TUNED DAMPING METHOD FOR SEISMIC ISOLATION OF PILE FOUNDATIONS IN DEEP-WATER LONG-SPAN CONTINUOUS RIGID FRAME BRIDGE

Abstract

Some embodiments of the disclosure disclose a tuned damping method for seismic isolation of pile foundations in a deep-water long-span continuous rigid frame bridge, which relates to the technical field of shock absorption for bridge engineering structures. It solves the problem that the pile foundation seismic isolation of deep-water long-span continuous rigid frame bridges lacks a theoretical quantitative calculation method, making it impossible to fully exert the effect of pile foundation isolation. The present disclosure includes: simplifying a deep-water long-span continuous rigid frame bridge model into a 2-degree-of-freedom dynamical model; setting an optimization objective function based on the 2-degree-of-freedom dynamical model; performing parameter optimization based on the optimization objective function; and determining structural seismic parameters of the deep-water long-span continuous rigid frame bridge based on parameters optimized in step 3.

Claims

1. A tuned damping method for seismic isolation of pile foundations in a deep-water long-span continuous rigid frame bridge, comprising the following steps: step 1: simplifying a deep-water long-span continuous rigid frame bridge model into a 2-degree-of-freedom dynamical model; step 2: constructing an optimization objective function based on the 2-degree-of-freedom dynamical model; step 3: performing parameter optimization based on the optimization objective function to obtain an optimal frequency ratio f, and based on the optimal frequency ratio f to calculate a total stiffness correction value k.sub.1 of a 2-degree-of-freedom pile foundation integration; step 4: calculating corrected equivalent moments of inertia I.sub.41 and I.sub.51 for a pile foundation based on the total stiffness correction value k.sub.1 of the 2-degree-of-freedom pile foundation integration, and inferring a corresponding pile foundation layout and an optimal diameter based on the corrected equivalent moments of inertia I.sub.41 and I.sub.51.

2. A tuned damping method for seismic isolation of pile foundations in a deep-water long-span continuous rigid frame bridge according to claim 1, wherein the step 1 comprises: step 1.1: simplifying the deep-water long-span continuous rigid frame bridge model into a 3-degree-of-freedom dynamical model; step 1.2: simplifying the 3-degree-of-freedom dynamical model into the 2-degree-of-freedom dynamical model.

3. A tuned damping method for seismic isolation of pile foundations in a deep-water long-span continuous rigid frame bridge according to claim 2, wherein the step 1.1 comprises: letting anti-thrust stiffness of a pile group in the deep-water long-span continuous rigid frame bridge model be k.sub.p, and solving by means of a displacement method or finite element simulation method to obtain an equivalent moment of inertia I.sub.p: $\begin{array}{cc}{I}_p=\frac{k_p{L}_p^3}{12E_p}& \left(1\right)\end{array}$ where, L.sub.p is a pile foundation length, and E.sub.p is the elastic modulus of pile foundation; basing on the equivalent moment of inertia I.sub.p to simplify the deep-water long-span continuous rigid frame bridge model into the 3-degree-of-freedom dynamical model, with parameters comprising a pier column stiffness, a pile foundation stiffness, and a mass parameters, and specific formulas for the parameters are as follows: the pier column stiffness comprises a pier column longitudinal stiffness k.sub.2 and a pile column transverse stiffness k.sub.3: $\begin{array}{cc}\begin{array}{cc}{k}_2=\frac{3E_2{I}_2}{L_2^3}+k& k_3=\frac{3E_3{I}_3}{L_3^3}+k\end{array}& \left(2\right)\end{array}$ the pile foundation stiffness comprises a pier foundation longitudinal stiffness k.sub.4 and a pile foundation transverse stiffness k.sub.5, which can be calculated using a force method or a finite element method; main girder bending restraint correction term: $\begin{array}{cc}k=\frac{6E_2{I}_2\left(6E_1{I}_1{I}_2+E_2{I}_2{L}_1\right)}{L_2^3\left(3E_1{I}_1{L}_2+2E_2{I}_2{L}_1\right)}-\frac{3E_2{I}_2}{L^3}& \left(3\right)\end{array}$ the mass parameters comprises a pier top mass and a pier bottom mass: the pier top mass: $\begin{array}{cc}\begin{array}{cc}{m}_1=\frac{m_b}{2}+\frac{m_{p2}}{2}& m_2=\frac{m_b}{2}+\frac{m_{p3}}{2}\end{array}& \left(4\right)\end{array}$ the pier bottom mass: $\begin{array}{cc}{m}_3=\frac{m_c}{2}+\frac{m_{p2}}{2}+\frac{m_{p4}}{2}& \left(5\right)\end{array}$ $m_4=\frac{m_c}{2}+\frac{m_{p3}}{2}+\frac{m_{p5}}{2}$ where, I.sub.2 and I.sub.3 are moments of inertia of cross-sections of pier columns, I.sub.4 and I.sub.5 are calculated according to equation (1), m.sub.1 is the sum of main girder mass and pier column mass on a left span of a bridge, m.sub.2 is the sum of main girder mass and pier column mass on a right span of the bridge, m.sub.3 is the sum of pier column mass and pile foundation mass on the left span of the bridge, m.sub.4 is the sum of pier column mass and pile foundation mass on the right span of the bridge, m.sub.b is total mass of a main girder, m.sub.p2 and m.sub.p3 are masses of bridge piers respectively, m.sub.p4 and m.sub.p5 are masses of pile foundations respectively, L.sub.1 is the length of the main girder, L.sub.2 and L.sub.3 are the lengths of the bridge piers, L.sub.4 and L.sub.5 are the lengths of the pile foundations, E.sub.1 is the elastic modulus of the main girder, E.sub.2 and E.sub.3 are the elastic modulus of the bridge piers, and L.sub.4 and L.sub.5 are the elastic modulus of the pile foundations; a mass matrix M and a stiffness matrix K of the 3-degree-of-freedom dynamical model are: $\begin{array}{cc}M=\left[\begin{array}{ccc}{m}_1+m_2& 0& 0\\ 0& m_3& 0\\ 0& 0& m_4\end{array}\right]& \left(6\right)\end{array}$ $\begin{array}{cc}K=\left[\begin{array}{ccc}{k}_2+k_3+2k& k_{2}k& k_{3}k\\ k_{2}k& k_4+k_2+k& 0\\ k_{3}k& 0& k_5+k_3+k\end{array}\right].& \left(7\right)\end{array}$

