Overhanging Beam - Uniformly Distributed Load Over Supported Span

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ob 3A

  

Overhanging Beam - Uniformly Distributed Load Over Supported Span formulas

\(\large{ R = V \;\;=\;\;  \frac{w\; L }{2}      }\) 

\(\large{ V_x \;\;=\;\;   w \; \left( \frac{L}{2} - x  \right)       }\) 

\(\large{ M_{max} \;  \left(at \;center \right)   \;\;=\;\;  \frac{w\; L^2 }{8}      }\) 

\(\large{ M_x \;\;=\;\;  \frac{w\; x }{2} \; \left( L - x  \right)    }\)

\(\large{ \Delta_{max} \; \left(at \;center \right)  \;\;=\;\;  \frac{5\;w\; L^4 }{384\; \lambda \; I}      }\)

\(\large{ \Delta_x   \;\;=\;\;   \frac{w\; x }{24\; \lambda\; I}   \;  \left( L^3 - 2\;L\;x^2 + x^3  \right)         }\)

\(\large{ \Delta_{x_1}   \;\;=\;\;   \frac{ -\; w \; L^3 \;x_1 }{24\; \lambda\; I}      }\)

Symbol English Metric
\(\large{ \Delta }\) = deflection or deformation \(\large{in}\) \(\large{mm}\)
\(\large{ x }\) = horizontal distance from reaction to point on beam \(\large{in}\) \(\large{mm}\)
\(\large{ w }\) = load per unit length \(\large{\frac{lbf}{in}}\) \(\large{\frac{N}{mm}}\)
\(\large{ M }\) = maximum bending moment \(\large{lbf-in}\) \(\large{N-mm}\)
\(\large{ V }\) = maximum shear force \(\large{lbf}\) \(\large{N}\)
\(\large{ \lambda  }\)   (Greek symbol lambda) = modulus of elasticity \(\large{\frac{lbf}{in^2}}\) \(\large{Pa}\)
\(\large{ R }\) = reaction load at bearing point \(\large{lbf}\) \(\large{N}\)
\(\large{ I }\) = second moment of area (moment of inertia) \(\large{in^4}\) \(\large{mm^4}\)
\(\large{ L }\) = span length of the bending member \(\large{in}\) \(\large{mm}\)

 

diagrams

  • 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.

 

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Tags: Beam Support Equations