A method for calculating the hydrodynamic pressure of a variable cross-section liquid storage tank based on modal synthesis method

Abstract

The present application belongs to the field of structural dynamic response analysis and monitoring of civil and hydraulic engineering, and provides a variable cross-section liquid storage tank hydrodynamic pressure calculation method based on modal synthesis method. From the perspective of seismic analysis of building structures, only when the mass does not change with the change of excitation acceleration, the distribution of hydrodynamic pressure can present a stable decoupling form. The direct method of decoupling is to extract dynamic characteristics through modal analysis, and to obtain the decoupling form of mass by using generalized single degree of freedom system, and to simplify the decoupling form of mass into a fast evaluation method of hydrodynamic pressure peak value based on the characteristics of analysis domain. The structural modal can reflect the sloshing characteristics of the liquid storage tank, which has practical engineering significance for quickly evaluating the vulnerable areas of the structure. The method of the present application starts from the physical meaning of the finite element expression formula, obtains the relationship between equivalent mass and equivalent area moment, obtains the equivalent hydrodynamic pressure, is suitable for nonlinear horizontal cross-section liquid storage structure, and has important practical value.

Description

Technical Field

[0001] This invention relates to the field of structural dynamic response analysis and monitoring in civil engineering and hydraulic engineering, and in particular to a method for calculating the dynamic water pressure of a variable cross-section liquid storage tank based on the modal synthesis method. Background Technology

[0002] In the field of civil and hydraulic structural design, especially in the seismic analysis of liquid storage tanks, aqueducts, reservoirs, dams, etc., the influence of liquid-solid coupling is a necessary issue to be considered in structural design. The US nuclear power seismic code ASCE4-16 states that the additional effects of liquid in vertical liquid storage tanks on the structure under strong earthquakes should be considered in nuclear power plant design. Under dynamic loads, the liquid also generates hydrodynamic pressure on the vessel walls in addition to hydrostatic pressure. This hydrodynamic pressure varies with time and space within a finite domain, and its expression is quite complex.

[0003] There are three main methods for calculating the hydrodynamic pressure of liquid storage tanks in engineering practice: one is based on experimental research on a series of sloshing characteristics of liquid storage tanks; the second is based on high-precision simulation methods such as the CAS method to isolate the interaction force between the liquid and the tank wall; and the third is a lumped parameter model that characterizes the fluid dynamics with added mass. Currently, the simplified model method commonly used in civil engineering and hydraulic engineering is the Housner added mass model based on the US nuclear power seismic designation ASCE4-16.

[0004] Methods for experimentally studying the sloshing characteristics of liquid storage tanks present certain challenges and lack repeatability. Furthermore, experiments are significantly affected by environmental conditions and equipment status, leading to variations and uncertainties in the results. The timing, location, and intensity of earthquakes are difficult to predict and reproduce. Even under identical seismic conditions, the sloshing patterns of liquid storage tanks are influenced by connecting and transmission devices, resulting in non-repeatability.

[0005] Methods based on high-precision simulation of stripping interaction forces, such as the CAS method, can provide more detailed results. However, the modeling process is complex, requiring consideration of the interaction between the liquid and the vessel wall, which increases the complexity and computational difficulty of the model. Secondly, the analysis process of this method is closely coupled to the input seismic motion, making it unsuitable for engineering calculations under complex conditions.

[0006] The lumped-parameter model, which characterizes fluid dynamics by adding mass, only yields a stable analytical solution for the Housner model at regular cross-sections under rigid walls. Furthermore, the lumped-mass model can only reflect overall effects and has significant limitations in local stress analysis. This simplification cannot accurately reflect the response at individual points on the liquid-solid boundary.

[0007] Under dynamic loads, the hydrodynamic pressure in a liquid storage container needs to be determined by comprehensively considering factors such as the liquid's kinematic characteristics and the container wall geometry. Hydrodynamic pressure can be divided into pulse pressure and relative flow pressure, each of which can be described by the product of mass and acceleration. If the mass and excitation acceleration are not decoupled, the coupled mass will change with the seismic motion time history at each time step, leading to a huge computational burden for each analysis step, which is impractical in engineering.

