A study of the accuracy of electronic digital computer calculations of ground motions and response spectra from. The dispersion of the results of ground motion calculations is greater than that of response spectrum calculations. In some engineering studies, the estimation of ground movements during strong earthquakes is useful.
A committee established by the Earthquake Engineering Research Institute in 1962 addressed the issue of digital accuracy.
NOMENCLATURE
2), [S] is a 3 X 3 matrix of the coefficients and duration s, and (R) is the column vector of the three integrals C: shown on the right. The inverse of the matrix [S] was very sensitive to the value of the record duration, s, chosen in the adjustment procedure. A reasonable method to standardize the adjustment of. acceleration records and calculation of velocity and displacement would be the following:. I).
The results of the above recommended procedure on the four decks used in this study are shown in Figs.
BERK ELEY CALTECH
16 numerically with zero initial conditions for x and x, following the maximum values of I~ I. At the same time, the maximum values of lxl and of. l2nw x 0 + w 0 xi can be obtained to give the relative displacement response spectrum, Sd:. and absolute acceleration response spectrum, . The response of an undamped oscillator increases over time during excitation, so that the spectrum values often appear towards the end of the record. 4 DEPENDENCE OF VELOCITY SPECTRA ON THE LENGTH OF THE INTEGRATION STEP. ii) Comparison of velocity spectra, spectra, PS , shown in fig.
For those periods in which the value of the spectrum depends significantly on the length of the recording, the approximate time.
OD seconds
I DAM PED
PROBABILITY DENSITY DISTRIBUTIONS OF THE MAXIMUM RES:PONSE OF SIMPLE OSCILLATORS TO RANDOM EXCITATION
One of the most important problems in earthquake engineering is determining the behavior of a structure subjected to a given strength. There are a number of reasons for choosing to focus on the maximum relative speed of a simple structure with one degree of freedom, out of all possible interesting responses. It is possible to characterize an earthquake and its effect on a simple one-degree-of-freedom structure by means of the.
0, and· n. Curves of response spectra, in terms of n and T, are available for all recorded earthquakes with strong motion <1). The eight components of the four strongest ground motions recorded up to 1959 have been used by G. TJ::iese distributions have been of increasing interest at the present time to those design engineers who are able to benefit from a prior knowledge of the probability of survival, or failure, of a structure.
How do the structural parameters of mass, damping and stiffness influence the average value of the maximum stress that a simple structure experiences at a certain critical point? What is the probability that the maximum stress is a certain multiple of the ensemble average of the maximum and how do the mass, damping and stiffness influence these probabilities. The specific nature of the excitation has not yet been specified, although it will be such that the basic acceleration will ultimately be one of the following: white noise, as first used by Housner(b) and later by other authors, a Gaussian non- whitening process with a specified power spectral density, as used by Housner and Jennings(lb); and actual data of strong motion earthquakes (l).
Appropriate choices will need to be made for the damping and period of the structure to provide a full range of useful dispersion curves. The power spectral density of an earthquake record is roughly related to the square of the undamped response spectrum, and the probability.
This behavior of the mean is expected from the shape of existing earthquake response spectra. Saturation of the equipment in the undamped cases caused an artificial upper limit at a value of approx. 50 volts, indicated in fig. In this section, the expression for the probability density is derived following Rosenblueth and Bustamante's(9) work on probability of survival and extends their results to obtain curves with which they can be compared in the previous section.
6 and the probability of finding r between r and r + dr at time t is 2rru(r, t) r dr, taking into account the radial symmetry in the formulation of the problem. The upper integration limit is chosen so that the asymptotic expansion for the ·sum of the series makes a negligible contribution to the rest of the integral. The formulation of the problem does not allow the use of the parameter r used in Eq.
