Beam Fixed at Both Ends - Concentrated Load at Center

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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.

  

febe 2A

Beam Fixed at Both Ends - Concentrated Load at Center formulas

\( R \;=\; V \;=\; P\;/\;2 \) 

\( M_{max} \; (at\; center\; and\; ends ) \;=\;  P \;L\;/\;8 \) 

\( M_x \; ( x <  \frac {L}{2} )  \;=\; (P\;/\;8) \; ( 4\;x - L )  \) 

\( \Delta_{max}  \; (at\; center )  \;=\; P\;L^3\;/\;192\; \lambda\; I  \)

\( \Delta_x \; ( x < \frac {L}{2} )  \;=\; (P\;x^2\;/\;48\; \lambda\; I) \; ( 3\;L - 4\;x )   \)

\( x \; ( point\; of \; contraflexure )  \;=\;  L\;/\;4 \) 

Symbol English Metric
\( BM \) = bending moment \(lbf\;/\;sec\) \(kg-m\;/\;s\)
\( \Delta \) = 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) \(in^4\) \(mm^4\)
\( SF \) = shear force \(lbf\;/\;in^2\) \(MPa\)
\( L \) = span length of the bending member \(in\) \(mm\)
\( P \) = total concentrated load \(lbf\) \(N\)

 

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Tags: Beam Support