Beam Fixed at Both Ends - Uniformly Distributed Load

on 24 April 2018. Posted in Structural Engineering

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 - Uniformly Distributed Load formulas
\( R \;=\; V \;=\; w\; L\;/\;2 \) \( V_x \;=\; w \; [\; (L\;/\;2) - x \;] \) \( M_{max} \; (at \;ends ) \;=\; w\; L^2\;/\;12 \) \( M_1 \; (at\; center ) \;=\; w\; L^2\;/\;24 \) \( M_x \;=\; (w\;/\;12) \; ( 6\;L\;x - L^2 - 6\;x^2 ) \) \( M_{max} \; (at \;center ) \;=\; w\; L^4\;/\;384\; \lambda\; I \) \( \Delta_x \;=\; (w\; x^2\;/\;24\; \lambda\; I) \; ( L - x ) ^2 \) \( x \; (points \;of \;contraflexure ) \;=\; (\sqrt{3} - 3 )\; L \)
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	\(in\)	\(mm\)
\( w \) = Load per Unit Length	\(lbf\;/\;in\)	\(N\;/\;m\)
\( L \) = 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)	\(in^4\)	\(mm^4\)

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