Simplified calculation method of dynamic response of high-speed railway subgrade under cyclic dynamic load

Abstract

A simplified calculation method of high-speed railway subgrade dynamic response under cyclic dynamic load, comprising: regarding the track structure as an Euler-Bernoulli beam resting on the subgrade, obtaining the displacement control equation of the beam according to the force balance of the beam; discretizing the displacement control equation by using the finite difference method, and obtaining the stiffness matrix and the mass matrix of the beam according to the boundary conditions; constructing the relationship between the subgrade reaction and the subgrade deformation based on the state evolution constitutive model; assuming a group of initial subgrade reaction to be substituted into the iterative calculation process for calculation; through iterative calculation until the condition that |[Q]new-[Q]old|≤0.01[Q]old is established, the final deformation of the subgrade is obtained. The reasoning idea of the method is clear, the involved mechanics theory knowledge is simple and easy to understand, the overall calculation method is simple and easy to implement and accurate, and has good popularization and application value.

Description

Technical Field

[0001] This invention relates to a simplified calculation method for the dynamic response of high-speed railway subgrade under cyclic dynamic loads, belonging to the field of high-speed railway technology. Background Technology

[0002] With the development of China's social economy and the improvement of people's living standards, China's high-speed rail technology has also been continuously developing and has reached the world's leading level. To ensure the safety and stability of high-speed trains operating at high speeds, the requirements for settlement control of high-speed rail subgrades are extremely strict. The long-term cyclic dynamic load of high-speed trains has a significant and undeniable impact on the settlement of high-speed rail subgrades; therefore, long-term settlement deformation of high-speed rail subgrades is a crucial consideration in engineering projects. The load of high-speed trains is transferred to the subgrade through concrete structures such as the track slab. Considering the subgrade-track slab coupled response and using this to calculate the long-term settlement of the subgrade under cyclic dynamic loads is more consistent with actual engineering conditions. Currently, the conventional method for considering the subgrade-track slab coupled response is numerical simulation, which requires enormous computer resources and the prediction results are overly dependent on the model parameter values. In addition, some scholars have used the finite element method to couple the interaction between the track and the subgrade, but the reasoning and calculation process of this method is complex, requiring a significant amount of time and effort and is not easily understood or used by engineers.

[0003] Therefore, this paper proposes a reliable and simple method for calculating the settlement of high-speed railway subgrade-track coupling response under cyclic dynamic load. The settlement of track and subgrade is obtained through simple theoretical derivation and calculation, making it easier to apply to engineering practice. This is a technical problem with great engineering application value. Summary of the Invention

[0004] The purpose of this invention is to address the dynamic response problem of high-speed railway subgrade under long-term cyclic dynamic loads from high-speed trains. A simplified calculation method for the dynamic response of high-speed railway subgrade under cyclic dynamic loads is proposed, employing state evolution constitutive calculation to improve calculation accuracy.

[0005] This invention provides a simplified calculation method for the dynamic response of high-speed railway subgrade under cyclic dynamic loads, and the technical solution is as follows:

[0006] Step 1: Consider the upper track structure of the high-speed railway subgrade (concrete base, CA mortar, track slab) as an Euler Bernoulli beam resting on the subgrade. Assume that the beam and subgrade remain in contact and deform in coordination under train loads. Based on the beam's force equilibrium, the displacement control equation can be obtained:

[0007]

[0008] In the formula: EI is the bending stiffness of the beam; m is the mass of the beam per unit length; w is the deformation of the beam; x is the coordinate on the beam; t is time; Q(x,t) is the subgrade reaction force; P(x,t) is the external load (force provided by the track fasteners), and the time history curve formula for the fastener force is:

[0009] P(t)=a0+a1cos(ωt)+b1sin(ωt)+a2cos(2ωt)+b2sin(2ωt)

[0010] +a3cos(3ωt)+b3sin(3ωt); (2)

[0011] In the formula: a0, a1, a2, a3, b1, b2, b3 are parameters related to the train axle load, and ω is a parameter related to the train speed. The values ​​of the relevant parameters are shown in the table below:

[0012]

[0013]

[0014] Step 2: Discretize the above displacement control equations using the finite difference method to obtain the following formula:

[0015] [K][w]-[w][M]=[P]-[O]; (3)

[0016] In the formula: [K] is the stiffness matrix of the beam; [w] is the displacement matrix of the beam; [M] is the mass matrix of the beam; [P] is the external force matrix, which can be obtained through the fastener force equation; [Q] is the roadbed reaction force matrix.

