1 Introduction and objectives

This paper describes the validation of an integrated design and simulation approach via the comparison of the results of tensile tests conducted on filament wound pipes prior to the development of the whole research, with other 45 specimens.

The tensile tests revealed that large hoop and helical displacements are caused by exceeding a loading threshold and that the damage behavior varies with the angular variation, temperature, and number of bundles used in the fabrication of these structural elements (Fig. 1a, b).

Fig. 1
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a Set of specimens tested at room temperature. b Detail of displacement for specimens tested at 85 °C

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The first paper in this series covered the preliminary analysis of acoustic emission behavior for the same sample typology [1]. In this work, the correlation between the three winding angles was investigated, and the variation of their elastic properties was used as a control parameter throughout the research project.

The approach developed in this work assumes that the composite structures are formed by assembling their raw materials, which are created individually, rather than treating these structures as homogeneous and isotropic single materials.

2 Literature review

Pipes in composite structures are generally fabricated via filament winding. This industrial process requires control of the roving slippage, angular path variation and inversion, head tension, and rotation of the mandrel throughout the lamination process. In this work, the angular path control is analyzed.

The calculation models most commonly used in the design of composite structures are classical laminate theory (CLT) and the rule of mixtures (RM), both of which have been discussed extensively in the literature [2–9].

Several published studies have investigated tubular specimens produced by filament winding, which used finite-element software to propose ways to predict the failures in the matrix and in the filaments [10].

Previous works have verified the behavior of pipes manufactured by filament winding. In one study, these pipes, manufactured with a winding angle of ±55°, were subjected to biaxial compression [11]. In another study, several theoretical models were compared with the experimental results obtained for specimens manufactured by filament winding at ±55° and then, subjected to biaxial compression and internal pressure tests [12].

For a more detailed description of the failure analysis criteria, please refer to the relevant studies in the literature [13–19].

Rousseau et al. [20] experimentally verified the effects and occurrences of progressive failure leading to fractures in pipes. The loss of stiffness was verified by comparing the deformed and undeformed geometries. In particular, the geometry of the roving formed a diamond shape in which the filaments intersected the preceding filaments, and this diamond was different in the deformed state. As a result, the successive deformation after each variation of the load level changed. Other studies conducted similar analyses in pipes wound at ±60° and ±45° [21].

The literature also contains studies on the effects of the environment. Aggressive environmental effects were found to cause a loss of adherence between the fiber and matrix in pipes wound at ±60° and subjected to internal pressure at different temperatures [22].

Some researchers have analyzed the shear stresses induced by tensile stress acting on a pipe and the correlation of these stresses with the torsional stress, internal pressure, and circumferential stress in pipes wound at ±45° [23] or the behavior of pipes wound at ±54.7° with different wall thicknesses [24].

Hu et al. [25] conducted a study that is somewhat similar to that presented here. However, they only considered pipes wound at ±55°, and their finite-element analysis (FEA) did not utilize a parametric model that considers the raw materials individually.

Based on this literature review, it is clear that although filament wound pipes have been studied extensively, these studies differ in terms of the raw materials, fiber orientation, or nature of the polymeric matrix considered [26–36].

3 Materials and methods

3.1 Specimens

The specimens were fabricated with three different typologies, winding the pipes at ±50.0°, ±52.5°, and ±55.0°. For each typology, 15 samples were produced with the same geometrical parameters: 300 mm of total length, 180 mm span length, 50.8 mm internal diameter, and 3.2 mm wall thickness.

Three specimens for each winding angle were randomly sampled, and the fiber volume fraction V fr was obtained experimentally based on ASTM D3171 [37] procedure A (Tables 1, 2).

Table 1 Density according to the experimental results
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Table 2 Fiber ratio according to the Table 1
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Equation 1 was used to calculate the number of layers needed to achieve the target design thickness, which was used as an input in the filament winding process (Table 3).

Table 3 Total number of paths to complete one layer as a function of the mandrel radius
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$$N = \frac{{2\pi {R_{\rm avg}}\tan \alpha}}{w}$$
(1)

where N = total number of paths to complete the closing equivalent layer of roving, w = band width, R avg = (D e − t)/2, where t = wall thickness and D e = external diameter of the pipe, α = winding angle.

