Spectrum Compatible Accellerograms in Earthquake Engineering Universidad de La Serena, Chile / Technische Universität Dresden, Germany

Spectrum Compatible Accelerograms in Earthquake Engineering December 2002 by M. Thiele [email protected]

Spectrum Compatible Accelerograms in Earthquake Engineering Universidad de La Serena, Chile / Technische Universität Dresden, Germany

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1

Introduction 1.1 Preface (Abstract) 2 Basics 2.1 Basics of Response Spectra 2.2 Response to general Dynamic Loading 2.3 Pseudo-Velocity Response Spectra 2.4 Selection of Design Earthquakes 2.4.1 2.4.2

3

General Conclusions The Code in Chile

Synthesis of Artificial Accelerograms 3.1 Synthesis through Sums of Harmonic Functions 3.2 Filtering of white Noise 3.3 Scaling of existing Accelerograms 3.3.1

Filtering of accelerograms

3.4 Modeling of Strong Ground Motion through Simulation 3.5 Fuzzy variables in earthquake loading 4 The Program RSCA 4.1 Introduction 4.2 The main Program and the control Dialog 4.3 File Formats 4.3.1 4.3.2 4.3.3

4.4 4.4.1 4.4.2 4.4.3 4.4.4 4.4.5 4.4.6 4.4.7

5

Input Accelerograms Files Output Accelerograms Files Input target Spectrum Files

Generating Accelerograms Generating a target spectrum The different types of Intensity Functions Synthesis through Sums of Harmonic Functions Filtering of white Noise Scaling and Filtering of existing Accelerograms Applying a maximum Acceleration Other software to create response spectrum compatible accelerograms

Comparison of different Accelerogram types 5.1 Introduction 5.2 The models 5.2.1 5.2.2

5.3 5.4 5.4.1 5.4.2 5.4.3

5.5 5.5.1 5.5.2 5.5.3

5.6

Frame structure Mixed frame and shear wall structure

Used Accelerograms theoretical comparison (observations) cut’n copy Selective filtering Summation of harmonics and white noise

Results of the calculations frame structure mixed frame and shear wall structure Comparison of the results

Conclusions

4 4 5 5 6 7 8 8 10

14 14 15 18 19

21 21 23 23 23 25 25 25 25

26 26 28 29 30 31 34 34

36 36 36 36 36

38 42 42 42 42

43 43 43 44

44

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6

Appendix 6.1 Maple Worksheets 6.1.1 6.1.2

6.2 6.2.1 6.2.2 6.2.3 6.2.4

6.3 6.3.1 6.3.2

6.4

the Duhamel Integral Scaling of accelerograms

Selected Passages of Source code from the Program RSCA generate spectrum (RSCA_gen_spectrum.f90 spek.for duhamel.for) search optimum (RSCA_linear_scaling_of_frequency_montecarlo.f90) selective filtering (RSCA_apply_selectiv_filtering_FFTn.f90) cut’n copy (RSCA_alter_duration_cut_n_copy.f90)

floor plans of the models frame structure mixed frame and shear wall structure

Literature

45 45 45 47

58 58 60 64 67

73 73 74

75

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For the work on this project I used text passages and pictures of diverse literature with out a permit, especially those of [1] and [4]. Therefore I would ask to consider this carefully when using parts of this project in further studies. I would also like to show appreciation for the tremendous amount of help I received from Prof. M. Durán, Prof. W. Graf and other Members of the “Departamento de Ingeniería Civil en Obras Civiles” at the “Universidad de La Serena” and the “Lehrstuhl für Statik” at the “Technische Universität Dresden”. I highly appreciate the fact that I was permitted to carry out my work at the “Universidad de La Serena” in Chile which has not always been easy for the people involved.

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1

Introduction

1.1 Preface (Abstract) Response spectra have been used for many years to describe the effects of an earthquake on a structure and are still widely used by many to design structures. But nowadays it is also very common to use the time history method together with advanced FEM software to do nonlinear calculations on structures. The methods based on response spectra could e.g. not predict the amount of damage a structure suffers during an earthquake although there have been numerous promising attempts by introducing various factors to describe the ductility. Since non-linear calculations with the time history method depend on accelerograms, the question that comes next is what kind of accelerogram should be used to describe a design earthquake. And here a good choice would obviously be an accelerogram that produces a response spectrum that fits the one of an average earthquake or one of the numerous design spectra that are to be found in different codes around the world. That would enable the structural engineer to do nonlinear calculations which still satisfy the code or to compare the time history methods with the code. Also especially since the code of some countries such as Chile or Peru nowadays requires calculations with spectrum compatible accelerograms for e.g. earthquake isolators, there is now a great interest in understanding the processes involved in creating spectrum compatible accelerogram and developing tools that are able to provide such accelerograms in an easy and convenient form. Furthermore, it would be desirable to include all the effects of a real earthquake in this artificially created accelerogram in view of the fact that the time history method is not only based on the response of a structure but also on the evolution of the earthquake over time. An accelerogram that has a lot of high frequency content could, for example, over its entire duration, produce a significantly different response in a non-linear calculation than an accelerogram whose frequency content shifts from high to low during its duration. There are different ways to generate such accelerograms: by filtering white noise, by summing the harmonic components with random phase, by scaling existing accelerograms or by simulating fault rupture processes. Although it is obvious that the last one would produce the most accurate results, it has not been included in my investigations since it is not very practical. During my project I shall try to investigate and compare the different kinds of compatible response spectra with each other and with the Chilean code, theoretically and on a given structure. To do that I will use Maple worksheets and a program I developed just for this purpose. Furthermore, I will try to discuss the introduction of fuzzy variables to strong motion recordings to better describe the effects of uncertainties, e.g. the dampening of a structure.

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2

Basics

2.1 Basics of Response Spectra For a better understanding of the processes it is important to know what response spectra are exactly and how they are derived from an accelerogram. Basically, a response spectrum shows the maximum response of a simple oscillator, such as a normalized SDOF system, over its natural period or its natural frequency and is called a pseudo-velocity response spectrum. The idea is that a given MDOF structure will have the same response on its eigen frequencies as the SDOF systems with the same natural frequencies. Most response spectra show the response of the velocity but obviously one can also construct spectral displacements or acceleration. But the simple relationship between these three quantities has made it more common to show them all in a single plot on four-way log paper (Fig. 2-1).

Fig. 2-1 response spectra for El Centro earthquake, 1940

One very inconvenient part of the response spectra is that they also depend on the damping of the structure, which is normally very hard to grasp, and contains a lot of uncertainty. Therefore it might be better to introduce this variable as a fuzzy variable especially before obtaining an accelerogram from a given response spectrum.

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2.2

Response to general Dynamic Loading The response of a SDOF system to specific ground accelerations v&&g (t ) may be best expressed by means of the Duhamel integral (Eq. 2-1). v(t ) =

1 mω D

t

∫ p(τ )e

−ξω ( t −τ )

sinω D (t − τ )dτ

0

(Eq. 2-1) But for the numerical processing of common accelerograms it is more convenient to use a sum based on this integral where v(t ) = A (t ) sinωt − B (t ) cos ωt

(Eq. 2-2)

∆τ 1 A ∑ (t ) mω 2

(Eq. 2-3)

∆τ 1 B B (t ) = ∑ (t ) mω 2

(Eq. 2-4)

A (t ) =

Fig. 2-2

A  ( t ) = ∑ ∑ (t − ∆τ ) + p(t − ∆τ ) cosϖ D (t − ∆τ )e −ξϖ∆τ + p(t ) cosϖ D t   A

(Eq. 2-5)

B  = ( t ) (Eq. 2-6) ∑ ∑ (t − ∆τ ) + p(t − ∆τ )sinϖ D (t − ∆τ )e −ξϖ∆τ + p(t )sinϖ D t   For further investigation of this method one should read chapter 6 and 7 in [1] where all the theoretical background is discussed and take a look on my Maple worksheet <1>. B

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2.3

Pseudo-Velocity Response Spectra

The maximum response S v (ξ , ω ) relative to the ground is called the pseudo-velocity response of the ground motion v&&g (t ) . As indicated, S v depends not only on the groundmotion history but also on the frequency of vibration and the dampening of the oscillator. Thus for any given earthquake record, by assuming a specific value of damping in the structure it is possible to calculate values of S v for a full range of vibration frequencies. A graph showing these spectral velocity-response values plotted as a function of frequency (or the reciprocal quantity, period of vibration) is called a pseudo-velocity response spectrum of the earthquake motion.

Fig. 2-3 pseudo velocity response spectrum, El Centro earthquake, 05.18.1940 (NS component)

Plots of the spectral displacement and acceleration obviously could be constructed in a form similar to the pseudo-velocity spectrum, but as said before, the simple relationships existing between these three quantities has made it more common to present them all in a single plot on four-way log paper (Fig. 2-1). On this graph, the abscissas represent the logarithm of the period of vibration, the ordinates show the logarithm of the pseudo-velocity response, and log S a and log S d are represented by distances measured at 45° to the base. It is evident that the response spectra provide a much more meaningful measure of the intensity of an earthquake motion than any single quantity, e.g., the peak acceleration, does. In fact these response spectra show directly to what extent any given real SDOF structure (with specified period of vibration and dampening) would respond to this ground motion. The only limitation in its application is that the response is assumed to be linear elastic because such behavior is inherent in the Duhamel integral (Eq. 2-1). But since they have been used for many years in earthquake design and are the basis of many design codes it would be desirable to be able to create accelerograms that match a specific response spectrum. Such accelerograms could be used to do non-linear calculations that satisfy a special design code or just to compare non-linear calculations

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with old style linear methods or such methods that introduce various factors accounting for the non-linearity. 2.4

Selection of Design Earthquakes

2.4.1 General Conclusions To the structural designer the only purpose in studying seismology is to enable him to predict the characteristics of the earthquake input for which his structures should be designed. The earthquake loading is unique among the types of loads that he must consider because a great earthquake would generally cause greater stress and deflections in various critical components of his structure than all the other loadings combined, yet the probability of such an earthquake occurring within the expected life of his structure is very low. In order to deal effectively with this combination of extreme loading and low probability, a strategy based on dual design criteria usually is adopted. 1. A moderate earthquake which reasonably may be expected at the building site during the life of the structure is taken as the basis of design. The building should be proportioned to resist this intensity of ground motion without significant damage to the basic structure. 2. The most severe earthquake which possibly could occur at the site is applied as a test of the structural safety. Because this earthquake is very unlikely to occur within the life of the structure, the designer is economically justified in permitting it to cause significant structural damage; however, collapse and loss of life must be avoided.

Fig. 2-4 smoothed average earthquake response spectra for acceleration response

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Fig. 2-5 smoothed average earthquake response spectra for velocity response

In order to establish the ground-motion characteristics of the design earthquake and of the maximum probable earthquake for any given building site, it is necessary first to study the earthquake history of the region for as long a period as any type of seismic information is available. Because earthquakes occur very infrequently, the statistical data which can be compiled give at best only a crude estimate of the seismicity of the site. Other supporting evidence should be obtained from field geological studies, which serve to locate potentially active faults and to identify the tectonic characteristic of the local geological structure. One of the simplest ways to define the expected ground motion is to make use of the accelerogram of a past earthquake which had the proper magnitude and was recorded at an appropriate distance. However, there may be drastic differences between the records of earthquakes having similar magnitudes and distances, and the structural responses produced by such records may vary even more widely. Thus, the use of a single record to define a design earthquake leaves considerable uncertainty as to the significance of the response it produces. Obviously an “average” earthquake would be a more meaningful design input, and the most effective way of describing an average earthquake is by means of its response spectra. For example, Housner [3] developed the design spectrum shown in (Fig. 2-5) by computing the response spectra for two components each of four different earthquake records and then normalizing, averaging, and smoothing the resulting curves.

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2.4.2 The Code in Chile The Chilean code includes a part about response spectra to be used in calculations which is described in the following. The spectra to be used are given by the equation: Sa =

IA0α R*

(Eq. 2-7)

where I describes the type of building. category of building A B C D

I 1.2 1.2 1.0 0.6

Tab. 2-1

A - governmental buildings - communal buildings - public buildings (police stations, facilities for electricity and telecommunication, post offices) - such that are important in a catastrophe (hospitals, first aid stations, fire departments, garages for rescue vehicles and equipment) B - buildings of great value (libraries, museums) - such with frequent culminations of people as: - convention centers for 100 or more people - open air stadiums and tribunes for more than 2000 people - schools, kindergartens, buildings of universities - prisons or other locations for imprisoning - shops with more than 500m2 per level or such higher than 12m - malls with halls of more than 3000m2 without the area of parking lots C - buildings for private use or such of public use that are not included in A and B - constructions of any kind whose collapsing would endanger other constructions of the type A, B and C D - isolated or private constructions that are not inhabited and are not included in any of the other categories A0 is the effective maximum acceleration which is determined by (Tab. 2-2) and based on the different seismic zones of Chile (Fig. 2-x).

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seismic zone

A0 0.20g 0.30g 0.40g

1 2 3 Tab. 2-2

The α value is called the factor of amplification and is determined by (Eq. 2-8). An illustration of the effect that have the different soil types have on this function can be seen in (Fig. 2-7). T  1 + 4,5 n   To  α= 3  Tn  1 +    To 

p

(Eq. 2-8)

where Tn is the period for which the response is calculated and To and p are parameters depending on the type of soil of the foundation. Type of soil I II III IV

To [s] 0.15 0.30 0.75 1.20

p 2.0 1.5 1.0 1.0

Tab. 2-3 type of soil

For the definition of the different soil types see also Tabla 4.2 of [13] or my translation in (Tab. 2-4).

