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Collapse loads of externally pressurised composite torispherical and hemispherical domes
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Advances in Engineering Software 15 (1992) 145-154
; ii:1
Collapse loads of externally pressurised composite torispherical and hemispherical domes J. Mistry, F. Levy-neto & Y-S. Wu University of Liverpool, Mechanical Engineering Department (Applied Mechanics), PO Box 147, Liverpool, UK Experimental work has been carried out to investigate how the collapse loads of externally pressurised composite carbon fibre/epoxy resin torispherical and hemispherical domes vary with the number of reinforcing fibre layers. The experimental results are compared with those calculated by finite element and BOSOR 4 computer programs and it is shown that correlation between the experimental and theoretical collapse loads can be improved with a material degradation model for the stiffness analysis following a First Ply failure of the composite. components in other directions in assessing the failure index of the likely damage zone. However, the material degradation model of reducing the stiffness by a degradation factor (DF) applied to appropriate ply properties, as proposed by Tsai 7 and Roy & Tsai, 8 remains a simple, reasonably accurate and easy to implement scheme. Dome like composite structures are employed in marine environments as submersibles, as radomes and as sonar domes for warship equipments. These structures are expected to withstand external pressurisation resulting from underwater explosions or hydrodynamic loads or deep sea pressures. The likely mode of failure is either by material failure or by buckling. 9 The buckling failure may be due to sudden snap-through collapse at the point of zero stiffness of a non-linear load-deflection path or at a bifurcation point before the snap-through condition has been reached. The buckling mode of failure is not a part of the present investigation but its presence should be recognised and its occurrence during the post-FPF stage is a possibility. For several years the University of Liverpool has been engaged in an extended research program on the buckling of plastic domes. From the numerous experimental work carried out, it had been observed that for carbon and hybrid fibre reinforced plastic (FRP) domes with diameter to thickness (D/t) ratio less than 100, the experimental results of the final collapse loads were consistently higher than the theoretical FPF predictions. This was surprising because the geometrical imperfections in the manufactured models should have led to results less than the predicted values or very close to them. When the results of 25 domes tested by Blachut et al. lo were examined, it was seen that for domes with
INTRODUCTION Design of composite structures based on first ply failure (FPF) is generally regarded to lead to conservative constructions because of the total disregard of the residual strength of structures until last ply failure (LPF) and collapse occurs. ~ This path is often followed because of the lack of information concerning the postF P F behaviour. Many investigators have proposed empirical material degradation models, which offer the best solution for the collapse load predictions in absence of accurate models to represent the mechanism of material failure following the FPF. Early proposals of reducing in-plane transverse or matrix moduli to zero 2 are also considered to lead to conservative estimates for composites prepared with woven reinforcing fabrics and angle ply laminates. However, such models may be satisfactory for filament wound structures where the fibre reinforcement lay-out is basically uni-directional. 3 Petit & Waddoups 4 suggested the use of a large negative tangent modulus, for a stress component which had reached its ultimate strength, and continuing with the applied load until that stress component reduces to zero. This model helps to predict the non-linear behaviour following the initial failure. An alternative proposal of step-wise reduction in stiffness based on the ratio of the residual strength to the ultimate strength has been put forward by Chiu. 5 Pandey 6 proposed that the stiffness reduction factor for any stress direction (longitudinal or transverse) should also reflect the contribution made by stress Advances in Engineering Software 0965-9978/92/$05.00 © 1992 Elsevier Science Publishers Ltd. 145
146
J. Mistry, F. Levy-neto, Y-S. Wu
D/t ~ 80 the range of ratio of experimental failure pressure to the F P F pressure (Pexp/PFP~) was 0"89 to 1.4, whereas for D/t ~ 120 this range was 0"68 to 1.2. A similar trend was present in the results of large 800 mm diameter domes of Blachut et al. 11 Here Pexp/PFPFwas in the range of 0.88 to 1"09 for 20 ply domes, of 1.04 to 1.19 for 30 ply domes and of 1.23 to 1.41 for 36 ply domes. There is a strong indication here that the postFPF strength increases considerably with the thickness of the domes. In the light of the above observation it was felt that there is a need for assessing LPF loads of composite domes in order to improve the correlation between the experimental results and theoretical predictions. Hence, the main aim of the present investigation was to compare the LPF predictions, based on a material degradation model (similar to Tsai's), with the experimental results obtained with tests on carbon fibre reinforced plastic (FRP) torispherical and hemispherical domes. In particular a common material degradation factor of 0.3 was applied to the ply properties El, E2 and G~2, because woven fabric reinforcements exhibit equal stiffness properties in both longitudinal and in-plane transverse directions. The effect of the material degradation on the Poisson's ratios was ignored. The subscripts used here for the elastic properties follow the conventional notation of 1 for the longitudinal and 2 for the in-plane transverse direction.
t. I
t~
~
---
D
~
(o) Torisphere
(b) Hemisphere
Fig. I. Geometrical parameters of the dome profiles.
DIMENSIONS OF THE DOMES All the domes were prepared with carbon fibre woven fabrics and epoxy resin as the matrix, using a hand lay-up and vacuum I~ag technique. To minimise creasing of the outer layer during their manufacture, the domes employed male moulds of torispherical and hemispherical profiles. 12 The geometrical parameters of the two profiles are shown in Fig. 1. Two types of dome support condition were considered. Eight of the domes were rigidly mounted, with their ends inserted into a 12 mm groove and glued to a base plate using the same epoxy resin as the matrix of the domes. Two of the torispherical domes were 'elastically' mounted, in which the ends were joined to filament wound glass FRP cylinders. In order to match the domes to the cylinders, 200mm diameter torispherical domes were utilised. The arrangement of the joint and end support of the cylinders are shown in Fig. 2. Recommended methods of joining two composite plates 13't4 involve either single or symmetrical double lap joints. The diameter of the cylinders at the edge where the domes were to be attached was reduced in size as shown in Fig. 2 and the two parts were then joined as a double lap joint with the aid of a 45 mm wide E-glass/ epoxy resin band of an overall thickness denoted by t~ as shown in the figure.
The thickness of the domes varied both circumferentially and meridionally. The variation of the thickness was assessed by its measurement at regular spacing along 12 equi-angular meridians. This survey helped to identify the meridian with the minimum average thickness. The torispheres have two distinct geometrical regions, the spherical cap and the knuckle. The
Dome
I
t I (mm)
t~(mm)
CTIOA
8.44
6.08
CTIOB
7.06
6.06
I00 Fig. 2. Dome/cylinder combination.
