Deflection Monitoring Method for Simply Supported Girder Bridges Using Strain Response under Traffic Loads

Abstract

Deflection measurements are usually used as a key index in civil engineering for performing structural assessments of bridge safety. However, owing to technical or cost issues, it may be difficult to implement long-term monitoring of bridge deflection, especially for short- or medium-span bridges. Therefore, this study presents a novel method for measuring the deflection of simply supported girder bridges. In the proposed method, the strain measurement was implemented under traffic loading at only one position, such as middle span, and then the strain distribution along the girder was reconstructed to calculate the girder deflection with basic structural mechanical theory. To implement the method, the theory was constructed based on the displacement reciprocal theorem at first to assess the strain distribution along the girder from the strain measurement at some position during traffic loads passing across the bridge. Second, a strain measurement method, namely long-gauge fibre Bragg grating (FBG) sensing technology, was introduced to take strain measurements for a concrete bridge. Third, various finite element (FE) bridge models were developed to validate the proposed method’s accuracy, the results from which indicated that the method accurately implemented deflection measurement with an approximately 5% calculation error. In addition, the influence of some key parameters, such as vehicle type, vehicle speed, and structural damage, was investigated. The simulation results revealed that damage to the hinge joint in the middle location could significantly influence the proposed method’s accuracy such that the error may exceed 10%. Finally, on-site experiments were conducted on a simply supported girder bridge to further validate the proposed method’s accuracy, and an approximately 8% deflection assessment error was found. Considering the additional advantages of FBG sensing technology, the proposed method can also be effective for long-term deflection measurements of short- or medium-span bridges.

1. Introduction

Simply supported girder bridges are typical types of highway bridges in China. Often referred to as hinge joints, transverse connections are used to link precast girders for vehicle load bearing [1]. However, the hinge joint may break easily under repeated traffic loads. In addition, concrete cracks, steel erosion, and concrete degradation can also influence bridge safety [2,3,4]. Therefore, effective measurement of key parameters for evaluating a bridge’s structural performance is crucial.

Deflection is often used as a key indicator for assessing the structural performance of actual bridges [5,6]. One reason is that it is easily used to implement the structural performance assessment according to the design code. Another reason is that it can be measured by simple methods, such as geodetic techniques, linear variable differential transformers (LVDT), a global positioning system (GPS), and so on [7,8].

For geodetic techniques, total station [9] and static level [10,11] are widely used in deflection measurement due to their high accuracy (up to 0.1 mm) and stability. However, they cannot be utilized to monitor dynamic deflection under traffic loads because their systems’ operation is time-consuming and laborious. Another method, linear variable differential transformer (LVDT), has a very high deflection measurement accuracy of about 0.01 mm. However, the system requires additional supports at the bridge bottom to support such sensors, while most bridge bottoms are unsuitable for constructing such support systems, especially for long-term monitoring, due to river and traffic factors [12,13]. The global positioning system (GPS) method is convenient to install on bridges and is applied on many large-scale bridges. However, the deflection measurement accuracy is insufficient for short- or medium-span bridges, because its static deflection measurement accuracy is only about 20 mm, whereas the large deflection of simply supported girder bridges under vehicle loads is often several millimeters [14,15,16]. Furthermore, GPS is limited partly by multi-path and cycle slips, low frequency of data, as well as the need for good satellite coverage. Acceleration sensors are convenient to install at a relatively low cost, but they are not a good choice for monitoring bridge deflection. One reason is that they cannot be easily used for static deflection measurement when the main parts of bridge deflection under traffic loads are from the static loading of the vehicle dead-weight. Another reason is that the deflection assessment accuracy is easily influenced by bridge vibration and cumulative amplification error [17]. Image methods [18,19], such as digital image correlation (DIC) [20], can achieve millimeter-level accuracy or better in experimental environments, and is considered an expectable method in bridge deflection measurement. However, the accuracy depends largely on the high requirements of pixels and complicated systems, which makes the deflection monitoring system expensive.

Owing to the limitations of the aforementioned methods, strain sensor-based deflection measurement methods with low cost and easy application have been developed [21,22]. In these methods, bridge deflection is calculated using the strain–deflection relationship. However, to acquire accurate measurements, many strain sensors are frequently required, resulting in markedly increased deflection testing costs. In recent years, distributed long-gauge strain sensing technology has been developed to improve deflection measurement systems [23,24,25,26]. However, long-gauge strain sensors are still needed to cover most of the beam length. Therefore, the systems are still complicated and expensive. Besides the sensors, another factor, namely the strain–deflection relationship, will determine the performance of these indirect methods. The strain–deflection relationship is usually determined by the bridge structure mode. If the difference between the mode and actual structure is small, deflection assessment accuracy can be good. If the difference is large, the method may fail. Finally, factors such as temperature, loading, and damage, causing strain change, may also influence the performance of the indirect methods based on strain input. Therefore, the methods should be further studied for actual application.

This study presents a novel method for the deflection monitoring of simply supported girder bridges. The proposed method requires only a single long-gauge strain sensor to be installed at the location where deflection measurement is needed. The long-gauge strain sensor enables the collection of the temporal response of strain when a vehicle crosses the bridge. The temporal response of strain results can then be converted into the spatial distribution of strain along the bridge length. The converted strain distribution enables the easy calculation of deflection based on material and structural mechanics. Finally, the proposed method’s accuracy is validated using finite element (FE) simulations and on-site experiments.

