Railway bridge earthquake response calculation method based on friction pendulum support

Abstract

The invention discloses a railway bridge earthquake response calculation method based on a friction pendulum support, which comprises the following two steps of: firstly, proposing an earthquake reduction and isolation support response simplification analysis method which comprises displacement response simplification calculation of a main beam, each support and a pier / platform; carrying out optimal calculation on the equivalent damping ratio of the equivalent linearization system; an equivalent linearization analysis method is simplified; and the equivalent linearization analysis method is explicit. Then providing a pier internal force response simplified analysis method which comprises the following steps: simplifying the pier internal force response into calculating an earthquake effect according to a rigid body; and calculating the combined shear force and bending moment of the pier bottom according to the sum of absolute values. According to the method, the operation workload is reduced, the calculation efficiency is improved, and a structure engineer can conveniently carry out a large amount of design optimization work.

Description

Technical Field

[0001] The invention relates to the field of railway bridge engineering structure vibration reduction technology, and in particular to a railway bridge earthquake response calculation method based on friction pendulum bearings. 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 domestic and foreign codes is an implicit solution process that requires iterative calculations, which is inconvenient in design and use. Although the first-order vibration mode analysis method based on the equivalent linearization principle can better estimate the support displacement response of the seismic isolation system, it often seriously underestimates the pier bottom bending moment, especially for bridges with large substructure volumes. 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 a friction pendulum support.

[0005] In order to achieve the above object, 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] The simplified analysis of the response of seismic isolation bearings includes: simplified calculation of displacement response of main beams, 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, each support and pier / platform satisfies the formula:

[0010]

[0011] In the formula,

[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 bearing at the i-th pier / abutment;

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

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

[0016] The design of railway bridges is controlled by stiffness, and the stiffness of the substructure far exceeds the equivalent stiffness of the friction pendulum isolation bearing, that is,

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

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

[0019] d i ≈d.

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

[0021]

[0022] In the formula,

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

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

[0025] ξ eff,i represents the equivalent damping ratio of the damper of 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 linearization system is simplified and adjusted to:

[0028]

[0029] In the formula,

[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 of 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] In the formula,

[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 of 0.05;

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

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

[0042] Establishment of the friction pendulum bearing seismic isolation system S for railway bridges based on power function D1 ~d, that is:

[0043] d=aS D1 b ;

[0044] In the formula, 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 Computational 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 self-vibration of the pier body. n and pier bottom bending moment M n for:

[0051]

[0052] In the formula,

[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 centroid 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;

[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 pier bottom 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 the nonlinear time domain analysis; the simplified solution method considering the combination of the first-order vibration mode and the pier body natural vibration mode 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 A 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 (5% damping ratio) selected in the present invention;

[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 condition in the present invention;

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

[0067] Figure 7 It is a comparison between the nonlinear time history of the pier bottom bending moment in the present invention and 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 embodiments are only used to explain the present invention and are not used to limit the scope of the present invention. The present invention is described in more detail by way of example with reference to the accompanying drawings in the following paragraphs. The advantages and features of the present invention will become clearer according to the following description. It should be noted that the accompanying drawings are all in a very simplified form and are not in precise proportions, and are only used to facilitate and clearly assist in explaining 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 those commonly understood by those skilled in the art of the present invention. The terms used herein in the specification of the present invention are only for the purpose of describing specific embodiments and are not intended to limit the present invention. The term "and / or" used herein includes any and all combinations of one or more related listed items.