4. A tuned damping method for seismic isolation of pile foundations in a deep-water long-span continuous rigid frame bridge according to claim 2, where in the step 1.2 comprises: assuming the deep-water long-span continuous rigid frame bridge model to be a symmetric structure, that is: E.sub.2I.sub.2=E.sub.3I.sub.3, E.sub.4I.sub.4=E.sub.5I.sub.5, m.sub.1=m.sub.2, m.sub.3=m.sub.4, L.sub.2=L.sub.3, and L.sub.4=L.sub.5, and further simplifying the 3-degree-of-freedom dynamical model into the 2-degree-of-freedom dynamical model, with the mass matrix M and stiffness matrix K thereof as shown in equations (8) and (9) respectively: $\begin{array}{cc}M=\left[\begin{array}{cc}{M}_{}& 0\\ 0& M_{}\end{array}\right]& \left(8\right)\end{array}$ $\begin{array}{cc}K=\left[\begin{array}{cc}{k}_{}& -k_{}\\ -k_{}& k_{}+k_{}\end{array}\right]& \left(9\right)\end{array}$ where, $\begin{array}{cc}\begin{array}{cc}{L}_{}=\frac{L_2+L_3}{2},& L_{}=\frac{L_4+L_5}{2}\end{array}& \left(10\right)\end{array}$ $\begin{array}{cc}\begin{array}{cc}{M}_{}=m_1+m_2,& M_{}=m_3+m_4\end{array}& \left(11\right)\end{array}$ $\begin{array}{cc}{k}_{}=4\left(k_2+k_3\right)& \left(12\right)\end{array}$ $\begin{array}{cc}{k}_{}=k_4+k_5& \left(13\right)\end{array}$ Where, k.sub. is total stiffness of a 2-degree-of-freedom pier column integration, and k.sub. is total stiffness of the 2-degree-of-freedom pile foundation integration.

5. A tuned damping method for seismic isolation of pile foundations in a deep-water long-span continuous rigid frame bridge according to claim 1, wherein the step 2 comprises: step 2.1: constructing motion equations for a 2-degree-of-freedom system based on the 2-degree-of-freedom dynamical model; step 2.2: constructing an optimization objective function based on the motion equations for the 2-degree-of-freedom system.

6. A tuned damping method for seismic isolation of pile foundations in a deep-water long-span continuous rigid frame bridge according to claim 5, wherein the step 2.1 comprises: considering damping of the bridge piers and the pile foundations, and enabling the bridge structure to undergo random vibration under the action of P(t), wherein y.sub.1 and y.sub.2 represent 2-degree-of-freedom displacement time history functions relative to a foundation, and motion equations for the 2-degree-of-freedom dynamical model can be obtained by means of a direct equilibrium method as follows: $\begin{array}{cc}{m}_{}{\stackrel{.Math.}{y}}_{}+c_{}\left({\stackrel{.}{y}}_{}-{\stackrel{.}{y}}_{}\right)+k_{}\left(y_{}-y_{}\right)=P\left(t\right)& \left(14\right)\end{array}$ $m_{}{\stackrel{.Math.}{y}}_{}+c_{}{\stackrel{.}{y}}_{}+c_{}\left({\stackrel{.}{y}}_{}-{\stackrel{.}{y}}_{}\right)+k_{}{y}_{}+k_{}\left(y_{}-y_{}\right)=0$ where c.sub.i is a damping coefficient; y.sub.i is the relative displacement of a mass point; {dot over (y)}.sub.i is the relative velocity of the mass point; .sub.i is a relative acceleration of the mass point; P(t) is an inertial force caused by ground motion, and P(t) is a random excitation which can be decomposed into the superposition of a series of harmonic components as follows: $\begin{array}{cc}P\left(t\right)={}_{-}^{}{p}_0{e}^{it}d& \left(15\right)\end{array}$ for any frequency component , P.sup.()(t)=p.sub.0e.sup.it is a harmonic excitation, and resulting displacements y.sub.1.sup.()(t) and y.sub.2.sup.() (t) are also harmonic variables and can be expressed as follows: $\begin{array}{cc}{y}_1^{\left(\right)}\left(t\right)=Y_1{e}^{it},y_2^{\left(\right)}\left(t\right)=Y_2{e}^{it}& \left(16\right)\end{array}$ substituting P.sup.()(t) and equation (16) into equation (14) to obtain $\begin{array}{cc}-m_{}{}^2{Y}_1+c_{}i\left(Y_1-Y_2\right)+k_{}\left(Y_1-Y_2\right)=p_0& \left(17\right)\end{array}$ $-m_{}{}^2{Y}_2+c_{}i{Y}_2+c_{}i\left(Y_2-Y_1\right)+k_{}{Y}_{}+k_{}\left(Y_2-Y_1\right)=0$ solving to obtain amplitudes of y.sub.1(t) and y.sub.2(t): $\begin{array}{cc}\begin{array}{cc}{Y}_1=& \frac{p_0\left(k_{}+k_{}{m}_{}{}^2+c_{}i+c_{}i\right)}{k_{}{m}_{}{}^2{k}_{}{k}_{}+k_{}{m}_{}{}^2+k_{}{m}_{}{}^2+c_{}{c}_{}{}^2{m}_{}{m}_{}{}^4-}\\ & k_{}{c}_{}i{k}_{}{c}_{}i+m_{}{c}_{}{}^{3}i+m_{}{c}_{}{}^{3}i+m_{}{c}_{}{}^{3}i\end{array}& \left(18\right)\end{array}$ $\begin{array}{cc}\begin{array}{cc}{Y}_2=& \frac{p_0\left(k_{}+c_{}i\right)}{k_{}{m}_{}{}^2{k}_{}{k}_{}+k_{}{m}_{}{}^2+k_{}{m}_{}{}^2+c_{}{c}_{}{}^2{m}_{}{m}_{}{}^4-}\\ & k_{}{c}_{}i{k}_{}{c}_{}i+m_{}{c}_{}{}^{3}i+m_{}{c}_{}{}^{3}i+m_{}{c}_{}{}^{3}i\end{array}& \left(19\right)\end{array}$ substituting into equations (16) to obtain y.sub.1.sup.()(t) and y.sub.2.sup.()(t), and summing y.sub.1.sup.()(t) and y.sub.2.sup.()(t) over a frequency domain to obtain y.sub.1(t) and y.sub.2(t).

