Chapter 7 less HW solution

Chapter 7 less HW solution - Chapter 7 Behavior of Thin...

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Unformatted text preview: Chapter 7 Behavior of Thin Shells 1. Deformation of a Shell Element Let ABCD shown in Fig. 7-1 represents an infinitesimal element of thin shell cut out by two pairs of adjacent planes normal to the middle surface. The thickness of the shell is h and is assumed constant. In bending of the shell, linear elements such as AD and BC , that are normal to the middle surface of the shell, are assumed to remain straight and be normal to the middle surface after deformation. In the case of simple bending, the lateral faces of the element ABCD rotate only with respect to their lines of intersection with the middle surface. If ' c r shown in Fig. 7- 2(a) and ' y r denote the radii of curvature after deformation, the unit elongations of a thin lamina (shown in Fig. 7-1) at a distance z from the middle surface are derived as follows: consider first the bending part. From Fig. 7-2(a), it is obvious that Fig. 7-1 Infinitesimal shell element x r y r dy D dx z h C y z A B x 1 Fig. 7-2 Bending and membrane stretching ' ' x x ds r r θ θ = = (a) Hence, it follows immediately from equation (a) and ' ' x x ds ds r r θ θ = = (b) By definition, the bending strain at a lamina a distance z from the middle surface is ' ' ' ' ' ' x dd ed ed ed ed ed ed ed ε-- = - = - = (c) a g b a d x r h ' x r ' θ θ i f e ' d ' b b c d x r ' x r ' θ θ f e ' d ' b c ds ( 29 b ( 29 a 2 ( 29 ( 29 ' x x x ds ed r z r z r θ =- =- ( 29 ( 29 ' ' ' ' x x x ds ed r z r z r θ =- =- Substituting these into equation (c), gives ( 29 ( 29 ( 29 ' 1 1 ' ' 1 1 ' 1 1 x x x x x x x x x x x x x ds ds z z r z r z r r r r z ds z z r r r z r r r ε----- + = = = -- --- (d) Likewise, 1 1 ' 1 y y y y z z r r r ε = -- - (e) In addition to rotation, the lateral sides of the element are displaced parallel to themselves due to stretching of the middle surface. If the unit membrane stretching of the middle surface in the x and y directions are denoted by 1 2 and ε ε , respectively, the membrane stretching of the lamina as shown in Fig. 7-2(b) is x ed ei ei ε- = (f) 1 or and x x ds fh ds r hc ds r θ θ ε = = = = (g) ( 29 1 x x x x ds z ei r z r z ds z ds r r θ θ θ =- =- =- =- (h) Likewise, ( 29 1 1 1 ' x z ed ds r ε = +- (i) 3 Substituting equations ((h) and (i) into equation (f), yields ( 29 ( 29 1 1 1 1 1 1 1 1 1 1 1 ' ' ' ' 1 1 1 1 x x x x x x x x x x x x z z z z z z z ds ds r r r r r r r z z z z ds r r r r ε ε ε ε ε ε +---- +-- + +- = = =- ---- 1 1 1 1 1 1 1 1 1 1 1 1 with 1 ' 1 ' 1 1 1 1 1 x x x x x x x x z z z z z z r r r r r r r r ε ε ε ε ε ε + =-- =-- + -- ---- ; (j) Likewise, 2 2 2 2 1 1 1 1 with 1 1 ' 1 1 1 y y y y y z z z r r r r ε ε ε ε ε =-- + -- -- ; (k) In the theory of thin-walled shell, the thickness,...
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This note was uploaded on 09/24/2011 for the course CIVL 7690 taught by Professor Staff during the Summer '10 term at Auburn University.

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Chapter 7 less HW solution - Chapter 7 Behavior of Thin...

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