# Overhanging Beam - Point Load on Beam End

Written by Jerry Ratzlaff on . Posted in Structural

## Overhanging Beam - Point Load on Beam End formulas

 $$\large{ R_1 = V_1 = \frac{P\;a}{L} }$$ $$\large{ R_2 = V_1 + V_2 = \frac{ P }{L} \; \left( L + a \right) }$$ $$\large{ V_2 = P }$$ $$\large{ M_{max} \; }$$  at   $$\large{ \left( R_2 \right) = P\;a }$$ $$\large{ M_x \; }$$  (between supports)    $$\large{ = \frac{P\;a\;x}{L} }$$ $$\large{ M_{x_1} \; }$$  (for overhang)    $$\large{ = P \left( a - x_1 \right) }$$ $$\large{ \Delta_x \; }$$  (between supports)    $$\large{ = \frac{ -\;P\;a\;x }{6\; \lambda \;I\;L} \; \left( L^2 - x^2 \right) }$$ $$\large{ \Delta_{x_1} \; }$$  (overhang)    $$\large{ = \frac{ P\;x_1 }{6\; \lambda\; I } \; \left( 2\;a\;L + 3\;a\;x_1 - x_{1}{^2} \right) }$$ $$\large{ \Delta_{max} \; }$$  for overhang at   $$\large{ \left(x_1 = a \right) = \frac{ P\;a^2 }{3\; \lambda \;I } \; \left( L + a \right) }$$ $$\large{ \Delta_{max} \; }$$  between supports at   $$\large{ \left( x = \frac{L}{\sqrt{3}} \right) = \frac{ -\;P\;a\;L^2 }{9\; \sqrt{3} \; \lambda\; I } = 0.06415 \; \frac{ P\;a\;L^2 }{ \lambda\; I } }$$

### 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