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Geometrical analysis of large elastic deflections of axially compressed cylindrical and conical shells
GEOMETRICAL ANALYSIS OF LARGE ELASTIC DEFLECTIONS OF AXIALLY COMPRESSED CYLINDRICAL AND CONICAL SHELLS
Abstract-Thii paper presents an analysis of the post-buckiing be&&our of isotropic q&m&ii and conical shells under axial compression. The starting point thepaper is Lagrange’s variational principle, the application of which consists in assuming a kinematically admissible strain and displacement field. The field is determined considering the geometry of quasi-isometric deformations of the shell after buckling That permits solving the problem with no limitation on the magnitude of the displacements.
of
1. INTRODUCTION
Large deflections of elastic shells have usually been analysed on the basis of DomeIl’s non-linear equations of elastic shells 11% 2] by means of the Ritz or Gale&in methods. The solutions obtained in this way showed considerable discrepancies when compared with the results of experiments. Recently, using Donnell equations, Yamaki and Kodama [3] obtained results which are in good agreement with experiment They assumed a deflection function satisfying completely clamped boundary conditions and applied the Galerkin method. However it can be observed that for large values of the deflection (of the order of ten times the thickness) this agreement is not good. Also thit agreement with experimental data presented in [3] is not so good for longer shells. Xn this paper rather short shells defined by the parameter 2==,/(l -v2)J?/Rh d 500 are considered. In our opinion these and other discrepances (for instance associated with shells of simply supported edges) are attributed, first of all, to the approximate nature of the equations employed. At present there is no theory of shells precise enough and yet simple in ~mpu~tions. Thus these problems need still to be considered. In this paper a non-&s&al method of analysis of the post-buckling behaviour of cylindrical and conical shells under axial compression is presented. The characteristic mode of deflections observed in experiments on shells under axial compression involves a diamond pattern built of nearly flat triangular areas jointed together along regions of large curvature [2,4]. It is known that for typical materials used in structures the admissible elastic strain rate is very small. For example, for steel with Young’s modulus E = 2 x 106 kg cmq2 and yield stress R,=4 x lo3 kg cm*, one obtains e = 2 x 10-j. This means that the elastic deformations are character&d by a variation in the metric tensor of its middle surface smaller than 0.2 per cent If such deformation is associated with considerable change of shape of the surface and still remains elastic, this shape must be very similar to the i~met~~ly transformed surface i.e. the surface deformed in such a manner that its first metric tensor remains unchanged. In this paper, the post-buckling deflection pattern is approximated, not a series of functions, but by certain class of surfaces, quasi-isometric to the primary one. In general, it is not possible to create the isometrically transformed surface which would simultaneously be regular everywhere. Therefore the real deformed surface F of the shell can be approximated by the surface F isometric to the initial one, except for a certain region S small compared to the entire surface F. The area S is selected so that the configuration f: presents a regular surface.
274
STANISLAWLUKASIEWICZ and WALERIANSZYSZKOWSKI
Since the Gaussian curvature of the isomet~~lly deformed surface must be equal to the curvature of the initial surface, the surfaces F and S are defined by a hnite number of geometrical parameters whose values can be obtained with the help of Lagrange’s variational principle. The calculation of strain energy and the work of the external loads in the structure can be done on the basis of geometrical considerations.
2. GEOMETRY
OF
THE
DEFORMED
SURFACE
Models of pure isometric surfaces representing the surface of the deformed shell are given in Fig. 1. For cylindrical shells the following geometrical parameters describe the surfaces (Figs, 2 and 3):
Fig. 2.
Fig. 1.
Fig. 3.
Geometrical analysis of large elastic deflections of axially compressed cylindrical and conical shells 275
n-number of rhom~idal elements along the ~r~umferen~. This corresponds to the number of waves in this direction. I = a/b--ratio of the breadth to the height of the rhomboidal element. 26,-plane angle between plane triangles measured along the oblique edge. 26,-plane angle between plane triangles measured along the horixontal edge. After some simple geometrical considerations we obtain sin a2 = Atgx/2n cos 26 1= [ 1 -(l + 21*)g2x/2n]/[ 1f tg%/2n].
(I)
The model of the conical shell is shown in Fig. 4.
Fig. 4.
