A method for monitoring deformation of a tower of an offshore wind turbine

Abstract

The application provides a kind of offshore wind turbine tower deformation monitoring method, belong to the ocean engineering structure monitoring technical field based on computer data processing;Through the data collection of strain sensor pasted in the specific position of the inner surface of segmented tower structure, avoid the influence of sensor by bad external environment.Different from iFEM which needs a large number of measuring points, this method uses sparse discrete strain data, combines the distance between measuring points and neutral axis to convert discrete strain into discrete curvature, and then generates continuous curvature function through interpolation, which solves the problem of difficult construction of DST matrix in traditional modal method and large cumulative error in Ko displacement theory.Finally, through curvature displacement inversion method and recursive splicing, the overall outer surface deformation curve of segmented tower structure is derived.The application not only reduces the number of sensors required and the complexity of layout, but also overcomes the dependence of non-contact method on environment and the limitations of modal method and Ko displacement theory on segmented variable cross-section tower.

Description

Technical Field

[0001] This invention belongs to the field of marine engineering structure monitoring technology based on computer data processing, and particularly relates to a method for monitoring the deformation of offshore wind turbine towers. Background Technology

[0002] Offshore wind turbine towers, as typical flexible and tall structures, possess the following significant characteristics: high flexibility, low natural frequency, and cross-sectional properties that vary along the height. As wind turbine power and height continue to increase, the flexibility of the tower continues to grow, placing higher demands on the accuracy and robustness of monitoring systems. Furthermore, modern large offshore wind turbine towers typically employ a modular design, composed of several steel cylinder segments of varying diameters and wall thicknesses connected by high-strength bolts via annular flanges. This segmented, variable cross-section construction not only leads to nonlinear variations in the tower's diameter, wall thickness, and stiffness along the height but also introduces significant discontinuities in geometric and mechanical properties at the flange connections. This further increases the difficulty of developing accurate deformation reconstruction algorithms. Therefore, traditional point-measurement-based monitoring methods are no longer suitable for the segmented, variable cross-section tower structures of offshore wind turbines. Currently, there is no mature algorithm specifically designed for such segmented, discontinuous structures that can accurately invert their overall deformation curves.

[0003] As ultra-flexible structures, wind turbine towers require structural health monitoring (SHM) for ensuring structural integrity and predicting fatigue life. However, existing monitoring technologies have significant limitations when applied to complex wind turbine towers: 1. Methods based on lasers, vision, photogrammetry, digital images, synthetic aperture radar interferometry, and GPS sensors are existing technologies. These methods have the advantages of being non-contact and allowing for remote measurement. However, these methods require favorable environmental conditions, which limits their ability to monitor the deformation of structures in harsh environments in real time. 2. Modal Method (MM): The accuracy of MM is affected by many factors, including mode selection, model building, sensor placement, and structural material properties. Most importantly, accurate MM monitoring heavily relies on the accuracy of the displacement-strain transformation (DST) matrix. Because modes are difficult to distinguish under large deformations in high-rise structures, it is challenging to construct a DST matrix; therefore, MM is not suitable for monitoring segmented variable cross-section tower structures of offshore wind turbines. 3. Ko displacement theory: This theory is based on the Euler-Bernoulli beam theory and obtains the displacement equation of the structure by integrating and summing the curvature function. However, reconstructing displacement using the Ko displacement theory introduces cumulative errors, leading to an amplification of the final predicted displacement error. Therefore, the Ko displacement theory is not suitable for monitoring segmented variable cross-section tower structures of offshore wind turbines. 4. Inverse Finite Element Method (iFEM): iFEM minimizes the weighted least squares function between the actual strain and the theoretical strain, enabling real-time reconstruction of the displacement, strain, and stress of a structure. It offers high accuracy, is unaffected by external loads, and relies solely on the strain-displacement relationship during calculation, requiring no material properties or load information. This makes it well-suited for complex structures. However, iFEM requires a large number of strain measurement points, and its programming and element construction are highly complex, thus limiting its application to more complex structures. Summary of the Invention

