1、NASA SP-288 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NASA SP-288 VIBRATION 11 - OF SHELLS Arthur W. Leissa Ohio State University Columbus, Ohio 0 Scientific and Technical Information OfFce NATIONAL AERONAUTICS AND SPACE ADMINISTRATION Provided
2、 by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402 Price $5.20 domestic postpaid or $4.75 GPO Bookstore Stock Number 3300-0422 Libraiy of Congress Catalogue Car
3、d Number 77-186367 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-Preface This monograph is the second in a series dedicated to the organization and summarization of knowledge existing in the field of continuum vibrations. The first monograph, entit
4、led Vibration of Plates, was published in 1969, also by the National Aeronautics and Space Administration. The objectives of the present work are the same as those of the previous one, namely, to provide (1) A comprehensive presentation of available results for free vibration frequencies and mode sh
5、apes which can be used by the design or development engineer. (2) A summary of known results for the researcher, facilitating comparison of future theoretical and experimental results, and delineating by implication those problems which need further study. The scope of the present monograph is also
6、the same as that of the previous one in that (1) Materials are assumed to be linearly elastic. (2) Structures were not included in this study, although some attention has been given to the accuracy of representing a stiffened shell as an orthotropic shell for purpose:s of vibration analysis. The key
7、 to a comprehensive monograph such as this is organization. Careful organization not only makes the completed work more understandable and useful to the reader, but also facilitates the writing. Although much of the organization can be seen from the Contents, I will attempt to explain it further bel
8、ow. Shells have all the characteristics of plates along with an additional one- curvature. Thus we have cylindrical (noncircular, as well as circular), conical, spherical, ellipsoidal, paraboloidal, toroidal, and hyperbolic paraboloidal shells as practical examples of various curvatures. The plate,
9、on the other hand, is the special limiting case of a shell having no curvature. So called “curved plates“ found in the literature are, in reality, shells. Thus, the primary classifier of the field of shell vibrations is chosen to be curvature. For a given curvature (say circular cylindrical, for exa
10、mple) the available literature is divided as to whether complicating effects such as anisotropy, initial stresses, variable thickness, large deflections, nonhomogeneity, shear deformation and rotary inertia, and the effects of surrounding media are present or not. The next subdivision of organizatio
11、n is boundary shape. Thus, a circular cylindrical shell can be open or closed, have boundaries which are parallel to the principal coordinates or not, and have cut- outs or not. Once the boundary shape is determined, attention is given to the possible types of fixity that can exist along each edge (
12、i.e., the boundary con- ditions). Finally, attention is given to such special considerations as point sup- ports or added point masses. Thus, for each type of curvature, the organization 3f the previous monograph Vibration of Plates is followed. Provided by IHSNot for ResaleNo reproduction or networ
13、king permitted without license from IHS-,-,-PREFACE In addition to having the added complexity of curvature, shells are more complicated than plates because their bending cannot, in general, be separated from their stretching. Thus, a “classical“ bending theory of shells is governed by an eighth ord
14、er system of governing partial differential equations of motion, while a corresponding plate bending theory is only of the fourth order. This added complexity enters into the problem not only by means of more complex equations of motion, but through the boundary conditions as well. The classical ben
15、ding theory of plates requires only two conditions to be specified along an edge, while a corresponding shell theory requires four specified conditions. To demonstrate the significance of the latter point, consider a flat panel (i.e., a plate) which is simply supported along two of its opposite edge
16、s. The num- ber of possible problems which can then arise, considering all combinations of “simple“ boundary conditions which can exist on the remaining two edges, is 10. For a cylindrically curved panel (i.e., a shell) the corresponding number is 136! To complicate matters further, whereas all acad
17、emicians will agree on the form of the classical, fourth order equations of motion for a plate, such agree- ment does not exist in shell theory. Numerous different shell theories have been derived and are used. Thus, if analytical results for frequencies and mode shapes of a given shell configuratio
