A calculation method for seismic response of railway bridges based on friction pendulum bearings

Abstract

The present invention discloses a method for calculating the seismic response of railway bridges based on friction pendulum supports, which includes two major steps: first, a simplified analysis method for the response of seismic isolation supports is proposed, including simplified calculation of the displacement response of the main beam, each support, and pier / platform; optimization calculation of the equivalent damping ratio of the equivalent linearized system; simplification of the equivalent linearization analysis method; and explicitization of the equivalent linearization analysis method. Then, a simplified analysis method for the internal force response of the pier is proposed, including simplifying the internal force response of the pier to calculate its seismic effect as a rigid body; and calculating the combined shear force and bending moment at the pier bottom by summing the absolute values. The present invention reduces the computational workload, improves computational efficiency, and facilitates structural engineers to carry out a large amount of design optimization work.

Description

Technical Field

[0001] The present invention relates to the field of railway bridge engineering structure vibration reduction technology, and in particular to a railway bridge seismic response calculation method based on friction pendulum supports. Background Art

[0002] Friction pendulum bearing isolation systems are widely used in the seismic isolation design of high-intensity railway bridges. The use of nonlinear time domain analysis methods is time-consuming and labor-intensive, and is very unfriendly to structural engineers.

[0003] The equivalent linearization analysis method recommended in relevant international and domestic standards uses an implicit solution process that requires iterative calculations, making it inconvenient during design and use. While the first-order modal analysis method based on the equivalent linearization principle can effectively estimate the support displacement response of the seismic isolation system, it often significantly underestimates the bending moment at the pier base, especially for bridges with large substructures. Summary of the Invention

[0004] The present invention aims to solve the deficiencies of the prior art and provides a method for calculating the seismic response of a railway bridge based on friction pendulum supports.

[0005] In order to achieve the above-mentioned purpose, the present invention adopts the following technical solutions:

[0006] A method for calculating the seismic response of railway bridges based on friction pendulum bearings, including simplified analysis of the response of seismic isolation bearings and simplified analysis of the internal force response of piers;

[0007] Simplified analysis of seismic isolation bearing response includes: simplified calculation of displacement response of main beam, bearings and piers / platforms; simplified calculation of equivalent damping ratio of equivalent linearized system; simplified equivalent linearized analysis method; explicit equivalent linearized analysis method;

[0008] The simplified analysis of the internal force response of the bridge pier includes: calculating its seismic effect as a rigid body; and calculating the combined shear force and bending moment at the pier bottom by adding up the absolute values.

[0009] The displacement response of the main beam, supports and piers / platforms satisfies the formula:

[0010]

[0011] Where,

[0012] d is the seismic displacement response of the main beam of the bilinear displacement-dependent damper isolation system;

[0013] d i is the design displacement of the support at the i-th pier / abutment;

[0014] k p,i is the anti-thrust stiffness of the i-th pier / abutment;

[0015] k eff,i is the equivalent stiffness of the damper at the i-th pier / abutment;

[0016] Railway bridge design is controlled by stiffness, and the stiffness of the substructure far exceeds the equivalent stiffness of the friction pendulum isolation bearing, i.e.

[0017] k p,i >>k eff,i ;

[0018] The displacement response of the main beam, supports and piers / platforms is simplified as follows:

[0019] d i ≈d.

[0020] The calculation formula of the equivalent damping ratio of the equivalent linearized system is:

[0021]

[0022] Where,

[0023] ξ p,i is the structural damping ratio of the i-th pier / abutment, which is 0.05 for reinforced concrete structures;

[0024] k p,i is the anti-thrust stiffness of the i-th pier / abutment;

[0025] ξ eff,i represents the equivalent damping ratio of the damper at the i-th pier / abutment;

[0026] Q d,i is the characteristic strength of the damper at the i-th pier / abutment;

[0027] Considering the damping contribution of the main beam, the calculation formula of the equivalent damping ratio of the equivalent linearized system is simplified and adjusted to:

[0028]

[0029] Where,

[0030] ξ str is the damping ratio of the structural components other than the seismic isolation measures, which is 0.05 for reinforced concrete structures and 0.03 for steel structures. The simplified formula for the friction pendulum bearing seismic isolation system when applying the equivalent linearization analysis method is:

[0031]

[0032]

[0033] S D1 =S 0.05 ·T eff ;

[0034]

[0035] Where,

[0036] S D1 is the acceleration response spectrum value corresponding to T = 1s;

[0037] ξ is the damping ratio including the structure and seismic isolation measures;

[0038] μ and R are the friction coefficient and equivalent radius of the friction pendulum support, respectively;

[0039] S 0.05 is the response spectrum value corresponding to the equivalent period of the seismic isolation bridge with a damping ratio equal to 0.05;

[0040] η is the equivalent damping ratio correction coefficient.

