Infinite ground non-uniform distribution building vibration prediction method based on local inverted T-shaped structure

Abstract

The invention discloses an infinite non-uniform distribution building vibration prediction method based on a local inverted T-shaped structure, and the method comprises the steps: building a 2.5-dimensional finite element boundary element model of a soil layer structure in a ground inertial reference system, and enabling the x axis, y axis and z axis of the inertial reference system to coincide with the longitudinal, transverse and vertical directions of a track respectively; and coupling the pile group with the 2.5-dimensional finite element boundary element model so as to predict vibration.

Description

Technical Field

[0001] The invention relates to the technical field of building vibration prediction, in particular to a method for predicting building vibration with infinite non-uniform distribution based on a local inverted T-shaped structure. Background Art

[0002] When studying environmental vibration problems, theoretical models have fast calculation speed and accurate results, but the applicable conditions are relatively harsh and the scope is narrow. For example, the layered earth theory, the semi-analytical model of dynamic interaction between two tunnels in a layered half-space, and the 2.5-dimensional finite element and 2.5-dimensional boundary element methods are superior to simulating infinitely long tracks, tunnel structures and infinitely large earth, but it is difficult to model and simulate the finite and irregular house structures along the line. The substructure method can combine the advantages and disadvantages of multiple methods and more comprehensively consider the coupling relationship between systems. For example, 3D boundary elements are used to analyze the earth structure, and finite element models are used to analyze the house structure, and the interaction between the two is considered. However, the coupling process between the structures involved in the operation is relatively complicated, and the 3D boundary element method requires users to have a high level of mathematics and programming, so it is rarely used in engineering. The method of combining finite elements with artificial boundaries is often relatively simple to apply and can be applied to various complex geometric structures. It is widely used in engineering, but the calculation speed is significantly reduced when the model is large, and the derivation of artificial boundaries is often based on a series of assumptions. When the conditions are not strictly met, its calculation accuracy is difficult to guarantee.

[0003] Artificial boundary conditions can be broadly divided into global artificial boundaries and local artificial boundaries. Global artificial boundaries are often more accurate, but require higher computer storage and computing speed. Local boundary conditions only consider the interaction between adjacent nodes, and the calculation speed is faster, but it may also bring more errors.

[0004] In order to solve the environmental vibration problem caused by trains, a global artificial boundary condition is derived based on the idea of ​​substructure method, combined with finite element theory and 2.5DFEM-BEM, named global coordinated artificial boundary. Global means that this method considers the node coupling on the boundary, and coordinated means that this method has high accuracy and can coordinate the stress-strain relationship on the boundary. This method has the advantages of substructure method, which can comprehensively consider the coupling between the earth and the structure and improve the calculation accuracy; this method has the advantages of artificial boundary conditions, can be used in combination with finite elements, broadens the scope of application, and is suitable for various complex models. Summary of the invention

[0005] In order to solve the problems existing in the prior art, the purpose of the present invention is to provide a method for predicting building vibration of infinite non-uniform distribution based on a local inverted T-shaped structure.

[0006] To achieve the above object, the technical solution adopted by the present invention is: a method for predicting building vibration of infinite ground non-uniform distribution based on a local inverted T-shaped structure, comprising the following steps:

[0007] Step 1: Establish a 2.5-dimensional finite element boundary element model of the soil structure in the inertial reference system on the ground, where the x-axis, y-axis and z-axis of the inertial reference system coincide with the longitudinal, transverse and vertical directions of the track respectively;

[0008] Step 2: Couple the pile group with the 2.5D finite element boundary element model to predict the vibration.

