BME $330-$ HW $3$

P$1$: Calculate $I_{},I_{yy}$ and $I_{xy}$ for the following shape about its centroid $($all in mm $).$ Once you

obtain $I=\left[\begin{array}{cc}{I}_{}& I_{xy}\\ I_{yx}& I_{yy}\end{array}\right],$ use MTALB eig$()$ function or Mohr's circle method to find the principal

moment of area $I=\left[\begin{array}{cc}{I}_1& 0\\ 0& I_2\end{array}\right]$ and the principal direction.

$I_x={\displaystyle \underset{i=1}{\overset{n}{}}}{I}_x^i+{\displaystyle \underset{i=1}{\overset{n}{}}}{A}_i(\frac{?}{\text{b}\text{a}\text{r}}(y{)}_i-(\frac{?}{\text{b}\text{a}\text{r}}(y)){)}^2$

$I_y={\displaystyle \underset{i=1}{\overset{n}{}}}{I}_y^i+{\displaystyle \underset{i=1}{\overset{n}{}}}{A}_i({\stackrel{}{x}}_i-(\stackrel{}{x}){)}^2$

$I_{xy}={\displaystyle \underset{i=1}{\overset{n}{}}}{I}_{xy}^i+{\displaystyle \underset{i=1}{\overset{n}{}}}{A}_i({\stackrel{}{x}}_i-(\stackrel{}{x}))(\frac{?}{\text{b}\text{a}\text{r}}(y{)}_i-(\frac{?}{\text{b}\text{a}\text{r}}(y)))$

Hint: First, divide into $3$ sections. Second, find the overall centroid. Third, calculate the $I_{},I_{yy}$

and $I_{xy}$ of each section. Fourth, assemble the final I using parallel$-$axis theorem.

P$2$: A beam has a cross$-$section as shown in P$1.$ A shear load at $100$ N passes through the

centroid of the cross$-$section at an angle of $30$ degrees from the $x$ axis. Calculate the normal

stress ${}_{zz}$ experienced at a location $10$ cm away from the force application point of the beam.

Find the inclination angle of the neutral axis and compare to the principal direction from P$1.$