ABSTRACT
Since the response spectrum is specific to a given earthquake in a given location, its direct application is not very helpful, so design spectra are used instead. They are obtained by statistical criteria from a set of response spectra. A design spectrum seeks to safely enveloping the effects of earthquakes with a particular probability of occurrence in a given seismic zone. Design spectra are used in frequency-domain analysis.
Yet, in many situations, such as in non-linear analysis, time-domain analysis is imposing. The accelerations imposed to the structure’s foundation by an earthquake are the most direct input data to characterize and use in the seismic analysis of structures. They can be real or artificially-generated event logs. Considering this fact, this paper will demonstrate the generation of a time-history graph of accelerations from a design spectrum, using a JAVA computer language programming.
An artificial earthquake is characterized by an accelerogram that is compatible with a design spectrum, i.e., the response spectrum that will be obtained from the former should be approximately equal to the latter. From the statistical properties of actual earthquakes recorded, with intensity and frequency content data as a function of time, the models for the simulation of these movements are determined.
A stationary process can be expanded in a harmonic series, which is why various methods are based on the reproduction of seismic time-history graphs of accelerations by superposition of harmonic components in a typical frequency range of actual earthquakes. Equation (1) for the generation of artificial time-history graphs of accelerations was proposed by Levy and Wilkinson (1976): () u ¨ g ( t ) = F ( t ) ∑ i = 1 N A i s e n ( ω i t + φ i ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315207681/cd556cd4-4dcf-4efe-8e29-56fc67b8bfbd/content/eq163.tif"/>
where u ¨ g ( t ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315207681/cd556cd4-4dcf-4efe-8e29-56fc67b8bfbd/content/eq164.tif"/> = ground acceleration time history of the artificial earthquake generated; F(t) = envelope function responsible for giving a non-stationary feature to the time-history graphs of accelerations; N = number of terms considered in the harmonic series; Ai = amplitude of each term of the harmonic series, determined by an iterative process; ωi = corresponding frequency considered in the harmonic series; and φi = corresponding phase angle, randomly generated.
The envelope function used to model the acceleration time-history graphs must be time-dependent and represent the characteristic phases of actual earthquakes in the time: start growing from rest; region of intense movement where the maximum intensity of the movement is reached; and decreasing intensity reaching rest again. In this work it will be used a simple trapezoidal model, as defined in the Equation (2). () F ( t ) = { a m a x t t 1 , 0 ≤ t ≤ t I a m a x , t I ≤ t ≤ t I I a m a x ( t − t f ) ( t I I − t f ) , t I I ≤ t ≤ t f https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315207681/cd556cd4-4dcf-4efe-8e29-56fc67b8bfbd/content/eq165.tif"/>
where amax = maximum amplitude of acceleration, which normalizes the envelope function; tI = time that corresponds to the end of the growing stretch; tII = time that corresponds to the beginning of the descending stretch; and tf = time the earthquake ends.
The maximum amplitude of the time-history graph is estimated from the design feature acceleration at the location of interest. The initial value of the amplitudes of harmonic series is estimated by: () A i , 0 = 4 S ( ω i ) δ ω i https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315207681/cd556cd4-4dcf-4efe-8e29-56fc67b8bfbd/content/eq166.tif"/>
where A i,0 = initial amplitude of the ith term of the harmonic series; δωi = band of influence of frequency ωi , in rad/s, given by: () δ ω i = ω i − ω i − 1 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315207681/cd556cd4-4dcf-4efe-8e29-56fc67b8bfbd/content/eq167.tif"/> S(ωi ) = power spectrum of the acceleration of a stationary Gaussian process, given by: () S ( ω i ) = ξ π ω i [ S a T ( ω i ) ] 2 1 l n [ − π ω i t f ln [ 1 − P ] ] https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315207681/cd556cd4-4dcf-4efe-8e29-56fc67b8bfbd/content/eq168.tif"/>
where S a T ( ω i ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315207681/cd556cd4-4dcf-4efe-8e29-56fc67b8bfbd/content/eq169.tif"/> = design spectral acceleration to the frequency ωi and P = probability of exceedance admitted.
522It will be demonstrated here a program application developed in JAVA language for generating artificial time-history graphs of accelerations from a design spectrum set in Eurocode 8 (EN-1998-1). Afterwards, the response spectrum will be recovered from the artificial time-history graphs generated aiming at a comparison with the design spectrum used as a reference.
As an illustrative example, an artificial earthquake (A) was generated, which simulates a suitable earthquake for seismic design in the territory of Portugal. The earthquake A was regarded as being of moderate magnitude, so the type 2 defined in the European standard was considered. The simulation was made for a ground type A and seismic zone A, and therefore it was adopted a design ground acceleration of 1.6 m/s2.
In the program developed, the following inputs were used: tf = 15 s for the total duration, tI = 2.5 s and tII = 5 s, as defined before. The frequency components used to generate and to adjust the time-history graphs of accelerations are those contained between f1 = 0.2 Hz and fN = 33 Hz. It was stipulated a total of 400 frequency values distributed in a non-uniform way in the domain.
The time-step adopted was δt = 0, 005 s, so the earthquake was discretized in time with 3000 points in total. The power spectrum was calculated with a probability of exceedance of P = 15% and the maximum number of iterations of the setting was set to 20.
To use this simulated accelerograms in structural design, Eurocode 8 (EN-1998) specifies a minimum of three accelerograms that should be used in the analysis. Figure 1 shows only one of the graphs generated by the program developed as an example. Examples of an artificial accelerogram generated from the design spectrum. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315207681/cd556cd4-4dcf-4efe-8e29-56fc67b8bfbd/content/fig295_1.tif"/>
From the graphs generated it is possible to recover the response spectrum and compare it to the reference design spectrum. This comparison is shown in Figure 2 and it is possible to conclude that the results are in accordance with the design spectrum used as an input. Comparison between the recovered response spectrum and the design spectrum. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315207681/cd556cd4-4dcf-4efe-8e29-56fc67b8bfbd/content/fig295_2.tif"/>
