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FAILURE CRITERIA FOR USE IN THE DESIGN ENVIRONMENT1
Composites Science and Technology 58 (1998) 1095±1105 Crown copyright # 1998 Published by Elsevier Science Ltd. All rights reserved Printed in Great Britain PII: S0266-3538(96)00144-3 0266-3538/98 $Ðsee front matter
FAILURE CRITERIA FOR USE IN THE DESIGN ENVIRONMENT* G. C. Eckold AEA Technology, Harwell, Oxfordshire OX11 ORA, UK (Received 8 August 1995; revised 6 January 1996; accepted 13 September 1996)
x , y, s
Abstract This paper is concerned with the use of composite failure criteria in a design situation. The example chosen is from the process industry which employs composite materials in demanding applications and has requirements for simple, pragmatic and auditable procedures. This is distinct from other environments such as the aerospace industry or in R&D where there are resources available to undertake more sophisticated analyses. In the context of this failure prediction exercise the methods currently adopted by the process sector have been applied, albeit with some modi®cation, since materials of construction are dierent. Attempts have also been made to predict ultimate strengths, although this is not the prime purpose of a design code, and it is envisaged that there will be some discrepancies owing to the fact that such issues as material non-linearity and dierences in tensile and compressive properties are ignored. The more interesting relationship to note will be that between the derived design allowables and the initiation of failure since the basis of chemical process plant design is that normal service loads should be kept within these limits. Crown copyright # 1998 Published by Elsevier Science Ltd. All rights reserved
1 INTRODUCTION The ability to predict failure is a key aspect in the successful design of an engineering structure. It is not that a component will be designed to the limits of ultimate performance because, except in all but rare circumstances, this would clearly be foolhardy, but knowledge of the limit is necessary for a judgement to be made with regard to factors of safety. The ®rst step in establishing a basis for design is to consider what is meant by failure. For a metallic structure it will almost certainly be related to the onset and development of yield or rupture. With composites, on the other hand, the situation can be more complex and is more closely associated with the particular needs of the component of concern. For example, in a tension element it could be breakage through the failure of ®bre reinforcement, for a pipe under internal pressure it could be `weeping' of ¯uid through the development and propagation of microcracks; for a vehicle suspension member (leaf spring) it could be gross delamination leading to the collapse of bending stiness; and for a stressed aircraft skin it could be instability initiated through the delamination of plies at a free edge. As a result, it is unlikely that a single approach to failure prediction will meet the needs of all potential applications and materials of construction, and this should be a consideration at the outset of any design process. The attributes of a failure criterion in the design environment are somewhat dierent from those required for R&D purposes. In the latter it is absolute predictive capability which is a goal, whereas in the former, needs are more pragmatic, speci®cally:
Keywords: design environment, process industry, design code NOTATION Eo Fi, Fij Fx, Fy, Fxx, Fxy, Fyy, Fss R i, j
Axial, hoop and shear stress components
Unidirectional laminate modules Strength parameters for quadratic interaction criterion Strength terms for quadratic interaction criterion (ASME RTP-1) Design factor (ASME RTP-1) Design strain Winding angle Stress components
. A basis on which the designer can ascribe some form of physical (or quasi-physical) meaning. Abstract mathematical formulations are not favoured as they can be dicult to rationalise in engineering terms.
This article represents the author's contribution to a worldwide exercise to con®rm the state-of-the-art for predicting failure in composites, organised by Hinton and Soden.28 1095
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. Implementation by the use of easily measured material constants. This is particularly important given that in a true component design situation, validation testing, often undertaken by third parties will have to be carried out. . Capability of providing a predictive method of known accuracy. Although absolute accuracy is desirable it is not necessarily a prerequisite. A known level of error can be accommodated in design, albeit with the introduction of an added level of conservatism. . Known range of applicability which extends to the materials of concern and duties under consideration. Application across the boundaries of industrial sectors is not necessarily of interest. . Ease of use. Sophisticated analyses can be employed, but they must be presented as a series of charts or tables. There may be exceptional circumstances for `specials' and here computer programs, FEA, etc may be needed. However, for what are considered to be standard components, procedures which can be easily used and audited by all parties are necessary. Given a suitable criterion, its function in the design situation is to allow an evaluation to be made as to the risk of failure. Based on this, an assessment is then required on what value is appropriate for factors of safety and this will be based on both technical and commercial considerations. The net result will be a component or structure which is judged to be ®t for the purpose. Failure criteria for composites are many and varied.1±6 Indeed, the profusion of such theories has been a feature of composites literature. In their simplest form they are similar, in principle, to those used for isotropic materials. Examples here include variants of maximum stress/strain and distortional energy theories. The prime dierences being that stress components parallel and transverse to the ®bre are considered as opposed to those in the principal directions. In some cases, criteria consist of a number of expressions, each of which is related to a separate stress/strain term. This allows some consideration of the mode of failure, such as ®bre fracture or transverse cracking to be undertaken. For others, where this is considered too elementary a view, stress terms are combined in an attempt to take into account their interaction. Typical amongst the latter is the quadratic interaction criterion7 which has the form:
Fij i j Fi i 1
1
where Fij and Fi are strength parameters and i and j are components of stress. For implementation, say for plane stress in two dimensions, six strength parameters are required. Of these, ®ve can be shown to be conventional tensile, compressive or shear strength terms which can be measured in a conventional test programme, but the last is more dicult to obtain, since a biaxial test is
necessary, which is not easy to perform. A further apparent diculty arises as a consequence of the way in which individual terms of eqn (1) interact. For example, it has been reported8 that under certain circumstances eqn (1) would predict that a reduction in one strength value would lead to an increase in overall composite strength in a desirable mode of loading. This clearly cannot be correct. It should not be construed that this observation is intended to single out this approach for particular criticism but to demonstrate that, basic diculties remain. It is cited because it is one of the most common criteria described in textbooks on the subject and is incorporated within most ®nite element codes which cater for anisotropic materials. The validation of failure theories with experimental data remains a vexed issue. For uniaxial loads and simple laminates correlation can be good. However, for other situations, some no more complex than biaxial loading on a simple cylinder with angle ply or quasiisotropic laminates, the situation is often unsatisfactory. Good agreement can be achieved, but often only with selected datasets. One reason for this may be that one of the basic assumptions used in composite stress analysis, that of linear behaviour up to failure (non-linearity due to ply discount notwithstanding), is not strictly correct. Comparatively recent work based on damage mechanics and micromechanics has shown that the initiation and propagation of damage, such as transverse cracks, can have a profound eect on structural response.9±12 These eects are akin to plasticity in metals where local redistribution of stress through local stiness change can result in signi®cant improvements in observed ultimate performance. It may be that continued work in these areas will provide the missing ingredient in failure theory development. The purpose of this paper is not to advocate any one particular approach to composite failure analysis, but to describe how one particular industrial sector, the process-plant industry, has addressed the issue. This industry uses composites in an environment which is both demanding in terms of operating conditions and competitive given that there are alternative material systems that can be used for most duties. It has been necessary to come to terms with the uncertainties described in the preceding paragraphs and to produce methods which are able to be presented in a format suitable for incorporation within formal design codes and which have and are being used with apparent success. Comparison with experimental data as part of the failure-prediction initiative is a valuable exercise in that it will assist in evaluating whether or not the degree of conservatism embodied within the methods is appropriate. 2 DESIGN METHODS Composite process plant is manufactured from a number of material types, but is almost invariably based on
Failure criteria for use in the design environment glass reinforcement in random mat, woven cloth or unidirectional form. There are numerous design standards13±19 for both pipe and vessels and the two documents which stand out as being the most comprehensive are BS4994.1987 and the more recent ASME RTP-1. Indeed, these are the only documents which provide explicit guidance on the use of anisotropic materials which in the case of the process plant usually take the form of ®lament wound angle ply or unidirectional laminates. In design, the overall objective is to produce a laminate which at operational loads will not be subject to microcracking at any time during its service life. This is considered important so as to prevent ingress of process ¯uid into the laminate whereupon it may precipitate rapid deterioration. It should be noted that in many cases the chemical environment can be very aggressive. Concentrated mineral acids, alkalis and oxidising agents are not unusual. Furthermore, loads tend to be applied continuously over extended timescales and creep rupture is a mode of failure which must be prevented. As a result, design margins, ie the ratio of shortterm static strength to the design allowable, tend to be in the region of 8±10. This is based on early work with chopped-strand-mat laminates which indicated that an upper design strain of 0.2% is appropriate20 and has been substantiated by subsequent creep rupture data from laminates exposed to various chemical media.21,22 2.1 BS4994.1987 The design basis in BS499413 is one of limited strain and a detailed procedure is given for the calculation of an allowable value based on the consideration of the method of manufacture, fatigue, operating temperature, enviromnent and curing procedure. The minimun design margin (de®ned above) is 8, although consideration is being given to reducing this to 6. From the derived strain value an allowable load is determined by the use of the tabulated modulus values, although a supplier may use higher measured values by agreement (note that BS4994 employs the concept of unit load and modulus expressed as kg/m2 of glass, but for the purposes of this paper the discussion is presented in con-
Fig. 1. Variation of modulus with winding angle.