2. Theory of Deflection Calculation Based on Strain Measurement Input

2.1. Deflection Calculation Based on Distributed Strain Measurement Input

In material mechanics, the strain ε(x) can be calculated from the known section’s bending moment M(x):

$$\epsilon (x)=\frac{M(x)y}{E{I}_z}$$

(1)

where y is the distance from the strain measurement location to the neutral axis, E is the elastic modulus, and I_z is the moment of inertia. Deflection can then be calculated using the strain distribution measurement along the beam based on classical structural mechanics:

$$\Delta ={\displaystyle \int \frac{\epsilon (x)\stackrel{\_\_\_}{M}}{y}}ds$$

(2)

where $\overline{M}$ is the virtual bending moment obtained by implementing a one-unit load at the beam location that requires deflection measurement. Therefore, beam deflection can be calculated when the distributed strains of different sections ε(x) are known.

2.2. Assessment Method of Deflection Based on Strain Measurements at One Location under TrafficLoad

2.2.1. Displacement Reciprocal Theorem

The displacement reciprocal theorem is a universal theorem that explains the force–displacement relationship of a beam under Hooke’s law and small deformation. As shown in Figure 1, this theorem is as follows:

$${\delta }_{12}={\delta }_{21}$$

(3)

If location 2 is further fixed in the beam middle and location 1 is moved, Equation (3) remains true. δ₁₂ is the deflection at location 1 when the load F lies at the middle location, and δ₂₁ is the deflection at the middle location when F moves to location 1. Therefore, the deflection at the beam middle when the moving load F acts on x equals that at x when the static load F acts on the middle location.

Furthermore, the deflection distribution along the beam length when the static load F acts at the middle location is equivalent to the measured deflection at the beam middle when the load F moves across the beam. Because curvature is the second-order integral of deflection, the law also applies to it. In other words, the curvature distribution along the beam length when the static load F acts at the middle location is equivalent to the measured curvature at the beam middle when the load F moves across the beam. This law is true for strain since it can be directly derived from the curvature. Therefore, this study further assumes that the strain distribution when traffic loads act at the beam middle is equivalent to the strain response when traffic loads move across the beam.

2.2.2. Static Strain Distribution Obtained with One Strain Sensor under Traffic Loads

(1)

Uniaxial vehicle load

When a vehicle wheelbase is markedly shorter than the bridge span, the vehicle load can be simplified into a uniaxial load. As shown in Figure 2a, the bending moment M(t) at the simply supported beam middle can be expressed as:

$$M(t)=\left\{\begin{array}{cc}F\cdot \frac{vt}{2}& 0\le t<\frac{L}{2v}\\ F\cdot \frac{L-vt}{2}& \frac{L}{2v}\le t\le \frac{L}{v}\end{array}\right.$$

(4)

where L is the beam’s total length; t is the time; F is the vehicle load; and v is the vehicle velocity. Considering the vehicle location x(t) = vt, M(t) can be converted into an equivalent bending moment distribution at the middle location M(x(t)):

$$M(x(t))=\left\{\begin{array}{cc}F\cdot \frac{x(t)}{2}& 0\le x(t)<\frac{L}{2}\\ F\cdot \frac{L-x(t)}{2}& \frac{L}{2}\le x(t)\le L\end{array}\right.$$

(5)

As shown in Figure 2b, when the uniaxial load is applied at the middle location, the bending moment distribution under the static load of the whole beam M(x) can be expressed as:

$$M(x)=\left\{\begin{array}{cc}F\cdot \frac{x}{2}& 0\le x<\frac{L}{2}\\ F\cdot \frac{L-x}{2}& \frac{L}{2}\le x\le L\end{array}\right.$$

(6)

Comparing Equations (5) and (6) shows that M(x) and M(x(t)) have the same form when x(t) and x mean the same location. The static strain distribution ε(x) can be obtained by Equation (7):

$$\epsilon (x)=\frac{M(x)y}{E{I}_z}=\frac{M(x(t))y}{E{I}_z}=\epsilon (x(t))$$

(7)

where ε(x(t)) is the equivalent strain distribution at the middle location. Therefore, the equivalent strain distribution can be substituted for the static strain distribution in Equation (2) to calculate deflection.

(2)

Biaxial vehicle load

Most short- and medium-sized bridges cannot accommodate the simplification of a vehicle load into a uniaxial load. The whole process for a vehicle load passing over a simply supported beam can be divided into five stages (Figure 3), where F₁ is the front axle load, F₂ is the rear axle load, and m is the wheelbase.

Similarly, the bending moment M(t) of the middle location of a simply supported beam can be expressed as:

$$M(t)=\left\{\begin{array}{ll}\frac{F_1}{2}\cdot vt\hfill & 0\le t<\frac{m}{v}\hfill \\ \frac{F_1}{2}\cdot vt+\frac{F_2}{2}\cdot (vt-m)\hfill & \frac{m}{v}\le t<\frac{L}{2v}\hfill \\ \frac{F_1}{2}\cdot (L-vt)+\frac{F_2}{2}\cdot (vt-m)\hfill & \frac{L}{2v}\le x(t)<\frac{L+2m}{2v}\hfill \\ \frac{F_1}{2}\cdot (L-vt)+\frac{F_2}{2}\cdot (L-vt+m)\hfill & \frac{L+2m}{2v}\le x(t)<\frac{L}{v}\hfill \\ \frac{F_2}{2}\cdot (L-vt+m)\hfill & \frac{L}{v}\le x(t)\le \frac{L+m}{v}\hfill \end{array}\right.$$

(8)

Let x(t) be the location of the combined force F₁ + F₂ (Figure 4) of the front axle F₁ and rear axle F₂ at time t. For brevity, let $\frac{F_1}{F_1+F_2}=\alpha$, $\frac{F_2}{F_1+F_2}=\beta$. Then, x(t) can be represented as:

$$x(t)=vt-\frac{F_2}{F_1+F_2}m=vt-\beta m$$

(9)

