[Solved] For A Rectangular Cross Section Beam, Subjected To Transvers

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Concept:

Shear stresses in beam

F1 Krupalu 13-12-21 Savita D1

Let,

l → length, b → breath, A → Area, y̅ → distance from reach of axis

Shear stress in beam = \(\rm τ=\frac{FA\overline{y}}{ZI}\)

Area \(=b\left(\frac{d}{2}-y\right), \overline{y}=\frac{1}{2}\left(\frac{d}{2}-y\right)+y=\frac{1}{2}\left(\frac{d}{2}+y\right)\)

Z = b \(I=\frac{bd^3}{12}\)

\(\rm τ=\frac{FA\overline{y}}{ZI}\)

\(\rm τ=\frac{F.b\left(\frac{d}{2}-y\right).\frac{1}{2}\left(\frac{d}{2}+y\right)}{\left(\frac{bd^3}{12}\times b\right)}\) parabolic

y = 0 for τmax

\(\rm \tau_{max}=\frac{bF}{bd^3}\left(\frac{d^2}{4}\right)\)

\(\rm \tau_{max}=\frac{3}{2}\frac{F}{bd}\)

∴ \(\rm \frac{F}{bd}=\tau_{Avg}\)

\(\rm \tau_{max}=\frac{3}{2}\tau_{Avg}\)

\(\rm\frac{\tau_{max}}{\tau_{Avg}}=\frac{3}{2}\)

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