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Table 1. BS4994 factors for calculating longitudinal and circumferential design stresses Fibre angle to longitudinal axis 0<15 15 <75 75 <90
Longitudinal factor
Circumferential factor
1.0 0.5 0
0 0.5 1.0
ventional stress terms). For ®lament-wound laminates a further factor is applied to take into account the eects of anisotropy.23 Figure 1 shows the variation in axial modulus as a function of winding angle as given in the standard. The data shown are for E-glass ®bres and a vinylester resin.24 Most of the process plant is fabricated with styrenated resin systems, e.g. polyester, vinylester, etc, and the graph was originally calculated by using properties appropriate to these materials and simple laminate theory. Modulus values would be low if an epoxybased system is to be considered. There is scope for adjusting the data to take the variation in ®bre fraction into account since, in the document itself, the information is given per unit weight of glass. In Fig. 1 the data corresponds to a glass content of 75% by weight (60% by volume). Also shown are measured modulus values from nozzle cutouts (material removed when a circular hole is cut for the attachment of a nozzle) taken from large GRP tanks and the agreement is good given that a design code should be based on lower bound or deviated data. The factors which are applied to these modulus values to determine allowable stresses are shown in Table 1. At 0.2% strain the allowables as calculated from this method are shown in Fig. 2. The discontinuities arise as a result of the step changes in the applied factor for 15 and 75 laminates. There are two key assumptions in this process: . strains in all directions (axial, transverse, shear) are equally important and should be limited by the same upper limit; . transverse strains for high winding angles (>75 ) should be further limited to a lower
Fig. 2. Allowable stresses for laminates.
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G. C. Eckold overriding maximum of 0.1%. This is because of the low absolute value of transverse strength and the fact that at high angles the transverse direction is not supported to any great extent by the reinforcement.
For transverse strains there is an inconsistency in the document because the data in Table 1 indicate that the design value should be zero as opposed to the 0.1% limit. The BS4994 procedure is still workable, however, as the intent is that the transverse strength should be ignored in terms of structural contribution when deriving the basic laminate construction, but then to go back and check that the overall strain is less than the 0.1% level. The calculation on which the Table 1 parameters were based again employed simple laminate analysis to evaluate strains in each of the primary material directions.23 This results in a continuous design curve (as an alternative to Table 1) for allowables an example of which, for 0.2% strain, is also shown in Fig. 2. One of the most signi®cant question marks over the method is whether or not shear and normal strains should be limited by the same value. A criticism of the current BS4994 approach is that the procedure does not allow the full bene®ts of the reinforcement for combined loading. This is not a issue for storage tanks as the loading is dominated by the static head pressure. Axial loading is small and variable. It would not be practical to advocate that all possible load permutations are considered, as there would be little added bene®t. The exception is for pressure vessels where there is a 2:1, hoop:axial pressure stress. It is straightforward to generate an additional curve for this case and this is also shown in Fig. 3. Note that it is the allowable stress in the axial direction which is shown. Again the 0.1% transverse strain limit has been imposed to ensure consistency with the uniaxial case. With this approach a simpli®ed biaxial load envelope such as that shown in Fig. 4 can be developed. The values on the ordinate and abscissa are given by the uniaxial curve in Fig. 2 and the combined 2:1 point is given by the biaxial plot in Fig. 3. Also shown in Fig. 4 are the results of a netting analysis where the stress values at the point where the loads combine to act in the ®bre direction are given by:
Fig. 3. Allowable stresses for laminates.
x Eo " cos2 y Eo " sin2
2
where x and y are axial and hoop stress components for the 2:1 condition, " is the design strain (0.2%), Eo is the unidirectional laminate modulus and is the winding angle. As can be seen the netting analysis provides a more conservative result for the winding angle used for the calculation (55 to the axial direction), although the curves are of a similar shape. For other laminate con®gurations this will not be the case as the values given by eqn (2) will not be for the 2:1, hoop:axial stress ratio. This is not a major issue for process plant as the main design conditions are static head pressure (uniaxial load) and internal pressure which is addressed by the above discussion. A point of interest is that the approach of evaluating allowables on the basis of limiting maximum strain does not apparently result in the 55 winding angle being optimum for a 2:1 load case as is customarily assumed (see Fig. 3). A somewhat higher angle is indicated. The corollary to this is that a 2:1 loading situation is not the best for a 55 angle ply. There is some supporting evidence for this from biaxial test results where the performance of such a laminate is noted to be better at a stress ratio of nearly 3:1.25 The physical signi®cance of this has not been explained, but clearly the assumptions within the netting analysis are in diculty. It is worth noting that the stress ratio at which the in-plane shear stress, in the material's coordinate axis becomes zero, is 2.8 (the winding angle at which the shear stress becomes 0 for a 2:1 loading condition is 60 ). 2.2 ASME RTP-1 The ASME document18 describes a two-tier approach and the supplier is free to chose which to use for the design. In the simpler of the methods the determination of allowables is carried out as follows: . in the hoop direction the allowable stress is calculated by using a maximum strain limitation of 0.1% regardless of the winding angle;
Fig. 4. Stress envelope for 55 winding angle.
