- 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.
Simple Beam - Load Increasing Uniformly to One End formulas |
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\( R_1 \;=\ V_1 \;=\; \dfrac{ W }{ 3 }\) \( R_2 \;=\ V_2 \;=\; \dfrac{ 2 \cdot W }{3 }\) \( V_x \;=\ \dfrac{ W }{ 3 } - \dfrac{ W \cdot x^2 }{ L^2 }\) \( M_{max} \; (at \; x = L\;/\; \sqrt{3}\; ) \;=\; \dfrac{ 2 \cdot W \cdot L }{ 9\cdot \sqrt{3} }\) \( M_x \;=\; \dfrac{ W \cdot x }{ 3 \cdot L^2 } \cdot ( L^2 - x^2 ) \) \( \Delta_{max} \; ( \;at \; x = L\; \sqrt{1 - ( 8/15 )^{\frac{1}{2} } } \;) \;=\; 0.01304 \cdot \dfrac{ W \cdot L^3 }{ \lambda \cdot I } \) \( \Delta_x \;=\; \dfrac{ W \cdot x }{ 180 \cdot \lambda \cdot I \cdot L^2 } \cdot ( 3 \cdot x^4 - 10 \cdot L^2 \cdot x^2 + 7 \cdot L^4 ) \) |
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| Symbol | English | Metric |
| \( R \) = reaction load at bearing point | \(lbf\) | \(N\) |
| \( V \) = maximum shear force | \(lbf\) | \(N\) |
| \( M \) = maximum bending moment | \(lbf-in\) | \(N-mm\) |
| \( \Delta \) = deflection or deformation | \(in\) | \(mm\) |
| \( W \) = total load or \( w\;L\;/\;2 \) | \(lbf\) | \(N\) |
| \( w \) = highest load per unit length of UIL | \(lbf\;/\;in\) | \(N\;/\;m\) |
| \( L \) = span length of the bending member | \(in\) | \(mm\) |
| \( x \) = horizontal distance from reaction to point on beam | \(in\) | \(mm\) |
| \( \lambda \) (Greek symbol lambda) = modulus of elasticity | \(lbf\;/\;in^2\) | \(Pa\) |
| \( I \) = second moment of area (moment of inertia) | \(in^4\) | \(mm^4\) |
