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A design chart for the plastic collapse of corrugated cylinders under external pressure
Ocean Engineering 28 (2001) 263–277
A design chart for the plastic collapse of corrugated cylinders under external pressure C.T.F. Ross *, A. Terry, A.P.F. Little Department of Mechanical Engineering, University of Portsmouth, Portsmouth PO1 3DJ, UK Received 29 August 1999; accepted 24 November 1999
Abstract The paper presents a theoretical and an experimental investigation into the plastic collapse of circular steel corrugated cylinders under external hydrostatic pressure. The experimental investigation gives a detailed study of 9 steel corrugated cylinders which were tested to destruction. Six of these cylinders failed by plastic non-symmetric bifurcation buckling and three failed by plastic axisymmetric deformation. The results of these tests were used, together with the results obtained from previous tests, to present a design chart for the plastic collapse of these vessels. The design chart was obtained by a semi-empirical approach, where the thinness ratios of the vessels were plotted against their plastic knockdown factors. The process of using the design chart is to calculate the theoretical elastic instability pressure for a perfect vessel by the finite element method and also to calculate the thinness ratio for this vessel. Using the appropriate value of the thinness ratio, the plastic knockdown factors are obtained from the design chart. To obtain the actual collapse pressure of the vessel, the theoretical elastic instability pressure for a perfect vessel is divided by the plastic knockdown factor. This work is of importance in ocean engineering. A large safety factor must also be introduced. 2000 Elsevier Science Ltd. All rights reserved. Keywords: Buckling; Inelastic; Corrugated cylinders; External pressure; Design charts
1. Introduction The earth’s oceans cover some 70% of its surface and reaches depths of up to 11.52 km (7.16 miles). Although there has been one small vessel which has been * Corresponding author. Tel.: +44-1705-842318; fax: +44-1705-842351. E-mail address: [email protected] (C.T.F. Ross). 0029-8018/01/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 9 - 8 0 1 8 ( 0 0 ) 0 0 0 0 7 - X
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C.T.F. Ross et al. / Ocean Engineering 28 (2001) 263–277
Fig. 1.
Submarine pressure hull.
able to reach these depths, research is needed on how to manufacture larger vessels strong enough to survive the great pressures at these depths. These vessels would be used for retrieving oil and gas deposits from the seabed that lie at depths of up to 3 miles. Most submarine pressure hulls take the form of a cylinder/cone/dome construction, surrounded by casing to improve the hydrodynamic streamlining as shown in Fig. 1. Pressure hulls constructed in this way can fail either through axisymmetric yield or by non-symmetric bifurcation buckling, known as shell instability and as shown in Fig. 2. Under external hydrostatic pressure, the pressure required to cause the shell instability of a thin walled circular cylinder might only be a small fraction of
Fig. 2. Shell instability of thin walled circular cylinders.
C.T.F. Ross et al. / Ocean Engineering 28 (2001) 263–277
Fig. 3.
265
Ring stiffened vessel.
that to cause axisymmetric yield (Ross, 1984). This mode of failure is undesirable, as it is structurally inefficient. 1.1. Ring stiffened vessels The introduction of circumferential ring stiffeners spaced at suitable distances apart, as shown in Fig. 3, is one way in which the structural efficiency can be improved. Vessels such as these are used for the construction of submarine pressure hulls, torpedoes, water-borne ballistic missiles and aircraft. If the ring stiffeners are not strong enough, the entire ring-shell combination could buckle bodily as shown in Fig. 4. This form of buckling is known as general instability.
Fig. 4.
General instability of ring stiffened circular cylinders.
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1.2. Previous research of corrugated vessels In 1987, Ross (1987) presented an alternative design to the ring-stiffened vessel as shown in Fig. 5. The pressure hull was a circumferentially corrugated vessel. Ross showed that the corrugated vessel was structurally more efficient than a ring stiffened equivalent. In 1991, Yuan et al. (1991) showed that the corrugated cylinder could be made more structurally efficient by increasing the cone angles to certain optimum values. It should be noted that if the cone angles were too large the vessel would fail axisymmetrically, as shown by Liang et al. (1993). These reported studies were of a theoretical nature and experimental work to verify the results was carried out by Ross (1990), Ross and Humphries (1993), Ross and Heigl (1995) and Ross et al. (1996). In 1999, Ross and Waterman (1999) carried out more experimental work and used this and existing results to produce a design chart. Experimental investigations have shown that like unstiffened and ring stiffened circular cylinders, the vessels were prone to suffer plastic knockdown due to initial out-of-circularity. To allow for this, Ross and Palmer (1993) presented a thinness ratio, which was based on the thinness ratio of Windenburg and Trilling (1934); the formula for which is given by the following equation.
冑
l⫽[(Lb/D1)2/(t1/D1)3]0.25⫻ (sYP/E)
(1)
where: Lb is the length between bulkheads; D1 is the equivalent diameter; t1 is the equivalent thickness; sYP is the yield stress and E is the Young’s modulus of elasticity.
2. Theoretical analysis To determine by analytical methods the uniform external pressure required to cause the elastic instability of thin walled conical shells is very difficult and for these cases it is more convenient to use the finite element method. If the cone is of small apical angle then buckling will take place in a lobar manner as shown in Fig. 2 but if the angle of the cone is large the vessel can buckle axisymmetrically.
