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Cantilever Beam with Uniform Distributed Load

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Find support reactions and maximum shear and bending moment for a cantilever beam with UDL.

Concept

For a cantilever beam fixed at one end and free at the other, carrying a uniform load: The fixed support resists all downward forces with a vertical reaction R_{y} . The fixed support prevents rotation by providing a reaction moment M_{A} . R_{y} = w L M_{A} = \frac{w L ^{2}}{2}

A cantilever beam of length L is fixed at the left end and carries a uniform distributed load w over its entire span. Find: Vertical reaction and reaction moment at the fixed support Maximum shear force and its location Maximum bending moment and its location

L = 10 m w = 5 kN / m (downward)

Vertical equilibrium: R_{y} = w L . Moment equilibrium about the fixed end: the resultant of the UDL acts at midspan with magnitude w L, so M_{A} = w L \cdot (L /2) = w L^{2} /2 . R_{y} = (5 kN/m) (10 m) = 50 kN ↑ M_{A} = \frac{( 5 ) ( 1 0 ^{2} )}{2} = 250 kN \cdot m (CCW)

For a cantilever with UDL, the shear is maximum at the fixed support (equal to R_{y}). The bending moment is maximum (most negative) at the fixed support. V_{m a x} = 50 kN at x = 0 M_{m a x} = - 250 kN \cdot m at x = 0

R_{y} = 50 kN ↑ M_{A} = 250 kN \cdot m (CCW) V_{m a x} = 50 kN at fixed support M_{m a x} = - 250 kN \cdot m at fixed support

R_{y} = w L (vertical reaction) M_{A} = \frac{w L ^{2}}{2} (reaction moment at fixed support) where w = uniform load, L = beam span

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