7. A tuned damping method for seismic isolation of pile foundations in a deep-water long-span continuous rigid frame bridge according to claim 6, where in the step 2.2 comprises: letting a pier bottom bending moment as an indicator to evaluate an overall structural response: $\begin{array}{cc}M\left(t\right)=k_{}{L}_{}\left(y_1\left(t\right)-y_2\left(t\right)\right)& \left(20\right)\end{array}$ for any frequency component , $\begin{array}{cc}{M}^{\left(\right)}\left(t\right)=k_{}{L}_{}\left(y_1^{\left(\right)}\left(t\right)-y_2^{\left(\right)}\left(t\right)\right)& \left(21\right)\end{array}$ substituting equations (18) and (19) into equation (21) to obtain M.sup.()(t), and expressing M.sup.()(t) in the form of a transfer function H() as follows: $\begin{array}{cc}M\left(t\right)={}_{-}^{}H\left(\right)P_0{e}^{it}d& \left(22\right)\end{array}$ where, $\begin{array}{cc}H\left(\right)=k_{}{L}_{}\frac{k_{}-m_{}{}^2+{\mathrm{ic}}_{}}{\begin{array}{c}-k_{}{m}_{}{}^2+k_{}{k}_{}-k_{}{m}_{}{}^2-k_{}{m}_{}{}^2-c_{}{c}_{}{}^2+m_{}{m}_{}{}^4+\\ i\left(k_{}{c}_{}+k_{}{c}_{}-m_{}{c}_{}{}^3-m_{}{c}_{}{}^3-m_{}{c}_{}{}^3\right)\end{array}}& \left(23\right)\end{array}$ when a variance .sub.m.sup.2 of M(t) is minimized, M(t) is also minimized, and let the variance .sub.m.sup.2 be: $\begin{array}{cc}{}_m^2=E\left[{.Math.M\left(t\right).Math.}^2\right]& \left(24\right)\end{array}$ substituting equation (23) into equation (24) and performing non-dimensionalization to obtain a non-dimensional form I of the variance .sub.m.sup.2: $\begin{array}{cc}I=\frac{1}{2}{}_{-}^{}{.Math.H\left(g\right).Math.}^{2}dg& \left(25\right)\end{array}$ where, g represents a ratio $\left(\frac{}{{}_{}}\right)$ of a seismic excitation to a structural frequency; and H(g) is the non-dimensional form of the transfer function H(): $\begin{array}{cc}H\left(g\right)=\frac{\left(f^2-g^2\right)+2i\frac{c_{}}{c_c}\mathrm{gf}}{\begin{array}{c}\left(-\frac{1}{}{g}^2+\left(g^2-f^2\right)\left(g^2-1\right)+\frac{4c_{}{c}_{}}{c_c{c}_c}{\mathrm{fg}}^2\right)+\\ i\left(2\frac{c_{}}{c_c}g\left(f^2-\frac{1}{}{g}^2-g^2\right)+2\frac{c_{}}{c_c}\mathrm{fg}\left(1-g^2\right)\right)\end{array}}& \left(26\right)\end{array}$ where, $\begin{array}{cc}=\frac{m_{}}{m_{}}{}_{}^2=\frac{k_{}}{m_{}}{}_{}^2=\frac{k_{}}{m_{}}f=\frac{{}_{}}{{}_{}}g=\frac{}{{}_{}}{c}_c=2m_{}{}_{}{c}_c=2m_{}{}_{}\frac{k_{}}{k_{}}=f^2& \left(27\right)\end{array}$ in equation (26), represents a mass ratio of a pile cap to a main girder equivalent mass point, .sub. represents a natural vibration frequency of the main girder and the bridge pier, .sub. represents a natural vibration frequency of the pile cap and the pile foundation, f represents a frequency ratio between the pile foundation and the pier column, g represents a frequency ratio between an external load excitation and the pier column, c.sub.c represents critical damping of the bridge pier, and c.sub.c represents critical damping of the pile foundation; organizing equation (25) as: $\begin{array}{cc}H\left(g\right)=\frac{B_0+{\mathrm{igB}}_1-g^2{B}_2}{A_0+{\mathrm{igA}}_1-g^2{A}_2-{\mathrm{ig}}^3{A}_3+g^4{A}_4}& \left(28\right)\end{array}$ where, $$\begin{gathered} \begin{array}{cc}{B}_0=f^2;B_1=2\frac{c_{}}{c_c}f;B_2=1;A_0=f^2;A_1=2\frac{c_{}}{c_c}{f}^2+2\frac{c_{}}{c_c}f \\ {A}_2=\frac{1}{}+f^2+1-\frac{4c_{}{c}_{}}{c_c{c}_c}f;A_3=2\frac{1}{}\frac{c_{}}{c_c}+2\frac{c_{}}{c_c}+2\frac{c}{c_c}f;A_4=1& \left(29\right)\end{array} \end{gathered}$$ substituting equation (28) into equation (25) for integration, letting ${}_1=\frac{c_1}{c_c}\mathrm{and}{}_2=\frac{c_2}{c_c},$ and obtaining the non-dimensional form I of the variance .sup.2, which serves as the optimization objective function: $\begin{array}{cc}I=\frac{\left[-A_0{A}_1{A}_4{B}_2^2-A_0{A}_3{A}_4\left(B_1^2-2B_0{B}_2\right)+A_4{B}_0^2\left(A_1{A}_4-A_2{A}_3\right)\right]}{2A_0{A}_4\left(A_0{A}_3^2+A_1^2{A}_4-A_1{A}_2{A}_3\right)}=\frac{M\left(,f,{}_1,{}_2\right)}{N\left(,f,{}_1,{}_2\right)}.& \left(30\right)\end{array}$

