- Email: [email protected]
Plastic buckling of complete toroidal shells of elliptical cross-section subjected to internal pressure
Thin-Walled Structures 34 (1999) 135–146 www.elsevier.com/locate/tws
Plastic buckling of complete toroidal shells of elliptical cross-section subjected to internal pressure A. Combescurea, G.D. Galletlyb,* LMT ENS Cachan, 61 bd. du Pre´sident Wilson, 94235 Cachan Cedex, France Department of Engineering, The University of Liverpool, Brownlow Hill, Liverpool L69 3GH, UK a
b
Abstract The plastic bifurcation buckling pressures of 60 internally-pressurised, perfect, complete toroidal shells of elliptical cross-section are given in the present paper, assuming elastic, perfectly plastic, material behaviour. The shell buckling programs employed in the computations were BOSOR 5 and INCA. Denoting the major-to-minor axis ratio by k, the numerical results show that the plastic buckling pressures are considerably lower than their elastic counterparts in the range 1.25 ⱕ k ⱕ 1.5 and are approximately equal to them for k ⫽ 2.5. A limited study of the effects of non-axisymmetric initial geometric imperfections on the buckling pressures of the shells was also carried out using the INCA code. For the four cases studied the postbuckling behaviour was stable. This means that designers can use the buckling pressures given herein for perfect shells as a basis for their initial designs. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Buckling; Internal pressure; Elliptic toroids; Plasticity; Imperfections
Notation a b d0 k
semi-major axis of ellipse (see Fig. 2) semi-minor axis of ellipse initial geometric imperfection amplitude (along the normal) ⫽ a/b
* Corresponding author. Tel: ⫹ 44-(0)151-794-4812; fax: ⫹ 44-(0)151-794-4848; e-mail: [email protected] 0263-8231/99/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 2 6 3 - 8 2 3 1 ( 9 9 ) 0 0 0 0 6 - 3
136
p pcr t ur E R yp
A. Combescure, G.D. Galletly / Thin-Walled Structures 34 (1999) 135–146
internal pressure buckling pressure thickness of shell radial (horizontal displacement) of point A on the shell (see Figs. 3 and 4) modulus of elasticity distance from axis of rotation to centre of ellipse Poisson’s ratio yield point of material
1. Introduction Complete (or closed) toroidal shells (see Fig. 1) have been used, to a limited extent, in aerospace, nuclear and subsea applications. The cross-sections of the toroidal shells employed have been circular, elliptical, D-shaped, etc. An elastic analysis of perfect elliptic toroidal shells under external pressure showed that they could be stronger or weaker than perfect circular toroidal shells, depending on the geometry— see [1] for further details. It was also noted [1] that perfect elastic toroidal shells of elliptical cross-section (see Fig. 2) could buckle under internal pressure. Although there is, as yet, no experimental evidence of this, a comparison of the numerical predictions of the BOSOR 5 [2] and INCA [3] shell buckling programs showed very good agreement [4]. This was the case for both the magnitudes of the internal buckling pressures and the numbers of circumferential waves at buckling. Some qualitative evidence for the occurrence of buckling in internally-pressurised elliptical toroidal shells is given in Figs. 3 and 4. The cross-section of a circular toroidal shell is shown in Fig. 3, both in the undeformed state and after the appli-
Fig. 1.
A typical complete toroidal shell of elliptical cross-section.
A. Combescure, G.D. Galletly / Thin-Walled Structures 34 (1999) 135–146
Fig. 2.
137
Geometry of a toroidal shell with an elliptical cross-section.
Fig. 3. Deformed (and unbuckled) shape of a steel toroidal shell having a circular cross-section (b/t ⫽ 100, R/b ⫽ 4, p ⫽ 9MPa).
cation of internal pressure. Looking at points A and B, it may be seen that, following the application of internal pressure, they both move farther away from the axis of rotation. That is to say, the hoop strains are tensile in both cases and buckling of the shell is unlikely. In Fig. 4, the similar states for an elliptical toroidal shell are shown. This time, the hoop strain is compressive at point A and tensile at point B. Thus, buckling may occur near point A but is unlikely near point B. Whether buckling will occur depends on the magnitudes of k, b/t, R/b, yp and p.
138
A. Combescure, G.D. Galletly / Thin-Walled Structures 34 (1999) 135–146
Fig. 4. Deformed shape prior to buckling of a steel toroidal shell having an elliptical cross-section (k ⫽ 1.2, b/t ⫽ 100, R/b ⫽ 4).
Some of the elastic buckling pressures in [4] were fairly high and it is necessary, in practical design problems, to take account of the yield point of the shell wall. This has been done in the present paper, the main aim being to calculate the plastic bifurcation buckling pressures of perfect elliptical toroids for selected ranges of the geometrical parameters and yield stresses. The shell buckling programs used in the calculations were again BOSOR 5 and INCA. As might be expected, taking the yield stress into account reduces the critical buckling pressure in many cases. The Figures presented later illustrate this point. A limited investigation into the effects of initial geometric imperfections on the buckling pressures of the shells is also included in the paper. The post-buckling behaviour of the internally-pressurised elliptic toroidal shells was studied via initially imperfect shell structures which were produced by adding an initial imperfection, having the shape of the non-linear buckling mode of the perfect structure, to the perfect shape. The amplitude of the imperfection is a variable and its sign is not important in the present case. This is because the buckling modes are decomposed into Fourier series. The INCA code was used for the analysis of the imperfect shells. This code has a special finite element (known as COMU) [3] which permits one to carry out a full non-linear analysis of axisymmetric structures with non-axisymmetric initial imperfections. The Riks method was used for the post-buckling analysis and the displacement response was expanded in a Fourier series. If the initial imperfection had n circumferential waves, the response was expanded in the modes 0, n, 2n and 3n. Plasticity was evaluated at 21 points regularly spaced around the circumference (over a total angle of 2/n). As was the case with the analysis of the perfect shells, plasticity, large displace-
A. Combescure, G.D. Galletly / Thin-Walled Structures 34 (1999) 135–146
139
ments and follower pressure effects were taken into account when analysing the imperfect shells.
2. Scope of the paper The cross-section of a typical elliptical toroidal shell studied in the paper is shown in Fig. 2. The notation used is as follows: a ⫽ semi-major axis of ellipse; b ⫽ semiminor axis of ellipse; k ⫽ a/b; t ⫽ thickness; R ⫽ distance from axis of rotation to centre of ellipse. A prolate ellipse (i.e. one which has its major axis parallel to the axis of rotation) has k > 1.0 and an oblate ellipse has k ⬍ 1.0. In this paper, only prolate ellipses are studied and the ranges of the parameters are as follows: R/b ⫽ 4 and 10, b/t ⫽ 50, 100 and 200, 1.25 ⱕ k ⱕ 2.50, yp ⫽ 350 MPa. Some results (for b/t ⫽ 100) are also given for yp ⫽ 210 MPa. The material of the shell wall is steel (E ⫽ 210 000 MPa, ⫽ 0.3) and it is assumed to have elastic–perfectly plastic behaviour. The loading is a uniform internal follower (live) pressure, the shell thickness is uniform and there are no initial stresses and no initial geometric imperfections (but see below for a limited study of imperfect shells). With the BOSOR 5 program, the elliptical cross-sections were divided into four 90° segments with 97 mesh points in each segment. With the INCA program, the perfect ellipses were modelled using 200 two-noded flat shell elements (50 in each 90° segment). Seven integration points across the thickness were employed in the INCA program and five in BOSOR. For the imperfection-sensitivity investigation, four elliptical cross-sections were chosen. These were as follows: Ref. No.
k
R/t
R/b
perfect pcr (MPa)
perfect n
E200 P50 E100 P100
2.5 1.25 2.5 1.25
200 50 100 100
10 4 10 10
0.032 4.27 0.16 1.0
19 12 15 102
In the above tabulation, E indicates elastic buckling and P plastic buckling. For each shell, three initial imperfection amplitudes were chosen. These were: d0 ⫽ 0.01, 0.1 and 1.0. The imperfect ellipses were modelled using 200 COMU elements regularly spaced around the cross-section. Each node has 16 degrees of freedom (16 ⫽ 4 Fourier modes times 4 degrees of freedom per node). A typical computation lasted 10 min of CPU time on an HP workstation.
140
A. Combescure, G.D. Galletly / Thin-Walled Structures 34 (1999) 135–146
3. Numerical results and discussion The internal buckling pressures, pcr, and the corresponding circumferential wave numbers at buckling, n, for b/t ⫽ 50, 100 and 200 are given in Table 1 for R/b ⫽ 4 and R/b ⫽ 10. As may be seen, the internal buckling pressures given by BOSOR 5 and INCA agree very well with each other. This is also the case, in general, for the wave numbers, n. However, with this non-linear problem, the n-value obtained sometimes depends on the load level at which the calculations are started and the load increment employed. The values of pcr for b/t ⫽ 100 are shown graphically in Fig. 5(a) and (b) for yp ⫽ 210 MPa and 350 MPa. The corresponding values for b/t ⫽ 50 and 200, yp ⫽ 350 MPa are given in Fig. 6(a) and (b) and Fig. 7(a) and (b), respectively. As is evident from these figures, the plastic internal buckling pressures are much lower than their elastic counterparts for the smaller values of k ( ⱕ 1.5). It was noted in [4] that the internal buckling pressures of elastic elliptic toroids were, approximately, proportional to the square of the shell thickness, t. By comparing the values of pcr given in Table 1, it will be seen that the plastic internal buckling pressures of elliptic toroids are, more-or-less, proportional to the shell thickness for the smaller values of k ( ⱕ 1.5). With elastic elliptic toroids, it was mentioned [4] that, at k ⫽ 1.78, there was a Table 1 A comparison of the BOSOR 5 and INCA plastic buckling pressures (MPa) for some internally-pressurised elliptic toroidal shells (yp ⫽ 350 MPa) k
b/t ⫽ 50, yp ⫽ 350 MPa 1.25 1.5 1.75 2.0 2.5 b/t ⫽ 100, yp ⫽ 350 MPa 1.25 1.5 1.75 2.0 2.5 b/t ⫽ 200, yp ⫽ 350 MPa 1.25 1.50 1.75 2.0 2.5 a
Elastic buckling.
R/b ⫽ 4
R/b ⫽ 10
BOSOR 5
INCA
BOSOR 5
INCA
4.33 (10) 3.075 (14) 2.65 (10) 2.42 (10) 2.02 (6)
4.29 (12) 3.10 (13) 2.66 (11) 2.25 (13) 2.0 (7)
2.06 (60) 1.29 (74) 1.075 (62) 1.05 (12) 0.83 (8)
2.065 (62) 1.30 (62) 1.08 (61) 0.965 (26) 0.81 (12)
2.09 1.54 1.33 1.09 0.46
2.10 1.54 1.32 1.10 0.42
1.01 (128) 0.65 (105) 0.55 (88) 0.39 (18)a 0.155 (14)a
0.98 (102) 0.655 (110) 0.536 (89) 0.40 (18)a 0.16 (15)a
0.505(182) 0.305(148) 0.165(30) 0.078(20)a 0.031(18)a
0.502(180) 0.309(149) 0.165(30) 0.078(20)a 0.031(19)a
(22) (35) (14) (11) (8)
1.05(40) 0.775(20) 0.505(19) 0.230(13)a 0.093(11)a
(20) (25) (26) (11) (9)a
1.04(36) 0.76(20) 0.51(19) 0.24(13)a 0.096(11)a
A. Combescure, G.D. Galletly / Thin-Walled Structures 34 (1999) 135–146
Fig. 5.
141
Plastic buckling pressures of steel internally-pressurised elliptic toroidal shells (b/t ⫽ 100).
Fig. 6. Plastic buckling pressures of steel internally-pressurised elliptic toroidal shells (b/t ⫽ 50).
sharp change in the critical value of n (e.g. with R/b ⫽ 10, b/t ⫽ 100). As shown in the foregoing reference, this was associated with a change in the shape of the buckling mode in the meridional direction. In the present plastic buckling case, a similar situation seems to happen. For instance, with R/b ⫽ 10, b/t ⫽ 100, yp ⫽
142
Fig. 7.
A. Combescure, G.D. Galletly / Thin-Walled Structures 34 (1999) 135–146
Plastic buckling pressures of steel internally-pressurised elliptic toroidal shells (b/t ⫽ 200).
Fig. 8. Some buckling mode shapes for internally-pressurised elliptic toroidal shells (R/b ⫽ 10, b/t ⫽ 100, yp ⫽ 350 MPa).
A. Combescure, G.D. Galletly / Thin-Walled Structures 34 (1999) 135–146
143
350 MPa, the BOSOR 5 program gives pcr ⫽ 0.53 MPa at both k ⫽ 1.77, n ⫽ 87 and k ⫽ 1.79, n ⫽ 28. The two corresponding buckling modes are shown in Fig. 8(a) and (b). A three-dimensional view of the buckling modes can be obtained using the INCA program. One example (for k ⫽ 2.25) is shown in Fig. 8(c). The results of the initial geometric imperfection investigation are shown in Figs. 9–12. In these figures the ordinates are the applied pressures divided by the bifurcation buckling pressures of the corresponding perfect shells. The abscissae are the normalised radial (horizontal) displacements of the innermost point of the shell (point A in Fig. 4). The normalisation factors are 500t, 1000t and 2000t and the relevant factor is shown on the corresponding figure. The total radial displacement, ur, of point A is divided into two parts, i.e. an axisymmetric component (mode O) and a non-symmetric component representing the sinusoidal buckling mode (n ⫽ 12, 19, 15 or 102). Figs 9–12 give the responses for the shells E200, E100, P100 and P50. From the aforementioned figures, it will be observed that, in all cases, the postbuckling behaviour is of the stable kind. The four sets of curves show an inward axisymmetric radial (horizontal) displacement of point A. This axisymmetric component is dominant in the plastic buckling case but is smaller in the elastic case. It will also be observed that the axisymmetric displacements do not differ much when the initial imperfection, d0, is smaller that 0.1t. The initial slope of the non-axisymmetric component of the radial displacement (mode n) varies with the initial imperfection amplitude.
Fig. 9. Shell E200: normalised radial displacement, ur, at point A versus the normalised applied pressure. Three values of the initial imperfection d0.
144
A. Combescure, G.D. Galletly / Thin-Walled Structures 34 (1999) 135–146
Fig. 10. Shell E100: normalised radial displacement, ur, at point A versus the normalised applied pressure. Three values of the initial imperfection d0.
Fig. 11. Shell P100: normalised radial displacement, ur, at point A versus the normalised applied pressure. Three values of the initial imperfection d0.
A. Combescure, G.D. Galletly / Thin-Walled Structures 34 (1999) 135–146
145
Fig. 12. Shell P50: normalised radial displacement, ur, at point A versus the normalised applied pressure. Three values of the initial imperfection d0.
4. Future work It would be good to have experimental verification of the present theoretical results, but making near-perfect models is likely to be expensive using metallic models. As the theoretical studies have shown that initial geometric imperfections are not very important, perhaps a relatively cheap form of constructing complete elliptic toroids can be devised. The effects of thickness imperfections should also be studied.
5. Conclusions Good agreement was found between the plastic buckling predictions of the BOSOR 5 and INCA codes for complete, perfect, elliptic toroidal shells subjected to internal pressure. This was the case for both the buckling pressures and the corresponding modes. The imperfection investigation of initial shapes showed that, for the range of parameters studied, the post-buckling behaviour of the shells seems to be of the stable kind. This imperfection analysis is relatively simple and efficient to carry out using the INCA code. As these elliptic toroidal shells have been shown to be imperfection insensitive (for this loading), a designer can use the buckling pressures given herein for perfect shells as a basis for his initial design.
146
A. Combescure, G.D. Galletly / Thin-Walled Structures 34 (1999) 135–146
References [1] Galletly GD, Blachut J. Stability of complete circular and non-circular toroidal shells. Proceedings of the Institution of Mechanical Engineers (London) Part C 1995;209:245–55. [2] Bushnell D. BOSOR 5—Program for buckling of elastic–plastic complex shells of revolution including large deflections and creep. Computers and Structures 1976;6:211–39. [3] Combescure A. Static and dynamic buckling of large thin shells. Nuclear Engineering and Design 1986;92:332–54. [4] Galletly GD. Elastic buckling of complete toroidal shells of elliptical cross-section subjected to uniform internal pressure. Presented at the Thin-Walled Structures Conference, December 1996, University of Strathclyde, Glasgow. Thin-Walled Structures 1998;30:23–34.
Plastic buckling of complete toroidal shells of elliptical cross-section subjected to internal pressure A. Combescurea, G.D. Galletlyb,* LMT ENS Cachan, 61 bd. du Pre´sident Wilson, 94235 Cachan Cedex, France Department of Engineering, The University of Liverpool, Brownlow Hill, Liverpool L69 3GH, UK a
b
Abstract The plastic bifurcation buckling pressures of 60 internally-pressurised, perfect, complete toroidal shells of elliptical cross-section are given in the present paper, assuming elastic, perfectly plastic, material behaviour. The shell buckling programs employed in the computations were BOSOR 5 and INCA. Denoting the major-to-minor axis ratio by k, the numerical results show that the plastic buckling pressures are considerably lower than their elastic counterparts in the range 1.25 ⱕ k ⱕ 1.5 and are approximately equal to them for k ⫽ 2.5. A limited study of the effects of non-axisymmetric initial geometric imperfections on the buckling pressures of the shells was also carried out using the INCA code. For the four cases studied the postbuckling behaviour was stable. This means that designers can use the buckling pressures given herein for perfect shells as a basis for their initial designs. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Buckling; Internal pressure; Elliptic toroids; Plasticity; Imperfections
Notation a b d0 k
semi-major axis of ellipse (see Fig. 2) semi-minor axis of ellipse initial geometric imperfection amplitude (along the normal) ⫽ a/b
* Corresponding author. Tel: ⫹ 44-(0)151-794-4812; fax: ⫹ 44-(0)151-794-4848; e-mail: [email protected] 0263-8231/99/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 2 6 3 - 8 2 3 1 ( 9 9 ) 0 0 0 0 6 - 3
136
p pcr t ur E R yp
A. Combescure, G.D. Galletly / Thin-Walled Structures 34 (1999) 135–146
internal pressure buckling pressure thickness of shell radial (horizontal displacement) of point A on the shell (see Figs. 3 and 4) modulus of elasticity distance from axis of rotation to centre of ellipse Poisson’s ratio yield point of material
1. Introduction Complete (or closed) toroidal shells (see Fig. 1) have been used, to a limited extent, in aerospace, nuclear and subsea applications. The cross-sections of the toroidal shells employed have been circular, elliptical, D-shaped, etc. An elastic analysis of perfect elliptic toroidal shells under external pressure showed that they could be stronger or weaker than perfect circular toroidal shells, depending on the geometry— see [1] for further details. It was also noted [1] that perfect elastic toroidal shells of elliptical cross-section (see Fig. 2) could buckle under internal pressure. Although there is, as yet, no experimental evidence of this, a comparison of the numerical predictions of the BOSOR 5 [2] and INCA [3] shell buckling programs showed very good agreement [4]. This was the case for both the magnitudes of the internal buckling pressures and the numbers of circumferential waves at buckling. Some qualitative evidence for the occurrence of buckling in internally-pressurised elliptical toroidal shells is given in Figs. 3 and 4. The cross-section of a circular toroidal shell is shown in Fig. 3, both in the undeformed state and after the appli-
Fig. 1.
A typical complete toroidal shell of elliptical cross-section.
A. Combescure, G.D. Galletly / Thin-Walled Structures 34 (1999) 135–146
Fig. 2.
137
Geometry of a toroidal shell with an elliptical cross-section.
Fig. 3. Deformed (and unbuckled) shape of a steel toroidal shell having a circular cross-section (b/t ⫽ 100, R/b ⫽ 4, p ⫽ 9MPa).
cation of internal pressure. Looking at points A and B, it may be seen that, following the application of internal pressure, they both move farther away from the axis of rotation. That is to say, the hoop strains are tensile in both cases and buckling of the shell is unlikely. In Fig. 4, the similar states for an elliptical toroidal shell are shown. This time, the hoop strain is compressive at point A and tensile at point B. Thus, buckling may occur near point A but is unlikely near point B. Whether buckling will occur depends on the magnitudes of k, b/t, R/b, yp and p.
138
A. Combescure, G.D. Galletly / Thin-Walled Structures 34 (1999) 135–146
Fig. 4. Deformed shape prior to buckling of a steel toroidal shell having an elliptical cross-section (k ⫽ 1.2, b/t ⫽ 100, R/b ⫽ 4).
Some of the elastic buckling pressures in [4] were fairly high and it is necessary, in practical design problems, to take account of the yield point of the shell wall. This has been done in the present paper, the main aim being to calculate the plastic bifurcation buckling pressures of perfect elliptical toroids for selected ranges of the geometrical parameters and yield stresses. The shell buckling programs used in the calculations were again BOSOR 5 and INCA. As might be expected, taking the yield stress into account reduces the critical buckling pressure in many cases. The Figures presented later illustrate this point. A limited investigation into the effects of initial geometric imperfections on the buckling pressures of the shells is also included in the paper. The post-buckling behaviour of the internally-pressurised elliptic toroidal shells was studied via initially imperfect shell structures which were produced by adding an initial imperfection, having the shape of the non-linear buckling mode of the perfect structure, to the perfect shape. The amplitude of the imperfection is a variable and its sign is not important in the present case. This is because the buckling modes are decomposed into Fourier series. The INCA code was used for the analysis of the imperfect shells. This code has a special finite element (known as COMU) [3] which permits one to carry out a full non-linear analysis of axisymmetric structures with non-axisymmetric initial imperfections. The Riks method was used for the post-buckling analysis and the displacement response was expanded in a Fourier series. If the initial imperfection had n circumferential waves, the response was expanded in the modes 0, n, 2n and 3n. Plasticity was evaluated at 21 points regularly spaced around the circumference (over a total angle of 2/n). As was the case with the analysis of the perfect shells, plasticity, large displace-
A. Combescure, G.D. Galletly / Thin-Walled Structures 34 (1999) 135–146
139
ments and follower pressure effects were taken into account when analysing the imperfect shells.
2. Scope of the paper The cross-section of a typical elliptical toroidal shell studied in the paper is shown in Fig. 2. The notation used is as follows: a ⫽ semi-major axis of ellipse; b ⫽ semiminor axis of ellipse; k ⫽ a/b; t ⫽ thickness; R ⫽ distance from axis of rotation to centre of ellipse. A prolate ellipse (i.e. one which has its major axis parallel to the axis of rotation) has k > 1.0 and an oblate ellipse has k ⬍ 1.0. In this paper, only prolate ellipses are studied and the ranges of the parameters are as follows: R/b ⫽ 4 and 10, b/t ⫽ 50, 100 and 200, 1.25 ⱕ k ⱕ 2.50, yp ⫽ 350 MPa. Some results (for b/t ⫽ 100) are also given for yp ⫽ 210 MPa. The material of the shell wall is steel (E ⫽ 210 000 MPa, ⫽ 0.3) and it is assumed to have elastic–perfectly plastic behaviour. The loading is a uniform internal follower (live) pressure, the shell thickness is uniform and there are no initial stresses and no initial geometric imperfections (but see below for a limited study of imperfect shells). With the BOSOR 5 program, the elliptical cross-sections were divided into four 90° segments with 97 mesh points in each segment. With the INCA program, the perfect ellipses were modelled using 200 two-noded flat shell elements (50 in each 90° segment). Seven integration points across the thickness were employed in the INCA program and five in BOSOR. For the imperfection-sensitivity investigation, four elliptical cross-sections were chosen. These were as follows: Ref. No.
k
R/t
R/b
perfect pcr (MPa)
perfect n
E200 P50 E100 P100
2.5 1.25 2.5 1.25
200 50 100 100
10 4 10 10
0.032 4.27 0.16 1.0
19 12 15 102
In the above tabulation, E indicates elastic buckling and P plastic buckling. For each shell, three initial imperfection amplitudes were chosen. These were: d0 ⫽ 0.01, 0.1 and 1.0. The imperfect ellipses were modelled using 200 COMU elements regularly spaced around the cross-section. Each node has 16 degrees of freedom (16 ⫽ 4 Fourier modes times 4 degrees of freedom per node). A typical computation lasted 10 min of CPU time on an HP workstation.
140
A. Combescure, G.D. Galletly / Thin-Walled Structures 34 (1999) 135–146
3. Numerical results and discussion The internal buckling pressures, pcr, and the corresponding circumferential wave numbers at buckling, n, for b/t ⫽ 50, 100 and 200 are given in Table 1 for R/b ⫽ 4 and R/b ⫽ 10. As may be seen, the internal buckling pressures given by BOSOR 5 and INCA agree very well with each other. This is also the case, in general, for the wave numbers, n. However, with this non-linear problem, the n-value obtained sometimes depends on the load level at which the calculations are started and the load increment employed. The values of pcr for b/t ⫽ 100 are shown graphically in Fig. 5(a) and (b) for yp ⫽ 210 MPa and 350 MPa. The corresponding values for b/t ⫽ 50 and 200, yp ⫽ 350 MPa are given in Fig. 6(a) and (b) and Fig. 7(a) and (b), respectively. As is evident from these figures, the plastic internal buckling pressures are much lower than their elastic counterparts for the smaller values of k ( ⱕ 1.5). It was noted in [4] that the internal buckling pressures of elastic elliptic toroids were, approximately, proportional to the square of the shell thickness, t. By comparing the values of pcr given in Table 1, it will be seen that the plastic internal buckling pressures of elliptic toroids are, more-or-less, proportional to the shell thickness for the smaller values of k ( ⱕ 1.5). With elastic elliptic toroids, it was mentioned [4] that, at k ⫽ 1.78, there was a Table 1 A comparison of the BOSOR 5 and INCA plastic buckling pressures (MPa) for some internally-pressurised elliptic toroidal shells (yp ⫽ 350 MPa) k
b/t ⫽ 50, yp ⫽ 350 MPa 1.25 1.5 1.75 2.0 2.5 b/t ⫽ 100, yp ⫽ 350 MPa 1.25 1.5 1.75 2.0 2.5 b/t ⫽ 200, yp ⫽ 350 MPa 1.25 1.50 1.75 2.0 2.5 a
Elastic buckling.
R/b ⫽ 4
R/b ⫽ 10
BOSOR 5
INCA
BOSOR 5
INCA
4.33 (10) 3.075 (14) 2.65 (10) 2.42 (10) 2.02 (6)
4.29 (12) 3.10 (13) 2.66 (11) 2.25 (13) 2.0 (7)
2.06 (60) 1.29 (74) 1.075 (62) 1.05 (12) 0.83 (8)
2.065 (62) 1.30 (62) 1.08 (61) 0.965 (26) 0.81 (12)
2.09 1.54 1.33 1.09 0.46
2.10 1.54 1.32 1.10 0.42
1.01 (128) 0.65 (105) 0.55 (88) 0.39 (18)a 0.155 (14)a
0.98 (102) 0.655 (110) 0.536 (89) 0.40 (18)a 0.16 (15)a
0.505(182) 0.305(148) 0.165(30) 0.078(20)a 0.031(18)a
0.502(180) 0.309(149) 0.165(30) 0.078(20)a 0.031(19)a
(22) (35) (14) (11) (8)
1.05(40) 0.775(20) 0.505(19) 0.230(13)a 0.093(11)a
(20) (25) (26) (11) (9)a
1.04(36) 0.76(20) 0.51(19) 0.24(13)a 0.096(11)a
A. Combescure, G.D. Galletly / Thin-Walled Structures 34 (1999) 135–146
Fig. 5.
141
Plastic buckling pressures of steel internally-pressurised elliptic toroidal shells (b/t ⫽ 100).
Fig. 6. Plastic buckling pressures of steel internally-pressurised elliptic toroidal shells (b/t ⫽ 50).
sharp change in the critical value of n (e.g. with R/b ⫽ 10, b/t ⫽ 100). As shown in the foregoing reference, this was associated with a change in the shape of the buckling mode in the meridional direction. In the present plastic buckling case, a similar situation seems to happen. For instance, with R/b ⫽ 10, b/t ⫽ 100, yp ⫽
142
Fig. 7.
A. Combescure, G.D. Galletly / Thin-Walled Structures 34 (1999) 135–146
Plastic buckling pressures of steel internally-pressurised elliptic toroidal shells (b/t ⫽ 200).
Fig. 8. Some buckling mode shapes for internally-pressurised elliptic toroidal shells (R/b ⫽ 10, b/t ⫽ 100, yp ⫽ 350 MPa).
