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Applied Structural Mechanics Fundamentals of Elasticity Part 8

Tham khảo tài liệu 'applied structural mechanics fundamentals of elasticity part 8', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 19 Cl 10 Coupled Disk-Plate Problems Solution Fig. B-37 Clamped circular plate under uniform pressure We proceed from von KARMAN s equations in operator notation 10.13 AA w Ộ4 w la IX. IX AA c - E Ộ4 w w . lb This system of nonlinear partial differential equations which is coupled in w and o is rewritten in tensor notation according to 10 J4 wK k K Í wL Ltf. 2b The problem is most conveniently solved in polar coordinates for which reason we transform the covariant derivatives in the equation by means of 2.40 K K r s13 wl 2 111 e12 s21 WI 1 I13 c 4- e21e12w 13 21 s21 E21 wjn 2J 13b The metric tensors theừ determinant and the CHRISTOFFEL symbols required for the further treatment of the problem have already been determined for cylindrical coordinates see 2.5 . If we now substitute the permutation symbols 2.20b and the covariant derivatives 2.34b e.g. w u w ill etc. into the right-hand sides and if we take into consideration that the dependence on the angle coordinate 2 tp vanishes in the derivatives owing to the axisymmetrical shape and loads the equations reduce with Ẹ1 r and d dr to Exercise B-10-2 197 In order to solve this nonlinear system of differential equations we first integrate 4b . This yields r r b dr dr c r 1 21nf 2C r I ậ- . s Eq. 5 contains terms with three constants which can be determined from the following conditions. First we use the disk stresses according to 8.7 o St 1 0 r t r r 6a d ÍPV t rr 6b Substitution of equation 5 into 6a b yields dr I C1 1 I 2 In - - 2 C2 4-----------------------------------------2 rr 7a VV - 2l7. r f-r-dr d4r Cl 3 1 21ni 1 2C3 - ậ- Owing to the condition that the stresses in the middle of the disk r 0 cannot become infinite finitness condition the constants Cị and c3 must vanish. The remaining constant can be calculated from the boundary condition u a 0 . 8 For this purpose we proceed from the material law 8.9 and then replace the expansion E by the radial displacement u using the corresponding linearized .