Modeling method of multilayer frame structure inerter type lumped parameter model for interaction of soil and structural power

Abstract

The invention provides a modeling method for an inertial lumped parameter model of a multi-layer frame structure for interaction of soil and structural power. The modeling method comprises the following steps: standardizing and describing foundation impedance of the multi-layer frame structure of an SSI effect; fitting the impedance of the multi-layer frame structure by utilizing a Legendre polynomial, decomposing the multi-layer frame structure into a partial fractional expansion form, comparing a decomposed first-order term and a decomposed second-order term with a first-order discrete element model and a second-order discrete element model respectively, and identifying undetermined coefficients of each spring, each inerter and each damper in the inerter type lumped parameter model. And finally, establishing a time domain kinetic equation of the multi-layer frame structure under the interaction of the soil and the structural power according to the Alembert principle, and solving a motion control equation by using a Wilson-theta method to obtain the time domain response of the multi-layer frame structure.

Description

Technical Field

[0001] The present invention relates to a modeling method for a multi - layer frame structure inertial - capacitance lumped - parameter model for soil - structure dynamic interaction, belonging to the field of structural dynamics modeling. Background Art

[0002] Most of the current research and application on structural dynamic elastoplastic analysis adopt the rigid foundation assumption, without considering the influence of soil - structure interaction (SSI). Usually, the structure and the foundation are calculated separately. In the Technical Specification for Concrete Structures of Tall Buildings JGJ3 - 2010, the constraint effect of the underground soil on the horizontal deformation of the structure is not considered. In the Code for Seismic Design of Buildings GB50011 - 2010, only for structures in high - intensity seismic regions and located on soft soil foundations, the influence of soil - structure interaction on the horizontal seismic shear force is approximately considered. In fact, soil - structure interaction exists widely, and considering this factor will significantly affect the seismic performance of the structure. Historical earthquake damage investigation and analysis show that the damage of the superstructure is closely related to the foundation conditions. Especially for deep and soft sites, the influence on the seismic response of the structure is more significant and cannot be ignored. At present, the full - method and the sub - structure method are the most commonly used SSI analysis methods. The full - method uses special boundaries to simulate the unbounded soil and analyzes the soil and the structure together. In contrast, the sub - structure method analyzes the soil and the structure separately, and usually uses impedance to represent the behavior of the soil. Impedance represents the force - displacement relationship of the interaction between the foundation and the soil. However, the impedance of the foundation is highly dependent on frequency, resulting in complex analysis and unable to be directly used for the time - domain analysis of structures and the treatment of the nonlinear dynamic response of the system.

[0003] Therefore, a series of simplified lumped - parameter models composed of mass, springs and dampers have emerged to effectively simulate the unbounded soil. For example, in the patent application No. 2016109842574, titled: A Method for Constructing an Equivalent Time - Domain Model Considering the Dynamic Interaction between Soil and Wind Turbine System, the undetermined coefficients of each spring and damper in the model are determined by a recursive method. However, the model proposed in this patent does not contain an inertial - capacitance. Under the premise of achieving the same effect of simulating the unbounded soil, the use of inertial - capacitance can reduce the number of components in the simplified model. The paper titled: Luan Maotian, Lin Gao. A Two - Degree - of - Freedom Lumped - Parameter Model for Foundation Dynamic Impedance [J]. Journal of Dalian University of Technology, 1996, (04): 109 - 114. This paper proposed a lumped - parameter model containing 8 constant parameters, but this model cannot be extended according to the requirements of fitting accuracy, and its application is limited. Therefore, the present invention proposes an inertial - capacitance - type simplified lumped - parameter model with strong applicability and few parameters to effectively simulate the unbounded soil for soil - structure dynamic interaction analysis. Summary of the Invention

[0004] Object of the Invention. To solve the problem that the foundation impedance is highly dependent on frequency and cannot be directly used for time-domain analysis of structures and the nonlinear dynamic response of processing systems. In view of the structural vibration problem under horizontal excitation, the present invention provides a modeling method for an inertial-capacitive lumped parameter model of a multi-story frame structure considering soil-structure dynamic interaction, which has high calculation accuracy and strong practicability.