Substituting Equation (9) into Equation (8) yields an equivalent bending moment distribution at the middle location M(x(t)) as follows:

$$M(x(t))=\left\{\begin{array}{ll}\frac{F_1}{2}(x(t)+\beta m)\hfill & -\beta m\le x(t)<\alpha m\hfill \\ \frac{(F_1+F_2)}{2}\cdot x(t)\hfill & \alpha m\le x(t)<\frac{L}{2}-\beta m\hfill \\ \frac{(F_2-F_1)x(t)}{2}-\alpha F_{2}m+\frac{F_{1}L}{2}\hfill & \frac{L}{2}-\beta m\le x(t)<\frac{L}{2}+\alpha m\hfill \\ \frac{(F_1+F_2)}{2}\cdot (L-x(t))\hfill & \frac{L}{2}+\alpha m\le x(t)<L-\beta m\hfill \\ \frac{F_2}{2}(L-x(t)+\alpha m)\hfill & L-\beta m\le x(t)\le L+\alpha m\hfill \end{array}\right.$$

(10)

For comparison, the locations of the front and rear axles are placed in reverse order, and the combined force is located at the middle location (Figure 5). The bending moment distribution of the whole beam under static biaxial vehicle load M(x) can be expressed as follows:

$$M(x)=\left\{\begin{array}{ll}\frac{(F_1+F_2)}{2}\cdot x\hfill & 0\le x<\frac{L}{2}-\beta m\hfill \\ \frac{(F_2-F_1)x}{2}-\alpha F_{2}m+\frac{F_{1}L}{2}\hfill & \frac{L}{2}-\beta m\le x<\frac{L}{2}+\alpha m\hfill \\ \frac{(F_1+F_2)}{2}\cdot (L-x)\hfill & \frac{L}{2}+\alpha m\le x<L\hfill \end{array}\right.$$

(11)

Comparing Equations (10) and (11) reveals that M(x(t)) equals M(x) when x = x(t) and x∈[αm, L − βm]. If the beam stiffness EI and height of the neutral axis y remain unchanged, then ε(x(t)) = ε(x). Therefore, the equivalent strain distribution ε(x(t)) can be substituted for the static strain distribution ε(x) into Equation (2) to calculate deflection directly. When x∈[−βm, αm] and x∈[L − βm, L + αm], a modified method is needed to improve accuracy.

The deflection Δ₁ calculated from the static strain distribution when x∈[0, αm] and x∈[L − βm, L] and the virtual bending moment $\overline{M}(x)$ are as follows:

$${\Delta }_1={\displaystyle {\int }_0^{\alpha m}\frac{\epsilon (x)\overline{M}(x)}{y}}dx+{\displaystyle {\int }_{L-\beta m}^L\frac{\epsilon (x)\overline{M}(x)}{y}}dx$$

(12)

$$\overline{M}(x)=\left\{\begin{array}{cc}\frac{1}{2}x& 0\le x<\frac{L}{2}\\ \frac{1}{2}(L-x)& \frac{L}{2}\le x\le L\end{array}\right.$$

(13)

The deflection Δ₂ calculated from the equivalent strain distribution when x ∈ [−βm, αm] and x ∈ [L − βm, L + αm] is:

$$\begin{array}{cc}\hfill {\Delta }_2& ={\displaystyle {\int }_{-\beta m}^{\alpha m}\frac{\epsilon (x(t))\overline{M^{\prime }}(x(t))}{y}}dx(t)+{\displaystyle {\int }_{L-\beta m}^{L+\alpha m}\frac{\epsilon (x(t))\overline{M^{\prime }}(x(t))}{y}}dx(t)\hfill \\ & ={\displaystyle {\int }_0^m\frac{\epsilon (x(t)-\beta m)\overline{M^{\prime }}(x(t)-\beta m)}{y}}dx(t)+{\displaystyle {\int }_{L-m}^L\frac{\epsilon (x(t)+\alpha m)\overline{M^{\prime }}(x(t)+\alpha m)}{y}}dx(t)\hfill \end{array}$$

(14)

where $\overline{M}(x)$ is the modified virtual bending moment. Let Δ₁ = Δ₂, then the following is obtained:

$$\left\{\begin{array}{l}{\displaystyle {\int }_0^m\frac{\epsilon (x(t)-\beta m)\overline{M^{\prime }}(x(t)-\beta m)}{y}}dx(t)={\displaystyle {\int }_0^{\alpha m}\frac{\epsilon (x)\overline{M}(x)}{y}}dx\hfill \\ {\displaystyle {\int }_{L-m}^L\frac{\epsilon (x(t)+\alpha m)\overline{M^{\prime }}(x(t)+\alpha m)}{y}}dx(t)={\displaystyle {\int }_{L-\beta m}^L\frac{\epsilon (x)\overline{M}(x)}{y}}dx\hfill \end{array}\right.$$

(15)

When x ∈ [0, αm] and x = x(t), then

$$\frac{\epsilon (x)}{\epsilon (x(t)-\beta m)}=\frac{M(x)}{M(x(t)-\beta m)}=\frac{\frac{(F_1+F_2)}{2}\cdot x}{\frac{F_1}{2}x(t)}=\frac{F_1+F_2}{F_1}=\frac{1}{\alpha }$$

(16)

When x ∈ [L − βm, L] and x = x(t), then

$$\frac{\epsilon (x)}{\epsilon (x(t)-\beta m)}=\frac{M(x)}{M(x(t)-\beta m)}=\frac{\frac{(F_1+F_2)}{2}\cdot x}{\frac{F_1}{2}x(t)}=\frac{F_1+F_2}{F_1}=\frac{1}{\alpha }$$

