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Page 1: composite pressure hulls for deep ocean submersibles

Composite Structures 32 (199.5) 331-343 Ekvier Science Limited Printed in Great Britain

0263-8223I95lS9.50 0263-8223(95)00028-3

Composite pressure hulls for deep ocean submersibles

Derek Graham DRA, Dunfennline, Fife KY11 2XR, Scotland UK

DRA Dunfermline first became involved in the development of a composite pressure hull for a deep ocean submersible with the NERC Autosub project and this work has continued under the European MAST II programme. This paper describes the analysis which has been carried out in support of an extensive programme of model testing. An anisotropic Lame solution has been developed for the analysis of cylindrical components under external pressure and this was used to evaluate some of the popular failure criteria. Extensive use was also made of the Finite Element method.

INTRODUCTION composites and metal matrix composites have great potential but are considered technically

Tethered remotely operated vehicles (ROVs) unfeasible at present. This leaves glass or car- are routinely used in the off-shore industry but bon fibre reinforced polymers as the only there are many potential applications for a com- current option which will provide a pressure pletely autonomous underwater vehicle (AUV) hull with a low enough weight to displacement with full ocean depth capability. DRA Dun- ratio to allow the required payload to be car- fermline’s fn-st involvement in the development ried. Most authors appear to favour the of such a vehicle came with the NERC (Natural theoretically stronger carbon fibre composites, Environment Research Council) AUTOSUB see for example Smith’ and Stachiw & Frame.2 project. This was originally prompted by con- Experimental work for both the AUTOSUB tern about climatic changes such as ‘global and the European Marine Science and Tech- warming’ and the need to understand fully the nology ‘MAST II’ programmes has been carried influence of the oceans in climatic behaviour. out using the high pressure test facilities at The ultimate objective is to develop a free DRA Dunfermline. A range of small scale swimming robotic instrument which is capable (typically about one-sixth) fibre wound speci- of repeated dives to full ocean depth, while con- mens have been tested under hydrostatic tinuously monitoring its surroundings. Other pressure as cylinders in the Ultra-High Pressure applications include the exploration and exploi- Vessel or as short ring specimens in a specially tation of resources from the ocean bed or developed ‘ring tester’. Over a period of time a perhaps under ice covered polar seas which are range of specimens have been produced by dif- inaccessible by other means. ferent manufacturers. Glass fibres and a variety

Submersibles which are capable of reaching of carbon fibres have been used in different great depths have of course been around for epoxy resins, specimen wall thickness has been some time. The pressure hulls of These vessels varied and alternative winding configurations are typically constructed of high strength steels have been employed. All specimens were fully or alloys of aluminium or titanium. This gives strain gauged and the resulting data was used to high weight to displacement ratios which makes provide measured values of material elastic them unsuitable for use in an autonomous vehi- properties and strengths. The main topic of this cle with limited energy carrying capability, paper is the associated analysis, which has been which has long endurance requirements. New carried out in parallel with the programme of materials such as solid glass, ceramics, ceramic testing.

331

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332 D. Graham

ANALfYSIS

A theoretical solution has been developed for an externally pressurised, thick walled compo- site cylinder with uniform, non-zero axial strain. This was initially developed for a cylinder con- sisting of orthotropic layers, described by Graham & Anderson,3 but has been extended to include shear coupling and thermal effects in individual laminae in a similar manner to that described by Tzeng & Chien.4 Extensive use has also been made of the Finite Element Method to investigate the fully three dimensional behav- iour of test cylinders. The package ASAS-NL’ was used.