Fig. 2-7 factor of amplification

Fig. 2-6 seismic zones

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Type of soil Description I Rock: Material natural, with a velocity of propagation of the shear waves at site with equal or more than 900m/s or such with high compression resistance in one direction of intact samples (without cracks) equal or more than 10 Mpa and RQD (specific laboratory test) equal or higher than 50%. II a.) Soil with v s equal or higher than 400m/s within the first 10m and increasing with depth. b.) Compact gravel, consistently dry with γ d equal or higher than 20kN/m3, or an index of density ID(RD) (relative density) equal or higher than 75%, or a compaction of more than 95% of the modified Proctor value. c.) Compact sand, with ID(RD) higher than 75%, or an index of standard penetration higher than 40 (normalized by the effective pressure load of 0.10 MPa), or an compaction of more than 95% of the modified Proctor value. d.) Compact cohesive soil with not drained shear resistance S u equal or higher than 0.10 MPa (simple compression resistance qu equal or higher than 0.20 MPa) in samples without cracks. In each case the indicated conditions have to be independent of the location of the groundwater and the layer has to be at least 20m. If the layer above rock is less than 20m the soil has to be defined as type I. III a.) Sand that is permanently not saturated with ID(RD) in between 55 and 75% or N greater than 20 (without normalization of the effective pressure of the load of 0.10 MPa) b.) Gravel or sand that is not saturated with a level of compaction less than 95% of the modified Proctor value. c.) Cohesive soil with S u in between 0.025 and 0.10 MPa ( qu between 0.05 and 0.20 MPa) independent from the level of groundwater. d.) Saturated sand with N in between 20 and 40 (normalized by the effective pressure load of 0.10 MPa). The minimal thickness of the layer is: 10m. If the thickness of the layer is less than 10m and above rock or a layer compost of soil type II, the soil has to be classified as type II. IV Saturated cohesive soil with an S u equal or less than 0.025 MPa ( qu equal or less than 0.05 MPa). The minimal thickness of the layer is: 10m. If the thickness of the layer is less than 10m and above a layer compost of soil type I, II or III, the soil has to be classified as type III. Tab. 2-4 definition of soil types (traduced from Tabla 4.2 of the “Norma Chilena Oficial”, NCh 433.Of96)

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The so called factor of reduction R * is determined by R* = 1 +

T*

(Eq. 2-9)

T* 0.10To + Ro

where T * is the period of the mode with the greatest mass movements in direction of the analysis and Ro can be determined by (Tab. 2-5). structural system

Material

R

Ro

structural steel

7

11

reinforced concrete

7

11

structural steel

7

11

reinforced concrete

7

11

reinforced concrete or masonry if it satisfies condition A2) if it does not satisfy condition A2)

6 4

9 4

5.5

7

4

4

4

4

3

3

2

-

frame structures

wood structures with walls or bracings

masonry reinforced masonry - composed of concrete blocks or units with similar geometry without holes in the units or where all holes will be closed and masonry of plastered walls - composed of brick with or without holes and masonry of concrete blocks or units with similar geometry with holes where not all holes will be closed

Any other type of structure or material that is not classified in one of the categories above. 1)

The values indicated for structural steel and reinforced concrete in this table adopt the compliance of the determinations of appendix B (“Norma Chilena Oficial”, NCh 433.Of96). 2) Condition A: The walls of reinforced concrete shall take at least 50% of the shear force of each floor. 3) They do not do the spectral modal analysis for this type of structure or material. Therefore no Ro value has been given. Tab. 2-5 maximum values of the factors to modify the response1) (traduced from Tabla 5.1 of the “Norma Chilena Oficial”, NCh 433.Of96)

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3

Synthesis of Artificial Accelerograms

3.1 Synthesis through Sums of Harmonic Functions A quite simple but effective method to derive artificial spectrum-compatible accelerograms is by summing harmonic functions with a random phase. This method has been described in [4] as well as in [5]. The accelerogram is in this method as indicated derived through the following summation, n

v&&(t ) = I (t )∑ Ai sin(ω i t + φ i )

(3-1)

i =1

where φ i , ω i and Ai are the phase, frequency and amplitude and I (t ) is a function of the intensity and duration of the earthquake (Fig 3-1). The target function which represents the spectrum of choice is composed of logarithmic deployed sampling points that satisfy the condition ∆f ≤ 2ξ f where ξ is the damping of the target spectrum and ∆f the increment at the sampling point with the frequency f . This ensures that enough virtual SDOF systems are available to include all frequency portions of the response spectrum in the target function. The accelerations at the frequencies of the sampling points of the target function are used (multiplied by a factor) as the initial amplitudes of the sinus function. Their summation results in an initial accelerogram. This accelerogram is now to be multiplied with the function of intensity I (t ) . After that a new response spectrum is derived and the comparison between it and the target function will give a modified

(3-2)

Fig. 3-1 different possible intensity functions

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factor for the amplitudes of the sinus functions by means of the following: Ai ,new = Ai ,old

S v ,shall S v ,is

(3-3)

A new accelerogram can be created and a new iteration cycle has started. Normally, 2 or 3 iterations are sufficient for a good approximation of the target function. Admittedly, there are difficulties if the frequency range is too wide or the ordinates of the target function are for a short period very small. This could be explained by the fact that, for physical reasons, not every given target spectrum has a realizable accelerogram. In practice one can avoid these difficulties by restricting the bandwidth of the considered frequency range without compromising the usability of this method. 3.2 Filtering of white Noise A very frequently used method to generate artificial accelerograms is based on the hypothesis that the source of the ground motion is a random sequence of impulses generated at some distance and propagated to the point of observation through the basement rock structure. In view of the irregular manner in which slippage undoubtedly occurs along a fault, strong ground motion at some distance from the fault might be considered as the superposition of short-duration random pulses arriving randomly in time. Therefore, since accelerograms usually have a phase of nearly constant intensity during the period of most severe oscillation, one might consider modeling this phase with a white-noise process of limited duration. Housner [6], Rosenblueth [7], Bycroft [8], Thomson [9] and others considered this possibility in their earlier investigations. Another good description of this method can be found in [4] or in chapter 28 of [1].

Fig. 3-2 scheme of spectral processes

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Here I will only give a short summary of the basic principles. Only one component of the acceleration is simulated through filtering of white-noise and modulation with a deterministic intensity function to simulate the non-stationary character of real accelerograms. For simplicity I will also forgo the modeling of the shifting of the frequency content to long wave components (the filtering is time invariant). Following the several steps of this method are listed: 1. The two sided spectral density of the site as well as the time step ∆t for the accelerogram are chosen. The white-noise is basically the marginal case of a broad band process whose spectral density function is a constant over all frequencies (Fig. 3-2) 2. A sample function is generated for the chosen S 0 and ∆t through Gaussian distributed time series (mean value is zero and variance is one) multiplied by a factor: c=

2πS 0 ∆t

(3-4)

3. Using the FOURIER-Transformation, the time series is transformed into the frequency domain and modified by applying the filter transfer functions of high, low or band pass filters. Mostly the FFT (Fast Fourier Transformation) method is used for the transformation since this is a very effective numerical algorithm. But one has to take into account that only frequencies up to the nyquistfrequency

π are considered by this method. Therefore it has to be ensured that ∆t

only frequencies lower than this nyquist-frequency are involved in the process.

Fig. 3-3 scheme of high and low pass filter

4. High or low pass filters are best described by the scheme in (Fig. 2-3). While a high pass filter filters the low frequencies of the signal, the low pass filter filters the high frequency parts of the signal. A band pass filter is a combination of

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both filters so that only specific frequencies are allowed to pass. A simple high pass filter of 1. order is H (iϖ ) =

ϖ 2 + iϖϖ H ϖ H2 + ω 2

(3-5)

where H (iω ) is called the transfer function and ϖ H is the characteristic frequency of the filter up to which the frequency content is filtered. The transfer function of a proper low pass filter is equally given through H (iϖ ) =

ϖ 2 − iϖϖ H . ϖ H2 + ω 2

(3-6)

But in practice normally a 2. order transfer function is used, as for example the so- called Kanai-Tajimi-filter: 1+ H (iω ) =

ϖ2 ω3 2 4 1 i 2 ( ) ξ − − ξ 0 0 ϖ 02 ω 03

 ϖ 2 1 −    ϖ 02 

  

2

2

2   + 4ξ 2  ϖ 0  2  ϖ 0 

  

.

(3-7)

2

This filter allows the dampening of the unwanted frequencies through ξ 0 and gives therefore more adequate results since total filtering of certain frequencies obviously cannot represent an actual earthquake. 5. Through inverse transformation into the time domain a stationary acceleration function is derived and after modulating with the deterministic intensity function (Fig. 3-1) one gets the transient acceleration. From this the response spectrum is derived to evaluate if it suits the conditions of the site.

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3.3 Scaling of existing Accelerograms Another interesting but rather seldom used method to generate artificial accelerograms is through scaling existing accelerograms until they fit the desired conditions. Also it has been suggested by few and I later found some software that is able to do such scaling, I could not find much about the theory of scaling accelerograms to create spectrum compatible response spectra. A good source of information has been [10] and it has been mentioned in [1], but no procedures have been described. Starting with an accelerogram, such as one of the few available recorded strong motion earthquakes, the process of obtaining an accelerogram that is compatible with a given smooth spectrum curve as defined above, involves three distinct processes: 1. Scalar multiplication of the acceleration amplitudes 2. An overall frequency content manipulation 3. Filtering of the unwanted responses Multiplying the earthquake record by a constant factor would result in multiplying the corresponding response spectrum curve by the same ratio. This scalar multiplication can be achieved very simply during the computation of the response spectrum curve data. The purpose of an overall change in the frequency content is to shift the relative position of the response spectrum curve with respect to the smooth response spectrum curve. This shift is desirable so that the response spectrum curve can be positioned relative to the smooth curve such that a better fit is achieved by the subsequent filtering processes. This operation is also easily handled during the computations by simply changing the digitized accelerogram time interval. The resulting relative shift for the acceleration response spectrum curve is in the same ratio as the change in the time interval. Thus, as shown in (Fig. 3-4), a change of the digitized time interval from 0.01 to 0.015 sec would shift the response spectrum curve to the right by 50% and the sharp peak would occur at the 0.53-sec period instead of at the 0.35-sec period.

Fig. 3-4 shifting process

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3.3.1 Filtering of accelerograms The final step in the process involves a more selective frequency content manipulation. The earthquake motion is filtered such that the high response portions of the curves may be lowered, or the low response portions of the curves may be raised, or both. The filtering methods used are similar to those described in chapter 3.2. and the effect is again illustrated in (Fig. 3-5)

Fig. 3-5 filtering of high period response

However these filtering methods provide only a relatively rough approximation of the desired spectrum and are relatively hard to be used in an automated process. Therefore I developed an alternate filtering method that is based on the idea that the results of an FFT on the accelerogram are nothing more than the factors of amplitudes and phases of harmonic functions. Knowing that and assuming that the amplitude of a certain frequency will most likely effect the SDOF system with this natural frequency (period) by much greater means than all the other SDOF systems, one could assume that modifying these factors directly will, Fig. 3-6 original and target response spectra

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after retransformation, result in an accelerogram that more closely matches the desired response spectrum. Given the existing spectrum of any accelerogram and a target spectrum, as for example shown in (Fig. 3-6), one can derive a function of factors that are to be multiplied with the according spectral powers of the accelerogram in the frequency domain (Fig. 3-8). This transfer function is derived by simply dividing the target spectrum Fig. 3-7 transfer function and the original spectrum. In order to not disturb the actual accelerogram by to much one might consider flattening this transfer function since the original is kind of oscillating in nature and could disturb the character of the original spectrum of the accelerogram (Fig. 3-7). After that one has to derive the real and imaginary parts of the accelerogram in the frequency domain from this new amplitude spectrum and the original phase spectrum and can than perform an inverse FFT to obtain Fig. 3-8 frequency domain a new accelerogram. Therefore only the real part of the results of the inverse FFT is to be used. After that a new response spectrum is calculated (Fig. 3-9) to evaluate the approximation of the target response spectrum and eventually one has to perform a couple more iteration cycles to obtain a good match. However, it shall be noted that obviously this filter falsifies the original accelerogram by a certain amount. The first usage, especially, will change the character Fig. 3-9 resulting response spectra of the waves by a great amount. If for some reason the filtering does not produce satisfying results one should consider to use another earthquake record since not every record is equally qualified to achieve an approximation of a certain response spectrum. This of course has only been a short summary of the basic principles of this method and to investigate it further one should take a look at my Maple worksheet “6.1.2

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Scaling of accelerograms”. One might also notice that the smoothed intensity function of the original accelerogram is not entirely the same in the resulting accelerogram. This effect can be reduced by modulating the resulting accelerogram with the normalized smoothed intensity function of the original accelerogram. Another very important point of scaling an accelerogram is the duration of it, since the final results of a non-linear, non-elastic calculation depends a lot on the duration of the earthquake. It has been suggested that this may be done by truncating or duplicating portions of the record. 3.4 Modeling of Strong Ground Motion through Simulation Several techniques have been described to derive artificial earthquake records through the simulation of fault rupture processes. A lot of information about this can be found in [2]. But these methods and their evaluation requires a rather complex computational process and would unfortunately exceed the time I have been given to complete this project. Because of the various variables with uncertainties introduced in these methods it might be worth looking into for generating even more meaningful fuzzy accelerograms. 3.5 Fuzzy variables in earthquake loading An interesting observation I made during my work on this project was that a response spectrum of an accelerogram is mostly fairly oscillating. Therefore I consider the often-used response spectra that are composed of rather few sampling points quite inaccurate. The effect of this is illustrated in (Fig. 3-10). A better mapping of the response spectrum would be of course when using a great amount of sampling points. The attained response spectra however are because of their Fig. 3-10 oscillating character of response spectra oscillating character very inconvenient for calculations. One could now, for example, gain a response spectrum that is good for calculations and also represents the actual oscillating spectra by generating a smoothed curve from its data. Yet, an even more meaningful representation of the response spectra would be a fuzzy response spectrum. Such a fuzzy response spectrum could, for example, look like (Fig. 3-11), where the period would be a crisp parameter and the response a fuzzy value. Fuzzy mathematics is basically a concept where a number no longer has just a crisp value but a number of Fig. 3-11 probable fuzzy spectrum

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possible values, or rather an affiliation function, and their corresponding probability of existence. A more detailed explanation of this would, however, exceed the means of this text, but for further insights on fuzzy logic and fuzzy algebra one might take a look at [14]. Further more it is assumed that there might exist methods that allow a reasonable use of this kind of fuzzy loading, see [15] and [16]. A quite more complicated thing than creating such a fuzzy spectrum would be to create a fuzzy accelerogram that produces a fuzzy response spectrum compatible to a fuzzy target response spectrum and can be used in fuzzy time history analysis. To do this on a scientific level one would at first have to define a fuzzy Duhamel integral and for this purpose fuzzy trigonometric functions (e.g. by Taylor series with fuzzy variables). Besides the acceleration the dampening and eventually even the period should also be introduced here as fuzzy variable in such a integration. Though the later would produce a fuzzy mapping of fuzzy values. Even though I did not have the time to actually calculate such fuzzy response spectra, I would expect that they will have the same oscillating character. One might therefore consider introducing a second order fuzziness to describe this effect properly. However, I would find it more convenient and satisfying just smoothening these fuzzy spectra and describing them with first order fuzzy values. Assuming we had the algorithms to provide us with the fuzzy response spectra and we would generalize the fuzzy values of the accelerogram to consist of linear membership functions, it would be possible to use the same methods as described in “3.3.1 Filtering of accelerograms”. Though to achieve this it might be necessary to also develop fuzzy FFT transformation algorithms.

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4

The Program RSCA

4.1 Introduction As a part of my work on artificial spectrum compatible accelerograms I programmed an application that would provide me with the different forms of accelerograms. This application is named RSCA 0.9 (Response Spectrum Compatible Accelerograms) and is entirely programmed in FORTRAN using the Compaq Visual Fortran Compiler 6.5 in the Microsoft Developer Environment 6.0. Other than the standard FORTRAN routines, I used the QuickWin routines of the Compaq Visual Fortran for graphic display, a fast Fourier transformation by J.H. Glassman and a routine to derive the Duhamel integral by M. Durán. Due to lack of time I did not implement error handling routines as to yet. Therefore mistreatment of the application, for example trying to create an accelerogram without having created a target response spectrum, will result in termination. 4.2 The main Program and the control Dialog The application basically consists of a main program area where the entire graphic is displayed and that allows handling and saving of all the different created diagrams and a control Dialog where all the values are set and the actions are taken. Like all QuickWin applications the main program area has these general characteristics: -

-

Window contents can be copied as bitmaps or text to the Clipboard for printing or pasting to other applications. In Fortran QuickWin applications, any portion of the window can be selected and copied. Vertical and horizontal scroll bars appear automatically, if needed. The base name of the application's .EXE file appears in the window's title bar. Closing the application window terminates the program.

In addition, the standard Fortran QuickWin application has a status bar and menu bar. The status bar at the bottom of the window reports the current status of the window program (for example, running or input pending). The menus are more or less self explanatory, but for further details on how to use this MDI (Multiple Document Interface) see also the “Visual Fortran: Programmer's Guide” [12]. The control dialog (Fig. 4-1) is the heart of the application and allows setting all major values and taking all the action needed to create or modify accelerograms. The more general controls which are not described in the further text shall be described in the following: -

-

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Input File: Specifies the name and location of the accelerogram to be loaded. The button “Load” will load and display the input accelerogram. The values “points”, “time step” and “first sampling point” will be automatically set when loading a new accelerogram but can be changed before proceeding with the manipulation of it. Output File: Specifies the name and location of the output accelerogram. The button “Save” will save the output accelerogram to this location. To use this function an output accelerogram must have been created. Exit: Will exit the control dialog only.

