A Displacement-Based Seismic Isolation and Vibration Reduction Design Method for Lead-Core Dampers

Abstract

This invention is a displacement-based lead-core damper seismic isolation design method, comprising the following steps: determining the applicable bridge system and establishing a full-bridge stress analysis model; determining the design displacement u. b ; Calculate the equivalent stiffness K of the lead-core damper e And equivalent damping ratio ζ e The preliminary arrangement scheme of the lead-core dampers was determined; the equivalent stiffness K1 and equivalent damping ratio ζ1 of the piers with lead-core dampers were calculated; if fixed supports were retained, the equivalent stiffness K2 and equivalent damping ratio ζ2 of the fixed piers were calculated; the bridge system was equivalent to a single-degree-of-freedom mechanical model, and the damping ratio ζ of the entire bridge was calculated. w And the damping influence coefficient η; based on the full bridge seismic isolation and reduction analysis model, calculate the horizontal force at the top of each pier and the displacement u of the bridge system; compare u b and u, if u b If u > u, then increase the number of dampers. b This reduces the number of dampers. This invention enables seismic isolation and damping design of lead-core dampers without the aid of computer programs, quickly determining the number of lead-core dampers, and is convenient and quick to operate.​

Description

Technical Field

[0001] This invention relates to the field of bridge seismic isolation technology, and in particular to a displacement-based lead-core damper seismic isolation design method. Background Technology

[0002] Bridges, as key engineering projects in the transportation system and an important component of lifeline engineering, are among the first to be damaged in an earthquake. Damage to bridges inevitably leads to traffic disruptions, making disaster relief efforts difficult or significantly delaying, and causing substantial losses. Therefore, seismic isolation and mitigation design for bridges is essential.

[0003] Lead-core dampers are widely used in bridges as seismic isolation devices. Determining the number and location of these dampers requires seismic isolation design. Since lead-core dampers need to undergo a certain displacement to form a hysteresis loop and dissipate energy, the design should ensure that the dampers displace to their design displacement for sufficient energy dissipation, while avoiding excessive displacement to guarantee the overall safety of the bridge system. Therefore, the number of dampers needs to be repeatedly adjusted during the design process to achieve the ideal displacement value.

[0004] Currently, the main seismic isolation and vibration reduction design methods are response spectrum method and time history analysis method. Although the response spectrum method considers the spectral characteristics and can obtain the maximum response of the structure, it cannot reflect the time history of the structure under seismic action and the duration effect of the seismic action. The time history analysis method is the main analysis method used in the design code. It has high calculation accuracy, but the calculation process is relatively complex and requires computer programs for modeling and calculation. Since the number of dampers needs to be repeatedly adjusted for seismic isolation and vibration reduction design using lead-core dampers, the repeated adjustment of the model is time-consuming and laborious when using traditional design methods for time history analysis. Summary of the Invention

[0005] The present invention aims to address the shortcomings of the prior art by providing a displacement-based design method for lead-core dampers for seismic isolation and damping.

[0006] To achieve the above objectives, the present invention adopts the following technical solution: a displacement-based lead-core damper seismic isolation design method, comprising the following steps:

[0007] S1. Determine the applicable bridge system and establish a full-bridge stress analysis model;

[0008] S2. Analyze the seismic resistance requirements of the bridge and determine its design displacement u based on the full-bridge stress model and the parameters of the lead-core damper. b ;

[0009] S3, based on the design displacement u b Calculate the equivalent stiffness K of the lead-core damper e And equivalent damping ratio ζ e ;

[0010] S4. The arrangement scheme of the lead-core dampers is initially determined, and the bridge system is equivalent to a linear model.

[0011] S5. Assuming that each pier has n dampers installed, calculate the equivalent stiffness K1 and equivalent damping ratio ζ1 of the pier with lead-core dampers installed according to the series and parallel stiffness theory.

[0012] S6. If the fixed support is retained, the lead core damper does not need to be installed on the fixed pier. Only the equivalent stiffness K2 and equivalent damping ratio ζ2 of the fixed pier need to be calculated.