[0071] The present invention will be further described below in conjunction with the accompanying drawings and embodiments:

[0072] A method for calculating the seismic response of railway bridges based on friction pendulum bearings, 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 the response of seismic isolation bearings includes: simplified calculation of the displacement response of the main beam, each bearing and pier / platform; simplified calculation of the equivalent damping ratio of the equivalent linearization system; simplified equivalent linearization analysis method; and 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 of seismic isolation bearing response

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

[0077] The displacement response of the main beam, supports and piers / platforms shall 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 bearing at the i-th pier / abutment; k p,i is the thrust stiffness of the ith pier / abutment; k eff,i is the equivalent stiffness of the damper at the i-th pier / platform;

[0080] Railway bridge design is controlled by stiffness, and sturdy 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 that:

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

[0088]

[0089] In the formula, 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, but not the damping contribution of the main beam, formula (4) is simplified and adjusted to:

[0091]

[0092] In the formula, 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;

[0093] 3) Simplification of equivalent linearization analysis method

[0094] Since the railway standard 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:

[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 seismic isolation measures for structural components), which can be taken as 0.05 for reinforced concrete structure and 0.03 for steel structure; ξ is the structural damping ratio (including structure and seismic isolation measures).

[0097] Solving 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 of 0.05;

[0100] η is the equivalent damping ratio correction coefficient, 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 requires 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 ), and substitute into equation (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 seismic 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 self-vibration can ignore the influence of the superstructure and bearings. Considering the high stiffness of the lower structure of the railway bridge, 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 bottom of the pier, which are mainly caused by the self-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 pier bottom respectively.

[0115] Considering that the first-order vibration mode of the friction pendulum seismic isolation system of railway bridges is generally different from the vibration mode period dominated by the pier body's natural vibration, the shear force and bending moment at the bottom of the combined pier 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 high-speed railway double-track simply supported beam bridge with a speed of 350 km / h is selected. The main beam is a box-shaped section, the single span span is 31.5m, the single span main beam weight is 8220kN, the second phase dead load is 4160kN, the bridge pier is a round-end solid pier, and the pier height is 12m. The bored cast-in-place pile group foundation has a pile diameter of 1.25m, a pile length of 48m, and a pile number of 10, arranged in a plum blossom shape. The peak acceleration of the rare earthquake at the bridge site is 0.3g, the characteristic period is 0.65s, and the horizontal seismic response spectrum is as follows Figure 3As shown in Figure 2, nonlinear time history analysis was performed using finite element software.

[0120] Different pier heights and foundation stiffness are considered in the structural parameters, among which the geometric dimension sampling parameter of the pier column h∈[0.0m,17.0m] corresponds to the total pier height H∈[3.0m,20.0m]; the adjustment coefficient sampling λ∈[0.5,5.0]. The friction pendulum seismic isolation bearing contains two design parameters, the bearing equivalent gyration radius parameter sampling R∈[3.0m,6.0m], and the friction coefficient parameter sampling μ∈[0.03,0.06]. The sampling of seismic motion parameters mainly considers the change in the intensity of seismic motion, and the peak acceleration of rare earthquakes PGA∈[0.15g,0.8g].

[0121] According to the above sampling parameters and value ranges, the Latin hypercube sampling method was used, with a total of 200 samplings. For each sampling, the structural model was reconstructed by modifying the pier node coordinates and section properties, foundation spring stiffness coefficients, and friction pendulum bearing unit constitutive parameters in the model. Then, the input intensity of 12 seismic waves was adjusted according to the PGA sampling, and nonlinear time history analysis was carried out to calculate the structural seismic response under all sampling conditions. At the same time, based on the simplified calculation method established above, the displacement of the seismic isolation system support and the internal force response of the pier under each sampling condition were calculated and compared with the nonlinear time history analysis results calculated by the model.

[0122] Figure 4 The figure shows the comparison of the analysis results of the relative displacement of the bearings for the above 200 sampling conditions based on nonlinear time history analysis and simplified explicit calculation method. It can be seen that for the same seismic isolation design parameters and seismic action intensity, the average results are in good agreement with the analysis results of the explicit method. This shows that the simplified explicit calculation method based on this calculation theory can give good prediction results for the displacement response of the seismic isolation bearings for each sampling condition.

[0123] For railway bridges, friction pendulum bearing isolation system is used. Figure 5 The figure shows the ratio of the girder displacement to the support deformation using the nonlinear time history analysis method for 200 sample conditions with different pier heights and foundation stiffnesses. It can be seen that the maximum deviation of the ratio does not exceed 10%, and the average deviation is about 4%. This confirms that this simplified treatment is reasonable and feasible.