In the appendix there is a description of the eigenvalues v and an explanation of how Eq. The probability density function f(R) is obtained by differentiating Eq. 2. 26 partly as to R; this is out of. Disc; using the probability density function equations Rosenblueth and Bustanmante's calculation{ 9 ) of the expected value of the maximum undamped response from Eq.
This different mathematical formulation of the problem thus yields the same differential equations as Rosenblueth and Bustamante's approach, under similar assumptions. A table showing the zeros of J. 0 and the values of J. The calculations are made in terms of the dimensionless quantity a given by.
I Rosenb lueth
FOURIER SPECTRA OF GROUND MOTION RECORDS
In recent years, many studies have been carried out on the role of the surface material's physical properties in relation to Before proceeding with theoretical studies of the amplification of sine waves and the magnification of undamped spectra of. The study involves simultaneous recording at two different but reasonably close locations of ground motion caused by earthquakes that are sufficiently far apart to be effectively equidistant from each location.
A meaningful comparison can then be made of the frequency components present in the vibration of bedrock granite and overlying soil layers. The first part of this chapter describes the USCGS data acquisition and digitization techniques with particular reference to the seismometer and digital reader. The next section contains the derivation of the expression for the Fourier spectrum of the ground acceleration in terms of the Fourier spectrum of the digitized record.
The reasons for choosing Fourier analysis of the data are related to the response characteristics of the instruments and to. The Fourier spectrum of the record and reasons are given for the final choice of one. Tests were also carried out to check the accuracy of the digitization procedure and to find the dependence of the Fourier spectra on the duration of the recording used.
Ten relevant earthquakes were recorded at these two locations, although only two records were available for this analysis. With these two sites on the granite base, an opportunity was given to verify the similarity of the underlying ground motion.
These tests consisted of laboratory calibrations of seismometer constants and field calibrations of the system magnification. Three methods were considered for the numerical calculation of the Fourier transform of the digitized record. A brief description of the methods follows, still using the frequency w rather than the period T for simplicity.
Therefore, the envelope, Ei {t), of the relative velocity curve is approximated as closely as possible. The value of Ei {t) at the end of the record is therefore obtained by substituting Ai and A. 1 was a check of the explanation of method (2) in Part D of this chapter for the calculation of Fourier spectra.
Method (2) explained how the envelope of such a response at the end of the excitation was related to the Fourier spectral value. This second trace processing should thus correspond to the first only when the maximum response is reached near the end of the record. Since these curves are ground acceleration spectra, the vertical unit is velocity, mm.
The question that subsequently arose was the accuracy of setting up the tracks in the reading machine prior to digitization. The dependence of the Fourier spectra of ground acceleration on duration was investigated using the same E-W component of.
SITE 2
1/4 and each end is ordered with half the amount. of the end ordinate and its neighbor. From these smooth curves it is possible to infer little about the magnification of earth movement through the alluvium layers in the Wheeler Ridge area. This means that the movement of the earth in the bedrock granite in the place of the alluvium is no more precisely known than is shown by Fig. 3.16, and comparisons of the spectra in Fig.
In light of the results described in this section, a standard method for calculating Fourier amplitude spectra of records of this type can be recommended. i). The general appearance of the Fourier spectra is discussed in particular with regard to their oscillatory nature. 34;Properties of Strong Ground Motion Earthquakes, 11 Bulletin of the Seismolo ical Societ of America, Vol.
34;Ground Displacement Computed from Strong-Motion Accelerograms, n Bulletin of the Seismological Association of America, Vol. For proof, we can use the Cauchy test using the mth root of the mth term. The rationale for this step is based on the validity of the corresponding step for the undamped case and the description of the equation reduction in the next section.
The effect of the error in velocity, or the difference between v(t) and e(t), is much more apparent than the difference between I\. 56 can be removed from the ground speed by minimizing the mean square value of the expression. When a complete registration is available, use can be made of the fact that the final ground speed must be zero.
In this case, it would be expected that any adjustment process should cause the ground speed to behave towards the end of the record.
