Nuclear power plant multi-span small branch pipe fatigue evaluation model based on vibration speed

Abstract

The invention particularly relates to a nuclear power plant multi-span small branch pipe fatigue evaluation model based on vibration velocity, which comprises the following steps: for nuclear power plant multi-span small branch pipes, selecting one of the small branch pipes, and solving a small branch pipe vibration displacement solution; solving a second derivative of the vibration displacement solution of the small branch pipe along the pipe length and solving a first derivative of the vibration displacement solution of the small branch pipe along the time to obtain a curvature function of a deflection curve of the small branch pipe and a speed solution of the small branch pipe; establishing a relation between a small branch pipe speed solution and a deflection curve curvature function by combining a small branch pipe vibration characteristic equation; according to a material mechanics theory, establishing a relationship between section stress and a velocity solution of the small branch pipe; and the vibration speed limit value of the fatigue limit of the small branch pipe is obtained by combining the material S-N curve of the small branch pipe, the relationship of the velocity solution and the curvature function of the deflection curve and the relationship of the section stress and the velocity solution. The multi-span small branch pipe fatigue evaluation model of the nuclear power plant is closer to the small branch pipe of the nuclear power plant in an actual state, and provides support for the fatigue evaluation of the small branch pipe of the nuclear power plant.

Description

Technical Field

[0001] The invention relates to the technical field of fatigue evaluation of multi-span small branch pipes in nuclear power plants, and in particular to a fatigue evaluation model of multi-span small branch pipes in nuclear power plants based on vibration velocity. Background Art

[0002] In a nuclear power plant, a small branch pipe refers to a pipe with a nominal diameter of 2 inches, which is used to connect to the main pipe to perform functions such as measurement, drainage, exhaust and bypass. In recent years, there have been many cases of small branch pipe fracture failure in nuclear power plants at home and abroad. A study by the American Electric Power Research Institute pointed out that vibration fatigue is the main cause of fracture failure of small branches. Regarding the fracture failure of small branches caused by vibration fatigue, there are differences in the standards of different countries in the world. The ASME standard led by the United States focuses on velocity peak evaluation; the French EDF standard tends to use the effective value of velocity. The above standards are derived on the basis of high-energy pipeline models and extended downward to be compatible with small branches. In 2021, Hainan Nuclear Power Co., Ltd. took the lead in compiling the first domestic guidance document for vibration assessment of small branches in nuclear power plants, NB / T 20612-2021 "Vibration Testing and Assessment of Small Branches in Nuclear Power Plants", but some problems have not been solved in practical applications, such as inaccurate assessments caused by model generalization. Summary of the invention

[0003] Based on this, it is necessary to address the model generalization problem existing in the vibration fatigue evaluation process of small branches in existing nuclear power plants, and provide a fatigue evaluation model for multi-span small branches in nuclear power plants based on vibration velocity. This model is closer to the small branches in nuclear power plants under actual conditions, and provides support for the fatigue evaluation of small branches in nuclear power plants.

[0004] In order to solve the above problems, the present invention provides a fatigue evaluation model for multi-span small branch pipes in nuclear power plants based on vibration velocity, comprising:

[0005] Step 1: For a multi-span small branch pipe in a nuclear power plant, select one of the small branch pipes and solve the vibration displacement solution of the small branch pipe;

[0006] Step 2, respectively, calculate the second-order derivative of the vibration displacement solution of the small branch pipe along the pipe length and the first-order derivative with respect to time, and obtain the curvature function of the small branch pipe deflection curve and the velocity solution of the small branch pipe respectively;

[0007] Step 3, combining the vibration characteristic equation of the small branch pipe, establish the relationship between the velocity solution of the small branch pipe and the curvature function of the deflection curve;

[0008] Step 4: According to the material mechanics theory, establish the relationship between the stress and velocity solution of the small branch section;

[0009] Step 5: Combining the material SN curve of the small branch pipe, the relationship between the velocity solution and the curvature function of the deflection curve, and the relationship between the cross-sectional stress and the velocity solution, the vibration velocity limit of the fatigue limit of the small branch pipe is obtained.

