**2. Theoretical Background**

#### *2.1. Identification Theory of Influence Lines*

According to the strain influence line theory [28] and material mechanics for a simply supported beam, as shown in Figure 1, the strain of the mid-span point *C* can be expressed as below: 

$$\varepsilon\_c = \begin{cases} \frac{\text{Plx}}{2EI} & 0 < \text{x} < \frac{L}{2} \\ \frac{\text{Plh}L}{2EI} \left(1 - \frac{\text{x}}{L}\right) & \frac{L}{2} < \text{x} < L \end{cases} \tag{1}$$

where *x* is the distance between the moving load P and the beam end *A*, *L* is the beam span, *h* is the height of the neutral axis, *I* is the section inertia moment of point *C*, and *E* is the elastic modulus.

Generally, the moving load on the bridge is a multi-axle vehicle load, so the measured strain response can be seen as the superposition of multiple concentrated loads [29]. Figure 2 shows a three-axle vehicle load as a sample. The strain equation of mid-span section under vehicle load can be expressed as below:

$$
\varepsilon(\mathbf{x}) = \varepsilon\_1(\mathbf{x} - \mathbf{x}\_1) + \varepsilon\_2(\mathbf{x} - \mathbf{x}\_2) + \varepsilon\_3(\mathbf{x} - \mathbf{x}\_3) \tag{2}
$$

where *x* is the distance between the vehicle's first axis and the left end of the bridge, *x*1, *x*2, and *x*3 are the distances between each axle and the first axle, *P*1, *P*2, and *P*3 are the axle loads, respectively.

**Figure 2.** Mid-span strain influence line under moving load.

From the above case of three-axle vehicle, the strain response of the mid-span beam under the *n*-axle vehicle load can be expressed as below:

$$\varepsilon(\mathbf{x}) = \sum\_{i=1}^{n} \varepsilon\_{\mathcal{U}}(\mathbf{x} - \mathbf{x}\_{\mathcal{U}}) \tag{3}$$

By introducing Equation (3) into Equation (1), the following equation can be obtained:

$$
\varepsilon\_{\rm ll}(\mathbf{x} - \mathbf{x}\_{\rm n}) = P\_{\rm n} f(\mathbf{x} - \mathbf{x}\_{\rm n}) \tag{4}
$$

in which:

$$f(\mathbf{x}) = \begin{cases} \frac{h\mathbf{x}}{2EI} & 0 < \mathbf{x} < \frac{L}{2} \\ \frac{hL}{2EI} \left(1 - \frac{\mathbf{x}}{L}\right) & \frac{L}{2} < \mathbf{x} < L \end{cases} \tag{5}$$

When the multi-axle vehicle passes through the bridge, the area enclosed by the mid-span strain function and the *x* axis can be expressed as:

$$A = \int\_{-\infty}^{+\infty} \mathfrak{e}(\mathfrak{x})d\mathfrak{x} \tag{6}$$

By introducing Equation (3) into Equation (6), the following equation can be obtained:

$$A = \int\_{-\infty}^{+\infty} \sum\_{i=1}^{n} P\_{\text{n}} f(\mathbf{x} - \mathbf{x}\_{\text{n}}) d\mathbf{x} = \sum\_{i=1}^{n} P\_{\text{n}} \int\_{-\infty}^{+\infty} f(\mathbf{x} - \mathbf{x}\_{\text{n}}) d\mathbf{x} \tag{7}$$

Consequently, the total weight *P* of the vehicle can be expressed as:

$$P = \sum\_{i=1}^{n} P\_{\mathfrak{n}} = \frac{A}{\int\_{-\infty}^{+\infty} f(\mathbf{x} - \mathbf{x}\_{\mathfrak{n}}) d\mathbf{x}} = \frac{A}{\mathfrak{n}}\tag{8}$$

in which:

$$\mathfrak{a} = \int\_{-\infty}^{+\infty} f(\mathfrak{x} - \mathfrak{x}\_{\mathfrak{n}}) d\mathfrak{x} \tag{9}$$

where *α* is the mid-span strain integral coefficient, it is related to the envelope area of the strain influence line. The *α* can be calibrated by Equation (9) when a known vehicle load passes through the bridge. Then, the *α* can be used to identify the vehicle load.

#### *2.2. Moving Load Identification Method Considering the Load Transverse Distribution*

When the vehicle load acts on the bridge, the load is not only transmitted in the longitudinal direction, but also in the horizontal direction. Therefore, the force analysis of the bridge under the vehicle load is a space calculation problem. Then, the internal force analysis of the bridge section can be carried out through the influence surface. The influence surface of the bridge internal force can be expressed by a two-valued function *η(x*, *y)*, then the internal force value of section *a* can be expressed as *S* = *P* · *η*(*<sup>x</sup>*, *y*), in which *S* is the internal force value of the section, and *P* is the vehicle load. In addition, *η(x*, *y)* can be separated into the product of two single-valued functions by the separation variable method. That is *η*(*<sup>x</sup>*, *y*) = *η*1(*x*)·*η*2(*y*), in which *η1*(*x*) is the internal force influence line of the beam section, and *η2(y)* is the change curve of the load ratio when the unit load acts in different positions along the horizontal direction. Then, the internal force value *P*' of the beam section can be expressed as *P* = *P* · *η*2(*y*), equivalent to assigned the load to the beam along the horizontal direction when the load *P* acts on point *<sup>a</sup>*(*<sup>x</sup>*, *y*).

