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$)(2$ pt$.)$ 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 failure criterion with $|S_{\text{u}\text{l}\text{t}}^c{?}_c|=S_{ul{t}_t}=400$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 strain gauge on top of the beam 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 gauges. Note: the material's elastic modulus is $200$ GPa $,$ and the Poisson's ratio is $0\mathrm{.}3.$