[0009] As a further improvement of the present invention, the step 1 is specifically as follows:

[0010] The motion differential equation of the wavenumber domain finite element boundary element model of the soil structure is established, where n represents the number of nodes; the corresponding node force vector is F(x, t), and the 2.5-dimensional finite element equation is expressed as:

[0011]

[0012] Where, M, K0 and K2 are 3n×3n symmetric matrices, M and K2 are positive definite matrices, K0 is a non-negative matrix, and K1 is an antisymmetric matrix; all matrices are constants and are independent of time and longitudinal x;

[0013] Performing Fourier transform on time and longitudinal direction, we get the equation with N nodes on the boundary:

[0014] -ω 2 Mq(β)+(K0+iβK1+β 2 K2)q(β)=F(β)

[0015] In the formula, the circular frequency ω is expressed as:

[0016] ω=Ω-βc

[0017] Calculate the excitation load e i(Ω-βc)t The Fourier transformed soil displacement matrix q(β) caused by the Fourier transform is then used to obtain the actual soil displacement matrix. Since there is no moving load effect c = 0;

[0018] According to the general equation of the boundary element method, the boundary integral equation is established by taking points on the boundary Γ; the equation with N nodes on the boundary is discretized to obtain the following boundary element equation:

[0019] Hu(β)=Gp(β)+B(β)

[0020] In the formula, the displacement and driving force of node i can be expressed as and p(β) i ,at this time:

[0021]

[0022] H and G are 3n×3n matrices related to β; It is expressed as:

[0023]

[0024] As a further improvement of the present invention, the step 2 is specifically as follows:

[0025] Define a boundary surface Γ1 for the 3D finite element model and the layered earth model, i.e., the 2.5D finite element-boundary element model; the volume formed by the interface is a rectangular solid, which contains the layered soil and the pile group structure; calculate the force of the cylinder and express it as a vector The vector dimension depends on the mesh density used in the soil structure; the node displacement vectors of the boundary surface are expressed as

[0026] The finite element model is established through a finite structure, called the T-type structure, that is, the target structure, and the following equations are established for the T-type structure:

[0027]

[0028] In the formula, is a nodal force vector acting on any position P0 of the T-structure, except the boundary nodes; and are two flexibility matrices, calculated using the finite element model of the T-type structure; is the force vector of the boundary node of the T-shaped structure, It is called the interface flexibility matrix of the T-type structure. is the flexibility matrix under the action of the excitation point P0;

[0029] After removing the T-shaped structure, a hollow ground with holes is left. Assume that the space is filled with soil, so that the ground becomes a layered structure. Near the boundary surface, a plane conformal to the boundary surface is defined, and the plane is represented as Γ2. The two boundary surface nodes Γ1 and Γ2 correspond one to one.

[0030] Assume that there are virtual nodal forces acting on the boundary surface Γ2 of the layered earth. The nodal forces are expressed as Indicates that Ι means that the force here is a virtual force, which makes the boundary node Γ1 of the layered ground vibrate as So the boundary force is Layered foundation under simple harmonic excitation The response under action is calculated using layered earth theory or 2.5-dimensional finite element boundary element method, and the equation is as follows:

[0031]

[0032] Where, the layered earth flexibility matrix is Calculated by transfer matrix method or 2.5D finite element boundary element method;

[0033] The filling soil is taken out from the layered earth. The filling soil is a cylindrical elastic body with a boundary Γ1 and a free top surface, which is affected by two node force vectors, namely and Since the boundary surfaces Γ1 and Γ2 are conformal surfaces, the filling soil is at the boundary surface Γ1, and the node force vector The finite element vibration equation is as follows:

[0034]

[0035] In the formula, the flexibility matrix Calculated by the finite element model of the filling soil.

[0036] The beneficial effects of the present invention are:

[0037] The present invention gives full play to the advantages of the substructure method and comprehensively considers the coupling between the earth and the structure, thereby significantly improving the calculation accuracy. This is crucial for accurately analyzing the performance of engineering structures in complex environments. At the same time, the method combines artificial boundary conditions with finite element analysis, which greatly broadens the scope of application and enables it to adapt to various complex models and different types of engineering projects. This flexibility not only improves the adaptability of the model, but also provides engineers with a more accurate design basis, helps to optimize the design process, and reduces engineering risks and costs. In addition, by improving calculation efficiency and reliability, it promotes technological progress in related fields and injects new impetus into the development of the industry. Therefore, the present invention will bring significant economic benefits and social value in practical applications, and at the same time lay a solid foundation for future engineering practice. BRIEF DESCRIPTION OF THE DRAWINGS

[0038] Figure 1 It is a schematic diagram of the structure of the viaduct pier-soil body in an embodiment of the present invention;