[0041] The explicit process of the equivalent linearization analysis method is:

[0042] Establishing a railway bridge friction pendulum bearing isolation system S based on power function D1 The relationship between ~d, that is:

[0043] d=aS D1 b ;

[0044] Where a and b are the coefficients of the power function, through two groups (d i , S D1 ) Relationship points are obtained;

[0045] When d=0.1m and d=0.4m are selected, and according to and Computed and Substitute d = aS D1 b Find the values ​​of the coefficients a and b of the power function:

[0046]

[0047] Calculate the seismic effect as a rigid body:

[0048] The shear force V1 and bending moment M1 at the pier bottom under the first-order vibration mode are:

[0049]

[0050] The shear force V at the bottom of the pier is mainly caused by the natural vibration of the pier body. n and pier bottom bending moment M n for:

[0051]

[0052] Where,

[0053] W is the total dead load weight of the single-span superstructure;

[0054] H is the calculated height, which is the distance from the center of mass of the main beam to the bottom of the pier in the transverse direction of the bridge, and the distance from the center of the support to the bottom of the pier in the longitudinal direction of the bridge;

[0055] PGA is the peak acceleration of the design earthquake;

[0056] m i and h i are the mass of the pier along the height and the height from the bottom of the pier respectively.

[0057] Calculate the combined pier bottom shear force V by adding the absolute values p and bending moment M P for:

[0058] V p =V1+V n ;

[0059] M P =M1+M n .

[0060] The beneficial effects of the present invention are as follows: the simplified explicit calculation method of the present invention has good consistency with the average result of nonlinear time domain analysis; the simplified solution method considering the combination of the first-order vibration mode and the natural vibration mode of the pier body can well estimate the shear force and bending moment response of the pier bottom of each sampling sample, reduce the calculation workload, improve the calculation efficiency, and facilitate structural engineers to carry out a large amount of design optimization work. BRIEF DESCRIPTION OF THE DRAWINGS

[0061] Figure 1 is a flow chart of the present invention;

[0062] Figure 2 Schematic diagram of the steps for comparing and verifying the present invention with the traditional analysis model;

[0063] Figure 3 is the response spectrum comparison selected in the present invention (5% damping ratio);

[0064] Figure 4 The nonlinear time history of the deformation of the seismic isolation bearing of the present invention is compared with the results of the simplified explicit method;

[0065] Figure 5 is the ratio of the main beam displacement to the support deformation of each sampling working condition in the present invention;

[0066] Figure 6 This is a comparison of the nonlinear time history of pier bottom shear force in the present invention and the results of the simplified method.

[0067] Figure 7 The nonlinear time history of the pier bottom bending moment in the present invention is compared with the results of the simplified method;

[0068] The following is a detailed description of the embodiments of the present invention with reference to the accompanying drawings. DETAILED DESCRIPTION

[0069] The principles and features of the present invention are described below in conjunction with the accompanying drawings. The examples given are only used to explain the present invention and are not intended to limit the scope of the present invention. The following paragraphs describe the present invention in more detail by way of example with reference to the accompanying drawings. The advantages and features of the present invention will become more apparent from the following description. It should be noted that the drawings are all in a very simplified form and are not in exact proportions, and are only used to facilitate and clearly illustrate the purpose of the embodiments of the present invention.

[0070] Unless otherwise defined, all technical and scientific terms used herein have the same meaning as commonly understood by those skilled in the art to which this invention pertains. The terms used in this specification of the present invention are for the purpose of describing specific embodiments only and are not intended to limit the present invention. The term "and / or" as used herein includes any and all combinations of one or more of the associated listed items.

[0071] The present invention will be further described below with reference to the accompanying drawings and examples:

[0072] A calculation method for the seismic response of railway bridges based on friction pendulum supports, such as Figure 1 As shown, it includes simplified analysis of seismic isolation bearing response and simplified analysis of pier internal force response.

[0073] The simplified analysis of seismic isolation bearing response includes: simplified calculation of displacement response of main beam, each bearing and pier / platform; simplified calculation of equivalent damping ratio of equivalent linearized system; simplified equivalent linearization analysis method; explicit equivalent linearization analysis method.

[0074] The simplified analysis of the internal force response of the bridge pier includes: calculating its seismic effect as a rigid body; and calculating the combined shear force and bending moment at the pier bottom by adding up the absolute values.

