Question

If a beam of rectangular cross-section is subjected to a vertical shear force V, the shear force carried by the upper one-third of the cross section is

• A. Zero
• B. $\frac{7V}{27}$
• C. $\frac{8V}{27}$
• D. $\frac{V}{3}$

Answer 1

The correct option is B $\frac{7V}{27}$


Upper one third means y varies from d/6 to d/2

Shear stress at distance y from NA,

$\tau =\frac{VA\overline{y}}{Ib}$

$shearforce=\int shearstress\times Area$

$=\underset{d/6}{\overset{d/2}{\int }}\frac{VA\overline{y}}{Ib}.b.dy$

$=V\underset{d/6}{\overset{d/2}{\int }}\frac{(\frac{d}{2}-y).b.\frac{1}{2}(\frac{d}{2}+y)dy}{\frac{b{d}^3}{12}}$

$=\frac{6V}{d^3}\underset{d/6}{\overset{d/2}{\int }}(\frac{d^2}{4}-y^2)dy$

$=\frac{6V}{d^3}{[\frac{d^2}{4}y-\frac{y^3}{3}]}_{d/6}^{d/2}$

$=\frac{6V}{d^3}\times \frac{7d^3}{162}=\frac{7}{27}V$