A tuning vibration reduction method for pile foundation isolation of deep-water long-span continuous rigid frame bridge

Abstract

The invention discloses a tuning shock absorption method for pile foundation isolation of a deep-water large-span continuous rigid frame bridge, relates to the technical field of shock absorption of bridge engineering structures, and solves the problem that "pile foundation isolation" of a deep-water large-span continuous rigid frame bridge lacks a theoretical quantitative calculation method, so that the effect of "pile foundation isolation" cannot be fully exerted; the invention comprises the following steps: simplifying a deep-water large-span continuous rigid frame bridge model into a 2-DOF dynamic model; setting an optimization objective function based on the 2-DOF dynamic model; performing parameter optimization based on the optimization objective function; determining the structural seismic parameters of the deep-water large-span continuous rigid frame bridge based on the parameters obtained by optimizing in step 3; based on the basic principle of tuning shock absorption, the invention proposes a reasonable calculation method for pile foundation-cap-pier design parameters, so as to improve the shock absorption efficiency of the pile foundation isolation system.

Description

Technical Field

[0001] The invention relates to the technical field of bridge engineering structure shock absorption, and in particular to a tuning shock absorption method for pile foundation seismic isolation of a deep-water large-span continuous rigid frame bridge. Background Art

[0002] Seismic isolation design is an important technical approach for seismic design of bridges. It achieves the purpose of reducing the internal force response of the structure by presetting seismic isolation elements and extending the structural period. In conventional seismic isolation design, bearings are usually used as seismic isolation elements. In deep-water large-span continuous rigid frame bridges, since the main piers and main beams are connected by a fixed connection, it is not convenient to use seismic isolation bearings to achieve seismic isolation design. However, deep-water large-span continuous rigid frame bridges generally use high pile caps, the unconstrained piles are long, and the pile foundation has weak lateral stiffness, which can extend the natural vibration period of the structure, forming "pile foundation seismic isolation."

[0003] The engineering community has a certain understanding of the seismic isolation effect of pile foundations. Through the reasonable matching of superstructure mass-bridge pier stiffness-cap mass-pile foundation stiffness, more seismic energy can be converted into structural kinetic energy, further exerting the effect of "pile foundation seismic isolation", but its discussion is still limited to the qualitative level. How to effectively utilize this beneficial effect, optimize structural design, and give full play to the seismic isolation effect of pile foundations still lacks theoretical research and quantitative calculation methods. Summary of the invention

[0004] In order to solve the problems existing in the above-mentioned prior art, the present invention provides a tuning vibration reduction method for pile foundation isolation of deep-water large-span continuous rigid frame bridges, which solves the problem of lack of theoretical quantitative calculation method for "pile foundation isolation" of deep-water large-span continuous rigid frame bridges.

[0005] A tuning and damping method for pile foundation isolation of a deep-water large-span continuous rigid frame bridge comprises the following steps:

[0006] Step 1: Simplify the deep-water long-span continuous rigid frame bridge model into a 2-DOF dynamic model;

[0007] Step 2: Set the optimization objective function based on the 2-DOF dynamics model;

[0008] Step 3: Based on the optimization objective function, perform parameter optimization to obtain the optimal frequency ratio f, and calculate the modified pile foundation stiffness k according to the optimal frequency ratio f β1 ;

[0009] Step 4: Based on the modified pile foundation stiffness k β1 Calculate the modified pile foundation equivalent inertia moment I 41 ,I 51 , according to I 41 ,I 51 The corresponding pile arrangement and optimal diameter are inferred.

[0010] Preferably, the step 1 comprises:

[0011] Step 1.1: Simplify the deep-water long-span continuous rigid frame bridge model into a 3-DOF dynamic model;

[0012] Step 1.2: Simplify the 3-DOF model into a 2-DOF model.

[0013] Preferably, the step 1.1 comprises:

[0014] Assume that the anti-thrust stiffness of the pile group of the deep-water long-span continuous rigid frame bridge model is k p And by solving it through displacement method or finite element simulation method, the equivalent moment of inertia I can be obtained. p for:

[0015]

[0016] Where: L p 、E p are the pile length and elastic modulus respectively;

[0017] Based on the equivalent moment of inertia, the deep-water long-span continuous rigid frame bridge is simplified into a 3-DOF dynamic model. Its parameters include pier column stiffness, pile foundation stiffness and mass parameters. The specific formulas of the parameters are as follows:

[0018] The pier column stiffness includes the pier column stiffness along the bridge and the pier column stiffness across the bridge:

[0019]

[0020] The pile foundation stiffness includes the pile foundation longitudinal stiffness k4 and the pile foundation transverse stiffness k5, which can be calculated by force method or finite element method.

