Beam Fixed at One End - Concentrated Load at Any Point

Written by Jerry Ratzlaff on . Posted in Structural

feoe 3Aformulas that use Beam Fixed at One End - Concentrated Load at Center

\(\large{ R_1 = V_1 = \frac {P\;b^2} {2\;L^3} \; \left(  a + 2\;L \right)   }\)   
\(\large{ R_2 = V_2 = \frac {P\;a} {2\;L^3} \; \left(  3\;L^2 - a^2 \right)   }\)   
\(\large{ M_1  }\)  (at point of load)  \(\large{ =  R_1 \;a  }\)   
\(\large{ M_2  }\)  (at fixed end)  \(\large{ =  \frac {P\;a\;b} {2\;L^2}  \; \left(  a +L  \right)   }\)  
\(\large{ M_x   \; }\)  when \(\large{    \left(  x < a   \right)   =   R_1\; x    }\)  
b\(\large{ M_x   \; }\)  when \(\large{    \left(  x > a  \right)   =  R_1 \;x -  P\; \left( x - a  \right)  }\)  
\(\large{ \Delta_{max}   \; }\)  when  \(\large{ \left(  a < .414\;L  \right)  \; }\)  at  \(\large{ L \;  \frac {  L^2\; +\; a^2  } {   3\;L^2 \;-\; a^2  } =  \frac {P\;a} {3\; \lambda\; I }\;  \frac { \left( L^2 \;-\; a^2 \right) ^3 } {  \left( 3\;L^2 \;- \;a^2 \right) ^2 }   }\)  
\(\large{ \Delta_{max}   \; }\)  when  \(\large{ \left(  a > .414\;L  \right)  \; }\)  at  \(\large{ L \;\sqrt{  \frac { a } {  2\;L \;+\; a  }  }  =  \frac {P\;a\;b^2} {6\; \lambda\; I }  \; \sqrt{  \frac {  a } {  2\;L \;+ \;a }    }   }\)  
\(\large{ \Delta_a \; }\)  (at point of load)  \(\large{ =  \frac { P\;a^2 \;b^3} {12\; \lambda\; I \;L^3}\;  \left( 3\;L + a  \right)     }\)  
\(\large{ \Delta_x   \; }\)  when \(\large{    \left(  x < a   \right)   =   \frac { P\;b^2\; x} {12 \;\lambda\; I \;L^3}\;  \left( 3\;a\;L^2 - 2\;L\;x^2  - a\;x^2  \right)    }\)  
\(\large{ \Delta_x   \; }\)  when \(\large{    \left(  x > a   \right)   =  \frac { P\;a} {12\; \lambda\; I \;L^3}   \;  \left( L  - x  \right)^2   \; \left( 3\;L^2 \;x - a^2 \;x   -  2\;a^2 \;L \right)    }\)  

Where:

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

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

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

\(\large{ P }\) = total concentrated load

\(\large{ R }\) = 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