[0005] Technical Solution. To achieve the above object, the present invention proposes a modeling method for an inertial-capacitive lumped parameter model of a multi-story frame structure for soil-structure dynamic interaction, and the method includes the following steps:

[0006] Step 1. Normalize the foundation impedance of the multi-story frame structure for soil-structure dynamic interaction;

[0007] Step 2. Fit the foundation impedance of the multi-story frame structure with Legendre polynomials;

[0008] Step 3. Use an inertial-capacitive lumped parameter model to simulate the foundation impedance of the multi-story frame structure obtained by fitting, and identify the parameters in the model;

[0009] Step 4. After the foundation impedance of the multi-story frame structure is simulated using an inertial-capacitive lumped parameter model, perform time-history response analysis.

[0010] Further, the method of Step 1 is as follows:

[0011] Set the foundation impedance of the multi-story frame structure for soil-structure dynamic interaction to be the ratio of the force to the displacement at the excitation frequency ω, and normalize it as follows:

[0012] S(a0) = K s [k(a0) + ia0c(a0)] (1)

[0013] In the formula, K s is the static stiffness of the foundation of the multi-story frame structure, k(a0) and c(a0) are the normalized stiffness and damping of the soil, a0 is the normalized frequency, denoted as a0 = ωd / V s , V s is the shear wave velocity of the soil, and d

[0014] is the characteristic length of the foundation of the multi-story frame structure.

[0015] Set the Legendre polynomial L n (x) of the nth order polynomial to be defined as follows:

[0016]

[0017] nL n (x) = (2n - 1)xL n-1(x)-(n - 1)L n-2 (x) (3)

[0018] where \(z = xi\), an imaginary number x represents the coefficient of the imaginary part, and n represents the order;

[0019] The foundation impedance of the multi - layer frame structure is fitted using Legendre polynomials and is expressed as follows:

[0020]

[0021] where \(t=\frac{a_0i}{a}\ 0max , a 0max is the maximum frequency of fitting, p n and q n are the coefficients of the numerator and denominator in Equation (4) respectively; To ensure the double - asymptotic property of the foundation impedance fitting of the multi - layer frame building, the following two conditions are satisfied:

[0022]

[0023] D and R are respectively:

[0024]

[0025]

[0026] where is the dimensionless damping coefficient when \(a = \ 0max ;

[0027] The least - squares method is used to fit and solve the undetermined coefficients p n and q n in Equation (4). After substituting the obtained coefficients into Equation (4), we get:

[0028]

[0029] where \(s = ia_0\), \(\eta n and \(\rho n represent the coefficients of the numerator and denominator respectively.

[0030] Furthermore, the specific steps of Step 3 are as follows:

[0031] Construct inertia - capacitance lumped - parameter models independent of frequency, including a first - order discrete - element model and a second - order discrete - element model respectively;

[0032] The first - order discrete - element model includes a spring with a coefficient of \(-\kappa K s , and there is also a parallel element, which consists of a spring with a coefficient of \(-\kappa K s and a coefficient of connected in series, where κ and γ are the dimensionless spring coefficient and dimensionless damping coefficient in the first-order discrete element model, and the impedance of the first-order discrete element model is expressed as the interaction force R a and displacement u a ratio:

[0033]

[0034] The second-order discrete element model includes a spring with coefficient κ′K s , an inertance with coefficient , and a damper with coefficient connected in series, and then connected in parallel with a spring with coefficient -κ′K s . κ′, δ′ and γ′ are the dimensionless spring coefficient, inertance coefficient and damping coefficient in the second-order discrete element model respectively. The impedance of the second-order discrete element model is expressed as the interaction force R a and displacement u a ratio:

[0035]

[0036] Write Equation (8) in the form of partial fraction expansion:

[0037]

[0038] where f m is the pole and C m is the corresponding residue;

[0039] If f m is a complex pole, it appears in conjugate form, and the corresponding residue C m also appears in conjugate complex form. If f m is a real pole, the corresponding residue C m is also real. Assuming there are G complex poles, Equation (11) is further expressed as:

[0040]

[0041] In the formula, represents the complex residue, * represents the conjugate. Combine the last two terms in Equation (12) to get:

[0042]