Theoretical solution

For a single lamina with one axis of material symmetry the constitutive relations in cylindrical coordinates are

-cl, Cl2 Cl3 0 0 Cl6

Cl2 c22 c23 0 0 C26

Cl3 c23 c33 0 0 C36

0 0 0 c44 c 45 0

0 0 0 c45 c55 0

-Cl, C26 C36 0 0 C66

X E,-ci,AT “. (1)

where the suffices z, 8 and r denote the axial, circumferential and radial directions, respec- tively. Assuming that all shear strain components vanish, the remaining strain com- ponents relate to the displacement field thus

au U i3W

El=% , r se=- and E,=-=E a.2 z

(2)

where E, is the assumed constant axial strain. For the condition of axisymmetry and uniform axial behaviour the appropriate equilibrium equation is

aor cr--tJe -+ =O. ar Y

(3)

Using eqns (1) and (2), eqn (3) can be written in terms of the displacements as

r2 a% au -+r --A2u=@(r,AT,E,) i3r2 ar

(4)

where

C p=22 C 33

and qb(r, AT,.?,) is the forcing function. The general solution of eqn (4) is

u=c&+pr-‘+Y(r,AT,&) (9

where a and /3 are unknown coefficients and Y(r, AT,&) is the known particular solution. Equations (l), (2) and (5) can be used to express strains and stresses in terms of the coef- ficients M. and p, for example the radial stress becomes

o,=car~-‘-dBr_~-‘+ee,+gAT (6)

where c, d and e are functions of the stiffness coefficients and g is a function of the stiffness coefficients and the coefficients of thermal expansion. The coefficients a and /? are found for each layer by applying the boundary condi- tions cr,=O on the inner surface and rsr= -P on the outer surface, and the compatibility of radial stress and displacement at each interface between layers. Once these coefficients have been determined, the radial displacements are obtained from eqn (5), strains are obtained from eqns (5) and (2) and stresses are obtained from equations similar to eqn (6). This analysis has been programmed in FORTRAN for a PC and provides a rapid solution to the problem of an externally pressurised filament wound cylin- der.

Input data required by the program includes the nine elastic constants and three coefficients of thermal expansion of a single lamina in the lamina coordinate system (e.g., El is the Young’s modulus in the fibre direction and E2 the transverse direction, etc.) as well as the radius, thickness and details of the lay-up of the cylinder. Loading consists of the external pres- sure, a constant axial strain and a temperature difference, any of which can be set to zero. Imposed axial strain was chosen as an input since this was directly measured during tests,

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Composite pressure hulls 333

however, as a check, the resulting axial load and equivalent end pressure were included in the output for comparison with the actual applied pressure. To provide a more general analysis tool it is intended to provide the facility to specify both internal pressure and axial load as input conditions.

The lamina elastic properties are trans- formed, using the lay-up information, to give the stiffness matrix C in eqn (l), in cylindrical coordinates. The stresses and strains can then be calculated, in cylindrical coordinates, as described above. For each lamina the stresses can be transformed from the global cylindrical coordinate system to the local lamina axes using

611 m2

i ii

n2 2mn 022 = n2 m2 -2mn Cl2 -mn mn (m”-n”) 1 0,

X i i fJB (7)

bze

where m =cosO and n =sin 19. The through thick- ness stress, c33 is assumed to remain equal to the radial stress or. In theory the stresses in lamina coordinates can be used in conjunction with a failure criterion to predict the failure pressure of a cylinder, provided buckling does not occur. Some of the popular failure criteria have been investigated.

The maximum stress or maximum strain cri- teria can be useful in situations where stresses are highly unidirectional but for general loading some form of polynomial criterion is generally considered more suitable. A number of possibil- ities have been considered and programmed for comparison. The first, and simplest, was a ver- sion of the Azzi-Tsai criterion, given in eqn (8).

dl fJllC22 0222 62 -_- -

S12 SlS2 + s22 +s= 1 (8)

where Sl is the strength in the fibre direction, S 2 the strength transverse to the fibres and S 12 the in plane shear strength. If these strengths are different in tension and compression then the appropriate value is used depending on the sign of each stress component. The Tsai-Hill criterion accounts for a fully three-dimensional stress system and is shown in eqn (9) (from Ochoa & Reddy6).

