Monitoring and model updating of an FRP pedestrian

Life-Cycle of Engineering Systems: Emphasis on Sustainable Civil Infrastructure – Bakker, Frangopol & van Breugel (Eds), pp.756-763, ISBN 978-1-138-02847-0 IALCCE2016, Oct. 16-19, 2016, July Delft, The Netherlands.

Monitoring and model updating of an FRP pedestrian truss bridge G. Hayashi, C.W. Kim, Y. Suzuki, K. Sugiura Dept. of Civil and Earth Resources Eng., Kyoto University, Kyoto 615-8540, Japan

P.J. McGetrick

School of Planning, Architecture & Civil Eng., Queen’s University Belfast, Belfast BT9 5AG, UK

H. Hibi Hibi Co. LTD., Yoro, Gifu 503-133, Japan

ABSTRACT: This paper presents the results of real bridge field experiment, carried out on a fiber reinforced polymer (FRP) pedestrian truss bridge of which nodes are reinforced with stainless steel plates. The aim of this paper is to identify the dynamic parameters of this bridge by using both conventional techniques and a model updating algorithm. In the field experiment, the bridge was instrumented with accelerometers at a number of locations on the bridge deck, recording both vertical and transverse vibrations. It was excited via jump tests at particular locations along its span and the resulting acceleration signals are used to identify dynamic parameters, such as the bridge mode shape, natural frequency and damping constant. Pedestrianinduced vibrations are also measured and utilized to identify dynamic parameters of the bridge. For a complete analysis of the bridge, a numerical model of the FRP bridge is created whose properties are calibrated utilizing a model updating algorithm. Comparable frequencies and mode shapes to those from the experiment were obtained by the FE models considering the reinforcement by increasing elastic modulus at every node of the bridge by stainless steel plate. Moreover, considering boundary conditions at both ends as fixed in the model resulted in modal properties comparable/similar to those from the experiment. This study also demonstrated that the effect of reinforcement and boundary conditions must be properly considered in an FE model to analyze real behavior of the FRP bridge. 1 INTRODUCTION Fiber Reinforced Polymer (FRP) is considered as a new kind of structural material in civil engineering as it has strength comparable to that of steel with better corrosion resistance. However, FRP bridges are more easily vibrated than steel bridges since the elastic modulus of FRP is smaller than that of steel. Another weak point of FRP is larger variance of its material properties than structural steel, such as the elastic modulus, etc. The elastic modulus assumed in the design stage would differ from the elastic modulus of the constructed bridge following the design; this sometimes leads to structural behavior that differs from that predicted by FE analysis at the design stage. Therefore, it is extremely important to estimate material properties of FRP structures and update associated FE models considering updated material properties as well as mechanical properties. This study thus aims firstly, to investigate dynamic characteristics of an FRP pedestrian bridge and secondly, to build an FE bridge model based on its dynamic properties identified from a field experiment. In the field experiment, accelerations of the bridge were measured under pedestrian walking excitation as well as vertical drop by a volunteer at

particular locations along its span. The multivariate autoregressive (AR) model (Kim et al. 2012) was used to identify modal properties of the bridge, such as mode shape, natural frequency and damping constant. The FE model of the bridge is updated by comparison with experimental modal parameters. The vibration serviceability of the bridge was also investigated. The updating of the stiffness of FRP member and reinforcement parts was carried out using the Cross-Entropy method (e.g. McGetrick et al. 2015). 2 LINEAR SYSTEM MODEL 2.1 Estimation of multivariate AR model parameter This study focuses on estimation of bridge properties using a parameter taken from AR coefficients since a linear dynamic system can be idealized using the AR model as shown in Equation 1. p

y (k )   Gi  yk  i   e(k )

(1)

i 1

where y (k ) denotes the output of the system, Gi is the i-th order AR coefficients matrix and e(k ) is

the noise (or error) term. The autocorrelation function of y (k ) is defined in Equation 2.