[0010] In the present invention, step 1, for a multi-span small branch pipe of a nuclear power plant, one of the small branch pipes is selected to solve the vibration displacement solution of the small branch pipe, including the following steps:

[0011] Step 11: For a multi-span small branch pipe in a nuclear power plant, select one span of the small branch pipe and establish a vibration differential equation of the small branch pipe under free and undamped vibration conditions;

[0012] Step 12: Based on the transverse vibration theory of uniform cross-section beams and the boundary conditions of small branches, solve the vibration displacement solution of the small branch.

[0013] In the present invention, in step 11, under the condition of free and undamped vibration, the vibration differential equation of the small branch pipe is:

[0014]

[0015] Among them, E is the elastic modulus of the small branch; I is the moment of inertia of the small branch section; y is the lateral vibration displacement of the small branch pipeline; x is the axial coordinate of the small branch pipeline; ρ is the material density of the small branch; A is the cross-sectional area of ​​the small branch; and t is time.

[0016] In the present invention, in step 12, the boundary condition of the small branch pipe is a simple support constraint at both ends; the vibration displacement solution of the small branch pipe is:

[0017] y n (x,t)=A n cos(k n x)sin(ω n t) (2)

[0018] Among them, y n (x, t) is the vibration displacement solution of the small branch pipe; ω n is the nth order natural frequency of the small branch, k n is the nth-order vibration mode wave number of the small branch pipe, n=1,2,...; x is the axial coordinate of the small branch pipe; t is time; A n is a dimensionless constant.

[0019] In the present invention, step 2, respectively taking the second-order derivative of the vibration displacement solution of the small branch pipe along the pipe length and the first-order derivative with respect to time, and respectively obtaining the curvature function of the small branch pipe deflection curve and the small branch pipe velocity solution, comprises the following steps:

[0020] Step 21, calculate the second-order partial derivative of the vibration displacement solution of the small branch pipe along the x direction to obtain the curvature function of the deflection curve of the small branch pipe; according to the curvature function of the deflection curve of the small branch pipe, obtain the maximum curvature of the small branch pipe at all positions along the x direction at all times;

[0021] Step 22, calculate the time first-order derivative of the vibration displacement solution of the small branch pipe to obtain the velocity solution of the small branch pipe; according to the velocity solution of the small branch pipe, obtain the maximum velocity of the small branch pipe at all positions along the x direction at all times.

[0022] In the present invention, in step 21, the curvature function of the small branch deflection curve is:

[0023]

[0024] The maximum curvature of the small branch at all positions along the x direction at all times is:

[0025]

[0026] Among them, y″ n (x, t) is the curvature function of the small branch deflection curve; y″ nmax is the maximum curvature of the small branch at all positions along the x direction at all times; ω n is the nth order natural frequency of the small branch, k n is the nth-order vibration mode wave number of the small branch pipe, n=1,2,...; x is the axial coordinate of the small branch pipe; A n is a dimensionless constant; t is time.

[0027] In the present invention, in step 22, the velocity of the small branch is solved as follows:

[0028] v i (x,t)=ω n A n cos(k n x)cos(ω n t) (8)

[0029] The maximum velocity of the small branch at all positions along the x direction at all times is:

[0030] v imax =ω n A n cos(k n x) (9)

[0031] Among them, ν i (x, t) is the velocity solution of the small branch; ν imax is the maximum velocity of the small branch at all positions along the x direction at all times; ω n is the nth order natural frequency of the small branch, k n is the nth-order vibration mode wave number of the small branch pipe, n=1,2,...; x is the axial coordinate of the small branch pipe; A n is a dimensionless constant; t is time.

[0032] In the present invention, step 3, combining the vibration characteristic equation of the small branch pipe to establish the connection between the curvature function of the small branch pipe deflection curve and the velocity solution, comprises the following steps:

[0033] Step 31, assuming that the small branch vibrates at the 7th-order vibration mode, the vibration mode characteristic length of the small branch satisfies formulas (3) and (4):

[0034] sin(k n l)=0(3)

[0035]

[0036] Substituting formula (2) into formula (1), the nth order natural frequency of the small branch pipe is obtained as:

[0037]

[0038] Step 32: Substitute formula (5) into formula (9) and combine with formula (8) to establish the relationship between the maximum velocity and the maximum curvature of the small branch at all times at all positions along the x direction:

[0039]

[0040] Among them, ν imax is the maximum velocity of the small branch at all positions along the x direction at all times; y″ nmax is the maximum curvature of the small branch at all positions along the x direction at all times; l is the characteristic length of the vibration mode of the small branch; ω n is the nth order natural frequency of the small branch, k n is the nth-order vibration mode wave number of the small branch pipe, n=1,2,...; E is the elastic modulus of the small branch pipe; I is the moment of inertia of the small branch pipe section; ρ is the material density of the small branch pipe; A is the cross-sectional area of ​​the small branch pipe; A n is a dimensionless constant; x is the axial coordinate of the small branch pipe.