[0004] To address the shortcomings of existing technologies, this invention proposes a method for monitoring the deformation of segmented variable cross-section towers of offshore wind turbines based on discrete strain on the inner surface. This method collects data by sparsely attaching strain sensors at specific locations on the inner surface of the segmented structure, avoiding the influence of harsh external environments on the sensors. Unlike iFEM, which requires a large number of measurement points, this method utilizes sparse discrete strain data and combines the distance from the measurement points to the neutral axis to transform discrete strain into discrete curvature, then generates a continuous curvature function through interpolation. This step effectively solves the problems of difficult DST matrix construction in traditional modal methods and accumulated errors in Ko displacement theory. Finally, through curvature displacement inversion and recursive stitching, the overall outer surface deformation curve of the segmented structure is efficiently derived. This invention not only significantly reduces the number of sensors required and the complexity of their deployment, but also overcomes the environmental dependence of non-contact methods and the limitations of modal methods and Ko displacement theory in segmented variable cross-section towers.

[0005] The present invention provides a method for monitoring the deformation of offshore wind turbine towers, comprising the following processes: S1, based on the offshore wind turbine tower structure, divide the tower into sections and variable cross-section parameter information; on the inner surface of a single segment of the segmented structure, collect discrete strain data in real time; By combining the distance from the corresponding measuring point to the neutral axis, the discrete curvature data set of the inner surface of the segmented structure is calculated and obtained using the Euler-Bernoulli beam theory formula; S2, the discrete curvature data set obtained in step S1 is processed using a piecewise interpolation method to generate a continuous inner surface curvature function covering the entire height of the segmented structure; S3. Based on the inversion of the continuous curvature function of the inner surface of the segmented structure, the deformation curve of the inner surface of the segmented structure is obtained. Combined with the wall thickness function of the segmented structure along the height, the deformation curve of the outer surface of the segmented structure is derived according to the geometric relationship of the structure. S4. Based on the end point of the deformation curve of the outer surface of the segmented structure and the initial position wall thickness of the next segment, obtain the starting point of the deformation of the inner surface of the next segment. Repeat the S1-S3 process to construct the deformation curve of the outer surface of the next segment structure; S5. Based on the deformation curves of the outer surface of each segment, the deformation curves of the outer surface of each segment are recursively spliced ​​together. Starting from the first segment at the bottom, the end point of the deformation curve of the previous segment becomes the starting point of the deformation curve of the next segment, and so on, to construct the deformation curve of the outer surface of the overall structure.

[0006] Preferably, in step S1, the tower is divided into sections and the cross-section parameter information is determined according to the design and installation data of the offshore wind turbine tower structure; strain sensors are attached to the inner surface of a single section of the segmented structure at specific positions along its height, and sparse discrete strain data are collected in real time by a multi-element data acquisition instrument.

[0007] Preferably, in step S1, the discrete curvature data set of the inner surface of the segmented structure is calculated and obtained using the Euler-Bernoulli beam theory formula. The specific calculation method is as follows: ; in, Let represent the strain at the i-th measuring point of the segmented structure. Let be the inner surface curvature of the i-th measurement point of the segmented structure. Let be the distance from the i-th measurement point of the segmented structure to the neutral axis.

[0008] Preferably, the specific process of S2 includes: Based on the discrete curvature data set obtained in S1, a continuous curvature function for the entire height of the inner surface of the segmented structure is generated by interpolating a subset of the discrete curvature data set using a piecewise cubic spline interpolation method. The true curvature values ​​of the check points in the preset discrete curvature data set are then selected. The curvature value at the check point is calculated by comparing it with the continuous curvature function of the full height of the inner surface of the generated segmented structure. Construct a relative error function : ; The calculated error The interpolation function is compared with a preset, user-adjustable accuracy threshold. If it is within the error accuracy threshold range, the current inner surface interpolation function is considered to have sufficient accuracy, and the process proceeds to S3. If it is not within the error accuracy threshold range, the adjustment loop is activated. Based on the preset optimization strategy, the number of interpolation points of the discrete curvature data is increased according to the interpolation method, and the spacing between the interpolated curvature data is reduced until the error is within the error accuracy threshold range. Then, an accurate and continuous inner surface curvature function is regenerated.

