# Overhanging Beam - Point Load Between Supports at Any Point

Written by Jerry Ratzlaff on . Posted in Structural

## Overhanging Beam - Point Load Between Supports at Any Point formulas

 $$\large{ R_1 = V_1 \; }$$  max. when   $$\large{ \left( a < b \right) = \frac{P\;b}{L} }$$ $$\large{ R_2 = V_2 \; }$$  max. when   $$\large{ \left( a > b \right) = \frac{P\;a}{L} }$$ $$\large{ M_{max} \; }$$  (at point of load)  $$\large{ = \frac{P\;a\;b}{L} }$$ $$\large{ M_x \; \left( x < a \right) = \frac{P\;b\;x}{L} }$$ $$\large{ \Delta_{x_1} = \frac{ P\;a\;b\;x_1 }{6\; \lambda\; I\;L } \; \left( L + a \right) }$$ $$\large{ \Delta_a \; }$$  (at point of load)    $$\large{ = \frac{ P\;a^2\; b^2 }{3\; \lambda\; I\;L} }$$ $$\large{ \Delta_x \; }$$  when  $$\large{ \left(x < a \right) = \frac{ P\;b\;x }{6\; \lambda\; I\;L} \; \left( L^2 - b^2 \;-\; x^2 \right) }$$ $$\large{ \Delta_x \; }$$  when  $$\large{ \left(x > a \right) = \frac{ P\;a \; \left( L\; - \;x \right) }{6\; \lambda\; I\;L} \; \left( 2\;L\;x\; - \;x^2\; -\; a^2 \right) }$$ $$\large{ \Delta_{max} \; }$$  at   $$\large{ \left( x = \sqrt{ \frac{ a \; \left(a \;+\; 2\;b \right) }{3} } \right) \; }$$  when  $$\large{ \left(a > b \right) = \frac{ P\;a\;b \; \left( a \;+\; 2\;b \right) \; \sqrt{ 3\;a \; \left( a \;+\; 2\;b \right) } } {27\; \lambda \;I\;L } }$$

### 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