Arch dam deformation monitoring model modeling method

Abstract

The invention discloses an arch dam deformation monitoring model modeling method, which comprises the following steps of: 1) deducing a deformation expression of a vertical cantilever beam under the action of a water pressure load by taking the vertical cantilever beam as a modeling research object, and constructing an arch dam deformation water pressure component model; 2) decomposing the temperature load borne by the arch dam, deducing an arch dam deformation expression under the action of the uniform temperature and the equivalent linear temperature difference, and constructing an arch dam deformation temperature component model; (3) representing the time effect of arch dam deformation by adopting a linear combination of a linear function and a logarithmic function, and constructing an arch dam deformation monitoring model by synthesizing the obtained arch dam deformation water pressure component model and the arch dam deformation temperature component model; 4, a multiple linear regression model is built based on the arch dam deformation monitoring model.The multiple linear regression model is used for predicting the arch dam deformation value.The multiple linear regression model built through the method is suitable for concrete arch dam deformation behavior rule analysis, and compared with a traditional model, the multiple linear regression model has better mechanical interpretation performance during arch dam deformation rule statistical analysis.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of safety monitoring of hydraulic structures, and particularly relates to a modeling method for an arch dam deformation monitoring model. BACKGROUND

[0002] The deformation value of an arch dam is the most intuitive physical quantity for monitoring the operating characteristics of the arch dam, and monitoring the deformation of the arch dam is the most important means for ensuring whether the arch dam can safely operate. In a traditional arch dam deformation monitoring statistical model, only the equal replacement of independent variables and effect variables is considered for three components of water pressure, temperature and time effect, and the structural characteristics that the horizontal arch ring and the cantilever beam of the arch dam are in consistent displacement under complex conditions are ignored. The deformation prediction result is dependent on the statistical law of the monitoring data, and the explanatory result lacks sufficient support, and the prediction accuracy of the arch dam deformation is affected to a certain extent. SUMMARY

[0003] The application aims at the shortage of the prior art, and provides a modeling method for an arch dam deformation monitoring model. The arch dam space structure is analyzed, the modeling method for the arch dam deformation monitoring model is researched from two perspectives of the horizontal arch ring and the vertical cantilever beam, the deformation characteristics of the arch dam under the action of two main loads of water pressure and temperature are researched, and the characteristic factors of the arch dam deformation monitoring model are determined. On the basis of respectively deriving the deformation expression of the arch dam under the action of the water pressure and temperature loads, the multivariate linear regression statistical model of the arch dam deformation is established, so as to realize the prediction of the deformation value of the arch dam.

[0004] In order to achieve the above object, the application adopts the following technical scheme.

[0005] A modeling method for an arch dam deformation monitoring model comprises the following steps.

[0006] Step S1, the arch dam space is regarded as being composed of a horizontal arch ring and a vertical cantilever beam, and the water pressure load in front of the dam is respectively distributed to the arch ring and the vertical cantilever beam. The water pressure load borne by the vertical cantilever beam is obtained by using a quadratic curve fitting, and the specific expression is a i is a fitting coefficient, H is the water depth in front of the dam, i=1,2, the vertical cantilever beam is selected as the modeling research object in combination with the vertical line arrangement characteristics of the arch dam deformation monitoring facility, the deformation expression of the vertical cantilever beam under the action of the water pressure load is derived, and thus the arch dam deformation water pressure component model is constructed.

[0007] In the derivation of the deformation expression of the vertical cantilever beam under the action of the water pressure load, the horizontal displacement of any point of the arch dam is regarded as being composed of the structure displacement of the dam body and the dam foundation, and the dam foundation displacement is divided into the corner displacement and the normal displacement. Therefore, the radial displacement of the arch dam is divided into three parts: the displacement δ 1H, the bending moment and shear force generated by the static water pressure acting on the dam body are transmitted to the dam foundation surface and make it rotate to cause displacement δ 2H , and the displacement δ caused by the rotation of the dam foundation surface due to the water pressure acting on the dam body 3H ;

[0008] Step S2, the temperature of the arch dam body is divided into three parts: uniform temperature t m , equivalent linear temperature difference t d , and nonlinear temperature difference t n ; the deformation effect of the uniform temperature load on the vertical cantilever beam is regarded as the deflection generated by the vertical cantilever beam bearing uniform load; the deformation generated by the equivalent linear temperature difference on the vertical cantilever beam is equivalent to the effect of the vertical cantilever beam bearing bending moment; since the influence of the nonlinear temperature difference on the arch dam is mostly on the surface of the dam body, the influence of the nonlinear temperature difference on the deformation of the arch dam is ignored, the deformation expression of the arch dam under the action of uniform temperature and equivalent linear temperature difference is derived, and thus the temperature component model of the arch dam deformation is constructed;

[0009] Step S3, the displacement generated by the arch dam under the influence of static water pressure, temperature load environmental variables is divided into three components: water pressure component δ H , temperature component δ T , and aging component δ θ ; the water pressure component and the temperature component are calculated by the model built in step S1 and step S2, and the aging component is calculated by the Zebaoerding-Thomson model, and thus the arch dam deformation monitoring model is constructed;

[0010] Step S4, on the basis of the arch dam deformation monitoring model constructed in step S3, a multiple linear regression model is established to predict the deformation value of the arch dam.

[0011] Specifically, the calculation process of the displacement δ 1H caused by the deformation of the dam body under the action of static water pressure in step S1 is as follows:

[0012] Step S11, considering the bending moment and shear force of the vertical cantilever beam under the action of water pressure load, the displacement δ 1H caused by the deformation of the dam body under the action of static water pressure is calculated, and the calculation formula is:

[0013]

[0014] In the above formula, and respectively represent the displacement of the vertical cantilever beam caused by the deformation of the dam body under the action of bending moment and the displacement of the vertical cantilever beam caused by the deformation of the dam body under the action of shear force;

[0015]

[0016] In the above formula, M1 and V1 are the cantilever beam bending moment and the cantilever beam shear force in the unit force state respectively; M p and V p are the cantilever beam bending moment and the cantilever beam shear force in the displacement state respectively; E c is the elastic modulus of the dam concrete; G c is the shear modulus of the dam concrete; I is the section moment of inertia; A is the cantilever beam cross-sectional area; k is the section shape coefficient, and k = 1.2 is taken for the common rectangular section;

[0017] Step S12, calculating the displacement of the vertical cantilever beam caused by the deformation of the dam under the bending moment of the static water pressure

[0018] Step S121, calculating the cantilever beam bending moment M1(x) in the unit force state:

[0019] M1(x) = h - d - x (3)

[0020] In the above formula, h is the dam height; d is the distance from the dam top to the observation point; x is the integral variable defined for calculating the bending moment of the observation point, and the physical meaning is the distance from the dam bottom to any force analysis point;

[0021] Step S122, calculating the structure bending moment M p (x) in the displacement state:

[0022]

[0023] In the above formula, H is the water depth; γ0 is the water unit weight; b is the cantilever beam cross-sectional width, and x1 is the integral variable defined for calculating the bending moment at x, and the physical meaning is the same as x. In the calculation of M p (x), x is regarded as a constant and x1 is regarded as a variable; a1, a2, and a3 are the regression coefficients of the water pressure load;