In this case the geometrical relations are more complex _ _ [k - cos n/2n] sin a cos~(p--62)-[k-cos
a] sin x/n
[k cos (n/n) - I J sin 01 cos (tp+ii;)= Ck cos a - 11 sin (7t/n) cos 26I = [l -$& tg (n/n) cos (cp-S;)](l
(2) $
tg2 (n/n)[sitG (u,-S;)
+(QJ2 -A, cos (cpA”,) ctg (@t))-1’2 where v, is the angle between the base and generator of the undeformed cone, and
a=;cos v, k=
sin a/Af [l +(sin a/A)2]1’2
A,= 2 sin a/[k -cos a]. The pattern of the pure isometrically deformed shell can be defined by two parameters (A,n) only. In the case of the conical shell the values of the angles a,, S, are constant along the generator. This condition results from a geometrical assumptions explained in detail in [S, 6,7] and requires the height of the rhomboidal elements to be dierent in every layer and equal to bi= &
exp [ - ib cos a/2R]( 1- exp [ - ib CQSa/R])
(3)
276
STANISLAW
LUKASI!&VICZ
and
WALERIANSZY~KOWSKI
where: R-radius of the lower base of the cone, &height of the rhomboidal element of the corresponding cylindrical shell with radius R (4+42), i-number of layers counted from the base of the cone. The formula (3) for the conical shell becomes the formula for the cylindrical shell if cp=n/2. Therefore in further considerations we will deal with the conical shell only. It can be shown that if I=&=ctg
6 -
1-cosa sin a/n
then 6, = S; = 8; = n/2. This corresponds to the case of complete collapse of the structure i.e. when its height becomes equal to zero. This means that Iz< 1,. The areas of strongly bent surfaces S which are observed in real deformation patterns are assumed to be sectors of conical and cylindrical surfaces (Fig. 5). These areas are called ridges. The following notations have been applied: b,-the breadth of the oblique ridge measured for level i=O (at the lower base of the cone), b,-the breadth of the horizontal ridge for i =0, L = R/cos a-the length of the generator.
Fig. 5.
The following dimensionless coefficients are introduced: pi = 2bi cos /l/L sin a,
p2 = 2bJ2L sin a,
where /I is denoted in Fig. 5. These values are very close to each other (7) so without making a large error it can be assumed that 2b, cos cp
~1=p2=p=yp&--p0sB
where ~1is the third parameter characterising the deformed surface. The region surrounding points of intersection of the ridges is called an apex (Fig. 6). In this area fragments of flat triangles and conical ridges should be fitted to make the surface of the apex regular. The Gaussian curvature in this region is different from zero. It is assumed that the form of the apex is rectangular (Fig 6). The dimensions L, and Lz can easily be determined as functions of parameters 1, n and p The deflections in this area are presented in Fig. 7. The apex is jointed to the horizontal ridges along the sides AD and BC and to the oblique ridges along the sides AB and CD. The following expression can be assumed for the deflections in the
Geometrical analysis of large elastic deflections of axially compressed cylindrical and conical shells
277
Fig. 6.
Fig. 7.
apex : w = R sin (2m/L,) + B cos (m/Li) cos (27ry/L,)+ C cos (xy/L,)
(4)
where e-W A=~(tgS;--tg6’,)--i-
(5)
c=: tg[arcsin
(si* e-sin
f)-
s
tg q] e_iQ‘I”’
The geometrical relation d= L In k, results from Fig. 4. Thus; exp (-d/L)= l/k. The function w fulfils the geometric boundary conditions imposed by the deformed surface. The surface su~ounding the apex is an i~met~~Ily transfo~ed surface which means no additional membrane stresses. The stresses resulting from the change of the first matrix tensor of the surface occur only within the apex element. Their values can be determined from the equation of non-linear shallow shell theory
-&w, w)- A,w
(5)
where QIis the classical stress function. Boundary conditions for b, require the stress along the edges to disappear. The right-hand-side of (5) is a known function of the parameters R, 1 and JL In order to solve (5) the method of finite differences has been applied.
3. STRAIN
ENERGY
The energy connected with the deformation of the shell can be determined by calculating the energy in one segment. It can be shown that this energy is identical in each segment.
STAN~SLAWLu~~rsw~czand WALERIANSZYSZKOW~KI
278
Because triangular regions and the regions of the ridges are isometrically transformed regions of the initial surface, the energy stored in them depends only on the variation of their curvatures. This energy can be found from the relation
where S is the area of the isometrically transformed surface and Axi. AxP AXijare changes of curvatures in the orthogonal system of coordinates Xi,i The changes of curvatures. can be calculated as a difference of curvatures of the deformed and initial surface. For instance, considering the triangular element of a cylindrical shell and taking the x-direction along the generator, we have A.x,=O-
f=
- f,
Ax,=Ax,,=O.
Introducing this into (a), we find Dab-F
UP=TF.
Where F is the area of the ridges and apices within one segment (Fig. 5). Introducing the coefficients p and rZwe find 2x2 U,=D p
1+3L’ 1-_p~+2j4z(l+12)
. 1
I
The energy stored in triangular areas of conical shell equals U,=D
;sinrptgrpln
where
$A!-
cosa
sina+cosa-l/k sin 2#I
($z+[l+(~~~‘2)(l-+)
1- -p2(A2+ k+l/k k+l
sin’ a+l)(l+l/k).
