Download MIME 5690 Exam 1 Spring 2008

MIME 5690
Exam 1
Spring 2008
This is a take-home exam. Please return your exam by Monday, April 7, 2008 at 10 AM.
Sign the honor pledge: I have not given nor received any help in this exam.
Name ________________, Signature __________________
1) A structural member is subjected to a tensile stress that is normal with mean value
50,000 psi and standard deviation 12,000 psi. The strength is also normal with mean
120,000 psi and standard deviation 18,000 psi. The stress and strength are
statistically independent. Calculate the probability of failure of the structural member.
2) Find the mean value and variance of the following variable: Y  X1X 22 X 31 / 3 where
X1, X 2 and X 3 are normal independent random variables with mean values 1, 1.5 and
0.8 and standard deviations 0.1, 0.2 and 0.15. Use the approximate equations that we
studied in class.
3) A structural member is subjected to a tensile stress that is normal with mean value
90,000 psi and standard deviation 5,000 psi. The strength follows a Weibull
distribution with scale parameter 100,000 psi, location parameter 150,000 psi and
shape parameter 2. The stress and strength are statistically independent. Using
standard Monte Carlo simulation with 50,000 replications estimate the probability of
failure, the standard deviation of this estimate and 95% confidence bounds of the this
estimate.
4) Answer the following true-false questions. You do not need to justify your answers,
just say if a statement is true or false. However, you can write something if you think
a question is vague or ambiguous.
a) The hazard function h(t) is greater or equal to the probability density function of the
time to failure (T-F)
b) The hazard function of a system whose time to failure follows an exponential
probability distribution is constant as a function time. (T-F)
c) The mean value of the sum of two random variables is always equal to the sum of the
mean values. (T-F)
d) The mean value of the product of two random variables is always equal to the product
of the mean values. (T-F)
e) The variance of the sum of two variables, which have zero correlation, is equal to the
sum of the variances. (T-F)
f) The variance of the sum of two variables can be less than the sum of their variances.
(T-F)
g) The variance of a variable Y=g(X), can be approximated by the product of the square
of the derivative of g(X) at the mean value multiplied by the variance of X. This
approximation does not work well if function g(X) is highly nonlinear and the
standard deviation of X is large compared to its mean value. (T-F)
h) Consider the following equation for the probability of failure of a rod in which there
is only axial stress, S, and whose ultimate strength is, Su, is:




P( F )   FSu ( s) f S ( s)ds   [1  FS ( s)] f Su ( s)ds
The above equation is not true if the stress and the strength are statistically dependent. (TF)
j) The safety index of a component is different than the safety factor of this component.
(T-F)