[0041] In the present invention, step 4, based on the material mechanics theory, establishes the relationship between the stress and velocity solution of the small branch pipe cross section, including the following steps:

[0042] Step 41. According to the material mechanics theory, the cross-sectional stress at the position of maximum deflection of the small branch pipe under the nth order vibration mode is:

[0043] σ x =Eε x (11)

[0044] Step 42: Assuming that the small branch pipe satisfies the plane assumption and the small displacement assumption during the vibration process, the relationship shown in formulas (12) and (13) is obtained:

[0045]

[0046] Substituting formula (12) and formula (13) into formula (11) yields formula (14):

[0047]

[0048] Substituting formula (10) into formula (14), we can obtain the relationship between the maximum cross-sectional stress and the maximum velocity at all times at the maximum deflection position of the small branch under the nth-order vibration mode:

[0049]

[0050] Among them, σ x is the cross-sectional stress in the x direction at the position of maximum deflection of the small branch pipe; y″ nmax is the maximum curvature of the small branch at all positions along the x direction at all times; σ xmax is the maximum cross-sectional stress in the x-direction at all times at the position of maximum deflection of the small branch; ν max is the maximum velocity of the small branch; ε x is the strain in the x direction of the small branch section; D is the outer diameter of the small branch pipe; E is the elastic modulus of the small branch pipe; is the rotation angle of the small branch section; A is the cross-sectional area of ​​the small branch; I is the moment of inertia of the small branch section; ρ is the material density of the small branch; E is the elastic modulus of the small branch.

[0051] In the present invention, step 5, combining the relationship between the material SN curve of the small branch pipe, the velocity solution and the curvature function of the deflection curve, and the relationship between the cross-sectional stress and the velocity solution, obtains the vibration velocity limit of the fatigue limit of the small branch pipe, including the following steps:

[0052] In the fatigue evaluation of small branch pipes, the fatigue strength reduction coefficient C2K2 of small branch pipes is introduced:

[0053]

[0054] Substituting formula (15) into formula (16), the vibration velocity limit of the fatigue limit of the small branch pipe is obtained as follows:

[0055]

[0056] Among them, σ xmax is the maximum cross-sectional stress in the x-direction at all times at the position of maximum deflection of the small branch pipe; σ max is the maximum stress of the small branch section; ν max is the maximum velocity of the small branch; C2 is the secondary stress index of the small branch, K2 is the local stress index of the small branch, S el is the SN curve fatigue limit of the small branch material, C2, K2 and S el The value of refers to the ASME BPVC-III specification; D is the outer diameter of the small branch pipe; A is the cross-sectional area of ​​the small branch pipe; I is the moment of inertia of the small branch pipe section; ρ is the density of the small branch pipe material; and E is the elastic modulus of the small branch pipe.

[0057] Beneficial technical effects of the present invention:

[0058] The fatigue evaluation model for multi-span small branch pipes in nuclear power plants of the present invention is based on the lateral vibration theory of equal-section beams, follows the plane assumption and the small displacement assumption, and is derived according to the material mechanics and vibration dynamics theory. It solves the generalization problem of the current vibration fatigue evaluation model for small branch pipes in nuclear power plants, realizes a vibration matching evaluation of small branch pipes in nuclear power plants that is closer to the actual state, and provides a basis for fatigue evaluation of small branch pipes in nuclear power plants. BRIEF DESCRIPTION OF THE DRAWINGS

[0059] Figure 1 A schematic diagram of an embodiment of a fatigue evaluation model for multi-span small branch pipes in a nuclear power plant according to the present invention;

[0060] Figure 2 Schematic diagram of the cross-sectional changes of the small branch pipe under transverse bending vibration. DETAILED DESCRIPTION

[0061] Unless otherwise defined, all technical and scientific terms used herein have the same meaning as commonly understood by technicians in the technical field to which this application belongs; the terms used in the specification of the application are only for the purpose of describing specific embodiments and are not intended to limit this application; the terms "including" and "having" and any variations thereof in the specification and claims of this application and the above-mentioned figure descriptions are intended to cover non-exclusive inclusions.

