Accurate horizontal displacement calculation method for vertical line of arch dam

Abstract

The invention belongs to the technical field of arch dam monitoring, and particularly provides an arch dam vertical line accurate horizontal displacement calculation method which comprises the steps that the radial azimuth angle of each vertical line measuring point is calculated; determining a rotation angle of a local coordinate system of adjacent vertical line measuring points; obtaining a rotation matrix according to the rotation angle obtained in the step 2; and converting the displacement of the previous measuring point to the local coordinate system of the current measuring point, and then performing algebraic superposition to obtain a monitoring result of the horizontal displacement of the to-be-measured point. According to the method, the horizontal displacement observed quantity of the previous elevation vertical line measuring point is converted to the current local coordinate system according to the rotation angle of the current measuring point local coordinate system, then the horizontal displacement observed value of the current measuring point is calculated through algebraic superposition, and accurate calculation of the displacement of the vertical line measuring point of the arch dam is achieved.

Description

Technical Field

[0001] The invention belongs to the technical field of arch dam monitoring, and in particular relates to a method for calculating the precise horizontal displacement of a vertical line of an arch dam. Background Art

[0002] Horizontal displacement is an important indicator for reflecting the operating performance of dam structures and implementing dam safety monitoring, and the vertical line method is the main method for observing the horizontal displacement of dams. The vertical line method horizontal displacement monitoring uses a stainless steel wire with a diameter of 1.5~2 mm, which is fixed at one end and vertically tensioned, as the reference line. The horizontal projection distance of different elevation measuring points along the vertical line relative to the vertical fixed point is measured as follows Figure 1 As shown in the figure, the horizontal displacement value of each measuring point can be calculated with an accuracy of sub-millimeter level. Therefore, ensuring the accuracy of vertical line monitoring results is of great practical significance to dam safety management.

[0003] In the vertical displacement observation, the vertical line measures the relative displacement relative to the hanging point, and the inverted line measures the relative displacement relative to the anchor point. The anchor point is a relatively fixed point, which can also be called absolute displacement. Therefore, in engineering, the vertical line and the inverted line are generally used together, and the top observation point of the inverted line and the bottom observation point of the vertical line are placed at the same elevation (which can be called the "associated measurement point" of the vertical and inverted lines, as shown below Figure 2 a), the absolute displacement of the lowest point of the vertical line is obtained through the inverted vertical line, and then the relative displacement of each observation point on the vertical line is converted into absolute displacement.

[0004] Dams can be classified into linear gravity dams and curved arch dams according to their shapes. For linear gravity dams, the vertical monitoring results are usually calculated using the algebraic superposition method. Specifically, a certain measuring point on the vertical line can measure and read the radial and tangential displacement increments at the location, and then the radial and tangential displacement increments of different elevation measuring points are directly algebraically accumulated to obtain the radial and tangential displacements of the measuring point, as follows Figure 2 Assume that a vertical line has an inverted observation point (the inverted observation value is the absolute displacement of the point), n The vertical observation point is i The absolute displacement calculation formula of a vertical measuring point is as follows: ; In the formula, and Respectively x and y Directional inverted observations; and Respectively represent i Vertical measuring points x and y Direction observations; and Respectively represent i Vertical measuring points x and y Absolute displacement in direction.

[0005] Due to the special curved surface structure design of the arch dam, the diameter and tangent of the measuring points at different elevations and different positions have different rotation angles. Therefore, in order to accurately calculate the displacement of the vertical measuring points at different elevations of the arch dam, the influence of different rotation angles should be considered. Summary of the invention

[0006] The technical problem to be solved by the present invention is to provide a method for calculating the accurate horizontal displacement of the vertical line of an arch dam, so that the horizontal displacement observation value of the previous vertical line measuring point is converted to the current local coordinate system according to the rotation angle of the local coordinate system of the current measuring point, and then the horizontal displacement observation value of the current measuring point is calculated by algebraic superposition, so as to realize the accurate calculation of the displacement of the vertical line measuring point of the arch dam.

[0007] In order to solve the above technical problems, the technical solution adopted by the present invention is: a method for calculating the precise horizontal displacement of the vertical line of an arch dam, comprising the following steps: Step 1: Calculate the radial azimuth of each vertical measuring point; Step 2: Determine the rotation angle of the local coordinate system of the adjacent vertical measuring point according to the radial azimuth calculated in step 1; Step 3: Obtain a rotation matrix according to the rotation angle obtained in step 2; Step 4: Convert the displacement of the previous measuring point to the local coordinate system of the current measuring point, and then perform algebraic superposition to obtain the displacement monitoring result of the measuring point.

