Calculation method and device for beam end displacement considering bearing friction under train action

Abstract

The present invention relates to a method and device for calculating the beam end displacement considering the bearing friction under the action of a train. The method includes the steps of: establishing a finite element model according to the bridge information and determining the numerical curve of the influence of the beam end displacement under the action of a moving unit force based on the finite element model, where the numerical curve of the influence of the beam end displacement reflects the relationship between the longitudinal displacement at the beam end and the position of the moving unit force on the bridge; determining the static beam end displacement corresponding to the train moving on the bridge according to the physical information of the train and the numerical curve of the influence of the beam end displacement; determining the longitudinal restoring force stiffness and equivalent mass of the bridge according to the dynamic behavior of the bridge structure itself; establishing an equilibrium displacement equation considering the bearing friction effect based on the static beam end displacement corresponding to the train moving on the bridge, the longitudinal restoring force stiffness and the equivalent mass, and solving the longitudinal displacement of the beam end based on the equilibrium displacement equation. The accuracy and efficiency of calculating the longitudinal displacement of the beam end can be improved.

Description

Technical Field

[0001] The present invention relates to the technical field of bridge analysis, and particularly to a method and device for calculating the beam end displacement considering the bearing friction under the action of a train. Background Art

[0002] With the development of China's economic technology, the construction of super-large-span railway bridges has developed rapidly. For example, the Changtai Yangtze River Bridge under construction carries two intercity railways, and the main bridge span reaches 1176m. For super-large-span railway bridges, the calculated longitudinal displacement of the beam end caused by the train crossing the bridge is generally too large, resulting in a relatively large specification of the beam end expansion device. However, the large-specification beam end expansion device endangers the safety of train operation due to its non-sliding longitudinally.

[0003] Regarding the calculation method of the beam end displacement caused by the train crossing the bridge, the static analysis method is adopted in the related technology, that is, through the influence line principle, the train loading position when the extreme value of the beam end displacement is determined, and the train load is applied for static solution to obtain the maximum and minimum values of the beam end displacement. The disadvantage of the static analysis method is that it fails to consider the inertial force generated by the acceleration and the hysteresis effect of the bearing friction resistance, resulting in an overestimated calculation result of the beam end displacement and not conforming to the actual displacement value of the project. Summary of the Invention

[0004] The embodiments of the present invention provide a method and device for calculating the beam end displacement considering the bearing friction under the action of a train, so as to improve the accuracy and efficiency of the calculation of the longitudinal displacement of the beam end.

[0005] In a first aspect, the embodiments of the present invention provide a method for calculating the beam end displacement considering the bearing friction under the action of a train, which is characterized by including the following steps:

[0006] Establish a finite element model according to the bridge information and determine the numerical curve of the influence of the beam end displacement under the action of a moving unit force based on the finite element model. The numerical curve of the influence of the beam end displacement reflects the relationship between the longitudinal displacement at the beam end and the position of the moving unit force on the bridge;

[0007] Determine the static displacement of the beam end corresponding to the train moving on the bridge according to the physical information of the train and the numerical curve of the influence of the beam end displacement;

[0008] Determine the longitudinal restoring force stiffness and equivalent mass of the bridge according to the dynamic behavior of the bridge structure itself;

[0009] Establish an equilibrium displacement equation considering the bearing friction effect based on the static displacement of the beam end corresponding to the train moving on the bridge, the longitudinal restoring force stiffness, and the equivalent mass, and solve the longitudinal displacement of the beam end based on the equilibrium displacement equation.

[0010] In some embodiments, the physical information of the train includes: the length of the train, the uniform load of the train, the traveling speed of the train, and the traveling time of the train; determining the static displacement of the beam end corresponding to the train moving on the bridge according to the physical information of the train and the numerical curve of the influence of the beam end displacement includes the steps:

[0011] Obtaining the static displacement of the beam end corresponding to the train moving on the bridge based on the first formula, the first formula includes:

[0012]

[0013] where D(t) is the static displacement of the beam end corresponding to the train moving on the bridge, l is the length of the train, t is the traveling time of the train, q is the uniform load of the dynamic load, L is the length of the bridge, v is the traveling speed of the train, and y(t) is the numerical curve of the influence of the beam end displacement.