8. A tuned damping method for seismic isolation of pile foundations in a deep-water long-span continuous rigid frame bridge according to claim 1, where in the step 3 comprises: optimizing the variance by using an enumeration method, finding an optimal frequency ratio f corresponding to each mass ratio when the variance is minimized, and performing curve fitting to obtain an explicit expression for the optimal frequency ratio f: $\begin{array}{cc}f=1.108e^{-0.02755}-1.116e^{-0.4368}& \left(31\right)\end{array}$ based on the optimal frequency ratio f, calculating a total stiffness correction value k.sub.1 of a 2-degree-of-freedom pile foundation integration: $\begin{array}{cc}{k}_1=\frac{m_{}{k}_{}{f}^2}{m_{}}.& \left(32\right)\end{array}$

9. A tuned damping method for seismic isolation of pile foundations in a deep-water long-span continuous rigid frame bridge according to claim 8, where in the optimization process using the enumeration method comprises the following steps: (1) determining the mass ratio ; (2) traversing the frequency ratio f; (3) calculating the variance I(f, ); (4) minimizing the variance I; (5) obtaining the corresponding I(f.sub.i, .sub.i); and (6) updating .sub.i=.sub.i+; and simultaneously obtaining the relationship f(u) between the optimal frequency ratio and the mass ratio.

10. A tuned damping method for seismic isolation of pile foundations in a deep-water long-span continuous rigid frame bridge according to claim 1, where in the step 4 comprises: calculating the mass ratio according to equation (27), substituting the mass ratio into equation (31) to obtain an optimal frequency ratio f of the pile foundation, according to equation (32) to obtain the total stiffness correction value k.sub.1 for the 2-degree-of-freedom pile foundation integration, calculating corrected bridge pier and pile foundation stiffness k.sub.41 and k.sub.51, back-calculating the equivalent moments of inertia I.sub.41 and I.sub.51 of the pile foundation according to equation (1), and based on the equivalent moments of inertia I.sub.41 and I.sub.51, inferring the corresponding pile foundation layout and the optimal diameter, where calculation formulas for k.sub.41 and k.sub.51 are as follows: $\begin{array}{cc}\begin{array}{c}{k}_{41}=\frac{k_1{k}_4}{k_{}}\\ k_{51}=\frac{k_1{k}_5}{k_{}}\end{array}.& \left(33\right)\end{array}$

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0056] FIG. 1 illustrates a mechanical model of a tuned mass damper according to Embodiment 1.

[0057] FIG. 2 illustrates a high-pile cap continuous rigid frame bridge and an equivalent model thereof according to Embodiment 1.

[0058] FIG. 3 (a) illustrates a pile group foundation being simplified to a one-dimensional structure according to Embodiment 1.

[0059] FIG. 3 (b) illustrates a simplified planar model of the pile group foundation according to Embodiment 1.

[0060] FIG. 4 illustrates a 3-degree-of-freedom simplified model for a continuous rigid frame bridge according to Embodiment 1.

[0061] FIG. 5 illustrates a 2-degree-of-freedom simplified model for a continuous rigid frame bridge according to Embodiment 1.

[0062] FIG. 6 illustrates a 2-degree-of-freedom system model according to Embodiment 1.

[0063] FIG. 7 illustrates a I(f, ) three-dimensional diagram according to Embodiment 1.

[0064] FIG. 8 illustrates a flowchart of calculating an optimal frequency ratio according to Embodiment 1.

[0065] FIG. 9 illustrates a graph of the optimal frequency ratio according to Embodiment 1.

[0066] FIG. 10 illustrates a flow diagram of a tuned damping method for seismic isolation of pile foundations in a deep-water long-span continuous rigid frame bridge.

DETAILED DESCRIPTION OF THE EMBODIMENTS

[0067] In order to make the objectives, technical solutions and advantages of embodiments of the present application clearer, the technical solutions in the embodiments of the present application will be described clearly and completely below with reference to the accompanying drawings in the embodiments of the present application. Apparently, the embodiments described are some of, rather than all of, the embodiments of the present application. Thus, the following detailed description of the embodiments of the present disclosure, as represented in the drawings, is not intended to limit the scope of the present application as claimed, but is merely representative of the selected embodiments of the present application. Based on the embodiments of the present application, all other embodiments obtained by those of ordinary skill in the art without creative work fall within the scope of protection of the present application.

Embodiment 1

[0068] A specific embodiment of the present disclosure will be described below in conjunction with FIGS. 1 to 10.

[0069] A core component for mass-tuned damping is a tuned mass damper (TMD), and a mechanical model thereof is composed of mass blocks, a damper, and springs, as shown in FIG. 1. When the TMD is added to an original system, the original system vibrates under dynamic loads, and the TMD generates an opposing force to counteract the vibration of the original system, thereby reducing the dynamic response.