[0021] Main beam bending constraint correction

[0022]

[0023] Pier top quality:

[0024]

[0025] Pier bottom quality:

[0026]

[0027] Where I2 and I3 are the moments of inertia of the cross section of the pier column, I4 and I5 are calculated according to formula (1), m1 is the sum of the mass of the main beam and the pier column of the left span, m2 is the sum of the mass of the main beam and the pier column of the right span, m3 is the sum of the mass of the pier column and the pile foundation of the left span, m4 is the sum of the mass of the pier column and the pile foundation of the right span, and m b is the total mass of the main beam, m p2 , mp3 are the masses of the bridge piers, m p4 、m p5 are the mass of the pile foundation, L1 is the length of the main beam, L2 and L3 are the lengths of the piers, L4 and L5 are the lengths of the pile foundation, E1 is the elastic modulus of the main beam, E2 and E3 are the elastic modulus of the piers, and L4 and L5 are the elastic modulus of the pile foundation;

[0028] The mass matrix M and stiffness matrix K of the 3-DOF model are:

[0029]

[0030] Preferably, step 1.2 includes: assuming that the deep-water long-span continuous rigid frame bridge model is a symmetrical structure, that is, E2I2=E3I3, E4I4=E5I5, m1=m2, m3=m4, L2=L3, L4=L5, then further simplifying the 3-DOF model to a 2-DOF model, whose mass matrix M and stiffness matrix K are shown in equations (8) and (9), respectively:

[0031]

[0032] in,

[0033]

[0034] M α =m1+m2,M β =m3+m4 (11)

[0035] k α =4(k2+k3) (12)

[0036] k β =k4+k5 (13)

[0037] Among them, k α is the total stiffness of the 2-DOF pier integration, k β Total stiffness integrated for the 2-DOF pier.

[0038] Preferably, the step 2 comprises:

[0039] Step 2.1: Construct the motion equation of the 2-DOF system based on the 2-DOF model;

[0040] Step 2.2: Construct the optimization objective function based on the motion equation of the 2-DOF system.

[0041] Preferably, step 2.1 includes: considering the damping of the pier and pile foundation, the structure performs random vibration under the action of P(t), y1 and y2 are expressed as displacement time history functions of two degrees of freedom relative to the foundation, and the motion equation of the two-degree-of-freedom model can be obtained by the direct equilibrium method as follows:

[0042]

[0043] where c i —structural damping coefficient; y i —Relative displacement of particles; — relative velocity of particles; —Relative acceleration of the particle, P(t) is the inertial force caused by ground motion, which is a random excitation and can be decomposed into a superposition of a series of simple harmonic components, namely:

[0044]

[0045] For any frequency component θ, P (θ) (t) = p0e iθt For simple harmonic excitation, the displacement y1 (θ) (t) and y2 (θ) (t) is also a simple harmonic variable and can be expressed as follows:

[0046] y1 (θ) (t) = Y1e iθt ,y2 (θ) (t) = Y2e iθt (16)

[0047] P (θ) (t) and equation (16) are substituted into equation (14) to obtain

[0048] -m α θ 2 Y1+c α iθ(Y1-Y2)+k α (Y1-Y2)=p0

[0049] -m β θ 2 Y2+c β iθY2+c α iθ(Y2-Y1)+k β Y β +k α (Y2-Y1)=0 (17)

[0051] The amplitudes of y1(t) and y2(t) are obtained as follows:

[0052]

[0053] Substituting into equation (16) we can get y1 (θ) (t) and y2 (θ) (t), and then y1 (θ) (t) and y2 (θ)(t) are summed in the frequency domain to obtain y1(t) and y2(t).

[0054] Preferably, the step 2.2 includes: setting the pier bottom bending moment as an indicator for evaluating the overall response of the structure to:

[0055] M(t)=k α L α (y1(t)-y2(t)) (20)

[0056] For any frequency component θ, we have

[0057] M (θ) (t) = k α L α (y1 (θ) (t)-y2 (θ) (t)) (21)

[0058] Substituting equations (18) and (19) into equation (21), we can obtain M (θ) (t), and express it in the form of transfer function H(θ):

[0059]

[0060] in,

[0061]

[0062] When the variance of M(t) When it is minimum, M(T) is also minimum, and the variance is set to:

[0063] σ m 2 =E[|M(t)| 2 ] (twenty four)

[0064] Substituting equation (23) into the dimensionless form, we can obtain the variance The dimensionless form I:

[0065]

[0066] Where g is the ratio of earthquake excitation to structural frequency H(g) is the dimensionless form of the transfer function H(θ):

[0067]

[0068] in,

[0069]

[0070] In formula (26), μ represents the mass ratio of the equivalent mass point between the cap and the main beam, ω αrepresents the natural vibration frequency of the main beam and pier, ω β represents the natural vibration frequency of the cap and pile foundation, f represents the frequency ratio of the pile foundation to the pier column, f represents the frequency ratio of the external load excitation to the pier column, c cα represents the critical damping of the bridge pier, c cβ represents the critical damping of pile foundation;

