Overhanging Beam - Point Load on Beam End
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Overhanging Beam - Point Load on Beam End formulas |
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\(\large{ R_1 = V_1 \;\;=\;\; \frac{P\;a}{L} }\) \(\large{ R_2 = V_1 + V_2 \;\;=\;\; \frac{ P }{L} \; \left( L + a \right) }\) \(\large{ V_2 \;\;=\;\; P }\) \(\large{ M_{max} \; \left( at \;R_2 \right) \;\;=\;\; P\;a }\) \(\large{ M_x \; \left(between\; supports \right) \;\;=\;\; \frac{P\;a\;x}{L} }\) \(\large{ M_{x_1} \; \left(for \;overhang \right) \;\;=\;\; P \left( a - x_1 \right) }\) \(\large{ \Delta_x \; \left(between\; supports \right) \;\;=\;\; \frac{ -\;P\;a\;x }{6\; \lambda \;I\;L} \; \left( L^2 - x^2 \right) }\) \(\large{ \Delta_{x_1} \; \left(overhang \right) \;\;=\;\; \frac{ P\;x_1 }{6\; \lambda\; I } \; \left( 2\;a\;L + 3\;a\;x_1 - x_{1}{^2} \right) }\) \(\large{ \Delta_{max} \; \left(for\;overhang\; at\; x_1 = a \right) \;\;=\;\; \frac{ P\;a^2 }{3\; \lambda \;I } \; \left( L + a \right) }\) \(\large{ \Delta_{max} \; \left( between\; supports\; at \;x = \frac{L}{\sqrt{3}} \right) \;\;=\;\; \frac{ -\;P\;a\;L^2 }{9\; \sqrt{3} \; \lambda\; I } \;\;=\;\; 0.06415 \; \frac{ P\;a\;L^2 }{ \lambda\; I } }\) |
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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.
Tags: Beam Support Equations