[0039] Figure 2 It is a schematic diagram of a 2.5-dimensional finite element-boundary element model in an embodiment of the present invention;

[0040] Figure 3 A schematic diagram of the coupling between the pier and the soil in an embodiment of the present invention;

[0041] Figure 4 This is a schematic diagram of a bridge pile structure model in an embodiment of the present invention;

[0042] Figure 5Schematic diagram of a T structure (soil-pile-soil interaction model) in an embodiment of the present invention;

[0043] Figure 6 Schematic diagram of pile-soil interaction model verification (single-layer soil) in an embodiment of the present invention;

[0044] Figure 7 Schematic diagram of multi-layer soil-pile-soil interaction model verification (multi-layer soil) in an embodiment of the present invention;

[0045] Figure 8 A schematic diagram of comparison measurement points in an embodiment of the present invention;

[0046] Fig. 9 Schematic diagram of acceleration response comparison of the top of the pier in the embodiment of the present invention;

[0047] Fig.10 Schematic diagram of the total value of the Z vibration level of the acceleration at the measuring point in the embodiment of the present invention;

[0048] Fig.11 It is a beam-pile equivalent comparison diagram in an embodiment of the present invention. DETAILED DESCRIPTION

[0049] The embodiments of the present invention are described in detail below with reference to the accompanying drawings.

[0050] Example

[0051] A method for predicting building vibrations of an infinitely large non-uniformly distributed structure based on a local inverted T-shaped structure, comprising:

[0052] 1. Soil-pile interaction model based on perfect matching boundary:

[0053] The viaduct pier is composed of pier body, abutment and piles. Figure 1 As shown. Without loss of generality, this embodiment only studies the pile-soil system response of a single pier under unit excitation. The interaction between the soil layer structure and the pile group foundation is the basis for calculating the environmental vibration of the high-speed railway bridge section. The conventional finite element model is quite large and the calculation efficiency is extremely low. Based on this, this embodiment introduces a new dynamic interaction model of the soil layer-pile group structure that utilizes the mutual coupling of 2.5-dimensional finite element-boundary element soil layer structure and 3D finite element pile-soil.

[0054] 1.1, 2.5 dimensional finite element-boundary element method:

[0055] The 2.5-dimensional finite element boundary element model of the soil structure is established in the inertial reference system of the ground, and the x-axis, y-axis and z-axis coincide with the longitudinal, transverse and vertical directions of the track respectively. Figure 2 shown.

[0056] The motion differential equation of the wave number domain finite element-boundary element model of the soil structure is established, where n represents the number of nodes. The corresponding node force vector is F(x,t) in N / m. The 2.5-dimensional finite element equation can be expressed as:

[0057]

[0058] Where M, K0 and K2 are 3n×3n symmetric matrices, M and K2 are positive definite matrices, K0 is a non-negative matrix, and K1 is an antisymmetric matrix. All these matrices are constants and are independent of time and longitudinal x.

[0059] Performing Fourier transform on time and longitudinal direction, the equation with N nodes on the boundary can be expressed as:

[0060]

[0061] In the formula, the circular frequency ω can be expressed as:

[0062] ω=Ω-βc

[0063] Calculate the excitation load e i(Ω-βc)t The Fourier transformed soil displacement matrix q(β) caused by the Fourier transformation can then be used to obtain the actual soil displacement matrix (since there is no moving load effect c=0).

[0064] According to the general equation of the boundary element method, the boundary integral equation is established by taking points on the boundary Γ. Discretizing the equation with N nodes on the boundary, the following boundary element equation is obtained:

[0065] Hu(β)=Gp(β)+B(β)

[0066] In the formula, the displacement and driving force of node i can be expressed as and p(β) i ,at this time:

[0067]

[0068] H and G are 3n×3n matrices related to β; It can be expressed as:

[0069]

[0070] 2. Coupling of pile groups and 2.5D finite element-boundary element model:

[0071] The whole calculation process is divided into two steps (schematic diagram as shown in Figure 3), where a boundary surface (Γ1) is defined for the 3D finite element model and the layered earth model (2.5D finite element-boundary element model). The volume formed by the interface is a rectangular solid, which contains the layered soil and the pile group structure. In the first step, the force of the cylinder is calculated and expressed as a vector The vector dimension depends on the mesh density used for the soil structure. The node displacement vectors on the boundary surface are expressed as

