Cantilever Beam - Uniformly Distributed Load and Variable End Moments
Cantilever Beam - Uniformly Distributed Load and Variable End Moments formulas |
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\(\large{ R = V \;\;=\;\; w\;L }\) \(\large{ V_x \;\;=\;\; w\;x }\) \(\large{ M_{max} \; \left(at\; fixed \;end \right) \;\;=\;\; \frac{w\; L^2}{3} }\) \(\large{ M_1 \; \left(at \;free \;end \right) \;\;=\;\; \frac {w \;L^2} {6} }\) \(\large{ M_x = \frac{ w }{6} \; \left( L^2 - 3\;x^2 \right) }\) \(\large{ \Delta_{max} \; \left(at\; free \;end \right) \;\;=\;\; \frac{w\; L^4}{24\; \lambda\; I} }\) \(\large{ \Delta_x \;\;=\;\; \frac{w \; \left[ L^2\; - \; \left( L\; - \;x \right)^2 \right]^2 }{24 \;\lambda\; I} }\) |
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| 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.
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Tags: Beam Support Equations