[0070] For a deep-water long-span continuous rigid frame bridge, the mass is mainly distributed in a main girder and a pile cap. The mass of the pile cap can be regarded as the first mass M1, and the mass of the main girder can be regarded as the second mass M2. An unrestrained pile foundation in water can be regarded as a first spring K1, and a bridge pier can be regarded as a second spring K2. Therefore, a high-pile cap continuous rigid frame bridge can be equivalent to a TMD system, as shown in FIG. 2. By applying the principles of mass-tuned damping, a reasonable stiffness ratio between the pile foundation and the bridge pier, as well as a reasonable mass ratio between the main girder and the pile cap, can be sought to reduce the seismic response of the structure. Based on the optimal stiffness or mass ratio, the seismic optimization design of the pile foundation seismic isolation system bridge can be guided.

[0071] A tuned damping method for seismic isolation of pile foundations in a deep-water long-span continuous rigid frame bridge specifically includes the following steps: [0072] Step 1: simplifying a mechanical model; [0073] step 1.1: simplifying a deep-water long-span continuous rigid frame bridge model into a 3-degree-of-freedom dynamical model;

[0074] In a direction perpendicular to seismic wave input, pile foundations are parallel, so the bending stiffness and axial stiffness thereof are the sum of the bending and axial stiffnesses of the pile foundations in the direction perpendicular to the seismic wave input. (as shown in FIG. 3-a). Therefore, a pile foundation structure with a two-dimensional distribution can be simplified into a one-dimensional planar structure. Since the stiffness of a pile cap is much greater than the stiffness of a pile foundation, the pile cap can be considered a rigid body. Thus, a simplified planar model of a pile group foundation is shown in FIG. 3-b.

[0075] Considering the axial deformation of the pile foundations, there are 3 key displacements. Even with n pile foundations along a seismic wave input direction, the number of basic unknown displacement quantities remains 3. Based on this, the anti-thrust stiffness k.sub.p of a pile group and the equivalent moment of inertia I.sub.p of the pile group can be solved:

[00030]$\begin{array}{cc}{I}_p=\frac{k_p{L}_p^3}{12E_p}& \left(1\right)\end{array}$

[0076] where, L.sub.p and E.sub.p represent the pile foundation length and the elastic modulus, respectively; k.sub.p is the anti-thrust stiffness of the pile group; based on the simplified planar model of the pile group foundation, the simplified mechanical model for the continuous rigid frame bridge is shown in FIG. 4.

[0077] Stiffness parameters in the figure include the pier column stiffness, the pile foundation stiffness, and the mass parameters, which are calculated according to formulas of structural dynamics methods as follows: [0078] the pier column stiffness includes a pier column longitudinal stiffness and a pier column transverse stiffness:

[00031]$\begin{array}{cc}\begin{array}{cc}{k}_2=\frac{3E_2{I}_2}{L_2^3}+k& k_3=\frac{3E_3{I}_3}{L_3^3}+k\end{array}& \left(2\right)\end{array}$ [0079] the pile foundation stiffness comprises pile foundation longitudinal stiffness k.sub.4 and pile foundation transverse stiffness k.sub.5, which can be calculated using a force method or a finite element method;

Main Girder Bending Restraint Correction Term:

[00032]$\begin{array}{cc}k=\frac{6E_2{I}_2\left(6E_1{I}_1{L}_2+E_2{I}_2{L}_1\right)}{L_2^3\left(3E_1{I}_1{L}_2+2E_2{I}_2{L}_1\right)}-\frac{3E_2{I}_2}{L^3}& \left(3\right)\end{array}$

Pier Top Mass:

[00033]$\begin{array}{cc}\begin{array}{cc}{m}_1=\frac{m_b}{2}+\frac{m_{p2}}{2}& m_2=\frac{m_b}{2}+\frac{m_{p3}}{2}\end{array}& \left(4\right)\end{array}$

Pier Bottom Mass:

[00034]$\begin{array}{cc}{m}_3=\frac{m_c}{2}+\frac{m_{p2}}{2}+\frac{m_{p4}}{2}& \left(5\right)\end{array}$$m_4=\frac{m_c}{2}+\frac{m_{p3}}{2}+\frac{m_{p5}}{2}$

[0080] where, I.sub.2 and I.sub.3 are moments of inertia of cross-sections of pier columns, I.sub.4 and I.sub.5 are equivalent moments of inertia of the pile foundations and are calculated according to equation (1), m.sub.1 is the sum of main girder mass and pier column mass on a left span of a bridge, m.sub.2 is the sum of main girder mass and pier column mass on a right span of the bridge, m.sub.3 is the sum of pier column mass and pile foundation mass on the left span of the bridge, m.sub.4 is the sum of pier column mass and pile foundation mass on the right span of the bridge, m.sub.b is the total mass of the main girder, m.sub.p2 and m.sub.p3 are masses of bridge piers, respectively, m.sub.p4 and m.sub.p5 are masses of pile foundations, L.sub.1 is the length of the main girder, L.sub.2 and L.sub.3 are the lengths of the bridge piers, L.sub.4 and L.sub.5 are the lengths of the pile foundations, E.sub.1 is the elastic modulus of the main girder, E.sub.2 and E.sub.3 are the elastic modulus of the bridge piers, and L.sub.4 and L.sub.5 are the elastic modulus of the pile foundations;

[0081] Since axial stiffness of the main girder L.sub.1 is infinite, a 4-mass-point model shown in FIG. 4 is essentially a 3-degree-of-freedom dynamical model. The first degree of freedom corresponds to the horizontal displacement of m.sub.1+m.sub.2, the second degree of freedom corresponds to the horizontal displacement of m.sub.3, and the third degree of freedom corresponds to the horizontal displacement of m.sub.4. A mass matrix M and a stiffness matrix K of the model are:

[00035]$\begin{array}{cc}M=\left[\begin{array}{ccc}{m}_1+m_2& 0& 0\\ 0& m_3& 0\\ 0& 0& m_4\end{array}\right]& \left(6\right)\end{array}$$\begin{array}{cc}K=\left[\begin{array}{ccc}{k}_2+k_3+2k& k_{2}k& k_{3}k\\ k_{2}k& k_4+k_2+k& 0\\ k_{3}k& 0& k_5+k_3+k\end{array}\right]& \left(7\right)\end{array}$ [0082] step 1.2: simplifying the 3-degree-of-freedom dynamical model into a 2-degree-of-freedom dynamical model;

[0083] Clearly, in the 3-degree-of-freedom dynamical model shown in FIG. 4, the motion of mass points m.sub.3 and m.sub.4 only occurs in two directions: in the same direction or the opposite directions. When moving in the same direction, an anti-symmetric vibration mode occurs, and when moving in the opposite directions, a symmetric vibration mode occurs. Since longitudinal seismic excitation is anti-symmetric, it only excites the anti-symmetric vibration mode. Therefore, under the longitudinal seismic action, the symmetric vibration mode does not contribute to the structural response. For simplicity in the derivation and from an engineering application perspective, it is assumed that the structure is symmetric, that is: E.sub.2I.sub.2=E.sub.3I.sub.3, E.sub.4I.sub.4=E.sub.5I.sub.5, m.sub.1=m.sub.2, m.sub.3=m.sub.4, L.sub.2=L.sub.3, and L.sub.4=L.sub.5, and the 3-degree-of-freedom dynamical model in FIG. 4 can be further simplified into a 2-degree-of-freedom dynamical model shown in FIG. 5, with the mass matrix M and the stiffness matrix K thereof as shown in equations (8) and (9) respectively:

[00036]$\begin{array}{cc}M=\left[\begin{array}{cc}{M}_{}& 0\\ 0& M_{}\end{array}\right]& \left(8\right)\end{array}$$\begin{array}{cc}K=\left[\begin{array}{cc}{k}_{}& -k_{}\\ -k_{}& k_{}+k_{}\end{array}\right]& \left(9\right)\end{array}$

where,

[00037]$\begin{array}{cc}{L}_{}=\frac{L_2+L_3}{2},L_{}=\frac{L_4+L_5}{2}& \left(10\right)\end{array}$$\begin{array}{cc}\begin{array}{cc}{M}_{}=m_1+m_2,& M_{}=m_3+m_4\end{array}& \left(11\right)\end{array}$$\begin{array}{cc}{k}_{}=4\left(k_2+k_3\right)& \left(12\right)\end{array}$$\begin{array}{cc}{k}_{}=k_4+k_5& \left(13\right)\end{array}$ [0084] where, k.sub. is the total stiffness of a 2-degree-of-freedom pier column integration, and k.sub. is total stiffness of the 2-degree-of-freedom pile foundation integration. [0085] step 2: constructing an optimization objective function based on the 2-degree-of-freedom dynamical model; [0086] step 2.1: constructing motion equations for a 2-degree-of-freedom system based on the 2-degree-of-freedom model; [0087] the 2-degree-of-freedom system model is shown in FIG. 6. Considering the damping of the bridge pier and the pile foundation, the structure undergoes random vibration under the action of P(t), where y.sub.1 and y.sub.2 represent the 2-degree-of-freedom displacement time history functions relative to the foundation.

[0088] Motion equations for the 2-degree-of-freedom dynamical model can be obtained by means of a direct equilibrium method:

[00038]$\begin{array}{cc}{m}_{}{\stackrel{.Math.}{y}}_{}+c_{}\left({\stackrel{.}{y}}_{}-{\stackrel{.}{y}}_{}\right)+k_{}\left(y_{}-y_{}\right)=P\left(t\right)& \left(14\right)\end{array}$$m_{}{\stackrel{.Math.}{y}}_{}+c_{}{\stackrel{.}{y}}_{}+c_{}\left({\stackrel{.}{y}}_{}-{\stackrel{.}{y}}_{}\right)+k_{}{y}_{}+k_{}\left(y_{}-y_{}\right)=0$

[0089] where c.sub.i is the damping coefficient; y.sub.i is the relative displacement of the mass point; {dot over (y)}.sub.i is the relative velocity of the mass point; .sub.i is the relative acceleration of the mass point; P(t) is an inertial force caused by ground motion, and a random excitation, which can be decomposed into the superposition of a series of harmonic components as follows:

[00039]$\begin{array}{cc}P\left(t\right)={}_{-}^{}{p}_0{e}^{it}d& \left(15\right)\end{array}$

[0090] for any frequency component , P.sup.()(t)=p.sub.0e.sup.it is a harmonic excitation, resulting displacements y.sub.1.sup.()(t) and y.sub.2.sup.()(t) are also harmonic variables and can be expressed as follows:

[00040]$\begin{array}{cc}{y}_1^{\left(\right)}(t)=Y_1{e}^{it},y_2^{\left(\right)}\left(t\right)=Y_2{e}^{it}& \left(16\right)\end{array}$

[0091] Substituting P.sup.()(t) and equation (17) into equation (15) to obtain

[00041]$\begin{array}{cc}-m_{}{}^2{Y}_1+c_{}i\left(Y_1-Y_2\right)+k_{}\left(Y_1-Y_2\right)=p_0& \left(17\right)\end{array}$$-m_{}{}^2{Y}_2+c_{}i{Y}_2+c_{}i\left(Y_2-Y_1\right)+k_{}{Y}_{}+k_{}\left(Y_2-Y_1\right)=0$

[0092] solving to obtain amplitudes of y.sub.1(t) and y.sub.2(t):

