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1.
A numerical scheme is developed in the paper for calculating torsional, vertical, horizontal, coupling and rocking impedances in frequency domain for axial-symmetric foundations embedded in layered media. In the scheme, the whole soil domain is divided into interior and exterior domains. For the exterior domain, the analytic solutions with unknown coefficients are obtained by solving three-dimensional (3D) wave equations in cylindrical coordinates satisfying homogeneous boundary conditions. For the interior domain, the analytical solutions are also obtained by solving the same 3D wave equations satisfying the homogeneous boundary conditions and the prescribed boundary conditions. The prescribed conditions are the interaction tractions at the interfaces between embedded foundation and surrounding soil. The interaction tractions are assumed to be piecewise linear. The piecewise linear tractions at the bottom surface of foundation will be decomposed into a series of Bessel functions which can be easily fitted into the general solutions of wave equations in cylindrical coordinates. After all the analytic solutions with unknown coefficients for both interior and exterior domains are found, the variational principle is employed using the continuity conditions (both displacements and stresses) at the interfaces between interior and exterior domains, interior domain and foundation, and exterior domain and foundation to find impedance functions. 相似文献
2.
Gin-Show Liou 《地震工程与结构动力学》1994,23(2):193-210
A systematic procedure is presented for generating dynamic stiffness matrices for two independent circular foundations on an elastic half-space medium. With the technique reported in References 1–3, the analytic solution of three-dimensional (3D) wave equations satisfying the prescribed traction due to the vibration of one circular foundation can be found. Since there are two analytic solutions for two prescribed tractions due to the vibrations of two circular foundations, the principle of superposition must be used to obtain the total solution. The interaction stresses (prescribed tractions) are assumed to be piecewise linear in the r-directions of both cylindrical co-ordinates for the two circular foundations. Then, the variational principle and the reciprocal theorem are employed to generate the dynamic stiffness matrices for the two foundations. In the process of employing the variational principle, a co-ordinate transformation matrix between two cylindrical co-ordinate systems is introduced. Some numerical results of dynamic stiffness matrices for the interaction of two identical rigid circular foundations are presented in order to show the effectiveness and efficiency of the present method, and some elaborations for its future extensions are also discussed. 相似文献
3.
This paper examines the axisymmetric torsional vibrations of an elastic pile and a hemispherical foundation embedded in a homogeneous elastic half-space. The embedded foundation–half-space system is decomposed into an extended half-space and a fictitious foundation. The deformations of the fictitious system are specified by an admissible function containing a set of generalized coordinates. The Lagrangian equations of motion are used to determine these coordinates associated with the assumed displacement function. Numerical results are presented for torsional impedance of an elastic pile and a hemisphere to illustrate the effects of relative flexibility and geometry. By employing certain simplifications on the pile–half-space system an approximate closed form solution is presented for the torsional impedance of an elastic pile. 相似文献
4.
A numerical procedure is proposed to investigate the transient response of a group of rigid strip foundations resting on an elastic, homogeneous half-space subjected to either external forces or seismic motions. A fundamental solution is presented for uniform strip loadings with Heaviside function time-dependence applied on the half-space. In the procedure, each of the foundations is discretized into subelements. The tractions between the half-space and the subelements are assumed constant at every time step. The through-soil coupling effects between the foundations are studied numerically. 相似文献
5.
A half-space finite element and a consistent transmitting boundary in a cylindrical coordinate system are developed for analysis of rigid circular (or cylindrical) foundations in a water-saturated porous layered half-space. By means of second-order paraxial approximations of the exact dynamic stiffness for a half-space in plane-strain and antiplane-shear conditions, the corresponding approximation for general three-dimensional wave motion in a Cartesian coordinate system is obtained and transformed in terms of cylindrical coordinates. Using the paraxial approximations, the half-space finite element and consistent transmitting boundary are formulated in a cylindrical coordinate system. The development is verified by comparison of dynamic compliances of rigid circular foundations with available published results. Examination of the advantage of the paraxial condition vis-á-vis the fixed condition shows that the former achieves substantial gain in computational effort. The developed half-space finite element and transmitting boundary can be employed for accurate and effective analysis of foundation dynamics and soil–structure interaction in a porous layered half-space. 相似文献
6.