Failure criteria for use in the design environment . in the axial direction the allowable stress is calculated by dividing the ultimate strength by a factor of 10 again regardless of winding angle. For the second method an altogether more complex approach is taken. Within the code are appendices which give a description of the mechanics of laminate theory and the user is invited to carry out a full analysis. It is arguable as whether or not this is appropriate for a design code as if not correctly interpreted its use will cause problems. For example, the calculation of the coupling [B] matrix is described, but no guidance is given as to what to do with it. Coupling may give rise to twist in an unconstrained ¯at plate, but such deformations cannot exist in a closed cylindrical surface! Once the stress analysis is complete the assessment continues by using the quadratic interaction criterion described in the proceeding section. It is presented in a modi®ed form as follows: R2 Fxx x2 2Fxy x y Fyy y2 Fss s2
R Fx x Fy y ÿ 1 0
3
Measured values can be used for the strength terms (Fxx, Fxy, etc.) and if these are not available, strains are given, which vary with the direction and mode of loading, to be used in conjunction with the relevant stiness (Table 2). The coecient, R, is the design factor for the laminate and is speci®ed as being between 8 and 12 for the inner surface adjacent to the process ¯uid, depending on the criticality of service etc. Away from the surface a lower ®gure may be used, but for simple membrane stresses there is no bene®t to be gained. Allowable stresses as calculated by the ASME method are compared with BS4994 values in Fig. 2. 2.3 Performance-based approach It is worth noting that there is another approach to process plant design, but one which is really only suitable for products manufactured on a commodity basis such as standard pipework systems. Here the objective is to measure long-term performance of typical components (not laminates) under conditions similar to those experienced in service. Creep±rupture testing of pipework components over durations >10 000 h is one of the established methods.26 The key dierence with the approach is that knowledge of how laminates behave is not necessary as actual components are tested. The supplier then has a responsibility to establish quality Table 2. ASME RTP-1 strain values Stress component Longitudinal tension Longitudinal compression Transverse tension Transverse compression Shear
Strain value (%) 2.00 1.20 0.15 0.80 1.50
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assurance and quality control procedures which ensure that future manufactured items are, for all practical purposes, identical to those which have been quali®ed. Another problem is that testing usually takes place under a single load condition, most commonly internal pressure with closed ends, and the use of the arising data for other situations is not straightforward. There has been some progress in addressing this aspect of design through making assumptions regarding the shape of the failure envelope.27 The strains at the design stresses tend to be higher (0.3±0.4%) than those calculated by the design-by-rule approach described in the vessel standards. This is appropriate given that the latter employ short term static strength values as a basis and some conservatism is necessary. 3 FAILURE PREDICTION The procedures described in the preceding sections are intended for the evaluation of design allowables for GRP process-plant materials and to be commensurate with the needs of a design environment. This poses certain diculties when attempting to oer predictions of behaviour for the materials included within this failure exercise. (Details of the lamina properties and lay-up con®gurations and loading of the laminates analyzed are provided in Ref. 29.) These problems have been accommodated as follows: . Material properties. Tabulated material properties in BS4994 are intended for resin systems of the polyester type and their use for epoxy systems would not be appropriate. However, the standard does allow alternative values to be used provided they are based on test data. In this exercise the assupplied property values for unidirectional laminae have been used together with simple laminate analysis to give o-axis information. Table 3 gives the values as calculated by this process. A diculty also arises in the speci®cation of an allowable maximum strain. For polyesters this is currently 0.2%, but this may be considered low for epoxy systems. The results from performance-based testing, Table 3. Calculated modulus values Laminate E-Glass/LY556/HT907/DY063 30 angle ply Hoop Axial E-glass/MY750/HY917/DY063 55 angle ply Hoop Axial Cross-ply 45 angle ply
Calculated modulus (GPa)
15.8 30.5 23.5 15.4 31.1 17.7
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G. C. Eckold
most of which relates to epoxy components, indicates that values in the range 0.3±0.4% are satisfactory. A strain limit of 0.4% has been used in all of the calculations described in the following sections, except for transverse tension where the 0.1% limit is maintained. The selection of 0.40% is, in a sense arbitrary, but it provides a basis for this exercise. . Prediction of ultimate strength. The design code is not concerned with the prediction of ultimate performance, and, furthermore, there is no mechanism for doing so. For the purposes of this paper the method which has been adopted is that which would be taken by an engineer given the as supplied properties and the general approach to laminate design described in BS4994. Calculations have been performed at selected points and straight lines have been drawn between them, as opposed to trying to use a formal composite failure criterion, which will be addressed by other contributors to this exercise. . Thermal stresses. Most GRP process vessels are fabricated by using room-temperature cured
Fig. 5. Stress envelope for 0 laminate (E-glass/MY750/ HY917/DY063).
Fig. 6. Stress envelope for 0 laminate (E-glass/LY556/ HY907/DY063). Note that structural instability will govern design in compressive quadrants.
materials. Sheer size eectively precludes systems which need elevated temperatures during processing and as a result the eects of thermal stresses are not relevant. However, in the following calculations a simple manual correction has been applied by using given thermal expansivities and the stated stress-free temperatures. . Compressive properties. BS4994 makes no attempt to dierentiate between tensile and compressive strengths. The reality is that if there is a signi®cant compressive loading component, e.g. vacuum, it will be structural instability which governs the design. This is especially true for glass-reinforced composites where modulus values are relatively low. The introduction of additional properties would be an unnecessary complication. In this exercise no attempt has been made to allow for differences in strengths in the two modes of loading. . Non-linear properties. Non-linearity, either through a material characteristic or ply discount, does not feature in the calculation method, since allowables are generally based on low strains prior to any form of initial failure. No attempt has been
Fig. 7. Stress envelope for 0 laminate (E-glass/LY556/ HT907/DY063).
Fig. 8. Stress envelope for 30 laminate (E-glass/LY556/ HT907/DY063).
Failure criteria for use in the design environment made to accommodate this in the following predictions. Calculated stress/strain curves have been limited (arbitrarily) to 0.5% strain. . Carbon-®bre materials. CFRPs do not yet feature in the catalogue of material systems employed by the process industry. A basic premise behind the arguments expressed in this paper is that design methods should be appropriate for the industry and materials for which they are intended. More advanced materials (and their associated cost) require, in principle, more sophisticated design methods if the advantages of their use are to be realised. On this basis no attempt has been made to oer predictions for CFRP laminates. 3.1 Unidirectional laminate
Fig. 9. Stress envelope for 30 /90 laminate (E-glass/ LY556/HT907/DY063).
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Predictions for biaxial stress envelopes for unidirectional laminae are shown in Figs 5 and 6. In Fig. 5 the envelope depicting ultimate strength is based on the as supplied strength values which are then treated as per BS4994. The design envelope was derived by using a strain limitation of 0.1% in the transverse direction and 0.4% (as discussed above) for the longitudinal direction. The ratios between ultimate strength and design are broadly in line with the intent of the code. The quadrant where there is compressive stress is, again, treated by following prevailing design code practice. The prediction will be conservative where loads are primarily uniaxial (compression) and in the transverse direction owing to the higher strengths in this direction. The converse will be true for compressive loads which will be predominantly in the ®bre direction. However, as has already been discussed, this is unlikely to be a design issue as structural instability will almost certainly become the governing concern. The envelopes in Fig. 6 have been calculated using a similar process, but here it is expected that shear non-linearity will have a marked eect on the results. As the shear modulus falls a greater proportion of the load will be carried in other directions and this may have the eect of reducing observed strength. On this basis it would be expected that the simple linear calculation would represent an upper bound. The combinations of load as represented by Fig. 6 do not feature in process-plant design where stresses are primarily membrane or, in the case of local features such as nozzles, discontinuities, lifting brackets etc, bending. Although pipework systems do experience some torsional loading it is generally the case that these are small. Large in-plane shear stresses are not seen. In BS7159 the combination of normal and shear stresses is considered using the conventional Tresca criterion. However, it must be said that this was probably adopted directly from the equivalent steel codes and its applicability to composites, especially those which are anisotropic, was not thoroughly considered.
Fig. 10. Stress envelope for 55 laminate.
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Fig. 11. Stress envelope for 55 laminate under a stress ratio of 1:0.
To adopt this approach in this exercise to shear loadings would perhaps be pushing the codes beyond their intent.
In terms of design, the approach taken for a hybrid laminate construction is to consider each type of layer in turn and to establish which has the lowest allowable strain. This value is then used as a basis for further evaluation. The ®rst stage in the analysis is to determine the behaviour of each laminate layer and this is shown in Figs 7 and 8. These were derived as follows: Unidirectional envelope:
3.2 (90/30 /90) laminate The use of combined winding angles is not an uncommon occurrence in storage/pressure vessels where an angle-ply laminate (usually >45 ) is used to support basic loadings in the axial direction and additional hoop winding is applied to accommodate pressure loads. In some design examples the two cases are kept entirely separate, i.e. angle-ply laminates are used to carry axial load only and 90 windings are employed to carry circumferential load only. Whilst this may be construed to be conservative, the fact that the role of each layer is unambiguous means that the design is `conceptually' simple and this has the bene®t that approval by third parties, who will be competent in terms of process plant but not necessarily well versed in composites, will be straightforward. The rami®cation of time saved in this part of the procurement process can outweigh the penalty of added material cost.
From these results, associated modulus values (Table 3) and the layer thicknesses (90 ÿ0.35 mm,
Fig. 12. Stress/strain curve for 55 laminate under a load ratio of 2:1.