[0008] Currently, there is a lack of a rapid method for assessing the peak hydrodynamic pressure of nonlinear horizontal cross-section liquid storage structures. From the perspective of seismic analysis of building structures, the distribution of hydrodynamic pressure can only exhibit a stable decoupled form when the mass does not change with the excitation acceleration. A direct method of decoupling is to extract dynamic characteristics through modal analysis and derive the mass decoupling equation using a generalized single-degree-of-freedom system. This mass decoupling equation is then simplified into a rapid assessment method for the peak hydrodynamic pressure based on the characteristics of the analysis domain. Structural modes can reflect the sloshing characteristics of liquid storage tanks, which has practical engineering significance for rapidly assessing vulnerable areas of the structure. Therefore, exploring and developing a rapid assessment method for the peak hydrodynamic pressure of nonlinear horizontal cross-section liquid storage structures is of significant research value. Summary of the Invention

[0009] To address the problems existing in the prior art, this invention provides a method for calculating the dynamic water pressure of variable cross-section liquid storage tanks based on modal synthesis, thereby greatly improving the speed of estimating the dynamic water pressure at variable cross-section nodes, which is of significance for preliminary dynamic seismic analysis.

[0010] To achieve the above objectives, the technical solution of the present invention is as follows: a method for calculating the dynamic water pressure of a variable cross-section liquid storage tank based on the modal synthesis method, comprising the following steps;

[0011] S1: Based on finite element discretization, obtain the equivalent mass borne by each discrete element of the variable cross-section liquid storage tank. Expression, dynamic water pressure Expression and hydrodynamic pressure expression;

[0012] S2: Based on equivalent mass Expression, dynamic water pressure Expression and hydrodynamic pressure The expression yields the area and the integration variable r. 2 The relationship between them;

[0013] S3: Based on the principle of variable cross-section equivalence, obtain the first-order and second-order equivalent area moments of the variable cross-section;

[0014] S4: Based on the parameter relationships obtained in S2, substitute them into the equivalent area moment of the variable cross-section obtained in S3 to obtain the equivalent pulse quality on each discrete element. The equivalent hydrodynamic pressure F is finally obtained. U ;

[0015] based on The correction coefficient proposed in the simplified formula is then multiplied by the formula based on... The expression yields the node's equivalent force along the radius outside the normal. The components on the surface are used to obtain the equivalent pulse mass of each discrete unit, and then the dynamic water pressure value is obtained.

[0016] The equivalent mass borne by each discrete unit of the variable cross-section liquid storage tank Expression, dynamic water pressure Expression and hydrodynamic pressure The specific expression is as follows:

[0017] Assume the wall of the variable cross-section liquid storage tank is rigid, and the liquid is incompressible, non-viscous, and non-rotational; in the Laplace equation with the velocity potential function Ф(x,y,z,t) as the variable and the boundary conditions of the liquid-solid interface and the free liquid surface, the normal velocity on the liquid-solid interface Ω1 remains consistent, and the pressure on the free liquid surface Ω2 is equal to zero.

[0018] The dynamic water pressure is the sum of the product of the convective mass and the liquid acceleration response, and the product of the impulse mass and the liquid acceleration response relative to the liquid tank; the liquid acceleration response is numerically equivalent to the seismic acceleration excitation.

[0019] The dynamic water pressure of the liquid relative to the liquid storage tank is specifically expressed as in equation (1);

[0020]

[0021] Among them, F U F represents the pulsed hydrodynamic pressure component representing the movement of the liquid together with the tank wall; S This represents the pressure component of the flowing water that is related to the free vibration of the liquid. This represents the pressure of the flowing water at each order in the modal decomposition expression; Represents the generalized coordinates of each order in the modal decomposition expression; This represents the relative acceleration of the liquid storage tank's swaying motion; M represents the acceleration of the liquid storage tank wall. I M represents the equivalent pulse quality on the discrete element. nc The equivalent convective mass of the nth mode;

[0022] From the structural dynamics equilibrium equations Substitution u nThis represents the relative displacement of the liquid relative to the sloshing motion of the liquid storage tank; from this, we can derive... ω represents the equivalent mass of the nth mode; ω represents the structural angular frequency. The equivalent hydrodynamic pressure of the nth mode;

[0023]

[0024]

[0025] Based on potential flow theory, the following equation is derived.

[0026]

[0027]

[0028]

[0029] M total Ψ represents the equivalent total mass of the liquid; g represents the acceleration due to gravity; n It is a variable cross-section characteristic function; Let ρ be the velocity vector of the particles on the wall of the liquid storage tank; ρ is the density of the liquid. This is the unit vector in the direction of the load. Let be the unit vector of the outer normal to the sidewall of the liquid-solid interface.

