COMREL Sample "Fatigue": Problem Description

Time-variant Problem with COMREL-TI: Use of Conditions in Failure FunctionsPreface

In this example we not only illustrate how to solve a time variant problem with COMREL-TI but also demonstrate a number of additional features of the symbolic processor which may be useful in more complicated cases. Additionally, we present some remarks on stochastic modelling for the special problem of crack propagation. Crack growth and crack instability problems are frequently encountered in structural reliability. The fatigue example provided with COMREL also addresses some peculiarities when performing such analyses. The basic formulation is for time-invariant analysis by COMREL-TI but it can easily be modified for a COMREL-TV application by assigning a scalar process to the load variable S. Note, however, that the failure criterion then should also be modified to describe crack instability. Crack growth is defined by Paris-Ergodan´s law:

(1)

with material constants m and C. S is the applied random stress range. Y(a) is a geometry factor usually depending on crack length a and containing , see e.g. [1]. For this example, Y(a) is the constant 1.0 and consequently Y^m = p^m/2. Eq. (1) can be integrated:

(2 a) (2 b)

with k = (2 - m)/2. For the time invariant case at hand (stationary loading) we compute N(t) from t ( is upcrossing rate of the mean of the load process denoted as parameter ‘Rate’, t is parameter ‘Time’). k is negative for any m > 2. Therefore, the expression (2b) can become singular for some t and makes no sense for times larger than this ‘explosion’ time. If the stress variation is a narrow band Gaussian process, the stress range is Rayleigh distributed and its expected value can be computed from:

(3)

Failure is defined if the crack depth a(t) exceeds a critical limit a_crit:

V = {a_crit - a(t) <= 0} (4 a)

Abbreviating C Y^m E[S^m] N(t) as ‘RHS’ (Right Hand Side) and using eqs.(2) the failure domain is

(4 b)

or

(4 c)

Note the ‘check on a(t) explosion’ in the numerical formulation in failure function. This formulation is possible because fatigue crack growth follows a monotonic function. Then, excedance of a_crit by a(t) at time t is equivalent to the event T < t with T the first passage time. The actual formulation of the failure function together with several auxiliary user defined functions is given below.

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