# Beam Fixed at Both Ends - Uniformly Distributed Load

Written by Jerry Ratzlaff on . Posted in Structural

## beam fixed at both ends - Uniformly Distributed Load formulas

 $$\large{ R = V = \frac {w\; L} {2} }$$ $$\large{ V_x = w \; \left( \frac {L} {2} - x \right) }$$ $$\large{ M_{max} }$$ (at ends)  =  $$\large{ \frac {w\; L^2} {12} }$$ $$\large{ M_1 }$$ (at center)  =  $$\large{ \frac {w\; L^2} {24} }$$ $$\large{ M_x = \frac {w}{12} \; \left( 6\;L\;x - L^2 - 6\;x^2 \right) }$$ $$\large{ M_{max} }$$ (at center)  $$\large{ = \frac {w\; L^4} {384\; \lambda\; I} }$$ $$\large{ \Delta_x = \frac {w\; x^2} {24\; \lambda\; I} \; \left( L - x \right) ^2 }$$ $$\large{ x }$$ (points of contraflexure)  $$\large{ = 0.2113 \; L }$$

### 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