(17)

$\overline{M}(x)$ can then be further expressed as:

$$\overline{M^{\prime }}(x)=\left\{\begin{array}{l}\begin{array}{ll}\overline{M}(x+\beta m)\cdot \frac{1}{\alpha }\hfill & -\beta m\le x<(\alpha -\beta )m\hfill \\ 0\hfill & (\alpha -\beta )m\le x<\alpha m\hfill \\ \overline{M}(x)\hfill & \alpha m\le x<L-\beta m\hfill \\ 0\hfill & L-\beta m\le x<L+(\alpha -\beta )m\hfill \\ \overline{M}(x-\alpha m)\cdot \frac{1}{\beta }\hfill & L+(\alpha -\beta )m\le x<L+\alpha m\hfill \end{array}\end{array}\right.$$

(18)

Therefore, if the virtual bending moment $\overline{M}(x)$ is represented as above, deflection calculation errors can be eliminated.

(3)

Multi-axle vehicle load

The proposed method also applies to vehicles with at least three axles. Here, the three-axle vehicle is taken as an example to explain the calculation process. As shown in Figure 6, F₁, F₂, and F₃ are the loads for the 3 axles, respectively; $m_1$ and $m_2$ are the wheelbases; p is the combined force location and $p=\frac{m_1{F}_2+(m_1+m_2)F_3}{F_1+F_2+F_3}$.

The equivalent bending moment distribution at the middle location M(x(t)) can further be expressed as:

$$M(x(t))=\left\{\begin{array}{ll}\frac{F_1}{2}(x(t)+p)\hfill & -p<x(t)\le -p+m_1\hfill \\ \frac{(F_1+F_2)(x(t)+p)-F_2{m}_1}{2}\hfill & -p+m_1<x(t)\le -p+m_1+m_2\hfill \\ \frac{F_1+F_2+F_3}{2}\cdot x(t)\hfill & -p+m_1+m_2<x(t)\le \frac{L}{2}-m_1-m_2+p\hfill \\ \frac{(F_1+F_2-F_3)}{2}\cdot x(t)+F_3(\frac{L}{2}-m_1-m_2+p)\hfill & \frac{L}{2}-m_1-m_2+p<x(t)\le \frac{L}{2}-m_1+p\hfill \\ \frac{F_1+F_2+F_3}{2}\cdot (L-x(t))-F_1(\frac{L}{2}-x(t)+p)\hfill & \frac{L}{2}-m_1+p<x(t)\le \frac{L}{2}+p\hfill \\ \frac{F_1+F_2+F_3}{2}\cdot (L-x(t))\hfill & \frac{L}{2}+p<x(t)\le L-p\hfill \\ \frac{(F_2+F_3)(L+m_1+m_2-p-x(t))-F_2{m}_2}{2}\hfill & L-p<x(t)\le L-p+m_1\hfill \\ \frac{F_3(L+m_1+m_2-p-x(t))}{2}\hfill & L-p+m_1<x(t)\le L+m_1+m_2-p\hfill \end{array}\right.$$

(19)

The bending moment distribution under static biaxial vehicle load M(x) can be expressed as follows:

$$M(x)=\left\{\begin{array}{ll}\frac{F_1+F_2+F_3}{2}\cdot x\hfill & 0<x\le \frac{L}{2}-m_1-m_2+p\hfill \\ \frac{(F_1+F_2-F_3)}{2}\cdot x+F_3(\frac{L}{2}-m_1-m_2+p)\hfill & \frac{L}{2}-m_1-m_2+p<x\le \frac{L}{2}-m_1+p\hfill \\ \frac{F_1+F_2+F_3}{2}\cdot (L-x)-F_1(\frac{L}{2}-x+p)\hfill & \frac{L}{2}-m_1+p<x\le \frac{L}{2}+p\hfill \\ \frac{F_1+F_2+F_3}{2}\cdot (L-x)\hfill & \frac{L}{2}+p<x\le L\hfill \end{array}\right.$$

(20)

M(x(t)) still equals M(x) when x = x(t) and x ∈ [m₁ + m₂ − p, L − p]. Aiming to eliminate errors, $\overline{M^{\prime }}(x)$ can also be expressed as:

$$\overline{M^{\prime }}(x)=\left\{\begin{array}{ll}\overline{M}(x+p)\cdot \frac{1}{\eta }\hfill & -p\le x<m_1+m_2-2p\hfill \\ 0\hfill & m_1+m_2-2p\le x<m_1+m_2-p\hfill \\ \overline{M}(x)\hfill & m_1+m_2-p\le x<L-p\hfill \\ 0\hfill & L-p\le x<L+m_1+m_2-2p\hfill \\ \overline{M}(x-m_1-m_2+p)\cdot \frac{1}{\gamma }\hfill & L+m_1+m_2-2p\le x<L+m_1-p\hfill \\ \overline{M}(x-m_1-m_2+p)\cdot \frac{1}{\lambda }\hfill & L+m_1-p\le x<L+m_1+m_2-p\hfill \end{array}\right.$$

(21)

The parameters are expressed as:

$$\left\{\begin{array}{c}\frac{1}{\eta }=\frac{F_1+F_2+F_3}{F_1}\\ \frac{1}{\gamma }=\frac{F_1+F_2+F_3}{F_2+F_3}\\ \frac{1}{\lambda }=\frac{F_1+F_2+F_3}{F_3}\end{array}\right.$$

(22)