1 1 1 - -_-

S+ ~22 ~32 611g22

1 1 1 - -_-

z+S32

1 1 1

(9

Again, tensile or compressive strengths are used as appropriate. The most general polynomial criterion is that of Tsai & WU,~ given by

Fioi+FqaiQj=l

where i,j= 1,2,. . .6 and

(10)

1 1 1 1 F1=---_----

SlT Sic ; F2~------ *

S2T s2c )

1 1 F3~------ -

s37- s3c ’

1 1 F11=

SlrSlc ; F22=

s27S2c ;

1 F33=

s37S3c ;

1 1 1 F&$=-

S232 ; Fss=-

s132 ; F@j=- *

s122 ’ (11)

F12= -1 1

2 Js 17Sl~S27S2~

Fi3= -1 1

2 Jsl~sl~s3*s3~

F23= _’ 1

2 &2TS2cS3TS3c

Page 4: composite pressure hulls for deep ocean submersibles

334 D. Graham

Subscripts T and C denote strengths in ten- sion and compression. This was used in two forms; its two-dimensional plane stress form (1) and its full three-dimensional form (2). Although this criterion is widely and success- fully used, one should be aware of its limitations. It does not differentiate between the very different modes of failure which occur in composite materials and also leads to the physically unlikely situation where failure under purely tensile stresses is dependent on the com- pressive strengths of the material and vice-versa. Hashing has attempted to overcome these difficulties by considering tensile and compressive, fibre and matrix modes separately. His criteria have not yet been included in the computer program but were considered, in some cases, for comparison.

Finite element analysis

Finite element analysis was employed for several reasons. It was used to provide some verification of the theoretical model, to inves- tigate axial variations due to end effects and to investigate overall buckling behaviour. A typical test cylinder is shown in Fig. 1. The ends were overwound to reduce stress levels near the sup- ports and ensure that failure occurred within the ‘test section’. Figure 2 shows three finite element idealisations of a test cylinder. The first uses 8-noded axisymmetric elements and pro- vides a rapid and detailed analysis of the stress distribution throughout the cylinder. It does not, however, give any information about non- axisymmetric buckling which was observed to occur in some of the thinner cylinders. The sec- ond model uses 8-noded thick shell elements, with variable thickness, to provide a three- dimensional analysis. The length to thickness ratio of the shell elements became rather low for the thicker-walled cylinders and so the third model, which uses 20-noded brick elements, was developed. This model is fully three-dimen- sional but is significantly more costly in terms of computer memory and CPU time.

Axisymmetric elements were used to model a slice of cylinder under plane strain conditions to give a direct comparison with the analytical solution described in the previous section and also to compare with the predicted stresses in the ‘test section’ of the specimens. Figure 3 shows circumferential and longitudinal stress

250

1

END REINFORCEMENT

Fig. 1. Typical test cylinder dimensions.

distributions, calculated through the thickness, for a 20 mm thick cylinder of tenax carbon fibres in epoxy resin, fibre wound at &55”. These cylinders were wound at DRA Fort Hal- stead and detailed material property data was supplied.g For comparison the stress distribu- tions were also calculated for a similar cylinder of steel. Although the total forces must be the same, the anisotropic nature of the CFRP cylin- der gives a very different distribution of stress with surprising results. In the isotropic cylinder the longitudinal stress is constant through the thickness while the circumferential stress decreases monotonically from the inner surface. Conversely, in this particular CFRP cylinder both longitudinal and circumferential stresses have maximum values on the outer surface with minima at some internal location. The form of the stress distribution in an anisotropic cylinder is dependent on the degree of anisotropy of the cylinder and this in turn is dependent on the properties of the material used and also the lay- up pattern. Thus Fig. 3 illustrates only one possible example from an effectively infinite range of possibilities. In this case differences of up to almost 20% exist between the isotropic and anisotropic solutions. Agreement is excel- lent, within l%, between the theoretical solution and the FE solution under plane strain conditions. Stresses calculated at the mid-bay of the supported cylindrical specimen differ by up to 1.6% for the circumferential stresses and up to 2.2% for the longitudinal stresses although this may be influenced by the fact that a coarser