[0024] For the convenience of calculation, the expression f(x) related to x in formula (4) is separated out for solving:

[0025]

[0026] Step S123, calculating the section moment of inertia I(x):

[0027]

[0028] In the above formula, T C is the dam top cantilever beam width, and T B is the dam bottom cantilever beam width;

[0029] Step S124, substituting formula (3) and formula (4) into the bending moment term expression of formula (2) has:

[0030]

[0031] Let’s make a substitution to simplify the integration process of formula (7):

[0032]

[0033] After substitution, the integral expression of the bending moment term in formula (1) is:

[0034]

[0035] In the above formula, κ is the substitution coefficient; A1, A2, A3, A4, A5, A6 are the simplified symbols of the corresponding expressions in formula (9), and their specific expression contents are as follows:

[0036]

[0037]

[0038] Step S13, calculate the shear term of the cantilever beam deformation Step S131, the shear size of the unit force state structure is:

[0039] V1(x) = 1 (10)

[0040] Step S132, the calculation formula of the displacement state structure shear is:

[0041]

[0042] To facilitate calculation, separate the expressions related to x in formula (11) and solve them:

[0043]

[0044] Step S133, the calculation formula of the cross-sectional area is:

[0045]

[0046] Step S134, substitute formula (11) to (13) into the bending moment term expression in formula (2):

[0047]

[0048] In the above formula, B1, B2, B3, B4 are the simplified symbols of the corresponding expressions in formula (14), and their specific expression contents are as follows:

[0049]

[0050] B3 = -H2 a2κ-2HT B a2κ 2 -Ha1κ-T B 2 a2κ 3 -T B a1κ 2 ;

[0051]

[0052] Step S14, the bending moment term of the vertical cantilever beam deformation combined with step S12 and step S13 and the shear force term The calculation result is the displacement δ caused by the deformation of the dam body under the action of hydrostatic pressure 1H :

[0053]

[0054] Specifically, the calculation process of the displacement δ caused by the bending moment and shear force generated after the dam body is subjected to the action of hydrostatic pressure in step S1 and the rotation of the dam foundation surface 2H :

[0055] Step S15, calculating the dam foundation rotation displacement θ caused by the force system on the foundation surface of the arch dam:

[0056] θ=Mα+Vα2 (16)

[0057] In the above formula, α is the rotation deformation of the foundation surface due to a unit bending moment, α=α'sin 3 φ+δ'sinφcos 2 φ, α2 is the rotation deformation of the foundation surface due to a unit radial shear force, α2=α"sin 2 φ, α' is the rotation deformation of the foundation surface normal plane caused by a unit moment, α" is the rotation deformation caused by a unit radial shear force, φ is the angle between the tangent of the bank slope and the vertical line, considering the influence of the bank slope, for the cantilever beam of the riverbed section M is the bending moment on the foundation normal surface, V is the shear force parallel to the foundation surface;

[0058]

[0059] The dam foundation rotation displacement θ caused by the force system on the foundation surface of the arch dam is represented by simultaneously solving formula (16)-formula (18):

[0060]

[0061] In the above formula, μ is the Poisson's ratio of the dam foundation; E r is the elastic modulus of the dam foundation;

[0062] Step S16, the displacement δ caused by the bending moment and shear force generated by the dam body under the action of static water pressure transferred to the dam foundation surface and made it rotate 2H ;

[0063] Since the deformation of the dam foundation surface belongs to the small deformation of linear elastic structure, θ ≈ tanθ is used to calculate the displacement δ caused by the bending moment and shear force generated by the dam body under the action of static water pressure transferred to the dam foundation surface and made it rotate 2H has:

[0064] δ 2H = (h-d) tanθ = θ (h-d) (20).

[0065] Specifically, the calculation process of the displacement δ caused by the dam foundation surface rotating under the action of the reservoir water pressure in step S1 is as follows: 3H

[0066] Step S17, according to the basic principle of Boussinesq solution, assuming that the dam width is 2C and the uniform load length is b-x0, the foundation settlement w(x) of the dam at x position is represented as:

[0067]

[0068] In the above formula, C is half the width of the reservoir; p is the static water pressure, p = γ0H; b is the coordinate of the transition point of the reservoir bottom flat slope and variable slope, x0 is the distance from the center of the dam to the upstream dam surface;

[0069] Since θ ≈ tanθ in the small deformation of linear elastic structure, the rotation angle θ(x0) of the dam foundation at x = x0 position is represented as:

[0070]

[0071] In the above formula,

[0072] Further, the displacement δ caused by the dam foundation surface rotating under the action of the reservoir water pressure is represented as: 3H

[0073]

[0074] Specifically, the process of constructing the arch dam deformation water pressure component model in step S1 is as follows:

[0075] Through the analysis of the three parts of the radial displacement of the arch dam cantilever beam, it is found that the deformation of the arch dam under the action of the reservoir water pressure is linearly related to the fourth power of the water depth H in front of the dam, and the expression of the arch dam deformation water pressure component model δ H is:

[0076] ​​

[0077] In the above formula, a i is a fitting coefficient, i = 1, 2, 3, 4; H is water depth.

[0078] Specifically, the derivation process of the arch dam deformation expression under the action of uniform temperature and equivalent linear temperature difference in step S2 is as follows:

[0079] Step S21, calculate the deformation under the action of uniform temperature load;

[0080] The uniform temperature load t m The deformation effect of the cantilever beam is regarded as the deflection of the cantilever beam under uniform load, and the uniform load on the cantilever beam is in the form of uniform load on the arch ring, which is determined by analyzing the expression of the uniform load on the arch ring under the consistency condition of the displacement of the conjugate points of the arch beam and the arch ring:

[0081] Considering that the stress condition of the uniform temperature of the equal-section arch ring is similar to that of the uniform water pressure, assuming that the outer radius of the equal-section arch ring is R, the average radius is r, and the thickness of the arch ring is T, the radial displacement Δr' and the axial force N generated under the action of the uniform load p are represented as:

[0082]

[0083] In the above formula, E is the elastic modulus of the dam concrete; N, M, and V are the axial force, bending moment, and shear force of the dam, respectively;

[0084] If uniform temperature change occurs in the arch ring, the radial displacement Δr' and the axial force N of the arch ring are represented as:

[0085]

[0086] In the above formula, α is the linear expansion coefficient of the concrete;

[0087] Let the radial displacements Δr' of the arch ring under the two types of loads be equal, then:

[0088]

[0089] According to the expression of the uniform load p, the deformation of the cantilever beam under the action of the uniform load p is further derived as:

[0090]

[0091] In the above formula, and respectively represent the bending moment term and the shear force term of the vertical cantilever beam under the action of the uniform load p to deform the dam;

[0092]

[0093] In the above formula, M1 and V1 are the structural bending moment and the structural shear force in the unit force state, respectively; M p and V p are the structural bending moment and the structural shear force in the displacement state, respectively; E c is the elastic modulus of the dam concrete; G c is the shear modulus of the dam concrete; I is the moment of inertia; A is the cross-sectional area; k is the cross-sectional shape coefficient, and k = 1.2 is taken for a common rectangular cross-section;

