tanks_731.pdf

ASIAN JOURNAL OF CIVIL ENGINEERING (BUILDING AND HOUSING) VOL. 12, NO. 6 (2011) PAGES 731-750

NONLINEAR ANALYSIS OF ABOVEGROUND ANCHORED STEEL TANKS USING ENDURANCE TIME METHOD

M. Alembagheri and H.E. Estekanchi* Department of Civil Engineering, Sharif University of Technology, P.O. Box 11155-9313,

Tehran, Iran

Received: 20 September 2010, Accepted: 5 April 2011

ABSTRACT

Aboveground steel tanks are widely utilized in residential and industrial areas, and assurance of their seismic safety is a very important engineering consideration. In this paper, nonlinear response of these tanks using a new dynamic pushover procedure called Endurance Time (ET) method along with conventional nonlinear response history (RHA) is studied. The comparison between the results of RHA and ET method shows reasonable consistency. Based on the level of accuracy observed in this research, it is concluded that ET method can effectively be used in predicting seismic performance of anchored tanks. By considerably reducing the number of required response history analyses for estimating the response at various excitation levels, ET method has a good potential in paving the way for practical application of response history based analysis and design procedures for thin walled structures such as steel tanks.

Keywords: Endurance time method; response history analysis; aboveground anchored tank; nonlinear analysis; spectral acceleration; impulsive and convective states

1. INTRODUCTION

Liquid storage tanks are important multipurpose structures widely used in urban and industrial areas. Fluid-structure interaction and nonlinear behavior, complicate seismic response of this type of structures. Anchored type of tanks may have large capacity and may include various functionalities such as supplying water for drinking or fire fighting, storage of chemicals for oil industry, etc. In addition to direct economic losses, damage of these tanks may cause significant indirect losses such as water cut, fire and environmental pollutions. Therefore, assurance of seismic performance of this type of tanks is necessary, and as a result, considerable research effort has been focused on this field during last decades.

Almost every occurrence of large earthquakes during last decades has led to serious * E-mail address of the corresponding author: [email protected] (H. Estekanchi)

M. Alembagheri and H.E. Estekanchi

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damages to aboveground steel tanks [1]. Cooper and Wachholz [2] described many examples of damages of the steel tanks due to the earthquakes of 1964 Alaska, 1989 Loma Prieta, 1992 Landers, 1994 Northridge and 1995 Kobe. First calculation of hydrodynamic pressures on dams in seismic regions was performed in early 1930. Westergard [3] presented first proposal in 1933. He calculated hydrodynamic pressure on rigid wall of a dam with infinite reservoir due to harmonic excitation. In 1934, Hoskins and Jacobsen [4] conducted experimental and analytical vibration of rectangular tanks with rigid shell on nonflexible base. Because of the shell and floor rigidity, they concentrate solely on dynamic response of the contained liquid. Later, Jacobsen [5] in 1949 developed hydrodynamics in tanks. Housner [6] in 1954 indicated that part of liquid in tank has a long period movement due to dynamic excitation, which is called "Convective fluid" and forms upper section of fluid. Other part, which forms lower section of fluid, has a rigid vibration with tank shell, which is called "Impulsive fluid". The impulsive fluid, which vibrates with the same acceleration as input excitations, has more significant effect on the base shear and overturning moment. The convective fluid, on the other hand, affects the wave height of fluid on liquid surface. Useful formulas for calculating the impulsive and convective masses of fluid and the height of tank at which these masses operate have also been provided [6].

