Simulation and Experiment Study on the Bridge Deflection Measuring Method Based on Secant Inclination

Deflection is the most direct indicator that reflects the bearing capacity of the bridge and the overall stiffness. There are many ways to measure the deflection of Bridges, and the inclination angle method is the most commonly used indirect method, but the existing theory of inclination angle method is relatively complicated. Based on the facts of the bridge small inclination, this ar-ticle proposes the method of obtaining the bridge deflection by the inclination of the secant line constructed from the adjacent measurement points. Firstly, according to the bending deformation curve of general simply supported beam, the deflection calculation formula of each measuring point is derived based on the assumption of small deformation and the inclination Angle of measuring point. Secondly, a large commercial finite element soft-ware ANSYS 10.0 is used to carry out numerical simulation on the simp-ly-supported beam under concentrated load in mid-span, and the deflection results of the numerical simulation are compared and verified with the theoretical results of the proposed method. Finally, the measured deflection results of the simply-supported beam model under mid-span load are compared with the theoretical results of the proposed method. The verification results show that if the actual model is consistent with the theoretical model, the proposed method has good accuracy.


Introduction
Bridge is an important part of the highway and railway. The safety of the bridge How to cite this paper: Wu, Y. and Li, J. is related to the normal performance of road transportation functions and the safety of the lives and property of the people. Therefore, the safety and health monitoring of the bridge are particularly important [1]. Deflection is an important indicator that reflects the overall stiffness of the bridge and the bearing capacity. It is an important parameter reflecting the health status of the bridge. Therefore, measurement of bridge deflection is very important to evaluate the bearing capacity and safety of the bridge [2].
The methods of bridge deflection measurement can be divided into two categories: direct and indirect method. There are many direct measurement methods, including displacement, full station instrument, GPS, foundation radar measurement system (such as IBIS•S, FastGBSAR) [3], connecting pipeline, level instrument [4], photoelectric imaging, laser, ground microwave interference [5], machine vision [6], theodolite, measuring robot [7], GNSS (Global Navigation Satellite System) [8], digital image correlation (DIC) [9], etc. However, direct measurement methods generally require a fixed reference point, and some methods or operations are difficult, some methods are not accurate enough, some methods or prices are expensive, so people have developed indirect measurement methods.
The parameters of indirect measurement methods are generally strain [5], acceleration [10] [11], and inclination [2] [6]. The deflection is generally obtained from strain by modal vibration method, that is, the corresponding strain modular vibration type is obtained from the measured dynamic strain, and the displacement modular vibration type is then obtained from the relationship between the strain and displacement. Then the coordinates of displacement mode are calculated. At this point, the dynamic deflection of bridge can be obtained from the displacement modular vibration and displacement modal coordinates. The quadratic integration of acceleration with respect to time can also give the bridge deflection. These two methods above are generally used for measurement of dynamic deflection. The strain modular method requires fairly many measurement points, and there are some difficulties in identifying the modular vibration [6]. The modular vibration type is inaccurate [12] or the number of mode identification is insufficient, which will affect the deflection measuring accuracy [13]. The quadratic integration method of acceleration will produce errors, and it is difficult to determine the initial conditions [14]. Bridge deflection can also be obtained from inclination angle by curve fitting and the numerical integration of inclination with respect to coordinates, or by mode superposition method of structural dynamics. In recent years, the inclination method has been increasingly applied to the deflection measurement of the bridge [2].
However, whether curve fitting method or mode superposition method is used to obtain deflection from inclination Angle, these methods are still complicated. In this paper, based on the fact that the inclination Angle of Bridges is generally small, a method to obtain the deflection of Bridges by the inclination Angle and simple trigonometric function is proposed and verified by numerical simulation and test. The experimental results show that the proposed method has good accuracy as long as the actual model agrees with the theoretical model.

Review of Bridge Deflection Measurement by Inclination Angle
According to the literature investigated by the author, at present, there are mainly integration method, summation method, fitting method and reconstruction method to measure bridge deflection through inclination Angle.

Integral Method
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: Assuming that the deformation is small, there approximately is: 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:

Summation Method
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]:

Fitting Method
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 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
Reconstruction method is a method to determine the deflection of a measuring Y. Wu, J. Li 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.