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-

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-

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Generate spectrum of output file: This button creates a target spectrum by the means of the given values and a spectrum of the output accelerogram. Both spectra will be displayed in the spectrum window. To use this function an accelerogram must have been loaded or created. Generate spectrum of input file: This button creates a target spectrum and a spectrum of the input accelerogram. Both spectra will be displayed in the spectrum window. To use this function an accelerogram must have been loaded or created. Smooth spectrum: This checkbox can be chosen if the created Spectrum should be smoothed after its creation. This is especially important when using a great amount of sampling points and applying the “selective filter”. Since the response spectrum is sometimes of a quiet oscillating nature, not using the smoothening could disturb the original frequency spectrum too much, because the transfer function of the filter is directly derived from the difference in target spectrum and response spectrum. However, severe smoothening of the response spectrum also hinders a close fitting of the real response spectrum to the target spectrum. The two values behind the checkbox define the grade of smoothening. Where the first value says from how many sampling points the average should be taken and the second value says how often this smoothening should be done. Correct intensity: This checkbox can be chosen if the intensity function of an accelerogram should be corrected by using the smoothed intensity function of the old accelerogram, after applying the “selective filter”. For the use of the smoothening parameter see the explanations in “Smooth spectrum” above.

Fig. 4-1 RSCA Dialog Box

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-

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Draw output accelerogram: With this button an accelerogram will be drawn with the data of the actual output accelerogram. The output accelerogram is always the manipulated accelerogram and further manipulations will be done on this accelerogram. A plot of the input accelerogram is automatically created when creating or loading a new accelerogram. To use this function an accelerogram must have been loaded or created. Plot spectra: This button will just plot the actual spectra. To use this function a target spectrum and response spectrum must have been created. Highest period: Here is the highest period specified that is used when plotting the spectra, e.g. by using the “plot spectra” function.

4.3 File Formats It is necessary for the program to deal with several input data that describe, for example, the target spectrum or an input accelerogram that is to be scaled. 4.3.1 Input Accelerograms Files When reading an input accelerogram, the program expects a file where the data is grouped in lines with each having 8 sampling points. The Files may have an arbitrary header and the program starts reading the file from a line where the first character is an “(“. This line should specify the format of the data in FORTRAN format as e.g. ”(8F10.5)”. The total length of the lines should not exceed 80 characters. In the following two lines the program expects the number of input lines and the time step between two sampling points. The unit of the sampling points should be [g] since the program expects this. But because of the nature of a scaling process, the units of the input accelerograms would have an effect on the output results. For now only this form of input data is provided but one could easily modify the routines in “RSCA_load_accelerogram.f90” to load any kind of data. The subroutine “RSCA_load_accelerogram” returns the data of the accelerogram to the global arrays “inputaccelerogram” and “outputaccelerogram” and the time step and number of sampling points to the global variables “g_timestep” and “g_accpoints”. 4.3.2 Output Accelerograms Files Since I worked with SAP to do the comparison of the different accelerogram types the format of the output accelerograms is designed to be readable by SAP. For specification see the manuals of SAP. Basically, each line of the output file contains two values, where the first is the time and the second the according acceleration. 4.3.3 Input target Spectrum Files The application allows the generation of the target spectrum for the parameters of the Chilean norm. But in order to also permit target spectra of other norms or of any thinkable nature the application allows the reading of the target spectrum from a file. The first line of such a file should contain the number of data lines and the following lines contain the data. Each data line has two values where the first is the time and the second is the value of the spectrum. The unit should be for example “cm/s” since the target spectrum used by the application is always the pseudo velocity target spectrum and the program expects its data in this unit. The unit of the output accelerogram

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depends upon the unit used by the target spectrum and would be “cm2” if “cm/s” is used. 4.4 Generating Accelerograms There are 4 different types of spectrum compatible artificial accelerograms that can be created with this application and each of them has different qualities. But before one can create a spectrum compatible accelerogram he of course has to create a target spectrum and eventually define the kind of intensity function to be used. 4.4.1

Generating a target spectrum In the section “target response spectrum” (Fig. 4-2) one can define and create a proper target spectrum that is used to further manipulate or create a spectrum compatible accelerogram. As to now there are two different ways to create an accelerogram and these ways can be chosen by checking either “use file” ore “use chilean norm”. When using an input file for the spectra the file has to be specified in the field below the checkbox. For the file format see the explanations in “4.3.3 Input target Spectrum Files”. When using the Chilean norm the following parameters have to be defined. -

building category: A; B; C or D seismic zone: 1; 2 or 3 type of soil: 1; 2; 3 or 4 natural period: in seconds Ro: as in (Tab. 2-4)

For further information on the meaning of these Values see also “2.4.2 The Code in Chile”. Fig. 4-2 target spectrum Also the amount of sampling points for the target spectrum and the highest period used in the target spectrum should be chosen with careful consideration. Normally 100 sampling points are sufficient here and higher numbers will only result in longer computing times for the response spectrum since the amount of sampling points of the target spectrum and the actual response spectrum have to be the same. Also higher periods than the default 5s for the two spectrums can be used but since earthquake loading of these low frequencies is normally not the authoritative loading it is not recommendable. Also it has been shown that the different generating and filtering algorithms used by the program cannot provide accelerograms whose response spectra accurately fit the target spectrum for very low frequencies. The dampening that can be chosen refers to the dampening that is used when computing the Duhamel integral and is normally by default set to 0.05 (5%). It will directly effect the response spectrum and therefore also the resulting accelerogram when generating a spectrum compatible accelerogram. Furthermore, it can be chosen if the deployment of the sampling points in time should be logarithmic or linear. This is important because normally the responses to the high

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frequencies of an earthquake are more important and a higher density of the sampling points in this region of the spectrum would more accurately represent the needs of the engineer. If the logarithmic deployment is chosen one has also to define a time for the first sampling point. All remaining sampling points will deployed between this first sampling point and the sampling point for the highest frequency based on a logarithm with the basis: 1  

 Thigh  n   b =  T  low 

(Eq. 4-1)

where n is the amount of sampling points, Thigh the highest period and Tlow the lowest period of the spectrum. This time for the first sampling point represents also the highest frequency that is taken into account when doing the calculations. In order to obtain a good approximation of the wave character of the signal one should use a frequency (period) that allows at least 10 or more sampling points of the accelerogram in one period. Therefore this value is set to 10 times the value of the time step value of the input accelerogram when loading a new accelerogram. Fig. 4-3 same time step but different first time step The effect of this is illustrated in (Fig. 4-3). However it has become clear that a smaller time step helps to decrease this effect. This is due to the fact that the response to high frequencies is also low. Sometimes it can even be helpful to choose a smaller first period than time step in order to maintain the wave character of an accelerogram, since the waveform depends a lot on the high frequency content. When using the linear deployment, the sampling points will be linearly deployed between the lowest and the highest period. This is useful if not the higher but the lower frequencies are of interest to the engineer.

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4.4.2

The different types of Intensity Functions The application allows defining certain parameters of the target accelerogram. The more general ones are: -

Fig. 4-4 target accelerogram

points: the amount of points of the accelerogram to be created time step: the time between two sampling points of the accelerogram duration: the duration time of the artificial earthquake

These values are automatically calculated when loading an input accelerogram. But when creating a new accelerogram one should set these values. The duration however depends directly on the chosen time step and when using this value one should press the “calculate new time step” button to calculate the appropriate time step that belongs to this duration and

amount of sampling points. The “calculate new time step” button can also be used to adjust the length of an input accelerogram to a certain value, but one should consider that this will also result in a frequency shift of the input accelerogram. Other than that the kind of intensity function that is used when creating an accelerogram can be chosen. As for now the program provides two different kinds of intensity functions, the “hybrid intensity function” and the “exponential intensity” function, which are described in “3.1 Synthesis through Sums of Harmonic Functions” the parameters of this functions should be set as given by the formulas in these sections. The intensity function of an input accelerogram is for now only available for plotting. The actually chosen intensity function can be plotted by pressing the “plot intensity function” button in the “plot options” section of the control dialog. It shall be noted that the intensity function of the exponential method is not normalized and does not need to be since a later scaling of the created accelerogram will have the same effect and is necessary anyway.

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4.4.3 Synthesis through Sums of Harmonic Functions Before creating an accelerogram with the “summation of harmonic functions” button it is necessary to create a target spectrum and define some values for the target accelerogram. When pressing the Fig. 4-5 accelerogram created through summation of harmonics “summation of harmonic functions” button the program will take the current first time step and duration settings and calculates automatically an appropriate time step and the amount of sampling points needed for the accelerogram. The algorithm uses for each sampling point of the target spectrum one harmonic component to generate the accelerogram. It is therefore essential to the result that one uses here enough sampling points to adequately simulate an earthquake. For the amplitude of the harmonic component the program uses the value of the target spectrum. Again a linear scaling of amplitudes is done on the accelerogram after this. Fig. 4-6 target and response spectrum The first time a summation of harmonics is performed the program will also do one enhancement automatically to fit the response spectrum better to the target spectrum. The enhancement works by dividing the actual spectrum with the target spectrum and multiplying these values with the factors of amplitudes used for each corresponding harmonic component. A further enhancement can be achieved by pressing the “enhance summation” button until the target and the response spectra’s are sufficiently matched. In (Fig. 4-6) and (Fig. 4-5) one can see the results of this method to create artificial accelerograms and it shall be noted that for the response spectrum in (Fig. 4-6) no smoothening was used and the intensity function was a hybrid one.

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4.4.4

Filtering of white Noise

By pressing the “generate white noise” button a general broad band white noise will be produced. This white noise has no wave character at all and a lot of high frequency content that is not needed. Therefore the resulting response spectra does not match any of the required target response spectra or the needs of an engineer. But after applying the “selective filter” function the high frequent content will be filtered out and a wave Fig. 4-7 white noise before and after filtering character can be obtained. Before using this selective filter one has to assure that a target spectrum as well as a response spectrum has been created. The kind of results that can be accomplished by this method can be seen in (Fig. 4-8) and (Fig. 4-9). It is remarkable how close the results seem to be with the ones achieved by the summation of harmonic functions. For the theory behind this selective filter one should read “3.3 Scaling of existing Accelerograms” or take a Fig. 4-8 accelerogram created with white noise look at my Maple worksheet “6.1.2 Scaling of accelerograms”. The further use of this function and its effects will be explained in “4.4.5.3 Filtering of Accelerograms using a selective Filter”. The filtering of white noise method is the for the most part a method used to generate artificial accelerograms. It has for example extensively been Fig. 4-9 target and response spectrum after filtering of white described in [1], [4] and [8]. noise

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4.4.5 Scaling and Filtering of existing Accelerograms A very important part of my work has been to determine if it is possible to alter existing accelerograms in a way that a new accelerogram, whose response spectrum fits the requirements, can be achieved. Basically this is possible by two different approaches. The one is pure scaling of the accelerogram by altering time step and amplitudes and the other is filtering of the existing accelerogram. The first method ensures that the initial accelerogram is altered as little as possible but since messing with the time step value changes the duration of the accelerogram it is impossible to predict the duration of the output accelerogram. To alter the duration one could try to use the “cut’n copy” method as suggested in [1] but programming and algorithms that do exactly this have shown that this method immensely alters the response spectrum. The second method works better and one can achieve a quite close match of the actual and the target response spectrum. But observation shows that this method also destroys the wave character of the initial spectrum. While the initial accelerogram has a rather angularly waveform the achieved output accelerogram has always a smooth wave-like character which is obviously an effect of the FFT transformation.

Fig. 4-10 response and target spectra’s during different phase’s while scaling the frequency content.

4.4.5.1 Scaling of Accelerograms There are basically two different options for just scaling an accelerogram. The less destructive one is the linear scaling of amplitudes of the accelerogram. By pressing the “linear scaling of amplitudes” button the program will at first calculate a target and a response spectrum, as well as the area below these two functions, by the given values. Using the areas a linear scaling factor will be calculated and applied to the accelerogram. It is important to know that the areas can be differently weighted, logarithmically and linear and that a value should be set for the period up to which the areas are calculated. The logarithmic weighting of the areas will only work if also the logarithmic deployment of sampling points is chosen otherwise it will be just a linear weighting. One can try different settings to achieve an optimal fitting. A linear scaling of the amplitudes of the accelerogram is automatically done before or after the most other operations. The more severe method to scale the accelerogram is by also scaling the time step value and will of course result in a change of the duration of the

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accelerogram. By pressing the “linear scaling of frequency content” button the program will do an optimizing of the area between target and response spectrum. Again the “highest period for weighting” value and the kind of weighting apply. For the optimizing algorithms a modified Monte Carlo simulation is used. The application has a start window in which it tries to find an optimal area by randomly choosing values. If the value found is on the boundary of the search window the center of the search window will be moved to this value and a new search is started. If the value is inside of the search window the search window will be made smaller and its center will also be moved to the value. This is done until the dimension of the search window under runs a certain value. The algorithm always looks for an optimum value when scaling both amplitudes and time step. The minimal area between the two functions is not always the kind of scaling we had in mind, especially when using high period for the weighting of the area one can receive odd results. The initial response spectrum should also resemble the target spectrum up to a certain degree since otherwise a severe change in the duration could occur. The amazing results sometimes achieved with this method are shown in (Fig. 4-10) nevertheless it shall be noted that this scaling also changed the duration of the earthquake from 116.440s to 285.894s. 4.4.5.2 Using the cut´n copy method to alter the duration This profound change in duration could be compensated by using a kind of cut and copy method by pressing the “alter duration using cut´n copy” button. This subroutine will tear the entire accelerogram apart into pieces with each containing one full cycle. This means each piece has data between three zero points. After that the new accelerogram is put together by copying these pieces one to another and parenthesizing pieces randomly and close to its original position until the desired length is reached. If a shortening of the accelerogram is required some pieces will be randomly erased. However useful this method seems it has become clear by observation that greater alterations will result in a rigorous change of the response spectrum. Therefore I developed an algorithm that checks a certain amount of copying operations before applying them on their effect to the response spectrum and chooses the one that has produced the least degradation or even caused an improvement. The idea behind this is to copy the portions of the accelerogram with frequency content that is wanted in the response spectrum or erase such with a negative influence. That this can be a quite effective method to further enhance an accelerogram can be seen in (). The initial accelerogram with a duration of 116s had after the scaling a duration of 66s and was after this altered with the cut´n copy method to produce an accelerogram with a duration of 100s. Nevertheless, since it is necessary for the evaluation of each coping or erasing operation to calculate a response spectrum, this is also quite a time consuming process. Therefore the percentage of calculation done as well as an estimation of time is shown in the “target and response spectrum” window. The estimation of time however is just a guess and will not produce satisfactory results if the altering in duration is very large. Also one can set the maximum amount of checking that is done before choosing a copying or erasing operation by defining the “tries for searching a good cut” value. Also one should carefully consider the “highest period of weighting” value since it is used when calculating the difference area between the two functions and that again is used to evaluate if the function fits the target spectrum or not.