During a period before Alaskan earthquake of 1964, the issue was considered to be resolved, but the Alaska earthquake caused destruction of many tanks designed considering available theories, and this incident prompted further studies in behavior and dynamic characteristics of flexible-shell tanks. Edwards [7] accomplished first computer analysis of tanks in 1969 using the finite element method. This analysis was done on an anchored liquid storage tank with the height to the diameter ratio less than one. This study involved interaction between liquid and elastic shell. Mentioned researches considered tanks as linear elastic and completely anchored. Because of further development of finite element method and computer software, numerical tank analysis, which was mainly done by experimental and analytical methods, gained momentum. Veletsos and Yang [8, 9] in 1977 indicated that the flexibility of shell causes the impulsive fluid to accelerate several times greater than the input excitation acceleration. They also showed that the calculated base shear and overturning moment in tanks by assuming rigid shells were unreliable. The Rayleigh-Ritz method was used for obtaining natural frequencies of shell filled with liquid. Liu [10] in 1981 determined nonlinear effects of large deformations and dynamic buckling of tank shell. Haroun and Housner [11] in 1982 used boundary integral theory to find the added mass matrix of fluid, and result of this method led to development of an advanced seismic design process for tanks. Balendra et al. [12] used direct finite element method for studying flexible tanks with this difference that the independent variable in liquid was dynamic pressure range. Malhotra [13, 14] and Hemdan [15] further advanced the available research results in this area.

As a result of these researches, various mechanical models for tank analysis were created. The most famous of them are Housner’s equivalent two-mass model [6], Haroun’s three-mass model [16], and Malhotra’s simplified method [14] which is one of the benchmark analytical methods and is primarily applied for finding initial answers and controlling accuracy of results of numerical models. In addition, various design charts have been provided for determination of the impulsive and convective frequencies, the base shear, the

NONLINEAR ANALYSIS OF ABOVEGROUND ANCHORED STEEL...

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overturning moment on foundation and the stress resultant in shell. The primary design codes such as API650 (Appendix E) [17] and Eurocode8 (Section 4) [18] were formed based on researches of Wosniac and Mitchell [19], and Malhotra et. al. [20] respectively.

2. THEORETICAL BACKGROUND

Malhotra's schematic model is presented in Figure 1. As can be seen from this figure, in classical analysis of tanks, contained fluid is divided into two sections, the impulsive and the convective, and both are connected to the tank shell with a spring and damper at a specific height. Common formulation in Codes is based on this model and its relations.

Figure 1. Malhotra's mechanical model [20]. Predominant periods of tanks are calculated using Eqs. (1) and (2):

Timp=Ci (1) Tcon=Cc (2) where Ci and Cc are two coefficients, the impulsive and convective effective masses, mi and mc are a fraction of the total fluid mass, ml, and the heights at which theses masses act, hi and hc are a fraction of the fluid height H, and are obtained from related diagrams [19]. The total base shear is computed using Eq. (3).

Q= (mi +mw +mr) Sei +mcSec (3)

Formulas for calculating the overturning moments and the maximum vertical

displacement of fluid surface are presented in related references [19, 20].

3. ENDURANCE TIME METHOD

The Endurance Time (ET) method’s objective is to provide a relatively comprehensive


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method based on time history analysis for evaluation of structural systems under dynamic excitations due to earthquakes. Basic concepts of this method can be explained by a hypothetical shaking table experiment. Suppose that three different prototypes, say there are three tanks with different designs, with unknown seismic resistance and dynamic characteristics are available, and investigation of their seismic performance is intended. All three structures are placed on a shaking table. The experiment begins by exerting an intensifying random acceleration function. Assume that at 5th second, the vibration intensity of the table is low and all of structures are stable (Figure 2(a)). By gradually increasing the intensity, suppose that the structure 1 collapses at 10th second (Figure 2(b)). With more increasing of the intensity, the structure 3 in this example collapses at 15th second (Figure 2(c)) and finally the structure 2 in higher vibration intensity at 20th second collapses (Figure 2(d)). Based on this hypothetical experiment, one can judge that the structure 2, which exhibited more durability than others has the best seismic performance, and the structure 1, which collapsed first has the weakest performance [21].

(a)

(b)

(c)

(d)

Figure 2. Shaking table hypothetical test, (a) at 5th second, (b) at 10th second, (c) at 15th second, (d) at 20th second.