[0062] Reference to "embodiments" herein means that a particular feature, structure, or characteristic described in conjunction with the embodiments may be included in at least one embodiment of the present application. The appearance of the phrase in various locations in the specification does not necessarily refer to the same embodiment, nor is it an independent or alternative embodiment that is mutually exclusive with other embodiments. It is explicitly and implicitly understood by those skilled in the art that the embodiments described herein may be combined with other embodiments.

[0063] The technical solution of the present invention is clearly and completely described below in conjunction with the accompanying drawings and specific embodiments.

[0064] refer to Figure 1 , showing a schematic diagram of an embodiment of a fatigue evaluation model for multi-span small branches in a nuclear power plant based on vibration velocity; the fatigue evaluation model for multi-span small branches in a nuclear power plant assumes that the small branches in a nuclear power plant are mainly subjected to transverse bending vibration, and considering that bending moment and shear force can be transmitted along the length of the pipe, the fatigue evaluation model for multi-span small branches in a nuclear power plant is derived from the transverse vibration theory of equal-section beams; the fatigue evaluation model for multi-span small branches in a nuclear power plant is suitable for fatigue life evaluation of transverse bending vibration of multi-span small branches in a nuclear power plant.

[0065] The fatigue evaluation model for multi-span small branch pipes in a nuclear power plant described in this embodiment includes:

[0066] Step 1: For a multi-span small branch pipe in a nuclear power plant, select one of the small branch pipes and solve the vibration displacement solution of the small branch pipe;

[0067] Step 2, respectively, calculate the second-order derivative of the vibration displacement solution of the small branch pipe along the pipe length and the first-order derivative with respect to time, and obtain the curvature function of the small branch pipe deflection curve and the velocity solution of the small branch pipe respectively;

[0068] Step 3, combining the vibration characteristic equation of the small branch pipe, establish the relationship between the velocity solution of the small branch pipe and the curvature function of the deflection curve;

[0069] Step 4: According to the material mechanics theory, the curvature function of the deflection curve of the small branch pipe is proportional to the bending moment. The bending moment of the small branch pipe is divided by the cross-sectional bending modulus to obtain the cross-sectional stress of the small branch pipe, thereby establishing the relationship between the cross-sectional stress of the small branch pipe and the velocity solution;

[0070] Step 5: Combining the material SN curve of the small branch pipe, the relationship between the velocity solution and the curvature function of the deflection curve, and the relationship between the cross-sectional stress and the velocity solution, the vibration velocity limit of the fatigue limit of the small branch pipe is obtained.

[0071] In this embodiment, step 1, for a multi-span small branch pipe in a nuclear power plant, one of the small branch pipes is selected to solve the vibration displacement solution of the small branch pipe, including the following steps:

[0072] Step 11: For a multi-span small branch pipe in a nuclear power plant, select a small branch pipe with a span length of L and a cross-sectional area of ​​A, and establish a vibration differential equation of the small branch pipe under free undamped vibration conditions;

[0073] Step 12: Based on the transverse vibration theory of uniform cross-section beams and the boundary conditions of small branches, solve the vibration displacement solution of the small branch.

[0074] In this embodiment, in step 11, under the condition of free and undamped vibration, the vibration differential equation of the small branch pipe is:

[0075]

[0076] Among them, E is the elastic modulus of the small branch; I is the moment of inertia of the small branch section; y is the lateral vibration displacement of the small branch pipeline; x is the axial coordinate of the small branch pipeline; ρ is the material density of the small branch; A is the cross-sectional area of ​​the small branch; and t is time.

[0077] In this embodiment, in step 12, the boundary condition of the small branch pipe is a simple support constraint at both ends; the vibration displacement solution of the small branch pipe is:

[0078] y n (x,t)=A n cos(k n x)sin(ω n t)(2)

[0079] Among them, y n(x, t) is the vibration displacement solution of the small branch pipe; ω n is the nth order natural frequency of the small branch, k n is the nth-order vibration mode wave number of the small branch pipe, n=1,2,...; x is the axial coordinate of the small branch pipe; t is time; A n is a dimensionless constant.

