An analytical continuum model for axially loaded end-bearing piles in inhomogeneous soil

An approximate static solution is derived for the elastic settlement and load-transfer mechanism in axially loaded end-bearing piles in inhomogeneous soil obeying a power law variation in shear modulus with depth. The proposed generalised formulation can handle different types of soil inhomogeneity by employing pertinent eigenexpansions of the dependent variables over the vertical coordinate, in the form of static soil “modes”, analogous to those used in structural dynamics. Contrary to available models for homogeneous soil, the associated Fourier coefficients are coupled, obtained as solutions to a set of simultaneous algebraic equations of equal rank to the number of modes considered. Closed-form solutions are derived for the (1) pile head stiffness; (2) pile settlement, axial stress, and side friction profiles leading to actual, depth-dependent Winkler moduli, (3) displacement and stress fields in the soil; and (4) average, depth-independent Winkler moduli to match pile head settlement. The predictive power of the model is verified via comparisons against finite element analyses. The applicability to inhomogeneous soil of an existing regression formula for the average Winkler modulus is explored.     

Nonstationary seismic responses of structure with nonlinear stiffness subject to modulated Kanai–Tajimi excitation

A simple structure under earthquake excitation is modeled as a single‐degree‐of‐freedom system with nonlinear stiffness subject to modulated Kanai–Tajimi excitation. The nonstationary responses including the nonstationary probability densities of the system responses and the statistical moments are obtained in semi‐analytical form. By applying the stochastic averaging method based on the generalized harmonic functions, the averaged Fokker–Planck–Kolmogorov(FPK) equation governing the nonstationary probability density of the amplitude is derived. Then, the solution of the FPK equation is approximately expressed by a series expansion in terms of a set of properly selected basis functions with time‐dependent coefficients. According to the Galerkin method, the time‐dependent coefficients are solved from a set of linear first‐order differential equations. Thus, the nonstationary probability densities of the amplitude and the state responses as well as the statistic moments of the amplitude are obtained. Finally, two types of the modulating functions, i.e. constant function and exponential function, are considered to give some semi‐analytical formulae. The proposed procedures are checked against the Monte Carlo simulation. The effects of the structure natural frequency and the intensity of the excitation as well as the ground stiffness on the system responses are discussed. It should be pointed out that the proposed method is good for broadband excitation and light damping. Copyright © 2011 John Wiley & Sons, Ltd.     

Discussion of ‘A new approach of selecting real input ground motions for seismic design: The most unfavourable real seismic design ground motions’ by C‐H Zhai and L‐L Xie

This discussion consists of two parts. The first part raises a few comments and questions on the method presented in the above paper. The second part proposes a measure for identifying resonant accelerograms in a set of earthquake records without the need for pre‐processing of the records or inclusion of the structure dynamic analysis. Copyright © 2008 John Wiley & Sons, Ltd.