# Cantilever Beam - Load at Free End Deflection Vertically with No Rotation

Written by Jerry Ratzlaff on . Posted in Structural Engineering

## Cantilever Beam - Load at Free End Deflection Vertically with No Rotation formulas

 $$\large{ R = V = P }$$ $$\large{ M_{max} \; }$$   (at both end)   $$\large{ = \frac{P \;L}{2} }$$ $$\large{ M_x = P \; \left( \frac{ L }{2} - x \right) }$$ $$\large{ \Delta_{max} \; }$$   (at free end)   $$\large{ = \frac{P\; L^3}{12 \;\lambda\; I} }$$ $$\large{ \Delta_x = \frac{P \; \left( L \;-\;x \right)^2 }{12\; \lambda\; I} \; \left( L + 2\;x \right) }$$

### Where:

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

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

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

$$\large{ P }$$ = total concentrated load

$$\large{ R }$$ = 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