A Study of the Return Period and Design Load by Means of the Monte Carlo Method

Abstract

Statistical characteristics of maximum values during N years observation, which are one of the most important factors in determining design load, are investigated by means of the Monte Carlo method. A model time series of annual maximum values of a hypothetical meteorological element is made by means of Monte Carlo method, under the assumption that the annual maximum is independent on each other.
A set of 4 figures of random number is assumed to express an excess probability of an annual maximum value. This excess probability is converted to the annual maximum value by taking a proper excess probability function. Such sets of random numbers are selected from random number tables, and a sufficiently long model time series of annual maximum values is made.
It is found that unexpectedly large values often appear in the curve of the model time series. This is neither error nor exception; it can be explained theoretically.
Next, the frequency distribution of maximum values in a period of N years for the above model time series is obtained and its characteristics are investigated. The linear logarithmic relationship between the maximum value within an observed period and the observation length N is shown to be correct in some cases, but in others it is not correct. It is dependent on the functional form of excess probability.
Hazen, Fuller, Gringorten and others have derived different equations which express the relation between the return period and N years observational data. According to the present investigation, this difference is explained by the difference of excess probability of annual maximum values. In the case of maximum wind speed and maximum daily amount of precipitation, Hazen's formula may be applicable.
N-year value, i. e., a value for N years return period is calculated by samples of the model time series, assuming that Hazen's formula is correct and the numerical factors of the population are unknown. The errors of estimated values are calculated by comparing them with the given value of the population. The error increases with the increase of the return period and decrease of the number of observations. An empirical formula for estimating errors of calculation is derived.
The life length of a construction, which is made by taking a design load equal to the observed maximum load in the past N years multiplied by safety factor a, is investigated. Variance of such life length is quite large and its expectancy is calculated to be infinity. When the risk of destruction is equal, the safety factor decreases with the increase of observational length. An empirical formula to estimate the degree of safety of the construction is obtained.