Deflection distribution estimation of the main arch of arch bridges based on long-gauge fiber optic sensing technology

Abstract

Deflection of the main arch of arch bridges is one of the main indices for supporting the alignment after construction and evaluating the structural performance. The existing sensing technology and analysis method for deflection monitoring have developed, but it is still difficult to monitor the deflection of the main arch of a long-span arch bridge with great height difference between measuring points. On the contrary, in recent years, with the outstanding advantages of fiber optic sensing technologies, a long-gauge fiber Bragg grating sensing technology has been used in structural health monitoring due to its characteristics, including reflecting the macro and micro information and being connected into network. For these reasons, the long-gauge fiber Bragg grating sensing technology is proposed to develop a method to monitor the deflection of the main arch of arch bridges. A curvature load method for deflection distribution estimation using strain measurements is proposed. It deduces the expression of the complex relation between the strain and the deformation on the main arch element and then separates the coupled strain on the element through the specific sensor layout. A series of simulation tests of the deck arch bridge, half-through bridge, and through arch bridge was conducted. It is concluded that the proposed method can not only be applied to these long-span arch bridges but also can identify the static and dynamic deflections of the main arch effectively.

Keywords

coupled strain separation curvature load method deflection distribution estimation long-gauge FBG sensing main arch of arch bridges

Introduction

Deflection is one of the most intuitive and convenient parameters in structural health monitoring. If the bridge deflection can be accurately measured and analyzed, the safety analysis of the bridge structure will be more direct, accurate, and guaranteed, which will promote the development of bridge monitoring system undoubtedly. As the main stress component of the long-span arch bridge, the deflection monitoring of the main arch is of great significance to the alignment control after construction and structural performance evaluation. However, the measurement accuracy of the main arch is often difficult to meet the requirements of engineering monitoring because of the great variation of the height difference between the monitoring points on the main arch.

There are two groups for measuring structural deformation: direct measurement and indirect measurement. Many types of displacement sensors have been used in the direct measurement method for deflection measurement, which involves the contact and non-contact sensors. As a contact sensor, the linear variable differential transducer can only measure deformation accurately if a fixed base and a complex installation setup are available (Joh et al., 2018; Liu et al., 2018; Ozdagli et al., 2018). Some non-contact sensors have also been developed for direct measurement of structural displacement. For the non-contact sensor, Global Positioning System (GPS) is used for dynamic displacement measurement in two orthogonal directions (Chan and Xu, 2005); when the deformation is caused by an earthquake or typhoon or when the measured values are off by normal observation, the GPS is designed to identify the location (Fujino et al., 2000); Robotic Total Stations (RTSs) and Global Navigation Satellite System (GNSS) techniques, as an integrated manner, may provide efficient solutions for measuring three-dimensional (3D) displacements of structures (Peppa et al., 2018; Psimoulis and Stiros, 2013; Scaioni et al., 2018; Zhong et al., 2018); an unmanned aerial vehicle (UAV) with 3D digital image correlation was developed to monitor the health of bridges (Ellenberg et al., 2015; Park et al., 2015a, 2015b; Reagan et al., 2018; Ribeiro et al., 2014). The non-contact sensors overcome the disadvantages of contact sensors, but their application is still affected by many factors. For example, the measurement accuracy of GPS depends on many factors such as satellite coverage, atmospheric effect, and processing methods (Chan et al., 2006; Chan and Xu, 2005); the monitoring network and a proper site are necessary for RTS and GNSS to minimize the signal reflection from the surroundings (Scaioni et al., 2018); it is so important to use UAVs equipped with position stabilization feature for the possibility of drifts during the flight, which should be considered. The required permission shall be obtained in advance, when the projects are located at urban areas (Akbar et al., 2019; Esmaeili et al., 2019). On the other aspect, the indirect method has been implemented to estimate the structural deflection. A curvature approach for vertical displacement using curvature measurements is proposed based on the geometric relationship between curvature and vertical displacement (Rodrigues et al., 2011; Yau et al., 2013). An improved conjugated beam method (ICBM) for distributed structural deformation estimation is based on fiber optic sensor (Shen et al., 2010a; Zhang et al., 2017). However, there are still limitations in the indirect measurement method. For example, the conjugate beam method can estimate deflection accurately and is good at robustness, but the conjugate beam method at this moment is just suitable for a simple structure.

