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Investigation of the Site Effect on Seismic Ground Motions Using Elasto-Plastic Constitutive Model
MSc of geotechnical engineering, Mazandaran University, Babol, Iran Hadi15012002@yahoo.com and Associate professor, department of civil engineering, |
ABSTRACT
The seismic response of a site depends strongly on its characteristics. This issue is considered in 2800 Iran code by classifying sites to 4 groups and 4 design spectra have presented. It suggests the use of shear wave velocity of the site as a means for characterizing different soil types. The common method in earthquake engineering for modeling wave transmission in layered sites and dynamic soil-structure interaction is "equivalent-linear" method. Since this method is widely used, and the fully nonlinear method embodied in FLAC is not, in this study pointing out some of the differences between the two methods’ results in site effect estimations. The equivalent linear method uses linear properties for each element that remain constant throughout the history of shaking. Also, the interference and mixing phenomena that occur between different frequency components in a nonlinear material are missing from an equivalent linear analysis. During plastic flow, it is commonly accepted that the strain-increment tensor is related to some function of the stress tensor, giving rise to the "flow rule" in plasticity theory. However, elasticity theory (as used by the equivalent-linear method) relates the strain tensor (not increments) to the stress tensor. Plastic yielding, therefore, is modeled somewhat inappropriately. Whereas, A proper plasticity formulation is used in all the built-in models, whereby plastic strain increments are related to stresses. Because of the wide use of equivalent linear method and these deficiencies, it is necessary to compare the site response estimated by this method and fully nonlinear method. In this paper, the site response is estimated by elasto-plastic constitutive model.
Keywords: fully nonlinear method; equivalent linear method; site effect; dynamic analysis; response spectra; FLAC; Iran 2800 code.
INTRODUCTION
It is widely observed that earthquakes generate different damages depending on the sites where buildings are located. In fact, the earthquake damage pattern of buildings located on a sediment filled site is consistently higher if compared with that exhibited by similar buildings located on rock sites even when there is no substantial difference in the location of the source of the earthquake and in the propagation path of the earthquake. Such effect is well known among the seismologists as a site effect and it appears to be linked to the propagation of seismic waves in the shallow layers of the earth crust. In other words, the seismic waves are amplified, with respect to those propagating through hard rocks, in certain frequency ranges, if they propagate into sediments [12].
ANALITICAL METHODS OF SITE EFFECT INVESTIGATION
In fact, what is investigated, as site effect is the local amplification of the seismic waves that can differ from one site to another [17].
The issue of the effects of site on seismic waves has found out by researchers in early part of 20th century [18]. They assessed changes of characteristics of waves by studying effects of earthquake at different parts of the site and different local stratification. This investigation has continued up to now. After happening earthquakes in all around of the world, large amount of researches are carried out to know how a site affects arrival seismic waves to bedrock [18]. Importance of site effects, mainly, find out in Caracas 1960 earthquake by observing different patterns of damages at buildings. After Mexico City 1985 earthquake, more attention paid to this issue. Mexico city has built on a valley of soft alluvial at hundred kilometers distance from an active fault where reported damages in far fields from fault was higher with respect to near fields.
In weak earthquakes, because of the low intense of motions, strains in deposit layers are small, thus soil behavior, mainly, remain in linear range. In this situation, magnification factor of the peak acceleration of bedrock will be several times more than the magnification factor of strong motion that causes soil to show nonlinear behavior [2]. Increasing of the peak acceleration of bedrock changes the dynamic period of deposits. Therefore, peak acceleration of bedrock has great influence on seismic response of the site [15]. When intensity of motion increases the shear strain level, nonlinear behavior of deposit causes to decrease shear modulus and increase the damping ratio. In this case, maximum values on magnification curve changes, furthermore, location of the dynamic periods that peak values have occurred, change in comparison with initial magnification curve [9].
Frequency content of the earthquake is an important factor in site effect investigations. Frequency content of earthquake, introduces by predominate period of the acceleration, indicate that most of the power of input waves is in what frequency range. Frequency characteristics of surface seismic motion and its strength are function of frequency characteristics of magnification curve of deposit and bedrock motion. Resonance will occur if predominate period of bedrock seismic motion be equal or near to natural period of site and, consequently, amplitude of motions on surface will increase considerably in this frequency range [6].