[0043] The real coefficients h m0 , h m1 , g m0 and g m1 take the values:

[0044]

[0045] In the formula, and are the real and imaginary parts of the complex poles, and are the real and imaginary parts of the residues corresponding to the complex poles;

[0046] By comparing the first term in Equation (13) with Equation (9), γ and κ in the first-order discrete element model are obtained:

[0047]

[0048] By comparing the second term in Equation (13) with Equation (10), γ′, κ′ and δ′ in the second-order discrete element model are obtained:

[0049]

[0050] Furthermore, the specific steps of Step 4 are as follows:

[0051] Set the multi-story frame structure to be discretized into N s segments. The moment of inertia and lumped mass of the i-th segment are I si and m si respectively. The horizontal damping and shear stiffness of the i-th segment are c si and k si respectively. The height from the i-th particle to the foundation is denoted as h i . The moment of inertia and mass of the foundation are I f and m f respectively; The horizontal and rocking impedances of the multi-story frame structure foundation are represented by the Legendre polynomial lumped parameter model with N h and N r degrees of freedom. According to the dynamic equilibrium conditions of the multi-story frame structure and the lumped parameter models of horizontal and rocking, the time-domain dynamic equation is established using D'Alembert's principle as follows:

[0052]

[0053] Solve Equation (18) using the Wilson-θ numerical integration method to obtain the time-domain response of the multi-story frame structure considering the soil-structure dynamic interaction under horizontal excitation;

[0054] The expression of the mass matrix [M] in Equation (18) is as follows:

[0055]

[0056] The sub-matrix [M s is:

[0057]

[0058] Sub - matrix [M h is:

[0059]

[0060] Wherein, and and represent that the lumped - parameter model in the horizontal direction contains o first - order discrete - element models and w second - order discrete - element models respectively;

[0061] Sub - matrix is:

[0062]

[0063] Wherein, z h represents the inertance coefficient in the lumped - parameter model in the horizontal direction;

[0064] Sub - matrix [M r is:

[0065]

[0066] Wherein, and and represent that the lumped - parameter model in the rocking direction contains o first - order discrete - element models and w second - order discrete - element models respectively;

[0067] Sub - matrix is:

[0068]

[0069] Wherein, z r represents the inertance coefficient in the lumped - parameter model in the rocking direction; The expression of the damping matrix [C] in Equation (18) is as follows:

[0070]

[0071] Wherein,

[0072]

[0073] Sub - matrix and are:

[0074]

[0075] Wherein, c h represents the damping coefficient in the lumped - parameter model in the horizontal direction;

[0076]

[0077] Sub - matrix and is:

[0078]

[0079] In the formula, c r represents the damping coefficient in the lumped parameter model of the rocking direction; the expression of the stiffness matrix [K] in Equation (18) is as follows:

[0080]

[0081] Wherein,

[0082]

[0083] The sub-matrix and are:

[0084]

[0085] The sub-matrix and are:

[0086]

[0087] In the formula, k h represents the stiffness coefficient in the lumped parameter model of the horizontal direction;

[0088]

[0089] The sub-matrix and are:

[0090]

[0091] The sub-matrix and are:

[0092]

[0093] In the formula, k r represents the stiffness coefficient in the lumped parameter model of the rocking direction; in Equation (18), {u} is the generalized displacement vector of the system, and [F] is the external excitation load acting on the system.

[0094] Beneficial effects: Compared with the prior art, the technical solution of the present invention has the following beneficial technical effects:

[0095] (1) The present invention fits the foundation impedance of a multi - layer frame structure based on Legendre polynomials and performs derivation in the form of partial - fraction expansion. The derived lumped - parameter model can be extended for operation according to the requirement of accuracy, and can reflect the variation of the exact solution with frequency in a relatively wide frequency range.

[0096] (2) The present invention introduces an inertance into the lumped - parameter model. Compared with the traditional lumped - parameter model constructed by springs and dampers, it can reduce the number of components in the model, making the calculation and application more convenient.

[0097] (3) The present invention compiles a general matrix module program for solving the time - history response of a multi - layer frame structure under seismic action considering the soil - structure dynamic interaction for the inertance - type lumped - parameter model based on Legendre polynomials, further improving the versatility of the inertance - type lumped - parameter model.