[0080] In this embodiment, step 2, respectively taking the second-order derivative of the vibration displacement solution of the small branch along the pipe length and the first-order derivative with respect to time, and respectively obtaining the curvature function of the small branch deflection curve and the small branch velocity solution, comprises the following steps:

[0081] Step 21, calculate the second-order partial derivative of the vibration displacement solution of the small branch pipe along the x direction to obtain the curvature function of the deflection curve of the small branch pipe; according to the curvature function of the deflection curve of the small branch pipe, obtain the maximum curvature of the small branch pipe at all positions along the x direction at all times;

[0082] Step 22, calculate the time first-order derivative of the vibration displacement solution of the small branch pipe to obtain the velocity solution of the small branch pipe; according to the velocity solution of the small branch pipe, obtain the maximum velocity of the small branch pipe at all positions along the x direction at all times.

[0083] In this embodiment, in step 21, the curvature function of the small branch deflection curve is:

[0084]

[0085] From the curvature function of the deflection curve of the small branch, we know that the curvature of the deflection curve of the small branch changes with time, and the maximum curvature of the small branch at all positions along the x direction at all times is obtained:

[0086]

[0087] Among them, y″ n (x, t) is the curvature function of the small branch deflection curve; y″ nmax is the maximum curvature of the small branch at all positions along the x direction at all times; ω n is the nth order natural frequency of the small branch, k n is the nth-order vibration mode wave number of the small branch pipe, n=1,2,...; x is the axial coordinate of the small branch pipe; A n is a dimensionless constant; t is time.

[0088] In this embodiment, in step 22, the velocity solution of the small branch pipe is:

[0089] v i (x,t)=ω n A n cos(k n x)cos(ω n t) (8)

[0090] From the solution of the velocity of the small branch, we know that the velocity of the small branch changes with time, and the maximum velocity of the small branch at all positions along the x direction at all times is obtained:

[0091] v imax =ω n A n cos(k n x) (9)

[0092] Among them, ν i (x, t) is the velocity solution of the small branch; ν imax is the maximum velocity of the small branch at all positions along the x direction at all times; ω n is the nth order natural frequency of the small branch, k n is the nth-order vibration mode wave number of the small branch pipe, n=1,2,...; x is the axial coordinate of the small branch pipe; A n is a dimensionless constant; t is time.

[0093] In this embodiment, step 3, combining the vibration characteristic equation of the small branch pipe to establish the relationship between the curvature function of the small branch pipe deflection curve and the velocity solution, includes the following steps:

[0094] Step 31, establishing the vibration characteristic equation of the small branch pipe;

[0095] Step 32: Establish the relationship between the maximum velocity of the small branch and the maximum curvature function of the deflection curve.

[0096] In this embodiment, in step 31, for ease of understanding and without loss of generality, it is assumed that the small branch vibrates in a 7th-order vibration mode, and the characteristic length of the vibration mode of the small branch satisfies formulas (3) and (4):

[0097] sin(k n l)=0 (3)

[0098]

[0099] Substituting formula (2) into formula (1), the nth order natural frequency of the small branch pipe is obtained as:

[0100]

[0101] Where l is the characteristic length of the small branch vibration mode; ω n is the nth order natural frequency of the small branch, k n is the nth-order vibration mode wave number of the small branch, n=1,2,...; E is the elastic modulus of the small branch; I is the moment of inertia of the small branch section; ρ is the material density of the small branch; A is the cross-sectional area of ​​the small branch.

[0102] In this embodiment, in step 32, formula (5) is substituted into formula (9), and combined with formula (8), the relationship between the maximum velocity and the maximum curvature of the small branch at all times at each position along the x direction is established as follows:

[0103]

[0104] Among them, ν imax is the maximum velocity of the small branch at all positions along the x direction at all times; y″ nmax is the maximum curvature of the small branch at all positions along the x direction at all times; E is the elastic modulus of the small branch; I is the moment of inertia of the small branch section; ρ is the material density of the small branch; A is the cross-sectional area of ​​the small branch; k n is the nth-order vibration mode wave number of the small branch pipe, n=1,2,......; A n is a dimensionless constant; x is the axial coordinate of the small branch pipe.

