Two-dimensional structure-foundation seismic response calculation method containing effective transmission boundary conditions

Abstract

The invention discloses an effective transmission boundary condition-containing two-dimensional structure-foundation seismic response calculation method, which comprises the following steps of: 1, intercepting a complete structure and a foundation part under the structure, and discretizing the structure-foundation into finite element grids, namely a main grid, a free field grid and a transmission layer boundary grid; 2, establishing a kinetic equation of the main grid, the free field grid and the transmission layer grid; and 3, solving the kinetic equations of the main grid, the free field grid and the transmission layer grid by using a Newmark explicit / implicit mixed integral formula to obtain the displacement, the speed and the acceleration of B and I in the structure-foundation model corresponding to each time step. According to the method, the radiation waves of the structure-foundation interaction finite model can be transmitted, the influence of the false reflection waves of the truncation boundary on the calculation of the structure-foundation finite model is eliminated, and the calculation precision of the two-dimensional structure-foundation seismic response is improved.

Description

Technical Field

[0001] The invention belongs to the field of civil engineering finite element calculation, and specifically is a two-dimensional structure-foundation seismic response calculation method including effective transmission boundary conditions. Background Art

[0002] The foundation is an infinite medium. In the structure-foundation system, the foundation is infinite relative to the structure, but the computer can only process finite models, which means that the infinite foundation must be truncated to establish a finite model of the structure-foundation to approximate the infinite foundation. Therefore, appropriate boundary conditions must be imposed on the truncation boundary, otherwise false reflection waves will be generated at the truncation boundary, polluting the numerical analysis of the structure-foundation finite model.

[0003] At present, the most commonly used method is to set an artificial transmission boundary (also called an absorption boundary or a non-reflection boundary) at the truncation boundary of the foundation model, the purpose of which is to allow the radiation waves inside the structure-foundation model to pass through the truncation boundary. Although the theoretical basis of the above method is clear, the calculation of the equivalent input load is very complicated, especially for processing three-dimensional models. The existing truncation boundary processing technology is complex and the effect is poor. Summary of the invention

[0004] In order to address the deficiencies of the above-mentioned technologies, the present invention proposes a two-dimensional structure-foundation seismic response calculation method including an effective transmission boundary condition, so as to effectively transmit the radiation waves in the structure-foundation model from the model truncation boundary without reflecting them into the model again, thereby accurately calculating the response of the structure-foundation under the action of an earthquake and achieving the effect of simulating an infinite site.

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

[0006] The invention provides a method for calculating the seismic response of a two-dimensional structure-foundation including an effective transmission boundary condition, which comprises the following steps:

[0007] Step 1, intercepting the structure-foundation model including: the complete structure on the ground and part of the foundation under the ground, wherein the bottom of the structure-foundation model is the horizontal boundary, and the left and right sides are the vertical boundaries;

[0008] Step 2, take a point in the structure-foundation model as the coordinate origin o, take the right direction of the parallel foundation surface or bottom boundary as the positive direction of the x-axis, rotate 90° counterclockwise as the positive direction of the y-axis, and thus establish the xoy plane rectangular coordinate system;

[0009] Step 3, perform finite element discretization processing on the complete structure on the ground and the foundation part under the ground, obtain a number of discrete nodes in the xoy plane coordinate system and correspondingly form a main grid m, a free field grid f and a transmission layer grid l; wherein all nodes in the main grid m are composed of a bottom transmission boundary node set B, a side transmission boundary node set b2 and an internal node set I;

[0010] Step 4, decompose the nodes in the free field grid f into the bottom transmission node set B and the internal node set I; decompose the nodes in the transmission layer grid l into the internal node set b1 and the side boundary node set b2; wherein, ;

[0011] Step 5, respectively establish the dynamic equations of the main grid m, free field grid f and transmission layer grid l of the structure-foundation model;

[0012] Step 6, constructing a Newmark numerical integration formula for solving the dynamic equations of the main grid m, the free field grid f, and the transmission layer grid l;

[0013] Step 7: Use the Newmark integral formula to solve the dynamic equations of the main grid m, free field grid f, and transmission layer grid l, and the displacement, velocity, and acceleration of B and I in the structure-foundation model corresponding to each time step.

[0014] The method for calculating the seismic response of a two-dimensional structure-foundation including an effective transmission boundary condition described in the present invention is also characterized in that step 5 includes the following steps:

[0015] Step 5.1, construct the main grid m dynamic equation of the structure-foundation model using equation (1):

[0016] [ M II O O M d BB ] m { u ¨ r I u ¨ r B } m + [ C II O O C tran _ d BB ] m { u ˙ r I u ˙ r B } m + [ K II K IB K BI K BB + K tran BB ] m { u r I u r B } m = { R I ( t ) R B ( t ) } m − [ M II O O M d BB ] m { I x a ¨ i 0 x ( t ) + I y a ¨ i 0 y ( t ) } − [ O O O C tran _ d BB ] m { I x a ˙ i 0 x ( t ) + I y a ˙ i 0 y ( t ) } − [ K Ib 2 K Bb 2 ] m { u r b 2 } m (1)

