# Simple Beam - Concentrated Load at Any Point

Written by Jerry Ratzlaff on . Posted in Structural Engineering

## Simple Beam - Concentrated Load 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)  $$\large{ = \frac{ P\;a\;b }{ L } }$$ $$\large{ M_x \; }$$  when $$\large{ \left( x < b \right) = \frac{ P\;b\;x }{ L } }$$ $$\large{ \Delta_a \; }$$  (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:

$$\large{ \Delta }$$ = deflection or deformation

$$\large{ a, b }$$ = distance to point load

$$\large{ x }$$ = horizontal distance from reaction to point on beam

$$\large{ M }$$ = maximum bending moment

$$\large{ V }$$ = maximum shear force

$$\large{ I }$$ = moment of inertia

$$\large{ \lambda }$$   (Greek symbol lambda) = modulus of elasticity

$$\large{ R }$$ = reaction load at bearing point

$$\large{ L }$$ = span length of the bending member

$$\large{ P }$$ = total concentrated load