In recent years, fiber Bragg grating (FBG) sensors have developed rapidly in the field of structural monitoring (Kim et al., 2012; Xu et al., 2015; Ye et al., 2013). Long-gauge FBG sensor has been experimentally studied in the laboratory and used extensively in various civil engineering structures (Guo and Zhang, 2014; Li and Wu, 2008; Zhang et al., 2015, 2017). Compared with the traditional accelerometer, long-gauge FBG sensor can determine the equivalent dynamic displacement using distributed strain measurement, and it reflects the micro and macro information of the structure and can be connected into a network. Therefore, a deflection distribution estimation method of the main arch of arch bridges based on long-gauge fiber optic sensing technology is proposed in this article. First, the long-gauge FBG sensing technology and its merit of revealing local and global structural features are presented. Second, mechanical analysis of the arch bridges is performed to reveal the relationship between main arch deformation and strain, and the axial strain and bending strain are separated from the measured strain by a special sensor layout. Finally, a curvature load method is proposed to estimate the deflection distribution of the main arch of arch bridges using long-gauge strain measurements. Numerical examples of static and dynamic tests are then investigated to verify the effectiveness of the proposed method for deflection distribution estimation of the main arch of arch bridges.

Long-gauge FBG sensing technology

With the absence of suitable sensors for accurate distributed monitoring, a concept of long-gauge sensing and its corresponding sensor networks have been developed (Geiger et al., 1995; Liang et al., 2005; Spillman and Huston, 2000; Zhou and Zhang, 2019). As shown in Figure 1(a), handling of an embedded tube inside which a bare optic fiber with FBG is sleeved and fixed at two ends to ensure the measured value represents the average strain over the gauge length. Two materials of different stiffness are utilized to package long-gauge FBG sensors for enhancing the measurement sensitivity of strain responses (Li and Wu, 2009). By designing FBG sensors with a long-gauge, which can be extended from 20 to 200 cm, the measured strain will be the average strain over the long-gauge length. The long-gauge FBG sensor can accurately output static and dynamic structural strain response. The measured strain is related to the corner of structure so that it can reveal the overall (angular measurement) and local information (strain measurement; Shen et al., 2010b, 2013; Zhang et al., 2017). An Euler–Bernoulli beam element with two local degrees of freedom (DOFs) at each node, one for vertical displacement ν and the other for rotation θ, is shown in Figure 1(a). The long-gauge strain measurement in element m can be expressed as

\begin{matrix} μ_{m} = \frac{h_{m}}{L_{m}} \\ ε_{m} (t) = μ_{m} [θ_{0} (t) - θ_{p} (t)] \end{matrix}

(1)

where the subscript m represents the mth long-gauge sensing unit, L_m is the long-gauge length, h_m is the distance from the sensor to the neutral axis of the beam, ε_m is macro-strain measured by the long-gauge sensor, and θ_o(t) and θ_p(t) represent the rotations of each node at time t.

Figure 1.

Concepts of long-gauge sensor and area-distributed monitoring: (a) long-gauge FBG sensor and (b) area-distributed monitoring.

Based on the above description, the long-gauge FBG sensor has the features unique to measuring both local (strain) and global information (rotation) of the structure. As shown in Figure 1(b), multiple long-gauge sensing units are connected in series to cover a key sensing area, and multiple key areas are connected to form a distributed sensing network for area-distributed structural monitoring.

Theory basis of the proposed method

In order to obtain the deformation distribution of the main arch using the strain data from long-gauge FBG sensors, it is necessary to establish the complex relationship between the strain and the deformation of the main arch element for identifying the deflection distribution of the main arch. By analyzing the relation between deflection and internal force under load, this article deduces the relational expression between strain and deformation on the main arch of arch bridges.

