Problem $6(15$ points$)$

Determine the following for the beam cross$-$section $(x^{\text{'}}-y$ plane$)$ shown in the figure below:

The centroid $y-$coordinate $(\frac{?}{\text{b}\text{a}\text{r}}(y))$ for the beam's cross$-$sectional area $(5$ points$)$

The moment of inertia about the $x^{\text{'}}$ axis, $I_{x^{\text{'}}}(5$ points$)$

The moment of inertia about the $y$ axis, $I_y(5$ points$)$

Problem $7(20$ points$)$

Consider the beam in the figure below has the same cross$-$section as Problem $6.$

Draw the shear and moment diagrams $(10$ points$)$

What is the magnitude of the peak moment in the beam and where along the length does it act? $(3$ points$)$

If the peak moment is q$($iented about the $x^{\text{'}}$ axis $($on Problem $6),$ what is the peak axial stress in the beam in MPa? $(3$ pyss$)$

Where is the peak stress located $($top or bottom of the beam$)?(2$ points$)$

Is the peak stress tensile or compressive? $(2$ points$)$

Note: $1lb=4\mathrm{.}448N$;$1ft=0\mathrm{.}3048m$

Parts $($I$)$ and $(2)$ can be solved in ft$-$lb units. Only part $(3)$ requires a unit conversion.

The figure in Problem $6$ is in the correct view of the beam cross$-$section $($i$.$e$.,$ vertical axis is $y-$axis$)$