- See Article - Beam Design Formula
Cantilever Beam - Load at Free End Deflection Vertically with No Rotation formulas |
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\( R \;=\; V \;=\; P \) \( M_{max} \; (at\; both\; end ) \;=\; \dfrac{ P \cdot L }{ 2 } \) \( M_x \;=\; P \cdot \left( \dfrac{ L }{ 2 } - x \right) \) \( \Delta_{max} \; (at\; free \;end ) \;=\; \dfrac{ P\cdot L^3 }{ 12 \cdot \lambda\cdot I } \) \( \Delta_x \;=\; \left( \dfrac{ P \cdot ( L - x )^2 }{ 12\cdot \lambda\cdot I } \right) \cdot ( L + 2\cdot x ) \) |
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| Symbol | English | Metric |
| \( \Delta \) = deflection or deformation | \(in\) | \(mm\) |
| \( x \) = horizontal distance from reaction to point on beam | \(in\) | \(mm\) |
| \( M \) = maximum bending moment | \(lbf-in\) | \(N-mm\) |
| \( V \) = maximum shear force | \(lbf\) | \(N\) |
| \( \lambda \) (Greek symbol lambda) = modulus of elasticity | \(lbf\;/\;in^2\) | \(Pa\) |
| \( R \) = reaction load at bearing point | \(lbf\) | \(N\) |
| \( I \) = second moment of area (moment of inertia) | \(in^4\) | \(mm^4\) |
| \( L \) = span length of the bending member | \(in\) | \(mm\) |
| \( P \) = total concentrated load | \(lbf\) | \(N\) |
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.
