Cantilever Beam - Concentrated Load at Any Point
Cantilever Beam - Concentrated Load at Any Point formulas |
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\(\large{ R = V \;\;=\;\; P }\) \(\large{ M_{max} \; \left(at\; fixed\; end \right) \;\;=\;\; P\;b }\) \(\large{ M_x \; \left(when \; x > a \right) \;\;=\;\; P \; \left( x - a \right) }\) \(\large{ \Delta_{max} \; \left(at\; fixed\; end \right) \;\;=\;\; \frac {P\; b^2} {6\; \lambda\; I} \; \left( 3\;L - b \right) }\) \(\large{ \Delta_a \; \left(at \;point\; of \;load \right) \;\;=\;\; \frac {P\; b^3} {3 \;\lambda\; I} }\) \(\large{ \Delta_x \; \left(when \; x < a \right) \;\;=\;\; \frac {P\; b^2 } {6\; \lambda\; I} \; \left( 3\;L - 3\;x - b \right) }\) \(\large{ \Delta_x \; \left(when \; x > a \right) \;\;=\;\; \frac {P\; \left( L\; - \;x \right)^2 } {6 \;\lambda\; I} \; \left( 3\;b - L + x \right) }\) |
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| Symbol | English | Metric |
| \(\large{ \Delta }\) = deflection or deformation | \(\large{in}\) | \(\large{mm}\) |
| \(\large{ x }\) = horizontal distance from reaction to point on beam | \(\large{in}\) | \(\large{mm}\) |
| \(\large{ M }\) = maximum bending moment | \(\large{lbf-in}\) | \(\large{N-mm}\) |
| \(\large{ V }\) = maximum shear force | \(\large{lbf}\) | \(\large{N}\) |
| \(\large{ \lambda }\) (Greek symbol lambda) = modulus of elasticity | \(\large{\frac{lbf}{in^2}}\) | \(\large{Pa}\) |
| \(\large{ R }\) = reaction load at bearing point | \(\large{lbf}\) | \(\large{N}\) |
| \(\large{ I }\) = second moment of area (moment of inertia) | \(\large{in^4}\) | \(\large{mm^4}\) |
| \(\large{ L }\) = span length of the bending member | \(\large{in}\) | \(\large{mm}\) |
| \(\large{ P }\) = total concentrated load | \(\large{lbf}\) | \(\large{N}\) |
diagrams
- 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.
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Tags: Beam Support Equations