In the analysis of contact mechanics problems, determination of stress field in mechanical elements is essential. Between the stress components the von Mises stress is more important, because it is used in the investigation of yield criteria and fatigue fracture of elements. The aim of this study is to present formulas for determining the magnitude and position of maximum von Mises stress. For this purpose, the effect of various material properties, element geometries and loading conditions on these two parameters are investigated. By applying Hertzian contact stress and von Mises relations, the magnitude and position of maximum von Mises stress are determined. The von Mises stress is assumed to be a function of material properties, geometry of the element and loading conditions and finally two formulas are presented for the calculation of the magnitude and position of maximum von Mises stress. The results of these presented formulas are in close agreement with the literature. The error is less than 1% for depth prediction and less than 6% for stress value prediction, which confirms the accuracy of the presented formulas.

The maximum earthquake magnitude plays a crucial role in different aspects of seismic hazard and risk assessments. Previous work by Salamat et al. [1] shows the divergence of the confidence interval of the maximum possible earthquake magnitude Mmax for high levels of confidence 1-a, in different seismotectonic zones of Iran. For this, Mmax is replaced by the maximum expected earthquake magnitude mt that is calculated for different predefined future time intervals Tf. In this work, the frequentist and Bayesian approaches are applied to calculate the upper bound of the confidence interval of mt. The frequentist confidence intervals are calculated for the level of confidence 1-a = 95% and 99%, and future time intervals Tf = 30, 50 years. In the Bayesian approach, the posterior distributions of the maximum expected earthquake magnitude are calculated for Tf = 30, 50 years and 90% confidence level. The stationary Poisson process in time and Gutenberg Richter relation are assumed as a statistical model for the magnitude distribution. In order to estimate mt in each seismotectonic zone, three different scenarios of Mmax= 8. 5, 9. 0, 9. 5 are assumed. In order to find the influence of the declustering, all calculations are applied for both original and declustered catalogs. The results show, as long as the length of the time interval is short or moderate, different values of Mmax have a minor effect on the estimation of the maximum expected earthquake magnitude mt.

According to the standard No.2800, as well as most of the seismic hazard documents and analyses, Ahvaz city is reported as a region with moderate seismic hazard. These documents and information do not match with the fault condition of the region. In this paper, a "comprehensive study" of the seismicity of the Ahvaz region is presented in the field of deterministic and probabilistic seismic hazard analysis. Based on the results of this study, it seems that a fundamental review of the seismic hazard of Ahvaz is necessary. The earthquake risk analysis determines the maximum acceleration above g0.7 for an unspecified period of time and the possible risk analysis gives the maximum acceleration 0.16 g to 0.29 g for a planned earthquake and 0.33 g to 0.61g for a severe earthquake. In this way, it is suggested to include the city of Ahvaz among the cities with high seismic hazard to ensure the safety of the citizens. This study has opened a parenthesis in the field of earthquake risk in Ahvaz city and the "comprehensive study" statement is incomplete due to the lack of necessary geological information. In order to reach more reliable results and close this parenthesis in terms of financial issues, it is strongly recommended that in this field, more detailed geological and geophysical research focusing on detailed methods of paleoseismology with emphasis on accurate determination the specification of Ahvaz fault should be made.

