Load combinations and FBC processes

Load combination reliability problems usually distinguish permanent actions, resistance variables, and variable actions that fluctuate in time. The Ferry-Borges-Castanheta (FBC) model represents a variable action as a rectangular-wave stochastic process: the process is constant during a basic interval \(\tau\), and a new independent value is drawn for each successive interval. This model is a standard basis for probabilistic load combination analysis in structural reliability texts [Thoft-Christensen] [Madsen2006] [Ditlevsen2007] [Melchers1999].

If \(F_Q(q)\) is the distribution of the action value in one basic interval and \(T\) is the reference period, the maximum over that period has distribution

(1)\[F_{Q,\max,T}(q) = F_Q(q)^r, \qquad r = \frac{T}{\tau}\]

where \(r\) is the number of basic intervals in the reference period. Equivalently, when a code or statistical model supplies a maximum distribution over duration \(T\), the corresponding maximum over a shorter duration \(d\) can be written as

(2)\[F_{Q,\max,d}(q) = F_{Q,\max,T}(q)^{d/T}\]

provided both maxima arise from the same FBC process assumptions. The recurrence count belongs to the underlying stochastic process, not to the load-combination factor itself.

Turkstra’s rule is a practical approximation for combining variable actions: each variable action is taken as the leading action in turn, usually as a maximum over the reference period, while the other variable actions are taken as companion values over a representative interval. In an FBC setting a companion action can be the point-in-time value or the maximum over the leading action’s basic interval. The modeling distinction is important: the FBC process defines the distribution of each action over time; Turkstra’s rule defines which distributions are placed together in each reliability case. This is the convention followed in Sørensen’s notes and common load combination examples [Sorensen2004] [Faber2009].

In PySTRA, FBCProcess exposes the process distributions: point_in_time() returns the basic-interval parent distribution, and maximum(duration=...) returns a maximum distribution for the requested duration. turkstra() then uses those process objects to create explicit named leading-action cases. The result is still an ordinary LoadCombination; the generated cases simply make the FBC and Turkstra assumptions visible in the model.

Use this method: Calibrating code factors · Load combinations and FBC processes · Code calibration and load combinations

For coordinate conventions, see Notation and probability spaces.