[0009] Preferably, the specific process of S3 includes: Based on the continuous curvature function of the segmented inner surface obtained from S2 and the displacement inversion method, the continuous inner surface curvature function is transformed into a series of tangent circular arc segments, each with a length of [missing information]. And it is assumed that it has a constant radius of curvature. , The inner surface deformation curve of the segmented structure is reconstructed and derived by recursively calculating the coordinates of the center and endpoints of each arc segment, which is the reciprocal of the curvature of the segmented structure. The outer surface deformation curve is obtained by geometric transformation in combination with the wall thickness of the segmented structure.

[0010] Preferably, the specific process of S4 includes: Based on the deformation curve of the outer surface of the i-th segment of the previous segmented structure obtained by S3, the lateral displacement at the endpoint of the connection is extracted. and cross section rotation angle This is set as the initial geometric condition for the deformation curve of the outer surface of the (i+1)th segment of the next segmented structure; subsequently, it is combined with the initial wall thickness parameters of the next segmented structure. The corresponding starting point lateral displacement of the deformation of the inner surface of the next segment structure is derived. and cross section rotation angle The starting point of this inner surface deformation constitutes the initial boundary condition for the calculation of the inner surface deformation of the next segmented structure, which is used to iteratively execute S1 to S3 to realize the segment-by-segment recursive solution of the outer surface deformation curve.

[0011] Preferably, the specific process of S5 includes: Based on the derived deformation curves of the outer surface of each segment, the deformation curves of each segment are recursively spliced ​​together, starting from the first segment at the bottom, ensuring the displacement of the endpoint of the deformation curve of the i-th segment. and corner The displacement used as the starting point of the deformation curve of the outer surface of the (i+1)th segment of the structure and corner Ultimately, a globally continuous outer surface deformation curve representing the overall deformation trend of the tower is constructed.

[0012] Compared with the prior art, the present invention has the following beneficial effects: 1. High precision and accurate reproduction of real deformation: By strictly deriving the deformation curve of the outer surface from the deformation curve of the inner surface, the approximate error of the traditional thin-walled beam theory in the case of segmented variable cross section, thick wall or large deformation scenarios is fundamentally avoided, so as to more realistically reproduce the overall deformation state of the tower. 2. Strong applicability and adept at handling complex structures: This method is directly designed for complex structures with "segmented variable cross sections", solving the problem that existing discrete measurement techniques cannot effectively handle complex geometries; 3. Good engineering practicality and sensor protection: Placing strain sensors on the inner surface of the tower can protect sensitive measuring equipment from direct corrosion by the harsh marine environment, improve the long-term durability and reliability of the monitoring system, ensure the continuity and stability of data, and at the same time not interfere with the normal operation of the tower exterior. Attached Figure Description

[0013] To more clearly illustrate the technical solutions of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the following description is only one embodiment of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0014] Figure 1 This is a load-condition deformation diagram of the multi-segment variable cross-section offshore wind turbine tower according to an embodiment of the present invention.

[0015] Figure 2 This is a flowchart of the overall process of the present invention.

[0016] Figure 3 This describes the overall deformation curve construction process of the multi-segment variable cross-section offshore wind turbine tower according to an embodiment of the present invention.

[0017] Figure 4 This is a comparison chart of the overall deformation curve of the offshore wind turbine tower obtained by monitoring the tower structure in this invention and the overall deformation curve of the tower extracted by Ansys. Detailed Implementation

[0018] The present invention will be further described below with reference to embodiments. Obviously, the described embodiments are only some, not all, of the embodiments of the present invention. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention without inventive effort are within the scope of protection of the present invention.

[0019] This embodiment models a 17MW offshore wind turbine tower. The tower has a fixed base, a height of 128.9m, a bottom outer diameter of 10m, and a top outer diameter of 6.198m. The tower is divided into 26 segments. The tower is made of Q345 steel with a yield strength of 345MPa, an elastic modulus of 210GPa, a density of 7850kg / m³, and a Poisson's ratio of 0.3. The tower's self-weight is applied by defining gravitational acceleration. The mass of the rotor nacelle assembly at the top of the tower is 1,145,803 kg, which is applied as a concentrated force at the top. The base of the tower is fixed. For the maximum actual load, a maximum horizontal force Fx = 8350 kN is applied to the top of the tower, along with a bending moment Mz = -42100 kN·m. The maximum actual load considers the maximum design load, the integrity of the offshore wind turbine structure, and the safety level. The load-deformation diagram of the tower is shown below. Figure 1 As shown.

