Beam Fixed at Both Ends - Concentrated Load at Center

• See Article  -  Beam Design Formulas
• Tags: Beam Support

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.

  

Beam Fixed at Both Ends - Concentrated Load at Center formulas
\( M_{max} \; (at\; center\; and\; ends ) \;=\;  \dfrac{P \cdot L}{8} \)  \( M_x \; ( x <  \frac {L}{2} )  \;=\; \dfrac{P \cdot L}{8} \cdot ( 4\: x - L )  \)  \( \Delta_{max}  \; (at\; center )  \;=\; \dfrac{P \cdot L^3}{192\; \lambda \cdot I } \) \( \Delta_x \; ( x < \frac {L}{2} )  \;=\; (3 \; L - 4 \; x ) \cdot \dfrac{P \cdot x^2}{48 \; \lambda \cdot I} \;   \) \( x \; ( point\; of \; contraflexure )  \;=\;  \dfrac{L}{4} \)
Symbol	English	Metric
\( R \) = Reaction Load at Bearing Point	\(lbf\)	\(N\)
\( V \) = Maximum Shear Force	\(lbf\)	\(N\)
\( M \) = Maximum Bending Moment	\(lbf - in\)	\(N - mm\)
\( \Delta \) = Deflection or Deformation	\(in\)	\(mm\)
\( x \) = Horizontal Distance from Reaction to Point on Beam	\(in\)	\(mm\)
\( P \) = Total Concentrated Load	\(lbf\)	\(N\)
\( L \) = Span Length of the Bending Member	\(in\)	\(mm\)
\( \lambda  \)   (Greek symbol lambda) = Modulus of Elasticity	\(lbf\;/\;in^2\)	\(Pa\)
\( I \) = Second Moment of Area (Moment of Inertia)	\(in^4\)	\(mm^4\)