[0098] (4) The lumped - parameter model proposed in the present invention uses inertance, reduces the number of components in the model, compiles a general matrix module program for solving the time - history response of a multi - layer frame structure under seismic action considering the soil - structure dynamic interaction, and can be extended for operation according to the requirement of fitting accuracy, making it more convenient to apply. Brief Description of the Drawings

[0099] Figure 1 The first seven - order curves of Legendre polynomials;

[0100] Figure 2 The first - order and second - order discrete - element models based on inertance and Legendre polynomials;

[0101] Figure 3 Verification of the inertance - type lumped - parameter model;

[0102] Figure 4 The inertance - type lumped - parameter model describing soil - structure dynamic interaction;

[0103] Figure 5 Time - domain response diagram of a multi - layer frame structure considering soil - structure dynamic interaction;

[0104] Figure 6 It is the flowchart of the method of the present invention. Detailed Embodiment

[0105] The present invention proposes a method for modeling an inertance - type lumped - parameter model for soil - structure dynamic interaction analysis. The specific implementation process is as Figure 6 shown. The following combines the drawings and specific cases to elaborate on the embodiments of the present invention in detail, so that the advantages and features of the present invention can be more easily understood by those skilled in the art.

[0106] The case is a 5-story frame building on homogeneous elastic soil. The modal damping ratio of each structural mode is assumed to be 2%. To demonstrate obvious soil-structure interaction effects, a high column stiffness of 180,000 N / m and a soft soil shear velocity of 150 m / s are intentionally selected. The specific parameters of the 5-story frame structure and the soil are shown in Table 1.

[0107] Table 1 Parameters of the 5-story frame structure and the soil

[0108]

[0109] Step 1, the horizontal impedance and rocking impedance of the foundation of the 5-story frame structure of the soil and the structure are normalized as follows:

[0110] S h (a0) = K hs [k h (a0) + ia0c h (a0)]

[0111] S r (a0) = K rs [k r (a0) + ia0c r (a0)]

[0112] In the formula, K s is the static stiffness of the foundation of the 5-story frame structure, k(a0) and c(a0) are the stiffness and damping of the normalized soil, a0 is the normalized frequency, denoted as a0 = ω / V s , V s is the shear wave velocity of the soil, and d is the characteristic length of the foundation of the 5-story frame structure. In this case, taking the embedded square foundation as an example, the horizontal and rocking vibration impedances corresponding to each frequency are shown in Table 2.

[0113] Table 2 Horizontal and rocking vibration impedances of the embedded square foundation

[0114]

[0115]

[0116] Step 2, recursively generate the first seven Legendre polynomials in turn, and the curves correspond as Figure 1 .

[0117]

[0118] Using the least squares method, use the Legendre polynomials to fit the data of the horizontal vibration impedance in Table 2. Figure 3 (a) shows the fitting results when n = 2. The required formula for fitting is as follows:

[0119]

[0120] where \(t = \frac{i a_0}{a}\) 0max , \(a\) 0max is the maximum frequency to be fitted; according to equations (6) and (7) in the invention content, \(R = 3a\) 0max \(\delta / 5\), \(D = p^2 / 2 - q^2 / 2\); further arranging gives:

[0121]

[0122] Expand the above equation into first - order and second - order terms using formula (13), compare them with the first - order and second - order discrete element models respectively, and then according to formulas (16) and (17), the dimensionless spring, damper and inertance coefficients in the first - order discrete element model can be obtained, as shown in Table 3.

[0123] For the rocking impedance in Table 2, use Legendre polynomials for fitting. When \(n = 3\), from Figure 3 (b) it can be seen that the Legendre polynomials can well reflect the variation of the rocking vibration impedance with frequency. Similarly, the lumped - parameter model of the rocking vibration impedance can be further obtained, and the corresponding dimensionless spring, damper and inertance coefficients in the second - order discrete element model are shown in Table 3.