[0017] In formula (1), [ M II ] m , [ M d BB ] m are the mass matrices of the internal node set I and the bottom transmission boundary node set B in the main grid m, respectively. The subscript d indicates the diagonalization of the matrix; [ K II ] m , [ K BB ] m are the stiffness matrices of the internal node set I and the bottom transmission boundary node set B in the main grid m respectively; [ K IB ] m The stiffness matrix of the node force of the internal node set I caused by the unit displacement of the bottom transmission node set B; [ K BI ] m The stiffness matrix of the node force of the bottom transmission boundary node set B caused by the unit displacement of the internal node set I in the main grid m; [ K Ib 2 ] m is the stiffness matrix of the node forces of the internal node set I caused by the unit displacement of the side transmission boundary node set b2 in the main grid m; [ K Bb 2 ] m The stiffness matrix of the node force of the bottom transmission boundary node set B caused by the unit displacement of the side transmission boundary node set b2 in the main grid m; [ C II ] m is the damping matrix of the internal node I in the main grid m; [ C tran _ d BB ] m is the transmission damping matrix after diagonalization of the bottom transmission boundary node set B in the main grid m; [ K tran BB ] m is the transmission stiffness matrix of the bottom transmission boundary node set B in the main grid m; O is the zero matrix; , , are the acceleration, velocity and displacement of the internal node set I in the main grid m relative to the incident wave at the bottom boundary, , , are the acceleration, velocity and displacement of the bottom transmission boundary node set B relative to the bottom boundary incident wave, is the displacement of the side boundary node set b2; , are the node loads of the internal node set I and the bottom transmission boundary node set B in the main grid at time t; , represents the acceleration and velocity of the shear wave S earthquake input in the main grid m at time t; , represents the P-wave P earthquake acceleration and velocity input in the main grid at time t; t is time, , Respectively represent the vector sets in the x direction and the y direction, and have:

[0018] (2.a)

[0019] (2.b)

[0020] In formula (2.a) and formula (2.b), the superscript T represents the matrix transpose;

[0021] Step 5.2, construct the bottom transmission boundary unit damping matrix using equation (3);

[0022] [ c tran ] = [ 1 3 r c 2 h 0 1 6 r c 2 h 0 0 1 3 r c 1 h 0 1 6 r c 1 h 1 6 r c 2 h 0 1 3 r c 2 h 0 0 1 6 r c 1 h 0 1 3 r c 1 h ] (3)

[0023] In formula (3), ρ is the mass density of the structure-foundation model, c1 is the longitudinal wave velocity of the structure-foundation model, c2 is the shear wave velocity of the structure-foundation model, and h is the side length of the grid;

[0024] Step 5.3, construct the bottom transmission boundary unit stiffness matrix using equation (4);

[0025] [ k tran ] = [ 0 − 1 2 r c 2 2 0 1 2 r c 2 2 − 1 2 r ( c 1 2 − 2 c 2 2 ) 0 1 2 r ( c 1 2 − 2 c 2 2 ) 0 0 − 1 2 r c 2 2 0 1 2 r c 2 2 − 1 2 r ( c 1 2 − 2 c 2 2 ) 0 1 2 r ( c 1 2 − 2 c 2 2 ) 0 ] (4)

[0026] Step 5.4, use equation (6) to establish the dynamic equation of the free field grid f:

[0027] [ M d II O O M d BB ] f { u ¨ r I u ¨ r B } f + [ O O O C tran BB ] f { u ˙ r I u ˙ r B } f + [ K II K IB K BI K BB ] f { u r I u r B } f = − [ M d II O O M d BB ] f { I x a ¨ i 0 x ( t ) + I y a ¨ i 0 y ( t ) } − [ O O O C tran BB ] f { I x a ˙ i 0 x ( t ) + I y a ˙ i 0 y ( t ) } (6) In formula (6), [ M d BB ] f , [ M d II ] f are the mass matrices of the bottom transmission boundary node set B and the internal node set I in the free field grid f, respectively; [ K BB ] f , [ K II ] f are the stiffness matrices of the bottom transmission boundary node set B and the internal node set I in the free field grid f, respectively; , , are the acceleration, velocity and displacement of the internal node set I in the free field grid f relative to the incident wave at the bottom boundary, , , are the acceleration, velocity and displacement of the bottom transmission boundary node set B in the free field grid f relative to the bottom boundary incident wave; [ C tran BB ] f is the damping matrix of the bottom transmission boundary node set B in the free field grid f, and:

[0028] [ C tran BB ] f = [ r c 2 0 0 r c 1 ] (7)

[0029] Step 5.5, use equation (8) to establish the dynamic equation of the transmission layer grid l:

[0030] [ M d b 2 b 2 ] l { u ¨ o b 2 } l + [ C d b 2 b 2 ] l { u ˙ o b 2 } l + [ K b 2 b 2 + K tran b 2 b 2 ] l { u o b 2 } l = − [ K tran b 2 b 1 ] l { u o b 1 } l (8)

[0031] In formula (8), [ M d b 2 b 2 ] l is the mass matrix of the node set b2 in the transmission layer grid l; [ C d b 2 b 2 ] l is the damping matrix of the node set b2 in the transmission layer grid l; [ K b 2 b 2 ] l is the stiffness matrix of the node set b2 in the transmission layer grid l; [ K tran b 2 b 2 ] l is the transmission stiffness matrix of the node force of node set b2 in the transmission layer grid l; [ K tran b 2 b 1 ] l is the transmission stiffness matrix of the node force of node set b2 caused by the unit displacement of node set b1 in the transmission layer grid l; is the displacement of the node set b1 in the transmission layer grid l in the radiation wave; , , are the displacement, velocity and acceleration of the side transmission boundary node b2 respectively;

[0032] Step 5.6, use equations (9), (10), (11) and (12) to respectively establish the left unit transmission damping matrix of the transmission boundary node set b2: [ c tran ] L , right unit transmission damping matrix [ c tran ] R , the transmission stiffness matrix of the left unit [ k tran ] L And the transmission stiffness matrix of the right element [ k tran ] R :

[0033] [ c tran ] L = [ 1 3 r c 1 h 0 1 6 r c 1 h 0 0 1 3 r c 2 h 0 1 6 r c 2 h 1 6 r c 1 h 0 1 3 r c 1 h 0 0 1 6 r c 2 h 0 1 3 r c 2 h ] (9)

[0034] [ c t ] R = [ c t ] L (10)

