. In order to minimize calibration time, some convergence criteria should be assigned to determine when the calibration of FE model is complete. Based on the experience of the authors, the calibration is considered to be complete when: (1) the value of objective function becomes lower than 5%;

15

and (2) during each of the last ten iterations, the relative reduction of objective function is less than 1%.

Application A representative bridge in Butler County, Ohio, was selected as a case study to illustrate the calibration procedure and static response based objective function discussed in this paper. BUT732-1043, shown in Figure 2, was constructed in 1951 and consists of three spans measuring 18.3 m, 22.9 m, and 18.3 m (60ft, 75ft, and 60ft). The bridge has no skew, is 11 m (36ft) wide, and consists of 5 girders (W840 × 210 interior and W920 × 289 exterior (W33 × 141 and W36 × 194)) spaced at 2.54 m (8.33ft).

Figure 3 shows a comparison of experimental and analytical deflections for girders 2 and 3 when a series of unit loads is applied along girder 2. The differences between experimental and analytical results are minimized from the nominal model to the calibrated model. Four load cases were defined for analysis with 38 required displacement values in each load case. Therefore, there are a total of 4×38 = 152 values required to evaluate the BGCI objective function. As is shown in Table 1, the value of deflection based objective function is reduced from 11.38% to 3.09% during calibration.

Figure 4 shows a comparison of experimental and analytical UILs for a strain gage on the bottom flange of girder 3 in span 1 for a load in the left lane. The difference between the experimental and analytical results is minimized from the nominal model to the calibrated model. For each strain gage, UIL data are recorded for loads in each lane. Confidence coefficients are defined as

16

1.0 for five sets of data and 0 for the other data. As specified in the previous section, the sum of c1 and c2 is 1. The ratio of c1 to c2 is unity if all data points are included in the calculation of MAC. It is clear that a different result of MAC will be given when a fraction of the full set of data is included. In this situation, more confidence should be given to the maximum value comparison than the shape correlation. Thus, the ratio of c1 to c2 will have a larger value when a fraction of the full set of data is included. Based on the specific situation of this bridge, scaling factor c1 is defined as 3/4 and c2 is defined as 1/4. As is shown in Table 1, the value of the UIL objective function is reduced from 14.57% to 5.81% during calibration.

During the calibration, deflection and strain-based objective functions are minimized simultaneously by using the objective function shown as Equation (2). For the purposes of illustration within this paper, objective functions calculated from Equation (8) and (14) are considered to have equal weight. Therefore, a value of 0.5 was used for weighting factors w1 and

w2 in Equation (2).

Finally, it is shown in Table 1 that the value of static response based

objective function, O.F.Static, decreases from 12.98% to 4.45% during calibration.

This

represents an improvement of 65.7% from the nominal FE model to the calibrated FE model.

Summary This paper presents the development of objective functions based on static response indices that can be used in calibration procedures for finite element models. A global objective function is constructed by combining two functions with weighting factors and is used to quantify the differences between experimental and analytical results. Objective functions based on deflection and strain responses are carefully developed in both structural and mathematic aspects. The

17

deflection-response-based objective function is generated with respect to the Bridge Girder Condition Indicator while the strain-response-based objective function is generated with respect to Unit Influence Lines and important issues regarding these two important indices are discussed. By providing a quantitative measure of the differences between experimental and analytical responses, the challenge of calibrating a finite element model is reduced from a process dominated by subjective assessments to a fundamental optimization problem that is well suited for computerized automation.

Following the developed strategy, a finite element model for a sample bridge was calibrated to match experimental data. The nominal model of the bridge was created using measured or assumed parameters. The bridge was initially assumed to be partially composite based on design drawings. During the calibration, several critical parameters of the FE model were adjusted with goal of minimizing the static objective function. The calibration was iterated until convergence was achieved. An evaluation of the results shows the FE model is optimized from an error of 12.98% at the nominal level to an error of 4.45% at the calibrated level.

Acknowledgements The authors gratefully acknowledge the sponsors of this research: the National Science Foundation, the Federal Highway Administration, the American Society for Nondestructive Testing, and the Ohio Department of Transportation.

18

Appendix I. References Aktan, A. E., Farhey, D. N., Helmicki, A. J., Brown, D.L., Hunt, V.J., Lee, K. L., and Levi, A. (1997). “Structural identification for condition assessment: experimental arts” J. of Struct.

Engrg., ASCE, 123(12), 1674 – 1684. Aktan, A. E., Catbas, N., Turer, A., and Zhang, Z. (1998a). “Structural identification: analytical aspects.” J. of Struct. Engrg., ASCE, 124(7), 817 – 829. Aktan, A. E., Helmicki, A. J., and Hunt, V. J. (1998b) “Issues in heath monitoring for intelligent infrastructure.” Smart Mater. Struct. Engrg., ASCE, 7(5), 674 – 692. Allemang, R. J., and Brown, D. L. (1982). “A correlation coefficient for modal vector analysis.”

