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Beam Deflection

Beam Deflection Formula

Load caseDiagramMaximum beam deflection
Cantilever with point load at free endCantilever with point load at free end\(\delta_{\max} = \dfrac{PL^3}{3EI}\)
Cantilever with point load at distance \(a\) from the fixed endCantilever with point load at distance \(a\) from the fixed end\(\delta_{\max} = \dfrac{Pa^2(3L-a)}{6EI}\)
Cantilever with uniformly distributed loadCantilever with uniformly distributed load\(\delta_{\max} = \dfrac{wL^4}{8EI}\)
Cantilever with triangular load, maximum at fixed endCantilever with triangular load, maximum at fixed end\(\delta_{\max} = \dfrac{wL^4}{30EI}\)
Cantilever with triangular load, maximum at free endCantilever with triangular load, maximum at free end\(\delta_{\max} = \dfrac{11wL^4}{120EI}\)
Cantilever with applied end momentCantilever with applied end moment\(\delta_{\max} = \dfrac{ML^2}{2EI}\)

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Second Moment of Area

GeometryDiagramFormula / Formulas
Rectangle\(I_y = \dfrac{w h^3}{12}\)
\(I_z = \dfrac{h w^3}{12}\)
Circular hollow section\(I_y = \dfrac{(D^4 - d^4)\cdot \pi}{64}\)
\(I_z = \dfrac{(D^4 - d^4)\cdot \pi}{64}\)
I-section\(I_y = \dfrac{w h^3}{12} - \dfrac{(w - t_w)\cdot (h - 2t_w)^3}{12}\)
\(I_z = \dfrac{h w^3}{12} - \dfrac{(w - t_w)^3 \cdot (h - 2t_w)}{12}\)
Rectangular hollow section\(I_y = \dfrac{W H^3 - w h^3}{12}\)
\(I_z = \dfrac{H W^3 - h w^3}{12}\)
Circle\(I_y = \dfrac{\pi D^4}{64}\)
\(I_z = \dfrac{\pi D^4}{64}\)

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