[0035] [ k tran ] L = [ 0 − 1 2 r ( c 1 2 − 2 c 2 2 ) 0 1 2 r ( c 1 2 − 2 c 2 2 ) − 1 2 r c 2 2 0 1 2 r c 2 2 0 0 − 1 2 r ( c 1 2 − 2 c 2 2 ) 0 1 2 r ( c 1 2 − 2 c 2 2 ) − 1 2 r c 2 2 0 1 2 r c 2 2 0 ] (11)

[0036] [ k tran ] R = [ 0 1 2 r ( c 1 2 − 2 c 2 2 ) 0 − 1 2 r ( c 1 2 − 2 c 2 2 ) 1 2 r c 2 2 0 − 1 2 r c 2 2 0 0 1 2 r ( c 1 2 − 2 c 2 2 ) 0 − 1 2 r ( c 1 2 − 2 c 2 2 ) 1 2 r c 2 2 0 − 1 2 r c 2 2 0 ] (12).

[0037] Further, step 6 includes the following steps:

[0038] Step 6.1, define all nodes in the structure-foundation model to be divided into a pure structure node set and a structure-foundation interface node set, wherein the pure structure node set is an implicit integration node set, denoted by i; the pure foundation node set is an explicit integration node set, denoted by e;

[0039] Step 6.2: Use equation (13) to process the dynamic equations of the main grid m, free field grid f, and transmission layer grid l of the structure-foundation model to obtain the discrete nth time independent variable t n = nΔt, where Δt is the time step and n is the number of the time step:

[0040] [ M ii 0 0 M d oh ] { u ¨ n i u ¨ n e } + [ C ii 0 0 C d oh ] { u ˙ n i u ˙ n e } + [ K ii K that is K oh K oh ] { u n i u n e } = { F n i F n e } (13)

[0041] In formula (13), , are the mass matrices of the implicit integration node set i and the explicit integration node set e respectively; , are the damping matrices of the implicit integration node set i and the explicit integration node set e respectively; , are the stiffness matrices of the implicit integration node set i and the explicit integration node set e respectively; is the stiffness matrix of the node force of implicit integration node set i caused by unit displacement of explicit integration node set e; is the stiffness matrix of the node force of the explicit integration node set e caused by the unit displacement of the implicit integration node set i; , are the nth time independent variable t n The generalized nodal loads of the implicit integration node set e and the explicit integration node set i; , , are the nth time independent variable t n The acceleration, velocity and displacement of the implicit integration node set i are as follows; , , are the nth time independent variable t n The following explicitly integrates the acceleration, velocity, and displacement of the node set e.

[0042] Step 6.3, use equation (15) and equation (16) to establish the Newmark implicit integration formula;

[0043] (15)

[0044] (16)

[0045] In formula (15) and formula (16), , , are the n-1th time independent variable t n-1 The acceleration, velocity and displacement of the implicit integration node set i are as follows, γ and β are two parameters in the Newmark implicit integration formula;

[0046] Use equations (17) and (18) to establish the Newmark explicit integral formula;

[0047] (17)

[0048] (18)

[0049] In formula (17) and formula (18), , , are the n-1th time independent variable t n-1 The following explicitly integrates the acceleration, velocity, and displacement of the node set e.

[0050] Further, step 7 includes the following steps:

[0051] Step 7.1, the total calculation time is assumed to be t a and the time step is , and according to the input seismic wave type, select the shear wave S seismic acceleration input in the main grid at time t ,speed Or the P-wave P seismic acceleration input in the main grid at time t ,speed ; Set the elastic modulus of the structure-foundation model to E, the mass density to ρ, the Poisson's ratio to μ and the side length of the grid to h;

[0052] Step 7.2: The known n-th time independent variable t in the master grid m n Displacement of lower b2 As the boundary value, the Newmark implicit integral formula and explicit integral formula established in step 6 are used to solve the dynamic equation of the master grid m, and the t nth time independent variable t in the master grid m is obtained. n The acceleration of b1 ,speed , displacement ;

[0053] Step 7.3, using the Newmark explicit integral formula to solve the dynamic control equation of the free field grid f, obtain the nth time independent variable t in the free field grid f n The acceleration of b1 ,speed , displacement and the n+1th time variable t n+1 The acceleration of b2 ,speed , displacement ; where the motion of the free field grid f at b1 and b2 is the same;

[0054] Step 7.4, use equations (19.a), (19.b), and (19.c) to obtain the nth time independent variable t in the transmission layer grid l nThe acceleration after b1 filtering ,speed , displacement ;

[0055] (19.a)

[0056] (19.b)

[0057] (19.c)

[0058] Step 7.5, , , As a known value, the dynamic equation of the transmission layer grid l is solved using the Newmark explicit integral formula to obtain the nth time independent variable t in the transmission layer grid l. n The acceleration of b2 ,speed , displacement , and then according to the nth time independent variable t n Downward acceleration ,speed , displacement Use Newmark's explicit integral formula to calculate the n+1th time independent variable t n+1 Displacement of lower b2 ;

[0059] Step 7.6, use equation (20) to get the n+1th time independent variable t in the main grid m n+1 Displacement of lower b2 ;

[0060] (20)

[0061] Step 7.7, output the nth time independent variable t n Next , , , , , ; Assign the value of n+1 to n, if When , stop the calculation, otherwise, return to step 7.2 and execute sequentially.

[0062] The electronic device of the present invention includes a memory and a processor, wherein the memory is used to store a program that supports the processor to execute the two-dimensional structure-foundation seismic response calculation method, and the processor is configured to execute the program stored in the memory.

[0063] The present invention provides a computer-readable storage medium, wherein a computer program is stored on the computer-readable storage medium, and the computer program executes the steps of the two-dimensional structure-foundation seismic response calculation method when the computer program is executed by a processor.

