# Cantilever Beam - Uniformly Distributed Load and Variable End Moments

on . Posted in Structural Engineering

### diagram Symbols

• Bending moment diagram (BMD)  -  Used to determine the bending moment at a given point of a structural element.  The diagram can help determine the type, size, and material of a member in a structure so that a given set of loads can be supported without structural failure.
• Free body diagram (FBD)  -  Used to visualize the applied forces, moments, and resulting reactions on a structure in a given condition.
• Shear force diagram (SFD)  -  Used to determine the shear force at a given point of a structural element.  The diagram can help determine the type, size, and material of a member in a structure so that a given set of loads can be supported without structural failure.
• Uniformly distributed load (UDL)  -  A load that is distributed evenly across the entire length of the support area.

### Cantilever Beam - Uniformly Distributed Load and Variable End Moments formulas

$$R = V \;=\; w\;L$$

$$V_x \;=\; w\;x$$

$$M_{max} \; (at\; fixed \;end ) \;=\; w\; L^2\;/\;3$$

$$M_1 \; (at \;free \;end ) \;=\; w \;L^2\;/\;6$$

$$M_x \;=\; ( w \;/\;6) \; ( L^2 - 3\;x^2 )$$

$$\Delta_{max} \; (at\; free \;end) \;=\; w\; L^4\;/\;24\; \lambda\; I$$

$$\Delta_x \;=\; [\;w \; ( L^2 - ( L - x )^2 )^2 \;] \;/\;24 \;\lambda\; I$$

Symbol English Metric
$$\Delta$$ = deflection or deformation $$in$$ $$mm$$
$$x$$ = horizontal distance from reaction to point on beam $$in$$ $$mm$$
$$w$$ = load per unit length $$lbf\;/\;in$$ $$N\;/\;m$$
$$M$$ = maximum bending moment $$lbf-in$$ $$N-mm$$
$$V$$ = maximum shear force $$lbf$$ $$N$$
$$\lambda$$   (Greek symbol lambda) = modulus of elasticity $$lbf\;/\;in^2$$ $$Pa$$
$$R$$ = reaction load at bearing point $$lbf$$ $$N$$
$$I$$ = second moment of area (moment of inertia) $$in^4$$ $$mm^4$$
$$L$$ = span length of the bending member $$in$$ $$mm$$

Tags: Beam Support