The energy connected with the ridges can be found in a similar way [S]. The membrane energy stored in the apex area is equal to I&=&
j-Jr [(A4)2 -(l +v)L&
(7)
41 dF
where 0 is the stress function found from (5). The last relation can be transformed to the from U,,,=60(1 -v2)
R 2 L,L,
h
F J,(4 n, PL,4. 0 The function J#, n, H cp) has been determined by approximating the results of numerical computations. This expression is the following:
.
It was proved that the error of this approximation does not exceed five percent. The work of the external loads A, is determined as the product of the axial force and the shortening of the shell which can be found considering the geometry of the deformations. Then A,= 2zR2ha, ~~{~sinrp-(cosa-t)sin((p+6i)
_~ sin a sin cp k+ 1 tg (6; + ‘;)/2 k
(6; + w/2
1
_ 1 k2 -k2(1 -m) k2-1
(8)
Geometrical analysis of large elastic deflections of axially compressed cylindricaI and conical shells
279
where : m-the number of segments along the generator, cr,---the mean compressive stress at the lower base of the cone in the direction of generator. 4. NUMERICAL
The functional W= U -A, tions
ANALYSIS
AND
DISCUSSION
is a function of three free parameters A, n and p The condi-
give the set of three non-linear algebraic equations defining the value of the load P, = a,R/ E/z (see Fig, 8) as a function of the number of waves n. It is found that for a constant value of R/k sin v, there exists a certain value n= rzmXbelow which the set (9) still has a real solution. This means that a stable configuration of the surface for n>n,, is impossible. Presenting p0 as a function of AH/H we obtain the relation shown in Fig. 9 for R/h sin C-Q = 500. Here AH is the shortening of the shell and H is the height. The behaviour of the shell during the deformation process can be described as follows. At the critical value of shortening the initial axisymmetrical form of the shell becomes unstable and it jumps into a near stable equilibrium position which is connected with the local defo~ation pattern. Then the load decreases. With the increase of shortening and decrease of the parameter the equilibrium state of buckling shell also becomes unstable at a certain critical load. The shell jumps into a new similar buckling pattern. The number n is reduced by one. The lower critical load, usually measured in experimental investigations, has the value to which the load jumps at the moment of buckling. That value corresponds to RN, see Fig. 10. The following fact should be stressed. The lines in this figure run close to each other for different 9. This indicates the possibility of a transformation of the results obtained for cylindrical
Fig. 8.
Fig. 9.
0.16
0
Fig. IO.
shells to conical ones. Such conclusions were drawn in a few experimental works, [8] but were not proved theoretically as yet. The classical solutions obtained for a cylindrical shell, by analysing the von KarmanDonnell equations gives the value of the lower critical load which can be expressed by the formula p= fiEh/R where 6 is a constant coefhcient, Various authors give different value of the coefhcient fi ranging from 0.3 to 0.043 [S]. l[t should be noted here that the coefficient $ obtained in the above mentioned papers has a value independent of the ratio h/R. But experiments performed by many investigators indicate a tendency for j to decrease as R/h increases. The results obtained in the present paper are in good agreement with the above-mentioned behaviour of the shell observed during experiments. ff we assume that the number of waves in the ~rc~ferential direction n is so large that n2 % I, a few interesting conclusions can be obtained. Then the relations
Geometrical analysis of iarge etastic deikctions
ofaxially compressed
cylindrical and conical she&s
281
(3) take the following form
and the functional W can be stated to be
where
2, B and c are known functions of the parameters p and A.The numerical analysis gives the relations shown in Fig. 11. If P denotes the total axial compressive load we have for the
cylindrical shell
(12) and for the cone
The graph in Fig. 11 shows that
hence P- 2nEhi sin2 qq$‘).
t131
The relation $14)enables to utilize the relations ~~=~/R~~~~~describing the behaviour of the c~~~n~~~ shell, for imputation of the force carried by the conical shell. This means
282
SZANISLAWLu~lrslnvicz
and WALEMAN Sz~sz~mvs~~
that the equivalent radius ofthe conical shell is R, sin cp,where R, is the radius of the Iuwer base of the cone. In the case of a cylindrical shell the parameter q can be interpreted as a quantity characterising the value of the maximum deflection {Fig. 12). If we introduce the parameter (14) the function p=f(<) has the form shown in Fig. 13. The dotted line represents the curve obtained by Volmir [9] on the basis of the von Karman-Donnell equations.
Fig. 13.
