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1.
The interaction forces of the linear unbounded soil in a non-linear soil-structure-interaction analysis can be calculated recursively, starting directly from the dynamic-stiffness coefficients in the frequency domain. Two possibilities of choosing a recursive equation are discussed.
- (i) The recursive equation in the frequency domain. For each frequency, the interaction force at a specific time station is expressed as a function of the corresponding interaction force at the previous time station and of the displacements at the current time station and at the two most recent past time stations. This recursive evaluation of the convolution integral. which can be derived using the z-transformation, is rigorous. By using interpolation in the frequency domain, an approximate procedure results, which leads to a significant reduction in computational effort.
- (ii) The recursive equation in the time domain. By approximating the dynamic-stiffness coefficients as the ratios of two polynomials in frequency using a curve-fitting technique based on the least-squares method and by applying the partial-fraction expansion and using the z-transformation, the recursive coefficients can be determined explicitly. Alternatively, the ratio of two polynomials can also be transformed to an ordinary differential equation together with the initial conditions.
2.
The interaction forces representing the contribution of the linear unbounded soil to the equations of motion of a nonlinear soil-structure-interaction analysis are specified in the form of convolution integrals. They can be evaluated recursively in the time domain. In this procedure, the forces at a specific time are computed from the displacements at the same time and from the most recent forces and most recent past displacements. It is, in principle, only approximate. When the dynamic-stiffness coefficients can be expressed as the ratios of two polynomials in frequency, the appropriately chosen recursive equations are exact. Two possibilities of choosing a recursive equation are discussed.
- (i) The impulse-invariant method, where the unknown recursive coefficients are calculated by solving a system of equations which are established by equating the rigorous and recursive formulations for a discretized unit impulse displacement.
- (ii) In the segment approach, the dynamic-stiffness coefficients in the time domain are interpolated piecewise. Applying the z-transformation analytically then results in an explicit recursive equation without solving a system of equations.
3.
John P. Wolf 《地震工程与结构动力学》1991,20(1):11-32
A systematic procedure to develop a consistent lumped-parameter model with real frequency-independent coefficients to represent the unbounded soil is developed. Each (modelled) dynamic-stiffness coefficient in the frequency domain is approximated as a ratio of two polynomials, which is then formulated as a partial-fraction expansion. Each of these terms is represented by a discrete model, which is the building block of the lumped-parameter model. A second-order term, for example, leads to a discrete model with springs and dampers with two internal degrees of freedom, corresponding to two first-order differential equations, or, alternatively, results in a discrete model with springs, dampers and a mass with one internal degree of freedom, corresponding to one second-order differential equation. The lumped-parameter model can easily be incorporated in a general-purpose structural dynamics program working in the time domain, whereby the structure can even be non-linear. A thorough evaluation shows that highly accurate results are achieved, even for dynamic systems with a cutoff frequency. 相似文献
4.
Green's influence functions are derived for a linearly distributed load acting on part of a layered elastic halfplane on a line which is inclined to the horizontal. Using these Green's functions as fundamental solutions in the boundary-element method, the dynamic-stiffness matrices of the unbounded soil with excavation, of the excavated part and of the free field are calculated. The indirect boundary-element method using distributed loads and no offset leads to more accurate results than the weighted-residual technique and the direct boundary-element method. At the natural frequencies of the undamped excavated part built-in along the structure-soil interface, the spring coefficients associated with the dynamic-stiffness matrices of the excavated part and of the free field will become infinite. If the dynamic-stiffness matrix of the soil with excavation is calculated as the difference of that of the free field and that of the excavated part, the difference of two large numbers will arise in the vicinity of these frequencies. A consistent discretization must then be used. In particular, the dynamic-stiffness matrix of the embedded part cannot be determined by the finite-element method in this case. A parametric study is performed for the dynamic-stiffness matrix of the free field for a rectangular foundation embedded in a halfplane and in a layer built-in at its base; the aspect ratio and the damping of the soil are varied. 相似文献
5.