[0094] Step S22, calculating the radial deformation of the cantilever beam under the action of uniform temperature;

[0095] is the radial deformation of the cantilever beam under the action of uniform load p in formula (28), and the dam height and water depth are h and H, respectively, the distance from the target point of the radial displacement to be solved to the dam top is d, and the bending moment term and the shear force term in formula (28) are solved, respectively;

[0096] Step S221, solving the bending moment term;

[0097] According to the basic principle of the unit force method, the unit force is applied to the observation point as the unit force state, and the actual stress state is the displacement state, and the bending moment term in formula (28) is solved;

[0098] The bending moment size of the unit force state cantilever beam M1(x) is:

[0099] M1(x) = h-d-x = z-x (30)

[0100] In the above formula, z is the elevation of the measuring point;

[0101] The bending moment size of the displacement state cantilever beam M pt (x) is:

[0102]

[0103] In the above formula, b is the width of the cantilever beam;

[0104] The moment of inertia I t (x) of the cantilever beam is:

[0105]

[0106] Substitute formula (30)-formula (32) into the expression of the bending moment term in formula (29) to obtain:

[0107]

[0108] Step S222, solving the shear force term;

[0109] The shear force size of the unit force state cantilever beam is Q1(x) = 1;

[0110] The magnitude of the shear force of the cantilever beam under displacement state is:

[0111]

[0112] The cross-sectional area of ​​the cantilever beam is A(x) = bT;

[0113] Q1(x) and Q p Substituting (x) and A(x) into the expression for the shear term in formula (29), we get:

[0114]

[0115] Step S223, the bending moment term combining steps S221 and S222. and shear term The calculation formula yields the displacement expression for a cantilever beam under a uniformly distributed load p:

[0116]

[0117] Step S23: Calculate the deflection of the cantilever beam under the action of an equivalent linear temperature difference;

[0118] The effect of equivalent linear temperature difference on the deflection of a cantilever beam is similarly considered as deformation under a certain bending moment. For the rectangular bending beam, the approach used in the arch ring calculation is the same, considering the temperature change along the upstream face of the arch ring as (t...). m -t d The downstream surface temperature change is (t) m +t d Temperature gradient 2t d / T is equivalent to the bending moment M generated in each section of the arch ring, then:

[0119]

[0120] The unit force method is still used to solve for the radial displacement of the cantilever beam under bending moment M. Obtain the deflection of the cantilever beam caused by the equivalent linear temperature difference. The expression is:

[0121]

[0122] Step S24: Calculate the deflection of the cantilever beam under temperature load;

[0123] Combining the deflection of the cantilever beam under uniform temperature and equivalent linear temperature difference obtained in steps S22 and S23, the radial deformation δ of the cantilever beam under temperature load is obtained. T for:

[0124]

[0125] Specifically, the process of constructing the arch dam deformation temperature component model in step S2 is as follows:

[0126] Step S25: Setting the periodicity term for temperature load;

[0127] Throughout the entire life cycle of an arch dam, the temperature load it experiences is complex and periodically changing. Therefore, it is necessary to consider the periodicity of the temperature load in the deformation of the arch dam. Based on previous research experience, the periodic term of the deformation effect of an arch dam under temperature load is... Represented as:

[0128]

[0129] In the above formula, B 1i B 2i t represents the regression coefficient; t is the cumulative time from the start of data collection to the observation date; B0 is the constant term.

[0130] Step S26: Construct a model of the deformation temperature components of the arch dam;

[0131] Considering the uniform temperature and equivalent linear temperature difference effects of the temperature load on the arch dam, and combining the periodic characteristics of the temperature load, the expression for the temperature component model of the arch dam deformation is constructed by multivariate power series expansion and merging like terms:

[0132]

[0133] In the above formula, B jkmn is the regression coefficient; H is the water depth in front of the dam; z is the elevation of the measuring point.

[0134] Specifically, the construction process of the arch dam deformation monitoring model in step S3 is as follows:

[0135] Step S31: Divide the displacement of the arch dam under the influence of environmental variables such as hydrostatic pressure and temperature load into three components: water pressure component δ H Temperature component δ T and the time-dependent component δ θ Then we have:

[0136] δ=δ H +δ T +δ θ (42)

[0137] Step S32: Considering the deformation of the dam concrete and foundation rock mass under external loads, and the time-dependent components of joints and fissures in the foundation rock mass under water pressure, the time-dependent components of the arch dam deformation are characterized by a linear combination of linear and logarithmic functions. The mathematical expression is as follows:

[0138] δ θ =c1θ+c2lnθ (43)

[0139] In the above formula, c1 and c2 are regression coefficients, and the time factor θ = t / 100, where t is the cumulative time from the start of data collection to the observation date.

[0140] Step S33: Based on the obtained deformation relationship of the cantilever beam of the arch dam under water pressure and temperature load, and combined with formula (42), construct the arch dam deformation monitoring model. The mathematical expression is as follows:

[0141]

[0142] Compared with the prior art, the present invention has the following beneficial effects:

[0143] 1. This invention proposes to analyze the deformation law of arch dams using the principle of load distribution between arch beams and cantilever beams. By reasonably considering the load distribution and corresponding displacement characteristics of the arch ring and cantilever beam, the deformation law of the arch dam under reservoir water pressure and temperature-induced loads is analyzed. The water pressure load distributed by the arch beam is applied to the cantilever beam and the arch ring respectively. Considering the deformation of the cantilever beam and the dam foundation, the radial displacement of the arch dam from the perspective of the cantilever beam is derived. The pure arch method is used to calculate the radial displacement of the arch dam from the perspective of the arch ring, and then the mathematical expression of the radial deformation of the arch dam under the action of reservoir water pressure is derived. Based on this, a water pressure component model of arch dam deformation is established. Compared with the equivalent substitution of traditional statistical models, the method of this invention has a clear mechanical derivation process and is more mechanically interpretable.

[0144] 2. This invention decomposes the temperature load on an arch dam along its thickness into uniform temperature, equivalent linear temperature difference, and nonlinear temperature difference. Ignoring the nonlinear temperature difference that affects local deformation and stress on the dam surface, the deformation law of the arch dam under temperature load is analyzed. From the perspectives of the horizontal arch ring and the vertical cantilever beam, the radial displacement expression of the arch dam under uniform temperature and equivalent linear temperature difference is derived separately, establishing the functional relationship of the temperature components of the arch dam deformation. Based on this, a new mathematical and statistical model suitable for analyzing the deformation behavior of concrete arch dams is constructed. Compared with traditional statistical models that use harmonic functions for fitting, this invention, through derivation using mechanical methods, makes it more mechanically interpretable in the statistical analysis of arch dam deformation laws. Attached Figure Description

[0145] Figure 1 This is a flowchart of a modeling method for arch dam deformation monitoring according to the present invention;

[0146] Figure 2 This is a diagram showing the layout of measuring points at the Ertan arch dam in an example of the present invention;

[0147] Figure 3 These are the fitting curves of the TCN08 measuring point of the Ertan Arch Dam in the embodiments of the present invention under three working conditions of 190 / 200 / 210 days;