[0072] pass Figure 3 The finite element model of the finite structure in (b) is established. This structure is called the T-type structure (target structure). The following equations can be established for the T-type structure:

[0073]

[0074] In the formula, is a nodal force vector acting on any position P0 of the T-shaped structure (except the boundary nodes), and There are two flexibility matrices that can be calculated using the T-type structure finite element model. is the force vector of the boundary node of the T-shaped structure, It is called the interface flexibility matrix of the T-type structure. It is the flexibility matrix under the action of the excitation point P0.

[0075] Now, after removing the T-shaped structure, a hollow ground is left. Suppose that this space is filled with soil, making the ground a layered structure ( Figure 3 (c) in the figure). In the vicinity of the boundary surface defined above, planes conforming to the boundary surface are defined, and these planes are denoted as Γ2. Here, the mesh density of Γ1 and Γ2 can be the same or different, but for the convenience of calculation, this paper considers a one-to-one correspondence between the nodes of the two boundary surfaces. On this basis, it is assumed that there are virtual nodal forces acting on the boundary surface Γ2 of the layered earth. These nodal forces are expressed as Indicates (Ι indicates that the force here is virtual), which is a method of tuning displacement, making the boundary node Γ1 of the layered ground vibrate as well. So the boundary force is Layered foundation under simple harmonic excitation The response under action can be effectively calculated using layered earth theory (Haskell-Thomson transfer matrix method) or 2.5D finite element-boundary element method, the equation is as follows:

[0076]

[0077] Where, the layered earth flexibility matrix is It can be calculated by transfer matrix method or 2.5D finite element-boundary element method.

[0078] Now, take out from the layered earth Figure 3 The filled soil is a cylindrical elastic body with boundary Γ1 (and free top surface) and is affected by two nodal force vectors, namely and Since the boundary surfaces Γ1 and Γ2 are conformal surfaces, the filling soil is at the boundary surface Γ1, and the node force vector The finite element vibration equation is as follows:

[0079]

[0080] In the formula, the flexibility matrix It can be calculated by filling the soil finite element model.

[0081] Based on this, combined with Figure 4 The bridge pile structure model shown in FIG. 1 is a finite element model of pile-soil interaction, as shown in FIG. Figure 5 shown.

[0082] 3. Numerical verification of soil model:

[0083] In order to verify the soil layer-pile group interaction model of this embodiment, the T-shaped structure is taken as a rectangular body for comparison with the 2.5D finite element-boundary element model for the convenience of calculation. Figure 6 The coupling verification of a single-layer soil is given (the elastic modulus of the soil is 7.0 MPa, the Poisson's ratio is 0.2, and the density is 2400 kg / m 3 , loss factor 0.1), a rectangular load is used, where the side length of the rectangle is 1.8m, the load size is 1N, and the length, width and height of the model are 60×60×30m 3 .

[0084] Depend on Figure 6 It can be seen that due to certain numerical errors in the matrix processing process, there is a certain difference between the model established below 50 Hz and the analytical solution of the layered earth. However, with the increase of frequency and the attenuation of vibration, the model established in this paper and the analytical solution of the single-layered earth above 50 Hz have good consistency in real part, imaginary part and amplitude. Therefore, the boundary flexibility matrix established in this embodiment can be applied to the simulation of the interaction between pile groups and single-layer soil.

[0085] Since the soil is basically layered, and there is rarely a single-layer soil, in order to further compare the comparison results of the model of this embodiment in layered soil, Figure 7 The verification results of multi-layer soil-pile-soil interaction are given. The multi-layer soil parameters are shown in Table 1.

[0086] Depend on Figure 7 It can be seen that:

[0087] (1) Due to certain numerical errors in the matrix processing process, the numerical errors of multi-layer soil layers are slightly larger than those of single-layer soil layers, but the overall trends are basically the same.