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Composite pressure hulls 335

FJSMGEWFEMVIEW 2.3~05.D D.R.A. DUNFERMLINE (CRAY) SDEc94

zGiGiiq pGcGEq

3 4

Fig. 2. Finite element idealisations of test cylinders.

mesh was employed. In any event it would appear that the stresses at the centre of the ‘test section’ are reasonably close to those in a ‘long’ cylinder.

Figure 4 shows the stress distributions throughout a typical cylindrical specimen, 20 mm thick in this case, calculated using the axi- symmetric FE model, and it can be seen that the dominant stresses, i.e. longitudinal and cir- cumferential (Figs 4.2 and 4.3), are reduced in the thickened end sections. More detailed inspection reveals that the stresses are all con- stant, to within a few percent, over a central span of about 80 mm, just over half the ‘test

section’. The analysis reveals two potential stress concentrations, in particular of interlami- nar shear (Fig. 4.4). The most severe is at the inner edge where the specimen is supported but this is very localised. Perhaps of more concern is the stress concentration on the outer surface where the end ‘overwinding’ tapers onto the ‘test section’. There has been no direct experi- mental evidence that failure initiates in the region of this stress concentration but it is diffi- cult to establish the exact mode of failure since hydrostatic collapse normally results in substan- tial destruction of the specimens.

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336 D. Graham

-220

-230

-240 3

8 -250

-260

-270

-280

-290 62.5 67.5 72.5 77.5 82.5

radius (mm)

*@l%xlretical 4CFRP plane strain cylinder

+CFRP supported cylinder -cSteelplane strain cylinder

(a)

’ c ’ E a ’ ’ ’ ’ ’ 62.5 67.5 72.5 77.5 82.5

radius (mm)

+Tbeoretical 4CF’RP plane E&R&I cyliuder

-c CFRP supported cylinder -t Steel plane strain cylinder

(b)

Fig. 3. Comparison of predicted stresses for 20 mm thick cylinders: (a) circumferential stresses; (b) longitudinal stresses.

EXPERIMENTAL RESULTS

Some early tests carried out for the NERC AUTOSUB project proved to be somewhat dis- appointing. Theoretical work by Smith et a1.l’ favoured a polar wound CFRP cylinder, with a 2 : 1 distribution of circumferential and longitu- dinal fibres, from a range of options which included glass or carbon fibres in epoxy resin and polar and helical winding configurations. A number of specimen cylinders, which were approximately 1/6th scale with internal diameter of 125 mm and a nominal thickness of 12 mm, were obtained from two manufacturers. When

hydrostatically loaded, these cylinders failed at lower than expected pressures and there was evidence to suggest that some of the cylinders were tending to buckle. Mechanical testing of offcuts from the specimens,‘r gave a Young’s modulus, in the longitudinal direction, of less than half that which had been calculated and used in the original analysis. This raised serious doubts about the quality of the manufacturing process as well as the ability to provide reliable analyses.

Collaboration with DRA Fort Halstead, who have considerable experience in the manufac- ture of filament wound components, resulted in

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Composite pressure hulls 337

FEMGEN/pEMvIEw 2.3-0S.D D.RA. DuNwlMLINE (CRAW 6DEC94

izFzG&q -1

Fig. 4. Axisymmetric stress distributions.

the supply of a number of cylinders of varying materials, thicknesses and winding configura- tions, to be tested as part of the MAST II program. From a small initial batch the helically wound carbon fibre/epoq composite was selec- ted for further investigation. XAS high strength (low modulus) carbon fibres and epoxy resin were used and the winding angle of the cylin- ders was f55”. Cylinders of nominally 6 mm, 8 mm, 14 mm and 20 mm wall thickness and 125 mm internal diameter were supplied and tested. Details of the experimental results and support- ing FE calculations are described by Graham & AndersonI More recently a number of 20 mm thick helically wound cylinders manufactured

using Tenax carbon fibres and epoxy resin have been tested. These were seen as a suitable replacement after the discontinuation of the supply of XAS fibres. All measured collapse pressures for the 455” carbon fibre wound cylinders (both XAS and Tenax) are shown in Fig. 5 which will be discussed in more detail in the following sections.

