# Simple Beam - Load Increasing Uniformly to Center

Written by Jerry Ratzlaff on . Posted in Structural

### Simple Beam - Load Increasing Uniformly to Center Formula

(Eq. 1)  $$\large{ R = V_{max} = \frac {W } {2} }$$

(Eq. 2)  $$\large{ V_x \; }$$  when $$\large{ \left( x < \frac {L}{2} \right) = \frac {W} {2L^2} \left( L^2 - 4x^2 \right) }$$

(Eq. 3)  $$\large{ M_{max} }$$  (at center)  $$\large{ = \frac {W L} {6} }$$

(Eq. 4)  $$\large{ M_x \; }$$  when $$\large{ \left( x < \frac {L}{2} \right) = Wx \left( \frac {1}{2} - \frac {2x^2}{3L^2} \right) }$$

(Eq. 5)  $$\large{ \Delta_{max} }$$  (at center)  $$\large{ = \frac {W L^3} {60 \lambda I } }$$

(Eq. 6)  $$\large{ \Delta_x = \frac {W x} {480 \lambda I L^2} \left( 5L^2 - 4x^2 \right)^2 }$$

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

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

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

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

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

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

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