Cantilever Beam - Uniformly Distributed Load and Variable End Moments

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cb 3A

  

Cantilever Beam - Uniformly Distributed Load and Variable End Moments formulas

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

\(\large{ V_x \;\;=\;\;  w\;x    }\) 

\(\large{ M_{max} \;  \left(at\; fixed \;end \right)   \;\;=\;\;  \frac{w\; L^2}{3}  }\) 

\(\large{ M_1 \;  \left(at \;free \;end \right)  \;\;=\;\;  \frac {w \;L^2} {6}  }\)

\(\large{ M_x   =   \frac{ w }{6}  \;  \left( L^2 - 3\;x^2 \right)      }\)

\(\large{ \Delta_{max} \;  \left(at\; free \;end \right)  \;\;=\;\;  \frac{w\; L^4}{24\; \lambda\; I}  }\)

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

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 \(\large{\frac{lbf}{in}}\) \(\large{\frac{N}{m}}\)
\(\large{ M }\) = 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