Extreme submarine NOT FOR REPRODUCTION

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1 Extreme submarine Ross, Spahiu, Brown and Little present new charts for use in the design of small submarines to descend to the bottom of the deep ocean. Carl T.F. Ross Astrit Spahiu Graham X. Brown Who should read this paper? The paper should be read by designers and manufacturers of cylindrical structures such as submarines or autonomous underwater vehicles that may be subject to external hydrostatic pressure. Under external pressure, circular cylinders normally collapse at pressures which are often a small fraction of those to cause the same vessels to collapse under uniform internal pressure. This mode of failure is called shell instability. Why is it important? The work reported here is innovative in that it presents a new design chart for circular cylinders collapsing under external hydrostatic pressure based on both theoretical and experimental data. The design chart consists of two straight lines, rather than the more complex curves typical of well-known codes. The design chart also allows the designer to design much shorter and thicker vessels than the standard codes do. The authors contend that this will enable engineers to design pressure vessels that can dive to a much greater depth than previously known. Currently, most large submarines can only dive to a maximum depth of about 400 m (0.5 mi.), but the maximum depth of the oceans is some 9 times deeper than this. About the authors Carl Ross is Professor of Structural Dynamics at the University of Portsmouth, UK, where he has been employed since His research interests include the statics, stability, and dynamics of submarine pressure hulls. Astrit Spahiu was a mechanical engineering student at the University of Portsmouth, UK, where he researched on the collapse of model submarine pressure hulls under external hydrostatic pressure. Currently he works for Pall Europe, Ltd., Portsmouth, UK. Andrew P.F. Little Graham Brown is Chief Mechanical Engineer at Sonardyne International Ltd., Yateley, Hampshire, UK. His interests are in the design, construction, and testing of pressure vessels under external hydrostatic pressure. Andrew Little is Principal Lecturer at the University of Portsmouth in the UK. His interests lie in the statics, stability, and dynamics of submarine pressure hulls, which he has been researching since The Journal of Ocean Technology Reviews & Papers Copyright Journal of Ocean Technology 009

2 BUCKLING OF NEAR-PERFECT THICK-WALLED CIRCULAR CYLINDERS UNDER UNIFORM EXTERNAL HYDROSTATIC PRESSURE Carl T.F. Ross 1, Astrit Spahiu 1, Graham X. Brown, and Andrew P.F. Little 1 1 University of Portsmouth, Portsmouth, United Kingdom ( Sonardyne International Ltd., Yateley, Hampshire, United Kingdom ( ABSTRACT The aim of this study was to produce design charts to predict inelastic collapse pressures for thick-walled circular cylinders under uniform external pressure because the existing charts were out of the range for shorter and thicker vessels. Both theoretical and experimental investigations were carried out on 15 stainless steel models, which were tested to destruction and reported for the first time. A theoretical investigation was also carried on other models, tested by previous researchers, to give more points and more credibility to the design chart. The theoretical investigation was based on an analytical method because previous work proved that, in general, it was superior to numerical methods for this particular problem. It was hoped that the details from the current series of models, together with the new design chart, would enable some smaller submarines to descend to the bottom of the Mariana s Trench (11.5 km or 7.16 mi); one of the models collapsed at a pressure of about 1000 bar, which was equivalent to a submarine diving to a depth of about 10 km (6. mi). The analytical solution adopted the von Mises buckling analysis via a home produced computer program called MisesNP, which also calculated the Windenburg thinness ratio (λ). By plotting the reciprocal thinness ratio against the plastic knockdown factor (PKD), where the PKD was obtained by dividing the theoretical buckling pressure by the corresponding experimentally obtained buckling pressure for each vessel, a useful design chart was produced. INTRODUCTION Research has found that the oceans contain large quantities of precious metals and minerals and Dickens et al. [1997] have estimated that there are about 10,000 billion tonnes of frozen methane hydrates buried underground in the deep oceans. In monetary terms, the total value of this gas is about $7,500 trillion. Some people believe that we should leave this methane where it is, but the authors of this paper believe that the temptation for humankind to mine this fossil fuel will be much too great. Currently a large submarine can only dive to a depth of about 400 m (0.5 mi) and this is one of the reasons why the present study has been conducted. Submarine pressure hulls are usually composed of a combination of circular Copyright Journal of Ocean Technology 009 Maritime and Port Security, Vol. 4, No.,

