- See Article - Beam Design Formulas
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 - Uniformly Distributed Load and Variable End Moments formulas |
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\( R = V \;=\; w\cdot L \) \( V_x \;=\; w\cdot x \) \( M_{max} \; (at\; fixed \;end ) \;=\; \dfrac{ w\cdot L^2 }{ 3 } \) \( M_1 \; (at \;free \;end ) \;=\; \dfrac{ w \cdot L^2 }{ 6 } \) \( M_x \;=\; \dfrac{ w }{ 6 } \cdot ( L^2 - 3\cdot x^2 ) \) \( \Delta_{max} \; (at\; free \;end) \;=\; \dfrac{ w\cdot L^4 }{ 24\cdot \lambda\cdot I } \) \( \Delta_x \;=\; \dfrac{ w \cdot ( L^2 - ( L - x )^2 )^2 }{ 24 \cdot \lambda \cdot I } \) |
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| Symbol | English | Metric |
| \( \Delta \) = deflection or deformation | \(in\) | \(mm\) |
| \( x \) = horizontal distance from reaction to point on beam | \(in\) | \(mm\) |
| \( w \) = load per unit length | \(lbf\;/\;in\) | \(N\;/\;m\) |
| \( M \) = 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 | \(in\) | \(mm\) |