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A further enhancement of the algorithms could be achieved by not copying a piece directly behind itself but by copying it into a region somewhere around the original piece. Especially when altering the duration by great amounts this would help to make the accelerogram look more random. 4.4.5.3 Filtering of Accelerograms using a selective Filter Another interesting method to adjust the frequency content of the accelerogram is by filtering unwanted frequencies or enhancing wanted ones. It has been suggested to use the Kanai-Tajimi-filter (Eq. 3-7) to do this but for several reasons I found this filter inapplicable for the purpose of generating a specific response spectrum. Therefore I have developed a filter with a transfer function that is more customizable. The theory of this filter has been described in “3.3 Scaling of existing Accelerograms”. This filter should be used also to change the frequency content of the white noise accelerograms. To apply the filter one has to press the “selective filtering” button and the new accelerogram will be calculated and its new spectrum is automatically calculated and plotted. The filter will not produce an absolute fitting in the first run but a notable enhancement and it can be used several times until an acceptable result is obtained. A result of Fig. 4-11 spectra’s before and after using this filter can be seen in (Fig. 4-11). In this applying the selective filter example the filter has been applied 10 times. One could also see in (Fig. 4-12) that the signal now has shifted from high frequency content to low frequency content while the general structure of the accelerogram has been maintained. To further alter the duration the cut’n copy algorithm could, as explained in “4.4.5.2 Using the cut´n copy method to alter the duration”, be used. A side effect of the filter will be that the intensity functions of the accelerogram change slightly. To counter affect this, it is possible to correct the intensity function of the new accelerogram using the intensity function of the old accelerogram by checking “correct intensity”. The values behind this checkbox define how the intensity function will be derived which works basically like the smoothening of a spectrum and has been explained previously. The intensity function can and should be checked by plotting it using the “plot intensity function” button (the option “of input accelerogram” in the “target Fig. 4-12 accelerograms before and after filtering accelerogram deff.” has to be chosen).

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And it shall be noted that using a not or inadequately smoothed intensity function will counter affect the filter. 4.4.6 Applying a maximum Acceleration Sometimes it can be very helpful if one can apply a maximum acceleration to an accelerogram, especially if the produced accelerogram has a few peaks that stick out very unnaturally. The easiest way to set a maximum acceleration would be to just set all values that exceed the maximum value back to this maximum. However, since I found this method rather crude I developed an algorithm that performs a somewhat softer adjustment. It determines the area in which the maximum acceleration is exceeded and within this the maximum value. After that this maximum value and all values that exceed the maximum acceleration are downscaled so that the maximum value will be just the maximum acceleration. In order to avoid points of discontinuity a certain Fig. 4-13 maximum acceleration area before and after the area of exceeding are also scaled with a steadily decreasing or increasing factor. The basic concept of this algorithm is illustrated in (Fig. 4-12). I found this algorithm to work quite good though when the maximum acceleration underflows a certain value, a fitting to a target spectrum will be more difficult or even impossible and the accelerogram will, depending on the method used to generate it, look quite unnatural. In addition, I made sure that the correction of intensity algorithm is applied whenever possible after the maximum acceleration has been applied to maintain a natural intensity function of the accelerogram. 4.4.7 Other software to create response spectrum compatible accelerograms It has come to my attention, unfortunately only after I had already finished the work on my own software that there are two more programs that would be able to scale existing earthquake records in a way that they fit certain response spectra. The first of them, SYNTH, by Naumoski, is available for about ca. $270. This program generates time-histories matching any target spectrum. A target spectrum in digital form and an accelerogram is required as input. The spectrum of the input accelerogram is computed and compared with the target spectrum for prescribed periods (all happens in time domain). Then the ratio between the target and the computed spectral ordinates is calculated and the ordinates of the computed spectrum are iteratively suppressed/raised by reducing/increasing the Fourier coefficients at according Period by the calculated ratios. Therefore only the Fourier amplitude spectrum is changed during the iteration process so that the phases of the Fourier components of the final accelerogram are the same as those of the initial accelerogram. I could not take a closer look at this software but it seems it uses the same methods to filter the initial accelerogram that I have been using for my software. The second and far more interesting program SIMQKE-II was developed 1997 by E.H. Vanmarcke, Department of Civil Engineering and Operations Research, Princeton University; G.A. Fenton, Department of Applied Mathematics, Technical University of

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Nova Scotia, Halifax, Nova Scotia, Canada; and E. Heredia, Instituto de Ingenieria, Universidad Autonoma de Mexico, Coyoacan, Mexico DF. This program is also principally based on the same methods as my software but far more advanced. The source code and the executables can be downloaded at http://nisee.berkeley.edu/software/simqke2/.

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5

Comparison of different Accelerogram types

5.1 Introduction In order to check what results the different artificial accelerograms would produce, I did calculations on two different models. The building types are typical for Chile and the basic dimensions have been kindly provided by T. Guendelman from IEC Ingeniería S.A. The models are very simple and do not represent any real existing building. For the purpose of this investigation however it is not necessary to use a full design optimized model since this normally will not have an effect when comparing results of the same model. Both models are of concrete design, will have 20 stories and the same dimensions whenever possible. A simple 3D sketch can be seen in (Fig. 5-1). More detailed plans with dimensions can be found in the appendix. For the calculations the SAP2000 Nonlinear Version 7.40 was used. 5.2

The models Fig. 5-1 3D view of models

5.2.1 Frame structure This type of structure is very ductile and the first modal modes will therefore have rather high periods. For that reason it should turn out that an earthquake with a fairly low frequency content will have more severe effects on the structure than an equal earthquake with a somewhat higher frequency content. 5.2.2 Mixed frame and shear wall structure This type of structure has in addition to the basic frame configuration of model 1, shear walls in the core which makes the whole structure stiff. Therefore the frequencies of the first modal modes will be somewhat higher and the effects of an earthquake with a high frequency content should be more severe. The modal mass participation as well as the periods of the first 40 modes for the two models, which had been used for the calculations, can be seen in (Tab. 5-1) and (Tab. 5-2).

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Period 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

2.0034 1.7511 1.2434 0.7304 0.6537 0.4836 0.4068 0.3684 0.2886 0.2790 0.2662 0.2539 0.2280 0.2208 0.2072 0.1991 0.1914 0.1879 0.1753 0.1726 0.1712 0.1650 0.1614 0.1532 0.1520 0.1453 0.1436 0.1411 0.1400 0.1283 0.1234 0.1203 0.1111 0.1099 0.1095 0.1078 0.1066 0.1059 0.1033 0.1020

Induvidual Mode [%] Cumulative Sum [%] UX UY UZ UX UY UZ 0.00 77.60 0.00 0.00 77.60 0.00 79.05 0.00 0.00 79.05 77.60 0.00 0.00 0.00 0.00 79.05 77.60 0.00 0.00 16.16 0.00 79.05 93.75 0.00 15.27 0.00 0.00 94.31 93.75 0.00 0.00 0.00 0.00 94.31 93.75 0.00 0.00 3.58 0.00 94.31 97.33 0.00 3.34 0.00 0.00 97.66 97.33 0.00 0.00 0.00 0.00 97.66 97.33 0.00 0.00 1.29 0.00 97.66 98.62 0.00 0.00 0.00 74.13 97.66 98.62 74.13 1.13 0.00 0.00 98.79 98.62 74.13 0.07 0.00 0.00 98.86 98.62 74.13 0.00 0.11 0.00 98.86 98.72 74.13 0.00 0.32 0.00 98.86 99.04 74.13 0.00 0.00 0.00 98.86 99.04 74.13 0.32 0.00 0.00 99.18 99.04 74.13 0.00 0.00 1.20 99.18 99.04 75.33 0.00 0.00 0.00 99.18 99.04 75.33 0.04 0.00 0.00 99.23 99.04 75.33 0.00 0.24 0.00 99.23 99.28 75.33 0.00 0.00 0.13 99.23 99.28 75.46 0.00 0.07 0.00 99.23 99.35 75.46 0.25 0.00 0.00 99.48 99.35 75.46 0.00 0.00 0.00 99.48 99.35 75.46 0.00 0.00 0.08 99.48 99.35 75.54 0.00 0.01 0.00 99.48 99.37 75.54 0.00 0.00 0.00 99.48 99.37 75.54 0.00 0.17 0.00 99.48 99.53 75.54 0.16 0.00 0.00 99.63 99.53 75.54 0.00 0.00 0.00 99.63 99.53 75.54 0.00 0.10 0.00 99.63 99.63 75.54 0.04 0.00 0.00 99.68 99.63 75.54 0.03 0.00 0.00 99.71 99.63 75.54 0.00 0.00 0.00 99.71 99.63 75.54 0.00 0.00 8.05 99.71 99.63 83.59 0.00 0.00 0.00 99.71 99.63 83.59 0.00 0.09 0.00 99.71 99.72 83.59 0.00 0.00 0.00 99.71 99.72 83.59 0.00 0.00 0.00 99.71 99.72 83.59

Tab. 5-1 modal mass participation model 1

Mode

Period 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

1.5382 1.0285 0.839 0.4423 0.3026 0.2674 0.2208 0.2126 0.1957 0.1859 0.1611 0.1595 0.1437 0.1392 0.1368 0.1364 0.1336 0.1102 0.1049 0.1038 0.1036 0.0989 0.098 0.0956 0.0951 0.0929 0.0917 0.0886 0.086 0.0833 0.0818 0.0814 0.0794 0.0778 0.0775 0.0753 0.0751 0.0743 0.0743 0.0712

Induvidual Mode [%] UX UY UZ 0.2231 70.497 0.0285 65.317 0.2338 0.0046 0.0074 0.0014 0.0001 0.001 15.718 0.0022 0.0032 0.0006 0.0088 21.116 0.0001 0.0046 0 4.8407 2.5498 0.0006 0.5046 62.997 0.0001 0 0.0273 0.3958 0.0138 1.2618 0.0158 0.6106 0.0842 0 0.0186 0.0003 0 0.0513 15.123 0.0179 0.0058 0.0029 0.352 2.1661 0.2933 6.0307 0.1315 0.0097 0.0157 0.0834 0.0099 0.0028 0.006 0.0087 1.8515 0.049 0.0246 0.0135 0.0005 0.0005 0.6205 0.1355 0.0781 0.0497 0.24 0.0008 0.0027 1.2099 0.001 0 0.0003 0.0023 0.0688 0.0111 2.1008 0.303 0.0001 1.7643 0.6321 0 0.2511 0.0018 0.0047 0.507 0.4629 0.0001 0.0014 0.0011 0.0797 0.0112 0.1263 0.0143 0.0302 0.0946 0.0083 0.0206 0 0.0019 0.277 0.1282 0.0001 0.0036 0.0024 0 0.0118 0 0.427 0.4292 0 0.2617 0.007 0.0274 0.1939 0.3701 0.0419 0.1414 0.2961 0.0443 0 0

Cumulative Sum [%] UX UY UZ 0.2231 70.497 0.0285 65.54 70.731 0.0331 65.548 70.732 0.0333 65.549 86.45 0.0355 65.552 86.451 0.0443 86.668 86.451 0.0489 86.668 91.292 2.5987 86.669 91.796 65.596 86.669 91.796 65.623 87.065 91.81 66.885 87.081 92.421 66.969 87.081 92.439 66.97 87.081 92.491 82.092 87.098 92.496 82.095 87.45 94.663 82.389 93.481 94.794 82.398 93.497 94.877 82.408 93.5 94.883 82.417 95.351 94.932 82.441 95.365 94.933 82.442 95.985 95.068 82.52 96.035 95.308 82.521 96.038 96.518 82.522 96.038 96.519 82.524 96.106 96.53 84.625 96.409 96.53 86.389 97.042 96.53 86.64 97.043 96.534 87.147 97.506 96.535 87.149 97.507 96.614 87.16 97.634 96.629 87.19 97.728 96.637 87.211 97.728 96.639 87.488 97.856 96.639 87.491 97.859 96.639 87.503 97.859 97.066 87.932 97.859 97.327 87.939 97.886 97.521 88.309 97.928 97.663 88.605 97.973 97.663 88.605

Tab. 5-2 modal mass participation model 2

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5.3 Used Accelerograms Since the two models are quite different in their modal frequencies I found it adequate to use two equally different target response spectra to create the artificial earthquake records. And since the soil type is the factor that influences the frequency content, I have chosen to create every accelerogram as well for soil type I as for soil type II. The other values to create the spectra have been selected according to the Chilean norm where the Ro = 11, seismic zone 3 and building type A. The maximum acceleration had been set to 0.03g for soil type I records and 0.05g for soil type IV records. For each model 4 different kinds of accelerogram have been created: one using the “summation of harmonic functions” method, one using the “white noise” method, two using the “selective filtration” method and two using the “cut´n copy” method. Since the Chilean norm contains the modal frequency of the mode with the highest modal mass participation in direction of the calculations it became necessary to create two accelerograms for each of these, one for the calculations in X direction and one for those in Y direction. All together there are 24 different accelerograms for each model. However, to be able to compare the results, I used the same accelerograms on both models. Plots of them are shown together with the according spectra in (Tab. 5-3-Tab. 5-8). Two different accelerograms served as input accelerograms for the filtering and the cut´n copy. One was the record of the N10E component at the LLOLLEO station of the earthquake from 03.03.1985 in Chile and the other accelerogram was the N90W component at the SCT1 station of the earthquake from 09.19.1985 in Mexico. These records are quite different in nature to show the capability of the different methods. While the first one has quite a high frequency content the second one has a rather low frequency content. The two spectra and accelerograms of the records can be seen in (Fig. 5-2) and (Fig. 5-3) the differences of the units are due to the fact that the input information was given in different units, this however is not important to the results as has been noted earlier in the explanations of the program RSCA. More earthquake records could be downloaded from http://peer.berkeley.edu/smcat/ , a database that includes most of the freely available strong motion records.

Fig. 5-2 LLOLLEO N10E Chile 03.03.1985

Fig. 5-3 SCT1 N90W Mexico 09.19.1985

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name

Spectra

Accelerogram

cnc1Xa

cnc1Ya

cnc4Xa

cnc4Ya

Tab. 5-3 artificial accelerograms from the Chilean record using the cut’n copy method

Name

Spectra

Accelerogram

cnc1Xb

cnc1Yb

cnc4Xb

cnc4Yb

Tab. 5-4 artificial accelerograms from the Mexican record using the cut’n copy method

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Name

Spectra

Accelerogram

filt1Xa

filt1Ya

filt4Xa

Filt4Ya

Tab. 5-5 artificial accelerograms from the Chilean record using the selective filtering method

Name

Spectra

Accelerogram

filt1Xb

filt1Yb

filt4Xb

Filt4Yb

Tab. 5-6 artificial accelerograms from the Mexican record using the selective filtering method

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Name

Spectra

Accelerogram

Sum1X

Sum1Y

Sum4X

Sum4Y

Tab. 5-7 artificial accelerograms from the Chilean record created using summation of harmonics

Name

Spectra

Accelerogram

Wn1X

Wn1Y

Wn4X

Wn4Y

Tab. 5-8 artificial accelerograms from the Chilean record created using filtration of white noise

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5.4

theoretical comparison (observations)

5.4.1 cut’n copy Although the simple scaling of accelerograms will sometimes achieve quite usable results and will have the least affect on the accelerogram, it will also result in a server change of duration. The cut’n copy method to stretch the accelerogram in time will however have a quite heavy impact on the accelerogram as can be seen in (Tab. 5-3) and (Tab. 5-4). Also the time needed to do such a cut’n copy operation for a real accelerogram exceeds the limits of practical use. Therefore, I would for now consider this method as impractical. On the one hand this is of course due to the nature of the cut’n copy process but on the other hand I am quite sure that a few enhancements of the algorithms could improve this method a lot. For example copying pieces not behind each other but randomly at a position near its original would maintain a more natural look of the accelerogram. Or doing the evaluation of the copying not by checking them randomly but through a comparison of the FFT transformation of each piece would obviously speed up the process. The fitting of the response spectra to the target response spectra that can be achieved with this method depends a lot on the input record and if a bad one is chosen a sufficient fitting cannot be attained (e.g. ‘cnc1Xb’, ‘cnc1Yb’, ‘cnc4Xb’, ‘cnc4Yb’) 5.4.2 Selective filtering This method provided clearly the better results when using existing strong motion records as input accelerograms. However, it has been observed that due to the nature of the filtering process sometimes the waveform character of the accelerogram can suffer. Although I would not expect this to have a severe influence on the results of a calculation this could be counteracted by changing the “first time step” value to a very low setting (below nyquist frequency). This however may result in very large files since the “time step” value should always be higher than the “first time step” value as explained in “4.4.1 Generating a target spectrum”. Other than that this method has been shown to be quite practical and a very fast way to generate artificial accelerograms from an input record. The generated accelerograms maintain the character of the original records as far as can possibly be seen in (Tab. 5-5) and (Tab. 5-6). Though I have not checked what effects this filter has on the frequency shifting during time that occurs in many strong motion records. The fitting of the response spectra to the target response spectra that can be achieved through this method does not depend upon the input record. Though it is obvious that choosing a record whose frequency content is close to the desired frequency content will help to maintain the character of the original record. 5.4.3 Summation of harmonics and white noise The results using one of these two methods appear to be quite the same even though I found the results through filtering of white noise to appear a little more natural. The fitting of the response spectra to the target response spectra that can be achieved are also comparable and should be sufficient for the use of an engineer.

