Beam Fixed at One End - Concentrated Load at Center

Written by Jerry Ratzlaff on . Posted in Structural

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Beam Fixed at One End - Concentrated Load at Center formulas

\(\large{ R_1 = V_1 = \frac {5\;P} {16}  }\)   
\(\large{ R_2 = V_2 = \frac {11\;P} {16}  }\)   
\(\large{ M_{max}  }\)  (at fixed end)  \(\large{ =  \frac {3\;P\;L} {16}  }\)   
\(\large{ M_1  }\)  (at point of load)  \(\large{ =  \frac {5\;P\;L} {32}  }\)  
\(\large{ M_x   \; }\)  when \(\large{    \left(  x < \frac {L}{2}    \right)   =   \frac  { 5\;P\;x} {16}    }\)  
\(\large{ M_x   \; }\)  when \(\large{    \left(  x > \frac {L}{2}    \right)   =  P\; \left(  \frac { L} {2}  - \frac { 11\;x} {16}  \right)  }\)  
\(\large{ \Delta_{max}   \; }\)  at  \(\large{  \left( x = L\; \sqrt { \frac {1}{5} } = .4472\;L  \; \right)   =  \frac {P\;L^3} {48\; \lambda\; I \;\sqrt {5}  }  =  .009317 \; \frac { P\;L^3} { \lambda\; I}    }\)  
\(\large{ \Delta_x \; }\)  (at point of load)  \(\large{ =  \frac { 7\;P\;L^3} {768\; \lambda\; I}  }\)  
\(\large{ \Delta_x   \; }\)  when \(\large{    \left(  x < \frac {L}{2}    \right)   =   \frac  { P\;x} {96 \;\lambda\; I} \; \left( 3\;L^2 - 5\;x^2  \right)    }\)  
\(\large{ \Delta_x   \; }\)  when \(\large{    \left(  x > \frac {L}{2}    \right)   =  \frac  { P} {96 \;\lambda\; I}\;  \left( x - L  \right)^2 \; \left( 11\;x - 2\;L  \right)   }\)  

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