[0013] S7. Treat the bridge system as a single-degree-of-freedom mechanical model, establish a full-bridge seismic isolation and damping analysis model using lead-core dampers, and calculate the damping ratio ζ of the entire bridge. w and damping influence coefficient η;

[0014] S8. Based on the full-bridge seismic isolation analysis model, calculate the horizontal force at the top of each pier and the displacement u of the bridge system.

[0015] S9, Comparison u b and u, if u b If u > u, then increase the number of dampers. b Then reduce the number of dampers and repeat process S5 to S8 until u and u b The difference meets the requirements;

[0016] S10. Determine the arrangement scheme of the lead-core damper through S1 to S9, establish a full-bridge analysis model using finite element software for time history analysis, verify the rationality of the arrangement scheme, and finally determine the arrangement scheme of the lead-core damper to complete the seismic isolation and damping design.

[0017] In particular, in S1, the bridge system is divided into beam bridges, arch bridges, rigid frame bridges, cable-stayed bridges, and composite system bridges. The stress mode and seismic isolation arrangement of each system are different and are determined according to the specific bridge type.

[0018] Specifically, in S2, the parameters of the lead-core damper are selected based on the bilinear restoring force model, including: the initial stiffness K of the lead-core damper. u Post-yield stiffness K d The ratio of initial stiffness to yield stiffness r, and the yield displacement u. y And the design displacement u b It should be ensured that the lead-core damper forms a hysteresis loop to dissipate energy sufficiently, while also meeting the displacement requirements of the main beam.

[0019] Specifically, in S3, the equivalent stiffness K of a single lead-core damper e And equivalent damping ratio ζ e Based on the equivalent linearization theory, the calculation formula is as follows: ​

[0020]

[0021] In the formula, μ is the displacement ductility coefficient, and

[0022] Specifically, in S5, the lead-core dampers are connected in parallel, and the effect of multiple lead-core dampers is equivalent to the superposition of the effect of a single lead-core damper. The new lead-core dampers are connected in series with the movable pier and the beam. According to the series and parallel theory, the equivalent stiffness K1 of the pier is calculated by the following formula:

[0023]

[0024] The equivalent damping ratio ζ1 is calculated using the modal strain energy theory, and the formula is as follows:

[0025]

[0026] In the formula, ζ s K is the elastic damping ratio of the bridge pier. P This refers to the horizontal stiffness of the bridge pier itself.

[0027] Specifically, in S6, the equivalent stiffness K2 and equivalent damping ratio ζ2 of the fixed pier are calculated using the following formulas:

[0028]

[0029] In the formula, μ P This is the displacement ductility ratio of the bridge pier.

[0030] Specifically, in S7, a simplified single-degree-of-freedom mechanical model is used, assuming that the pier mass is simulated using lumped mass, with half of the mass concentrated in the main girder and the other half in the pier base. The main girder is a rigid body, and the damping ratio ζ of the entire bridge is... w Calculate using the following formula:

[0031]

[0032] c = ∑c i ;

[0033]

[0034] K=ΣK i ;

[0035]

[0036] In the formula, c is the damping of the entire bridge, c i K i M i ζ iLet M0 be the damping, equivalent stiffness, mass, and equivalent damping ratio of the i-th pier, respectively; M0 be the mass of the main girder; M be the total mass of the bridge system; and K be the overall stiffness of the bridge system.

[0037] Specifically, in S8, the horizontal force at the pier top is calculated using the following formula:

[0038] F = α·β·η·M;

[0039]

[0040] The displacement u of the bridge system is calculated using the following formula:

[0041]

[0042] In the formula, α is the peak ground acceleration; β is the dynamic amplification factor of the structure, calculated based on the natural period of the structure and the dominant period of the earthquake; T g The characteristic period of the seismic wave is determined based on the selected seismic wave; T e Let be the natural period of the structure.

