# Cantilever Beam - Uniformly Distributed Load and Variable End Moments

Written by Jerry Ratzlaff on . Posted in Structural

### Cantilever Beam - Uniformly Distributed Load and Variable End Moments Formula

$$\large{ R = V = wL }$$

$$\large{ V_x = wx }$$

$$\large{ M_{max} \; }$$   (at fixed end)   $$\large{ = \frac{w L^2}{3} }$$

$$\large{ M_1 \; }$$   (at free end)   $$\large{ = \frac {w L^2} {6} }$$

$$\large{ M_x = \frac{ w }{6} \left( L^2 - 3x^2 \right) }$$

$$\large{ \Delta_{max} \; }$$   (at free end)   $$\large{ = \frac{w L^4}{24 \lambda I} }$$

$$\large{ \Delta_x = \frac{w \left( L^2 - \left( L - x \right)^2 \right)^2 }{24 \lambda I} }$$

Where:

$$\large{ I }$$ = moment of inertia

$$\large{ L }$$ = span length of the bending member

$$\large{ M }$$ = maximum bending moment

$$\large{ R }$$ = reaction load at bearing point

$$\large{ V }$$ = shear force

$$\large{ w }$$ = load per unit length

$$\large{ W }$$ = total load from a uniform distribution

$$\large{ x }$$ = horizontal distance from reaction to point on beam

$$\large{ \lambda }$$   (Greek symbol lambda) = modulus of elasticity

$$\large{ \Delta }$$ = deflection or deformation