# Simple Beam - Load Increasing Uniformly to One End

Written by Jerry Ratzlaff on . Posted in Structural

## Simple Beam - Load Increasing Uniformly to One End formulas

 $$\large{ R_1 = V_1 = \frac{W }{3} }$$ $$\large{ R_2 = V_2 = \frac{2\;W }{3} }$$ $$\large{ V_x = \frac{W}{3} - \frac{W\;x^2}{L^2} }$$ $$\large{ M_{max} \; }$$  at $$\large{ \left( x = \frac{L}{ \sqrt{3} } = 0.5774\;L \right) = \frac{ 2\;W\;L }{ 9\; \sqrt{3} } = 0.1283 \;W\;L }$$ $$\large{ M_x = \frac{W \;x}{3\;L^2} \; \left( L^2 - x^2 \right) }$$ $$\large{ \Delta_{max} \; }$$  at $$\large{ \left( x = L\; \sqrt{1 - \frac{8}{15} } = 0.5193\;L \right) = 0.01304 \; \frac{ W \;L^3}{ \lambda \;I} }$$ $$\large{ \Delta_x = \frac{W \;x}{ 180\; \lambda \;I \;L^2 } \; \left( 3\;x^4 - 10\;L^2\;x^2 + 7\;L^4 \right) }$$

### Where:

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

$$\large{ w }$$ = highest load per unit length of UIL

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

$$\large{ M }$$ = 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

$$\large{ W }$$ = total load or wL/2