[0030] The area and integral variable r, integral variable r 2 The specific relationships between them are as follows;

[0031] The characteristic function Ψ corresponding to a certain variable cross section problem n (r,y,θ)=[N]u n Substituting cosθ into equations (4)-(6), we get r represents the radius in cylindrical coordinates; y represents the height coordinate in cylindrical coordinates; and θ represents the angle in cylindrical coordinates. The eigenvectors representing the characteristic equation;

[0032]

[0033] [N] is a shape function. The physical meaning of is the mass within the integration domain. This represents the area controlled by each finite element node in the Ω² domain, which satisfies the condition when ρ = 1. The form of formula (7); Let Ω be the first-order area moment within the two-dimensional area domain Ω2;

[0034]

[0035] In formula (8) For finite element node equivalent force, the finite element node equivalent force is numerically equal to the force acting on a unit area; the force acting on a unit area is a uniformly distributed load, then F q =PA i When P = 1, it satisfies The form of formula (8); It is the second-order area moment within the two-dimensional area domain Ω2;

[0036]

[0037] Equation (9) This is represented as the node's equivalent force along the outer radius normal of Ω1. The projection on; in equation (9) The physical meaning is half of the integral of the perimeter within the Ω1 domain; The integral of the hydrodynamic pressure on the inner wall element of the Ω1 domain.

[0038] In step S3, based on the principle of variable cross-section equivalence, the first-order and second-order equivalent area moments of the variable cross-section are obtained, as follows:

[0039] S31. For discontinuous variable cross-section liquid storage tanks, based on the area S or circumference L of any cross-sectional shape, obtain the equivalent radius r of the equivalent circle. Or r = L / 2π; calculate the first-order and second-order equivalent area moments directly based on the equivalent radius r;

[0040] S32. For continuously variable cross-section liquid storage tanks, according to the material mechanics formula for variable cross-section deflection... Solving for the equivalent radius of the equivalent circle yields M. p (x) represents the bending moment of the liquid storage tank under actual load; Let I(x) represent the bending moment of the liquid storage tank under unit load; H represent the height of each section of the liquid storage tank; I(x) represent the moment of inertia function; the variable cross-section liquid storage tank is divided into i equal-length sections, and the second-order area moments of the variable cross-section liquid storage tank of equal length are represented as I1, I2, I3, ..., I... i ;

[0041] A column with a constant cross-section exhibits the same deflection displacement as a column with a continuously variable cross-section under the same external conditions. Let Δ1 = Δ2, then the equivalent second-order area moment I of the continuously variable cross-section liquid section is obtained. eq . Will I eq Substituting into the formula for calculating the second-order area moment, we can obtain the equivalent radius r of the equivalent circle, and then directly calculate the first-order equivalent area moment based on the equivalent radius r.

[0042] f(x) = x 3Taking / I(x) as an example, according to Simpson's Rule, the variable cross-section column is divided into several segments of equal length, and the second-order area moments of the variable cross-section column of equal length are represented as I1, I2, I3, and I4.

[0043]

[0044]

[0045] Suppose a column with a uniform cross-section has the same deflection and displacement as a continuously variable cross-section column under the same external conditions. Let the second-order area moment of the equivalent column's cross-section be I. eq The formula for deflection of constant cross section in mechanics of materials is derived.

[0046] Let Δ1 = Δ2, then the equivalent second-order area moment of the continuously variable cross-section liquid section can be obtained.

[0047]

[0048] Will I eq Substituting into the second-order area moment formula makes The equivalent radius r of the equivalent circle can be obtained, and the first-order equivalent area moment can be directly calculated based on the equivalent radius r.

[0049] The equivalent quality of each discrete unit is as follows;

[0050]

[0051]

[0052] As a kind of based Additional mass correction factor, This represents the area controlled by each finite element node; the overall idea is to equate the variable cross section to a circular cross section, and to quickly evaluate the equivalent pulse quality on the element using the above simplified formula. Equivalent convective mass of the nth mode This is represented as the node's equivalent force along the outer radius normal of Ω1. Projection onto; through the above simplified formula Obtain the equivalent pulse quality on the discrete unit Equivalent convective mass of the nth mode The physical meaning is the component of dynamic water pressure over the total mass at each order, multiplied by For equivalent convection mass