[0014] In some embodiments, determining the longitudinal restoring force stiffness and equivalent mass of the bridge according to the dynamic behavior of the bridge structure itself includes the steps:

[0015] Based on the finite element model, applying a longitudinal force to the main beam and recording the longitudinal deformation of the main beam;

[0016] Determining the frequency of the first-order longitudinal drift through modal analysis;

[0017] Calculating the longitudinal restoring force stiffness and equivalent mass based on the longitudinal force, the longitudinal deformation, and the frequency of the first-order longitudinal drift.

[0018] In some embodiments, calculating the longitudinal restoring force stiffness and equivalent mass based on the longitudinal force, the longitudinal deformation, and the frequency of the first-order longitudinal drift includes the steps:

[0019] Calculating the longitudinal restoring force stiffness and equivalent mass based on the second formula, the second formula includes: K = F / Δ, M = K / ω 2 , where F is the longitudinal force, Δ is the longitudinal deformation, ω is the frequency of the first-order longitudinal drift, K is the longitudinal restoring force stiffness, and M is the equivalent mass.

[0020] In some embodiments, establishing an equilibrium displacement equation considering the bearing friction effect based on the static displacement of the beam end corresponding to the train moving on the bridge, the longitudinal restoring force stiffness, and the equivalent mass includes the steps:

[0021] Establishing an equilibrium displacement equation considering the bearing friction effect based on the third formula, the third formula includes:

[0022]

[0023]

[0024] Among them, n is the number of bearings distributed between the bridge pier and the bridge tower, f is the bearing friction resistance, u is the bearing friction coefficient, G i is the vertical force of the i-th bearing, and B(t) is the longitudinal displacement of the beam end to be determined.

[0025] On the other hand, the embodiment of the present invention provides a device for calculating the beam end displacement considering the bearing friction under the action of a train, which is characterized in that it includes:

[0026] A beam end static displacement determination module, which is used for:

[0027] Establish a finite element model according to the bridge information and determine the numerical curve of the influence of the beam end displacement under the action of a moving unit force based on the finite element model. The numerical curve of the influence of the beam end displacement reflects the relationship between the longitudinal displacement at the beam end and the position of the moving unit force on the bridge;

[0028] Determine the corresponding beam end static displacement when the train moves on the bridge according to the physical information of the train and the numerical curve of the influence of the beam end displacement;

[0029] A beam end displacement calculation module, which is used for:

[0030] Determine the longitudinal restoring force stiffness and equivalent mass of the bridge according to the dynamic behavior of the bridge structure itself;

[0031] Based on the corresponding beam end static displacement when the train moves on the bridge, the longitudinal restoring force stiffness and the equivalent mass, establish an equilibrium displacement equation considering the bearing friction effect and solve the longitudinal displacement of the beam end based on the equilibrium displacement equation.

[0032] In some embodiments, the physical information of the train includes: the length of the train, the uniform load of the train, the traveling speed of the train, and the traveling time of the train;

[0033] The beam end static displacement determination module is further used for:

[0034] Obtain the corresponding beam end static displacement when the train moves on the bridge based on the first formula, and the first formula includes:

[0035]

[0036] Among them, D(t) is the corresponding beam end static displacement when the train moves on the bridge, l is the length of the train, t is the traveling time of the train, q is the uniform load of the dynamic load, L is the length of the bridge, v is the traveling speed of the train, and y(t) is the numerical curve of the influence of the beam end displacement.

[0037] In some embodiments, the beam end displacement calculation module is further configured to:

[0038] Based on the finite element model, apply a longitudinal force to the main beam and record the longitudinal deformation of the main beam;

[0039] Determine the frequency of the first-order longitudinal drift through modal analysis;

[0040] Calculate the longitudinal restoring force stiffness and the equivalent mass based on the longitudinal force, the longitudinal deformation, and the frequency of the first-order longitudinal drift.