Dynamic response of site can be investigated using different kinds of experimental techniques or numerical methods [17].
Using microtremors [14] and spectral ratio techniques [11] are the most import experimental methods. In recent years, the use of microtremore has increased among the seismological society as an alternative way to estimate the fundamental frequencies of a given site. After the technique was made popular by Nakamura there has been an intensive line of research on the effectiveness of this method to determine site effects.
The calculation of the site effects using ambient noise followed the same procedure as that for the earthquakes. The signal recorded previous to the P-wave arrival time was used [14].
Spectral ratio techniques included horizontal to vertical spectral ratio (HVSR) and sediment to bedrock spectral ratio. The H/V spectral ratio technique based on recording of ambient noise with only one three-component station to calculate horizontal to vertical spectral ratios relatively not time consuming and is inexpensive. Several experimentations confirm the good agreement between classical techniques and H/V spectral ratio technique in obtaining the fundamental soil frequency, although questions are still unsolved about the validity of the ground motion amplification obtained by this technique [11].
Numerical methods are second group in analytical methods. For estimating dynamic response, soil layers are expressed with mathematical model. Appropriate model is depending on the geometric condition and soil characteristics. By using a suitable constitutive model, reaction of soil to earthquake motion is estimated. Numerical methods include mass-spring model, one dimensional thin layers model, and finite element model have developed. Based upon each numerical model, some software is available. FLAC [7] uses mass-spring model, SHAKE [16] and EERA [4] use thin layers model and plaxis [5] uses finite element method. Also, depend on strain level, suitable model must be chosen. These models include linear model, equivalent linear model and nonlinear model [13].
When expected strains are small, using elastic-linear model is reasonable. When the strains are at medium level, soil behaves elasto-plasticaly. In this case, soil behavior can be expressed by equivalent linear model based on visco-elastic conception. For large strain level (more than 10-2) one needs to a composite method to specify stress-strain variation in loading, unloading and reloading process [13].
The "equivalent-linear" method is common in geotechnical earthquake engineering for modeling wave transmission in layered sites and dynamic soil-structure interaction [1, 8, 10, 19, 20]. Since this method is widely used, and the fully nonlinear method embodied in FLAC is not, in this study pointing out some of the differences between the two methods.
The equivalent-linear method is distinguished by the following characteristics [7]:
This method uses linear properties for each element that remain constant throughout the history of shaking and are estimated from the mean level of dynamic motion. During quiet periods in the excitation history, elements will be over-damped and too soft; during strong shaking, elements will be underdamped and too stiff. However, there is a spatial variation in properties that corresponds to different levels of motion at different locations.
Also, the interference and mixing phenomena that occur between different frequency components in a nonlinear material are missing from an equivalent linear analysis.
This method does not directly provide information on irreversible displacements and the permanent changes that accompany liquefaction, since oscillatory motion only is modeled. These effects may be estimated empirically, however.
During plastic flow, it is commonly accepted that the strain-increment tensor is related to some function of the stress tensor, giving rise to the "flow rule" in plasticity theory. However, elasticity theory (as used by the equivalent-linear method) relates the strain tensor (not increments) to the stress tensor. Plastic yielding, therefore, is modeled somewhat inappropriately.
Finally, the material constitutive model is built into the method: it consists of a stress-strain curve in the shape of an ellipse. Although this pre-choice relieves the user of the need to make any decisions, the flexibility to substitute alternative shapes is removed. However, the effects of a different shape to the curve are partially allowed for by the iteration procedure used in the method. It should be pointed out that a frequency-independent hysteresis curve in the form of an ellipse is physically impossible, since the continuous change in slope prior to reversal implies pre-knowledge (and rate information is not available to the model because the model is defined as being rate-independent).
In contrast, characteristics of the fully nonlinear method should be compared to the corresponding points listed above [7]:
Fully nonlinear method follows any prescribed nonlinear constitutive relation. If a hysteretic-type model is used and no extra damping is specified, then the damping and tangent modulus are appropriate to the level of excitation at each point in time and space, since these parameters are embodied in the constitutive model.