  Ey k  s y

 k 

R ( s )  E y k y T k  s  T

(2)

where E[] is the arithmetic mean. From Equation 2, the Yule-Walker equation shown in the Equation 3 is obtained.  R0 R  1    R p 1

R1 R0  R p2

 R p 1   G 1  R 1     R   R p  2  G 2  2                 R 0  G p  R p 

(3)

AR coefficients are obtained by solving Equation 3. 2.2 Stabilization Diagram (SD) Plotting identified frequency versus model order yields a SD (Brincker & Ventura 2015). The estimated frequency can vary with AR coefficient. Frequencies which are physically meaningful for the structure have been confirmed with the SD. In other words, in the case of frequencies which have physical meaning, the same frequencies are identified regardless of AR coefficients on the SD. Conversely, frequencies identified with large variations in accuracy are considered to be physically meaningless. The damping constants and mode shapes follow similar trends. 2.3 Stability Criteria (SC) In this study, the stability criteria (SC) are introduced to filter out unstable vibration characteristics as described above. According to existing researches (e.g. Cauberghe et al. 2004), applying the SC results in highly stable identified modal parameters. Assuming that frequency fm, damping ratio ξm, mode shape φm are obtained in the m-th order multivariate AR coefficient, only those satisfying the following Equations 4 to 7, are extracted as locally stable parameters.

f m p  f   f m  f m p  f 

(4)

 m p      m   m p   

(5)

MACl  MACm, m p 

(6)

T

MAC(a,b) 



a b T a

a

 

2

T b

b

(8)

As described above, SC evaluate the degree of precision of the identified frequency for each AR coefficient. SC allow identified parameters to be regarded as locally stable vibration characteristics. Here, locally stable frequency is supposed to satisfy the SC. 3 FIELD EXPERIMENT AND MODAL PROPERTIES 3.1 Target bridge and field experiments The bridge investigated in this study is an FRP through truss bridge constructed in 2013, shown in Figure 1. Although FRP is the main structural material, structural stainless steel, SUS304, was also used to reinforce nodal points of the truss to improve both strength and vibration serviceability. The span length of the bridge is 17,500 mm and its width is 2150 mm, as shown in Figure 2. The gross weight of the bridge is 5612 kg. In the experiment only the acceleration was measured. Figure 2 shows the acceleration sensor layout and excitation position for testing. To investigate the vibration of the bridge, accelerations of 18 measurements points in the vertical direction and 22 points in the transverse direction were measured under pedestrian loadings i.e. walking excitation and a simple vertical drop test by a volunteer at particular locations along its span. It is noteworthy that in the first field experiment the sensors were instrumented in four and two stages due for measurements of transverse and vertical vibrations respectively due to shortage of sensors. For the second and third field experiments the sensors were instrumented in three and two stages for measurements of transverse and vertical vibrations respectively also due to shortage of sensors. Acceleration data for the span center is shown in Figure 3. The sampling frequency was 200 Hz. Details of the pedestrian excitations are as follows: (a) a volunteer of weight 70kg dropping from a chair of height 40 cm; (b) pedestrian walking approximately in-sync with a metronome set to 2Hz.

p   p s ,  ,1,1,  , p s

(7) where fε is a pre-selected frequency deviation tolerance; ξε a pre-selected damping deviation tolerance; MACl is the modal assurance criteria (MAC) between φm and φm+p; MACl is a pre-selected MAC lower bound; ps is a pre-selected number of MAR order to be evaluated. Herein, these parameters are adopted empirically as fε = 0.1 Hz, ξε = 1%, MACl = 0.95. MAC is defined by the following equation.



Figure 1 FRP through truss bridge.

30

F3

4375

17500

4375

4375

2288

4375

F1 F2

2288

: Accelerometer (Horizontal direction) : Accelerometer (Vertical direction)

4 Acceleration m/s2

10

F1～F3 : Position of drop tests W1～W2 : Position of walking tests

Figure 2 Sensor layout and excitation position.

2 0 -2 10 Time s

a)

15

original stable identified 0 0 2 4 6 8 10 12 14 16 18 20 22 24 Frequency (Hz) Figure 5 Stabilization Diagram.

b) 2nd field experiment

5

20

5

a) 1st field experiment

-4 0

25 AR Order

W1 W2

1250 1250

(Unit : mm)

15

20

Table 1 Experimental scenarios. Location of Case Excitation Excitation (Longitudinal) F1 Center F2 Center Impact F3 L/4 W1 W2

Walking

Location of Excitation (Transverse) Center Edge Center Center Edge

N.A.

