Cantilever Beam - Uniformly Distributed Load and Variable End Moments

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• 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.

 

Cantilever Beam - Uniformly Distributed Load and Variable End Moments formulas
\( R = V \;=\;  w\;L  \)   \( V_x \;=\;  w\;x   \)  \( M_{max} \; (at\; fixed \;end )   \;=\;  w\; L^2\;/\;3  \)  \( M_1 \; (at \;free \;end )  \;=\;  w \;L^2\;/\;6  \) \( M_x   \;=\;  ( w \;/\;6)  \; ( L^2 - 3\;x^2 )  \) \( \Delta_{max} \; (at\; free \;end)  \;=\;  w\; L^4\;/\;24\; \lambda\; I  \) \( \Delta_x   \;=\;  [\;w \; (  L^2 -  ( L - x )^2 )^2 \;] \;/\;24 \;\lambda\; I   \)
Symbol	English	Metric
\( \Delta \) = deflection or deformation	\(in\)	\(mm\)
\( x \) = horizontal distance from reaction to point on beam	\(in\)	\(mm\)
\( w \) = load per unit length	\(lbf\;/\;in\)	\(N\;/\;m\)
\( M \) = 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)	\(in^4\)	\(mm^4\)
\( L \) = span length of the bending member	\(in\)	\(mm\)