[0041] In some embodiments, the beam end displacement calculation module is further configured to:

[0042] Calculate the longitudinal restoring force stiffness and the equivalent mass based on a second formula, where the second formula includes: K = F / Δ, M = K / ω 2 , where F is the longitudinal force, Δ is the longitudinal deformation, ω is the frequency of the first-order longitudinal drift, K is the longitudinal restoring force stiffness, and M is the equivalent mass.

[0043] In some embodiments, the beam end displacement calculation module is further configured to:

[0044] Establish an equilibrium displacement equation considering the bearing friction effect based on a third formula, where the third formula includes:

[0045]

[0046]

[0047] where n is the number of bearings distributed between the piers and the bridge towers, f is the bearing friction force, u is the bearing friction coefficient, G i is the vertical force of the i-th bearing, and B(t) is the longitudinal displacement of the beam end to be determined.

[0048] The beneficial effects brought by the technical solution provided by the present invention include:

[0049] The embodiments of the present invention provide a method and device for calculating the beam end displacement considering bearing friction under the action of a train. Since the inertial force generated by the acceleration and the constraint effect of the bearing friction on the main beam are considered, the calculated beam end displacement considering the bearing friction effect is in good agreement with the actual value, making the calculation of the longitudinal displacement of the beam end accurate and efficient when a train crosses a super-long-span railway bridge, conforming to the actual displacement value of the project, reducing the specification of the beam end expansion device, and improving the durability of the expansion device. Description of the Drawings

[0050] To more clearly illustrate the technical solutions in the embodiments of the present invention, the following will briefly introduce the accompanying drawings required for the description of the embodiments. Obviously, the accompanying drawings in the following description are only some embodiments of the present invention. For those of ordinary skill in the art, without creative efforts, other accompanying drawings can be obtained based on these drawings.

[0051] Figure 1 It is a schematic flow chart of a method for calculating the beam end displacement considering the bearing friction under the action of a train provided by an embodiment of the present invention;

[0052] Figure 2 It is a schematic elevation structure diagram of a cable-stayed bridge provided by an embodiment of the present invention;

[0053] Figure 3 It is a schematic diagram of the numerical curve of the influence of the beam end displacement under the action of a moving unit force provided by an embodiment of the present invention;

[0054] Figure 4 It is a schematic diagram of the static displacement curve of the beam end provided by an embodiment of the present invention;

[0055] Figure 5 It is a schematic diagram of the beam end displacement curve provided by an embodiment of the present invention

[0056] Figure 6 It is a schematic structural diagram of a device for calculating the beam end displacement considering the bearing friction under the action of a train provided by an embodiment of the present invention. Detailed implementation manners

[0057] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the following will clearly and completely describe the technical solutions in the embodiments of the present invention with reference to the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are some, but not all, of the embodiments of the present invention. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative efforts belong to the scope of protection of the present invention.

[0058] As Figure 1 shown, an embodiment of the present invention provides a method for calculating the beam end displacement considering the bearing friction under the action of a train, which is characterized in that it includes the steps:

[0059] S100: Establish a finite element model according to the bridge information and determine the numerical curve of the influence of the beam end displacement under the action of a moving unit force based on the finite element model. The numerical curve of the influence of the beam end displacement reflects the relationship between the longitudinal displacement at the beam end and the position of the moving unit force on the bridge;

[0060] S200: Determine the static displacement of the beam end corresponding to the train moving on the bridge according to the physical information of the train and the numerical curve of the influence of the beam end displacement;

[0061] S300: Determine the longitudinal restoring force stiffness and equivalent mass of the bridge according to the dynamic behavior of the bridge structure itself;

[0062] S400: Based on the static displacement of the beam end corresponding to the train moving on the bridge, the longitudinal restoring force stiffness and equivalent mass, establish an equilibrium displacement equation considering the bearing friction effect and solve the longitudinal displacement of the beam end based on the equilibrium displacement equation.

[0063] It should be noted that in S100, the bridge information includes the length information of the bridge, etc. When determining the numerical curve of the influence of the beam end displacement under the action of a moving unit force based on the finite element model, the relationship between the longitudinal displacement at the beam end and the position x of the moving unit force on the bridge can be obtained by performing a moving unit force loading calculation, and the numerical curve of the influence of the beam end displacement y(x) can be determined accordingly.

