Author: Tanase, Nicoleta
Date published: December 1, 2011
Journal code: SBTC
(ProQuest: ... denotes formulae omitted.)
The frequency content of a ground motion can be represented by power spectrum or power spectral density. The characteristics of the seismic events are known only after the earthquakes had occurred and for this reason probabilistic models are being used as seismic input data. The power spectral density (PSD) of ground motion is widely used in building analysis to determine the structural response in the form of displacement, velocity and acceleration.
For single degree of freedom (SDOF)systems, the equation of motion can be written
where m is the mass of the system, c is damping coefficient, k is the elastic constant and ... are the displacement, velocity and relative acceleration, and pt) is a stationary stochastic process represented by the power spectral density function, s p.
The process p(t) is considered ergodic. The Fourier transform of the input motion p(t) is p(ω) , and the problem can be translated into frequency domain as follows:
x(ω) = h(ω)*p(ω) (2)
where h(ω) is the transfer function, ... is the ground acceleration.
The maximum response of the SDOF is
The mean of maximum displacement response value is
The maximum displacement response having p non-exceedance probability is given by
For a stationary ergodic normal process p(t) , v is the average number of positive-slope crossings.
The dimensionless stochastic indicators of frequency content of the ground motion used in this paper are e (Cartwright & Longuet-Higgins) and q (Vanmarcke).
where ... is the spectral moment of order i . (8)
2. Power Spectral Density and Frequency Content of a Ground Motion
The predominant period of a ground motion primarily depends on the soil type, but it also depends on the fault and path characteristics and on the epicentral distance (especially in near fault regions).
Figure 1 shows that for a SDOF system having the natural period To the maximum value of PSD of the acceleration response corresponds to its natural circular frequency ....
The following subchapters refer to the PSDF of ground acceleration determined for several components of important seismic events recorded on different soil types. The PSDF was used for determining the mean maximum displacement and the maximum displacement having p non-exceedance probability during the time history of the record in the cases of SDOF systems having various natural periods of vibration. The results are presented in the third chapter in graphical and tabular form.
2.1. Vrancea Earthquakes, March 4th 1977, August 31st 1986 and May 30th 1990
The first earthquake record that unveiled the long predominant period of the soil (-1.6 seconds) of Romania's national capital city was the accelerogram from INCERC Bucharest station, during the Vrancea earthquake of March 4, 1977.
As shown in figure 2, the record of 1986 earthquake also revealed a long period of soil vibration, shorter than the one observed during the 1977 seismic event.
The long predominant periods of the soil in the South and the East of Bucharest area during the severe Romanian earthquake from 1977 (M^sub GR^ = 7.2) and the moderate one from 1986 ( M^sub GR^ = 7.0) were explained by the alluvional deposits of sand, gravel, loess and clay. The 1977 accelerogram from INCERC Bucharest station showed the narrowest frequency band of a seismic event recorded in Romania. 
The earthquake produced on May 30, 1990 (M^sub GR^ = 6.7 ) has not shown the same long period, fact that could be explained by the nonlinear behavior of the soil, and the source mechanism (magnitude, type of rupture, etc.). .
2.2. El Centro, Imperial Valley, May 18th 1940
In Imperial Valley region during El Centro earthquake (M^sub GR^ = 7.1), small and stiffer buildings were damaged. This fact was due to the short predominant period of the ground motion (Figure 3) as a function of soil type. In U.S. standards, the soil with predominant period of vibration from 0.4 to 1.0 second are classified as type I (bed rock).
2.3. Tangshan, China, July 28 th 1976
Tangshan earthquake had a magnitude of 7.8 on Richter scale and hypocenter of 1 1 km depth, causing the highest number of deaths (242,400 deaths) of the earthquakes of the 20th century. 
The EW component of acceleration record from Beijing Hotel station revealed a wide frequency band earthquake characterized by the dimensionless stochastic indicators e = 0.89 and q = 0.54 .