3. Strain Measurement Based on Optical Fibre Sensors

3.1. Strain Measurement Principle

Owing to their high precision and serializability, fibre Bragg grating (FBG) strain sensors are often used in engineering to measure the strain of structures. As shown in Figure 7, an FBG sensor comprises a cladding layer, coating layer, fibre core, and grating region. When input light passes through the fibre grating, the grating will reflect back some light satisfying the Bragg condition, whereas the remaining transmitted light will continue to propagate along the fibre core. If some strain ε or temperature T change happens upon the grating region, the distance of grating Λ will change. Then, the central wavelength of the reflected light will change. A linear relationship between wavelength variation and strain is given as follows:

$${\lambda }_B={\alpha }_{\epsilon }\epsilon +{\alpha }_{\mathrm{T}}\mathrm{T}$$

(23)

where ${\lambda }_B$ is the central wavelength change in reflected light; ${\alpha }_{\epsilon }$ is the strain sensitivity coefficient, which is 1.2 pm/με generally; α_T is the temperature sensitivity coefficient.

3.2. Long-Gauge Optical Fibre Sensor

The size of a bare FBG sensor is approximately 2 cm and may not be ideal for monitoring concrete structures, due to the nonuniform material characteristics of concrete. Moreover, a steel material is often used to package the bare FBG sensor since it is brittle and difficult to use without treatment. However, the bonding surface between the inner optical fibre and outer steel is not reliable over the long term. Therefore, the authors’ research team has proposed a new long-gauge FBG sensor using a fibre-reinforced polymer (FRP) package, as shown in Figure 8, with a gauge length of up to 1 m. Owing to strain uniformity along the optical fibre within the sensing range, the strain will reflect the average deformation of the monitored structure throughout this range, thereby resulting in a significant reduction in the negative influence on strain measurement caused by material inhomogeneity and local damage, such as concrete cracks. Furthermore, the long-term performance of FRP and the bonding surface between optical and reinforced fibres are excellent.

3.3. Strain Sensing Performance of the Sensor

Because the long-gauge FBG sensor is cross-sensitive to temperature and strain, the sensors were subjected to tensile and temperature tests [25]. The test results shown in Figure 9 reveal that the wavelength–strain relationship is highly linear. The sensor has good repeatability since the correlation of the average linear fitting exceeds 0.999. The strain sensing coefficient is 1.18 pm/με, which is very close to the theoretical value of 1.2 pm/με. The linearity and repeatability of the sensor’s wavelength and temperature are also good, with the correlation of the average linear fitting exceeding 0.99.

4. Study on the FE Bridge Model

4.1. Introduction to the Bridge Model

One typical highway bridge is selected as an example to assess the performance of the proposed method. The bridge comprises 12 prefabricated hollow prestressed reinforced concrete (PRC) girders with a span of 20 m. The girders are combined with each other with one hinge joint made of reinforced concrete (RC). The dimensions are as shown in Figure 10. Then the finite element (FE) model of the simply supported girder bridge is established using Midas FEA (Figure 11). In the FE model, the girders, and hinge joints are simulated with solid elements. The constraints at both ends are simply supported by imposing three degrees of freedom on the left (Dx, Dy, and Dz) and two degrees of freedom on the right (Dx and Dz). In this paper, the steel rebar and prestressed rebar are not included in the model for simplification, as the elastic model is sufficient to verify the proposed method. The detailed information of the material properties of FE models are listed in

Table 1. Material properties of FE models.

Elastic Mode for Girder	Compressive Strength for Girder	Tensile Strength for Girder	Elastic Mode for Joints	Compressive Strength for Joints	Tensile Strength for Girder
32.5 GPa	19.1 Mpa	1.71 Mpa	30 Gpa	14.3 Mpa	1.43 Mpa

.

In this study, the loads of one typical truck are placed on the surface of the middle girder (the sixth one). The front axle load of the vehicle is 25 kN, whereas the rear axle load is 100 kN. The wheelbase is 4 m. The driving speed is 30 km/h. As shown in Figure 12, static and vehicle moving loads are applied respectively, in which the dotted lines represent the combined force.

4.2. Extraction and Processing of Strain Data

The girder is divided into 20 measurement units, each of which is 1 m long. The strain measurement by long-gauge FBG sensors is simulated to extract the average strain of each unit, called the macro-strain. The simulated sensor distribution is shown in Figure 13. Here, there is no actual sensor model in the FE model, while the strain is just extracted at the position at which the designed sensor lies. For each sensor, the average strain within the gauge length is obtained from the model directly, which is taken as the measured macro-strain.

Apart from truck weight-induced strain, dynamic strain contains some additional vibration response, which inevitably causes interference and error. Therefore, in actual signal processing, signal denoising is required to obtain an accurate quasi-static strain response.

Traditional denoising methods, such as the fast Fourier transform (FFT), cannot well distinguish the useful signal within a high frequency from the high frequency part caused by noise, especially for the case of non-stationary signals whose frequency changes with time. Luckily, the improvement of a wavelet denoising method can be used to distinguish the useful signal. In the wavelet denoising method, the traditional basis function with an infinite length is replaced by a finite length attenuation wavelet basis (Figure 14). The change makes it easy to know not only the frequency component of the signal, but also its specific location in the time domain.

The basic idea of wavelet denoising is to remove noise components and leave key information by decomposition and selective reconstruction. The specific expression of wavelet denoising is as follows.

$$WT(a,\tau )=\frac{1}{\sqrt{a}}{\displaystyle {\int }_{-\infty }^{\infty }f(t)\Psi }(\frac{t-\tau }{a})dt$$

(24)

where τ is time shift, a is the scaling factor, Ψ(t) is the wavelet basis function.