3 cylinders, cones, and domes, especially the former. Under external pressure, circular cylinders normally collapse at pressures, which are often a small fraction of those to cause the same vessels to collapse under uniform internal pressure. This mode of failure is called shell instability [Ross, 001; Bryan, 1888; von Mises, 1914; Windenburg and Trilling, 1934; Bryant, 1954], or lobar buckling, Figure 1: Shell instability (The TVR Series). Figure : Ring-stiffened circular cylinders. Figure 3: General instability of ringstiffened circular cylinders. where the vessel collapses around its circumference in the form of a number of circumferential waves or lobes; it is shown in Figure 1, where it can be seen that the number of lobes was six for the models shown. It is an undesirable mode of failure, as it is structurally inefficient and one way to improve its structural efficiency is to ring-stiffen its flank, as shown in Figure. If the ring-stiffeners are not strong enough, the entire ring-shell combination can collapse bodily, as shown in Figure 3 [Ross, 001; Bryant, 1954; Nash, 1995]. This mode of failure is known as general instability. Another mode of failure is known as axisymmetric deformation [Ross, 001; Ross, 1999], where the circular cylindrical shell Figure 4: Axisymmetric collapse. implodes inwards, keeping its circular form while collapsing, as shown in Figure 4. In this paper, we will only consider the shell instability and axisymmetric modes of failure. THEORETICAL ANALYSES Axisymmetric Failure As described earlier, one mode of failure is called axisymmetric deformation. For un-stiffened thin-walled circular cylinders [Ross, 1999] under uniform pressure, the membrane principal stresses for this mode of failure are given by: σ H = Hoop Stress = pr/h (1) σ L = Longitudinal Stress = pr/(h) where P = Pressure R = Internal Radius h = Wall Thickness We will need the Hoop Stress formula of equation (1) to show how the Windenburg thinness ratio [Windenburg and Trilling, 1934] is derived. 86 The Journal of Ocean Technology Reviews & Papers Copyright Journal of Ocean Technology 009

4 Shell Instability Previous researchers have found that shell instability [Ross, 001; Bryan, 1888; von Mises, 1914; Windenburg and Trilling, 1934] is the most important failure mode of pressure vessels under external pressure because thinwalled circular cylinders have little or no resistance to this mode of failure. One of the first buckling analyses by this mode of failure, where the end boundary conditions corresponded to that of simple-supports, was presented in 1914 by von Mises [1914], as follows: p = where Eh R n 1 1 π R + L + n πr L πr + L n = the number of circumferential waves or lobes that the vessel buckles into L = unsupported length of the circular cylinder E = Young s modulus μ = Poisson s ratio Another simpler formula for elastic shell instability is from the David Taylor Model Basin [Windenburg and Trilling, 1934]; that is as follows: p =.4E ( 1 µ ) h L L h 0.45 R R 1 (3) 4 However, the above theories are for thin nearperfect vessels that buckle elastically. In practice, many shorter and thicker vessels buckle inelastically at pressures that are a small fraction of the predictions of elastic theory. A thick-walled circular cylinder was defined by Wilson [1956] as one where h/r > 1/30. For the present series, h/r=1/9.3. Attempts to analyse the thicker and shorter types of vessels by so-called exact theories have not been successful. This is because many models give rogue results where vessels, which one would expect to have a higher h R ( µ ) n πr + L () buckling resistance than similar thinner models, do not always follow the expected common sense behavioural patterns. In the present paper, the problem of reduced buckling pressures due to inelastic instability is addressed with the aid of the Windenburg thinness ratio λ ; this has been successfully achieved in previous publications [Ross, 001; Little et al., 008; Ross et al., 008]. In the case of the present paper, this facility has been extended so that shorter and thicker vessels are catered for. The importance of this is that thicker walled submarine pressure hulls can dive deeper into the oceans. One attempt to numerically analyse initially imperfect thick-walled circular cylinders was by Bosman et al. [1993]. However, the initial out-of-circularity of the model of Bosman et Copyright Journal of Ocean Technology 009 Maritime and Port Security, Vol. 4, No.,

