Overhanging Beam - Uniformly Distributed Load

Written by Jerry Ratzlaff on . Posted in Structural

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Overhanging Beam - Uniformly Distributed Load formulas

\(\large{ R_1 = V_1 = \frac{w}{2\;L}  \;  \left( L^2 - a^2  \right)     }\)   
\(\large{ R_2 = V_2 + V_3 =  \frac{ w }{2\;L} \;  \left( L + a  \right)^2     }\)   
\(\large{ V_2  =   w\;a   }\)   
\(\large{ V_3  =  \frac{ w }{2\;L}  \; \left( L^2 + a^2  \right)    }\)  
\(\large{ V_x    \; }\)  (between supports)    \(\large{  =     R_1 - w\;x  }\)  
\(\large{ V_{x_1}    \; }\)  (for overhang)    \(\large{  =    w \; \left(  a - x_1 \right)    }\)  
\(\large{ M_x    \; }\)  (between supports)    \(\large{  =    \frac{ w\;x }{2\;L}  \; \left( L^2 - a^2 - x\;L  \right)    }\)  
\(\large{ M_{x_1}    \; }\)  (overhang)    \(\large{  =    \frac{ w }{2}  \; \left( a - x_1  \right)^2    }\)  
\(\large{ M_1  \; }\)  at   \(\large{ \left[  x = \frac{L}{2}    \left(   1 - \frac{a^2}{L^2}   \right)   \right]   =    \frac{ w }{8\; L^2}  \;  \left(  L + a  \right)^2  \;   \left(  L - a  \right)^2          }\)  
\(\large{ M_2    \; }\)  at   \(\large{ \left( R_2 \right)  =    \frac{ w\;a^2 }{2}     }\)  
\(\large{ \Delta_x  \; }\)  (between supports)   \(\large{   =   \frac { w \;x} { 24 \;\lambda \;I \;L}  \;  \left(  L^4  - 2\;L^2\;x^2  + L\;x^3  - 2\;a^2\;L^2 + 2\;a^2\;x^2   \right)    }\)  
\(\large{ \Delta_{x_1}  \; }\)  (for overhang)   \(\large{   =   \frac { w\; x_1} { 24 \;\lambda\; I }  \;  \left(  4\;a^2\;L  - L^3  + 6\;a^2\;x_1 - 4\;a\;x_{1}{^2} +  x_{1}{^3}  \right)    }\)  

Where:

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

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

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

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

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

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

\(\large{ W }\) = total load from a uniform distribution

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

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

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

 

Tags: Equations for Beam Support