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Beam Fixed at One End - Concentrated Load at Center

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Beam Fixed at One End - Concentrated Load at Center formulas

\(\large{ R_1 = V_1 \;\;=\;\; \frac {5\;P} {16}  }\) 

\(\large{ R_2 = V_2 max \;\;=\;\; \frac {11\;P} {16}  }\) 

\(\large{ M_{max} \; \left(at \;fixed \;end \right)  \;\;=\;\;  \frac {3\;P\;L} {16}  }\) 

\(\large{ M_1  \; \left(at\; point\; of \;load \right) \;\;=\;\;  \frac {5\;P\;L} {32}  }\)

\(\large{ M_x   \;  \left(  x < \frac {L}{2}    \right)   \;\;=\;\;   \frac  { 5\;P\;x} {16}    }\)

\(\large{ M_x   \;  \left(  x > \frac {L}{2}    \right)   \;\;=\;\;  P\; \left(  \frac { L} {2}  - \frac { 11\;x} {16}  \right)  }\)

\(\large{ \Delta_x \;   \left(at\; point\; of\; load \right)  \;\;=\;\;  \frac { 7\;P\;L^3} {768\; \lambda\; I}  }\)

\(\large{ \Delta_x  \;  \left(  x < \frac {L}{2}    \right)   \;\;=\;\;   \frac  { P\;x} {96 \;\lambda\; I} \; \left( 3\;L^2 - 5\;x^2  \right)    }\)

\(\large{ \Delta_x   \;  \left(  x > \frac {L}{2}    \right)   \;\;=\;\;  \frac  { P} {96 \;\lambda\; I}\;  \left( x - L  \right)^2 \; \left( 11\;x - 2\;L  \right)   }\)

\(\large{ \Delta_{max}  \;  \left[ at \; x = L \; \left( \frac{1}{5} \right)^{\frac{1}{2}}  \right]   \;\;=\;\;  \frac{P\;L^3}{48\; \lambda\; I \; \left( 5 \right)^{\frac{1}{2}}  }    }\)
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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