[0071] Formula (25) can be rearranged as:

[0072] in,

[0073]

[0074] Substitute equation (28) into equation (25), integrate, and let The variance σ can be obtained 2 The dimensionless form I of is used as the objective function for optimization:

[0075]

[0076] Preferably, step 3 comprises: optimizing the variance by enumeration method, finding the optimal frequency ratio f corresponding to each mass ratio μ when the variance is minimum, and performing curve fitting to obtain a display expression of the optimal frequency ratio f:

[0077] f=1.108e -0.02755μ -1.116e -0.4368μ (31)

[0078] Calculate the total stiffness correction value k of the 2-DOF pile foundation integration according to the optimal frequency ratio f β1 :

[0079]

[0080] Preferably, the optimization process of the enumeration method is as follows:

[0081] (1) Determine the mass ratio μ i ;

[0082] (2) ergodic frequency ratio f;

[0083] (3) Calculate the variance I(f,μ);

[0084] (4) Minimize the variance I;

[0085] (5) The corresponding I(f i ,μ i );

[0086] (6) Update μ i =μ i +Δμ; and at the same time, the optimal frequency ratio and mass ratio relationship f(u) is obtained.

[0087] Preferably, step 4 comprises:

[0088] The mass ratio μ calculated by formula (27) is substituted into formula (31) to obtain the optimal frequency ratio f of the pile foundation. Substituting each parameter into formula (32) obtains the total stiffness correction value k of the 2-DOF pile foundation integration: β1 , calculate the corrected pier pile foundation stiffness k 41 , k 51 , the equivalent moment of inertia of the pile foundation I is calculated by formula (1) 41 ,I 51 , according to I 41 ,I 51 Infer the corresponding pile arrangement and optimal diameter, where k 41 , k 51 The calculation formula is as follows:

[0089]

[0090] The beneficial effects of the present invention include:

[0091] Based on the basic principle of tuned shock absorption, the present invention proposes a reasonable method for calculating the design parameters of pile foundation-cap-pier column, so as to improve the shock absorption efficiency of the pile foundation isolation system. By adopting the optimized pile foundation diameter obtained after random excitation optimization, the response of the pier column, pile foundation and main beam is greatly reduced, and the seismic response of the continuous rigid frame bridge is effectively controlled. The method for determining the design parameters of the pile foundation and pier is changed from relying on experience to determining through structural dynamics calculation. The parameters are more accurate and the parameter determination speed is faster, which significantly shortens the design time and reduces the design cost. BRIEF DESCRIPTION OF THE DRAWINGS

[0092] Figure 1 This is the mechanical model of the tuned mass damper involved in Example 1.

[0093] Figure 2 This is the high pile cap continuous rigid frame bridge involved in Example 1 and its equivalent model.

[0094] Figure 3 (a) is the pile group foundation involved in Example 1 simplified into a one-dimensional structure.

[0095] Figure 3 (b) is a simplified plane model of the pile group foundation involved in Example 1.

[0096] Figure 4 This is a simplified model of the continuous rigid frame bridge with three degrees of freedom involved in Example 1.

[0097] Figure 5 This is a simplified model of the continuous rigid frame bridge with two degrees of freedom involved in Example 1.

[0098] Figure 6 This is the two-degree-of-freedom system model involved in Example 1.

[0099] Figure 7 This is the three-dimensional graph of I(f, μ) involved in Example 1.

[0100] Figure 8 This is a flow chart of the optimal frequency ratio calculation involved in Example 1.

[0101] Fig. 9 This is the optimal frequency ratio curve diagram involved in Example 1.

[0102] Fig.10 The present invention is a flow chart of a tuned vibration reduction method for pile foundation isolation of a deep-water large-span continuous rigid frame bridge. DETAILED DESCRIPTION

[0103] In order to make the purpose, technical scheme and advantages of the embodiments of the present application clearer, the technical scheme in the embodiments of the present application will be clearly and completely described below in conjunction with the drawings in the embodiments of the present application. Obviously, the described embodiments are only part of the embodiments of the present application, rather than all of the embodiments. Therefore, the following detailed description of the embodiments of the present application provided in the drawings is not intended to limit the scope of the application for protection, but merely represents the selected embodiments of the present application. Based on the embodiments of the present application, all other embodiments obtained by those skilled in the art without making creative work belong to the scope of protection of the present application.

[0104] Example 1

[0105] The following is combined with Figure 1-10 The specific embodiments of the present invention are described in detail;

[0106] The core component of mass tuned vibration reduction is tuned mass damper (TMD), whose mechanical model is composed of mass block, damper and spring. Figure 1 When TMD is added to the original system, the original system vibrates under the dynamic load, and TMD generates an opposite force to offset the vibration of the original system and reduce the dynamic response.

