Beam Fixed at Both Ends - 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 Both Ends - Concentrated Load at Any Point formulas |
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\( R_1 = V_1 \; (max.\; when \; a < b ) \;=\; (P\;b^2\;/\;L^3) \; ( 3\;a + b ) \) \( R_2 = V_2 \; (max.\; when \; a > b ) \;=\; (P\;a^2\;/\;L^3) \; ( a + 3\;b ) \) \( M_1 \; (max.\; when \; a < b ) \;=\; P\;a\;b^2\;/\;L^2 \) \( M_2 \; (max. \;when \; a > b ) \;=\; P\;a^2\;b\;/\;L^2 \) \( M_a \; (at \;point \;of \;load ) \;=\; 2\;P\;a^2\;b^2\;/\;L^3 \) \( M_x \; ( x < a ) \;=\; (R_1 \; x) - (P\;a\;b^2\;/\;L^2) \) \( \Delta_{max} \; (at \; x = \frac{2\;a\;L}{3\;a \;+\; b} \;when \;a > b ) \;=\; 4\;P\;a^3\;b^2\;/\; [\;3 \;\lambda\; I \; ( 3\;a \;+ \;b )^2 \;] \) \( M_a \; (at \;point \;of \;load ) \;=\; 2 \;P\;a^3\;b^3\;/\;6\; \lambda\; I \;L^3 \) \( \Delta_x \; ( x < a ) \;=\; (2 \;P\;b^2\;x^2\;/\;12\; \lambda \;I \;L^3) \; ( 3\;a\;L - 3\;a\;x - b\;x ) \) \( x \; ( point\; of\; contraflexure\;between\;supports ) \;=\; a\;b^2\;P\;/\;L^2\;R_1 \) |
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| Symbol | English | Metric |
| \( \Delta \) = Deflection or Deformation | \(in\) | \(mm\) |
| \( x \) = Horizontal Distance from Reaction to Point on Beam | \(in\) | \(mm\) |
| \( M \) = Maximum Bending Moment | \(lbf-in\) | \(N-mm\) |
| \( V \) = Maximum Shear Force | \(lbf\) | \(N\) |
| \( \lambda \) (Greek symbol lambda) = Modulus of Elasticity | \(lbf\;/\;in^2\) | \(Pa\) |
| \( I \) = Moment of Inertia | \(in^4\) | \(mm^4\) |
| \( R \) = Reaction Load at Bearing Point | \(lbf\) | \(N\) |
| \( L \) = Span Length Under Consideration | \(in\) | \(mm\) |
| \( a, b \) = Span Length Under Consideration | \(in\) | \(mm\) |
| \( P \) = Total Concentrated Load | \(lbf\) | \(N\) |
Tags: Beam Support