# Cantilever Beam - Load Increasing Uniformly to One End

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 - Load Increasing Uniformly to One End formulas

$$R \;=\; V \;=\; W$$

$$V_x \;=\; W\; (x^2\;/\;L^2 )$$

$$M_{max} \; \left(at\; fixed \;end \right) \;=\; W\; L\;/\;3$$

$$M_x \;=\; W\;x^3 \;/\;3\;L^2$$

$$\Delta_{max} \; \left(at\; free\; end \right) \;=\; W\; L^3\;/\;15\; \lambda\; I$$

$$\Delta_x \;=\; (W\;x^2\;/\;60\; \lambda \;I \;L^2) \; ( x^5 + 5\;L^4 x + 4\;L^5)$$

### C B - Load Increasing Unif to One End - Solve for R

$$\large{R = \frac{w\;L}{2} }$$

 load per unit length, w span length, L

### C B - Load Increasing Unif to One End - Solve for Vx

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

 load per unit length, w span length, L distance from reaction, x

### C B - Load Increasing Unif to One End - Solve for Mmax

$$\large{ M_{max} = \frac{\frac{w\;L}{2} \; L}{3} }$$

 load per unit length, w span length, L

### C B - Load Increasing Unif to One End - Solve for Mx

$$\large{ M_x = \frac{ \frac{w\;L}{2} \;x^3 }{3\;L^2} }$$

 load per unit length, w span length, L distance from reaction, x

### C B - Load Increasing Unif to One End - Solve for Δmax

$$\large{ \Delta_{max} = \frac{\frac{w\;L}{2} \; L^3}{15\; \lambda\; I} }$$

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

### C B - Load Increasing Unif to One End - Solve for Δx

$$\large{ \Delta_x = \frac{\frac{w\;L}{2} \;x^2}{60\; \lambda \;I \;L^2} \; \left( x^5 + 5\;L^4 x + 4\;L^5 \right) }$$

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

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 $$in$$ $$mm$$
$$W$$ = total load $$(w\;L\;/\;2)$$ $$lbf$$ $$N$$
$$w$$ = highest 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