[0124] Table 3 Spring, damper and inertance coefficients of the inertance - type lumped - parameter model

[0125]

[0126] To prove the feasibility of applying the inertance - type lumped - parameter model to the seismic analysis of soil - structure interaction systems, a 5 - story frame building provided by the case was evaluated. The inertance - type lumped - parameter model describing the dynamic interaction between soil and structure is as Figure 4 shown. Using D'Alembert's principle, the time - domain dynamic equation of the interaction between soil and structure is established as follows:

[0127]

[0128] where \([M]\), \([K]\), \([C]\) are the generalized mass matrix, stiffness matrix and damping matrix of the system respectively, \(\{u\}\) is the displacement column vector of each mass point of the system; \([F]\) is the external excitation acting on the system. Combining the structure and soil parameters in Table 1 and the lumped - parameter model parameters obtained in step three, the values of the matrix are as follows:

[0129]

[0130]

[0131] \(x = [x s5 \(x s4 \(xs3 x s2 x s1 u h0 u h1 u h2 u h3 u r0 u r1 u r2 u r3 u r4 T

[0132]

[0133] Under the excitation of the El - centro wave, using the Wilson - θ method to solve the above formula, the time - domain response diagram of the top layer of the 5 - story frame building structure of the dynamic interaction between soil and structure under horizontal excitation is obtained, as Figure 5 shown.

[0134] The above - mentioned are only the embodiments of the present invention, and do not limit the patent scope of the present invention accordingly. All equivalent structural or equivalent process transformations made by using the content of the specification and drawings of the present invention, or directly or indirectly applied in other related technical fields, are equally included in the patent protection scope of the present invention.​

Claims

1. A modeling method for an inertial mass type lumped parameter model of a multi-story frame structure for soil-structure dynamic interaction, characterized in that, The method includes the following steps: Step 1, normalize the foundation impedance of a multi-story frame structure considering soil-structure dynamic interaction; Step 2, fit the foundation impedance of the multi-story frame structure using Legendre polynomials; Step 3, simulate the fitted foundation impedance of the multi-story frame structure using an inertial capacitance lumped parameter model and identify the parameters in the model; Step 4, after the foundation impedance of the multi-story frame structure is simulated using the inertial capacitance lumped parameter model, conduct time-history response analysis.

2. A modeling method for an inertial mass type lumped parameter model of a multi-layer frame structure for soil-structure dynamic interaction according to claim 1, characterized in that The method of Step 1 is as follows: Set the foundation impedance of the multi-story frame structure considering soil-structure dynamic interaction as the ratio of the force with excitation frequency ω to the displacement, and normalize it as follows: S(a0) = K s [k(a0) + ia0c(a0)] (1) where K s is the static stiffness of the foundation of a multi-story frame structure, k(a0) and c(a0) are the stiffness and damping of the normalized soil mass, a0 is the normalized frequency, denoted as a0 = ωd / V s , V s is the shear wave velocity of the soil mass, and d is the characteristic length of the foundation of the multi-story frame structure.

3. A modeling method for an inertial mass type lumped parameter model of a multi - layer frame structure for soil - structure dynamic interaction according to claim 1, characterized in that, The method of Step 2 is as follows: Set the Legendre polynomial \(L\) n (x) is defined as an \(n\)th order polynomial as follows: nL n (x) = (2n - 1)xL n-1 (x) - (n - 1)L n-2 (x) (3) where \(z = xi\), \(i\) is an imaginary number \(x\) represents the coefficient of the imaginary part, and \(n\) represents the order; Fit the foundation impedance of the multi-story frame structure using Legendre polynomials, which is expressed as follows: where \(t = a_{0i} / a\) 0max , \(a\) 0max is the maximum frequency of fitting, \(p\) n and \(q\) n are the coefficients of the numerator and denominator in Equation (4) respectively; to ensure the double-asymptotic property of the fitting of the foundation impedance of the multi-story frame building, the following two conditions are satisfied: D and R are respectively: wherein, is the dimensionless damping coefficient at a 0max ; Use the least squares method to calculate the undetermined coefficients p n and q n in Equation (4) by least squares fitting. Substitute the obtained coefficients into Equation (4) to get: where s = ia0, η n and ρ n represent the coefficients of the numerator and denominator respectively.