Using a nonlinear material law, interference and mixing of different frequency components occur naturally. Irreversible displacements and other permanent changes are modeled automatically. A proper plasticity formulation is used in all the built-in models, whereby plastic strain increments are related to stresses and the effects of using different constitutive models may be studied easily.
Since the equivalent linear method is widely used in site effect investigations and fully nonlinear method is not, it is worth pointing out some of the differences between response spectra calculated by these methods.
Site Characteristics and Modeling Circumstances
In most of the codes including 2800 code of Iran, site effect has considered. The 2800 Code of Iran suggests the development of response spectra based on four various soil types. It suggests the use of shear wave velocity of the site as a means for characterizing different soil types. This measure takes into account the various layer thicknesses up to a depth of 30 meters [3].
In this study, the effects of soil layer at depth of 30 meters on the seismic response of a site are studied. Dynamic analysis is performed by FLAC (Fast Lagrangian Analysis of Continua) which is software that capable to modeling wave transmission and dynamic analysis by fully nonlinear method. The constitutive model that is used in this study is Mohr-Colomb plasticity.
For applying tectonic circumstances of near, middle and far fields, 24 different ground strong motions are chosen that each field has eight accelerogram. For missing the effect of soil layers, it is necessary that these motions had been recorded on stiff layers. All of the accelerograms that are used in this study recorded on stiff soil. These accelerograms are recorded during real earthquakes and in this study have applied to the bedrock. In selection of earthquakes, has tried to choose accelerograms that cover wide range of frequencies. Specifications of selected accelerograms in this study are as Tables 1 to 3.
Table 1. Specifications of strong motions used for near field analyses
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| Distance to | Magnitude | Country | Station | Number |
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| fault(km) | (M) | Country | Station | Number |
| 12 | 7.1 | IRAN | Rudbar-Tarom | Accg.1 |
| 10 | 5 | IRAN | Naghan | Accg.2 |
| 7 | 6.1 | IRAN | Naghan | Accg.3 |
| 12 | 7.3 | IRAN | Tabas-e-Golshan | Accg.4 |
| 23.5 | 6.6 | USA | Sanfernando | Accg.5 |
| 22.7 | 6.7 | USA | Northridge | Accg.6 |
| 36.1 | 6.7 | USA | Northridge | Accg.7 |
| 32 | 6.8 | IRAN | Abbar | Accg.8 |
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In Table 1 collection of near field accelerograms used in this study has shown. In this table, the fault distance to place that the earthquakes has recorded is from 0-40 km.
Table 2. Specifications of strong motions used for middle field analyses
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| Distance to | Magnitude | Country | Station | Number |
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| fault(km) | (M) | Country | Station | Number |
| 12 | 7.1 | IRAN | Rudbar-Tarom | Accg.1 |
| 10 | 5 | IRAN | Naghan | Accg.2 |
| 7 | 6.1 | IRAN | Naghan | Accg.3 |
| 12 | 7.3 | IRAN | Tabas-e-Golshan | Accg.4 |
| 23.5 | 6.6 | USA | Sanfernando | Accg.5 |
| 22.7 | 6.7 | USA | Northridge | Accg.6 |
| 36.1 | 6.7 | USA | Northridge | Accg.7 |
| 32 | 6.8 | IRAN | Abbar | Accg.8 |
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In Table 2 and 3, collections of middle field and far field earthquakes have shown, respectively. The fault distance to place that the earthquakes have recorded is from 40-70 and 70-110 km for middle and far field earthquakes, respectively. On the other hand, the analyses have classified in four groups base upon the Iran 2800 code.
Iran 2800 code has classified soils based on shear wave velocity in them to four classes. In analyses of this study, site classification of Iran 2800 code has employed with the aim to summarize analyses in particular groups to utilize results in engineering works.