J
5. Final remarks and conclusions The method presented here enables one to achieve results which are in good agreement with experiments, The cardinal virtue ofthis method is the possibility of solving the problem for arbitrary large values of displacements. The displacement fieid is described without ~~~li~~tion by simple geometrical formulae which are valid in the whole range of possible deflections and rotations [lOJ. The maximum deflection depends on the geometrical parameters only and reaches the value -$ = aR/n2 sin qr, (Fig. 14). For the cylindrical shell rp3 n/2 and then we obtain f_ N +(rr2/n2)R. If, for example n - 5 we find the maximum possible deflection f_=O.l3R which is a relatively large value. This feature of the method makes it very useful in analysis of large deflections which can not be determined correctly by the known equations of theory of shells. Boundary conditions have not been discussed in this paper because in our opinion their influence is
Geometrical analysis of large elastic deflections of axially compressed cylindrical and conical shells
283
it Fig. 14.
important for short shells only. Many experimental investigations [S, 91 have proved that the boundary conditions have considerable influence on the upper critical load. However, if we consider the case of large deflections, its effect is small [ 11,12,13]. The change ofthe lower critical load (pi) with the length of the shell obtained experimentally is presented in Fig. 14 [S]. It follows that the effect of the length is negligible if this length is longer than the wave length of the deflection in axial direction. That means that the shell should be longer than Lc&G
[-m-j
where L, is the length of the generator of the cone and a, L are parameters characterising the deformation pattern of the shell. For cylindrical shells this condition takes the form
The limit values for R/h=200 and R/h = 800 are marked in Fig. 15 by vertical lines. It follows that the critical loads do not depend in this region on the length of the shell and the boundary conditions. The above results are obtained on the assumption that the buckling process remains purely elastic. In the case when plastic regions appear the behaviour of the shell is changed. Some preliminary considerations in this matter have been given in [6].
REFERENCES 1. N. J. Hoff, W. A. Madsen and J. Mayers, Post-buckling equilibrium of axially compressed circular cylindrical shells. AJAA J. No. 14, (1966). 2. Y. Yoshimura, On the mechanism of buckling of circular cylindrical shell under axial compression. Repts. Inst. Sci. and Tech. Univ. Tokyo, No. 5, (1951). 3. N. Yamaki and S. Kodama, Post-buckling behavior of circular cylindrical shells under compression. Int. J. Non-linear Mech. 11,99, (1976). 4. W. F. Thielemann and M. E. Esslinger, On the post-buckling equilibrium and stability of thin-walled circular cylinders under axial compression, Theory of Thin Shells. Springer, Berlin, pp. 26&293 (1968). 5. St. Lukasiewicz and W. Szyszkowski, Geometrical analysis of the post-buckling behaviour of the cylindrical shell. Proc. Symp. IASS, Kielce, (1973). 6. St. Lukasiewicz and W. Szyszkowski, Geometrical methods in non-linear theory of shells. Proc. Symp. Shell Structures, Krakow, Poland, (1974). 7. W. Szyszkowski, PhD. Thesis. Technical University of Warsaw, (1973). 8. V. I. Weingarten, E. I. Morgan and P. Seide, Elastic stability of thin walled cylindrical and conical shells under axial compression. AIAA J. (1965). 9. A. S. Volmir, Stability ofDeformable System (in Russian) Moscow, (1%7). 10. A. V. Pogorelow, Geometrical Methods in Non-linear Theory ojshells (in Russian), Moscow (1967). 11. E. I. Grigoluk and V. V. Kabanov, Stability ofCylindrical Shells (in Russian), Moscow (1%7). 12. R. L. De Neufville, Influence of geometry on the number of buckles in cylinder. AIAA J. (1965).
284
STANISLAW ~JKAS~CZ
and
WALERIANSZYSZKOWSKI
13. R. L. De Neufville and I. I. Connor, Post-buckling behavior of thin cylinders. J. Engng Me&. Division, ASCE, (1968). 14. A, P. Coppa, Inextensional buckling configurations of conical shells. AIAA J. 5 No. 4. (1967).
Resume : Cet article donne une analyse du comportement de coques cylindriques et coniques isotropes compression axiale.
apres flambage soumises a une
Le point de depart de ce papier est le principe des variations de Lagrange don I’application consiste a supposer un champ de contraintes et de deplacements cinematiquement admissible. On determine le champ en considerant la geometrie des deformations quasi isometriques de la coque apres f lambage. Cece permet de resoudre le probleme sans limitation sur I’ordre de grandeur des deplacements.
Zusamnengassung: Diese Arbeit beschreibt die analytische Behandlung des Verhaltens nach der Knickung von isotropischen ZylinderAusgehend und Kegelschalen unter achsialer Druklast. vom Lagrangeschem Variationsprinzip wird ein kinematisch zulassiges Dehnungsund Verstzungsfeld angenomen. Das Feld wird durch Betrachtung der Geometrieder quasiisotropischen Verformungen der Schale nach der Knickung bestirmnt. Dadurch wird es moglich, die Aufgabe zu losen. ohne die Grosse der Versetzungen zu begrenzen.