A. Combescure, G.D. Galletly / Thin-Walled Structures 34 (1999) 135–146
143
350 MPa, the BOSOR 5 program gives pcr ⫽ 0.53 MPa at both k ⫽ 1.77, n ⫽ 87 and k ⫽ 1.79, n ⫽ 28. The two corresponding buckling modes are shown in Fig. 8(a) and (b). A three-dimensional view of the buckling modes can be obtained using the INCA program. One example (for k ⫽ 2.25) is shown in Fig. 8(c). The results of the initial geometric imperfection investigation are shown in Figs. 9–12. In these figures the ordinates are the applied pressures divided by the bifurcation buckling pressures of the corresponding perfect shells. The abscissae are the normalised radial (horizontal) displacements of the innermost point of the shell (point A in Fig. 4). The normalisation factors are 500t, 1000t and 2000t and the relevant factor is shown on the corresponding figure. The total radial displacement, ur, of point A is divided into two parts, i.e. an axisymmetric component (mode O) and a non-symmetric component representing the sinusoidal buckling mode (n ⫽ 12, 19, 15 or 102). Figs 9–12 give the responses for the shells E200, E100, P100 and P50. From the aforementioned figures, it will be observed that, in all cases, the postbuckling behaviour is of the stable kind. The four sets of curves show an inward axisymmetric radial (horizontal) displacement of point A. This axisymmetric component is dominant in the plastic buckling case but is smaller in the elastic case. It will also be observed that the axisymmetric displacements do not differ much when the initial imperfection, d0, is smaller that 0.1t. The initial slope of the non-axisymmetric component of the radial displacement (mode n) varies with the initial imperfection amplitude.
Fig. 9. Shell E200: normalised radial displacement, ur, at point A versus the normalised applied pressure. Three values of the initial imperfection d0.
144
A. Combescure, G.D. Galletly / Thin-Walled Structures 34 (1999) 135–146
Fig. 10. Shell E100: normalised radial displacement, ur, at point A versus the normalised applied pressure. Three values of the initial imperfection d0.
Fig. 11. Shell P100: normalised radial displacement, ur, at point A versus the normalised applied pressure. Three values of the initial imperfection d0.
A. Combescure, G.D. Galletly / Thin-Walled Structures 34 (1999) 135–146
145
Fig. 12. Shell P50: normalised radial displacement, ur, at point A versus the normalised applied pressure. Three values of the initial imperfection d0.
4. Future work It would be good to have experimental verification of the present theoretical results, but making near-perfect models is likely to be expensive using metallic models. As the theoretical studies have shown that initial geometric imperfections are not very important, perhaps a relatively cheap form of constructing complete elliptic toroids can be devised. The effects of thickness imperfections should also be studied.
5. Conclusions Good agreement was found between the plastic buckling predictions of the BOSOR 5 and INCA codes for complete, perfect, elliptic toroidal shells subjected to internal pressure. This was the case for both the buckling pressures and the corresponding modes. The imperfection investigation of initial shapes showed that, for the range of parameters studied, the post-buckling behaviour of the shells seems to be of the stable kind. This imperfection analysis is relatively simple and efficient to carry out using the INCA code. As these elliptic toroidal shells have been shown to be imperfection insensitive (for this loading), a designer can use the buckling pressures given herein for perfect shells as a basis for his initial design.
146
A. Combescure, G.D. Galletly / Thin-Walled Structures 34 (1999) 135–146
References [1] Galletly GD, Blachut J. Stability of complete circular and non-circular toroidal shells. Proceedings of the Institution of Mechanical Engineers (London) Part C 1995;209:245–55. [2] Bushnell D. BOSOR 5—Program for buckling of elastic–plastic complex shells of revolution including large deflections and creep. Computers and Structures 1976;6:211–39. [3] Combescure A. Static and dynamic buckling of large thin shells. Nuclear Engineering and Design 1986;92:332–54. [4] Galletly GD. Elastic buckling of complete toroidal shells of elliptical cross-section subjected to uniform internal pressure. Presented at the Thin-Walled Structures Conference, December 1996, University of Strathclyde, Glasgow. Thin-Walled Structures 1998;30:23–34.