[0064] It can be understood that in the embodiment of the present invention, the numerical curve of the influence of the beam end displacement under the action of a moving unit force is determined by using the finite element, and then combined with the physical information of the train, the static displacement value of the beam end at the moment when the train moves on the bridge is determined. According to the dynamic mechanical behavior of the bridge structure itself, the equivalent mass and restoring force stiffness are determined, and an equilibrium displacement equation considering the bearing friction effect is established by using the dynamics theory. The equilibrium displacement solved by the equilibrium displacement equation is the longitudinal displacement of the beam end considering the bearing friction effect. Thus, the calculated beam end displacement considering the bearing friction effect is in good agreement with the actual value, making the longitudinal displacement calculation of the beam end accurate and efficient when the train crosses the bridge on the super-long-span railway bridge, meeting the actual displacement value of the project, reducing the specification of the beam end expansion device, and improving the durability of the expansion device. At the same time, the calculation time is only a few minutes, improving the calculation efficiency of engineering personnel.

[0065] In some embodiments, the physical information of the train in step S200 includes: the length of the train, the uniform load of the train, the traveling speed of the train, and the traveling time of the train, etc. S200 can obtain the static displacement of the beam end corresponding to the train moving on the bridge according to the first formula, and the first formula includes:

[0066]

[0067] where D(t) is the static displacement of the beam end corresponding to the train moving on the bridge, l is the length of the train, t is the traveling time of the train, q is the uniform load of the train, L is the length of the bridge, v is the traveling speed of the train, and y(t) is the numerical curve of the influence of the beam end displacement.

[0068] In some embodiments, step S300 includes:

[0069] S310: Based on the finite element model, apply a longitudinal force on the main beam and record the longitudinal deformation of the main beam;

[0070] S320: Determine the frequency of the first longitudinal drift of the bridge through modal analysis;

[0071] S330: Calculate the longitudinal restoring force stiffness and equivalent mass based on the longitudinal force, the longitudinal deformation, and the frequency of the first longitudinal drift of the bridge.

[0072] Further, S330 can calculate the longitudinal restoring force stiffness and equivalent mass based on the second formula, and the second formula includes: K = F / Δ, M = K / ω 2 , where F is the longitudinal force, Δ is the longitudinal deformation, ω is the frequency of the first longitudinal drift of the bridge, K is the longitudinal restoring force stiffness, and M is the equivalent mass.

[0073] Further, S400 can establish an equilibrium displacement equation considering the bearing friction effect based on the third formula, and the third formula includes:

[0074]

[0075] where n is the number of bearings distributed between the piers and the bridge towers, f is the bearing friction force, u is the bearing friction coefficient, G i is the vertical force of the i-th bearing, and B(t) is the longitudinal displacement of the beam end to be determined.

[0076] In a specific embodiment, taking a double-tower five-span railway cable-stayed bridge as an example, the bridge elevation layout is as Figure 2 shown. This cable-stayed bridge includes four parts: the main girder 1, the bridge towers 2, the stay cables 3, and the piers 6. Longitudinal movable bearings 5 are arranged between the main girder 1 and the piers 6 and the bridge towers 2. The span layout of this cable-stayed bridge is (140 + 490 + 1176 + 490 + 140) m, and the total length of the whole bridge is 2436 m.

[0077] Based on the method as Figure 1 shown, the specific calculation process is as follows:

[0078] Based on S100, establish a finite element calculation model of the bridge, and the longitudinal length of the bridge is L = 2436 m. Conduct a moving unit force loading calculation to obtain the numerical curve y(x) of the relationship between the longitudinal displacement at the beam end and the position x of the moving unit force on the bridge.

[0079] Based on S200, the length of the train is l = 550 m, the uniform load of the train is q = 64 kN / m, and the train running speed is set to v = 150 km / h. The time for the train to get on the bridge is 0, and at the running time to the t-th moment, the train runs to the bridge position x = vt. According to the x = vt relationship, transform the numerical curve y(x) into the numerical curve of the beam end displacement influence y(t), as Figure 3As shown, the time for the train to leave the bridge is (L + l) / v = 71.6 s. At the same time, the static displacement D(t) of the beam end can be calculated according to the first formula, as Figure 4 shown.