[0105] In this embodiment, step 4, according to the material mechanics theory, the curvature function of the deflection curve of the small branch is proportional to the bending moment, and the bending moment of the small branch is divided by the cross-sectional bending modulus to obtain the cross-sectional stress of the small branch, thereby establishing the relationship between the cross-sectional stress of the small branch and the velocity solution, including the following steps:

[0106] Step 41. According to the material mechanics theory, the cross-sectional stress at the position of maximum deflection of the small branch pipe under the nth order vibration mode is:

[0107] σ x =Eε x (11)

[0108] Step 42 Figure 2 A schematic diagram of the cross-sectional change of the small branch under transverse bending vibration is given. Under the influence of the vibration mode, the curvature of the deflection curve at the maximum deflection position of the small branch changes, and the change corresponds to the angle of rotation of the cross section at that location. And the x-direction strain is generated; assuming that the small branch meets the plane assumption and small displacement assumption during the vibration process, the relationship shown in formulas (12) and (13) is obtained:

[0109]

[0110] Substituting formula (12) and formula (13) into formula (11) yields formula (14):

[0111]

[0112] Substituting formula (10) into formula (14), we can obtain the relationship between the maximum cross-sectional stress in the x-direction and the maximum velocity at all times at the maximum deflection position of the small branch under the nth-order vibration mode:

[0113]

[0114] Among them, σ x is the cross-sectional stress in the x direction at the position of maximum deflection of the small branch pipe; y″ nmax is the maximum curvature of the small branch at all positions along the x direction at all times; σ xmax is the maximum cross-sectional stress in the x-direction at all times at the position of maximum deflection of the small branch; ν max is the maximum velocity of the small branch; ε x is the strain in the x direction of the small branch section; D is the outer diameter of the small branch pipe; E is the elastic modulus of the small branch pipe; is the rotation angle of the small branch section; A is the cross-sectional area of ​​the small branch; I is the moment of inertia of the small branch section; ρ is the material density of the small branch; E is the elastic modulus of the small branch; ν max is the maximum velocity of the small branch.

[0115] In this embodiment, step 5, combining the material SN curve of the small branch pipe, the relationship between the velocity solution and the curvature function of the deflection curve, and the relationship between the cross-sectional stress and the velocity solution, obtains the vibration velocity limit of the fatigue limit of the small branch pipe, including the following steps:

[0116] In the fatigue evaluation of small branch pipes, the fatigue strength reduction coefficient C2K2 of small branch pipes is introduced:

[0117]

[0118] Substituting formula (15) into formula (16), the vibration velocity limit of the fatigue limit of the small branch pipe is obtained as follows:

[0119]

[0120] Among them, σ xmax is the maximum cross-sectional stress in the x-direction at all times at the position of maximum deflection of the small branch pipe; σ max is the maximum stress of the small branch section; ν max is the maximum velocity of the small branch; C2 is the secondary stress index of the small branch, K2 is the local stress index of the small branch, S el is the SN curve fatigue limit of the small branch material, C2, K2 and S el The value of refers to the ASME BPVC-III specification; D is the outer diameter of the small branch pipe; A is the cross-sectional area of ​​the small branch pipe; I is the moment of inertia of the small branch pipe section; ρ is the density of the small branch pipe material; and E is the elastic modulus of the small branch pipe.

[0121] The above-mentioned embodiments only express several implementation methods of the present invention, and the description thereof is relatively specific and detailed, but it cannot be understood as limiting the scope of the patent of the present invention. It should be pointed out that, for ordinary technicians in this field, several variations and improvements can be made without departing from the concept of the present invention, which all belong to the protection scope of the present invention. Therefore, the protection scope of the patent of the present invention shall be subject to the attached claims.

Claims

1. A fatigue evaluation model for multi-span small branch pipes in nuclear power plants based on vibration velocity, characterized in that: include: Step 1: For a multi-span small branch pipe in a nuclear power plant, select one of the small branch pipes and solve the vibration displacement solution of the small branch pipe; Step 2, respectively, calculate the second-order derivative of the vibration displacement solution of the small branch pipe along the pipe length and the first-order derivative with respect to time, and obtain the curvature function of the small branch pipe deflection curve and the velocity solution of the small branch pipe respectively; Step 3, combining the vibration characteristic equation of the small branch pipe, establish the relationship between the velocity solution of the small branch pipe and the curvature function of the deflection curve; Step 4: According to the material mechanics theory, establish the relationship between the stress and velocity solution of the small branch section; Step 5: Combining the material SN curve of the small branch pipe, the relationship between the velocity solution and the curvature function of the deflection curve, and the relationship between the cross-sectional stress and the velocity solution, the vibration velocity limit of the fatigue limit of the small branch pipe is obtained.