5 al. was some 50 times of that of the present series and, whereas they obtained good results, it cannot be considered here. Derivation of the Windenburg Thinness Ratio Experiments on circular section tubes of intermediate and shorter lengths when the thinness ratio, namely λ [Ross, 001; Windenburg and Trilling, 1934], has a value of less than about 1.3, have shown that they fail somewhere in-between the pressures of equation (1) and (3). Windenburg and Trilling [Ross, 001; Windenburg and Trilling, 1934] argued that if we equated equations (1) and (3), we can get a thinness ratio relating these two modes of failure, which will enable us to precisely predict the collapse pressures for intermediate circular cylinders. They called this their thinness ratio λ. (Note: Windenburg and Trilling [1934] squared λ in the above calculation, so that for most intermediate length vessels the value of λ would be approximately one.) It should be noted that in the above analyses, the general instability mode of failure was not considered. MODELS AND SPECIMENS Material Properties The material selected to carry out this study was Duplex Stainless Steel (DSS) UNS S31803, which was produced by extrusion. It is a kind of Cr ferrite-austenite, supplied by Forfab Limited, Scotland, via Sterling Tubes. The chemical composition is given in Table 1. Now if we examine equation (3), we can see in the denominator on the right hand side of equation () that L/d is much larger than 0.45*(h/d) 0.5, thus if we neglect 0.45*(h/d) 0.5 and assume that µ = 0.3, we can simplify equation (3) to the form: Pcr =.6*E*(h/d).5 / (L/d), (4) where d = R. Equating (1) and (4), we get σyp*(h) / d =.6E*(h/d).5 / (L/d) Or σyp*h/d = λ * E*(h/d).5 / (L/d) Or λ =(L/d) / (h/d) -1.5 * (σyp / E) Or λ = [(L/d) / (h/d) 3 ] 0.5 * (σyp / E) 0.5 Alloying elements affect properties and the microstructure of DSS in various ways, thus each must be understood in order to maximise the effectiveness and to prevent the alloying element from becoming harmful and instead being beneficial to the marine application. The Present Series of Models Fifteen Duplex stainless steel tube specimens with a wall thickness of 3.07 mm were tested; their geometrical properties are shown in Table. From Table, it can be seen that the ratio of the initial out-of-circularity to the wall thickness of these vessels varied from h to 0.09 h, where h = the wall thickness. That is, the vessels were nearly geometrically Table 1: Chemical Composition of SAF05 Duplex Stainlesss Steel (%). C Si Mn Cr Ni Mo N Note: The data was provided by the suppliers in their Material Test Certificate. 88 The Journal of Ocean Technology Reviews & Papers Copyright Journal of Ocean Technology 009

6 Models Overall Unsupported Mean Out of O/D Length L o Length L Radius Circularity (e) AS 1a AS 1b AS a AS b AS 3a AS 3b AS 4a AS 4b AS 5a AS 5b AS 6a AS 6b AS 7a AS 7b AS Table : Models' Data (mm). perfect when compared with the charts of references 11 and 1, where the vessels had a maximum initial out-of-roundness of up to 0.16 h, or more than five times those of the present series. Thus, it is fair to refer to the present series as being of near perfect circularity, based on their e/h values, where e = initial out-of-circularity. The initial out-of-circularity was measured by a Mitutoya Co-ordinate Measuring Machine, where the out-of-circularity was defined as the maximum deviation from an inward point to an outward point, on a mean circumference. The mean circumference was obtained from a least squares fit at mid-bay. The material was supplied in the form of a long extruded tube of length of about 1300 mm; this tube was cut to the required lengths, as shown in Figure 5. Outer Diameter = 60 mm Wall Thickness = 3.07 mm To seal each tube due to the effects of external hydrostatic pressure, two end bungs were manufactured. The end bungs were made of mild steel and were fitted with O-ring nitrile gaskets to seal against any water ingress that may have occurred while the specimens were under external hydrostatic pressure; see Figure 6. The chosen unsupported length L and the actual overall length L o, of each model is shown in Figure 7, where it can be seen that the value of L was considered to be between the O-ring gaskets. Tensile Testing The tensile strength of the material was obtained with two uniaxial specimens and its Copyright Journal of Ocean Technology 009 Maritime and Port Security, Vol. 4, No.,