Fig. 5. Corrugated pressure hull.
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267
It must be emphasised that as these vessels are sensitive to initial geometric imperfections, they can suffer inelastic instability at buckling pressures considerably less than those causing elastic instability. This is where the importance of experimentally obtained results is crucial, to enable a plastic reduction factor to be obtained for a particular vessel. This plastic reduction factor must be divided into the theoretical elastic buckling pressure for the vessel in question, to obtain the reduced inelastic buckling pressure. Pcr/Pexp = plastic reduction factor = PKD Pcr = theoretical buckling pressure, based on the finite element method Pexp = experimentally obtained buckling pressure The theoretical solution based on small deflection elastic theory uses the element first developed by Ross in 1974 of a truncated cone with two nodal circles at its ends, as shown in Fig. 6. Each node has four degrees of freedom, namely (u0, v, w0, and q), making a total of eight degrees of freedom per element, where: u0 is the displacement in the x° direction; v is the displacement in the circumferential direction; w0 is the displacement in the z° direction; and, q is the rotational displacement. 2.1. Objectives To enable the authors to complete this project successfully, five main objectives were set out and completed in the following order: 1. To evaluate the experimental study on nine circular corrugated cylinders tested to destruction under hydrostatic pressure. Vessels were of varying lengths, with less than 13 corrugations. 2. To calculate theoretical elastic buckling pressures (Pcr) of these vessels, using finite element analysis. 3. To determine the plastic knockdown factor for each vessel due to initial outof-circularity. 4. To calculate the thinness ratio l1 for each vessel.
Fig. 6.
Truncated conical shell element.
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5. To create a design chart for these corrugated vessels together with results from other vessels.
3. Experimental procedure 3.1. Vessels Nine steel corrugated cylindrical models were tested to destruction. These models were AST1, AST2, AST3, AST4, AST5, AST6, AST7, AST8 and AST9, as shown in Fig. 7. Each vessel was carefully measured to determine its initial out-of-circularity, which was defined as the difference between the maximum inward and outward deviations of the external surface at the mid point of the cylinder. This was achieved with the use of a Mitutoyo BN706 co-ordinate measuring machine, together with a touch-trigger probe. The longitudinal profiles were of a sinusoidal nature, but for the theoretical analysis, it was found convenient to represent each corrugation with two truncated finite element conical shell elements as shown in Fig. 6; previous work by Ross and Humphries (1993), Ross and Heigl (1995), Ross et al. (1996) and Ross and Waterman (1999) has found this to be a satisfactory procedure. The geometrical details are
Fig. 7.
The AST series.
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269
Table 1 Geometrical details of AST series (mm) Model
AST1 AST2 AST3 AST4 AST5 AST6 AST7 AST8 AST9
Corrugations (N)
1 2 3 5 6 8 9 11 12
Out of circularity
0.15 0.16 0.07 0.08 0.18 0.15 0.08 0.15 0.16
Fig. 8.
Length between bulkheads (Lb) 3.57 7.14 10.71 17.85 21.42 28.56 32.13 39.27 42.84
Sketch of cylinder.
listed in Table 1; a sketch of the cylinder is shown in Fig. 8. Fig. 9 shows details of a typical corrugation, where: N is the number of corrugations; t is the wall thickness =0.25 mm; Ls is the length of corrugation =3.57 mm; Ro = RE = external radius of cylinder; and, Ri = RI = internal radius of cylinder. Young’s modulus and Poisson’s ratio were not measured, because the cylinders were made from typical carbon steels, so that the assumed values for these constants were reasonable. To determine the value for Yield stress, three specimens of length
Fig. 9.
Details of a typical corrugation (mm).
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150 mm by width 15 mm, were cut from the vessel. Material and tensile tests were performed on the Lloyd’s Materials testing machine. Fig. 10 shows the graph of load against extension. Material properties were found to be as follows: Young’s modulus = E =200 GPa (assumed); Poisson’s ratio = v =0.3 (assumed); Yield stress = syp =405 MPa (measured). The cylinders were blocked off at their ends by end closure plates. These closure plates were simply push-fitted onto the ends of the cylinders and sealed with silicone. The silicone was initially in a semiliquid form, which hardened after a few hours of exposure to the atmosphere, creating a watertight seal. 3.2. Pressure system The models were tested in the test tank shown in Fig. 11. Water was used as the pressure raising fluid. A Tangye Hydra hand pump was used to apply pressure to the tank. Line losses were negligible as the hose was only about 2 m in length. 3.3. Experimental procedure The pressure tank was partially filled with water prior to placing a vessel into it. The vessel which was previously made watertight, as described earlier, was placed in the pressure tank as shown in Fig. 11. An O ring seal was placed in the O ring groove of the tank. The lid was carefully placed onto the tank, washers and nuts were fitted and tightened evenly. Care was taken not to overtighten the tank top to maintain elasticity of the O ring seal. Prior to raising the water pressure, the bleed screw situated in the lid of the test tank was left open so that any trapped air could be pumped out. When the air had
Fig. 10.