[0017] The stiffness matrix [K] and the mass matrix [M] are as follows:

[0018]

[0019] In the formula: l is the difference length of the beam in the finite difference method, and h is the difference length of time.

[0020] Step 3: At this point, in equation (3) obtained by the finite difference method, the displacement matrix [w] of the beam and the reaction force matrix [Q] of the subgrade are both unknown. Due to the interaction of forces, the reaction force provided by the subgrade is equal to the external force on the subgrade, thus the relationship between the subgrade reaction force matrix [Q] and the displacement matrix [w] can be constructed.

[0021] Since the roadbed will deform when it is subjected to external forces, the roadbed deformation ws under the long-term action of external forces can be calculated by using the state evolution constitutive equation. Furthermore, based on the assumption that the concrete base and the roadbed are always in contact, the displacement matrix [w] of the beam can be equated with the roadbed deformation matrix [ws].

[0022] Step 4: The above-mentioned state evolution constitutive equations can effectively simulate the creep of subgrade fill material under static and dynamic loading conditions. Furthermore, the influence of historical strain on subgrade deformation can be considered to improve calculation accuracy. The state evolution constitutive equations are as follows:

[0023]

[0024] In the formula: σ, γ represents the dynamic stress on the surface of the subgrade and its first derivative with respect to time, which can be obtained by observing the surface load of the subgrade bed; E is the compression modulus of the subgrade, which is measured through basic geotechnical tests; γ, D represents the roadbed strain and its first derivative with respect to time (hereinafter referred to as strain rate); The fluidization parameters in the model and their first derivatives with respect to time; γ c , These are characteristic strain and model parameters, typically Or even smaller, the specific values ​​of the three parameters can be obtained by measuring the historical strain of the subgrade (field monitoring data) or the triaxial test of the subgrade fill material.

[0025] The state evolution constitutive equation is discretized using the finite difference method, resulting in equation (5):

[0026]

[0027] Since the external force on the subgrade is equal to the subgrade reaction force, the stress on the subgrade can be obtained from the subgrade reaction force matrix [Q], and the strain of the subgrade can be obtained through the state evolution constitutive equation. Finally, the subgrade deformation ws is calculated according to the subgrade deformation ws = γ × H, where H is the height of the subgrade. Based on the assumption that the concrete base and the subgrade always maintain contact and coordinate deformation, the displacement matrix [w] of the beam is equivalent to the subgrade deformation matrix [ws].

[0028] Step 5: Based on the above derivation, we can first assume a set of subgrade reaction force matrices [Q]old, calculate the subgrade deformation matrix [ws] using the method in Step 3, and then obtain the beam displacement matrix [w]. Substituting the beam displacement matrix [w] into Equation (3) yields a new set of subgrade reaction force matrices [Q]new. Then, we determine whether |[Q]new-[Q]old|≤0.01[Q]old holds true. If not, we continue to substitute the new subgrade reaction force matrix [Q]new into the method in Step (4) to calculate the beam displacement matrix [w], and then substitute it into Step 2 to obtain a new set of subgrade reaction force matrices [Q]new. We can repeat the above steps for iterative calculation until |[Q]new-[Q]old|≤0.01[Q]old holds true. When this condition holds true, the beam displacement matrix [w] obtained under this iterative step is the displacement under the final subgrade-track coupling response.

[0029] The beneficial effects of this invention are that it treats the superstructure of the high-speed railway subgrade (concrete base, CA mortar, track slab) as an Euler-Bernoulli beam resting on the subgrade. The finite difference method is used to discretize the beam's displacement control equations. A state evolution constitutive equation that considers historical strain is used to calculate the subgrade deformation ws under cyclic loading. Furthermore, by assuming an initial subgrade reaction matrix [Q], the settlement of the subgrade under the final subgrade-track coupled response is iteratively calculated. This method has a clear reasoning, involves simple and easy-to-understand mechanical theories, and its overall calculation method is simple, easy to implement, and accurate, possessing significant potential for widespread application. Attached Figure Description

[0030] Figure 1 This is a flowchart of the method described in this invention;

[0031] Figure 2 This is a flowchart of the iterative calculation process of the present invention;

[0032] Figure 3 This is a time history curve of the fastener force in an embodiment of the present invention;

[0033] Figure 4 This is the roadbed settlement curve finally calculated in the embodiment of the present invention. Detailed Implementation

[0034] The iterative calculation process of a simplified calculation method for the dynamic response of high-speed railway subgrade under cyclic dynamic load in this embodiment is as follows: Figure 1 As shown, the specific steps are described below with reference to the calculation values ​​of a specific embodiment:

[0035] Step 1: Consider the track structure as an Euler-Bernoulli beam resting on the roadbed. Based on the force equilibrium of the beam, obtain the displacement governing equation of the beam:

[0036]

[0037] Where: the bending stiffness of the track structure EI=1.13×103GN·m2; the mass of the track structure per unit length m=4.74×103kg / m2.