7 Figure 5: Test models and end bungs with O-Ring nitrile gaskets. Young s modulus was obtained with a circular ring specimen; see Figure 8. Figures 9 and 10 show the load-extension relationships for two uniaxial tensile tests from which the yield stress and the ultimate tensile strength were determined. Figure 11 shows the load-deflection relationship for the ring specimen, which was used to obtain the Young s modulus of Duplex stainless steel. The ring specimen was loaded diametrically in compression and the resulting diametrical deflection was automatically recorded. The relationship can be seen in Figure 11. From this relationship, the experimental Young s modulus was calculated from Roark s formula, Table 17 and [Young, 1989]. Whereas Figures 9 and 10 are Load-Deflection relationships and it may have been preferable Figure 6: Details of the end bungs (mm). Figure 7: Assembly view of the models with the end bungs. 90 The Journal of Ocean Technology Reviews & Papers Copyright Journal of Ocean Technology 009

8 Figure 8: Specimens used to obtain the mechanical properties. Figure 9: Uniaxial tensile test results (first test). Tensile test to find yield stress and UTS; δ = 4 mm. Figure 10: Uniaxial tensile test results (second test). Tensile test to find yield stress and UTS; d = 4 mm. Copyright Journal of Ocean Technology 009 Maritime and Port Security, Vol. 4, No.,

9 Figure 11: Non-destructive O Ring test to obtain E. O-Ring testing results to find Young Modulus E; δ = 1 mm. Experimental Data Manufacturer s Data (1) & () Young Modulus E (GPa) Ultimate Tensile Strength σ UTS (MPa) Yield Stress σ YP (MPa) Poisson s Ratio µ (Assumed) Table 3: Mechanical Properties of Duplex Stainless Steel. to have plotted Stress-Strain relationships, Figures 9 are 10 were automatic computer outputs from the tensile testing machine. The measured and the manufacturer s values of the material properties are given in Table 3. THEORETICAL INVESTIGATION MisesNP: Buckling Predictions The computer program MisesNP calculates the elastic buckling pressure for perfect circular cylinders subjected to uniform external pressure and simply supported at their ends, together with the Windenburg thinness ratio λ [Windenburg and Trilling, 1934]. This program can be used on most personal computers and at present it is supplied free of charge. The buckling pressures are based on the David Taylor Marine Basin (DTMB) formula and also the von Mises formula [Windenburg and Trilling, 1934]. The following input data was used for this program: Unsupported Length; from Table Shell Thickness: 3.07 mm Mean Radius; from Table Young Modulus E; from Table 3 Poisson s Ratio; from Table 3 Yield Stress σ yp ; from Table 3 9 The Journal of Ocean Technology Reviews & Papers Copyright Journal of Ocean Technology 009

10 Figure 1: Output of the computer program MisesNP. NOT FOR REPRODUCTION A screen dump of the output of MisesNP is shown in Figure 1 and the results are shown in Table 4; it is a very easy program to use and the instructions for using it are given in [Ross, 001]. EXPERIMENTAL INVESTIGATION The main aim of the experimental investigation was to determine the collapse pressures of the Duplex stainless models. The models were sealed when the end bungs were fitted together with their O-ring gaskets. Equipment Used The specimens were tested in a high-pressure tank supplied by Sonardyne Limited (Yateley, Hampshire, United Kingdom) with a maximum working pressure of 100 bar; see Figure 13. The equipment was pressurised by a hand-driven hydraulic pump; thus, line losses were negligible. The hydrostatic pressure was measured by a Bourdon Tube pressure gauge. Experimental Procedure 1. Each specimen was sealed off with the end bungs.. Each model was submerged in turn in the pressure tank, which was filled with water. 3. The tank s closure plate was screwed down. 4. The trapped air in the tank was pumped out through the closure plate s bleed hole and then the bleed hole was sealed. Model MisesNP DTMB No Lobes P cr1 (MPa) P cr (MPa) λ 1/ λ AS1a AS1b ASa ASb AS3a AS3b AS4a AS4b AS5a AS5b AS6a AS6b AS7a AS7b AS Table 4: Theoretical results for the present series. Copyright Journal of Ocean Technology 009 Maritime and Port Security, Vol. 4, No.,