[00042]$\begin{array}{cc}\begin{array}{cc}{Y}_1=& \frac{p_0\left(k_{}+k_{}{m}_{}{}^2+c_{}i+c_{}i\right)}{k_{}{m}_{}{}^2{k}_{}{k}_{}+k_{}{m}_{}{}^2+k_{}{m}_{}{}^2+c_{}{c}_{}{}^2{m}_{}{m}_{}{}^4-}\\ & k_{}{c}_{}i{k}_{}{c}_{}i+m_{}{c}_{}{}^{3}i+m_{}{c}_{}{}^{3}i+m_{}{c}_{}{}^{3}i\end{array}& \left(18\right)\end{array}$$\begin{array}{cc}\begin{array}{cc}{Y}_2=& \frac{p_0\left(k_{}+c_{}i\right)}{k_{}{m}_{}{}^2{k}_{}{k}_{}+k_{}{m}_{}{}^2+k_{}{m}_{}{}^2+c_{}{c}_{}{}^2{m}_{}{m}_{}{}^4-}\\ & k_{}{c}_{}i{k}_{}{c}_{}i+m_{}{c}_{}{}^{3}i+m_{}{c}_{}{}^{3}i+m_{}{c}_{}{}^{3}i\end{array}& \left(19\right)\end{array}$

[0093] substituting into equations (16) to obtain y.sub.1.sup.()(t) and y.sub.2.sup.()(t), and summing over a frequency domain to obtain y.sub.1(t) and y.sub.2(t). [0094] step 2.2: optimizing an objective function;

[0095] The optimization should aim to minimize the overall dynamic response of the structure. Key indicators that can reflect the overall dynamic response of the structure include parameters such as the pier bottom bending moment and the main girder displacement. The pier bottom bending moment and the main girder displacement are correlated, and optimization targets thereof are the stiffness ratio between the pile foundation and the bridge pier, as well as the mass ratio between the pile cap and the main girder. Therefore, by optimizing the objective function with parameters such as the pier bottom bending moment and the main girder displacement, a reasonable matching of main girder mass-bridge pier stiffness-pile cap mass-pile foundation stiffness can be achieved, thereby optimizing the design of the pile foundation seismic isolation system. In the seismic design of bridges, the pier bottom bending moment is often the key parameter controlling the design. In this embodiment, the pier bottom bending moment is used as the indicator to evaluate the overall response of the structure. As shown in FIG. 6, the pier bottom bending moment:

[00043]$\begin{array}{cc}M\left(t\right)=k_{}{L}_{}\left(y_1\left(t\right)-y_2\left(t\right)\right)& \left(20\right)\end{array}$

[0096] Similarly, for any frequency component ,

[00044]$\begin{array}{cc}{M}^{\left(\right)}\left(t\right)=k_{}{L}_{}\left(y_1^{\left(\right)}\left(t\right)-y_2^{\left(\right)}\left(t\right)\right)& \left(21\right)\end{array}$

[0097] substitute equations (19) and equations (20) into equation (22) to obtain M.sup.()(t), which is expressed in the form of a transfer function H() as:

[00045]$\begin{array}{cc}M\left(t\right)={}_{-}^{}H\left(\right)P_0{e}^{it}d& \left(22\right)\end{array}$

where,

[00046]$\begin{array}{cc}H\left(\right)=k_{}{L}_{}\frac{k_{}-m_{}{}^2+i{c}_{}}{\begin{array}{c}-k_{}{m}_{}{}^2+k_{}{k}_{}-k_{}{m}_{}{}^2-k_{}{m}_{}{}^2-\\ c_{}{c}_{}{}^2+m_{}{m}_{}{}^4+\\ i\left(k_{}{c}_{}+k_{}{c}_{}-m_{}{c}_{}{}^3-m_{}{c}_{}{}^3-m_{}{c}_{}{}^3\right)\end{array}}& \left(23\right)\end{array}$

[0098] Since seismic motion is a zero-mean random process, M(t) is also a zero-mean random process, when the variance .sub.m.sup.2 of M(t) is minimized, M(t) is also minimized, and according to the definition of the variance, it can be obtained that

[00047]$\begin{array}{cc}{}_m^2=E\left[{.Math.M\left(t\right).Math.}^2\right]& \left(24\right)\end{array}$

[0099] substituting equation (23) into equation (24) and performing non-dimensionalization to obtain a non-dimensional form I of the variance .sub.m.sup.2:

[00048]$\begin{array}{cc}I=\frac{1}{2}{}_{-}^{}{.Math.H\left(g\right).Math.}^{2}dg& \left(25\right)\end{array}$

[0100] where, g represents a ratio

[00049]$\left(\frac{}{{}_{}}\right)$

of a seismic excitation to a structural frequency; and H(g) is the non-dimensional form of the transfer function H():

[00050]$\begin{array}{cc}H\left(g\right)=\frac{\left(f^2-g^2\right)+2i\frac{c_{}}{c_c}\mathrm{gf}}{\begin{array}{c}\left(-\frac{1}{}{g}^2+\left(g^2-f^2\right)\left(g^2-1\right)+\frac{4c_{}{c}_{}}{c_c{c}_c}{\mathrm{fg}}^2\right)+\\ i\left(2\frac{c_{}}{c_c}g\left(f^2-\frac{1}{}{g}^2-g^2\right)+2\frac{c_{}}{c_c}\mathrm{fg}\left(1-g^2\right)\right)\end{array}}& \left(26\right)\end{array}$

where,

[00051]$\begin{array}{cc}=\frac{m_{}}{m_{}}{}_{}^2=\frac{k_{}}{m_{}}{}_{}^2=\frac{k_{}}{m_{}}f=\frac{{}_{}}{{}_{}}g=\frac{}{{}_{}}{c}_c=2m_{}{}_{}{c}_c=2m_{}{}_{}\frac{k_{}}{k_{}}=f^2& \left(27\right)\end{array}$

[0101] in equation (26), p represents a mass ratio of a pile cap to a main girder equivalent mass point, .sub. represents a natural vibration frequency of the main girder and the bridge pier, .sub. represents a natural vibration frequency of the pile cap and the pile foundation, f represents a frequency ratio between the pile foundation and the pier column, g represents a frequency ratio between an external load excitation frequency and the pier column, c.sub.c represents critical damping of the bridge pier, and c.sub.c represents critical damping of the pile foundation.

