Predicting multi-span bridge maximum deflection: Numerical simulation and validation

Deflection is one of the main parameters often required to reflect and monitor the overall health state of bridge structures. Several approaches have been developed to monitor deflection accurately and economically. This research proposes a theoretical method to estimate the maximum deflection of the longest section of a seven-span continuous bridge using stress data collected through strain gauges. Firstly, initial data were collected from field tests under various controlled loadings to obtain the maximum deflections and stresses. Then, the stress-deflection regression equations are derived to reconstruct deflection from estimated gauges. Finally, the maximum deflection values are compared with analytical results using finite element simulations to verify the method’s effectiveness.


Introduction
Deflection is an essential indicator of a bridge's overall health and safety.Monitoring vertical deflection provides valuable information about the bridge's condition.It helps identify potential problems: excessive component stress, fatigue cracking, or other damage [1].There are various ways to monitor bridge conditions; one uses GPS, which is relatively expensive despite producing accurate data [2].Less costly alternatives may also provide the necessary information to determine deflection on the bridge.A few studies adopted methods for empirically fitting curves using predefined deflection forms.Zhang et al. [3] reconstructed the estimated bridge deflection of cable-stayed bridges through inclinometer data using the finite-element model combined with the partial least-square regression method.Sun et al. [4] proposed a novel method for bridge displacement estimation based on inclinometer data by developing a hybrid monitoring algorithm that uses the finite-element model and the monitored data.In addition, placing a tiltmeter or strain gauge is also an effective method of monitoring deflection [5].Penuntun et al. [6] proposed a new method depending on the Lagrange Interpolation Half approach to obtain deflection data from rotational data; they examined the deflection of a cable-stayed bridge using tiltmeters and approved the accuracy of the proposed method.In order to implement the idea of hybrid monitoring identified above.Suangga et al. [7] employed a particular stress-deflection relationship to predict bridge maximum deflection by utilizing strain data of a simple span bridge.Their findings indicated that the deflection outcomes from the suggested static equation agree well with the 2D Finite Element Method (FEM) analysis results.In this study, the stressdeflection relationship will be applied to load test data of a continuous multi-span steel box girder bridge presented in Brinissat et al. [8], to predict the maximum analytical deflections.This approach will use the regression analysis method, and based on the test loadings and a 3D FEM, the maximum deflections will be correlated accurately.

Analytical approach of beam maximum deflection
A simplified analytical formulation for maximum beam deflection based on section proprieties and stress follows in this section.Consider an idealized uniform elastic beam with constant depth 'h,' breadth IOP Publishing doi:10.1088/1757-899X/1304/1/012016 2 'b,' and span 'L,' as shown in Figure 1.The beam is assumed to be loaded with a uniform distributed load 'w' over the entire span, with different support conditions, as shown in Table 1.The maximum bending stress that occurs at the center of the beam in the extreme fiber follows as: Here σmax and Mmax represent the maximum stress and the maximum bending moment at the crosssection, respectively, ymax is the distance from the neutral axis to the extreme fiber, and I is the moment of Inertia of the beam.Where: (2) Equation ( 1) can be combined with equation ( 2) and the maximum bending moment formula to express the maximum deflection at the center of the beam shown in Table 1

Case study: The Szapáry Bridge
The structure this research considers is the Szapáry bridge, a multi-span steel box girder bridge located in Hungary.The bridge is 758 meters long; it is composed of a single-cell steel box girder weighing 8,327 tons, orthotropic deck plates, and skewed girders with two tracks, each with two lanes, as shown in Figure 2.

Monitoring parameters based on FE Model
A Finite-Element model of the bridge superstructure was simulated in Midas Civil software to represent the condition of the right independent track of the continuous steel box girder bridge, as shown in Figure 3.One of the significant challenges with these bridges is the substantial deflection in the midspan, which impacts bridge structure safety.Therefore, this research assessed deflections and stresses at the mid-span of the longest span (3 rd span) created by numerous load cases, presented in Table 2.

Monitoring parameters based on in-situ test
The deflection and the stress data were based on real-time monitoring of the critical section in the steel box girder to ensure that the bridge could be considered safe and under control.
The midspan section of the longest span provided locations for the stress measurement.Strain gauges produced data at six locations in the section, as shown in Figure 4.The top and the bottom steel plate each contained three measuring points (left, middle, and right).Gauges measured the longitudinal strain of each measuring point, and then the data were collected and analyzed.The relationship between stress and strain allowed for the conversion from strain to longitudinal stress at each measuring point.Two rotary resistance meters fixed on the deck of every span mid-section recorded the deflection measurement.The maximum deflection and stress data from the measured 13 load cases appear in Table 3.

Monitoring parameters based on a theoretical formula
The developed formula (deflection-stress) is based on interpolation between the recorded data using a multiple regression solver.The interpolation technique was processed to obtain the maximum deflection for each reference point, the sixth point in the middle of the largest span.Table 4 presents the equations and goodness of fit for each location using linear and quadratic regression methods.An example of the theoretical approach that was applied to develop the linear regression equation is presented in Figure 5.

Validation of the theoretical formula
In order to validate the approach, the differences between the measured and the predicted deflections are computed using Equation 3: The analysis of available data presented in Tables 5 and 6, shows that the average difference between measured deflection and deflections from regression equations is 7.61 mm for the linear regression and 7.42 mm for the quadratic regression.Figure 6 provides a comparison chart that summarizes the measured, analyzed, and calculated maximum deflection results.By examining this chart, it is clear that the regression results demonstrate a high degree of accuracy when compared to both the full FEM model and the field test measurements.This suggests that the generated theoretical formula proved its ability to provide adequate maximum deflection values.In light of the results, it can be considered an economical method for bridge deflection estimation.

Conclusions
Regression formulas predict the maximum deflection of a bridge deck accurately and reliably.Higherorder formulas may produce more accurate results, but the proposed method already provides a good prediction.This approach gives the engineer another method to verify field test data in a simple and rapid format for the bridge monitoring process.Using numerical simulations makes it possible to analyze further and validate the outcomes.
Nevertheless, strain gauge sensors demonstrated their high potential to measure the stress and estimate the maximum deflections of a large bridge structure.This method provides an economic estimation system in structural health monitoring.

Figure 3 .
Figure 3. Finite element model of the bridge superstructure.

Figure 4 .
Figure 4. Locations of the strain gauge measurement points (FEM X-sec).

Figure 6 .
Figure 6.Deflections from the field, FEM, and best-fit regressions of stress.

Table 1 .
below : Maximum deflection of beams with different support systems.

Table 2 .
Arrangement of the load cases.

Table 3 .
Stresses and maximum deflections from recorded data.

Table 4 .
Theoretical formula of maximum deflection as a function of stress.

Table 5 .
Comparison of maximum deflections from the field test and linear regression equations.

Table 6 .
Comparison of maximum deflections from the field test and quadratic regression equations.