11 Figure 13: High Pressure Tank (material of construction: Stainless Steel). 5. The hydrostatic pressure was raised by a hand-driven hydraulic pump connected by a flexible hose to the tank. 6. The collapse pressures of each model were noted; each model s failure was accompanied by a bang and a corresponding drop in the hydrostatic pressure. 7. The tank was depressurised via the bleed valve located in the closure plate after each model collapsed. 8. After each model collapsed, it was removed from the test tank and examined closely by the naked eye. 9. The material of the models, when observed by the naked eye, did not appear to have suffered from visual fractures or other microstructure failures during the experiment. Experimental Results Table 5 gives the experimentally obtained collapse pressures Pexp, together with other observations. Although strain gauges were not used, the vessels appeared to fail plastically. This type of failure was assumed in the experiments because of the massive plastic deformations that took place when each vessel collapsed. If the vessels collapsed elastically, then they may have regained their shape, partially or fully, because of the hydrostatic pressure drop that was accompanied when each vessel collapsed. It is true some vessels that collapse elastically suffer from post-failure plastic deformation, but in general this plastic deformation would normally be relatively small due to the experimental procedure. 94 The Journal of Ocean Technology Reviews & Papers Copyright Journal of Ocean Technology 009

12 Model Overall Length Lo (mm) Unsupported Length L (mm) Pressure P exp (MPa) Pressure P exp (bar) Post-Failure Remarks AS 1a Perfect Axisymetric Failure AS 1b Perfect Axisymetric Failure AS a Plastic Buckling Partially Axisymetric AS b Lobar Buckling Partially Axisymetric AS 3a Inelastic Buckling AS 3b Inelastic Buckling AS 4a Inelastic Buckling AS 4b Inelastic Buckling AS 5a One Sided Failure over its length AS 5b Inelastic Buckling AS 6a One Sided Failure over its length AS 6b One Sided Failure over its length AS 7a One Sided Failure over its length AS 7b One Sided Failure over its length AS One-sided Failure over its length Table 5: Experimental results for the present series. The collapsed models are shown in Figure 14, where it can be seen that many suffered considerable plastic deformation. DESIGN CHARTS This section brings together the theoretical and experimental results from the proceedings of present series with the results from other experiments to present a design chart that can cater for a wider geometry than previous design charts. Similar design charts for geometrically imperfect circular cylinders [Ross, 001; Little et al., 008; Ross et al., 008] have been produced elsewhere. Design Chart for the Present Series First the Plastic Knockdown Factor (PKD) must be calculated in order to produce the design chart. This is defined between the relationship of theoretical and experimental buckling pressures presented on previous sections through investigations. Hence: P PKD = P cr 1 exp Pcr1 = von Mises Theoretical Buckling Pressure Pexp = Experimental Buckling Pressure The plastic knockdown factors are listed in Table 6 together with the thinness ratios Copyright Journal of Ocean Technology 009 Maritime and Port Security, Vol. 4, No.,

13 Figure 14: Plan view of the collapsed AS Series of models together with the end bungs and the nitrile O Rings. Model Unsupported P cr1 MisesNP P exp Λ 1/λ Length (mm) (MPa) (MPa) PKD AS1a AS1b ASa ASb AS3a AS3b AS4a AS4b AS5a AS5b AS6a AS6b AS7a AS7b AS Table: 6 Plastic Knockdown Factors for the present series. 96 The Journal of Ocean Technology Reviews & Papers Copyright Journal of Ocean Technology 009

14 Table 7: Theoretical and Experimental results of Sturm. Table 8: Collapse of ring reinforced cylinders by Ross and Reynolds. * Denotes Reynolds inelastic lobar buckling of circular cylindrical shells under external hydrostatic pressure. calculated from the computer program MisesNP. Results Obtained from Other Geometrically Perfect Circular Cylinders Collapse pressures of Sturm s thin-walled circular cylinders The work carried out in Sturm s study [Sturm, 1941] in Table 7 was on eight carefully manufactured models. The models were manufactured through extrusion. They were made very precisely in aluminium alloy; their out-of-circularity was not given. Collapse pressures of other models The results carried by Ross [1965] and Reynolds [1960] on machined stiffened models are given in Table 8. These models were found to have failed by inelastic shell instability. Models 1 to 3 were machined from aluminum alloy and their initial out-of-circularity was less than 0.08 mm. Models U1 and U were machined at the David Taylor Model Basin; they were made from high-strength steel, but their initial out-of-circularity was not given. Table 9 shows the results obtained from Ross Model 7 [Ross, 1965], which was compared with the experimental buckling pressure of this isotropic model and the David Taylor Model Basin formula. The model was machined from mild steel and its initial out-of-circularity was 0.1 mm. Copyright Journal of Ocean Technology 009 Maritime and Port Security, Vol. 4, No.,