Acceleration m/s2

0.8

Table 2 Temperature data from experimental periods. Spring Summer Winter Air temperature (28°C) (34°C) (9°C)

0.4 0

-0.4 -0.8 0

Bridge surface 5

10

15 Time s

20

25

N.A. N.A.

In open Shade

70°C 36°C

28°C 10°C

30

b) Figure 3 Example of acceleration response (vertical) at span center: a) Impact load; b) Walking load.

Table 3 Stable vertical vibration modes, natural frequencies and damping ratios from all three experiments. Temperature st

1 Bending Torsional A Frequency Torsional B 2nd Bending Damping ratio

1st Bending Torsional A Torsional B 2nd Bending

9°C

28°C

34°C

8.99Hz N.A. 12.90Hz 16.63Hz

8.48Hz 9.93Hz 12.69Hz 16.03Hz

8.30Hz 9.96Hz 12.71Hz 16.09Hz

1.93% N.A. 1.13% 1.37%

2.19% 1.91% 1.71% 1.85%

2.18% 1.97% 1.63% 1.80%

Figure 4 End view of the experimental bridge.

Table 1 outlines the experimental scenarios. To activate the second bending mode and torsional mode, a vertical drop test at L/4 and an eccentric drop test at 1/4 of the transverse width from the span center were also performed. Three experimental campaigns were executed in winter, spring and summer respectively to investigate changes in modal properties due to temperature changes. Table 2 shows temperature data for each experiment.

3.2 Identification of modal properties Stabilized modal properties were identified utilizing both a stabilization diagram (SD) and a multivariate AR model (Kim et al. 2013). Plotting frequencies of locally stable modes versus AR model order yields a stabilization diagram as shown in Figure 5. It has been observed that meaningful structural modes show nearly the same frequency value when the model order is over-specified, but the spurious modes do not. This statement may also be true for

the damping ratios and mode shapes. Based on the above statements, the structural modes can be identified in a statistical manner: the modes appearing frequently throughout a wide range of model orders, i.e., the statistically dominant mode, are identified as structural modes. The vertical broken lines in Figure 5 indicate the identified structural modes. The frequencies for the first two bending modes and corresponding mode shapes are shown in Figure 6. The first torsional mode was observed both near 10 Hz and 13 Hz since the corresponding modes shapes were similar. One mode may be relevant to the vertical torsional mode, but the other might be relevant to transverse vibrations of the upper-chord. However, the modal properties relevant to the transverse direction have not been comprehensively examined yet. Therefore, only vertical bending modes are discussed in this study. The modal identification demonstrates that the frequency corresponding to the first bending mode is around 8.6 Hz on average; this is larger than the frequency of pedestrian excitation (1.5-2.3Hz) specified in the design guideline (JRA 1979), demonstrating that no resonance will occur under these conditions. 3.3 Temperature effect The identified frequency and vertical vibration modes are summarized in Table 3. Two similar torsional modes were observed from the data measured in spring and summer, but only the Torsional-B mode was identified from the data measured in winter. Thus this study assumed Torsion-B mode as the first torsional mode. However, the reason for the

identification of a specific mode like the TorsionalA mode is not clear yet; this needs comprehensive examination including investigations on the influence of transverse vibrations. A noteworthy point in Table 3 is that the frequency corresponding to the bending mode tends to decrease with increasing temperature and vice versa. Changes in boundary condition with respect to temperature change could be a cause of this change in frequency, especially for the bending mode. 4 EIGENVALUE ANALYSIS 4.1 Analysis Model An FE model of the FRP pedestrian bridge was built, based on the structural and material parameters outlined in the original design documents, utilizing commercial structural analysis software ABAQUS, and eigenvalue analysis was carried out in order to estimate the natural frequencies and mode shapes of the bridge. The beam element was adopted to model the bridge as shown in Figure 7. For boundary conditions in the FE analysis, both Fixed-Fixed and Fixed-Roller conditions were considered. The elastic modulus of FRP was assumed as 23 GPa following the document on bridge design. To investigate the influence of reinforcement by stainless steel at each nodal point of the bridge, the elastic modulus of SUS304 was assumed as 200 GPa. The crosssectional properties and elastic moduli are summarized in Table 4. The cross-sections of the FRP and stainless steel are assumed to consist of uniform material to allow calculation of the equivalent elastic modulus. The equivalent elastic modulus was calculated using the formula shown in Equation 9.

