Overhanging Beam - Point Load on Beam End

on . Posted in Structural Engineering

ob 5A

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.

 

 

 

 

Overhanging Beam - Point Load on Beam End formulas

\( R_1 \;=\; V_1 \;=\; P\;a\;/\;L \) 

\( R_2 \;=\; V_1 + V_2 \;=\; ( P \;/\;L)  \; ( L + a ) \) 

\( V_2  \;=\;   P   \) 

\( M_{max}  \; ( at \;R_2 )  \;=\; P\;a   \)

\( M_x  \; (between\; supports )  \;=\; P\;a\;x\;/\;L \) 

\( M_{x_1}  \; (for \;overhang ) \;=\;  P\; ( a - x_1 )  \)

\( \Delta_x  \; (between\; supports )  \;=\; ( -(P\;a\;x) \;/\;6\; \lambda \;I\;L)  \; ( L^2 - x^2 )  \)

\( \Delta_{x_1}  \; (overhang ) \;\;=\;\; ( P\;x_1 \;/\;6\; \lambda\; I ) \; ( 2\;a\;L + 3\;a\;x_1 - x_{1}{^2} )   \)

\( \Delta_{max}  \; ( for\;overhang\; at\;  x_1 = a )   \;=\;  ( P\;a^2 \;/\;3\; \lambda \;I ) \; ( L + a )  \)

\( \Delta_{max}  \;  ( between\; supports\; at \;x = \frac{L}{\sqrt{3}}  )    \;=\;   \frac{ -(P\;a\;L^2) }{9\; \sqrt{3} \; \lambda\; I }  \;\;=\;\;  0.06415 \; ( P\;a\;L^2 \;/\; \lambda\; I )  \)

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 \(lbf\) \(N\)

 

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Tags: Beam Support