The member having a rectangular cross section, Fig. $6-28$a$,$ is

designed to resist a moment of $40N*m.$ In order to increase its

strength and rigidity, it is proposed that two small ribs be added at

its bottom, Fig. $6-28$b$.$ Determine the maximum normal stress in the

member for both cases.

SOLUTION

Without Ribs. Clearly the neutral axis is at the center of the cross

section, Fig. $6-28$a$,$ so $\frac{?}{\text{b}\text{a}\text{r}}(y)=c=15mm=0\mathrm{.}015m.$ Thus,

$I=\frac{1}{12}b{h}^3=\frac{1}{12}(0\mathrm{.}060m)(0\mathrm{.}030m{)}^3=0\mathrm{.}135({10}^{-6})m^4$

Therefore the maximum normal stress is

${}_{\text{m}\text{a}\text{x}}=\frac{Mc}{\text{I}}=\frac{(40(N)*m)(0\mathrm{.}015(m))}{0\mathrm{.}135({10}^{-6})m^4}=4\mathrm{.}44$MPa

With Ribs. From Fig. $6-28$b$,$ segmenting the area into the large

main rectangle and the bottom two rectangles $($ribs$),$ the location $\frac{?}{\text{b}\text{a}\text{r}}(y)$ of

the centroid and the neutral axis is determined as follows:

$\frac{?}{\text{b}\text{a}\text{r}}(y)=\frac{(tilde(y))A}{A}$

$=\frac{[0\mathrm{.}015(m)](0\mathrm{.}030(m))(0\mathrm{.}060(m))+2[0\mathrm{.}0325(m)](0\mathrm{.}005(m))(0\mathrm{.}010(m))}{(0\mathrm{.}03(m))(0\mathrm{.}060(m))+2(0\mathrm{.}005(m))(0\mathrm{.}010(m))}$

$=0\mathrm{.}01592m$

This value does not represent $c.$ Instead

$c=0\mathrm{.}035m-0\mathrm{.}01592m=0\mathrm{.}01908m$

Using the parallel$-$axis theorem, the moment of inertia about the

neutral axis is

$I=[\frac{1}{12}(0\mathrm{.}060(m))(0\mathrm{.}030(m){)}^3+(0\mathrm{.}060(m))(0\mathrm{.}030(m))(0\mathrm{.}01592(m)-0\mathrm{.}015(m){)}^2]$

$+2[\frac{1}{12}(0\mathrm{.}010(m))(0\mathrm{.}005(m){)}^3+(0\mathrm{.}010(m))(0\mathrm{.}005(m))(0\mathrm{.}0325(m)-0\mathrm{.}01592(m){)}^2]$

$=0\mathrm{.}1642({10}^{-6})m^4$

Therefore, the maximum normal stress is

${}_{\text{m}\text{a}\text{x}}=\frac{Mc}{\text{I}}=\frac{40(N)*m(0\mathrm{.}01908(m))}{0\mathrm{.}1642({10}^{-6})m^4}=4\mathrm{.}65$MPa

NOTE: This surprising result indicates that the addition of the ribs to

the cross section will increase the maximum normal stress rather than

decrease it$,$ and for this reason they should be omitted.