- 1.
Time-to-rupture data (from a creep study) are shown for steel rods that were held at a fixed level of stress at very high temperatures until rupture occurred. The time-to-failure data are shown in hours. Find the lognormal *t*_{50} and *σ* that describes the data.

**Answers**:

*t*_{50} = 49.8 h,

*σ* = ln(

*t*_{50}/

*t*_{16}) = 0.22

- 2.
Given the lognormal distribution (*t*_{50} = 49.8 h, *σ* = 0.22) from Problem 1,

- (a)
What is the expected time for 0.1 % of the steel rods to rupture?

- (b)
What is the expected time for 99.9 % of the steel rods to rupture?

**Answers**: (a)

*t*_{0.1 %} = 25.2 h, (b)

*t*_{99.9 %} = 98.3 h

- 3.
Given the lognormal distribution (*t*_{50} = 49.8 h, *σ* = 0.22) from Problem 1, what fraction of failures occur between 35 and 55 h?

**Answer**: 0.620

- 4.
Using the time-to-rupture data in problem 1, find the Weibull distribution that gives the best fitting to the data. What are the values for *t*_{63} and the Weibull slope *β*?

**Answer**:

*t*_{63} = 55.2 h,

*β* = 5.4

- 5.
Given the Weibull distribution (*t*_{63} = 55.2 h, *β* = 5.4) from Problem 4,

- (a)
What is the expected time for 0.1 % of the steel rods to rupture?

- (b)
What is the expected time for 99.9 % of the steel rods to rupture?

**Answers**: (a)

*t*_{0.1 %} = 15.4 h, (b)

*t*_{99.9 %} = 79.0 h

- 6.
Given the Weibull distribution (*t*_{63} = 55.2 h, *β* = 5.4) from Problem 4, what fraction of the failures occurred between 35 and 55 h?

**Answer**: 0.543

- 7.
Using the normal distribution in Chap. 5, fit the data shown in Table 7.1.

- (a)
What are the values of *t*_{50} and sigma for the normal distribution?

- (b)
Compare your normal fit to the lognormal-fit shown in Fig. 7.3. Which distribution gives the better fitting, normal or lognormal?

**Answers**:

- (a)
*t*_{50} = 1,573 h, *σ* = 576 h,

- (b)
Lognormal distribution gives a better fitting to this data set.

- 8.
The following time-to failure data was collected and found to have two failure mechanisms in the time-to-failure data.

- (a)
Perform a lognormal plot of the above data.

- (b)
Find the point of inflection which separates the two mechanisms.

- (c)
Replot the data for the two mechanisms.

- (d)
What are the (*t*_{50}, *σ*) values for the two mechanisms?

**Answers**:

(b) Point of inflection: F = 0.42,

(d) Mechanism A: *t*_{50} = 27.3 h, *σ* = 0.4,

Mechanism B: *t*_{50} = 131.1 h, *σ* = 0.17.

9. Using the time-to failure data (shown in the table in Problem 8):

- (a)
Perform a Weibull plot of the above data.

- (b)
Find the point of inflection which separates the two mechanisms.

- (c)
Replot the data for the two mechanisms.

- (d)
What are the (*t*_{63}, *β*) values for the two mechanisms?

**Answers**:

(b) Point of inflection: *F* = 0.42,

(d) Mechanism A: *t*_{63} = 32.3 h, *β* = 2.99 Mechanism B: *t*_{63} = 139.6 h, *β* = 6.29.