XAS +55” cylinders

The first cylinders to be tested were 14 mm thick since these were expected to be most rep- resentative of a full scale hull capable of reaching ocean depth, corresponding to hydro-

Page 8: composite pressure hulls for deep ocean submersibles

338 D. Graham

A A’

“r

0 / I I 1

0 5 10 I5 20 25

Wall Thickness (mm)

Fig. 5. Collapse pressures of hydrostatically tested cylinders.

static pressure of 60 MPa, with reasonable safety factors. The cylinders were strain gauged internally with twelve right angle pairs, 30 apart, at mid-length. Cylinders were loaded axially, at relatively low levels to avoid non- linearity or damage, to provide a measure of axial stiffness, and then hydrostatically loaded to destruction. In total, five of these cylinders were tested after an initial batch of three had produced some surprising results. Circumferen- tial strains to failure of three of them are shown in Fig. 6. Cylinder (a) apparently shows an n=2 type collapse while (b) shows an n=3 collapse. Cylinder (c) gives no indication of overall buck- ling but it is thought that some localised failure may be responsible for the ‘spikes’ observed in the later traces, see Ref. 12. The other two cylinders gave no indication of buckling although one of them may have given a falsely low pressure because of leakage, although not at the lowest observed collapse pressure. There was considerable scatter in the observed col- lapse pressures which ranged from 65 MPa to 90 MPa, with only the highest of these giving a satisfactory safety factor.

Using laminate material properties supplied by the manufacturer,13 both three-dimensional FE models predicted elastic buckling pressures of the order of 200 MPa, two to three times

2 4 6 8 10 12 14 16 18 20 22 24 2

Strain Gauge Number

Strain Gauge Number

C>

1 3 5 7 9 11 13 IS 17 19 21 23 1

Strain Gauge Number

Fig. 6. Circumferential strains from three 14 mm thick XAS cylinders.

greater than the observed collapse pressures. The predicted buckling pressures and associated mode shapes, with alternative boundary condi- tions, are summarised in Table 1. In all cases, the buckling pressures for n=2 and n=3 mode shapes were within a few percent of each other.

A parametric studyI revealed that, for this specimen type, the in-plane material properties were the most influential regarding buckling, i.e. E,, E. and vze in cylindrical coordinates. A 20% change in one of these properties could alter the predicted buckling pressure by any- thing from about 10% to almost 35% and it is just about conceivable that combined effects could lead to errors of the magnitude observed. However, the in-plane properties could be deduced, at least approximately, from the strain data from the end load and hydrostatic tests and were found to agree reasonably well with

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Composite pressure hulls 339

Table 1. Predicted buckling pressures (MPa)

Element type Clamped ends Simply supported ends

Shell Brick

242.5 (n=3) 197.4 (n =2)

229.6 (n = 2) 188-7 (n=2)

the manufacturers data. It was certainly felt that the errors were extremely unlikely to be large enough to explain the observed discrepancy and some thinner walled specimens, which were designed to buckle elastically, were supplied.