2. The fatigue evaluation model for multi-span small branch pipes in nuclear power plants according to claim 1, characterized in that: Step 1, for a multi-span small branch pipe in a nuclear power plant, select one of the small branch pipes and solve the vibration displacement solution of the small branch pipe, including the following steps: Step 11: For a multi-span small branch pipe in a nuclear power plant, select one span of the small branch pipe and establish a vibration differential equation of the small branch pipe under free and undamped vibration conditions; Step 12: Based on the transverse vibration theory of uniform cross-section beams and the boundary conditions of small branches, solve the vibration displacement solution of the small branch.

3. The fatigue evaluation model for multi-span small branch pipes in nuclear power plants according to claim 2, characterized in that: In step 11, under the condition of free and undamped vibration, the vibration differential equation of the small branch pipe is: Among them, E is the elastic modulus of the small branch; I is the moment of inertia of the small branch section; y is the lateral vibration displacement of the small branch pipeline; x is the axial coordinate of the small branch pipeline; ρ is the material density of the small branch; A is the cross-sectional area of ​​the small branch; and t is time.

4. The fatigue evaluation model for multi-span small branch pipes in nuclear power plants according to claim 3, characterized in that: In step 12, the boundary condition of the small branch is a simple support constraint at both ends; the vibration displacement solution of the small branch is: y n (x,t)=A n cos(k n x)sin(ω n t)(2) Among them, y n (x, t) is the vibration displacement solution of the small branch pipe; ω n is the nth order natural frequency of the small branch, k n is the nth-order vibration mode wave number of the small branch pipe, n=1,2,...; x is the axial coordinate of the small branch pipe; t is time; A n is a dimensionless constant.

5. The fatigue evaluation model for multi-span small branch pipes in nuclear power plants according to claim 4, characterized in that: Step 2, respectively taking the second-order derivative of the vibration displacement solution of the small branch pipe along the pipe length and the first-order derivative with respect to time, and obtaining the curvature function of the small branch pipe deflection curve and the small branch pipe velocity solution, including the following steps: Step 21, calculate the second-order partial derivative of the vibration displacement solution of the small branch pipe along the x direction to obtain the curvature function of the deflection curve of the small branch pipe; according to the curvature function of the deflection curve of the small branch pipe, obtain the maximum curvature of the small branch pipe at all positions along the x direction at all times; Step 22, calculating the time first-order derivative of the vibration displacement solution of the small branch pipe to obtain the velocity solution of the small branch pipe; According to the velocity solution of the small branch, the maximum velocity of the small branch at all positions along the x direction at all times is obtained.

6. The fatigue evaluation model for multi-span small branch pipes in nuclear power plants according to claim 5, characterized in that: In step 21, the curvature function of the small branch deflection curve is: The maximum curvature of the small branch at all positions along the x direction at all times is: Among them, y″ n (x, t) is the curvature function of the small branch deflection curve; y″ nmax is the maximum curvature of the small branch at all positions along the x direction at all times; ω n is the nth order natural frequency of the small branch, k n is the nth-order vibration mode wave number of the small branch pipe, n=1,2,...; x is the axial coordinate of the small branch pipe; A n is a dimensionless constant; t is time.

7. The fatigue evaluation model for multi-span small branch pipes in nuclear power plants according to claim 6, characterized in that: In step 22, the velocity solution of the small branch is: v i (x,t)=ω n A n cos(k n x)cos(ω n t)(8) The maximum velocity of the small branch at all positions along the x direction at all times is: v imax =ω n A n cos(k n x)(9) Among them, ν i (x, t) is the velocity solution of the small branch; ν imax is the maximum velocity of the small branch at all positions along the x direction at all times; ω n is the nth order natural frequency of the small branch, k n is the nth-order vibration mode wave number of the small branch pipe, n=1,2,...; x is the axial coordinate of the small branch pipe; A n is a dimensionless constant; t is time.

