Question

Prove that the following function can be used as an Airy stress function. Find the stresses in the range x > 0, and -d < y <d; d being a constant. Elaborate on the stress distributions, what kind of physical example could it correspond to? 0 = (3F/4d)(xy - (xy?/3d)) + (P/4d)y2

Answer 1

solve the biharmonic equation carefully

Given stress airy function o is o ( 35 ) bay (xyz 3d2 + Fyz p 40 X70 -dc y cd Los above to be stress ain function it must satisfy biharmonic equation le. 240 / 2 d 3F xy P 9 3F xy3 + ud (322) ud ud vad u 3 ad + 2 ap any2 344 Co-oto) +2 Co-o+0) + duh Co-oto) so. Hence above equation Satisfies biharmonic can be used stress airy equation and function Now finding Stresses P (2 a2g + бях - was (6xy aya ча 3F ny + l ad 2d3 Syy 226 2012 0 Txy 220 2x 29 (3-5.600) - 3F y ² - 3F id Ud 3 i. бух 3F xy p W + ad 2d3 x=0 at - P ad = - 3F x + 1 ..at 2 d² 2d at y=-d - 1 3F l nt ad 2d² Tyy everywhere 3F Exy y²- 3F ud Yd3 3F at y=td 3 F = 0 TA ud . Physical model will y=d P 'Shear stress on the plane Albi 2 zd y=-d Above is the stress distribution around rectangular bür. be infinite length rectangular bur It may Stretched at model of end.