In this study of the dynamics of building-soil interaction, the soil is modeled by a linear elastic half-space, and the building structure by an n-degree-of-freedom oscillator. Both earthquake response and steady-state response to sinusoidal excitation are examined. By assuming that the interaction system possesses n + 2 significant resonant frequencies, the response of the system is reduced to the superposition of the responses of damped linear oscillators subjected to modified excitations. The results are valid even though the interaction systems do not possess classical normal modes. For the special cases of single-story systems and the first modes of n-story systems, simplified approximate formulas are developed for the modified natural frequency and damping ratio, and for the modified excitation. Example calculations are carried out by the approximate and more exact analysis for one-story, two-story and ten-story interaction systems.

The results show that interaction tends to decrease all resonant frequencies, but that the effects are often significant only for the fundamental mode for many n-story structures and are more pronounced for rocking than for translation. If the fixed-base structure has damping, the effects of interaction on the earthquake responses are not always conservative, and an increase or decrease in the response can occur, depending on the parameters of the system.

Dynamics of building-soil interaction

Most seismic codes specify empirical formulas to estimate the fundamental vibration period of buildings. Evaluated first in this paper are the formulas specified in present U.S. codes using the available data on the fundamental period of buildings “measured” from their motions recorded during eight California earthquakes, starting with the 1971 San Fernando earthquake and ending with the 1994 Northridge earthquake. It is shown that, although the code formulas provide periods that are generally shorter than measured periods, these formulas can be improved to provide better correlation with the measured data. Subsequently, improved formulas for estimating the fundamental periods of reinforced concrete (R/C) and steel moment-resisting frame buildings are developed by regression analysis of the measured period data. Also recommended are factors to limit the period calculated by a rational analysis, such as Rayleigh’s method.

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Period formulas for moment-resisting frame buildings

Most seismic codes specify empirical formulas to estimate the fundamental vibration period of buildings. Evaluated first in this paper are the formulas specified in present U.S. codes using the available data on the fundamental period of buildings “measured” from their motions recorded during eight California earthquakes, starting with the 1971 San Fernando earthquake and ending with the 1994 Northridge earthquake. It is shown that current code formulas for estimating the fundamental period of concrete shear wall buildings are grossly inadequate. Subsequently, an improved formula is developed by calibrating a theoretical formula, derived using Dunkerley's method, against the measured period data through regression analysis. Also recommended is a factor to limit the period calculated by a “rational” analysis, such as Rayleigh's method.

/pdf/period-formulas-for-concrete-shear-wall-buildings-59cebsdvr2.pdf

Period formulas for concrete shear wall buildings

In this, the second of a two-part paper, the analysis of the apparent frequency of a seven-story reinforced-concrete hotel building in Van Nuys, Calif., is extended to consider its time-dependent changes, both short and long term. The instantaneous apparent frequency is measured by two methods: windowed Fourier analysis and zero-crossings analysis. The results show that it changes from earthquake to earthquake and during a particular earthquake. The results also suggest ''self healing'' believed to result from settlement of the soil with time and dynamic compaction from aftershock shaking. Implications of such high variability of the system frequency on structural health monitoring, control of response, as well as on the design codes are discussed. Nonlinear response of the foundation soil acts as a sink of the incident seismic wave energy. It is suggested that it could be exploited in future designs to serve as a powerful and inexpensive energy-dissipation mechanism. reinforced-concrete hotel in Van Nuys, Calif. (VN7SH) by Fourier analysis. We also described an approximate model in which the soil-structure interaction phenomena are viewed in the simplest possible form, via a rigid foundation model. Be- cause the Fourier transform of the response is evaluated from the entire time history of response, the results in Part I gave an insight only into the overall changes of the apparent system frequency from one earthquake to another. The trend that emerged from the results of the Fourier analysis is a reduction of the apparent system frequency with increasing levels of shaking at the site. In this part, we study the time and ampli- tude-dependent changes of by time-frequency analysis. The ˜ f description of the building, recorded strong motion data, earth- quake damage, and full-scale ambient vibration tests con- ducted following the Northridge earthquake of 1994 were pre- sented in Part I and will not be repeated here.

Apparent Periods of a Building. II: Time-Frequency Analysis

The aim of this work is to estimate the fundamental translational frequencies and relative damping of a large number of existing buildings, performing ambient vibration measurements. The first part of the work is devoted to the comparison of the results obtained with microtremor measurements with those obtained from earthquake recordings using four different techniques: horizontal-to-vertical spectral ratio, standard spectral ratio, non-parametric damping analysis (NonPaDAn) and half bandwidth method. We recorded local earthquakes on a five floors reinforced concrete building with a pair of accelerometers located on the ground and on top floor, and then collected microtremors at the same location of the accelerometers. The agreement between the results obtained with microtremors and earthquakes has encouraged extending ambient noise measurements to a large number of buildings. We analysed the data with the above-mentioned methods to obtain the two main translational frequencies in orthogonal directions and their relative damping for 80 buildings in the urban areas of Potenza and Senigallia (Italy). The frequencies determined with different techniques are in good agreement. We do not have the same satisfactory results for the estimates of damping: the NonPaDAn provides estimates that are less dispersed and grouped around values that appear to be more realistic. Finally, we have compared the measured frequencies with other experimental results and theoretical models. Our results confirm, as reported by previous authors, that the theoretical period–height relationships overestimate the experimental data. Copyright © 2008 John Wiley & Sons, Ltd.

Empirical estimate of fundamental frequencies and damping for Italian buildings

The measured periods of vibrations of a large number of buildings are used to compare the merits of existing formulas and of equations derived using a rational approach. It is concluded that no single, simple, empirical equation will give reasonably accurate estimates for the periods of buildings having shear wall characteristics. The calculated periods of steel frame buildings are found to be proportional to the square root of the number of stories rather than directly proportional as is usually assumed. However, observed periods of such buildings indicate that the steel frame alone would contribute only about 25% of the effective stiffness. The dependence of the period on the flexibility of the floor girders is also demonstrated using a digital computer. Calculations show that these buildings behave essentially as if they had rigid floor girders, although the girders of the steel frames themselves are very flexible, so far as the period of vibration is concerned. Additional studies of the natural periods of actual structures are recommended.

A. G. Brady

Papers

Natural Periods of Vibration of Buildings