- See Article Link - Beam Design Formulas
Beam Fixed at One End - Concentrated Load at Center formulas |
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\( R_1 \;=\; V_1 \;=\; \dfrac{ 5 \cdot P }{ 16 }\) \( R_2 \;=\; V_2 max \;=\; \dfrac{ 11 \cdot P }{ 16 }\) \( M_{max} \; (at \;fixed \;end ) \;=\; \dfrac{ 3 \cdot P \cdot L }{ 16 }\) \( M_1 \; (at\; point\; of \;load ) \;=\; \dfrac{ 5\cdot P\cdot L }{ 32 }\) \( M_x \; ( x < \frac {L}{2} ) \;=\; \dfrac{ 5 \cdot P \cdot x }{ 16 }\) \( M_x \; ( x > \frac {L}{2} ) \;=\; P\cdot \left( \dfrac{ L }{ 2 } - \dfrac{ 11 \cdot x }{ 16 } \right) \) \( \Delta_x \; (at\; point\; of\; load ) \;=\; \dfrac{ 7\cdot P \cdot L^3 }{ 768 \cdot \lambda \cdot I }\) \( \Delta_x \; ( x < \frac {L}{2} ) \;=\; \dfrac{ P \cdot x }{ 96 \cdot \lambda \cdot I } \cdot ( 3 \cdot L^2 - 5 \cdot x^2 ) \) \( \Delta_x \; ( x > \frac {L}{2} ) \;=\; \dfrac{ P }{ 96 \cdot \lambda \cdot I } \cdot ( x - L )^2 \cdot ( 11 \cdot x - 2 \cdot L ) \) \( \Delta_{max} \; ( at \; x = L \; \left( \frac{1}{5} \right)^{\frac{1}{2}} ) \;=\; \dfrac{ P\cdot L^3 }{ 48 \cdot \lambda \cdot I \cdot \left( 5 \right)^{\frac{1}{2}} }\) |
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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 | \(in\) | \(mm\) |
| \( P \) = total concentrated load | \(lbf\) | \(N\) |
| \( L \) = span length of the bending member | \(in\) | \(mm\) |
| \( x \) = horizontal distance from reaction to point on beam | \(in\) | \(mm\) |
| \( \lambda \) (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.