[0080] Based on S300, in the finite element model established on S100, a longitudinal force F = 1000 N is applied to the main beam, and the longitudinal deformation Δ of the main beam is recorded as Δ == 7.894×10 -6 = m; perform modal analysis to determine that the frequency of the first-order longitudinal drift is ω = 0.708 rad / s. According to the second formula, the longitudinal recovery stiffness K of the bridge structure is calculated as K = 126666616 N / m and the equivalent mass M = 2.521×10 8 kg.

[0081] Based on S400, under the action of the dead load, the vertical force of the main pier support 5.1 is 64302 kN, the vertical force of the auxiliary pier support 5.2 is 48915 kN, the vertical force of the side pier support 5.3 is 38332 kN, the friction coefficient u of the support is 0.03, and the friction resistance f of the full-bridge support is 9092 kN. According to the third formula, the displacement B(t) of the beam end considering the friction effect of the support (i.e., the longitudinal displacement of the beam end) is solved, as Figure 5 shown.

[0082] Figure 4 and Figure 5 By comparison, the maximum value of the static displacement of the beam end is 0.1 m, and the longitudinal displacement of the beam end considering the friction effect of the support is 0.07 m, with a decrease of 30%.

[0083] Through the refined calculation of the displacement of the beam end when the train passes through the bridge in the embodiment of the present invention, it is beneficial to reduce the specification of the beam end expansion device, improve the service life of the beam end expansion device, and ensure the formation safety of the train.

[0084] On the other hand, as Figure 6 shown, the embodiment of the present invention also provides a device for calculating the displacement of the beam end considering the friction of the support under the action of the train, which includes:

[0085] A static displacement determination module of the beam end, which is used for:

[0086] Establish a finite element model according to the bridge information and determine the numerical curve of the influence of the beam end displacement under the action of a moving unit force based on the finite element model, and the numerical curve of the influence of the beam end displacement reflects the relationship between the longitudinal displacement at the beam end and the position of the moving unit force on the bridge;

[0087] Determine the static displacement of the beam end corresponding to the train moving on the bridge according to the physical information of the train and the numerical curve of the influence of the beam end displacement;

[0088] A beam end displacement calculation module, which is used for:

[0089] Determine the longitudinal restoring force stiffness and equivalent mass of the bridge according to the dynamic behavior of the bridge structure itself;

[0090] Based on the static displacement of the beam end corresponding to the train moving on the bridge, the longitudinal restoring force stiffness and equivalent mass, establish an equilibrium displacement equation considering the bearing friction effect, and solve the longitudinal displacement of the beam end based on the equilibrium displacement equation.

[0091] In some embodiments, the physical information of the train includes: the length of the train, the uniform load of the train, the traveling speed of the train, and the traveling time of the train;

[0092] The beam end static displacement determination module is further configured to:

[0093] Obtain the static displacement of the beam end corresponding to the train moving on the bridge based on the first formula, and the first formula includes:

[0094]

[0095] where D(t) is the static displacement of the beam end corresponding to the train moving on the bridge, l is the length of the train, t is the traveling time of the train, q is the uniform load of the dynamic load, L is the length of the bridge, v is the traveling speed of the train, and y(t) is the numerical curve of the beam end displacement influence.

[0096] In some embodiments, the beam end displacement calculation module is further configured to:

[0097] Based on the finite element model, apply a longitudinal force on the main beam and record the longitudinal deformation of the main beam;

[0098] Determine the frequency of the first-order longitudinal drift through modal analysis;

[0099] Calculate the longitudinal restoring force stiffness and equivalent mass based on the longitudinal force, longitudinal deformation, and the frequency of the first-order longitudinal drift.

[0100] In some embodiments, the beam end displacement calculation module is further configured to:

[0101] Calculate the longitudinal restoring force stiffness and equivalent mass based on the second formula, and the second formula includes: K = F / Δ, M = K / ω 2 , where F is the longitudinal force, Δ is the longitudinal deformation, ω is the frequency of the first-order longitudinal drift, K is the longitudinal restoring force stiffness, and M is the equivalent mass.