[0020] To address the aforementioned problems, this invention proposes a method for monitoring the deformation of segmented variable cross-section tower structures in offshore wind turbines. The overall process is as follows: Figure 2 As shown, it includes the following steps: S1. Real-time acquisition of discrete strain data at measurement points on the inner surface of the segmented structure and obtaining discrete curvature data in conjunction with wall thickness; the tower is divided into 23 segments based on the tower segmentation and variable cross-section parameter information according to the offshore wind turbine tower structure design and installation data. In this embodiment, horizontal force and bending moment are applied to the top of the tower, which will cause the tower to bend mainly in the XZ plane in the direction of action, where Z is the height axis. FBG fiber optic strain sensors are attached at specific positions (1 / 4, 2 / 4, 3 / 4, and 4 / 4 height) along the intersection of the main bending plane of a single segment of the segmented structure and the inner surface. Sparse discrete strain data are acquired in real time by a multivariate data acquisition instrument. The sparse discrete strain data at the four height positions acquired in real time by the multivariate data acquisition instrument are denoted as ε1, ε2, ε3, and ε4; combined with the distance from the measurement point position to the neutral axis, Based on the relationship between strain and curvature: ; The strain data is converted into curvature data to obtain a discrete curvature dataset of the segments. ; in, Let represent the strain at the i-th measuring point of the segmented structure. Let be the inner surface curvature of the i-th measurement point. This represents the distance from the point to the neutral axis in a segmented structure.

[0021] S2, based on the subset of discrete curvature data from measurement points in S1, is interpolated to the continuous curvature function of the inner surface and verified using the curvature values ​​from the verification points: S21, based on the discrete curvature dataset obtained from the measurement at specific measurement points. ,from A subset is selected as the interpolation base point. In this embodiment, three data points located at 1 / 4, 2 / 4, and 4 / 4 height are selected: By interpolating the three data points using a piecewise cubic spline interpolation method, a piecewise continuous inner surface curvature function is generated. Select data points that were not involved in interpolation. The check points are obtained based on the generated continuous inner surface curvature function. curvature value Compare it with the directly measured curvature value at that location. Compare the results. Calculate the relative error: ; S22, the calculated error E rror The interpolation function is compared with a preset, adjustable accuracy threshold. If it falls within the threshold range, the current inner surface interpolation function is considered to have sufficient accuracy to accurately describe the curvature distribution of segment 1, and the next step can proceed. If it falls outside the threshold range, it indicates that the initial interpolation base points are insufficient to accurately describe the curvature distribution under the load condition. At this point, an adjustment loop is activated, and the processing unit, according to a preset optimization strategy and the interpolation method, expands the number of interpolation points for discrete curvature data and reduces the spacing between interpolated curvature data until the error is within the accuracy threshold range. A verified, accurate, continuous inner surface curvature interpolation function is then regenerated. Then proceed to S3.

[0022] S3, the inner surface deformation curve is obtained based on the numerical inversion of the inner surface continuous curvature function, and the outer surface deformation curve is obtained by combining the wall thickness: Based on the validated continuous curvature function and displacement inversion method of the inner surface of the segmented structure in S2, the segmented structure is divided into N sufficiently small micro-segments, with the starting point h=0 of segment 1 being a fixed support. In the Cartesian coordinate system (x is the horizontal deflection, z is the vertical height), the starting point Pinner,0=(x0, z0) =(0,0), and the initial tangent angle is... = 0 (i.e., vertically upward). The continuous inner surface curvature function obtained from S2 is transformed into a series of tangent circular arc segments, each with a length of... The curvature of segment k is: ; It has a constant radius of curvature: ; The central angle corresponding to this arc segment is: ; The center of the arc lie in Located on the law line, at a distance of The coordinates of the arc are: ; The coordinates of the endpoint of the k-th infinitesimal segment are: ; By recursively calculating the coordinates of the center and endpoints of each arc segment, the endpoints of N points are determined. By connecting and reconstructing, the continuous inner surface deformation curve of the segmented structure can be derived. ; and combined with the wall thickness of the segment varying with height The endpoints of N points on the inner surface deformation curve are obtained. The point corresponding to the distance t1(hk) extrapolated along the normal direction. : ; All Connecting the points yields the deformation curve of the outer surface of the segmented structure. .