[0064] Compared with the prior art, the present invention has the following beneficial effects:

[0065] 1. The present invention fully considers the influence of the truncated boundary in the finite element analysis of the structure-foundation seismic response on the radiation wave of the structure-foundation model, introduces the transmission boundary condition, and establishes a transmission layer boundary model (main grid, free field grid, transmission boundary layer grid) to solve the radiation wave (outgoing wave) problem in the structure-foundation model, and uses the internal node b1 and boundary node b2 of the transmission boundary layer and the main grid, free field grid, and transmission layer grid to calculate in parallel to transmit the motion component of the radiation wave (outgoing wave) in the model from the truncated boundary of the model to the outside, so that the radiation wave will not be falsely reflected into the structure-foundation model again. Compared with the traditional transmission boundary scheme, the calculation method of the present invention is not only more accurate, but also the boundary setting method is relatively simpler.

[0066] 2. The present invention is not only applicable to calculations under vertical incident plane wave earthquakes, but also to inclined incident plane wave earthquakes; in addition to limiting linear material behaviors near the truncation boundary, other nonlinear material behaviors can be accommodated inside the main grid. Therefore, the present invention is more universal than traditional artificial boundaries. BRIEF DESCRIPTION OF THE DRAWINGS

[0067] Figure 1 The figure is a schematic diagram of the setting of the transmission layer boundary model in the structure-foundation finite element model;

[0068] Figure 2a It is a schematic diagram of a small model of the structure-foundation finite element model;

[0069] Figure 2b It is a schematic diagram of the large model of the structure-foundation finite element model;

[0070] Figure 3 It is a flow chart of the implementation of the present invention;

[0071] Figure 4a is the normal stress σ in the x direction at the monitoring point A in the small model and the large model xx Plot of changes over time;

[0072] Figure 4b is the normal stress σ in the y direction at the monitoring point A in the small model and the large model yy Plot of changes over time;

[0073] Figure 4cis the shear stress σ at monitoring point A in the small model and the large model xy Graph of changes over time. DETAILED DESCRIPTION

[0074] In this embodiment, a method for calculating the seismic response of a two-dimensional structure-foundation including an effective transmission boundary condition is provided, characterized in that it comprises the following steps:

[0075] Step 1: intercept the structure-foundation model including: the complete structure on the ground and part of the foundation under the ground, wherein the bottom of the structure-foundation model is the horizontal boundary, and the left and right sides are the vertical boundaries, such as Figure 1 shown.

[0076] Step 2: Take a point in the structure-foundation model as the coordinate origin o, take the positive direction of the x-axis parallel to the foundation surface or the bottom boundary to the right as the positive direction of the y-axis, and rotate 90° counterclockwise as the positive direction of the y-axis, so as to establish the xoy plane rectangular coordinate system, as shown in Figure 2a and Figure 2b shown.

[0077] Step 3, perform finite element discretization on the complete structure on the ground and the foundation part under the ground, obtain a number of discrete nodes in the xoy plane coordinate system and correspondingly constitute the main grid m, free field grid f and transmission layer grid l. For the complete structure on the ground, rods, beams, columns or solid elements can be used according to actual conditions; for the foundation part under the ground, square four-node bilinear elements are used; Among them, all nodes in the main grid m are composed of the bottom transmission boundary node set B, the side transmission boundary node set b2 and the internal node set I.

[0078] Step 4, decompose the nodes in the free field grid f into the bottom transmission node set B and the internal node set I; decompose the nodes in the transmission layer grid l into the internal node set b1 and the side boundary node set b2; wherein, In this example, according to steps 1 to 3, a larger wide-deep foundation and a smaller narrow-shallow foundation are selected and combined with the structure to form a large model and a small model respectively. Figure 2a and Figure 2b As shown, if the calculation results of the node motion in the structure in the large and small models are consistent, it means that the calculation method of the present invention can effectively transmit the radiation waves caused by the structure.

[0079] Step 5, establish the dynamic equations of the main grid m, free field grid f and transmission layer grid l of the structure-foundation model respectively:

[0080] Step 5.1, construct the main grid m dynamic equation of the structure-foundation model using equation (1):

[0081] [ M II O O M d BB ] m { u ¨ r I u ¨ r B } m + [ C II O O C tran _ d BB ] m { u ˙ r I u ˙ r B } m + [ K II K IB K BI K BB + K tran BB ] m { u r I u r B } m = { R I ( t ) R B ( t ) } m − [ M II O O M d BB ] m { I x a ¨ i 0 x ( t ) + I y a ¨ i 0 y ( t ) } − [ O O O C tran _ d BB ] m { I x a ˙ i 0 x ( t ) + I y a ˙ i 0 y ( t ) } − [ K Ib 2 K Bb 2 ] m { u r b 2 } m (1)

[0082] In formula (1), [ M II ] m , [ M d BB ] m are the mass matrices of the internal node set I and the bottom transmission boundary node set B in the main grid m, respectively. The subscript d indicates the diagonalization of the matrix; [ K II ] m , [ K BB ] m are the stiffness matrices of the internal node set I and the bottom transmission boundary node set B in the main grid m respectively; [ K IB ] m The stiffness matrix of the node force of the internal node set I caused by the unit displacement of the bottom transmission node set B; [ K BI ] m The stiffness matrix of the node force of the bottom transmission boundary node set B caused by the unit displacement of the internal node set I in the main grid m; [ K Ib 2 ] m is the stiffness matrix of the node forces of the internal node set I caused by the unit displacement of the side transmission boundary node set b2 in the main grid m; [ K Bb 2 ] m The stiffness matrix of the node force of the bottom transmission boundary node set B caused by the unit displacement of the side transmission boundary node set b2 in the main grid m; [ C II ] m is the damping matrix of the internal node I in the main grid m; [ C tran _ d BB ] m is the transmission damping matrix after diagonalization of the bottom transmission boundary node set B in the main grid m; [ K tran BB ] m is the transmission stiffness matrix of the bottom transmission boundary node set B in the main grid m; O is the zero matrix; , , are the acceleration, velocity and displacement of the internal node set I in the main grid m relative to the incident wave at the bottom boundary, , , are the acceleration, velocity and displacement of the bottom transmission boundary node set B relative to the bottom boundary incident wave, is the displacement of the side boundary node set b2; , They are the node loads of the internal node set I and the bottom transmission boundary node set B in the main grid at time t; , represents the acceleration and velocity of the shear wave S earthquake input in the main grid m at time t; , represents the P-wave P earthquake acceleration and velocity input in the main grid at time t; t is time, , Respectively represent the vector sets in the x direction and the y direction, and have:

[0083] (2.a)

[0084] (2.b)

[0085] In formula (2.a) and formula (2.b), the superscript T represents matrix transpose.

[0086] Step 5.2, construct the bottom transmission boundary unit damping matrix using equation (3);

[0087] [ c tran ] = [ 1 3 r c 2 h 0 1 6 r c 2 h 0 0 1 3 r c 1 h 0 1 6 r c 1 h 1 6 r c 2 h 0 1 3 r c 2 h 0 0 1 6 r c 1 h 0 1 3 r c 1 h ] (3)

[0088] In formula (3), ρ is the mass density of the structure-foundation model, c1 is the longitudinal wave velocity of the structure-foundation model, c2 is the shear wave velocity of the structure-foundation model, and h is the side length of the grid.

[0089] Step 5.3, construct the bottom transmission boundary unit stiffness matrix using equation (4);

[0090] [ k tran ] = [ 0 − 1 2 r c 2 2 0 1 2 r c 2 2 − 1 2 r ( c 1 2 − 2 c 2 2 ) 0 1 2 r ( c 1 2 − 2 c 2 2 ) 0 0 − 1 2 r c 2 2 0 1 2 r c 2 2 − 1 2 r ( c 1 2 − 2 c 2 2 ) 0 1 2 r ( c 1 2 − 2 c 2 2 ) 0 ] (4)

[0091] Step 5.4, the free field can be abstracted as a vertical cylinder of unit cross-sectional area, and the free field grid f is obtained by discretizing it with two-node linear units. The dynamic equation of the free field grid f is established using equation (6):

[0092] [ M d II O O M d BB ] f { u ¨ r I u ¨ r B } f + [ O O O C tran BB ] f { u ˙ r I u ˙ r B } f + [ K II K IB K BI K BB ] f { u r I u r B } f = − [ M d II O O M d BB ] f { I x a ¨ i 0 x ( t ) + I y a ¨ i 0 y ( t ) } − [ O O O C tran BB ] f { I x a ˙ i 0 x ( t ) + I y a ˙ i 0 y ( t ) } (6) In formula (6), [ M d BB ] f , [ M d II ] f are the mass matrices of the bottom transmission boundary node set B and the internal node set I in the free field grid f, respectively; [ K BB ] f , [ K II ] f are the stiffness matrices of the bottom transmission boundary node set B and the internal node set I in the free field grid f, respectively; , , are the acceleration, velocity and displacement of the internal node set I in the free field grid f relative to the incident wave at the bottom boundary, , , are the acceleration, velocity and displacement of the bottom transmission boundary node set B in the free field grid f relative to the bottom boundary incident wave; [ C tran BB ] f is the damping matrix of the bottom transmission boundary node set B in the free field grid f, and:

[0093] [ C tran BB ] f = [ r c 2 0 0 r c 1 ] (7)

[0094] Step 5.5, copy a transmission layer from the left and right sides of the structure-foundation model. The transmission layer is discretized using a square four-node bilinear unit with a side length of h. The dynamic equation of the transmission layer grid l is established using equation (8):

[0095] [ M d b 2 b 2 ] l { u ¨ o b 2 } l + [ C d b 2 b 2 ] l { u ˙ o b 2 } l + [ K b 2 b 2 + K tran b 2 b 2 ] l { u o b 2 } l = − [ K tran b 2 b 1 ] l { u o b 1 } l (8)

[0096] In formula (8), [ M d b 2 b 2 ] l is the mass matrix of the node set b2 in the transmission layer grid l; [ C d b 2 b 2 ] l is the damping matrix of the node set b2 in the transmission layer grid l; [ K b 2 b 2 ] l is the stiffness matrix of the node set b2 in the transmission layer grid l; [ K tran b 2 b 2 ] l is the transmission stiffness matrix of the node force of node set b2 in the transmission layer grid l; [ K tran b 2 b 1 ] l is the transmission stiffness matrix of the node force of node set b2 caused by the unit displacement of node set b1 in the transmission layer grid l; is the displacement of the node set b1 in the transmission layer grid l in the radiation wave; , , They are the displacement, velocity and acceleration of the side transmission boundary node b2, which are the radiation waves at the side boundary of the foundation.

[0097] Step 5.6, use equations (9), (10), (11) and (12) to respectively establish the left unit transmission damping matrix of the transmission boundary node set b2: [ c tran ] L , right unit transmission damping matrix [ c tran ] R , the transmission stiffness matrix of the left unit [ k tran ] L And the transmission stiffness matrix of the right element [ k tran ] R :

[0098] [ c tran ] L = [ 1 3 r c 1 h 0 1 6 r c 1 h 0 0 1 3 r c 2 h 0 1 6 r c 2 h 1 6 r c 1 h 0 1 3 r c 1 h 0 0 1 6 r c 2 h 0 1 3 r c 2 h ] (9)

[0099] [ c tran ] R = [ c tran ] L (10)

