Simple Beam - Uniformly Distributed Load and Variable End Moments

Written by Jerry Ratzlaff on . Posted in Structural

sb 12Dformulas that use Simple Beam - Uniformly Distributed Load and Variable End Moments

\(\large{ R_1 = V_1  = \frac { w\;L }  { 2 }  +  \frac { M_1\; - \;M_2 }  { L }   }\)   
\(\large{ R_2 = V_2  =  \frac { w\;L }  { 2 }  -  \frac { M_1\; - \;M_2 }  { L }    }\)   
\(\large{ V_x   =   w \;  \left(   \frac { L }  { 2 } - x  \right)     +  \frac { M_1 \;-\; M_2 }  { L } }\)   
\(\large{ a }\)  (inflection points)  \(\large{ =  \sqrt{  \frac { L^2 } { 4 }  -  \left( \frac { M_1 \;+ \;M_2 }  { w } \right)  +  \left(  \frac { M_1 \;+\; M_2 }  { w\;L }  \right)^2    }  }\)  
\(\large{ M_x   =  \frac { w\;x } { 2 } \;  \left( L - x  \right)  +      \left(  \frac { M_1\; - \;M_2 } { L }  \right)  x -M_1    }\)  
\(\large{ M_3 }\)  at  \(\large{    \left(  x =  \frac { L } { 2 }  +  \frac { M_1\; - \;M_2 }  { w\;L }  \right)     =    \frac { w\;L^2 } { 8 }  -  \frac { M_1 \;+ \;M_2 }  { 2 }  +  \frac  {  \left(  M_1 \;- \;M_2  \right)^2 }  { 2\;w\;L^2 }            }\)  
\(\large{ \Delta_x   =    \frac { w\;x } { 48\; \lambda\; I }  \; \left[  x^3  - \;  \left(  2\;L + \frac { 4\;M_1 } { w\;L }  -   \frac { 4\;M_2 } { w\;L }  \right)  x^2   +     \frac { 12\;M_1 } { w }  + L^3  + \frac { 8\;M_1 \;L } { w }    -  \frac { 4\;M_2\; L } { w }    \right]  }\)  

Where:

\(\large{ I }\) = moment of inertia

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

\(\large{ M }\) = maximum bending moment

\(\large{ R }\) = reaction load at bearing point

\(\large{ V }\) = maximum shear force

\(\large{ w }\) = load per unit length

\(\large{ x }\) = 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