Using 48016 synthetic maximum Wood-Anderson amplitudes read from waveforms of 2650 events recorded by stations of Iranian Seismological Center (IRSC, irsc. ut. ac. ir), Iranian National Seismograph Network (INSN, www. iiees. ac. ir) and temporary seismic networks belong to Institute for Advanced Studies in Basic Sciences (IASBS, iasbs. ac. ir), the empirical attenuation curve ( 0  log A ) for local magnitude of Iran has been calculated as follows: 0 log (1. 556 0. 06) log (0. 001637 0. 0009) ( 100) 3 100 R A R              where R is hypocentral distance in km and 0 A is maximum displacement amplitude of shear wave in millimeter. The empirical attenuation relationship is valid for hypocentral distances equal or smaller than 800 km. ML amplitude is the maximum amplitude observed on a Wood-Anderson (W-A) seismogram. We manually picked the maximum amplitudes on the shear window of synthetic W-A seismograms having S/N of larger than 5. We calculated synthetic W-A seismograms by removing the instrument response of each record and convolving the resulting signal with the response of the standard W-A torsion seismograph. We assumed a static magnification of 2080 for the W-A instrument. The selected ML amplitudes are belonging to events at hypocentral distance of 10 to 800 km. Except for the Makran and South Caspian Basin regions, the ray coverage of the ML amplitude covers properly the whole Iranian Plateau. To reduce the scatter of magnitude residues and ensure a reliable calculation of the attenuation curve, the selected events belong to 45 precisely relocated seismic clusters with location uncertainties of 5 km or less. The cluster approach produces redundancy in amplitudes arriving from a cluster to a given station. The redundancy will facilitate easy recognition and removal of possible outliers and thus provide a reliable estimate for the magnitude station correction. The magnitude station corrections attempts to absorb the regional attenuation difference relative to that dictated by average attenuation relationship derived in this work. The calculated attenuation curve shows a larger geometrical spreading for hypocentral distances closer than 100 km, representing a super-spherical geometrical spreading, and a smaller value for intrinsic attenuation for distances farther than 200 km once compared with the currently used ML relationship of Hutton and Boore (1987). Excluding amplitudes with hypocentral distances smaller than 60 km results in a geometrical spreading coefficient close to spherical spreading, emphasizing the importance of near distances data on accurate estimation of the geometrical spreading value. The difference in the attenuation parameters between our results and those of Hutton and Boore (1987) relationship clearly indicates the crustal disparity of Iranian Plateau and southern California. This necessitates using the new attenuation relationship for Iran. We calculated the local magnitude empirical attenuation relationship by inverting the amplitude data set for the geometrical spreadin]g and intrinsic attenuation. We did not consider magnitude station corrections in our inversion to avoid any tradeoff between the station corrections and attenuation parameters. We have shown that the magnitude residuals calculated by our local magnitude empirical relationship do not vary systemically versus hypocentral distance or magnitude. Due to the cluster-wise approach in selection of our events and partially because of the precise location of the selected events, the standard deviation of magnitude residues is about 0. 19, significantly smaller than those reported by others. We calculated the station corrections by averaging the magnitude residual in each station. The station corrections vary between-0. 44 to 0. 32. Generally, stations located in Zagros, Alborz and north west of Iran have negative station correction representing amplitude amplification in these regions relative to central Iran and north east of Iran. The new attenuation relationship provides better estimates for the attenuation parameters and especially provides precise magnitudes at close hypocentral distances. By time, the expansion of Iranian seismic networks reduces the average distance spacing of Iranian seismic stations and thus usage of better local magnitude formula such as ours becomes more important.

magnitude has played a particular role in the realistic description of global seismicity. Most studies in earthquake seismology use magnitude data as a guide to the strength of an earthquake. So biases in magnitude estimates, caused by any effect, directly affect the result of any study in which magnitude data is used. In this study, efficiency of using different formulae and depth-distance calibration terms are examined. Applications of the MsR-P formula and new depth-distance terms to the ISC dataset, show that the estimated Ms and mb values are independent of distance, and provide unbiased estimates of Ms and mb in comparison with commonly-used Prague formula and Gutenberg-Richter terms. Comparison of standard deviation of Ms values for single events using the MsR-P and MsPrague formulae show that the MsR-P standard deviations are consistently smaller than those of Prague formula. Also standard deviations of estimated mb values using the new depth-distance terms are smaller than standard deviations of estimated values using Gutenberg-Richter terms. Estimated Ms and mb values using MsR-P formula and the new depth-distance terms reduce overlap in Ms:mb criterion for underground explosions and earthquakes. The study reported here confirms the need to modify the formula for Ms calculation and depth-distance correction terms for mb calculation, which are used by global agencies such as ISC and NEIC.

