Beams: maximum moment

Find the maximum moment of the above beam which is subjected to triangular vertical load.

SOLUTION:

ΣΜ_Β = 0 → Α_Υ –(1/2)ql(l/3)=0 → Α_Υ = (1/6)ql

ΣF_Y = 0 → ql/6 + B_Υ –(1/2)ql=0 → B_Υ = (1/3)ql

Assuming the equilibrium in distance x from the A end:

ΣΜ_X = 0 →  (1/6)qlx – (1/2)(x/l)qx(x/3) - M_x = 0 → M_x = (ql/6)x(1 - x²/l²)

ΣF_Y = 0 → (1/6)ql –(1/2)(x/l)qx - Q_x=0 → Q_x =(ql/6)(1 - 3x²/l²)

Maximum M means zero Q:

Q_X = 0 → (ql/6)(1 - 3x²/l²) = 0 → x = l/ √ 3

maxM = (ql/6)(l/ √ 3 )(1 - l²/(3l²)) → maxM = (ql²/ 9√ 3 )