[0043] The beneficial effects of this invention are as follows: This invention enables seismic isolation and reduction design of lead-core dampers without the aid of computer programs, quickly determines the number of lead-core dampers, and is convenient and quick to operate; compared with conventional seismic isolation and reduction design, this method allows for free selection of the displacement generated by the lead-core dampers during an earthquake, and the target performance can be explicitly set by the owner, in addition to being given by the specifications, making the target achievable and satisfying both the specifications and the target performance requirements; this invention uses simplified calculations to ensure the speed and efficiency of seismic isolation and reduction design, and uses time history analysis for dual verification to ensure the reliability and stability of the calculation results. Attached Figure Description

[0044] Figure 1 This is a flowchart of the design method of the present invention;

[0045] Figure 2 This is a diagram of the polygonal restoring force model of the design method of this invention;

[0046] Figure 3 This is a layout diagram of a four-span continuous beam bridge in Embodiment 1 of the present invention;

[0047] Figure 4 This is a model diagram of the restoring force of the lead-core damper in Embodiment 1 of the present invention;

[0048] Figure 5 This is a schematic diagram of the support arrangement in Embodiment 1 of the present invention;

[0049] Figure 6 This is a diagram of the full-bridge single-degree-of-freedom model in Embodiment 1 of the present invention;

[0050] Figure 7 This is a time history curve of the displacement of the pier top under seismic waves in Embodiment 1 of the present invention;

[0051] The following will describe in detail, with reference to the accompanying drawings, embodiments of the present invention. Detailed Implementation

[0052] The present invention will be further described below with reference to embodiments:

[0053] like Figure 1 As shown, a displacement-based lead-core damper seismic isolation design method includes the following steps:

[0054] S1. Determine the applicable bridge system and establish a full-bridge stress analysis model. Bridge systems are divided into beam bridges, arch bridges, rigid frame bridges, cable-stayed bridges, and composite system bridges, etc. The stress modes and seismic isolation arrangements of each system are different and should be determined according to the specific bridge type.

[0055] S2. Analyze the seismic resistance requirements of the bridge and determine its design displacement u based on the full-bridge stress model and the parameters of the lead-core damper. b ;

[0056] The parameters of the lead-core damper are selected based on the bilinear restoring force model, such as... Figure 2 As shown, this includes: the initial stiffness K of the lead-core damper. u Post-yield stiffness K d The ratio of initial stiffness to yield stiffness r, and the yield displacement u. y And the design displacement u b It should be ensured that the lead-core damper forms a hysteresis loop to dissipate energy sufficiently, while also meeting the displacement requirements of the main beam.

[0057] S3, based on the design displacement u b Calculate the equivalent stiffness K of the lead-core damper e And equivalent damping ratio ζ e ;

[0058] The equivalent stiffness K of a single lead-core damper e And equivalent damping ratio ζ e Based on the equivalent linearization theory, the calculation formula is as follows:

[0059]

[0060] In the formula, μ is the displacement ductility coefficient, and

[0061] S4. The arrangement scheme of the lead-core dampers is initially determined, and the bridge system is equivalent to a linear model.

[0062] S5. Assuming that each pier has n dampers installed, calculate the equivalent stiffness K1 and equivalent damping ratio ζ1 of the pier with lead-core dampers installed according to the series and parallel stiffness theory.

[0063] The lead-core dampers are connected in parallel, and the effect of multiple lead-core dampers is equivalent to the superposition of the effect of a single lead-core damper. The new lead-core dampers are connected in series with the movable pier and the beam. According to the series and parallel theory, the equivalent stiffness K1 of the pier is calculated by the following formula:

[0064]

[0065] The equivalent damping ratio ζ1 is calculated using the modal strain energy theory, and the formula is as follows:

[0066]

[0067] In the formula, ζ s K is the elastic damping ratio of the bridge pier. P This refers to the horizontal stiffness of the bridge pier itself.

[0068] S6. If the fixed support is retained, the lead core damper does not need to be installed on the fixed pier. Only the equivalent stiffness K2 and equivalent damping ratio ζ2 of the fixed pier need to be calculated.

[0069] The equivalent stiffness K2 and equivalent damping ratio ζ2 of the fixed pier are calculated by the following formulas:

[0070]

[0071] In the formula, μ P This is the displacement ductility ratio of the bridge pier.