We propose an effective and reliable time-domain substructure technique which takes soil-structure interaction effects into account and uses the flexibility coefficients of unbounded soil obtained in the frequency domain. Compared with previous methods, the frequency points to calculate flexibility coefficients, and computational loads in the calculation of time-domain interaction forces, are reduced. In the formulation, we have assumed the flexibility coefficient to be a periodic function, obtained within the bandlimited frequency range, which also includes the predominant frequencies of the structure and incident wave. Then we simulate the periodic flexibility coefficients using discrete impulse responses in the time-domain analyses. However, the real and imaginary parts of the bandlimited flexibility coefficients do not form a Hilbert transform pair; the discrete impulse responses should be modified to be causal for the time-domain analyses. We present various discrete impulse responses which have been obtained from only the real part, only the imaginary part and from both the real and imaginary parts of the frequency-domain flexibility coefficients by FFT with causality conditions. Through a numerical example the relationship between the modified discrete impulse responses and the validity of the time-domain substructure method is presented. 相似文献
6.
To represent a cylindrical rigid foundation vibrating in horizontal, vertical, rocking or torsional motions embedded in a soil layer resting on rigid rock, a lumped-parameter model is described. The coupling between the horizontal and rocking degrees of freedom is considered. For each degree of freedom eight frequency-independent real coefficients determine the springs, dashpots and the mass of the lumped-parameter model with two internal degrees of freedom. These coefficients are specified for various ratios of the radius of the foundation to the depth of the layer and lateral contact ratios. To derive the mechanical properties of the lumped-parameter model a systematic procedure of curvefitting of the dynamic-stiffness coefficient up to, in general, twice the fundamental frequency of the layer is applied, capturing the fact that below the (horizontal) fundamental frequency (cutoff frequency) no radiation of energy occurs. The lumped-parameter model can be used to represent the soil in a standard finite-element program for structural dynamics working in the time domain, whereby the structure can exhibit non-linear behaviour. Stability of the unbounded soil-layer model and of the total system is guaranteed. A hammer foundation with partial uplift of the anvil is analysed for illustration. 相似文献
7.
A systematic procedure to construct the (symmetric) static-stiffness, damping and mass matrices representing the unbounded medium is presented addressing the unit-impulse response matrix corresponding to the degrees of freedom on the structure–medium interface. The unit-impulse response matrix is first diagonalized which then permits each term to be modelled independently from the others using expansions in a series of Legendre polynomials in the time domain. This leads to a rational approximation in the frequency domain of the dynamic-stiffness coefficient. Using a lumped-parameter model which provides physical insight the property matrices are constructed. 相似文献
8.
Approximate dynamic-stiffness coefficients of a disk on the surface of a single layer on a half-space may be calculated using cone models. This concept is generalized to the case of a horizontally stratified site consisting of many layers on a homogeneous half-space. After constructing the so-called ‘backbone cone’ determining the radii of the disks at all interfaces, the dynamic-stiffness matrices of the layers (modelled as cone frustums) and the dynamic-stiffness coefficient of the underlying half-space (modelled as a cone) are assembled to that of the site. The dynamic-stiffness matrix of a layer is a complex-valued function of frequency because radiation of energy in the horizontal direction is considered. In this model of the layered half-space the properties of the cone reproduce themselves (cloning). The advantages of using cone models are also present for the layered half-space; in particular, no transformation to the wave-number domain is performed. 相似文献
9.
In the damping-solvent extraction method, to calculate the dynamic-stiffness matrix of an unbounded medium, a finite region of the medium, adjacent to the structure is analysed in the first step, whereby hysteretic material damping is introduced artificially as a solvent. This leads to the dynamic-stiffness matrix of the damped bounded medium, which is assumed in the second step to be equal to that of the damped unbounded medium. In the third step, the effect of the material damping on the dynamic-stiffness matrix is eliminated, i.e. the damping solvent is extracted, resulting in the dynamic-stiffness matrix of the unbounded medium. The damping-solvent extraction method permits an efficient calculation of the dynamic-stiffness matrix of an unbounded medium by analysing the adjacent bounded medium only, which exhibits the same dynamic characteristics as the (bounded) structure. The familiar standard finite-element method is sufficient for the analysis and the hysteretic damping is introduced by multiplying the elastic moduli by 1 + 2i£. The introduced hysteretic material damping, the solvent, is extracted at the end of the analysis for each coefficient of the dynamic-stiffness matrix and for each frequency independently of the others by a very concise equation based on a Taylor expansion. The method is evaluated thoroughly for dynamic soil-structure interaction and for seismic reservoir-dam interaction using stringent simple cases with analytical solutions available and is also applied to practical examples, by calculating the dynamic-stiffness matrix of a semi-infinite wedge and an embedded foundation. 相似文献
10.
