f

_{b} = stress at c

Let

f

_{1} = stress at y

_{1}
f

_{2} = stress at y

_{2}
$f_b = \dfrac{Mc}{I}$

$M = \dfrac{f_b \, I}{c}$

At y_{1}:

$f_1 = \dfrac{My_1}{I} = \dfrac{\dfrac{f_b \, I}{c}y_1}{I}$

$f_1 = \dfrac{f_b \, y_1}{c}$

At y_{2}:

$f_2 = \dfrac{My_2}{I} = \dfrac{\dfrac{f_b \, I}{c}y_2}{I}$

$f_2 = \dfrac{f_b \, y_2}{c}$

At (barred y'):

$f\,' = \dfrac{f_1 + f_2}{2} = \dfrac{\dfrac{f_b \, y_1}{c} + \dfrac{f_b \, y_2}{c}}{2}$

$f\,' = \dfrac{f_b}{c} \left( \dfrac{y_1 + y_2}{2} \right)$

$f\,' = \dfrac{f_b}{c}\,\bar{y}\,'$

$F = f\,' \, A' = \dfrac{f_b}{c}\,\bar{y}\,' \, A'$

$F = (f_b \, / \, c) \, A' \, \bar{y}\,'$ (*okay!*)

$M = F \bar{y}\,'$

$M = [ \, (f_b \, / \, c) \, A' \, \bar{y}\,' \, ] \, \bar{y}\,'$

$M = (f_b \, / \, c) \, A' \, \bar{y}\,'^2$

but $\,\, A' \, \bar{y}\,'^2 = I'$

thus,

$M = ( \, f_b \, / \, c \, ) \, I'$

$M = f_b \, I' \, / \, c$ (*okay!*)