torsion3 - 04.01.2010 Burak AYCAN(1631506 Monday Orçun...

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Unformatted text preview: 04.01.2010 Burak AYCAN (1631506) Monday Orçun ESER (1503135) TORSION Prandt’s Stress Function τ zx = 𝜕∅ 𝜕? τ zy = 𝜕∅ 𝜕? and 𝜕 2 ∅ 𝜕? 2 + 𝜕 2 ∅ 𝜕? 2 = − 2 ¡𝜃 Bcs ∅ = 0 on the boundaries ¢ = 2 ∅???? = 2( 𝐴??£ ????? ∅ ????? ) 𝜏 ¤£? = ¢? 𝐽 twist  ∅ = 𝜃¥ = ¢¥ 𝐽¡ PRANDTL’S MEMBRANE ANALOGY Consider an edge-supported homogeneous membrane, given its boudnary contour by a hole cut in a plate.The shape of the hole is the same as that of the twisted bar to be studied; the sizes need not be identical. ∅ satisfies the same equation from describes the deflection of a membrane (or soap film) subjected to a pressure ( ∅ ==> ? : ?¡? ¡???¡? ?? ?¡? ¢??? ???£ ) Equation for the deflection z; consider equilibrium of an infinitesmall element ABCD; Let the tensile forces per unit membrane length to be denoted by S=const β + d β dx dx ?? = 0 ( −¡?¢ ) ?? ?¢ + ( ¡?? )( ?? ?¢ + ? 2 ? ?¢ 2 ?¢ ) − ( ¡?¢ ) ?? ?? + ¡?¢ ( ?? ?? + ? 2 ? ?? 2 ?? ) + £?¢?? = 0 Leading to ? 2 ? ?¢ 2 + ? 2 ? ?? 2 = −£ ¡ [1] This is again Poisson’s equation. Upon comparison of poisson’s equation (Ugural 6.9), and (Ugural 6.8) The quantities shown in following table are aboserved to be analogous.The Membrane subject to the conditions outlined, thus represent the ∅ surface.In view of the surface....
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torsion3 - 04.01.2010 Burak AYCAN(1631506 Monday Orçun...

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