[0102] For the convenience of calculation, equation (25) is organized as:

[00052]$\begin{array}{cc}H\left(g\right)=\frac{B_0+{\mathrm{igB}}_1-g^2{B}_2}{A_0+{\mathrm{igA}}_1-g^2{A}_2-{\mathrm{ig}}^3{A}_3+g^4{A}_4}& \left(28\right)\end{array}$

where,

[00053]$\begin{array}{cc}{B}_0=f^2;B_1=2\frac{c_{}}{c_c}f;B_2=1;A_0=f^2;A_1=2\frac{c_{}}{c_c}{f}^2+2\frac{c_{}}{c_c}f{A}_2=\frac{1}{}+f^2+1-\frac{4c_{{}^C}}{c_c{c}_c}f;A_3=2\frac{1}{}\frac{c_{}}{c_c}+2\frac{c_{}}{c_c}+2\frac{c_{}}{c_c}f;A_4=1& \left(29\right)\end{array}$

[0103] equation (28) is substituted into equation (25) and then integrated, and let

[00054]${}_1=\frac{c_1}{c_c}\mathrm{and}{}_2=\frac{c_2}{c_c},$

and a formula expression for the non-dimensional form I of the variance .sup.2 is obtained, which is the optimization objective function:

[00055]$\begin{array}{cc}\begin{array}{c}I=\frac{\left[\begin{array}{c}-A_0{A}_1{A}_4{B}_2^2-A_0{A}_3{A}_4\left(B_1^2-2B_0{B}_2\right)+\\ A_4{B}_0^2\left(A_1{A}_4-A_2{A}_3\right)\end{array}\right]}{2A_0{A}_4\left(A_0{A}_3^2+A_1^2{A}_4-A_1{A}_2{A}_3\right)}\\ =\frac{M\left(,f,{}_1,{}_2\right)}{M\left(,f,{}_1,{}_2\right)}\end{array}& \left(30\right)\end{array}$ [0104] Step 3: parameter optimization

[0105] For equation (30), since the damping ratios .sub.1 and .sub.2 of a concrete structure can be taken as 0.05, I is only related to the frequency ratio f and the mass ratio . A three-dimensional graph thereof can be plotted, as shown in FIG. 7. From the three-dimensional graph of I(f, ), it can be seen that for a constant mass ratio , the variance decreases first and then increases as the frequency ratio f increases. Therefore, for a constant mass ratio, there is a minimum value of the variance, and the corresponding frequency ratio at this minimum value is the optimal frequency ratio. By optimizing a variance equation, the optimal frequency ratio f corresponding to each mass ratio can be determined.

[0106] For a continuous rigid-frame bridge, since the ranges of variation for and f are not large and the velocity requirement is not high, the optimal frequency ratio for the pile foundation and the bridge pier can be determined by means of an enumeration method. The non-dimensional form I of the variance .sup.2 has two variables and f. By treating f as a function of , the problem is transformed into a two-dimensional optimization problem. On this basis, a relational expression for the optimal frequency ratio and the mass ratio can be determined.

[0107] The optimization process using the enumeration method is as follows: [0108] (1) determining the mass ratio .sub.i; [0109] (2) traversing the frequency ratio f; [0110] (3) calculating the variance I(f, ); [0111] (4) minimizing the variance I; [0112] (5) obtaining the corresponding I(f.sub.i, .sub.i); [0113] (6) updating .sub.i=.sub.i+; and simultaneously obtaining the relationship f(u) between the optimal frequency ratio and the mass ratio.

[0114] Through the optimization process mentioned above, the optimal frequency ratio f corresponding to each mass ratio when the variance is minimized is determined and plotted in FIG. 9. Through curve fitting based on the above data, an explicit expression for the optimal frequency ratio f is obtained.

[00056]$\begin{array}{cc}f=1.108e^{-0.02755}-1.116e^{-0.4368}& \left(31\right)\end{array}$

[0115] A correlation coefficient of a fitting curve R2=0.993, indicates a high correlation between the two. This represents a calculation formula for the optimal frequency ratio of the pile foundation and the bridge pier with the pier bottom bending moment as an optimization objective; after the optimal frequency ratio f is obtained, the total stiffness correction value k.sub.1 for the 2-degree-of-freedom pile foundation integration is calculated using the following formula:

[00057]$\begin{array}{cc}{k}_1=\frac{m_{}{k}_{}{f}^2}{m_{}}.& \left(32\right)\end{array}$ [0116] Step 4: calculation of the pile diameter;

[0117] calculating the ratio of masses m.sub. and m.sub. according to equation (27) to obtain =0.66, substituting the mass ratio into equation (31) to obtain the optimal frequency ratio f of the pile foundation, substituting parameters into equation (32) to obtain the total stiffness correction value k.sub.1 of the 2-degree-of-freedom pile foundation integration, calculating corrected bridge pier and pile foundation stiffness k.sub.41 and k.sub.51, back-calculating I.sub.41 and I.sub.51 according to equation (1), and inferring the corresponding pile foundation layout and the optimal diameter based on I.sub.41 and I.sub.51. Calculation formulas for k.sub.41 and k.sub.51 are as follows:

[00058]$\begin{array}{cc}{k}_{41}=\frac{k_1{k}_4}{k_{}}& \left(33\right)\end{array}$$k_{51}=\frac{k_1{k}_5}{k_{}}$

[0118] The above embodiments only express specific implementations of the present application, and are described in more detail, but are not to be construed as a limitation to the scope of protection of the present application. It is to be noted that several variations and modifications can also be made by those of ordinary skill in the art without departing from the concepts the technical solutions of the present application, which all fall within the scope of protection of the present application.