Figure 6 Mode shapes and natural frequencies obtained from experiments (red line: right side, blue line: left side).

Ec  E f 

As Es  (1 np)E f Af

(9)

where Ec is the equivalent elastic modulus of composite members comprising GFRP and stainless steel. Ef is the elastic modulus of FRP members. Es is the elastic modulus of stainless steel, Af is the cross-sectional area of FRP members and As is the cross-sectional area of stainless steel. np= (AsEs)/(AfEf). Figure 7 Basic FE analysis model for model updating.

4.2 Eigenvalue analysis and FE model updating

a)

b) Figure 8 Bending modes from eigenvalue analysis: a) 1st bending mode; b) 2nd bending mode. Table 4 Cross-sectional properties of reinforced parts. UC at joint

LC at joint

DB at joint

UC

Cross - sectional area of SUS304 member (mm2)

2200

1080

900

1120

Cross - sectional area of FRP member（mm2)

2688

2688

2345

2688

Elastic modulus of SUS304 (GPa)

200

Elastic modulus of FRP (GFRP) (GPa)

23

Elastic modulus of Composite member 186.69 103.36 99.76 106.33 (GPa) UC: Upper Chord LC: Lower Chord DB: Diagonal Bracing

The results of eigenvalue analysis are shown in Table 5 and Figure 8. Therein Un_R-FR and Un_R-FF models indicate the model without considering reinforcement at each nodal point, with Fixed-Roller and Fixed-Fixed as the boundary conditions respectively. The Joint_R-FR and Joint_R-FF models indicate the model considering reinforcement at each nodal point, with Fixed-Roller and Fixed-Fixed as the boundary conditions respectively. Finally, ALL_RFR and ALL_R-FF models indicate the model considering reinforcement at each nodal point and upper chord members with Fixed-Roller and Fixed-Fixed as the boundary conditions respectively. The eigenvalue analysis demonstrates that the ALL_R-FR model provides the most comparable vertical bending modes and frequencies to those obtained from the experiment. It also shows that a change in boundary condition greatly influences analytical frequencies. In other words, in the FE analysis of the FRP pedestrian bridge, any reinforcement of structural members should be considered, which is different from what is usually considered in FE models of steel highway bridges. In steel bridges, the local reinforcement has little influence on structural behavior since the mass ratio of the reinforcement to the entire bridge is negligible; the mass ratio of the FRP bridge is not negligible. On the other hand, the elastic modulus of steel is very high compared to FRP (see Table 4). Therefore, FE models of FRP bridges must consider reinforcement members if it is present in the structure.

Table 5 Eigenvalue analysis results. FE model Un_R-FR Un_R-FF

Reinforcement Not considered

Boundary Condition Fixed Roller Fixed Fixed

1st mode Freq. (Hz) 5.26 6.16

2nd mode Freq. (Hz) 12.65 13.69

Joint_R-FR Joint_R-FF

Joint only

Fixed Fixed

Roller Fixed

6.58 7.88

15.37 16.70

ALL_R-FR ALL_R-FF

Joint + Upper Chord

Fixed Fixed

Roller Fixed

7.21 9.43

16.12 17.13

8.48 8.99 8.30

16.03 16.63 16.09

Experiment in Spring (28°C) Experiment in Winter (9°C) Experiment in Summer (34°C)

fication results. Then, the new distribution Uɤ is defined based on this sample. The parameter learning stage was performed by repeating this operation. In this research, the Gaussian distribution is used as the distribution g(x). The objective function O was set as follows:

9.2 Frequency Hz

8.8 8.4

O   ai ( f exp i  f i ) 2

8 7.6

Freq. in Summer Freq. in Winter

7.2 0

10 20 30 40 Spring constant [GPa]

a)

50

18

(11)

where ai is a weighting to consider importance of certain mode; fexpi is the natural frequency of the identified vibration modes from the experiment; fi is natural frequency of the identified vibration modes from eigenvalue analysis. Herein, the weighting ai is adopted as ai = 1, which means that in this study all the considered frequencies should satisfy same tolerance in the model update.