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In ET method, structure is subjected to an artificial acceleration function with increasing intensity, and its response is evaluated based on its performance against increasing levels of exerted forces and displacements. Advantages of this method in brief are: uniform applicability to various structure types, direct use of step-by-step time history analysis procedure, and significant reduction in computational demand when multi level seismic investigation is involved. Application of this new method in framed structures has produced promising results [22].

Response spectrum of ET's accelerograms used in this study increase or decrease proportionally with time, i.e. at the time equal to half of the target time (ttarget), the intensity will be half of the template spectrum value, and at the time equal to twice the target time, the intensity will be twice the template spectrum value. Therefore, the target acceleration response of ET's accelerograms is defined using Eq. (4) [23].

Sat(T,t)=Sac (T).t /ttarget (4)

where Sac(T) is the template acceleration spectrum, Sat(T,t) is the target response acceleration in time t, and T is period time.

In order to investigate consistency, and determination of ET method potential in predicting nonlinear response of tanks subjected to earthquakes, a series of accelerograms (Series e) are used which consists of three accelerograms. These accelerograms are created based on 200 target points in the periods between 0 to 5 second, and 20 target points in the periods between 6 to 50 second, for considering the effect of higher periods. In these accelerograms, template spectrum corresponds to the average spectrum from seven earthquakes on stiff soils [24].

In this method, selection of the appropriate acceleration function is very important. The acceleration functions in "Series e" of ET method are based on the average spectrum of seven real ground motions, which are extracted from FEMA440 [25], and recorded on soils similar to type 2 of the Iranian Seismic Standard (No. 2800) [26]. Properties of these earthquakes are listed in Table 1. These records have been scaled so that the area under their spectrums (with 5% damping) being equal to the area under the design spectrum of Iranian Seismic Standard. The intensifying (scaling) factors are also listed in Table 1. The sample intensifying acceleration function "ETA20e01" is shown in Figure 3. Each of these functions has 2048 points with 0.01-second steps, thus the length of the accelerograms of this series is 20.48 second [24].

Table 1: Earthquake records specifications used in creating "Series e" accelerograms of ET method


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Figure 3. A typical ET acceleration function (ETA20e01). Average of response spectrums of the seven scaled actual earthquakes and the three ET

accelerograms at the target time of 10th second are shown in Figure 4. Objective is comparing average of maximum values, resulted from the scaled actual earthquakes and the ET records until 10th second, and examining the reliability of ET method in predicting results of the response history analysis of anchored tanks with a reasonable accuracy. Because of linear and time proportional relation of ET spectrums, as can be seen in Figure 4, relative compatibility exists between average of response spectrums of the seven un-scaled actual earthquakes and average of response spectrums of the three ET accelerograms at fifth second (half of target time). Thus, average of maximum results from un-scaled actual records and ET records until fifth second are also compared.

Figure 4. Average spectra of actual earthquakes (ground motions) and ET records.


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It should be mentioned that in present study all tanks are analyzed considering nonlinear material and geometric effects. Linear analysis of anchored tanks is investigated in another research [27].

4. FINITE ELEMENT MODEL

In the present study, three tanks with various dimensions and geometric shapes are considered. These tanks are named A, D32H12, and D8H12, in which the tank A is similar to that one studied by Barton and Parker [28] for comparison purposes. The two other tanks are designed using App. E of API650 Code [17] and 12th chapter of Iranian Oil Industries Seismic Design Guide (Publication No. 038) [29]. The tanks' properties are shown in Figure 5, and listed in Table 2. Full fixity is considered for all of tanks and tanks' shell is considered flexible.

A general-purpose Finite Element program has been used for the purpose of this study [30]. Tanks' shell is modeled by quadrilateral shell elements with reduced integration formulation, and fluid is modeled by cubic eight-node continuum elements with reduced integration formulation. Analyses are done via explicit dynamic analysis using the direct integration method. Typical element configuration of tanks' shell and contained fluid are shown in Figure 6.

The impulsive and convective periods of vibration of tanks are obtained from Eqs. 2 and 3 in section 2. Fluid in these tanks is water and shell is made from steel. Material properties of water and steel used in this research are listed in Table 3. Preliminary investigation of linear and nonlinear behavior of steel showed the close similarity between results except for stresses of steel after yielding.