Analysis of the relationship between deformation and internal force of arch

As a curved beam, a curved bar on the main arch is taken as the research object to analyze the relationship between the deflection and the internal force of the main arch (Chen and Zuo, 1988). An infinitesimal section ds is taken from the plane curved bar, whose displacement and deformation are shown in Figure 2 under load.

Figure 2.

Displacement and deformation of the infinitesimal section ds.

According to the differential geometric relationship, the horizontal and longitudinal displacement can be expressed respectively as

\begin{matrix} du = θ ds (1 + ε_{N}) \sin ϕ - ε_{N} ds \cos ϕ \\ dv = θ ds (1 + ε_{N}) \cos ϕ + ε_{N} ds \sin ϕ \end{matrix}

(2)

where du and dv are the horizontal and longitudinal displacement of the infinitesimal section ds, respectively. ϕ is the angle between the curved bar and the x-axis before it deforms. θ is the change of the ϕ after curved bar deforms. ε_N is the strain caused by the axial force N_x in the infinitesimal section ds. Under the load, the length of the curved bar changes from ds to ds(1 + ε_N). The first derivative of u and v with respect to x in equation (2) is expressed in equations (3) and (4), respectively

\frac{du}{dx} = θ \tan ϕ (1 + ε_{N}) - ε_{N}

(3)

\frac{dv}{dx} = θ (1 + ε_{N}) + ε_{N} \cdot \tan ϕ = ε_{N} y^{'} + θ (1 + ε_{N})

(4)

where $\tan ϕ$ in equation (4) is equal to $y^{'}$ . The second derivative of v with respect to x in equation (4) is expressed as

\frac{d^{2} v}{d x^{2}} = \frac{d θ}{dx} (1 + ε_{N}) + y^{'} \frac{d ε_{N}}{dx} + ε_{N} y^{″}

(5)

If material nonlinearity is not taken into account, the relation between deformation and internal force considering the effect of shear deformation is

\frac{d θ}{ds} = - \frac{M_{x}}{E I_{x}} + \frac{d}{ds} (\frac{a_{s} Q}{G A_{x}})

(6)

where E and G are the elastic modulus and shear modulus of the material, respectively. M_x and Q are the flexural moment and shear force on the section, respectively. I_x and A_x are the moment of inertia and cross-sectional area of the section, respectively. a_s is the shear coefficient of the section.

The positive directions of the internal force M_x and N_x are shown in Figure 3. The horizontal component of N_x is H = N_x·cosϕ, thus equation (6) can be further rewritten as

\frac{d θ}{dx} = \frac{1}{\cos ϕ} \cdot \frac{d θ}{ds} = - \frac{M_{X}}{E I_{x} \cos ϕ} + \frac{d}{ds} (\frac{a_{s} Q}{G A_{x}}) \frac{1}{\cos ϕ}

(7)

where N_x is the axial force of section x and H is the level component of N_x.

Figure 3.

Internal force of the infinitesimal section ds.

Substituting equation (7) into equation (5), we get

\begin{matrix} \frac{d^{2} v}{d x^{2}} = [- \frac{M_{x}}{E I_{x} \cos ϕ} + \frac{d}{ds} (\frac{a_{s} Q}{G A_{x}}) \frac{1}{\cos ϕ}] \\ (1 + \frac{N_{x}}{E A_{x}}) + {(\frac{N_{x}}{E A_{x}})}^{'} y^{'} + \frac{N_{x}}{E A_{x}} y^{″} \end{matrix}

(8)

If the effect of shear deformation on bending is ignored, equation (8) can be further deduced as follows

\frac{d^{2} v}{d x^{2}} = - \frac{M_{x}}{E I_{x} \cos ϕ} (1 + \frac{N_{x}}{E A_{x}}) + {(\frac{N_{x}}{E A_{x}})}^{'} y^{'} + \frac{N_{x}}{E A_{x}} y^{″}

(9)

When the main arch of arch bridges has a constant section, A_x = A and I_x = I, thus equation (9) is written as

\frac{d^{2} v}{d x^{2}} = - \frac{M_{x}}{EI \cos ϕ} (1 + \frac{N_{x}}{EA}) + {(\frac{N_{x}}{EA})}^{'} y^{'} + \frac{N_{x}}{EA} y^{″}

(10)

It is known from Figures 2 and 3 that

\begin{matrix} N_{x} = H \cdot \sec ϕ \\ \frac{d N_{x}}{dx} = H \cdot \sec ϕ \cdot \tan ϕ \frac{d ϕ}{dx} \\ \frac{d^{2} y}{d x^{2}} = \sec^{2} ϕ \cdot \frac{d ϕ}{dx} \end{matrix}

(11)

Substituting equation (11) into equation (10), we get

\begin{matrix} \frac{d^{2} v}{d x^{2}} = - \frac{M_{x}}{EI \cos ϕ} - \frac{M_{x} N_{x}}{E^{2} IA \cos ϕ} + \frac{H}{EA \cos ϕ} \\ \cdot \frac{d^{2} y}{d x^{2}} (1 + \sin^{2} ϕ) \end{matrix}

(12)

The second term on the right side in equation (12) has little effect on the vertical displacement. So, the relationship between the flexural moment M_x, axial force N_x, and vertical displacement v can be finally established through equation (13)

\frac{d^{2} v}{d x^{2}} = - \frac{M_{x}}{EI \cos ϕ} + \frac{N_{x}}{EA} \cdot \frac{d^{2} y}{d x^{2}} (1 + \sin^{2} ϕ)

(13)

Analysis of the relationship between the deformation and strain of arch

In order to obtain the deflection distribution from strain data, it is necessary to establish the relationship between the deformation and the stain for the main arch. Based on the above theoretical derivation, the measured strain data obtained from the long-gauge FBG sensors are mainly caused by flexural moment and axial force, as shown in equation (14)

ε_{N} = \frac{N_{x}}{EA} = \frac{H}{EA \cos ϕ}

(14a)

ε_{M} = \frac{M_{x}}{EI} h

(14b)

ε = ε_{M} + ε_{N} = \frac{M_{x}}{EI} h + \frac{H}{EA \cos ϕ}

(14c)

where ε_M and ε_N are the axial strain component and flexural strain component on the element of the main arch, respectively. h is the distance from the sensor to the neutral axis of the arch.

Substituting equation (14c) into equation (13), equation (13) can be further deduced

\frac{d^{2} v}{d x^{2}} = - \frac{ε_{M}}{h \cdot \cos ϕ} + ε_{N} \cdot \frac{d^{2} y}{d x^{2}} (1 + \sin^{2} ϕ)

(15)

It is concluded from equation (15) that not only the flexural moment but also the axial force contribute to the displacement of the main arch. Especially when the axial force accounts for a large proportion in the coupled force, the effect of the axial force on the displacement becomes more obvious.

Deflection distribution estimation of arch bridges

Based on the complex relationship between strain and deformation, a curvature load method is proposed for identifying the deflection distribution on the main arch of arch bridges. Considering the fact that the collected strain data are coupled strain and cannot be used in the proposed method, the sensor optimal placement of the main arch is presented to effectively identify the deflection distribution.

Curvature load method for deflection distribution estimation

The ICBM has been proposed to identify the structural deflection distribution (Zhang et al., 2017). It is assumed that the density q(x) of the distributed load on the conjugate beam at any point x is equal to the value of $ε (x) / h$ on the section of the actual beam at the corresponding point. The curvature load method is proposed for deflection distribution estimation of the main arch based on this theory.

The curved load method (CLM) proposed in this article takes the change curvature k(x) along the span as the uniformly distributed load $q_{i}^{'} (x)$ and makes a simply supported beam with the same span as the conjugate beam corresponding to the main arch. Because the curvature change of the main arch is mainly caused by the bending strain and axial strain of the structure, as described in Figure 4, the uniformly distributed load $q_{i}^{'} (x)$ consists of two parts, as shown in equation (16)

q_{i}^{'} = q_{mi}^{'} + q_{ni}^{'}

(16)

Figure 4.