[0088] (2) The model established in this embodiment differs from the analytical solution of the multi-layered soil earth in the low frequency band below 40 Hz, which may be caused by the accumulation of errors in the calculation of the soil layer interface after the soil layer is stratified. However, with the increase of frequency and the attenuation of vibration, the consistency is better above 40 Hz.

[0089] Table 11 Soil layer parameters

[0090]

[0091] 2.5D Finite Element-Boundary Element Method:

[0092] At present, the research on the interaction between pier-pile group and soil mainly includes the following approaches: beam-rod layered earth model (① separate the soil and piles, consider the soil as a complete half-space or layered half-space, and subtract the original pile parameters from the soil parameters to make it equivalent to a virtual pile; ② calculate the soil dynamic flexibility matrix and the pile dynamic flexibility matrix separately. It is worth noting that the contact area between the pile and the soil is considered when calculating the soil dynamic flexibility matrix; ③ make the boundary conditions between the soil and the pile consistent, and solve the interaction between the soil and the pile.), static equivalent model (considering the consistency of stiffness, using the static equivalent method, the pile-soil is equivalent to the pure soil parameters between the original pile and the soil, and the damping equivalent method is consistent with the stiffness equivalent method) and three-dimensional finite element model (in the three-dimensional model, the pile is considered as a beam and shares nodes with the soil structure). Therefore, this embodiment compares the top (measurement point 1), bottom (measurement point 2), abutment (measurement point 3), 5m (measurement point 4), 10m (measurement point 5), 20m (measurement point 6), and 30m (measurement point 7) of the pier with these algorithms, as shown in the schematic diagram. Figure 8 shown.

[0093] In order to explore the differences between the three pile-soil interaction calculation algorithms, Fig. 9 The comparison diagram of the acceleration amplitude at the top of the pier is given after adding unit excitation at the top of the pier. Fig. 9 It can be seen that:

[0094] (1) It is not appropriate to simply couple the beam model with the 3D solid finite element model, mainly because the shape of the pile and the vibration of the soil conforming to it are ignored when coupling the beam model with the 3D finite element, which will amplify the vibration of the top of the pier. In addition, since the values ​​of the coupling points are singular, the pile-soil interaction numerical model is greatly affected by the finite element soil mesh, and the difference will increase as the mesh size decreases (the area of ​​soil in contact with the beam decreases).

[0095] (2) The static equivalent model can simulate the pile-soil interaction well below 10 Hz, but above 10 Hz, the static equivalent model overestimates the equivalent stiffness of the pile-soil, which will lead to a lower calculation result.

[0096] In order to further compare the differences brought by these three equivalent methods, Fig.10 The total value of the acceleration Z vibration level at different measuring points under the action of unit force is given. Fig.10 It can be seen that:

[0097] (1) Simply coupling the beam model with the three-dimensional solid finite element model will overestimate the overall vibration of the pier and underestimate the environmental vibration caused by the pile soil.

[0098] (2) Using the static equivalent model at 10 Hz will result in an underestimation of the ambient vibrations at the top of the bridge pier and caused by the high-speed railway bridge section, with a maximum difference of about 2 dBZ per unit force.

[0099] If the beam-pile model is considered as soil connected to the pile, a soil hole is reserved in the finite element model of this embodiment and three degrees of freedom are coupled between the soil and the pile, such as Fig.11 As shown in (a) in the figure, the comparison results between the beam-rod model and the three-dimensional solid finite element model are as follows: Fig.11 As shown in (b) in .

[0100] Depend on Fig.11 It can be seen that when the beam-pile model considers the coupling of soil holes, the calculation results are basically consistent with the solid pile-soil coupling model, but this is contrary to the original intention of using the beam-rod model to simulate piles in the finite element; in the three-dimensional finite element, the beam-rod model is used to avoid the increase in the difficulty of soil meshing due to the pile holes, thereby reducing the model's degrees of freedom. In addition, when the piles are densely packed, each node of each pile needs to consider the coupling of the degrees of freedom of the pile and the soil hole finite element, and the operation is also cumbersome.

[0101] The above-mentioned embodiments only express the specific implementation of the present invention, and the description thereof is relatively specific and detailed, but it cannot be understood as limiting the scope of the present invention. It should be pointed out that, for ordinary technicians in this field, several variations and improvements can be made without departing from the concept of the present invention, which all belong to the protection scope of the present invention.