Fig. 13. Stress/strain curve for 0 /90 laminate under a load ratio of 0:1.
. from treatment for the unidirectional laminate described in the preceding section. 30 envelope: . axial and hoop strengths were determined by the ultimate transverse (0.197%) or shear (0.38%) strains as appropriate; . strength at the 2:1 ratio was determined by netting analysis limited by the ultimate longitudinal strain (2.13%).
Failure criteria for use in the design environment 30 ÿ1.65 mm) the various contributions of each layer type can be assessed. The results of this calculation are shown in Fig. 9. The envelope for ultimate strength takes into account redistribution of stress after the `weaker' layer has failed. For the design envelope all values were determined by using a design-strain limitation of 0.4% (0.1% for transverse tension) generally in accordance with the methods described in Section 2.1. For the axial directions the design values look particularly pessimistic. This is due to the 0.1% transverse strain value. In principle this could be increased if it were felt that cracks in this layer could be tolerated. The values for the compressive quadrants were evaluated by assuming that the simple use of the equivalent tensile data was appropriate. The comments on structural instability made in material Section 3.1 for unidirectional material are equally applicable for the more complex laminates.
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It is acknowledged that the ultimate envelope may represent an upper bound as a consequence of material non-linearity which will result in local redistribution of load giving rise to higher strains in other directions. Discrepancies will also occur because the compression strengths have not been used. For the design envelope the important considerations are not limited to its relationship to static strength, but more importantly how it compares with the initiation of damage. It is essential that there is a sucient margin here so that eects such as long term loading, the chemical environment and variability of material properties can be accommodated. 3.3 Angle ply (55 ) laminate Figure 10 shows the biaxial stress envelope for a 55 laminate. Calculations were carried out as follows.
Fig. 14. Stress/strain curve for 45 laminate made of GRP material (sy/sx=1/ÿ1).
Fig. 15. Stress/strain curve for 45 laminate made of GRP material (sy/sx=1/1).
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Ultimate strength envelope: . axial and hoop strengths were determined by the ultimate transverse (0.246%) or shear (4%) strains as appropriate; . strength at the 2:1 ratio was determined by netting analysis limited by the ultimate longitudinal strain (2.13%).
areas where there is scope to back up more rigorous approaches with extensive testing programmes. Participation in this failure prediction exercise will be of bene®t in putting this and other methods into an overall design and engineering context.
Design strength envelope:
REFERENCES
. Comments with regard to the ultimate envelope being an upper bound because of material nonlinearity and the eects of the higher compressive strengths are as per previous predictios.
1. Azzi, V. D. and Tsai, S. W., Anisotropic strength of composites. Exp. Mech., 1965, 5, 283. 2. Norris, C. B., Strength of orthotropic materials subjected to combined stresses. Forest Products Lab, FPL 1816, 1950. 3. Puppo, A. H. and Evensen, H. A., Strength of anisotropic materials under combined stresses. AIAA J., 1972, 10, 468. 4. HuÈtter, U., Schelling, H. and Krauss, H., An experimental study to determine the failure envelope of composite materials with tubular specimens under combined loads and comparison between several classical criteria. AGARD-CP-163, 1975. 5. Gol'denblat, I. I. and Kopnov, V. A., Strength of glass reinforced plastics in the complex stress state. Mekhanika Polimerov, 1965, 1, 70. 6. Huang, C. L. and Kimser, P. G., A criterion for strength for orthotropic materials. Fibre Sci. Technol., 1975, 8, 103. 7. Tsai, S. W. and Wu, E. M., A general theory of strength for anisotropic materials. J. Compos. Mater., 1971, 5, 58± 82. 8. Hart-Smith, L. J., The role of biaxial stresses in discriminating between meaningful and illusory composite failure theories. Compos. Struct., 1993, 25, 3±20. 9. Talreja, R., Stiness properties of composite laminates with matrix cracking and interior delamination. Engng Fract. Mech., 1986, 25, 751±762. 10. Joshi, G. P. and Frantziskonis, G., Damage evolution in laminated advanced composites. Compos. Struct., 1991, 17, 127±139. 11. McCartney, L. N., The prediction of non uniform cracking in biaxially loaded cross ply laminates. NPL Report DMM(A) 142, 1994. 12. Eckold, G. C., Hancox, N. L. and Lee, R. J., Application of micromechanics in the prediction of damage initiation and growth in structural composites. Deformation and Fracture of Composites, Surrey, March 1995. 13. Speci®cation for design and construction of vessels and tanks in reinforced plastics, BS4994. BSI, 1984. 14. Speci®cation for reinforced plastics pipes, ®ttings and joints for process plants, BS6464. BSI, 1984. 15. Code of practice for design and construction of glassreinforced plastics (GRP) piping systems for individual plants or sites, BS7159. BSI, 1989. 16. Speci®cation for low pressure ®breglass line pipe, API 15LR. API, 1990. 17. Speci®cation for high pressure ®breglass line pipe, API 15HR. API, 1988. 18. Reinforced thermoset plastic corrosion resistant equipment, ASME RTP-1. ASME, 1992. 19. Pressure vessels in glass ®bre reinforced thermosetting plastics. AD-Merkblatt N1. 20. Smith, T. R. and Owen, M. J., The progressive nature of fatigue damage in glass reinforced plastics. In Proc. 6th Int. Resins and Plastics Conf. of the British Plastics Federation. British Plastics Federation, London, 1968, paper 27. 21. Roberts, R. C., Reinforced Plastics Congress. BPF, 1978, p. 145.
Stress/strain curves, to 0.5% strain, are shown in Figs 11 and 12. These are based on simple laminate calculations using the provided material constants. It is anticipated that at high strains signi®cant non-linearity will be noticeable. This is not necessarily relevant in design if it is concerned solely with allowables limited by lower strains. 3.4 Cross-ply laminate Figure 13 shows a stress/strain curve for the cross-ply laminate subjected to uniaxial loading. Again it is linear and no attempt has been made to identify the `knee' which occurs as a consequence of transverse cracking. 3.5 Angle ply (45 ) laminate Figures 14 and 15 show the stress/strain characteristics for the 45 laminates. These were calculated as for those described above and similar comments apply. 4 CONCLUSIONS This paper considers the attributes of composite failure criteria necessary for use in the design environment where the need for reliable prediction of performance must be balanced by simplicity and ease of use. The prime purpose of a failure theory is as a tool in the assessment of whether or not a design is `®t for purpose'. It may be that calculations for ultimate strength are necessary, but more likely it will be some other facet of behaviour which provides a baseline for design. This will certainly be the case where loads are applied continuously and over a long term. This is not to say that knowledge of maximum load capability is not important, as it can provide a backdrop against which upset conditions can be considered and give con®dence to the user with respect to performance in service. The industrial sector which has been chosen as an example, process plant manufacture, has at its disposal a number of design standards for use which, in the main, meet their needs. However, it is acknowledged that their treatment of composite materials behaviour can be criticised as not being technically robust and certainly would not be considered satisfactory in other
Failure criteria for use in the design environment 22. Hogg, P. J., Hull, D. and Legg, M. J., Composite Structures, ed. I. M. Marshall. Applied Science Publishers, London, 1981, p. 106. 23. Eckold, G. C., A design method for ®lament wound GRP vessels and pipework. Composites, 1985, 16, 41±47. 24. AEA Technology, Internal Report, 1995. 25. Tolhoek, P., Composite Materials in the Oshore Industry. Aberdeen, November 1995. 26. Practice for obtaining hydrostatic or pressure design basis for ®breglass pipe and ®ttings, ASTM D2992.
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27. Eckold, G. C., A performance based design methodology for GRP pipework and ®ttings. Proc. Inst. Mech. Eng., 1995, 209, 41±50. 28. Hinton, M. J. and Soden, P. D., Predicting failure in composite laminates: the background to the exercise. Compos. Sci. Technol., 1998, 58(7), 1001. 29. Soden, P. D., Hinton, M. J. and Kaddour, A. S., Lamina properties, lay-up con®gurations and loading conditions for a range of ®bre reinforced composite laminates. Compos. Sci. Technol., 1998, 58(7), 1011.
