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.

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SUMMARYClosed-form expressions and comprehensive numerical solutions are presented for the transfer functions of surfacesupported, rigid, rectangular foundations excited by horizontally polarized, incoherent shear waves for which the motions are parallel to one of the foundation sides. The free-field ground motion is specified stochastically in terms of a local power spectral density function and an orthotropic incoherence function which decays exponentially with the square of the excitation frequency and the separation distance. The response quantities examined include the lateral and torsional components of the foundation motion. Displayed graphically, the results elucidate the effects and relative importance of the numerous parameters involved. For vertically incident incoherent wave fields, the lateral transfer function of a rectangular foundation is related to that of a judiciously selected square foundation, and the interrelationship of the results is examined.

The modal parameters of numerous modes for a stay cable are usually required in engineering practice. The application of conventional stochastic subspace identification techniques without delicate cautions, however, has been found to be unsuccessful in this case. Aiming to attack such a difficulty, this study establishes a new methodology based on the covariance type of stochastic subspace identification for extensively identifying the modal parameters of a stay cable. Several details of choosing the parameters in performing stochastic subspace identification are first discussed. An important discovery is that the lower limit for setting the time lag parameter can be decided by the ratio of the fundamental period of cable to the sampling time increment for a valid identification with the conventional stabilization diagram. Inspired by the aforementioned criterion, an alternative stabilization diagram is further proposed to more conveniently distinguish stable modal parameters of cable. A hierarchical sifting process including three stages is then developed to systematically and automatically extract reliable modal parameters from the alternative stabilization diagram. Demonstrated by analyzing the ambient vibration measurements for three stay cables of Chi-Lu Bridge, the feasibility of this new approach is verified with successfully obtaining the modal frequencies, damping ratios, and mode shape ratios for almost all the cable modes in the examined frequency range. Another interesting finding is that the modal frequencies and damping ratios of bridge deck can also be effectively identified from the ambient vibration signals of cable.

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