Abstract-Thii paper presents an analysis of the post-buckiing be&&our of isotropic q&m&ii and conical shells under axial compression. The starting point thepaper is Lagrange’s variational principle, the application of which consists in assuming a kinematically admissible strain and displacement field. The field is determined considering the geometry of quasi-isometric deformations of the shell after buckling That permits solving the problem with no limitation on the magnitude of the displacements.
of
1. INTRODUCTION
Large deflections of elastic shells have usually been analysed on the basis of DomeIl’s non-linear equations of elastic shells 11% 2] by means of the Ritz or Gale&in methods. The solutions obtained in this way showed considerable discrepancies when compared with the results of experiments. Recently, using Donnell equations, Yamaki and Kodama [3] obtained results which are in good agreement with experiment They assumed a deflection function satisfying completely clamped boundary conditions and applied the Galerkin method. However it can be observed that for large values of the deflection (of the order of ten times the thickness) this agreement is not good. Also thit agreement with experimental data presented in [3] is not so good for longer shells. Xn this paper rather short shells defined by the parameter 2==,/(l -v2)J?/Rh d 500 are considered. In our opinion these and other discrepances (for instance associated with shells of simply supported edges) are attributed, first of all, to the approximate nature of the equations employed. At present there is no theory of shells precise enough and yet simple in ~mpu~tions. Thus these problems need still to be considered. In this paper a non-&s&al method of analysis of the post-buckling behaviour of cylindrical and conical shells under axial compression is presented. The characteristic mode of deflections observed in experiments on shells under axial compression involves a diamond pattern built of nearly flat triangular areas jointed together along regions of large curvature [2,4]. It is known that for typical materials used in structures the admissible elastic strain rate is very small. For example, for steel with Young’s modulus E = 2 x 106 kg cmq2 and yield stress R,=4 x lo3 kg cm*, one obtains e = 2 x 10-j. This means that the elastic deformations are character&d by a variation in the metric tensor of its middle surface smaller than 0.2 per cent If such deformation is associated with considerable change of shape of the surface and still remains elastic, this shape must be very similar to the i~met~~ly transformed surface i.e. the surface deformed in such a manner that its first metric tensor remains unchanged. In this paper, the post-buckling deflection pattern is approximated, not a series of functions, but by certain class of surfaces, quasi-isometric to the primary one. In general, it is not possible to create the isometrically transformed surface which would simultaneously be regular everywhere. Therefore the real deformed surface F of the shell can be approximated by the surface F isometric to the initial one, except for a certain region S small compared to the entire surface F. The area S is selected so that the configuration f: presents a regular surface.
274
STANISLAWLUKASIEWICZ and WALERIANSZYSZKOWSKI
Since the Gaussian curvature of the isomet~~lly deformed surface must be equal to the curvature of the initial surface, the surfaces F and S are defined by a hnite number of geometrical parameters whose values can be obtained with the help of Lagrange’s variational principle. The calculation of strain energy and the work of the external loads in the structure can be done on the basis of geometrical considerations.
2. GEOMETRY
OF
THE
DEFORMED
SURFACE
Models of pure isometric surfaces representing the surface of the deformed shell are given in Fig. 1. For cylindrical shells the following geometrical parameters describe the surfaces (Figs, 2 and 3):
Fig. 2.
Fig. 1.
Fig. 3.
Geometrical analysis of large elastic deflections of axially compressed cylindrical and conical shells 275
n-number of rhom~idal elements along the ~r~umferen~. This corresponds to the number of waves in this direction. I = a/b--ratio of the breadth to the height of the rhomboidal element. 26,-plane angle between plane triangles measured along the oblique edge. 26,-plane angle between plane triangles measured along the horixontal edge. After some simple geometrical considerations we obtain sin a2 = Atgx/2n cos 26 1= [ 1 -(l + 21*)g2x/2n]/[ 1f tg%/2n].
(I)
The model of the conical shell is shown in Fig. 4.
Fig. 4.