FAILURE CRITERIA FOR USE IN THE DESIGN ENVIRONMENT* G. C. Eckold AEA Technology, Harwell, Oxfordshire OX11 ORA, UK (Received 8 August 1995; revised 6 January 1996; accepted 13 September 1996)
x , y, s
Abstract This paper is concerned with the use of composite failure criteria in a design situation. The example chosen is from the process industry which employs composite materials in demanding applications and has requirements for simple, pragmatic and auditable procedures. This is distinct from other environments such as the aerospace industry or in R&D where there are resources available to undertake more sophisticated analyses. In the context of this failure prediction exercise the methods currently adopted by the process sector have been applied, albeit with some modi®cation, since materials of construction are dierent. Attempts have also been made to predict ultimate strengths, although this is not the prime purpose of a design code, and it is envisaged that there will be some discrepancies owing to the fact that such issues as material non-linearity and dierences in tensile and compressive properties are ignored. The more interesting relationship to note will be that between the derived design allowables and the initiation of failure since the basis of chemical process plant design is that normal service loads should be kept within these limits. Crown copyright # 1998 Published by Elsevier Science Ltd. All rights reserved
1 INTRODUCTION The ability to predict failure is a key aspect in the successful design of an engineering structure. It is not that a component will be designed to the limits of ultimate performance because, except in all but rare circumstances, this would clearly be foolhardy, but knowledge of the limit is necessary for a judgement to be made with regard to factors of safety. The ®rst step in establishing a basis for design is to consider what is meant by failure. For a metallic structure it will almost certainly be related to the onset and development of yield or rupture. With composites, on the other hand, the situation can be more complex and is more closely associated with the particular needs of the component of concern. For example, in a tension element it could be breakage through the failure of ®bre reinforcement, for a pipe under internal pressure it could be `weeping' of ¯uid through the development and propagation of microcracks; for a vehicle suspension member (leaf spring) it could be gross delamination leading to the collapse of bending stiness; and for a stressed aircraft skin it could be instability initiated through the delamination of plies at a free edge. As a result, it is unlikely that a single approach to failure prediction will meet the needs of all potential applications and materials of construction, and this should be a consideration at the outset of any design process. The attributes of a failure criterion in the design environment are somewhat dierent from those required for R&D purposes. In the latter it is absolute predictive capability which is a goal, whereas in the former, needs are more pragmatic, speci®cally:
Keywords: design environment, process industry, design code NOTATION Eo Fi, Fij Fx, Fy, Fxx, Fxy, Fyy, Fss R i, j
Axial, hoop and shear stress components
Unidirectional laminate modules Strength parameters for quadratic interaction criterion Strength terms for quadratic interaction criterion (ASME RTP-1) Design factor (ASME RTP-1) Design strain Winding angle Stress components
. A basis on which the designer can ascribe some form of physical (or quasi-physical) meaning. Abstract mathematical formulations are not favoured as they can be dicult to rationalise in engineering terms.
This article represents the author's contribution to a worldwide exercise to con®rm the state-of-the-art for predicting failure in composites, organised by Hinton and Soden.28 1095
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. Implementation by the use of easily measured material constants. This is particularly important given that in a true component design situation, validation testing, often undertaken by third parties will have to be carried out. . Capability of providing a predictive method of known accuracy. Although absolute accuracy is desirable it is not necessarily a prerequisite. A known level of error can be accommodated in design, albeit with the introduction of an added level of conservatism. . Known range of applicability which extends to the materials of concern and duties under consideration. Application across the boundaries of industrial sectors is not necessarily of interest. . Ease of use. Sophisticated analyses can be employed, but they must be presented as a series of charts or tables. There may be exceptional circumstances for `specials' and here computer programs, FEA, etc may be needed. However, for what are considered to be standard components, procedures which can be easily used and audited by all parties are necessary. Given a suitable criterion, its function in the design situation is to allow an evaluation to be made as to the risk of failure. Based on this, an assessment is then required on what value is appropriate for factors of safety and this will be based on both technical and commercial considerations. The net result will be a component or structure which is judged to be ®t for the purpose. Failure criteria for composites are many and varied.1±6 Indeed, the profusion of such theories has been a feature of composites literature. In their simplest form they are similar, in principle, to those used for isotropic materials. Examples here include variants of maximum stress/strain and distortional energy theories. The prime dierences being that stress components parallel and transverse to the ®bre are considered as opposed to those in the principal directions. In some cases, criteria consist of a number of expressions, each of which is related to a separate stress/strain term. This allows some consideration of the mode of failure, such as ®bre fracture or transverse cracking to be undertaken. For others, where this is considered too elementary a view, stress terms are combined in an attempt to take into account their interaction. Typical amongst the latter is the quadratic interaction criterion7 which has the form:
Fij i j Fi i 1
1
where Fij and Fi are strength parameters and i and j are components of stress. For implementation, say for plane stress in two dimensions, six strength parameters are required. Of these, ®ve can be shown to be conventional tensile, compressive or shear strength terms which can be measured in a conventional test programme, but the last is more dicult to obtain, since a biaxial test is
necessary, which is not easy to perform. A further apparent diculty arises as a consequence of the way in which individual terms of eqn (1) interact. For example, it has been reported8 that under certain circumstances eqn (1) would predict that a reduction in one strength value would lead to an increase in overall composite strength in a desirable mode of loading. This clearly cannot be correct. It should not be construed that this observation is intended to single out this approach for particular criticism but to demonstrate that, basic diculties remain. It is cited because it is one of the most common criteria described in textbooks on the subject and is incorporated within most ®nite element codes which cater for anisotropic materials. The validation of failure theories with experimental data remains a vexed issue. For uniaxial loads and simple laminates correlation can be good. However, for other situations, some no more complex than biaxial loading on a simple cylinder with angle ply or quasiisotropic laminates, the situation is often unsatisfactory. Good agreement can be achieved, but often only with selected datasets. One reason for this may be that one of the basic assumptions used in composite stress analysis, that of linear behaviour up to failure (non-linearity due to ply discount notwithstanding), is not strictly correct. Comparatively recent work based on damage mechanics and micromechanics has shown that the initiation and propagation of damage, such as transverse cracks, can have a profound eect on structural response.9±12 These eects are akin to plasticity in metals where local redistribution of stress through local stiness change can result in signi®cant improvements in observed ultimate performance. It may be that continued work in these areas will provide the missing ingredient in failure theory development. The purpose of this paper is not to advocate any one particular approach to composite failure analysis, but to describe how one particular industrial sector, the process-plant industry, has addressed the issue. This industry uses composites in an environment which is both demanding in terms of operating conditions and competitive given that there are alternative material systems that can be used for most duties. It has been necessary to come to terms with the uncertainties described in the preceding paragraphs and to produce methods which are able to be presented in a format suitable for incorporation within formal design codes and which have and are being used with apparent success. Comparison with experimental data as part of the failure-prediction initiative is a valuable exercise in that it will assist in evaluating whether or not the degree of conservatism embodied within the methods is appropriate. 2 DESIGN METHODS Composite process plant is manufactured from a number of material types, but is almost invariably based on
Failure criteria for use in the design environment glass reinforcement in random mat, woven cloth or unidirectional form. There are numerous design standards13±19 for both pipe and vessels and the two documents which stand out as being the most comprehensive are BS4994.1987 and the more recent ASME RTP-1. Indeed, these are the only documents which provide explicit guidance on the use of anisotropic materials which in the case of the process plant usually take the form of ®lament wound angle ply or unidirectional laminates. In design, the overall objective is to produce a laminate which at operational loads will not be subject to microcracking at any time during its service life. This is considered important so as to prevent ingress of process ¯uid into the laminate whereupon it may precipitate rapid deterioration. It should be noted that in many cases the chemical environment can be very aggressive. Concentrated mineral acids, alkalis and oxidising agents are not unusual. Furthermore, loads tend to be applied continuously over extended timescales and creep rupture is a mode of failure which must be prevented. As a result, design margins, ie the ratio of shortterm static strength to the design allowable, tend to be in the region of 8±10. This is based on early work with chopped-strand-mat laminates which indicated that an upper design strain of 0.2% is appropriate20 and has been substantiated by subsequent creep rupture data from laminates exposed to various chemical media.21,22 2.1 BS4994.1987 The design basis in BS499413 is one of limited strain and a detailed procedure is given for the calculation of an allowable value based on the consideration of the method of manufacture, fatigue, operating temperature, enviromnent and curing procedure. The minimun design margin (de®ned above) is 8, although consideration is being given to reducing this to 6. From the derived strain value an allowable load is determined by the use of the tabulated modulus values, although a supplier may use higher measured values by agreement (note that BS4994 employs the concept of unit load and modulus expressed as kg/m2 of glass, but for the purposes of this paper the discussion is presented in con-
Fig. 1. Variation of modulus with winding angle.