[0038] Assuming that the force of each fastener is equal along the x-direction, and based on the train axle load of 170kN and speed of 350km / h, the time history curve formula for the fastener force is taken as follows:

[0039] (2)

[0040] Figure 3 This is the time history curve of the above fastener force.

[0041] Step 2: Discretize the above displacement control equations using the finite difference method to obtain the following formula:

[0042] [K][w]-[w][M]=[P]-[O]; (3)

[0043] The stiffness matrix [K] and the mass matrix [M] are as follows:

[0044]

[0045] The track structure is 20m long, and the differential length of the track structure is taken as l = 0.2m; the total duration of the fastener force is 1s, and the time differential length is taken as h = 0.001s.

[0046] Step 3: At this point, in equation (3) obtained by the finite difference method, the displacement matrix [w] of the track structure and the subgrade reaction matrix [Q] are both unknown. Due to the interaction of forces, the reaction force provided by the subgrade is equal to the external force on the subgrade, thus the relationship between the subgrade reaction matrix [Q] and the displacement matrix [w] can be constructed.

[0047] Since the roadbed will deform when it is subjected to external forces, the roadbed deformation ws under the long-term action of external forces can be calculated by using the state evolution constitutive equation. Furthermore, based on the assumption that the concrete base and the roadbed are always in contact, the displacement matrix [w] of the beam can be equated with the roadbed deformation matrix [ws].

[0048] Step 4: The constitutive equation for the state evolution is:

[0049]

[0050] In the formula: σ represents the dynamic stress on the surface of the roadbed. Since the roadbed reaction matrix [Q] represents the reaction force received by the roadbed per unit length, dividing the roadbed reaction matrix [Q] by the corresponding track structure width of 3.1m yields the dynamic stress σ on the surface of the roadbed. It can also be calculated that the compressive modulus of the roadbed is E = 140 MPa; γ c =2.14×10 -4 ,

[0051] The state evolution constitutive equation is discretized using the finite difference method, resulting in equation (5):

[0052]

[0053] In the formula: the initial strain of the subgrade γ0 = 0; the initial fluidization parameter D0 = 0.05.

[0054] Since the external force on the subgrade is equal to the subgrade reaction force, the stress on the subgrade can be obtained from the subgrade reaction force matrix [Q]. The strain of the subgrade can be obtained through the state evolution constitutive equation. Finally, the subgrade deformation ws can be calculated from the subgrade deformation ws = γ × H, where H is the height of the subgrade. Based on the assumption that the concrete base and the subgrade always maintain contact and coordinate deformation, the displacement matrix [w] of the beam is equivalent to the subgrade deformation matrix [ws].

[0055] Step 5: Based on the above derivation, first assume that a set of subgrade reaction force matrix [Q]old is 1 / 2 of the external force matrix [P] ([P] can be obtained from equation (2)), that is, [Q]old = 1 / 2 [P]. Calculate the subgrade matrix [ws] using the methods in steps 3 and 4, and then obtain the beam displacement matrix [w]. Substitute the beam displacement matrix [w] into equation (3) to obtain a new set of subgrade reaction force matrix [Q]new. Then determine whether |[Q]new-[Q]old|≤0.01[Q]old holds true. If not, continue to substitute the new subgrade reaction force matrix [Q]new into the method in step 3 to calculate the beam displacement matrix [w], and then substitute it into step 2 to obtain a new set of subgrade reaction force matrix [Q]new. Repeat the above steps for iterative calculation until |[Q]new-[Q]old|≤0.01[Q]old holds true. When this condition is met, the beam displacement matrix [w] obtained under this iterative step is the displacement under the final roadbed-track coupled response.

[0056] Figure 4 The final calculated roadbed settlement curve for this example is derived from... Figure 4 It can be seen from this that, Figure 3 Under the action of fastener force, the elastic deformation of the roadbed is consistent with the form of fastener force, while the plastic deformation of the roadbed gradually increases with the increase of the fastener force application time.

[0057] The embodiments described above represent only one implementation of the present invention, and while the descriptions are specific and detailed, they should not be construed as limiting the scope of the present invention. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of the present invention, and these all fall within the scope of protection of the present invention. Therefore, the scope of protection of this patent should be primarily defined by the appended claims.