Table 3. Specifications of strong motions used for far field analyses
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| Distance to | Magnitude | Country | Station | Number |
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| fault(km) | (M) | Country | Station | Number |
| 70 | 7 | IRAN | Geshm Island | Accg.1 |
| 55 | 6.6 | IRAN | Ghaen | Accg.2 |
| 61 | 6.8 | IRAN | Khezri | Accg.3 |
| 50 | 5.9 | IRAN | Maharloo | Accg.4 |
| 57.6 | 6 | USA | N.Palm Springs | Accg.5 |
| 41 | 6.5 | USA | FUC0226 | Accg.6 |
| 58 | 6.3 | USA | FUC02244 | Accg.7 |
| 58 | 6.3 | USA | FUC02245 | Accg.8 |
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In this study, dynamic analyses has performed on standard homogen sites that their thickness were 30 meters and soil layers were on top of the bedrock at depth of 30 meters. Specific weight of soils were 20 kN/m3 and the underground water level has neglected. Shear wave velocity in site type(1) is 800 m/s, soil type (2) 550m/s, soil type (3) 275m/s and soil type (4) 150m/s.
In this study, eight accelerograms have applied to each site in each field and then the response spectra have calculated. Then, these response spectra normalized and the average of eight response spectra in each field for each site reported. Findings are then compared to the normalized response spectra estimated by equivalent linear method and 2800 code of Iran.
Comparison of Fully Nonlinear Method, Equivalent Linear Method and 2800 Code of Iran Response Spectra
This section is concentrated on the site response of the performed analyses on different types of soils. The normalized response spectra estimated by fully nonlinear method is presented and compared to response spectra of equivalent linear method and 2800 code of Iran.
Site type 1
In Figure 1 normalized response spectra of site type 1 estimated by fully nonlinear and equivalent linear methods and 2800 code of Iran have shown.
Because of the large stiffness of site type 1, its natural period, in comparison with other site types is small. As be seen, natural period estimated by fully nonlinear method is 0.16 sec.
Figure 1a. Comparison of normalized response spectra of site type 1- near field
The natural period calculated by fully nonlinear method for site type 1 is equal in all three fields (Fig. 1A, 1B and 1C). On the other hand, equivalent linear method has estimated different natural periods for this site in near, middle and far fields and therefore with changes in frequency content of dynamic loads, scatter values for natural period has estimated.
Figure 1b. Comparison of normalized response spectra of site type 1– middle field
In resonance period, calculated values by equivalent linear method in all three fields with respect to fully nonlinear method are underestimated and consequently unsafe.
In other periodic domains, equivalent linear method’s results are overestimated in comparison with fully nonlinear method in all three fields.
Because fully nonlinear method in estimating of spectra uses mass-spring model and equivalent linear method uses thin layer model, moreover, fully nonlinear method works in time domain and equivalent linear method in frequency domain, observed that, two method’s results have some differences.
Figure 1c. Comparison of normalized response spectra of site type 1– far field
In all periods, outside resonance period of site, fully nonlinear method’s spectra with respect to 2800 code of Iran, is in allowable domain of code. In resonance period of site, 2800 code of Iran spectra is underestimated and consequently unsafe.
Site type 2
In figure (2), normalized response spectra of site type 1 estimated by equivalent linear and fully nonlinear methods and 2800 code of Iran have shown.
Figure 2a. Comparison of normalized response spectra of site type 2– near field
As be seen, in figure (2A) there is two maximum points in both method’s spectra. Second maximum point related to the initial natural period of site and the first related to the
Figure 2b. Comparison of normalized response spectra of site type 2– middle field
resonance period of site after changing in shear modulus because of nonlinear behavior of site. When strains left elastic domain, shear modulus decreases and consequently natural period of the site changes. Figure (3) shows stress-strain curve of site type 2 under near field earthquakes during nonlinear analysis. changing of shear modulus is shown in this figure. As be seen in figure (2A) this behavior of soil has modeled by two methods but there is not exact correlation in their results.
Figure 2c. Comparison of normalized response spectra of site type 2– far field
After densification of site and changing shear modulus, estimated resonance period by fully nonlinear method is 0.07 sec and by equivalent linear method 0.1 sec. In near field there is good correlation between two method’s spectra in periods higher than 1.04 sec but not exact correlation in lower periods. In near field analyses, equivalent linear method’s results are overestimated with respect to fully nonlinear method’s but the resonance amplitude calculated by fully nonlinear method is higher. Normalized response spectra of middle and far field of site type 2 are shown in figures (2B) and (2C). As be seen, fully nonlinear method has estimated same natural period in all three fields for site type 2 but by equivalent linear method scatter values has reported for three fields. In middle and far field analyses, equivalent linear spectra is over estimated with respect to the fully nonlinear spectra but in resonance period, values estimated by fully nonlinear method are higher and in this case equivalent linear method’s results aren’t safe.