Frequency Hz

17.6

5.2 Definition of design variables

17.2 16.8 16.4 Freq. in Winter

16 0 b)

10 20 30 40 Spring constant GPa

50

Figure 9 Sensitivity analysis results to decide initial spring constant of the roller support: a) 1st bending mode; a) 2nd bending mode.

5 FE MODEL UPDATING 5.1 Cross-entropy method The cross-entropy method is a parameter learning method that is similar to the distribution estimation algorithm. First, a sample distribution Uɤ is defined by N samples of the design variables such as mean value and standard deviation of elastic modulus in this study. Next, a distribution having the minimum cross-entropy distance DCE is selected from distribution set G, such as Gaussian or Bernoulli distributions etc. in which the cross-entropy distance DCE is defined by Equation 10.





DCE g U    g ( x) log

U  ( x) g ( x)

dx

(10)

where, g(x) is a probability density function. N samples having design variables such as elastic moduli are created on the basis of this distribution. Then, to create the input data using this sample, an analysis (e.g. eigenvalue analysis of the FRP bridge in this study) is carried out N times. The elite, or fittest sample is extracted as that which has the smallest error between the analysis results and the identi-

The supports of the FRP pedestrian bridge studied had been placed directly on the base plate. This study considers a longitudinal spring at roller supports. This virtual spring is a linear spring in the longitudinal direction and contributes to the freedom of movement at the support. Thus, spring constant was a design variable adopted in order to consider the uncertainty at the bridge support. In addition, the elasticity of the FRP members and joint parts were also selected as design variables in order to consider the uncertainty in the elastic modulus of the joint parts. In this analysis, Kspr is a virtual spring constant, Ef is the elastic modulus of FRP member, Er is the elastic modulus of the joint parts, lower chord member and diagonal members of the upper chord member. EUC_j is the elastic modulus of the joint parts of the upper chord member. The initial value of the virtual spring constant is discussed in the next section. 5.3 Study of initial virtual spring constant The nominal value of the virtual spring constant does not exist. Therefore, a sensitivity analysis was performed to determine a reasonable initial value for the spring constant for model updating as shown in Figure 9 in which the vertical axis of the graph is the frequency of vibration while the horizontal axis is the virtual spring constant. The virtual spring was adopted in the ALL_FR model. The virtual spring constant is varied from 0GPa to 50GPa in this sensitivity analysis. The target is the same vertical modes, 1st and 2nd bending modes, in the previous chapter (8.48Hz and 16.03Hz of temperature of 28°C in Fig. 6). In the sensitivity analysis, first, the validity of the analysis is considered. The boundary conditions are fixed-roller conditions when the virtual spring constant is zero. This result is consistent with the fre-

5.4 Discussion of FE model updating Figure 10 illustrates the convergence of the elastic modulus of the FRP member by the cross-entropy method. Table 6 shows the initial and updated frequencies and elastic moduli of the FRP member while Table 7 shows the updated results for elastic moduli and the spring constant. As a result of the eigenvalue analysis using the optimum elastic modulus, the updated FE model reproduces the identified mode. These results indicate the possibility of identifying the physical properties of the members from the bridge vibration characteristics. However, observations from Table 7 showed that the model update resulted in increase in the elastic moduli of the joint parts and the reinforcement parts and the spring constant increased, but decrease in the elastic modulus of the FRP members. It is noteworthy that in this study only the vertical vibration mode of the lower chord member was considered as the control parameter in the cost function for the FE model updating. Therefore, it needs to consider transverse vibration modes, esp. those of the upper chord members, in the cost function to discuss. 6 CONCLUSIONS In this study the modal properties of an FRP pedestrian bridge were identified to clarify its dynamic characteristics. The effect of temperature on frequency was also investigated to gain an understanding of seasonal effects on modal properties of the FRP bridge. Finally, the most appropriate FE model was created based on a comparison of analytical modal properties with those obtained from the experiments. Observations made during this study are summarized as follows; 1) The bridge will not undergo resonance since its natural frequencies were far from code specified resonance frequencies. 2) The frequency corresponding to the bending mode tends to decrease with increasing temperature and vice versa.

