Analytical Solution of HL Safety Index
β should be easily obtained through an iteration process satisfying the conditions given in equation (2.18), so too the design point.
By using the technique of Lagrange multipliers, Shinozuka  found that the minimum distance represented by the HL safety index is
where the gradient is defined as
An excellent first-order interpretation of this expression (AII.1) was given by Ang and Tang , which is demonstrated as follows.
In general, for a nonlinear limit state function G(x) involving ...