[0072] S7. Treat the bridge system as a single-degree-of-freedom mechanical model, establish a full-bridge seismic isolation and damping analysis model using lead-core dampers, and calculate the damping ratio ζ of the entire bridge. w and damping influence coefficient η;

[0073] A simplified single-degree-of-freedom mechanical model is adopted, assuming that the pier mass is simulated using lumped mass, with half of the mass concentrated in the main girder and the other half in the pier base. The main girder is a rigid body, and the damping ratio ζ of the entire bridge is assumed. w Calculate using the following formula:

[0074]

[0075] c = ∑c i ;

[0076]

[0077] K = ∑K i ;

[0078]

[0079] In the formula, c is the damping of the entire bridge, c i K i M i ζ i Let M0 be the damping, equivalent stiffness, mass, and equivalent damping ratio of the i-th pier, respectively; M0 be the mass of the main girder; M be the total mass of the bridge system; and K be the overall stiffness of the bridge system.

[0080] S8. Based on the full-bridge seismic isolation and damping analysis model, calculate the horizontal force at the top of each pier and the displacement u of the bridge system.

[0081] The horizontal force at the top of the pier is calculated using the following formula:

[0082] F = α·β·η·M;

[0083]

[0084] The displacement u of the bridge system is calculated using the following formula:

[0085]

[0086] In the formula, α is the peak ground acceleration; β is the dynamic amplification factor of the structure, calculated based on the natural period of the structure and the dominant period of the earthquake; T g The characteristic period of the seismic wave is determined based on the selected seismic wave; T e Let be the natural period of the structure.

[0087] S9, Comparison u b and u, if u b If u > u, then increase the number of dampers. b Then reduce the number of dampers and repeat process S5 to S8 until u and u b The difference meets the requirements;

[0088] S10. Determine the arrangement scheme of the lead-core damper through S1 to S9, establish a full-bridge analysis model using finite element software for time history analysis, verify the rationality of the arrangement scheme, and finally determine the arrangement scheme of the lead-core damper to complete the seismic isolation and damping design.

[0089] Example 1

[0090] like Figure 3 As shown, taking a four-span continuous beam as an example, the present invention is used for the design of lead-core dampers for seismic isolation.

[0091] S1. Determine the applicable bridge system and establish a full-bridge stress analysis model;

[0092] Taking a four-span continuous beam bridge with a span of (32+48+48+32)m as an example, the site characteristic period is T. g ​=0.45s, peak ground acceleration is 0.57g, bridge span arrangement is as follows Figure 3 As shown.

[0093] S2. Analyze the seismic resistance requirements of the bridge and determine its design displacement u based on the full-bridge stress model and the parameters of the lead-core damper. b ;

[0094] The parameters of the lead-core damper selected in this embodiment are as follows: initial stiffness K u =7879kN / m, yield displacement u y =0.06m, stiffness K after yielding d = 317kN / m, restoring force model as follows Figure 4 As shown. To fully utilize the energy dissipation capacity of the lead-core damper and ensure that the main beam displacement of this continuous beam bridge meets the requirements under seismic loading, the design displacement u is taken. b =0.23m.

[0095] S3, based on the design displacement u b Calculate the equivalent stiffness K of the lead-core damper e And equivalent damping ratio ζ e ;

[0096] For this embodiment, the displacement ductility coefficient is calculated as follows: μ = u b / u y =0.23 / 0.06=3.819, the ratio of initial stiffness to yield stiffness r=K d / K u =317 / 7879=0.0402.

[0097] The equivalent stiffness and equivalent damping are calculated as follows:

[0098]

[0099] S4. The arrangement scheme of the lead-core dampers is initially determined, and the bridge system is equivalent to a linear model.

[0100] In this embodiment, the fixed support on the intermediate pier P3 is retained, while the remaining piers are equipped with movable supports and one lead-core damper support, arranged as follows: Figure 5 As shown.

[0101] S5. Based on the series and parallel stiffness theory, calculate the equivalent stiffness K1 and equivalent damping ratio ζ1 of the pier with lead-core dampers installed. In this embodiment, the horizontal stiffness of the pier is K. P =2.803×10 5 kN / m.