An approximate numerical procedure for calculation of the harmonic force-displacement relationships for a rigid foundation of arbitrary shape placed on an elastic half-space is presented. This procedure is used to evaluate the vertical, rocking and horizontal compliance functions for rigid rectangular foundations and the vertical compliance for a rigid square foundation with an internal hole. Several comparisons between the results obtained by the proposed approach and other methods are also presented. 相似文献
7.
A version of the global–local finite element method is presented for studying dynamic steady-state soil–structure interaction wherein the soil medium extends to infinity. Herein, only axisymmetric behaviour is considered. In this approach, conventional finite elements are used to model the structure and some portion of the surrounding soil medium considered to be homogeneous and isotropic. A complete set of outgoing waves in the form of spherical harmonics for the entire space is used to represent the behaviour in the half-space beyond the finite element mesh and these are termed the global functions. Full traction and displacement continuity is enforced at the finite element mesh interface with the outer region. On the free surface of the half-space in the outer field, traction-free surface conditions are enforced by demanding that a sequence of integrals of the weighted-average tractions must vanish. Numerical examples are presented for the response of different shaped foundations, resting on the free surface or at various submerged levels, due to a normal seismic plane compressional wave. Plots of differential scattering cross-sections show the angular distribution of the energy (its directional nature) of the scattered field. 相似文献
8.
Asymmetric steady-state structure-media interaction due to obliquely incident body waves is investigated via a version of the global local finite element method. In the present version, a local region that houses an axisymmetric structure is modelled by conventional finite elements, while the behaviour in the remaining portion of the homogeneous semi-infinite medium is presented by the spherical harmonics that are the eigensolutions of the entire space problem. The solution scheme involves (1) full displacement and traction continuity along the boundary between the local and the exterior regions and (2) satisfaction of the traction-free requirement on the surface of the half-space beyond the discretized region by virtue of a sequence of integral constraints of the non-zero weighted surface tractions of the spherical harmonics. The numerical results presented are for a perfectly bonded rigid circular foundation resting on the surface of the half-space and subjected to obliquely incident body waves. Dependence of the displacement response of the footing upon incident angles and dimensionless wave numbers is thoroughly studied. 相似文献
9.
A time domain boundary element in a cylindrical co-ordinate system is developed for the analysis of wave propagation in a layered half-space. The field quantities (displacements and tractions) are expressed as products of Fourier series in the circumferential direction and as linear polynomials in the other spatial directions. An integral equation is written for each layer as an independent domain, and these equations are then assembled into a general equation by virtue of compatibility and equilibrium conditions between the interfaces. Examples of three-dimensional wave propagation in the layered half-spaces due to various forms of surface and inner-domain excitations are reported to demonstrate the accuracy and versatility of the method. 相似文献
10.
The precise integration method (PIM) is proposed for the dynamic response analysis of rigid strip footing resting on arbitrary anisotropic multi-layered half-space. In the frequency domain, the governing equation of wave motion is converted into dual vector form of first-order ordinary differential equations which is solved by PIM. Each layer is divided into a large number (say, 2N) of mini-layers of equal thickness, within which characteristic matrices are assumed to vary following the Taylor series expansion to the fourth order. As a result, any desired accuracy of the displacements and stresses can be achieved by PIM. In addition, dual vector form equation makes it quite easily to combine two adjacent mini-layers into a new one. Each pass of combination reduces the total number of mini-layers by a half. The computational effort for the evaluation of the dynamic impedance of rigid strip footing can be reduced to a great extent. Numerical examples are provided to validate the efficiency and accuracy of the proposed approach. 相似文献
11.
