The integration method [15] is a relatively traditional and simple method, because the first derivative of the deflection y with respect to its position coordinate x is equal to the tangent of the inclination Angle θ, namely:

d y d x = tan θ (1)

Assuming that the deformation is small, there approximately is:

tan θ = θ (2)

Therefore, as long as the inclination curve is fitted from the measured inclination Angle and the inclination Angle is integrated, the deflection of the bridge can be obtained, namely:

y = ∫ 0 x θ d x (3)

The summation method [16] is actually a discrete form of the continuous integral method. Change the length dx of the Infinitesimal element in Equation (3) of the above integration method into the distance Δx_i of the adjacent measuring points, and change the inclination θ at any position x into the inclination θ_i of secant line of the adjacent measuring points, then (3) becomes the following discrete summation form [17]:

y j = ∑ i = 1 j θ i Δ x i (4)

Although the integration method is simple and direct, its integration results are easily disturbed by test datum. Therefore, in order to ensure the accuracy, a large number of measuring points need to be arranged [18]. Therefore, people seek other inclination angle method. According to the results of literature survey, fitting method is the most popular inclination method for bridge deflection measurement. In 2002, Yang Xueshan et al. [19] proposed a fitting method to measure bridge deflection by inclination Angle. The fitting method uses a linear combination of a set of basis functions to fit the curve of the bridge, and its basic form is shown as follows:

y ( x ) = A ( x ) ∑ j = 1 n X j g j ( x ) (5)

where, A(x) represents the boundary value condition, X_j is the combination coefficient, and g_j(x) is the basis function.

The determination of combination coefficient and basis function is the key of fitting method.

The combination coefficient is basically determined by the following two methods [20].

1) Solve linear equation set directly

When the measurement points are set such that the number of equations m is equal to the number of unknowns n, then the set of equations has a unique solution, and the combination coefficient X_j can be obtained by directly solving the set of equations.

2) The least square method

When the measurement points are set so that the number of equations m is greater than the number of unknowns n and the rank of the coefficient matrix of the set of equations is different from that of the augmented matrix, the combined coefficient X_j can be solved by the least square method.

Because of the measurement error, more measuring points are generally arranged, and the least square method is used to determine the combination coefficient.

The basis functions come in a variety of forms. In early stage, it was common to use power function [21] or displacement mode (or shape function) [22] determined by unit characteristic load as basis function. In order to improve the accuracy and stability [23] and reduce the number of measuring points [20], spline curve was proposed as the basis function. In order to adjust the shape of the deflection curve and further improve the fitting accuracy, people later adopted rational spline curve as the basis function [23]. In order to avoid singularity of coefficient matrix and get rid of restrictions on the placement of measurement points, Liu Yan put forward an improved spline fitting method with no measurement point restrictions based on the comprehensive application of traditional cubic spline curves, least-squares method, and integral method [24]. In order to adapt to the complex bridge with varying stiffness [25], naturally meet the structural boundary conditions and make use of the orthogonal characteristics of the mass matrix of the vibration mode, people use the natural vibration mode of the bridge as the basis function to fit the deflection curve [20]. However, the mode shape function needs to be calculated and determined by establishing a finite element model, which will be quite different from the actual model [25]. In order to solve this problem, yao jingchuan et al. proposed the fitting method of simplified mode shape function [26] [27] as the basis function.

Reconstruction method is a method to determine the deflection of a measuring point by determining the relationship between the deflection of a measuring point and the inclination of a certain point [18]. Because these two quantities have different relationships with different structures and loads, the relationship between these two quantities is relatively easy to determine for simple structures and loads, but it is difficult to determine for complex cases.

In this test, a simply supported steel beam is used to simulate the bridge. The section size of the steel beam is 6 mm × 20 mm, and the beam span is 9600 mm. There are a total of five measuring points A, B, C, D and E, and the distance between the measuring points is 2400 mm, as shown in Figure 3. Among them, A, B, D and E are arranged to measure the inclination Angle, B, C and D are arranged to measure the deflection, and C is arranged to load. Inclinometer is HUOTO brand DC111 model digital display high-precision electronic inclinometer box, its resolution is 0.05˚, accuracy is ±0.02˚, repeatability is 0.1˚. The load is P = 43.9N. The test site device for measuring the Angle of point A and deflection of point B is shown in Figure 4.

By substituting 240 mm distance between measuring points in the test into Δx in formula (2), and the measured inclination Angle of each measuring point into θ_i in formula (2), the theoretical increment Δf_i of deflection of each measuring point can be obtained. By substituting the result of formula (2) into formula (3), the obtained result is used as the theoretical value of deflection of each measuring point. The comparison results of theoretical and measured deflection of each measuring point are shown in Table 2. From the comparison results, it can be seen that there is a certain error between the theoretical value of deflection and the measured value at point B at 1/4 span and point C in the middle span. Among them, the error at point B at 1/4 span is small, which is close to the error value in Table 1 of Section 3. The error at point C in the middle span is big, which is much bigger than the error value in Table 1 in Section 3.

According to the test results in Table 2 of Section 5.2, it can be seen that there are the following characteristics:

1) The test data of symmetric points are not completely symmetric

In the test beam, the point A of left support and the point E of right support is symmetrical, and the point B at left 1/4 span and the point D at the right 1/4 is also symmetrical, but no matter the measured values of Angle or the deflection at point B and point D are not completely symmetrical, which suggests that there may exist supporting conditions that are not absolutely symmetrical, support and geometry of steel beam that is not absolutely smooth, quality and geometry of materials is not absolutely even, loading position for each time is not absolutely the same, and the disturbance that comes from different loading position. All the factors mentioned above could affect the measurement result and lead to the deviation of the measurement data.

2) There exist errors in the measurement itself

According to the measurement datum in Table 2, the deflection readings obtained from two measurements of the same point C are not exactly the same, which indicates that there may be errors in the measurement itself (of course, there may also be deviations caused by the disturbance of the test operation to the conditions of two tests). In addition, because the inclination box has a certain size, the measured inclination Angle is not strictly speaking the Angle of the measuring point, but only the Angle of the two ends of the inclination box near the measuring point.

3) The mid-span error is larger than that at 1/4 span

According to the comparison results in Table 2, the theoretical deflection values of point B and point D at the 1/4 span are very close to the measured values with very small errors, while there is a certain gap between the theoretical deflection values of point C at the middle span and the measured values with large errors, which is mainly caused by the error in using the tangent inclination approximation to replace the secant inclination Angle.

Due to the limitations of the test model and the size of the inclinometer, it is difficult to measure the inclination Angle of the line between two neighbor measurement points (secant inclination Angle) in the model test, so it is approximately replaced by the tangent inclination Angle of the measurement point, which is generally larger than the secant inclination Angle, as shown in Figure 5, resulting in a larger theoretical value.

As can be seen from Figure 5, for the measurement point A of the support, the tangent inclination a1 is close to the secant inclination a2. Therefore, the tangent inclination a1 is approximately used to replace the secant inclination a2 to calculate the deflection of the measurement point B, and the error will be relatively

small. However, for measuring point B at 1/4 span, there is a large difference between tangent Angle b1 and secant Angle b2. Therefore, the tangent Angle b1 is set to approximate the secant Angle b2 to calculate the deflection of measuring point C, and the error will be relatively large. According to visual estimation, the secant inclination Angle of point B at 1/4 span is about 1/2 of the tangent inclination Angle. Therefore, when the secant inclination Angle of point B is approximately set to 1/2 of its tangent inclination Angle, that is, b1 = b2/2, the theoretical deflection of mid-span measurement point C calculated by this method has a great reduction in its error, as shown in Table 3.