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Table 1. BS4994 factors for calculating longitudinal and circumferential design stresses Fibre angle to longitudinal axis 0<15 15 <75 75 <90
Longitudinal factor
Circumferential factor
1.0 0.5 0
0 0.5 1.0
ventional stress terms). For ®lament-wound laminates a further factor is applied to take into account the eects of anisotropy.23 Figure 1 shows the variation in axial modulus as a function of winding angle as given in the standard. The data shown are for E-glass ®bres and a vinylester resin.24 Most of the process plant is fabricated with styrenated resin systems, e.g. polyester, vinylester, etc, and the graph was originally calculated by using properties appropriate to these materials and simple laminate theory. Modulus values would be low if an epoxybased system is to be considered. There is scope for adjusting the data to take the variation in ®bre fraction into account since, in the document itself, the information is given per unit weight of glass. In Fig. 1 the data corresponds to a glass content of 75% by weight (60% by volume). Also shown are measured modulus values from nozzle cutouts (material removed when a circular hole is cut for the attachment of a nozzle) taken from large GRP tanks and the agreement is good given that a design code should be based on lower bound or deviated data. The factors which are applied to these modulus values to determine allowable stresses are shown in Table 1. At 0.2% strain the allowables as calculated from this method are shown in Fig. 2. The discontinuities arise as a result of the step changes in the applied factor for 15 and 75 laminates. There are two key assumptions in this process: . strains in all directions (axial, transverse, shear) are equally important and should be limited by the same upper limit; . transverse strains for high winding angles (>75 ) should be further limited to a lower
Fig. 2. Allowable stresses for laminates.
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G. C. Eckold overriding maximum of 0.1%. This is because of the low absolute value of transverse strength and the fact that at high angles the transverse direction is not supported to any great extent by the reinforcement.
For transverse strains there is an inconsistency in the document because the data in Table 1 indicate that the design value should be zero as opposed to the 0.1% limit. The BS4994 procedure is still workable, however, as the intent is that the transverse strength should be ignored in terms of structural contribution when deriving the basic laminate construction, but then to go back and check that the overall strain is less than the 0.1% level. The calculation on which the Table 1 parameters were based again employed simple laminate analysis to evaluate strains in each of the primary material directions.23 This results in a continuous design curve (as an alternative to Table 1) for allowables an example of which, for 0.2% strain, is also shown in Fig. 2. One of the most signi®cant question marks over the method is whether or not shear and normal strains should be limited by the same value. A criticism of the current BS4994 approach is that the procedure does not allow the full bene®ts of the reinforcement for combined loading. This is not a issue for storage tanks as the loading is dominated by the static head pressure. Axial loading is small and variable. It would not be practical to advocate that all possible load permutations are considered, as there would be little added bene®t. The exception is for pressure vessels where there is a 2:1, hoop:axial pressure stress. It is straightforward to generate an additional curve for this case and this is also shown in Fig. 3. Note that it is the allowable stress in the axial direction which is shown. Again the 0.1% transverse strain limit has been imposed to ensure consistency with the uniaxial case. With this approach a simpli®ed biaxial load envelope such as that shown in Fig. 4 can be developed. The values on the ordinate and abscissa are given by the uniaxial curve in Fig. 2 and the combined 2:1 point is given by the biaxial plot in Fig. 3. Also shown in Fig. 4 are the results of a netting analysis where the stress values at the point where the loads combine to act in the ®bre direction are given by:
Fig. 3. Allowable stresses for laminates.
x Eo " cos2 y Eo " sin2
2
where x and y are axial and hoop stress components for the 2:1 condition, " is the design strain (0.2%), Eo is the unidirectional laminate modulus and is the winding angle. As can be seen the netting analysis provides a more conservative result for the winding angle used for the calculation (55 to the axial direction), although the curves are of a similar shape. For other laminate con®gurations this will not be the case as the values given by eqn (2) will not be for the 2:1, hoop:axial stress ratio. This is not a major issue for process plant as the main design conditions are static head pressure (uniaxial load) and internal pressure which is addressed by the above discussion. A point of interest is that the approach of evaluating allowables on the basis of limiting maximum strain does not apparently result in the 55 winding angle being optimum for a 2:1 load case as is customarily assumed (see Fig. 3). A somewhat higher angle is indicated. The corollary to this is that a 2:1 loading situation is not the best for a 55 angle ply. There is some supporting evidence for this from biaxial test results where the performance of such a laminate is noted to be better at a stress ratio of nearly 3:1.25 The physical signi®cance of this has not been explained, but clearly the assumptions within the netting analysis are in diculty. It is worth noting that the stress ratio at which the in-plane shear stress, in the material's coordinate axis becomes zero, is 2.8 (the winding angle at which the shear stress becomes 0 for a 2:1 loading condition is 60 ). 2.2 ASME RTP-1 The ASME document18 describes a two-tier approach and the supplier is free to chose which to use for the design. In the simpler of the methods the determination of allowables is carried out as follows: . in the hoop direction the allowable stress is calculated by using a maximum strain limitation of 0.1% regardless of the winding angle;
Fig. 4. Stress envelope for 55 winding angle.
