# Cantilever Beam - Concentrated Load at Any Point

Written by Jerry Ratzlaff on . Posted in Structural

## formulas that use Cantilever Beam - Concentrated Load at Any Point

 $$\large{ R = V = P }$$ $$\large{ M_{max} \; }$$   (at fixed end)   $$\large{ = Pb }$$ $$\large{ M_x \; }$$ when  $$\large{ \left( x > a \right) = P \left( x - a \right) }$$ $$\large{ \Delta_{max} \; }$$   (at free end)   $$\large{ = \frac {P b^2} {6 \lambda I} \left( 3L - b \right) }$$ $$\large{ \Delta_a \; }$$   (at point of load)   $$\large{ = \frac {P b^3} {3 \lambda I} }$$ $$\large{ \Delta_x \; }$$ when  $$\large{ \left( x < a \right) = \frac {P b^2 } {6 \lambda I} \left( 3L - 3x - b \right) }$$ $$\large{ \Delta_x \; }$$ when  $$\large{ \left( x > a \right) = \frac {P \left( L - x \right)^2 } {6 \lambda I} \left( 3b - L + x \right) }$$

### 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