[0102]

[0103] S6. The yield displacement of the fixed pier is 0.075mm. No lead-core damper is installed on the fixed pier. Only the equivalent stiffness K2 and equivalent damping ratio ζ2 of the fixed pier are calculated.

[0104]

[0105] S7. Treat the bridge system as a single-degree-of-freedom mechanical model, establish a full-bridge seismic isolation and damping analysis model using lead-core dampers, and calculate the damping ratio ζ of the entire bridge. w Calculate the damping influence coefficient ratio η;

[0106] The single-degree-of-freedom mechanical model in this embodiment is as follows: Figure 6 As shown, it is assumed that the mass of the bridge pier is simulated using concentrated mass, with half of the mass concentrated in the main beam and the other half concentrated at the bottom of the pier, and the main beam is a rigid body.

[0107] The formula for calculating the damping value of each bridge pier is as follows: Substituting, we get:

[0108] Movable block:

[0109] Fixed pier:

[0110] The full-bridge damping value is c = ∑c i =4c1+c2=5160591;

[0111] The total mass of the bridge is

[0112] The overall stiffness of the bridge is K = ∑K i =3.835×10 4 kN / m;

[0113] The full bridge damping ratio is:

[0114]

[0115] The damping influence coefficient of the entire bridge is:

[0116]

[0117] S8. Based on the full-bridge seismic isolation and damping analysis model, calculate the horizontal force at the top of each pier and the displacement u of the bridge system.

[0118] The first-order natural frequency of the bridge system in this embodiment is:

[0119]

[0120] The power amplification factor is:

[0121] The horizontal force at the top of the pier is:

[0122] F=α·β·η·M=0.57×9.8×0.379×0.671×6915399=9843kN;

[0123] The displacement u of the bridge system is:

[0124]

[0125] S9, Comparison u b and u, if u>u b If u < u b Then reduce the number of dampers and repeat process S4 to S7 until u and u b The difference meets the requirements;

[0126] Since in this embodiment u>u b Therefore, the number of dampers needs to be increased. Based on the stress characteristics of the continuous beam in this case, piers P2 and P4 bear a larger load from the main beam. Therefore, the number of lead-core dampers at piers P1 and P5 remains unchanged, while the number of lead-core dampers at piers P2 and P4 is increased, repeating steps S5-S8. Calculations show that when piers P1 and P5 have one lead-core damper and piers P2 and P4 have three lead-core dampers, u = u b =0.23m, which meets the requirements.

[0127] S10. The above calculations can determine the arrangement scheme of the lead-core damper. The finite element software is used to establish a full-bridge analysis model for time history analysis to verify the rationality of the arrangement scheme.

[0128] A full bridge model was created for time history analysis, and the time history curves of pier top displacements were calculated as follows: Figure 7 As shown, the maximum displacement is 0.221m, which is less than 10% different from the simplified calculation result. The final arrangement of the lead core damper was determined, and the seismic isolation design was completed.

[0129] This invention enables seismic isolation and reduction design of lead-core dampers without the aid of computer programs, quickly determining the number of lead-core dampers, and is convenient and fast to operate. Compared with conventional seismic isolation and reduction design, this method allows for free selection of the displacement generated by the lead-core dampers during an earthquake. The target performance is not only based on the specifications but can also be explicitly set by the owner, resulting in strong target achievability and meeting both specification requirements and target performance requirements. This method uses simplified calculations to ensure rapid and efficient seismic isolation and reduction design, while employing time history analysis for dual verification to ensure the reliability and stability of the calculation results.

[0130] The present invention has been described above by way of example. Obviously, the specific implementation of the present invention is not limited to the above-described manner. Any improvements made by adopting the inventive concept and technical solution of the present invention, or direct application to other occasions without modification, are all within the protection scope of the present invention.

