# Simple Beam - Concentrated Load at Any Point

Written by Jerry Ratzlaff on . Posted in Structural

### Simple Beam - Concentrated Load at Any Point Formula

(Eq. 1)  $$\large{ R_1 = V_1 }$$  max. when  $$\large{ \left( a < b \right) = \frac {Pb} {L} }$$

(Eq. 2)  $$\large{ R_2 = V_2 }$$  max. when  $$\large{ \left( a > b \right) = \frac { Pa} {L} }$$

(Eq. 3)  $$\large{ M_{max} \; }$$  (at point of load)  $$\large{ = \frac { Pab } { L } }$$

(Eq. 4)  $$\large{ M_x \; }$$  when $$\large{ \left( x < b \right) = \frac { Pbx } { L } }$$

(Eq. 5)  $$\large{ \Delta_a \; }$$  (at point of load)  $$\large{ = \frac { Pa^2b^2 } { 3 \lambda I L } }$$

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

(Eq. 7)  $$\large{ \Delta_x \; }$$  when $$\large{ \left( x > a \right) = \frac { Pa \left( L - x \right) } { 6 \lambda I L } \left( 2Lx - x^2 - a^2 \right) }$$

(Eq. 8)  $$\large{ \Delta_{max} \; }$$  at  $$\large{ \left( x = \sqrt{ \frac{ a \left( a + 2b \right) }{3} } \right) }$$    when  $$\large{ \left( a > b \right) = \frac { Pab \left( a + 2b \right) \sqrt{ 3a \left( a + 2b \right) } } { 27 \lambda I L } }$$

Where:

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

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

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

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

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

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

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

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

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

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