Proc., Int. Modal Analysis Conf., Society for Experimental Mechanics, Bethel, Conn., 110 – 116. Chajes, M. J., Mertz, D. R., and Commander, B. (1997). “Experimental load rating of a posted bridge.” J. of Bridge Engrg., ASCE, 2(1), 1 – 10. Chen, H. L., Spyrakos, C. C., and Venkatesh, G. (1995). “Evaluating structural deterioration by dynamic response.” J. of Struct Engrg., ASCE, 121(8), 1197 – 1204. Choo, T., Linzell, D.G., Lee, J., and Swanson, J.A., "Response of a Continuous, Skewed, High Performance Steel, Semi-Integral Abutment Bridge During Deck Placement," Journal of

Constructional Steel Research, Elsevier, Vol. 61, Num. 5, Pgs. 567-586, May 2005. Chowdhury M. R. and Ray J. C. (1995). “Further consideration for nonlinear finite – element analysis.” J. of Struct. Engrg., ASCE, 121(9), 1377 – 1379. Doebling, S., Farrar, C., Prime, M., and Shevitz, D. (1996). “ Damage identification and health monitoring of structural and mechanical systems from changes in their vibration characteristics: A literature review.” Rep. No. LA-13070-MS, Los Alamos National Laboratories, LosAlamos, N.M. Ewins, D. J. (2000). Modal testing: theory, practice and application. Research Studies Press Ltd. Letchworth, Hertfordshire, England.

19

Farhey, D. N., Naghavi, R., Levi, A., Thakur, A. M., Pickett, M.A., Nims, D.K., and Aktan, A. E. (2000). “Deterioration assessment and rehabilitation design of existing steel bridge.” J. of

Bridge Engrg., ASCE, 5(1), 39 – 48. Farrar, C. R., and Jauregui, D. V. (1998). “Comparative study of damage identification algorithms applied to a bridge: Part I, experimental” Smart Mater. Struct. Engrg., ASCE, 7(5), 704 – 719. Goble, G. G. et al. (1991). “Load prediction and structural response.” Final Rep. Submitted to the

FHWA by Department of CAE, Univ. Colorado, Boulder, Colo. Kim, J. T., Ryu, Y. S., Cho, H. M., and Stubbs, N. (2003). “Damage identification in beam-type structures: frequency-based method vs mode-shape-based method”, Engrg. Struct. 25(2003), 57 – 67. Helmicki, A., Hunt, V., Lenett, M., and Turer, A. (2000). “Structural Identification and Statistical Database Development for a Class of Bridges: Towards Type-Specific Management.”

Proceedings of the 2000 American Control Conference, Chicago, IL, June. Helmicki, A., Hunt, V., Lenett, M. (2002). “Non-destructive testing and evaluation of the ironton-russell bridge,” Proceedings of ASNT Structural Materials Technology V: An NDT

Conference, Cincinnati, Ohio, 139 – 146. Hunt, V., Turer, A., Gao, Y., Levi, A., Helmicki, A. (1998). "Instrumented Monitoring and Nondestructive Evaluation of Highway Bridges," Proceedings of Structural Materials

Technology III - An NDT Conference, San Antoinio, TX, Technomic Publishing, Lancaster, Pennsylvania, pp. 472-481, April. Hunt, V. (2000). “Nondestructive evaluation and health monitoring of highway bridges”. Doctoral Dissertation, University of Cincinnati, Cincinnati, Ohio. Hunt, V., Helmicki, A. (2002). “Field testing and condition evaluation of a steel-stringer bridge for superload,” Proceedings of ASNT Structural Materials Technology V: An NDT Conference, Cincinnati, Ohio, 56 – 62.

20

Kambli, S., Helmicki, A., Hunt, V. (2002). “Dedicated real time monitoring of a steel-stringer highway bridge,” Proceedings of ASNT Structural Materials Technology V: An NDT

Conference, Cincinnati, Ohio, 283 – 290. Kayser, C.R., Swanson, J.A., and Linzell, D.G., (2006a) " Fatigue Resistance of HPS-485W (70W) Continuous Plate with Punched Holes," Journal of Bridge Engineering, ASCE, InReview Kayser, C.R., Swanson, J.A., and Linzell, D.G., (2006b) " Characterization of the Material Properties of HPS-485W (70W) TMCP for Bridge Girder Applications," Journal of Bridge

Engineering, ASCE, Jan. 2006, In-Press Law, S. S., Shi, Z. Y., and Zhang L. M. (1998). “Structural damage detection from incomplete and noisy modal test data”, J. of Engrg. Mech., ASCE, 124(11), 1280 – 1288. Lenett, M. S. (1998). “Global condition assessment using modal analysis and flexibility”, Doctoral Dissertation, College of Engineering, University of Cincinnati, Cincinnati, Ohio. Lenett, M., Hunt, V., Helmicki, A., Brown, D., and Catbas, F.N., (1999) “Condition Assessment of Commissioned Infrastructure Using Modal Analysis and Flexibility,” Proceedings of

International Modal Analysis Conference XVII, Kissimmee, FL, pp. 1251-1259, February, 1999. Li. Z. (2003) Investigation of Several Issues Related to Three Dimensional Finite Element Bridges Condition Evaluation, MS Thesis College of Engineering, University of Cincinnati, Cincinnati, OH. Padur, D.C., Wang, X., Türer, A., Swanson, J.A., Helmicki, A.J., and Hunt, V.J., (2002) "Non Destructive Evaluation/Testing Methods - 3D Finite Element Modeling of Bridges," Proceedings: NDE/NDT for Highways and Bridges, ASNT, Cincinnati, OH, Sept 10-13, 2002 Pothisiri, T., and Hjelmstad, K. D. (2003). “Structural damage detection and assessment from modal response.” J. of Engrg. Mech., ASCE, 129(2), 135 – 145. Salawu O. S. S and Williams C.(1995). “Bridge assessment using force – vibration testing.” J. of

Struct Engrg., ASCE, 121(2), 161 – 173.

