Simple Beam - Load Increasing Uniformly to Center
- See Article Link - Beam Design Formulas
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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\(\large{ R = V_{max} = \frac{W}{2} }\) \(\large{ V_x \; \left( x < \frac{L}{2} \right) = \frac{W}{2\;L^2} \; \left( L^2 - 4\;x^2 \right) }\) \(\large{ M_{max} \; \left(at \;center\right) = \frac{W \;L}{6} }\) \(\large{ M_x \; \left( x < \frac{L}{2} \right) = W\;x \; \left( \frac{1}{2} - \frac {2\;x^2}{3\;L^2} \right) }\) \(\large{ \Delta_{max} \; \left(at \;center\right) = \frac{W \;L^3} {60 \; \lambda \;I } }\) \(\large{ \Delta_x \; \left( x < \frac{L}{2} \right) = \frac{W\; x}{480\; \lambda \;I \;L^2} \; \left( 5\;L^2 - 4\;x^2 \right)^2 }\) |
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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 |
\(\large{ R }\) = reaction load at bearing point | \(\large{lbf}\) | \(\large{N}\) |
\(\large{ V }\) = maximum shear force | \(\large{lbf}\) | \(\large{N}\) |
\(\large{ M }\) = maximum bending moment | \(\large{lbf-in}\) | \(\large{N-mm}\) |
\(\large{ \Delta }\) = deflection or deformation | \(\large{in}\) | \(\large{mm}\) |
\(\large{ W }\) = total load or \(\large{ \frac{w\;L}{2} }\) | \(\large{lbf}\) | \(\large{N}\) |
\(\large{ w }\) = highest load per unit length of UIL | \(\large{\frac{lbf}{in}}\) | \(\large{\frac{N}{m}}\) |
\(\large{ L }\) = span length of the bending member | \(\large{in}\) | \(\large{mm}\) |
\(\large{ x }\) = horizontal distance from reaction to point on beam | \(\large{in}\) | \(\large{mm}\) |
\(\large{ \lambda }\) (Greek symbol lambda) = modulus of elasticity | \(\large{\frac{lbf}{in^2}}\) | \(\large{Pa}\) |
\(\large{ I }\) = second moment of area (moment of inertia) | \(\large{in^4}\) | \(\large{mm^4}\) |
Tags: Beam Support Equations