Question 12.5: Assume a cantilever beam with rectangular cross-section. Obt...
Assume a cantilever beam with rectangular cross-section. Obtain an expression of the distance of the neutral axis as measured from the centroid of the section as a function of x for the loading shown in Figure 12.12.
Let us draw the free-body diagram of the beam section considered at a distance of x measured from the fixed end of the beam as shown in Figure 12.13.
Thus, normal stress at any point ( y, z) on the beam section is given by
\sigma_{x x}=\left\lgroup \frac{P}{A} \right\rgroup+\left\lgroup -\frac{M_x y}{\bar{I}_{z z}} \right\rgroup
=\frac{P}{b h}-\frac{12 P(L-x)}{b h^3} y
For NA, we must have \sigma_{x x}=0
\Rightarrow \frac{P}{b h}\left[1-\frac{12(L-x)}{h^2} y\right]=0
\Rightarrow \quad 1-\frac{12(L-x)}{h^2} y=0
\Rightarrow \quad y=\frac{h^2}{12(L-x)}
which is the required distance of NA as measured from the centroid of the beam cross-section.