[0148] Figure 4These are the fitting curves of the TCN09 measuring point of the Ertan Arch Dam in the embodiments of the present invention under three working conditions of 190 / 200 / 210 days;

[0149] Figure 5 These are the fitting curves of the TCN10 measuring point of the Ertan Arch Dam under three working conditions of 190 / 200 / 210 days in the embodiment of the present invention;

[0150] Figure 6 These are the prediction curves of the three measuring points TCN08 / TCN09 / TCN10 of the Ertan Arch Dam in this embodiment of the invention under three working conditions of 190 / 200 / 210 days;

[0151] Figure 7 This is an evaluation of the prediction effect of three measuring points (TCN08 / TCN09 / TCN10) of the Ertan Arch Dam under three working conditions of 190 / 200 / 210 days in this embodiment of the invention. Detailed Implementation

[0152] To facilitate understanding and implementation of the present invention by those skilled in the art, the various steps of the method proposed in this invention are described in detail below. It should be understood that these embodiments are for illustrative purposes only and are not intended to limit the scope of the invention. Furthermore, it should be understood that after reading the teachings of this invention, those skilled in the art can make various modifications or alterations to the invention, and these equivalent forms also fall within the scope defined by the appended claims.

[0153] Example

[0154] like Figure 1 As shown, this invention discloses a modeling method for arch dam deformation monitoring, comprising the following steps:

[0155] Step S1: The arch dam space is considered to consist of a horizontal arch ring and a vertical cantilever beam. The water pressure load in front of the dam is distributed to the arch ring and the vertical cantilever beam respectively. The water pressure load on the vertical cantilever beam is obtained by fitting a quadratic curve, and the specific expression is as follows: a i The fitting coefficients are H, the water depth in front of the dam is i = 1, 2. Based on the vertical arrangement characteristics of the arch dam deformation monitoring facilities, the vertical cantilever beam is selected as the modeling research object. The deformation expression of the vertical cantilever beam under water pressure load is derived, thereby constructing the arch dam deformation water pressure component model.

[0156] The derivation of the deformation expression of a vertical cantilever beam under water pressure load considers the horizontal displacement of any point in the arch dam as consisting of the structural displacement of both the dam body and the dam foundation. The dam foundation displacement is further divided into angular displacement and normal displacement. Therefore, the radial displacement of the arch dam is divided into three parts: the displacement δ caused by the deformation of the dam body due to hydrostatic pressure. 1HThe displacement δ caused by the bending moment and shear force generated by the dam body under hydrostatic pressure being transmitted to the dam foundation surface and causing it to rotate. 2H And the displacement δ caused by the rotation of the dam foundation surface due to the pressure of the reservoir water on the dam body. 3H ;

[0157] Step S2: Decompose the temperature of the arch dam body into three parts: uniform temperature t m Equivalent linear temperature difference t d and nonlinear temperature difference t n The deformation effect of uniform temperature load on the vertical cantilever beam is regarded as the deflection generated by the vertical cantilever beam bearing a uniformly distributed load; the deformation of the vertical cantilever beam caused by the equivalent linear temperature difference is regarded as the effect of the bending moment generated by the vertical cantilever beam bearing a bending moment; since the influence of nonlinear temperature difference on the arch dam is mostly on the dam surface, the influence of nonlinear temperature difference on the arch dam deformation is ignored. The deformation expression of the arch dam under uniform temperature and equivalent linear temperature difference is derived, thereby constructing the temperature component model of arch dam deformation.

[0158] Step S3: Divide the displacement of the arch dam under the influence of environmental variables such as hydrostatic pressure and temperature load into three components: water pressure component δ H Temperature component δ T and the time-dependent component δ θ The water pressure and temperature components are calculated using the models established in steps S1 and S2, and the aging component is calculated using the Zebaer-Thomson model, thereby constructing an arch dam deformation monitoring model.

[0159] Step S4: Based on the arch dam deformation monitoring model constructed in step S3, establish a multiple linear regression model to predict the arch dam deformation value.

[0160] Specifically, the displacement δ caused by the deformation of the dam body due to the hydrostatic pressure in step S1. 1H The calculation process is as follows:

[0161] Step S11: Considering the effects of bending moment and shear force on the vertical cantilever beam under water pressure load, calculate the displacement δ caused by the deformation of the dam body due to hydrostatic pressure. 1H The calculation formula is:

[0162]

[0163] In the above formula, and These respectively represent the displacement caused by the deformation of the dam body due to the bending moment of the vertical cantilever beam under hydrostatic pressure, and the displacement caused by the deformation of the dam body due to the shear force of the vertical cantilever beam under hydrostatic pressure;

[0164]

[0165] In the above formula, M1 and V1 are the bending moment and shear force of the cantilever beam under unit force conditions, respectively; M p and V p These represent the bending moment and shear force of the cantilever beam under displacement conditions, respectively; E c G represents the elastic modulus of the dam concrete. c denoted as , where is the shear modulus of the dam concrete; I is the moment of inertia of the section; A is the cross-sectional area of ​​the cantilever beam; and k is the section shape factor, commonly used for rectangular sections, where k = 1.2.

[0166] Step S12: Calculate the displacement caused by the deformation of the dam body due to the bending moment of the vertical cantilever beam under hydrostatic pressure.

[0167] Step S121: Calculate the magnitude of the bending moment M1(x) of the cantilever beam under unit force:

[0168] M1(x)=hdx (3)

[0169] In the above formula, h is the height of the dam; d is the distance from the top of the dam to the observation point; x is an integral variable defined to calculate the bending moment at the observation point, and its physical meaning is the distance from the bottom of the dam to any stress analysis point;

[0170] Step S122: Calculate the magnitude M of the bending moment of the structure under displacement state. p (x):

[0171]

[0172] In the above formula, H is the water depth; γ0 is the unit weight of water; b is the cross-sectional width of the cantilever beam; and x1 is an integral variable defined to calculate the bending moment at position x, with the same physical meaning as x. p (x) In the calculation, x is regarded as a constant and x1 is regarded as a variable; a1, a2, and a3 are the regression coefficients of water pressure load;

[0173] To facilitate calculation, the expression f(x) related to x in formula (4) is separated and solved:

[0174]

[0175] Step S123: Calculate the moment of inertia I(x) of the cross section:

[0176]

[0177] In the above formula, T C T is the width of the cantilever beam on the dam crest. B The width of the cantilever beam at the bottom of the dam;

[0178] Step S124: Substituting formulas (3) and (4) into the moment term expression of formula (2), we get:

[0179]

[0180] To simplify the integration process of formula (7), let us substitute:

[0181]

[0182] After substitution, the integral expression of the bending moment term in formula (1) is as follows:

[0183]

[0184] In the above formula, κ is the substitution coefficient; A1, A2, A3, A4, A5, and A6 are the abbreviations of the corresponding expressions in formula (9), and their specific expressions are as follows:

[0185]

[0186]

[0187] Step S13: Calculate the shear force term of the cantilever beam deformation.

