The following problems must be completed by hand except where noted. You are highly encouraged to verify your results in Excel. However, an Excel spreadsheet will not be submitted and all work must be shown on paper. Final submission is in the form of a pdf document.
Problem 1
A technician at a steel mill records the results of a tension coupon test on three different random steel plates per day over a period of 5 days. The yield strength in megapascals is a measure of the largest stress the material can support before permanent deformation occurs.
Day Yield Strength (MPa)
1 350 367 345
2 339 329 360
3 343 353 359
4 325 380 346
5 351 359 341
Assume that each measurement is xi and the total number of measurements in the sample is n. You may not use Excel to complete your calculations.
a) Calculate the Yield Strength sum as ∑𝑥!.
b) Determine the mean Yield Strength.
c) Calculate the sum ∑𝑥!” for the sample.
d) Determine the standard deviation of the sample.
e) Group the data into approximately equal bins and construct a frequency distribution table. The Yield Strength bins should be <335, 336-345, 346-355, 356-365, 366-375 and >375.
f) Draw a histogram of the distribution. You may present this chart using Excel output.
Problem 2
Strain gauges were mounted on a structure and recorded the strain (e) versus stress (s) as load was applied to the structure. Two different locations were recorded.
Location 1 Location 2
Strain 𝜖 Stress 𝜎 (Mpa) Stress 𝜎 (Mpa)
0 90 208
0.0001 132 229
0.0002 154 244
0.0003 180 224
0.0004 206 247
0.0005 221 282
0.0006 231 330
0.0007 273 312
0.0008 250 311
0.0009 305 364
We are interested in obtaining the linear relationship between the strain (e) and stress (s). For Location 1, analysis showed the material stress s followed the relationship
𝜎=214000𝜖+108
This relationship shows there is a residual stress 𝜎#$ of 108 MPa already existing on the structure, when the applied strain e is zero. The material modulus of elasticity, E, is 214 000 MPa.
You are expected to develop a similar relationship for Location 2. You should consider strain as the independent variable X, and stress as the dependent variable Y.
a) Calculate ∑𝑋!.
b) Calculate ∑𝑌!.
c) Calculate ∑𝑋!𝑋!.
d) Calculate ∑𝑌!𝑌!.
e) Calculate ∑𝑋!𝑌!.
f) Using the least squares approach, calculate the slope m and y-intercept b for the straight line which best fits the data for Location 2. You may NOT use Excel.
g) Calculate the coefficient of correlation, r, for the line found in step f. You may NOT use Excel.