Strategies for finding the design point under bounded random variables

Reliability analysis of built concrete dam

“…The results presented as β > 8 and P f < 6 x 10 -16 are related to the computational representation of the standard Gaussian cumulative distribution function (Φ) and its inverse (Φ -1 ). According to Beck and Ávila [33], these functions have no closed analytic form, and polynomial approximations are often employed. In the StRAnD Mathematica version, from β > 8 instability occurs, so an operational safety limit is implemented, which corresponds to β = 8, for which Φ (-8) ≈ 6x10 - 16 .…”

Reliability-based design of an earth and concrete dam.

“…The Hasofer-Lind reliability index is given by β = [(y*) T (y*)] 1/2 , and the failure probability estimate is given by Equation (2.7). The accuracy of the FORM approximation depends on the degree of nonlinearity of the limit state in Y space (Beck and Ávila da S., 2016). The solution is exact only for linear functions of independent Gaussian random variables (Kroetz et al, 2018;Melchers and Beck, 2018).…”

“…The optimization problem in Equation (2.11) can be solved using various optimization algorithms, such as the gradient projection method, augmented Lagrangian method, and sequential quadratic programming method (Beck and Ávila da S., 2016;Val et al, 1996). The Hasofer-Lind Rackwitz-Fiesler (HLRF) algorithm (Abraham M. Rackwitz and Fiessler, 1978) is based on the Newton method; it is very efficient when it converges, and this occurs for most well-behaved problems (linear functions of independent Gaussian random variables) (Liu and Der Kiureghian, 1991).…”

“…These two properties are sufficient to ensure the unconditional convergence of the algorithm (Beck, 2019(Beck, , 2017Beck and Ávila da S., 2016;Zhang and Der Kiureghian, 1997). The probability of failure (Pf) in Equation (2.2) can also be evaluated directly by means of "crude" MCS.…”