[0053] The equivalent pulse mass and convective mass at each finite element node are solved by the ratio of the control area of ​​the finite element node to the area of ​​the discrete element; for the hydrodynamic pressure analysis of liquid storage tanks of arbitrary shape, the distributed equivalent convective mass and equivalent pulse mass are obtained by equations (10) and (11); the pulse hydrodynamic pressure is obtained by multiplying the equivalent pulse mass by the excitation acceleration;

[0054] The pulse dynamic water pressure of the liquid relative to the liquid storage tank is specifically expressed as in equation (12);

[0055]

[0056] Compared with the prior art, the present invention has the following beneficial effects:

[0057] This invention discloses a method for calculating the dynamic water pressure of variable cross-section liquid storage tanks based on modal synthesis. It utilizes the governing equations with the velocity potential function Ф(x,y,z,t) as variables and the boundary conditions of the structural interface and free liquid surface to obtain the equivalent dynamic water pressure formula. By considering the correction coefficient of dynamic water pressure, it can more quickly evaluate the dynamic water pressure of liquid storage tanks with variable cross-section characteristics based on the free liquid surface region. This method has practical engineering significance for the rapid evaluation of structurally vulnerable areas. Attached Figure Description

[0058] Figure 1 This is a flowchart illustrating a method for calculating the dynamic water pressure of a variable cross-section liquid storage tank based on modal synthesis, according to the present invention.

[0059] Figure 2(a) is a schematic diagram of the irregular liquid storage tank in this invention;

[0060] Figure 2(b) is a front view schematic diagram of the irregular liquid storage tank in this invention.

[0061] Figure 3(a) is a schematic diagram of the finite element model of a nuclear power plant building structure;

[0062] Figure 3(b) is a schematic diagram of the discrete elements of a finite element model of a nuclear power plant structure.

[0063] Figure 4 This is a graph of the Pn / Mn ​​coefficients distributed along the equivalent radius.

[0064] Figure 5 It is a comparison chart of the dynamic water pressure time history curves at a distance of 3m from the free liquid surface. Detailed Implementation

[0065] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. The embodiments of the present invention described and shown in the accompanying drawings can generally be arranged and designed in various different configurations.

[0066] Therefore, the following detailed description of the embodiments of the invention provided in the accompanying drawings is not intended to limit the scope of the claimed invention, but merely to illustrate selected embodiments of the invention. All other embodiments obtained by those skilled in the art based on the embodiments of the invention without inventive effort are within the scope of protection of the invention.

[0067] It should be noted that similar labels and letters in the following figures indicate similar items. Therefore, once an item is defined in one figure, it does not need to be further defined and explained in subsequent figures.

[0068] From the perspective of seismic analysis of building structures, the distribution of hydrodynamic pressure can only exhibit a stable decoupled form when the mass does not change with the excitation acceleration. A direct method of decoupling is to extract dynamic characteristics through modal analysis and derive the mass decoupling equation using a generalized single-degree-of-freedom system. This mass decoupling equation is then simplified into a rapid assessment method for peak hydrodynamic pressure based on the characteristics of the analysis domain. Structural modes can reflect the sloshing characteristics of liquid storage tanks, which has practical engineering significance for rapidly assessing vulnerable areas of the structure.

[0069] The specific embodiments of the present invention will now be described in detail with reference to the technical solutions and accompanying drawings.

[0070] See Figures 1 to 4 This invention discloses a method for calculating the dynamic water pressure of a variable cross-section liquid storage tank based on modal synthesis, comprising the following steps:

[0071] This example establishes a finite element model of a nuclear power plant building, as shown in Figures 3(a) and 3(b). The model has a PCS liquid storage tank with variable cross-section at the top, the geometric dimensions of which are shown in Table 1. The total height is 11.799 m, the water level is 9.8 m, and the total water mass is 3000 kg. Figure 4 It can be seen that the elevation position where the cross-section of the liquid storage tank changes is -9.594m down from the top of the liquid storage tank.

[0072] Table 1 Material Parameters

[0073]

[0074] Step 1: Obtain the expression for the hydrodynamic pressure value borne by each discrete element of the variable cross section based on finite element discretization;

[0075] Assume the container walls are rigid, and the liquid is incompressible, inviscid, and irrotational. Apply the Laplace equation with the velocity potential function Ф(x,y,z,t) as variables, and define the boundary conditions for the liquid-solid interface and the free surface. The normal velocity at the liquid-solid interface Ω1 must remain constant, and the pressure at the free surface Ω2 must be zero.

