- 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.
Beam Fixed at One End - Concentrated Load at any point formulas |
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\( R_1 \;=\; V_1 \;=\; \dfrac{ P \cdot b^2 }{ 2 \cdot L^3 } \cdot ( a + 2 \cdot L ) \) \( R_2 \;=\; V_2 \;=\; \dfrac{ P \cdot a }{ 2 \cdot L^3 } \cdot ( 3 \cdot L^2 - a^2 ) \) \( M_1 \; (at\; point\; of \;load ) \;=\; R_1 \cdot a \) \( M_2 \; (at\; fixed \;end ) \;=\; \dfrac{ P \cdot a \cdot b }{ 2 \cdot L^2 } \cdot ( a + L ) \) \( M_x \; ( x < a ) \;=\; R_1 \cdot x \) \( M_x \; ( x > a ) \;=\; R_1 \cdot x - [\; P\; ( x - a ) \;] \) \( \Delta_{max} \; ( at \;x = L \; \frac{ L^2 \;+\; a^2 }{ 3\;L^2 \;-\; a^2 } \; when\; a < 0.414 \;L ) \;=\; \dfrac{ P \cdot a }{ 3 \cdot \lambda \cdot I } \cdot \dfrac{ ( L^2 - a^2 ) ^3 }{ ( 3 \cdot L^2 - a^2 ) ^2 } \) \( \Delta_{max} \; ( at \;x = L \;\sqrt{ \frac{ a }{ 2\;L \;+\; a } } \; when\; a > 0.414 \;L ) \;=\; \dfrac{ P \cdot a \cdot b^2 }{ 6 \cdot \lambda \cdot I } \cdot \sqrt{ \dfrac{ a }{ 2 \cdot L + a } } \) \( \Delta_a \; (at\; point\; of\; load ) \;=\; \dfrac{ P \cdot a^3 \cdot b^2 }{ 12 \cdot \lambda \cdot I \cdot L^3 } \cdot ( 3 \cdot L + b ) \) \( \Delta_x \; ( x < a ) \;=\; \dfrac{ P \cdot b^2 \cdot x }{ 12 \cdot \lambda \cdot I \cdot L^3 } \cdot ( 3 \cdot a \cdot L^2 - 2 \cdot L \cdot x^2 - a \cdot x^2 ) \) \( \Delta_x \; ( x > a ) \;=\; \dfrac{ P \cdot a }{ 12 \cdot \lambda \cdot I \cdot L^3 } \cdot ( L - x )^2 \cdot ( 3 \cdot L^2 \cdot x - a^2 \cdot x - 2 \cdot a^2 \cdot L ) \) |
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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) | \(in^4\) | \(mm^4\) |