Collapse loads of externally pressurised composite torispherical and hemispherical domes
147
Table 1. Geometrical parameters of the domes at the meridian of the minimum average thickness
Base support
Stacking sequence of the fibres (degs)
Mean diameter (mm)
Rs -~-
CH4A CH4B CH4C
rigid
[0/45];
224
CH6A CH6B CH6C
rigid
[0/60/-60]s
rigid
[0/45/0/45]~
Specimen
Ib or ts (ram)
ta or t k
1-100 1-121 1'231
1"321
0.5
1.330 1.356
225
0.5
1"693 1'708 1"723
1-919 1'986 2-004
2"769
3-008
226
0.6
2"776
3"064
2"980
3.040
2"860
2'910
r
~
CT8A 0.24
CT8B CT10A elastic
[0/18/36/54/72]s
208
1.0
CT10B
(ram)
0.15
*The subscript s refers to symmetrical stacking sequence relative to the middle surface. hemispheres were considered to be divided similarly into two regions, each bearing an angle of 45 ° as shown in Fig. 1. The average thicknesses of these two regions at the meridian of the minimum average thickness are denoted by ts and tk for the torispheres and by t a and tb for the hemispheres. The theoretical model used these thicknesses as uniform thicknesses for the respective regions. The other geometrical parameters of the domes tested were as given in Table 1. Four groups of the domes were tested in the experimental program. Within each group, two or three specimens of almost similar geometrical constructions were manufactured in order to check the repeatability of the results. The two groups, series CH4 and CH6, had four and six layers respectively of the reinforcing fibres laid in symmetrical orientations about the middle surface. These two groups formed a set of domes with hemispherical profiles. The other two groups, CT8 and CT10, had torispherical profiles and involved symmetrically stacked eight and ten layers respectively.
THEORETICAL CONSIDERATIONS The dome shapes used in the investigation were essentially axially symmetric thin shell structures. Theoretical analysis of the shell structures can be carried out with the aid of numerical methods such as finite differences and finite elements. In fact both methods were employed in this investigation. BOSOR 4 computer program, 15 which uses the finite difference approximations in the variational or energy equations of the axisymmetric shells, was used to calculate the buckling loads and to provide computer plotted results of the domes. For the F P F and LPF analyses, a special finite element (FE) program ~6 was utilised. This program allowed one to check material failure of the shell at any point using seven different criteria. It is also
capable of predicting the buckling loads thus it was possible, by comparing its results with those of the BOSOR 4 program, to ensure that there were no modelling errors. The seven criteria of failure used in the program were Maximum stress, Maximum strain, Tsai-Hill, 17 Hoffman, 18 Tsai-Wu stress, Tsai-Wu strainl9 and Owen. 20 However, throughout this investigation an estimate of the failure load was always based on the most conservative F P F prediction of all the criteria. The FE program employed a combined strategy of load increments and Lagrangian polynomial (of order three) interpolation over the calculations of the last four load increments to predict the load at which maximum failure index (FI) in the structure reached its specified value according to a chosen failure criterion. At this point the layer with the maximum FI was deemed to have failed. The first occurrence o f this was the FPF. In order to ensure that the predicted value was the converged solution, the maximum FI was recalculated at the new load and this data was fed into the interpolation procedure after discarding the data remote from the last step or the new load. The new prediction was usually within 1% of its previous value unless a large load increment was selected in the first instant. In that event, the interpolation procedure was repeated until convergence was obtained. The layer of the element was now recorded as failed and its material properties were adjusted using the degradation factor DF. The new material model was expected to have a new equilibrium path. To establish a point on this path at the current load, the interpolation and load increment method described above were restarted from 90% of the previous failure load. This is illustrated by a typical non-linear load-deflection characteristic behaviour of a shell segment as shown in Fig. 3. The non-linearity is attributed to the large deflection geometric non-linearity of the shell. In the figure the primary equilibrium path is due to the initial intact material and the subsequent secondary equi-
148
J. M&try, F. Levy-neto, Y-S. Wu
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(resultant path with a p e ~ strength)
(steady collapse path)
--Primary
equilibrium path Secondary equilibriura paths of the shell w~th degraded material . . . . Resultant post-PFP path ~< Axi-symmetric collapse (snap-through buckling) cor~di~ions
....
,__
Deflection Fig. 3. Pseudo-post FPF paths.
librium paths result from the consideration of the degradation of the material as other plies at the same or other locations go through their failure cycles. The difference in deflection between primary and secondary paths and between secondary paths are deliberately exaggerated to demonstrate the possible post-FPF behaviours of the shell segment. Consider C1 as a point corresponding to the FPF load. The deflections of the shell segment at the FPF load on the secondary equilibrium path was obtained by decreasing the load by 10% to the point ci first. The point cii , which is at the same load level as the previous failure load, was obtained with the help of the normal iterative algorithm described above. The damage zone may expand locally without any increase in strength at the point cii or it may lead to stress redistribution in a manner such that there is an increase in strength to a point such as ciii. Depending on the position of the FPF load on the primary equilibrium path, it was possible to predict three different types of resultant post-FPF paths. A steady collapse path such as A1 - A 2 (Fig. 3) may be obtained with no significant increase in strength. The other possibility was the path B~ - B2, a resultant path with a peak strength. Though the points on the secondary equilibrium paths are all stable conditions, the peak strength on the resultant path is in fact the point of the axisymmetric collapse or snap-through buckling. The third possibility was that the resultant path may become tangential to the axisymmetric collapse point of a secondary equilibrium path and, again, the likely mode of failure is the snap-through buckling. During the analysis, if any damage zone spreads throughout the thickness of the shell then a conventional burst failure was expected but if the resultant path reached a peak load then the shell
structure was presumed to have failed by axisymmetric collapse. The behaviour of the dome following the snapthrough buckling, whether it occurs prior to FPF or in the post-FPF domain, often results in large unstable displacements. Another factor that can influence the behaviour of domed structures is the manner in which they are supported. The support condition can be described in terms of displacement components in axial, circumferential and radial directions together with a rotation about the circumference. In the case of a fully clamped
DEFORMED STRUCTURE LOAD STEP 1, LOAD =
1.000E+O0
PRESTRESS
INDIC =0
~0.
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~10.
60.
_I
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100.
__ 120.
R
Fig. 4. Deformation of the clamped torispherical dome.
Collapse loads of externally pressurised composite torispherical and hemispherical domes
149
CTIOA
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0
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situation all these displacements and the rotation were assumed to be zero. However, it may be difficult to achieve some of these conditions in practice. Galletly & Mistry 21 had shown that if the constraint in the
CTIOA
INDIC =
l.O00E+O0 PRESTRESS
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........ ~_._......
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-TOO.
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Fig. 5. Principal stress resultants of the clamped torispherical dome.
