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MIME 5690
Exam 2
4/25/2001
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1. Two nominally identical rods with circular cross sections are connected in series as
shown below. An axial force, P, is applied at the two ends A and B. The axial force is
normal with mean EP and standard deviation P. The ultimate strengths of rods 1 and 2 are
normal, independent, identically distributed random variables with mean value ESu and
standard deviation Su. The diameters of the rods are both deterministic, and are both equal
to d. The system fails if any of the rods fails (this occurs when the stress in a rod due to the
applied load P exceeds the ultimate strength Su).
Su1, d
P
A
rod 1
Su2, d
rod 2
B
P
a) Derive an expression for the reliability of the system of the two rods.
b) Using the expression derived in a) compute the reliability of the system when the
following data is given: EP=2.356106 N, P=2.356105 N, ESu=4108 Pa, Su=4107 Pa,
d=0.1 m.
2) A beam of circular cross section is subjected to a toque Q and an axial load P. Both Q
and P are normal, independent random variables with mean values EQ and EP and standard
deviations Q and P. The beam has diameter, d, which is normal with mean value Ed and
standard deviation d. Consider that the beam fails when the equivalent von Mises stress,
eq, exceeds the yield stress, SY. The yield stress, SY, is normally distributed with mean ESY
and standard deviation SY.
2.1) Develop the equations for estimating the failure probability of the beam using FORM.
These equations must include the following:
a) The performance function, g, expressed as a function of the random variables
b) The formulation of the optimization problem for finding the most probable failure
point, and the safety index.
c) The probability of failure and the reliability.
Solve the above problem for the following problem parameters:
Parameter
Mean
Standard
deviation
SY (psi)
86000
8000
T (lbsin)
100000
10000
d (in)
2.5
0.01255
P (lb)
10000
1000
2.2) Find the mean value of the diameter of the beam Ed so that the reliability is 0.99. The

standard deviation of the diameter is  d   Ed , where =0.015.
3
3 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) 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,




P( F )   FSu ( s) f S ( s)ds   [1  FS ( s)] f Su ( s)ds
The above equation is only true if the stress and the strength are statistically dependent. (TF)
b) If the correlation coefficient of two random variables is zero, then the random variables
are always statically independent. (T-F)
c) If the correlation coefficient of two random variables is zero and the joint probability
density function of the random variables is normal, then the random variables are
always statically independent (T-F)
d) The joint probability density function of the random variables in a reliability assessment
problem is normal and the performance function, g, is linear. Then the method for
estimation of the probability of failure by linear expansion of the performance function
about the mean values of the random variables yields the exact value of the probability
of failure. (T-F)
e) Consider two random variables whose joint probability density function is normal. If
we know the mean values, the standard deviations of the random variables and their
correlation coefficients then we know the joint probability density function of these
random variables. (T-F)
f) In question e) we can find an appropriate transformation of the random variables, which
yields a new set of independent random variables. (T-F)
g) Consider a system of components in series. The failures of the components can be
dependent or independent. The reliability of the system is always equal to the product
of the reliabilities of the components. (T-F)
h) The reliability of the system in question g) is always less or equal to the minimum of the
reliabilities of the components. (T-F)
i) Consider a system of components connected in parallel. The system survives if at least
one component survives. One way to increase reliability is to increase the number of
the components in the system. (T-F)