18、n are presented, strictly speaking, the shell theory used in the calculations must be specified. For the sake of separating and defining clearly the various shell theories commonly found in the shell furthermore, certain manu- facturing processes naturally yield shells of es- sentially constant thic
19、kness. Shells may be regarded as generalizations of a flat plate; conversely, a flat plate is a special case of a shell having no curvature. The terminology “curved plate“ is used occasionally in the litera- ture-usually referring to a shell having small changes in slope of the undeformed middle sur
20、- face. In this work the “shallow shell“ will be used to describe this type of shell. This chapter presents the fundamental equa- tions of thin shell theory in their most simple, consistent form. Thus the material is assumed to be linearly elastic, isotropic, and homogene- ous; displacements are ass
21、umed to be small, thereby yielding linear equations; shear defor- mation and rotary inertia effects are neglected; and the thickness is taken to be constant. Inas- much as this work is aimed at the vibration of shells, it should also be said that the vibration results predicted analytically are assu
22、med to be for a shell in a vacuum (although experimental results will generally be given in air) and that vibrations will occur with respect to zero values of static initial stress in the shell. These compli- cating features will be discussed (in those cases for which information is available) in su
23、bsequent chapters dealing with special configurations of shells. A large number of differing sets of equations have been arrived at by various academicians, all purporting to describe the motion of a given shell. This state of affairs is in contrast with the thin plate theory, wherein a single fourt
24、h order differential equation of motion is universally agreed upon. Furthermore, there is considerable argument in the literature as to whether the differences between the various thin shell theories are sig- nificant or not (cf., refs. 1.1 through 1.8). In chapter 2 some attempt will be made to com
25、- pare the results for free vibration frequencies and mode shapes arising from various thin shell theories in the case of circular cylindrical shells, especially for one particular set of boundary conditions. The main purpose of this chapter is to present straightforward derivations of the sets of e
26、qua- tions of various thin shell theories. It will be seen that differences in the theories result from slight differences in simplifying assumptions and/or the exact point in a derivation where a given assumption is used. Only those theories which are obtainable from Loves postulates (see sec. 1.3)
27、 by using a differential element bf the middle surface, have been derived for shells of arbitrary curvature, and which have been ap- plied in the literature to shell vibration problems will be considered in this chapter. Among the thin shell theories which will be derived in this chapter are those a
28、ttributed to Donne11 (refs. 1.9 and 1.10), Mushtari (refs. 1.11 and 1.12), Love (refs. 1.13 and 1.14), Timoshenko (ref. 1.15) Reissner (ref. 1.16) Naghdi and Berry (ref. 1.17), Vlasov (refs. 1.18 and 1.19) Sanders (ref. 1.20) Byrne (ref. 1.21) Fliigge (refs. 1.22 and 1.23), Goldenveizer (ref. 1.24)
29、Lurye (ref. 1.25) Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-VIBRATION OF SHELLS I and Novozhilov (ref. 1.26). However, not all of the theories listed above are independent. Many of the theories use certain sets of equations in com- mon, and som
30、e are generalizations or duplica- tions of another. Numerous other theories are available in the literature. Some are derived by expansion of the displacements and stresses in power series in the thickness coordinate z. Others are derived by asymptotic integration. The fol- lowing authors have origi
31、nated some of the gen- eral theories for arbitrary curvature not included in this chapter: Aron (ref. 1.27) Basset (ref. 1.28), Epstein (ref. 1.29) Trefftz (ref. 1.30) Synge and Chien (refs. 1.31 and 1.32), Lamb (ref. 1.33), Osgood and Joseph (ref. 1.34) Hay- wood and Wilson (ref. 1.35), Iioiter (re
32、f. 1.36), Cohen (refs. 1.37 and 1.38), and ICnowles and Reissner (refs. 1.39 and 1.40). Writings which are particularly good from the standpoint of com- parison of various thin shell theories include ref- erences 1.1, 1.4, 1.7, 1.17, and 1.41 through 1.47. 1.1 BRIEF OUTLINE OF THE THEORY OF SURFACES
33、 The deformation of a thin shell will be com- pletely determined by the displacements of its middle surface. Certain relationships relating to the behavior of a surface will be summarized in this section. More useful information can be found in the numerous texts dealing with differ- ential geometry
