# Simple Beam - Load Increasing Uniformly to Center

Written by Jerry Ratzlaff on . Posted in Structural

## Simple Beam - Load Increasing Uniformly to Center formulas

 $$\large{ R = V_{max} = \frac{W}{2} }$$ $$\large{ V_x \; }$$  when $$\large{ \left( x < \frac{L}{2} \right) = \frac{W}{2\;L^2} \; \left( L^2 - 4\;x^2 \right) }$$ $$\large{ M_{max} }$$  (at center)  $$\large{ = \frac{W \;L}{6} }$$ $$\large{ M_x \; }$$  when $$\large{ \left( x < \frac{L}{2} \right) = W\;x \; \left( \frac{1}{2} - \frac {2\;x^2}{3\;L^2} \right) }$$ $$\large{ \Delta_{max} }$$  (at center)  $$\large{ = \frac{W \;L^3} {60 \; \lambda \;I } }$$ $$\large{ \Delta_x = \frac{W\; x}{480\; \lambda \;I \;L^2} \; \left( 5\;L^2 - 4\;x^2 \right)^2 }$$

### 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