DEFORMED STRUCTURE L O A D STEP I, LOAD =
/
I T50.
200.
R
Fig. 6. Deformation of the torispherical dome/cylinder combination.
Fig. 7. Principal stress resultants of the tofispherical dome/ cylinder combination. circumferential direction was relaxed then a marked change in the natural frequency of dome/cylinder combinations was possible. Similar sensitivity can be exhibited by the support condition on the final collapse loads. Figure 4 shows the deformed shape of the clamped torispherical dome of the series CT10 compared with the undeformed shape as calculated by the BOSOR 4 program. The predicted variation of the meridional force per unit length (N~0), circumferential force per unit length (N20) and bending moment per unit length (M~0) for an external pressure of 2.87MPa are shown in the Fig. 5. From these, it can be seen that the maximum bending stress is as expected at the clamped end of the dome. Any changes to the clamping condition is likely to affect this distribution. This can be seen clearly, if the dome is attached to a cylinder and the support condition is provided at the far end of the cylinder. The deformed and undeformed shapes of this case are shown in Fig. 6 and its main stress resultants (Ni0, N20 and M~0 ) in Fig. 7. The effect of removing the clamped condition from the dome end results in migration of the peak bending stress to the middle of the knuckle region. There is a peak bending moment at the clamped end of the cylinder, but due to the relatively higher thickness of the cylinder than that of the dome, a material failure was unlikely at this end. This case was also included in the study in order to isolate the effects of the end conditions on the final collapse loads.
J. Mistry, F. Levy-neto, Y-S. Wu
150 Port for ~
oil
/
Transparent
=~ome
/ ~
~
~
cover
4--
'.
2m
~ 1
~ Oil-filled~r~ure chamber ~ ~
//////// xxx xxxxxxxxx
~rl0B d o m e
J
I
0.5
I
1.0
[
1.5
Relatlv¢ Volumetric Change (~)
Fig. 8. Schematic diagram of the external pressure loading of the clamped domes. EXPERIMENTAL P R O C E D U R E The rigidly mounted domes were tested in a small pressure vessel in the form of a thick cylinder closed at one end and its other end was closed by the dome itself such that the pressurising fluid of the pressure vessel provided the external pressure loading of the domes. The arrangement is shown by the schematic diagram of Fig. 8. This arrangement allowed an easy inspection of the concave side of the dome during the progress of an experiment. At the failure load, the shell wall showed signs of weeping initially, and eventually at the final collapse it developed cracks. As the pressure vessel was too small in size to mount the dome/cylinder combinations, it was found necessary to test them within a large hyperbaric chamber. Since the inspection of the domes was impossible, the failure of the domes was detected by recording changes in internal volume of the dome/cylinder combination. The internal volume was charged with oil at atmospheric pressure. A small hole through the base support plate was connected to a small tube which passed through a monitor port of the chamber. Its external end was open and allowed the compression of the dome/cylinder
5 _
3_
J
~
2
1
a] n o i s e
~
CTIOAdome 0.5
1.0
1.5
2.0
Relative Volumetric Change (X)
Fig. 9. Variation of the external pressure with the volumetric changes of the CTIOA dome/cylinder combination.
Fig. 10. Variation of the external pressure with the volumetric changes of the CT10B dome/cylinder combination. combination to be measured in terms of the outflow of the oil. The pressure and relative volumetric changes recorded for the domes CT10A and CT10B are shown in Figs 9 and 10. Both tests recorded an initial loud cracking noise before the final collapse. Without the aid of expensive instrumentation or without making subjective judgements, it was impossible to identify the first ply failure by this noise alone because of the continuous low level acoustic emission throughout the tests.
C O M P A R I S O N OF THE EXPERIMENTAL RESULTS WITH THEORY In order to predict the failure loads, flat test specimens were prepared alongside the domes in identical environmental conditions and using the same manufacturing process. In all 16 specimens (6 for compressive, 5 for tensile and 5 for in-plane shear properties) were prepared. These test specimens were employed to determine the basic elastic and strength properties of the plies in longitudinal and in-plane transverse directions according to the ASTM Standards D303976, D3518-76 and D3410-87. 22'23 The statistical mean and standard deviation (as a percentage of the mean) of the elastic and strength properties were as given in Table 2. Results of the experiments carried out on the domes and theoretical prediction of their failure loads (FPF, LPF and buckling) using the above stiffness and strength data are compared with each other in Table 3. The theoretical buckling pressures were derived by assuming that the material remained intact. The figures quoted in the table were the lowest buckling pressures found from a search over a wide range of circumferential wave numbers. It is widely recognised that thin shell like structures tend to be sensitive to the geometrical imperfections, as a result, any prediction based on presumed perfect geometry usually works out to be an over-estimation of
Collapse loads of externally pressurised composite torispherical and hemispherical domes
151
Table 2. Mechanical properties of the woven carbon fibre/epoxy test specimens
Properties
Mean
Elastic moduli, E1 & E2 (GPa) In-plane shear modulus, Gl2 (GPa) Poisson's ratio, ulz Ultimate strengths in tension, XIT & XzT (MPa) Ultimate strengths in compression, X~c & X:c (MPa) Ultimate in-plane shear strength, X~2 (MPa) Maximum strains in tension, ~r & ~x (%) Maximum strains in compression, elC & e2c (%) Maximum in-plane shear strain, 712 (%)
66.45 5.00 0.05 617.6 465.4 78.2 0.93 0.71 1-86
the buckling or material strength. When one considers a ratio of the experimental result to its theoretical prediction Pexp/Ptheory t h e n the ratio is expected to be less than unity. This is seen from the set of results for the CH4 series domes. Both the LPF and FPF load predictions are of similar magnitude and the Pexp/Ptheory ratio is found in the range of 0.740 and 0"884. This set of domes involve only four layers of the reinforcing fibres and it is the thinnest set. The second set of domes, CH6 series, have the same geometrical profile as the previous series but the number of reinforcing fibre layers is six. In this case the FPF load predictions are in close agreement with the experimental failure loads except for CH6A dome, which in fact recorded a Pexp/Ptheoryratio of 1.149. The F P F load is starting to underestimate the actual strength of the dome. This factor is becoming more evident with the thicker torispherical domes CT8 and CT10. The strength is underestimated by almost 54% for the dome CT10A. On the other hand, the LPF load predictions show better correlation with the experiments as the
Standard deviation (%) 3.60 0.80 4.00 6.64 8.81 7.93 7.50 11.27 10.75
number of layers increase. With the increase in the number of layers, the thickness also increases and the effective bending resistance increases significantly. In fact for homogeneous materials, the bending resistance is proportional to the cube of the overall thickness. Hence, any failure of the outer layers of the composite shells does not necessarily lead to an immediate collapse but there is a substantial residual strength as shown by the post-FPF strength of the CT10 domes. Other interesting information that can be gained from the results concerns the resultant post-FPF paths. All CH4 series domes followed a C1-C2 path (see Fig. 3), in which the resultant path becomes tangential to a secondary equilibrium path at its snap-through buckling condition. In the case of the CH4A dome, the lowest buckling load is found for the wave number zero, (i.e. elastic snap-through buckling condition) and its value of 5.40 MPa was closest to the experimental and theoretical material failure loads. Though axisymmetric collapse was predicted, the actual mode of failure was by means of cracks near the clamped edge (see Fig. 11)
~
~
Fig. 11. Dome CH4A after failure with cracks near the clamping edge.
