SUMMARYBased on the approximation by polynomial-fraction, a series of systematic lumped-parameter models are developed in this paper for e ciently representing the dynamic behaviour of unbounded soil. Concise formulation is ÿrst employed to represent the dynamic exibility function of foundation with a ratio of two polynomials. By deÿning an appropriate quadratic error function, the optimal coe cients of the polynomials can be directly solved from a system of linear equations. Through performing partial-fraction expansion on this polynomial-fraction and designing two basic discrete-element models corresponding to the partial fractions, systematic lumped-parameter models can be conveniently established by connecting these basic units in series. Since the systematic lumped-parameter models are conÿgured without introducing any mass, the foundation input motion can be directly applied to these models for their applications to the analysis of seismic excitation. The e ectiveness of these new models is strictly validated by successfully simulating a semi-inÿnite bar on an elastic foundation. Subsequently, these models are applied for representing the dynamic sti ness functions for di erent types of foundation. Comparison of the new models with the other existing lumped-parameter models is also made to illustrate their advantages in requiring fewer parameters and featuring a more systematic expansion.
SUMMARYThe concept of polynomial-fraction approximation is explored in this article to develop a nested type of systematic lumped-parameter model for unbounded soil. Based on the optimal coe cients determined from the exibility formulation, the reciprocal of the polynomial-fraction is ÿrst taken to represent the dynamic sti ness function of foundation and then decomposed into a linear polynomial and another polynomial-fraction. The nested division introduced in this study is operated to generate a nested form for this decomposed polynomial-fraction, which directly corresponds to a nested discrete-element model. The nested type of lumped-parameter model is then easily constructed by connecting this nested discrete-element model in series with another simple discrete-element model corresponding to the linear polynomial.
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