Linear Elastic Theory of Thin Shells by J. E. Gibson and B. G. Neal (Auth.)

By J. E. Gibson and B. G. Neal (Auth.)

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17. 50) in which At to A4 are arbitrary constants which are to be determined so that the boundary conditions of the problem are satisfied. By suitable choice of the origin of the coordinate axes, these constants may be reduced to two in number as follows. Choosing the origin O at the centre of the vessel as shown in Fig. e. have identical values at ±x. 50) for w must be symmetrical about the origin, but the only two terms 38 Linear Elastic Theory of Thin Shells in this equation that are symmetrical about the origin are sin/fosinh/fa; and cos/fa; cosh/fa;.

18). 8. Curvature Displacement Equations We must now consider the displacements of the middle surface in relation to the changes of curvature χ^ and χ2 in the x and y directions. 6) is xi = ( i / r ; - i / i g Now rx is the radius of curvature of the straight line generator which is infinite and therefore \jrx = l/oo = 0. After bending, the middle surface (along the direction of x) which was originally along OA is displaced to some position O'Ä due to the radial deflexion w in the z direction, as shown in Fig.

16. 38) 4 on substituting for 4/? 29). 39) in which C1 to C4 are arbitrary constants which must be chosen to satisfy the boundary conditions. These arbitrary constants, however, may be reduced to two in number if it is assumed that the tank is relatively tall and that the radial deflexion w is small in magnitude. As the term exp/fa; becomes large for large values of x, excessive radial deflexions would occur unless the term in Bending Theory of Closed Circular Cylindrical Shells square brackets associated with it, namely^[C1cosj5x+C2sinjßA:], vanishes.

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