In this case the geometrical relations are more complex _ _ [k - cos n/2n] sin a cos~(p--62)-[k-cos
a] sin x/n
[k cos (n/n) - I J sin 01 cos (tp+ii;)= Ck cos a - 11 sin (7t/n) cos 26I = [l -$& tg (n/n) cos (cp-S;)](l
(2) $
tg2 (n/n)[sitG (u,-S;)
+(QJ2 -A, cos (cpA”,) ctg (@t))-1’2 where v, is the angle between the base and generator of the undeformed cone, and
a=;cos v, k=
sin a/Af [l +(sin a/A)2]1’2
A,= 2 sin a/[k -cos a]. The pattern of the pure isometrically deformed shell can be defined by two parameters (A,n) only. In the case of the conical shell the values of the angles a,, S, are constant along the generator. This condition results from a geometrical assumptions explained in detail in [S, 6,7] and requires the height of the rhomboidal elements to be dierent in every layer and equal to bi= &
exp [ - ib cos a/2R]( 1- exp [ - ib CQSa/R])
(3)
276
STANISLAW
LUKASI!&VICZ
and
WALERIANSZY~KOWSKI
where: R-radius of the lower base of the cone, &height of the rhomboidal element of the corresponding cylindrical shell with radius R (4+42), i-number of layers counted from the base of the cone. The formula (3) for the conical shell becomes the formula for the cylindrical shell if cp=n/2. Therefore in further considerations we will deal with the conical shell only. It can be shown that if I=&=ctg
6 -
1-cosa sin a/n
then 6, = S; = 8; = n/2. This corresponds to the case of complete collapse of the structure i.e. when its height becomes equal to zero. This means that Iz< 1,. The areas of strongly bent surfaces S which are observed in real deformation patterns are assumed to be sectors of conical and cylindrical surfaces (Fig. 5). These areas are called ridges. The following notations have been applied: b,-the breadth of the oblique ridge measured for level i=O (at the lower base of the cone), b,-the breadth of the horizontal ridge for i =0, L = R/cos a-the length of the generator.
Fig. 5.
The following dimensionless coefficients are introduced: pi = 2bi cos /l/L sin a,
p2 = 2bJ2L sin a,
where /I is denoted in Fig. 5. These values are very close to each other (7) so without making a large error it can be assumed that 2b, cos cp
~1=p2=p=yp&--p0sB
where ~1is the third parameter characterising the deformed surface. The region surrounding points of intersection of the ridges is called an apex (Fig. 6). In this area fragments of flat triangles and conical ridges should be fitted to make the surface of the apex regular. The Gaussian curvature in this region is different from zero. It is assumed that the form of the apex is rectangular (Fig 6). The dimensions L, and Lz can easily be determined as functions of parameters 1, n and p The deflections in this area are presented in Fig. 7. The apex is jointed to the horizontal ridges along the sides AD and BC and to the oblique ridges along the sides AB and CD. The following expression can be assumed for the deflections in the
Geometrical analysis of large elastic deflections of axially compressed cylindrical and conical shells
277
Fig. 6.
Fig. 7.
apex : w = R sin (2m/L,) + B cos (m/Li) cos (27ry/L,)+ C cos (xy/L,)
(4)
where e-W A=~(tgS;--tg6’,)--i-
(5)
c=: tg[arcsin
(si* e-sin
f)-
s
tg q] e_iQ‘I”’
The geometrical relation d= L In k, results from Fig. 4. Thus; exp (-d/L)= l/k. The function w fulfils the geometric boundary conditions imposed by the deformed surface. The surface su~ounding the apex is an i~met~~Ily transfo~ed surface which means no additional membrane stresses. The stresses resulting from the change of the first matrix tensor of the surface occur only within the apex element. Their values can be determined from the equation of non-linear shallow shell theory
-&w, w)- A,w
(5)
where QIis the classical stress function. Boundary conditions for b, require the stress along the edges to disappear. The right-hand-side of (5) is a known function of the parameters R, 1 and JL In order to solve (5) the method of finite differences has been applied.
3. STRAIN
ENERGY
The energy connected with the deformation of the shell can be determined by calculating the energy in one segment. It can be shown that this energy is identical in each segment.
STAN~SLAWLu~~rsw~czand WALERIANSZYSZKOW~KI
278
Because triangular regions and the regions of the ridges are isometrically transformed regions of the initial surface, the energy stored in them depends only on the variation of their curvatures. This energy can be found from the relation
where S is the area of the isometrically transformed surface and Axi. AxP AXijare changes of curvatures in the orthogonal system of coordinates Xi,i The changes of curvatures. can be calculated as a difference of curvatures of the deformed and initial surface. For instance, considering the triangular element of a cylindrical shell and taking the x-direction along the generator, we have A.x,=O-
f=
- f,
Ax,=Ax,,=O.
Introducing this into (a), we find Dab-F
UP=TF.
Where F is the area of the ridges and apices within one segment (Fig. 5). Introducing the coefficients p and rZwe find 2x2 U,=D p
1+3L’ 1-_p~+2j4z(l+12)
. 1
I
The energy stored in triangular areas of conical shell equals U,=D
;sinrptgrpln
where
$A!-
cosa
sina+cosa-l/k sin 2#I
($z+[l+(~~~‘2)(l-+)
1- -p2(A2+ k+l/k k+l
sin’ a+l)(l+l/k).