8. The fatigue evaluation model for multi-span small branch pipes in nuclear power plants according to claim 7, characterized in that: Step 3, combining the vibration characteristic equation of the small branch pipe to establish the connection between the curvature function of the small branch pipe deflection curve and the velocity solution, including the following steps: Step 31, assuming that the small branch vibrates at the 7th-order vibration mode, the vibration mode characteristic length of the small branch satisfies formulas (3) and (4): sin(k n l)=0(3) Substituting formula (2) into formula (1), the nth order natural frequency of the small branch pipe is obtained as: Step 32: Substitute formula (5) into formula (9) and combine with formula (8) to establish the relationship between the maximum velocity and the maximum curvature of the small branch at all times at all positions along the x direction: Among them, ν imax is the maximum velocity of the small branch at all positions along the x direction at all times; y″ nmax is the maximum curvature of the small branch at all positions along the x direction at all times; l is the characteristic length of the vibration mode of the small branch; ω n is the nth order natural frequency of the small branch, k n is the nth-order vibration mode wave number of the small branch pipe, n=1,2,...; E is the elastic modulus of the small branch pipe; I is the moment of inertia of the small branch pipe section; ρ is the material density of the small branch pipe; A is the cross-sectional area of ​​the small branch pipe; A n is a dimensionless constant; x is the axial coordinate of the small branch pipe.

9. The fatigue evaluation model for multi-span small branch pipes in nuclear power plants according to claim 8, characterized in that: Step 4, based on the material mechanics theory, establish the relationship between the stress and velocity solution of the small branch section, including the following steps: Step 41. According to the material mechanics theory, the cross-sectional stress at the position of maximum deflection of the small branch pipe under the nth order vibration mode is: s x =Ee x (11) Step 42: Assuming that the small branch pipe satisfies the plane assumption and the small displacement assumption during the vibration process, the relationship shown in formulas (12) and (13) is obtained: Substituting formula (12) and formula (13) into formula (11) yields formula (14): Substituting formula (10) into formula (14), we can obtain the relationship between the maximum cross-sectional stress in the x-direction and the maximum velocity at all times at the maximum deflection position of the small branch under the nth-order vibration mode: Among them, σ x is the cross-sectional stress in the x direction at the position of maximum deflection of the small branch pipe; y″ nmax is the maximum curvature of the small branch at all positions along the x direction at all times; σ xmax is the maximum cross-sectional stress in the x-direction at all times at the position of maximum deflection of the small branch; ν max is the maximum velocity of the small branch; ε x is the strain in the x direction of the small branch section; D is the outer diameter of the small branch pipe; E is the elastic modulus of the small branch pipe; is the rotation angle of the small branch section; A is the cross-sectional area of ​​the small branch; I is the moment of inertia of the small branch section; ρ is the material density of the small branch; E is the elastic modulus of the small branch; ν max is the maximum velocity of the small branch; σ xmax is the maximum cross-sectional stress of the small branch.

10. The fatigue evaluation model for multi-span small branch pipes in nuclear power plants according to claim 9, characterized in that: Step 5, combining the material SN curve of the small branch pipe, the relationship between the velocity solution and the curvature function of the deflection curve, and the relationship between the cross-sectional stress and the velocity solution, to obtain the vibration velocity limit of the fatigue limit of the small branch pipe, including the following steps: In the fatigue evaluation of small branch pipes, the fatigue strength reduction coefficient C2K2 of small branch pipes is introduced: Substituting formula (15) into formula (16), the vibration velocity limit of the fatigue limit of the small branch pipe is obtained as follows: Among them, σ xmax is the maximum cross-sectional stress in the x-direction at all times at the position of maximum deflection of the small branch pipe; σ max is the maximum stress of the small branch section; ν max is the maximum velocity of the small branch; C2 is the secondary stress index of the small branch, K2 is the local stress index of the small branch, S el is the SN curve fatigue limit of the small branch material, C2, K2 and S el The value of refers to the ASME BPVC-III specification; D is the outer diameter of the small branch pipe; A is the cross-sectional area of ​​the small branch pipe; I is the moment of inertia of the small branch pipe section; ρ is the density of the small branch pipe material; and E is the elastic modulus of the small branch pipe.

Images