Simple Beam - Concentrated Load at Any Point
Simple Beam - Concentrated Load at Any Point formulas |
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\(\large{ R_1 = V_1 \; \left( max.\; when \;\; a < b \right) \;\;=\;\; \frac{P\;b}{L} }\) \(\large{ R_2 = V_2 \; \left( max.\; when \;\; a > b \right) \;\;=\;\; \frac{ P\;a}{L} }\) \(\large{ M_{max} \; \left(at \;point \;of \;load \right) \;\;=\;\; \frac{ P\;a\;b }{ L } }\) \(\large{ M_x \; \left( x < a \right) \;\;=\;\; \frac{ P\;b\;x }{ L } }\) \(\large{ \Delta_a \; \left(at \;point \;of \;load \right) \;\;=\;\; \frac{ P\;a^2\;b^2 }{ 3\; \lambda \;I \;L } }\) \(\large{ \Delta_x \; \left( x < a \right) \;\;=\;\; \frac{ P\;b\;x }{ 6\; \lambda \;I \;L } \; \left( L^2 - b^2 - x^2 \right) }\) \(\large{ \Delta_{max} \; \left(at \; x = \sqrt{ \frac{ a\; \left( a \;+\; 2\;b \right) }{3} } \; when \; a > b \right) \;\;=\;\; \frac{ P\;a\;b \; \left( a \;+\; 2\;b \right) \; \sqrt{ 3\;a \; \left( a \;+\; 2\;b \right) } } { 27\; \lambda \;I \;L } }\) |
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| Symbol | English | Metric |
| \(\large{ \Delta }\) = deflection or deformation | \(\large{in}\) | \(\large{mm}\) |
| \(\large{ a, b }\) = distance to point load | \(\large{in}\) | \(\large{mm}\) |
| \(\large{ x }\) = 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