The energy connected with the ridges can be found in a similar way [S]. The membrane energy stored in the apex area is equal to I&=&
j-Jr [(A4)2 -(l +v)L&
(7)
41 dF
where 0 is the stress function found from (5). The last relation can be transformed to the from U,,,=60(1 -v2)
R 2 L,L,
h
F J,(4 n, PL,4. 0 The function J#, n, H cp) has been determined by approximating the results of numerical computations. This expression is the following:
.
It was proved that the error of this approximation does not exceed five percent. The work of the external loads A, is determined as the product of the axial force and the shortening of the shell which can be found considering the geometry of the deformations. Then A,= 2zR2ha, ~~{~sinrp-(cosa-t)sin((p+6i)
_~ sin a sin cp k+ 1 tg (6; + ‘;)/2 k
(6; + w/2
1
_ 1 k2 -k2(1 -m) k2-1
(8)
Geometrical analysis of large elastic deflections of axially compressed cylindricaI and conical shells
279
where : m-the number of segments along the generator, cr,---the mean compressive stress at the lower base of the cone in the direction of generator. 4. NUMERICAL
The functional W= U -A, tions
ANALYSIS
AND
DISCUSSION
is a function of three free parameters A, n and p The condi-
give the set of three non-linear algebraic equations defining the value of the load P, = a,R/ E/z (see Fig, 8) as a function of the number of waves n. It is found that for a constant value of R/k sin v, there exists a certain value n= rzmXbelow which the set (9) still has a real solution. This means that a stable configuration of the surface for n>n,, is impossible. Presenting p0 as a function of AH/H we obtain the relation shown in Fig. 9 for R/h sin C-Q = 500. Here AH is the shortening of the shell and H is the height. The behaviour of the shell during the deformation process can be described as follows. At the critical value of shortening the initial axisymmetrical form of the shell becomes unstable and it jumps into a near stable equilibrium position which is connected with the local defo~ation pattern. Then the load decreases. With the increase of shortening and decrease of the parameter the equilibrium state of buckling shell also becomes unstable at a certain critical load. The shell jumps into a new similar buckling pattern. The number n is reduced by one. The lower critical load, usually measured in experimental investigations, has the value to which the load jumps at the moment of buckling. That value corresponds to RN, see Fig. 10. The following fact should be stressed. The lines in this figure run close to each other for different 9. This indicates the possibility of a transformation of the results obtained for cylindrical
Fig. 8.
Fig. 9.
0.16
0
Fig. IO.
shells to conical ones. Such conclusions were drawn in a few experimental works, [8] but were not proved theoretically as yet. The classical solutions obtained for a cylindrical shell, by analysing the von KarmanDonnell equations gives the value of the lower critical load which can be expressed by the formula p= fiEh/R where 6 is a constant coefhcient, Various authors give different value of the coefhcient fi ranging from 0.3 to 0.043 [S]. l[t should be noted here that the coefficient $ obtained in the above mentioned papers has a value independent of the ratio h/R. But experiments performed by many investigators indicate a tendency for j to decrease as R/h increases. The results obtained in the present paper are in good agreement with the above-mentioned behaviour of the shell observed during experiments. ff we assume that the number of waves in the ~rc~ferential direction n is so large that n2 % I, a few interesting conclusions can be obtained. Then the relations
Geometrical analysis of iarge etastic deikctions
ofaxially compressed
cylindrical and conical she&s
281
(3) take the following form
and the functional W can be stated to be
where
2, B and c are known functions of the parameters p and A.The numerical analysis gives the relations shown in Fig. 11. If P denotes the total axial compressive load we have for the
cylindrical shell
(12) and for the cone
The graph in Fig. 11 shows that
hence P- 2nEhi sin2 qq$‘).
t131
The relation $14)enables to utilize the relations ~~=~/R~~~~~describing the behaviour of the c~~~n~~~ shell, for imputation of the force carried by the conical shell. This means
282
SZANISLAWLu~lrslnvicz
and WALEMAN Sz~sz~mvs~~
that the equivalent radius ofthe conical shell is R, sin cp,where R, is the radius of the Iuwer base of the cone. In the case of a cylindrical shell the parameter q can be interpreted as a quantity characterising the value of the maximum deflection {Fig. 12). If we introduce the parameter (14) the function p=f(<) has the form shown in Fig. 13. The dotted line represents the curve obtained by Volmir [9] on the basis of the von Karman-Donnell equations.
Fig. 13.
