Cantilever Beam - Concentrated Load at Any Point

on . Posted in Structural Engineering

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.

 

cb 4A

Cantilever Beam - Concentrated Load at Any Point formulas

\( R = V \;=\;  P  \) 

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

\( M_x  \; (when \; x > a )  \;=\;  P \; ( x - a )  \) 

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

\( \Delta_a \; (at \;point\; of \;load )  \;=\;  P\; b^3\;/\;3 \;\lambda\; I \)

\( \Delta_x \; (when \; x < a ) \;=\;  (P\; b^2 \;/\;6\; \lambda\; I) \; ( 3\;L - 3\;x - b)   \)

\( \Delta_x \; (when \; x > a ) \;=\;  [\;P\; ( L - x )^2  \;/\;6 \;\lambda\; I)\;]  \; ( 3\;b - L + x )   \)

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

 

Piping Designer Logo Slide 1

Tags: Beam Support