The basic steps of wavelet denoising are as follows:

• Wavelet decomposition. Generally, a wavelet basis type and the layer N of wavelet decomposition are selected first, and then N layer decomposition of the noisy signal is performed to obtain the wavelet coefficient of each layer. The wavelet basis function used in this paper is Sym8, which has already been used in many application scenarios in signal processing and data compression. Sym8 can decompose the data signal into multiple sub-signals, reducing the data noise while retaining the important features of the signal. It can retain the important features of the signal as much as possible and remove the noise interference. Here N is set as 4.
• Threshold processing. The wavelet coefficients from the first layer to the N layer are processed with appropriate thresholds, and the noise is removed with its de-correlation (Figure 15). The main principle is to set a threshold λ. When the wavelet coefficient is greater than λ, it retains its original value. When the wavelet coefficient is less than λ, it is directly set to 0. In this way, the original features are well preserved and the noise removal signal is smoother.
• Wavelet reconstruction. The processed signal is reconstructed and the denoised signal is obtained. In Figure 16, the signal S(t) is decomposed by four-layer wavelet, in which W_CD1, W_CD2, W_CD3, and W_CD4 are the high-frequency wavelet coefficients of each layer wavelet decomposition, while W_CA1, W_CA2, W_CA3, and W_CA4 are the low-frequency wavelet coefficients. Here, noise is mainly contained in the low-frequency wavelet coefficient, so it is considered that W_CA1, W_CA2, W_CA3, and W_CA4 are less than λ, and then can be set as 0. Finally, the denoised signal is obtained.

In this study, the signal processing method of wavelet transform is applied to denoise the dynamic strain. Figure 17 depicts the typical results of signal denoising.

4.3. Results and Analysis

Owing to the symmetry of the studied bridge, half of the girders are selected for analysis. The denoised strain responses are transformed into the equivalent strain distribution based on the method described in Section 2, which is then compared with the static strain distribution directly obtained from the FE simulation (Figure 18). The sampling frequency is set as 200 Hz. The findings show that the proposed method is effective despite some minor discrepancies, particularly at the two ends, which occur due to the partial and full exertion of moving biaxial and static loads on the girder and the inability of the denoising process to delete all dynamic responses.

The deflection at the girder midpoint can then be calculated by inputting the strain results into Equation (2).

Table 2. Results of deflection of different girders.

Girder No.	Actual Deflection (mm)	Deflection by Static Strain Distribution		Deflection by Equivalent Strain Distribution
		Deflection (mm)	Error (%)	Deflection (mm)	Error (%)
1	0.83	0.86	3.39	0.88	5.63
2	0.83	0.85	2.61	0.86	3.34
3	0.85	0.85	0.00	0.85	0.62
4	0.85	0.85	0.00	0.86	0.84
5	0.86	0.86	0.00	0.86	0.11
6	0.86	0.86	0.00	0.86	−0.05

lists the deflection calculation results; notably, the actual deflection is derived directly from the FE simulation. Most deflection errors are below 5%, indicating that the proposed method has a high degree of accuracy. Moreover, the accuracy based on the input of the equivalent strain distribution slightly exceeds that based on the input of the static strain distribution. However, the highest error occurs at the side girder for both strain input types, but the reason for this is unclear. This may be caused by the FE simulation itself, as the side girder has an unrestrained side, making the FE calculation more complicated.

4.4. Parameter Influence Analysis

For actual bridges, some parameters can influence the proposed method’s effectiveness. Therefore, parameters such as vehicle type, vehicle speed, and structural damage are investigated in this study.

(1)

Vehicle type

Based on different axle weights, wheelbases, and number of axles, vehicle types can be roughly divided into the following categories: biaxial van, biaxial truck, three-axle truck, and multi-axle trailer. Figure 19 shows the axle weights and wheelbases of the different vehicle types.

The vehicle parameters are incorporated into the FE model. The static and equivalent strain distributions can then be obtained, as shown in Figure 20. Compared with the other two types of vehicles, the degree of coincidence between the two types of strain results produced by a biaxial van and a biaxial truck is greater in the figures. At the two ends of the figures, notably for the multi-axle trailer, there are obvious variances due to the partial and total exertion of moving axle and static loads on the girder. This situation is more obvious when the wheelbases are long. Due to their longer wheelbases, multi-axle trailers have the highest error.

Similarly, deflection is calculated using the two types of strain results (

Table 3. Results of deflection of different vehicle models.

Vehicle Type	Actual Deflection (mm)	Deflection by Static Strain Distribution		Deflection by Equivalent Strain Distribution
		Deflection (mm)	Error (%)	Deflection (mm)	Error (%)
Biaxial van	0.86	0.86	0.00	0.86	−0.03
Biaxial truck	0.81	0.82	1.12	0.82	1.92
Three-axle truck	1.44	1.46	1.35	1.45	0.81
Multi-axle trailer	1.40	1.40	0.00	1.32	−5.62

), with only girder 6#’s results included. The results indicate that vehicle type influences the proposed method’s accuracy. However, the error caused by vehicle type differences can be controlled since the highest error is approximately 5%.

(2)

Vehicle speed

Four typical speeds are included in the FE model to investigate how vehicle speed affects deflection measurement accuracy: 30, 60, 90, and 120 km/h. The difference between both strain types increases as vehicle speed increases, as shown in Figure 21. The main reason is that a higher vehicle speed will cause the bridge to vibrate more severely. In addition, the denoising process can only delete a portion of the vibration strain response because the strain curves are no longer smooth and fluctuate slightly after denoising.

