$($a$)$ Overvie

$($b$)$ Zoom$-$it

at the botti

the shaft

$($c$)$ Zoom$-$in "top" view of the solid

Particle at the bottom of the

middle of the shaft

Figure $1$: The shaft subject to a torque and a venaing moment $($induced by two point loadings$).$

$P$ of a single load $=2000N,L=3m,T=1000Nm,r=0\mathrm{.}05m.$

Consider a circular solid shaft that is subject to two point loads $($each by $P)$ and a torque $(T)$ throughout the shaft. Each point loading $(P)$ is located $\frac{L}{3}$ away from a thin bearing. The point loading, $P,$ induces the bending moment $($and$,$ thus, normal stress$)$ throughout the shaft, and the torque, $T,$ leads to shear stress at a cross section throughout the shaft $($see Fig. $1($a$)).$

Then, consider a solid particle at the bottom of the middle of the shaft. Its stress state is shown in Fig. $1($b$).$ Please note that we are ignoring the shear stress at the solid particle that is attributed to the shear force induced by the point loading.

The stress state of this solid particle in the view from the top is our $2$D stress state of interest $($see Fig. $1($c$)).$

Problem $1(20$ pts$)$ Draw the shear force diagram $($SFD$)$ and bending moment diagram $($BMD$)$ of the shaft, which are induced by two point loads.

Problem $2(20$ pts$)$ Compute all the stresses mentioned above at the solid particle shown in Fig. $1($c$).$ Use the following formula and values for computing stresses $($ and $).$ Then, draw the Mohr's circle for this solid.

$=\frac{Tr}{J},=\frac{Mr}{\text{I}},J=\frac{r^4}{2},I=\frac{r^4}{4}$

$Pof$ a single load $-2000N,L=3m,T=1000Nm,r=0\mathrm{.}05m.$

Problem $3(25$ pts$)$ Compute the maximum tensile stress $(($:${}_1$