Problem 574

In the beam section shown in Fig. P-574, prove that the maximum horizontal shearing stress occurs at layers h/8 above or below the NA.

Solution 574

[math]f_v = \dfrac{VQ}{Ib}[/math]

Where
[math]b = x[/math]

[math]Q = \frac{1}{2}xy\,(\frac{1}{2}h - \frac{2}{3}y) = \frac{1}{2}xy \left( \dfrac{3h - 4y}{6} \right)[/math]

[math]Q = \frac{1}{12}xy(3h - 4y)[/math]

[math]f_v = \dfrac{V \, [ \, \frac{1}{12}xy(3h - 4y) \,]}{Ix} = \dfrac{V}{12I}(3hy - 4y^2)[/math]

[math]\dfrac{df_v}{dy} = \dfrac{V}{12I}(3h - 8y) = 0[/math]

[math]3h = 8y[/math]
[math]y = \frac{3}{8}h[/math]

Location of maximum horizontal shearing stress:
[math]d = \frac{1}{2}h - y = \frac{1}{2}h - \frac{3}{8}h[/math]
[math]d = \frac{1}{8}h \,\, [/math]           answer