[0188] Step S131, the magnitude of the structural shear force under unit force state is:

[0189] V1(x)=1 (10)

[0190] Step S132, the formula for calculating the structural shear force in the displacement state is expressed as follows:

[0191]

[0192] To facilitate calculation, the expression related to x in formula (11) is separated and solved:

[0193]

[0194] Step S133, the formula for calculating the cross-sectional area is expressed as follows:

[0195]

[0196] Step S134: Substituting formulas (11)-(13) into the expression for the bending moment term in formula (2), we get:

[0197]

[0198] In the above formula, B1, B2, B3, and B4 are the abbreviations of the corresponding expressions in formula (14), and their specific expressions are as follows:

[0199]

[0200] B3 = -H 2 a2κ-2HT B a2κ 2 -Ha1κ-T B 2 a2κ 3 -T B a1κ 2 ;

[0201]

[0202] Step S14, the bending moment term of the vertical cantilever beam deformation combining steps S12 and S13. and shear term The calculation results show the displacement δ caused by the deformation of the dam body due to hydrostatic pressure. 1H :

[0203]

[0204] In step S1, the bending moment and shear force generated by the hydrostatic pressure on the dam body are transmitted to the dam foundation surface, causing it to rotate and resulting in displacement δ. 2H The calculation process is as follows:

[0205] Step S15: Calculate the dam foundation rotation displacement θ caused by the surface force system of the arch dam:

[0206] θ=Mα+Vα2 (16)

[0207] In the above formula, α is the angular deformation of the base surface due to the unit bending moment, α=α'sin 3 φ+δ'sinφcos 2 φ, α2 are the angular deformations of the base surface caused by the unit radial shear force, α2=α”sin 2 φ and α' represent the angular deformation in the normal plane of the foundation surface caused by a unit moment, α” represents the angular deformation caused by a unit radial shear force, and φ is the angle between the tangent and the vertical of the bank slope. Considering the influence of the bank slope, for the cantilever beam in the riverbed section... M is the bending moment on the foundation normal plane, and V is the shear force parallel to the foundation surface;

[0208]

[0209] Combining formulas (16) and (18), the dam foundation rotation θ caused by the surface force system of the arch dam is expressed as:

[0210]

[0211] In the above formula, μ is the Poisson's ratio of the dam foundation; Er The elastic modulus of the dam foundation;

[0212] Step S16: Calculate the displacement δ caused by the bending moment and shear force generated by the dam body under hydrostatic pressure being transmitted to the dam foundation surface and causing it to rotate. 2H ;

[0213] Since the deformation of the dam foundation surface is a small deformation of a linear elastic structure, θ≈tanθ is used to calculate the displacement δ caused by the transmission of bending moment and shear force generated by the hydrostatic pressure to the dam foundation surface and causing it to rotate. 2H have:

[0214] δ 2H =(hd)tanθ=θ(hd) (20).

[0215] Specifically, in step S1, the displacement δ caused by the rotation of the dam foundation surface due to the reservoir water pressure on the dam body. 3H The calculation process is as follows:

[0216] Step S17: Based on the fundamental principle of Boussinesq solution, let the dam width be 2C and the uniformly distributed load length be b-x0. Then, the foundation settlement w(x) of the dam body at position x is expressed as:

[0217]

[0218] In the above formula, C is half the width of the reservoir; p is the hydrostatic pressure, p = γ0H; b is the coordinate of the inflection point between the flat slope and the slope change at the bottom of the reservoir; and x0 is the distance from the centroid of the dam to the upstream dam face.

[0219] Since θ≈tanθ in small deformations of a linear elastic structure, the dam foundation rotation angle θ(x0) at the position x=x0 can be expressed as:

[0220]

[0221] In the above formula,

[0222] Furthermore, the displacement δ caused by the rotation of the dam foundation surface due to the pressure of the reservoir water on the dam body. 3H Represented as:

[0223]

[0224] Specifically, the process of constructing the deformation hydraulic pressure component model of the arch dam in step S1 is as follows:

[0225] Through three-part radial displacement analysis from the perspective of the cantilever beam of the arch dam, it is found that the deformation of the arch dam under reservoir water pressure is linearly related to the fourth power of the water depth H in front of the dam. Therefore, the water pressure component model of the arch dam deformation is δ H The expression is:

[0226]

[0227] In the above formula, a i The fitting coefficients are i = 1, 2, 3, 4; H is the water depth in front of the dam.

[0228] Specifically, the derivation process of the expression for the arch dam deformation under uniform temperature and equivalent linear temperature difference in step S2 is as follows:

[0229] Step S21: Calculate the deformation under uniform temperature load;

[0230] A uniform temperature load t m The deformation effect of the cantilever beam is considered as the deflection caused by the cantilever beam bearing a uniformly distributed load. Combining the consistency condition of displacement at the conjugate points of the arch beam, the uniformly distributed load on the cantilever beam is adopted as the uniformly distributed load of the arch ring. The expression for the uniformly distributed load of the arch ring is then analyzed and determined:

[0231] Considering the similarity between the stress conditions of a uniform cross-section arch ring under uniform temperature and under uniformly distributed water pressure, let the outer radius of the uniform cross-section arch ring be R, the average radius be r, and the thickness of the arch ring be T. Then, the radial displacement Δr' and axial force N generated under the uniformly distributed load p are expressed as:

[0232]

[0233] In the above formula, E is the elastic modulus of the dam concrete; N, M, and V are the axial force, bending moment, and shear force acting on the dam body, respectively.

[0234] If a uniform temperature change occurs within the arch ring, then the radial displacement Δr' and axial force N of the arch ring are expressed as:

[0235]

[0236] In the above formula, α is the coefficient of linear expansion of concrete;

[0237] If we assume that the radial displacement Δr' of the arch ring is equal under both types of loads, then we have:

[0238]

[0239] Based on the obtained expression for the uniformly distributed load p, the deformation of the cantilever beam under the uniformly distributed load p is further derived.

[0240]

[0241] In the above formula, and These represent the bending moment and shear force terms, respectively, that cause the dam body to deform under a uniformly distributed load p on a vertical cantilever beam.

[0242]

[0243] In the above formula, M1 and V1 are the structural bending moment and structural shear force under unit force conditions, respectively; M p and V p These represent the structural bending moment and structural shear force under displacement conditions, respectively; E c G represents the elastic modulus of the dam concrete. c denoted as , where is the shear modulus of the dam concrete; I is the moment of inertia of the section; A is the cross-sectional area of ​​the cantilever beam; and k is the section shape factor, commonly used for rectangular sections, where k = 1.2.

[0244] Step S22: Calculate the radial deformation of the cantilever beam under uniform temperature.

[0245] To calculate the radial deformation of the cantilever beam under uniformly distributed load p in formula (28), the dam height and water depth are set as h and H respectively, and the distance from the target point of the radial displacement to be solved to the top of the dam is d. The bending moment term and shear force term in formula (28) are solved respectively.

[0246] Step S221: Solve for the bending moment term;

[0247] Based on the basic principle of the unit force method, the bending moment term in formula (28) is solved by taking the unit force acting on the observation point as the unit force state and the actual force state as the displacement state.

