SPRING 1998

CIV 360/548: Risk Assessment and Management

Prof. Erik VanMarcke

ASSIGNMENT #2

In simple, single-component reliability analysis of systems, one seeks to evaluate the probability of failure, p_f, defined as the probability that the "load" L exceeds the load-carrying capacity or "resistance" R, where L and R have the same dimensions and are assumed herein to be independent random variables. In other applications, the quantity L may represent the "demand" on a system, and R its "capacity", and in the case of a financial institution, L may be (cumulative) liabilities and R (cumulative) assets.

The probability of failure, p_f = P[L > R], can be expressed in terms of the "margin of safety", Y = R - L,

p_f = P[Y < 0],

or in terms of the "factor of safety", Z = R/L,

p_f = P[Z < 1],

The quantities Y and Z are "derived" random variables, dependent on random load L and random resistance R.

Question 1: Express the probability density funtions (pdf's) of Y and Z in terms of the pdf's of L and R. (Hint: see course notes packet, pp. 11/12; first obtain the cumulative probability distribution function, then take its derivative.)

Question 2: If L and R are both Gaussian (normal), with known means and variances, what is the pdf of Y? State the mean and the variance of Y in terms of the means and variances of L and R.

Question 3: If L and R are both lognormally distributed, with known means and variances, what is the pdf of Z? Likewise express the mean and the variance of Z. (Hint: see page 21 in course notes packet for information about the lognormal distribution).

Two useful measures of "the degree of safety" are:

• the central safety factor "s", defined as the quotient of the mean of R and the mean of L), namely: s = E[R]/E[L] 
• the reliability index "beta", defined as the quotient of the mean E[Y] and the standard deviation of Y.
  Question 4: Show that p_f has a one-to-one monotonic relationship to "beta" if the safety margin Y has a Gaussian distribution. Sketch the relationship between p_f and "beta", for values of "beta" ranging between 0 and 4. (Hint: use the table of the standard normal cumulative distribution given on page 20 of the course notes packet).

  Question 5 (required for students registered in CIV 548, optional for CIV-360 students): Develop/propose a measure of safety similar to reliability index "beta" for the case when L and R are both lognormally distributed; denote it by "beta prime". Sketch the relationship between p_f and the (alternative) reliability index, "beta-prime".  

PROBLEM 2 (Continuation)

In the following, you may assume that the safety margin Y follows a Gaussian distribution and use (the sketch of) the relationship between p_f and "beta" produced in answer to Problem 1, Question 4.

(a) Risk Assessment

You are given the following information in a "design" situation referred to as "the basic case":

• E[R] = m
• E[L] = 0.5 m
• V_R = 0.3
• V_L = 0.15

(Recall, V denotes the "coefficient of variation", defined as the ratio of the standard deviation to the mean.)

Question 6: Evaluate the central safety factor "s", the reliability index "beta", and the probability of failure "p_f".

(b) Risk Management

Often, in practice, little or nothing can be done about the loads or "demands", but the failure probability can be reduced by changing the probability distribution of the resistance R, either by increasing its mean or by reducing its variability. To illustrate the effects, re-evaluate the quantities s, beta, and p_f for the following cases:

(i) Increase the mean resistance to E[R] = 1.2 m, while keeping all other values the same as in "the basic case".

(ii) Through quality control, decrease the coefficient of variation of the resistance to V_R = 0.15, while keeping all other values the same as in "the basic case".

Question 7: Which of these two measures reduces the basic-case p_f the most (or increases the basic-case reliability index the most)? Show your calculations.

Note the inadequacy of a common deterministic approach in which safety is measured by the "central safety factor" alone, and decisions hinge on whether the outcome is either "s > 1" (safe) or "s < 1" (unsafe).