Scheme of curvature load method theory.

Substituting equation (15) into equation (16), we get

q_{i}^{'} = q_{mi}^{'} + q_{ni}^{'} = - \frac{ε_{M}}{h \cdot \cos ϕ} + ε_{N} \cdot \frac{d^{2} y}{d x^{2}} (1 + \sin^{2} ϕ)

(17)

According to static equilibrium, the reaction at the left support of the conjugate beam is

R^{'} = \frac{L_{m}}{n} \sum_{i = 1}^{n} {q^{'}}_{i} (n - i + 0.5)

(18)

The bending moment of the midpoint in each element of the conjugate beam (i.e. the deflection of the same point in the arch bridges) is

y_{i} = \frac{{L^{2}}_{m} \sum_{j = 1}^{i} {q^{'}}_{j} (i - j) + {q^{'}}_{i} {L^{2}}_{m}}{8 - R^{'} L_{m} (i - 0.5)} (i = 1, 2, \dots, n)

(19)

where L_m is the length of element and i is the number of elements.

Sensor layout

As the main member of arch bridges, the strain distribution of the main arch is complicated, and there is no linear relationship. The traditional sensor in bridge monitoring mainly includes strain gauge (strain or stress monitoring) and deflectometer (deflection monitoring), which usually lay out at the L/4, 3L/4, top, and bottom of the arch. The sensor layout at an unfavorable loading position or equivalence points of the structure is not targeted and cannot meet the accuracy requirement of deformation distribution estimation of the arch structure.

Based on the advantages of long-gauge FBG sensor mentioned above, as given in Figure 5, a sensor layout research idea for deformation distribution estimation of the main arch of arch bridges is proposed. The sensor layout research idea mainly includes two parts: top section and bottom section. The characteristic curve of coupled strain of the main arch was fitted by the top section sensor, so as to separate the axial strain and bending strain for the deformation distribution estimation of the arch structure. The sensor located on the bottom section is mainly used to determine the target function of the deformation estimation error less than the limited. The sensors of the top section and bottom section will be arranged symmetrically in the main arch (Zhang et al., 2016).

Figure 5.

Research idea of the sensor layout.

The main arch is mainly compression bending, and the total strain can be approximated to the coupled value of the axial strain and bending strain. Before estimating the deflection of the structure, the axial strain ε_N and bending strain ε_M should be separated from coupled strain measured by long-gauge FBG sensor. Assuming that the material nonlinearity of the main arch is not taken into account, it can be considered that the neutral axis of the arch rib of a common symmetric structure under load is half the height of the section. According to the linear superposition principle of the Mechanics of Materials, based on the strain of main arch

\begin{matrix} ε_{Ni} = \frac{ε_{U i} + ε_{Di}}{2} \\ ε_{UMi} = ε_{Ui} - ε_{Ni} \\ ε_{DMi} = ε_{Di} - ε_{Ni} \end{matrix}

(20)

where ε_Ni is the axial strain of element i, ε_UMi and ε_DMi are the bending strains of the top and bottom section of element i, and ε_Ui and ε_Di are the coupled strains of the cross-section’s top and bottom of element i, respectively.

According to the above research idea of layout based on long-gauge FBG sensor, as shown in Table 1, the layout scheme sensor of arch bridges is classified by its bearing mode. The effectiveness of the scheme is proved in the subsequent finite element method (FEM).

Table 1.

Long-gauge FBG sensor layout scheme.

Arch bridge type	Layout scheme based on long-gauge FBG sensor
Deck arch bridge	Arch foot section, the arch rib section corresponding to column and the middle of column
Half-through arch bridge	Range of hanger: quarter span sections Arch foot section, the intersection of arch rib and beam
Through arch bridge	Arch foot section, eighth section

FBG: fiber Bragg grating.