6"80 6"04 6" 11
10-14 9-97
4"41 3"79
CH6A 'CH6B CH6C
CT8A CT8B
CT10A CT10B
5-40 (0) a 6"79 (9) 6.89 (8) 13.63 (8) 13.72 (0) 13.40 (0) 16.93 (6) 16"83 (6) 6.408 (2) 5.880 (3)
Tsai-Wu strain Tsai-Wu strain Tsai-Wu stress Tsai-Wu strain Tsai-Wu stress Tsai-Wu stress Owen/Tsai-Hill Tsai-Wu strain Owen/Tsai-Hill Owen/TsaioHill
4.04 4-75 4'31 8.72 8-97 8-33 9.64 8-66 4.21 3-98
3-96 4"17 4'08 5"92 6"03 6'41 8.56 8.59 2.87 2-69
a
Theoretical buckling pressure (MPa)
Failure criterion b
Theoretical pressure at LPF (MPa)
Theoretical pressure at F P F (MPa)
Figures in round brackets refer to the circumferential wave number of the buckling mode. b Most conservative F P F predicting criterion.
3"50 3-52 3"54
Final pressure at failure, Pexp (MPa)
CH4A CH4B CH4C
Specimen
1.537 1.409
l- 185 1.161
1.149 1'002 0.953
0.884 0.844 0-868
C1 C2 C1-C2 C1 C2 A1 A2 A1-A2 A1 A2 B1 B2 B1 B2 B1-B2 B1-B2
0.780 0-673 0-733 1.052 1.151 1-048 0-952
Theoretical resultant post-FPF path 0-866 0.740 0.821
Pexp PLPF
Pexp
PFPF
Table 3. Comparison between experimental failure pressures and predicted FPF and LPF pressures
i
Collapse loads of externally pressurised composite torispherical and hemispherical domes
153
Fig. 12. Dome CH6A after failure with a large diametral crack. suggesting that the mode of failure was other than axisymmetric. This dome had only four layers and with almost negligible post-FPF load bearing capacity. The failure was found to be mainly governed by the material failure at the location of the minimum thickness. Additionally, when compared to the other domes, the experimental collapse loads of this series were less than the predicted FPF loads. The theoretical buckling loads of the hemispherical domes with six layers and for the remainder of the domes were found to be about twice the material failure loads. In the case of the CH6 series, the resultant path is that of a steady collapse A 1 - A 2 of Fig. 3. The torispherical domes, CT8 and CT10 series, had considerable post-FPF strength before their final collapse. These two series showed a B1-B2 resultant post-FPF path. The failure of the CH6, CT8 and CT10 domes was typified by large cracks spanning a wide surface area. In the case of CH6A dome, a diametral crack was formed and almost split it into two parts as shown in Fig. 12.
CONCLUSIONS As the number of layers in the composites increases there is a need for a material degradation model to assess the post-FPF strength. Theoretical analyses of the domes using the material degradation factor 0"3 applied to E~, E2 and GI2 have shown that more reliable estimates of the final collapse loads can be made compared to the FPF loads. This model appears to be satisfactory for the F R P domes made from woven carbon fibres because of their equal stiffness properties in the fibre and transverse directions.
ACKNOWLEDGEMENT The authors wish to acknowledge the University of Liverpool for allowing the use of the BOSOR 4 program and its laboratory and computing facilities.
REFERENCES 1. Tsai, S.W. Strength characteristics of composite materials, NASA CR-224, Washington, DC, USA, 1965. 2. Dillard, D.A. & Brinson, H.F. A numerical procedure for predicting creep and delayed failure in laminated composites, ,4STM-STP-813, Baltimore, USA, 1983. 3. Rosato, D.V. & Grove, C.S. Filament Winding: Its Development, Manufacture, Application and Design, Interscience Publishers, New York, USA, 1964. 4. Petit & Waddoups. A method of predicting the nonlinear behaviour of laminated composites, J. Composite Materials, 1969, 3, 2-19. 5. Chiu, K.D. Ultimate strengths of laminated composite, J. of Composite Materials, 1987, 3, 578-582. 6. Pandey, A.K. A non-linear computational model for the strength and failure of composite plates and shells, PhD. dissertation, Virginia Polytechnic Institute & State University, Blackberg, USA, 1987. 7. Tsai, S.W. Composite Design, Think Composites Publ., Dayton, 1988. 8. Roy, A.K. & Tsai, S.W. Design of thick composite cylinders, J. of Pressure Vessel Technology, 1988, 110, 255-261. 9. Levy, F., Galletly, G.D. & Mistry, J. Buckling of composite torispherical and hemispherical domes, CADCOMP-90 Conference, Brussels, Belgium, 1990. 10. Blachut, J., Galletly, G.D. & Levy-neto, F. Towards optimum CFRP enclosures, J. of Mechanical Engineering Science, 1991, 205, 329-342. 11. Blachut, J., Galletly, G.D. & Gibson, A.G. CFRP domes subjected to external pressure, Marine Structures, 1990, 3, 149-173.
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J. Mistry, F. Levy-neto, Y-S. Wu
12. Levy-neto, F. & Mistry, J. The behaviour of repaired composite domes subjected to external pressure, Composites, 1992, 23, 271-277. 13. Vinson, J.R. & Sierakowski, R.L. The Behaviour of Structures Composed of Composite Materials, Martinus Nijhoff Publ., 1986. 14. Chamis, C.C. & Murthy, P.L.N. Simplified procedures for designing adhesively bonded composite joints, J. of Reinforced Plastics and Composites, 1991, III, 29-41. 15. Bushnell, D. Computerized analysis of shells - - Governing equations, Computers and Structures, 1984, 18, 471-536. 16. Mistry, J. Theoretical investigation into the effect of the winding angle of the fibres on the strength of filament wound GRP pipes subjected to combined external pressure and axial compression, Composite Structures, 1992, 20, 83-90. 17. Azzi, V.D. & Tsai, S.W. Anisotropic strength of components, Experimental Mechanics, 1965, 5, 286-288.