[0075] (1) Simplified analysis method for seismic isolation bearing response

[0076] 1) Simplified calculation of displacement response of main beam, supports and piers / platforms.

[0077] The displacement responses of the main beam, supports and piers / platforms must meet the following requirements:

[0078]

[0079] Where d is the seismic displacement response of the main beam of the bilinear displacement-dependent damper isolation system; d iis the design displacement of the support at the i-th pier / abutment; k p,i is the anti-thrust stiffness of the i-th pier / abutment; k eff,i is the equivalent stiffness of the damper at the i-th pier / abutment;

[0080] Railway bridge design is controlled by stiffness, and thick solid piers are often used. The stiffness of the substructure is very large, generally far exceeding the equivalent stiffness of the friction pendulum isolation bearing, that is,

[0081] k p,i >>k eff,i ;(2)

[0082] Further, we can deduce:

[0083] d i ≈d. (3)

[0084] 2) Simplified calculation of equivalent damping ratio of equivalent linearized system.

[0085] The calculation formula of the equivalent damping ratio of the equivalent linearized system is:

[0086]

[0087] Where, ξ p,i is the structural damping ratio of the i-th pier / platform, which can be 0.05 for reinforced concrete structures; k p,i is the anti-thrust stiffness of the i-th pier / abutment; ξ eff,i represents the equivalent damping ratio of the damper at the ith pier / platform, which can be calculated as follows:

[0088]

[0089] Where Q d,i is the characteristic strength of the damper at the i-th pier / abutment.

[0090] Because formula (4) only considers the damping contribution of the pier column and does not consider the damping contribution of the main beam, formula (4) is simplified and adjusted to:

[0091]

[0092] Where, ξ str is the structural damping ratio (structural components excluding seismic isolation measures), which can be taken as 0.05 for reinforced concrete structures and 0.03 for steel structures;

[0093] 3) Simplification of equivalent linearization analysis method

[0094] Since the standard railway beam is a simply supported structure with a simple structural system, the coupling effect between different pier positions is relatively small. Therefore, the friction pendulum bearing isolation system used in my country's railway bridges can be simplified to the following when applying the equivalent linearization analysis method:

[0095]

[0096] μ and R are the friction coefficient and equivalent radius of the friction pendulum support respectively; S D1 is the acceleration response spectrum value corresponding to T = 1s, calculated by formula (10); T eff Calculated by formula (8); ξ str is the structural damping ratio (excluding structural components other than seismic isolation measures), which can be taken as 0.05 for reinforced concrete structures and 0.03 for steel structures; ξ is the structural damping ratio (including structure and seismic isolation measures).

[0097] Solve for S D1 and T eff Iterative calculation is required.

[0098] S D1 =S 0.05 ·T eff ;(10)

[0099] S 0.05 is the response spectrum value corresponding to the equivalent period of the seismic isolation bridge with a damping ratio equal to 0.05;

[0100] η is the equivalent damping ratio correction coefficient, which is specified in the seismic design code for highway bridges:

[0101]

[0102] 4) Explicit equivalent linearization analysis method

[0103] Equations (7) to (11) are implicit solution processes, which require iterative calculations.

[0104] The railway bridge friction pendulum bearing isolation system S can be established based on the power function D1 ~d relationship, that is:

[0105] d=aS D1 b ;(12)

[0106] Where a and b are the coefficients of the power function, which can be simply obtained by two sets of (d i , S D1 ) is obtained by selecting d = 0.1m and d = 0.4m, and calculating according to formula (7), formula (8) and formula (9) and (hereinafter referred to as and ), substitute into formula (12) to obtain the coefficients a and b of the power function:

[0107]

[0108] (2) Simplified analysis method of pier internal force response

[0109] For the friction pendulum bearing isolation system of railway bridges, the first-order vibration mode of the control bearing displacement response can ignore the influence of the pier mass, and the vibration mode dominated by the pier body's natural vibration can ignore the influence of the superstructure and bearings. Considering the high stiffness of the railway bridge substructure, it is simplified to calculate its seismic effect as a rigid body.

[0110] The shear force and bending moment at the pier bottom under the first-order vibration mode are:

[0111]

[0112] The shear force and bending moment at the pier bottom, which are mainly caused by the natural vibration of the pier body, are:

[0113]

[0114] Where W is the total dead load weight of the single-span superstructure; H is the calculated height, which should be the distance from the center of mass of the main beam to the bottom of the pier in the transverse direction and the distance from the center of the support to the bottom of the pier in the longitudinal direction; PGA is the peak acceleration of the design earthquake; m i and h i are the mass of the pier along the height and the height from the bottom of the pier respectively.

