Overhanging Beam - Uniformly Distributed Load
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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 formulas |
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\( R_1 \;=\; V_1 \;=\; (w\;/\;2\;L) \; ( L^2 - a^2 ) \) \( R_2 \;=\; V_2 + V_3 \;=\; ( w \;/\;2\;L) \; ( L + a )^2 \) \(V_2 \;=\; w\;a \) \( V_3 \;=\; ( w \;/\;2\;L) \; ( L^2 + a^2 ) \) \( V_x \; (between\; supports ) \;=\; R_1 - w\;x \) \(V_{x_1} \; (for \;overhang ) \;=\; w \; ( a - x_1 ) \) \( M_x \; (between\; supports ) \;=\; (w\;x \;/\;2\;L) \; ( L^2 - a^2 - x\;L ) \) \( M_{x_1} \; (overhang ) \;=\; ( w \;/\;2) \; ( a - x_1 )^2 \) \( M_1 \; [\;at\; x = \frac{L}{2} (1 - \frac{a^2}{L^2} )\;] \;=\; (w \;/\;8\; L^2) \; (L + a)^2 \; (L - a)^2 \) \( M_2 \; (at\; R_2 ) \;=\; w\;a^2 \;/\;2 \) \( \Delta_x \; (between \;supports ) \;=\; \frac { w \;x} { 24 \;\lambda \;I \;L} \; ( L^4 - 2\;L^2\;x^2 + L\;x^3 - 2\;a^2\;L^2 + 2\;a^2\;x^2 ) \) \( \Delta_{x_1} \; (for\; overhang ) \;=\; \frac { w\; x_1} { 24 \;\lambda\; I } \; ( 4\;a^2\;L - L^3 + 6\;a^2\;x_1 - 4\;a\;x_{1}{^2} + x_{1}{^3} ) \) |
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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 | \(in\) | \(mm\) |
Tags: Beam Support