Table 2: Geometric and dynamic properties of studied tanks


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Figure 5. Typical longitudinal section of tanks.

Figure 6. Typical finite element configuration, (a) Shell, (b) Fluid.

Table 3: Material properties for water and steel


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Verification of models has been performed using the physical principles which govern the behavior of fluids. For this purpose, the models are loaded by self-weight, and responses in tanks' floor, shell, and contained fluid were compared with the theory. In addition, the tanks were accelerated with a constant horizontal acceleration and the fluid oscillation (convective period, displacement pattern, and monolithic performance of fluid's mesh and distortions of it) and energy changes were investigated. Also, tanks were accelerated in vertical direction with a constant acceleration as same pattern as horizontal one, and pressure change in the fluid was verified. In all of these verifications, consistency between the computer analyses results and the theory were observed. These investigations proved the acceptable performance of computer models.

5. ANALYSIS RESULTS

In ET method the response history analysis results are presented as the maximum absolute value of considered variables from t=0 up to the desired time. In Figures 7 to 10, some ET diagrams for various tanks are shown. As can be seen in these figures, in general, most responses increase in a relatively linear manner complying to the rate of excitation intensification with time. Von-Mises stresses in Figure 9 are obviously limited by the yielding level. In addition, as can be seen in Figure 10, the fluid wave height exhibits some time lag resulting in an increasing rate of amplification.

Figure 7. ET diagram of Max. shear stress in tank A.


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Figure 8. ET diagram of Max. relative horizontal deformation in tank A.

Figure 9. ET diagram of Max. von-Mises stress in tank D32H12.

Figure 10. ET diagram of Max. fluid wave height in tank D8H12.


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Tables 4 and 5 summarize the results of analyses of the tank A subjected to the ground motions and ET acceleration records respectively. The difference percentage of the base shear between Barton and Parker [24] and the present study is 8.7%. The base shear in this case using Eq. (3) is 3.74 MN that differs from FE results by 5.27%.

Table 4: Analysis results of tank A under seven scaled actual records

Table 5: Analysis results of tank A under three ET records (Series e) until target time (10th

second)

Table 6 shows the summary results of analysis of the studied tanks. The average

Difference that is indicated in this table is computed using Eq. (5).

Average Difference = (RET – REQ) / (REQ). 100 (5)

where RET is the average of maximum values resulted from ET records until the target time (10th second), and REQ is the average of maximum values resulted from the scaled earthquake records. As can be seen in Table 6, in most cases, difference percentages are below 20%. The differences in results between the two accelerogram groups (the actual earthquakes and ET accelerograms) are justifiable considering differences in the spectral accelerations, but these differences are reasonable taking into account the random nature of earthquakes. These differences indicate the reasonable accuracy of using ET accelerograms


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for estimating the average results of ground motions.

Table 6: Summary results of analyses of studied tanks

6. BASE SHEAR BASED SCALING

Base shear scaling can be used as a tool for improving the accuracy of estimations made by ET method. Because of 16.47% difference percentage in results of the base shear of the tank A, an equivalent time in ET is considered in which the average of base shears in both ET and actual records being nearly equal. By creating trend lines in ET base shear diagrams in accordance to Figure11 the time 9.24 second is obtained. Then values of other parameters in this time are read from ET diagrams, and compared with results from seven scaled actual earthquake records. Comparison between the difference percentages of averages before and after the base shear balancing is shown in Figure 12. As can be seen because of the relation of the base shear to the impulsive fluid, differences in maximum stresses and relative deformations, which are in relation to the impulsive fluid, decrease significantly, but the maximum fluid wave height does not change considerably.


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7. EVALUATION OF RESULTS

The difference percentages in Table 6 are due to differences in the spectral accelerations at the impulsive and convective periods of tanks that are mentioned in that table. As can be seen, almost all of results' difference percentages are below 20%, and indicates reasonable accuracy of using ET accelerograms instead of the ground motions that Codes required.

