Tham khảo tài liệu 'aircraft structures 1 2011 part 5', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 146 Bending of thin plates Fig. Calculation of shear strain corresponding to bending defledion. dw dy ốy and the angle DC2C is dw dx. Thus CjD is equal to dw dw _ axayiy and the angle DAC representing the shear strain corresponding to the bending displacement vv is dw dw dx dy so that the work done on the element by the shear force NxvSx is -V 1 . c dw dw l axay Similarly the work done by the shear force NXyôy is 1 T . dw dw 2NxySyfafy and the total work done taken over the complete plate is 1 ffl f _A7 dwdw 7- I 2NXV dxdy 2 Jo Jo y ỠX dy It follows immediately that the potential energy of the NXy loads is 1 ta f _ Qw dw . . _ . 544 and for the complete in-plane loading system we have from Eqs and a potential energy of V -I ư HS 2 sy 2N X SI 5-45 2 Jo Jo dx J dy J y dx dy J Energy method for the bending of thin plates 147 We are now in a position to solve a wide range of thin plate problems provided that the deflections are small obtaining exact solutions if the deflected form is known or approximate solutions if the deflected shape has to be guessed . Considering the rectangular plate of Section simply supported along all four edges and subjected to a uniformly distributed transverse load of intensity qQ we know that its deflected shape is given by Eq. namely v2 . v2 . . miĩx . niĩy w z a b m ln l The total potential energy of the plate is from Eqs and Substituting in Eq. for w and realizing that cross-product terms integrate to zero we have 2 2_4 _ _ _ .mmri. itnnx . 7niry 7m- ĩx 7niĩy 2 1 zz 5-75 sin sin -A- cos cos -A-crlr a b a b v2 v2 J miĩX mry qũ y 2- sin a s n b fdxd7 The term multiplied by 2 1 zz integrates to zero and the mean value of sin2 or cos2 over a complete number of half waves is thus integration of the above expression yields m2 n2 2 4ab a2 b2 q nm itmn m 5 71 1 3 5 From the principle of the stationary value of the total potential energy we have d u v D iSab m2 ỵ n2 2 n .
