Dynamic stiffness of deep foundations with inclined piles

The influence of inclined piles on the dynamic response of deep foundations and superstructures is still not well understood and needs further research. For this reason, impedance functions of deep foundations with inclined piles, obtained numerically from a boundary element–finite element coupling model, are provided in this paper. More precisely, vertical, horizontal, rocking and horizontal–rocking crossed dynamic stiffness and damping functions of single inclined piles and 2 × 2 and 3 × 3 pile groups with battered elements are presented in a set of plots. The soil is assumed to be a homogeneous viscoelastic isotropic half‐space and the piles are modeled as elastic compressible Euler–Bernoulli beams. The results for different pile group configurations, pile–soil stiffness ratios and rake angles are presented. Copyright © 2010 John Wiley & Sons, Ltd.     

Dynamic stiffness of rigid rectangular foundations on the half-space

The dynamic soil–structure interaction of a rigid rectangular foundation with the subsoil represents a mixed-boundary value problem. This problem is formulated in terms of a system of coupled Fredholm integral equations of the first kind. The subsoil is modelled by a homogeneous, linear-elastic and isotropic half-space which is perfectly bonded to the rigid, rectangular foundation. An approximate solution for the resultant loads between the foundation and the half-space due to a unit forced displacement or rotation is obtained using the Bubnov–Galerkin method. Using this method the displacement boundary value conditions are exactly satisfied and the contact stress distributions between the foundation and the half-space are approximated by series expansions of Chebyshev polynomials. This method provides a simple means of studying the soil-structure interaction of rectangular foundations with different inertia properties.     

Nonlinear rocking stiffness of foundations

The response of surface foundations to large overturning moments is studied under undrained conditions. Rigid circular, strip, and rectangular footings of various aspect ratios are considered, with the soil modeled as an inelastic homogeneous deposit, characterized by an elastic (small-strain) shear modulus G_o, an undrained shear strength S_u, and a G/G_o versus γ curve appropriate for medium-plasticity clays. Three stages of foundation performance, ranging from the initial elastic fully-bonded response, to the nearly-elastic but nonlinear response with the foundation partially detaching and uplifting from the soil, and finally to the ultimate stage where full mobilization of soil bearing failure mechanisms develop. Simple to use formulas or charts are developed for all stages of response in terms of dimensionless parameters, prominent among which is the static factor of safety against bearing-capacity failure under purely-vertical loading.     

In-plane foundation-soil interaction for embedded circular foundations

Foundation soil interaction is studied using an analytical two-dimensional model, for circular foundations embedded in a homogeneous elastic half-space and for incident plane P- and SV- and for surface Rayleigh waves. The scattered waves are expanded in complete series of cyclindrical wave functions. A detailed analysis is presented of the foundation response to unit amplitude incident waves as a function of the type of incident waves and angle of incidence, the depth of the embedment and the foundation mass per unit length.It is shown that free-field translations and point rotation approximate well the foundation input motion only for very long incident waves. For shorter incident waves, those in general overestimate the foundation input motion. Neglecting the rotation of the foundation input motion (which is usually done in practice) may eliminate a major contribution to the base excitation of buildings and may cause nonconservative estimates of the forces in these buildings. Incident waves appear as ‘longer’ to a shallow foundation than to a deeper foundation. Therefore, deeper foundations are more effective in reflecting and scattering the short incident waves.     

Impedance matrices for circular foundation embedded in layered medium

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.     

Dynamic interaction between adjacent foundations

The results of a parametric study on the dynamic interaction between adjacent foundations are presented. Addressed in this study are the effects of distance, direction of alignment, embedment and structural inertia. Several conclusions are drawn regarding the significance of each of these factors.     

Modal analysis of circular flexible foundations under vertical vibration

A methodology using modal analysis is presented to evaluate dynamic displacements of a circular flexible foundation on soil media subjected to vertical vibration. The interaction effects between the foundation and the underlying soil are represented using modal soil impedance functions determined by an efficient procedure developed. The displacements of the foundation can then be easily solved by modal superposition. Comparing with existing solutions, the presented method is found to provide accurate results with less computational effort using only a few vibration modes. In addition, parametric studies for modal responses of the flexible foundation indicate that the response of the foundation are significantly influenced by relative stiffness between the foundation and the soil medium, load distributions, vibration frequency range, and the foundation mass. Besides, justification for flexible foundations to be considered as rigid are investigated.     

