# Cantilever Beam - Uniformly Distributed Load

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 formulas

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

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

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

$$M_x \;=\; w\;x^2 \;/\;2$$

$$\Delta_{max} \; (at\; free\; end ) \;=\; w\; L^4\;/\;8 \;\lambda\; I$$

$$\Delta_x \;=\; (w\;/\;48\; \lambda\; I) \; ( x^4 - 4\;L^3\;x - 3\;x^4 )$$

### C B - Uniformly Distributed Load - Solve for R

$$\large{ R = V = w \; L }$$

 load per unit length, w span length, L

### C B - Uniformly Distributed Load - Solve for Vx

$$\large{ V_x = w \; x }$$

 load per unit length, w distance from reaction, x

### C B - Uniformly Distributed Load - Solve for Mmax

$$\large{ M_{max} \; \left(at\; fixed \;end \right) = \frac{w\; L^2}{2} }$$

 load per unit length, w span length, L

### C B - Uniformly Distributed Load - Solve for Mx

$$\large{ M_x = \frac{ w \; x^2 }{2} }$$

 load per unit length, w distance from reaction, x

### C B - Uniformly Distributed Load - Solve for Δmax

$$\large{ \Delta_{max} \; \left(at\; free\; end \right) = \frac{w\; L^4}{8 \;\lambda\; I} }$$

 load per unit length, w span length, L modulus of elasticity, λ second moment of area, I

### C B - Uniformly Distributed Load - Solve for Δx

$$\large{ \Delta_x = \frac{w}{48\; \lambda\; I} \; \left( x^4 - 4\;L^3\;x - 3\;x^4 \right) }$$

 load per unit length, w modulus of elasticity, λ second moment of area, I distance from reaction, x span length, L

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

Tags: Beam Support