Graph of tensile test.
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Fig. 11.
271
Test tank and model.
been pumped out and water flowed from the bleed hole, the screw to the bleed hole was tightened, so that the bleed hole was sealed and the experiment was ready to commence. At least two operators were required to carry out the tests; one to operate the hand driven pump and record the buckling pressure and the other to observe for any leaks in the system. Each model was tested to destruction by raising the pressure in steady increments. The pressure gauge was vigilantly monitored to ensure that the correct buckling pressure could be logged. Fig. 12 shows the vessels in their collapsed state. AST 4-9 collapsed in an inelastic lobar manner. AST 1, 2, and 3 collapsed axisymmetrically; these were the vessels of smallest overall length. The experimentally obtained results are shown in Table 2. It can be seen from Table 2 that the general trend is that as Lb increases the experimentally obtained buckling pressure decreases. AST6 and AST8 do not follow this pattern,: the buckling pressure for AST8 should be higher than that of AST9 and the buckling pressure for AST6 should be higher than AST7. However, this is typical of vessels of this type, where occasionally low experimental buckling pressures are observed, which cannot be described as rogue results. The result also cannot be explained by the out of circularity measurement as this is larger for AST9, but the buckling pressure is still higher than both the shorter vessels AST8 and AST6. It is because of results like these that the present study has been conducted, and why a traditional non-linear finite element is unlikely to prove successful.
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Fig. 12.
The collapsed models.
Table 2 Experimental buckling pressures (Pexp) Model
Out of circularity
Lb (mm)
Pressure at collapse (bar)
AST1 AST2 AST3 AST4 AST5 AST6 AST7 AST8 AST9
0.15 0.16 0.07 0.08 0.18 0.15 0.08 0.15 0.16
3.57 7.14 10.71 17.85 21.42 28.56 32.13 39.27 42.84
13.1 12.8 12.7 10.8 8.8 6.9 8.0 5.7 7.1
4. Determination of the design chart 4.1. Calculations of Pcr, the elastic buckling pressures To calculate the theoretical buckling of the vessels, a computer program called CONEBUCK (1984), based on finite element analysis, was used. The element adopted in the program was the thin walled truncated conical element of Ross (1990).
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273
This element has been described earlier in Section 2 and the element was shown in Fig. 6. The theoretical buckling pressures Pcr obtained by this program are shown in Table 3. It should be noted that the buckling pressures are based on elastic theory for perfect vessels. For imperfectly shaped vessels, which buckle inelastically, the experimental buckling pressure may be a small fraction of the theoretical values. 4.2. Calculation of the thinness ratios The design procedure is to first calculate theoretical pressure Pcr and thinness ratio l. The theoretical pressure is then used to calculate the actual collapse pressure by dividing Pcr by the plastic knockdown factor PKD, (Pcr/Pexp), which is obtained from the design curve, where 1/l is plotted against Pcr/Pexp. To obtain the design pressure it will be necessary to divide the actual collapse pressure by a large safety factor. The thinness ratio l is calculated from Eq. (1) as shown earlier. The equation uses an equivalent wall thickness t1 to enable the corrugated cylinder to be represented as an unstiffened circular cylinder. The thickness t1 is such that the flexural stiffness of the equivalent unstiffened vessel is the same as the corrugated one; this is calculated using the formula below [Lb(t1)3]/12⫽INAN⫹⌺[(Lft3)/12]⫻2
(2)
where: INA = tw × (D /12) × 2; t = equivalent wall thickness; tw = width of a leg corrugation (see Fig. 13); tw = t/sina; Ds = depth of a leg of the corrugation; NA = neutral axis of swedge centroid; Lf = Length of flat part of vessel. The model AST9 is now used as an example of how to calculate the thinness ratio l and the Plastic Knockdown Factor. 3 s
1
4.3. A typical calculation for l and PKD (for AST9) Ds =0.55mm N =12 Ls =3.57 Table 3 Theoretical buckling pressures (Pcr) Model
Lb (mm)
Pcr (bar)
AST1 AST2 AST3 AST4 AST5 AST6 AST7 AST8 AST9
3.57 7.14 10.71 17.85 21.42 28.56 32.13 39.27 42.84
315.8 68.9 40.6 25.2 19.5 15 13.5 11.2 10
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Fig. 13. Swedge leg (assumed to be a parallelogram).