15 Lobar Buckling, Thin-Walled Cylinder, Ross [1965] Model Λ 1/λ DTMB formula Experimental PKD (Pcr/Pexp) Table 9: Buckling Pressure of Model Number 7. Buckling Pressures with Boundary Conditions, Ross and Johns [1971] Model λ 1/λ von Mises Experimental PKD (Pcr/Pexp) TVR TVR TVR Table 10: Buckling Pressures for TVR Series; see Figure 1. Ross et al. [1995] Model λ 1/λ von Mises Experimental Table 11: Buckling Pressures based on shell Instability. PKD (Pcr/Pexp) In another study, Ross and Johns [1971] believed that boundary conditions play an important role in the experimentally obtained buckling pressures for many vessels. Fixed boundaries are found to produce larger collapse pressures than those with simply-supported edges. This proof comes from testing to destruction (see Table 10) of three machine-stiffened circular cylinders under uniform external pressure (see Figure 1), where TVR-1 had the largest ring stiffeners at the ends of the shell and TVR-3 had the smallest ring stiffeners at the ends of the shell. The models of Tables 10 to 1 were carefully machined from mild steel and their initial out-of-circularity was less than 0.13 mm, where the initial out-of-circularity was measured with the aid of a Talyrond Machine. The theoretical and experimental buckling pressures obtained by Ross et al. [1995] based on the shell instability of circular cylinders is given in Table 11. According to Ross [001], there may be a connection between plastic shell instability and plastic axisymmetric collapse of machined circular cylinders under uniform external pressure. The results from theoretical and experimental investigations of buckling pressures showed this (see Table 1). 98 The Journal of Ocean Technology Reviews & Papers Copyright Journal of Ocean Technology 009

16 Plastic Axisymmetric Buckling of Thin-Walled Circular Cylinders, Ross [1996] Model Λ 1/λ von Mises Experimental PKD (Pcr/Pexp) Table 1: Plastic axisymmetric buckling pressures of machined cylinders. Inelastic and elastic stability of ring stiffened shells, Hom and Couch [1961] Model λ 1/λ DTMB PKD Experimental formula (Pcr/Pexp) Table 13: Results of Hom and Couch [Ross and Johns, 1971]. In Table 13 the results are listed from ten machined stiffened circular cylinders tested to destruction under external hydrostatic pressure by Hom and Couch [1961]. The cylinders were designed with geometries so that they would collapse in the lobar buckling range and they were machined from strain hardened steel. The models of Table 13 were machined by the DTMB, but their initial out-of-circularity was not given. Table 14 gives the results of Seleim and Kennedy [1990]. These models were machined in aluminium alloy and their maximum initial out-of-circularity was 0.9 mm. Updated Design Chart Including the Present Series The updated design chart, which allows for shorter and thicker vessels to be analysed, is shown in Figure 15. This design chart adopts the results of the present series of models together with those of similar near-perfect circular cylindrical shells that collapsed under external hydrostatic pressure. The linearity of the chart in the plastic zone, where the PKD is much greater than unity, appears to indicate that it will prove a very useful design tool for industry. Copyright Journal of Ocean Technology 009 Maritime and Port Security, Vol. 4, No.,

17 Model 1/λ Pcr MisesNP (MPa) Pexp (MPa) PKD (Pcr/Pexp) Table 14: Results of imperfection sensitivity of stiffened cylinders [Ross et al., 1995]. Figure 15: Design chart for geometrically perfect circular cylindrical shells. The design chart was obtained from models made from mild steel, high-tensile steel, stainless steel, and aluminium alloys; thus, the chart should be suitable for structural design in such metals. The elastic portion of the chart is to the left of the vertical line and the plastic portion is above the long inclined line; that is, where the reciprocal thinness ratio is greater than 0.8. How to Use the Design Charts Determine the design diving depth of a circular cylindrical section of a submarine pressure hull with the following particulars: Mean diameter = 14m Wall thickness = 7.5 cm Unsupported length L = 1m Young s modulus = E =00 GPa Yield stress = σyp = 500 MPa Poisson s ratio = ν =0.3 Safety Factor = SF= 3 Inputting the above data into the computer program MisesNP, we get the following: Theoretical buckling pressure = Pcr = 6.1 MPa, with n = 13 lobes & λ = Therefore 1/λ = 1.48 and, from Figure 15, Plastic Knockdown Factor = PKD = 4.. Therefore the predicted collapse buckling pressure = 6.1 bar/4. = 6.4 bar. Design buckling pressure = 6.4/SF = 6.4/3 = 0.8 bar. Design Diving depth = 08 m = 68 ft. To enable the vessel to dive to a greater depth, it will be efficient to decrease the value of the unsupported length between frames (i.e. L should be made smaller). 100 The Journal of Ocean Technology Reviews & Papers Copyright Journal of Ocean Technology 009

