Simple Beam - Uniform Load Partially Distributed at Each End

Simple Beam - Uniform Load Partially Distributed at Each End Formula

\(\large{ R_1 = V_1 = \frac{ w_1 \;a \;  \left(  2\;L \;-\; a  \right) \; + \;w_2 \;c^2  }{ 2\;L }     }\)

\(\large{ R_2 = V_2 = \frac{  w_2 \;c  \; \left(  2\;L \;-\; c  \right)  \;+\; w_1\; a^2  }{ 2\;L }    }\)

\(\large{ V_x   \; }\)  when \(\large{  \left(  x < a \right)  =  R_1 - w_1 \;x  }\)     

\(\large{ V_x  \; }\)  when \(\large{  \left[   a < x <  \left(  a + b  \right) \right]  =  R_1 - w_1 \;a   }\)

\(\large{ V_x   \; }\)  when \(\large{   \left[  x >  \left(  a + b  \right)  \right] =  R_2 - w_2  \left(  1 - x  \right)    }\)

\(\large{ M_{max} \; }\)   at  \(\large{ \left(  x = \frac{R_1}{w_1}  \right)  }\)  when  \(\large{ \left(  R_1 < w_1 \;a  \right)   =  \frac{ R_{1}{^2} }{ 2\;w_1  }   }\)

\(\large{ M_{max} \; }\)   at  \(\large{ \left(  x = L - \frac{R_2}{w_2}  \right)  }\)  when  \(\large{ \left(  R_2 < w_2 \;c  \right)   =  \frac{ R_{2}{^2} }{ 2\;w_2  }   }\)

\(\large{ M_x  \; }\)  when \(\large{  \left(  w < a \right)     =  R_1 \;x  - \frac{ w_1 \;x^2}{ 2 }      }\)

\(\large{ M_x  \; }\)  when \(\large{  \left[  a < x <  \left(  a + b  \right)  \right]  =  R_1 x -  \frac{ w_1 \;a}{ 2 }  \;  \left(  2\;x - a  \right)   }\)

\(\large{ M_x  \; }\)  when \(\large{  \left[ x >  \left(  a + b  \right)   \right]  =  R_2  \; \left(  L - x  \right)  -   \frac{ w_2  \; \left(  L - x  \right)^2   }{ 2 }    }\)

Where:

\(\large{ x }\) = horizontal distance from reaction to point on beam

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

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

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

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

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

\(\large{ a, b, c }\) = width and seperation of UDL