Cantilever Beam - Concentrated Load at Any Point

on 05 May 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.

Cantilever Beam - Concentrated Load at Any Point formulas
\( R = V \;=\; P \) \( M_{max} \; (at\; fixed\; end ) \;=\; P\;b \) \( M_x \; (when \; x > a ) \;=\; P \; ( x - a ) \) \( \Delta_{max} \; (at\; fixed\; end ) \;=\; (P\; b^2\;/\;6\; \lambda\; I) \; ( 3\;L - b ) \) \( \Delta_a \; (at \;point\; of \;load ) \;=\; P\; b^3\;/\;3 \;\lambda\; I \) \( \Delta_x \; (when \; x < a ) \;=\; (P\; b^2 \;/\;6\; \lambda\; I) \; ( 3\;L - 3\;x - b) \) \( \Delta_x \; (when \; x > a ) \;=\; [\;P\; ( L - x )^2 \;/\;6 \;\lambda\; I)\;] \; ( 3\;b - L + x ) \)
Symbol	English	Metric
\( \Delta \) = deflection or deformation	\(in\)	\(mm\)
\( x \) = horizontal distance from reaction to point on beam	\(in\)	\(mm\)
\( M \) = 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	\(in\)	\(mm\)
\( P \) = total concentrated load	\(lbf\)	\(N\)

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