21

Toksoy, T., and Aktan, A. E. (1994). “Bridge condition assessment by modal flexibility.” Experimental Mechanics, Scan Electron Microsc., 34(3), 271 – 278. Turer, A., and Aktan, A.E. (1999). "Issues in Superload Crossing of Three Steel-Stringer Bridges in Toledo, Ohio," Transportation Research Record, No. 1688, TRB, National Research Council, Washington, D.C., pp. 87-96. Turer, A. (2000). “Condition evaluation and load rating of steel stringer highway bridges using field calibrated 2D-Grid and 3D-FE models.” Doctoral Dissertation, University of Cincinnati, Cincinnati, Ohio. Wang, X., Kangas, S., Padur, D., Liu, L., Swanson, J., Helmicki, A., Hunt, V. (2004). “Overview of a modal test based condition assessment procedure.” submitted to J. of Bridge Engrg., ASCE. Wang, X. (2005) “Structural Condition Assessment of Steel Stringer Highway Bridges,” Doctoral Dissertation, College of Engineering, University of Cincinnati, Cincinnati, Ohio Wang, X, Swanson, J.A., Helmicki, A.J., Hunt V.J., (2006) “Development of Dynamic Response Based Objective Functions for Finite Element Modeling of Bridges,” Engineering, ASCE, In-Press

22

Journal of Bridge

Appendix II. Notation

The following symbols are used in this paper: BGCIError = deflection response based objective function c1, c2 = scaling factors [f] = modal flexibility matrix

Fconf-i,j = confidence coefficient for ith strain gage of jth test lane {F}BGCI = unit force vector applied to the accelerometer locations over a girder

O.F.static = static response based objective function Tanh(x) = hyperbolic tangent function uanal,ij-k = analytical stress coefficient at the same location as uexp,ij-k uexp,ij-k = the kth peak stress coefficient in the experimental UIL vector for the ith strain gage of the jth test lane

uanal,ij-max = analytical stress coefficient at the same location as uexp,ij uexp,ij-max = stress coefficient with the maximum magnitude in the experimental UIL vector for the ith strain gage of the jth test lane

uexp,j-max = the strain value with the maximum magnitude from all gage locations due to a loading in the jth lane {u}anal,ij = analytical UIL vector for the ith strain gage of the jth test lane {u}exp,ij = experimental UIL vector for the ith strain gage of the jth test lane

UILError = strain response based objective function w1 and w2 = weighting factors ∆anal,ij = analytical vertical deflection at the jth degree of freedom in the ith load case ∆exp,ij = experimental vertical deflection at the jth degree of freedom in the ith load

23

case

∆exp-max = experimental vertical deflection with the maximum magnitude for all load cases {∆}BGCI = BGCI deflected shape

ωk2 = square of the rth circular frequency (rad/s)2 {ψ}r = rth mode shape/eigenvector matrix

24

Appendix III. List of Tables Table 1 – Comparison of Calibration Results

Appendix IV. List of Figures Figure 1 – Absolute Value of Hyperbolic Tangent Function Figure 2 - BUT-732-1043 Bridge Figure 3 – Bridge Condition Indicator of Modal Test and FE Models Figure 4 – Unit Influence Lines of Truckload Test and FE Models

Table 1 Comparison of Calibration Results O.F.

BCGI_Error

UIL_Error

O.F.Static

Nominal Model

11.38%

14.57%

12.98%

Calibrated Model

3.09%

5.81%

4.45%

FE Models

25

1 0.8 0.6 0.4 0.2

-4

-2

2

4

Figure 1 – Absolute Value of Hyperbolic Tangent Function

26

(a) General Plan

(b) General Elevation Figure 2 - BUT-732-1043 Bridge

27

0

Deflections (cm)

-0.005 -0.01 -0.015 -0.02 -0.025 Girder 2

Girder 3

-0.03 0

20

40

60

80

100

Length of Girder (m) Modal Test

Nominal Model

Calibrated Model

Figure 3 – Bridge Girder Condition Indicators of Modal Test and FE Models

28

120

350 Truckload Test

300

Calibrated Model

250 Stress(kPa)

200 150 100 Nominal Model

50 0 -50 -100 0

10

20

30

40

Distance(m) Truckload Test

Nominal Model

Calibrated Model

Figure 4 – Unit Influence Lines of Truckload Test and FE Models

29

50

60