Simple Beam - Uniformly Distributed Load and Variable End Moments
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- Tags: Beam Support
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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\( R_1 \;=\; V_1 \;=\; ( w\;L \;/\; 2 ) + (M_1 - M_2 \;/\; L) \) \( R_2 \;=\; V_2 \;=\; ( w\;L \;/\; 2 ) - (M_1 - M_2 \;/\; L) \) \( V_x \;=\; w \; [\;( L \;/\; 2 ) - x\;] + ( M_1 - M_2 \;/\; L ) \) \( b \; (inflection\; points) \;=\; \sqrt{ ( L^2 \;/\; 4 ) - ( M_1 + M_2 \;/\; w ) + ( M_1 + M_2 \;/\; w\;L )^2 } \) \( M_x \;=\; [\;( w\;x \;/\; 2 ) \; ( L - x )\;] + [\;( M_1 - M_2 \;/\; L ) x \;] - M_1 \) \( M_3 \; ( at\; x = \frac{ L }{ 2 } + \frac{ M_1 - M_2 }{ w\;L } ) \;=\; ( w\;L^2 \;/\; 8 ) - ( M_1 + M_2 \;/\; 2 ) + [\; (M_1 - M_2)^2 \;/\; 2\;w\;L^2 \;] \) \( \Delta_x \;=\; ( \frac{w\;x }{ 48\; \lambda\; I} ) \; [ \; x^3 - [\;( 2\;L + \frac { 4\;M_1 } { w\;L } - \frac { 4\;M_2 } { w\;L } ) x^2 \;] + \frac { 12\;M_1 } { w } + L^3 + \frac { 8\;M_1 \;L } { w } - \frac { 4\;M_2\; L } { w } \;] \) |
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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) | \(in^4\) | \(mm^4\) |
Tags: Beam Support