Rocking response of flexible circular foundations on layered media

The analysis of the response of a flexible circular foundation on layered media due to an arbitrarily distributed vertical loading is presented. The analysis is based on the ‘ring method’ approach, i.e. discretization of the foundation in a set of concentric rings. The arbitrarily distributed loading is expanded in the circumferential direction in a Fourier series. The influence coefficient matrix of soil for each element of the series is evaluated utilizing the stiffness matrix approach. The stiffness matrix of the foundation is obtained from the finite difference energy method approach. Numerical examples illustrate the influence of several soil-foundation parameters on the rocking response of a foundation. Results are presented in terms of displacement and soil reaction distributions and impedance functions point to significantly different responses of flexible and rigid foundations.     

Dynamic interaction between flexible strip foundations

This paper is concerned with the dynamic response of a system of flexible strip foundations resting in smooth contact with a homogeneous isotropic viscoelastic half space. An arbitrary number of foundations with different flexibilities and geometries subjected to time-harmonic distributed loadings are considered in the formulation. The response of each strip foundation is governed by the classical plate theory and its transverse deflection is represented by an admissible function containing a set of generalized co-ordinates. A coupled variational-Green's function scheme is employed to establish the equations of motion of the strip foundation system. The numerical stability and convergence of the solution scheme are established. The influence of the foundation flexibility, distance between adjacent foundations and frequency of motion on the response of the foundation system is investigated in the numerical study.     

Dynamic response of axisymmetric embedded foundations

The boundary element method is used to obtain dynamic stiffness functions of rigid cylindrical foundations embedded in a uniform or layered viscoelastic half-space. Dynamic stiffness functions of hemispherical foundations embedded in a uniform half-space are also computed. The direct integral equation formulation is used in combination with the complete space point load fundamental solution that is integrated numerically along the azimuthal coordinate. The approach is easy to implement because of the simplicity of the fundamental solution. The numerical results obtained by this method for cylindrical and hemispherical foundations are very close to corresponding published results obtained by different procedures. A parametric study shows the important effects of the Poisson's ratio on the dynamic stiffness functions of cylindrical foundations embedded in a uniform viscoelastic half-space. The effect of the bedrock compliance on the stiffness functions is also shown in the case of cylindrical foundations embedded in a soil layer that rests on a bedrock.     

Cone model for two surface foundations on layered soil

In this paper, the cone model is applied to the vibration analysis of two foundations on a layered soil half space. In the analysis, the total stress field in the subsoil is divided into the free-field and the scattering field. Seed＇s simplified method is adopted for the free-field analysis, while the cone model is proposed for analyzing the dynamic scattering stress wave field. The shear stress field and the compressive stress field in the layered stratum with two scattering sources are calculated by shear cone and compressive cone, respectively. Furthermore, the stress fields in the subsoil with two foundations are divided into six zones, and the P wave and S wave are analyzed in each zone. Numerical results are provided to illustrate features of the added stress field for two surface foundations under vertical and horizontal sinusoidal force excitation. The proposed cone model may be useful in handling some of the complex problems associated with multi-scattering sources.     

Dynamic stiffness of parabolic cables

A closed-form expression for the in-plane horizontal stiffness of a viscously damped, uniform, inclined cable in harmonic motion is presented. The cable is presumed to be deflected in a parabolic profile at its position of static equilibrium, and all dynamic displacements are assumed to be small. The stiffness expression is valid for an arbitrary angle of inclination of the cable chord in the range between zero and 90 degrees. In addition, a simpler solution, valid over a narrower range of the parameters, is included and its accuracy examined. Comprehensive numerical data are presented and discussed, with particular emphasis on explaining the physical significance of the results and providing insight into the action of the cable and into the parameters that control it. Finally, a simple, single-degree-of-freedom model is proposed which reproduces with good accuracy the salient features of the response of the prototype cable over a wide range of conditions.     

