Simple Beam - Load Increasing Uniformly to One End

Written by Jerry Ratzlaff on . Posted in Structural

Simple Beam - Load Increasing Uniformly to One End Formula

(Eq. 1)  $$\large{ R_1 = V_1 = \frac {W } {3} }$$

(Eq. 2)  $$\large{ R_2 = V_2 = \frac {2W } {3} }$$

(Eq. 3)  $$\large{ V_x = \frac {W} {3} - \frac {Wx^2} {L^2} }$$

(Eq. 4)  $$\large{ M_{max} \; }$$  at $$\large{ \left( x = \frac {L}{ \sqrt {3} } = 0.5774L \right) = \frac { 2WL } { 9 \sqrt{3} } =0.1283 WL }$$

(Eq. 5)  $$\large{ M_x = \frac {W x} {3L^2} \left( L^2 - x^2 \right) }$$

(Eq. 6)  $$\large{ \Delta_{max} \; }$$  at $$\large{ \left( x = L \sqrt {1 - \frac{8}{15} } = 0.5193L \right) = 0.01304 \frac { W L^3} { \lambda I} }$$

(Eq. 7)  $$\large{ \Delta_x = \frac {W x} { 180 \lambda I L^2 } \left( 3x^4 - 10L^2x^2 + 7L^4 \right) }$$

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