To calculate the dynamic-stiffness matrix in the time domain (unit-impulse response functions) of the unbounded medium, the infinitesimal finite element cell method based solely on the finite element formulation and working exclusively in the time domain is developed. As in the cloning algorithm, the approach is based on similarity of the unbounded media corresponding to the interior and exterior boundaries of the infinitesimal finite element cell. The derivation can be performed exclusively in the time domain, or alternatively in the frequency domain. At each time station a linear system of equations is solved. The consistent-boundary method to analyse a layered medium in the frequency domain and the viscous-dashpot boundary method are special cases of the infinitesimal finite element cell method. The error is governed by the finite element discretization in the circumferential direction, as the width of the finite-element cell in the radial direction is infinitesimal. The infinitesimal finite element cell method is thus ‘exact in the finite-element sense’. This method leads to highly accurate results for a vast class of problems, ranging from a one-dimensional spherical cavity to a rectangular foundation embedded in a half-plane. 相似文献
11.
Starting from a weighted-residual formulation, the various boundary-element methods, i.e. the weighted-residual technique, the indirect boundary-element method and the direct boundary-element method, are systematically developed for the calculation of the dynamic-stiffness matrix of an embedded foundation. In all three methods, loads whose analytical response in the unbounded domain can be determined are introduced acting on the continuous soil towards the region to be excavated. In the weighted-residual technique and in the indirect boundary-element method, a weighting function is used; in the latter case, it is selected as the Green's function for the surface traction. In the direct boundary-element method, the surface traction along the structure-soil interface is interpolated. The same type of boundary matrices which have a clear physical interpretation are identified in the three formulations, each of which is illustrated with a simple static example. The indirect boundary-element method leads to the most accurate results. The guaranteed symmetry and the fact that the displacement arising from the applied loads can easily be calculated and compared to the prescribed displacement makes the indirect boundary-element method especially attractive for calculating the dynamic-stiffness matrix of the soil. Instead of calculating the dynamic-stiffness matrix of the embedded foundation with the boundary-element method, it can be determined as the difference of those of the regular free field and of the excavated part. The calculation of the former does not require the Green's function for the surface traction. The dynamic stiffness of the excavated part can be calculated by the finite-element method. 相似文献
12.
Hydrodynamic-stiffness matrix based on boundary elements for time-domain dam-reservoir-soil analysis
For a reservoir with an arbitrary shape of the upstream dam face and of the bottom including an adjacent regular part of constant depth extending to infinity, the hydrodynamic-stiffness matrix in the frequency domain for a displacement formulation is derived using the boundary-element method. The fundamental solution takes the boundary condition at the free surface into account. The analytical solution of the semi-infinite reservoir is used to improve the accuracy. To be able to transform the hydrodynamic-stiffness matrix from the frequency to the time domain, the singular part consisting of its asymptotic value of ω ∞ is split off. It consists of an imaginary linear term in ω which can be interpreted as a damper with a coefficient per unit area equal to the product of the mass density and the wave velocity. This also applies for a reservoir bottom of arbitrary shape. The remaining regular part of the stiffness matrix is transformed numerically. The corresponding interaction force-displacement relationship involves convolution integrals. This boundary-element solution agrees well with analytical results and with those of other numerical procedures based on a time-stepping method. The method is also applied to an actual earthquake acting on a reservoir with an irregular part with an inclined bottom and a regular part extending to infinity. The results of the analysis in the time domain coincide with those determined in the frequency domain. 相似文献
13.
To calculate the dynamic-stiffness matrix of the unbounded soil, a multi-cell cloning algorithm based solely on the finite-element formulation is developed. A non-linear system of equations has to be solved iteratively once for the whole range of frequency. For a specific frequency, two- and three-cell cloning lead to highly accurate results, which is demonstrated by examples of increasing complexity. 相似文献
14.
15.