[0008] In the preferred solution, in step 1, the radial azimuth of each vertical line measuring point is calculated by a direct method: when designing the vertical line, two control points are designed: A.B Determine the x direction of the local coordinate system of the vertical measuring point, and the radial azimuth of the vertical measuring point is x Azimuth of direction , the calculation method is as follows: Assume two control points A.B The coordinates are , then the coordinate increment between points A and B is , for: ; 1) When When , the azimuth is in the first quadrant. x Azimuth of direction Calculated according to the following formula: ; In the formula, represents the inverse tangent function; 2) When When , the azimuth is in the second quadrant. x Azimuth of direction Calculated according to the following formula: ; In the formula, Indicates taking the absolute value; 3) When When , the azimuth is in the third quadrant. x Azimuth of direction Calculated according to the following formula: ; 4) When When , the azimuth is in the third quadrant. x Azimuth of direction Calculated according to the following formula: .

[0009] In the preferred solution, the radial azimuth of the vertical measuring point is calculated by the indirect method: the slope of the tangent direction of the measuring point is obtained by taking the derivative of the parabola equation of the arch dam centerline. K , and then the semi-center angle of the arch dam at the vertical measuring point is obtained according to the geometric relationship, and then the radial azimuth of the vertical measuring point is obtained according to the azimuth of the arch dam centerline and the obtained semi-center angle of the arch dam.

[0010] In the preferred solution, in step 1, it is assumed that the center line of the arch dam is a parabola, and its equation is set as: ; In the formula, For the parabola equation Y The intercept on the axis; R represents the radius of curvature; Derivative the above formula to get the slope of the tangent line at the measuring point K : ; ; In the formula, x Determined based on the distance from the vertical measuring point to the center line of the dam.

[0011] In the preferred solution, in step 1, the semi-center angle of the arch dam at the vertical measuring point is The calculation formula is: ; In the formula, To find the semi-central angle, The inclination of the tangent line at the vertical measuring point.

[0012] In the preferred solution, in step 1, the radial azimuth of the vertical measuring point is calculated according to the semi-center angle of the arch dam and the azimuth of the centerline of the arch dam. That is, the calculation formula is: ; In the formula, is the azimuth of the arch dam centerline.

[0013] In the preferred solution, in step 2, the relative rotation angle of adjacent vertical line measuring points is determined according to the radial azimuth of each vertical line measuring point. , the calculation formula is: ; In the above formula, and Respectively i and i -The radial azimuth of a vertical measuring point in the unified arch dam coordinate system.

[0014] In the preferred solution, in step 3, the rotation matrix The expression is: .

[0015] In the preferred solution, in step 4, the monitoring result of the displacement of the measured point is the horizontal displacement change of each vertical measuring point of the arch dam. , the expression is: ; In the above formula, k Indicates the number of points to be tested.

[0016] The invention provides a method for calculating the precise horizontal displacement of a vertical line of an arch dam, which fully considers the structural characteristics of the arch dam and the reality that the local coordinate systems of vertical line measuring points at different elevations are not unified and there are rotation angles, and proposes a strict local coordinate transformation of the horizontal displacement of adjacent vertical line measuring points at elevations, so that the observed horizontal displacement of the previous vertical line measuring point is transformed to the current local coordinate system according to the rotation angle of the local coordinate system of the current measuring point, and then the observed horizontal displacement of the current measuring point is calculated by algebraic superposition, so as to realize the precise calculation of the displacement of the vertical line measuring points of the arch dam. BRIEF DESCRIPTION OF THE DRAWINGS

[0017] The present invention will be further described below in conjunction with the accompanying drawings and embodiments: Figure 1 This is a schematic diagram of horizontal displacement observation using the vertical line method; Figure 2 It is a schematic diagram of the layout of different elevation measuring points of the vertical line; Figure 3 Schematic diagram of radial azimuth calculation for the vertical measuring point of the right arch dam. DETAILED DESCRIPTION

[0018] In order to make the purpose, technical solution and advantages of the present invention more clearly understood, the present invention is further described in detail below in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not used to limit the present invention.

[0019] The process of calculating the precise horizontal displacement of vertical measuring points at different elevations of an arch dam according to the present invention is as follows: The key to calculating the accurate horizontal displacement of vertical measuring points at different elevations of an arch dam is to determine the rotation angle of the local coordinate system of adjacent vertical measuring points. Therefore, the radial azimuth of the vertical measuring points is calculated first.

[0020] Step 1: Calculate the radial azimuth of each vertical measuring point.

[0021] One is the direct method.