The majority of the residential buildings in Tangshan were old, unreinforced or reinforced brick and stone masonry structures. The newest ones were brick masonry structures or reinforced concrete frames built in the '60s and having several stories. In the most populated city area over 95% of multi-storey masonry buildings, and single-story ones collapsed. Most of Tangshan' s industrial reinforced structures concrete or masonry were severely damaged.
It was found that in areas with rocky terrain (North of the city of Tangshan, near the mountains) properly constructed buildings suffered slight damage.  Spectral analysis of the records obtained during Tangshan earthquake revealed several peak periods of 0.4, 0.6 and 1.4 seconds. The collapse and severe damage of a large number of buildings were explained by the poor quality of design or construction. 80% of residential buildings in Tangshan had heavy roofs (with loads up to 400 kg/m2), many did not have enough structural walls to withstand lateral forces induced by earthquake, and frame joints of reinforced concrete structures were not properly detailed.
The first Chinese seismic code was published in 1955 and was essentially a translation of the USSR building code and it had no mandatory character. Tangshan earthquake determined Chinese authorities to develop in 1978 a new seismic design code, with mandatory provisions and a new seismic zoning of the country.
2.4. Ulcinj, Montenegro, April 15th 1979
1979 Montenegro earthquake had its epicenter on the Adriatic coast between the towns of Bar and Ulcinj and had the local magnitude Ml = 7.0. Hotel Albatros station in Ulcinj is placed on hard ground (rock). Cities and villages within a radius of several tens of kilometers have suffered huge losses, buildings of historic importance collapsed, churches and monasteries suffered important damage, and also the administrative and residential buildings were affected.
2.5. Michoacán Earthquake, Mexico City, September 25th 1985
The long predominant period (over 2.0 seconds) is explained by the soil conditions: noncompressible sandy clay with intercalations of gravel and mud, volcanic ash and a high water content, total thickness 100 m (former Lake Texcoco basin sediments). 
Tall structures were affected by the Michoacán earthquake as an effect of the long predominant period of vibration that characterizes the site. This effect was caused by later called Mexico City effect which consists in the amplification of seismic waves passing through soft soil saturated with water. Another factor also leaded to a large number of collapsed buildings - the soil liquefaction.
The short predominant period (T < 1.0 s) of soil in the CUIP station area can be explained by the fact that, unlike the first record that was made in Mexico Valley on soft soil, this one was obtained on hard ground where rigid structures can be affected by the earthquake.
On September 25, 1985 the devastating effect of Rayleigh waves known to occur during a large earthquake (M > 7.0 -7.5) was observed. These waves have a long period (12 to 18 seconds) and a special feature: they affect tall structures (with heights exceeding 17 meters) located at a distance of more than 150 km and less than 550 km from the epicenter.
Mexico City was located at 400 km from the epicenter of the magnitude Mw 8.0 earthquake. Only tall structures in Mexico City (height over 17 m) were severely damaged, while structures up to 3 stories suffered minor damage or no damage at all.
Figure 7 presents the PSDs of ground acceleration of horizontal components recorded at stations located on soft soil during Vrancea earthquake from 1977 and 1985 Michoacán earthquake. Similarity in the amplification of the horizontal acceleration due to passing of seismic waves through inhomogeneous layers of soft soil is observed.
2.6. Kobe Earthquake, Japan, January 17th 1995
In the epicentral area a variability of ground motions was noticed indicating a complex interaction with important effects of seismic wave directivity and local ground conditions. Peak ground accelerations of 0.8g were recorded in the epicenter on alluvial deposits; while peak ground accelerations on hard ground were about 0.3g. Most casualties were caused by the collapse of wooden houses having heavy ceramic roof covering, these structures being very vulnerable to horizontal ground motion.