[0023] S4: Based on the endpoint of the deformation curve of the outer surface of the previous structural segment and the wall thickness of the next structural segment, obtain the starting point of the deformation of the inner surface of the next structural segment. Repeat the S1-S4 process to construct the deformation curve of the outer surface of the next structural segment. The endpoint of the outer surface curve of segment 1 structure obtained based on S3 and the tangent angle at that point And the starting wall thickness of the next structural segment The starting point of the inner surface deformation curve of the next segment structure is obtained. : ; Repeat steps S1 to S3 to obtain the outer surface deformation curve of the next segment structure.

[0024] S5, Construct the overall outer surface deformation curve: Based on the 26 segmented structural outer surface deformation curves derived in S4, obtain a set of segmented curves. The deformation curves of each segment's outer surface are recursively spliced ​​together, starting from the bottom first segment, with the end point of the previous segment's deformation curve becoming the starting point of the next segment's deformation curve, and so on. Figure 3 The figure shows the deformation curve of the outer surface of the overall structure.

[0025] Numerical simulation and result analysis: To verify the accuracy of the deformation inversion algorithm of this invention, a 17MW tower model was imported into ANSYS for simulation analysis. Based on the design and installation data of the offshore wind turbine tower structure, the tower was divided into segments and variable cross-section parameters, resulting in 23 segmented structures. Fiber grating strain sensors were attached at 1 / 4, 2 / 4, 3 / 4, and 4 / 4 heights of each segment. The strain at these heights was collected using a multi-element acquisition system. Combined with the distance of each measuring point from the neutral axis, discrete curvature data of the segmented structures were obtained. The cubic spline interpolation method was used to interpolate the curvature at 1 / 4, 2 / 4, and 4 / 4 of the tower segment height, generating a continuous inner surface curvature function for the tower segment structure. The accuracy of the interpolated inner surface curvature function was verified by comparing the curvature values ​​at the check points with those calculated using the continuous inner surface curvature function. The inner surface curvature function of each segment was obtained, and the outer surface curvature function of each segment was derived based on the wall thickness. Then, the outer surface curvature function of the overall structure was fitted according to the principle of consistent curvature at the segments. The overall tower deformation curve obtained based on this algorithm was rigorously compared with the Ansys finite element (FEM) simulation results. Figure 4 As shown, the deformation trends and values ​​of the two curves exhibit a high degree of consistency.

[0026] The comparison results strongly verify the effectiveness and accuracy of the continuous surface curvature determination method and corresponding inversion algorithm proposed in this invention, showing that the method can accurately reconstruct the overall deformation response of large segmented, variable cross-section marine structures under specific loads.

[0027] The above description is merely a preferred embodiment of this application and is not intended to limit this application. Various modifications and variations can be made to this application by those skilled in the art. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of this application should be included within the protection scope of this application.

[0028] While the specific embodiments of the present invention have been described above, they are not intended to limit the scope of protection of the present invention. Those skilled in the art should understand that various modifications or variations that can be made by those skilled in the art without creative effort based on the technical solutions of the present invention are still within the scope of protection of the present invention.

Claims

1. A method for monitoring the deformation of an offshore wind turbine tower, characterized in that, Includes the following processes: S1, based on the offshore wind turbine tower structure, divide the tower into sections and variable cross-section parameter information; on the inner surface of a single segment of the segmented structure, collect discrete strain data in real time; By combining the distance from the corresponding measuring point to the neutral axis, the discrete curvature data set of the inner surface of the segmented structure is calculated and obtained using the Euler-Bernoulli beam theory formula; S2, the discrete curvature data set obtained in step S1 is processed using a piecewise interpolation method to generate a continuous inner surface curvature function that covers the entire height of the segmented structure; S3. Based on the inversion of the continuous curvature function of the inner surface of the segmented structure, the deformation curve of the inner surface of the segmented structure is obtained. Combined with the wall thickness function of the segmented structure along the height, the deformation curve of the outer surface of the segmented structure is derived according to the geometric relationship of the structure. S4. Based on the end point of the deformation curve of the outer surface of the segmented structure and the initial position wall thickness of the next segment, obtain the starting point of the deformation of the inner surface of the next segment. The specific process includes: Based on the deformation curve of the outer surface of the i-th segment of the previous segmented structure obtained by S3, the lateral displacement at the endpoint of the connection is extracted. and the corner of the end section This is set as the initial geometric condition for the deformation curve of the outer surface of the (i+1)th segment of the next segmented structure; subsequently, it is combined with the initial wall thickness parameters of the next segmented structure. The corresponding starting point lateral displacement of the deformation of the inner surface of the next segment structure is derived. and the starting section angle The starting point of this inner surface deformation constitutes the initial boundary condition for calculating the inner surface deformation of the next segment structure. Repeat the S1-S3 process to construct the deformation curve of the outer surface of the next segment structure, thereby realizing the segment-by-segment recursive solution of the outer surface deformation curve; S5. Based on the deformation curves of the outer surface of each segment, the deformation curves of the outer surface of each segment are recursively spliced ​​together. Starting from the first segment at the bottom, the end point of the deformation curve of the previous segment becomes the starting point of the deformation curve of the next segment, and so on, to construct the deformation curve of the outer surface of the overall structure.