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5.5 Results of the calculations Although the results of the calculations are all close by the variation is still remarkable, especially when the response spectrum could not be fitted well. Nevertheless it is astonishing how close these results vary around the results from the response spectrum analysis. 5.5.1 cnc1Xa cnc1Xb filt1Xa filt1Xb sum1X wn1X

frame structure min. -0.019780 -0.011150 -0.019570 -0.017560 -0.022220 -0.017320

max. abs. average 0.017910 0.018845 0.011640 0.011395 0.016780 0.018175 0.012160 0.014860 0.017530 0.019875 0.017120 0.017220

average -0.017933 0.015523 0.016728 total average 0.017450 spec1X

0.017089

% 108.0 65.3 104.2 85.2 113.9 98.7

cnc1Ya cnc1Yb filt1Ya filt1Yb sum1Y wn1Y

97.9 102.1

average -0.018696 0.018242 0.018469 total average 0.020400 spec1Y

Tab. 5-9 displacement in X for soil type I

cnc4Xa cnc4Xb filt4Xa filt4Xb sum4X wn4X

min. -0.101000 -0.113700 -0.141200 -0.124500 -0.129700 -0.134000

max. abs. average 0.082420 0.091710 0.112400 0.113050 0.127800 0.134500 0.087170 0.105835 0.135700 0.132700 0.122200 0.128100

average -0.124017 0.111282 0.117649 total average 0.126450 spec4X

0.122050

cnc1Xa cnc1Xb filt1Xa filt1Xb sum1X wn1X

0.019434

% 113.8 51.7 80.9 101.0 106.7 89.2

95.0 105.0

Tab. 5-10 displacement in Y for soil type I

cnc4Ya cnc4Yb filt4Ya filt4Yb sum4Y wn4Y

min. -0.143100 -0.122900 -0.138400 -0.161500 -0.163300 -0.136400

max. abs. average 0.142200 0.142650 0.175800 0.149350 0.153800 0.146100 0.128900 0.145200 0.144800 0.154050 0.117600 0.127000

96.4 103.6

average -0.144267 0.143850 0.144058 total average 0.142710 spec4Y

0.143384

% 100.0 104.7 102.4 101.7 107.9 89.0

100.5 99.5

Tab. 5-12 displacement in Y for soil type IV

mixed frame and shear wall structure min. -0.013250 -0.006087 -0.008994 -0.010180 -0.010040 -0.010890

max. abs. average 0.013930 0.013590 0.006164 0.006126 0.009757 0.009376 0.009470 0.009825 0.011970 0.011005 0.008982 0.009936

average -0.009907 0.010046 0.009976 total average 0.010920 spec1X

0.010448

% 124.5 56.1 85.9 90.0 100.8 91.0

cnc1Ya cnc1Yb filt1Ya filt1Yb sum1Y wn1Y

95.5 104.5

average -0.014465 0.014913 0.014689 total average 0.015920 spec1Y

Tab. 5-13 displacement in X for soil type I cnc4Xa cnc4Xb filt4Xa filt4Xb sum4X wn4X

max. abs. average 0.023910 0.023205 0.011230 0.010553 0.016930 0.016495 0.017870 0.020605 0.021310 0.021765 0.018200 0.018190

% 72.5 89.4 106.4 83.7 104.9 101.3

Tab. 5-11 displacement in X for soil type IV

5.5.2

min. -0.022500 -0.009876 -0.016060 -0.023340 -0.022220 -0.018180

min. -0.042410 -0.078260 -0.067150 -0.063570 -0.064280 -0.068180

max. abs. average 0.040400 0.041405 0.074930 0.076595 0.070180 0.068665 0.072560 0.068065 0.073310 0.068795 0.069490 0.068835

average -0.063975 0.066812 0.065393 total average 0.074940 spec4X

0.070167

min. -0.016980 -0.006560 -0.016000 -0.015000 -0.016920 -0.015330

max. abs. average 0.017250 0.017115 0.008679 0.007620 0.015030 0.015515 0.013540 0.014270 0.017520 0.017220 0.017460 0.016395

96.0 104.0

Tab. 5-14 displacement in Y for soil type I min. -0.106400 -0.130700 -0.131900 -0.132800 -0.114200 -0.110300

max. abs. average 0.085560 0.095980 0.134000 0.132350 0.117200 0.124550 0.137600 0.135200 0.119500 0.116850 0.102500 0.106400

% 55.3 102.2 91.6 90.8 91.8 91.9

cnc4Ya cnc4Yb filt4Ya filt4Yb sum4Y wn4Y

93.2 106.8

average -0.121050 0.116060 0.118555 total average 0.124640 spec4Y

Tab. 5-15 displacement in X for soil type IV

0.015305

% 107.5 47.9 97.5 89.6 108.2 103.0

0.121598

% 77.0 106.2 99.9 108.5 93.8 85.4

97.5 102.5

Tab. 5-16 displacement in Y for soil type IV

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5.5.3 Comparison of the results The diagram of (Fig. 5-4) shows the average difference in percent of the results from the response spectrum analysis to the results that could be achieved through time history analysis. 20.0

10.0

0.0

-10.0

-20.0

-30.0

-40.0

-50.0

cut´n copy A

cut´n copy B

filtration A

filtration B

summation

white noise

all

-5.2

-22.1

-3.9

-6.2

3.5

-6.3

soil type 1

13.4

-44.8

-7.9

-8.6

7.4

-4.5

soil type 2

-23.8

0.6

0.1

-3.8

-0.4

-8.1

Fig. 5-4 average variation around the results of the response spectrum analysis

5.6 Conclusions The outcome of my calculations show that when using spectrum compatible accelerograms for time history analysis the results will be close to those of traditional response spectrum analysis as long as a sufficient fitting of the target spectrum could be achieved. Since the results however vary by some 10%, I would recommend doing either a statistically satisfying amount of calculations with different accelerograms fitted to the same target spectrum or introduce fuzzy values to the problem. The observed variation is very likely produced by the oscillating character of a response spectrum, which has been described in “3.5 Fuzzy variables in earthquake loading”. This oscillating character however cannot be suppressed with the algorithms I have used and I would not expect that this is possible without damaging the initial accelerogram by too much.

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6 Appendix 6.1 6.1.1

Maple Worksheets the Duhamel Integral

restart Reading the data of the accelerogram from a file Accfile := fopen( READ, TEXT ); Accinp := readdata( Accfile, 2 ) ; fclose( 0 )

with( plots ) ; polygonplot( Accinp , labels = [ "Time [s]", "Acceleration [cm/s^2]"], labeldirections = [ HORIZONTAL, VERTICAL ], axes = framed )

Obtaining the timestep dt := Accinp − Accinp 2, 1

1, 1

dt := .01000000

Amount of data (timespteps) N := nops( Accinp ) N := 1024

The dampening of the SDOF system (e.g. 5%) d := .5e-1 The mass of the SDOF system (e.g. 1kg) m := 1 Natural period of SDOF system (e.g. 2s) T := 2 Circular frequency 2 evalf( π ) ω := T ω := 3.141592654

,

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Conversion of data AccDur := arraz( 1 .. N ) ; for n to N do AccDurn := op( 2, op( n, Accinp ) ) end do Applying the Duhamel Integral SA := 0; SB := 0; vr := array( 1 .. N ) ; for i from 2 to N do SA := ( SA + AccDuri − 1 cos( ω ( i dt − dt ) ) ) e SB := ( SB + AccDuri − 1 sin( ω ( i dt − dt ) ) ) e vri :=

( −d ω dt )

( −d ω dt )

+ AccDuri cos( ω i dt ) ; + AccDuri sin( ω i dt ) ;

dt ( SA sin( ω i dt ) − SB cos( ω i dt ) ) 2mω

end do ; vr1 := 0

The Duhamel Integral for a SDOF system with the dampening of 5% and a natural period of 2 seconds plot( [ seq( [ n dt , vrn ], n = 1 .. N ) ], color = black, labels = [ "Time [s]", "Velocity [cm/s]"], labeldirections = [ HORIZONTAL, VERTICAL ] )

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6.1.2

Scaling of accelerograms

restart Loading of a external Fortran routine to calculate the spectrum extspekt := define_external( 'SPEC1', AUTO, FORTRAN, N::integer4, NSPC::integer4, TP::float4, FS::float4, DT::float4, XI::float4, ACC::ARRAY( float4 ), T::ARRAY( float4 ), PRV::ARRAY( float4 ), PRA::ARRAY( float4 ), LIB = "C:\\Program Files\\Microsoft Visual Studio\\MyProjects\\spek\\Debug\\spek.dll") Reading from the file , Accfile := fopen( READ, TEXT ); Accinp := readdata( Accfile, 2 ) ; fclose( 0 ) Choosing a time step DT := .1e-1 The dampening of the SDOF systems to 5% XI := .5e-1 Amount of sampling points N := nops( Accinp )

N := 1024

Preparation of the data AccDur := rtable( 1 .. N, datatype = float4 ) ; for n to N do AccDurn := Accinp

end do n, 2

A plot of the initial accelerogram with( plots ) ; listplot ( AccDur, labels = [ "time [s]", "acceleration [cm/s^2]"], labeldirections = [ HORIZONTAL, VERTICAL ], axes = framed )

Amount of sampling points for the response spectrum NSCH := 1000 Highest used period TP := 10 Time value of first sampling point (for example equal the chosen time step)

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FS := DT Array of the time values of the sampling points T := rtable( 1 .. NSCH + 1, datatype = float4 )

Arrays for the response values of the sampling points (velocity and acceleration) PRV := rtable( 1 .. NSCH, datatype = float4 ) ; PRA := rtable( 1 .. NSCH, datatype = float4 )

Generating the time values of the response spectrum (logarithmic deployed)  1   NSCH

  

TP  ; for i to NSCH do Ti + 1 := Ti fakt end do T1 := FS; fakt :=    FS  Alternatively it is possible to use linear deployed time values TP i for i to NSCH do Ti := end do NSCH Calculating and plotting of the initial response spectrum extspekt( N, NSCH, TP, FS, DT, XI, AccDur, T, PRV, PRA ) ; pointplot ( [ seq( [ Tn, PRVn ], n = 1 .. NSCH ) ], color = black, connect = true, labels = [ "period [s]", "response [cm/s]"], labeldirections = [ HORIZONTAL, VERTICAL ], axes = framed )

Generation of the desired target spectrum (the one used here is entirely invented and does not represent any existing code PRVZ := rtable( 1 .. NSCH, datatype = float4 ) ; for i to NSCH do PRVZi := e

( −T .75 ) i

.75 ( Ti − T1 )

1.50

end do

A plot of the initial response spectrum together with the target response spectrum with( plots ) ; display( pointplot ( [ seq( [ Tn, PRVZn ], n = 1 .. NSCH ) ], connect = true ), pointplot ( [ seq( [ Tn, PRVn ], n = 1 .. NSCH ) ], connect = true ), labels = [ "period [s]", "response [cm/s]"], labeldirections = [ HORIZONTAL, VERTICAL ], axes = framed )

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Linear scaling of the accelerogram to better fit the desired target spectrum. Here is the equivalent of the area below the functions (weighted by the deployment of the time values) used to derive a factor to be used for scaling the accelerogram. lfak1 := 0; lfak2 := 0; for i to NSCH do lfak1 := lfak1 + PRVZi ; lfak2 := lfak2 + PRVi end do ; lfak :=

lfak1 lfak2

lfak := 2.056726525

Applying the factor to the accelerogram for i to N do AccDuri := lfak AccDuri end do Calculating and plotting of the new response spectra together with the target response spectra extspekt( N, NSCH, TP, FS, DT, XI, AccDur, T, PRV, PRA ) ; display( pointplot ( [ seq( [ Tn, PRVn ], n = 1 .. NSCH ) ], color = black, connect = true ), pointplot ( [ seq( [ Tn, PRVZn ], n = 1 .. NSCH ) ], connect = true ), labels = [ "period [s]", "response [cm/s]"], labeldirections = [ HORIZONTAL, VERTICAL ], axes = framed )

Area between the functions as a measure of fitting (in percent of the area of the target function)

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- 50 -

divsum1 := 0; sumz := 0; for i to NSCH do divsum1 := divsum1 + PRVZi − PRVi ; sumz := sumz + PRVZi end do ; 100 divsum1 sumz 59.16652150

A procedure to do simple linear fitting, returning the area between the functions in percent of the target function divsum := proc (DTT) local lfak1, lfak2, i, lfak, divsum2 , sumz; global N, NSCH, TP, FS, DT, XI, AccDur, T, PRV, PRA; extspekt( N, NSCH, TP, FS, DT×DTT, XI, AccDur, T, PRV, PRA ) ; lfak1 := 0; lfak2 := 0; for i to NSCH do lfak1 := lfak1 + PRVZ[ i ] ; lfak2 := lfak2 + PRV[ i ] end do ; lfak := lfak1/lfak2; for i to N do AccDur[ i ] := lfak×AccDur[ i ] end do ; extspekt( N, NSCH, TP, FS, DT×DTT, XI, AccDur, T, PRV, PRA ) ; divsum2 := 0; sumz := 0; for i to NSCH do divsum2 := divsum2 + abs( PRVZ[ i ] − PRV[ i ] ) ; sumz := sumz + PRVZ[ i ] end do ; 100×divsum2/sumz end proc Optimizing for the area between the function through linear scaling of the frequency content (by changing the time step). Optimization is achieved by simply scanning for an optimum. dold := 0; DTT := 1; d := divsum( DTT ) ; dold := d; while d ≤ dold do DTTl := DTT ; DTT := DTT 1.2 ; dold := d ; d := divsum( DTT ) DTTo − DTTu end do ; + DTTu; DTTn := 2 DTTo := DTT; dn := divsum( DTTn ) ; DTTu := 0; DTTn_m1 := 0; dold := d; DTTn_p1 := DTTo;

Spectrum Compatible Accelerograms in Earthquake Engineering Universidad de La Serena, Chile / Technische Universität Dresden, Germany

- 51 -

dn_m1 := 1; dn_p1 := 2; while .1 < dn_m1 − dn_p1 or .1 < DTTn_m1 − DTTn_p1 do DTTn − DTTu ; DTTn_m1 := DTTn − 2 DTTo − DTTn ; DTTn_p1 := DTTn + 2 dn_m1 := divsum( DTTn_m1 ) ; dn_p1 := divsum( DTTn_p1 ) ; if dn < dn_m1 then DTTu := DTTn_m1 else DTTn := DTTn_m1 ; dn := divsum( DTTn ) end do end if ; if dn < dn_p1 then DTTo := DTTn_p1 else DTTn := DTTn_p1 ; dn := divsum( DTTn ) end if New optimized time step DT := DTTn DT DT := .04001685120