Figure 11. ET diagram of base shear in tank A.

Figure 12. Average differences in tank A before and after base shear base scaling.

As can be expected, the base shear, stresses and relative deformations are better related to

the impulsive state, and the fluid wave height is related to the convective state. For example, in the tank A, 12% difference in spectral acceleration at the convective period, is seen in


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results of the maximum fluid wave height. In addition, since the spectral acceleration at the impulsive period in the record LPLOB is greater than the record LPSTG, the base shear and stresses in the record LPLOB is greater as expected. However, the spectral acceleration at the convective period in the record LPSTG is greater than the record LPLOB, and therefore the maximum fluid wave height in the record LPSTG is greater.

The impulsive and convective masses of fluid, which are presented in Table 7, have a significant influence on the results. For example, because of low amount of the impulsive mass in the tank D32H12 relative to the convective mass and high effect of the convective mass on responses, the base shears and stresses in this tank do not follow out the spectral acceleration changes at the impulsive period. In addition, using linear acceleration spectrums, the first mode, and its period cause dissimilarity of difference percentages between results and spectral accelerations. Effective impulsive and convective masses of first mode of tanks are presented in Table 7 that shows more significant influence of higher modes in responses.

Table 7: Mass of shell and fluid, and convective and impulsive masses of fluid in studied tanks

In order to further investigation of the influence of spectral accelerations at the impulsive

and convective modes and their relationship to the results, correlation of several response parameters to respective spectral acceleration at the impulsive and convective periods are studied as shown in Figures 13 to 15. As can be seen from Figures 13 and 14, the base shear, maximum von-Mises and shear stress have a strong correlation with the impulsive spectral acceleration. However, in the tank D3H12, considering Table 7, high amount of convective mass influences the results, and their level of scattering. In accordance to Figure 15, the maximum fluid wave height has stronger correlation with the convective spectral acceleration. Consequently, for better accuracy of estimation obtained from ET records, for parameters such as the base shear and stresses, difference in spectral accelerations in the vicinity of the impulsive period (periods between 0.2 to 1.5 times the first mode period which are recommended by Codes) should be minimizes. For parameters such as fluid wave height, spectral accelerations near the convective period should be matched.


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Figure 13. Base shear (parameter) diagram vs. (a) the impulsive and (b) the convective spectral

accelerations in tank A.

Figure 14. Max. von-Mises stress (parameter) diagram vs. (a) the impulsive and (b) the convective spectral accelerations in tank D8H12.

Figure 15. Max. fluid wave height (parameter) diagram vs. (a) the impulsive and (b) the convective spectral accelerations in tank A.


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In accordance to Figure 16 (a), difference between average acceleration spectrum of ground motions and ET records in higher impulsive modes in the tank A are less than 30.4% (that corresponds to the first mode). Thus, differences in parameters related to impulsive modes in this tank, are, on average, less than 30.4%. Meanwhile in Figure 16 (b), while spectral intensities of ground motions an ET records match closely for the first mode, the related responses are actually higher in ET analysis in the tank D8H12 due to the effects of higher modes. Therefore, for reducing differences between results of the ET and ground motions, an equivalent time in ET should be selected such that the difference between acceleration spectrums in ET in the vicinity of effective periods becomes minimum, rather than at the exact period itself.

Figure 16. Average of acceleration spectra of seven scaled actual records and three ET records until 10th second, (a) tank A, (b) Tank D8H12.

8. ESTIMATING MAXIMUM STRESSES AND STRESS DISTRIBUTION

In this section, the accuracy of ET analysis in estimating maximum von-Mises stresses, which are resulted from analysis of the studied tanks, are investigated. This is done by studying correlation diagrams that demonstrate relation of the von-Mises stress at various elements of tanks' shell in two methods. These diagrams are shown in Figures 17 and 18.

As can be seen from Figure 17, in general, the maximum von-Mises stresses resulted from two analysis procedures have good correlation and consistency. In all of tanks distribution of points in the diagrams is so that trend line in each diagram is lying approximately on line Y=X. This implies good level of accuracy achieved in predictions by ET method and correct selection of target time. The results of ET analyses in the tank D32H12 are somewhat less than those by the ground motions, and in the tank D8H12 are somewhat more than those from the analysis by ground motions. In addition, compatibility of results in the tank D32H12 is less than two other tanks. This can be attributed to the effect of relatively high convective fluid mass in this tank that affects the results of von-Mises stress.

Correlation diagrams of the maximum von-Mises stress for the tank D8H12 between earthquakes and their average, and for ET records and their average, are separately shown in Figure 18. As can be seen in this figure, scattering of results for ground motions is greater


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than ET records, and ET records have an appropriate consistency. This observation also holds for two other tanks. It seems that for the preliminary design of the anchored tanks’ shell, even one ET record can be used in analysis instead of using three ET records depending on the required level of accuracy.

Figure 17. Correlation diagram of Max. von-Mises stresses at various elements between earthquakes and ET records with their trend line, (a) Tank A, (b) Tank D32H12, (c) Tank

D8H12.

Figure 18. Correlation diagram of Max. von-Mises stresses with their average at various

elements in tank D8H12, (a) Earthquakes, (b) ET analysis.


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9. CONCLUSIONS

In this paper, nonlinear response of aboveground steel tanks using a new dynamic pushover procedure called Endurance Time (ET) method is studied. The results are also compared to those from conventional nonlinear response history (RHA). It is shown that the average seismic response of tanks subjected to a set of ground motions at various excitation levels can be estimated with reasonable accuracy using intensifying ET accelerograms. If the average of acceleration spectrums of ET accelerograms is approximately coincident with the average of acceleration spectrums of earthquake records, then they provide relatively accurate estimation of the seismic response of anchored tanks. This estimation can be improved by applying the base shear based scaling especially in the case of parameters that are directly related to the impulsive vibration mode. More reasonable target time that results in better consistency between acceleration spectrums of ground motions and ET records, especially in the vicinity of the impulsive and convective periods will help to further improve accuracy of ET estimates. It is shown that differences in the spectral spectrums of ground motions and ET records at the impulsive and convective periods of tanks influence the results corresponding to impulsive or convective mode of vibration rather independently. Because of low scattering of maximum von-Mises stresses in ET analyses, it seems that even one ET analysis is sufficient for estimating stresses in anchored tanks at various excitation levels with reasonable accuracy. It is observed that local yielding of steel has negligible effect on responses except on maximum stresses in steel. It is concluded that ET method provides a useful method for estimating response parameters of anchored tanks at different levels of excitation, and can reduce the required computational effort as compared to complete response history analysis.

Acknowledgments: The authors acknowledge the support of Sharif University of Technology Research Council, and Iranian Oil Building And Engineering Co. (contract NIOEC2/ER-CV/CON-EX05-00) for support for this research.

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6. Housner GW. Dynamic pressure on accelerated fluid containers, Bulletin of the Seismological Society of America, 47(1954) 15-35.

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9. Veletsos AS, Yang JY. Earthquake response of liquid storage tanks, Proceeding of the Second Engineering Mechanics Specialty Conference, ASCE, Raleigh, 1977, pp. 1-24.

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Nomenclature mc convective effective mass mi impulsive effective mass mw wall (shell) mass mr roof mass hi height in which impulsive mass acts hc height in which convective mass acts E modulus of elasticity of shell's material ET Endurance Time method RHA Response History Analysis FE Finite Element method H height of contained fluid ρ fluid density R tank radius ( = D/2) h relative thickness of wall (shell) L wall height l roof height t time ttarget time at which excitation spectrum in ET method matches the target spectrum

by scale 1.0 Timp (Ti) impulsive period Tcon (Tc) convective period Sa spectral acceleration Sei elastic spectrum acceleration of impulsive mode assuming 2% damping Sec elastic spectrum acceleration of convective mode assuming 0.5% damping Sat target response acceleration Sac Code's design acceleration spectrum STDEV standard deviation g ground acceleration

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