The problem of bending of an isotropic compressible rectangular block into a parabolic shell has been solved. It is shown the deformation can be maintained by applying forces and couples to the edges of the shell only

Deshmukh, A.

English

Defence Science Journal

BENDING OF AN ISOTROPIC COMPRESSIBLE RECTANGULAR BLOCK INTO A PARABOLIC SHELL A. DESHMUKH ' Mumtaz Science College, Hyderabad (Received 8 November 1969 ; revised 26 August 1970) The problem of bending of an i8otropic compressible rectangular block into a parabolic shell has been solved. It is shown that the deformation can be maintained by applj ing force8 and couplel~ to the edge8 of the shell only. Bending of rectangular plates into cylindrical shells has been considered by various authorsla -on the basis of linear stress strain relations by referring the components of finite strain to the strained state of the body. The exact solutions obtained for compressible bodies have not been very large. In the present paper a solution is obtained for the problem of bending of an isotropic compressible rectangular bolck into a parabolic shell. The strain tensor has been calculated directly from the metric tensors of the strained and unstrained states of the body without any reference to the displacements. We use Seth's stress-strain relations2 uij = X I S i j +  2 p e i j  (1) The notation used by Green & Zerna4 have been followed. D E F O R N A T I O N  OF THE B L O C K  i Let a rectangular block in the undeformed state be bounded by the planes XI = a,, xl = a, (a, > al), x2 = & b and x3 = f d. Then it is bent into a part of parabolic shell whose internal and externaI boundaries are the confocal parabolas. with the edges r ]  = & a. Let yi-axes coincide with the %%-axes and the curvilin~r coordinates Bi  ia the deformed state be a system of orthogonal curvilinear coordinates ( 5, 1 , z ) ;  so that Since $he deformation is symmetric about the xl-axis, and supposing that there is no extension in the x3-direction, we get , The metric tensors for the strained and matrained states are given 0 1 df whem,ft = - a' d t  and P' = - ; Then dq The equations of ,equilibrium ta be sa.$kfied~ttre \ . a ~ 2 3  B -  9 -+  (3  Ta2 - Tll)-= 0 al;l t2 + q2 . (10) . .  x & i d  rlbein&* identically -.satisfied. On substituting (8) in (9) and (10) and solving, both these equations give the same solution where d and Aiis an arbitrary constant. h i - 2 ~  Since f and P are functionsiflf C and q respectively& eqaatioa .(ll) will be consistent r d y  if q is l m d l l  w h  t b t  72 crtn be- neglected grid F = B'l (12) Then, we have t.2 - f - - B 2 + D ( f ) 2 - - d  (13) Substituting (l2)'and (13) in (8), we obta;in B O U N D A R Y  C O N D I T I O N S  If the block is bent by applying forces to the edges only, we must have u,, = 0 when 5 = ti i = 1, 2 . ( '6 )  which gives I Solving these equations we obtain On the straight edges 7 = f a , the distribution of tractions gives rise to a Porce E; and a couple MI given by The force E", required to keep the length of the block constant in the direction of the axis of the shell and the couple M2 in the axial plane applied to the plane ends of the shell per unit area between q and 7 + 6~ are given by 110 801. J., VOL. 21, APRIL 1971 " R E F E R E N C E S  1. RIVXLIH, R. S., Prm. Roy. Soc., 195A (1949). 463. . ~ 2. SETH, B. R., Phil. Tran. Roy. Soc. fLondon),234A (1935), 231. 3. RAO, B. K., SARMA, K.8. & LAKSAMINARAYANA, G.,Proc. N6t .  Acmd. Sc;. (Indie),@ (1Q63),455. 4. Cfaneni, A. E. & ZEBNA, W., "Theoretioal Elestioity" (Oxford University Press, Oxford), 1964. 

Bending of an Isotropic Compressible Rectangular Block into a Parabolic Shell

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Bending of an Isotropic Compressible Rectangular Block into a Parabolic Shell

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