One cylinder of nominally 6 mm thick and three of nominally 8 mm thick were tested and all buckled with an n = 3 mode shape. The 6 mm thick cylinder was loaded until it had clearly begun to buckle and then unloaded. This was repeated three times with the onset of buckling observed at about 29 MPa each time before the cylinder was loaded to collapse at about 31 MPa. The 8 mm thick cylinders all failed at around 56 MPa with very little scatter. Using the manufacturer’s laminate properties with the measured in-plane properties, see Ref. 12, the shell element FE model predicted collapse in an n=3 mode at pressures within 3% of the observed collapse pressures. The n =3 buckling curve is shown in Fig. 5 along with the experi- mental data. This excellent agreement gave confidence in the material properties data and FE idealisation but did not help to explain the observed discrepancies in the results from the 14 mm thick cylinders.

The three cylinders of nominally 20 mm wall thickness failed at just over 100 MPa, again with very little scatter, and with no indication of overall buckling. Graham & Anderson2 applied a crude material failure analysis to the thicker cylinders based on the Azzi-Tsai criterion, eqn (8). S 1 and S2 were taken to be equal and assumed to apply to the laminate rather than individual laminae, S 12 was taken to be zero. The overall failure stress for the laminate was estimated to be about 450 MPa. This is not particularly rigorous and loses relevance as the cylinder thickness increases and the state of stress becomes triaxial. However it may be use- ful to provide a first estimate of failure pressure and is plotted against the experimental data in Fig. 5. The reason for the scatter in the data from the 14 mm thick cylinders is not known but has been discussed at some length in Ref. 12. There will be a ‘transition range’ of thick-

nesses where the dominant mode of failure changes from elastic buckling to material failure and it is clear from the current data that, for this particular material and specimen type, it lies somewhere between 8 mm and 20 mm.

Tenax f55” cylinders and rings

To date, three 20 mm thick, 125 mm internal diameter, cylinders of this material have been tested, collapsing just below 100 MPa, slightly lower than the equivalent XAS cylinders. These experimental data are also included in Fig. 5. A number of 20 mm thick, 125 mm internal diam- eter, ring specimens have been tested using the method described by King & Bird.14 This test operates on the assumption that a state of plane stress exists in the specimen, i.e. no axial load is introduced, and this was verified for a variety of aluminium specimens. Under these conditions it would be possible to measure circumferential modulus and strength as well as in-plane Pois- son‘s ratio of a composite specimen from a single test. However consideration of experi- mental data by Graham” showed that a significant amount of axial loading was intro- duced to 3” (76.2 mm) long composite specimens. This means that the ring test, as such, cannot be used to measure circumferential modulus and strength directly but can still pro- vide useful data relating failure if appropriate strain measurements are made and elastic prop- erties are determined elsewhere, e.g. from lamina data.

Ferguson et ~1.~ have characterised the mate- rial properties for this particular material in some detail giving lamina in-plane and through thickness stiffnesses and strengths, in both ten- sion and compression in most cases. This information was used in conjunction with exper- imental strain data and the analysis described earlier to ‘test’ the failure criteria described by eqns (8)-(11) when applied to hydrostatic and ring tests. Unfortunately Ferguson ef aZ.9 do not report a value for the compressive strength of the lamina in the fibre direction. In the first instance this was assumed to be equal to the tensile strength in this direction, in common with many quoted strength data for unidirec- tional CFRP composites.

Figure 7 shows the values of each criterion, through the thickness, at the collapse pressure. Bearing in mind that a value of unity predicts failure, it is apparent that some rather unreli-

Page 10: composite pressure hulls for deep ocean submersibles

340 D. Graham

1

0

-1

-2

-3

-4

-5

-6

62.5 67.5 72.5

radius (mm)

82.5

4&zi-Tsai +Tsai-Hill 4Tsai-Wu(1) 4Tsa.i.Wu(2)

(a)

25

62.5 67.5 72.5 77.5 82.5

4AzzLTsai +Tsai-Hill 4Ts&Wu(l) 4Ts&Wu(2)

(b)

Fig. 7. Values of failure criteria at collapse pressure: (a) hydrostatic test; (b) ring test.