34、, the theory of surfaces, and shell theory (cf., refs. 1.19, 1.24-1.26, and 1.42). 1.1.1 Coordinate System Let the equation of the undefrmed middle surface be given in terms of two independent parameters a and p by the radius vector + -, = (ff,P) (1.1) Equation (1.1) determines the geometric prop- e
35、rties of the surface and yields a method for finding points on the surface. Suppose that the parameter a is kept at a fixed value ao, while p changes. In this case equation (1.1) deter- mines a space curve on the surface. Such curves are called p curves, and the set of all values a. within a given i
36、nterval corresponds to a family of /3 curves. In an analogous manner one can introduce the concept of a curves (fig. 1.1). Assume that the parameters a and p always vary within a definite region, and that a one- to-one correspondence exists between the points of this region and points on the portion
37、 of the surface of interest. Denote The vectors Za and $ are tangent to the a and p curves, respectively. The length of these vec- tors will be denoted by I Consequently it follows that ,/A and in moving from the first point to the second point is , -+ -+ dr =r, da+r, dp (1.6) From equations (1.3) (
38、1.4), (1.5) and (1.6) the square of the differential of the arc length on the surface is + -+ dr-dr = ds2 = A2 da2 +2AB cos xda!d/?+B2dp2 (1.7) The right-hand side of equation (1.7) is the “first quadratic form of the surface.“ This form determines the infinitesimal lengths, the angle between the cu
39、rves, and the area on the surface, i.e., the intrinsic geometry of the surface. Hom- ever, the first quadratic form does not determine a surface by itself. The terms A2, AB cos X, and B2 are called the “first fundamental quantities.“ 1.1.3 Second Quadratic Form The concept of the second quadratic fo
40、rm arises when one considers the problem of find- ing the curvature of a curve which lies on the -+ -+ surface. Let r = r(s) be the vectorial equation of a curve on the surface (s is the arc length from a certain origin). Denoting the unit vector along the tangent to the curve by .i, then According
41、to Frenets formula (ref. 1.48) the derivative of this vector is where lip is the curvature of the curve, and is the unit vector of the principal normal to the curve. Substituting for + from equation (1.8) into equation (1.9) one obtains where Let cp be the angle between the normal to the surface t,
42、and the principal normal to the curve under consideration I?; then cos cp =c. (1.11) If both sides of equation (1.10) are scalar-multi- plied by t, one obtains cos - cp L da2+2M da dp+N dp2 - (1.121 P ds2 where -+ L = r,.t, 1 The expression (L da2+2M da dp+N dB2) is called the “second quadratic form
43、“ of the surface and the quantities L, M, and N are the coeffi- cients of the form. The second quadratic form is thus related to the curvatures of the curves on the surface. From equation (1.12) one can obtain the nor- mal curvatures of the surface; i.e., the curva- tures of the curves obtained by i
44、ntersecting the surface with normal planes. For the curve gen- erated by a normal plane, t, and fl are either par- allel (cp=O) or have opposite directions (cp=r). Since a “plane“ curve always leaves its tangent in the direction of vector 8 and if one takes its outer normal as the positive normal to
45、 the sur- face, cp = r results. Thus from equations (1.7) and (1.12) the normal curvature is To obtain the curvatures of the a curves and the p curves take p= constant and a = constant respectively, thus Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-
46、,-VIBRATION OF SHELLS 1.1.4 Gauss Derivative Formulas At this point assume that the curves a = con- stant and = constant are lines of principal cur- vature of the undeformed middle surface. The coordinates a, B are then called principal coordi- nates. Weatherburn (ref. 1.49) shows that the necessary
47、 and sufficient conditions for the para- metric curves to be lines of principal curvature on a surface are that cos x=O (1.16a) The condition given by equation (1.16a) is that of orthogonality satisfied by all lines of principal curvature, while M = 0 is the necessary and suffi- cient condition that
48、 the parametric curves form a conjugate system (i.e., through each point on the surface passes a unique curve of each family of curves). The second derivatives of; with respect to the + -+ parameters may be expressed in terms of r,r, and $. Remembering that L, Atn= 1 one obtains the following expres
49、sions for the derivatives of the basic vec- tors (ref. 1.421 I 1aB B J %,a= - -ta-$n A aa R 1 .I .6 Gauss Characteristic Equation 7 n #,I The four fundamental quantities for principal coordinates A, B, L, and N are not functionally independent, but are connected by three differ- ential relations. One of these, due to Gauss, is an expression for (LN) in terms of A and B and their der