Figure 3. Stress- strain curve during nonlinear analysis under near field earthquakes-site type 2
In all three fields and in whole periodic domain without resonance period, fully nonlinear spectra, is in allowable domain of 2800 code of Iran.
In resonance period of site, 2800 code of Iran is underestimated and consequently unsafe.
Site type 3
In figure (4), normalized response spectra of site type 1 estimated by equivalent linear and fully nonlinear methods and 2800 code of Iran have shown.
At near and middle fields, the power of applied earthquakes are enough high to cause site behaves elasto-plastically. Existing of two maximum point at near and middle field’s spectra indicates this issue. Second maximum point related to initial natural period of site and the first related to the resonance period of site after changing in shear modulus.
Figure 4a. Comparison of normalized response spectra of site type 3– near field
Figure (5) shows stress-strain curve of site type 3 under near field earthquakes during nonlinear analysis. This figure shows that because of the nonlinear behavior of site under dynamic loads, the shear modulus has decreased.
Figure (4A) shows that fully nonlinear method has estimated initial natural period of site type 3, 0.45 sec where equivalent linear method has estimated 0.41 sec. there is suitable correlation between two method’s results.
Figure 4b. Comparison of normalized response spectra of site type 3– middle field
When the site under applied loads behaved nonlinearly, its natural period changes. In this case, natural period estimated by fully nonlinear method is 0.14 sec and by equivalent linear method 0.20 sec. with respect to the estimated near values in elastic domain, differences growth as system leave elastic domain.
As be seen, near field spectra in figure (4A) at periods higher than 0.60 sec and far field spectra at periods higher than 1.00 sec, by two methods, have good correlation.
Figure 4c. Comparison of normalized response spectra of site type 3– far field
Figure (4c) shows that by both methods, near values have estimated for initial resonance period of site type 3, but by departure from elastic domain, differences between two method’s spectra growth so that after changing of shear modulus, resonance period estimated by fully nonlinear method became 0.14 sec and by equivalent linear method 0.2 sec. Also, stress-strain curve depicted during nonlinear analysis under middle field earthquakes has shown in figure (6). This curve indicates that the site under middle field earthquakes has showed nonlinear behavior and the shear modulus has decreased.
In far field (fig (4c), two method’s spectra have high
Figure 5. Stress- strain curve during nonlinear analysis under near field earthquakes-site type 3
correlation and the results are very near. Existing of one maximum point indicate that site has behaved elastically. Stress-strain curve (fig (7)) verifies that the strains have remained in small strain level and shear modulus didn’t have changed during analysis.
Figure 6. Stress- strain curve during nonlinear analysis under middle field earthquakes-site type 3
Using appropriate curve for relating shear modulus and damping ratio to strain in equivalent linear method analyses for site type 3 caused that the estimated response in this case be reasonable and consequently correlation between two method’s spectra be more than other sites.
Figure 7. Stress- strain curve during nonlinear analysis under far field earthquakes-site type 3
In all three fields (figure (4A), (4B) and (4C)) and in whole periodic domains without resonance period, fully nonlinear spectra are in allowable domain of 2800 code of Iran.
In resonance period of site, 2800 code of Iran is underestimated and consequently unsafe.
Site type 4
In figure (8), normalized response spectra of site type 4 estimated by equivalent linear, fully nonlinear methods and 2800 code of Iran have shown.
Site 4 is the softest site that has investigated in this research. As be seen in figures (8A), (8B) and (8C), different maximum points in spectra indicate that the site has showed nonlinear behavior.
Figure 8a. Comparison of normalized response spectra of site type 4 – near field
By fully nonlinear method, initial natural period of site has estimated 0.80 sec in all three fields but equivalent linear method has estimated 0.8 sec for near field (fig (8A)), 1sec for middle field (fig (8B)) and 0.9 sec for far field (fig (8C)). It was seen that by changing in period content of dynamic loads, scatter results has reported by equivalent linear method.
