The influence of manufacturing variances on the

Composite Structures 133 (2015) 853–862

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Composite Structures journal homepage: www.elsevier.com/locate/compstruct

The influence of manufacturing variances on the progressive failure of filament wound cylindrical pressure vessels Brian Ellul ⇑, Duncan Camilleri Department of Mechanical Engineering, Faculty of Engineering, University of Malta, Msida MSD2080, Malta

a r t i c l e

i n f o

Article history: Available online 21 July 2015 Keywords: Numerical modelling Progressive ply failure analysis Manufacturing variances Voids and imperfections Filament wound pressure vessels

a b s t r a c t Developing modelling methodologies to characterise the post first-ply failure behaviour is an ongoing tough challenge. In this study the results of a progressive failure algorithm based on a sudden mode-dependant degradation methodology are presented and applied to filament wound cylindrical pressure vessels subject to an internal pressure undergoing first ply-failure, post failure and ultimate failure. In this case, the material properties are degraded according to the failure mode detected from the homogenised stresses at lamina level. The numerical models were corroborated with experimental tests results. Influences due to manufacturing divergences from near perfect samples such as material thickness and voids are also investigated. The results show that the laminated pressure vessels investigated are able to sustain considerable higher pressures beyond the predicted first-ply failure of approximately three times as much. However the progression of failure and pressure vessel response was found to be highly dependent on manufacturing divergences. This property of fibre reinforced composites gives rise to the importance of successfully modelling the post first-ply failure of these novel materials to enable the engineer to carry out analyses of structural integrity beyond their design loads and possibly exploit their advantages and increase confidence in their design and application. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction Fibre reinforced composites became widely used in various industries, from relatively simple civil applications such as waste management piping to the state-of-the-art aerospace applications. This wide range of application is attributed to their high customisability which leads to other advantages like high strength to weight ratio resulting in lower material costs. On the other hand, considering that they are tailored to cater for specific needs, the design phase is perhaps the most crucial factor for the successful implementation of these novel materials. Modelling of composites proves to be a tough challenge for engineers and scientists since they are composed out of different materials and phase constituents that interact together to give a global response. The numerous publications on this subject, which are summarised in several review publications and the extensive research published after the world-wide composite failure exercise carried out in the early 2000’s [1], are a proof of the tough task. As an example, this challenge is reflected in standards related to the

⇑ Corresponding author. E-mail address: [email protected] (B. Ellul). http://dx.doi.org/10.1016/j.compstruct.2015.07.059 0263-8223/Ó 2015 Elsevier Ltd. All rights reserved.

design of filament wound pipes or pressure vessels where they stipulate relatively high partial safety factors [2]. In view of this, adequate modelling of these complex systems is necessary to exploit their advantages and increase confidence in their design and application. Standards such as the EN 13923:2005 [3] defines a design method for filament wound fibre reinforced polymer pipes. The design load is based on the lowest pressure which produces the first ply failure according to a predefined failure criterion irrespective of the failure mode and location. A safety factor of 4 or more is applied to the first ply failure load to ultimately stipulate the corresponding design load leading to a very conservative design. It is widely known that composites fail progressively thus their structural performance is not fully compromised after the first predicted failure. The mechanical performance beyond the first ply failure load can be analysed using a progressive failure analysis (PFA). A PFA is an iterative process where the plies of a laminate fail progressively. During the failure progression, the stresses are redistributed to compensate for the loss in structural strength. This procedure can be solved analytically for simple structures while other more robust methods, such as the finite element method (FEM), are used for more complex problems. Numerous publications tackling the solution to the tough challenges

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B. Ellul, D. Camilleri / Composite Structures 133 (2015) 853–862

Nomenclature Symbol DV

Description cross-sectional area expansion at mid-span of the pipe, % mij (i, j = 1, 2, 3) Poisson’s ratio in the principal coordinate system 1, 2, 3 shown in Fig. 3 h off-axis angle, ° ri , rj (i, j = 1, 2, 6) stresses in the material coordinate system, Pa cv coefficient of variation Ei (i = 1, 2, 3) Young’s modulus in the principal coordinate system 1, 2, 3 shown in Fig. 3, Pa F i , F ij (i, j = 1, . . ., 6) material strength tensors found experimentally, Pa Gij (i, j = 1, 2, 3) shear moduli in the principal coordinate system 1, 2, 3 shown in Fig. 3, Pa Hi (i = 1, 2, 6) Hashin’s failure indices kn random number which is multiplied with all the material constants of layer, n S shear strength, Pa std() standard deviation of the variable in the brackets t average pipe wall thickness, mm t new new randomly generated pipe wall thickness, mm t CSM ; t DR thickness of the layer denoted by the subscript, mm new t new new randomly generated thickness of the layer CSM ; t DR denoted by the subscript

