# 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

### Point Load Between Supports at Any Point Formula

$$\large{ R_1 = V_1 \; }$$  max. when   $$\large{ \left( a < b \right) = \frac{Pb}{L} }$$

$$\large{ R_2 = V_2 \; }$$  max. when   $$\large{ \left( a > b \right) = \frac{Pa}{L} }$$

$$\large{ M_{max} \; }$$  (at point of load)  $$\large{ = \frac{Pab}{L} }$$

$$\large{ M_x \; \left( x < a \right) = \frac{Pbx}{L} }$$

$$\large{ \Delta_{x_1} = \frac{ Pabx_1 }{6 \lambda IL } \left( L + a \right) }$$

$$\large{ \Delta_a \; }$$  (at point of load)    $$\large{ = \frac{ Pa^2 b^2 }{3 \lambda IL} }$$

$$\large{ \Delta_x \; }$$  when  $$\large{ \left(x < a \right) = \frac{ Pbx }{6 \lambda IL} \left( L^2 - b^2 - x^2 \right) }$$

$$\large{ \Delta_x \; }$$  when  $$\large{ \left(x > a \right) = \frac{ Pa \left( L - x \right) }{6 \lambda IL} \left( 2Lx - x^2 - a^2 \right) }$$

$$\large{ \Delta_{max} \; }$$  at   $$\large{ \left( x = \sqrt{ \frac{ a \left(a + 2b \right) }{3} } \right) \; }$$  when  $$\large{ \left(a > b \right) = \frac{ Pab \left( a + 2b \right) \sqrt{ 3a \left( a + 2b \right) } } {27 \lambda IL } }$$

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