Lb =42.84 a
=tan−1[(2×Ds)/Ls] =tan−1[(2×0.55)/3.57]
a
=17.125
tw
=t/sina =0.25/sin17.125
tw
=0.849 mm
INA =tw×(D3s /12)×2 =0.849×(0.553/12)×2 INA =0.0235 As this series has no flat lengths on the vessel from Eq. (2): [Lb(t1)3]/12 =INA×N t1
=[(INA×N×12)/Lb]1/3 =[(.0235×12×12)/42.84]1/3
t1
=0.429
Using Eq. (1) the thinness ratio can now be calculated: l1
冑
=[(Lb/D1)2/(t1/D1)3]0.25× (sYP/E)
冑
=[(42.84/99)2/(0.429/99)3]0.25× (410×106/200×109) l1
=1.752
PKD =Pcr/Pexp =10/7.1 PKD =1.408
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Table 4 Calculations for thinness ratio and PKD Model AST1 AST2 AST3 AST4 AST5 AST6 AST7 AST8 AST9
e/t1 0.349 0.373 0.163 0.186 0.419 0.349 0.186 0.349 0.373
l 0.506 0.716 0.876 1.315 1.239 1.431 1.518 1.678 1.752
1/l 1.976 1.397 1.141 0.884 0.807 0.699 0.659 0.596 0.57
PKD 24.1 5.39 3.2 2.16 2.21 2.17 1.67 1.95 1.41
The results for the calculations for the remaining vessels are given in Table 4. Table 5 shows thinness ratios and PKD for other vessels, [9], where e = measured outof-circularity. Models excluded from Tables 4 and 5 were partially corrugated vessels and vessels with values of e/t1⬎2.91. Models AST1, AST2 and AST3 collapsed axisymmetrically and were therefore analyzed by the non-linear finite element computer program, named: “Stresses in ring-stiffened axisymmetric thin walled conical shells”. This gave poor results. To obtain the design chart, the results for this series has been combined with the provisional design chart of Ross and Waterman (1999), by plotting 1/l1 against PKD in Fig. 14 to create a more comprehensive chart other results (5 to 8) were also used.
5. Conclusions The apparent rogue results of AST7 and AST8 from this study and the GAW and MBL series from previous research, appear to indicate that traditional non-linear Table 5 Thinness ratios for other series Model GAW1 GAW2 GAW3 GAW4 GAW5 GAW6 DF MBS MBL CA
e/t1 1.02 2.91 1.53 0.42 2.75 2.09 0.185 0.185 0.365 0.46
l 1.483 1.716 2.015 2.275 2.508 2.722 3.57 2.67 3.53 2.73
1/l 0.674 0.583 0.496 0.44 0.4 0.367 0.28 0.37 0.28 0.37
PKD 2.77 2.23 2.39 1.48 1.36 1.35 1.15 1.61 1.69 1.45
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Fig. 14.
Design chart.
finite element solutions are not suitable. The theoretical method presented in this project seems to be preferable for the design of vessels such as these; it allows for the effects of plastic knockdown due to initial imperfections. To obtain the design pressure it is advisable to divide the actual collapse pressures obtained with the aid of the design charts, by a large safety factor. The results have shown that although the larger vessels fail by general instability as the length and number of corrugations decrease the vessels tend to fail in an axisymmetric manner. Vessels whose values of 1/l were less than 1 failed by general instability. The three vessels, which had the largest values of 1/l actually failed by plastic axisymmetric deformation. Ross and Johns (1998) have shown that there is a link between
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these to be work chart
277
two modes of failure for circular cylinders. Consequently there is also likely a similar link for corrugated circular cylinders. As a result more experimental is required for shorter vessels with greater values of 1/l, so that the design can be used with a greater degree of reliability.
Acknowledgements The authors would like to thank Miss V Harvey for the care and devotion she showed in typing this manuscript.
References Liang, C.-C., Yang, M.-F., Chen, H.-W., 1993. Elastic-plastic axisymmetric failure of swedge-stiffened cylindrical pressure hulls under external pressure. J. Ship Res. 37, 176–188. Ross, C.T.F., 1974. Lobar buckling of thin-walled cylindrical and truncated conical shells under external pressure. J. Ship Res. 18, 272–277. Ross, C.T.F., 1984. Finite Element Programs for Axisymmetric Problems in Engineering. Ellis Horwood, Chichester. Ross, C.T.F., 1987. A novel submarine pressure hull design. J. Ship Res. 31, 186–188. Ross, C.T.F., 1990. Pressure Vessels Under External Pressure. Routledge, London. Ross, C.T.F., Palmer, A., 1993. General instability of swedge-stiffened circular cylinders under uniform external pressure. J. Ship Res. 37, 77–85. Ross, C.T.F., Humphries, M., 1993. The buckling of corrugated circular cylinders under uniform external pressure. J. Thin-Walled Structures 17, 259–271. Ross, C.T.F., Heigl, T., 1995. Buckling of corrugated axisymmetric shells under uniform external pressure. ASME Conf on “Structured Dynamics and Vibration” PD-70, 199–205. Ross, C.T.F., Johns, T., 1998, Plastic axisymmetric collapse of thin-walled circular cylinders and cones under uniform external pressure. J. Thin-Walled Structures 30, 35–54. Ross, C.T.F., Lilleland, S.E., Richards, W.D., Little, A.P.F., 1996, Buckling and vibration of circumfrentially corrugated cylinders under uniform external pressure, ASME Conf. On “Engineering technology”, Book IV, 276–282. Ross, C.T.F., Waterman, G.A. 1999, Inelastic instability of circular corugated cylinders under external hydrostatic pressure to be published in Ocean Engineering. Windenburg, D.F., Trilling, C., 1934. Collapse by instability of thin cylindrical shells under external pressure. Trans. ASME 11, 819–825. Yuan, K.-Y., Liang, C.C., Ma, Y.C., 1991. Investigation of the cone angle of a novel swedged-stiffend pressure hull. J. Ship Res. 35, 83–86.