18. Hoffman, O. The brittle strength of orthotropic materials, J. of Composite Materials, 1967, 1, 200-206. 19. Tsai, S.W. & Wu, E.M. A general theory of strength of anisotropic materials, J. of Composite Materials, 1971, 5, 58-80. 20. Owen, M.J. Biaxial failure of GRP - - mechanisms, modes and theories, Proceedings of the 2nd Int. Conf. on Composite Structures, ed. I.H. Marshall, Elsevier Applied Science, 1983. 21. Galletly, G.D. & Mistry, J. The free vibrations of cylindrical shells with various end closures, Nuclear Engineering and Design, 1974, 30, 249-268. 22. ASTM Standards D3039-76 and D3518-76, Annual
Book of Standards, Part 36, Plastic Fibre Composites, 1977. 23. ASTM Standard D3410-87, Compressive properties of unidirectional or cross-ply fibre-resin composites, ASTM, 1987.
Advances in Engineering Software 15 (1992) 145-154
; ii:1
Collapse loads of externally pressurised composite torispherical and hemispherical domes J. Mistry, F. Levy-neto & Y-S. Wu University of Liverpool, Mechanical Engineering Department (Applied Mechanics), PO Box 147, Liverpool, UK Experimental work has been carried out to investigate how the collapse loads of externally pressurised composite carbon fibre/epoxy resin torispherical and hemispherical domes vary with the number of reinforcing fibre layers. The experimental results are compared with those calculated by finite element and BOSOR 4 computer programs and it is shown that correlation between the experimental and theoretical collapse loads can be improved with a material degradation model for the stiffness analysis following a First Ply failure of the composite. components in other directions in assessing the failure index of the likely damage zone. However, the material degradation model of reducing the stiffness by a degradation factor (DF) applied to appropriate ply properties, as proposed by Tsai 7 and Roy & Tsai, 8 remains a simple, reasonably accurate and easy to implement scheme. Dome like composite structures are employed in marine environments as submersibles, as radomes and as sonar domes for warship equipments. These structures are expected to withstand external pressurisation resulting from underwater explosions or hydrodynamic loads or deep sea pressures. The likely mode of failure is either by material failure or by buckling. 9 The buckling failure may be due to sudden snap-through collapse at the point of zero stiffness of a non-linear load-deflection path or at a bifurcation point before the snap-through condition has been reached. The buckling mode of failure is not a part of the present investigation but its presence should be recognised and its occurrence during the post-FPF stage is a possibility. For several years the University of Liverpool has been engaged in an extended research program on the buckling of plastic domes. From the numerous experimental work carried out, it had been observed that for carbon and hybrid fibre reinforced plastic (FRP) domes with diameter to thickness (D/t) ratio less than 100, the experimental results of the final collapse loads were consistently higher than the theoretical FPF predictions. This was surprising because the geometrical imperfections in the manufactured models should have led to results less than the predicted values or very close to them. When the results of 25 domes tested by Blachut et al. lo were examined, it was seen that for domes with
INTRODUCTION Design of composite structures based on first ply failure (FPF) is generally regarded to lead to conservative constructions because of the total disregard of the residual strength of structures until last ply failure (LPF) and collapse occurs. ~ This path is often followed because of the lack of information concerning the postF P F behaviour. Many investigators have proposed empirical material degradation models, which offer the best solution for the collapse load predictions in absence of accurate models to represent the mechanism of material failure following the FPF. Early proposals of reducing in-plane transverse or matrix moduli to zero 2 are also considered to lead to conservative estimates for composites prepared with woven reinforcing fabrics and angle ply laminates. However, such models may be satisfactory for filament wound structures where the fibre reinforcement lay-out is basically uni-directional. 3 Petit & Waddoups 4 suggested the use of a large negative tangent modulus, for a stress component which had reached its ultimate strength, and continuing with the applied load until that stress component reduces to zero. This model helps to predict the non-linear behaviour following the initial failure. An alternative proposal of step-wise reduction in stiffness based on the ratio of the residual strength to the ultimate strength has been put forward by Chiu. 5 Pandey 6 proposed that the stiffness reduction factor for any stress direction (longitudinal or transverse) should also reflect the contribution made by stress Advances in Engineering Software 0965-9978/92/$05.00 © 1992 Elsevier Science Publishers Ltd. 145
146
J. Mistry, F. Levy-neto, Y-S. Wu
D/t ~ 80 the range of ratio of experimental failure pressure to the F P F pressure (Pexp/PFP~) was 0"89 to 1.4, whereas for D/t ~ 120 this range was 0"68 to 1.2. A similar trend was present in the results of large 800 mm diameter domes of Blachut et al. 11 Here Pexp/PFPFwas in the range of 0.88 to 1"09 for 20 ply domes, of 1.04 to 1.19 for 30 ply domes and of 1.23 to 1.41 for 36 ply domes. There is a strong indication here that the postFPF strength increases considerably with the thickness of the domes. In the light of the above observation it was felt that there is a need for assessing LPF loads of composite domes in order to improve the correlation between the experimental results and theoretical predictions. Hence, the main aim of the present investigation was to compare the LPF predictions, based on a material degradation model (similar to Tsai's), with the experimental results obtained with tests on carbon fibre reinforced plastic (FRP) torispherical and hemispherical domes. In particular a common material degradation factor of 0.3 was applied to the ply properties El, E2 and G~2, because woven fabric reinforcements exhibit equal stiffness properties in both longitudinal and in-plane transverse directions. The effect of the material degradation on the Poisson's ratios was ignored. The subscripts used here for the elastic properties follow the conventional notation of 1 for the longitudinal and 2 for the in-plane transverse direction.
t. I
t~
~
---
D
~
(o) Torisphere
(b) Hemisphere
Fig. I. Geometrical parameters of the dome profiles.
DIMENSIONS OF THE DOMES All the domes were prepared with carbon fibre woven fabrics and epoxy resin as the matrix, using a hand lay-up and vacuum I~ag technique. To minimise creasing of the outer layer during their manufacture, the domes employed male moulds of torispherical and hemispherical profiles. 12 The geometrical parameters of the two profiles are shown in Fig. 1. Two types of dome support condition were considered. Eight of the domes were rigidly mounted, with their ends inserted into a 12 mm groove and glued to a base plate using the same epoxy resin as the matrix of the domes. Two of the torispherical domes were 'elastically' mounted, in which the ends were joined to filament wound glass FRP cylinders. In order to match the domes to the cylinders, 200mm diameter torispherical domes were utilised. The arrangement of the joint and end support of the cylinders are shown in Fig. 2. Recommended methods of joining two composite plates 13't4 involve either single or symmetrical double lap joints. The diameter of the cylinders at the edge where the domes were to be attached was reduced in size as shown in Fig. 2 and the two parts were then joined as a double lap joint with the aid of a 45 mm wide E-glass/ epoxy resin band of an overall thickness denoted by t~ as shown in the figure.
