Cantilever Beam - Concentrated Load at Any Point

Written by Jerry Ratzlaff on . Posted in Structural

Cantilever Beam - Concentrated Load at Any Point formulas

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

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