Simple Beam - Load Increasing Uniformly to Center
- 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 - Load Increasing Uniformly to Center formulas |
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\( R = V_{max} \;=\; W\;/\;2 \) \( V_x \; [ \; x < (L\;/\;2) \;] \;=\; (W\;/\;2\;L^2) \; ( L^2 - 4\;x^2 ) \) \( M_{max} \; (at \;center) \;=\; W \;L\;/\;6 \) \( M_x \; [\; x < (L\;/\;2) \;] \;=\; W\;x \; [\; (1/2) - (2\;x^2\;/\;3\;L^2) \;] \) \( \Delta_{max} \; (at \;center) \;=\; W \;L^3 \;/\; 60 \; \lambda \;I \) \( \Delta_x \; [\; x < (L\;/\;2) \;] \;=\; (W\; x \;/\;480\; \lambda \;I \;L^2) \; ( 5\;L^2 - 4\;x^2 )^2 \) |
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S B - Load Increasing Unif to Center - Solve for R\(\large{ R = \frac{ \frac{w\;L}{2} }{2} }\)
S B - Load Increasing Unif to Center - Solve for Vx\(\large{ V_x = \frac{ \frac{w\;L}{2} }{2\;L^2} \; \left( L^2 - 4\;x^2 \right) }\)
S B - Load Increasing Unif to Center - Solve for Mmax\(\large{ M_{max} = \frac{ \frac{w\;L}{2} \;L}{6} }\)
S B - Load Increasing Unif to Center - Solve for Mx\(\large{ M_x = \frac{w\;L}{2} \; x \; \left( \frac{1}{2} - \frac {2\;x^2}{3\;L^2} \right) }\)
S B - Load Increasing Unif to Center - Solve for Δmax\(\large{ \Delta_{max} = \frac{ \frac{w\;L}{2} \;L^3} {60 \; \lambda \;I } }\)
S B - Load Increasing Unif to Center - Solve for Δx\(\large{ \Delta_x = \frac{ \frac{w\;L}{2} \; x}{480\; \lambda \;I \;L^2} \; \left( 5\;L^2 - 4\;x^2 \right)^2 }\)
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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 | \(in\) | \(mm\) |
\( W \) = total load or \( w\;L\;/\;2 \) | \(lbf\) | \(N\) |
\( w \) = highest load per unit length of UIL | \(lbf\;/\;in\) | \(N\;/\;m\) |
\( L \) = 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