[0248] The magnitude of the bending moment M1(x) of the cantilever beam under unit force:

[0249] M1(x)=hdx=zx (30)

[0250] In the above formula, z is the elevation of the measuring point;

[0251] Magnitude M of bending moment of cantilever beam under displacement state pt (x):

[0252]

[0253] In the above formula, b is the width of the cantilever beam;

[0254] Moment of inertia of cantilever beam I t (x):

[0255]

[0256] Substituting equations (30)-(32) into the expression for the bending moment term in equation (29), we get:

[0257]

[0258] Step S222: Solve for the shear force term;

[0259] The magnitude of the shear force of the cantilever beam under unit force conditions is Q1(x) = 1;

[0260] The magnitude of the shear force of the cantilever beam under displacement state is:

[0261]

[0262] The cross-sectional area of ​​the cantilever beam is A(x) = bT;

[0263] Q1(x) and Q p Substituting (x) and A(x) into the expression for the shear term in formula (29), we get:

[0264]

[0265] Step S223, the bending moment term combining steps S221 and S222. and shear term The calculation formula yields the displacement expression for a cantilever beam under a uniformly distributed load p:

[0266]

[0267] Step S23: Calculate the deflection of the cantilever beam under the action of an equivalent linear temperature difference;

[0268] The effect of equivalent linear temperature difference on the deflection of a cantilever beam is similarly considered as deformation under a certain bending moment. For the rectangular bending beam, the approach used in the arch ring calculation is the same, considering the temperature change along the upstream face of the arch ring as (t...). m -t d The downstream surface temperature change is (t) m +t d Temperature gradient 2t d / T is equivalent to the bending moment M generated in each section of the arch ring, then:

[0269]

[0270] The unit force method is still used to solve for the radial displacement of the cantilever beam under bending moment M. Obtain the deflection of the cantilever beam caused by the equivalent linear temperature difference. The expression is:

[0271]

[0272] Step S24: Calculate the deflection of the cantilever beam under temperature load;

[0273] Combining the deflection of the cantilever beam under uniform temperature and equivalent linear temperature difference obtained in steps S22 and S23, the radial deformation δ of the cantilever beam under temperature load is obtained. T for:

[0274]

[0275] Step S25: Setting the periodicity term for temperature load;

[0276] Throughout the entire life cycle of an arch dam, the temperature load it experiences is complex and periodically changing. Therefore, it is necessary to consider the periodicity of the temperature load in the deformation of the arch dam. Based on previous research experience, the periodic term of the deformation effect of an arch dam under temperature load is... Represented as:

[0277]

[0278] In the above formula, B 1i B 2i t represents the regression coefficient; t is the cumulative time from the start of data collection to the observation date; B0 is the constant term.

[0279] Step S26: Construct a model of the deformation temperature components of the arch dam;

[0280] Considering the uniform temperature and equivalent linear temperature difference effects of the temperature load on the arch dam, and combining the periodic characteristics of the temperature load, the expression for the temperature component model of the arch dam deformation is constructed by multivariate power series expansion and merging like terms:

[0281]

[0282] In the above formula, B jkmn is the regression coefficient; H is the water depth in front of the dam; z is the elevation of the measuring point.

[0283] Step S31: Divide the displacement of the arch dam under the influence of environmental variables such as hydrostatic pressure and temperature load into three components: water pressure component δ H Temperature component δ T and the time-dependent component δ θ Then we have:

[0284] δ=δ H +δ T +δ θ (42)

[0285] Step S32: Considering the deformation of the dam concrete and foundation rock mass under external loads, and the time-dependent components of joints and fissures in the foundation rock mass under water pressure, the time-dependent components of the arch dam deformation are characterized by a linear combination of linear and logarithmic functions. The mathematical expression is as follows:

[0286] δ θ =c1θ+c2lnθ (43)

[0287] In the above formula, c1 and c2 are regression coefficients, and the time factor θ = t / 100, where t is the cumulative time from the start of data collection to the observation date.

[0288] Step S33: Based on the obtained deformation relationship of the cantilever beam of the arch dam under water pressure and temperature load, and combined with formula (42), construct the arch dam deformation monitoring model. The mathematical expression is as follows:

[0289]

[0290] The following example uses the deformation monitoring of the Ertan arch dam. Figure 2 The diagram shown is a layout of the measuring points for the Ertan arch dam. Figure 2 TCN01-TCN20 are radial deformation measuring points of the arch dam, and EX1-EX7 are horizontal deformation measuring points of the arch dam. These points further verify the effectiveness of the arch dam deformation monitoring model established according to the method of this invention. Figures 3-5 As shown, the deformation fitting curves of the arch dam corresponding to three measuring points under three working conditions using the method of the present invention are in high agreement with the measured values, and Figure 6 The predicted curves closely match the measured values, demonstrating the effectiveness of the method described in this invention. Furthermore, to quantitatively analyze the scientific effectiveness of the method, statistical analysis of the prediction performance indicators was conducted for three working conditions at three measurement points. Figure 7 As can be seen from the radar chart, the R2 / MAE / MAPE / RMESE indicators of the method of the present invention are all better than those of the traditional statistical model, which shows that the prediction effect of the method of the present invention is better than that of the traditional statistical model.

[0291] The above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention in any other way. Any person skilled in the art may make changes or modifications to the above-disclosed technical content to create equivalent embodiments. However, any simple modifications, equivalent changes, and modifications made to the above embodiments based on the technical essence of the present invention without departing from the scope of the present invention shall still fall within the protection scope of the present invention.

Claims

1. A modeling method for arch dam deformation monitoring, characterized in that, Includes the following steps: Step S1: The arch dam space is considered to consist of a horizontal arch ring and a vertical cantilever beam. The water pressure load in front of the dam is distributed to the arch ring and the vertical cantilever beam respectively. The water pressure load on the vertical cantilever beam is obtained by fitting a quadratic curve, and the expression is: a i The fitting coefficients are H, the water depth in front of the dam is i = 1, 2. Based on the vertical arrangement characteristics of the arch dam deformation monitoring facilities, the vertical cantilever beam is selected as the modeling research object. The deformation expression of the vertical cantilever beam under water pressure load is derived, thereby constructing the arch dam deformation water pressure component model. The derivation of the deformation expression of a vertical cantilever beam under water pressure load considers the horizontal displacement of any point in the arch dam as consisting of the structural displacement of both the dam body and the dam foundation. The dam foundation displacement is further divided into angular displacement and normal displacement. Therefore, the radial displacement of the arch dam is divided into three parts: the displacement δ caused by the deformation of the dam body due to hydrostatic pressure. 1H The displacement δ caused by the bending moment and shear force generated by the dam body under hydrostatic pressure being transmitted to the dam foundation surface and causing it to rotate. 2H And the displacement δ caused by the rotation of the dam foundation surface due to the pressure of the reservoir water on the dam body. 3H ; Step S2: Decompose the temperature of the arch dam body into three parts: uniform temperature t m Equivalent linear temperature difference t d and nonlinear temperature difference t n The deformation effect of uniform temperature load on the vertical cantilever beam is regarded as the deflection generated by the vertical cantilever beam bearing a uniformly distributed load; the deformation of the vertical cantilever beam caused by the equivalent linear temperature difference is regarded as the effect of the bending moment generated by the vertical cantilever beam bearing a bending moment; since the influence of nonlinear temperature difference on the arch dam is mostly on the dam surface, the influence of nonlinear temperature difference on the arch dam deformation is ignored. The deformation expression of the arch dam under uniform temperature and equivalent linear temperature difference is derived, thereby constructing the temperature component model of arch dam deformation. Step S3: Divide the displacement of the arch dam under the influence of environmental variables such as hydrostatic pressure and temperature load into three components: water pressure component δ H Temperature component δ T and the time-dependent component δ θ The water pressure and temperature components are calculated using the models established in steps S1 and S2, and the aging component is calculated using the Zebaer-Thomson model, thereby constructing an arch dam deformation monitoring model. Step S4: Based on the arch dam deformation monitoring model constructed in step S3, establish a multiple linear regression model to predict the arch dam deformation value.

