Simple Beam - Concentrated Load at Any Point

Written by Jerry Ratzlaff on . Posted in Structural

sb 8DSimple Beam - Concentrated Load at Any Point Formula

\(\large{ R_1 = V_1 }\)  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

 

Tags: Equations for Beam Support