[0022] The direct method is to design two control points through the vertical line. A.B Determine the x direction of the local coordinate system of the vertical measuring point, and the radial azimuth of the vertical measuring point is x Azimuth of direction , and thus the azimuth angle can be calculated directly through the coordinates of the two points. Assuming that the two control points A.B The coordinates are , then the coordinate increment between points A and B is , for: ; 1) When When , the azimuth is in the first quadrant (measurement coordinate system, not mathematical Cartesian coordinate system), then the azimuth (Unit: radians) can be calculated according to the following formula: ; In the formula, Represents the inverse tangent function.

[0023] 2) When When , the azimuth is in the second quadrant (measurement coordinate system, not mathematical Cartesian coordinate system), then the azimuth (Unit: radians) can be calculated according to the following formula: ; In the formula, Indicates taking the absolute value.

[0024] 3) When When , the azimuth is in the third quadrant (measurement coordinate system, not mathematical Cartesian coordinate system), then the azimuth (Unit: radians) can be calculated according to the following formula: ; 4) When When , the azimuth is in the third quadrant (measurement coordinate system, not mathematical Cartesian coordinate system), then the azimuth (Unit: radians) can be calculated according to the following formula: ; The vertical line measuring point can be calculated by the above method x Azimuth of direction , y The azimuth of the direction is x Add the azimuth angle You can get it.

[0025] In addition to the above direct method for calculating the azimuth, an indirect method for calculating the azimuth is also proposed.

[0026] The indirect method is a method of calculating the radial azimuth of the vertical measuring point through the surface design equation of the arch dam when the installation direction coordinates of the vertical measuring point are unknown. Assuming that the center line of an arch dam is a parabola, its equation is set as: ; In the formula, For the parabola equation Y The intercept on the axis; R Indicates the radius of curvature. To obtain the radial azimuth of each vertical measuring point, the semi-center angle between the normal direction of each vertical measuring point and the central axis of the arch dam can be obtained according to the above formula. Specifically, the above formula is first derived to obtain the slope of the tangent direction of the point: K : ; Then, the semi-center angle to be determined is obtained based on the geometric relationship, and then the radial azimuth of the vertical line measuring point can be obtained based on the azimuth of the arch dam centerline and the semi-center angle of the arch dam obtained in the previous step.

[0027] Taking the calculation of the half center angle of the right arch dam as an example, Figure 3 A schematic diagram of the coordinates of the right arch dam is given. Represents the arch dam coordinate system, black Represents the Cartesian coordinate system, which is convenient for derivative calculations, and the relative angle relationship remains unchanged. To find the semi-central angle, The inclination of the tangent line at the vertical measuring point.

[0028] According to the arch dam design parameters, the curvature radius of the right arch dam can be obtained R , therefore, according to the formula, the slope of the tangent line is: ; in,x Determined by the distance from the vertical measuring point to the center line of the dam, and then according to the geometric relationship, the semi-center angle can be obtained : .

[0029] Finally, according to the semi-center angle of the arch dam at the vertical measuring point and the azimuth of the arch dam centerline The radial azimuth of the vertical line point can be calculated : .

[0030] Step 2: Determine the rotation angle of the local coordinate system of adjacent vertical measuring points.

[0031] After the radial azimuth of different vertical measuring points is calculated, the rotation angle of adjacent vertical measuring points is calculated. First, the azimuth of each vertical measuring point is calculated according to the previous section, and then the relative rotation angle of adjacent vertical measuring points is determined. : ; In the above formula, and Respectively i and i -The radial azimuth of a vertical measuring point in the unified arch dam coordinate system.

[0032] Step 3: Get the rotation matrix according to the rotation angle obtained in step 2. The expression is: .

[0033] Step 4: Convert the displacement of the previous measuring point to the local coordinate system of the current measuring point, and then perform algebraic superposition to obtain the displacement monitoring results of the measuring point to be measured, so as to obtain a more accurate horizontal displacement change of each vertical measuring point of the arch dam. , the expression is: .

[0034] Example Description In order to illustrate the calculation effect of the present invention on the horizontal displacement of the vertical line of the arch dam, the vertical horizontal displacement monitoring system of a dam section of a certain arch dam (composed of 1 inverted vertical line and 5 vertical lines) is taken as an example to compare the horizontal displacement difference obtained by the traditional approximate calculation method and the rigorous calculation method of the present invention.

[0035] Table 1 below gives the results of the vertical horizontal displacement of a certain arch dam in July 2023 calculated using the traditional approximate method.

[0036] Table 2 gives the vertical horizontal displacement of a certain arch dam in July 2023 calculated by the method of the present invention.

[0037] Table 3 gives the difference in the vertical horizontal displacement of a certain arch dam in July 2023 calculated by the traditional method and the method of the present invention.