3.2 Theoretical composite properties

The theoretical analysis was conducted using the well-known modified RM (mRM) and CLT approaches. First, the fiber volume fraction V fr was obtained experimentally [37]. Next, V fr was input into the mRM to calculate the modulus in direction 1, E 11, the modulus in direction 2, E 22, the shear modulus in plane 1–2 G 12, and the Poisson’s ratio in plane 1–2 ν 12, shown in Table 4. Finally, the theoretical design stress (S d) and strain (ε) of the specimens were determined through CLT, as shown in Table 5.

Table 4 Elastic properties found through the mRM
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Table 5 Elastic properties found through CLT
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3.3 CAD and simulation procedures

The literature has demonstrated that convergent results can be reached by modeling the raw materials separately and assembling them into a single complex model that includes the interactions among these sub-models [38]. Hence, this approach was used to create a virtual model for export to the FEA tool [39] to perform the simulation with the same boundary conditions as used in the experiments.

The parametric assembly was developed through SOLIDWORKS® [40], as shown in Fig. 2a–d. Here, a parallelogram cross section envelope was used to account for the hexagonal distribution of the dry filaments forming one composite roving-epoxy bundle.

Fig. 2
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a–c Stacked sequence. d Study of possible geometric arrangements of filaments, according to Eq. 2

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The average roughness of the fiber filaments is 3.02 e−3 mm [41–43]; in this work, this value was multiplied by three to estimate the mean roughness of the filaments, corresponding to a 9.1 e−3 mm gap between each filament (Fig. 2d).

The interface thickness between two overlapping layers was taken as 9.0 e−2 mm, which was calculated as a function of the fiber volume fraction (55 %). This geometric distribution resulted in a bandwidth of 23.34 mm, which is very close to the actual bandwidth of 23.4 mm used in the specimen production.

This process was based on the equivalent fiber volume fraction (V fre) for a bundle of unitary length given by Eq. 2. This model was developed by other researchers in a partially concluded study.

$$V_{\rm fre} = \frac{\sum\nolimits_{k = 1}^{n} D_{fk}}{\sum\nolimits_{k = 1}^{n} S_e}$$
(2)

where D fk  = diameter of the kth fiber filament, with D > 0, S e  = envelope area of the polymer under consideration, with S e  ≥ D fk  × 0.85.

The mechanical and physical properties of each constituent material, i.e., the fiber strands and polymer, were input individually into the simulation tool [39], and the envelope area was selected as a function of V fre using the geometrically calculated distances between each filament and the edges of the surrounding envelopes.

First, it was assumed that all the filaments were homogeneous, perfectly aligned relative to the winding machine axis, and perfectly hexagonally spaced. Next, the filamentary cross-sectional area needed to fill 55 % of a circumscribed rectangle was calculated via an iterative process. Here, the distance between the filaments was modified to achieve the cross-sectional area needed by the polymer to satisfy the relationship V fre + V pr = 1, where V pr is the polymer volume fraction.

A detailed view of the cross section of the computer-aided drafting (CAD) model proposed with the polymeric film passing through the strands is shown in Fig. 3a and b.

Fig. 3
figure 3

a Cross section of the strands. b Detail of strands separation and matrix

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The FEA tool considered the direction in which ε 1 was most positive as the major strain and the direction in which ε 3 was most negative as the minor strain. Therefore, the output in terms of the strain intensity is given as

$${\varepsilon_I} = \hbox{max} \left(\left| \varepsilon_{1} - \varepsilon_{2}\right|,\left|\varepsilon_{2}-\varepsilon_{3}\right|,|\varepsilon_{3}-\varepsilon_{1}|\right)$$
(3)

where ε I = maximum absolute value of strain intensity achieved.

The Lagrangian method was used to describe the friction and adhesion between the faces in contact within the materials as well as to predict the loss of adhesion between the contacting faces [41]. This method is based on the theory of Coulomb, which establishes that the maximum limit of friction should be less than or equal to the shear stress between the surfaces in contact.

The selected approach for the application of the forces between the faces in contact is described by Eq. 4, which addresses the problem of tangential contact friction between two faces.

Fig. 4
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a Hexahedral mesh. b Mesh detail

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$${\rm d}{\tau_I} = {\rm{}}{t_i}\left({\mu + P\frac{\partial\mu}{\partial P}} \right){\rm d}P + \left({{\delta_{ij}} -{t_i}{t_j}} \right)\frac{{{\tau_{\lim}}}}{{\left\| {u^{\rm pred}}\right\|}}{\rm d}{u_j} + {t_i}{t_j}\frac{P}{\Delta t}\frac{\partial \mu}{{\partial \left\| {\dot u} \right\|}}{\rm d}{u_j}$$
(4)

where \(\left\| {{u^{\rm pred}}} \right\|\) is the equivalent slip, which can be predicted for any given sub-step during the iteration process [42], and \(\dot u\) is equivalent slip rate at a specific iteration number. Additionally, t i, j is the unitary vector in the slip direction, P is the contact normal pressure, μ is the friction coefficient calculated as \({\tau_i} = {\mu_i}P + b\)(i = 1,2), τ is the shear stress, and δ ij is the displacement due to the shear effect with respect to the plane under consideration.

Other models can also be used in simulation tools, such as those created through mathematical models; however, these models are often restricted to two dimensions (2D) by hardware limitations [43–48].

Table 6 shows the initial values of the mechanical and physical parameters input into the library of the FEA tool. After inputting these values, the mesh for each material was created, limiting the larger edge of the elements to no more than 1.8 mm.

Table 6 Material properties used as the input data for simulation
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It should be observed that the longitudinal cross-sectional area of the bundles forming the thickness of pipes, S Lb, and the decomposed loading, P θ refer to the local coordinates as shown in Fig. 7, rather than axial loading, P, and the cross-sectional area S.

Figure 4a and b shows the mesh for the specimen laminated with an angular variation of ±50°, and (c) shows the notation used by both CAD and FEA, with two strands hidden for internal visualization.

3.4 Experimental procedure

The tensile tests were performed in a servo-hydraulic universal testing machine (MTS®, model 810) with a 500-kN load cell using computational strain evaluation and displacement control.

Three specimens were randomly sampled from the 45 fabricated samples and conditioned in an environment with controlled temperature and relative humidity. After 48 h, the specimens were equipped with a pair of axial–radial strain gages: the first placed in each pipe at the zero-degree circumferential mark and the second phased 180° from the first. Next, the specimens were monotonically loaded at a displacement velocity of 0.0106 mm s−1 until reaching 6 MPa or 12 MPa, as shown in Fig. 5a and b, respectively.

Fig. 5
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a Total population of specimens. b General testing setup

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The elastic properties were determined from the 6 MPa loading for each specimen, and plots of the average stress versus recorded strain (Figs. 6, 7, 8a, b) were used to calculate the other elastic properties (Table 7).

Fig. 6
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a Stress–strain curve for each specimen. b Mean for all SPs ±50º

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Fig. 7
figure 7

a Stress–strain curve for each specimen. b Mean for all SPs of ±52.5º

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Fig. 8
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a Stress–strain curve for each specimen. b Mean for all SPs ±55º

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Table 7 Mechanical properties found by testing the three winding angles
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Equation 5 was used to decompose the global stress (Y-axis) into the local axial and tangential loadings along each infinitesimal angle over the turned θ degree path. In this calculation, one-half turn of a helical model, with a known winding angle θ (Fig. 9), was considered for each roving bundle wound over the mandrel, according to Eq. 1 above.

Fig. 9
figure 9

Geometry used to calculate the normal loading of the bundles, by winding angle

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$${\sigma_{T\theta}} = \frac{{{\sigma_A}{w_b}\tan \theta}}{{2{D_{\rm avg}}}}$$
(5)

where σ Tϑ  = resultant transversely stress for specific winding angle, σ A  = axial stress, considering the Y-axis as the axial direction for CAD and FEA, w b  = bundle width, θ = angle that the element or laminate makes with the axial direction, D avg = D e –½t θ , where t θ is the wall thickness of each winding angle.

As the torsional moments were acting along the turned path, the Bernoulli–Navier principle was abandoned, and it was assumed that a flexural–torsional effect acted on the cross section.

4 Results and discussion

The fibers subjected to tensile stress had a strong tendency to straighten, especially with large displacements, inducing rotational moments around the local X-axis, i.e., transverse to the winding path. Because the X-axis passes through the wall in the direction tangential to the winding path, when the interlacing prevents the layers from moving (Fig. 9), localized failure occurs once the limit of rupture is reached. The failure then propagates throughout the laminate, leading to the collapse of the bundle interfaces.

The simulation results were closest to the experimental results, whereas the theoretical results were very different from either of these sets of results, as we will see later. The theoretical results differed substantially from the simulation results because the former treated the material as a perfectly bonded whole, while the latter treated it as a set of individually stacked layers.

Although the limit values were similar to the experimental results, the strains in plane 2–1 were inhomogeneous, with larger variations in the intermediate loading level.

The simulated models reveal a spreading of the strains, which formed niches along the pipes, with some points of high strain localized along the pipe length, as shown in Figs. 10, 11, and 12a–c. This behavior was observed throughout the laminate, which may indicate that the stretching and movement restrictions caused localized displacements and concentration of stress, especially in intersections between layers as indicated in same figures by capital letter ‘D’. Other indications in the same direction are the strains due to axial loading been occurred in simulation, where two consecutive bundles presented jumps in the strain level, when they were near from overlapping region, as shown in Fig. 13.

Fig. 10
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a Strain in direction 1. b Strain in direction 2. c Total elongation (±50º)

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Fig. 11
figure 11

a Strain in direction 1. b Strain in direction 2. c Total elongation (±52.5º)

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Fig. 12
figure 12

a Strain in direction 1. b Strain in direction 2. c Total elongation (±55º)

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Fig. 13
figure 13

Strains in overlapping points between two consecutive bundles due to axial loading

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The quasi-linear behavior of the experimental results was verified through the plotted stress–strain curves. In contrast, according to the FEA, quasi-linear behavior only occurred around 1.6 MPa, above which the behavior became non-linear, as seen in Fig. 14a–d.

Fig. 14
figure 14

a Strains from FEA and tests for ±50º. b Strains from FEA and tests for ±52.5º. c Strains from FEA and tests for ±55º. d Strains from FEA for each angular variation

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As the filaments are stretched by traction, they are progressively compressed laterally. This behavior can be explained by the load level difference between experiments, which achieved 12.5 % of the average ultimate limit stress of pipes. Therefore, there were innumerous points at which the strands interacted with one another, and the resulting internal stresses changed from traction to compression. In Fig. 15, some of the strands are hidden to visualize the inner layers and the concentration of stress on the strand edges.

Fig. 15
figure 15

Interlaminar stress effects (traction and compression) on the layers

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The moduli E 11, E 22, and G 12 and the strains ε 12 and ε 21 obtained via mRM versus CLT and experimentally (Tests) versus FEA were compared for each winding angle as shown in Figs. 16, 17, 18 and 19a and b.

Fig. 16
figure 16

a Compared values of E 11 for RMm versus CLT, and b Tests versus FEA, for all typologies with the mean and the SD

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Fig. 17
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a Compared values of E 22 for RMm versus CLT, and b Tests versus FEA, for all typologies with the mean and the SD

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Fig. 18
figure 18

a Compared values of G 12 for RMm versus CLT, and b Tests versus FEA, for all typologies with the mean and the SD

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Fig. 19
figure 19

a Values of ε 12 for Tests versus FEA, for all typologies with the mean and the SD. b Values of ε 21 for Tests versus FEA, for all typologies with the mean and the SD

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The plot of the strains for mRM and CLT is not presented herein because the parameters were theoretical values obtained from the literature and then averaged. The mRM and CLT provided higher strain values than the Tests and FEA (Table 8).

