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Main-shock Probabilistic Seismic Hazard AnalysisEquation Chapter 2 Section 1 Whether a structure can resist a given level of ground motion while still having an expected level of performance or not is the main goal of most of engineering analysis on earthquake. However it is necessary to decide what level of ground shaking should be used to perform this analysis. Before the calculation of risk of a structure after occurrence of earthquake, the first step is to determine the annual probability (or frequency, rate) of exceeding some level of earthquake ground shaking at a site of interest. The worst-case intensity is not quiet necessary in practical engineering. Probabilistic Seismic Hazard Analysis is a method to compute results of probabilities of occurrence, including all earthquake events and data of ground motion. It is very useful to determine those intensity levels which has much small frequency. There are five steps in PSHA, which are shown below:

1. Find and identify all earthquake sources which can cause damaging ground motions. 2. Characterize the distribution of magnitudes related with potential earthquakes. 3. Characterize the distribution of source-to-site distances related with potential

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Characterize earthquake magnitudes The distribution of these earthquake sizes in a region generally follows the following distribution:

log〖λ_m 〗=a-bm (2.1)

λ_m is the rate of earthquakes with magnitudes greater than m. Parameter a value indicates the overall rate of earthquakes in a region. Parameter b value indicates the relative ratio of small and large magnitudes (typical b values are approximately equal to 1). This equation results from the Gutenberg-Richter law because Gutenberg and Richter first studied observations of earthquake magnitudes. This equation can also lead to the cumulative distribution function (CDF) for the magnitudes of earthquakes. There is also some limit on the upper bound of earthquake magnitudes in specific region. So the cumulative distribution function for magnitude M, given as follows

F_M (m)=(1-〖10〗^(-b(m-m_min)))/(1-〖10〗^(-b(m_max-m_min)) ) m_min<m_(j+1). This equation shows the probabilities related with all magnitudes between m_j and m_(j+1) to the discrete magnitude m_j .

Table 2. 1: Magnitude probabilities with truncated Gutenberg-Richter distribution, where m_min=5, m_msc=8, a=1 and b=1.

m_j F_M (m_j ) P(M=m_j )

5.00 0 0.438

5.25 0.438 0.246

5.50 0.685