A procedure which involves a non‐linear eigenvalue problem and is based on the substructure method is proposed for the free‐vibration analysis of a soil–structure system. In this procedure, the structure is modelled by the standard finite element method, while the unbounded soil is modelled by the scaled boundary finite element method. The fundamental frequency, and the corresponding radiation damping ratio as well as the modal shape are obtained by using inverse iteration. The free vibration of a dam–foundation system, a hemispherical cavity and a hemispherical deposit are analysed in detail. The numerical results are compared with available results and are also verified by the Fourier transform of the impulsive response calculated in the time domain by the three‐dimensional soil–structure–wave interaction analysis procedure proposed in our previous paper. The fundamental frequency obtained by the present procedure is very close to that obtained by Touhei and Ohmachi, but the damping ratio and the imaginary part of modal shape are significantly different due to the different definition of damping ratio. This study shows that although the classical mode‐superposition method is not applicable to a soil–structure system due to the frequency dependence of the radiation damping, it is still of interest in earthquake engineering to evaluate the fundamental frequency and the corresponding radiation damping ratio of the soil–structure system. Copyright © 2001 John Wiley & Sons, Ltd. 相似文献
16.
The contribution of the (linear) unbounded soil to the basic equation of motion of a non-linear analysis of soil-structure interaction consists of convolution integrals of the displacement-force relationship in the time domain and the history of the interaction forces. The former is calculated using the indirect boundary-element method, which is based on a weighted-residual technique and involves Green's functions. As an example of a non-linear soil-structure-interaction analysis, the partial uplift of the basemat of a structure is examined. As the convolution integrals have to be recalculated for each time step, the computational effort in this rigorous procedure is substantial. A reduction can be achieved by simplifying the Green's function by ‘concentrating’ the region of influence. Alternatively, assuming a specified wave pattern, a coupled system of springs and dashpots with frequency-independent coefficients can be used as an approximation. 相似文献
17.
A discrete model to represent the unbounded soil (halfspace) in a soil–structure interaction analysis in the time domain is developed. For each dynamic degree of freedom of the foundation node, the discrete model consists of a mass M0 which is attached to a rigid support with a spring K and with a damper C0. In addition, a free node with the mass M1 is introduced, which is connected to the foundation node with a damper C1. All coefficients are frequency-independent. The discrete model is semi-empirical. It is based on a semi-infinite truncated cone, whereby, after enforcing the static stiffness, the remaining parameters are modified to achieve an optimal fit of the dynamic-stiffness coefficient in the frequency domain. The spring K is equal to the static stiffness. The coefficients appearing in the equations for the dampers C0, C1 and the masses M0, M1 are specified (assuming a homogeneous halfspace) for the disc, the embedded cylinder, the rectangle (also embedded) and the strip. A square on a layer whose stiffness increases with depth resting on a homogeneous halfspace is also treated. For an embedded foundation, eccentricities arise. Material damping increases the damper C0 and the mass M0. 相似文献
18.
A systematic procedure to construct a consistent global lumped-parameter model consisting of springs, dashpots and possibly masses with frequency-independent coefficients connecting the degrees of freedom of the nodes of any structure-medium interface for the unbounded medium is presented. The dynamic-stiffness matrix is first diagonalized which then permits each term to be modelled independently from the others. Physical insight is thus provided. Alternatively, the (symmetric) static-stiffness and damping matrices and possibly mass matrix of the unbounded medium can be established directly. 相似文献
19.
Exact representation of unbounded soil contains the single output–single input relationship between force and displacement in the physical or transformed space. This relationship is a global convolution integral in the time domain. Rational approximation to its frequency response function (frequency‐domain convolution kernel) in the frequency domain, which is then realized into the time domain as a lumped‐parameter model or recursive formula, is an effective method to obtain the temporally local representation of unbounded soil. Stability and identification for the rational approximation are studied in this paper. A necessary and sufficient stability condition is presented based on the stability theory of linear system. A parameter identification method is further developed by directly solving a nonlinear least‐squares fitting problem using the hybrid genetic‐simplex optimization algorithm, in which the proposed stability condition as constraint is enforced by the penalty function method. The stability is thus guaranteed a priori. The infrequent and undesirable resonance phenomenon in stable system is also discussed. The proposed stability condition and identification method are verified by several dynamic soil–structure‐interaction examples. Copyright © 2009 John Wiley & Sons, Ltd. 相似文献