Table 8 A comparison of the mechanical properties obtained through all analyses
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For the pair mRM–CLT, the largest standard deviation—SD for the modulus E 11 was found for the angular variation of ±55°, for which the mean value was ±2.81 GPa. Meanwhile, for the pair Tests–FEA, the SD for E 11 was highest for the angular variation of ±50°, for which the mean value was ±0.54 GPa. In the first case, the SD was 20 % higher than the average, while in the second case, the SD was 2.5 % less than the average.

For E 22, the highest variation occurred for the pair Tests–FEA and an angular variation of ±55°; the value, ±1.52 GPa, corresponded to a 3.8 % deviation from the average. In contrast, the pair mRM–CLT presented the lowest SD of all moduli, with a value of ±0.014°GPa for the angular variation of ±52.5°.

Among the results obtained, the highest convergence was obtained for the G 12 modulus for the pair Tests–FEA, with an SD of zero for the angular variation ±52.5°. Meanwhile, the most divergent was the E 11 of the ±55° for the pair mRM–CLT (Figs. 15, 16).

Regarding the strains, the SDs were all lower than 1.0 mm/mm−5. The highest SD (±7.1 mm/mm−6) was obtained for ε 12 from the typologies ±50° and ±55°, while the lowest (±2.1 mm/mm−6) was obtained for ε 21 from the typology ±50°.

The results of these various analyses were somewhat consistent regarding the angular variation but did not follow a regular pattern, as seen in Figs. 18 and 19.

5 Conclusions and remarks

In the analysis of filament wound elements, the standards [49, 50] are unlikely to provide accurate results. This is because the recommended directions are not coincident with the winding angles: there is missing information regarding the actual strain in the direction of the bundles. To address this problem, other analyses can be used to determine this correlation experimentally rather than theoretically. In this work, experimental, simulation, and theoretical analysis techniques were compared in the context of automated filament winding lamination.

The results showed that the results obtained from the theoretical models, computational simulation, and experimental analysis of this type of structure varied slightly.

The analyses based on the mRM and CLT, which consider perfectly adhered laminates, use the Kirchhoff hypothesis wherein the fibers normal to the middle plane of the laminate remain normal during the deformation and retain their initial length. Under transverse strain, the plate deflections are significantly smaller than the thickness of the plate under consideration [51], which implies that

$${e_{xz}} = {e_{yz}} = {e_z} = 0 \quad {\rm and} \quad {s_{xz}},{s_{yz,}}{s_z} < < {s_{xy}},{s_{y,}}{s_x}.$$

The boundary conditions of each theoretical analysis may have contributed to the difference between the pair of theoretical results with respect to the results of the tests and FEA. Moreover, the most meaningful observation was the convergence and greatest similarity between the results of the tests and the FEA regarding the G 12 modulus, whereas the greatest variation occurred for the pair mRM–CLT for the E 11.

The variations of the moduli and respective Poisson ratios followed the expected behavior: the distance between the values increased as the winding angle changed when the laminates were rotated in direction 2 (x in this work).

In turn, the experimental analysis conducted based on the strain gage positioning standards may not be the best suited for curved surfaces produced by filament winding.

The moduli values followed a pattern for most of the winding angles and analysis methods; however, additional testing is required to confirm whether this pattern is real. All the other results are summarized for each analysis method and angular variation and revealed the same trends as described for the moduli [52].

Additional tests should be performed with a larger number of strain gages, some aligned along the path of the strands and others perpendicular to this path. In both this study and the literature, significant differences were found between the results of tensile (or compressive) tests for 180- and 600-mm-long specimens [1, 24, 52].

Software packages exist for specific application to composites; however, they are still too expensive for use by small- and medium-sized companies in emerging countries [53].

Finally, similar analyses should be performed using other winding angles to validate the approach for the filament winding fabrication and integrated CAD.

This research was supported by the “Brazilian Federal Agency for the Support and Evaluation of Graduate Education (CAPES)” of the Ministry of Education of Brazil.