tailieunhanh - Applied Structural Mechanics Fundamentals of Elasticity Part 7

Tham khảo tài liệu 'applied structural mechanics fundamentals of elasticity part 7', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 166 9 Plates and with DO 2 w yy x o - 1 aTO1 An 4- Bj ự siu-5 - n 1 the moments at y 0 yield the expression do Mxx KU tD1 z 11 1 1- A -2vBa 2 n KX sin - d. Here we apply the same assumptions as in the case of the deflection functions he. 1 X - 0 2 X - 1 1 - V Aj - 2 V Bj 1 - V Aj - 2 V Bj 2 0 ii 2 - 0 . Mxx re - sin K 1. I V Op . Mxx K l v ctT0r The curves for the dimensionless bending moments Mxx X D K 1 vJotrjOj at y 0 are presented in Fig. B-22. With vanishing stiffness of the tubes X 0 at the boundaries the plate deforms without stresses. Fig. B-22 Dimensionless bending moments MkxC .0 Exercise E-9-4 1E7 Exercise B-9-4 A thin rectangular plate with all boundaries clamped dimensions a b thickness t is subjected to a uniformly distributed load with the intensity p Po Fig B-23 Determine the deflection function by means of the RITZ method using the trigonometrical double series as an approximation function oc DO w m 1 n 1 2 m 7T X 1 - cos -a 2 n Try b2 with amn as free coefficients. Po Solution For this task we first write the internal potential strain energy of a rectangular plate according to expressed in terms of the approximation function w b a ni 2 KJJ XX 1 -2 1 - - 1 0 0 Ft3 _ with the plate stiffness K A. and the notation ò ò X . ò òy . 12 1- V2 For the external potential potential of the external forces we have in terms of w r p0 w dx dy . 0 0 ne 2 When applying the RITZ method the approximation function has to fulfill at least the geometrical boundary conditions. These are w 0 y w x 0 y 0 w a y w x a y - 0 w X 0 w y X 0 0 w x b w y X b 0 . 168 9 Plates The principle of virtual displacements see is now used as a necessary condition for determining the unknown coefficients amn an 0 For the given approximation function 1 - cos m J n we now iAleu late the derivatives w XT1 X 1 2IÍ1K . 2rnKX T sin 7 mn a a n 1n 1 Ỉ 2nKy I 1 - COS 1 X b in - 1 n 1 2mroi 1 - COE- -- niu . 2nKy b- Bln b w rJCJC 00 2 mrot in 2 III 1 n 1