2. The modeling method for arch dam deformation monitoring according to claim 1, characterized in that, The displacement δ caused by the hydrostatic pressure on the dam body during step S1 1H The calculation process is as follows: Step S11: Considering the effects of bending moment and shear force on the vertical cantilever beam under water pressure load, calculate the displacement δ caused by the deformation of the dam body due to hydrostatic pressure. 1H The calculation formula is: In the above formula, and These respectively represent the displacement caused by the deformation of the dam body due to the bending moment of the vertical cantilever beam under hydrostatic pressure, and the displacement caused by the deformation of the dam body due to the shear force of the vertical cantilever beam under hydrostatic pressure; In the above formula, M1 and V1 are the bending moment and shear force of the cantilever beam under unit force conditions, respectively; M p and V p These represent the bending moment and shear force of the cantilever beam under displacement conditions, respectively; E c G represents the elastic modulus of the dam concrete. c denoted as , where is the shear modulus of the dam concrete; I is the moment of inertia of the section; A is the cross-sectional area of ​​the cantilever beam; and k is the section shape factor, commonly used for rectangular sections, where k = 1.

2. Step S12: Calculate the displacement caused by the deformation of the dam body due to the bending moment of the vertical cantilever beam under hydrostatic pressure. Step S121: Calculate the magnitude of the bending moment M1(x) of the cantilever beam under unit force: M1(x)=hdx (3) In the above formula, h is the height of the dam; d is the distance from the top of the dam to the observation point; x is an integral variable defined to calculate the bending moment at the observation point, and its physical meaning is the distance from the bottom of the dam to any stress analysis point; Step S122: Calculate the magnitude M of the bending moment of the structure under displacement state. p (x): In the above formula, H is the water depth; γ0 is the unit weight of water; b is the cross-sectional width of the cantilever beam; and x1 is an integral variable defined to calculate the bending moment at position x, with the same physical meaning as x. p (x) In the calculation, x is treated as a constant and x1 is treated as a variable; a1 and a2 are the regression coefficients of the water pressure load; To facilitate calculation, the expression f(x) related to x in formula (4) is separated and solved: Step S123: Calculate the moment of inertia I(x) of the cross section: In the above formula, T C T is the width of the cantilever beam on the dam crest. B The width of the cantilever beam at the bottom of the dam; Step S124: Substituting formulas (3) and (4) into the moment term expression of formula (2), we get: To simplify the integration process of formula (7), let us substitute: After substitution, the integral expression of the bending moment term in formula (1) is as follows: In the above formula, κ is the substitution coefficient; A1, A2, A3, A4, A5, and A6 are the abbreviations of the corresponding expressions in formula (9), and their specific expressions are as follows: Step S13: Calculate the shear force term of the cantilever beam deformation. Step S131, the magnitude of the structural shear force under unit force state is: V1(x)=1 (10) Step S132, the formula for calculating the structural shear force in the displacement state is expressed as follows: To facilitate calculation, the expression related to x in formula (11) is separated and solved: Step S133, the formula for calculating the cross-sectional area is expressed as follows: Step S134: Substituting formulas (11)-(13) into the expression for the bending moment term in formula (2), we get: In the above formula, B1, B2, B3, and B4 are the abbreviations of the corresponding expressions in formula (14), and their specific expressions are as follows: B3=-H 2 a2κ-2HT B a2κ 2 -Ha1κ-T B 2 a2κ 3 -T B a1κ 2 ; Step S14, the bending moment term of the vertical cantilever beam deformation combining steps S12 and S13. and shear term The calculation results show the displacement δ caused by the deformation of the dam body due to hydrostatic pressure. 1H :

3. The modeling method for arch dam deformation monitoring according to claim 2, characterized in that, In step S1, the bending moment and shear force generated by the hydrostatic pressure on the dam body are transmitted to the dam foundation surface, causing it to rotate and resulting in displacement δ. 2H The calculation process is as follows: Step S15: Calculate the dam foundation rotation displacement θ caused by the surface force system of the arch dam: θ=Mα+Vα2(16) In the above formula, α is the angular deformation of the base surface due to the unit bending moment, α=α'sin 3 φ+δ'sinφcos 2 φ, α2 are the angular deformations of the base surface caused by the unit radial shear force, α2=α”sin 2 φ and α' represent the angular deformation in the normal plane of the foundation surface caused by a unit moment, α” represents the angular deformation caused by a unit radial shear force, and φ is the angle between the tangent and the vertical of the bank slope. Considering the influence of the bank slope, for the cantilever beam in the riverbed section... M is the bending moment on the foundation normal plane, and V is the shear force parallel to the foundation surface; Combining formulas (16) and (18), the dam foundation rotation θ caused by the surface force system of the arch dam is expressed as: In the above formula, μ is the Poisson's ratio of the dam foundation; E r The elastic modulus of the dam foundation; Step S16: Calculate the displacement δ caused by the bending moment and shear force generated by the dam body under hydrostatic pressure being transmitted to the dam foundation surface and causing it to rotate. 2H ; Since the deformation of the dam foundation surface is a small deformation of a linear elastic structure, θ≈tanθ is used to calculate the displacement δ caused by the transmission of bending moment and shear force generated by the hydrostatic pressure to the dam foundation surface and causing it to rotate. 2H have: δ 2H =(h-d)tanθ=θ(h-d) (20)。 4. The modeling method for arch dam deformation monitoring according to claim 3, characterized in that, In step S1, the displacement δ caused by the rotation of the dam foundation surface due to the pressure of the reservoir water on the dam body. 3H The calculation process is as follows: Step S17: Based on the fundamental principle of Boussinesq solution, let the dam width be 2C and the uniformly distributed load length be b-x0. Then, the foundation settlement w(x) of the dam body at position x is expressed as: In the above formula, C is half the width of the reservoir; p is the hydrostatic pressure, p = γ0H; b is the coordinate of the inflection point between the flat slope and the slope change at the bottom of the reservoir; and x0 is the distance from the centroid of the dam to the upstream dam face. Since θ≈tanθ in small deformations of a linear elastic structure, the dam foundation rotation angle θ(x0) at the position x=x0 can be expressed as: In the above formula, Furthermore, the displacement δ caused by the rotation of the dam foundation surface due to the pressure of the reservoir water on the dam body. 3H Represented as:

5. The modeling method for arch dam deformation monitoring according to claim 4, characterized in that, The process of constructing the deformation hydraulic pressure component model of the arch dam in step S1 is as follows: Through three-part radial displacement analysis from the perspective of the cantilever beam of the arch dam, it is found that the deformation of the arch dam under reservoir water pressure is linearly related to the fourth power of the water depth H in front of the dam. Therefore, the water pressure component model of the arch dam deformation is δ H The expression is: In the above formula, a i is the fitting coefficient, i = 1, 2, 3, 4; H is the water depth.