Table 6 Comparison of bending mode frequencies. 1st bending mode (Hz)

2nd bending mode (Hz)

Elasticity of FRP (GPa)

Identified from experiment

8.30

16.09

N.A.

Before FE model update

9.43

17.13

23.0

After FE model update

8.30

16.09

20.2

Table 7 Initial value and model updating results. Initial elastic modulus (GPa)

Optimal elastic modulus (GPa)

Ef

23

20.2

EUC_j

186

222

Er

103

114

Kspr

10

13.6

Ef: the elastic modulus of FRP member, EUC_j: the elastic modulus of the joint parts of the upper chord member, Er: the elastic modulus of the joint parts, lower chord member and diagonal members of the upper chord member, Kspr: virtual spring constant. FRP member After Iteration 23

4

3

x 10

2.8 2.6

Youngs modulus (MPa)

quency obtained for the ALL_FR condition outlined in Table 5. Furthermore, it can be defined as fixedfixed by increasing the virtual spring constant, which can then be assumed to approximate the frequency obtained for the ALL_FF model, confirming the validity of the sensitivity analysis results. Next, the virtual spring constant is defined based on the analysis results. In the 1st bending mode, the virtual spring constant is found to be in the range of 10 to 40 GPa. On the other hand, focusing on the 2nd bending mode, the virtual spring constant is found below 10 GPa. Therefore the initial virtual spring constant was set 10GPa.

2.4 2.2 2 1.8 1.6 1.4 1.2 1 0

5

10

15

20

Iteration

Figure 10 An example of the convergence of elastic modulus of the FRP member by the cross-entropy method.

3) The eigenvalue analysis demonstrated that changes in the boundary condition greatly influence analytical frequencies. 4) Analytical frequencies were greatly affected by the modeling of the reinforced members. It was demonstrated that FE models of FRP bridges

must consider reinforcement members if they are present in the structure. It is clear that proper modeling of boundary conditions and reinforcement of the FRP bridge have a direct effect on the accuracy of the analysis. Therefore, the next step for this study will focus on FE model updating algorithms considering all the possible control parameters in the object function for optimization. ACKNOWLEDGMENTS The authors would like to thank Yokkaichi City Government for providing the experiment bridge and their assistance in the field experiment. REFERENCES Brincker, R. and Ventura, C. 2015. Introduction to operational modal analysis, Wiley. Cauberghe, B., Guillaume, P., Verboven, P., Parloo, E., Vanlanduit, S. 2004. The secret behind clear stabilization diagrams: The influence of the parameter constraint on the stability of the poles, SEM X Int. Congress & Exposition on Experimental & Applied Mechanics, Costa Mesa, CA, USA. Dassault Systemes Simulia: ABAQUS Analysis User’s Manual, Version 6.12. JRA. 1979. Specification and commentary for crossing facilities, Japan Road Association. (in Japanese) Kim, C.W., Kawatani, M. and Hao, J. 2012. Modal parameter identification of short span bridges under a moving vehicle by means of multivariate AR model, Structure and Infrastructure Engineering. 8(5): 459-472. Kim, C.W., Chang, K.C., Kitauchi, S., McGetrick, P.J., Hashimoto, K. and Sugiura, K. 2013. Changes in modal parameters of a steel truss bridge due to artificial damages, Proc. 11th Int. Conf. on Structural Safety and Reliability, June 16-20, 2013, Columbia Univ., New York, USA: 3725-3731. Ljung, L. 1999. System Identification: Theory for the User, 2nd edition. Prentice Hall PTR. McGetrick, P. J., Kim, C. W., González, A. and OBrien, E. J. 2015. Experimental Validation of a Drive-By Monitoring System for Bridges, Structural Health monitoring, 14(4): 317-331.