Simple Beam - Uniform Load Partially Distributed at One End

Written by Jerry Ratzlaff on . Posted in Structural

sb 4DSimple Beam - Uniform Load Partially Distributed at One End Formula

\(\large{ R_1 = V_1 = \frac{w \;a}{2\;L}  \;  \left(  2\;L - a  \right)     }\)

\(\large{ R_2 = V_2 = \frac{ w \;a^2 }{2\;L}  }\)

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

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

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

\(\large{ M_x \; }\)  when \(\large{ \left(  x > a \right)  =     R_2  \; \left(  L - x  \right)   }\)

\(\large{ \Delta_x  \; }\)  when \(\large{ \left(  x < a \right)  =   \frac { w\; x} { 24\; \lambda \;I \;L}  \; \left[ a^2  \;  \left(  2\;L - a  \right)^2 - 2\;a\;x^2 \;  \left(  2\;L - a  \right)  + L\;x^3  \right]    }\)

\(\large{ \Delta_x  \; }\)  when \(\large{ \left(  x > a \right)   =  \frac{ w\; a^2  \; \left(  L \;-\; x  \right)  } { 24\; \lambda \;I \;L} \;  \left( 4\;x\;L - 2\;x^2 - a^2     \right)    }\)

Where:

\(\large{ \Delta }\) = deflection or deformation

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

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

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

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

\(\large{ \lambda  }\)   (Greek symbol lambda) = modulus of elasticity

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

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

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

\(\large{ a }\) = width of UDL

 

Tags: Equations for Beam Support