[0115] Considering that the first-order vibration mode of the friction pendulum isolation system of railway bridges is generally significantly different from the vibration mode dominated by the natural vibration of the pier body, the shear force and bending moment at the pier bottom after the combination are calculated by adding the absolute values:

[0116] V p =V1+V n ; 19)

[0117] M P =M1+M n (20) Specific embodiment 1:

[0119] like Figure 2 As shown in the figure, a double-track, simply supported girder bridge for a 350 km / h high-speed railway was selected. The main girder has a box-section, a single span of 31.5m, a single-span main girder weight of 8220kN, and a second-phase dead load of 4160kN. The bridge piers are solid, round-ended piers with a height of 12m. The bored cast-in-place pile group foundation has a pile diameter of 1.25m, a pile length of 48m, and 10 piles arranged in a quincunx pattern. The bridge site experiences a rare earthquake with a peak acceleration of 0.3g and a characteristic period of 0.65s. The horizontal seismic response spectrum is shown in the figure below. Figure 3As shown in Figure 2, nonlinear time history analysis was performed using finite element software.

[0120] Different pier heights and foundation stiffnesses are considered in the structural parameters. The geometric dimensions of the pier columns are sampled from h∈[0.0m, 17.0m], corresponding to a total pier height H∈[3.0m, 20.0m]. The adjustment coefficient is sampled from λ∈[0.5, 5.0]. The friction pendulum isolation bearing includes two design parameters: the equivalent radius of gyration R∈[3.0m, 6.0m] and the friction coefficient μ∈[0.03, 0.06]. Seismic motion parameters are sampled primarily to account for variations in seismic intensity, with the peak acceleration PGA∈[0.15g, 0.8g] for rare earthquakes.

[0121] Based on the aforementioned sampling parameters and value ranges, a Latin hypercube sampling method was used, with a total of 200 samples taken. For each sampling, the structural model was rebuilt by modifying the pier node coordinates and cross-sectional properties, foundation spring stiffness coefficients, and friction pendulum bearing unit constitutive parameters in the model. The input intensity of the 12 seismic waves was then adjusted based on the PGA sampling, and a nonlinear time-history analysis was performed to calculate the structural seismic response under all sampling conditions. Simultaneously, based on the simplified calculation method established above, the displacement of the seismic isolation system supports and the internal force response of the piers were calculated for each sampling condition and compared with the nonlinear time-history analysis results calculated from the model.

[0122] Figure 4 The figure shows a comparison of the relative displacement analysis results for the 200 sample conditions described above, using both nonlinear time-history analysis and the simplified explicit calculation method. It can be seen that for the same isolation design parameters and seismic intensity, the average results are highly consistent with those obtained using the explicit method. This demonstrates that the simplified explicit calculation method, based on this calculation theory, can provide excellent predictions of the displacement responses of the isolation supports for each sample condition.

[0123] For railway bridges, friction pendulum bearing isolation system is used. Figure 5 The figure shows the ratio of girder displacement to support deformation for 200 sample load cases with varying pier heights and foundation stiffnesses, using a nonlinear time-history analysis method. As can be seen, the maximum deviation in the ratio does not exceed 10%, and the average deviation is approximately 4%. This confirms the feasibility of this simplified approach.

[0124] Figure 6 The figure shows a comparison of the predicted shear force at the pier base using the average of 12 seismic wave nonlinear time-history analysis results for 200 sampled conditions and the simplified analysis method. It can be seen that the simplified analysis method, which considers the combination of the first-order mode and the pier's natural vibration modes, produces results very similar to the nonlinear time-history average, with an average deviation of approximately 3%.

[0125] Figure 7The figure shows a comparison of the average of the nonlinear time-history analysis results for 12 seismic waves for 200 sampled conditions and the predicted pier base bending moment using a simplified analysis method. It can be seen that the simplified analysis method, which considers the combination of the first-order mode and the pier's natural vibration modes, produces results very similar to the nonlinear time-history average, with an average deviation of approximately 5%.

[0126] It can be seen that the combination of considering the first-order vibration mode and the pier body's own vibration mode proposed in this paper can obtain better prediction results.

[0127] In summary, the present invention proposes a method for calculating the seismic response of railway bridges based on friction pendulum bearings, which has the following advantages:

[0128] First, a sampling analysis of operating conditions that systematically considers variations in parameters such as pier height, foundation stiffness, friction pendulum bearing radius of gyration, friction coefficient, and seismic intensity shows that the displacement response of the seismic isolation bearings based on a simplified explicit solution method is highly consistent with the nonlinear time history results.