Table 4. Results of deflection at different vehicle speeds.

Item	Actual Deflection	Deflection by Static Strain Distribution	Deflection by Equivalent Strain Distribution
			30 km/h	60 km/h	90 km/h	120 km/h
Deflection (mm)	0.86	0.86	0.86	0.86	0.87	0.92
Error (%)	/	0.00	−0.05	0.41	1.20	5.50

shows the calculated deflection results at different vehicle speeds. The results show that the calculated deflection error also increases as vehicle speed increases. However, the error caused by differences in vehicle speed can also be controlled since the highest error is 5.5% at 120 km/h.

(3)

Structural damage

A decrease in local stiffness induced by structural deterioration, such as damage to hinge joints and concrete cracks, will impact strain. Consequently, these damage types are used as examples to examine the influence of structural damage on the proposed deflection measurement method.

 (1)

Damage between the hinge joints

Partial hinge joint damage is typical damage of a simply supported girder bridge in China, which greatly decreases the entire bridge stiffness. In the FE model, hinge joint damage is simulated by decreasing the elastic modulus. The degree of hinge joint damage is calculated using the length of damaged hinge joints. The damage factor D is then defined as follows:

$$D=\frac{l}{L}$$

(25)

where l is the length of the damaged hinge joints, and L is the total length of the hinge joints. Notably, hinge joints are usually destroyed from the middle part outwards, due to the larger load response there. Therefore, damage to the hinge joint is assumed to start symmetrically from the middle unit to the two ends (Figure 22). The damage factor D of the 5# hinge joint is set from 0.1 to 1.0.

As illustrated in Figure 23, the development of hinge joint damage has a substantial effect on strain results. As damage accumulates, the disparity between the two strain distribution types widens.

Table 5. Deflection results for hinge joint damage developing from the middle part.

D	Actual Deflection (mm)	Static Strain Distribution		Equivalent Strain Distribution
		Deflection (mm)	Error (%)	Deflection (mm)	Error (%)
0.1	0.86	0.87	0.66	1.00	15.88
0.2	0.89	0.89	0.01	1.02	15.37
0.3	0.92	0.92	−0.70	1.09	17.63
0.4	0.97	0.95	−1.45	1.14	17.60
0.5	1.02	1.00	−2.24	1.18	15.97
0.6	1.08	1.04	−3.11	1.22	13.53
0.7	1.14	1.10	−4.06	1.26	10.46
0.8	1.22	1.16	−5.14	1.30	6.91
0.9	1.31	1.23	−6.34	1.35	2.78
1.0	1.76	1.65	−6.33	1.66	−5.78

shows the deflection results for hinge joint damage developing from the middle part. When the damage factor D is less than 0.5, which means that the damage only occurs at the small middle part, the deflection error calculated by equivalent strain distribution is the largest, reaching a peak value of about 18%. However, if the damage factor D exceeds 0.5, the calculation error decreases significantly as the damage develops, because the local stiffness of a bridge differs along the girder due to local hinge joint damage. In the proposed method, stiffness is assumed to be uniform or known along the girder. If the changed stiffness distribution is disregarded in the proposed method when converting the strain response to the equivalent strain distribution, an obvious error will occur as a result. Therefore, if damage to hinge joints has already occurred at the middle parts, especially where the strain sensor is installed, stiffness distribution should be modified to improve deflection calculation accuracy.

Due to a defective bearing system, the hinge joint occasionally suffers damage at the girder end. Therefore, some simulations are conducted to address such a case (Figure 24). The damage factor D is set from 0.05 to 0.45. The typical strain results are shown in Figure 25. The difference between the two strain distribution types also tends to widen as the damage develops, but not as clearly as identified above. This difference is quite small, indicating that the damaged hinge joint at the girder’s edge has little effect on the proposed method. This result is validated by the deflection calculation results (

Table 6. Deflection results for hinge joint damage developing from the girder’s edge.

D	Actual Deflection (mm)	Static Strain Distribution		Equivalent Strain Distribution
		Deflection (mm)	Error (%)	Deflection (mm)	Error (%)
0.05	0.87	0.88	1.12	0.88	1.54
0.10	0.87	0.88	1.10	0.88	1.51
0.15	0.87	0.88	1.07	0.88	1.42
0.20	0.87	0.88	1.04	0.88	1.27
0.25	0.87	0.88	1.00	0.88	0.94
0.30	0.87	0.88	0.96	0.87	0.36
0.35	0.87	0.88	0.95	0.86	−0.60
0.40	0.87	0.88	1.00	0.85	−2.18
0.45	0.88	0.89	1.13	0.84	−4.05

), which show the highest error of calculated deflection as about 4%. The primary reason is because the strain response is measured at the girder’s middle, which is minimally affected by local damage at the girder’s edge. Therefore, assuming that damage to the hinge joint only occurs on one side, the proposed method for measuring deflection is promising.

 (2)

Concrete cracks

Two types of concrete cracks at the bottom of actual bridge girders are longitudinal and transverse. Prestress-induced cracks are longitudinal, whereas bending moment-induced cracks are transverse. As seen in Figure 26, the FE model includes a longitudinal crack over the whole girder length and a transverse crack across the entire girder width. The width of the two cracks is 0.4 mm, whereas the depth is 50 mm. Crack simulation is achieved by removing entities of corresponding shapes.

The strain results are shown in Figure 27, and the deflection results are listed in

Table 7. Deflection results under crack conditions.