[0076] The dynamic water pressure p(x,y,z,t) can be understood as the product of the convective mass and the liquid acceleration response, plus the product of the impulse mass and the liquid acceleration response relative to the liquid tank. The liquid acceleration response reflects the response of the liquid with respect to the liquid tank, and is numerically equivalent to the seismic acceleration excitation. The liquid acceleration response relative to the liquid tank reflects the response of the liquid under sloshing action relative to the liquid tank, specifically expressed as equation (1).

[0077]

[0078] in, This represents the relative acceleration of the liquid storage tank swaying, and in the distributed mass model, it represents the acceleration of the equivalent spring. M represents the acceleration of the container wall. I M represents the equivalent pulse quality on the unit cell. ns Let be the equivalent convective mass of the nth mode. In the dynamic response analysis, considering the first mode can yield accurate results that meet engineering requirements.

[0079] From the structural dynamics equilibrium equations assumed These are generalized coordinate displacement parameters, which can be derived. It represents relative motion with respect to the liquid.

[0080]

[0081]

[0082] According to the potential flow theory, the following equation can be derived.

[0083]

[0084]

[0085]

[0086] Step two, based on The expression describes the relationship between the area controlled by the finite element nodes on the finite element relation analysis domain Ω2 and the integral variable r.

[0087] The characteristic function Ψ corresponding to a certain variable cross section problem n (r,y,θ)=[N]u n Substituting cosθ into equations (4)-(6), we can obtain...

[0088]

[0089]

[0090]

[0091] According to the finite element principle, [N] is a shape function, and the expression within the parentheses... The physical meaning of "partial" refers to the mass within the integration domain, as in the above equation. This represents the area controlled by each finite element node. It satisfies the condition when ρ = 1. Formula (7) takes the form of an integral variable r, which is the first-order area moment in the two-dimensional area domain Ω2.

[0092] based on The expression uses the equivalent force and integral variable r at the finite element nodes in the finite element relation analysis domain Ω2. 2 The relationship between them;

[0093] According to the finite element principle, the part within the parentheses in equation (5) This is expressed as the equivalent force at the finite element node. According to the finite element principle, the equivalent force at the node is numerically equal to the force acting on a unit area. Assuming the force on the unit area is a uniformly distributed load, then F... q =PA i When P = 1, it satisfies Formula (8) takes this form. Note that the integral sign in Formula (8) also contains the integration variable r. 2 The numerical meaning is the second-order area moment within the two-dimensional area domain Ω2.

[0094] based on The expression utilizes the equivalence of finite element relation nodes along the radial outward normal in the analysis domain Ω1. The relationship between projections on;

[0095] According to the finite element principle, the part within parentheses in equation (9) This is represented as the node's equivalent force along the outer radius normal of Ω1. The projection onto the surface. In equation (9) Physically, it represents half of the integral of the perimeter within the Ω1 domain. It is the integral of the dynamic water pressure on the inner wall element of the Ω1 domain.

[0096] Step 3, based on The correction coefficients proposed in the simplified formula provide a method for rapidly evaluating the equivalent mass of a unit with approximate physical meaning.

[0097] It should be noted that in equation (5) It can be recognized as a kind of based on The additional mass correction factor can be simplified to In form, This represents the area controlled by each finite element node. The overall approach is to equate the variable cross-section to a circular cross-section, and then use the approximate physical meaning of the simplified formula above to quickly evaluate the equivalent pulse quality on the element. Equivalent convective mass of the nth mode

[0098]

[0099] exist Figure 4 Only applies to containers with equivalent circular cross-sections that vary with radius. The value is given by the curve for other liquid storage tank cross-sections. The value needs to be calculated sequentially according to the steps above.

[0100] Step 4: Based on the corresponding equivalent principle formula of variable cross section, obtain the first-order area moment and the second-order area moment of the variable cross section.

[0101] For discontinuous variable cross-section liquid storage tanks, first calculate the area S or circumference L of any arbitrary cross-sectional shape, then use the formula to calculate the radius r of the circle. The formula for the equivalent radius of the equivalent circle is derived from the area. The formula for the equivalent radius of an equivalent circle derived from its circumference is r = L / 2π.

[0102] The modes and frequencies of a structure are inherent characteristics of the structure. To verify the correctness and accuracy of this distributed added mass model, the natural frequencies (Hz) of the first three modes of vibration commonly used in CAS numerical simulation in fluid-structure interaction analysis are compared with those of the model in Table 2.

