Cantilever Beam - Uniformly Distributed Load and Variable End Moments

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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.

 

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Cantilever Beam - Uniformly Distributed Load and Variable End Moments formulas

\( R = V \;=\;  w\;L  \)  

\( V_x \;=\;  w\;x   \) 

\( M_{max} \; (at\; fixed \;end )   \;=\;  w\; L^2\;/\;3  \) 

\( M_1 \; (at \;free \;end )  \;=\;  w \;L^2\;/\;6  \)

\( M_x   \;=\;  ( w \;/\;6)  \; ( L^2 - 3\;x^2 )  \)

\( \Delta_{max} \; (at\; free \;end)  \;=\;  w\; L^4\;/\;24\; \lambda\; I  \)

\( \Delta_x   \;=\;  [\;w \; (  L^2 -  ( L - x )^2 )^2 \;] \;/\;24 \;\lambda\; I   \)

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 \(in\) \(mm\)

 

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Tags: Beam Support