Simple Beam - Central Point Load and Variable End Moments

Written by Jerry Ratzlaff on . Posted in Structural

Simple Beam - Central Point Load and Variable End Momentssb 13D

Central Point Load and Variable End Moments Formula

\(\large{ R_1 = V_1  =  \frac { P }  { 2 }  +  \frac { M_1 - M_2 }  { L }   }\)

\(\large{ R_2 = V_2  =  \frac { P }  { 2 }  -  \frac { M_1 - M_2 }  { L }    }\)

\(\large{ M_3  }\)  (at center)  \(\large{   =  \frac { PL }  { 4 }  -  \frac { M_1 + M_2 }  { L }    }\)

\(\large{ M_x   \left(  x <  \frac{L} {2}    \right)   =   \left(     \frac { P }  { 2 }  +  \frac { M_1 - M_2 }  { L }  \right)  x - M_1    }\)

\(\large{ M_x   \left(  >  \frac{L} {2}    \right)   =  \frac {P}{2}  \left( L - x  \right)  +   \frac { \left(  M_1 - M_2  \right) x }  { L }  - M_1    }\)

\(\large{ \Delta_x   \left( x <  \frac{L} {2}    \right)    =    \frac { Px } { 48 \lambda I }         \left[        3L^2  -  4x^2 -    \frac {  8  \left( L - x  \right) }  { PL }      \left[ M_1 \left( 2L - x \right)   +  M_2 \left( L + x \right)  \right]       \right]  }\)

\(\large{ x }\)  (first point of contraflexure)  \(\large{ =  \frac { 2L M_1 }  { LP + 2M_1 - 2M_2 }    }\)

\(\large{ x }\)  (second point of contraflexure)  \(\large{ =  \frac {  L  \left( LP - 2M_1     \right)       }  { LP + 2M_1 + 2M_2 }    }\)

 

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