Claims

1. A method for predicting building vibrations of an infinitely large non-uniform distribution based on a local inverted T-shaped structure, characterized in that: The following steps are involved: Step 1: Establish a 2.5-dimensional finite element boundary element model of the soil structure in the inertial reference system on the ground, where the x-axis, y-axis and z-axis of the inertial reference system coincide with the longitudinal, transverse and vertical directions of the track respectively; Step 2: Couple the pile group with the 2.5D finite element boundary element model to predict the vibration.

2. The method for predicting building vibration of infinite non-uniform distribution based on local inverted T-shaped structure according to claim 1 is characterized in that: The step 1 is specifically as follows: The motion differential equation of the wavenumber domain finite element boundary element model of the soil structure is established, where n represents the number of nodes; the corresponding node force vector is F(x, t), and the 2.5-dimensional finite element equation is expressed as: Where, M, K0 and K2 are 3n×3n symmetric matrices, M and K2 are positive definite matrices, K0 is a non-negative matrix, and K1 is an antisymmetric matrix; all matrices are constants and are independent of time and longitudinal x; Performing Fourier transform on time and longitudinal direction, we get the equation with N nodes on the boundary: -oh 2 Mq(β)+(K0+iβK1+β 2 K2)q(β)=F(β) In the formula, the circular frequency ω is expressed as: ω=Ω-βc Calculate the excitation load e i(Ω-βc)t The Fourier transformed soil displacement matrix q(β) caused by the Fourier transform is then used to obtain the actual soil displacement matrix. Since there is no moving load effect c = 0; According to the general equation of the boundary element method, the boundary integral equation is established by taking points on the boundary Γ; the equation with N nodes on the boundary is discretized to obtain the following boundary element equation: Hu(β)=Gp(β)+B(β) In the formula, the displacement and driving force of node i can be expressed as and p(β) i ,at this time: H and G are 3n×3n matrices related to β; It is expressed as:

3. The method for predicting building vibration of infinite non-uniform distribution based on local inverted T-shaped structure according to claim 2 is characterized in that: The step 2 is specifically as follows: Define a boundary surface Γ1 for the 3D finite element model and the layered earth model, i.e., the 2.5D finite element-boundary element model; the volume formed by the interface is a rectangular solid, which contains the layered soil and the pile group structure; calculate the force of the cylinder and express it as a vector The vector dimension depends on the mesh density used in the soil structure; the node displacement vectors of the boundary surface are expressed as The finite element model is established through a finite structure, called the T-type structure, that is, the target structure, and the following equations are established for the T-type structure: In the formula, is a nodal force vector acting on any position P0 of the T-structure, except the boundary nodes; and are two flexibility matrices, calculated using the finite element model of the T-type structure; is the force vector of the boundary node of the T-shaped structure, It is called the interface flexibility matrix of the T-type structure. is the flexibility matrix under the action of the excitation point P0; After removing the T-shaped structure, a hollow ground with holes is left. Assume that the space is filled with soil, so that the ground becomes a layered structure. Near the boundary surface, a plane conformal to the boundary surface is defined, and the plane is represented as Γ2. The two boundary surface nodes Γ1 and Γ2 correspond one to one. Assume that there are virtual nodal forces acting on the boundary surface Γ2 of the layered earth. The nodal forces are expressed as Indicates that Ι means that the force here is a virtual force, which makes the boundary node Γ1 of the layered ground vibrate as So the boundary force is Layered foundation under simple harmonic excitation The response under action is calculated using layered earth theory or 2.5-dimensional finite element boundary element method, and the equation is as follows: Where, the layered earth flexibility matrix is Calculated by transfer matrix method or 2.5D finite element boundary element method; The filling soil is taken out from the layered earth. The filling soil is a cylindrical elastic body with a boundary Γ1 and a free top surface, which is affected by two node force vectors, namely and Since the boundary surfaces Γ1 and Γ2 are conformal surfaces, the filling soil is at the boundary surface Γ1, and the node force vector The finite element vibration equation is as follows: In the formula, the flexibility matrix Calculated by the finite element model of the filled soil.

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