encountered during the modelling of progressive failure of laminated composites can be found in [4,5]. There are various factors affecting the eventual failure of a structure such as micro scale defects in the material resulting in cracks or voids, inappropriate material selection, unpredicted loadings or perhaps defects induced during the manufacturing or installation. All these factors can introduce different types of defects which eventually will compromise the integrity of the structure. From a modelling point of view, these factors need to be simplified such that the problem can be dealt with more efficiently. This is one of the major purposes of progressive failure analyses where the complex failure process of laminated composites is simplified into an iterative process composed of two main parts; 1. The failure criterion which predicts when failure occurs. Depending on the type of the criterion, it may also define the type of failure. Extensive literature reviews on failure criteria were carried out by various authors such as [6–15]. 2. The material degradation model which defines how the material properties are degraded on the macro scale to simulate failure in the micro scale. A review on various degradation models can be found in [16]. The uncertainty of various parameters such as material composition, winding angles and wall thickness all contribute to the probability of failure of a composite structure [17]. The propagation of uncertainty in design parameters was studied by [18,19] where they presented a multi-scale nondeterministic progressive failure model. The micro and macro domains where connected via a set of coupling variables and the uncertainties related to these coupling variables were obtained from the uncertainties of the material and geometry parameters. 1.1. Scope of this study In this study a probabilistic nondeterministic progressive failure algorithm applied at the meso-scale utilising the Tsai–Wu failure

t no v oid;n t new no v oid;n Vv X, Y, Z

thickness of the layer n (n = CSM, DR) reduced by respective void content V v , mm new randomly generated tno_void,n, mm average void content obtained from the material characterisation test specimens, % unidirectional strengths. Subscripts c and t denotes in compression and tension respectively, Pa

Abbreviation CSM chopped strand mat DOF degrees of freedom DR direct roving FE finite element FEM finite element method FPF first ply failure FRP fibre reinforced polymers FSDT first-order shear deformation theory GDIS random number having a normal distribution LVDT linear variable differential transformer PFA progressive failure analysis RAND random number having a uniform distribution function SPF second ply failure

criterion [20] coupled with a sudden degradation model based on the Hashin’s failure indices [21], is presented. The uncertainties of the material properties obtained from the material characterisation tests are included in the model via two different methods; 1. Variability in the material properties of the initial structure. 2. Geometric imperfections of the initial structure. The aim of this paper is to analyse the effect of these two types of initial imperfections applied to cylindrical filament wound laminated pipes with closed ends. An asymmetric lay-up composed from three direct roving (DR) layers wound over a chopped strand mat (CSM) layer [CSM/82.7°/±54.3°] (angles are with respect to the axial direction) is analysed. The progressive failure algorithm is implemented into the FE package ANSYSÒ R15.0 [22] and the results of the simulations are corroborated with experimental tests carried out by the same authors [23]. This paper is sectioned in the following order. A description of the progressive failure algorithm used including a detailed description of the main components. This is followed with an explanation of the pipe specimens under investigation together with details of the FE models used in subsequent simulations. Finally the results are presented and discussed accordingly. 2. Progressive failure algorithm The progressive failure algorithm employed in this study is based on an iterative method specifically used to simulate the structural performance of composites based on the work done on laminated plates by the authors [24]. A generic flowchart of the progressive failure algorithm is shown in Fig. 1 (left). The algorithm begins with the setting up of the initial problem, which usually entails the creation of the structure including a set of boundary conditions and loadings. The initial problem is solved where the initial applied load is set below the predicted initial failure. After solving, the structure is checked against the chosen failure criterion for any possible failures. If failure is detected at any part of the structure, the corresponding material properties in that

855


START Problem setup n = # of elements plies

Add initial imperfections START

Solve analysis Problem setup

Solution converged ?

Solve analysis

Solution converged?

No

END Calculate the elements’ stress/strain after establishing equilibrium

Calculate the elements’ stress/strain after establishing equilibrium

Degrade material properties

No

END

x = element/ply #1

x=n ?

x=x+1

Yes

No Any failed ply/ element?

Laminate failure

Yes

Laminate failure

Yes

Increment load

No

Check element/ply x with Failure Criterion Yes

No

Any failures ?

Yes

No

Increment load

element / ply failed? Yes

Degrade material properties Fig. 1. (left) Flowchart illustrating the main components of the PFA and their connections between them. (right) Flowchart of the PFA used throughout this paper.

particular region, lamina and direction are degraded according to a degradation methodology. After degrading the material accordingly the problem is solved again with the degraded material at the same loading. The structure is checked for any failure once again and the material is degraded if more failures are detected. Once no failures are detected, the load is incremented and the problem is solved again iteratively. This process is repeated until the solution cannot converge which indicates the ultimate failure of the structure. A more detailed flowchart of the actual progressive failure algorithm as implemented in this study can be seen in Fig. 1 (right). The notable difference from the generic PFA is the inclusion of initial imperfections to the initial problem. More details on how these are implemented can be found in the Section 2.2. 2.1. Problem setup In the following section, a brief explanation of the tests carried out by Ellul et al. [23] is given for completeness. Furthermore, details regarding the finite element model used throughout the simulations presented in the paper are presented in Section 2.1.2. 2.1.1. The experimental test specimens The experimental tests referred to in this paper, carried out by Ellul et al. [23], consisted of four filament wound pipes with end domes having an asymmetric layup [CSM/82.7°/±54.3°] and pressurised at a rate approximately 0.5 bar/s. Apart from other measurements such as strain gauges, the expansion at the mid-span of the pipes during the pressurisation was recorded via three LVDTs equally distributed around the circumference. The setup is shown in Fig. 2 (right). The expansion was calculated by assuming that the deformation of the pipes during expansion was circular