[0124] Figure 6 The figure shows the comparison of the average value of the nonlinear time history analysis results based on 12 seismic waves for 200 sample conditions and the predicted results of the pier bottom shear force by the simplified analysis method. It can be seen that the simplified analysis method considers the combination of the first-order vibration mode and the pier body's natural vibration mode and gives a result that is very close to the average result of the nonlinear time history, and the average deviation is about 3%.

[0125] Figure 7The figure shows the comparison of the average value of the nonlinear time history analysis results based on 12 seismic waves and the prediction results of the pier bottom bending moment by the simplified analysis method for 200 sampling conditions. It can be seen that the simplified analysis method considers the combination of the first-order vibration mode and the pier body's natural vibration mode and gives a result very close to the average result of the nonlinear time history, which is about 5% higher on average.

[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 a railway bridge based on a friction pendulum support, which has the following advantages:

[0128] First, the analysis of sampling conditions that systematically considers the changes in parameters such as pier height, foundation stiffness, friction pendulum bearing gyration radius, friction coefficient, and seismic intensity shows that the displacement response of the seismic isolation bearing based on the simplified explicit solution method is well 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 pier body's natural vibration mode is well consistent 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 are directly applied to other occasions without improvement, they are all within the protection scope of the present invention.

Claims

1. A method for calculating the seismic response of a railway bridge based on a friction pendulum support, characterized in that: Including simplified analysis of seismic isolation bearing response and simplified analysis of pier internal force response; The simplified analysis of the response of seismic isolation bearings includes: simplified calculation of displacement response of main beams, 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; and calculating the combined shear force and bending moment at the pier bottom by adding up the absolute values.

2. The method for calculating the seismic response of a railway bridge based on a friction pendulum support according to claim 1 is characterized in that: The displacement response of the main beam, each support and pier / platform satisfies the formula: In the formula, 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 bearing at the i-th pier / abutment; k p,i is the thrust stiffness of the i-th pier / abutment; k eff,i is the equivalent stiffness of the damper at the i-th pier / platform; The design of railway bridges is controlled by stiffness, and the stiffness of the substructure far exceeds the equivalent stiffness of the friction pendulum isolation bearing, that is, k p,i >>k eff,i ; The displacement response of the main beam, supports and piers / platforms is simplified as: d i ≈d。 3. The method for calculating the seismic response of a railway bridge based on a friction pendulum support according to claim 2 is characterized in that: The calculation formula of the equivalent damping ratio of the equivalent linearized system is: In the formula, ξ p,i is the structural damping ratio of the ith pier / abutment, which is 0.05 for reinforced concrete structures; k p,i is the thrust stiffness of the i-th pier / abutment; ξ eff,i represents the equivalent damping ratio of the damper of 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 linearization system is simplified and adjusted to: In the formula, 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 a friction pendulum support 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 ; In the formula, 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 of 0.05; η is the equivalent damping ratio correction coefficient.

5. The method for calculating the seismic response of a railway bridge based on a friction pendulum support according to claim 4 is characterized in that: The explicit process of the equivalent linearization analysis method is: Establishment of the friction pendulum bearing seismic isolation system S for railway bridges based on power function D1 ~d, that is: daS D1 b ; In the formula, 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 Computational and Substitute d = aS D1 b Find the values ​​of the coefficients a and b of the power function:

6. The method for calculating the seismic response of a railway bridge based on a friction pendulum support according to claim 5, characterized in that: 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 self-vibration of the pier body. n and pier bottom bending moment M n for: In the formula, W is the total dead load weight of the single-span superstructure; H is the calculated height, which is the distance from the centroid 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 pier bottom respectively.

7. The method for calculating the seismic response of a railway bridge based on a friction pendulum support according to claim 6 is characterized in that: 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 。

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