A design chart for the plastic collapse of corrugated cylinders under external pressure C.T.F. Ross *, A. Terry, A.P.F. Little Department of Mechanical Engineering, University of Portsmouth, Portsmouth PO1 3DJ, UK Received 29 August 1999; accepted 24 November 1999
Abstract The paper presents a theoretical and an experimental investigation into the plastic collapse of circular steel corrugated cylinders under external hydrostatic pressure. The experimental investigation gives a detailed study of 9 steel corrugated cylinders which were tested to destruction. Six of these cylinders failed by plastic non-symmetric bifurcation buckling and three failed by plastic axisymmetric deformation. The results of these tests were used, together with the results obtained from previous tests, to present a design chart for the plastic collapse of these vessels. The design chart was obtained by a semi-empirical approach, where the thinness ratios of the vessels were plotted against their plastic knockdown factors. The process of using the design chart is to calculate the theoretical elastic instability pressure for a perfect vessel by the finite element method and also to calculate the thinness ratio for this vessel. Using the appropriate value of the thinness ratio, the plastic knockdown factors are obtained from the design chart. To obtain the actual collapse pressure of the vessel, the theoretical elastic instability pressure for a perfect vessel is divided by the plastic knockdown factor. This work is of importance in ocean engineering. A large safety factor must also be introduced. 2000 Elsevier Science Ltd. All rights reserved. Keywords: Buckling; Inelastic; Corrugated cylinders; External pressure; Design charts
1. Introduction The earth’s oceans cover some 70% of its surface and reaches depths of up to 11.52 km (7.16 miles). Although there has been one small vessel which has been * Corresponding author. Tel.: +44-1705-842318; fax: +44-1705-842351. E-mail address: [email protected] (C.T.F. Ross). 0029-8018/01/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 9 - 8 0 1 8 ( 0 0 ) 0 0 0 0 7 - X
264
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Fig. 1.
Submarine pressure hull.
able to reach these depths, research is needed on how to manufacture larger vessels strong enough to survive the great pressures at these depths. These vessels would be used for retrieving oil and gas deposits from the seabed that lie at depths of up to 3 miles. Most submarine pressure hulls take the form of a cylinder/cone/dome construction, surrounded by casing to improve the hydrodynamic streamlining as shown in Fig. 1. Pressure hulls constructed in this way can fail either through axisymmetric yield or by non-symmetric bifurcation buckling, known as shell instability and as shown in Fig. 2. Under external hydrostatic pressure, the pressure required to cause the shell instability of a thin walled circular cylinder might only be a small fraction of
Fig. 2. Shell instability of thin walled circular cylinders.
C.T.F. Ross et al. / Ocean Engineering 28 (2001) 263–277
Fig. 3.
265
Ring stiffened vessel.
that to cause axisymmetric yield (Ross, 1984). This mode of failure is undesirable, as it is structurally inefficient. 1.1. Ring stiffened vessels The introduction of circumferential ring stiffeners spaced at suitable distances apart, as shown in Fig. 3, is one way in which the structural efficiency can be improved. Vessels such as these are used for the construction of submarine pressure hulls, torpedoes, water-borne ballistic missiles and aircraft. If the ring stiffeners are not strong enough, the entire ring-shell combination could buckle bodily as shown in Fig. 4. This form of buckling is known as general instability.
Fig. 4.
General instability of ring stiffened circular cylinders.
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1.2. Previous research of corrugated vessels In 1987, Ross (1987) presented an alternative design to the ring-stiffened vessel as shown in Fig. 5. The pressure hull was a circumferentially corrugated vessel. Ross showed that the corrugated vessel was structurally more efficient than a ring stiffened equivalent. In 1991, Yuan et al. (1991) showed that the corrugated cylinder could be made more structurally efficient by increasing the cone angles to certain optimum values. It should be noted that if the cone angles were too large the vessel would fail axisymmetrically, as shown by Liang et al. (1993). These reported studies were of a theoretical nature and experimental work to verify the results was carried out by Ross (1990), Ross and Humphries (1993), Ross and Heigl (1995) and Ross et al. (1996). In 1999, Ross and Waterman (1999) carried out more experimental work and used this and existing results to produce a design chart. Experimental investigations have shown that like unstiffened and ring stiffened circular cylinders, the vessels were prone to suffer plastic knockdown due to initial out-of-circularity. To allow for this, Ross and Palmer (1993) presented a thinness ratio, which was based on the thinness ratio of Windenburg and Trilling (1934); the formula for which is given by the following equation.
冑
l⫽[(Lb/D1)2/(t1/D1)3]0.25⫻ (sYP/E)
(1)
where: Lb is the length between bulkheads; D1 is the equivalent diameter; t1 is the equivalent thickness; sYP is the yield stress and E is the Young’s modulus of elasticity.
2. Theoretical analysis To determine by analytical methods the uniform external pressure required to cause the elastic instability of thin walled conical shells is very difficult and for these cases it is more convenient to use the finite element method. If the cone is of small apical angle then buckling will take place in a lobar manner as shown in Fig. 2 but if the angle of the cone is large the vessel can buckle axisymmetrically.
