# Simple Beam - Uniformly Distributed Load

Written by Jerry Ratzlaff on . Posted in Structural

### Simple Beam - Uniformly Distributed Load Formula

$$\large{ R = V_{max} = \frac{w \; L}{2} }$$

$$\large{ V_x = w \; \left( \frac{L}{2} - x \right) }$$

$$\large{ M_{max} }$$  (at center)  $$\large{ = \frac{w \; L^2}{8} }$$

$$\large{ M_x = \frac{w \; x}{2} \; \left( L - x \right) }$$

$$\large{ \Delta_{max} }$$  (at center)  $$\large{ = \frac{5 \;w \;L^4}{384\; \lambda \;I} }$$

$$\large{ \Delta_x = \frac{w\; x}{24\; \lambda \;I} \; \left( L^3 - 2\;L\;x^2 + x^3 \right) }$$

Where:

$$\large{ \Delta }$$ = deflection or deformation

$$\large{ x }$$ = horizontal distance from reaction to point on beam

$$\large{ w }$$ = load per unit length

$$\large{ M }$$ = maximum bending moment

$$\large{ V }$$ = maximum shear force

$$\large{ \lambda }$$   (Greek symbol lambda) = modulus of elasticity

$$\large{ I }$$ = moment of inertia

$$\large{ R }$$ = reaction load at bearing point

$$\large{ L }$$ = span length of the bending member