3.0 Determining the Reliability Index by Monte-Carlo Simulation
Monte-Carlo simulation is a scheme employing random numbers and is used for solving deterministic problems where time does not play a substantive role. Thus, Monte-Carlo simulations are generally static and not dynamic (Law and Kelton, 1991). Random values of Q and R are generated according to basic statistical parameters. i.e., mean, coefficient of variation and an assigned distribution. The random values are then combined to form a limit state function, g, according to a pre-determined combination such as g = φR – γQ where φ is the resistance factor and g is the load factor. Based on the definition of failure, e.g., g < 0, the number of â€œfailingâ€ simulations is counted and the probability of failure determined as follows:
The load and resistance factors used with the Q and R values to determine the value of g can be varied until an owner-specified acceptable value of Pf is obtained, e.g., 1 failure in 5,000 simulations. Generally, for substructure design the value of resistance factor, φ, is varied for a given load factor. Once a value of Pf acceptable to the facility owner is determined, the chart in Figure 2.1 or Figure 2.2 may be used to determine the approximate reliability index, β.
3.1 Use of β value versus Pf value
The value of β determined based on the Monte-Carlo simulation technique is approximate because the relationships shown in Figure 2.1 or Figure 2.2 are strictly valid only for the case of a single load and a single resistance. In practice, multiple loads and a resistance are considered in evaluation of g (limit state) values and the actual distribution of g values may be intermediate between normal and log-normal patterns. Consider a scenario where a numerical simulation using Monte-Carlo technique gave a Pf of 1 in 5,000 based on a normally distributed dead load, a normally distributed live load, and a log-normally distributed resistance. For this scenario, the g values will be neither normally distributed nor log-normally distributed but will be intermediate between the two distributions. For Pf of 1 in 5,000, based on Figure 2.2, a β of 3.54 is obtained if g values are assumed to be normally distributed while a β of 3.4 is obtained if g values are assumed to be log-normally distributed. Since the actual distribution is intermediate between the normal and log-normal distributions the β value is between 3.4 and 3.54. Assigning and reporting a single value for such a scenario can be misleading. This is because most designers conventionally use the assumption of normal distribution to evaluate the meaning of β in terms of Pf. Referring to Figure 2.2, for the case of normal distribution, a β = 3.54 corresponds to Pf of 1 in 5,000 while a β = 3.4 corresponds to Pf of 1 in 3,000. Such a discrepancy, i.e., 1 in 5,000 versus 1 in 3,000 may be significant for critical facilities. Therefore, rather than using β value one can simply report Pf as 1 in 5,000. Alternatively, one can express Pf of 1 in 5,000 as 0.02%. Use of Pf in this manner can then enable a direct comparison between different numerical simulations using the Monte-Carlo technique. As a comparison, in this example, the Ps would be 99.98%.
Since the LRFD literature commonly uses reliability index, β, the remainder of this article continues to use the concept of reliability index for the sole purpose of discussions within the AASHTO-LRFD framework. Nevertheless, based on the above discussions, the reader should be cognizant of the shortcomings of using a β value in lieu of Pf (or Ps).
3.2 Comments on Use of Monte-Carlo Simulation in LRFD
The Monte-Carlo simulation technique is particularly useful when multiple load sources and resistances need to be evaluated. For example, in a drilled shaft analysis the loads may be comprised primarily of dead loads and live loads while resistances may be from side resistance and base resistance, each of which is mobilized at a different rate as a function of the vertical movement of the shaft. As noted earlier, the loads may be normally distributed while resistances may not be normally distributed at the component or system level. Such problems are not analytically tractable and the designer has to resort to numerical approximations. This is exactly what the designer should bear in mind, i.e., the Monte-Carlo simulation provides an approximation. Given that the simulation process is based on random numbers, it is not possible to get exactly the same answer for every simulation. If one does get exactly the same answer, then either it is a coincidence, or else there is something wrong with the random number generator, i.e., the random numbers are not truly random and/or an adequate number of simulations was not evaluated.
The large volume of numbers produced by Monte-Carlo simulation often creates a tendency to place greater confidence in the results of the simulation than is justified. The key to Monte-Carlo simulation in the context of LRFD is to critically evaluate the basic statistical parameters for the input variables. If any of the input variables is not a valid or justifiable representation of the data, the Monte-Carlo simulation data, no matter how impressive they appear, will provide little useful information about the actual limit state under consideration.
Considering the above discussion, the designer should not get carried away in trying to determine the â€œexactâ€ value of reliability index since that is simply not possible with a numerical simulation technique. Realistically, the best that the designer can expect is to determine the reliability index within Â±0.10. Thus, for example, if the target β value is 3.0, then the results for β = 3.0 Â± 0.10 should be considered to be satisfactory. No further refinement is warranted because of the inherent variability of the Monte-Carlo simulation technique based on random numbers as well as the difficulty in defining Pf in terms of β value as discussed in Section 3.1.