Numerical verification

Three types of arch bridges under multiple target load conditions with different loading modes are investigated to verify the proposed method for deflection distribution estimation of the main arch of arch bridges. Two static load conditions and two dynamic load conditions are used to verify the CLM. The deck arch bridge is 112 m long, 25 m wide, and 24 m high, with a rise span ratio of 1/5. The rectangular section parameters of the main arch are as follows: 2 m high, 8 m width, and C40 concrete. The elastic modulus E of the main arch is 32.599 GPa. The half-through arch bridge is 40 m long, 8 m wide, and 7.95 m high, with a rise span ratio of 1/4. The rectangular section parameters of the main arch are as follows: 0.5 m high, 0.5 m wide, and Q345 steel. The elastic modulus E of the main arch is 210 GPa. The through arch bridge is 64 m long, 7.8 m wide, and 12.8 m high, with a rise span ratio of 1/5. The I-section parameters of the main arch are as follows: 1.2 m high, 0.8 m wide, 0.5 m width of web, and C40 concrete. The elastic modulus E of the main arch is 34.554 GPa. In the simulation, the main arch element of each bridge was divided into 56, 40, and 32. The length of the long-gauge FBG sensor is the same as the element of each bridge. Considering the possible interference of various factors in the actual testing, the strain data collected by the long-gauge FBG sensors under various load conditions were extracted under 5% white noise (the white noise is proportional to the structural response).

Sensor layout

Sensor layout scheme of each bridge is investigated based on the layout scheme of long-gauge FBG sensor of Table 1 which has been described above. The sensors layout point of the main arch of each bridge in simulation, as shown in Table 2 and Figure 6, is located symmetrically at the top and bottom of element.

Table 2.

Sensor layout of main arch.

Arch bridge type	Element of sensors located
Deck arch bridge	1, 3, 7, 10, 14, 18, 22, 28, 35, 39,42, 47, 50, 54, 56
Half-through arch bridge	1, 5, 10, 15, 20, 24, 30, 36, 40
Through arch bridge	1, 4, 8, 12, 16, 20, 24, 28, 32

Figure 6.

Load conditions and sensor layout scheme of each arch bridge: (a) deck arch bridge, (b) half-through bridge, and (c) through arch bridge.

In order to contrast the accuracy of the traditional sensors layout scheme points (TP) and optimal sensors layout scheme points (OP) for deflection distribution estimation, the number of sensors in TP is same as OP. In this article, TP of the deck, half-through, and through arch bridge, respectively, is located at 14 equal points, 8 equal points, and 8 equal points.

Static load condition

Each numerical model is investigated under two different static load conditions: (P1) L/2 section load and (P2) uniformly distributed load. Each load is applied non-centrally and close to the analyzed main arch. In the static load condition (P1), a 300-kN force as a vehicle (1:2 axle ratio) is applied to L/2 section of lane, while 50-kN/m uniformly distributed load is applied in static load condition (P2).

The macro-strain (coupled strain) curves at the top section of the main arch were obtained by fitting, and then the CLM is used to estimate the deflection distribution of the main arch. Figures 7 and 8 display the comparison of the differences between TP and OP based on the estimation result of the main arch. (a1C)–(c1C) and (a2C)–(c2C) show that the fitting curve of the coupled strain of OP is better than that of TP. The TP of deck arch bridges is same as OP, so there is no difference in the recognition results of both of them. Estimation deflection distribution of each main arch under static load conditions is shown in (a1D)–(c1D) and (a2D)–(c2D). The estimation accuracy of OP is better than that of TP, which corresponds to the coupled strain of both of them. It is observed that the layout scheme of long-gauge FBG sensor and the ICBM proposed in this article can estimate the actual deformation distribution of arch bridges effectively under static load conditions.

Figure 7.

Coupled strain curves and displacement under L/2 load: (a1C) and (a1D) deck arch bridge, (b1C) and (b1D) half-through arch bridge, and (c1C) and (c1D) through arch bridge.

Figure 8.

Coupled strain curves and displacement under uniformly distributed load: (a2C) and (a2D) deck arch bridge, (b2C) and (b2D) half-through arch bridge, and (c2C) and (c2D) through arch bridge.

