A simplified fundamental period equation for RC buildings

Considering the huge differences in the prediction and organization of equations available in the literature, this paper aims at developing a reliable equation including mass and stiffness parameters. Microtremor (ambient vibration) measurements were taken from 23 RC buildings and their fundamental periods were compared to the dynamic analysis results. Building models were then calibrated to account for the infill wall effect. After that, 156 RC buildings were 3D modelled and their dynamic analysis results were used to calibrate the proposed fundamental period equation.

Measured natural periods of concrete shear wall buildings: insights for the design of Canadian buildings

Structural engineers routinely use rational dynamic analysis methods for the seismic analysis of buildings. In linear analysis based on modal superposition or response spectrum approaches, the overall response of a structure (for instance, base shear or inter-storey drift) is obtained by combining the responses in several vibration modes. These modal responses depend on the input load, but also on the dynamic characteristics of the building, such as its natural periods, mode shapes, and damping. At the design stage, engineers can only predict the natural periods using eigenvalue analysis of structural models or empirical equations provided in building codes. However, once a building is constructed, it is possible to measure more precisely its dynamic properties using a variety of in situ dynamic tests. In this paper, we use ambient motions recorded in 27 reinforced concrete shear wall (RCSW) buildings in Montréal to examine how various empirical models to predict the natural periods of RCSW buildings compare to the periods measured in actual buildings under ambient loading conditions. We show that a model in which the fundamental period of RCSW buildings varies linearly with building height would be a significant improvement over the period equation proposed in the 2010 National Building Code of Canada. Models to predict the natural periods of the first two torsion modes and second sway modes are also presented, along with their uncertainty.

Normal modes in seismic data — Revisited

At long distances from a seismic shot, the recorded signal is dominated by reflections and refractions within the water layer. This guided wave signal is complex and often is referred to as normal or harmonic modes. From the period equation, we derive a new approximate expression for the local minima in group velocity versus frequency. We use two data sets as examples: one old experiment where the seismic signal is recorded at approximately 13 km offset and another example using life of field seismic data from the Valhall Field. We identify four and five normal modes for the two examples, respectively. A fair fit is observed between the estimated and modeled normal mode curves. Based on the period equation for normal modes, we derive a simple, approximate equation that relates the traveltime difference between various modes directly to the velocity of the second layer. Using this technique for offsets ranging from 6 to 10 km (in step of 1 km), we find consistent velocity values for the second layer. We think that this method can be extended to estimate shallow lateral velocity variations if the method is applied for the whole field. We find that the simple equations and approximations used here offer a nice tool for initial investigations and understanding of normal modes, although a multilayered method is needed for detailed analysis. A comparison of three vintages of estimated normal mode curves for the Valhall field example representing seabed locations shifted by 1 km indicates that minor shifts in group velocity minima for the various modes are detectable.

Study of the leaking channel modes of in‐seam exploration seismology by means of synthetic seismograms

If the dispersion characteristics of coal‐seam channel guides do not extend smoothly at frequencies below cut‐off, the number of parameters required to process inseam seismic data increases. Unwanted modes become more difficult to suppress, the arriving pulses from targets become more difficult both to identify and to pulse compress. The connection between leaking and normal channel mode dispersion is established here by first synthesizing and then analyzing theoretical in‐seam seismograms. Calculation of synthetic seismograms is based on numerical evaluation of the spatial Fourier integral for elastic displacements in the complex wavenumber plane. Theoretical seismograms are presented for three‐layered models. Phase velocity characteristics are recovered from these signals and compared with those obtained from the zero loci of the period equation in the complex wavenumber plane. Under cut‐off the former method yields smooth extensions of the normal mode dispersion characteristics, in contrast to the velocity curves obtained from the period equation only. It is found that the dispersion characteristics obtained from analyzing the seismograms can be used to recompress the dispersed arrivals.

A fast, accurate method for computing group-velocity partial derivatives for Rayleigh and Love modes

A method for quickly and accurately calculating Rayleigh- and Love-mode group-velocity partial derivatives with respect to model parameters (m) is developed. The method requires computer codes that calculate C, U, and ∂C∂m|ω and employs numerical differentiation of ∂C∂m|ω to yield ∂U∂m|ω. The method is fast because ∂C∂m|ω and ∂U∂m|ω for all the model parameters can be obtained at a given frequency from only two solutions of the period equation. The accuracy of the method is established with two examples. For Love waves, the group-velocity partials computed by this method agree exactly with those obtained analytically by Novotny (1970). For Rayleigh waves, comparison with a “brute force” calculation of group-velocity partials showed agreement to the order of 0.00002. Systematic inversion of group-velocity data separately or in combination with phase-velocity data is computationally feasible using this method.