able predictons are achieved. The calculations were repeated with reduced values of the com- pressive strength in the fibre direction but this was found to make little difference, even when reduced by a factor of two. In the case of the hydrostatic test the Tsai-Wu criterion, eqns (10) and (ll), gives meaningless negative values. This appears to be because the solution is domi- nated by F2 and F3 which turn out to be relatively large and negative because of large differences between the transverse, and through thickness, strengths in tension and compression. This perhaps highlights the conceptual difficulty of using vastly different compressive and tensile strengths to predict failure in an entirely com- pressive stress field. The three dimensional formulation on the Tsai-Wu criterion, which is dominated by F3, also gives some negative values in the case of the ring test. Although the

ring test introduces some axial load to this type of specimen,” it is very much less than in the hydrostatic test and this allows tensile trans- verse stresses to develop in the inner laminae. This is the cause of the step observed in the Tsai-Hill prediction in Fig. 7(b). Although using measured tensile and compressive strengths when appropriate may be physically more real- istic, the high values observed do not instill a great deal of confidence in this particular pre- diction. Figure 8 shows some of the data contained in Fig. 7 with the more extreme pre- dictions removed. Although there is some reasonable correlation in the case of the ring test the best predictions are still out by a factor of 2.

As was mentioned in the previous paragraph, these quadratic failure criteria appeared to be relatively insensitive to the material strength in

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Composite pressure hulls 341

62.5 67.5 72.5 77.5 82.5

m%us (mm)

IMzzi-Tsai +Tsai-Hill 1

2.5

0 62.5 67.5 72.5 77.5

radius (mm)

/+&,&Tsai +Tsai-Hill +Tsai-Wu(1) 1

82.5

(b)

Fig. 8. Selected values of failure criteria at collapse pressure: (a) hydrostatic test; (b) ring test.

the fibre direction. This was becaue the stresses in the fibre direction, although up to a factor of ten times greater than the other stresses (e.g. transverse and through thickness), were still a fraction of this strengh and thus terms involving strength in the fibre direction tended to have little contribution towards failure. Solving for the unknown compressive strength in the fibre direction, assuming all other values were accu- rate, gave unlikely values ranging from about 60 MPa to 300 MPa. These criteria are much more sensitive to the transverse and through thick- ness strengths. Because they are much lower than the strength in the fibre direction it is pos- sible for relatively low levels of transverse and through thickness stress to dominate the solu- tion and, as observed above, large differences between compressive and tensile strengths can

lead to odd results in some formulations. The strength data used in these calculations are con- sidered to be very reliable although the test specimens used to obtain them varied geometri- cally from the +55” fibre wound cylinders considered here. This may have had some effect on the measured properties. While inaccurate strength data will obviously affect the accuracy of failure predictions, it does not really explain the wildly diverging results from alternative cri- teria.

The analysis has assumed linear behaviour using initial values of Young’s and shear moduli and Poisson’s ratios. Ferguson et al.9 show that the material exhibits non-linearities in compres- sion and is particularly non-linear in shear. These effects will inevitably influence the solu- tion and it is intended to include them in the

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342 D. Graham

analysis in the future. Thermal residual stresses are another factor which will influence predic- ted failure and these were modelled crudely using a uniform step change in temperature. This is not a particularly accurate representa- tion of reality, especially in thick section components where the temperature is likely to vary through the thickness at any given time during a cure cycle. This is a complex area and deserves further attention, particularly since residual stresses can be used to advantage in some cases.

The criteria of Hashin,’ which allow for dif- ferent modes of failure, have so far been considered only briefly, but with encouraging results. In the few cases investigated the results were at least as good as the best of the quad- ratic criteria. More recently Hart-Smith,16 who is very critical of the quadratic criteria, has developed a relatively simple prediction for laminates consisting of 0”, 90” and +45” layers, which also allows that failure can occur by dif- ferent modes. This type of approach is physically more realistic and is perhaps most likely to produce a reliable, general design tool.