Dynamic load condition

In order to verify the precision and the robustness of the CLM with dynamic test, a single impact and a continuous impact has been taken in this article. In this test, a single impact and a continuous impact were used to strike in the lane location the same as P1 of each arch bridge. The single impact loads of 3000, 600, and 400 kN were applied to each arch bridge under single impact test. In the test of continuous impact, the loads of (2000+3000+1000) kN, (1000+1500+600) kN, and (800+1000+500) kN were applied to each bridge. The main arch of each arch bridge was investigated to estimate deflection distribution at time 1 s under single impact as mentioned above, depicted in Figure 9. To observe the deflection responses under the continuous impacts clearly, a short interval of estimated deflection time history of middle span element of the main arch was extracted under the continuous impact test, as shown in Figure 10. It is evident that under the dynamic test, the estimated deflection time history by the proposed scheme OP is close to the FEM, compared with TP. In this section, the layout scheme of long-gauge FBG sensor and the ICBM proposed in this article can estimate the actual deformation distribution of the main arch of arch bridges effectively under dynamic load conditions.

Figure 9.

Estimated deflection distribution under single-impact condition at time 1 s: (a-s) deck arch bridge, (b-s) half-through arch bridge, and (c-s) through arch bridge.

Figure 10.

Estimated deflection time histories in the continuous impact condition: (a-c) deck arch bridge, (b-c) half-through arch bridge, and (c-c) through arch bridge.

The error analysis

The deformation distribution estimation by OP of the main arch of arch bridges under static load and dynamic load was calculated by CLM proposed in this article, and the results are compared with those of the TP. As shown in Figure 11(a), under the static load conditions of P1 (a1, b1, c1) and P2 (a2, b2, c2), the error at maximum deflection OP of deck arch bridge and half-through arch bridge is less than 5% and is better than TP. Due to the agreed same layout points of TP and OP, the error at maximum deflection OP of through arch bridge is the same as TP. The algorithm to identify the structure deformation distribution under dynamic load condition is based on the time superposition of static load condition, as shown in Figure 11(b), so the estimation error OP of element is the same as the static load condition, better than TP. According to the above analysis of the deflection results, the deflection maximum error OP is better than TP, as a result, the CLM proposed in this article can estimate deformation distribution of the main arch of arch bridges with higher precision under multi-operation.

Figure 11.

The error of the maximum displacement of the main arch: (a) static load condition and (b) dynamic load condition.

Conclusion

This article presents a deflection distribution estimation scheme for the main arch of arch bridges using long-gauge FBG sensors. Through analyzing the relation between deflection and internal force, the expression between strain and deformation is first established for the main arch element of arch bridge. Then, the CLM is proposed to identify the deflection distribution of the main arch. Finally, the sensor placement scheme is presented to obtain the flexural strain and axial strain used in the curvature load method, and it is able to guarantee the accuracy of the deformation distribution identification with limited number of sensors. Based on the research to date, the following conclusions can be drawn:

A curvature load method is developed to estimate the deflection distribution of the main arch of arch bridges based on the long-gauge FBG sensing technology, in which the expression between the deformation and strain of main arch element is established for analyzing the change of curvature caused by axial strain and bending strain that are used to estimate the deflection distribution of the main arch.

The optimal layout scheme of long-gauge FBG sensor for deformation distribution estimation proposed in this article has been presented. This sensor layout scheme can not only obtain the flexural strain and axial strain from measured strain but also provide guarantee for the accuracy of deflection distribution estimation of the main arch.

Numerical examples studied successfully verify that the proposed method can accurately identify the deflection of the main arch in both static and dynamic cases, and the error at maximum deflection is less than 5%, which meets the requirements of the actual project. This article has studied the deflection distribution estimation of arch bridges under the static and dynamic cases. When the structure is subjected to temperature loads, the idea of calculating the deflection from strain measurement is the same. However, the specific equations may be different, which will be studied in the future work.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Tongfei Sun

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