Fig. 5. Corrugated pressure hull.
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267
It must be emphasised that as these vessels are sensitive to initial geometric imperfections, they can suffer inelastic instability at buckling pressures considerably less than those causing elastic instability. This is where the importance of experimentally obtained results is crucial, to enable a plastic reduction factor to be obtained for a particular vessel. This plastic reduction factor must be divided into the theoretical elastic buckling pressure for the vessel in question, to obtain the reduced inelastic buckling pressure. Pcr/Pexp = plastic reduction factor = PKD Pcr = theoretical buckling pressure, based on the finite element method Pexp = experimentally obtained buckling pressure The theoretical solution based on small deflection elastic theory uses the element first developed by Ross in 1974 of a truncated cone with two nodal circles at its ends, as shown in Fig. 6. Each node has four degrees of freedom, namely (u0, v, w0, and q), making a total of eight degrees of freedom per element, where: u0 is the displacement in the x° direction; v is the displacement in the circumferential direction; w0 is the displacement in the z° direction; and, q is the rotational displacement. 2.1. Objectives To enable the authors to complete this project successfully, five main objectives were set out and completed in the following order: 1. To evaluate the experimental study on nine circular corrugated cylinders tested to destruction under hydrostatic pressure. Vessels were of varying lengths, with less than 13 corrugations. 2. To calculate theoretical elastic buckling pressures (Pcr) of these vessels, using finite element analysis. 3. To determine the plastic knockdown factor for each vessel due to initial outof-circularity. 4. To calculate the thinness ratio l1 for each vessel.
Fig. 6.
Truncated conical shell element.
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5. To create a design chart for these corrugated vessels together with results from other vessels.
3. Experimental procedure 3.1. Vessels Nine steel corrugated cylindrical models were tested to destruction. These models were AST1, AST2, AST3, AST4, AST5, AST6, AST7, AST8 and AST9, as shown in Fig. 7. Each vessel was carefully measured to determine its initial out-of-circularity, which was defined as the difference between the maximum inward and outward deviations of the external surface at the mid point of the cylinder. This was achieved with the use of a Mitutoyo BN706 co-ordinate measuring machine, together with a touch-trigger probe. The longitudinal profiles were of a sinusoidal nature, but for the theoretical analysis, it was found convenient to represent each corrugation with two truncated finite element conical shell elements as shown in Fig. 6; previous work by Ross and Humphries (1993), Ross and Heigl (1995), Ross et al. (1996) and Ross and Waterman (1999) has found this to be a satisfactory procedure. The geometrical details are
Fig. 7.
The AST series.
C.T.F. Ross et al. / Ocean Engineering 28 (2001) 263–277
269
Table 1 Geometrical details of AST series (mm) Model
AST1 AST2 AST3 AST4 AST5 AST6 AST7 AST8 AST9
Corrugations (N)
1 2 3 5 6 8 9 11 12
Out of circularity
0.15 0.16 0.07 0.08 0.18 0.15 0.08 0.15 0.16
Fig. 8.
Length between bulkheads (Lb) 3.57 7.14 10.71 17.85 21.42 28.56 32.13 39.27 42.84
Sketch of cylinder.
listed in Table 1; a sketch of the cylinder is shown in Fig. 8. Fig. 9 shows details of a typical corrugation, where: N is the number of corrugations; t is the wall thickness =0.25 mm; Ls is the length of corrugation =3.57 mm; Ro = RE = external radius of cylinder; and, Ri = RI = internal radius of cylinder. Young’s modulus and Poisson’s ratio were not measured, because the cylinders were made from typical carbon steels, so that the assumed values for these constants were reasonable. To determine the value for Yield stress, three specimens of length
Fig. 9.
Details of a typical corrugation (mm).
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150 mm by width 15 mm, were cut from the vessel. Material and tensile tests were performed on the Lloyd’s Materials testing machine. Fig. 10 shows the graph of load against extension. Material properties were found to be as follows: Young’s modulus = E =200 GPa (assumed); Poisson’s ratio = v =0.3 (assumed); Yield stress = syp =405 MPa (measured). The cylinders were blocked off at their ends by end closure plates. These closure plates were simply push-fitted onto the ends of the cylinders and sealed with silicone. The silicone was initially in a semiliquid form, which hardened after a few hours of exposure to the atmosphere, creating a watertight seal. 3.2. Pressure system The models were tested in the test tank shown in Fig. 11. Water was used as the pressure raising fluid. A Tangye Hydra hand pump was used to apply pressure to the tank. Line losses were negligible as the hose was only about 2 m in length. 3.3. Experimental procedure The pressure tank was partially filled with water prior to placing a vessel into it. The vessel which was previously made watertight, as described earlier, was placed in the pressure tank as shown in Fig. 11. An O ring seal was placed in the O ring groove of the tank. The lid was carefully placed onto the tank, washers and nuts were fitted and tightened evenly. Care was taken not to overtighten the tank top to maintain elasticity of the O ring seal. Prior to raising the water pressure, the bleed screw situated in the lid of the test tank was left open so that any trapped air could be pumped out. When the air had
Fig. 10.
Graph of tensile test.