Principle of Secant Inclination Deflection Measurement
The principle of measuring the bridge deflection by the secant inclination method is shown in Figure 1 In actual measurement, a linear chord can be erected between adjacent measuring points (i.e. each internode). One end of the chord is restrained by a hinge support, and the other end is restrained by a roller support to ensure that the chord can be free to expand and contract as the bridge deformed, without bending deformation to affect the accuracy of angle measurement. The inclinometer is fixed on the chord. Before the bridge is deformed, or when the chord is in a horizontal state, the inclinometer's reading is 0. After the bridge is deformed, the inclinometer's reading θ i is the inclination angle of the connecting line between the i + 1 th and the i-th measuring points, namely the inclination angle of the secant It is easy to know from Figure 1 that the deflection increment of the measuring point of i + 1 th relative to the i-th is: The deflection corresponding to the i-th measuring point is: The deflection f 0 of the initial measurement point at the left support can be directly measured by a displacement meter erected on the pier (the pier can be approximately regarded as a static reference point). After f 0 is measured, the measured inclination angle θ i of each chord, as well as formula (2) and formula (3) can be used to obtain the deflection of each measuring point. It can be seen from Figure 1 that the cross-section of the bridge will rotate after the deformation, so that the measuring point is not right under the point where before the bridge deforms, resulting in an error in the deflection calculated according to the above formula. A test of this error will made below by using numerical simulation method.

Verification of the Measured Deflection Value under Static Load
In this section, a large-scale commercial finite element software, ANSYS 10.0, will be used to numerically simulate a simply supported steel beam under static load to verify the accuracy and reliability of the proposed secant inclination deflection measurement theory. The span of the simply supported beam is l = 10 m, the width of the rectangular section is b = 0.2 m, the section height is h = 0.5 m, the steel's elastic modulus is E = 2 × 1011 N/m 2 , its Poisson's ratio is μ = 0.3, and its density is ρ = 7800 kg/m 3 . The plane 42 plane stress element is used in simulation model, and the finite element model is shown in Figure 2.
Here, three load cases are studied, namely the deflection of the concentrated load acting on the 1/2 span (that is, the middle span) and the 1/4 span, and the uniform load. The deflections at the 1/2 span and 1/4 span are investigated. Set the concentrated load P = 10,000,000 N and the uniform load q = 1,000,000 N/m. In the numerical simulation, the secant inclination is calculated with the coordinates between the two measuring points after the bridge is deformed. The analysis results are shown in Table 1. It can be seen from Table 1 that the maximum error is only 0.91%, which is less than 1%. Because the actual deflection-span ratio limit l/600 [7] is much smaller than the ratio l/16.7 of this example, the actual error should be much smaller than the error of this example.

Experimental Study on Deflection Measurement by Inclination Method
In order to verify the feasibility of the proposed deflection measurement method and to compare with the numerical simulation results, an experimental study on the single-span simply supported beam is conducted in this paper.

Test Device
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.

Test Results
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) 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.

Analysis and Discussion
According to the test results in Table 2 of Section 5.2, it can be seen that there are the following characteristics: Open Journal of Civil Engineering   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.

Conclusions and Future Work
The following conclusions can be drawn from the above research: 1) As long as the actual model is consistent with the theoretical model, the secant line inclination method proposed in this paper has high accuracy.
2) Bearing conditions, the flatness of supports and components, the uniformity of materials, and the measurement error may be the factors that affect the accuracy of the secant line inclination method of bridge deflection measurement. The influence of these factors should be considered and evaluated when the method is applied to the actual bridge deflection measurement.
3) The measurement operation can be simplified by approximating the inclination angle of measurement point to replace the inclination angle of the line between two neighbor measurement points (secant inclination Angle), which makes the measurement of bridge deflection easier. However, the influence of this approximate operation on the measurement accuracy needs to be considered and evaluated.  4) In this paper, numerical simulation and experimental verification are only carried out for simply-supported beams with mid-span concentrated loads. Future work can carry out verification research under other load forms (such as uniform load, moving load, and even random load). 5) This paper only studies the case of single-span simply supported Bridges, and future work can study other bridge forms, such as multi-span continuous Bridges, arch Bridges, cable-stayed Bridges and even suspension Bridges.