[0100] [ k tran ] L = [ 0 − 1 2 r ( c 1 2 − 2 c 2 2 ) 0 1 2 r ( c 1 2 − 2 c 2 2 ) − 1 2 r c 2 2 0 1 2 r c 2 2 0 0 − 1 2 r ( c 1 2 − 2 c 2 2 ) 0 1 2 r ( c 1 2 − 2 c 2 2 ) − 1 2 r c 2 2 0 1 2 r c 2 2 0 ] (11)

[0101] [ k tran ] R = [ 0 1 2 r ( c 1 2 − 2 c 2 2 ) 0 − 1 2 r ( c 1 2 − 2 c 2 2 ) 1 2 r c 2 2 0 − 1 2 r c 2 2 0 0 1 2 r ( c 1 2 − 2 c 2 2 ) 0 − 1 2 r ( c 1 2 − 2 c 2 2 ) 1 2 r c 2 2 0 − 1 2 r c 2 2 0 ] (12)

[0102] Step 6, constructing a Newmark numerical integration formula for solving the dynamic equations of the main grid m, the free field grid f, and the transmission layer grid l;

[0103] Step 6.1, define all nodes in the structure-foundation model and divide them into a pure structure node set and a structure-foundation interface node set, where the pure structure node set is an implicit integration node set, denoted by i; the pure foundation node set is an explicit integration node set, denoted by e.

[0104] Step 6.2: Use equation (13) to process the dynamic equations of the main grid m, free field grid f, and transmission layer grid l of the structure-foundation model to obtain the discrete nth time independent variable t n = nΔt, where Δt is the time step and n is the number of the time step:

[0105] [ M ii 0 0 M d oh ] { u ¨ n i u ¨ n e } + [ C ii 0 0 C d oh ] { u ˙ n i u ˙ n e } + [ K ii K that is K oh K ee ] { u n i u n e } = { F n i F n e } (13)

[0106] In formula (13), , are the mass matrices of the implicit integration node set i and the explicit integration node set e respectively; , are the damping matrices of the implicit integration node set i and the explicit integration node set e respectively; , are the stiffness matrices of the implicit integration node set i and the explicit integration node set e respectively; is the stiffness matrix of the node force of implicit integration node set i caused by unit displacement of explicit integration node set e; is the stiffness matrix of the node force of the explicit integration node set e caused by the unit displacement of the implicit integration node set i; , are the nth time independent variable t n The generalized nodal loads of the implicit integration node set e and the explicit integration node set i; , , are the nth time independent variable t n The acceleration, velocity and displacement of the implicit integration node set i are as follows; , , are the nth time independent variable t n The following explicitly integrates the acceleration, velocity, and displacement of the node set e.

[0107] Step 6.3, use equation (15) and equation (16) to establish the Newmark implicit integration formula;

[0108] (15)

[0109] (16)

[0110] In formula (15) and formula (16), , , are the n-1th time independent variable t n-1 The acceleration, velocity and displacement of the implicit integration node set i are as follows. γ and β are two parameters in the Newmark implicit integration formula.

[0111] Use equations (17) and (18) to establish the Newmark explicit integral formula;

[0112] (17)

[0113] (18)

[0114] In formula (17) and formula (18), , , are the n-1th time independent variable t n-1 The acceleration, velocity and displacement of the explicit integration node set e are as follows;

[0115] The values ​​of the parameters β and γ in the Newmark integral formula are determined according to the stability and accuracy requirements of the integral of the dynamic equation. In this embodiment, β=0.25 and γ=0.5 are taken in the Newmark implicit integral formula, which can make the integral of the dynamic equation unconditionally stable and use a larger time step Δt to improve the calculation efficiency; in the explicit integral formula, the time step Δt must satisfy that Δt does not exceed the critical time step Δt c , in order to ensure the stability of the time integral, when Δt= Δt c The time integral accuracy is the highest and the efficiency is the best.

[0116] Step 7, such as Figure 3 As shown in the figure, the Newmark integral formula is used to solve the dynamic equations of the main grid m, the free field grid f, and the transmission layer grid l. The displacement, velocity, and acceleration of B and I in the structure-foundation model corresponding to each time step are:

[0117] Step 7.1, the total calculation time is assumed to be t a and the time step is , and according to the input seismic wave type, select the shear wave S seismic acceleration input in the main grid at time t ,speed Or the P-wave P seismic acceleration input in the main grid at time t ,speed ; Set the elastic modulus of the structure-foundation model to E, the mass density to ρ, the Poisson's ratio to μ and the side length of the grid to h;

[0118] In this embodiment, the mass density of the structure-foundation is ρ =2×10 3 kg / m 3 , elastic modulus E = 2 × 10 9 Pa, Poisson's ratio μ = 0.2, square unit side length h = 10 m. Set the time step Δt = 0.008 s and calculate the total time t a = 0.8 s; input seismic wave is longitudinal wave, select input , , assuming that the longitudinal wave displacement The expression is: , the first-order derivative and the second-order derivative of the longitudinal wave displacement expression in time are calculated to obtain the longitudinal wave velocity and acceleration The expressions are .

[0119] Step 7.2: The known n-th time independent variable t in the master grid m n Displacement of lower b2 As the boundary value, the Newmark implicit integral formula and explicit integral formula established in step 6 are used to solve the dynamic equation of the master grid m, and the t nth time independent variable t in the master grid m is obtained. n The acceleration of b1 ,speed , displacement .

[0120] Step 7.3, using the Newmark explicit integral formula to solve the dynamic control equation of the free field grid f, obtain the nth time independent variable t in the free field grid f n The acceleration of b1 ,speed , displacement and the n+1th time variable t n+1 The acceleration of b2 ,speed , displacement ; where the motion of the free field grid f at b1 and b2 is the same;

[0121] Step 7.4, use equations (19.a), (19.b), and (19.c) to obtain the nth time independent variable t in the transmission layer grid l n The acceleration after b1 filtering ,speed , displacement ;

[0122] (19.a)

[0123] (19.b)

[0124] (19.c)

[0125] Step 7.5, , , As a known value, the dynamic equation of the transmission layer grid l is solved using the Newmark explicit integral formula to obtain the nth time independent variable t in the transmission layer grid l. n The acceleration of b2 ,speed , displacement , and then according to the nth time independent variable t n Downward acceleration ,speed , displacement Use Newmark's explicit integral formula to calculate the n+1th time independent variable t n+1 Displacement of lower b2 .

