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