# Beam Fixed at One End - Uniformly Distributed Load

Written by Jerry Ratzlaff on . Posted in Structural

## Beam Fixed at One End - Uniformly Distributed Load formulas

 $$\large{ R_1 = V_1 = \frac {3\; w \;L} {8} }$$ $$\large{ R_1 = V_1 = \frac {3\; w\; L} {8} }$$ $$\large{ V_x = R_1 - w\;x }$$ $$\large{ M_{max} = \frac {w\; L^2} {8} }$$ $$\large{ M_1 }$$   at  $$\large{ \left( x = \frac {3\;L}{8} \right) = \frac { 9\;w\;L^2 } {128} }$$ $$\large{ M_x = R_1\; x - \frac {w\; x^2} {2} }$$ $$\large{ \Delta_{max} = }$$  at   $$\large{ \left[ x = \frac {L} {16}\; \left( 1 + \sqrt {33} \right) \right] }$$  or  $$\large{ \left( x = 0.4215\;L \right) = \frac {w\; L^4} {185\; \lambda\; I} }$$ $$\large{ \Delta_x = \frac {w \;x} {48\; \lambda\; I} \; \left( L^3 - 3\;L\;x^2 + 2\;x^3 \right) }$$

### 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