[0102] In some embodiments, the beam end displacement calculation module is further configured to:

[0103] Establish an equilibrium displacement equation considering the bearing friction effect based on the third formula, and the third formula includes:

[0104]

[0105] Among them, n is the number of bearings distributed between the bridge pier and the bridge tower, f is the frictional resistance of the bearing, u is the coefficient of friction of the bearing, G i is the vertical force of the i-th bearing, and B(t) is the longitudinal displacement of the beam end to be determined.

[0106] It can be understood that the technical effects achievable by the above device embodiments are the same as those achievable by the method embodiments.

[0107] Those of ordinary skill in the art can understand that all or some of the steps in the methods disclosed above, and the functional modules / units in the systems and devices, can be implemented as software, firmware, hardware, and their appropriate combinations. In the hardware implementation, the division of the functional modules / units mentioned above does not necessarily correspond to the division of physical components; for example, a physical component can have multiple functions, or a function or step can be executed by several physical components in cooperation. Some or all physical components can be implemented as software executed by a processor, such as a central processing unit, a digital signal processor, or a microprocessor, or implemented as hardware, or implemented as an integrated circuit, such as an application-specific integrated circuit. Such software can be distributed on a computer-readable storage medium, which can include a computer-readable storage medium (or non-transitory medium) and a communication medium (or transitory medium).

[0108] It should be noted that in the present invention, relational terms such as "first" and "second" are only used to distinguish one entity or operation from another entity or operation, and do not necessarily require or imply any actual relationship or order between these entities or operations. Moreover, the terms "include", "comprise", or any other variant thereof are intended to cover non-exclusive inclusion, such that a process, method, article, or device that includes a series of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such process, method, article, or device. Without further limitation, an element defined by the statement "including one..." does not exclude the existence of additional identical elements in the process, method, article, or device that includes the element.

[0109] The above are only specific embodiments of the present invention, enabling those skilled in the art to understand or implement the present invention. Various modifications to these embodiments will be obvious to those skilled in the art, and the general principles defined herein can be implemented in other embodiments without departing from the spirit or scope of the present invention. Therefore, the present invention will not be limited to the embodiments shown herein, but rather will be accorded the widest scope consistent with the principles and novel features claimed herein.

Claims

1. A calculation method for the beam end displacement considering the bearing friction under the action of a train, characterized in that It includes the steps: Establish a finite element model according to the bridge information and determine the numerical curve of the influence of the beam end displacement under the action of a moving unit force based on the finite element model. The numerical curve of the influence of the beam end displacement reflects the relationship between the longitudinal displacement at the beam end and the position of the moving unit force on the bridge; Determine the static displacement of the beam end corresponding to the train moving on the bridge according to the physical information of the train and the numerical curve of the influence of the beam end displacement; Determine the longitudinal restoring force stiffness and equivalent mass of the bridge according to the dynamic behavior of the bridge structure itself; Establish an equilibrium displacement equation considering the bearing friction effect based on the static displacement of the beam end corresponding to the train moving on the bridge, the longitudinal restoring force stiffness and the equivalent mass, and solve the longitudinal displacement of the beam end based on the equilibrium displacement equation; The physical information of the train includes: the length of the train, the uniform load of the train, the traveling speed of the train, and the traveling time of the train. The step of determining the static displacement of the beam end corresponding to the train moving on the bridge according to the physical information of the train and the numerical curve of the influence of the beam end displacement includes: Obtain the static displacement of the beam end corresponding to the train moving on the bridge based on the first formula, and the first formula includes: ， Among them, is the static displacement at the beam end corresponding to the train moving on the bridge, is the length of the train, is the running time of the train, is the uniform load of the dynamic load, is the length of the bridge, v is the running speed of the train, is the numerical curve of the influence of the beam end displacement; The step of establishing an equilibrium displacement equation considering the bearing friction effect based on the static displacement of the beam end corresponding to the train moving on the bridge, the longitudinal restoring force stiffness and the equivalent mass includes: Establish an equilibrium displacement equation considering the bearing friction effect based on the third formula, and the third formula includes: ， ， Among them, n is the number of bearings distributed between the bridge piers and the bridge towers, f is the bearing friction resistance, is the bearing friction coefficient, is the i vertical force of the B ( t ) is the longitudinal displacement of the beam end to be determined.

