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Stability of an elastic thick walled tube under end thrust and external pressure
STABILITY OF AN ELASTIC THICK WALLED TUBE UNDER END THRUST AND EXTERNAL PRESSURE J. C. Department
of Mathematics.
University
PATTERSON
of Western
Australia,
Nedlands
6009 (WA), Australia
(Receiced 21 February 1974)
Abstract-The stability of a thick-walled Neo-Hookean tube under end thrust and external pressure is examined by the use of the theory of an infinitesimal deformatton supertmposed on a finite deformation, Loss of stability occurs when a non-trivial solution for the infinitesimal deformation is available. The resulting equations are solved numerically and the critical conditions presented graphically for axisymmetric and non-axisymmetric buckling. The effects of external pressure are shown. The effect of placing a simple end condition on the tube is also demonstrated. INTRODUCTION
effect of the application of hydrostatic pressure on the stability of elastic bodies is a question of considerable practical importance. Investigations in the past have mainly been concerned with the behaviour of thin bodies, and it is only during the last 15 yr that it has become clear that instability may be associated with other than slender bodies. The first attempt to consider the stability of thick-walled solids appears to be that of Wilkes [I] in 1955 who found conditions for the axially symmetric buckling of a tube under end thrust. In 1957, Green and Spencer [2] considered the stability of a tube under extension and torsion. Lubkin [3] and Sensenig [4] appear to be the first to include the effects of an applied external pressure, but with a particular form of strain energy. In 1967, Wesolowski [5] examined the buckling conditions for a thick-walled sphere, and, in 1969, Nowinski and Shahinpoor [6], those for a tube, both under external pressure. Both of these analyses contained uncertain numerical results. Wang and Ertepinar in 1972 [7] reconsidered both of these problems as special cases of the dynamic problem of calculating the frequencies of free vibrations. The use of an improved numerical technique provided results which appeared to clear up the problems of [5] and [6]. The analyses contained in [l. 2,5-71 are based on the same technique; that of superimposing an infinitesimal deformation on a finite one, the theory of which was developed by Green, Rivlin and Shield [8]. Instability of the finitely deformed solid is determined when non-trivial infinitesimal deformations, together with the finite deformation, satisfy the equilibrium equations and the unchanged boundary conditions. Such a technique is referred to as that of adjacent equilibria. The present paper is. in essence, an extension of the investigation of [l]. Specifically, non-axisymmetric modes of buckling of the tube under end thrust are found, and the effects of an external pressure on both the axisymmetric and non-axisymmetric modes are shown. The method of adjacent equilibria is used, and the resulting boundary value problem is solved by the method of finite differences. The
FORMULATION The tube under consideration is assumed to be composed of an isotropic, incompressible elastic material of the Neo-Hookean type. Other types of material, including compressible types, could have been used as easily. The inner and outer radii of the undeformed tube are Ai and A2 respectively, and its undeformed length is L. Under the action of end thrust and an external pressure cl, it is easy to show that the resulting finite deformation may be described in cylindrical polar coordinates as
38.5
J. C. PATTERSON
386
where (r, 0. -_)are the spatial and (R. 0, Z) the material coordinates of the tube, p a pressure function, and K and p. unknown constants to be evaluated by the boundary conditions. The parameter C1 is the constant from the expression for strain energy. The boundary conditions are
(2) where al and ~72are the deformed inner and outer radii respectively. and r” the normal component of stress. Equations (1) and (2) describe the initial finite deformation. If, on this initial deformation, a secondary infinitesimal deformation (.\“. .?. .t3) is superimposed, the equations of equilibrium resulting from the theory developed in [8] become
(3)
where (jr’, i2, m3) have been replaced by (u, c, w), and p’ is an unknown secondary pressure function. The variables R and p are obtained from (1). The incompressibility condition for the secondary deformation is &A
u
61:
z+;+g+Sz The boundary surface tractions
dw
=
0.
conditions on the secondary deformation require that are zero. This requirement becomes, after manipulation; 4C1 R2 du _p’+_-= ?A2 Pr
(4) the secondary
0
(5)
c;u
s++
At‘
= 0,
all at r = al, u2.
The derivation of equations (3), (4) and (5) is similar to the derivation of the comparable equations in [l], [2], [6], and [7], and will be omitted here. We note that for K = 0 (the no pressure case), equations (3) collapse onto equations (3.9) of [l], and for 1, = 1 (no extension or compression) onto (5) and (6) of [7]. Equations (3), (4) and (5) constitute the boundary value problem to be solved for the secondary deformations U, c, and w. If non-trivial solutions of (3), (4). and (5) exist for specified values of the parameters involved, then buckling occurs at those values. We seek solutions in the form u = L’= M‘= p’ =
f~(r)cosrd9cos k: fI(r) sin no cos kz f3(r) cos nOsin kz <(r)cos n0cos kz.
(6)
If, in (6) n = 0. and in (3) (4) and (5), K = 0, the resulting fourth order set of ordinary differential equations for fi,f2,f3, and 4 are those solved in detail in [l], whilst if in (6) k = 0. equations (3), (4), and (5) become independent of i. and the resulting plane problem
Stability
of an elastic thick walled tube under end thrust
and external
pressure
387
is that solved in detail in [6] and [7]. In this analysis we include the cases K = 0, and n = 0 as special cases of the more general result. It appears that only in the case K = 0 does the system (3), (4), and (5) have a solution in closed form, that found in [I]. On substitution of (6) and after elimination off*(r) and t(r), equations (3), (4). and (5) become the following sixth order ordinary differential system forit andf3(r); f;” = -I;.(F+*h)$+fi($-2j.+g(n2+k2i2R2))$-
+f1
(
R2 -7+y-7
2rn2i2
23
-f;(!t$+B”R’
+$(n2+k’i2R2)
)&-f;[n2(~-$)+k2r2(~+$)]&
+f3
(
n2J2 F++-
k2i2r2 1
(n2+k2i2R2)&,
-_ili+~-!$$+k2~~R~)
f;” = -f~(4ir-~)$-f~(
+f3 (
$
j
(7)
$ )
2n2A3r3 -7--
2r3/2k2 lG
+f,k
(
R2 2ir-T+F
1
F-f;‘k-f/k
1
2r3i2 )
( 2r4L3 -+_-R4
~(n2+i2k2R2)-4j.+
1
RZ
3R2
2;L2r2
r2
R2
(8)
with the boundary conditions; f;‘$+f;
(
F-5
)
F_!$_!$
i-f3(n2+k2jL2R2)&+$fi_ fy+f;_f~(l-n2) r -kf,+f;
= 0,
SOLUTION
+ kf; = 0
r2
fi = 0
( (9)
allatr=al,a2.
AND
RESULTS
The system of equations (7) (8). and (9) is non-dimensionalized and written in finite difference form. The result is 6(N+ 1) linear equations in the 6(N+ 1) unknowns fi, f;, f;‘, f3, f;, f;‘, at the (Nt 1) nodes of the finite difference scheme. For a non-trivial solution, we require that the determinant of the coefficient matrix vanishes. After considerable manipulation, the determinant may be reduced to one of order six, the vanishing of which provides a relation between the parameters n, k, K, A1/A2, and I which must be satisfied for buckling to occur. We choose specific values of n, A1/A2, and K, and plot the variation of i. with k for these specified cases. In the following figures, k and K are in non-dimensional form; k, = a2 k, K, = K/a:. The values of K, specified are representative values taken between 0 (corresponding to zero pressure), and the value for which the inside radius becomes zero. It is easy to show that this value is given by
Al 2
K, =
C-1 A2
2’
(10)
Figure 1 shows the variation of 3. with k. for the indicated values of K, for n = 1,2 and Al/A2 = 0.25: Fig. 2 is for AI/A2 with the value 0.5.
J.C. PATTERSON
388
k.
Fig. 1. The variation
of i with k. for Al/A2
= 0.25. with K. = 0, 0.03,and n = 1, 2
07x 0.6-
05-
0.31 0
IO
20
3.0
k.
Fig. 2. The variation
ofi
with k. for Al/A2
40
5.0
60
1 70
= 0.5. with K. = 0, 0.1, and n = 0, I. 2.
DISCUSSION
In the case of no applied external pressure (K. = 0), it is apparent that, if no restriction is placed on the value of k,, the tube is immediately unstable in the mode n = 1 for any 1, < 1. The value of /I for a particular k, is always greater in the n = 1 mode than in the n = 2 mode, hence, for either value of AI/Al, the tube will buckle into the n = 1 mode. The thinner tube is considerably less stable in both modes than the thick tube, as expected. The application of an external pressure, however, alters this behaviour. Using (lo), the maximum value allowed for K. if AI/A2 = 0.25 is 0.06, and if AI/A2 = 0.5, the maximum is 0.33. In Figs. 1 and 2, the K, values taken are 003 and 01 respectively, considerably less than these maxima. Figure 1 shows that, for ZL = 0.03, the basic behaviour of the thicker tube is unchanged, the stability being lessened in both modes, more so in the
Stability of an elastic thick walled tube under end thrust and external pressure
_._ 0 (
110
389
1
210
3lo
410
5(0
610
7.0
h
Fig. 3. The operating curves resulting from the requirement w = 0, z = 0, 1, for AI/L = 0.1, 0.5, and for m = 1, showing the intersection with the typical curves Al/A2 = 0.5, K. = 0.1.
n = 2 mode, the n = 1 mode remaining the most likely for all k, In the case of the thinner tube however, Fig. 2 shows that, for K. = 0.1, the n = 2 mode has become more likely than n = 1 for a range of values of k,. Hence, if the value of k, is unrestricted, the thinner tube under pressure will buckle into the n = 2 mode for any A< 1. This mode is also available for II > 1 (extension), although this behaviour has not been investigated. The behaviour of the tubes under pressure is then very dependent on the conditions, if any, placed on k,. If k, is specified in some manner, buckling will not occur until i is reduced to the value corresponding to the appropriate point on the uppermost of the I-k, curves of Figs. 1 or 2. In the case of AI/AI = 0.25, buckling will still occur in the n = 1 mode, but the type of buckling for Al/A2 = 0.5 will depend on the value of k. relative to the value at which the n = 1 and n = 2 curves intersect. For k. less than that value, the n = 2 mode will be taken up when 1 is reduced to the required value, for k, greater, the n = 1 mode will occur. The restrictions on k, are determined by the conditions specified at the ends of the tube. If the tube is very long in comparison with its radius, end conditions may be ignored, and the value of k. is unrestricted. If, on the other hand, the tube is of length 1 in the finitely deformed state, 1= AI., L being the initial length, k, is restricted in a manner which depends on the conditions at z = 0,1. If, for example, the tube is compressed between flat plates, we may demand that the secondary axial displacement be zero at the ends, w = 0,
z = 0,
1.
(11)
This condition results in specified values of k, or k., k=y
or
k.=(%)[$(1-&)rmn
m=l.2,3
,....
(12)
Equation (12) is a relation between k, and 1 for specified values of AZ/L, K., and m, which must be satisfied simultaneously with the curves of Figs. 1 and 2 if buckling is to occur and (11) to be satisfied. Equation (12) then provides an operating curve, the intersection of which with the A-k. curves provides the critical points of instability. Figure 3 shows the operating curves for AZ/L = O-5, 0.1, for m = 1, K. = 0.1, with the curves for Al/A2 = 0.5, K, = 0.1, n = 1,2. We note that, as the tube becomes long and thin (AZ/L + 0), the operating curve for a particular m moves closer to the 3,axis, hence increasing the critical value of A, making the tube more unstable, as expected.
J.C. PATTERSON
390
REFERENCES 1. E. W. Wilkes, On the stability of a circular tube under end thrust. Q. JI Xlrclr. ,4ppl. Marls. 8. 88.-100 I 1955). 2. A. E. Green and A. J. M. Spencer. The stability of a circularcylinderunder finite extension and torsion. J. AQUA. Ph.% 37, 316-338 (1959). 3. S. Lubkin, Determination of buckling criteria by minimization of total energy, fast. .%Zurll. Sci. New York Univ., Rep IMM-NYU241 (1957). 4. C. B. Sensenig, Instability of thick elastic solids. Contmuns. purr appl. rV~jtl~. 17. 451-491 (19641. 5. Z. Wesolowski, Stability of an elastic. thick-walled sphere uniformly loaded by an external pressure. Archum Mech. SfOSOW. 19. 3-.44 (1967). 6. J. L. Nowinski and M. Shahinpoor. Stability of an elastic circular tube of arbitrary wall thickness subjected to an external pressure. Inr. J. Non-Lifrear Mrclr, 4. 143--i%? (1969). 7. A. S. D. Wang and A. Ertepinar. Stability and vihrattons of elastic thick-walled c?Iindrical and sphericai shells subjected to pressure. Inr. J. Non-Litrear Me&. 7. 539-555 (1972). 8. A. E. Green, R. S. Rivlin and R. T. Shield. General theory of small elastic deformation superposed on finite elastic deformations. Proc. R. SW. A211 (1952).
R&urn&: On examine la stabilit.6 d’un tube n;o Hookeen 'aparoi Gpaisse soumis a une pouss&e et une pression externe en utilisant la thiorie d'une deformation infini&imale superposGe i une dCformation finie. La perte de stabilit; apparait lorsqu'il existe une solution non triviale pour la d&formation On r&oud num~riquement les gquations infinitCsimale. rksultantes et on donne graphiq~ement les conditions critiques pour un flambage ayant un axe de sym&zrie et non symhtrique. On montre les effets de la pression externe. On dgmontre egalement l'effet de mettre une condition simple 'a l'extremit; du tube.
Unter Verwendung der Theorie der infinitesimalen Verformungen, die endlichen Verformungen ijberlagert sind, wird die Stabilitst eines dickwandigen, neohooke'schen Rohrs unter Endschub und Aussendruck untersucht. StabilitEtsverlust tritt aug, wenn eine nuchttriviale Lb;sung fiir die infinitesimale Verformung vorhanden ist. Die sich erqebenden Bleichungen werden n&erisch gel&t und die dritiichen Bedingungen werden graphisch dargestellt fiir den Fall der achsensvrmnetrischen und nichtachsensvmmetrischen Knickuns. Die Einflb'sse des Aussendrucks werden aufgezefgt. Weiterhin wird die Wirkung einer einfachen Endbedingung fiir das Rohr demonstriert.
of Mathematics.
University
PATTERSON
of Western
Australia,
Nedlands
6009 (WA), Australia
(Receiced 21 February 1974)
Abstract-The stability of a thick-walled Neo-Hookean tube under end thrust and external pressure is examined by the use of the theory of an infinitesimal deformatton supertmposed on a finite deformation, Loss of stability occurs when a non-trivial solution for the infinitesimal deformation is available. The resulting equations are solved numerically and the critical conditions presented graphically for axisymmetric and non-axisymmetric buckling. The effects of external pressure are shown. The effect of placing a simple end condition on the tube is also demonstrated. INTRODUCTION
effect of the application of hydrostatic pressure on the stability of elastic bodies is a question of considerable practical importance. Investigations in the past have mainly been concerned with the behaviour of thin bodies, and it is only during the last 15 yr that it has become clear that instability may be associated with other than slender bodies. The first attempt to consider the stability of thick-walled solids appears to be that of Wilkes [I] in 1955 who found conditions for the axially symmetric buckling of a tube under end thrust. In 1957, Green and Spencer [2] considered the stability of a tube under extension and torsion. Lubkin [3] and Sensenig [4] appear to be the first to include the effects of an applied external pressure, but with a particular form of strain energy. In 1967, Wesolowski [5] examined the buckling conditions for a thick-walled sphere, and, in 1969, Nowinski and Shahinpoor [6], those for a tube, both under external pressure. Both of these analyses contained uncertain numerical results. Wang and Ertepinar in 1972 [7] reconsidered both of these problems as special cases of the dynamic problem of calculating the frequencies of free vibrations. The use of an improved numerical technique provided results which appeared to clear up the problems of [5] and [6]. The analyses contained in [l. 2,5-71 are based on the same technique; that of superimposing an infinitesimal deformation on a finite one, the theory of which was developed by Green, Rivlin and Shield [8]. Instability of the finitely deformed solid is determined when non-trivial infinitesimal deformations, together with the finite deformation, satisfy the equilibrium equations and the unchanged boundary conditions. Such a technique is referred to as that of adjacent equilibria. The present paper is. in essence, an extension of the investigation of [l]. Specifically, non-axisymmetric modes of buckling of the tube under end thrust are found, and the effects of an external pressure on both the axisymmetric and non-axisymmetric modes are shown. The method of adjacent equilibria is used, and the resulting boundary value problem is solved by the method of finite differences. The
FORMULATION The tube under consideration is assumed to be composed of an isotropic, incompressible elastic material of the Neo-Hookean type. Other types of material, including compressible types, could have been used as easily. The inner and outer radii of the undeformed tube are Ai and A2 respectively, and its undeformed length is L. Under the action of end thrust and an external pressure cl, it is easy to show that the resulting finite deformation may be described in cylindrical polar coordinates as
38.5
J. C. PATTERSON
386
where (r, 0. -_)are the spatial and (R. 0, Z) the material coordinates of the tube, p a pressure function, and K and p. unknown constants to be evaluated by the boundary conditions. The parameter C1 is the constant from the expression for strain energy. The boundary conditions are
(2) where al and ~72are the deformed inner and outer radii respectively. and r” the normal component of stress. Equations (1) and (2) describe the initial finite deformation. If, on this initial deformation, a secondary infinitesimal deformation (.\“. .?. .t3) is superimposed, the equations of equilibrium resulting from the theory developed in [8] become
(3)
where (jr’, i2, m3) have been replaced by (u, c, w), and p’ is an unknown secondary pressure function. The variables R and p are obtained from (1). The incompressibility condition for the secondary deformation is &A
u
61:
z+;+g+Sz The boundary surface tractions
dw
=
0.
conditions on the secondary deformation require that are zero. This requirement becomes, after manipulation; 4C1 R2 du _p’+_-= ?A2 Pr
(4) the secondary
0
(5)
c;u
s++
At‘
= 0,
all at r = al, u2.
The derivation of equations (3), (4) and (5) is similar to the derivation of the comparable equations in [l], [2], [6], and [7], and will be omitted here. We note that for K = 0 (the no pressure case), equations (3) collapse onto equations (3.9) of [l], and for 1, = 1 (no extension or compression) onto (5) and (6) of [7]. Equations (3), (4) and (5) constitute the boundary value problem to be solved for the secondary deformations U, c, and w. If non-trivial solutions of (3), (4). and (5) exist for specified values of the parameters involved, then buckling occurs at those values. We seek solutions in the form u = L’= M‘= p’ =
f~(r)cosrd9cos k: fI(r) sin no cos kz f3(r) cos nOsin kz <(r)cos n0cos kz.
(6)
If, in (6) n = 0. and in (3) (4) and (5), K = 0, the resulting fourth order set of ordinary differential equations for fi,f2,f3, and 4 are those solved in detail in [l], whilst if in (6) k = 0. equations (3), (4), and (5) become independent of i. and the resulting plane problem
Stability
of an elastic thick walled tube under end thrust
and external
pressure
387
is that solved in detail in [6] and [7]. In this analysis we include the cases K = 0, and n = 0 as special cases of the more general result. It appears that only in the case K = 0 does the system (3), (4), and (5) have a solution in closed form, that found in [I]. On substitution of (6) and after elimination off*(r) and t(r), equations (3), (4). and (5) become the following sixth order ordinary differential system forit andf3(r); f;” = -I;.(F+*h)$+fi($-2j.+g(n2+k2i2R2))$-
+f1
(
R2 -7+y-7
2rn2i2
23
-f;(!t$+B”R’
+$(n2+k’i2R2)
)&-f;[n2(~-$)+k2r2(~+$)]&
+f3
(
n2J2 F++-
k2i2r2 1
(n2+k2i2R2)&,
-_ili+~-!$$+k2~~R~)
f;” = -f~(4ir-~)$-f~(
+f3 (
$
j
(7)
$ )
2n2A3r3 -7--
2r3/2k2 lG
+f,k
(
R2 2ir-T+F
1
F-f;‘k-f/k
1
2r3i2 )
( 2r4L3 -+_-R4
~(n2+i2k2R2)-4j.+
1
RZ
3R2
2;L2r2
r2
R2
(8)
with the boundary conditions; f;‘$+f;
(
F-5
)
F_!$_!$
i-f3(n2+k2jL2R2)&+$fi_ fy+f;_f~(l-n2) r -kf,+f;
= 0,
SOLUTION
+ kf; = 0
r2
fi = 0
( (9)
allatr=al,a2.
AND
RESULTS
The system of equations (7) (8). and (9) is non-dimensionalized and written in finite difference form. The result is 6(N+ 1) linear equations in the 6(N+ 1) unknowns fi, f;, f;‘, f3, f;, f;‘, at the (Nt 1) nodes of the finite difference scheme. For a non-trivial solution, we require that the determinant of the coefficient matrix vanishes. After considerable manipulation, the determinant may be reduced to one of order six, the vanishing of which provides a relation between the parameters n, k, K, A1/A2, and I which must be satisfied for buckling to occur. We choose specific values of n, A1/A2, and K, and plot the variation of i. with k for these specified cases. In the following figures, k and K are in non-dimensional form; k, = a2 k, K, = K/a:. The values of K, specified are representative values taken between 0 (corresponding to zero pressure), and the value for which the inside radius becomes zero. It is easy to show that this value is given by
Al 2
K, =
C-1 A2
2’
(10)
Figure 1 shows the variation of 3. with k. for the indicated values of K, for n = 1,2 and Al/A2 = 0.25: Fig. 2 is for AI/A2 with the value 0.5.
J.C. PATTERSON
388
k.
Fig. 1. The variation
of i with k. for Al/A2
= 0.25. with K. = 0, 0.03,and n = 1, 2
07x 0.6-
05-
0.31 0
IO
20
3.0
k.
Fig. 2. The variation
ofi
with k. for Al/A2
40
5.0
60
1 70
= 0.5. with K. = 0, 0.1, and n = 0, I. 2.
DISCUSSION
In the case of no applied external pressure (K. = 0), it is apparent that, if no restriction is placed on the value of k,, the tube is immediately unstable in the mode n = 1 for any 1, < 1. The value of /I for a particular k, is always greater in the n = 1 mode than in the n = 2 mode, hence, for either value of AI/Al, the tube will buckle into the n = 1 mode. The thinner tube is considerably less stable in both modes than the thick tube, as expected. The application of an external pressure, however, alters this behaviour. Using (lo), the maximum value allowed for K. if AI/A2 = 0.25 is 0.06, and if AI/A2 = 0.5, the maximum is 0.33. In Figs. 1 and 2, the K, values taken are 003 and 01 respectively, considerably less than these maxima. Figure 1 shows that, for ZL = 0.03, the basic behaviour of the thicker tube is unchanged, the stability being lessened in both modes, more so in the
Stability of an elastic thick walled tube under end thrust and external pressure
_._ 0 (
110
389
1
210
3lo
410
5(0
610
7.0
h
Fig. 3. The operating curves resulting from the requirement w = 0, z = 0, 1, for AI/L = 0.1, 0.5, and for m = 1, showing the intersection with the typical curves Al/A2 = 0.5, K. = 0.1.
n = 2 mode, the n = 1 mode remaining the most likely for all k, In the case of the thinner tube however, Fig. 2 shows that, for K. = 0.1, the n = 2 mode has become more likely than n = 1 for a range of values of k,. Hence, if the value of k, is unrestricted, the thinner tube under pressure will buckle into the n = 2 mode for any A< 1. This mode is also available for II > 1 (extension), although this behaviour has not been investigated. The behaviour of the tubes under pressure is then very dependent on the conditions, if any, placed on k,. If k, is specified in some manner, buckling will not occur until i is reduced to the value corresponding to the appropriate point on the uppermost of the I-k, curves of Figs. 1 or 2. In the case of AI/AI = 0.25, buckling will still occur in the n = 1 mode, but the type of buckling for Al/A2 = 0.5 will depend on the value of k. relative to the value at which the n = 1 and n = 2 curves intersect. For k. less than that value, the n = 2 mode will be taken up when 1 is reduced to the required value, for k, greater, the n = 1 mode will occur. The restrictions on k, are determined by the conditions specified at the ends of the tube. If the tube is very long in comparison with its radius, end conditions may be ignored, and the value of k. is unrestricted. If, on the other hand, the tube is of length 1 in the finitely deformed state, 1= AI., L being the initial length, k, is restricted in a manner which depends on the conditions at z = 0,1. If, for example, the tube is compressed between flat plates, we may demand that the secondary axial displacement be zero at the ends, w = 0,
z = 0,
1.
(11)
This condition results in specified values of k, or k., k=y
or
k.=(%)[$(1-&)rmn
m=l.2,3
,....
(12)
Equation (12) is a relation between k, and 1 for specified values of AZ/L, K., and m, which must be satisfied simultaneously with the curves of Figs. 1 and 2 if buckling is to occur and (11) to be satisfied. Equation (12) then provides an operating curve, the intersection of which with the A-k. curves provides the critical points of instability. Figure 3 shows the operating curves for AZ/L = O-5, 0.1, for m = 1, K. = 0.1, with the curves for Al/A2 = 0.5, K, = 0.1, n = 1,2. We note that, as the tube becomes long and thin (AZ/L + 0), the operating curve for a particular m moves closer to the 3,axis, hence increasing the critical value of A, making the tube more unstable, as expected.
J.C. PATTERSON
390
REFERENCES 1. E. W. Wilkes, On the stability of a circular tube under end thrust. Q. JI Xlrclr. ,4ppl. Marls. 8. 88.-100 I 1955). 2. A. E. Green and A. J. M. Spencer. The stability of a circularcylinderunder finite extension and torsion. J. AQUA. Ph.% 37, 316-338 (1959). 3. S. Lubkin, Determination of buckling criteria by minimization of total energy, fast. .%Zurll. Sci. New York Univ., Rep IMM-NYU241 (1957). 4. C. B. Sensenig, Instability of thick elastic solids. Contmuns. purr appl. rV~jtl~. 17. 451-491 (19641. 5. Z. Wesolowski, Stability of an elastic. thick-walled sphere uniformly loaded by an external pressure. Archum Mech. SfOSOW. 19. 3-.44 (1967). 6. J. L. Nowinski and M. Shahinpoor. Stability of an elastic circular tube of arbitrary wall thickness subjected to an external pressure. Inr. J. Non-Lifrear Mrclr, 4. 143--i%? (1969). 7. A. S. D. Wang and A. Ertepinar. Stability and vihrattons of elastic thick-walled c?Iindrical and sphericai shells subjected to pressure. Inr. J. Non-Litrear Me&. 7. 539-555 (1972). 8. A. E. Green, R. S. Rivlin and R. T. Shield. General theory of small elastic deformation superposed on finite elastic deformations. Proc. R. SW. A211 (1952).
R&urn&: On examine la stabilit.6 d’un tube n;o Hookeen 'aparoi Gpaisse soumis a une pouss&e et une pression externe en utilisant la thiorie d'une deformation infini&imale superposGe i une dCformation finie. La perte de stabilit; apparait lorsqu'il existe une solution non triviale pour la d&formation On r&oud num~riquement les gquations infinitCsimale. rksultantes et on donne graphiq~ement les conditions critiques pour un flambage ayant un axe de sym&zrie et non symhtrique. On montre les effets de la pression externe. On dgmontre egalement l'effet de mettre une condition simple 'a l'extremit; du tube.
Unter Verwendung der Theorie der infinitesimalen Verformungen, die endlichen Verformungen ijberlagert sind, wird die Stabilitst eines dickwandigen, neohooke'schen Rohrs unter Endschub und Aussendruck untersucht. StabilitEtsverlust tritt aug, wenn eine nuchttriviale Lb;sung fiir die infinitesimale Verformung vorhanden ist. Die sich erqebenden Bleichungen werden n&erisch gel&t und die dritiichen Bedingungen werden graphisch dargestellt fiir den Fall der achsensvrmnetrischen und nichtachsensvmmetrischen Knickuns. Die Einflb'sse des Aussendrucks werden aufgezefgt. Weiterhin wird die Wirkung einer einfachen Endbedingung fiir das Rohr demonstriert.