[0103] Table 2. Natural frequencies (Hz) of the first three vibration modes.

[0104]

[0105]

[0106] The acceleration time history of the synthesized seismic wave was input using the standard spectrum RG1.60, with a duration of 27.99 s, a time step of 0.01 s, and a peak acceleration of 0.3 g. The calculated pulse mass and convective mass were applied, in the form of spring dampers and additional masses, to the sidewalls and sloping bottom of the liquid storage tank at the top of the AP1000 for dynamic calculation.

[0107] The dynamic water pressure at a location 3m above the free liquid surface in the middle of the liquid storage tank is extracted from the dynamic water pressure time history curve. The method proposed in this invention has good consistency with the equivalent dynamic water pressure generated by the CAS model.

[0108] Therefore, the present invention proposes a method for calculating the dynamic water pressure of variable cross-section liquid storage tanks based on modal synthesis. This method utilizes the governing equations with velocity potential function Ф(x,y,z,t) as variables and the boundary conditions of structural interfaces and free liquid surfaces to obtain the equivalent dynamic water pressure formula. By considering the correction coefficient of dynamic water pressure, it can more quickly evaluate the dynamic water pressure of liquid storage tanks with variable cross-section characteristics based on the free liquid surface region. This method has practical engineering significance for the rapid evaluation of structurally vulnerable areas.

[0109] The embodiments described above are merely illustrative of several implementation methods of this application, and while the descriptions are specific and detailed, they should not be construed as limiting the scope of this patent application. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of this application, and these all fall within the protection scope of this application. Therefore, the protection scope of this patent application should be determined by the appended claims.

Claims

1. A method for calculating the hydrodynamic pressure of a variable cross-section liquid storage tank based on the modal synthesis method, characterized by, Includes the following steps; S1: Based on finite element discretization, obtain the equivalent mass borne by each discrete element of the variable cross-section liquid storage tank. Expression, dynamic water pressure Expression and hydrodynamic pressure expression; S2: Based on equivalent mass Expression, dynamic water pressure Expression and hydrodynamic pressure The expression yields the area and the integration variable r. 2 The relationship between them; S3: Based on the principle of variable cross-section equivalence, obtain the first-order and second-order equivalent area moments of the variable cross-section; S4: According to the parameter relationship obtained in S2, substitute the variable cross-section equivalent area moment obtained in S3 to obtain the equivalent pulse mass on each discrete unit Finally, the equivalent dynamic water pressure F is obtained U ; based on The correction coefficient proposed in the simplified formula is then multiplied by the formula based on... The expression yields the node's equivalent force along the radius outside the normal. The components on the surface are used to obtain the equivalent pulse mass of each discrete unit, and then the dynamic water pressure value is obtained.

2. The variable cross-section liquid storage tank hydrodynamic pressure calculation method based on modal synthesis method according to claim 1, characterized by, Equivalent mass borne by each discrete unit of the variable cross-section liquid tank Expression, dynamic water pressure Expression and dynamic water pressure The expression is as follows: Assume the wall of the variable cross-section liquid storage tank is rigid, and the liquid is incompressible, non-viscous, and non-rotational; in the Laplace equation with the velocity potential function Ф(x, y, z, t) as the variable and the boundary conditions of the liquid-solid interface and the free liquid surface, the normal velocity on the liquid-solid interface Ω1 remains consistent, and the pressure on the free liquid surface Ω2 is equal to zero. The dynamic water pressure is the sum of the product of the convective mass and the liquid acceleration response, and the product of the impulse mass and the liquid acceleration response relative to the liquid tank; the liquid acceleration response is numerically equivalent to the seismic acceleration excitation. The dynamic water pressure of the liquid relative to the liquid storage tank is specifically expressed as in equation (1); Among them, F U F represents the pulsed hydrodynamic pressure component representing the movement of the liquid together with the tank wall; S This represents the pressure component of the flowing water that is related to the free vibration of the liquid. This represents the pressure of the flowing water at each order in the modal decomposition expression; Represents the generalized coordinates of each order in the modal decomposition expression; This represents the relative acceleration of the liquid storage tank's swaying motion; M represents the acceleration of the liquid storage tank wall. I M represents the equivalent pulse quality on the discrete element. nc The equivalent convective mass of the nth mode; from the structural dynamic force balance equation substituting u n represents the relative displacement of the liquid with respect to the liquid tank sloshing; from which it is derived that Mn represents the equivalent mass of the nth mode; ω represents the structural circular frequency; Pn represents the equivalent hydrodynamic pressure of the nth mode; Based on potential flow theory, the following equation is derived. M total represents the equivalent total mass of liquid; g represents the acceleration of gravity; Ψ n is a variable cross-section characteristic function; is the velocity vector of the liquid storage tank wall surface mass point; ρ is the liquid density; is the unit vector of the load action direction; is the unit vector of the outer normal of the liquid-solid interface side wall.