Tall buildings designed and constructed according to the Japanese seismic code introduced in 1981 behaved acceptably, with few exceptions, for example when significant changes of stiffness occurred on the building height. Figure 8 confirms the existence of predominant periods with values between 0.2 and 1.1 seconds and also that the damaged buildings were mostly short, while modern buildings built according to the new seismic code had minor damage.
2.7.Kocaeli and Duzce Earthquakes, Yarimca Station, 1999
Most buildings that collapsed during these two earthquakes were reinforced concrete moment frames with masonry infill. Four to 6-story buildings were severely damaged. 
Inspections of the affected buildings revealed poor seismic design or the lack of it, poor quality of building material used and low quality of construction works. The structures had very large displacements amplified by the effects of soft soil on the seismic waves. Also it was concluded that some buildings collapsed because of the soft story mechanism, poor detailing, and the usage of deep beams with flexible columns (strong beams - weak columns). 
After analyzing the NS component of the records from these two earthquakes obtained at Yarimca station, several values for frequency, circular frequency, predominant period and maximum normalized power spectrum density have been determined:
- f = 0,29 Hz, ω =1,80rad / s, T^sub C^ = 3,48s , NSD^sub max^ = 0,17 1 for Kocaeli earthquake;
- f = 0,17 Hz, ω = 1,07 rad /s, Tc = 5,85s , NPSD^sub max^ = 0,272 for Duzce earthquake.
It can be noted that not only very tall buildings are vulnerable to these earthquakes, but also buildings with natural period of vibration of 1.3-1.4 seconds because of the soft soil on which Yarimca station is placed.
2.8. Maule, Chile Earthquake, February 27th 2010
The analysis of ground motion records from 2010 Chilean earthquake indicates a rocky terrain having a short predominant period (<1 seconds, Figure 10.) which would lead to damages for rigid structures.
During this earthquake, old buildings having the strength capacity below the required level of modern design codes collapsed, but also newer buildings where the quality of works and materials was not consistent with the projects.
It was noticed that tall buildings designed according to seismic codes requirements have presented only minor cracks. An example is that of a residential complex in Santiago consisting in two 24-story reinforced concrete towers, having two basements and a slab-type foundation, which had only cracks in masonry partitions.
Also in the city of Talca located 250 km from Santiago the collapsed buildings were those with up to four stories (built from 1960 to 1970) adobe type or unreinforced masonry buildings. Highest buildings (20-story-tall), properly designed and constructed, showed only minor cracks.
3. Stochastic Seismic Response of SDOF systems
The mean of the maximum displacement responses has been determined for SDOF systems having £ = 5% and natural period of 0.9, 1.6, 2.0 and 2.5 seconds using equation (4). The influence of soil type over SDOF esponse was evident.
Figure 12 shows the maximum displacement having ? non-exceedance probability of a SDOF system characterized by the damping ratio ? = 5% and the natural period of vibration of 2.5 seconds. The response has been determined using equation (5). The probabilities considered were 10%, 50% and 90%. The results show the high amplitude of the displacement response of this SDOF located on soft soil, e.g. during 1985 Michoacán or 1977 Vrancea earthquakes.
The paper analyzes the frequency content of several important seismic events using power spectral density of the horizontal ground accelerations and presents the displacement response of SDOF systems having 5% damping ratio and various natural periods while it also emphasizes the importance of adequately design a structure so that it fits the soil type. Dimensionless indicators e (Cartwright & Longuet-Higgins) and q (Vanmarcke) were determined as stochastic measures of frequency content.
For the particular case of Bucharest city it is important learning from the past experience of other countries with similar soil conditions and also monitoring and studying the behavior of tall structures during the earthquakes from Vrancea source. Improved design solutions will always be needed as new information will appear on the effect of Vrancea earthquakes on Bucharest's building stock.
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NICOLETA TAÑASE -PhD student, Research Assistant, INCD URBAN-INCERC, INCERC Bucharest
Branch, Romania, e-mail: firstname.lastname@example.org