ENDCLOSURES

Shallow diving submarine pressure hulls are limited by failure due to buckling and typically consist of ring stiffened cylindrical, and some- times conical, sections with hemispherical or torispherical end closures. The collapse pres- sure of the end domes is very sensitive to shape imperfections, see Graham et a1.,17 and design is normally conservative. As a consequence the domes are normally much stiffer than the cylin- drical sections which leads to high local bending stresses near the connection. The buckling pres- sure of a cylinder is proportional to (t/r)3 and that of a sphere to (t/r)2 while the material failure pressure is proportional to t/r, where t is the thickness and Y the radius. Because of this, a pressure hull designed for depths up to 6000 m will be strength limited and the optimum design will be a monocoque construction. An interest- ing possibility is to design a cylindrical hull and end dome to be of equal radial stiffness and thus eliminate increased stresses due to local bending. This has been considered for a CFRP cylinder and titanium dome by Graham & Anderson” and Graham.” A membrane analy- sis equating circumferential strains gives the

following expression relating the relative thick- nesses of cylinder and dome.

tCFRP -= 2ETi [L”] (12)

tTi (l-v,) EO 2E.7 CFRP

Finite element analysis of dome/cylinder com- binations determined by eqn (12) showed that bending stresses could indeed be virtually elimi- nated although care would have to be exercised in the design of the connection to prevent local concentration, particularly of shear stress, in the composite material. However, suitable titanium domes proved to be of somewhat non-conserva- tive design and it is possible that domes of composite material may be more versatile in designing stiffness matched components. Two small scale CFRP domes, t=12 mm and r=125 mm, of quasi-isotropic construction are to be tested as part of the MAST II programme and FE models have recently been developed. Thick shell and three-dimensional brick models gave buckling pressures of 184 MPa and 181 MPa respectively in an y1= 5 mode as shown in Fig. 9, far in excess of the expected collapse pressure due to material failure. Composite domes pres- ent other problems however, for example it is difficult to maintain constant thickness over a complete hemisphere and a significant increase has been found to occur towards the edges of the MAST II specimens. This will have to be included in future analyses but an efficient deep diving pressure hull which develops only mem- brane stresses is a worthwhile goal.

CONCLUDING REMARKS

This paper describes a Lame type analysis for a thick, overall orthotropic cylinder which can be constructed of anisotropic layers. Loading con- sists of external pressure, a uniform axial strain and a uniform temperature difference. Dis- placement, stress and strain predictions have been found to be in excellent agreement with finite element analyses. FE analysis has also shown that the stresses predicted by analytical solution are close to those developed in the ‘test section’ of the cylindrical specimens used in the current experimental programme. The analyt- ical solution was used in conjunction with some of the popular failure theories but predictions of failure pressures were found to be greatly

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Composite pressure hulls 343

Fig. 9. Buckling mode shapes of dome predicted by both shell and brick FE models.

variable and, in many cases, differed greatly from observed collapse pressures. There are many refinements which can be made to the analysis, such as allowing for non linear elastic behaviour and residual stresses and failure cri- teria which account for different modes of failure may be more appropriate. The finite ele- ment method has been used to accurately predict the elastic buckling pressure of the thin- ner cylinders but at present the only reliable method of predicting the collapse of thick com- posite cylinders due to material failure appears to be by using empirical data.

ACKNOWLEDGEMENTS

The author would like to thank his colleagues at DRA Fort Halstead who provided specimens, materials data and a wealth of background information, DRA Dunfermline who carried out the experimental programme and provided the data and also the partners in the MAST II programme at NERC Institute of Oceano- graphic Sciences, IFREMER and the National Technical University of Athens for their con-

tributions. Most of the analytical work described in this paper was supported by the MOD under ongoing strategic research. Some early experimental work was carried out in con- nection with the NERC Autosub project. The major part of the experimental programme and some of the FE cylinder analysis was funded under the EC MAST II programme (Contract No. MAS2-CT92-0028).

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