Simple Beam - Central Point Load and Variable End Moments
- See Article Link - Beam Design Formulas
- 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 - Central Point Load and Variable End Moments formulas |
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\( R_1 \;=\; V_1 \;=\; ( P \;/\; 2 ) + ( M_1 - M_2 \;/\; L ) \)
\( R_2 \;=\; V_2 \;=\; ( P \;/\; 2 ) - ( M_1 - M_2 \;/\; L ) \) \( M_3 \; (at\; center ) \;=\; ( P\;L \;/\; 4 ) - ( M_1 + M_2 \;/\; L ) \) \( M_x \; ( x < \frac{L}{2} ) \;=\; [\; ( P \;/\; 2) + ( M_1 - M_2 \;/\; L ) \; x \;] - M_1 \) \( M_x \; ( > \frac{L}{2} ) \;=\; [\;(P\;/\;2) \; ( L - x ) \;] + [\; ( M_1 - M_2 ) \;x \;/\; L \;] - M_1 \) \( \Delta_x ( x < \frac{L}{2} ) \;=\; \frac{ P\;x }{ 48\; \lambda\; I } \; [ \; 3\;L^2 - 4\;x^2 - \frac{ 8\; ( L - x ) }{ P\;L } \; [ \;M_1 ( 2\;L - x ) + M_2\; ( L + x ) \; ] \;] \) \( x \; (first \;point \;of \;contraflexure ) \;=\; 2\;L\; M_1 \;/\; L\;P + 2\;M_1 - 2\;M_2 \) \( x \; (second\; point \;of\; contraflexure ) \;=\; [\; L \; ( L\;P - 2\;M_1 ) \;] \;/\; L\;P - 2\;M_1 + 2\;M_2 \) |
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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 | \(in\) | \(mm\) |
| \( x \) = horizontal distance from reaction to point on beam | \(in\) | \(mm\) |
| \( P \) = total concentrated load | \(lbf\) | \(N\) |
| \( L \) = span length of the bending member | \(in\) | \(mm\) |
| \( \lambda \) (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