- 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 - Uniformly Distributed Load and Variable End Moments formulas |
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\( R_1 \;=\; V_1 \;=\; \dfrac{ w \cdot L }{ 2 } + \dfrac{ M_1 - M_2 }{ L } \) \( R_2 \;=\; V_2 \;=\; \dfrac{ w \cdot L }{ 2 } - \dfrac{ M_1 - M_2 }{ L } \) \( V_x \;=\; w \cdot \left( \dfrac{ L }{ 2 } - x \right) + \dfrac{ M_1 - M_2 }{ L } \) \( b \; (inflection\; points) \;=\; \sqrt{ \dfrac{ L^2 }{ 4 } - \dfrac{ M_1 + M_2 }{ w } + \left(\dfrac{ M_1 + M_2 }{ w\cdot L } \right)^2 } \) \( M_x \;=\; \left( \dfrac{ w \cdot x }{ 2 } \cdot ( L - x ) \right) + \left( \dfrac{ M_1 - M_2 }{ L } \cdot x \right) - M_1 \) \( M_3 \; ( at\; x = \frac{ L }{ 2 } + \frac{ M_1 - M_2 }{ w\;L } ) \;=\; \dfrac{ w \cdot L^2 }{ 8 } - \dfrac{ M_1 + M_2 }{ 2 } + \dfrac{ (M_1 - M_2)^2 }{ 2 \cdot w \cdot L^2 } \) \( \Delta_x \;=\; \dfrac{w\cdot x }{ 48\cdot \lambda\cdot I } \cdot \left(\; x^3 - \left(\; (\; 2\cdot L + \dfrac{ 4\cdot M_1 }{ w \cdot L } - \dfrac{ 4\cdot M_2 }{ w\cdot L } \;) \cdot x^2 \;\right) + \dfrac{ 12\cdot M_1 }{ w } + L^3 + \dfrac{ 8\cdot M_1 \cdot L }{ w } - \dfrac{ 4 \cdot M_2\cdot L }{ w } \;\right) \) |
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| Symbol | English | Metric |
| \( R \) = reaction load at bearing point | \(lbf\) | \(N\) |
| \( V \) = maximum shear force | \(lbf\) | \(N\) |
| \( b \) = length to point load | \(in\) | \(mm\) |
| \( M \) = maximum bending moment | \(lbf - in\) | \(N - mm\) |
| \( \Delta \) = deflection or deformation | \(in\) | \(mm\) |
| \( w \) = load per unit length | \(lbf\;/\;in\) | \(N\;/\;m\) |
| \( L \) = 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) | \(in^4\) | \(mm^4\) |
