Defining "Level of Safety"-continue

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Using equation (3.2-3), we have

pc(d2|a) = 1 - PD(a)

(3.2-5)

Substituting (3.2-4) and (3.2-5) into (3.2-2) yields:

p(a) = p(a)PD(a) + p(a)[1 - PD(a)]

(3.2-6)

The first term of (3.2-6) is proportional to the probability density function of detected

damage size p0(a), or mathematically (Multiplication Rule):

p(a)pc(d1|a) = f(d1)g(a|d1)

(3.2-7)

in which, pc(d1|a) = PD(a), g(a|d1) = p0(a) and f(d1) is a constant. Therefore,

p(a)PD(a) = Cp0(a)

p(a) = Cp0(a)/PD(a)

(3.2-8)

where C is a normalizing constant that is determined by the condition of:

∞

∫ p(a)da = 1

(3.2-9)

Substituting (3.2-8) into (3.2-9) yields:

∞

p (a )

p0 (a)

∫ PD (a )

or C = 1 / ∫

C 0 da = 1

(3.2-10)

PD (a )

Therefore, according to (3.2-8):

∞

p (a )

p0 (a)

∫ PD (a ) da

p(a ) = 0

(3.2-11)

PD (a)

Note that the actual and detected damage size distributions are two distinct functions in this

equation.

Finally, "Level of Safety" is defined as 1- Probability(damage size ≥ ac | not detected ),

where ac is the critical damage size for failure of the structure: