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Buckling of a non-uniform, long cylindrical shell subjected to external hydrostatic pressure
Engineering Structures 24 (2002) 1027–1034 www.elsevier.com/locate/engstruct
Buckling of a non-uniform, long cylindrical shell subjected to external hydrostatic pressure J. Xue, M.S. Hoo Fatt ∗ Department of Mechanical Engineering, The University of Akron, Akron, OH 44325-3903, USA Received 18 September 2001; received in revised form 10 December 2001; accepted 12 February 2002
Abstract Analytical solutions for elastic buckling of a non-uniform, long cylindrical shell that is subjected to external hydrostatic pressure are presented in this paper. The non-uniform shell has two regions: one with a nominal thickness and the other with reduced thickness. Symmetric and anti-symmetric buckling modes have been found to occur depending on the relative thickness of the two different regions and the angular extent of the section with the reduced thickness. A diagram showing the ranges of relative thickness and angular extent of the section with the reduced thickness is given for the occurrence of symmetric and anti-symmetric modes. In general, the buckling pressure decreases when either the relative thickness or the angular extent of the section with the reduced thickness increase. Finite element solutions for the elastic buckling pressure and modes are also found using ABAQUS and the results from analytical solutions are found to be in close agreement with finite element predictions. 2002 Elsevier Science Ltd. All rights reserved. Keywords: Elastic buckling; Non-uniform, long cylindrical shell; Symmetric and anti-symmetric buckling modes
1. Introduction Buckling of long cylindrical shells subjected to external hydrostatic pressure has been a widely investigated phenomenon. This type of buckling analysis is used to predict collapse failure of long pressure vessels and pipelines when they are subjected to external over-pressure. Timoshenko and Gere [1] gave the classical solution of the buckling pressure for a very long cylindrical shell with uniform thickness under external hydrostatic pressure by considering that the cylinder is in plane strain and solving for the buckling load of a ring. The plane strain assumption that allows one to model shell collapse as ring collapse, is valid when length-to-diameter ratio of the shell is greater than 25. When the length-to-diameter ratio is less than 25, the shell must be treated as one with finite length since the stiffening effect of the end constraints are no longer negligible and will affect the pressure at which buckling occurs. Buckling solutions
Corresponding author. Tel.: +1-330-972-7731; fax: +1-330-9726027. E-mail address: [email protected] (M.S. Hoo Fatt). ∗
for finite length cylindrical shell under uniform lateral pressure and several boundary conditions were derived in Ref. [2], using the energy method, and in Ref. [3], using Flugge’s stability equations in coupled form. The above-mentioned buckling solutions deal with uniform cylindrical shells, i.e., shells with uniform or constant thickness. However, there are circumstances where thin-walled cylindrical shells can have varied thickness, such as a pipe with longitudinal stiffeners or a corroded pipeline where part of thickness has been reduced by corrosion. Few buckling analyses of non-uniform shells exist [4–7]. In these papers, the shells were assumed to be infinitely long so that they can be treated as rings. Furthermore, only symmetric buckling modes were considered in these papers even though variation in thickness of the shell may allow it to also undergo anti-symmetric buckling modes. If a fixed-end, thin arch undergoes anti-symmetric buckling when subjected to uniform pressure load, anti-symmetric buckling modes for a non-uniform, long cylinder can exist [1]. In this paper, exact solutions of elastic buckling pressure for a non-uniform, long cylindrical shell subjected to external hydrostatic pressure are derived. We assume that the shell is in plane strain and examine buckling of
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a ring that represents a transverse cross-section of the cylinder. We also assume the existence of symmetric and anti-symmetric buckling modes by determining appropriate boundary conditions in our buckling analysis. The mode corresponding to the lowest eigenvalue load shall determine the actual collapse mode of the shell. A parametric analysis will show how the geometry of the nonuniform shell influences the buckling mode. Finally, our analytical model will be compared to finite element predictions using ABAQUS.
2. Problem formulation Fig. 1 shows a ring representing the cross-section of the non-uniform, long cylindrical shell. It is composed of two regions with different thickness, and both regions have a common centerline or neutral axis. The ring has a nominal thickness t in Region 2. Region 1, which is subtended by angle 2b, has a thickness of t⫺d. We denote d and b as the depth and angular extent of the region where the thickness is reduced from the nominal value. The material of the non-uniform shell is linear elastic with Young’s modulus E and Poisson’ ratio n. When the non-uniform, long shell is subjected to external hydrostatic pressure, it will undergo elastic buckling. In order to simplify the problem, three assumptions are made: (1) the thickness of the ring is small compared to its mean radius, (2) the deformation of the ring is in the plane of its initial curvature, and (3) the extension of the center line of the non-uniform shell is insignificant. Based on these assumptions, the ring can be considered as two curved beams that can buckle under external pressure loading. If the ring has a reduced thickness only at the inner or outer surface, the neutral surfaces
in the two regions will be different. One of the hoop compressive forces, which is induced by the external pressure loading, will be eccentric with respect to the neutral axis, and this would produce an extra circumferential bending moment at the neutral axis. Such a load/geometric eccentricity has been considered in related work by one of the authors [8]. When(d / t)→0, i.e., uniform shell, the buckling mode of the shell is symmetric about the axis of q ⫽ 0. When (d / t)→1, the section with reduced thickness behaves like a circular arch shell with built-in ends. Timoshenko and Gere [1] showed that the buckling mode of circular arch shell with built-in ends subjected to external pressure is anti-symmetric about the axis of q ⫽ 0. Based on these observations, one can assume that the non-uniform shell can buckle in either a symmetric or an anti-symmetric mode. 3. Eigenvalue problem The differential equation for the radial deflection w for the non-uniform shell with the cross-section shown in Fig. 1, was given as follows [1]: 12(1⫺n2)MR2 d 2w for ⫺bⱕqⱕb, 2 ⫹ w ⫽ ⫺ dq E(t⫺d)3
(1)
12(1⫺n2)MR2 d 2w ⫹w⫽⫺ for bⱕqⱕ2π⫺b, 2 dq Et3 where Et3 / 12(1⫺n2) is the nominal bending rigidity (based on Region 2) of the cylindrical shell. Slope and deflection continuity at q ⫽ b, 2π⫺b are used to guarantee that the deflection is continuous and smooth. The circumferential bending moment M in the nonuniform cylindrical shell is M ⫽ pRw,
(2)
where p is the external hydrostatic pressure and R is the mean radius of the shell. Substituting Eq. (2) into Eq. (1) yields
冦
d2w ⫹ k21w ⫽ 0 for ⫺bⱕqⱕb dq2
d2w ⫹ k22w ⫽ 0 for bⱕqⱕ2π⫺b dq2
(3)
where
Fig. 1. Cross-section of non-uniform cylinder.
k21 ⫽ 1 ⫹
12(1⫺n2)pR3 , E(t⫺d)3
(4)
k22 ⫽ 1 ⫹
12(1⫺n2)pR3 Et3
(5)
and k1 and k2 are buckling load parameters and correspond to a given pressure p.
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6. Anti-symmetric buckling mode solution
The general solution for Eq. (3) is w⫽
再
Asink1q ⫹ Bcosk1q for ⫺bⱕqⱕb
Csink2q ⫹ Dcosk2q for bⱕqⱕ2π⫺b
1029
(6)
where A, B, C and D are unknown constants, which are determined by satisfying the boundary conditions. 4. Boundary conditions The boundary conditions for the symmetric and antisymmetric modes are different. Both modes require that the displacement and slope are continuous at q ⫽ b, 2π⫺b, but (a) the slope is zero at q ⫽ 0, π for the symmetric buckling and (b) the bending moment is zero at q ⫽ 0, π for anti-symmetric buckling. The actual buckling pressure and mode are obtained by choosing the lower buckling pressure when symmetric and anti-symmetric buckling modes are assumed independently.
The four boundary conditions are d 2w ⫽ 0 at q ⫽ 0,π dq2
and deflection and slope continuity at q ⫽ b. Eq. (13) is satisfied if B ⫽ 0 and C ⫽ ⫺D(cosk2π / sink2π). Substituting B and C into Eq. (6) gives the deflection equation
冦
Asink1q
w⫽
D
for ⫺bⱕqⱕb
. sink2(π⫺q) for bⱕqⱕ2π⫺b sink2π
(14)
The deflection and slope continuity conditions at q ⫽ b give Asink1b ⫽ D
5. Symmetric buckling mode solution
(13)
sink2(π⫺b) , sink2π cosk2(π⫺b) . sink2π
(15)
The four boundary conditions which are required to determine the four unknown constant A, B, C and D are
Ak1cosk1b ⫽ ⫺Dk2
(dw / dq) ⫽ 0 at q ⫽ 0,π
A non-trivial solution for Eqs. (15) and (16) requires that the determinant of these equations becomes zero:
(7)
and deflection and slope continuity at q ⫽ b. Eq. (7) is satisfied by taking A ⫽ 0 and C ⫽ Dtank2π. Substituting A and C into Eq. (6) yields
w⫽
冦
. cosk2(π⫺q) D for bⱕqⱕ2p⫺b cosk2π
(8)
Furthermore, deflection and slope continuity at q ⫽ b require Bcosk1b ⫽ D
cosk2(π⫺b) , cosk2π
⫺Bk1sink1b ⫽ Dk2
sink2(π⫺b) . cosk2π
(10)
(11)
The above equation is the eigenfunction to determine the critical value of uniform pressure for symmetric elastic buckling. By solving Eqs. (4), (5) and (11), the critical value of the uniform external pressure for symmetric elastic buckling is found as follows: pcr ⫽
冉冊
(k22⫺1)E t 3 , 12(1⫺v2) R
where k2 is the eigenvalue obtained from Eq. (11).
⫺k1tank2(π⫺b) ⫽ k2tank1b.
(17)
Eq. (17) is the eigenfunction for the anti-symmetric elastic buckling of the non-uniform cylinder. Note that Eqs. (17) and (11) are very similar, but not the same. The buckling pressure for anti-symmetric mode is still given by Eq. (12), but k2 is the eigenvalue of Eq. (17) and not the eigenvalue of Eq. (11).
(9)
Substituting Eq. (9) into Eq. (10) yields ⫺k1tank1b ⫽ k2tank2(π⫺b).
cosk2(π⫺b) sink2(π⫺b) ⫹ k1cosk1b ⫽0 sink2π sink2π
After some simplification, the above equation becomes
for ⫺bⱕqⱕb
Bcosk1q
k2sink1b
(16)
(12)
7. Elastic buckling of a circular arch with built-in ends The solutions for the elastic buckling of the non-uniform shell should converge to the solution of a circular arch shell with built-in ends as (d / t)→1. This is because when (d / t)→1, the stiffness of Region 2 is large compared to the stiffness of Region 1. Region 1, therefore, behaves like a uniformly compressed circular shell with built-in ends at q ⫽ b, 2π⫺b. The general solutions for the elastic buckling load and the mode shape of a circular arch with built-in ends subjected to uniform external pressure are given in Ref. [1] as ktanbcotkb ⫽ 1
(18)
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and
冉
冊
sinb sinkq ⫹ sinq , w ⫽ wa ⫺ sinkb
After some algebraic calculation, the radial deflection is written as following: (19)
where wa is the amplitude of the buckling mode and k is the buckling load parameter that is related to the buckling pressure pa as follows: pcr ⫽
Et3 (k2⫺1). 12R3(1⫺n2)
冉 冊 d t
3
⫽ (k22⫺1).
(21)
When (d / t)→1, the left-hand side of Eq. (21) approaches zero. Thus k2 ⫽ 1. Substituting k2 ⫽ 1 into Eq. (17) gives ⫺k1tan(π⫺b) ⫽ tank1b.
(22)
Using that tan(π⫺b) ⫽ ⫺tanb and cotk1b ⫽ 1 / tank1b yields k1tanbcotk1b ⫽ 1.
(23)
Eq. (23) is the exact eigenfunction of an arch with builtin ends, which is given in Eq. (18). This proves that as (d / t)→1, for any given value b, k1 ⫽ k, and k2 ⫽ 1. Substituting Eq. (15) into (14) gives the deflection profile of anti-symmetric buckling mode for a non-uniform cylinder. When (d / t)→1, k1→k and k2→1. The deflection equation is given as the following:
冦
Asinkq
w⫽
A
for ⫺bⱕqⱕb
sinkb sinq for bⱕqⱕ2π⫺b sinb
(24)
where A is constant. The clamped boundary condition can be composed by subtracting A(sinkb / sinb)sinq from Eq. (24):
冦
sinkb Asinkq⫺A sinq for ⫺bⱕqⱕb sinb w⫽ 0 for bⱕqⱕ2π⫺b
0
(20)
The buckling modes and the values of k1 and k2 of the non-uniform shell must converge to that of the circular arch shell with built-in ends. By comparing Eq. (8) to (19), one can immediately tell that as d/t approaches unity, the symmetric buckling mode cannot converge to that of a circular arch shell with built-in ends. As to the anti-symmetric buckling, the following mathematical proof shows that the values of k1 and k2 and the buckling mode converge to those of the circular arch shell with built-in ends. From Eqs. (4) and (5), one obtains the following equation: (k21⫺1) 1⫺
冦
(25)
冉
Asinkb sinb sinkq ⫹ sinq ⫺ ⫺ sinb sinkb w⫽
冊
for ⫺bⱕqⱕb
(26)
for bⱕqⱕ2π⫺b
If one chooses wa ⫽ ⫺(Asinkb / sinb) to be the amplitude of buckling mode, then one finds that the buckling mode for a non-uniform shell when (d / t)→1 is exactly the same as that of an arch with built-in ends.
8. Normalized buckling pressure Define a normalized pressure h ⫽ (pcr / pe), where pe ⫽ (Et3 / 4(1⫺n2)R3) is the elastic buckling pressure of a uniform shell with thickness t and radius R. The normalized bucking pressure with respect to k2 is given as h⫽
(k22⫺1) , 3
(27)
where k2 is determined either by Eq. (11) or (17). It can be seen from the Eqs. (11) and (17) that both the symmetric and the anti-symmetric elastic buckling mode are dependent on the geometry of the shell. The material property has no influence on the buckling mode. However, the buckling pressure does depend on material properties. The values of k1, k2 and h are calculated separately for symmetric and anti-symmetric buckling mode. Cases of different d/t ratio are calculated for b ⫽ 30, 45, 60, 90, 120, 135, 150° and the results of normalized buckling pressure are shown in Fig. 2. Two observations can be made from Fig. 2: 1. Symmetric and anti-symmetric modes depend on the ranges of b and d/t. For instance, when b ⫽ 30° and 0 ⬍ (d / t)ⱕ0.3, the normalized buckling pressure for the symmetric mode is lower than that for the antisymmetric mode. Thus the buckling mode in this region is symmetric. However, when b ⫽ 30°and 0 ⬍ (d / t)ⱕ0.9, the buckling mode is anti-symmetric. 2. One can see that for bⱕ90° and (d / t)→1, the normalized buckling pressure predicted by symmetric buckling mode is always lower than that predicted by anti-symmetric buckling mode. This should not be the case because only the solution from the anti-symmetric buckling mode converges to the solution of the arch with fixed ends. Thus we must modify the solution for the buckling pressure when (d / t)→1.
J. Xue, M.S. Hoo Fatt / Engineering Structures 24 (2002) 1027–1034
Fig. 2.
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Normalized buckling pressure obtained from symmetric mode (———) and anti-symmetric mode (-----).
9. Critical values of (d / t)∗,(d / t)∗∗and(d / t)∗∗∗ Fig. 3, which shows the buckling mode for different ranges of d/t and b, was obtained from a MATLAB [9] program. Each region shown in Fig. 3 is defined by critical values of (d / t):(d / t)∗ is a curve describing when the buckling mode changes from a symmetric mode to an anti-symmetric mode; (d / t)∗∗is a curve describing when the buckling mode changes from an anti-symmetric mode to a symmetric mode; and (d / t)∗∗∗is a curve describing when the buckling mode changes from a symmetric mode to an anti-symmetric mode. The equations for (d / t)∗ and (d / t)∗∗ are obtained by setting the buckling pressure predicted from a symmetric mode equal to that predicted by an anti-symmetric mode. The solution
for (d / t)∗∗∗ is also obtained by setting the buckling pressures for symmetric and anti-symmetric modes equal only when b ⬎ 90°. When bⱕ90° and (d / t)→1, our theory predicts a symmetric buckling mode even though it should be anti-symmetric, i.e., approach the buckling mode of a built-in arch. Thus a separate criterion should be established for the solution of (d / t)∗∗∗whenbⱕ90°. 10. Modified solution when bⱕ90° As argued earlier, the non-uniform shell should behave like a circular arch with built-in ends when (d / t)→1. The question is at what value should (d / t) be such that we can treat the non-uniform shell as built-in circular arch?
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Fig. 3.
Variation of the critical values of (d / t)∗,(d /t)∗∗and(d / t)∗∗∗ with respect to b.
We know that when the non-uniform shell behaves like a circular arch with built-in ends, the deflection in Region 2 should tend to zero. From Eqs. (14) and (15), the deflection for anti-symmetric buckling is
冦
for ⫺bⱕqⱕb
Asink1b sink2(π⫺q) for bⱕqⱕ2π⫺b sink2(π⫺b)
(28)
冦
⫺
syp ⫽ 410MPa.
Poisson’s ratio:
n ⫽ 0.3
Shell radius:
冋
Asink1b sink2(π⫺b) sink1q ⫹ sink2(π⫺q) ⫺ sink2(π⫺b) sink1b
册
for ⫺bⱕqⱕb
(29)
for bⱕqⱕ2π⫺b
0
We proved previously that when (d / t)→1, k1→k and k2→1 or that the buckling mode approaches that of a built-in arch. Comparing Eq. (26) to (29), one can see that the important term that causes the buckling mode of Eq. (29) to converge to that of Eq. (26) is the amplitude of sink1q in the square parentheses, i.e., the term ⫺(sink2(π⫺b) / sink1b). A criterion for determining (d / t)∗∗∗ is set by the following condition: (sinb / sinkb)⫺(sink2(π⫺b) / sink1b) ⱕ5% sinb / sinkb
|
Yield strength:
The geometry parameters of the non-uniform shell are
Subtracting (Asink1b / sink2(π⫺b))sink2(π⫺q) from Eq (28) gives the following:
w⫽
Consider a non-uniform shell made from steel with following properties: Young’s modulus: E ⫽ 207GPa
Asink1q
w⫽
11. Finite element analysis
|
(30)
where the 5% difference in the magnitude of the two terms is chosen arbitrarily. The buckling pressure when (d / t) ⬎ (d / t)∗∗∗ is obtained from the anti-symmetric mode solution.
R ⫽ 228.6mm
Shell thickness: t ⫽ 18.9mm
.
The ratio of thickness reduction d/t and angular extension b to be considered is (d / t) ⫽ 0.1,0.3, 0.5, 0.7 and 0.9 for b ⫽ 30,45,60,90,120,135and150°, respectively. The buckling pressure of the non-uniform shell is obtained from a ring, which has the same cross-section as the non-uniform shell, by replacing Young’s modulus E with E / (1⫺n2) in ABAQUS [10] code. The ring is discretized into a mesh with two-node, isoparametric, linear, Timoshenko beam elements. The mesh has 72 elements consisting of two parts: one part corresponds to Region 1 in Fig. 1, which has the reduced thickness; and the second part corresponds to Region 2, which has the nominal thickness. Three boundary conditions are imposed at the mid-point of Region 2: Degree 1 is zero, which confines the movement in the local x-direction; Degree 2 is zero, which confines the movement in the local y-direction; Degree 6 is zero, which restrict the ring from rotating about the z-axis. The reason that the above three boundary conditions are used is because (1) the minimum number of constraints or boundary conditions to prevent rigid body motion of the ring is three and (2) constraining only one point allows the structure to choose any buckling mode, whether symmetric or anti-
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ling mode predicted by analytical method are almost identical to that obtained from the finite element analysis. Both the analytical method and finite element analysis predict the same buckling mode variation. The buckling mode changes specified by (d / t)∗, ∗∗ ∗∗∗ in Fig. 3 also coincide with the buck(d / t) and(d / t) ling modes changes shown in Figs. 5–8. 13. Conclusion
Fig. 4. Comparison of analytical and FEA predictions for the variation of the elastic buckling pressure with d/t.
symmetric. Any other boundary conditions would overconstrain the structure and result in higher buckling pressures.
12. Comparison between analytical and FEA results The analytical solutions of the elastic buckling pressure as it varies with thickness reduction d/t for several b are plotted in Fig. 4. The analytical solutions and finite element predictions are within 5% of each other. Some of the buckling modes corresponding to these solutions are shown in Figs. 5–8. Buckling modes from finite element analysis are also plotted in Figs. 5–8 to compare with the analytical solutions. One can see that the buck-
Exact solutions for elastic buckling of a non-uniform, long cylindrical shell subjected to external hydrostatic pressure were derived in this paper. The non-uniform shell has two regions: one with a nominal thickness and the other with reduced thickness. The buckling pressure was found to decrease with the increasing value of the thickness reduction and angular extension. Symmetric and anti-symmetric buckling modes occurred depending on the relative thickness and the angular extent of the two sections. Criteria were established to determine ranges of relative thickness and angular extent that correspond to symmetric mode and anti-symmetric modes. Furthermore, the buckling modes and pressures that were predicted from ABAQUS were almost identical to the analytical solutions. The analytical predictions and finite element results for the bucking pressure were within 5% of each other. Solutions from this research can serve as a good point of reference in the offshore industry. For example, they can be used to address buckle propagation in corroded pipelines, where the pipelines are often treated as infinitely long cylinder and the propagation pressure is usu-
Fig. 5.
Buckling modes for b ⫽ 30°.
Fig. 6.
Buckling modes for b ⫽ 45°.
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Fig. 7.
Buckling modes for b ⫽ 90°.
Fig. 8.
Buckling modes for b ⫽ 135°.
ally calculated from a ring [11–13]. Our analysis can provide valuable technical information for monitoring pipeline performance in the offshore or oil industry, where corrosion is a threat. Technicians can design and apply suitable equipment, such as buckle arrestors, to prevent catastrophic pipeline failure when their buckling strength has been compromised by corrosion.
Acknowledgements The work reported herein was supported by the Department of Mechanical Engineering and a Faculty Research Grant from The University of Akron.
References [1] Timoshenko SP, Gere JM. Theory of elastic stability. New York: McGraw-Hill, 1961. [2] Pinna R, Ronalds BF. Hydrostatic buckling of shells with various boundary conditions. J Constr Steel Res 2000;56(1):1–16. [3] Vodenitcharova T, Ansourian P. Buckling of circular cylindrical shells subject to uniform lateral pressure. Eng Struct 1996;18(8):604–14.
[4] Bai Y, Hauch S. Analytical collapse of corroded pipes. In: Proceedings of the Eighth International Conference on Offshore and Polar Engineering. Montreal, Canada, 24–29 May 1998;2:182-8. [5] Hauch S, Bai Y. Use of finite element methods for the determination of local buckling strength. In: Proceedings of the 1998 International Conference on Offshore Mechanics and Arctic Engineering. Lisbon, Portugal; 5–9 July 1998. [6] Bai Y, Hauch S, Jenses JC. Local buckling and plastic collapse of corroded pipes with yield anisotropy. In: Proceedings of the Ninth International Offshore and Polar Engineering Conference. Golden, CO, 1999;2:74–81. [7] Hoo Fatt MS. Elastic-plastic collapse of non-uniform cylindrical shells subjected to uniform external pressure. Thin-Walled Struct 1999;35:117–37. [8] Xue J. Structural failure of non-uniform, infinitely long cylindrical shells subjected to uniform external pressure. Doctoral Dissertation, The University of Akron; May 2002. [9] MATLAB. The language of technical computing, Version 5.3.0.10183(R11); 21 January 1999. [10] HKS. ABAQUS user’s manual: theoretical manual, ABAQUS post manual and example problem, Version 4.9. Hibbit, Karlsson and Sorensen, Inc; 1999. [11] Palmer AC, Martin JH. Buckle propagation in submarine pipeline. Nature 1975;254:46–8. [12] Chater E, Hutchinson JW. On the propagation of bulges and buckles. J Appl Mech 1984;51:269–77. [13] Kyriakides S, Yeh MK, Roach D. On the determination of the propagation pressure of long circular tubes. Trans ASME J Press Vess Technol 1984;106:150–9.
Buckling of a non-uniform, long cylindrical shell subjected to external hydrostatic pressure J. Xue, M.S. Hoo Fatt ∗ Department of Mechanical Engineering, The University of Akron, Akron, OH 44325-3903, USA Received 18 September 2001; received in revised form 10 December 2001; accepted 12 February 2002
Abstract Analytical solutions for elastic buckling of a non-uniform, long cylindrical shell that is subjected to external hydrostatic pressure are presented in this paper. The non-uniform shell has two regions: one with a nominal thickness and the other with reduced thickness. Symmetric and anti-symmetric buckling modes have been found to occur depending on the relative thickness of the two different regions and the angular extent of the section with the reduced thickness. A diagram showing the ranges of relative thickness and angular extent of the section with the reduced thickness is given for the occurrence of symmetric and anti-symmetric modes. In general, the buckling pressure decreases when either the relative thickness or the angular extent of the section with the reduced thickness increase. Finite element solutions for the elastic buckling pressure and modes are also found using ABAQUS and the results from analytical solutions are found to be in close agreement with finite element predictions. 2002 Elsevier Science Ltd. All rights reserved. Keywords: Elastic buckling; Non-uniform, long cylindrical shell; Symmetric and anti-symmetric buckling modes
1. Introduction Buckling of long cylindrical shells subjected to external hydrostatic pressure has been a widely investigated phenomenon. This type of buckling analysis is used to predict collapse failure of long pressure vessels and pipelines when they are subjected to external over-pressure. Timoshenko and Gere [1] gave the classical solution of the buckling pressure for a very long cylindrical shell with uniform thickness under external hydrostatic pressure by considering that the cylinder is in plane strain and solving for the buckling load of a ring. The plane strain assumption that allows one to model shell collapse as ring collapse, is valid when length-to-diameter ratio of the shell is greater than 25. When the length-to-diameter ratio is less than 25, the shell must be treated as one with finite length since the stiffening effect of the end constraints are no longer negligible and will affect the pressure at which buckling occurs. Buckling solutions
Corresponding author. Tel.: +1-330-972-7731; fax: +1-330-9726027. E-mail address: [email protected] (M.S. Hoo Fatt). ∗
for finite length cylindrical shell under uniform lateral pressure and several boundary conditions were derived in Ref. [2], using the energy method, and in Ref. [3], using Flugge’s stability equations in coupled form. The above-mentioned buckling solutions deal with uniform cylindrical shells, i.e., shells with uniform or constant thickness. However, there are circumstances where thin-walled cylindrical shells can have varied thickness, such as a pipe with longitudinal stiffeners or a corroded pipeline where part of thickness has been reduced by corrosion. Few buckling analyses of non-uniform shells exist [4–7]. In these papers, the shells were assumed to be infinitely long so that they can be treated as rings. Furthermore, only symmetric buckling modes were considered in these papers even though variation in thickness of the shell may allow it to also undergo anti-symmetric buckling modes. If a fixed-end, thin arch undergoes anti-symmetric buckling when subjected to uniform pressure load, anti-symmetric buckling modes for a non-uniform, long cylinder can exist [1]. In this paper, exact solutions of elastic buckling pressure for a non-uniform, long cylindrical shell subjected to external hydrostatic pressure are derived. We assume that the shell is in plane strain and examine buckling of
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a ring that represents a transverse cross-section of the cylinder. We also assume the existence of symmetric and anti-symmetric buckling modes by determining appropriate boundary conditions in our buckling analysis. The mode corresponding to the lowest eigenvalue load shall determine the actual collapse mode of the shell. A parametric analysis will show how the geometry of the nonuniform shell influences the buckling mode. Finally, our analytical model will be compared to finite element predictions using ABAQUS.
2. Problem formulation Fig. 1 shows a ring representing the cross-section of the non-uniform, long cylindrical shell. It is composed of two regions with different thickness, and both regions have a common centerline or neutral axis. The ring has a nominal thickness t in Region 2. Region 1, which is subtended by angle 2b, has a thickness of t⫺d. We denote d and b as the depth and angular extent of the region where the thickness is reduced from the nominal value. The material of the non-uniform shell is linear elastic with Young’s modulus E and Poisson’ ratio n. When the non-uniform, long shell is subjected to external hydrostatic pressure, it will undergo elastic buckling. In order to simplify the problem, three assumptions are made: (1) the thickness of the ring is small compared to its mean radius, (2) the deformation of the ring is in the plane of its initial curvature, and (3) the extension of the center line of the non-uniform shell is insignificant. Based on these assumptions, the ring can be considered as two curved beams that can buckle under external pressure loading. If the ring has a reduced thickness only at the inner or outer surface, the neutral surfaces
in the two regions will be different. One of the hoop compressive forces, which is induced by the external pressure loading, will be eccentric with respect to the neutral axis, and this would produce an extra circumferential bending moment at the neutral axis. Such a load/geometric eccentricity has been considered in related work by one of the authors [8]. When(d / t)→0, i.e., uniform shell, the buckling mode of the shell is symmetric about the axis of q ⫽ 0. When (d / t)→1, the section with reduced thickness behaves like a circular arch shell with built-in ends. Timoshenko and Gere [1] showed that the buckling mode of circular arch shell with built-in ends subjected to external pressure is anti-symmetric about the axis of q ⫽ 0. Based on these observations, one can assume that the non-uniform shell can buckle in either a symmetric or an anti-symmetric mode. 3. Eigenvalue problem The differential equation for the radial deflection w for the non-uniform shell with the cross-section shown in Fig. 1, was given as follows [1]: 12(1⫺n2)MR2 d 2w for ⫺bⱕqⱕb, 2 ⫹ w ⫽ ⫺ dq E(t⫺d)3
(1)
12(1⫺n2)MR2 d 2w ⫹w⫽⫺ for bⱕqⱕ2π⫺b, 2 dq Et3 where Et3 / 12(1⫺n2) is the nominal bending rigidity (based on Region 2) of the cylindrical shell. Slope and deflection continuity at q ⫽ b, 2π⫺b are used to guarantee that the deflection is continuous and smooth. The circumferential bending moment M in the nonuniform cylindrical shell is M ⫽ pRw,
(2)
where p is the external hydrostatic pressure and R is the mean radius of the shell. Substituting Eq. (2) into Eq. (1) yields
冦
d2w ⫹ k21w ⫽ 0 for ⫺bⱕqⱕb dq2
d2w ⫹ k22w ⫽ 0 for bⱕqⱕ2π⫺b dq2
(3)
where
Fig. 1. Cross-section of non-uniform cylinder.
k21 ⫽ 1 ⫹
12(1⫺n2)pR3 , E(t⫺d)3
(4)
k22 ⫽ 1 ⫹
12(1⫺n2)pR3 Et3
(5)
and k1 and k2 are buckling load parameters and correspond to a given pressure p.
J. Xue, M.S. Hoo Fatt / Engineering Structures 24 (2002) 1027–1034
6. Anti-symmetric buckling mode solution
The general solution for Eq. (3) is w⫽
再
Asink1q ⫹ Bcosk1q for ⫺bⱕqⱕb
Csink2q ⫹ Dcosk2q for bⱕqⱕ2π⫺b
1029
(6)
where A, B, C and D are unknown constants, which are determined by satisfying the boundary conditions. 4. Boundary conditions The boundary conditions for the symmetric and antisymmetric modes are different. Both modes require that the displacement and slope are continuous at q ⫽ b, 2π⫺b, but (a) the slope is zero at q ⫽ 0, π for the symmetric buckling and (b) the bending moment is zero at q ⫽ 0, π for anti-symmetric buckling. The actual buckling pressure and mode are obtained by choosing the lower buckling pressure when symmetric and anti-symmetric buckling modes are assumed independently.
The four boundary conditions are d 2w ⫽ 0 at q ⫽ 0,π dq2
and deflection and slope continuity at q ⫽ b. Eq. (13) is satisfied if B ⫽ 0 and C ⫽ ⫺D(cosk2π / sink2π). Substituting B and C into Eq. (6) gives the deflection equation
冦
Asink1q
w⫽
D
for ⫺bⱕqⱕb
. sink2(π⫺q) for bⱕqⱕ2π⫺b sink2π
(14)
The deflection and slope continuity conditions at q ⫽ b give Asink1b ⫽ D
5. Symmetric buckling mode solution
(13)
sink2(π⫺b) , sink2π cosk2(π⫺b) . sink2π
(15)
The four boundary conditions which are required to determine the four unknown constant A, B, C and D are
Ak1cosk1b ⫽ ⫺Dk2
(dw / dq) ⫽ 0 at q ⫽ 0,π
A non-trivial solution for Eqs. (15) and (16) requires that the determinant of these equations becomes zero:
(7)
and deflection and slope continuity at q ⫽ b. Eq. (7) is satisfied by taking A ⫽ 0 and C ⫽ Dtank2π. Substituting A and C into Eq. (6) yields
w⫽
冦
. cosk2(π⫺q) D for bⱕqⱕ2p⫺b cosk2π
(8)
Furthermore, deflection and slope continuity at q ⫽ b require Bcosk1b ⫽ D
cosk2(π⫺b) , cosk2π
⫺Bk1sink1b ⫽ Dk2
sink2(π⫺b) . cosk2π
(10)
(11)
The above equation is the eigenfunction to determine the critical value of uniform pressure for symmetric elastic buckling. By solving Eqs. (4), (5) and (11), the critical value of the uniform external pressure for symmetric elastic buckling is found as follows: pcr ⫽
冉冊
(k22⫺1)E t 3 , 12(1⫺v2) R
where k2 is the eigenvalue obtained from Eq. (11).
⫺k1tank2(π⫺b) ⫽ k2tank1b.
(17)
Eq. (17) is the eigenfunction for the anti-symmetric elastic buckling of the non-uniform cylinder. Note that Eqs. (17) and (11) are very similar, but not the same. The buckling pressure for anti-symmetric mode is still given by Eq. (12), but k2 is the eigenvalue of Eq. (17) and not the eigenvalue of Eq. (11).
(9)
Substituting Eq. (9) into Eq. (10) yields ⫺k1tank1b ⫽ k2tank2(π⫺b).
cosk2(π⫺b) sink2(π⫺b) ⫹ k1cosk1b ⫽0 sink2π sink2π
After some simplification, the above equation becomes
for ⫺bⱕqⱕb
Bcosk1q
k2sink1b
(16)
(12)
7. Elastic buckling of a circular arch with built-in ends The solutions for the elastic buckling of the non-uniform shell should converge to the solution of a circular arch shell with built-in ends as (d / t)→1. This is because when (d / t)→1, the stiffness of Region 2 is large compared to the stiffness of Region 1. Region 1, therefore, behaves like a uniformly compressed circular shell with built-in ends at q ⫽ b, 2π⫺b. The general solutions for the elastic buckling load and the mode shape of a circular arch with built-in ends subjected to uniform external pressure are given in Ref. [1] as ktanbcotkb ⫽ 1
(18)
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and
冉
冊
sinb sinkq ⫹ sinq , w ⫽ wa ⫺ sinkb
After some algebraic calculation, the radial deflection is written as following: (19)
where wa is the amplitude of the buckling mode and k is the buckling load parameter that is related to the buckling pressure pa as follows: pcr ⫽
Et3 (k2⫺1). 12R3(1⫺n2)
冉 冊 d t
3
⫽ (k22⫺1).
(21)
When (d / t)→1, the left-hand side of Eq. (21) approaches zero. Thus k2 ⫽ 1. Substituting k2 ⫽ 1 into Eq. (17) gives ⫺k1tan(π⫺b) ⫽ tank1b.
(22)
Using that tan(π⫺b) ⫽ ⫺tanb and cotk1b ⫽ 1 / tank1b yields k1tanbcotk1b ⫽ 1.
(23)
Eq. (23) is the exact eigenfunction of an arch with builtin ends, which is given in Eq. (18). This proves that as (d / t)→1, for any given value b, k1 ⫽ k, and k2 ⫽ 1. Substituting Eq. (15) into (14) gives the deflection profile of anti-symmetric buckling mode for a non-uniform cylinder. When (d / t)→1, k1→k and k2→1. The deflection equation is given as the following:
冦
Asinkq
w⫽
A
for ⫺bⱕqⱕb
sinkb sinq for bⱕqⱕ2π⫺b sinb
(24)
where A is constant. The clamped boundary condition can be composed by subtracting A(sinkb / sinb)sinq from Eq. (24):
冦
sinkb Asinkq⫺A sinq for ⫺bⱕqⱕb sinb w⫽ 0 for bⱕqⱕ2π⫺b
0
(20)
The buckling modes and the values of k1 and k2 of the non-uniform shell must converge to that of the circular arch shell with built-in ends. By comparing Eq. (8) to (19), one can immediately tell that as d/t approaches unity, the symmetric buckling mode cannot converge to that of a circular arch shell with built-in ends. As to the anti-symmetric buckling, the following mathematical proof shows that the values of k1 and k2 and the buckling mode converge to those of the circular arch shell with built-in ends. From Eqs. (4) and (5), one obtains the following equation: (k21⫺1) 1⫺
冦
(25)
冉
Asinkb sinb sinkq ⫹ sinq ⫺ ⫺ sinb sinkb w⫽
冊
for ⫺bⱕqⱕb
(26)
for bⱕqⱕ2π⫺b
If one chooses wa ⫽ ⫺(Asinkb / sinb) to be the amplitude of buckling mode, then one finds that the buckling mode for a non-uniform shell when (d / t)→1 is exactly the same as that of an arch with built-in ends.
8. Normalized buckling pressure Define a normalized pressure h ⫽ (pcr / pe), where pe ⫽ (Et3 / 4(1⫺n2)R3) is the elastic buckling pressure of a uniform shell with thickness t and radius R. The normalized bucking pressure with respect to k2 is given as h⫽
(k22⫺1) , 3
(27)
where k2 is determined either by Eq. (11) or (17). It can be seen from the Eqs. (11) and (17) that both the symmetric and the anti-symmetric elastic buckling mode are dependent on the geometry of the shell. The material property has no influence on the buckling mode. However, the buckling pressure does depend on material properties. The values of k1, k2 and h are calculated separately for symmetric and anti-symmetric buckling mode. Cases of different d/t ratio are calculated for b ⫽ 30, 45, 60, 90, 120, 135, 150° and the results of normalized buckling pressure are shown in Fig. 2. Two observations can be made from Fig. 2: 1. Symmetric and anti-symmetric modes depend on the ranges of b and d/t. For instance, when b ⫽ 30° and 0 ⬍ (d / t)ⱕ0.3, the normalized buckling pressure for the symmetric mode is lower than that for the antisymmetric mode. Thus the buckling mode in this region is symmetric. However, when b ⫽ 30°and 0 ⬍ (d / t)ⱕ0.9, the buckling mode is anti-symmetric. 2. One can see that for bⱕ90° and (d / t)→1, the normalized buckling pressure predicted by symmetric buckling mode is always lower than that predicted by anti-symmetric buckling mode. This should not be the case because only the solution from the anti-symmetric buckling mode converges to the solution of the arch with fixed ends. Thus we must modify the solution for the buckling pressure when (d / t)→1.
J. Xue, M.S. Hoo Fatt / Engineering Structures 24 (2002) 1027–1034
Fig. 2.
1031
Normalized buckling pressure obtained from symmetric mode (———) and anti-symmetric mode (-----).
9. Critical values of (d / t)∗,(d / t)∗∗and(d / t)∗∗∗ Fig. 3, which shows the buckling mode for different ranges of d/t and b, was obtained from a MATLAB [9] program. Each region shown in Fig. 3 is defined by critical values of (d / t):(d / t)∗ is a curve describing when the buckling mode changes from a symmetric mode to an anti-symmetric mode; (d / t)∗∗is a curve describing when the buckling mode changes from an anti-symmetric mode to a symmetric mode; and (d / t)∗∗∗is a curve describing when the buckling mode changes from a symmetric mode to an anti-symmetric mode. The equations for (d / t)∗ and (d / t)∗∗ are obtained by setting the buckling pressure predicted from a symmetric mode equal to that predicted by an anti-symmetric mode. The solution
for (d / t)∗∗∗ is also obtained by setting the buckling pressures for symmetric and anti-symmetric modes equal only when b ⬎ 90°. When bⱕ90° and (d / t)→1, our theory predicts a symmetric buckling mode even though it should be anti-symmetric, i.e., approach the buckling mode of a built-in arch. Thus a separate criterion should be established for the solution of (d / t)∗∗∗whenbⱕ90°. 10. Modified solution when bⱕ90° As argued earlier, the non-uniform shell should behave like a circular arch with built-in ends when (d / t)→1. The question is at what value should (d / t) be such that we can treat the non-uniform shell as built-in circular arch?
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J. Xue, M.S. Hoo Fatt / Engineering Structures 24 (2002) 1027–1034
Fig. 3.
Variation of the critical values of (d / t)∗,(d /t)∗∗and(d / t)∗∗∗ with respect to b.
We know that when the non-uniform shell behaves like a circular arch with built-in ends, the deflection in Region 2 should tend to zero. From Eqs. (14) and (15), the deflection for anti-symmetric buckling is
冦
for ⫺bⱕqⱕb
Asink1b sink2(π⫺q) for bⱕqⱕ2π⫺b sink2(π⫺b)
(28)
冦
⫺
syp ⫽ 410MPa.
Poisson’s ratio:
n ⫽ 0.3
Shell radius:
冋
Asink1b sink2(π⫺b) sink1q ⫹ sink2(π⫺q) ⫺ sink2(π⫺b) sink1b
册
for ⫺bⱕqⱕb
(29)
for bⱕqⱕ2π⫺b
0
We proved previously that when (d / t)→1, k1→k and k2→1 or that the buckling mode approaches that of a built-in arch. Comparing Eq. (26) to (29), one can see that the important term that causes the buckling mode of Eq. (29) to converge to that of Eq. (26) is the amplitude of sink1q in the square parentheses, i.e., the term ⫺(sink2(π⫺b) / sink1b). A criterion for determining (d / t)∗∗∗ is set by the following condition: (sinb / sinkb)⫺(sink2(π⫺b) / sink1b) ⱕ5% sinb / sinkb
|
Yield strength:
The geometry parameters of the non-uniform shell are
Subtracting (Asink1b / sink2(π⫺b))sink2(π⫺q) from Eq (28) gives the following:
w⫽
Consider a non-uniform shell made from steel with following properties: Young’s modulus: E ⫽ 207GPa
Asink1q
w⫽
11. Finite element analysis
|
(30)
where the 5% difference in the magnitude of the two terms is chosen arbitrarily. The buckling pressure when (d / t) ⬎ (d / t)∗∗∗ is obtained from the anti-symmetric mode solution.
R ⫽ 228.6mm
Shell thickness: t ⫽ 18.9mm
.
The ratio of thickness reduction d/t and angular extension b to be considered is (d / t) ⫽ 0.1,0.3, 0.5, 0.7 and 0.9 for b ⫽ 30,45,60,90,120,135and150°, respectively. The buckling pressure of the non-uniform shell is obtained from a ring, which has the same cross-section as the non-uniform shell, by replacing Young’s modulus E with E / (1⫺n2) in ABAQUS [10] code. The ring is discretized into a mesh with two-node, isoparametric, linear, Timoshenko beam elements. The mesh has 72 elements consisting of two parts: one part corresponds to Region 1 in Fig. 1, which has the reduced thickness; and the second part corresponds to Region 2, which has the nominal thickness. Three boundary conditions are imposed at the mid-point of Region 2: Degree 1 is zero, which confines the movement in the local x-direction; Degree 2 is zero, which confines the movement in the local y-direction; Degree 6 is zero, which restrict the ring from rotating about the z-axis. The reason that the above three boundary conditions are used is because (1) the minimum number of constraints or boundary conditions to prevent rigid body motion of the ring is three and (2) constraining only one point allows the structure to choose any buckling mode, whether symmetric or anti-
J. Xue, M.S. Hoo Fatt / Engineering Structures 24 (2002) 1027–1034
1033
ling mode predicted by analytical method are almost identical to that obtained from the finite element analysis. Both the analytical method and finite element analysis predict the same buckling mode variation. The buckling mode changes specified by (d / t)∗, ∗∗ ∗∗∗ in Fig. 3 also coincide with the buck(d / t) and(d / t) ling modes changes shown in Figs. 5–8. 13. Conclusion
Fig. 4. Comparison of analytical and FEA predictions for the variation of the elastic buckling pressure with d/t.
symmetric. Any other boundary conditions would overconstrain the structure and result in higher buckling pressures.
12. Comparison between analytical and FEA results The analytical solutions of the elastic buckling pressure as it varies with thickness reduction d/t for several b are plotted in Fig. 4. The analytical solutions and finite element predictions are within 5% of each other. Some of the buckling modes corresponding to these solutions are shown in Figs. 5–8. Buckling modes from finite element analysis are also plotted in Figs. 5–8 to compare with the analytical solutions. One can see that the buck-
Exact solutions for elastic buckling of a non-uniform, long cylindrical shell subjected to external hydrostatic pressure were derived in this paper. The non-uniform shell has two regions: one with a nominal thickness and the other with reduced thickness. The buckling pressure was found to decrease with the increasing value of the thickness reduction and angular extension. Symmetric and anti-symmetric buckling modes occurred depending on the relative thickness and the angular extent of the two sections. Criteria were established to determine ranges of relative thickness and angular extent that correspond to symmetric mode and anti-symmetric modes. Furthermore, the buckling modes and pressures that were predicted from ABAQUS were almost identical to the analytical solutions. The analytical predictions and finite element results for the bucking pressure were within 5% of each other. Solutions from this research can serve as a good point of reference in the offshore industry. For example, they can be used to address buckle propagation in corroded pipelines, where the pipelines are often treated as infinitely long cylinder and the propagation pressure is usu-
Fig. 5.
Buckling modes for b ⫽ 30°.
Fig. 6.
Buckling modes for b ⫽ 45°.
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J. Xue, M.S. Hoo Fatt / Engineering Structures 24 (2002) 1027–1034
Fig. 7.
Buckling modes for b ⫽ 90°.
Fig. 8.
Buckling modes for b ⫽ 135°.
ally calculated from a ring [11–13]. Our analysis can provide valuable technical information for monitoring pipeline performance in the offshore or oil industry, where corrosion is a threat. Technicians can design and apply suitable equipment, such as buckle arrestors, to prevent catastrophic pipeline failure when their buckling strength has been compromised by corrosion.
Acknowledgements The work reported herein was supported by the Department of Mechanical Engineering and a Faculty Research Grant from The University of Akron.
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[4] Bai Y, Hauch S. Analytical collapse of corroded pipes. In: Proceedings of the Eighth International Conference on Offshore and Polar Engineering. Montreal, Canada, 24–29 May 1998;2:182-8. [5] Hauch S, Bai Y. Use of finite element methods for the determination of local buckling strength. In: Proceedings of the 1998 International Conference on Offshore Mechanics and Arctic Engineering. Lisbon, Portugal; 5–9 July 1998. [6] Bai Y, Hauch S, Jenses JC. Local buckling and plastic collapse of corroded pipes with yield anisotropy. In: Proceedings of the Ninth International Offshore and Polar Engineering Conference. Golden, CO, 1999;2:74–81. [7] Hoo Fatt MS. Elastic-plastic collapse of non-uniform cylindrical shells subjected to uniform external pressure. Thin-Walled Struct 1999;35:117–37. [8] Xue J. Structural failure of non-uniform, infinitely long cylindrical shells subjected to uniform external pressure. Doctoral Dissertation, The University of Akron; May 2002. [9] MATLAB. The language of technical computing, Version 5.3.0.10183(R11); 21 January 1999. [10] HKS. ABAQUS user’s manual: theoretical manual, ABAQUS post manual and example problem, Version 4.9. Hibbit, Karlsson and Sorensen, Inc; 1999. [11] Palmer AC, Martin JH. Buckle propagation in submarine pipeline. Nature 1975;254:46–8. [12] Chater E, Hutchinson JW. On the propagation of bulges and buckles. J Appl Mech 1984;51:269–77. [13] Kyriakides S, Yeh MK, Roach D. On the determination of the propagation pressure of long circular tubes. Trans ASME J Press Vess Technol 1984;106:150–9.