therefore the expansion in the cross-sectional area could be calculated from the three displacements recorded by the three LVDTs. These results are used and compared to the numerical models. 2.1.2. Finite element model The main types of elements available to model thin laminated shells are solids, layered shells or a hybrid element between a solid and a shell. While the solid and hybrid elements are more accurate to compute stress fields, shell elements are far more computationally efficient due to their formulation. However, shell elements impose certain restrictions and are specific to certain cases where they can be used, in particular it is assumed that the strain developed through the thickness is linear. Two formulations are commonly available for shell elements which are the classical Love–Kirchhoff [25] and the first order-shear deformation theories [26,27]. In this case, the first-order shear deformation theory (FSDT) was adopted which is particularly suited for thin shells without imposing a large amount of processing requirements. The 4-noded SHELL181 structural multi-layered elements were used to model the pipe. This element assumes a state of plane-stress and the kinematics supports membrane stretching. A cylinder with a diameter of 250 mm and 800 mm long was modelled in ANSYS [22] which correspond to the actual dimension of the pipes under investigation. A mesh sensitivity analysis was conducted and showed that a mesh with 40 elements in the circumferential direction and 20 elements in the axial direction are sufficient to capture the mechanical performance of the pipes. The overall thickness of the pipe wall was set equal to the ones reported in [23] whilst the thickness of the direct roving layers was kept constant at tDR ¼ 0:783 mm if not otherwise noted. The FE model was held in 3D space by a set of boundary conditions that impede rigid body motion whilst leaving the model to deform freely under the action

856


Pressure transducer

Air vent v

v

LVDTs placed around the mid-section DR layers CSM layer

Pressurised water from pump

Built-in end-dome

Fig. 2. (left) FE model in ANSYS R15.0 showing the mesh using element SHELL181 illustrating the four layer stacking. (right) The experimental setup showing on of the pipes ready to be pressurised.

of the applied loads. The model was loaded internally under a pressure load corresponding to the applied pressure which is varied during the progressive failure algorithm. Furthermore, a force on the edge of the pipe acting axially outward was applied to simulate the stress developed due to the built-in end-domes trying to rip apart the pipe during pressurisation. Fig. 2 shows the FE model in ANSYS with the loads together with the experimental setup. SHELL181 elements having 6 degrees of freedom (DOF) at each node, three transitional and three rotational, were employed. The rotational DOFs at the free edges were coupled together to simulate the stiffness offered by the end-domes at the edges. As post-processing, the Tsai–Wu failure index was calculated after each solution for each and every element and layer. If failure is detected, then the Hashin’s indices are calculated and the failure mode established. The material is degraded accordingly as listed in Section 2.4. Furthermore, the average expansion at the mid-span of the pipe was recorded after each solution to be able to obtain a plot of the expansion with respect to the applied internal pressure. 2.2. Initial imperfections It is widely known that imperfections arising from the manufacturing process are one of the main sources for damage initiation. This fact should not be underestimated when modelling composites since these novel materials are intrinsically more prone to defects. This fact is reflected in the certain standards related to fibre reinforced composite structures such as [3], where they stipulate relatively high safety factors during the design process. The reason behind this is the high uncertainties related to various aspects such as manufacturing procedures, raw material properties and their interaction. All these aspects affect the final performance of the composite structure. On the other hand, analytical or numerical models usually model a perfect structure thus over predicting the performance of the structure under investigation. In view of this, imperfections for failure initiation are applied on the initial structure before solving the first iteration and the algorithm continues as described in Section 2. In this study, two types of imperfections are implemented. The first imperfection methodology, referred to as ‘‘Type 1’’, involves the random modification of the layer thickness whilst the second type, ‘‘Type 2’’, involves the modification of the material stiffness of each layer. These are explained in the subsequent sections. 2.2.1. Ply thickness imperfections method (Type 1) With this type of imperfection, the thickness of some or all the layers for each and every element is modified randomly and independently. Three separate cases were analysed which are detailed hereunder. 2.2.1.1. Case A. As a preliminary implementation of the algorithm, the thickness of the chopped strand mat layer (tnew CSM ) was varied

randomly between ±2% and ±20% of the nominal CSM layer thickness (tCSM ) which were chosen arbitrary at the preliminary stage. The thickness of the other DR layers was kept fixed at tDR ¼ 0:783 mm each. This variation in CSM layer thickness was applied across all the elements defining the pipe. 2.2.1.2. Case B. In this case, the thickness of the CSM layer (tCSM ) was varied according to the thickness of the individual pipe under test. The following table list the average thicknesses (t) of each pipe and their corresponding standard deviation (stdðtÞ) and coefficient of variation (cv ). In this case, the CSM layer thickness was varied according to two types of distributions, the uniform distribution, similar to the one mentioned in Case A, and the normal distribution. For the uniform distribution, the simulations were similar to those in Case A but the limits for the new thickness of the CSM layers (tnew CSM ) were set equal to:

tCSM ð1 cv Þ 6 tnew CSM 6 t CSM c v For the normal distribution, the new thickness of the CSM layer was obtained by first generating a random variable having a normal probability density function with mean and standard deviation corresponding to the average pipe thickness (t) and standard deviation (stdðtÞ) listed in Table 1 respectively. The generated random number was set equal to the new thickness of the pipe (t new ). Secondly, the thickness of the direct roving layers was kept constant at t DR ¼ 0:783 mm each and subtracted from the newly generated thickness t new such that the final new thickness of the CSM layer (t new CSM ) is; new tnew 3t DR CSM ¼ t