J
5. Final remarks and conclusions The method presented here enables one to achieve results which are in good agreement with experiments, The cardinal virtue ofthis method is the possibility of solving the problem for arbitrary large values of displacements. The displacement fieid is described without ~~~li~~tion by simple geometrical formulae which are valid in the whole range of possible deflections and rotations [lOJ. The maximum deflection depends on the geometrical parameters only and reaches the value -$ = aR/n2 sin qr, (Fig. 14). For the cylindrical shell rp3 n/2 and then we obtain f_ N +(rr2/n2)R. If, for example n - 5 we find the maximum possible deflection f_=O.l3R which is a relatively large value. This feature of the method makes it very useful in analysis of large deflections which can not be determined correctly by the known equations of theory of shells. Boundary conditions have not been discussed in this paper because in our opinion their influence is
Geometrical analysis of large elastic deflections of axially compressed cylindrical and conical shells
283
it Fig. 14.
important for short shells only. Many experimental investigations [S, 91 have proved that the boundary conditions have considerable influence on the upper critical load. However, if we consider the case of large deflections, its effect is small [ 11,12,13]. The change ofthe lower critical load (pi) with the length of the shell obtained experimentally is presented in Fig. 14 [S]. It follows that the effect of the length is negligible if this length is longer than the wave length of the deflection in axial direction. That means that the shell should be longer than Lc&G
[-m-j
where L, is the length of the generator of the cone and a, L are parameters characterising the deformation pattern of the shell. For cylindrical shells this condition takes the form
The limit values for R/h=200 and R/h = 800 are marked in Fig. 15 by vertical lines. It follows that the critical loads do not depend in this region on the length of the shell and the boundary conditions. The above results are obtained on the assumption that the buckling process remains purely elastic. In the case when plastic regions appear the behaviour of the shell is changed. Some preliminary considerations in this matter have been given in [6].
REFERENCES 1. N. J. Hoff, W. A. Madsen and J. Mayers, Post-buckling equilibrium of axially compressed circular cylindrical shells. AJAA J. No. 14, (1966). 2. Y. Yoshimura, On the mechanism of buckling of circular cylindrical shell under axial compression. Repts. Inst. Sci. and Tech. Univ. Tokyo, No. 5, (1951). 3. N. Yamaki and S. Kodama, Post-buckling behavior of circular cylindrical shells under compression. Int. J. Non-linear Mech. 11,99, (1976). 4. W. F. Thielemann and M. E. Esslinger, On the post-buckling equilibrium and stability of thin-walled circular cylinders under axial compression, Theory of Thin Shells. Springer, Berlin, pp. 26&293 (1968). 5. St. Lukasiewicz and W. Szyszkowski, Geometrical analysis of the post-buckling behaviour of the cylindrical shell. Proc. Symp. IASS, Kielce, (1973). 6. St. Lukasiewicz and W. Szyszkowski, Geometrical methods in non-linear theory of shells. Proc. Symp. Shell Structures, Krakow, Poland, (1974). 7. W. Szyszkowski, PhD. Thesis. Technical University of Warsaw, (1973). 8. V. I. Weingarten, E. I. Morgan and P. Seide, Elastic stability of thin walled cylindrical and conical shells under axial compression. AIAA J. (1965). 9. A. S. Volmir, Stability ofDeformable System (in Russian) Moscow, (1%7). 10. A. V. Pogorelow, Geometrical Methods in Non-linear Theory ojshells (in Russian), Moscow (1967). 11. E. I. Grigoluk and V. V. Kabanov, Stability ofCylindrical Shells (in Russian), Moscow (1%7). 12. R. L. De Neufville, Influence of geometry on the number of buckles in cylinder. AIAA J. (1965).
284
STANISLAW ~JKAS~CZ
and
WALERIANSZYSZKOWSKI
13. R. L. De Neufville and I. I. Connor, Post-buckling behavior of thin cylinders. J. Engng Me&. Division, ASCE, (1968). 14. A, P. Coppa, Inextensional buckling configurations of conical shells. AIAA J. 5 No. 4. (1967).
Resume : Cet article donne une analyse du comportement de coques cylindriques et coniques isotropes compression axiale.
apres flambage soumises a une
Le point de depart de ce papier est le principe des variations de Lagrange don I’application consiste a supposer un champ de contraintes et de deplacements cinematiquement admissible. On determine le champ en considerant la geometrie des deformations quasi isometriques de la coque apres f lambage. Cece permet de resoudre le probleme sans limitation sur I’ordre de grandeur des deplacements.
Zusamnengassung: Diese Arbeit beschreibt die analytische Behandlung des Verhaltens nach der Knickung von isotropischen ZylinderAusgehend und Kegelschalen unter achsialer Druklast. vom Lagrangeschem Variationsprinzip wird ein kinematisch zulassiges Dehnungsund Verstzungsfeld angenomen. Das Feld wird durch Betrachtung der Geometrieder quasiisotropischen Verformungen der Schale nach der Knickung bestirmnt. Dadurch wird es moglich, die Aufgabe zu losen. ohne die Grosse der Versetzungen zu begrenzen.