Dunfermline, Five, 1956. 41. Bodner, S.R., "Analysis of the general instability of ring reinforced circular cylindrical shells of orthotropic shell theory", J. Appl.
LAMINATED CONICAL SHELLS CONSIDERATIONS FOR THE VARIATIONS OF THE STIFFNESS COEFFICIENTS Menahem Baruch*, Johann Arbocz**,Guo Qi Zhang*** Abstract The stiffness coefficients of a laminated conical shell, orthotropic, or in some more general mode, are usually assumed to be constant. It is shown that due to the geometry of the conical surface, in connection with the filament winding process necessary to build the laminated conical shell, this can never be achieved. To calculate the varying stiffness coefficients of a laminated conical shell the geometry of the shell and the winding process needed to build the shell must be taken into account. A method to calculate those stiffness coefficients is proposed. The geodesic paths, which are the practical ways to position the fibers, are especially examined. Of course stiffeners can also be placed on these paths. However, other paths are also examined. Stiffeners, for example, can be placed on these paths for optimization purposes. The stiffness coefficients of a laminated conical shell vary always and are functions of the coordinates of the shell. It is shown that by proper filament winding process these functions can be made to be functions of the longitudinal coordinate only. Introduction The laminated shell structures have found and continue to find more and more applications in different kinds of industries. Because of the hlgh stiffness to weight ratio and its capability of being tailored to meet design demands they are especially fitted for aerospace applications. The usefulness of the laminated shell structures depends on many factors where two of them, the manufacturing process and the analysis can be pinpointed as its cornerstones. To make the analyses of the structure possible the researchers have to assume and calculate the mechanical behavior of the microstructure.
* ** ***
Professor, Faculty of Aerospace Engineering, Technion - I.I.T., Haifa 32000, Israel. Professor, Faculty of Aerospace Engineering, Delft University of Technology, The Netherlands. Dr., Fokker Aircraft B.V., The Netherlands.
Copyright @ 1993 American Institute of Aeronautics and Astronautics, Inc. All rights reserved.
However, in some cases the manufacturing process cannot reproduce this microstructure. In other words many analyses have been performed on structures which cannot be practically realized. This paper will make an effort to connect between the analysis of the laminated shell structures and their theoretically and practically possible real microstructures for which a manufacturing process can be designed. Although the above expressed thoughts are general, this paper will focus on the filament winding manufacturing processes and the analysis of laminated conical shells. One can find in the literature a vast number of scientific papers connected with the vibration, buckling and post-buckling analysis of laminated circular cylindrical shells with different levels of sophistication. For example, Refs. 1 and 2 are based on the classical laminated shell theory which incorporates the Kirchhoff-Love hypotheses of straight inextensional normals for the entire shell package. References 3 and 4 are based on the first order Reissner-Bolle-Mindlin shear deformation theories and Refs. 5 and 6 are based on high order shear theories. For more information about analyses of laminated circular cylindrical shells see Refs. 7 and 8. The common feature of the analyses of the laminated cylindrical shells is that the mechanical coefficients of the shell were taken for granted to be constants. In other words the so-called A-B-D matrix is assumed to be not a function of the cylindrical coordinates. Can it be acheved, at least theoretically, by the filament winding manufacturing process? Fortunately, the answer is positive (see Refs. 9-22). Why the stiffness coefficients A-B-D of a circular cylindrical shell can be treated as the same coefficients of a rectangular plate? The reason is connected with the basic theoretical fact that a cylindrical surface can be cut out in the direction of the longitudinal coordinate and unfolded to become a rectangular flat surface (see Fig. 1). Hence, the classical approach for plates in which orthotropic laminae with different principal material axes are
2505
used to build a laminate (see Refs. 23-26) can be used to build also a cylindrical shell.
Fig. 1. A circular surface can be unfolded to become a rectangular flat surface.
By keeping the distance between the fibers constant the macrostructure of the lamina can be considered to be homogeneous and orthotropic with constant thickness. The principal material coordinates of the lamina coincide with the principal coordinates of the cylindrical shell (see Refs. 23-26). The same can be said for a helically wound fiber- reinforced lamina of a circular cylindrical shell, only now the principal material coordinates of the lamina do not coincide with the principal coordinates of the cylindrical shell (see Fig. 4 and Refs. 11,12, 15-17, 21-23,26).
A conical surface can theoretically also be cut up and unfolded to become a flat surface. However, this surface is not a rectangle (see Fig. 2). As we will see later this fact has important consequences connected with the stiffness coeff~cientsof a laminated conical shell.
Fig. 4. Helically wound fiber-reinforced circular cylindrical shell.
