# Models of Deterioration

1.1 Random Variable (RV) Deterioration Model

In case of a deteriorating system, the structural failure is defined as an event when the strength falls short of the applied stress. A corresponding limit state function is defined as

R(t) – 5 = [ – x (t)]- 5 = P – x (t) < 0 (1) where R(t) denotes the deteriorating resistance at time t, s the load effect, r0 the initial resistance, X(t) the cumulative deterioration at time t, and p = (r0 – s) > 0 is the available design margin or a failure threshold. For the sake of simplicity of the discussion, r0, s and p are assumed as deterministic constants. Thus the failure is the event of cumulative deterioration x(t) exceeding the threshold p.

The random variable (RV) model characterizes the randomness of the deterioration by a finite – dimension vector of time invariant random variables© as X(t; 0) . For example, consider a simple linear deterioration model as

X (t) = At (2)

where A is the deterioration rate, which is typically randomized to reflect the variability in a large population of similar components. Given the probability distribution of random rate, FA(a), the distribution of the amount of deterioration, X(t), is derived as FX(t)(x) = FA(x/t). The mean, variance and coefficient of variation (COV) of X(t) are expressed respectively as

According to the failure definition in Eq. (1), the cumulative probability distribution of the lifetime, T, can be written as

Ft (t) = P[T < t] = P[p / A < t] = P[A > (p /1)] = 1 – Fa [p /1] (4)

Depending on the probability distribution of A, the lifetime distribution can be derived analytically or computed numerically.

Suppose the deterioration rate is a gamma distributed random variable with probability density function given as

fA (a) = S e ~a/S, for a > 0 (5)

ST(jf)

where n and Sare the shape and scale parameter, respectively. We denote the gamma density function by ga(a;i], S) and its cumulative distribution function by GA(a;tf, S). From Eq.(2), the deterioration, X(t), is also gamma distributed with density function ga(х;Ц,8 t) . From Eq.(4), the lifetime (T = p /A) follows an inverted gamma distribution with the following density function:

and the cumulative distribution function is written as

FT (t) = 1 – GA(p/ tn, S) (7)

It can be shown that the moments of the lifetime are given as follows:

Since the gamma distributed deterioration rate has mean 8l) and COV – J 1/ n, the moments of lifetime distribution can be related with that of the degradation rate as