# Cantilever Beam - Uniformly Distributed Load and Variable End Moments

on . Posted in Structural Engineering

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

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

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

$$\large{ M_{max} \; \left(at\; fixed \;end \right) \;\;=\;\; \frac{w\; L^2}{3} }$$

$$\large{ M_1 \; \left(at \;free \;end \right) \;\;=\;\; \frac {w \;L^2} {6} }$$

$$\large{ M_x = \frac{ w }{6} \; \left( L^2 - 3\;x^2 \right) }$$

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

$$\large{ \Delta_x \;\;=\;\; \frac{w \; \left[ L^2\; - \; \left( L\; - \;x \right)^2 \right]^2 }{24 \;\lambda\; I} }$$

Symbol English Metric
$$\large{ \Delta }$$ = deflection or deformation $$\large{in}$$ $$\large{mm}$$
$$\large{ x }$$ = horizontal distance from reaction to point on beam $$\large{in}$$ $$\large{mm}$$
$$\large{ w }$$ = load per unit length $$\large{\frac{lbf}{in}}$$ $$\large{\frac{N}{m}}$$
$$\large{ M }$$ = 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 $$\large{in}$$ $$\large{mm}$$

## 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.