2. The method for monitoring the deformation of an offshore wind turbine tower as described in claim 1, characterized in that: In S1, the tower is divided into sections and variable cross-section parameter information according to the design and installation data of the offshore wind turbine tower structure; strain sensors are attached to the inner surface of a single section of the segmented structure at specific positions along its height, and sparse discrete strain data are collected in real time by a multi-element data acquisition instrument.

3. The method for monitoring the deformation of an offshore wind turbine tower as described in claim 1, characterized in that: In S1, the discrete curvature data set of the inner surface of the segmented structure is calculated and obtained using the Euler-Bernoulli beam theory formula. The specific calculation method is as follows: ； in, Let represent the strain at the i-th measuring point of the segmented structure. Let be the inner surface curvature of the i-th measurement point of the segmented structure. Let be the distance from the i-th measurement point of the segmented structure to the neutral axis.

4. The method for monitoring the deformation of an offshore wind turbine tower as described in claim 1, characterized in that: The specific process of S2 includes: Based on the discrete curvature data set obtained in S1, a continuous curvature function for the entire height of the inner surface of the segmented structure is generated by interpolating a subset of the discrete curvature data set using a piecewise cubic spline interpolation method. The true curvature values ​​of the check points in the preset discrete curvature data set are then selected. The curvature value at the check point is calculated by comparing it with the continuous curvature function of the full height of the inner surface of the generated segmented structure. Construct a relative error function : ； The calculated error The interpolation function is compared with a preset, user-adjustable accuracy threshold. If it is within the error accuracy threshold range, the current inner surface interpolation function is considered to have sufficient accuracy, and the process proceeds to S3. If it is not within the error accuracy threshold range, the adjustment loop is activated. That is, based on the preset optimization strategy, the number of points for interpolation of discrete curvature data is increased and the spacing between interpolated curvature data is reduced according to the interpolation method until the error is within the error accuracy threshold range, and an accurate and continuous inner surface curvature function is regenerated.

5. The method for monitoring the deformation of an offshore wind turbine tower as described in claim 1, characterized in that: The specific process of S3 includes: Based on the continuous curvature function of the segmented inner surface obtained from S2 and the displacement inversion method, the continuous inner surface curvature function is transformed into a series of tangent circular arc segments, each with a length of [missing information]. And it is assumed that it has a constant radius of curvature. radius of curvature The curve is the reciprocal of the curvature of the segmented structure. The continuous inner surface deformation curve of the segmented structure is reconstructed and derived by recursively calculating the coordinates of the center and endpoints of each arc segment. Combined with the wall thickness of the segmented structure, the outer surface deformation curve is obtained through geometric transformation.

6. The method for monitoring the deformation of an offshore wind turbine tower as described in claim 1, characterized in that: The specific process of S5 includes: Based on the derived deformation curves of the outer surface of each segment, the deformation curves of each segment are recursively spliced ​​together, starting from the first segment at the bottom, ensuring that the endpoint of the deformation curve of the i-th segment is laterally displaced. and the corner of the end section The lateral displacement used as the starting point of the deformation curve of the outer surface of the (i+1)th segment structure and the starting section angle Ultimately, a globally continuous outer surface deformation curve representing the overall deformation trend of the tower is constructed.

Images