The thickness of the domes varied both circumferentially and meridionally. The variation of the thickness was assessed by its measurement at regular spacing along 12 equi-angular meridians. This survey helped to identify the meridian with the minimum average thickness. The torispheres have two distinct geometrical regions, the spherical cap and the knuckle. The
Dome
I
t I (mm)
t~(mm)
CTIOA
8.44
6.08
CTIOB
7.06
6.06
I00 Fig. 2. Dome/cylinder combination.
Collapse loads of externally pressurised composite torispherical and hemispherical domes
147
Table 1. Geometrical parameters of the domes at the meridian of the minimum average thickness
Base support
Stacking sequence of the fibres (degs)
Mean diameter (mm)
Rs -~-
CH4A CH4B CH4C
rigid
[0/45];
224
CH6A CH6B CH6C
rigid
[0/60/-60]s
rigid
[0/45/0/45]~
Specimen
Ib or ts (ram)
ta or t k
1-100 1-121 1'231
1"321
0.5
1.330 1.356
225
0.5
1"693 1'708 1"723
1-919 1'986 2-004
2"769
3-008
226
0.6
2"776
3"064
2"980
3.040
2"860
2'910
r
~
CT8A 0.24
CT8B CT10A elastic
[0/18/36/54/72]s
208
1.0
CT10B
(ram)
0.15
*The subscript s refers to symmetrical stacking sequence relative to the middle surface. hemispheres were considered to be divided similarly into two regions, each bearing an angle of 45 ° as shown in Fig. 1. The average thicknesses of these two regions at the meridian of the minimum average thickness are denoted by ts and tk for the torispheres and by t a and tb for the hemispheres. The theoretical model used these thicknesses as uniform thicknesses for the respective regions. The other geometrical parameters of the domes tested were as given in Table 1. Four groups of the domes were tested in the experimental program. Within each group, two or three specimens of almost similar geometrical constructions were manufactured in order to check the repeatability of the results. The two groups, series CH4 and CH6, had four and six layers respectively of the reinforcing fibres laid in symmetrical orientations about the middle surface. These two groups formed a set of domes with hemispherical profiles. The other two groups, CT8 and CT10, had torispherical profiles and involved symmetrically stacked eight and ten layers respectively.
THEORETICAL CONSIDERATIONS The dome shapes used in the investigation were essentially axially symmetric thin shell structures. Theoretical analysis of the shell structures can be carried out with the aid of numerical methods such as finite differences and finite elements. In fact both methods were employed in this investigation. BOSOR 4 computer program, 15 which uses the finite difference approximations in the variational or energy equations of the axisymmetric shells, was used to calculate the buckling loads and to provide computer plotted results of the domes. For the F P F and LPF analyses, a special finite element (FE) program ~6 was utilised. This program allowed one to check material failure of the shell at any point using seven different criteria. It is also
capable of predicting the buckling loads thus it was possible, by comparing its results with those of the BOSOR 4 program, to ensure that there were no modelling errors. The seven criteria of failure used in the program were Maximum stress, Maximum strain, Tsai-Hill, 17 Hoffman, 18 Tsai-Wu stress, Tsai-Wu strainl9 and Owen. 20 However, throughout this investigation an estimate of the failure load was always based on the most conservative F P F prediction of all the criteria. The FE program employed a combined strategy of load increments and Lagrangian polynomial (of order three) interpolation over the calculations of the last four load increments to predict the load at which maximum failure index (FI) in the structure reached its specified value according to a chosen failure criterion. At this point the layer with the maximum FI was deemed to have failed. The first occurrence o f this was the FPF. In order to ensure that the predicted value was the converged solution, the maximum FI was recalculated at the new load and this data was fed into the interpolation procedure after discarding the data remote from the last step or the new load. The new prediction was usually within 1% of its previous value unless a large load increment was selected in the first instant. In that event, the interpolation procedure was repeated until convergence was obtained. The layer of the element was now recorded as failed and its material properties were adjusted using the degradation factor DF. The new material model was expected to have a new equilibrium path. To establish a point on this path at the current load, the interpolation and load increment method described above were restarted from 90% of the previous failure load. This is illustrated by a typical non-linear load-deflection characteristic behaviour of a shell segment as shown in Fig. 3. The non-linearity is attributed to the large deflection geometric non-linearity of the shell. In the figure the primary equilibrium path is due to the initial intact material and the subsequent secondary equi-
148
J. M&try, F. Levy-neto, Y-S. Wu
f~-~. / /
_
~/
/
~(~
~
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C~2
//
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(resultant path ~ h ~ ~e.~ ~ ~ e ...~-~O.~ ~ . ~ l , . , ~o[.Q
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/.pC"
/
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,"
/ ~ . ~ "
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~ ~/
///////////// B2
/
/:;;;;;.." /.,,',,,/;, ~-. ///'X"
A,
(resultant path with a p e ~ strength)
(steady collapse path)
--Primary
equilibrium path Secondary equilibriura paths of the shell w~th degraded material . . . . Resultant post-PFP path ~< Axi-symmetric collapse (snap-through buckling) cor~di~ions
....
,__
Deflection Fig. 3. Pseudo-post FPF paths.
librium paths result from the consideration of the degradation of the material as other plies at the same or other locations go through their failure cycles. The difference in deflection between primary and secondary paths and between secondary paths are deliberately exaggerated to demonstrate the possible post-FPF behaviours of the shell segment. Consider C1 as a point corresponding to the FPF load. The deflections of the shell segment at the FPF load on the secondary equilibrium path was obtained by decreasing the load by 10% to the point ci first. The point cii , which is at the same load level as the previous failure load, was obtained with the help of the normal iterative algorithm described above. The damage zone may expand locally without any increase in strength at the point cii or it may lead to stress redistribution in a manner such that there is an increase in strength to a point such as ciii. Depending on the position of the FPF load on the primary equilibrium path, it was possible to predict three different types of resultant post-FPF paths. A steady collapse path such as A1 - A 2 (Fig. 3) may be obtained with no significant increase in strength. The other possibility was the path B~ - B2, a resultant path with a peak strength. Though the points on the secondary equilibrium paths are all stable conditions, the peak strength on the resultant path is in fact the point of the axisymmetric collapse or snap-through buckling. The third possibility was that the resultant path may become tangential to the axisymmetric collapse point of a secondary equilibrium path and, again, the likely mode of failure is the snap-through buckling. During the analysis, if any damage zone spreads throughout the thickness of the shell then a conventional burst failure was expected but if the resultant path reached a peak load then the shell
structure was presumed to have failed by axisymmetric collapse. The behaviour of the dome following the snapthrough buckling, whether it occurs prior to FPF or in the post-FPF domain, often results in large unstable displacements. Another factor that can influence the behaviour of domed structures is the manner in which they are supported. The support condition can be described in terms of displacement components in axial, circumferential and radial directions together with a rotation about the circumference. In the case of a fully clamped
DEFORMED STRUCTURE LOAD STEP 1, LOAD =
1.000E+O0
PRESTRESS
INDIC =0
~0.
f
I
I
F
-[
~
--
20,
_~
O,
-20.