[0126] Step 7.6, use equation (20) to get the n+1th time independent variable t in the main grid m n+1 Displacement of lower b2 ;

[0127] (20)

[0128] Step 7.7, output the nth time independent variable t n Next , , , , , ; Assign the value of n+1 to n, if When , stop the calculation, otherwise, return to step 7.2 and execute sequentially. Set monitoring point A in the large model and the small model. After the calculation is completed, compare the normal stress σ in the x direction at point A. xx, normal stress in the y direction σ yy and the shear stress σ xy .like Figure 4a , Figure 4b , Figure 4c The stress components of the large model and the small model have the same changing trend over time, which shows that the calculation method of the present invention can effectively transmit the radiation waves in the model.

[0129] In this embodiment, an electronic device includes a memory and a processor, wherein the memory is used to store a program that supports the processor to execute the above method, and the processor is configured to execute the program stored in the memory.

[0130] In this embodiment, a computer-readable storage medium stores a computer program on the computer-readable storage medium, and the computer program executes the steps of the above method when executed by a processor.

Claims

1. A method for calculating the seismic response of a two-dimensional structure-foundation including effective transmission boundary conditions, characterized in that: The following steps are involved: Step 1, intercepting the structure-foundation model including: the complete structure on the ground and part of the foundation under the ground, wherein the bottom of the structure-foundation model is the horizontal boundary, and the left and right sides are the vertical boundaries; Step 2, take a point in the structure-foundation model as the coordinate origin o, take the right direction of the parallel foundation surface or bottom boundary as the positive direction of the x-axis, rotate 90° counterclockwise as the positive direction of the y-axis, and thus establish the xoy plane rectangular coordinate system; Step 3, perform finite element discretization processing on the complete structure on the ground and the foundation part under the ground, obtain a number of discrete nodes in the xoy plane coordinate system and correspondingly form a main grid m, a free field grid f and a transmission layer grid l; wherein all nodes in the main grid m are composed of a bottom transmission boundary node set B, a side transmission boundary node set b2 and an internal node set I; Step 4, decompose the nodes in the free field grid f into the bottom transmission node set B and the internal node set I; decompose the nodes in the transmission layer grid l into the internal node set b1 and the side boundary node set b2; wherein, ; Step 5, respectively establish the dynamic equations of the main grid m, free field grid f and transmission layer grid l of the structure-foundation model; Step 6, constructing a Newmark numerical integration formula for solving the dynamic equations of the main grid m, the free field grid f, and the transmission layer grid l; Step 7: Use the Newmark integral formula to solve the dynamic equations of the main grid m, free field grid f, and transmission layer grid l, and the displacement, velocity, and acceleration of B and I in the structure-foundation model corresponding to each time step.

