Problem $1(40$ points$)$

a$.$ For a biaxial state of stress at a critical point of a machine part the principal stresses are $S_A=30$ksi and $S_B=-15$ksi. Answer the following related questions $($each for $4$ pts $).$

i$.$ What is the maximum shear stress?

ii$.$ What is the value of $\frac{s_x+s_y}{2}.$

iii. On a Mohr circle, if $s_1$ makes an angle of $25$ the $+x-$axis, what would be the angle of the normal to the plane of $t_{\text{m}\text{a}\text{x}}$ and the corresponding $s_x$ and $t_{xy}?$

iv$.$ What would be the normal stresses acting on the planes of $t_{\text{m}\text{a}\text{x}}.$

v $.$ If the machine part is made of steel with $S_{yt}=45$ksi, determine graphically the FS using appropriate failure theory.

b$.$ Determine the stress concentration factor at the notch of the member shown. $(5$ pts$)$

c$.$ If the angle of twist of a hollow elliptical cross$-$section shaft is to be kept to $<mn>1</mn>$deg, what would be the FS if: shaft length $L=1m,$ applied torque $T=100$ N$.$m$,$ cross section outer dimensions $0\mathrm{.}05$ m $,$ depth $0\mathrm{.}01$ m $,$ and thickness $0\mathrm{.}0025m.G=79$GPa. $(7$ pts $)$

d$.$ The critical point $($ CP $)$ on a shaft subject to bending and torsion has a shear stress of $20000$ Pa $.$ The maximum shear stress at the CP was determined to be $100,000Pa.$ What is the bending moment acting at the section of the CP point if the diameter of the shaft cross section is $0\mathrm{.}1$ m $.(8$ pts