Now we will examine the simple case of a lamina with fibers oriented along the longitudinal coordinate of a cylindrical shell (see Fig. 3 and Refs. 20, 22 and 26).
For a conical shell the situation is much more complicated. Even for a lamina with fibers oriented along the longitudinal coordinate of the conical shell the material characteristics of the macrostructure of the lamina and hence the stiffness coefficients of the laminate A-B-D, cannot be kept constant. The reason for this is that the distance between the fibers changes along the longitudinal coordinate (see Fig. 5). Something similar happens with helically wound fiber-reinforced conical shells (see Fig. 6 and Refs. 10 and 17).
Fig. 3. Polar (whirling arm) filament winder.
Fig. 5. Fibers along the longitudinal coordinate of a conical shell.
Fig. 2. Unfolded conical surface.
In contrast to stflened cylindrical shells where the local variation of the mechanical properties of the shells depends on the designer, in conical shells they also depend strongly on the geometry of the shell. Here we will represent an example taken from Ref. 48. See Figs. 7 and 8 for a general description of a conical shell and the force and moment resultants acting on it.
Fig. 6. Variable pitch winding path. However, this fact has not prevented many researchers analyzing composite conical shells and even optimizing the structure by assuming the stflness coefficients A-B-D and the angle of the fibers in respect to the longitudinal coordinate, to be independent of the conical shell coordinates (see for example, Refs. 27-29). It must be emphasized that the assumed change of the thickness of the conical shell in Ref. 29 is arbitrary and has nothing to do with a possible way of building the shell.
Fig. 7. Notation and sign convention.
In this paper we will make an attempt to narrow the gap between the analysis and the possible theoretical and practical way of constructing a laminated conical shell. Some History
The material of thin metallic conical shells is homogeneous and isotropic, and for shells with constant thickness the local mechanical properties are independent of the coordinates of the shell. See, for example, Refs. 30-37. Stiffened cylindrical shells have also been analyzed by many researchers. See, for example, Refs. 38-44. However, the local stiffness coefficients, even for a cylindrical shell with stiffeners whch are theoretically "distributed" over the shell, can be a function of the coordinates of the cylinder. See, for example, Ref. 45 where Merent forms of longitudinal stiffeners and rings are analyzed for minimization purposes. One can see there that the local mechanical properties of the cylindrical shell are strong functions of the longitudinal coordinate of the cylinder. Clearly, this is true for any kind of stiffened shells. For a general description of the variation of the local mechanical properties of stiffened shells, see Refs. 46,47.
Fig. 8. Force and moment resultants for conical shell. In Fig. 9, taken from Ref. 48, the notation has
been changed to the notation used in this paper. One can see that the distance between the ring stiffeners depends on the designer. The coefficient F was assigned for optimization purposes. For F=O equal ring spacing is obtained while for conical shells under hydrostataic pressure 6=1.5 divides the shell into conical subshells with equal buckling load. However, the distance between the stringers, as the distaxe between the longitudinal fibers in a composite conical shell, is prescribed by the geometry of the shell itself. The
detail
where I,7 and T( are the unit vectors of the Cartesian coordinates X,Y,Z.
LA.
of rings
The position of a fiber will be defined if 0 is a known function of s, detail of stringers
-
The unit vector along the fiber is given by,
Fig. 9. Description of stiffeners. where, distance between the stringers is a linear function of the longitudinal coordinate s and depends on the basic parameters of the structure (see Fig. 7). The same can be said for the longitudinal fibers of a composite conical shell (see Fig. 5 ) , and more generally, the distance between the fibers of a helically wound fiber-reinforced circular conical shell is a strong function of the longitudinal coordinate s (see Fig. 6). This function will be described later. The stiffness coefficients A-B-D for a stiffened conical shell, as described in Ref. 49, are strong functions of the longitudinal coordinate s.
-
R,, = cosa 7 + (sina cos0 - ssina sin0 .O,, )5 +
+(sins sin8 +ssina cosO.O,,)K
(4)
and
The unit vector along coordinate s is simply given by,
the
longitudinal
Position of a Fiber on a Truncated Conical Shell The position of the point P that belongs to a fiber which lies on a truncated conical shell is defined by the radius-vector
because, as expected,
Ef (see Fig. 10). The angle between the fiber and the longitudinal coordinate is given by (see Fig. lo),
Fig. 10. Position of a fiber on a conical surface.
