The article "Withdrawal Strength of Threaded Nails" (D.

Chapter 4, Problem 7E

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QUESTION:

The article “Withdrawal Strength of Threaded Nails” (D. Rammer, S. Winistorfer, and D. Bender, Journal of Structural Engineering 2001:442–449) describes an experiment comparing the ultimate withdrawal strengths (in N/mm) for several types of nails. For an annularly threaded nail with shank diameter 3.76 mm driven into spruce-pine-fir lumber, the ultimate withdrawal strength was modeled as lognormal with \(\mu\) = 3.82 and \(\sigma\) = 0.219. For a helically threaded nail under the same conditions, the strength was modeled as lognormal with \(\mu\) = 3.47 and \(\sigma\) = 0.272.

a. What is the mean withdrawal strength for annularly threaded nails?

b. What is the mean withdrawal strength for helically threaded nails?

c. For which type of nail is it more probable that the withdrawal strength will be greater than 50 N/mm?

d. What is the probability that a helically threaded nail will have a greater withdrawal strength than the median for annularly threaded nails?

e. An experiment is performed in which withdrawal strengths are measured for several nails of both types. One nail is recorded as having a withdrawal strength of 20 N/mm, but its type is not given. Do you think it was an annularly threaded nail or a helically threaded nail? Why? How sure are you?

Equation transcription:

Text transcription:

\mu

\sigma

Questions & Answers

QUESTION:

The article “Withdrawal Strength of Threaded Nails” (D. Rammer, S. Winistorfer, and D. Bender, Journal of Structural Engineering 2001:442–449) describes an experiment comparing the ultimate withdrawal strengths (in N/mm) for several types of nails. For an annularly threaded nail with shank diameter 3.76 mm driven into spruce-pine-fir lumber, the ultimate withdrawal strength was modeled as lognormal with \(\mu\) = 3.82 and \(\sigma\) = 0.219. For a helically threaded nail under the same conditions, the strength was modeled as lognormal with \(\mu\) = 3.47 and \(\sigma\) = 0.272.

a. What is the mean withdrawal strength for annularly threaded nails?

b. What is the mean withdrawal strength for helically threaded nails?

c. For which type of nail is it more probable that the withdrawal strength will be greater than 50 N/mm?

d. What is the probability that a helically threaded nail will have a greater withdrawal strength than the median for annularly threaded nails?

e. An experiment is performed in which withdrawal strengths are measured for several nails of both types. One nail is recorded as having a withdrawal strength of 20 N/mm, but its type is not given. Do you think it was an annularly threaded nail or a helically threaded nail? Why? How sure are you?

Equation transcription:

Text transcription:

\mu

\sigma

ANSWER:

Solution 7E

Step1 of 6:

Let us consider a random variable X it presents the withdrawal strength for annularly threaded nails. And X follows lognormal distribution with mean  and standard deviation .

That is XN(3.82, 0.2192)

Also we have a random variable Y it presents the withdrawal strength for helically threaded nails. And Y follows lognormal distribution with mean  and standard deviation .

That is YN(3.47, 0.2722)

Here our goal is:

a).We need to find the mean withdrawal strength for annularly threaded nails.

b).We need to find the mean withdrawal strength for helically threaded nails.

c).We need to check for which type of nail is it more probable that the withdrawal strength will be greater than 50 N/mm.

d).We need to find the probability that a helically threaded nail will have a greater withdrawal strength than the median for annularly threaded nails.

e).We need to check whether it was an annularly threaded nail or a helically threaded nail? Why? How sure are you? When One nail is recorded as having a withdrawal strength of 20 N/mm.


Step2 of 6:

a).

The probability density function of lognormal distribution is given by

Where,

mean

standard deviation

Z = standard normal variable.

Now, the mean a withdrawal strength for annularly threaded nails is given by

               =

               =

          =

           = 46.7074

Hence, E(X) = 46.7072.


Step3 of 6:

b).

The probability density function of lognormal distribution is given by

Where,

mean

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