2. The calculation method of beam end displacement considering support friction under the action of a train according to claim 1, characterized in that The step of determining the longitudinal restoring force stiffness and equivalent mass of the bridge according to the dynamic behavior of the bridge structure itself includes: Based on the finite element model, apply a longitudinal force to the main beam and record the longitudinal deformation of the main beam; Determine the frequency of the first-order longitudinal drift through modal analysis; Calculate the longitudinal restoring force stiffness and equivalent mass based on the longitudinal force, the longitudinal deformation, and the frequency of the first-order longitudinal drift.

3. The calculation method of beam end displacement considering support friction under the action of a train according to claim 2, characterized in that, The step of calculating the longitudinal restoring force stiffness and equivalent mass based on the longitudinal force, the longitudinal deformation, and the frequency of the first-order longitudinal drift includes: Calculate the longitudinal bridge direction restoring force stiffness and the equivalent mass based on the second formula, where the second formula includes: , where is the longitudinal bridge direction force, is the longitudinal bridge direction deformation, is the frequency of the first-order longitudinal bridge direction drift, is the longitudinal bridge direction restoring force stiffness, is the equivalent mass.

4. A device for calculating the displacement at the beam end considering the friction of the bearing under the action of a train, characterized in that, It includes: A static displacement determination module of the beam end, which is used for: Establish a finite element model according to the bridge information and determine the numerical curve of the influence of the beam end displacement under the action of a moving unit force based on the finite element model. The numerical curve of the influence of the beam end displacement reflects the relationship between the longitudinal displacement at the beam end and the position of the moving unit force on the bridge; Determine the static displacement of the beam end corresponding to the train moving on the bridge according to the physical information of the train and the numerical curve of the influence of the beam end displacement; A beam end displacement calculation module, which is used for: Determine the longitudinal restoring force stiffness and equivalent mass of the bridge according to the dynamic behavior of the bridge structure itself; Establish an equilibrium displacement equation considering the bearing friction effect based on the static displacement of the beam end corresponding to the train moving on the bridge, the longitudinal restoring force stiffness and the equivalent mass, and solve the longitudinal displacement of the beam end based on the equilibrium displacement equation; The physical information of the train includes: the length of the train, the uniform load of the train, the traveling speed of the train, and the traveling time of the train; The static displacement determination module of the beam end is also used for: Obtain the static displacement at the beam end corresponding to the train moving on the bridge based on the first formula, and the first formula includes: ， Among them, is the static displacement at the beam end corresponding to the train moving on the bridge, is the length of the train, is the running time of the train, is the uniform load of the dynamic load, is the length of the bridge, v is the running speed of the train, is the numerical curve of the influence of the beam end displacement; The beam end displacement calculation module is further configured to: Establish an equilibrium displacement equation considering the support friction effect based on the third formula, and the third formula includes: ， ， Among them, n is the number of bearings distributed between the bridge pier and the bridge tower, f is the bearing friction resistance, is the bearing friction coefficient, is the i vertical force of the B ( t ) is the longitudinal displacement of the beam end to be determined.

5. The beam end displacement calculation device considering the bearing friction under the action of a train according to claim 4, characterized in that The beam end displacement calculation module is further configured to: Based on the finite element model, apply a longitudinal force to the main beam and record the longitudinal deformation of the main beam; Determine the frequency of the first-order longitudinal drift through modal analysis; Calculate the longitudinal restoring force stiffness and the equivalent mass based on the longitudinal force, the longitudinal deformation, and the frequency of the first-order longitudinal drift.

6. The beam end displacement calculation device considering the bearing friction under the action of a train according to claim 5, characterized in that The beam end displacement calculation module is further configured to: Calculate the longitudinal bridge direction restoring force stiffness and the equivalent mass based on the second formula, where the second formula includes: , where is the longitudinal bridge direction force, is the longitudinal bridge direction deformation, is the frequency of the first-order longitudinal bridge direction drift, is the longitudinal bridge direction restoring force stiffness, is the equivalent mass.

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