4. A modeling method for an inertial mass type lumped parameter model of a multi-layer frame structure for soil-structure dynamic interaction according to claim 1, characterized in that The details of Step 3 are as follows: Construct an inertial capacitance lumped parameter model independent of frequency, including a first-order discrete element model and a second-order discrete element model respectively; The first-order discrete element model includes a spring with a coefficient of -κK s and a parallel element, which consists of a spring with a coefficient of -κK s and a damper with a coefficient of connected in series. κ and γ are the dimensionless spring coefficient and dimensionless damping coefficient in the first-order discrete element model respectively. The impedance of the first-order discrete element model is expressed as the ratio of the interaction force R a to the displacement u a : The second-order discrete element model includes a spring with a coefficient of κ′K s , an inertance with a coefficient of , and a damper with a coefficient of , which are connected in series and then in parallel with a spring with a coefficient of -κ′K s . κ′, δ′, and γ′ are dimensionless spring coefficient, inertance coefficient, and damping coefficient in the second-order discrete element model respectively. The impedance of the second-order discrete element model is expressed as the ratio of the interaction force R a to the displacement u a : Write Equation (8) in the form of partial fraction expansion: where f m is a pole and C m is the corresponding residue; If f m is a complex pole, it appears in conjugate form, and the corresponding residue C m also appears in conjugate complex form. If f m is a real pole, the corresponding residue C m is also real; assuming there are G complex poles, formula (11) is further expressed as: where, represents the complex residue, * represents the conjugate, and combining the last two terms in Equation (12) gives: The real coefficients h in Equation (13) m0 , h m1 , g m0 and g m1 take values as follows: wherein, and are the real part and the imaginary part of the complex poles, and are the real part and the imaginary part of the residues corresponding to the complex poles; Compare the first term in Equation (13) with Equation (9) to obtain the identified γ and κ in the first-order discrete element model: Compare the second term in Equation (13) with Equation (10) to obtain the identified γ′, κ′ and δ′ in the second-order discrete element model:

5. A modeling method for an inertial mass type lumped parameter model of a multi-story frame structure for soil-structure dynamic interaction according to claim 1, characterized in that The details of Step 4 are as follows: Discretize the multi - layer frame structure into N s segments. The moment of inertia and lumped mass of the i - th segment are I si and m si respectively. The horizontal damping and shear stiffness of the i - th segment are c si and k si respectively. The height from the i - th mass point to the foundation is denoted as h i . The moment of inertia and mass of the foundation are I f and m f respectively. The horizontal and rocking impedances of the multi - layer frame structure foundation are represented by the Legendre polynomial lumped - parameter model with N h and N r degrees of freedom. According to the dynamic equilibrium conditions of the multi - layer frame structure and the lumped - parameter models of horizontal and rocking, the time - domain dynamic equations are established using D'Alembert's principle as follows: Use the Wilson-θ numerical integration method to solve Equation (18) to obtain the time-domain response of the multi-story frame structure considering soil-structure dynamic interaction under horizontal excitation; The expression of the mass matrix [M] in Equation (18) is as follows: Sub-matrix [M s is: Sub-matrix [M h is as follows: In the formula, and and represent that the lumped parameter model in the horizontal direction contains o first-order discrete element models and w second-order discrete element models respectively; Submatrix is as follows: where z h represents the inertance coefficient in the horizontal lumped parameter model; Sub-matrix [M r is as follows: In the formula, and and represent that the lumped parameter model of the sway direction contains o first-order discrete element models and w second-order discrete element models respectively; Submatrix is as follows: where z r represents the inertance coefficient in the lumped parameter model of the rocking direction; The expression of the damping matrix [C] in Equation (18) is as follows: Where, Sub-matrix and are as follows: where c h represents the damping coefficient in the lumped parameter model in the horizontal direction; Submatrix and are as follows: where c r represents the damping coefficient in the lumped parameter model of the sway direction; the expression of the stiffness matrix [K] in Equation (18) is as follows: Where, Sub-matrix and are as follows: Sub-matrix and are as follows: where k h represents the stiffness coefficient in the horizontal lumped parameter model; Sub-matrix and are as follows: Sub-matrix and are as follows: where k r represents the stiffness coefficient in the lumped parameter model of the sway direction; In Equation (18), {u} is the system generalized displacement vector and [F] is the external excitation load acting on the system.

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