- See Article - Beam Design Formulas
Overhanging Beam - Point Load on Beam End formulas |
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\( R_1 \;=\; V_1 \;=\; \dfrac{ P\cdot a}{L }\) \( R_2 \;=\; V_1 + V_2 \;=\; \dfrac{ P }{L} \cdot ( L + a ) \) \( V_2 \;=\; P \) \( M_{max} \; ( at \;R_2 ) \;=\; P\cdot a \) \( M_x \; (between\; supports ) \;=\; \dfrac{ P\cdot a\cdot x}{L }\) \( M_{x_1} \; (for \;overhang ) \;=\; P\cdot ( a - x_1 ) \) \( \Delta_x \; (between\; supports ) \;=\; \dfrac{ - \;(P \cdot a\cdot x) }{ 6\cdot \lambda \cdot I\cdot L } \cdot ( L^2 - x^2 ) \) \( \Delta_{x_1} \; (overhang ) \;\;=\;\; \dfrac{ P\cdot x_1 }{ 6\cdot \lambda\cdot I } \cdot ( 2\cdot a\cdot L + 3\cdot a\cdot x_1 - x_{1}{^2} ) \) \( \Delta_{max} \; ( for\;overhang\; at\; x_1 = a ) \;=\; \dfrac{ P\cdot a^2 }{ 3 \cdot \lambda \cdot I } \cdot ( L + a ) \) \( \Delta_{max} \; ( between\; supports\; at \;x = \frac{L}{\sqrt{3}} ) \;=\; \frac{ -(P\;a\;L^2) }{9\; \sqrt{3} \; \lambda\; I } \;\;=\;\; 0.06415 \cdot \dfrac{ P\cdot a\cdot L^2 }{ \lambda\cdot I } \) |
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| Symbol | English | Metric |
| \( \Delta \) = deflection or deformation | \(in\) | \(mm\) |
| \( x \) = horizontal distance from reaction to point on beam | \(in\) | \(mm\) |
| \( M \) = 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 | \(in\) | \(mm\) |
| \( P \) = total concentrated load | \(lbf\) | \(N\) |
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.