[0107] For a deep-water long-span continuous rigid frame bridge, the mass is mainly distributed in the main beam and the pedestal. The pedestal mass can be regarded as the first mass M1, the main beam can be regarded as the second mass M2, the unconstrained pile foundation in the water can be regarded as the first spring K1, and the bridge pier can be regarded as the second spring K2. Therefore, the high-pile pedestal continuous rigid frame bridge can be equivalent to a TMD system, such as Figure 2 , thereby utilizing the principle of mass-tuned shock absorption to seek a reasonable pile foundation-pier stiffness ratio and main beam-cap mass ratio to reduce the seismic response of the structure, and guiding the seismic optimization design of the "pile foundation isolation" system bridge based on the optimal stiffness ratio or mass ratio.

[0108] A tuning vibration reduction method for pile foundation isolation of a deep-water large-span continuous rigid frame bridge specifically comprises the following steps:

[0109] Step 1: Simplify the mechanical model;

[0110] Step 1.1: Simplify the deep-water long-span continuous rigid frame bridge model into a 3-DOF dynamic model;

[0111] In the direction perpendicular to the input direction of the seismic wave, each pile foundation is in parallel, so its bending stiffness and axial stiffness are the sum of the number of pile foundations perpendicular to the input direction of the seismic wave, such as Figure 3 As shown in (a), the two-dimensional pile foundation structure can be simplified to a one-dimensional plane structure; since the stiffness of the cap is much greater than that of the pile foundation, it can be regarded as a rigid body, so the simplified plane model of the pile group foundation is as follows: Figure 3 (b) as shown.

[0112] Considering the axial deformation of the pile foundation, there are 3 key displacements. Even if there are n pile foundations along the input direction of the seismic wave, the number of basic unknown quantities in the displacement method is still 3. The thrust stiffness k of the pile group is solved by p , equivalent moment of inertia of pile group I p for:

[0113]

[0114] Where: L p 、E p are the pile length and elastic modulus, k p is the anti-thrust stiffness of the pile group; based on the simplified plane model of the pile group foundation, the simplified mechanical model of the continuous rigid frame is obtained as follows: Figure 4 As shown;

[0115] The stiffness parameters in the figure include pier column stiffness, pile foundation stiffness, and mass parameters according to the structural dynamics method formula as follows:

[0116] The pier column stiffness includes the pier column stiffness along the bridge and the pier column stiffness across the bridge:

[0117]

[0118] The pile foundation stiffness includes the pile foundation longitudinal stiffness k4 and the pile foundation transverse stiffness k5, which can be calculated by force method or finite element method.

[0119] Main beam bending constraint correction

[0120]

[0121] Pier top quality:

[0122]

[0123] Pier bottom quality:

[0124]

[0125] Where I2 and I3 are the moments of inertia of the cross-section of the pier column, I4 and I5 are the equivalent moments of inertia of the pile foundation, calculated according to formula (1), m1 is the sum of the mass of the main beam and the pier column of the left span of the bridge, m2 is the sum of the mass of the main beam and the pier column of the right span of the bridge, m3 is the sum of the mass of the pier column and the pile foundation of the left span of the bridge, m4 is the sum of the mass of the pier column and the pile foundation of the right span of the bridge, and m b is the total mass of the main beam, m p2 , m p3 are the masses of the bridge piers, m p4 、m p5 are the mass of the pile foundation, L1 is the length of the main beam, L2 and L3 are the lengths of the piers, L4 and L5 are the lengths of the pile foundation, E1 is the elastic modulus of the main beam, E2 and E3 are the elastic modulus of the piers, and L4 and L5 are the elastic modulus of the pile foundation;

[0126] Since the axial stiffness of the main beam L1 is infinite, Figure 4 The 4-mass model shown is essentially a 3-DOF model. The first degree of freedom is the horizontal displacement of m1+m2, the second degree of freedom is the horizontal displacement of m3, and the third degree of freedom is the horizontal displacement of m4. The mass matrix M and stiffness matrix K of the model are:

[0127]

[0128] Step 1.2: Simplify the 3-DOF model into a 2-DOF model;

[0129] Obviously, Figure 4 In the 3-DOF model shown, the movement directions of the particles m3 and m4 are only in the same direction and in the opposite direction. Antisymmetric vibration modes appear when they move in the same direction, and positive symmetric vibration modes appear when they move in the opposite direction. Since the longitudinal seismic excitation is antisymmetric excitation, it will only excite antisymmetric vibration modes. Therefore, the positive symmetric vibration modes under the longitudinal seismic action have no contribution to the structural response. From the perspective of engineering application, in order to simplify the derivation, it is assumed that the structure is a symmetrical structure, that is, E2I2=E3I3, E4I4=E5I5, m1=m2, m3=m4, L2=L3, L4=L5, then Figure 4 The 3-DOF model can be further simplified as Figure 5 For the 2-DOF model shown in Figure 1, its mass matrix M and stiffness matrix K are shown in equations (8) and (9), respectively:

[0130]

[0131] in,

[0132]

[0133] m α=m1+m2,m β =m3+m4 (11)

[0134] k α =4(k2+k3) (12)

[0135] k β =k4+k5 (13)

[0136] Among them, k α is the total stiffness of the 2-DOF pier integration, k β Total stiffness integrated for the 2-DOF pier.

