Beam Fixed at Both Ends - Concentrated Load at Center

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febe 2A

   

Beam Fixed at Both Ends - Concentrated Load at Center formulas

\(\large{ R = V \;\;=\;\;  \frac {P} {2}  }\) 

\(\large{ M_{max} \; \left(at\; center\; and\; ends \right)  \;\;=\;\; \frac {P \;L} {8}  }\) 

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

\(\large{ \Delta_{max}  \; \left(at\; center \right)  \;\;=\;\; \frac {P\;L^3}{192\; \lambda\; I}   }\)

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

\(\large{ x \; \left( point\; of \; contraflexure \right)  \;\;=\;\;  \frac{L}{4}  }\) 

Symbol English Metric
\(\large{ BM }\) = bending moment \(\large{\frac{lbf}{sec}}\) \(\large{\frac{kg-m}{s}}\)
\(\large{ \Delta }\) = 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) \(\large{in^4}\) \(\large{mm^4}\)
\(\large{ SF }\) = shear force \(\large{\frac{lbf}{in^2}}\) \(\large{MPa}\)
\(\large{ L }\) = 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