MVFOSM: Mean-Value First-Order Second-Moment (MVFOSM)
In TesiproV: Reliability Analysis Methods for Structural Engineering

MVFOSM

R Documentation

Mean-Value First-Order Second-Moment (MVFOSM)

Description

Mean-Value First-Order Second-Moment (MVFOSM) method for the approximation of failure probabilities in structural reliability analysis.

The limit-state function is linearised at the mean values of the basic variables. The reliability index is

\beta = \frac{g(\mu)}{\sqrt{\sum_i \left(\frac{\partial g}{\partial x_i}\Big|_{\mu} \sigma_i\right)^2}}

and the probability of failure is approximated by p_f = \Phi(-\beta).

Failure is defined as g(x) < 0. If g(\mu) < 0, the reliability index is negative.

Only the mean values and standard deviations of the basic variables are used. The distribution types do not influence the result. The method is exact for linear limit-state functions of normal variables. For nonlinear limit-state functions the result depends on how the limit-state function is formulated (e.g. R - S or R/S - 1 give different values of \beta for the same failure domain).

Usage

MVFOSM(lsf, lDistr, h = 1e-04, isExpression = FALSE, debug.level = 0)

Arguments

`lsf`	Limit-state function, either a function or an expression, as indicated by `isExpression`. Supplied automatically by a SYS_ object (SYS_PROB or SYS_PARAM) when the method is run through a PROB_MACHINE. Do not provide it in the machine's options.
`lDistr`	List of distributions as returned by `getlDistr()` of the basic variables. Each variable must have a finite mean and standard deviation. Supplied automatically as well.
`h`	Step of the central finite differences, relative to the standard deviation of each variable (default 1e-4). Only used if `isExpression = FALSE`.
`isExpression`	Logical. If `TRUE`, `lsf` must be an expression and the derivatives are computed symbolically. Otherwise `lsf` must be a function.
`debug.level`	Verbosity level: 0 = silent, 1 = basic info, 2 = detailed output.

Details

When MVFOSM is run through a PROB_MACHINE (fCall = "MVFOSM"), lsf and lDistr are supplied automatically by SYS_PROB or SYS_PARAM and must not be part of the machine's options. Only h, isExpression and debug.level are passed as options. With isExpression = TRUE the system passes the symbolic expression of the limit-state function instead of the function.

The mean values and standard deviations are taken from the distribution objects (fields mean and Sd), not from the native distribution parameters. All basic variables must therefore have a finite mean and standard deviation. This excludes for example the log-Student-t distribution ("lt"), for which the moments are not defined. In this case the function stops with an error.

With isExpression = FALSE the partial derivatives are computed by central finite differences with the step h \sigma_i for variable i. The step is relative to the standard deviation, so that variables of very different magnitude (e.g. a section modulus in m^3 and a yield strength in kN/m^2) are handled equally. For a variable with \sigma_i = 0 the step h \max(|\mu_i|, 1) is used. With isExpression = TRUE the derivatives are computed symbolically with D.

Value

MVFOSM returns a list with the following elements:

method: Character string "MVFOSM".
beta: Reliability index.
pf: Probability of failure \Phi(-\beta).
design.point.x: Approximate design point in the original x-space, \mu_i + \alpha_i \beta \sigma_i.
alpha: Direction cosines (sensitivity factors).
runtime: Runtime as returned by proc.time().

Author(s)

References

Freudenthal, A. M. (1956). Safety and the probability of structural failure. Transactions of the American Society of Civil Engineers, 121, 1337-1397.

Nowak, A. S., & Collins, K. R. (2000). Reliability of Structures. McGraw-Hill.

TesiproV documentation built on Oct. 28, 2026, 5:13 p.m.