3. The variable cross-section liquid storage tank hydrodynamic pressure calculation method based on modal synthesis method according to claim 2, characterized by, The area and the integral variable r, the integral variable r 2 The relationship between them is as follows; Ψ n (r,y,θ) = [N]u n cosθ into equations (4)-(6), r represents the radius in the cylindrical coordinate system; y represents the height coordinate in the cylindrical coordinate system; and θ represents the angle in the cylindrical coordinate system; Ψ [N] is a shape function, has the physical meaning of mass in the integration domain, is expressed as the area controlled by each finite element node in the domain Ω2, which satisfies formula (7) in form; is the first order area moment in the two-dimensional area domain Ω2. In formula (8) The equivalent force of the finite element node is equal to the force received in the unit area in the numerical value. The force experienced by a unit area is the uniform load, F q = PA i When P = 1, it satisfies The form of equation (8); is the second area moment of the two-dimensional area domain Ω2. Equation (9) This is represented as the node's equivalent force along the outer radius normal of Ω1. The projection on; in equation (9) The physical meaning is half of the integral of the perimeter within the Ω1 domain; The integral of the hydrodynamic pressure on the inner wall element of the Ω1 domain.

4. The variable cross-section liquid storage tank hydrodynamic pressure calculation method based on modal synthesis method according to claim 3, characterized by, In step S3, based on the principle of variable cross-section equivalence, the first-order and second-order equivalent area moments of the variable cross-section are obtained, as follows: S31. For discontinuous variable cross-section liquid storage tanks, based on the area S or perimeter L of any cross-sectional shape, obtain the equivalent radius r of the equivalent circle. Or r = L / 2π; calculate the first-order and second-order equivalent area moments directly based on the equivalent radius r; S32, for a continuously variable cross-section liquid storage tank, according to the material mechanics variable cross-section deflection formula Solve to obtain the equivalent radius of the equivalent circle; M p (x) represents the bending moment of the liquid storage tank under actual load; The bending moment of the liquid storage tank under unit load is represented by H; the height of each section of the liquid storage tank is represented by I(x); the moment of inertia function is represented by I(x). Divide the variable cross-section liquid storage tank into i segments of equal length, and represent the second-order area moments of the variable cross-section liquid storage tank of equal length as I1, I2, I3, ..., I... i A column with a constant cross-section exhibits the same deflection displacement as a column with a continuously variable cross-section under the same external conditions. Let Δ1 = Δ2, then the equivalent second-order area moment I of the continuously variable cross-section liquid section is obtained. eq ; will I eq Substituting into the formula for calculating the second-order area moment, we obtain the equivalent radius r of the equivalent circle, and then directly calculate the first-order equivalent area moment based on the equivalent radius r.

5. The variable cross-section liquid storage tank hydrodynamic pressure calculation method based on modal synthesis method according to claim 4, characterized in that, The equivalent quality of each discrete unit is as follows; As a kind of based Additional mass correction factor, This represents the area controlled by each finite element node; This is represented as the node's equivalent force along the outer radius normal of Ω1. Projection onto; through the above simplified formula Obtain the equivalent pulse quality on the discrete unit Equivalent convective mass of the nth mode The physical meaning is the component of dynamic water pressure over the total mass at each order, multiplied by For equivalent convection mass The equivalent pulse mass and convective mass at each finite element node are solved by the ratio of the finite element node control area to the discrete element area. For the dynamic water pressure analysis of liquid storage tanks of arbitrary shape, the distributed equivalent convection mass and equivalent pulse mass are obtained from equations (10) and (11); the pulse dynamic water pressure is obtained by multiplying the equivalent pulse mass by the excitation acceleration. The pulse dynamic water pressure of the liquid relative to the liquid storage tank is specifically expressed as in equation (12);

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