New response spectrum together with target response spectrum display( pointplot ( [ seq( [ Tn, PRVn ], n = 1 .. NSCH ) ], color = black, connect = true ), pointplot ( [ seq( [ Tn, PRVZn ], n = 1 .. NSCH ) ], color = blue, connect = true ), labels = [ "period [s]", "response [cm/s]"], labeldirections = [ HORIZONTAL, VERTICAL ], axes = framed )

Deriving of a transfer function for the frequency domain   PRVZn  , n = 1 .. NSCH   ; freq_kor :=  seq   PRVn     PRVZn  , n = 1 .. NSCH   freq_kora :=  seq   PRVn   Initial transfer function pointplot( [ seq( [ Tn, freq_koran ], n = 1 .. NSCH ) ], color = black, connect = true,

labels = [ "period [s]", "factor of modulation [cm/s]"], labeldirections = [ HORIZONTAL, VERTICAL ], axes = framed )

Spectrum Compatible Accelerograms in Earthquake Engineering Universidad de La Serena, Chile / Technische Universität Dresden, Germany

- 52 -

Smoothening of the transfer function to prevent greater disturbances in the frequency content dw := 50; dw2 := 0; for i2 to 2 do dw3 := 0; korn := array( 1 .. NSCH ) ; dw if i < NSCH − then 2 for i to NSCH do dw1 := 0; dw dw1 for t from i to i + do dw1 := dw1 + freq_kort end do ; dw1 := 2 dw 2 end if ; dw if < i then 2 dw dw2 for t from i − to i do dw2 := dw2 + freq_kort end do ; dw2 := 2 dw 2 end if ; dw dw2 if i ≤ then for t to i do dw2 := dw2 + freq_kort end do ; dw2 := 2 i end if ; dw ≤ i then if NSCH − 2 for t from i to NSCH do dw1 := dw1 + freq_kort end do ; dw1 :=

dw1 NSCH + 1 − i

end if ; korni :=

dw1 + dw2 2

for i to NSCH do freq_kori := korni end do end do

end do ; freq_kor := array( 1 .. NSCH ) ; The smoothed transfer function together with the initial one

Spectrum Compatible Accelerograms in Earthquake Engineering Universidad de La Serena, Chile / Technische Universität Dresden, Germany

- 53 -

display( pointplot ( [ seq( [ Tn, freq_korn ], n = 1 .. NSCH ) ], color = black, connect = true, thickness = 3 ), pointplot ( [ seq( [ Tn, freq_koran ], n = 1 .. NSCH ) ], color = blue, connect = true ), labels = [ "period [s]", "factor of modulation"], labeldirections = [ HORIZONTAL, VERTICAL ], axes = framed )

Determination of the next higher power of 2 as amount of the sampling points for the FFT ( trunc( log ( NSCH ) ) + 1 ) NSCH2 := 2

2

; mFFT := trunc( log2( NSCH ) ) + 1

NSCH2 := 1024 mFFT := 10

One array for the real and one for the imaginary part of the accelerogram (FFT is performed with complex numbers) AccRe := array( 1 .. NSCH2 ) ; AccIm := array( 1 .. NSCH2 ) Transferring the accelerogram to complex numbers. The imaginary part is set to 0 and the real part of the sampling points exceeding NSCH is also set to 0 for n to NSCH2 do if n ≤ N then AccRen := AccDurn else AccRen := 0 end if ; AccImn := 0 end do Applying the FFT (transformation of the accelerogram in the frequency domain) FFT( mFFT, AccRe, AccIm ) Calculation of the amplitude spectrum from the complex result of the FFT up to the nyquist-frequency which is at NSCH/2 2 2 n NSCH2   , AccRen + AccImn , n = 1 .. Ampf :=  seq    2    DT N  The amplitude spectrum over frequency (highest frequency depends on the sampling rate) plot( Ampf, axes = framed, labels = [ "frequency [1/s]", "amplitude"], labeldirections = [ HORIZONTAL, VERTICAL ], axes = framed, color = black )

Spectrum Compatible Accelerograms in Earthquake Engineering Universidad de La Serena, Chile / Technische Universität Dresden, Germany

- 54 -

Calculation of the phase spectrum from the complex result of the FFT up to the nyquistfrequency Pha := [ seq( arctan( AccImn, AccRen ), n = 1 .. NSCH2 ) ] Phase spectrum listplot ( Pha, labels = [ "sampling point", "phase"], labeldirections = [ HORIZONTAL, VERTICAL ], axes = framed, connect = false )

Generation of the transfer function over the period (and frequency) for further calculations in the frequency domain korT := [ seq( [ Tn, freq_korn ], n = 1 .. NSCH ) ] ; 1 korF :=  seq  , freq_korn , n = 1 .. NSCH      Tn       Transferring the amplitude spectrum so it is over the period rather than over frequency 2 2 1 NSCH2   , AccRen + AccImn , n = 1 .. ampT :=  seq    ;    n  2     DT N       2 2 n NSCH2   ; , AccRen + AccImn , n = 1 .. ampF :=  seq    2     DT N ampT_TP :=  seq  1 , AccRe 2 + AccIm 2 , n = trunc DT N  + 1 .. NSCH2      TP  n n     n    2      DT N       Amplitude spectrum over period pointplot( ampT_TP, axes = framed, connect = true, labels = [ "period [s]", "amplitude"], labeldirections = [ HORIZONTAL, VERTICAL ], axes = framed )

Spectrum Compatible Accelerograms in Earthquake Engineering Universidad de La Serena, Chile / Technische Universität Dresden, Germany

- 55 -

Sorting of korT for interpolation on ampT korT_sort := [ seq( korTNSCH − n + 1, n = 1 .. NSCH ) ] Sorting the transfer function NSCH2  korfak := array 1 ..  ; 2  i2o := 1; NSCH2 for i1 to do 2 for i2 from i2o to NSCH − 1 while ampTi1, 1 < korT_sorti2, 1 do end do ; end do i2o := i2; k2 := korT_sorti2, 2 ; korfaki1 := k2

Data processing and a plot of the used transfer function over the period NSCH2  for i to trunc  do ampi := ampTi, 2 end do ; 2   NSCH2   pointplot   seq [ ampTn, 1, korfakn ], n = 1 ..  , color = black, connect = true, 2    labels = [ "frequency [1/s]", "factor of modulation"], labeldirections = [ HORIZONTAL, VERTICAL ], axes = framed  

Applying the transfer function on the amplitude spectrum

Spectrum Compatible Accelerograms in Earthquake Engineering Universidad de La Serena, Chile / Technische Universität Dresden, Germany

- 56 -

NSCH2   ampn := rtable 1 .. trunc , datatype = float4  ; 2     NSCH2  for i to trunc  do ampni := ampTi, 2 korfaki end do 2   The new and the old amplitude spectrum over the period display  NSCH2    pointplot   seq [ ampTn, 1, ampnn ], n = 1 ..  , color = blue, connect = true , 2     NSCH2  , color = red, connect = true , pointplot   seq [ ampTn, 1, ampn ], n = 1 ..    2     labels = [ "period [s]", "amplitude"], labeldirections = [ HORIZONTAL, VERTICAL ], axes = framed  

The new amplitude spectrum over the frequency listplot ( ampn, labels = [ "frequency [1/s]", "amplitude"], labeldirections = [ HORIZONTAL, VERTICAL ], axes = framed )

Transformation of the new amplitude spectrum and the old phase spectrum into complex numbers for inverse FFT AccnRe := rtable( 1 .. NSCH2, datatype = float4 ) ; AccnIm := rtable( 1 .. NSCH2, datatype = float4 ) ; NSCH2  for i to trunc  do AccnRei := ampni cos( Phai ) ; AccnImi := ampni sin( Phai ) 2   end do Mirroring of the data above the nyquist frequency

Spectrum Compatible Accelerograms in Earthquake Engineering Universidad de La Serena, Chile / Technische Universität Dresden, Germany

- 57 -

NSCH2  for i from trunc  to NSCH2 − 1 do 2   AccnRei := ampnNSCH2 − i cos( Phai ) ; AccnImi := ampnNSCH2 − i sin( Phai ) end do Inverse transformation into the time domain iFFT( mFFT, AccnRe, AccnIm ) Calculation and plot of the resulting spectrum together with the target spectrum extspekt( N, NSCH, TP, FS, DT, XI, AccnRe, T, PRV, PRA ) ; display( pointplot ( [ seq( [ Tn, PRVn ], n = 1 .. NSCH ) ], color = black, connect = true ), pointplot ( [ seq( [ Tn, PRVZn ], n = 1 .. NSCH ) ], color = red, connect = true ), labels = [ "period [s]", "response [cm/s]"], labeldirections = [ HORIZONTAL, VERTICAL ], axes = framed )

Data processing for further iteration processes for i to N do AccDuri := AccnRei end do The new accelerogram listplot ( AccDur, labels = [ "time [s]", "acceleration [cm/s^2]"], labeldirections = [ HORIZONTAL, VERTICAL ], axes = framed )

Spectrum Compatible Accelerograms in Earthquake Engineering Universidad de La Serena, Chile / Technische Universität Dresden, Germany

- 58 -

6.2 6.2.1

Selected Passages of Source code from the Program RSCA generate spectrum (RSCA_gen_spectrum.f90 spek.for duhamel.for)

‘RSCA_gen_spectrum.f90’: ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !

Copyright (C) 2003 Marko Thiele, [email protected] This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with this program; if not, write to the Free Software Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA.

SUBROUTINE RSCA_gen_spectrum(dlg,id, callbacktype ) use dflogm use RSCAGlobals implicit none include 'resource.fd' TYPE (dialog) dlg LOGICAL lret integer id, callbacktype if (callbacktype == dlg_clicked) then ! values from the dialog call RSCA_read_values(dlg,id,callbacktype) call RSCA_gen_target_spectrum(dlg,id,callbacktype) ! has sampling point array been used ? IF (.NOT. ALLOCATED(spectruma)) ALLOCATE(spectruma(1:g_specsamplpoints)) IF (.NOT. ALLOCATED(spectrumv)) ALLOCATE(spectrumv(1:g_specsamplpoints)) ! in case the amount of sampling points has been changed DEALLOCATE(spectruma) DEALLOCATE(spectrumv) ALLOCATE(spectruma(1:g_specsamplpoints)) ALLOCATE(spectrumv(1:g_specsamplpoints)) call SPEC1(g_accpoints,g_specsamplpoints,g_highperiod,g_timestep, g_dampening,inputaccelerogram,targetspectrumtime,spectrumv,spectruma) if (g_chousesmoothspec==.TRUE.) call RSCA_smoothcurve(g_specsamplpoints, spectruma,g_smoothspec1,g_smoothspec2) endif call RSCA_draw_spectrum_and_targetspectrum(dlg,id,callbacktype,spectruma) END SUBROUTINE RSCA_gen_spectrum

Spectrum Compatible Accelerograms in Earthquake Engineering Universidad de La Serena, Chile / Technische Universität Dresden, Germany

- 59 ‘spek.for’: SUBROUTINE SPEC1(N,NSPC,TP,DT,XI,ACC,T,PRV,PAA) DIMENSION ACC(N),PRV(NSPC),PAA(NSPC),T(NSPC) REAL MAX DO 25 I=1,NSPC W=6.2831853/T(I) if (T(I)==0) W=6.2831853/T(I+1) CALL DUHAMEL(N,ACC,TAU,DT,W,XI,SV,SA,TMAX) PRV(I)=SV PAA(I)=SA 25 CONTINUE RETURN END

‘duhamel.for’: SUBROUTINE DUHAMEL(N,Y,TAU1,DTAU,OMEGA,ETA,SV,SA,TMAX) DIMENSION Y(N),ERG(N) REAL KONST1,KONST2,KOEF B=OMEGA*SQRT(1.-ETA*ETA) A=ETA*OMEGA AB2=2.*A*B KONST1=1./(OMEGA*OMEGA) KONST2=A*A-B*B ERG(1)=0 TAUB=TAU1*b AUX1=SIN(TAUB) S=AUX1 C=COS(TAUB) SUA=0 SUB=0 SV=0 SA=0 SR=KONST2/B CR=2.*A NN=N-1 TAU=TAU1 DO 20 I=1,NN I1=I+1 TAU=TAU+DTAU TAUB=TAU*B C1=COS(TAUB) S1=SIN(TAUB) AUX2=S1 11 DY=Y(I1)-Y(I) KOEF=DY/DTAU*KONST1 E=EXP(-A*DTAU) AA=Y(I1)*(A*C1+B*S1)-Y(I)*E*(A*C+B*S)#KOEF*(KONST2*C1+AB2*S1-E*(KONST2*C+AB2*S)) S=-C S1=-C1 C=AUX1 C1=AUX2 BB=Y(I1)*(A*C1+B*S1)-Y(I)*E*(A*C+B*S)#KOEF*(KONST2*C1+AB2*S1-E*(KONST2*C+AB2*S)) SUA=SUA*E+AA SUB=SUB*E+BB DUH=(SUA*AUX2+SUB*S1)*KONST1 DUC=(SUA*AUX2-SUB*S1)*KONST1 IF(ABS(DUH).LE.SV)GO TO 15 SV=ABS(DUH) TMAX=TAU 15 SAI=SR*DUH+CR*DUC ERG(I1)=-SAI IF(ABS(SAI).GT.SA) SA=ABS(SAI) AUX1=AUX2 S=AUX2 C=-S1 20 CONTINUE RETURN END

Spectrum Compatible Accelerograms in Earthquake Engineering Universidad de La Serena, Chile / Technische Universität Dresden, Germany

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6.2.2 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !

search optimum (RSCA_linear_scaling_of_frequency_montecarlo.f90)

Copyright (C) 2003 Marko Thiele, [email protected] This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with this program; if not, write to the Free Software Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA.

