- 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 - Central Point Load and Variable End Moments formulas |
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\( R_1 \;=\; V_1 \;=\; \dfrac{ P }{ 2 } + \dfrac{ M_1 - M_2 }{ L } \)
\( R_2 \;=\; V_2 \;=\; \dfrac{ P }{ 2 } - \dfrac{ M_1 - M_2 }{ L } \) \( M_3 \; (at\; center ) \;=\; \dfrac{ P \cdot L }{ 4 } - \dfrac{ M_1 + M_2 }{ L } \) \( M_x \; ( x < \frac{L}{2} ) \;=\; \left(\; \dfrac{ P }{ 2} + \dfrac{ M_1 - M_2 }{ L } \cdot x \;\right) - M_1 \) \( M_x \; ( > \frac{L}{2} ) \;=\; \left( \dfrac{ P}{2} \cdot ( L - x ) \right) + \left( \dfrac{ ( M_1 - M_2 ) \cdot x }{ L } \right) - M_1 \) \( \Delta_x ( x < \frac{L}{2} ) \;=\; \dfrac{ P\cdot x }{ 48\cdot \lambda\cdot I } \cdot \left(\; 3\cdot L^2 - 4\cdot x^2 - \dfrac{ 8\cdot ( L - x ) }{ P\cdot L } \cdot ( \;M_1 \cdot ( 2\cdot L - x ) + M_2\cdot ( L + x ) \; ) \;\right) \) \( x \; (first \;point \;of \;contraflexure ) \;=\; \dfrac{ 2 \cdot L\cdot M_1 }{ L\cdot P + 2\cdot M_1 - 2\cdot M_2 } \) \( x \; (second\; point \;of\; contraflexure ) \;=\; \dfrac{ L \cdot ( L\cdot P - 2\cdot M_1 ) }{ L\cdot P - 2\cdot M_1 + 2\cdot M_2 } \) |
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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 | \(in\) | \(mm\) |
| \( x \) = horizontal distance from reaction to point on beam | \(in\) | \(mm\) |
| \( P \) = total concentrated load | \(lbf\) | \(N\) |
| \( L \) = span length of the bending member | \(in\) | \(mm\) |
| \( \lambda \) (Greek symbol lambda) = modulus of elasticity | \(lbf\;/\;in^2\) | \(Pa\) |
| \( I \) = second moment of area (moment of inertia) | \(in^4\) | \(mm^4\) |
