Beam Fixed at Both Ends - Concentrated Load at Any Point

Written by Jerry Ratzlaff on . Posted in Structural

Beam Fixed at Both Ends - Concentrated Load at Any Pointfebe 3A

Concentrated Load at Any Point Formula

\(\large{ R_1 = V_1  \; }\) max. when  \(\large{ \left( a < b \right)  = \frac {Pb^2} {L^3}  \left( 3a + b  \right)    }\)

\(\large{ R_2 = V_2  \; }\) max. when  \(\large{ \left( a > b \right)  = \frac {Pa^2} {L^3}  \left( a + 3b  \right)    }\)

\(\large{ M_1  \; }\) max. when  \(\large{ \left( a < b \right)  = \frac {Pab^2} {L^2}   }\)

\(\large{ M_2  \; }\) max. when  \(\large{ \left( a > b \right)  = \frac {Pa^2b} {L^2}   }\)

\(\large{ M_a  \; }\)  (at point of load)  \(\large{  = \frac {2Pa^2b^2} {L^3}   }\)

\(\large{ M_x  \; }\) when  \(\large{ \left( x < a \right) = \frac {Pab^2} {L^2}   }\)

\(\large{ \Delta_{max}  \; }\) when  \(\large{  \left(   a > b \right)  }\)   at    \(\large{  \left( x =  \frac {2aL}{3a + b}  \right)   = \frac {2Pa^3b^2} {3 \lambda I \left( 3a + b  \right)^2 }  }\) 

\(\large{ M_a  \; }\)  (at point of load)  \(\large{  = \frac {Pa^3b^3} {3 \lambda I L^3}   }\)

\(\large{ \Delta_x  \; }\) when  \(\large{ \left( x < a \right)  = \frac {Pb^2x^2} {6 \lambda I L^3}  \left( 3aL - 3ax - bx  \right)    }\)

 

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