It is important to note that since there are no theoretical bounds on the generated random variable the new thickness for the CSM layer could be zero or even negative. In view of this shortcoming, a lower limit was set such that when tnew CSM 6 0 the new thickness is set to

tnew CSM ¼ 0:1 t CSM : 2.2.1.3. Case C. Varying the thickness based on the thickness of the actual pipe means that the pipe needs to be manufacture beforehand to obtain the variation in the thicknesses. Furthermore the Table 1 Average thickness of the pipe walls together with the corresponding standard deviation and coefficient of variation.

Pipe Pipe Pipe Pipe

A B C D

Average thickness t [mm]

Standard deviation stdðtÞ [mm]

Coefficient of variation cv [%]

3.345 3.355 3.350 3.415

0.2149 0.3189 0.4292 0.2996

6.425 9.505 12.812 8.773

857

B. Ellul, D. Camilleri / Composite Structures 133 (2015) 853–862 Table 2 Average void content and its standard deviation with respect to the type of layer [23].

Direct roving (DR) Chopped strand mat (CSM)

Average void content Vv [%]

Standard deviation std (Vv) [%]

4.95 4.89

1.46 0.40

void content was not considered when calculating the material properties from the material characterisation tests. In this case, the average void content and its standard deviation of the test specimens used to obtain the material properties are taken into account. Table 2 lists the average void fraction and its standard deviation with respect to the layer type. The average void content (Vv) was used to reduce the thickness of the respective layer (tn where n = DR, CSM) such that;

t no v oid;n ¼ ð1 V v Þ t n

ðn ¼ DR; CSMÞ

where tno_void,n is the thickness of the layer without the voids having material type n. This reduction in layer thickness was applied throughout all the elements. The standard deviation of the void content std(Vv)n is used to calculate the upper and lower bounds between which the new random thickness of each respective layer tnew no v oid;n can vary randomly.

t no v oid;n ð1 stdðV v Þn 6 tnew no v oid;n 6 t no v oid;n stdðV v Þn where n ¼ DR; CSM In contrast with the previous two cases, the thicknesses of all the layers were varied in this case.

Fig. 3. Schematic of a fibre-reinforced composite showing the principal directions 1, 2 & 3 with respect to the fibre orientations and the off-axis angle h between the principal directions and the material’s coordinate system (x, y, z).

in Fig. 3 while the subscript denotes whether in compression (c) or in tension (t), the shear strength denoted by S, the tensor strength criterion is given by:

F i ri þ F ij ri rj ¼ 1 i; j ¼ 1; . . . ; 6

ð1Þ

where Fi and Fij are material strength tensors found experimentally while ri and rj are the stresses in the material coordinate system (Fig. 1). Under planes stress conditions Eq. (1) can be expressed as a polynomial in terms of the material strengths by;

F 1 r1 þ F 2 r2 þ F 6 r6 þ F 11 r21 þ F 22 r22 þ F 66 r26 þ 2F 12 r1 r2 ¼ 1

ð2Þ

by applying a uniaxial tensile stress to failure in the fibre direction, Eq. (2) reduces to:

F 1 X t þ F 11 X 2t ¼ 1

ð3Þ

Similarly, by applying a uniaxial compressive stress to failure in the fibre direction, Eq. (2) reduces to:

F 1 X c þ F 11 X 2c ¼ 1

ð4Þ

1 stdðV v ÞDR 6 kDR 6 1 þ stdðV v ÞDR

By solving Eqs. (3) and (4) simultaneously, F1 and F11 can be expressed in terms of the material strength parameters. Similarly, the other coefficients in Eq. (2) can be found by means of uniaxial tests with the exception of F12. Tsai–Wu [20] proposed an off-axis uniaxial test to evaluate F12. Considering the difficulty and expense to perform the required off-axis uniaxial tests, Narayanaswami and Adelman [28] proposed to set F12 = 0 and neglect the term F12 since it has minimal influence on the predicted strength [28]. Finally, the coefficients for the Tsai–Wu criterion, Eq. (2), in terms of the lamina strengths are:

1 stdðV v ÞCSM 6 kCSM 6 1 þ stdðV v ÞCSM

F1 ¼

2.2.2. Material stiffness modification method (Type 2) The imperfections are included by modifying the material properties of each element. For each element and layer type, namely the direct roving and chopped strand mat, a random number kn ðn ¼ DR; CSMÞ having a uniform distribution function is generated whose limits are calculated from the standard deviation of the void content (std(V v )) of the material characterisation specimens respectively listed in Table 2;

The generated random number kn is multiplied with all the material constants of the respective layer type. A new set of random numbers (kDR and kCSM ) is generated for each element therefore the material properties varies from element to element. 2.3. Failure criterion Numerous failure criterion theories have been put forward by various researchers [6–15]. These theories can be categorised according to their scale of application either macro or micro length scale. The failure criterion used in this paper is the Tsai–Wu [20] generalised quadratic failure criterion proposed by Tsai and Wu and perhaps it is the most popular failure criterion used to detect failure in laminates [13]. This criterion is applied at the macro properties of the lamina. Similar to most of the macro-mechanics based failure theories, this criterion treats the composite as a homogenous material where stiffness is predicted according to the average properties across the lamina with respect to the material parameters obtained from material characterisation tests. Considering the unidirectional strengths denoted by Xc, Xt, Yc, Yt, Zc and Zt where the capital letter denotes the material principal direction corresponding to directions 1, 2 & 3 respectively shown

1 1 þ ; Xt Xc 1 1 F2 ¼ þ ; Yt Yc F 6 ¼ 0;

F 66 ¼

1 Xt Xc 1 ¼ YtYc

F 11 ¼ F 22

ð5Þ

1 S2

Since the Tsai–Wu failure criterion is a generalised criterion, it lacks to link the predicted failure with the mode of failure of the lamina. In view of this drawback, the Hashin’s indices Hi (i = 1, 2, 6), proposed by Hashin [21] are used to distinguish between the modes of failure. The highest Hi index shown in Eq. (6) determines the mode of failure.

H1 ¼ F 1 r1 þ F 11 r21

H2 ¼ F 2 r2 þ F 22 r22

H6 ¼ F 66 r26

ð6Þ

where H1 and H2 represents fibre and matrix failure respectively while H6 represents fibre–matrix interface failure. 2.4. Material degradation model A sudden material degradation model was used throughout this study. This means that as soon as failure is detected with the Tsai–Wu criterion (Eq. (2)), the mode is identified according to the Hashin index (Eq. (6)) and consequently the respective material stiffness are reduced to near zero. The material properties were

858


Table 3 Reduction in material stiffness degradation with respect to the failure mode. Ei, Gij and mij (i, j = 1, 2, 3) are the Young’s moduli, shear moduli and the Poisson’s ratio respectively corresponding to the directions illustrated in Fig. 3. (–) means that the property was unaltered. Failure mode

E1

E2

E3

G12

G23

G13

m12

m23

m13

Fibre matrix Fibre–matrix

0 – –

0 0 –

0 0 –

0 – 0

0 – –

0 – 0

0 0 0

0 – –

0 0 0

not reduced to zero exactly for computational purposes otherwise the solution does not converge. Table 3 lists the modifications made to the material properties with respect to the failure mode. Fibre failure is the most server failure out of the three failure modes where all the stiffness constants are reduced to zero and occurs when H1 is the highest out of the three Hashin’s indices listed in equation (6). When H2 is the highest, the transverse (E2) and out of plane (E3) Young’s moduli are reduced to zero together with the Poisson’s ratios related to the fibre direction. When failure occurs between the fibres and matrix due to shear loading recognised through the H6 index, the shear moduli and the Poisson’s ratios related to the fibre direction are reduced to zero while the remaining properties are unmodified. Note that, if a layer is predicted to fail in either the matrix or interlaminar shear it can fail again until fibre failure is detected and all stiffness properties are reduced to zero. 3. Results and comments The results are categorised into four distinct sections. 1. A set of simulations where no imperfections were implemented which resembles a classical PFA.

2. Imperfections are introduced on the thickness of the CSM layer where it is varied randomly following a uniform distribution at two arbitrary levels i.e. ±2% and ±20% of the nominal CSM layer thickness. 3. Imperfections are applied on the thickness of the CSM layer based on the variation of the thickness of the pipes. These simulations were performed both using a uniform random thickness and normal distributed random thickness. 4. Two sets of PFA simulations for each pipe were carried out. In the first instance the thicknesses of all the layers were varied and in the other the material properties of all the layers were varied. The variations were based on the fluctuations of the void content of the specimens used during the material characterisation tests [23]. 3.1. No imperfections (classical PFA) A PFA simulation without any imposed initial imperfections was carried out for each pipe listed in Table 1. This means that the thickness of the direct roving layers was fixed at tDR ¼ 0:783 mm whilst the thickness of the chopped strand mat layer was set constant at tCSM ¼ t 3t DR where t is the average thickness of the pipe listed in Table 1. The experimental and PFA results of the mid-span expansion (DV) versus the applied pressure for all the pipes are illustrated in Fig. 4. From Fig. 4, one can notice that the expansion of the actual experimental pipes with respect to the applied internal pressure is linear until a particular pressure level corresponding to first ply failure. Following this a number of step changes are observed and the relationship between applied pressure and volume change is no longer linear. This reflects progressive failure. Similarly in the case of the numerical models step changes are also observed

1.4%

1.4%

1.2%

1.2%

Volumetric change [%]


Mid-span expansion vs. applied pressure

1.0% 0.8% 0.6% 0.4%

Pipe A

0.2% 0.0%

0.8% 0.6% 0.4%

Pipe B

0.2% 0.0%

0

5

10

15 20 25 Pressure [bar]

30

35

40

45

0

1.4%

1.4%

1.2%

1.2% Volumetric change [%]


1.0%

1.0% 0.8% 0.6% 0.4%

Pipe C

0.2% 0.0%

5

10


30

35

40

45

1.0% 0.8% 0.6% 0.4%

Pipe D

0.2% 0.0%

0

5

10


30

35

Experimental

40

45

0

5

10


30

35

40

45

PFA without imperfections

Fig. 4. Plots of the mid-span expansion (DV) versus the applied pressure (bar) for the four pipes listed in Table 1. Each plot compares the experimental results [23] with the corresponding classical PFA i.e. without any initial imperfections.

859


First and second ply failure load results summary Test specimen

A B C D

Wall thickness [mm] ± 80 lm

First ply failure (FPF) [bar] ± 0.05 bar Experimental [23]

Classical PFA

3.345 3.355 3.350 3.415

13.3 18.2 13.8 18.0

12.4 12.6 12.4 12.5

2nd Ply failure from PFA [bar] ± 0.05 bar

21.7 21.7 21.7 22.2

Fig. 5. Plot showing the results of the mid-span expansion (DV) versus the applied pressure (bar) for pipe A obtained from the experiments [23] and PFA without imperfections.

however apart from two to three significant step changes the relationship between the expansion and the applied pressure is relatively linearly smooth suggesting that ply failure occurs throughout the whole lamina at the same time. The first sudden drop in expansion occurred in the range of 12.4 bar and 12.7 bar for all the pipes. This slight drop in expansion is the first ply failure load and corresponds to the matrix failure of the three direct roving layers. Comparing these results with the experimental ones reported in [23] (see Table 4) one can notice that the FPF load from the PFA is slightly underestimated. The second and most pronounced expansion drop occurred between 21.7 bar and 22.2 bar except for pipe B where the most pronounced expansion drop occurred at 32.2 bar. This pronounced drop in expansion corresponds to the matrix failure of the chopped strand mat layer. It is difficult to compare the second ply failure (SPF) load obtained from the PFA with the experiment results since the actual pipes experience more gradual progressive failure. This behaviour can be seen more clearly in Fig. 5 which shows the expansion with respect to the applied pressure of pipe A. The SPF of the actual pipe A occurred at 19.4 bar whilst the SPF of the corresponding PFA occurs at 21.7 bar. Beyond the SPF, the PFA predicts a linear expansion with respect to the applied pressure until the final failure at 43.3 bar. Comparing this prediction with the experimental result, the expansion after the SPF at 19.4 bar of the actual pipe was not linear but rather jittery which is a clear indication that failure is progressing constantly. This region is referred to as the instable region as indicated in Fig. 5 where the performance of the pipe is unpredictable due to the progression of failures. This phenomenon can be seen in the remaining pipes which is a characteristic behaviour of failing composites [29].

3.2. Preliminary simulations – varying the thickness of the CSM (Case A) In the following numerical models, the thickness of the chopped strand mat layer was varied randomly. This modification was opted for more accurate representation of the instable region that otherwise was not captured with the classical PFA. At this preliminary stage, the thickness of the CSM layer of all the elements of pipe D was varied randomly with a uniform distribution. As detailed in Section 2.2.1, two simulations where carried out where the new thickness of the CSM layer was varied between the intervals ±2% and ±20% of the nominal thickness. The results of these simulations of pipe D are illustrated in Fig. 6. The FPF corresponds to the matrix failure of the three direct roving layers whilst the SPF corresponds to the matrix failure of the CSM layer which is similar to the simulations without imperfections. From Fig. 5, one can notice that the FPF load of the simulation where the thickness of the CSM layer was varied between ±20% of tCSM was lowered to 11.1 bar where the FPF load of the ±2% simulation remained approximately equal to the simulation without imperfections. Details of the FPF and SPF loads are listed in Table 5. The SPF load of the ±20% simulation was lowered such that it coincides with the experimental result. The behaviour of both the ±2% and ±20% simulations beyond the SPF load is instable which is in agreement with the experimental observations. The region ends until all the matrix of the CSM layer of all the elements fails progressively. This is clearly an improvement over the simulation without imperfections since the region where the pipe is failing progressive can be predicted. The region following the instable region corresponds to phase where the matrix of the CSM layer of all the elements failed. The simulations continue until the first fibre failure occurs thus rendering an un-converged solution indicating global failure of the pipe.

Applied Pressure [bar] vs. Vol. Change [%] 1.2%


Table 4 List of first and second ply failure loads obtained from the PFA without imperfections (termed ‘‘Classical PFA’’) compared with the experimental results [23].

1.0% 0.8% 0.6% 0.4% 0.2% 0.0% 0

5 Experimental

10

15

20 25 30 Applied pressure [bar]

No Imperfections

±2% of CSM tks

35

40

45

±20% of CSM tks

Fig. 6. Plot showing the results of the mid-span expansion (DV) versus the applied pressure (bar) of the actual pipe D with respect to the PFA without any imperfections and the PFA with two levels of imperfections (±2% and ±20%) applied on the thickness of the CSM layer.

Table 5 List of first and second ply failure loads for pipe D. Pipe D

FPF load [bar]

SPF load [bar]

Experimental No imperfections ±2% of tCSM ±20% of tCSM

18.0 12.5 12.4 11.1

20.5 22.2 22.0 20.4

860


1.40%

1.20%

1.20%

Mid-span expansion [%]


Mid-span expansion vs. applied pressure 1.40%

1.00% 0.80% 0.60% 0.40%

Pipe A

0.20% 0.00%

0.80% 0.60% 0.40%

Pipe B

0.20% 0.00%

0

5

10

15 20 25 30 Applied pressure [bar]

35

40

45

0

1.40%

1.40%

1.20%

1.20%



1.00%

1.00% 0.80% 0.60% 0.40% 0.20%

Pipe C

0.00%

5

10


35

40

45

1.00% 0.80% 0.60% 0.40% 0.20%

Pipe D

0.00% 0

5

10


35

Experimental

40

45

0

5

10

PFA without imperfections


GDIS

35

40

45

RAND

Fig. 7. Plots of the mid-span expansion (DV) versus the applied pressure (bar) for the four pipes listed in Table 1. Each plot compares the experimental results [23] with the corresponding PFA simulations with varying CSM layer thickness according to the respective pipe thickness. GDIS and RAND denote a normal and uniform distribution of the random number generated to calculate the random CSM layer thickness.

Accumulation of failures vs. applied pressure of pipe D (a) Normal distribution 100%

80% 70%

90%

70%

60%

60%

1st

fibre failure in the CSM layer at 22.8 bar

40% 30% 20% 10%

FPF load = 12.1 bar

50%

50% 40%

0.20%

30% 0.00%

20% 22

24

26

28

30

Matrix failure of the CSM layer

Matrix failure of the DR layers

80%

FPF load = 8.6 bar

% of total elements

90%

(b) Uniform distribution 100%

Matrix failure of the CSM layer

Matrix failure of the DR layers

10% 0%

0% 8

10

12

14

16

18

20

22

24

26

28

8

30

10

12

Applied pressure [bar] Unfailed

14

16

18

20

22

24

26

28

30

32

34

36

38

40

Applied pressure [bar] Matrix Failure

Fibre failure

Fig. 8. Area plots showing the accumulation and distributions of failure modes for pipe D according to the random number distribution used; (a) normal distribution and (b) uniform distribution.

3.3. Varying the thickness of the CSM based on the pipe thickness (Case B) In this section, the thickness of the CSM layer of all the elements was varied randomly using uniform and normal distributions, denoted as RAND and GDIS respectively in the following plots, as

detailed in Section 2.2.1.2. Fig. 7 shows the results of the mid-span expansion with respect to the applied internal pressure for all the pipes. It can be noted from Fig. 6 that the FPF load of the GDIS simulations is lower than the simulations with the uniform distribution random number. This is attributed to the fact that there are no

861


bounds on the generated thickness which means that the will be layers with very small thickness. This will induce premature failure and hence lower the FPF load. This can be seen even in the SPF load. The gradual variation in thickness for the normal distribution gave rise to a gradual failure progression when compared to the uniform distribution. This effect is clearly shown in Fig. 8 where the matrix failure of the DR layers started at 8.6 bar which marks the FPF load and ended at 13.6 bar (indicated by the first light blue shading in Fig. 8a). In the case of the uniform distribution, this failure occurred suddenly at 12.1 bar (indicated by the vertical dotted line in Fig. 8b). Furthermore, in the case of the normal distribution fibre failure was predicted to initiate at 22.8 bar. This is indicated in Fig. 8a in the small plot with enlarged scale since the fraction of fibre failures was too low to show in the full plot (0.16% of the total elements experienced fibre failure at the maximum load reached i.e. 30.9 bar). These fibre failures are the reason why the ultimate load is lower with respect to the simulation using the uniform distribution. Although the gradual failing is more realistic with respect to the experimental results, the simulations require more time to converge because the failures are distributed over a wider range of iterations. 3.4. Varying the thickness or material stiffness constants of all the layers based on Vv of the material characterisation specimens (Case C, Type 1 & 2) Following the results obtained in the previous section, the uniform distribution was opted for the following last set of simulations. Two simulations where carried out for each pipe, one in which the material property of each and every layer was varied and in the other, the thickness of every layer was varied. In each simulation, the thickness of the layer was reduced by the void fraction Vv listed in Table 2 respectively. Furthermore, the thickness or