[0137] Step 2: Set the optimization objective function based on the 2-DOF dynamics model;

[0138] Step 2.1: Construct the motion equation of the 2-DOF system based on the 2-DOF model;

[0139] The 2-DOF system model is as follows Figure 6 As shown, considering the damping of the piers and pile foundations, the structure vibrates randomly under the action of P(t), and y1 and y2 are expressed as time-history functions of displacements with two degrees of freedom relative to the foundation.

[0140] The motion equation of the 2-DOF model can be obtained by the direct balance method as follows:

[0141]

[0142] where c i —structural damping coefficient; y i —Relative displacement of particles; — relative velocity of particles; —Relative acceleration of the particle, P(t) is the inertial force caused by ground motion, which is a random excitation and can be decomposed into a superposition of a series of simple harmonic components, namely:

[0143]

[0144] For any frequency component θ, P (θ) (t) = p0e iθt For simple harmonic excitation, the displacement y1 (θ) (t) and y2 (θ) (t) is also a simple harmonic variable and can be expressed as follows:

[0145] y1 (θ) (t) = Y1e iθt ,y2 (θ) (t) = Y2e iθt (16)

[0146] P (θ)(t) and formula (17) into formula (15) we can get

[0147] -m α θ 2 Y1+c α iθ(Y1-Y2)+k α (Y1-Y2)=p0

[0148] -m β θ 2 Y2+c β iθY2+c α iθ(Y2-Y1)+k β Y β +k α (Y2-Y1)=0 (17)

[0150] The amplitudes of y1(t) and y2(t) are obtained as follows:

[0151]

[0152] Substituting into equation (16) we can get y1 (θ) (t) and y2 (θ) (t), and then summing them in the frequency domain to get y1(t) and y2(t).

[0153] Step 2.2: Optimize the objective function;

[0154] The optimization should aim to minimize the overall dynamic response of the structure. The main indicators that can reflect the overall dynamic response of the structure include parameters such as the pier bottom bending moment and main beam displacement. The pier bottom bending moment and main beam displacement are interrelated. The optimization objects are the stiffness ratio between the pile foundation and the pier, and the mass ratio between the abutment and the main beam. Therefore, by using the pier bottom bending moment and main beam displacement as the objective function for optimization, a reasonable match of the main beam mass-pier stiffness-abutment mass-pile foundation stiffness can be obtained, thereby achieving an optimized design of the pile foundation isolation system. In bridge earthquake resistance, the pier bottom bending moment is often the key parameter controlling the design. In this embodiment, the pier bottom bending moment is used as an indicator to evaluate the overall response of the structure. Figure 6 It can be seen that the bending moment at the pier bottom is:

[0155] M(t)=k α L α (y1(t)-y2(t)) (20)

[0156] Similarly, for any frequency component θ, we have

[0157] M (θ) (t) = k α L α (y1 (θ) (t)-y2 (θ)(t)) (21)

[0158] Substituting equations (19) and (20) into equation (22), we can obtain M (θ) (t), and express it in the form of transfer function H(θ) as

[0159]

[0160] in,

[0161]

[0162] Since the earthquake motion is a zero-mean random process, M(t) is also a zero-mean random process. When the variance of M(t) is When it is minimum, M(t) is also minimum. According to the definition of variance, we can get:

[0163] σ m 2 =E[|M(t)| 2 ] (twenty four)

[0164] Substituting equation (23) into the dimensionless form, we can obtain the variance The dimensionless form I:

[0165]

[0166] Where g is the ratio of earthquake excitation to structural frequency H(g) is the dimensionless form of the transfer function H(θ):

[0167]

[0168] in,

[0169]

[0170] In formula (26), μ represents the mass ratio of the equivalent mass point between the cap and the main beam, ω α represents the natural vibration frequency of the main beam and pier, ω β represents the natural vibration frequency of the cap and pile foundation, f represents the frequency ratio of the pile foundation to the pier column, g represents the frequency ratio of the external load excitation to the pier column, c cα represents the critical damping of the bridge pier, c cβ represents the critical damping of the pile foundation.

