Simple Beam - Uniform Load Partially Distributed at Each End

Simple Beam - Uniform Load Partially Distributed at Each End Formula

(Eq. 1)  \(\large{ R_1 = V_1 = \frac {  w_1 a   \left(  2L - a  \right)  + w_2 c^2  }  { 2L }     }\)

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

(Eq. 3)  \(\large{ V_x   \; }\)  when \(\large{  \left(  x < a \right)  =     R_1 - w_1 x  }\)     

(Eq. 4)  \(\large{ V_x  \; }\)  when \(\large{  \left[   a < x <  \left(  a + b  \right) \right]  =  R_1 - w_1 a   }\)

(Eq. 5)  \(\large{ V_x   \; }\)  when \(\large{   \left[  x >  \left(  a + b  \right)  \right] =  R_2 - w_2  \left(  1 - x  \right)    }\)

(Eq. 6)  \(\large{ M_{max} \; }\)   at  \(\large{ \left(  x = \frac {R_1}{w_1}  \right)  }\)  when  \(\large{ \left(  R_1 < w_1 a  \right)   =  \frac {   R_{1}{^2} }   { 2w_1  }   }\)

(Eq. 7)  \(\large{ M_{max} \; }\)   at  \(\large{ \left(  x = L - \frac {R_2}{w_2}  \right)  }\)  when  \(\large{ \left(  R_2 < w_2 c  \right)   =  \frac {   R_{2}{^2} }   { 2w_2  }   }\)

(Eq. 8)  \(\large{ M_x  \; }\)  when \(\large{  \left(  w < a \right)     =  R_1 x  - \frac { w_1 x^2} { 2 }      }\)

(Eq. 9)  \(\large{ M_x  \; }\)  when \(\large{  \left[  a < x <  \left(  a + b  \right)   \right]   =  R_1 x -  \frac { w_1 a} { 2 }    \left(  2x - a  \right)         }\)

(Eq. 10)  \(\large{ M_x  \; }\)  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:

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

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

\(\large{ M }\) = 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