Beam Fixed at One End - Concentrated Load at Any Point

Written by Jerry Ratzlaff on . Posted in Structural

feoe 3ABeam Fixed at One End - Concentrated Load at Center Formula

\(\large{ R_1 = V_1 = \frac {Pb^2} {2L^3}  \left(  a + 2L \right)   }\)

\(\large{ R_2 = V_2 = \frac {Pa} {2L^3}  \left(  3L^2 - a^2 \right)   }\)

\(\large{ M_1  }\)  (at point of load)  \(\large{ =  R_1 a  }\)

\(\large{ M_2  }\)  (at fixed end)  \(\large{ =  \frac {Pab} {2L^2}   \left(  a +L  \right)   }\)

\(\large{ M_x   \; }\)  when \(\large{    \left(  x < a   \right)   =   R_1 x    }\)

\(\large{ M_x   \; }\)  when \(\large{    \left(  x > a  \right)   =  R_1 x -  P \left( x - a  \right)  }\)

\(\large{ \Delta_{max}   \; }\)  when  \(\large{ \left(  a < .414L  \right)  \; }\)  at  \(\large{ L   \frac {  L^2 + a^2  } {   3L^2 - a^2  } =  \frac {Pa} {3 \lambda I }  \frac { \left( L^2 - a^2 \right) ^3 } {  \left( 3L^2 - a^2 \right) ^2 }   }\)

\(\large{ \Delta_{max}   \; }\)  when  \(\large{ \left(  a > .414L  \right)  \; }\)  at  \(\large{ L \sqrt{  \frac { a } {  2L + a  }  }  =  \frac {Pab^2} {6 \lambda I }   \sqrt{  \frac {  a } {  2L + a }    }   }\)

\(\large{ \Delta_a \; }\)  (at point of load)  \(\large{ =  \frac { Pa^2 b^3} {12 \lambda I L^3}  \left( 3L + a  \right)     }\)

\(\large{ \Delta_x   \; }\)  when \(\large{    \left(  x < a   \right)   =   \frac { Pb^2 x} {12 \lambda I L^3}  \left( 3aL^2 - 2Lx^2  - ax^2  \right)    }\)

\(\large{ \Delta_x   \; }\)  when \(\large{    \left(  x > a   \right)   =  \frac { Pa} {12 \lambda I L^3}     \left( L  - x  \right)^2    \left( 3L^2 x - a^2 x   -  2a^2 L \right)    }\)

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