-80.
-80.
-~oo.
- - _ -20.
I O.
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I
20.
~10.
60.
_I
I 80.
100.
__ 120.
R
Fig. 4. Deformation of the clamped torispherical dome.
Collapse loads of externally pressurised composite torispherical and hemispherical domes
149
CTIOA
-~(
0
•~ - 3 7 ~
~
~
.00
~_~ 0'.17
~-,,
~
0".3~ ~
0".52 t~l~;TH
0".70
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situation all these displacements and the rotation were assumed to be zero. However, it may be difficult to achieve some of these conditions in practice. Galletly & Mistry 21 had shown that if the constraint in the
CTIOA
INDIC =
l.O00E+O0 PRESTRESS
0
50.
........ ~_._......
--.-...~
I
-300. -~50.
-TOO.
I -50.
I
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O.
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l TOO.
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Fig. 5. Principal stress resultants of the clamped torispherical dome.
DEFORMED STRUCTURE L O A D STEP I, LOAD =
/
I T50.
200.
R
Fig. 6. Deformation of the torispherical dome/cylinder combination.
Fig. 7. Principal stress resultants of the tofispherical dome/ cylinder combination. circumferential direction was relaxed then a marked change in the natural frequency of dome/cylinder combinations was possible. Similar sensitivity can be exhibited by the support condition on the final collapse loads. Figure 4 shows the deformed shape of the clamped torispherical dome of the series CT10 compared with the undeformed shape as calculated by the BOSOR 4 program. The predicted variation of the meridional force per unit length (N~0), circumferential force per unit length (N20) and bending moment per unit length (M~0) for an external pressure of 2.87MPa are shown in the Fig. 5. From these, it can be seen that the maximum bending stress is as expected at the clamped end of the dome. Any changes to the clamping condition is likely to affect this distribution. This can be seen clearly, if the dome is attached to a cylinder and the support condition is provided at the far end of the cylinder. The deformed and undeformed shapes of this case are shown in Fig. 6 and its main stress resultants (Ni0, N20 and M~0 ) in Fig. 7. The effect of removing the clamped condition from the dome end results in migration of the peak bending stress to the middle of the knuckle region. There is a peak bending moment at the clamped end of the cylinder, but due to the relatively higher thickness of the cylinder than that of the dome, a material failure was unlikely at this end. This case was also included in the study in order to isolate the effects of the end conditions on the final collapse loads.
J. Mistry, F. Levy-neto, Y-S. Wu
150 Port for ~
oil
/
Transparent
=~ome
/ ~
~
~
cover
4--
'.
2m
~ 1
~ Oil-filled~r~ure chamber ~ ~
//////// xxx xxxxxxxxx
~rl0B d o m e
J
I
0.5
I
1.0
[
1.5
Relatlv¢ Volumetric Change (~)
Fig. 8. Schematic diagram of the external pressure loading of the clamped domes. EXPERIMENTAL P R O C E D U R E The rigidly mounted domes were tested in a small pressure vessel in the form of a thick cylinder closed at one end and its other end was closed by the dome itself such that the pressurising fluid of the pressure vessel provided the external pressure loading of the domes. The arrangement is shown by the schematic diagram of Fig. 8. This arrangement allowed an easy inspection of the concave side of the dome during the progress of an experiment. At the failure load, the shell wall showed signs of weeping initially, and eventually at the final collapse it developed cracks. As the pressure vessel was too small in size to mount the dome/cylinder combinations, it was found necessary to test them within a large hyperbaric chamber. Since the inspection of the domes was impossible, the failure of the domes was detected by recording changes in internal volume of the dome/cylinder combination. The internal volume was charged with oil at atmospheric pressure. A small hole through the base support plate was connected to a small tube which passed through a monitor port of the chamber. Its external end was open and allowed the compression of the dome/cylinder
5 _
3_
J
~
2
1
a] n o i s e
~
CTIOAdome 0.5
1.0
1.5
2.0
Relative Volumetric Change (X)
Fig. 9. Variation of the external pressure with the volumetric changes of the CTIOA dome/cylinder combination.
Fig. 10. Variation of the external pressure with the volumetric changes of the CT10B dome/cylinder combination. combination to be measured in terms of the outflow of the oil. The pressure and relative volumetric changes recorded for the domes CT10A and CT10B are shown in Figs 9 and 10. Both tests recorded an initial loud cracking noise before the final collapse. Without the aid of expensive instrumentation or without making subjective judgements, it was impossible to identify the first ply failure by this noise alone because of the continuous low level acoustic emission throughout the tests.