SUBROUTINE RSCA_linear_scaling_of_frequency_montecarlo(dlg,id, callbacktype ) use dflogm use dflib use RSCAGlobals implicit none include 'resource.fd' TYPE (dialog) dlg LOGICAL lret integer id, callbacktype,n,n2 REAL maxfakamp,maxfakfrq,minarea,fakamp,fakfrq,res,minfakamp,minfakfrq,ran REAL spectrum(g_specsamplpoints) REAL montwin,montwinfak,mont,montsen,minmontamp,maxmontamp,minmontfrq,maxmontfrq REAL allowminamp,allowminfrq,allowmaxamp,allowmaxfrq,montstop CHARACTER(256) text if (callbacktype == dlg_clicked) then ! values from the dialog call RSCA_read_values(dlg,id,callbacktype) ! !

search for optimum with evolutionary modified Monte Carlo simulation first window for Monte Carlo +- (has to be less than 1)

montwin=0.75 !

factor to reduce window if result is not at border

montwinfak=0.5 !

amount of simulations for each window

mont=1000 !

border sensitivitiy in %

montsen=0.75 !

end of iteration if window size is less than

montstop=0.1 DO WHILE(montwin>montstop) maxfakamp=2*montwin maxfakfrq=2*montwin minarea=1000000000 minmontamp=1 maxmontamp=1 minmontfrq=1 maxmontfrq=1

Spectrum Compatible Accelerograms in Earthquake Engineering Universidad de La Serena, Chile / Technische Universität Dresden, Germany

- 61 DO 21,n=1,mont ! transfer the initial spectrum to the spectrum for calculations DO 11, n2=1, g_specsamplpoints 11 spectrum(n2)=spectruma(n2) !

generate random numbers for calculations CALL RANDOM(ran) fakamp=maxfakamp*ran+(1-montwin) CALL RANDOM(ran) fakfrq=maxfakfrq*ran+(1-montwin)

!

do linear escalation with both numbers call scalespec(fakamp,fakfrq,spectrum)

!

obtain difference area between functions call diffarea(res,spectrum)

!

determine min and max factors for search if (fakamp < minmontamp) minmontamp=fakamp if (fakamp > maxmontamp) maxmontamp=fakamp if (fakfrq < minmontfrq) minmontfrq=fakfrq if (fakfrq > maxmontfrq) maxmontfrq=fakfrq

if (minarea>res) then minarea=res minfakamp=fakamp minfakfrq=fakfrq ! 21

draw new spectrum call RSCA_draw_spectrum_and_targetspectrum(dlg,id,callbacktype,spectrum) end if

! determine if border or not (if border than do no reduction of window size) ! allowed factors allowminamp=1-(1-minmontamp)*montsen allowminfrq=1-(1-minmontfrq)*montsen allowmaxamp=1+(maxmontamp-1)*montsen allowmaxfrq=1+(maxmontfrq-1)*montsen if ((minfakamp>allowminamp).AND.(maxfakampallowminfrq).AND. (maxfakfrq
Spectrum Compatible Accelerograms in Earthquake Engineering Universidad de La Serena, Chile / Technische Universität Dresden, Germany

- 62 SUBROUTINE diffarea(res,spectrum) use dflogm use RSCAGlobals REAL res,maxt,lfak1,lfak2,f11,f21,f12,f22,t12,t11,t22,t21,fl1,fl2,dfl REAL spectrum(g_specsamplpoints) !

limit area to be weighted maxt=1 DO WHILE((targetspectrumtime(maxt) .LE. g_highperiodcal).AND.(maxt.LT.g_specsamplpoints)) maxt=maxt+1 END DO lfak1=0 lfak2=0 ! calculating the area IF (g_chouselogweight==.FALSE.) THEN DO 41, n=1,maxt-1 f11=targetspectrum(n) f21=targetspectrum(n+1) t11=targetspectrumtime(n) t21=targetspectrumtime(n+1) fl1=((f11+f21)/2)*(t21-t11) f12=spectrum(n) f22=spectrum(n+1) t12=targetspectrumtime(n) t22=targetspectrumtime(n+1) fl2=((f12+f22)/2)*(t22-t12) dfl=abs(fl2-fl1)

41

51

lfak1=lfak1+dfl lfak2=lfak2+fl1 ELSE DO 51, n=1,maxt lfak1=lfak1+abs(targetspectrum(n)-spectrum(n)) lfak2=lfak2+targetspectrum(n) END IF res=100*lfak1/lfak2;

END SUBROUTINE diffarea

Spectrum Compatible Accelerograms in Earthquake Engineering Universidad de La Serena, Chile / Technische Universität Dresden, Germany

- 63 SUBROUTINE scalespec(fakamp,fakfrq,spectrum) use dflogm use RSCAGlobals REAL res,fakamp,fakfrq REAL spectrum(g_specsamplpoints),newspectrum(g_specsamplpoints), newtime(g_specsamplpoints) !

apply the factor for amplitude

61

DO 61, n=1, g_specsamplpoints spectrum(n)=spectrum(n)*fakamp

! !

81

now apply the factor of frequency (a little bit harder :( ) new time table DO 81, n=1, g_specsamplpoints newtime(n)=targetspectrumtime(n)*fakfrq CONTINUE

!

projecting the spectrum with the new timetable on the old timetable DO 71, n=1, g_specsamplpoints-1 tn1=newtime(n) tn2=newtime(n+1) fn1=spectrum(n) fn2=spectrum(n+1) ! find time points of old timetable between this one and project new value n2=1 DO WHILE ((targetspectrumtime(n2) .LE. tn2) .AND. (N2 .LE. g_specsamplpoints)) if (targetspectrumtime(n2) .GE. tn1) then timep=targetspectrumtime(n2) newspectrum(n2)=(((fn2-fn1)*(timep-tn1))/(tn2-tn1))+fn1 end if n2=n2+1 END DO 71 CONTINUE

91

DO 91, n=1, g_specsamplpoints spectrum(n)=newspectrum(n)

END SUBROUTINE scalespec

Spectrum Compatible Accelerograms in Earthquake Engineering Universidad de La Serena, Chile / Technische Universität Dresden, Germany

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6.2.3 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !

selective filtering (RSCA_apply_selectiv_filtering_FFTn.f90)

Copyright (C) 2003 Marko Thiele, [email protected] This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with this program; if not, write to the Free Software Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA.

SUBROUTINE RSCA_apply_selectiv_filtering(dlg,id, callbacktype ) use dflogm use RSCAGlobals implicit none include 'resource.fd' TYPE (dialog) dlg LOGICAL lret integer id, callbacktype,n,n2,mFFT REAL re,im,re2,im2,timefrq,frequenc,period,v,v1,v2,p1,p2,corfak,maxt CHARACTER(256) text REAL, ALLOCATABLE :: FFTphases(:),FFTamplitudes(:),frqtargetspectrum(:),frqspectrum(:) REAL, ALLOCATABLE :: intens1(:),intens2(:) COMPLEX, ALLOCATABLE :: F(:),WORK(:) if (callbacktype == dlg_clicked) then !

values from the dialog call RSCA_read_values(dlg,id,callbacktype)

!

generate arrays IF (.NOT. ALLOCATED(F)) ALLOCATE(F(0:g_accpoints-1)) IF (.NOT. ALLOCATED(WORK)) ALLOCATE(WORK(0:g_accpoints-1)) IF (.NOT. ALLOCATED(FFTphases)) ALLOCATE(FFTphases(0:g_accpoints-1)) IF (.NOT. ALLOCATED(FFTamplitudes)) ALLOCATE(FFTamplitudes(0:g_accpoints-1)) IF (.NOT. ALLOCATED(sumfak)) ALLOCATE(sumfak(1:g_specsamplpoints)) IF (.NOT. ALLOCATED(frqtargetspectrum)) ALLOCATE(frqtargetspectrum(1:g_accpoints/2)) IF (.NOT. ALLOCATED(frqspectrum)) ALLOCATE(frqspectrum(1:g_accpoints/2)) IF (.NOT. ALLOCATED(intens1)) ALLOCATE(intens1(1:g_accpoints)) IF (.NOT. ALLOCATED(intens2)) ALLOCATE(intens2(1:g_accpoints)) DEALLOCATE(F) DEALLOCATE(WORK) DEALLOCATE(FFTphases) DEALLOCATE(FFTamplitudes) DEALLOCATE(sumfak) DEALLOCATE(frqtargetspectrum) DEALLOCATE(frqspectrum) DEALLOCATE(intens1) DEALLOCATE(intens2) ALLOCATE(F(0:g_accpoints-1)) ALLOCATE(WORK(0:g_accpoints-1)) ALLOCATE(FFTphases(0:g_accpoints-1)) ALLOCATE(FFTamplitudes(0:g_accpoints-1)) ALLOCATE(sumfak(1:g_specsamplpoints)) ALLOCATE(frqtargetspectrum(1:g_accpoints/2)) ALLOCATE(frqspectrum(1:g_accpoints/2)) ALLOCATE(intens1(1:g_accpoints)) ALLOCATE(intens2(1:g_accpoints)) !

derive intensity function of old accelerogram if (g_chousecorint==.TRUE.) then

Spectrum Compatible Accelerograms in Earthquake Engineering Universidad de La Serena, Chile / Technische Universität Dresden, Germany

- 65 DO 9,n=1,g_accpoints intens1(n)=abs(outputaccelerogram(n)) CONTINUE

9

call RSCA_smoothcurve(g_accpoints,intens1,g_smoothacc1,g_smoothacc2) end if !

transmit accelerogram to new complex accelerogram

10

DO 10,n=0,g_accpoints-1 re=real(outputaccelerogram(n+1)) F(n)=cmplx(re,0.0) CONTINUE

! fast Fourier transformation call SPCFFT(F,g_accpoints,0,WORK,1.0) !

20 !

30

generate phase spectrum DO 20,n=0,g_accpoints-1 FFTphases(n)=ATAN2(imag(F(n)),real(F(n))) CONTINUE generate amplitude spectrum DO 30,n=0,g_accpoints-1 re=real(F(n)) im=imag(F(n)) FFTamplitudes(n)=sqrt(re**2+im**2) CONTINUE

!

derive highest period maxt=1 DO WHILE((targetspectrumtime(maxt) .LE. g_highperiodcal).AND.(maxt.LT.g_specsamplpoints)) maxt=maxt+1 END DO ! transfer target spectrum and initial spectrum to frequency target spectrum and initial spectrum DO 40,n=1,g_accpoints/2 timefrq=n*g_timestep frequenc=n/(g_timestep*g_accpoints) period=(g_timestep*g_accpoints)/n ! search for period in spectrum timetable and approximate a value for the frequency spectra if ((period <= targetspectrumtime(2)).OR.(period >= targetspectrumtime(g_specsamplpoints))) then if (period <= targetspectrumtime(2)) then frqspectrum(n)=spectruma(g_specsamplpoints) frqtargetspectrum(n)=targetspectrum(g_specsamplpoints) frqtargetspectrum(n)=0 else frqspectrum(n)=spectruma(1) frqtargetspectrum(n)=targetspectrum(1) frqtargetspectrum(n)=0 end if else DO 50,n2=1,g_specsamplpoints-1 p1=targetspectrumtime(n2) p2=targetspectrumtime(n2+1) if ((p1 < period).AND.(period <= p2)) then v1=targetspectrum(n2) v2=targetspectrum(n2+1) v=((v2-v1)*(period-p1)/(p2-p1))+v1 frqtargetspectrum(n)=v v1=spectruma(n2) v2=spectruma(n2+1) v=((v2-v1)*(period-p1)/(p2-p1))+v1 frqspectrum(n)=v end if CONTINUE

50 end if 40

CONTINUE

Spectrum Compatible Accelerograms in Earthquake Engineering Universidad de La Serena, Chile / Technische Universität Dresden, Germany

- 66 !

derive and apply corrector factors DO 60, n=0, g_accpoints/2-1 corfak=frqtargetspectrum(n+1)/frqspectrum(n+1) FFTamplitudes(n)=FFTamplitudes(n)*corfak FFTamplitudes(g_accpoints-n-1)=FFTamplitudes(g_accpoints-n-1)*corfak

60

CONTINUE

!

retransfer to complex array and mirroring above nyquist frequency DO 70, n=1, g_accpoints re=FFTamplitudes(n-1)*cos(FFTphases(n-1)) im=FFTamplitudes(n-1)*sin(FFTphases(n-1)) F(n-1)=CMPLX(re,im)

70

CONTINUE

! inverse fast Fourier transformation call SPCFFT(F,g_accpoints,1,WORK,1.0) !

80 !

89

extract real part of array to outputaccelerogram DO 80, n=0, g_accpoints-1 outputaccelerogram(n+1)=real(F(n)) CONTINUE derive intensity function of accelerogram if (g_chousecorint==.TRUE.) then DO 89,n=1,g_accpoints intens2(n)=abs(outputaccelerogram(n)) CONTINUE call RSCA_smoothcurve(g_accpoints,intens2,g_smoothacc1,g_smoothacc2) end if

!

90

apply corrected intensity function if (g_chousecorint==.TRUE.) then DO 90, n=1, g_accpoints outputaccelerogram(n)=outputaccelerogram(n)*((intens1(n))/(intens2(n))) CONTINUE end if call RSCA_linear_scaling_of_amplitude(dlg,id, callbacktype )

!

apply max acceleration if (g_chousemaxacc==.TRUE.) then call RSCA_apply_maximum_acceleration(g_accpoints,outputaccelerogram) ! generate the new spectrum call SPEC1(g_accpoints,g_specsamplpoints,g_highperiod,g_timestep,g_dampening,outputaccelerogr am,targetspectrumtime,spectrumv,spectruma) if (g_chousesmoothspec==.TRUE.) call RSCA_smoothcurve(g_specsamplpoints,spectruma,g_smoothspec1,g_smoothspec2) ! draw the new spectrum together with the target spectrum call RSCA_draw_spectrum_and_targetspectrum(dlg,id,callbacktype,spectruma) ! hate this but we have to do the whole correction thing all over again ! derive intensity function of accelerogram if (g_chousecorint==.TRUE.) then DO 1589,n=1,g_accpoints intens2(n)=abs(outputaccelerogram(n)) 1589 CONTINUE call RSCA_smoothcurve(g_accpoints,intens2,g_smoothacc1,g_smoothacc2) end if ! apply corrected intensity function if (g_chousecorint==.TRUE.) then DO 1590, n=1, g_accpoints

1590

outputaccelerogram(n)=outputaccelerogram(n)*((intens1(n))/(intens2(n))) CONTINUE end if end if end if

END SUBROUTINE RSCA_apply_selectiv_filtering

Spectrum Compatible Accelerograms in Earthquake Engineering Universidad de La Serena, Chile / Technische Universität Dresden, Germany

- 67 -

6.2.4 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !

cut’n copy (RSCA_alter_duration_cut_n_copy.f90)

Copyright (C) 2003 Marko Thiele, [email protected] This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with this program; if not, write to the Free Software Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA.

SUBROUTINE RSCA_alter_duration_cut_n_copy(dlg,id, callbacktype ) use dflogm use DFPORT use RSCAGlobals implicit none include 'resource.fd' TYPE (dialog) dlg LOGICAL lret integer id ,callbacktype, opt_samplpoints, act_samplpoints,act_samplpoints2,p,n,lmax,lmin,l integer flag1,tries,piece,c,c2,t,l1,l2,l3,diff_piece,n2,accpoints_before,tryes,tryes2,t2,av_copy ,old_accpoints,y CHARACTER(256) text real old,res,diff_min,percent,time_start,time_act,time_left REAL, ALLOCATABLE :: piece_data(:,:),accelerogram(:) REAL, ALLOCATABLE :: intens1(:),intens2(:) INTEGER, ALLOCATABLE :: piece_start(:),piece_length(:),piece_copy1(:),piece_copy2(:) INTEGER, ALLOCATABLE :: piece_erase1(:),piece_try(:),piece_erase2(:) ! resetting the random number generator using the momentary system time as pseudo random number integer seed real rnd character(8) seedtext character(6) seedtext2 call time(seedtext) seedtext2(1:1)=seedtext(8:8) seedtext2(2:2)=seedtext(7:7) seedtext2(3:3)=seedtext(5:5) seedtext2(4:4)=seedtext(4:4) seedtext2(5:5)=seedtext(2:2) seedtext2(6:6)=seedtext(1:1) READ (seedtext2, *) seed rnd=ran(seed) if (callbacktype == dlg_clicked) then !

!