[0171] In order to facilitate calculation, formula (25) is rearranged as follows:

[0172] in,

[0173]

[0174] Substitute equation (28) into equation (25), integrate, and let The variance σ can be obtained 2 The dimensionless form I of is the objective function of optimization:

[0175]

[0176] Step 3: Parameter Optimization

[0177] For equation (30), since the damping ratios ξ1 and ξ2 of the concrete structure can be taken as 0.05, I is only related to the frequency ratio f and the mass ratio μ. Its three-dimensional graph can be drawn as follows: Figure 7 As shown. From the three-dimensional graph of I(f, μ), it can be seen that when the mass ratio μ is constant, the variance first decreases and then increases with the increase of the frequency ratio f. Therefore, when the mass ratio is constant, the variance has a minimum value. The frequency ratio corresponding to this minimum value is the optimal frequency ratio. In this way, the optimal frequency ratio f corresponding to each mass ratio μ can be found by optimizing the variance equation.

[0178] For continuous rigid frame bridges, since the range of mass ratio μ and f is not large and the speed requirement is not high, the enumeration method is used to find the optimal frequency ratio of pile foundation piers. 2 The dimensionless form I has two variables μ and f. Considering f as a function of μ, it is transformed into a two-dimensional optimization problem. On this basis, the relationship between the optimal frequency ratio and mass ratio is obtained.

[0179] The optimization process of the enumeration method is as follows:

[0180] (1) Determine the mass ratio μ i ;

[0181] (2) ergodic frequency ratio f;

[0182] (3) Calculate the variance I(f,μ);

[0183] (4) Minimize the variance I;

[0184] (5) The corresponding I(f i ,μ i );

[0185] (6) Update μ i =μ i +Δμ; and at the same time, the optimal frequency ratio and mass ratio relationship f(u) is obtained.

[0186] Through the above optimization process, the optimal frequency ratio f corresponding to each mass ratio μ when the variance is minimum is found and plotted on Figure 8 By fitting the above data, we can get the display expression of the optimal frequency ratio f:

[0187] f=1.108e-0.02755μ -1.116e -0.4368μ (31)

[0188] The correlation coefficient of the fitting curve is R2 = 0.993, indicating that the correlation between the two is very high. This is the calculation formula for the optimal frequency ratio of pile foundation and bridge pier with the pier bottom bending moment as the optimization target. After obtaining the optimal frequency ratio f, the total stiffness correction value k of the 2-DOF pile foundation integration is calculated as follows: β1 :

[0189]

[0190] Step 4: Calculation of pile diameter;

[0191] According to formula (27), the mass m is calculated as β 、m α The ratio μ = 0.66. Substituting the mass ratio μ into formula (31), the optimal frequency ratio f of the pile foundation can be obtained. Substituting each parameter into formula (32) can obtain the total stiffness correction value k of the 2-DOF pile foundation integration. β1 , calculate the corrected pier pile foundation stiffness k 41 , k 51 , calculate I by formula (1) 41 ,I 51 , according to I 41 ,I 51 The corresponding pile arrangement and optimal diameter are inferred. 41 , k 51 The calculation formula is as follows:

[0192]

[0193] The above-mentioned embodiments only express the specific implementation methods of the present application, and the descriptions thereof are relatively specific and detailed, but they cannot be understood as limiting the protection scope of the present application. It should be pointed out that, for ordinary technicians in this field, several variations and improvements can be made without departing from the technical solution concept of the present application, and these all belong to the protection scope of the present application.