[0038]

[0039]

[0040]

[0041] From the above data, it can be seen that in the displacement calculation, due to the small radial and tangential (local coordinate system) rotation angles of adjacent vertical measuring points, and the small horizontal displacement change, the difference between the traditional approximate method and the method of the present invention is small, about 0.01 mm. However, for the cumulative displacement change of the horizontal method, the horizontal displacement change calculated by the traditional approximate method differs from the displacement change calculated by the rigorous method of the present invention by about 1.4 mm. For high-precision horizontal displacement deformation monitoring of large-scale projects, the traditional approximate method does not take the rotation angle into account, and the error in calculating the vertical displacement change cannot be ignored.

[0042] It will be easily understood by those skilled in the art that the above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions and improvements made within the spirit and principles of the present invention should be included in the protection scope of the present invention.

Claims

1. A method for calculating the precise horizontal displacement of a vertical line of an arch dam, characterized in that: The following steps are involved: Step 1: Calculate the radial azimuth of each vertical measuring point; Step 2: Determine the rotation angle of the local coordinate system of the adjacent vertical measuring point according to the radial azimuth calculated in step 1; Step 3: Obtain a rotation matrix according to the rotation angle obtained in step 2; Step 4: Convert the displacement of the previous measuring point to the local coordinate system of the current measuring point, and then perform algebraic superposition to obtain the displacement monitoring result of the measuring point.

2. The method for calculating the precise horizontal displacement of the vertical line of an arch dam according to claim 1 is characterized in that: In step 1, the radial azimuth of each vertical line measuring point is calculated by direct method: when designing the vertical line, two control points are designed A.B Determine the local coordinate system of the vertical line measurement point x Direction, the radial azimuth of the vertical measuring point is x Azimuth of direction , the calculation method is as follows: Assume two control points A.B The coordinates are , then the coordinate increment between points A and B is , for: The calculation method is: Assume two control points A.B The coordinates are , then the coordinate increment between points A and B is , for: ； 1) When When , the azimuth is in the first quadrant. x Azimuth of direction Calculated according to the following formula: ； In the formula, represents the inverse tangent function; 2) When When , the azimuth is in the second quadrant. x Azimuth of direction Calculated according to the following formula: ； In the formula, Indicates taking the absolute value; 3) When When , the azimuth is in the third quadrant. x Azimuth of direction Calculated according to the following formula: ； 4) When When , the azimuth is in the third quadrant. x Azimuth of direction Calculated according to the following formula: 。 3. The method for calculating the precise horizontal displacement of the vertical line of an arch dam according to claim 1 is characterized in that: In step 1, the radial azimuth of the vertical measuring point is calculated by an indirect method: the slope of the tangent direction of the measuring point is obtained by derivation of the parabola equation of the arch dam centerline. K , and then the semi-center angle of the arch dam at the vertical measuring point is obtained according to the geometric relationship, and then the radial azimuth of the vertical measuring point is obtained according to the azimuth of the arch dam centerline and the obtained semi-center angle of the arch dam.

4. The method for calculating the precise horizontal displacement of the vertical line of an arch dam according to claim 3 is characterized in that: In the step 1, it is assumed that the center line of the arch dam is a parabola, and its equation is set as: ； In the formula, For the parabola equation Y The intercept on the axis; R represents the radius of curvature; Derivative the above formula to get the slope of the tangent line at the measuring point K : ； ； In the formula, x Determined based on the distance from the vertical measuring point to the center line of the dam.

5. The method for calculating the precise horizontal displacement of the vertical line of an arch dam according to claim 3 is characterized in that: In the step 1, the semi-center angle of the arch dam at the vertical measuring point The calculation formula is: ； In the formula, To find the semi-central angle, The inclination of the tangent line at the vertical measuring point.

6. The method for calculating the precise horizontal displacement of the vertical line of an arch dam according to claim 3 is characterized in that: In the step 1, the radial azimuth of the vertical measuring point is calculated according to the semi-center angle of the arch dam and the azimuth of the centerline of the arch dam. That is, the calculation formula is: ； In the formula, is the azimuth of the arch dam centerline.

7. The method for calculating the precise horizontal displacement of the vertical line of an arch dam according to claim 1 is characterized in that: In the second step, the relative rotation angle of adjacent vertical line measuring points is determined according to the radial azimuth of each vertical line measuring point. , the calculation formula is: ； In the above formula, and Respectively i and i -The radial azimuth of a vertical measuring point in the unified arch dam coordinate system.

8. The method for calculating the precise horizontal displacement of the vertical line of an arch dam according to claim 1 is characterized in that: In step 3, the rotation matrix The expression is: 。 9. The method for calculating the precise horizontal displacement of the vertical line of an arch dam according to claim 1 is characterized in that: In step 4, the monitoring result of the displacement of the measured point is the horizontal displacement change of each vertical measuring point of the arch dam. , the expression is: ； In the above formula, k Indicates the number of points to be tested.

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