18 (Note: The use of the Safety Factor effectively gives us a lower bound.) DISCUSSION OF RESULTS The theoretical and experimental results are summarised in Table 6 for all models from the current series. The experimental results from Table 5 could be divided mainly into three categories. The shorter models AS1a and AS1b failed in the form of axisymmetric failure, while the slightly longer models ASa and ASb failed in the form of combined plastic shell instability and a plastic axisymmetric collapse. The mid-range models collapsed by an inelastic buckling mode. The remaining six models (AS5a, AS6a/b, AS7a/b, and AS8) suffered from one sided failure mode over their length (i.e. ovalling). Tested models from present series were found to have only minor geometrical imperfections overall; these models were thus nearly geometrically perfect. Previous work carried out by others suggests that such imperfections can influence experimental results. The theoretical investigation produced a consistent relationship between the length and the buckling pressure when applying the computer program MisesNP. Furthermore, the experimental results were also encouraging in terms of consigned buckling pressure results. In summary, from all the investigative proceedings considered in this paper, a useful design chart was successfully produced. CONCLUSIONS 1. The buckling resistance of the vessels was reduced because, in general, plastic shell instability took place.. All the models in this paper failed either axisymmetrically or by plastic or inelastic lobar buckling. 3. The design chart for use of cylindrical shells, including those made from Duplex Stainless Steel, was successfully produced. 4. The theoretical buckling pressures by MisesNP were higher than the experimentally determined ones because the theory was based on elastic theory; however, the models failed plastically at much-reduced pressures. 5. The present series of models produced additional points on the graph (see Figure 15), which gave the graph more credence than the previous one [Ross, 001]. It also allowed shorter and thicker vessels to be analysed than could be done previously so that submarine design can include vessels that can dive to much greater depths of water. 6. It is true that the British Ministry of Defence s (MOD) design charts [Defence Procurement Agency, 001] are used very successfully to design these vessels. However, when the AS series of models was plotted on the MOD chart, it showed that the MOD chart underestimated the collapse pressures of Models AS1a and AS1b by circa 50%. This is in contrast to the design chart of Figure 15, which underestimated the collapse pressures of Models AS1a & AS1b by only circa 5%. 7. In the case of BS5500 [British Standards Institution, 1980], it was even more conservative than the MOD chart when analysing short thick models, such as AS1a and AS1b. Here, it underestimated the collapse pressures of these vessels by circa 100%. Copyright Journal of Ocean Technology 009 Maritime and Port Security, Vol. 4, No.,