Claims

1. A tuning and damping method for pile foundation isolation of a deep-water long-span continuous rigid frame bridge, characterized in that: The following steps are involved: Step 1: Simplify the deep-water long-span continuous rigid frame bridge model into a 2-DOF dynamic model; solve the equivalent moment of inertia I of the pier pile foundation by displacement method or finite element simulation method. p , calculate the longitudinal stiffness of the piers and pile foundations and the transverse stiffness of the piers, and calculate the main beam bending constraint correction item; Calculate the mass of the top and bottom of the piers of the left and right spans of the bridge to form a 3-DOF model; combine the mass and stiffness of the piers and pile foundations, establish the mass and stiffness matrix, and establish a simplified 2-DOF dynamic model; The equivalent moment of inertia Ip is: The main beam bending constraint correction term is: The mass matrix M and stiffness matrix K of the 3-DOF dynamic model are: The mass matrix M and stiffness matrix K of the simplified 2-DOF dynamic model are: m α =m1+m2,m β =m3+m4 (11) k α =4(k2+k3) (12) kβ=k4+k5 (13) Where, L p 、E p are the pile length and elastic modulus, k p is the anti-thrust stiffness of the pile foundation, l2 and l3 are the section moments of inertia of the pier column section, and after obtaining the longitudinal stiffness k4 and transverse stiffness k5 of the pile foundation respectively, l4 and l5 are calculated according to formula (1), where m1 is the sum of the mass of the main beam of the left span and 1 / 2 of the pier column mass, m2 is the sum of the mass of the main beam of the right span and 1 / 2 of the pier column mass, m3 is the sum of the mass of 1 / 2 of the pier column mass of the left span and the pile foundation mass, m4 is the sum of the mass of 1 / 2 of the pier column mass of the right span and the pile foundation mass, and m b is the total mass of the main beam, m p2 , m p3 are the masses of the bridge piers, m p4 、m p5 are the mass of the pile foundation, L1 is the length of the main beam, L2 and L3 are the lengths of the pier, L4 and L5 are the lengths of the pile foundation, E1 is the elastic modulus of the main beam, and E2 and E3 are the elastic modulus of the pier; Step 2: Set the optimization objective function based on the 2-DOF dynamic model: Establish the motion equation of the 2-DOF model by the direct balance method, decompose the random excitation into a superposition of a series of simple harmonic components, solve the frequency domain solution of the motion equation, take the pier bottom bending moment as an indicator to evaluate the overall response of the structure, establish its transfer function H(θ) and variance calculation formula, take the minimum variance as the optimization goal, and take the dimensionless form of the variance I as the optimization objective function; The equation of motion for the 2-DOF model is: The displacement y1 under any frequency component θ (θ) (t) and y2 (θ) (t) is also a simple harmonic variable: y1 (θ) (t)=Y1e iθt ,y2 (θ) (t)=Y2e iθt (16) The indicators for evaluating the overall response of the structure are: M(t)=k α L α (y1(t)-y2(t)) (20) In the formula, c i -structural damping coefficient; y i -Relative displacement of particles; - Relative velocity of particles; - relative acceleration of the particle, i = α, β, P(t) is the inertial force caused by ground motion, Y1 and Y2 represent the amplitudes of y1 and y2; Step 3: Based on the optimization objective function, perform parameter optimization to obtain the optimal frequency ratio f, and calculate the total stiffness correction value k of the 2-DOF pile foundation integration according to the optimal frequency ratio f. β1 :By using the enumeration method to optimize the variance, the optimal frequency ratio f corresponding to each mass ratio μ when the variance is minimum is found, and the curve fitting is performed to obtain the display expression of the optimal frequency ratio f, where the optimal frequency ratio refers to the ratio of the pile foundation frequency to the pier column frequency that minimizes the pier bottom bending moment; the total stiffness correction value k of the 2-DOF pile foundation integration is calculated according to the optimal frequency ratio f β1 , where the total stiffness correction value k β1 Point: Press k β1 After the pile foundation stiffness is corrected, the ratio of the pile foundation frequency to the pier column frequency meets the optimal frequency ratio f, that is, the ratio of the integrated total stiffness of the pile foundation after optimization when the pier bottom bending moment is the smallest to the integrated total stiffness of the optimized pile foundation; The optimal frequency ratio f is: f=1.108e -0.02755μ -1.116e -0.4368μ (31) In the formula, μ represents the mass ratio of the equivalent mass points of the cap and the main beam; Calculate the total stiffness correction value k of the 2-DOF pile foundation integration according to the optimal frequency ratio f β1 : Step 4: Total stiffness correction value k based on 2-DOF pile foundation integration β1 Calculate the modified pile foundation equivalent inertia moment I 41 ,I 51 , according to I 41 ,I 51 Infer the corresponding pile arrangement and optimal diameter; Calculate the mass ratio μ of the equivalent mass point of the cap and the main beam and substitute it into formula (31) to obtain the optimal frequency ratio f of the pile foundation. Substitute each parameter into formula (32) to obtain the total stiffness correction value k of the 2-DOF pile foundation integration β1 , calculate the pile foundation stiffness k corresponding to the corrected bridge piers 41 , k 51 , the equivalent moment of inertia of the pier pile foundation I is calculated according to the equivalent moment of inertia 41 ,I 51 , according to I 41 ,I 51 Infer the corresponding pile arrangement and optimal diameter, where k 41 , k 51 The calculation formula is as follows:

2. The tuning and damping method for pile foundation seismic isolation of a deep-water long-span continuous rigid frame bridge according to claim 1 is characterized in that: Based on the equivalent moment of inertia, the deep-water long-span continuous rigid frame bridge is simplified into a 3-DOF dynamic model. Its parameters include the main pier column stiffness, the main pier pile foundation stiffness and mass parameters. The specific formulas of the parameters are as follows: The stiffness of a pier column includes the stiffness of the pier column along the bridge and the stiffness of the pier column transverse to the bridge: The main pier pile foundation stiffness includes the pile foundation longitudinal stiffness k4 and the pile foundation transverse stiffness k5, which can be calculated by force method or finite element method: The mass of the top of a pier: The quality of the bottom of a bridge pier: Where I2 and I3 are the moments of inertia of the cross section of the pier column. After obtaining the longitudinal stiffness k4 and transverse stiffness K5 of the pile foundation respectively, I4 and I5 are calculated according to formula (1). m1 is the sum of the mass of the main beam of the left span and 1 / 2 of the mass of the pier column, m2 is the sum of the mass of the main beam of the right span and 1 / 2 of the mass of the pier column, m3 is the sum of the mass of 1 / 2 of the pier column of the left span and the mass of the pile foundation, m4 is the sum of the mass of 1 / 2 of the pier column of the right span and the mass of the pile foundation, and m b is the total mass of the main beam, m p2 , m p3 are the masses of the bridge piers, m p4 、m p5 are the masses of the pile foundation, L1 is the length of the main beam, L2 and L3 are the lengths of the pier columns, L4 and L5 are the lengths of the pile foundation, E1 is the elastic modulus of the main beam, E2 and E3 are the elastic modulus of the pier columns.

3. The tuning and damping method for pile foundation seismic isolation of a deepwater long-span continuous rigid frame bridge according to claim 1 is characterized in that: The step 2 includes: considering the damping of the bridge pier and pile foundation, the structure performs random vibration under the action of P(t), y1 and y2 are expressed as displacement time history functions of two degrees of freedom relative to the foundation, and the motion equation of the two-degree-of-freedom model can be obtained by the direct equilibrium method as follows: where c i —structural damping coefficient; y i —Relative displacement of particles; — relative velocity of particles; —Relative acceleration of the particle i = α, β, P(t) is the inertial force caused by ground motion, which is a random excitation and can be decomposed into a superposition of a series of simple harmonic components, namely: For any frequency component θ, P (θ) (t) = p0e iθt For simple harmonic excitation, the displacement y1 generated under any frequency component θ (θ) (t) and y2 (θ) (t) is also a simple harmonic variable and can be expressed as follows: y1 (θ) (t)=Y1e iθt ,y2 (θ) (t)=Y2e iθt (16) Among them, Y1 and Y2 represent y1 (θ) (t) and y2 (θ) The amplitude of (t) P (θ) (t) and equation (16) are substituted into equation (14) to obtain -m α θ 2 Y1+c α iθ(Y1-Y2)+k α (Y1-Y2)=p0 -m β i 2 Y2+c β iθY2+c α iθ(Y2-Y1)+k β Y2+k α (Y2-Y1)=0 (17) Solve to get y1 (θ) (t) and y2 (θ) The amplitudes of (t) are: Substituting into equation (16) we can get y1 (θ) (t) and y2 (θ) (t), and then y1 (θ) (t) and y2 (θ) (t) are summed in the frequency domain to obtain y1(t) and y2(t).

4. The tuning and damping method for pile foundation isolation of a deepwater long-span continuous rigid frame bridge according to claim 3 is characterized in that: The step 2 also includes: setting the pier bottom bending moment as an indicator for evaluating the overall response of the structure to: M(t)=k α L α (y1(t)-y2(t)) (20) For any frequency component θ, we have M (θ) (t)=k α L α (y1 (θ) (t)-y2 (θ) (t)) (21) Substituting equations (18) and (19) into equation (21), we can obtain M (θ) (t), and express it in the form of transfer function H(θ): in, When the variance of M(t) When it is minimum, M(t) is also minimum, and the variance is set to: s m 2 =E[|M(t)| 2 ] (24) Where E is the expected value; Substituting equation (23) into the dimensionless form, we can obtain the variance The dimensionless form I: Where g is the ratio of earthquake excitation to structural frequency H(g) is the dimensionless form of the transfer function H(θ): in, In formula (26), μ represents the mass ratio of the equivalent mass point between the cap and the main beam, ω α represents the natural vibration frequency of the main beam and pier, ω β represents the natural vibration frequency of the cap and pile foundation, f represents the frequency ratio of the pile foundation to the pier column, c cα represents the critical damping of the bridge pier, c cβ represents the critical damping of pile foundation; Formula (25) can be rearranged as: in, <h2 style=";text-align:left;direction:ltr">B0=f<h2 style=";text-align:left;direction:ltr"> 2 <h2 style=";text-align:left;direction:ltr"> ;<h2 style=";text-align:left;direction:ltr"> <h2 style=";text-align:left;direction:ltr"> B2 = 1 = A0 = f<h2 style=";text-align:left;direction:ltr"> 2 <h2 style=";text-align:left;direction:ltr"> ;<h2 style=";text-align:left;direction:ltr"> <h2 style=";text-align:left;direction:ltr"> <h2 style=";text-align:left;direction:ltr"> A4=1 (29) Substitute equation (28) into equation (25), integrate, and let The variance σ can be obtained 2 The dimensionless form I of is used as the objective function for optimization:

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