2. A two-dimensional structure-foundation seismic response calculation method including effective transmission boundary conditions according to claim 1, characterized in that: Step 5 includes the following steps: Step 5.1, construct the main grid m dynamic equation of the structure-foundation model using equation (1): (1) In formula (1), , are the mass matrices of the internal node set I and the bottom transmission boundary node set B in the main grid m, respectively. The subscript d indicates the diagonalization of the matrix; , are the stiffness matrices of the internal node set I and the bottom transmission boundary node set B in the main grid m respectively; The stiffness matrix of the node force of the internal node set I caused by the unit displacement of the bottom transmission node set B; The stiffness matrix of the node force of the bottom transmission boundary node set B caused by the unit displacement of the internal node set I in the main grid m; is the stiffness matrix of the node forces of the internal node set I caused by the unit displacement of the side transmission boundary node set b2 in the main grid m; The stiffness matrix of the node force of the bottom transmission boundary node set B caused by the unit displacement of the side transmission boundary node set b2 in the main grid m; is the damping matrix of the internal node I in the main grid m; is the transmission damping matrix after diagonalization of the bottom transmission boundary node set B in the main grid m; is the transmission stiffness matrix of the bottom transmission boundary node set B in the main grid m; O is the zero matrix; , , are the acceleration, velocity and displacement of the internal node set I in the main grid m relative to the incident wave at the bottom boundary, , , are the acceleration, velocity and displacement of the bottom transmission boundary node set B relative to the bottom boundary incident wave, is the displacement of the side boundary node set b2; , are the node loads of the internal node set I and the bottom transmission boundary node set B in the main grid at time t; , represents the acceleration and velocity of the shear wave S earthquake input in the main grid m at time t; , represents the P-wave P earthquake acceleration and velocity input in the main grid at time t; t is time, , Respectively represent the vector sets in the x direction and the y direction, and have: (2.a) (2.b) In formula (2.a) and formula (2.b), the superscript T represents the matrix transpose; Step 5.2, construct the bottom transmission boundary unit damping matrix using equation (3); (3) In formula (3), ρ is the mass density of the structure-foundation model, c1 is the longitudinal wave velocity of the structure-foundation model, c2 is the shear wave velocity of the structure-foundation model, and h is the side length of the grid; Step 5.3, construct the bottom transmission boundary unit stiffness matrix using equation (4); (4) Step 5.4, use equation (6) to establish the dynamic equation of the free field grid f: (6) In formula (6), , are the mass matrices of the bottom transmission boundary node set B and the internal node set I in the free field grid f, respectively; , are the stiffness matrices of the bottom transmission boundary node set B and the internal node set I in the free field grid f, respectively; , , are the acceleration, velocity and displacement of the internal node set I in the free field grid f relative to the incident wave at the bottom boundary, , , are the acceleration, velocity and displacement of the bottom transmission boundary node set B in the free field grid f relative to the bottom boundary incident wave; is the damping matrix of the bottom transmission boundary node set B in the free field grid f, and: (7) Step 5.5, use equation (8) to establish the dynamic equation of the transmission layer grid l: (8) In formula (8), is the mass matrix of the node set b2 in the transmission layer grid l; is the damping matrix of the node set b2 in the transmission layer grid l; is the stiffness matrix of the node set b2 in the transmission layer grid l; is the transmission stiffness matrix of the node force of node set b2 in the transmission layer grid l; is the transmission stiffness matrix of the node force of node set b2 caused by the unit displacement of node set b1 in the transmission layer grid l; is the displacement of the node set b1 in the transmission layer grid l in the radiation wave; , , are the displacement, velocity and acceleration of the side transmission boundary node b2 respectively; Step 5.6, use equations (9), (10), (11) and (12) to respectively establish the left unit transmission damping matrix of the transmission boundary node set b2: , right unit transmission damping matrix , the transmission stiffness matrix of the left unit And the transmission stiffness matrix of the right element : (9) (10) (11) (12)。 3. A two-dimensional structure-foundation seismic response calculation method including effective transmission boundary conditions according to claim 2, characterized in that: Step 6 includes the following steps: Step 6.1, define all nodes in the structure-foundation model to be divided into a pure structure node set and a structure-foundation interface node set, wherein the pure structure node set is an implicit integration node set, denoted by i; the pure foundation node set is an explicit integration node set, denoted by e; Step 6.2: Use equation (13) to process the dynamic equations of the main grid m, free field grid f, and transmission layer grid l of the structure-foundation model to obtain the discrete nth time independent variable t n = nΔt, where Δt is the time step and n is the number of the time step: (13) In formula (13), , are the mass matrices of the implicit integration node set i and the explicit integration node set e respectively; , are the damping matrices of the implicit integration node set i and the explicit integration node set e respectively; , are the stiffness matrices of the implicit integration node set i and the explicit integration node set e respectively; is the stiffness matrix of the node force of implicit integration node set i caused by unit displacement of explicit integration node set e; is the stiffness matrix of the node force of the explicit integration node set e caused by the unit displacement of the implicit integration node set i; , are the nth time independent variable t n The generalized nodal loads of the implicit integration node set e and the explicit integration node set i; , , are the nth time independent variable t n The acceleration, velocity and displacement of the implicit integration node set i are as follows; , , are the nth time independent variable t n The following explicitly integrates the acceleration, velocity, and displacement of the node set e.

4. Step 6.3, use equations (15) and (16) to establish the Newmark implicit integration formula; (15) (16) In formula (15) and formula (16), , , are the n-1th time independent variable t n-1 The acceleration, velocity and displacement of the implicit integration node set i are as follows, γ and β are two parameters in the Newmark implicit integration formula; Use equations (17) and (18) to establish the Newmark explicit integral formula; (17) (18) In formula (17) and formula (18), , , are the n-1th time independent variable t n-1 The following explicitly integrates the acceleration, velocity, and displacement of the node set e.

5. A two-dimensional structure-foundation seismic response calculation method including effective transmission boundary conditions according to claim 3, characterized in that: Step 7 includes the following steps: Step 7.1, the total calculation time is assumed to be t a and the time step is , and according to the input seismic wave type, select the shear wave S seismic acceleration input in the main grid at time t ,speed Or the P-wave P seismic acceleration input in the main grid at time t ,speed ; Set the elastic modulus of the structure-foundation model to E, the mass density to ρ, the Poisson's ratio to μ and the side length of the grid to h; Step 7.2: The known n-th time independent variable t in the master grid m n Displacement of lower b2 As the boundary value, the Newmark implicit integral formula and explicit integral formula established in step 6 are used to solve the dynamic equation of the master grid m, and the t nth time independent variable t in the master grid m is obtained. n The acceleration of b1 ,speed , displacement ; Step 7.3, using the Newmark explicit integral formula to solve the dynamic control equation of the free field grid f, obtain the nth time independent variable t in the free field grid f n The acceleration of b1 ,speed , displacement and the n+1th time variable t n+1 The acceleration of b2 ,speed , displacement ; where the motion of the free field grid f at b1 and b2 is the same; Step 7.4, use equations (19.a), (19.b), and (19.c) to obtain the nth time independent variable t in the transmission layer grid l n The acceleration after b1 filtering ,speed , displacement ; (19.a) (19.b) (19.c) Step 7.5, , , As a known value, the dynamic equation of the transmission layer grid l is solved using the Newmark explicit integral formula to obtain the nth time independent variable t in the transmission layer grid l. n The acceleration of b2 ,speed , displacement , and then according to the nth time independent variable t n Downward acceleration ,speed , displacement Use Newmark's explicit integral formula to calculate the n+1th time independent variable t n+1 Displacement of lower b2 ; Step 7.6, use equation (20) to get the n+1th time independent variable t in the main grid m n+1 Displacement of lower b2 ; (20) Step 7.7, output the nth time independent variable t n Next , , , , , ; Assign the value of n+1 to n, if When , stop the calculation, otherwise, return to step 7.2 and execute sequentially.

6. An electronic device, comprising a memory and a processor, characterized in that: The memory is used to store a program that supports the processor to execute the two-dimensional structure-foundation seismic response calculation method described in any one of claims 1 to 4, and the processor is configured to execute the program stored in the memory.

7. A computer-readable storage medium having a computer program stored thereon, characterized in that: When the computer program is executed by a processor, the steps of the two-dimensional structure-foundation seismic response calculation method according to any one of claims 1 to 4 are executed.

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