1109

generate arrays for intensity functions IF (.NOT. ALLOCATED(intens1)) ALLOCATE(intens1(1:g_accpoints)) IF (.NOT. ALLOCATED(intens2)) ALLOCATE(intens2(1:g_accpoints)) DEALLOCATE(intens1) DEALLOCATE(intens2) ALLOCATE(intens1(1:g_accpoints)) ALLOCATE(intens2(1:g_accpoints)) derive intensity function of accelerogram if (g_chousecorint==.TRUE.) then DO 1109,n=1,g_accpoints intens1(n)=abs(outputaccelerogram(n)) CONTINUE call RSCA_smoothcurve(g_accpoints,intens1,g_smoothacc1,g_smoothacc2)

Spectrum Compatible Accelerograms in Earthquake Engineering Universidad de La Serena, Chile / Technische Universität Dresden, Germany

- 68 end if old_accpoints=g_accpoints !

values from the dialog call RSCA_read_values(dlg,id,callbacktype) accpoints_before=g_accpoints

!

determine new amount of sampling points needed for the new accelerogram (optimum) opt_samplpoints=int(g_duration/g_timestep)

!

counting the full periods (two zero passings) and determine the longest one old=outputaccelerogram(1) p=0 l=0 lmax=0 lmin=g_accpoints flag1=0 DO 10, n=1, g_accpoints if ( sign(1.0,outputaccelerogram(n)) .NE. sign(1.0,old)) then if (flag1 .NE. 0) then p=p+1 if (l>=lmax) lmax=l if (l<=lmin) lmin=l l=0 flag1=-1 end if flag1=flag1+1

10

end if l=l+1 old=outputaccelerogram(n) CONTINUE

!

generating the arrays necessary ALLOCATE(piece_start(0:p)) ALLOCATE(piece_length(0:p)) ALLOCATE(piece_erase1(0:p)) ALLOCATE(piece_erase2(0:p)) ALLOCATE(piece_copy1(0:p)) ALLOCATE(piece_copy2(0:p)) ALLOCATE(piece_try(0:p)) ALLOCATE(piece_data(0:lmax,0:p))

!

cut the accelerogram in pieces old=outputaccelerogram(1) p=0 l=0 flag1=0 piece_start(0:p)=1 piece_data(0:lmax,0:p)=0 DO 20, n=1, g_accpoints pice_data(l,p)=outputaccelerogram(n) if ( sign(1.0,outputaccelerogram(n)) .NE. sign(1.0,old)) then if (flag1 .NE. 0) then piece_length(p)=l piece_start(p+1)=n p=p+1 l=0 flag1=-1 end if flag1=flag1+1 end if piece_data(l,p)=outputaccelerogram(n) l=l+1 old=outputaccelerogram(n) CONTINUE piece_length(p)=l act_samplpoints=g_accpoints

!

20

if (opt_samplpoints>g_accpoints) then !===================================================================================== !==========================NEW ACCELEROGRAM IS LONGER================================= !===================================================================================== !

set all copy flags .TRUE.

Spectrum Compatible Accelerograms in Earthquake Engineering Universidad de La Serena, Chile / Technische Universität Dresden, Germany

- 69 piece_try(0:p)=.TRUE. piece_copy1(0:p)=1 piece_copy2(0:p)=1 time_start=TIMEF() tryes=g_tryes if (tryes>=p) tryes=p ! calculate average amount of times that copying of each piece is necessary av_copy=int(opt_samplpoints/act_samplpoints)+1 DO WHILE (act_samplpoints=av_copy) tryes2=tryes2-1 610 CONTINUE if (tryes2<=tryes) tryes=tryes2 DO 503,n2=1,tryes piece_copy2=piece_copy1 ! select randomly one piece that has not been erased or tried for erasing piece=int(ran(seed)*p) DO WHILE((piece_copy2(piece)>=av_copy).OR.(piece_try(piece)==.FALSE.)) piece=int(ran(seed)*p) END DO ! set the copy flag for this one to FALSE piece_copy2(piece)=piece_copy2(piece)+1 piece_try(piece)=.FALSE. ! reassemble accelerogram act_samplpoints2=act_samplpoints+piece_length(piece) IF (.NOT. ALLOCATED(accelerogram)) ALLOCATE(accelerogram(1:g_accpoints)) DEALLOCATE(accelerogram) ALLOCATE(accelerogram(1:act_samplpoints2)) c2=0 c=0 DO 304, n=0,p DO 405, t2=1, piece_copy2(n) DO 404, t=1,piece_length(n) c=c+1 accelerogram(c)=outputaccelerogram(c2+t) 404 CONTINUE 405 CONTINUE c2=c2+piece_length(n) 304 CONTINUE ! regenerate array for new intensity function IF (.NOT. ALLOCATED(intens2)) ALLOCATE(intens2(1:act_samplpoints2)) DEALLOCATE(intens2) ALLOCATE(intens2(1:act_samplpoints2)) ! derive intensity function of accelerogram if (g_chousecorint==.TRUE.) then DO 1289,n=1,act_samplpoints2 intens2(n)=abs(accelerogram(n)) 1289 CONTINUE call RSCA_smoothcurve(act_samplpoints2,intens2,g_smoothacc1,g_smoothacc2) end if ! apply corrected intensity funktion if (g_chousecorint==.TRUE.) then DO 1290, n=1, old_accpoints-1 y=int(n*act_samplpoints2/old_accpoints)+1 accelerogram(y)=accelerogram(y)*((intens1(n))/(intens2(y))) CONTINUE end if ! calculate a new response spectrum call SPEC1(act_samplpoints2,g_specsamplpoints,g_highperiod,g_timestep,g_dampening,accelerogra m,targetspectrumtime,spectrumv,spectruma) if (g_chousesmoothspec==.TRUE.) call RSCA_smoothcurve(g_specsamplpoints,spectruma,g_smoothspec1,g_smoothspec2) ! calculate the difference area (subroutine is in RSCA_linear_scaling_of_frequency_montecarlo.f90) 1290

Spectrum Compatible Accelerograms in Earthquake Engineering Universidad de La Serena, Chile / Technische Universität Dresden, Germany

- 70 call diffarea(res,spectruma) if (res
set all copy flags .TRUE. piece_try(0:p)=.TRUE. piece_erase1(0:p)=.TRUE. time_start=TIMEF() tryes=g_tryes tryes2=p if (tryes>p) tryes=p+1 DO WHILE (act_samplpoints>opt_samplpoints) ! do (g_tryes) tries (or less if no more pieces are available) and chouse the one that effects the response spectrum the least or enhances it diff_min=100 piece_try(0:p)=.TRUE. if (tryes2<=tryes) tryes=tryes-1 tryes2=tryes2-1 DO 50,n2=1,tryes piece_erase2=piece_erase1 ! select randomly one piece that has not been erased or tried for erasing piece=int(ran(seed)*p) DO WHILE((piece_erase2(piece)==.FALSE.).OR.(piece_try(piece)==.FALSE.)) piece=int(ran(seed)*p) END DO ! set the copy flag for this one to FALSE piece_erase2(piece)=.FALSE. piece_try(piece)=.FALSE. ! reassemble accelerogram act_samplpoints2=act_samplpoints-piece_length(piece)

Spectrum Compatible Accelerograms in Earthquake Engineering Universidad de La Serena, Chile / Technische Universität Dresden, Germany

- 71 IF (.NOT. ALLOCATED(accelerogram)) ALLOCATE(accelerogram(1:g_accpoints)) DEALLOCATE(accelerogram) ALLOCATE(accelerogram(1:act_samplpoints2)) c2=0 c=0 DO 30, n=0,p DO 40, t=0,piece_length(n)-1 c2=c2+1 IF (piece_erase2(n)==.TRUE.) then c=c+1 accelerogram(c)=outputaccelerogram(c2) END IF 40 CONTINUE 30 CONTINUE ! regenerate array for new intensity function IF (.NOT. ALLOCATED(intens2)) ALLOCATE(intens2(1:act_samplpoints2)) DEALLOCATE(intens2) ALLOCATE(intens2(1:act_samplpoints2)) ! derive intensity function of accelerogram if (g_chousecorint==.TRUE.) then DO 1189,n=1,act_samplpoints2 intens2(n)=abs(accelerogram(n)) 1189 CONTINUE call RSCA_smoothcurve(act_samplpoints2,intens2,g_smoothacc1,g_smoothacc2) end if ! apply corrected intensity function if (g_chousecorint==.TRUE.) then DO 1190, n=1, old_accpoints-1 y=int(n*act_samplpoints2/old_accpoints)+1 accelerogram(y)=accelerogram(y)*((intens1(n))/(intens2(y))) CONTINUE end if ! calculate a new response spectrum call SPEC1(act_samplpoints2,g_specsamplpoints,g_highperiod,g_timestep,g_dampening,accelerogra m,targetspectrumtime,spectrumv,spectruma) if (g_chousesmoothspec==.TRUE.) call RSCA_smoothcurve(g_specsamplpoints,spectruma,g_smoothspec1,g_smoothspec2) ! calculate the difference area (subroutine is in RSCA_linear_scaling_of_frequency_montecarlo.f90) call diffarea(res,spectruma) if (res
Spectrum Compatible Accelerograms in Earthquake Engineering Universidad de La Serena, Chile / Technische Universität Dresden, Germany

- 72 call RSCA_draw_spectrum_and_targetspectrum(dlg,id,callbacktype,spectruma) calculate percentage of progress percent=(100*real(accpoints_beforeact_samplpoints)/real(accpoints_before-opt_samplpoints)) WRITE (text, '(F10.2, A)') percent, "[%] of calculation done" call RSCA_draw_text(text,50,20,13) ! calculate the time left for completion time_left=(((100*(TIMEF()-time_start))/percent)-(TIMEF()-time_start))/60 !

WRITE (text, '(F10.1, A)') time_left, "[min] left until completion" call RSCA_draw_text(text,50,32,13) END DO !===================================================================================== !===================================================================================== !===================================================================================== end if !

transfer the accelerogram g_accpoints=act_samplpoints DEALLOCATE(outputaccelerogram) ALLOCATE(outputaccelerogram(1:g_accpoints)) DO 70, n=1, g_accpoints outputaccelerogram(n)=accelerogram(n) 70 CONTINUE ! apply max acceleration if (g_chousemaxacc==.TRUE.) then call RSCA_apply_maximum_acceleration(g_accpoints,outputaccelerogram) ! do the correction thing again ! regenerate array for new intensity function IF (.NOT. ALLOCATED(intens2)) ALLOCATE(intens2(1:g_accpoints)) DEALLOCATE(intens2) ALLOCATE(intens2(1:g_accpoints)) ! derive intensity function of accelerogram if (g_chousecorint==.TRUE.) then DO 11189,n=1,g_accpoints intens2(n)=abs(outputaccelerogram(n)) 11189 CONTINUE call RSCA_smoothcurve(g_accpoints,intens2,g_smoothacc1,g_smoothacc2) end if ! apply corrected intensity function if (g_chousecorint==.TRUE.) then DO 11190, n=1, old_accpoints-1 y=int(n*g_accpoints/old_accpoints)+1

11190

!

accelerogram(y)=accelerogram(y)*((intens1(n))/(intens2(y))) CONTINUE end if end if write values to the dialog WRITE (text,*) g_accpoints lret = DlgSet( dlg, IDC_EDIT_ACCPOINTS, TRIM(ADJUSTL(text)) )

endif !

call RSCA_linear_scaling_of_amplitude(dlg,id, callbacktype ) call RSCA_draw_accelerogram(g_accpoints,outputaccelerogram) ! generate the new spectrum call SPEC1(g_accpoints,g_specsamplpoints,g_highperiod,g_timestep,g_dampening,outputaccelerogr am,targetspectrumtime,spectrumv,spectruma) if (g_chousesmoothspec==.TRUE.) call RSCA_smoothcurve(g_specsamplpoints,spectruma,g_smoothspec1,g_smoothspec2) ! draw the new spectrum together with the target spectrum call RSCA_draw_spectrum_and_targetspectrum(dlg,id,callbacktype,spectruma)

END SUBROUTINE RSCA_alter_duration_cut_n_copy

Spectrum Compatible Accelerograms in Earthquake Engineering Universidad de La Serena, Chile / Technische Universität Dresden, Germany

- 73 -

6.3 6.3.1

floor plans of the models

frame structure

Spectrum Compatible Accelerograms in Earthquake Engineering Universidad de La Serena, Chile / Technische Universität Dresden, Germany

- 74 -

6.3.2

mixed frame and shear wall structure

Spectrum Compatible Accelerograms in Earthquake Engineering Universidad de La Serena, Chile / Technische Universität Dresden, Germany

- 75 -

6.4 [1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12] [13]

[14]

Literature Ray W. Clough, Joseph Penzien: Dynamics of Structures -International Student Edition, McGraw-Hill, Kogakusha, 1975 Bruce A. Bolt: Seismic Strong Motion Synthetics -Computational Techniques, Academic Press, 1987 G. W. Houser: Behavior of Structures during Earthquakes, Proc. ASCE, vol. 85, no. EM-4, October, 1959 K. Meskouris, W.B. Krätzig, A. Elanas, L. Heiny, I.F. Meyer: Mikrocomputerunterstützte Erdbebenuntersuchung von Tragwerken, SFB 151-Berichte Nr. 8, Februar 1988 Alex H. Barbat, Juan Miquel Canet: Estructuras Sometidas a Acciones Sísmicas, Cálculo por ordenador Segunda Edición, Centro Internacional de Métodos Numéricos en Ingeniería, Barcelona Artes Gráficas Torres, S.A., 1994 G. W. Housner: Properties of Strong Ground Motion Earthquakes, Bull. Seismol. Soc. Am., vol. 45, no. 3, pp. 197-218, July 1955 E. Rosenblueth/ J. Bustamante: Distribution of Structural Response to Earthquakes, Proc. Pap. 3173, J. Eng. Mech. Div. ASCE, vol. 88, no. EM3, pp. 75-106, June 1962 G. N. Bycroft: White Noise Representation of Earthquakes, Proc. Pap. 2434, J. Eng. Mech. Div. ASCE, vol. 86, no. EM2, pp. 1-16, April 1960 W. T. Thomson: Spectral Aspect of Earthquakes, Bull. Seismol. Soc. Am., vol. 49, pp. 91-98, 1959 Asadour H. Hadjian: Scaling of Earthquake Accelerograms-A Simplified Approach, Journal of the Structural Division, ASCE, Vol. 98, No. 2, February 1972, pp. 547-551 J. A. Glassman: A Generalization of the Fast Fourier Transform IEEE Transactions on Computers February, 1970 Visual Fortran: Programmer's Guide http://www.compaq.com/fortran/docs/index.html Norma Chilena Oficial: NCh 433.Of96 Diseño sísmico de edificios (earthquake resistant design buildings) Primera edición: 1996 H.H. Bothe: Fuzzy Logic, Einführung in Theorie und Anwendung Springer Verlag, 1993

Spectrum Compatible Accelerograms in Earthquake Engineering Universidad de La Serena, Chile / Technische Universität Dresden, Germany

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[15]

[16] [17]

[18]

[19]

[20]

[21]

[22]

[23]

[24]

Bernd Möller, Wolfgang Graf, Song Ha Nguyen: Modelling the life-cycle of a structure using fuzzy processes The Ninth International Conference on Computing in Civil and Building Engineering, April 3-5, 2002, Taipei, Taiwan Bernd Möller, Wolfgang Graf, Michael Beer, Wigand Stransky: Seismische Trakwerksanalyse mit Fuzzy-Grössen S. Zhang: Verfahren zur Ermittlung der Erdbebenlasten mit Berücksichtigung des stochastischen Charakters des Bebens Disertation: Bericht Nr.:I-4(1998), Institut für Mechanik (Bauwesen) Lehrstuhl I Stuttgart 1998 Martin Zsohar: Stochastische Grössen der Resonanzfrequenzen und der Verstärkung seismischer Wellen im horizontal geschichtetn zufälligen Medium Disertation: Technische Universität Dresden, Fakultät Bauingenieurwesen Dresden 1998 Joaquín Monge Espiñeira: El Sismo de Marzo 1985, Chile Acero Comercial S.A., Empresa del Grupo CAP Bronstein, Semendijajew, Musiol, Mühlig DeskTop Taschenbuch der Mathematik Verlag Harri Deutsch AG Lawrence J. Christiano, Terry J. Fitzgerald: The Band Pass Filter Working Paper 7257, http://www.nber.org/papers/w7257 National Bureau of Economic Research 1999, Cambridge I.D.Gupta, R.G.Joshi: Response Spectrum Superposition for Structures with Uncertain Properties Jornal of Engineering Mechanics, March 2001, pp233-241 Manish Shrikhande, Vinay K. Gupta: Synthesizing Ensembles of Spatial Correlated Accelerograms Jornal of Engineering Mechanics, November 1998, pp1185-1192 James M. Nau, William J. Hall: Scaling Methods for Earthquake Response Spectra Journal of Structural Engineering, Vol. 110, No. 7, July, 1984, pp1533-1548