Up to now the description of the position is a general one. However, it is well known that if the
fibers are not positioned along geodesic paths they will slip in search for exactly those paths (see Refs. 12-15, 17, 18, 21 and also Refs. 50-51). In spite of this, for theoretical and possibly optimization reasons, we will examine several nongeodesic paths, beginning with the simplest one and ending up with the geodesic path. The nongeodesic paths have also some practical importance. One must not forget that on a conical surface even the ring is not a geodesical line. However, there are new ideas connected with the filament winding process (see Refs. 10, 21, 50). There is some hope that with special techniques and especially with the introduction of thermoplastic matrices the geodesic constraints will be somewhat loosened. For stiffened conical shells the nongeodesic lines have much more practical importance. Helical stiffeners can be used for minimization purpose. For the stiffened cylindrical shells stiffeners in different directions, ring-stiffeners (Ref. 44), 45' ribstiffening and isogrid (60") rib grid (Refs. 52, 53) have been applied. In the stflened conical shells similar angles can be applied. One can imagine also some kind of conical isogrids where the cross section of the stiffeners can be changed to compensate for the growing distance between the stfleners in the longitudinal direction and so on. It seems that these possibilities have not been sufficiently exploited. 1. The circumferential coordinate 0 is a linear function of the longitudinal coordinate s
Note that the angle P grows with growing s and this, as will be seen later, may have important consequences. From Eq. (5) the elementary length of the fiber in t h s case is given by,
from where one obtains the length 1,
. 2 a . s2 )112 I t A = ( s / 2 ) ( l + a 2 sin
(14) . 2 a - s2 )112, +(1/2a sina)ln[asina.s+(l+2 sin
where A is a constant of integration. If the length is measured from the initial coordinate s l the constant A will be equal to the right hand side of Eq. (14) where s is replaced by s l . 2. The lengths of the fiber and circumferential coordinate change in a proportional way
In this case one obtains,
where q is the coefficient of proportionality. From Eqs. (15) and (5) one obtains,
This is the simplest possible case. Now the circumferential coordinate 0 is given by, Using Eq. (8) one obtains, where a is an appropriate constant.
. 2 a )112 cosp=(1-s2q 2 sin
From Eqs. (8) and (9) one obtains the angle between the fiber and the longitudinal coordinate as, . 2 a ) 112 c o s p = l / ( l + s2a2 sin
(1 1)
or
sinp = sq sina
(18)
where again the angle P grows with s. tanP = as sina
(12)
From Eq. (18) one can see that the coefficient of proportionality q is restricted by (see Fig. 7),
The length of the fiber is given simply by,
and hence (see Eq. (S)),
3. Constant angle between the fiber and the longitudinal coordinate
This case has interesting theoretical consequences. For p to be a constant one has to assume (see Eq. (8)),
where B is a constant of integration. B is equal to the right hand side of Eq. (28) where s is replaced by s l . Equations (5,26) yield,
s2 sin2 a.8,: = g2 where g is a constant. Equation (2 1) yields,
One can see that the length is a strong function of s. 5. Geodesic path
From Eq. (8), cosp = 1 I (l+g2)112 = const
(23)
This is the most important practical case because in the present state of the art the filament winding process is usually applied by using geodesic paths.
or, We will calculate the geodesic lines of a conical surface by using a simple isomorphism which is based on the fact that a conical surface can be unfolded to become a flat surface (see Fig. 2).
and
That is, the length is proportional to the longitudinal coordinate. We will see later that this case has other interesting theoretical properties. 4. Constant distance between the fibers
It will be shown later that in this case the distance between the fibers is kept approximately constant. For this case we assume, cosp = P I S where,
Fig. 11. Conical surface - flat surface isomorphism.
Figure 11 shows a conical surface and its flatted image. A point P on the conical surface is defined by the conical coordinates s and 8, and while the image of P in the unfolded surface is defined either by the coordinates s and qt or by the Cartesian coordinates x and y. The isomorphism between the two surfaces is simply,
A geodesical line on the conical surface will appear in its flat surface image as a straight line and hence, in the x,y system P(x,y) is defined by,
where k is a constant. But,
L z
Fig. 12. Distance between two fibers.
Substitution of Eqs. (32) into Eq. (31) after some manipulations yields,
where
After some manipulations and using Eq. (9) one obtains, sinp = sl sin
/s
(35)
where J31 is the initial winding angle. The change of in Fig. 14.