For the simply supported T-beam bridge, as shown in Figure 3, it is approximately assumed that *S* = *P* · *η*(*<sup>x</sup>*, *y*) ≈ *P* · *η*1(*x*) · *η*2(*y*), which neglects the spatial effect of the bridge and turns it into a plane problem. When a moving load acts on the bridge deck and its position changes with the *x* coordinate but *y* coordinate is constant, then the *P* · *η*2(*y*) is constant too. That is the direction of the load transverse distribution coefficient along

the beam span does not change. Therefore, each beam can be analyzed individually when analyzing the internal force influence line of the beam section, and the equivalent load of each beam can be obtained according to the load transverse distribution coefficient.

**Figure 3.** The internal force calculation under vehicle load.

Take the two-axle vehicle as an example to analyze the internal force (as shown in Figure 4). The axle weight is *P*11, *P*12, *P*21, and *P*22, respectively, and its action position is (*<sup>x</sup>*1, *y*1), (*<sup>x</sup>*1, *y*2), (*<sup>x</sup>*2, *y*1), (*<sup>x</sup>*2, *y*2), respectively. The *y* coordinate values of the four wheel loads are constant when the vehicle travels parallel to the *x* coordinate on the bridge, that is, the transverse distribution coefficient of each wheel load is constant. When analyzing the internal force of a single beam, the equivalent load *Pn*1 , *Pn*2 acting on it can be obtained by the following equation:

$$\begin{cases} P\_1^n = P\_{11} \cdot \eta\_2(y\_1) + P\_{12} \cdot \eta\_2(y\_2) \\\ P\_2^n = P\_{21} \cdot \eta\_2(y\_1) + P\_{22} \cdot \eta\_2(y\_2) \end{cases} \quad (n = 1, 2, 3, 4, 5) \tag{10}$$

**Figure 4.** Wheel load transverse distribution on the bridge.

The total weight of the vehicle can be expressed as:

$$P = P\_{11} + P\_{12} + P\_{21} + P\_{22} = \sum\_{n=1}^{5} \left( P\_1^n + P\_2^n \right) \tag{11}$$

According to the influence line theory, when the strain integral coefficient of one beam section is known, the total weight of the moving load can be calculated through the monitored strain integral value on the section.

Assuming that the above vehicle loads drive parallel to the *x* coordinate from one end to the other end of the beam, the measured strain integral values of the mid-span section of each beam bottom are *A*2, *A*3, *A*4 and *A*5, respectively. In addition, it is assumed that the mid-span strain integral coefficients of each beam bottom are *α*1, *α*2, *α*3, *<sup>α</sup>*4, and *α*5, respectively. The following equation can be obtained:

$$P = \sum\_{n=1}^{5} (P\_1^n + P\_2^n) = \sum\_{n=1}^{5} \frac{A\_n}{n\_n} \tag{12}$$

From Equation (12), it can be seen that the mid-span strain integral coefficient of each beam bottom must be obtained first in order to ge<sup>t</sup> the total weight *P* of the vehicle. The *P*, *A*1, *A*2, *A*3, *A*4, and *A*5 in Equation (12) can be obtained by test. Therefore, the essence of calculating the strain integral coefficient is to solve a five-element linear equation. Keeping the vehicle weight constant but changing the driving position (five different values of *y*1 and *y*2), the equation group can be obtained as below:

$$\begin{cases} \frac{A\_{11}}{a\_1} + \frac{A\_{12}}{a\_2} + \frac{A\_{13}}{a\_3} + \frac{A\_{14}}{a\_4} + \frac{A\_{15}}{a\_5} = P\\ \frac{A\_{21}}{a\_1} + \frac{A\_{22}}{a\_2} + \frac{A\_{23}}{a\_3} + \frac{A\_{24}}{a\_4} + \frac{A\_{25}}{a\_5} = P\\ \frac{A\_{31}}{a\_1} + \frac{A\_{32}}{a\_2} + \frac{A\_{33}}{a\_3} + \frac{A\_{34}}{a\_4} + \frac{A\_{35}}{a\_5} = P\\ \frac{A\_{41}}{a\_1} + \frac{A\_{42}}{a\_2} + \frac{A\_{43}}{a\_3} + \frac{A\_{44}}{a\_4} + \frac{A\_{45}}{a\_5} = P\\ \frac{A\_{51}}{a\_1} + \frac{A\_{52}}{a\_2} + \frac{A\_{53}}{a\_3} + \frac{A\_{54}}{a\_4} + \frac{A\_{55}}{a\_5} = P \end{cases} \tag{13}$$

In order to obtain the strain integral coefficient, its reciprocal can be calculated first:

$$\left\{\frac{1}{\mathfrak{a}}\right\} = \left\{A\right\}^{-1}\left\{P\right\} \tag{14}$$

After obtaining the strain integral coefficient by the above method, the vehicle load identification can be carried out subsequently. Moreover, it should be noted that the theoretical derivation above is aimed at the simply supported T-beam bridge, but the method is still applicable to the similar bridge types, such as box girder bridges.