Condition	Actual Deflection (mm)	Static Strain Distribution		Equivalent Strain Distribution
		Deflection (mm)	Error (%)	Deflection (mm)	Error (%)
Longitudinal crack	0.84	0.85	1.20	0.86	1.75
Transverse crack	0.84	0.84	0.15	0.87	2.87

. Notably, concrete cracks have a negligible effect on the proposed method since the equivalent strain is close to the static strain and the calculated deflection error is below 3%. In general, longitudinal cracks have little impact on a girder’s bending stiffness, but transverse cracks decrease bending stiffness. Fortunately, due to the long-gauge strain sensor, the transverse crack’s effect on the average bending stiffness within the long-sensing gauge will be minimal. Therefore, when transverse cracks are few, their effects on the proposed method’s accuracy can be ignored.

5. On-Site Experimental Study

5.1. Experimental Description

In this study, the experimental bridge is a standard simply supported girder bridge with a span of 20 m on China’s National Highway G204. The bridge is separated into two independent parts for two traffic directions: south and north. One side of the bridge is made of 12 precast girders and is separated into a pavement, a heavy lane, and a fast lane (Figure 28). Strain sensors are installed on the 7# girder’s bottom since this location experiences heavy traffic loads.

The 7# girder is divided into 20 monitoring units, each of which is 1 m long. The gauge length of the long-gauge FBG sensors (shown in Figure 8) used in the experiments is also 1 m, and then each sensor can cover each girder unit. Despite the difficulty in installing sensors due to space limits, 16 sensors were used in the actual experiments (Figure 29). The sensors were prepared in two groups, each of which had 8 FBG sensors (Figure 29a). For comparison, the strains of units without sensors were interpolated based on the strains detected by nearby sensors. In the experiments, the FBG sensors were adhered to the selected girder’s bottom surface with epoxy resin (Figure 30a). Furthermore, some displacement meters were installed with scaffolds to measure the actual deflection (Figure 30b).

Static and moving vehicle loads were applied in the experiments (Figure 31). A truck full of sand, about 25 t, was used to implement the loading tests. The wheelbase between the two axles was 4.2 m. When the static load was applied, the midpoint of the two axles was placed at the bridge midpoint. When the moving vehicle load was applied, the truck passed over the bridge at 30 and 60 km/h. Planned speeds of 90 and 120 km/h were also established, although these were difficult to achieve owing to traffic limits. The sampling frequency was 200 Hz.

5.2. Experimental Results and Analysis

Figure 32 shows the typical strain results obtained before and after denoising. Certain signal noise and dynamic signals can be observed via the denoising process. Figure 33 compares the strain distribution under static loading tests to the comparable strain distribution derived from the strain response. Speed has some effect on the accuracy of strain transformation since the difference between the two strain distribution types widens as driving speed increases. This effect occurs because increasing the speed causes the bridge to vibrate more and the denoising process can delete only a part of the dynamic strain response.

Table 8. Deflection results of on-site experiments.

Condition		Actual Deflection (mm)	Static Strain Distribution		Equivalent Strain Distribution
			Deflection (mm)	Error (%)	Deflection (mm)	Error (%)
South direction	30 km/h	0.71	0.68	−3.40	0.75	6.02
	60 km/h	0.71	0.68	−3.40	0.69	−2.48
North direction	30 km/h	0.62	0.63	1.81	0.57	−8.25
	60 km/h	0.62	0.63	1.81	0.59	−4.25

lists the results of the deflection at the girder’s midpoint calculated using Equation (2) and the strain distribution. The proposed method is deemed to have satisfactory deflection measurement accuracy because the largest relative error is roughly 8%. Meanwhile, the largest absolute error of the deflection measurement is 0.05 mm, which is small enough. Overall, these findings suggest that the proposed method is an effective technique for measuring deflection, notably for short- or medium-span concrete bridges, where deflection is often minimal (several millimetres or smaller than one millimetre).

6. Conclusions and Remarks

In this study, a novel method for quick deflection measurement of a simply supported girder bridge based on strain response under traffic loads was developed. The strain conversion theory was first constructed for the temporal strain response and spatial distribution of strain. The accuracy of the proposed deflection measurement method was then validated using FE simulations and on-site experiments of a typical bridge. Consequently, the following are the key conclusions:

(1)

The displacement reciprocal theorem enabled the development of a method for converting the temporal strain response under traffic loads to the spatial distribution of strain under the same static loads. Meanwhile, modified methods for decreasing the error caused by vehicle type differences were established.

(2)

The deflection measurement method relied largely on strain distribution input, which was measured using the proposed macro-strain sensor, a long-gauge FBG sensor. The high accuracy of the proposed method was validated by FE simulation results, which indicated a relative error of about 5%. However, when key parameters such as vehicle type, vehicle speed, and structural damage were considered, damage to the hinge joint in the middle location significantly affected the proposed method’s accuracy, as the deflection error could exceed 10%.

(3)

The results of on-site experiments validated the proposed method, as the maximum relative error of deflection measurement was about 8% and the absolute error was below 0.05 mm.

Although the proposed method is effective for measuring deflection, especially for short- and medium-span concrete bridges where deflection is often minimal (several millimetres or smaller than one millimetre), due to time constraints, future studies are required to expand its application. First, more on-site tests should be conducted and completed using different vehicle types to further prove the proposed method’s accuracy. Second, long-term performance, such as repeatability of strain measurement, sensor life, and the sensor–girder bonding surface should be given greater consideration. Third, material degradation should also be considered in the equations when the method is used for long-term monitoring.

Considering cost and efficiency, the proposed method is a good choice for measuring the deflection of short- and medium-span concrete bridges, although further research must be conducted in the future. In addition, certain benefits will be realized given the superior long-term performance of optical fibre sensing technology; hence, the proposed method has a promising future in practical applications.