Dynamic finite strip analysis of surface foundations

The finite strip method is used to study the dynamic response of surface strip foundations. This method is simple to use and versatile. The two-dimensional problem is effectively reduced to an equivalent one-dimensional problem. The choice of trigonometric series for the displacement functions results in the uncoupling of the terms of the series and a significant reduction in the semi-bandwidth of the complex set of equations, making the method well suited to small computers. Non-homogeneous soil profiles are treated as a matter of course. The finite strip solutions are shown to be in general agreement with other theoretical methods of analysis.     

Dynamic behaviour of structures with embedded foundations

A study of the dynamics of building-soil interaction is presented that includes embedding of the foundation and material damping. By considering buildings on rigid footings embedded into linear elastic soil with hysteretic damping, it is shown that the earthquake response of the building-foundation model may be found from the response to modified excitation of equivalent one-degree-of-freedom linear, viscously damped oscillators resting on rigid ground. For a single-storey building approximate formulas are developed for the modified natural frequency and damping ratio. Results show that the natural frequency and damping in the system increase with embedding. Effective damping also increases with internal friction in the soil. Ignoring these two factors may underestimate considerably the effective natural frequency and damping in the system. In spite of additional sources of energy dissipation provided by the soil, damping in the equivalent oscillator may be greater or smaller than that corresponding to the superstructure alone, depending upon the system parameters. For lightly damped superstructures, the peak amplitude of the steady-state overturning moment at the base of a building supported on flexible soil is significantly smaller than that corresponding to rigid ground. This result has practical implications for earthquake design.     

基于相似性导出基础的动力刚度矩阵

详细论述了基于相似性导出基础动力刚度矩阵的思想，包括克隆算法、无限元方法、一致无穷小有限元法和标度有限元法。这些方法，仅对地基－基础界面进行有限元离散，使问题的空间维数减少一维；但是，它们不需要基本解就可以满足无穷远辐射条件，从而避免了奇异积分的计算，减少了计算工作量。     

钢筋混凝土桥墩刚度和强度折减系数确定

目的是解析地预测钢筋混凝土桥墩在反复荷载作用下的非线性滞回特性。使用实验中得到的力一位移滞回曲线,对随轴压比,配筋率和配箍率的变化而变化的刚度和强度折减系数,进行了回归分析,并提出了其表达式。按照提出的理论力一位移滞回模型,能够预测现存钢筋混凝土桥墩的刚度和强度折减情况。     

Response of circular flexible foundations subjected to horizontal and rocking motions

A methodology using modal analysis is developed to evaluate dynamic vertical displacements of a circular flexible foundation resting on soil media subjected to horizontal and rocking motions. The influence of the soil reaction forces on the foundation is considered by introducing modal impedance functions, which can be determined by an efficient procedure with ring elements. The displacements of the foundation can then be easily solved by modal superposition. Parametric studies for modal responses of the flexible foundation indicate that the coupled response of the foundation is significantly influenced by relative stiffness among the foundation and the soil medium, vibration frequency range, foundation mass, and boundary contact conditions. The welded boundary condition should be considered to predict the coupling response while the relaxed boundary condition may be used to predict approximately the vertical displacements. As a foundation with a relative stiffness ratio more than three, it is found that the foundation can be considered as rigid to calculate coupling displacements. For a slightly flexible foundation, considerations of three modes are sufficient enough to obtain accurate foundation responses. Moreover, at low frequencies, the coupling effect due to higher mode can be neglected.     

Consistent lumped-parameter models for unbounded soil: Frequency-independent stiffness,damping and mass matrices

A systematic procedure to develop consistent (symmetric) stiffness, damping and mass matrices with real coefficients to represent any unbounded soil is developed. These property matrices are based on the lumped-parameter models of Reference 1. Either stiffness and damping matrices corresponding to first-order differential equations involving the internal degrees of freedom and those on the structure-soil interface result or, alternatively, in addition mass matrices are introduced, corresponding to second-order differential equations, which reduce the number of internal degrees of freedom by a factor 2. The stiffness, damping and mass matrices can easily be incorporated in a general-purpose structural dynamics program working in the time domain, whereby the structure can even be non-linear.