- See Article - Beam Design Formulas
Beam Fixed at Both Ends - Concentrated Load at Center formulas |
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\( M_{max} \; (at\; center\; and\; ends ) \;=\; \dfrac{P \cdot L}{8} \) \( M_x \; ( x < \frac {L}{2} ) \;=\; \dfrac{P \cdot L}{8} \cdot ( 4 \cdot x - L ) \) \( \Delta_{max} \; (at\; center ) \;=\; \dfrac{P \cdot L^3 }{ 192 \cdot \lambda \cdot I } \) \( \Delta_x \; ( x < \frac {L}{2} ) \;=\; (3 \cdot L - 4 \cdot x ) \cdot \dfrac{P \cdot x^2}{48 \cdot \lambda \cdot I} \; \) \( x \; ( point\; of \; contraflexure ) \;=\; \dfrac{L}{4} \) |
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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 | \(in\) | \(mm\) |
| \( x \) = Horizontal Distance from Reaction to Point on Beam | \(in\) | \(mm\) |
| \( P \) = Total Concentrated Load | \(lbf\) | \(N\) |
| \( L \) = Span Length of the Bending Member | \(in\) | \(mm\) |
| \( \lambda \) (Greek symbol lambda) = Modulus of Elasticity | \(lbf\;/\;in^2\) | \(Pa\) |
| \( I \) = Second Moment of Area (Moment of Inertia) | \(in^4\) | \(mm^4\) |
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.
