Simple Beam - Central Point Load and Variable End Moments

Written by Jerry Ratzlaff on . Posted in Structural

sb 13Dformulas that use Simple Beam - Central Point Load and Variable End Moments

\(\large{ R_1 = V_1  =  \frac { P }  { 2 }  +  \frac { M_1 \;- \;M_2 }  { L }   }\)   
\(\large{ R_2 = V_2  =  \frac { P }  { 2 }  -  \frac { M_1 \;-\; M_2 }  { L }    }\)   
\(\large{ M_3  }\)  (at center) \(\large{ =  \frac { P\;L }  { 4 }  -  \frac { M_1 \;+\; M_2 }  { L }    }\)   
\(\large{ M_x   \left(  x <  \frac{L} {2}    \right)   =   \left(     \frac { P }  { 2 }  +  \frac { M_1\; -\; M_2 }  { L }  \right)  x - M_1    }\)  
\(\large{ M_x   \left(  >  \frac{L} {2}    \right)   =  \frac {P}{2} \; \left( L - x  \right)  +   \frac { \left(  M_1 \;-\; M_2  \right) \;x }  { L } \; -\; M_1    }\)  
\(\large{ \Delta_x   \left( x <  \frac{L} {2}    \right)    =    \frac { P\;x } { 48\; \lambda\; I }   \;      \left[        3\;L^2  -  4\;x^2 -    \frac {  8\;  \left( L\; -\; x  \right) }  { P\;L }  \;    \left[ M_1 \left( 2\;L - x \right)   +  M_2\; \left( L + x \right)  \right]       \right]  }\)  
\(\large{ x }\)  (first point of contraflexure)  \(\large{ =  \frac { 2\;L\; M_1 }  { L\;P \;+ \;2\;M_1\; - \;2\;M_2 }    }\)  
\(\large{ x }\)  (second point of contraflexure)  \(\large{ =  \frac {  L \; \left( L\;P\; - \;2\;M_1     \right)       }  { L\;P \;+ \;2\;M_1 \;+\; 2\;M_2 }    }\)  

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

\(\large{ V }\) = 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

 

Tags: Equations for Beam Support