Martha Surez Javier Avils Francisco J. Snchez-Sesma 《Soil Dynamics and Earthquake Engineering》2002,22(8)
A simplified indirect boundary element method is applied to compute the impedance functions for L-shaped rigid foundations embedded in a homogeneous viscoelastic half-space. In this method, the waves generated by the 3D vibrating foundation are constructed from radiating sources located on the actual boundary of the foundation. The impedance functions together with the free-field displacements and tractions generated along the soil–foundation interface are used to calculate the foundation input motion for incident P, S and Rayleigh waves. This is accomplished by application of Iguchi's averaging method which, in turn, is verified by comparison with results obtained rigorously using the relation between the solutions of the basic radiation (impedance functions) and scattering (input motions) problems. Numerical results are presented for both surface-supported and embedded foundations. It is shown how the seismic response of L-shaped foundations with symmetrical wings differs from that of enveloping square foundations. The effects of inclination and azimuth of the earthquake excitation are examined as well. These results should be of use in analyses of soil–structure interaction to account for the traveling wave effects usually overlooked in practice. 相似文献
12.
R. Betti 《地震工程与结构动力学》1997,26(10):1005-1019
A boundary element formulation of the substructure deletion method is presented for the seismic analysis of the dynamic cross-interaction between multiple embedded foundations. This approach is particularly suitable for three-dimensional foundations of any arbitrary geometrical shape and spatial location, since it requires only the discretization of the foundations’ surfaces. The surrounding soil is represented by a homogeneous viscoelastic half-space while the foundations are assumed to be rigid and subjected to incoming SH-, P-, and SV-waves arbitrarily inclined in both the horizontal and vertical planes. The proposed methodology is tested for the case of two identical embedded square foundations for different values of the foundations’ embedment and distance. The effects of the cross-interaction are outlined in the components of the impedance matrix and of the foundation input motion. © 1997 John Wiley & Sons, Ltd. 相似文献
13.
Scattering of elastic waves by dipping layers of arbitrary shape embedded within an elastic half-space is investigated for a plane strain model by using a boundary method. Unknown scattered waves are expressed in the frequency domain in terms of wave functions which satisfy the equations of motion and appropriate radiation conditions at infinity. The steady state displacement field is evaluated throughout the elastic medium for different incident waves so that the continuity conditions along the interfaces between the layers and the traction-free conditions along the surface of the half-space are satisfied in the least-squares sense. Transient response is constructed from the steady state one through the Fourier synthesis. The results presented show that scattering of waves by dipping layers may cause locally very large amplification of surface ground motion. This amplification depends upon the type and frequency of the incident wave, impedance contrast between the layers, component of displacement which is being observed, location of the observation station and the geometry of the subsurface irregularity. These results are in agreement with recent experimental observations. 相似文献
14.
Chi Hsin Lin Vincent W. Lee Maria I. Todorovska Mihailo D. Trifunac 《Soil Dynamics and Earthquake Engineering》2010
The zero-stress boundary conditions at the surface of the half-space in the presence of surface and sub-surface cavities for in-plane, incident cylindrical P- and SV-waves have always posed challenging problems. The outgoing cylindrical P- and SV-waves can be represented by Hankel functions of radial distance coupled with the sine and cosine functions of angle. Together, at the half-space surface the P- and SV-wave functions are not orthogonal over the semi-infinite radial distance from 0 to infinity. Thus, to simultaneously satisfy the zero in-plane, normal, and shear stresses, an approximation of the geometry is often made. This paper presents an analytical formulation of the boundary-valued problem, where the Hankel wave functions are expressed in integral form, changing the representation from cylindrical to rectangular coordinates, so that the zero-stress boundary conditions at the half-space surface can be applied in a more straightforward way. 相似文献
15.
With a simplified model and Galerkin's weighted residual procedure, two simple differential equations of dynamic behavior of a bounded rectangular medium are established along the boundaries in the x- and y-direction in the medium. Solutions of these equations yield closed form expressions of soil stiffnesses for various cases of a partially embedded rigid foundation, including the stiffnesses per depth of foundation with rectangular base area and the stifnesses of strip foundation. The developed procedure provides the definition of the weight functions, which are used in Galerkin's method for weighted residual. In addition to these weight functions, their conjugators are also suitable for weight functions. When the soil depth is finite, the original weight functions fail to produce physically meaningful results in some frequency range but the conjugators do not fail at any frequencies. The developed equations to compute soil stiffnesses for embedded foundations are simple yet capable of calculating the responses close to those computed by the much more elaborated finite element method. 相似文献
16.