Failure criteria for use in the design environment . in the axial direction the allowable stress is calculated by dividing the ultimate strength by a factor of 10 again regardless of winding angle. For the second method an altogether more complex approach is taken. Within the code are appendices which give a description of the mechanics of laminate theory and the user is invited to carry out a full analysis. It is arguable as whether or not this is appropriate for a design code as if not correctly interpreted its use will cause problems. For example, the calculation of the coupling [B] matrix is described, but no guidance is given as to what to do with it. Coupling may give rise to twist in an unconstrained ¯at plate, but such deformations cannot exist in a closed cylindrical surface! Once the stress analysis is complete the assessment continues by using the quadratic interaction criterion described in the proceeding section. It is presented in a modi®ed form as follows: R2 Fxx x2 2Fxy x y Fyy y2 Fss s2
R Fx x Fy y ÿ 1 0
3
Measured values can be used for the strength terms (Fxx, Fxy, etc.) and if these are not available, strains are given, which vary with the direction and mode of loading, to be used in conjunction with the relevant stiness (Table 2). The coecient, R, is the design factor for the laminate and is speci®ed as being between 8 and 12 for the inner surface adjacent to the process ¯uid, depending on the criticality of service etc. Away from the surface a lower ®gure may be used, but for simple membrane stresses there is no bene®t to be gained. Allowable stresses as calculated by the ASME method are compared with BS4994 values in Fig. 2. 2.3 Performance-based approach It is worth noting that there is another approach to process plant design, but one which is really only suitable for products manufactured on a commodity basis such as standard pipework systems. Here the objective is to measure long-term performance of typical components (not laminates) under conditions similar to those experienced in service. Creep±rupture testing of pipework components over durations >10 000 h is one of the established methods.26 The key dierence with the approach is that knowledge of how laminates behave is not necessary as actual components are tested. The supplier then has a responsibility to establish quality Table 2. ASME RTP-1 strain values Stress component Longitudinal tension Longitudinal compression Transverse tension Transverse compression Shear
Strain value (%) 2.00 1.20 0.15 0.80 1.50
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assurance and quality control procedures which ensure that future manufactured items are, for all practical purposes, identical to those which have been quali®ed. Another problem is that testing usually takes place under a single load condition, most commonly internal pressure with closed ends, and the use of the arising data for other situations is not straightforward. There has been some progress in addressing this aspect of design through making assumptions regarding the shape of the failure envelope.27 The strains at the design stresses tend to be higher (0.3±0.4%) than those calculated by the design-by-rule approach described in the vessel standards. This is appropriate given that the latter employ short term static strength values as a basis and some conservatism is necessary. 3 FAILURE PREDICTION The procedures described in the preceding sections are intended for the evaluation of design allowables for GRP process-plant materials and to be commensurate with the needs of a design environment. This poses certain diculties when attempting to oer predictions of behaviour for the materials included within this failure exercise. (Details of the lamina properties and lay-up con®gurations and loading of the laminates analyzed are provided in Ref. 29.) These problems have been accommodated as follows: . Material properties. Tabulated material properties in BS4994 are intended for resin systems of the polyester type and their use for epoxy systems would not be appropriate. However, the standard does allow alternative values to be used provided they are based on test data. In this exercise the assupplied property values for unidirectional laminae have been used together with simple laminate analysis to give o-axis information. Table 3 gives the values as calculated by this process. A diculty also arises in the speci®cation of an allowable maximum strain. For polyesters this is currently 0.2%, but this may be considered low for epoxy systems. The results from performance-based testing, Table 3. Calculated modulus values Laminate E-Glass/LY556/HT907/DY063 30 angle ply Hoop Axial E-glass/MY750/HY917/DY063 55 angle ply Hoop Axial Cross-ply 45 angle ply
Calculated modulus (GPa)
15.8 30.5 23.5 15.4 31.1 17.7
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most of which relates to epoxy components, indicates that values in the range 0.3±0.4% are satisfactory. A strain limit of 0.4% has been used in all of the calculations described in the following sections, except for transverse tension where the 0.1% limit is maintained. The selection of 0.40% is, in a sense arbitrary, but it provides a basis for this exercise. . Prediction of ultimate strength. The design code is not concerned with the prediction of ultimate performance, and, furthermore, there is no mechanism for doing so. For the purposes of this paper the method which has been adopted is that which would be taken by an engineer given the as supplied properties and the general approach to laminate design described in BS4994. Calculations have been performed at selected points and straight lines have been drawn between them, as opposed to trying to use a formal composite failure criterion, which will be addressed by other contributors to this exercise. . Thermal stresses. Most GRP process vessels are fabricated by using room-temperature cured
Fig. 5. Stress envelope for 0 laminate (E-glass/MY750/ HY917/DY063).
Fig. 6. Stress envelope for 0 laminate (E-glass/LY556/ HY907/DY063). Note that structural instability will govern design in compressive quadrants.
materials. Sheer size eectively precludes systems which need elevated temperatures during processing and as a result the eects of thermal stresses are not relevant. However, in the following calculations a simple manual correction has been applied by using given thermal expansivities and the stated stress-free temperatures. . Compressive properties. BS4994 makes no attempt to dierentiate between tensile and compressive strengths. The reality is that if there is a signi®cant compressive loading component, e.g. vacuum, it will be structural instability which governs the design. This is especially true for glass-reinforced composites where modulus values are relatively low. The introduction of additional properties would be an unnecessary complication. In this exercise no attempt has been made to allow for differences in strengths in the two modes of loading. . Non-linear properties. Non-linearity, either through a material characteristic or ply discount, does not feature in the calculation method, since allowables are generally based on low strains prior to any form of initial failure. No attempt has been
Fig. 7. Stress envelope for 0 laminate (E-glass/LY556/ HT907/DY063).
Fig. 8. Stress envelope for 30 laminate (E-glass/LY556/ HT907/DY063).
Failure criteria for use in the design environment made to accommodate this in the following predictions. Calculated stress/strain curves have been limited (arbitrarily) to 0.5% strain. . Carbon-®bre materials. CFRPs do not yet feature in the catalogue of material systems employed by the process industry. A basic premise behind the arguments expressed in this paper is that design methods should be appropriate for the industry and materials for which they are intended. More advanced materials (and their associated cost) require, in principle, more sophisticated design methods if the advantages of their use are to be realised. On this basis no attempt has been made to oer predictions for CFRP laminates. 3.1 Unidirectional laminate
Fig. 9. Stress envelope for 30 /90 laminate (E-glass/ LY556/HT907/DY063).
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Predictions for biaxial stress envelopes for unidirectional laminae are shown in Figs 5 and 6. In Fig. 5 the envelope depicting ultimate strength is based on the as supplied strength values which are then treated as per BS4994. The design envelope was derived by using a strain limitation of 0.1% in the transverse direction and 0.4% (as discussed above) for the longitudinal direction. The ratios between ultimate strength and design are broadly in line with the intent of the code. The quadrant where there is compressive stress is, again, treated by following prevailing design code practice. The prediction will be conservative where loads are primarily uniaxial (compression) and in the transverse direction owing to the higher strengths in this direction. The converse will be true for compressive loads which will be predominantly in the ®bre direction. However, as has already been discussed, this is unlikely to be a design issue as structural instability will almost certainly become the governing concern. The envelopes in Fig. 6 have been calculated using a similar process, but here it is expected that shear non-linearity will have a marked eect on the results. As the shear modulus falls a greater proportion of the load will be carried in other directions and this may have the eect of reducing observed strength. On this basis it would be expected that the simple linear calculation would represent an upper bound. The combinations of load as represented by Fig. 6 do not feature in process-plant design where stresses are primarily membrane or, in the case of local features such as nozzles, discontinuities, lifting brackets etc, bending. Although pipework systems do experience some torsional loading it is generally the case that these are small. Large in-plane shear stresses are not seen. In BS7159 the combination of normal and shear stresses is considered using the conventional Tresca criterion. However, it must be said that this was probably adopted directly from the equivalent steel codes and its applicability to composites, especially those which are anisotropic, was not thoroughly considered.
Fig. 10. Stress envelope for 55 laminate.
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Fig. 11. Stress envelope for 55 laminate under a stress ratio of 1:0.
To adopt this approach in this exercise to shear loadings would perhaps be pushing the codes beyond their intent.
In terms of design, the approach taken for a hybrid laminate construction is to consider each type of layer in turn and to establish which has the lowest allowable strain. This value is then used as a basis for further evaluation. The ®rst stage in the analysis is to determine the behaviour of each laminate layer and this is shown in Figs 7 and 8. These were derived as follows: Unidirectional envelope:
3.2 (90/30 /90) laminate The use of combined winding angles is not an uncommon occurrence in storage/pressure vessels where an angle-ply laminate (usually >45 ) is used to support basic loadings in the axial direction and additional hoop winding is applied to accommodate pressure loads. In some design examples the two cases are kept entirely separate, i.e. angle-ply laminates are used to carry axial load only and 90 windings are employed to carry circumferential load only. Whilst this may be construed to be conservative, the fact that the role of each layer is unambiguous means that the design is `conceptually' simple and this has the bene®t that approval by third parties, who will be competent in terms of process plant but not necessarily well versed in composites, will be straightforward. The rami®cation of time saved in this part of the procurement process can outweigh the penalty of added material cost.
From these results, associated modulus values (Table 3) and the layer thicknesses (90 ÿ0.35 mm,
Fig. 12. Stress/strain curve for 55 laminate under a load ratio of 2:1.
Fig. 13. Stress/strain curve for 0 /90 laminate under a load ratio of 0:1.
