MIME 5690
Homework 8
Spring 2015
_____________________________________________________________________
Due 4/30/2015
1) The applied stress at the most critical location in a beam, S, has a probability density,
fS(s). The ultimate strength, Su, has a probability density, fSu(s). Derive an expression
for the probability density function of the difference between the ultimate stress and the
applied stress, Z=Su-S. Do not assume that the stress and the ultimate stress are normal
in this problem.
2) Derive an expression for the probability of failure in problem 1. Failure occurs when
the stress exceeds the ultimate stress. Note that the figure below is for illustration
purposes only. Do not measure any values from the figure.
3
7.508 10
ultimate stress
0.01
stress
f s( s )
f Su ( s )
8
5.733 10
0.005
interference area
0
200
200
300
400
500
s
600
700
800
700
3) In problem 2, an engineer looks at the figure and says that the probability of failure is
equal to the area under the interference area. Is this statement correct? Justify your
answer.
4) The equivalent von-Mises stress in a beam is S vm  S 2  3T 2 , where S is the normal
stress and T is the shear stress. Assume the axial and the shear stresses are statistically
independent. The means of the axial and shear stresses are E(S) and E(T), respectively,
while their standard deviations are S and T, respectively. Using the equations for the
approximations of the mean and the standard deviation of a function of many random
variables in the part of the course on reliability design, derive an expression for the
mean and standard deviation of von-Mises stress, and the probability that Svm could
exceed a limit SU.