Exercise $18\mathrm{.}1.$ An $x-y$ plane CAD drawing of a beam section is shown at right. The CAD program's analysis of the cross$-$section is as follows:

Area

Bounding box:

Centroid:

Moments of inertia:

X: $10266693333\mathrm{.}3333$

Y: $2266693333\mathrm{.}3333$

Products of inertia: XY: $,4000000000\mathrm{.}0000$

Radii of gyration:

$x$:$358\mathrm{.}2369$

$Y$:$168\mathrm{.}3261$

$Z$:$395\mathrm{.}8114$

Principal moments and $x-Y-Z$ directions about centroid:

a$.)$ Verify the $x-$ and $y-$centroid coordinates.

b$.)$ Verify the moments of inertia $(I_x,I_y,$ and $I_{xy})$ which are taken about the global $x-$ and $y-$axes rather than the centroidal $x-$ and $y-$axes.

c$.)$ The principal centroidal axes of the section are given as: $x^{\text{'}}=\mathrm{.}8575x-\mathrm{.}5145y$ and $y^{\text{'}}=\mathrm{.}5145x+\mathrm{.}8575y.$ The associated centroidal principal moments of inertia are: $I_{x^{\text{'}}}=2\mathrm{.}268*{10}^{9}m{m}^4$ and $I_{y^{\text{'}}}=5\mathrm{.}667*{10}^{8}m{m}^4.$ Sketch these axes on the section.

d$.)$ A moment of $15kNm$ is applied about the section$\text{'}$s centroidal $x-$axis. Resolve this into components about the principal axes.

e$.)$ Solve for the neutral axis on the section and sketch it$.$

f$.)$ Calculate the maximum tensile and compressive bending stresses on the section and identify the points where they occur.