P for a particular case is shown
and the measure of the distance between them must be defined. A natural measure of the distance is as follows (see Fig. 12). Draw a curve on the conical surface which is perpendicular to both fibers fl and f2. Indeed, it can be shown that the curve is perpendicular to all the family of the positions of the fibers that follow the same law of change. The curve will meet fiber 1 at point P3 and fiber 2 at point P5. The length of the curve between P3 and P5 is the distance, bs, between the two fibers. The distance between the fibers in the general case, although well defined mathematically, is a complicated function of s. However, the fibers have to be closely winded and hence the distance between them must be small. this fact would permit us to linearize the process in order to obtain reasonable approximation expressed in a practical form. Figure 13 shows a graphic linearization based on the small distance between the fibers. Now, bs o cscosp,
From Eq. (5) the length of the fiber is given by,
Finally,
S PCOSP~),bl = C bs o ~ ~ S C O/ (s1
~ C O S ~ ~
(3 8)
Distance Between Two Fibers
Two fibers on a circular cylindrical shell which follow the same law of change along the longitudinal coordinate are parallel and the distance between them is well defined (see Fig. 4). However, for a conical shell the fibers in general are not parallel
It can be shown, (see Ref. 49), that for the case, constant, expression (38) gives the exact solution and for the case 4 the distance between the fibers is approximately constant.
p
ts
= tlSl cos p1/
(scosp)
(40)
Following Refs. (23-25, 49) and using the fact that kth lamina is homogeneous orthotropic in the direction of the fiber one obtains,
Fig. 13. Graphical linearization.
Stiffness behaviour of a lamina In the winding process any fiber brings with itself the same amount of matrix (see Refs. 9-22, 5051). This means that in the direction of the fiber, Vf the volume fraction of the fiber and Vm - the volume fraction of the matrix are kept constant along the fiber direction. By the rule of mixture (see Refs. 23, 25) it follows that the apparent moduli of the lamina are constants and hence it is homogeneous orthotropic. However, this can not be said for the thickness of the lamina. From what was said before it follows that the amount of material for unit length of the fiber is kept constant, that is, bsts = bltl + const. (39)
-
where ts is the thickness of the lamina and t l is the thickness of the lamina for s = s l Equations (38,39) yield, (see Fig. 14), where, &;
= U,S
O = v,,-v 1 s + u,-Is Ys8 8
Fig. 14. Change of thickness and angle of inclination for a typical lamina of geodetical winding.
and
Discussions
where
and
The technological state of art of the filament winding is such that the fibers must be positioned on geodesic paths of the conical shell. As shown, the stiffness coefficients obtained for conical shells become a strong function of the longitudinal coordinate and this fact must be taken into account in the analysis of laminated conical shells. The other paths discussed in the report, on whlch one can position stiffeners for optimization purpose, do not change the trend, the stiffness coefficient will be again a function of the longitudinal coordinate. It can be shown that for a cylindrical shell all paths become identical. Let us assume that in the future the geodesic constraint would be override and other winding paths like the ones described in the report, would be possible. In these cases too the stiffness coefficients will vary as some function of the longitudinal coordinate. Let us also assume that in the future some way would be found to add a matrix to the fiber, during the winding of the lamina or after finishing it, in such a way that the thickness of the lamina would be constant. Now, the apparent modulus of elasticity and the winding angle will vary and the stiffness coefficients will be again some functions of the coordinate. The conclusion is that the stiffness coefficients at a laminated conical shell can not be constants. They are functions of the coordinates of the shell and by proper winding process can be made functions of the longitudinal coordinate only.
where Qij can be found in Refs. 23-25 or 49. It should be emphasized here that unlike the case of laminated cylindrical shells, where the winding angle Pk is usually a constant within one lamina, the winding angle Pk of a conical shell depends on the winding process and it is a function of s (coordinate in the longitudinal direction). That is to say, the conical geometry dictates the reinforcement trace in which the fiber orientation relative to the cone axis changes as the fiber is wound. Besides (see Eq. (40)) the thickness of the lamina also changes in the longitudinal direction. Therefore, the [A], [B] and [Dl matrices are strong functions of s.
The stiffness coefficients needed for the analysis of a laminated structure, and especially of a laminated conical shell, have to be calculated only after taking into account the process which can be used to build the structure.
Conclusion A method for the calculation of the varying stiffness coefficients of a laminated conical shell was proposed. The method takes into account the filament winding process needed to build the shell. In addition to the practical case of a geodesic path other paths were also examined. Stiffeners, for example, can be placed on these paths for optimization purposes. Although not necessary for
st~ffencrs,they of course can also be pos~tmnedon geodeslc paths
Applications, and DesiiyirM. J Wile! & Sons, Inc. 1964.
It was shown that the st~ffnesscoefficients of a lanunated con~cal shell T ary alnays and arc functions of the coord~natesof the shell By proper filament w n d ~ n gprocess these f~mct~ons can be made to be funct~onsof the longltud~nalcoordinate only
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