. from treatment for the unidirectional laminate described in the preceding section. 30 envelope: . axial and hoop strengths were determined by the ultimate transverse (0.197%) or shear (0.38%) strains as appropriate; . strength at the 2:1 ratio was determined by netting analysis limited by the ultimate longitudinal strain (2.13%).
Failure criteria for use in the design environment 30 ÿ1.65 mm) the various contributions of each layer type can be assessed. The results of this calculation are shown in Fig. 9. The envelope for ultimate strength takes into account redistribution of stress after the `weaker' layer has failed. For the design envelope all values were determined by using a design-strain limitation of 0.4% (0.1% for transverse tension) generally in accordance with the methods described in Section 2.1. For the axial directions the design values look particularly pessimistic. This is due to the 0.1% transverse strain value. In principle this could be increased if it were felt that cracks in this layer could be tolerated. The values for the compressive quadrants were evaluated by assuming that the simple use of the equivalent tensile data was appropriate. The comments on structural instability made in material Section 3.1 for unidirectional material are equally applicable for the more complex laminates.
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It is acknowledged that the ultimate envelope may represent an upper bound as a consequence of material non-linearity which will result in local redistribution of load giving rise to higher strains in other directions. Discrepancies will also occur because the compression strengths have not been used. For the design envelope the important considerations are not limited to its relationship to static strength, but more importantly how it compares with the initiation of damage. It is essential that there is a sucient margin here so that eects such as long term loading, the chemical environment and variability of material properties can be accommodated. 3.3 Angle ply (55 ) laminate Figure 10 shows the biaxial stress envelope for a 55 laminate. Calculations were carried out as follows.
Fig. 14. Stress/strain curve for 45 laminate made of GRP material (sy/sx=1/ÿ1).
Fig. 15. Stress/strain curve for 45 laminate made of GRP material (sy/sx=1/1).
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Ultimate strength envelope: . axial and hoop strengths were determined by the ultimate transverse (0.246%) or shear (4%) strains as appropriate; . strength at the 2:1 ratio was determined by netting analysis limited by the ultimate longitudinal strain (2.13%).
areas where there is scope to back up more rigorous approaches with extensive testing programmes. Participation in this failure prediction exercise will be of bene®t in putting this and other methods into an overall design and engineering context.
Design strength envelope:
REFERENCES
. Comments with regard to the ultimate envelope being an upper bound because of material nonlinearity and the eects of the higher compressive strengths are as per previous predictios.
1. Azzi, V. D. and Tsai, S. W., Anisotropic strength of composites. Exp. Mech., 1965, 5, 283. 2. Norris, C. B., Strength of orthotropic materials subjected to combined stresses. Forest Products Lab, FPL 1816, 1950. 3. Puppo, A. H. and Evensen, H. A., Strength of anisotropic materials under combined stresses. AIAA J., 1972, 10, 468. 4. HuÈtter, U., Schelling, H. and Krauss, H., An experimental study to determine the failure envelope of composite materials with tubular specimens under combined loads and comparison between several classical criteria. AGARD-CP-163, 1975. 5. Gol'denblat, I. I. and Kopnov, V. A., Strength of glass reinforced plastics in the complex stress state. Mekhanika Polimerov, 1965, 1, 70. 6. Huang, C. L. and Kimser, P. G., A criterion for strength for orthotropic materials. Fibre Sci. Technol., 1975, 8, 103. 7. Tsai, S. W. and Wu, E. M., A general theory of strength for anisotropic materials. J. Compos. Mater., 1971, 5, 58± 82. 8. Hart-Smith, L. J., The role of biaxial stresses in discriminating between meaningful and illusory composite failure theories. Compos. Struct., 1993, 25, 3±20. 9. Talreja, R., Stiness properties of composite laminates with matrix cracking and interior delamination. Engng Fract. Mech., 1986, 25, 751±762. 10. Joshi, G. P. and Frantziskonis, G., Damage evolution in laminated advanced composites. Compos. Struct., 1991, 17, 127±139. 11. McCartney, L. N., The prediction of non uniform cracking in biaxially loaded cross ply laminates. NPL Report DMM(A) 142, 1994. 12. Eckold, G. C., Hancox, N. L. and Lee, R. J., Application of micromechanics in the prediction of damage initiation and growth in structural composites. Deformation and Fracture of Composites, Surrey, March 1995. 13. Speci®cation for design and construction of vessels and tanks in reinforced plastics, BS4994. BSI, 1984. 14. Speci®cation for reinforced plastics pipes, ®ttings and joints for process plants, BS6464. BSI, 1984. 15. Code of practice for design and construction of glassreinforced plastics (GRP) piping systems for individual plants or sites, BS7159. BSI, 1989. 16. Speci®cation for low pressure ®breglass line pipe, API 15LR. API, 1990. 17. Speci®cation for high pressure ®breglass line pipe, API 15HR. API, 1988. 18. Reinforced thermoset plastic corrosion resistant equipment, ASME RTP-1. ASME, 1992. 19. Pressure vessels in glass ®bre reinforced thermosetting plastics. AD-Merkblatt N1. 20. Smith, T. R. and Owen, M. J., The progressive nature of fatigue damage in glass reinforced plastics. In Proc. 6th Int. Resins and Plastics Conf. of the British Plastics Federation. British Plastics Federation, London, 1968, paper 27. 21. Roberts, R. C., Reinforced Plastics Congress. BPF, 1978, p. 145.
Stress/strain curves, to 0.5% strain, are shown in Figs 11 and 12. These are based on simple laminate calculations using the provided material constants. It is anticipated that at high strains signi®cant non-linearity will be noticeable. This is not necessarily relevant in design if it is concerned solely with allowables limited by lower strains. 3.4 Cross-ply laminate Figure 13 shows a stress/strain curve for the cross-ply laminate subjected to uniaxial loading. Again it is linear and no attempt has been made to identify the `knee' which occurs as a consequence of transverse cracking. 3.5 Angle ply (45 ) laminate Figures 14 and 15 show the stress/strain characteristics for the 45 laminates. These were calculated as for those described above and similar comments apply. 4 CONCLUSIONS This paper considers the attributes of composite failure criteria necessary for use in the design environment where the need for reliable prediction of performance must be balanced by simplicity and ease of use. The prime purpose of a failure theory is as a tool in the assessment of whether or not a design is `®t for purpose'. It may be that calculations for ultimate strength are necessary, but more likely it will be some other facet of behaviour which provides a baseline for design. This will certainly be the case where loads are applied continuously and over a long term. This is not to say that knowledge of maximum load capability is not important, as it can provide a backdrop against which upset conditions can be considered and give con®dence to the user with respect to performance in service. The industrial sector which has been chosen as an example, process plant manufacture, has at its disposal a number of design standards for use which, in the main, meet their needs. However, it is acknowledged that their treatment of composite materials behaviour can be criticised as not being technically robust and certainly would not be considered satisfactory in other
Failure criteria for use in the design environment 22. Hogg, P. J., Hull, D. and Legg, M. J., Composite Structures, ed. I. M. Marshall. Applied Science Publishers, London, 1981, p. 106. 23. Eckold, G. C., A design method for ®lament wound GRP vessels and pipework. Composites, 1985, 16, 41±47. 24. AEA Technology, Internal Report, 1995. 25. Tolhoek, P., Composite Materials in the Oshore Industry. Aberdeen, November 1995. 26. Practice for obtaining hydrostatic or pressure design basis for ®breglass pipe and ®ttings, ASTM D2992.
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27. Eckold, G. C., A performance based design methodology for GRP pipework and ®ttings. Proc. Inst. Mech. Eng., 1995, 209, 41±50. 28. Hinton, M. J. and Soden, P. D., Predicting failure in composite laminates: the background to the exercise. Compos. Sci. Technol., 1998, 58(7), 1001. 29. Soden, P. D., Hinton, M. J. and Kaddour, A. S., Lamina properties, lay-up con®gurations and loading conditions for a range of ®bre reinforced composite laminates. Compos. Sci. Technol., 1998, 58(7), 1011.