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Applied Structural Mechanics Fundamentals of Elasticity Part 5

Tham khảo tài liệu 'applied structural mechanics fundamentals of elasticity part 5', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 106 9 Plates Material law - stress resultant-displacement relations - K w . aT a e 9Ji with K plate stiffness aQ0 aa components of the metric tensor in the mid-plane of the plate 0 temperature gradient E 0 7 1- v- a 7a ố 4- aa 5 a 7 V a.a0 plane elasticity tensor. Differential equation KAAw - p - QT 1 4 i JKA1 9.13 with the LAPLACE operator A given by 2.39 in terms of the applied curvilinear coordinates. Energy expressions - Cartesian coordinates ồ ồx x ò òy y from 6.16a A.9 I i nI 2 1 - 4- KaT 1 4- V w xx 4- wyv 50 Ị dxdy or Ui Jf 14K44 w.jy 2 - 2 1 - w w - Kot 1 I p wi t - Wyyjteldxdy . - Polar coordinates ồ òr r òỊòip ự ni JJ 4K W. 2 w.rrW.W v . w.j . 1 _ t . 2 _ _2 1 T1 2 4 1 1 - Ư w rr - - w . w_ 4- -i w I v rrff J- Tifl ự 9.33 I 0M J A 2 L 2 . 1 .2 W.W 7W.WW.r W.r r dự dr . - Curvilinear coordinates with a I aQ n JJ TKa E wl ejAdi di2 . 9.35 9.2 Analytical solutions for shear-rigid plates 107 9.2 Analytical solutions for shear-rigid plates a Cartesian coordinates - Simply supported plate strip d ò y 0 ò ò X Differential equation from 9.Ỉ3 AAw w T jr y A A.A. A. Solution 4_ TX a V1 Pn n FX n l w K z f 2x4 sin a 9-36 K n Th 7T a for _ _ V n TTX p x z Pn Sin A n a with a pn ẳ fp x sin Ra dx n 1 3 5 . . 0 - Rectangular plate with simply supported boundaries dimensions a b Fig. 9.3 Differential equation from 9.Ĩ3 è ồx fX ò òy w AZiW w 4- 2w w xxxx xxyy yyyy K Solution Double series expansion according to NAVIER X V V. niTTX . n Try _ICQ w x y ZZWmnSÌn a s n lo m n - in n Load p x y V J pmnsin x sin. 9 37 m n Fig. 9.3 Plate under a uniformly distributed load over a rectangular subdomain 108 9 Plates A plate subjected to a uniformly distributed load p0 over a rectangular subdomain as shown in Fig. 9.3 will be considered as an example of applica-tion. We first expand the constant load p0 in the y-direction p y Spnsil1 n n ry b with - n 7T Posin b vid jp sin dy 2 b_ b n 7T _ f n 7T Pol cos b V-ÍỈ 3 pn b . 5 V 4 d - c Po on 7F v n vi d _ V - 2 _ 2 .