Please be sure to turn in your three sheets $({8\mathrm{.}5}^{\text{'}\text{'}}{11}^{\text{'}\text{'}}$ or A$4)$ of handwritten notes with your name on them, along with the questions for the exam and all the answer sheets $($blank ones, $100).$ Once the exam begins, please separate the answer sheets from the questions and write your name on at least one side of each page. Please show all work on the answer sheets. Writing on the questions themselves will not be graded. You may use a standard calculator without emailing, texting, or internet$-$enabled access. Finally, please ensure you give numerical answers using the appropriate units. Good luck and do the best you can!

Question $1(10$ points$)$

You get the job at Company XYZ$,$ and your boss asks you to mount a strain rosette on a structure you can model as a cantilevered beam with a rectangular cross$-$section.

a$).)$ Draw diagrams for the shear force and bending moment along the beam.

b$)(2$ pt$.)$ Calculate the maximum normal stress due to bending with the following parameters: width $b=2cm,$ height thickness $h=1\mathrm{.}5cm,$ length $L=0\mathrm{.}3m,$ and concentrated force $P=50N.$

c$)(3$ pt $.)$ Assuming the material of the beam is brittle and all other stresses are zero at the location and orientation used to determine the maximum normal, draw an envelope on a ${}_1-{}_2$ plot using the maximum normal stress fallure criterion with $|S_{u_i}|=S_{ul{l}_l}=400$MPa. Is the state of stress within the envelope? If so$,$ what is the safety factor?

d$)(3$ pt $.)$ We use a rectangular rosette with three axial strain gauges oriented at $0$;$45,$ and $90$ in a plane. If we mount the sirain gauge on top of the bean with the $0-$gauge aligned with the $x-$direction, what are the strains measured by all three axial strain gauges? Please indicate the corresponding orientations of the strain ganges. Note the material's elastic modulus is $200$ GPa $,$ and the Poisson's ratio is $0\mathrm{.}3.$