# Simple Beam - Uniform Load Partially Distributed at Each End

Written by Jerry Ratzlaff on . Posted in Structural

### 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