Beam Fixed at One End - Concentrated Load at Any Point
- See Article - 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.
Beam Fixed at One End - Concentrated Load at any point formulas |
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\( R_1 \;=\; V_1 \;=\; (P\;b^2\;/\;2\;L^3) \; ( a + 2\;L ) \) \( R_2 \;=\; V_2 \;=\; (P\;a\;/\;2\;L^3) \; ( 3\;L^2 - a^2 ) \) \( M_1 \; (at\; point\; of \;load ) \;=\; R_1 \;a \) \( M_2 \; (at\; fixed \;end ) \;=\; (P\;a\;b\;/\;2\;L^2) \; ( a +L ) \) \( M_x \; ( x < a ) \;=\; R_1\; x \) \( M_x \; ( x > a ) \;=\; R_1 \;x - [\; P\; ( x - a ) \;] \) \( \Delta_{max} \; ( at \;x = L \; \frac{ L^2 \;+\; a^2 }{ 3\;L^2 \;-\; a^2 } \; when\; a < 0.414 \;L ) \;=\; (P\;a\;/\;3\; \lambda\; I ) \; \frac{ ( L^2 \;-\; a^2 ) ^3 }{ ( 3\;L^2 \;- \;a^2 ) ^2 } \) \( \Delta_{max} \; ( at \;x = L \;\sqrt{ \frac{ a }{ 2\;L \;+\; a } } \; when\; a > 0.414 \;L ) \;=\; (P\;a\;b^2\;/\;6\; \lambda\; I ) \; \sqrt{ a \;/\; 2\;L + a } \) \( \Delta_a \; (at\; point\; of\; load ) \;=\; ( P\;a^3 \;b^2\;/\;12\; \lambda\; I \;L^3) \; ( 3\;L + b ) \) \( \Delta_x \; ( x < a ) \;=\; ( P\;b^2\; x\;/\;12 \;\lambda\; I \;L^3) \; ( 3\;a\;L^2 - 2\;L\;x^2 - a\;x^2 ) \) \( \Delta_x \; ( x > a ) \;=\; ( P\;a\;/\;12\; \lambda\; I \;L^3) \; ( L - x )^2 \; ( 3\;L^2 \;x - a^2 \;x - 2\;a^2 \;L ) \) |
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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 | \(in\) | \(m\) |
| \( P \) = Total Concentrated Load | \(lbf\) | \(N\) |
| \( a, b \) = Length to Point Load | \(in\) | \(m\) |
| \( L \) = Span Length of the Bending Member | \(in\) | \(m\) |
| \( x \) = Horizontal Distance from Reaction to Point on Beam | \(in\) | \(m\) |
| \( \lambda \) (Greek symbol lambda) = Modulus of Elasticity | \(lbf\;/\;in^2\) | \(Pa\) |
| \( I \) = Eecond Moment of Area (Moment of Inertia) | \(in^4\) | \(mm^4\) |
Tags: Beam Support