Cantilever Beam - Concentrated Load at Any Point

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Cantilever Beam - Concentrated Load at Any Point formulas

\(\large{ R = V \;\;=\;\;  P  }\) 

\(\large{ M_{max} \; \left(at\; fixed\; end \right)  \;\;=\;\;  P\;b  }\) 

\(\large{ M_x  \;  \left(when \; x > a \right)   \;\;=\;\;   P \; \left( x - a \right)       }\) 

\(\large{ \Delta_{max} \; \left(at\; fixed\; end \right)  \;\;=\;\;  \frac {P\; b^2} {6\; \lambda\; I} \; \left( 3\;L - b \right)       }\)

\(\large{ \Delta_a \; \left(at \;point\; of \;load \right)  \;\;=\;\;  \frac {P\; b^3} {3 \;\lambda\; I}    }\)

\(\large{ \Delta_x \; \left(when \; x < a \right) \;\;=\;\;  \frac {P\; b^2 } {6\; \lambda\; I} \; \left( 3\;L - 3\;x - b  \right)       }\)

\(\large{ \Delta_x \; \left(when \; x > a \right) \;\;=\;\;  \frac {P\; \left( L\; - \;x \right)^2    } {6 \;\lambda\; I} \; \left( 3\;b - L + x \right)       }\)

Symbol English Metric
\(\large{ \Delta }\) = deflection or deformation \(\large{in}\) \(\large{mm}\)
\(\large{ x }\) = horizontal distance from reaction to point on beam \(\large{in}\) \(\large{mm}\)
\(\large{ M }\) = 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 \(\large{in}\) \(\large{mm}\)
\(\large{ P }\) = total concentrated load \(\large{lbf}\) \(\large{N}\)

 

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