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).

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 / T (lbsin) / SY (psi) / d (in) / P (lb)
Mean / 100000 / 86000 / 2.5 / 10000
Standard deviation / 10000 / 8000 / 0.01255 / 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 , where =0.015.

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,


The above equation is only true if the stress and the strength are statistically dependent. (T-F)

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)