To assess seismic hazard, it is essential to estimate the potential seismicity, particularly estimation of size of the largest earthquake that can be identified by a distinct fault or earthquake source. One of the methods applied to estimate magnitude of earthquake is use of empirical relationships between magnitude and fault parameters. Fault parameters for the earthquakes from early instrumental period to 2014 are compiled to develop a series of empirical relationships among Mw and Ms with surface rupture length, horizontal and vertical displacements. The objective of this study is provision of an accurate piece of information about previous earthquakes through studies and investigations as well as a comprehensive and through catalog of the recent earthquakes so as to represent precise empirical relationships between the magnitude and earthquake fault parameters. Moreover, the information about earthquakes with magnitudes greater than Mw and Ms ³ 5.5 were selected. In the beginning, the relationships between magnitudes and fault parameters were acquired, then the database on the basis of three slip types, consisting of strike-slip, normal and thrust, were manipulated and eventually relations were obtained. One also should not overlook the fact that three regression models, including SR, ISR and OR, were employed in this study. In SR and ISR methods, no error are considered for independent variables, whereas the mentioned error is taken into account in OR method. The OR method is obtained by minimization of the squares of the orthogonal distances to the best-fit line, whereas SR is derived by minimizing the squares of the vertical offset and also inverted standard least-squares regression (ISR) is derived by minimizing the squares of the horizontal offsets. However, roughly equal uncertainties for the two variables are regarded in the SR and OR methods. According to the obtained results, the best relationship between Mw and Ms with surface rupture length was established with correlation coefficients of 0.87 and 0.86, respectively. Also the relationships between Ms with maximum horizontal and vertical displacements with correlation coefficient of 0.63 and 0.62 in a respect way, are far better than the relationships between Mw with maximum horizontal and vertical displacements with correlation coefficient of 0.59 and 0.59. In addition, the relationships between magnitudes (Mw or Ms) and maximum horizontal displacements indicate a better fit than the relationships between the magnitudes and maximum vertical displacements. It is also worth mentioning that the best fit between Mw or Ms and surface rupture length was acquired with correlation coefficients of 0.87 and 0.86, respectively, by separating the database based on slip type. Whatever was mentioned about relationships between magnitudes and maximum horizontal and vertical displacements, is still valid by separating the database on the basis of slip type. According to the results of the direct relations, the SR and OR regression methods are far better than the ISR regression method with less error than ISR regression method. What’s more, in inverse relations, the ISR regression method estimates the coefficients with the lowest error rate in comparison with other methods. As an outcome, the findings were established from the current study are better than the global relationships for Iran and its adjacent regions. For example, in the light of the relationship between Ms and surface rupture length, the utilized global relation has been overestimated and then underestimated up to M=7.2 with respect to OR regression. The obtained relations have been simultaneously compared to two global relations within relationship between MS and surface rupture length. The results would seem to suggest that both global relations are overestimated and then underestimated up to M=6.5 in comparison with other regression methods.

In the 16th and 17th centuries the classical Greek notions of (discrete) numberand (continuous) magnitude (preserved in medieval Latin translations of Euclid’sElements) underwent a major transformation that turned them into continuous but measurablemagnitudes. This article studies the changes introduced in the classical notionsof number and magnitude by three influential Renaissance editions of Euclid’s Elements.Besides providing evidence of earlier discussions preparing notions and argumentseventually introduced in Simon Stevin’s Arithmétique of 1585, these editionsdocument the role abacus algebra and Renaissance views on the history of mathematicsplayed in bridging the gulf between discrete numbers and continuous magnitudes.

The availability of a large amount of the data recorded by the Iranian Seismic Telemetry Network (ISTN) has motivated this study to develop relations for the routine determination of ML scale for Central Alborz region of northern Iran. The ML is commonly used in engineering because it is determined within the frequency range (0.5-3 sec) of interest in most of such applications. For any comprehensive seismic hazard analysis, one needs a calibrated magnitude relationship as well as an earthquake catalog for the study region. It is a well-known fact that the regional geology has a great influence on magnitude relations. Therefore, for each seismic region a specific magnitude relation has to be developed. The ML scale is based on the arithmetic mean of horizontal components of the synthesized Wood–Anderson seismograms. We used both nonparametric and parametric methods for inversion. We used a large dataset of 3886 events including 62031 waveforms which recorded by Tehran, Semnan and Sari seismic networks during 02/03/1997 to 13/03/2011. These seismic networks comprise of 19 three-component stations. We calculated the associated synthesized Wood-Anderson seismogram for each SS-1 waveform which records the velocity. Based on Richter’s method, we used amplitudes which are arithmetic means of those of horizontal components.Richter’s ML formula first developed for southern California and Savage and Anderson introduced a nonparametric least-squares inversion method which has been used by others. In this method, the amplitudes recorded at arbitrary distances are linearly interpolated to yield values for the attenuation curve at some fixed distances. In this study, we used both methods.The resulting equations are -logA0=0.9819log (r/100)+0.0028 (r-100)+3.0 and-logA0=1.076log (r)+0.0029 (r)+0.5580 from parametric and non-parametric methods, respectively. Where r is hypocentral in kilometer and A0 is amplitude in millimeter. The two methods yielded very similar results. Unlike the parametric method, the nonparametric one does not impose any a priori assumption of the shape of the attenuation curve on the data and has the potential to detect hinges in the attenuation curve that are caused by structural boundaries such as Moho or geological variations affects on the attenuation curve. Thus the result obtained by nonparametric method was chosen as the final result.Bakun and Joyner (1984) give the following formula for the Q/f ratio: taking an average S-wave crustal velocity of VS=3.3 km/sec, the k value obtained by the non-parametric method, 0.0029, would imply a Q/f ratio of 150 in Central Alborz, Iran.