Buckling and Vibration Analysis for Stiffened Orthotropic Shells of Revolution

Abstract

The best known transformation is the "Prandtl stretching" defined, for small e, byx-+x: y-+sy: u-+u: v-+sv: s 2 = l/R ^ 0 (5) The transformation preserves the form of the equations and the coefficient of u yy in Eq. (1) becomes unity. This "stretching" has been thoroughly discussed elsewhere/ BirkhofT 2 suggests the following one parameter group:Under this transformation the form of Eqs. (1) and (2) is preserved with all terms having a multiplying factor of I//? 2 . Thus for ^ ^ 0 or oo the group defined by Eq. (6) leaves the boundarylayer equations invariant. Finally we discuss the general four-parameter group 2 described by x-»oo;: y-+/3y-u-^yu\ v-+dv (7) Birkhoff discusses the application to flow past an infinite wedge for which U = cx m . This analysis reduces the boundary-layer equations to the usual Falkner-Skan ordinary differential equation. We are interested in generalized flow around a cylinder of the form U = m sin xWe wish to know for what subgroups of (7) the boundary-layer equations are invariant with respect to variations in m. We also wish to preserve the form of 17. Transforming the equations by 2-0 1-8 1-6 1-4 /m" 1-0 OS 0-6 0-4 0-2 D TERRILL ~ SOLUTION (REF. 3) O SCHONAUER -SOLUTION (REF. 4)means of (7) and equating the coefficients of different terms in each equation it is found that a=l: )5=l/m 1/2 : y = m: d = m 1/2 (9) where m^Q, oo. Thus the required transformation is a oneparameter subgroup of (7) uniquely defined byx->x: y-+y/m l/2 : u^nnu: v-+m 1/2 v (10) It is worth noting that u y -»m(m) ll2 u y . These results have the following application. It is often useful to be able to compare the boundary-layer solutions of different workers who have used different values of m for the external flow. The transformation (10) gives the scaling that must be used on the coordinates and velocities to compare solutions that have different m values. Typical values are m = 1 for Terrill's solution (Ref. 3) and m = 2 for Schonauer's solution (Ref. 4). The velocity profiles, at x = 1.0, obtained from Refs. 3 and 4 are compared by plotting u/U vs y/m 1/2 on Fig. 1. It is seen that the two solutions fall on the same curve, confirming the prediction of Eq. (10).

Piecewise continuous eigenfunction solutions for laminated slabs

“…At present, there are essentially three large scale procedures which are capable of effectively evaluating the transcendental eigenvalue problems which arise from these equations. Namely: (i) the regular false determinant search a la Goldberg et al [16]; (ii) the modified Stodola method of Cohen [17] or lastly; (iii) the linear eigenvalue iteration of Svalbonas and Balderes [18,19]. Since the procedure described by (iii) yields an effective means of obtaining excellent starting points for evaluating the higher order values of )t,,n, it was adapted for the present purposes.…”

“…With this in mind, since the ~.~.~ basis is complete, a generalized version of the Gram Schmidt process [11] can be used to construct the required orthogonal basis. Since a total of six Xl edge conditions must be satisfied, in order to establish the appropriate starting basis, (18) …”