Simple Beam - Load Increasing Uniformly to Center

Written by Jerry Ratzlaff on 19 April 2018. Posted in Structural

\(\large{ R = V_{max} = \frac{W}{2} }\)

\(\large{ V_x \; }\) 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