[0129] Second, the simplified solution method for the internal force response of the pier considering the combination of the first-order vibration mode and the natural vibration mode of the pier body is in good consistency with the nonlinear time history results.

[0130] The present invention is described above by way of example in conjunction with the accompanying drawings. It is obvious that the specific implementation of the present invention is not limited to the above-mentioned method. As long as various improvements are made using the method concept and technical solution of the present invention, or they are directly applied to other occasions without improvement, they are all within the scope of protection of the present invention.

Claims

1. A method for calculating the seismic response of a railway bridge based on friction pendulum bearings, characterized in that: Including simplified analysis of seismic isolation bearing response and simplified analysis of pier internal force response; Simplified analysis of seismic isolation bearing response includes: simplified calculation of displacement response of main beam, bearings and piers / platforms; simplified calculation of equivalent damping ratio of equivalent linearized system; simplified equivalent linearized analysis method; explicit equivalent linearized analysis method; The simplified analysis of the internal force response of the bridge pier includes: calculating its seismic effect as a rigid body; calculating the combined shear force and bending moment at the pier bottom by adding the absolute values; Calculate the seismic effect as a rigid body: The shear force V1 and bending moment M1 at the pier bottom under the first-order vibration mode are: The shear force V at the bottom of the pier is mainly caused by the natural vibration of the pier body. n and pier bottom bending moment M n for: Where, W is the total dead load weight of the single-span superstructure; H is the calculated height, which is the distance from the center of mass of the main beam to the bottom of the pier in the transverse direction of the bridge, and the distance from the center of the support to the bottom of the pier in the longitudinal direction of the bridge; PGA is the peak acceleration of the design earthquake; m i and h i are the mass of the pier along the height and the height from the pier bottom respectively; Calculate the combined pier bottom shear force V by adding the absolute values p and bending moment M P for: V p =V1+V n ; M P =M1+M n 。 2. The method for calculating the seismic response of a railway bridge based on friction pendulum support according to claim 1 is characterized in that: The displacement response of the main beam, supports and piers / platforms satisfies the formula: Where, d is the seismic displacement response of the main beam of the bilinear displacement-dependent damper isolation system; d i is the design displacement of the support at the i-th pier / abutment; k p,i is the anti-thrust stiffness of the i-th pier / abutment; k eff,i is the equivalent stiffness of the damper at the i-th pier / abutment; Railway bridge design is controlled by stiffness, and the stiffness of the substructure far exceeds the equivalent stiffness of the friction pendulum isolation bearing, i.e. k p,i >>k eff,i ; The displacement response of the main beam, supports and piers / platforms is simplified as follows: d i ≈d。 3. The method for calculating the seismic response of a railway bridge based on friction pendulum bearings according to claim 2 is characterized in that: The calculation formula of the equivalent damping ratio of the equivalent linearized system is: Where, ξ p,i is the structural damping ratio of the i-th pier / abutment, which is 0.05 for reinforced concrete structures; k p,i is the anti-thrust stiffness of the i-th pier / abutment; ξ eff,i represents the equivalent damping ratio of the damper at the i-th pier / abutment; Q d,i is the characteristic strength of the damper at the i-th pier / abutment; Considering the damping contribution of the main beam, the calculation formula of the equivalent damping ratio of the equivalent linearized system is simplified and adjusted to: Where, ξ str The damping ratio of structural components other than seismic isolation measures is 0.05 for reinforced concrete structures and 0.03 for steel structures.

4. The method for calculating the seismic response of a railway bridge based on friction pendulum bearings according to claim 3 is characterized in that: The simplified formula of the friction pendulum bearing isolation system when applying the equivalent linearization analysis method is: S D1 =S 0.05 ·T eff ; Where, S D1 is the acceleration response spectrum value corresponding to T = 1s; ξ is the damping ratio including the structure and seismic isolation measures; μ and R are the friction coefficient and equivalent radius of the friction pendulum support, respectively; S 0.05 is the response spectrum value corresponding to the equivalent period of the seismic isolation bridge with a damping ratio equal to 0.05; η is the equivalent damping ratio correction coefficient.

5. The method for calculating the seismic response of a railway bridge based on friction pendulum bearings according to claim 4 is characterized in that: The explicit process of the equivalent linearization analysis method is: Establishing a railway bridge friction pendulum bearing isolation system S based on power function D1 The relationship between ~d, that is: daS D1 b ; Where a and b are the coefficients of the power function, through two groups (d i , S D1 ) Relationship points are obtained; When d=0.1m and d=0.4m are selected, and according to and Computed and Substitute d = aS D1 b Find the values ​​of the coefficients a and b of the power function:

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