Structure Response

Event though peak ground acceleration is the most commonly used parameter used in seismic hazard analysis, it may happen that this value, taken by itself, becomes meaningless. The main reason resides in the fact, that the damage edifices may undergo during an earthquake, highly depend on the dominant frequency of the radiated wave and the characteristics of the building, respectively. A way out of this shortcoming, at least to some part, is the use of so-called response spectra, which by now are at the base of most common codes for seismic regulation, for instance the US building code in the United States of America, the EC 8 in Europe, the DIN 4149 in Germany etc. In the response spectrum method the structure is, in a rough approximation, supposed to behave like a single mass oscillator with one degree of freedom,comparable to a spring or a pendulum. The characteristics of this model are its eigenfrequency fn (or 1 structure response= 2 2-structure-response) and the d 3 structure respo which depend on size, configuration and material makeup of the structure. Earthquake ground motion will excite the structure and cause it to “resonate” according to its natural frequency. Following the US building code and the EC 8 the 1st eigenfrequency of a building can be roughly estimated after

Here Ct is a factor between 0.05 and 0.085 depending on the characteristics of the structure, and H being its height in m. After this formula the 1st eigenfrequency of a three story building with a height of ca. 10 m and brick walls (i. e., Ct = 0.05) should be expected at around 3.5 Hz.

For the calculation of response spectra for a given ground motion first eigenfrequency and damping of the oscillator have to be selected. The next step consists in the compuation the oscillators displacement with respect to ground. The displacement response u(t) to a ground acceleration a(t) is given, for small damping values 3 structure respo by the Duhamel integral

Instead o the Duhamel integral (2.17) which is the exact mathematical formulation but slows down considerably the processing, SHAKYGROUND uses a fast recursive filter algorithm for the calculation of the displacement response. Note that only the maximum value of the oscillators response is kept. The procedure is repeated for a set of eigenfrequencies in order to obtain a “spectrum of maximum responses”. Standard seismic regulations usually require that response spectra are given in terms of acceleration. The correct way to obtain acceleration response spectra would be to differentiate twice the displacement response of the oscillator. For historical reasons, however, the velocity and acceleration response spectra are obtained by simply multiplying the maximum displacement by 1 structure response and 4 structure response The resulting spectra are called pseudo response spectra of velocity or acceleration. In fact, many seismic regulation codes were originally designed in times when computing capacities were low and response spectra were partly obtained from Fourier spectra of acceleration time series. In order to maintain compatibility with the older seismic codes the convention prescribes the use of pseudo response spectra instead of “true” velocity or acceleration response spectra.

The response spectra, though representing a highly simplified description of structure response, have proven to be an important tool for the understanding of the damages caused by earthquakes. First of all, they are at the base of standard calculation of the structure resistance with respect to dynamic loading. In particular, the structure response of towers or similar high buildings can by simulated in a straight forward way using acceleration response spectra. From the seismological point of view response spectra, contrary to mere peak ground acceleration, provide useful information about the frequency intervals where maximum hazard has to be expected. Particularly dangerous situations arise when the eigenfrequencies of the structures happen to be in the frequency band of major seismic radiation. In microzonation studies land use planners may identify sites where specific facilities would be at particular risk, emergency planners may recognize damage prone areas and develop their scenarios with respect to these information.