Claims

1. A displacement-based lead-core damper seismic isolation design method, characterized in that, Includes the following steps: S1. Determine the applicable bridge system and establish a full-bridge stress analysis model; S2. Analyze the seismic resistance requirements of the bridge and determine its design displacement based on the full-bridge force model and the parameters of the lead-core damper. The parameters of the lead-core damper are selected based on the bilinear restoring force model, including the initial stiffness of the lead-core damper. Stiffness after yielding The ratio of initial stiffness to yield stiffness r, and yield displacement. And design displacement It should be ensured that the lead-core damper forms a hysteresis loop to dissipate energy fully, while also meeting the displacement requirements of the main beam; S3, based on the design displacement u b Calculate the equivalent stiffness of the lead-core damper and equivalent damping ratio The equivalent stiffness of a single lead-core damper and equivalent damping ratio Based on the equivalent linearization theory, the calculation formula is as follows: ； ； In the formula, It is the displacement ductility coefficient, and ; S4. The arrangement scheme of the lead-core dampers is initially determined, and the bridge system is equivalent to a linear model. S5. Assuming the number of dampers installed on each pier is n, calculate the equivalent stiffness of the pier with lead-core dampers installed based on the series and parallel stiffness theory. and equivalent damping ratio The lead-core dampers are connected in parallel, and the effect of multiple lead-core dampers is equivalent to the superposition of the effect of a single lead-core damper. The new type of lead-core damper is connected in series with the movable pier and the beam. According to the series and parallel theory, the equivalent stiffness of the pier is... Calculate using the following formula: ； Equivalent damping ratio The calculation method using modal strain energy theory is as follows: ； In the formula, The elastic damping ratio of the bridge pier. The horizontal stiffness of the bridge pier itself; S6. If the fixed support is retained, there is no need to install a lead-core damper on the fixed pier; only the equivalent stiffness of the fixed pier needs to be calculated. and equivalent damping ratio Equivalent stiffness of fixed pier and equivalent damping ratio Calculate according to the following formulas: ； ； In the formula, The displacement ductility ratio of the bridge pier; S7. Treat the bridge system as a single-degree-of-freedom mechanical model, establish a full-bridge seismic isolation and damping analysis model using lead-core dampers, and calculate the damping ratio ζ of the entire bridge. w and damping influence coefficient η; S8. Based on the full-bridge seismic isolation analysis model, calculate the horizontal force at the top of each pier and the displacement u of the bridge system. S9, Comparison u b and u, if u b If u > u, then increase the number of dampers. b Then reduce the number of dampers and repeat process S5~S8 until u and u b The difference meets the requirements;​ S10. Determine the arrangement scheme of the lead-core dampers through S1~S9, establish a full-bridge analysis model using finite element software for time history analysis, verify the rationality of the arrangement scheme, and finally determine the arrangement scheme of the lead-core dampers to complete the seismic isolation and damping design.

2. The displacement-based lead-core damper seismic isolation design method according to claim 1, characterized in that, In S1, the bridge system is divided into beam bridges, arch bridges, rigid frame bridges, cable-stayed bridges, and composite system bridges. The stress mode and seismic isolation arrangement of each system are different and are determined according to the specific bridge type.

3. The displacement-based lead-core damper seismic isolation design method according to claim 1, characterized in that, In S7, a simplified single-degree-of-freedom mechanical model is used, assuming that the pier mass is simulated using lumped mass, with half of the mass concentrated in the main girder and the other half in the pier base. The main girder is a rigid body, and the damping ratio of the entire bridge is ζ. w Calculate using the following formula: ； ； ； ； ； ； In the formula, c is the damping of the entire bridge, c i K i M i ζ i Let M0 be the damping, equivalent stiffness, mass, and equivalent damping ratio of the i-th pier, respectively; M0 be the mass of the main girder; M be the total mass of the bridge system; and K be the overall stiffness of the bridge system.

4. The displacement-based lead-core damper seismic isolation design method according to claim 3, characterized in that, In S8, the horizontal force at the top of the pier is calculated using the following formula: ； ； The displacement u of the bridge system is calculated using the following formula: ； In the formula, This refers to the peak ground acceleration (PGA). The dynamic amplification factor of the structure is calculated based on the natural vibration period of the structure and the dominant seismic period. The characteristic period of the seismic wave is determined based on the selected seismic wave. Let be the natural period of the structure.

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