The response of a rigid rectangular foundation block resting on an elastic half-space has been determined by considering first the displacement functions for any position on the surface of an unloaded half-space due to a harmonic point force. The influence of the foundation has been taken into account by assuming a relaxed condition at the interface, i.e. a uniform displacement under the foundation and that the sum of the point forces must equal the total applied force. The three motions of vertical, horizontal and rocking have been considered and numerical values for the in-phase and the quadrature components of the displacement functions are presented for a Poisson's ratio of 0.25. The effect of the mass and inertia of the foundation can be allowed for by an impedance matching technique. Response curves and non-dimensional resonant frequency curves are given for a square and a rectangular foundation for different mass and inertia ratios and for several values of Poisson's ratio. These curves are for design purposes and are an addition to similar published curves for circular and infinitely long rectangular foundations. Some of the calculated results have been verified by a laboratory experiment. 相似文献
17.
Y. K. Chow 《地震工程与结构动力学》1987,15(5):585-594
A numerical method of analysis is presented for the determination of the steady-state vertical vibration of rigid foundations with arbitrary three-dimensional geometries resting on the surface of a layered soil medium. The method utilizes the flexibility concept applied to steady-state periodic problems and it is solved in the frequency domain. The accuracy of the method is verified by comparison with several published solutions for massless, smooth rigid rectangular foundations on a homogeneous, isotropic elastic half-space. Parametric solutions are presented to study the dynamic behaviour of massless, smooth rigid rectangular foundations on a homogeneous, elastic stratum. 相似文献
18.
Gin-Show Liou 《地震工程与结构动力学》1989,18(5):667-686
The total system studied in this paper is a layered soil stratum with a rigid bedrock and a cylindrical cavity on the surface. Analytic solutions for the layered medium with prescribed harmonic displacement time history on the surface of the cylindrical cavity are presented. The whole soil domain is divided into interior and exterior domains. The interior domain is the projection of the cylindrical cavity down to the rigid bedrock, whereas the exterior domain is then the soil medium complement to the interior domain. The displacement and stress fields in both domains are expanded as an infinite series of Fourier components with respect to the azimuth. For each Fourier component in the infinite series, the solutions for both domains are found independently by solving the general differential equations of wave propagation satisfying the boundary conditions of the top surface and the lower rigid boundary. Displacement and stress continuity conditions are then imposed on the vertical interface between the two domains using the formulation of a weighted residual. For the soil-structure interaction problem, the impedance matrix at the interface between the structure and the soil medium can be easily generated using the analytic solutions, which can then be combined with the finite element model of the structure. A simple example is presented to demonstrate the effectiveness of the procedure presented. 相似文献
19.
An alternative technique to obtain the dynamic response of a massless rigid circular foundation resting on a uniform elastic half-space when subjected to harmonic plane waves is presented. The technique relies on the use of an integral representation involving the free-field ground motion and the contact tractions obtained in the course of calculating the dynamic force–displacement relationship of the foundation for external forces. Tables listing the translational and rotational components of the response of the foundation for non-vertically incident SH, P, SV and Rayleigh waves are presented. 相似文献
20.
A direct boundary element procedure is presented to determine the impedance matrix for a three-dimensional foundation supported on an infinitely-long canyon of uniform cross-section cut in a homogeneous half-space. The uniform cross-section of the canyon permits analytical integration along the canyon axis leading to a series of two-dimensional boundary problems involving Fourier transforms of the full-space Green's functions. Solution of these two-dimensional boundary problems leads to a dynamic flexibility influence matrix which is inverted to determine the impedance matrix. The accuracy of the procedure is demonstrated by comparison with previous solutions for a surface-supported, square foundation and results obtained by a three-dimensional boundary element method (BEM) for a foundation of finite-width supported on an infinitely-long canyon. Compared with the three-dimensional BEM, the present method requires less computer storage and is more accurate and efficient. The foundation impedance matrix determined by this procedure can be incorporated in a substructure method for earthquake analysis of arch dams. 相似文献