# Cantilever Beam - Load Increasing Uniformly to One End

on . Posted in Structural Engineering

## Cantilever Beam - Load Increasing Uniformly to One End formulas

$$\large{ R = V \;\;=\;\; W }$$

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

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

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

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

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

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

## diagrams

• 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. 