Overhanging Beam - Point Load Between Supports at Any Point

Written by Jerry Ratzlaff on . Posted in Structural

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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

 

Tags: Equations for Beam Support