6. The modeling method for arch dam deformation monitoring according to claim 1, characterized in that, The derivation process of the expression for the arch dam deformation under uniform temperature and equivalent linear temperature difference in step S2 is as follows: Step S21: Calculate the deformation under uniform temperature load; A uniform temperature load t m The deformation effect of the cantilever beam is considered as the deflection caused by the cantilever beam bearing a uniformly distributed load. Combining the consistency condition of displacement at the conjugate points of the arch beam, the uniformly distributed load on the cantilever beam is adopted as the uniformly distributed load of the arch ring. The expression for the uniformly distributed load of the arch ring is then analyzed and determined: Considering the similarity between the stress conditions of a uniform cross-section arch ring under uniform temperature and under uniformly distributed water pressure, let the outer radius of the uniform cross-section arch ring be R, the average radius be r, and the thickness of the arch ring be T. Then, the radial displacement Δr' and axial force N generated under the uniformly distributed load p are expressed as: In the above formula, E c denoted as the elastic modulus of the dam concrete; N, M, and V represent the axial force, bending moment, and shear force acting on the dam body, respectively. If a uniform temperature change occurs within the arch ring, then the radial displacement Δr' and axial force N of the arch ring are expressed as: In the above formula, α is the coefficient of linear expansion of concrete; If we assume that the radial displacement Δr' of the arch ring is equal under both types of loads, then we have: Based on the obtained expression for the uniformly distributed load p, the deformation of the cantilever beam under the uniformly distributed load p is further derived. In the above formula, and These represent the bending moment and shear force terms, respectively, that cause the dam body to deform under a uniformly distributed load p on a vertical cantilever beam. In the above formula, M1 and V1 are the structural bending moment and structural shear force under unit force conditions, respectively; M p and V p These represent the structural bending moment and structural shear force under displacement conditions, respectively; E c G represents the elastic modulus of the dam concrete. c denoted as , where is the shear modulus of the dam concrete; I is the moment of inertia of the section; A is the cross-sectional area of ​​the cantilever beam; and k is the section shape factor, commonly used for rectangular sections, where k = 1.

2. Step S22: Calculate the radial deformation of the cantilever beam under uniform temperature. To calculate the radial deformation of the cantilever beam under uniformly distributed load p in formula (28), the dam height and water depth are set as h and H respectively, and the distance from the target point of the radial displacement to be solved to the top of the dam is d. The bending moment term and shear force term in formula (28) are solved respectively. Step S221: Solve for the bending moment term; Based on the basic principle of the unit force method, the bending moment term in formula (28) is solved by taking the unit force acting on the observation point as the unit force state and the actual force state as the displacement state. The magnitude of the bending moment M1(x) of the cantilever beam under unit force: M1(x)=hdx=zx (30) In the above formula, z is the elevation of the measuring point; Magnitude M of bending moment of cantilever beam under displacement state pt (x): In the above formula, b is the width of the cantilever beam; Moment of inertia of cantilever beam I t (x): Substituting equations (30)-(32) into the expression for the bending moment term in equation (29), we get: Step S222: Solve for the shear force term; The magnitude of the shear force of the cantilever beam under unit force conditions is Q1(x) = 1; The magnitude of the shear force of the cantilever beam under displacement state is: The cross-sectional area of ​​the cantilever beam is A(x) = bT; Q1(x) and Q p Substituting (x) and A(x) into the expression for the shear term in formula (29), we get: Step S223, the bending moment term combining steps S221 and S222. and shear term The calculation formula yields the displacement expression for a cantilever beam under a uniformly distributed load p: Step S23: Calculate the deflection of the cantilever beam under the action of an equivalent linear temperature difference; The effect of equivalent linear temperature difference on the deflection of a cantilever beam is similarly considered as deformation under a certain bending moment. For the rectangular bending beam, the approach used in the arch ring calculation is the same, considering the temperature change along the upstream face of the arch ring as (t...). m -t d The downstream surface temperature change is (t) m +t d Temperature gradient 2t d / T is equivalent to the bending moment M generated in each section of the arch ring, then: The unit force method is still used to solve for the radial displacement of the cantilever beam under bending moment M. Obtain the deflection of the cantilever beam caused by the equivalent linear temperature difference. The expression is: Step S24: Calculate the deflection of the cantilever beam under temperature load; Combining the deflection of the cantilever beam under uniform temperature and equivalent linear temperature difference obtained in steps S22 and S23, the radial deformation δ of the cantilever beam under temperature load is obtained. T for: 。 7. The modeling method for arch dam deformation monitoring according to claim 6, characterized in that, The process of constructing the arch dam deformation temperature component model in step S2 is as follows: Step S25: Setting the periodicity term for temperature load; Throughout the entire life cycle of an arch dam, the temperature load it experiences is complex and periodically changing. Therefore, it is necessary to consider the periodicity of the temperature load in the deformation of the arch dam. Based on previous research experience, the periodic term of the deformation effect of an arch dam under temperature load is... Represented as: In the above formula, B 1i B 2i t represents the regression coefficient; t is the cumulative time from the start of data collection to the observation date; B0 is the constant term. Step S26: Construct a model of the deformation temperature components of the arch dam; Considering the uniform temperature and equivalent linear temperature difference effects of the temperature load on the arch dam, and combining the periodic characteristics of the temperature load, the expression for the temperature component model of the arch dam deformation is constructed by multivariate power series expansion and merging like terms: In the above formula, B jkmn is the regression coefficient; H is the water depth in front of the dam; z is the elevation of the measuring point.

8. The modeling method for arch dam deformation monitoring according to claim 7, characterized in that, The construction process of the arch dam deformation monitoring model in step S3 is as follows: Step S31: Divide the displacement of the arch dam under the influence of environmental variables such as hydrostatic pressure and temperature load into three components: water pressure component δ H Temperature component δ T and the time-dependent component δ θ Then we have: d=d H +d T +d θ (42) Step S32: Considering the deformation of the dam concrete and foundation rock mass under external loads, and the time-dependent components of joints and fissures in the foundation rock mass under water pressure, the time-dependent components of the arch dam deformation are characterized by a linear combination of linear and logarithmic functions. The mathematical expression is as follows: d θ =c1θ+c2 lnθ (43) In the above formula, c1 and c2 are regression coefficients, and the time factor θ = t / 100, where t is the cumulative time from the start of data collection to the observation date. Step S33: Based on the obtained deformation relationship of the cantilever beam of the arch dam under water pressure and temperature load, and combined with formula (42), construct the arch dam deformation monitoring model. The mathematical expression is as follows:

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