Figure 8b. Comparison of normalized response spectra of site type 4 – middle field
In figure (8A) existing of several maximum points indicate that the site has showed nonlinear behavior. Referring to stress-strain curve depicted during nonlinear analysis under near field earthquakes (fig (9)) verify that the shear modulus has decreased. Also, the strain level and several times changing of shear modulus indicates that the site has showed degraded hystersis behavior. Two method’s spectra have good nearness in near field analysis.
In middle field, differences between two method’s spectra are growth. It seem that by increasing the frequency content of dynamic loads, the equivalent linear method’s precise decrease because, in spit of calculation of 0.8 sec for initial period of site in near field, different value has given in middle field. Existing of two maximum point in spectra in middle field indicate that nonlinear behavior has occurred. By referring to stress-strain curve (fig (10)) that depicted during nonlinear analysis under middle field earthquakes can see that the shear modulus has decreased.
Figure 8c. Comparison of normalized response spectra of site type 4 – far field
In far field, figure (8C), shows that the site type 4 has exhibited nonlinear behavior under far field earthquakes, too. Stress-strain curve (fig (11)) depicted in this case, verify this issue. In this case, far field spectra of two methods have exact correlation at periods higher than 1 sec but in lower periods and especially in resonance period of site, there is some differences between spectra.
Figure 9. Stress- strain curve during nonlinear analysis under near field earthquakes-site type 4
In spite of the fact that equivalent linear method has
Figure 10. Stress- strain curve during nonlinear analysis under middle field earthquakes-site type 4
modeled nonlinear behavior of site but there are differences between equivalent linear and fully nonlinear Method’s spectra. Two method’s correlation at near field in periods higher than 1 sec is good but at middle field, differences are more.
Figure 11. Stress- strain curve during nonlinear analysis under far field earthquakes-site type 4
In all three fields and in whole periodic domain without resonance period, fully nonlinear spectra, is in allowable domain of 2800 code of Iran.
The resonance period of the site is underestimated by 2800 code of Iran, which is consequently found to be on the unsafe side.
CONCLUSIONS
1. At all sites investigated in this study, site response was sensitive to distance to fault. In other words, changes in frequency content and acceleration level of the earthquake loads affect on the results of equivalent linear method. By increasing distance from fault differences between fully nonlinear and equivalent linear method’s results growth.
2. Equivalent linear method’s results are very sensitive to the relating curve of shear modulus and damping ratio to strain level. Changes in earthquake characteristics at near, middle and far fields and applying them to site in time domain causes to different stress path and using one relating curve for a site at different fields in dynamic analyses, cause unstably in equivalent linear method’s results but fully nonlinear method by using constitutive model and applying dynamic load in time domain, decrease this issues effect on estimating correct results. Effect of this issue on amplitude of spectra in resonance period is very important.
3. Nonlinear behavior of site outside elastic strains has modeled by two methods but there are differences between results by increasing strain level. Also, at middle and large strains, by increasing distance from fault, differences between results more increase.
4. Estimated response spectra by two methods in sites type 1, 2, 3 and 4 at near field in higher periods, have suitable correlation. In lower periods, in spite of nearness of estimated natural periods, differences between results have seen. The closest results were in near field spectra with respect to other fields.
5. Estimated response spectra by two methods in all sites studied in this paper at middle field analyses, have less correlation in whole periods. By investigating of expected specifications, was seen that equivalent linear method’s results at middle field analyses had the most differences with respect to fully nonlinear method.
6. Estimated response spectra by two methods in all sites studied in this paper at far field analyses have good correlation in higher periods. In lower periods differences between site 1 and 2 responses was great but at site type 3 there was good correlation between two method’s results. At site type 4, in spite of nearness of estimated natural periods, is seen that after nonlinear response of site, differences between results of two methods growth.
7. By comparison the fully nonlinear method and 2800 Iran code spectra is seen that in all three fields and in whole periodic domain without resonance period, fully nonlinear spectra are in allowable domain of 2800 code of Iran. The 2800 code of Iran underestimated the resonance period of the site, and is consequently unsafe.
ACKNOWLEDGMENT
The authos wish to thank Dr Janalizade for his cooperation in preparing of this paper.
REFERENCES
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