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Fig. 11.
271
Test tank and model.
been pumped out and water flowed from the bleed hole, the screw to the bleed hole was tightened, so that the bleed hole was sealed and the experiment was ready to commence. At least two operators were required to carry out the tests; one to operate the hand driven pump and record the buckling pressure and the other to observe for any leaks in the system. Each model was tested to destruction by raising the pressure in steady increments. The pressure gauge was vigilantly monitored to ensure that the correct buckling pressure could be logged. Fig. 12 shows the vessels in their collapsed state. AST 4-9 collapsed in an inelastic lobar manner. AST 1, 2, and 3 collapsed axisymmetrically; these were the vessels of smallest overall length. The experimentally obtained results are shown in Table 2. It can be seen from Table 2 that the general trend is that as Lb increases the experimentally obtained buckling pressure decreases. AST6 and AST8 do not follow this pattern,: the buckling pressure for AST8 should be higher than that of AST9 and the buckling pressure for AST6 should be higher than AST7. However, this is typical of vessels of this type, where occasionally low experimental buckling pressures are observed, which cannot be described as rogue results. The result also cannot be explained by the out of circularity measurement as this is larger for AST9, but the buckling pressure is still higher than both the shorter vessels AST8 and AST6. It is because of results like these that the present study has been conducted, and why a traditional non-linear finite element is unlikely to prove successful.
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Fig. 12.
The collapsed models.
Table 2 Experimental buckling pressures (Pexp) Model
Out of circularity
Lb (mm)
Pressure at collapse (bar)
AST1 AST2 AST3 AST4 AST5 AST6 AST7 AST8 AST9
0.15 0.16 0.07 0.08 0.18 0.15 0.08 0.15 0.16
3.57 7.14 10.71 17.85 21.42 28.56 32.13 39.27 42.84
13.1 12.8 12.7 10.8 8.8 6.9 8.0 5.7 7.1
4. Determination of the design chart 4.1. Calculations of Pcr, the elastic buckling pressures To calculate the theoretical buckling of the vessels, a computer program called CONEBUCK (1984), based on finite element analysis, was used. The element adopted in the program was the thin walled truncated conical element of Ross (1990).
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This element has been described earlier in Section 2 and the element was shown in Fig. 6. The theoretical buckling pressures Pcr obtained by this program are shown in Table 3. It should be noted that the buckling pressures are based on elastic theory for perfect vessels. For imperfectly shaped vessels, which buckle inelastically, the experimental buckling pressure may be a small fraction of the theoretical values. 4.2. Calculation of the thinness ratios The design procedure is to first calculate theoretical pressure Pcr and thinness ratio l. The theoretical pressure is then used to calculate the actual collapse pressure by dividing Pcr by the plastic knockdown factor PKD, (Pcr/Pexp), which is obtained from the design curve, where 1/l is plotted against Pcr/Pexp. To obtain the design pressure it will be necessary to divide the actual collapse pressure by a large safety factor. The thinness ratio l is calculated from Eq. (1) as shown earlier. The equation uses an equivalent wall thickness t1 to enable the corrugated cylinder to be represented as an unstiffened circular cylinder. The thickness t1 is such that the flexural stiffness of the equivalent unstiffened vessel is the same as the corrugated one; this is calculated using the formula below [Lb(t1)3]/12⫽INAN⫹⌺[(Lft3)/12]⫻2
(2)
where: INA = tw × (D /12) × 2; t = equivalent wall thickness; tw = width of a leg corrugation (see Fig. 13); tw = t/sina; Ds = depth of a leg of the corrugation; NA = neutral axis of swedge centroid; Lf = Length of flat part of vessel. The model AST9 is now used as an example of how to calculate the thinness ratio l and the Plastic Knockdown Factor. 3 s
1
4.3. A typical calculation for l and PKD (for AST9) Ds =0.55mm N =12 Ls =3.57 Table 3 Theoretical buckling pressures (Pcr) Model
Lb (mm)
Pcr (bar)
AST1 AST2 AST3 AST4 AST5 AST6 AST7 AST8 AST9
3.57 7.14 10.71 17.85 21.42 28.56 32.13 39.27 42.84
315.8 68.9 40.6 25.2 19.5 15 13.5 11.2 10
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Fig. 13. Swedge leg (assumed to be a parallelogram).
Lb =42.84 a
=tan−1[(2×Ds)/Ls] =tan−1[(2×0.55)/3.57]
a
=17.125
tw
=t/sina =0.25/sin17.125
tw
=0.849 mm
INA =tw×(D3s /12)×2 =0.849×(0.553/12)×2 INA =0.0235 As this series has no flat lengths on the vessel from Eq. (2): [Lb(t1)3]/12 =INA×N t1
=[(INA×N×12)/Lb]1/3 =[(.0235×12×12)/42.84]1/3
t1
=0.429
Using Eq. (1) the thinness ratio can now be calculated: l1
冑
=[(Lb/D1)2/(t1/D1)3]0.25× (sYP/E)
冑
=[(42.84/99)2/(0.429/99)3]0.25× (410×106/200×109) l1
=1.752
PKD =Pcr/Pexp =10/7.1 PKD =1.408
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Table 4 Calculations for thinness ratio and PKD Model AST1 AST2 AST3 AST4 AST5 AST6 AST7 AST8 AST9
e/t1 0.349 0.373 0.163 0.186 0.419 0.349 0.186 0.349 0.373
l 0.506 0.716 0.876 1.315 1.239 1.431 1.518 1.678 1.752
1/l 1.976 1.397 1.141 0.884 0.807 0.699 0.659 0.596 0.57
PKD 24.1 5.39 3.2 2.16 2.21 2.17 1.67 1.95 1.41
The results for the calculations for the remaining vessels are given in Table 4. Table 5 shows thinness ratios and PKD for other vessels, [9], where e = measured outof-circularity. Models excluded from Tables 4 and 5 were partially corrugated vessels and vessels with values of e/t1⬎2.91. Models AST1, AST2 and AST3 collapsed axisymmetrically and were therefore analyzed by the non-linear finite element computer program, named: “Stresses in ring-stiffened axisymmetric thin walled conical shells”. This gave poor results. To obtain the design chart, the results for this series has been combined with the provisional design chart of Ross and Waterman (1999), by plotting 1/l1 against PKD in Fig. 14 to create a more comprehensive chart other results (5 to 8) were also used.
5. Conclusions The apparent rogue results of AST7 and AST8 from this study and the GAW and MBL series from previous research, appear to indicate that traditional non-linear Table 5 Thinness ratios for other series Model GAW1 GAW2 GAW3 GAW4 GAW5 GAW6 DF MBS MBL CA
e/t1 1.02 2.91 1.53 0.42 2.75 2.09 0.185 0.185 0.365 0.46
l 1.483 1.716 2.015 2.275 2.508 2.722 3.57 2.67 3.53 2.73
1/l 0.674 0.583 0.496 0.44 0.4 0.367 0.28 0.37 0.28 0.37
PKD 2.77 2.23 2.39 1.48 1.36 1.35 1.15 1.61 1.69 1.45
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Fig. 14.
Design chart.
finite element solutions are not suitable. The theoretical method presented in this project seems to be preferable for the design of vessels such as these; it allows for the effects of plastic knockdown due to initial imperfections. To obtain the design pressure it is advisable to divide the actual collapse pressures obtained with the aid of the design charts, by a large safety factor. The results have shown that although the larger vessels fail by general instability as the length and number of corrugations decrease the vessels tend to fail in an axisymmetric manner. Vessels whose values of 1/l were less than 1 failed by general instability. The three vessels, which had the largest values of 1/l actually failed by plastic axisymmetric deformation. Ross and Johns (1998) have shown that there is a link between
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these to be work chart
277
two modes of failure for circular cylinders. Consequently there is also likely a similar link for corrugated circular cylinders. As a result more experimental is required for shorter vessels with greater values of 1/l, so that the design can be used with a greater degree of reliability.
Acknowledgements The authors would like to thank Miss V Harvey for the care and devotion she showed in typing this manuscript.
References Liang, C.-C., Yang, M.-F., Chen, H.-W., 1993. Elastic-plastic axisymmetric failure of swedge-stiffened cylindrical pressure hulls under external pressure. J. Ship Res. 37, 176–188. Ross, C.T.F., 1974. Lobar buckling of thin-walled cylindrical and truncated conical shells under external pressure. J. Ship Res. 18, 272–277. Ross, C.T.F., 1984. Finite Element Programs for Axisymmetric Problems in Engineering. Ellis Horwood, Chichester. Ross, C.T.F., 1987. A novel submarine pressure hull design. J. Ship Res. 31, 186–188. Ross, C.T.F., 1990. Pressure Vessels Under External Pressure. Routledge, London. Ross, C.T.F., Palmer, A., 1993. General instability of swedge-stiffened circular cylinders under uniform external pressure. J. Ship Res. 37, 77–85. Ross, C.T.F., Humphries, M., 1993. The buckling of corrugated circular cylinders under uniform external pressure. J. Thin-Walled Structures 17, 259–271. Ross, C.T.F., Heigl, T., 1995. Buckling of corrugated axisymmetric shells under uniform external pressure. ASME Conf on “Structured Dynamics and Vibration” PD-70, 199–205. Ross, C.T.F., Johns, T., 1998, Plastic axisymmetric collapse of thin-walled circular cylinders and cones under uniform external pressure. J. Thin-Walled Structures 30, 35–54. Ross, C.T.F., Lilleland, S.E., Richards, W.D., Little, A.P.F., 1996, Buckling and vibration of circumfrentially corrugated cylinders under uniform external pressure, ASME Conf. On “Engineering technology”, Book IV, 276–282. Ross, C.T.F., Waterman, G.A. 1999, Inelastic instability of circular corugated cylinders under external hydrostatic pressure to be published in Ocean Engineering. Windenburg, D.F., Trilling, C., 1934. Collapse by instability of thin cylindrical shells under external pressure. Trans. ASME 11, 819–825. Yuan, K.-Y., Liang, C.C., Ma, Y.C., 1991. Investigation of the cone angle of a novel swedged-stiffend pressure hull. J. Ship Res. 35, 83–86.