Claims

1. A simplified calculation method for the dynamic response of high-speed railway subgrade under cyclic dynamic load, characterized in that, Combining the coupled deformation of the roadbed and track structure, the deformation of the roadbed under cyclic loading is solved by iterative calculation. The specific steps of the method include: Step 1: Treat the track structure as an Euler-Bernoulli beam resting on the roadbed, and obtain the displacement control equation of the beam based on the force balance of the beam. Step 2: Discretize the displacement control equations using the finite difference method, and obtain the stiffness matrix and mass matrix of the beam based on the boundary conditions; Step 3: Based on the state evolution constitutive model, construct the relationship between subgrade reaction force and subgrade deformation; Step 4: Set a set of initial subgrade reactions and substitute them into the iterative calculation process for calculation; Step 5: Iterate through calculations until it is determined that |[Q]new-[Q]old|≤0.01[Q]old holds true, and obtain the final deformation of the roadbed; Where [Q]new is the new subgrade reaction force matrix, and [Q]old is the subgrade reaction force matrix; The specific method for combined roadbed-track structure coupling deformation is as follows: the beam and the roadbed are always kept in contact to coordinate deformation, and the displacement control equation of the beam is obtained based on the force balance of the beam. ；（1） In the formula: EI is the bending stiffness of the beam; m is the mass of the beam per unit length; w is the deformation of the beam; x is the coordinate on the beam; t is time; Q(x, t) is the subgrade reaction force; P(x, t) is the external load, i.e., the fastening force provided by the track fasteners, and the time history curve formula of the fastening force is: ；（2） In the formula: a0, a1, a2, a3, b1, b2, b3 are parameters related to the axle load of the train. These are parameters related to train speed; In step three, the reaction force provided by the roadbed is equal to the external force on the roadbed, thus establishing the relationship between the roadbed reaction force matrix [Q] and the displacement matrix [w]. Since the roadbed will deform when subjected to external forces, the roadbed deformation ws under long-term external force can be calculated by using the state evolution constitutive equation. Furthermore, based on the assumption that the concrete base and the roadbed are always in contact, the displacement matrix [w] of the beam is equivalent to the roadbed deformation matrix [ws].

2. The simplified calculation method for the dynamic response of high-speed railway subgrade under cyclic dynamic load as described in claim 1, characterized in that, The formulas for calculating the stiffness matrix and mass matrix in step two are as follows: ；（3） In the formula: [K] is the stiffness matrix of the beam; [w] is the displacement matrix of the beam; [M] is the mass matrix of the beam; [P] is the external force matrix, obtained through the fastener force equation; [Q] is the roadbed reaction force matrix; The stiffness matrix [K] and the mass matrix [M] are as follows: ； ； In the formula: l is the difference length of the beam in the finite difference method, and h is the difference length of time.

3. The simplified calculation method for the dynamic response of high-speed railway subgrade under cyclic dynamic load as described in claim 1, characterized in that, In the state evolution constitutive model described in step three, the stress-strain relationship of the constitutive model is as follows: ； ； In the formula: , The dynamic stress on the surface of the subgrade and its first derivative with respect to time are obtained through observation of the surface load of the subgrade bed; E is the compression modulus of the subgrade, obtained through basic geotechnical tests. , For the roadbed strain and its first derivative with respect to time; , These are the fluidization parameters in the model and their first derivatives with respect to time; , These are characteristic strain and model parameters, respectively.

4. The simplified calculation method for the dynamic response of high-speed railway subgrade under cyclic dynamic load as described in claim 3, characterized in that, In the iterative calculation process described in step four, a set of subgrade reaction force matrices [Q]old is first assumed. The subgrade deformation matrix [ws] is calculated using the method in step three, and then the beam displacement matrix [w] is obtained. At this point, the beam displacement matrix [w] is substituted into... A new subgrade reaction matrix [Q]new is obtained, and then it is determined whether |[Q]new-[Q]old|≤0.01[Q]old holds true; If this is not the case, then substitute the new subgrade reaction matrix [Q]new into the method in step three to calculate the beam displacement matrix [w], and then substitute it into... A new set of roadbed reaction force matrices [Q]new is obtained; Perform iterative calculations until |[Q]new-[Q]old|≤0.01[Q]old holds true; When |[Q]new-[Q]old|≤0.01[Q]old holds true, the obtained beam displacement matrix [w] is the displacement under the final roadbed-track coupled response.

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