# Cantilever Beam - Uniformly Distributed Load and Variable End Moments

Written by Jerry Ratzlaff on . Posted in Structural Engineering

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

 $$\large{ R = V = w\;L }$$ $$\large{ V_x = w\;x }$$ $$\large{ M_{max} \; }$$   (at fixed end)   $$\large{ = \frac{w\; L^2}{3} }$$ $$\large{ M_1 \; }$$   (at free end)   $$\large{ = \frac {w \;L^2} {6} }$$ $$\large{ M_x = \frac{ w }{6} \; \left( L^2 - 3\;x^2 \right) }$$ $$\large{ \Delta_{max} \; }$$   (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