Mid-span expansion vs. applied pressure (Pipe A) Experimental FPF 13.3 bar

1.4%


1.2% 1.0% 0.8%

4. Conclusions

0.6%

Instable region PFA FPF

0.4%

SPF

0.2% 0.0% 0

5

10

Experimental

material property of each layer was varied randomly where the limits were set equal to the std(Vv) of the respective layer type listed in Table 2. The results of the mid-span expansion with respect to the applied pressure for pipe A are shown in Fig. 9. The FPF load for both simulations occurred at approximately 11.6 bar which is lower than the reported FPF load of the experiment 13.3 bar. This is expected due to the inclusion of imperfections. On the other hand, the SPF predicted by both the simulations (20.4 bar) is in good approximation with the experimental one (19.4 bar). Table 6 lists the FPF, SPF and the corresponding failure modes of both simulations with respect to the PFA without imperfections and the experimental observations [23] of pipe A. Although only the FPF and SPF for pipe A are listed, the numerical results for the other three pipes were similar. The failure modes corresponding to both the FPF and the SPF are similar to the ones obtained from the PFA without imperfections. Analysing the expansion of both the simulations shown in Fig. 9, varying the material properties or the thicknesses of the layers gave relatively better results with respect to the results obtained from the simulations which were based on the variation of the pipes’ wall thicknesses using the normal distribution shown in Fig. 7. Furthermore, it was found that for the above mentioned cases, varying the material properties (Type 2) gave better predictions during the instable region with respect to Type 1 imperfections. The irregular variations in the mid-section expansion during the instable regions are attributed to the random variations included in the thickness or material properties. This irregular failure progression is difficult to predict accurately and this can be noted by comparing the experimental results. Although the pipes were manufactured with the same materials and the same machine, every pipe exhibited a slightly different failure progression. This randomness is featured in the results obtained with the proposed modifications to the PFA. Nonetheless, applying imperfections of either type predicted similar FPF loads, SPF loads and instability regions which are the salient objectives of a PFA.

15 20 25 30 Applied pressure [bar] Vary material properties

35

40

45

Vary layer thicknesses

Fig. 9. Plot of the mid-span expansion (DV) versus the applied pressure (bar) for pipe A with varying the thickness of the layers and material properties independently.

In this paper a probabilistic nondeterministic PFA applied to asymmetric filament wound pressure vessels was presented. A set of simulations using the classical PFA were run and it was shown that although the FPF load was predicted successfully, failure progression thereafter was not. This is attributed to the fact the geometry in the modelling software used is perfect and free from any defect. In reality, this is not the case and in fact this is shown in the numerous experiments on fibre reinforced polymers where the load–deflection relationships are not linearly smooth. In this paper this region is referred to as the instable region where the behaviour of the structure is irrational thus almost unpredictable. With the proposed modification, i.e. the inclusion of imperfections based on the void fractions of the specimens used in the material characterisation tests, the PFA is improved considerably since the instable region can be predicted. The strength capability of a pipe is determined from the FPF load according to the EN 13923:2005 [3] therefore knowing where

Table 6 List of first and second ply failure loads together with the corresponding failure modes of pipe A. Pipe A

FPF load [bar]

FPF failure mode

SPF load [bar]

SPF failure mode

Experimental No imperfections Vary material properties Vary layer thicknesses

13.3 12.4 11.6 11.7

N/A Matrix failure of the DR layers Matrix failure of the DR layers Matrix failure of the DR layers

19.4 21.7 20.4 20.4

N/A Matrix failure of the CSM layer Matrix failure of the CSM layer Matrix failure of the CSM layer

862


the structure is instable is essential from the optimisation point of view. For maximum performance efficiency, the FPF load should be approximately equal to the start of the instable region. In this way using the proposed modified PFA methodology will enable the engineer to design more efficient FRP structures utilising their full strength potential. References [1] Hinton MJ, Soden PD, Kaddour AS. Failure criteria in fibre-reinforced-polymer composites: the world-wide failure exercise. Amsterdam, The Netherlands: Elsevier Science Ltd; 2004. [2] Muc A, Fugiel T. Influence of physical and geometrical nonlinearities on failure analysis of composite pressure vessels. In: Proc. ICCM13, Beijing. 25–29 June 2001. [3] Filament-wound FRP pressure vessels. Materials, design, manufacturing and testing, EN 13923:2005. [4] Liu PF, Zheng JY. Recent developments on damage modeling and finite element analysis for composite laminates: a review. Mater Des 2010;31(8):3825–34. [5] Lapczyk I, Hurtado JA. Progressive damage modeling in fiber-reinforced materials. Compos Part A Appl Sci Manuf 2007;38(11):2333–41. [6] Rowlands RE. Strength (failure) theories and their experimental correlations. In: Sih GC, Skudra AM, editors. Handbook of composites. New York: Elsevier; 1985. p. 71–125. [7] Nahas MN. Survey of failure and post-failure theories of laminated fibre reinforced composites. J Compos Technol Res 1986;8(4):138–53. [8] Echaabi JF, Trochu F. Review of failure criteria of fibrous composite materials. Polym Compos 1996;17(6):786–98. [9] Hinton MJ, Soden PD. Predicting failure in composite laminates: the background to the exercise. Compos Sci Technol 1998;58(7):1001–10. [10] Soden PD, Hinton JM, Kaddour AS. A comparison of the predictive capabilities of current failure theories for composite laminates. Compos Sci Technol 1998;58(7):1225–54. [11] Hinton MJ, Kaddour AS, Soden PD. A comparison of the predictive capabilities of current failure theories for composite laminates, judged against experimental evidence. Compos Sci Technol 2002;62(12–13):1725–97. [12] Paris F. A study of failure criteria of fibrous composite materials. NASA/CR2001-210661; 2001. [13] Icardi U, Locatto S, Longo A. Assessment of recent theories for predicting failure of composite laminates. Appl Mech Rev 2007;60(1–6):76–86.

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