19 8. Many practising structural designers have complained to the present authors of the conservativeness of BS5500 when designing circular cylinders to withstand shell instability under external hydrostatic pressure. Many prefer the Design Chart of Ross [001] and have used it successfully over many years and have saved a lot of money. The present design chart extends Ross design chart to cater for shorter and thicker circular cylinders. 9. Moreover, the use of BS5500 to determine the buckling pressure of a circular cylinder, in comparison with using the method presented here, is very time consuming. This is because for BS5500 the von Mises buckling pressure has to be determined for every value of n from another set of charts in order to obtain the minimum value of the theoretical buckling pressure. This is in contrast to Ross computer program, namely MisesNP, which determines the minimum buckling pressure and the Windenburg thinness ratio in just a few seconds. Instructions on how to use this program are given in [Ross, 001]. ACKNOWLEDGEMENTS The authors would like to thank Mr. Jack Murray of Forfab Limited for supplying the material of construction of the models. REFERENCES Bosman, T.N., Pegg, N.G., and Kenning, P.J. [1993, May]. Experimental and Numerical Determination of Non-Linear Overall Collapse of Imperfect Pressure Vessel Compartments. (International Symposium on Naval Submarines 4, RINA, London, United Kingdom.) British Standards Institution [1980]. BS 5500 British Standard Specification for Unfired Fusion Welded Pressure Vessels, Issue 5. (United Kingdom: British Standards Institution.) Bryan, G.H. [1888]. Application of the Energy test to the collapse of a long thin pipe Under External Pressure. (Proceedings of the Philosophical Society, pp 87-9, Cambridge, United Kingdom.) Bryant, A.R. [1954]. Hydrostatic Pressure Buckling of Ring-Stiffened Tubes. (Report, Naval Construction Research Establishment, R-306, Dunfermline, Scotland.) Defence Procurement Agency [001]. SSP74 Design of Submarine Structures. (United Kingdom: Defence Procurement Agency, Sea Technology Group.) Dickens, G.R., Paull, C.K., Wallace, P., and the ODP Leg 164 Scientific Party [1997, January]. Direct measurement of in situ quantities in a large gas-hydrate reservoir. Nature, Vol 385. Hom, K. and Couch, P. [1961]. Hydrostatic tests of Inelastic and Elastic Stability of Ring-stiffened cylindrical shells Machined from Strain-hardening steel. (DTMB Report 1501, pp -11.) Little, A.P.F., Ross, C.T.F., Short, D. And Brown, G.X. [008]. Inelastic Buckling of Geometrically Imperfect Tubes under External Hydrostatic Pressure. Journal of Ocean Technology, Vol 3 No 1, pp Nash, W.A. [1995]. Hydrostatically Loaded Structures: The Structural Mechanics, Analysis and Design of Powered Submersibles. Exeter, United Kingdom: BPC Wheatons Limited. 10 The Journal of Ocean Technology Reviews & Papers Copyright Journal of Ocean Technology 009

20 Reynolds, T.E. [1960, August]. Inelastic Lobar Buckling of Cylindrical hells Under External Hydrostatic Pressure. (DTMB Report Number 139.) Ross, C.T.F. [1965]. The Collapse of Ringreinforced Cylinders Under Uniform External Pressure. Trans. Royal Institution of Naval Architects, Vol 107, pp Ross, C.T.F. and Johns, T. [1971]. The Effect of Stiffener size on Interframe Shell Instability of Ring-reinforced Circular cylinders. Journal of Ship Research, Vol 1, pp Ross, C.T.F., Haynes, P., Seers, A. and Johns, T. [1995]. Inelastic Buckling of Ringstiffened Circular Cylinders Under Uniform External Pressure. (International Conference Proceedings on Structural Dynamics and Vibration, American Society of Mechanical Engineers, P.D. Vol 70, pp ) Ross, C.T.F. [1996]. Plastic Axisymmetric Buckling of Thin-walled Circular cylinders Under Uniform External Pressure. (International Conference Proceedings on Engineering Technology, American Society of Mechanical Engineers, Book IV.) Ross, C.T.F. [1999]. Mechanics of Solids. Chichester, United Kingdom: Horwood Publishing Limited. Ross, C.T.F. [001]. Pressure Vessels: External Pressure Technology. Chichester, United Kingdom: Horwood Publishing Limited. Ross, C.T.F., Little, A.P.F., Brown, G.X. and Nagappan, A. [008]. Inelastic Shell Instability of Geometrically Imperfect Aluminium Alloy Circular Cylinders under Uniform External Pressure. Marine Technology, SNAME, Vol 45, No 3, pp Seleim, S.S. and Kennedy, J.B. [1990]. Imperfection Sensitivity of Stiffened Cylinders Subjected to External Pressure. Computers and Structures, Vol 34, No1, pp Sturm, G.R. [1941]. A Study of the Collapsing Pressure of Thin-walled Cylinders. (University of Illinois Bulletin Series, No 39, pp 1-77.) von Mises, R. [1914]. Der Kritische Aussendruck Zylindrischer Rohre. Zeitschrift Verienger Deutchscher Ingenieur, Vol 58, pp Wilson, L.B. [1956, December]. The Deformation under Uniform Pressure of a Circular Cylindrical Shell Supported by Equally Spaced Circular Ring Frames. (NCRE Report No. 337B, Dunfermline, United Kingdom.) Windenburg, D.F. and Trilling, C. [1934]. Collapse by Instability of cylindrical shells Under External Pressure. Trans. American Society of Mechanical Engineers, Vol 11, pp Young, W.C. [1989]. Roark s Formulas for Stress and Strain. New York, United States of America: McGraw-Hill Book Company. Copyright Journal of Ocean Technology 009 Maritime and Port Security, Vol. 4, No.,