- 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.
Overhanging Beam - Uniformly Distributed Load on Overhang formulas |
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\( R_1 \;=\; V_2 \;=\; \dfrac{ w\cdot a^2 }{ 2\cdot L }\) \( R_2 \;=\; V_1 + V_2 \;=\; \dfrac{ w\cdot a }{ 2 \cdot L } \cdot ( 2 \cdot L + a) \) \( V_2 \;=\; w \cdot a \) \( V_{x _1} \;=\; w \cdot ( a - x_1 ) \) \( M_{max} \; ( at\; R_2 ) \;=\; \dfrac{ w \cdot a^2 }{ 2 }\) \( M_x \; (between\; supports ) \;=\; \dfrac{ w \cdot a^2 \cdot x }{ 2\cdot L }\) \( M_{x_1} \; (for \;overhang ) \;=\; \dfrac{ w }{ 2 } \cdot ( a - x_1)^2 \) \( \Delta_x \; (between\; supports ) \;=\; \dfrac{ - w \cdot a^2\cdot x }{ 12\cdot \lambda\cdot I \cdot L } \cdot ( L^2 - x^2 ) \) \( \Delta_{max} \; (between\; supports \;at\; x = \frac{L}{\sqrt{3}} ) \;=\; \frac{ - \;w\; a^2 \;L^2 }{18 \; \sqrt{3} \; \lambda\; I } \;=\; 0.03208 \cdot \dfrac{ w \cdot a^2 \cdot L^2 }{ \lambda\cdot I } \) \( \Delta_{max} \; (for \;overhang \;at\; x_1 = a ) \;=\; \dfrac{ w\cdot x^3 }{ 24\cdot \lambda\cdot I } \cdot ( 4\cdot L + 3\cdot a ) \) \( \Delta_{x1} \; (for \;overhang ) \;=\; \dfrac{ w\cdot x_1 }{ 24\cdot \lambda\cdot I } \cdot ( 4\cdot a^2 \cdot L + 6\cdot a^2\cdot x_1 - 4\cdot a \cdot x_{1}{^2} + x_{1}{^3} ) \) |
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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 | \(in\) | \(mm\) |
