- See Article - Beam Design Formulas
Simple Beam - Uniform Load Partially Distributed at One End formulas |
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\( R_1 \;=\; V_1 \;=\; \dfrac{ w \cdot a }{ 2 \cdot L } \cdot ( 2 \cdot L - a ) \) \( R_2 \;=\; V_2 \;=\; \dfrac{ w \cdot a^2 }{ 2 \cdot L }\) \( V_x \; ( x < a ) \;=\; R_1 - w \cdot x \) \( M_{max} \; ( at \; x \;=\; R_1\;/\;w ) \;=\; \dfrac{ R_{1}{^2} }{ 2 \cdot w }\) \( M_x \; ( x < a ) \;=\; (R_1 \cdot x) - \dfrac{ w \cdot x^2 }{ 2 } \) \( M_x \; ( x > a ) \;=\; R_2 \cdot ( L - x ) \) \( \Delta_x \; ( x < a ) \;=\; \dfrac{ w\cdot x }{ 24 \cdot \lambda \cdot I \cdot L } \cdot [\; (\; a^2 \cdot ( 2 \cdot L - a )^2\;) - (\; 2 \cdot a \cdot x^2 \cdot ( 2 \cdot L - a )\;) + L \cdot x^3 \;] \) \( \Delta_x \; ( x > a ) \;=\; \dfrac{ w\cdot a^2 \cdot ( L - x ) }{ 24 \cdot \lambda \cdot I \cdot L } \cdot ( 4 \cdot x \cdot L - 2 \cdot x^2 - a^2 ) \) |
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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 | \(in\) | \(mm\) |
| \( w \) = load per unit length | \(lbf\;/\;in\) | \(N\;/\;m\) |
| \( a \) = width of UDL | \(in\) | \(mm\) |
| \( L \) = 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) | \(in^4\) | \(mm^4\) |
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.