C O M P A R I S O N OF THE EXPERIMENTAL RESULTS WITH THEORY In order to predict the failure loads, flat test specimens were prepared alongside the domes in identical environmental conditions and using the same manufacturing process. In all 16 specimens (6 for compressive, 5 for tensile and 5 for in-plane shear properties) were prepared. These test specimens were employed to determine the basic elastic and strength properties of the plies in longitudinal and in-plane transverse directions according to the ASTM Standards D303976, D3518-76 and D3410-87. 22'23 The statistical mean and standard deviation (as a percentage of the mean) of the elastic and strength properties were as given in Table 2. Results of the experiments carried out on the domes and theoretical prediction of their failure loads (FPF, LPF and buckling) using the above stiffness and strength data are compared with each other in Table 3. The theoretical buckling pressures were derived by assuming that the material remained intact. The figures quoted in the table were the lowest buckling pressures found from a search over a wide range of circumferential wave numbers. It is widely recognised that thin shell like structures tend to be sensitive to the geometrical imperfections, as a result, any prediction based on presumed perfect geometry usually works out to be an over-estimation of
Collapse loads of externally pressurised composite torispherical and hemispherical domes
151
Table 2. Mechanical properties of the woven carbon fibre/epoxy test specimens
Properties
Mean
Elastic moduli, E1 & E2 (GPa) In-plane shear modulus, Gl2 (GPa) Poisson's ratio, ulz Ultimate strengths in tension, XIT & XzT (MPa) Ultimate strengths in compression, X~c & X:c (MPa) Ultimate in-plane shear strength, X~2 (MPa) Maximum strains in tension, ~r & ~x (%) Maximum strains in compression, elC & e2c (%) Maximum in-plane shear strain, 712 (%)
66.45 5.00 0.05 617.6 465.4 78.2 0.93 0.71 1-86
the buckling or material strength. When one considers a ratio of the experimental result to its theoretical prediction Pexp/Ptheory t h e n the ratio is expected to be less than unity. This is seen from the set of results for the CH4 series domes. Both the LPF and FPF load predictions are of similar magnitude and the Pexp/Ptheory ratio is found in the range of 0.740 and 0"884. This set of domes involve only four layers of the reinforcing fibres and it is the thinnest set. The second set of domes, CH6 series, have the same geometrical profile as the previous series but the number of reinforcing fibre layers is six. In this case the FPF load predictions are in close agreement with the experimental failure loads except for CH6A dome, which in fact recorded a Pexp/Ptheoryratio of 1.149. The F P F load is starting to underestimate the actual strength of the dome. This factor is becoming more evident with the thicker torispherical domes CT8 and CT10. The strength is underestimated by almost 54% for the dome CT10A. On the other hand, the LPF load predictions show better correlation with the experiments as the
Standard deviation (%) 3.60 0.80 4.00 6.64 8.81 7.93 7.50 11.27 10.75
number of layers increase. With the increase in the number of layers, the thickness also increases and the effective bending resistance increases significantly. In fact for homogeneous materials, the bending resistance is proportional to the cube of the overall thickness. Hence, any failure of the outer layers of the composite shells does not necessarily lead to an immediate collapse but there is a substantial residual strength as shown by the post-FPF strength of the CT10 domes. Other interesting information that can be gained from the results concerns the resultant post-FPF paths. All CH4 series domes followed a C1-C2 path (see Fig. 3), in which the resultant path becomes tangential to a secondary equilibrium path at its snap-through buckling condition. In the case of the CH4A dome, the lowest buckling load is found for the wave number zero, (i.e. elastic snap-through buckling condition) and its value of 5.40 MPa was closest to the experimental and theoretical material failure loads. Though axisymmetric collapse was predicted, the actual mode of failure was by means of cracks near the clamped edge (see Fig. 11)
~
~
Fig. 11. Dome CH4A after failure with cracks near the clamping edge.
6"80 6"04 6" 11
10-14 9-97
4"41 3"79
CH6A 'CH6B CH6C
CT8A CT8B
CT10A CT10B
5-40 (0) a 6"79 (9) 6.89 (8) 13.63 (8) 13.72 (0) 13.40 (0) 16.93 (6) 16"83 (6) 6.408 (2) 5.880 (3)
Tsai-Wu strain Tsai-Wu strain Tsai-Wu stress Tsai-Wu strain Tsai-Wu stress Tsai-Wu stress Owen/Tsai-Hill Tsai-Wu strain Owen/Tsai-Hill Owen/TsaioHill
4.04 4-75 4'31 8.72 8-97 8-33 9.64 8-66 4.21 3-98
3-96 4"17 4'08 5"92 6"03 6'41 8.56 8.59 2.87 2-69
a
Theoretical buckling pressure (MPa)
Failure criterion b
Theoretical pressure at LPF (MPa)
Theoretical pressure at F P F (MPa)
Figures in round brackets refer to the circumferential wave number of the buckling mode. b Most conservative F P F predicting criterion.
3"50 3-52 3"54
Final pressure at failure, Pexp (MPa)
CH4A CH4B CH4C
Specimen
1.537 1.409
l- 185 1.161
1.149 1'002 0.953
0.884 0.844 0-868
C1 C2 C1-C2 C1 C2 A1 A2 A1-A2 A1 A2 B1 B2 B1 B2 B1-B2 B1-B2
0.780 0-673 0-733 1.052 1.151 1-048 0-952
Theoretical resultant post-FPF path 0-866 0.740 0.821
Pexp PLPF
Pexp
PFPF
Table 3. Comparison between experimental failure pressures and predicted FPF and LPF pressures
i
Collapse loads of externally pressurised composite torispherical and hemispherical domes
153
Fig. 12. Dome CH6A after failure with a large diametral crack. suggesting that the mode of failure was other than axisymmetric. This dome had only four layers and with almost negligible post-FPF load bearing capacity. The failure was found to be mainly governed by the material failure at the location of the minimum thickness. Additionally, when compared to the other domes, the experimental collapse loads of this series were less than the predicted FPF loads. The theoretical buckling loads of the hemispherical domes with six layers and for the remainder of the domes were found to be about twice the material failure loads. In the case of the CH6 series, the resultant path is that of a steady collapse A 1 - A 2 of Fig. 3. The torispherical domes, CT8 and CT10 series, had considerable post-FPF strength before their final collapse. These two series showed a B1-B2 resultant post-FPF path. The failure of the CH6, CT8 and CT10 domes was typified by large cracks spanning a wide surface area. In the case of CH6A dome, a diametral crack was formed and almost split it into two parts as shown in Fig. 12.
CONCLUSIONS As the number of layers in the composites increases there is a need for a material degradation model to assess the post-FPF strength. Theoretical analyses of the domes using the material degradation factor 0"3 applied to E~, E2 and GI2 have shown that more reliable estimates of the final collapse loads can be made compared to the FPF loads. This model appears to be satisfactory for the F R P domes made from woven carbon fibres because of their equal stiffness properties in the fibre and transverse directions.
ACKNOWLEDGEMENT The authors wish to acknowledge the University of Liverpool for allowing the use of the BOSOR 4 program and its laboratory and computing facilities.
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18. Hoffman, O. The brittle strength of orthotropic materials, J. of Composite Materials, 1967, 1, 200-206. 19. Tsai, S.W. & Wu, E.M. A general theory of strength of anisotropic materials, J. of Composite Materials, 1971, 5, 58-80. 20. Owen, M.J. Biaxial failure of GRP - - mechanisms, modes and theories, Proceedings of the 2nd Int. Conf. on Composite Structures, ed. I.H. Marshall, Elsevier Applied Science, 1983. 21. Galletly, G.D. & Mistry, J. The free vibrations of cylindrical shells with various end closures, Nuclear Engineering and Design, 1974, 30, 249-268. 22. ASTM Standards D3039-76 and D3518-76, Annual
Book of Standards, Part 36, Plastic Fibre Composites, 1977. 23. ASTM Standard D3410-87, Compressive properties of unidirectional or cross-ply fibre-resin composites, ASTM, 1987.