- See Article - Beam Design Formulas
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.
Overhanging Beam - Uniformly Distributed Load formulas |
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\( R_1 \;=\; V_1 \;=\; \dfrac{ w }{ 2 \cdot L } \cdot (L^2 - a^2) \) \( R_2 \;=\; V_2 + V_3 \;=\; \dfrac{ w }{ 2 \cdot L } \cdot ( L + a )^2 \) \(V_2 \;=\; w \cdot a \) \( V_3 \;=\; \dfrac{ w }{ 2 \cdot L } \cdot (L^2 + a^2) \) \( V_x \; (between\; supports ) \;=\; R_1 - w \cdot x \) \(V_{x_1} \; (for \;overhang ) \;=\; w \cdot ( a - x_1 ) \) \( M_x \; (between\; supports ) \;=\; \dfrac{ w\cdot x }{ 2 \cdot L } \cdot ( L^2 - a^2 - x\cdot L ) \) \( M_{x_1} \; (overhang ) \;=\; \dfrac{ w }{ 2 } \cdot ( a - x_1 )^2 \) \( M_1 \; [\;at\; x = \frac{L}{2} (1 - \frac{ a^2}{L^2} )\;] \;=\; \dfrac{ w }{ 8\cdot L^2 } \; (L + a)^2 \cdot (L - a)^2 \) \( M_2 \; (at\; R_2 ) \;=\; \dfrac{ w\cdot a^2 }{ 2 } \) \( \Delta_x \; (between \;supports ) \;=\; \dfrac{ w \cdot x }{ 24 \cdot \lambda \cdot I \cdot L } \cdot ( L^4 - 2\cdot L^2\cdot x^2 + L\cdot x^3 - 2\cdot a^2\cdot L^2 + 2\cdot a^2\cdot x^2 ) \) \( \Delta_{x_1} \; (for\; overhang ) \;=\; \dfrac{ w\cdot x_1 }{ 24 \cdot \lambda \cdot I } \cdot ( 4\cdot a^2\cdot L - L^3 + 6\cdot a^2\cdot x_1 - 4\cdot a\cdot x_{1}{^2} + x_{1}{^3} ) \) |
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| Symbol | English | Metric |
| \( \Delta \) = deflection or deformation | \(in\) | \(mm\) |
| \( x \) = horizontal distance from reaction to point on beam | \(in\) | \(mm\) |
| \( w \) = load per unit length | \(lbf\;/\;in\) | \(N\;/\;m\) |
| \( M \) = maximum bending moment | \(lbf-in\) | \(N-mm\) |
| \( V \) = maximum shear force | \(lbf\) | \(N\) |
| \( \lambda \) (Greek symbol lambda) = modulus of elasticity | \(lbf\;/\;in^2\) | \(Pa\) |
| \( R \) = reaction load at bearing point | \(lbf\) | \(N\) |
| \( I \) = second moment of area (moment of inertia) | \(in^4\) | \(mm^4\) |
| \( L \) = span length of the bending member | \(in\) | \(mm\) |
