Structural optimization of filament wound composite pipes

Abstract

An optimization procedure is developed for obtaining optimal structural design of filament wound composite pipes with minimum cost utilized in pressurized water and waste-water pipelines. First, the short-term and long-term design constraints dictated by international standards are identified. Then, proper computational tools are developed for predicting the structural properties of the composite pipes based on the design architecture of layers. The developed computational tools are validated by relying on experimental analysis. Then, an integrated design-optimization process is developed to minimize the price as the main objective, taking into account design requirements and manufacturing limitations as the constraints and treating lay-up sequence, fiber volume fraction, winding angle, and the number of total layers as design variables. The developed method is implemented in various case studies, and the results are presented and discussed.

Graphical abstract

Keywords

composite pipes / optimization / experimental validation / computational modeling / filament winding

1 Introduction

Glass Fiber Reinforced Polymer (GFRP) pipes play an important role in the infrastructure of countries. Various components of infrastructure, including oil and gas supply,water supply and waste systems, and other utilities, benefit from the unique features of GFRP pipeline systems for conveying potable water, wastewater, oil, gas, or other chemically reactive fluids. Excellent corrosion resistance, high stiffness, outstanding fatigue strength, and favorable hydraulic characteristics have made them a promising alternative to conventional piping systems.

The final price of implementing the GFRP pipeline system is governed not only by the cost of the produced pipes as the short-term indicator but also by the repair and maintenance expenses based on long-term considerations. From the long-term point of view, they are very competitive with other conventional and traditional piping systems mainly because they are not suffering from corrosion. Therefore, the final price of the produced GFRP pipes contributes significantly to the final cost of GFRP pipeline projects. The final price of the GFRP pipes is directly determined by their structural design since configurations of constructive layers define the quantity of raw material consumption.

The structural design pattern of GFRP pipes can fall into two categories: 1) trustworthy pipes with high safety factors but quite unfeasible due to the over-designed scheme; 2) unreliable pipes with economically attractive prices where pre-mature failure is experienced much earlier than design lifetime. Both scenarios are undesirable, and thus industrial producers prefer the first scenario and reduce the cost by decreasing the safety factors through trial-and-error design procedures. Industrial centers practically seek a cost-effective design scenario where all design requirements are fulfilled in parallel with maintaining the price at its minimum possible level. This is strategically a crucial issue to compete in the market with an acceptable profit margin. Consequently, the optimal design is to reach the ideal match among the lifetime, mechanical performance, and cost.

As evidenced by the literature review, aerospace and naval applications were the main areas of optimization studies on cylindrical composites [1]. Thus, the main target was to increase either critical buckling loads [2–12] or natural frequencies [13–20] and minimize the weight. Minimal studies have optimized the structural design of GFRP pipes [21–27]. Among them, a few studies have considered different sources of loadings rather than only internal pressure.

Minsch et al. [21] analyzed different aspects of the filament winding process and equipment technology and derived a decision matrix for process engineering to improve the selection of proper equipment for a particular design. A useful analytical tool was provided by Colombo and Vergani [22] for the optimal design of a composite pipe, minimizing the wall thickness by taking into account internal pressure and axial loads. Jin et al. [23] proposed a safety evaluation method based on pipe wall materials’ strength and fracture characteristics and suggested a formula to calculate an optimum pipe wall thickness for a pipe undergoing live and dead loads. Almeida et al. [24] proposed a genetic algorithm (GA) to optimize the stacking sequence to improve the strength of a cylindrical shell under internal pressure with and without manufacturing restrictions. Liu and Shi [25] established a model to calculate the winding-induced residual stresses in thick cylinders undergoing internal pressure and optimize the failure pressure. The influences of stacking sequence and fiber orientation on energy absorption in cylindrical composites were studied by Zhang et al. [26], and optimal ply angles were obtained based on finite element modeling and analysis. Two methodologies for optimization of a type III pressure vessel were proposed by Alcántar et al. [27], where weight minimization was the objective function.

The main objective of this article is to develop a systematic and comprehensive approach for optimizing the structural design of GFRP pipes to achieve not only the minimum level of raw material consumption but also proper mechanical performance.

2 Problem statement

It is intended to develop an optimization procedure for obtaining optimal structural design of GFRP pipes with minimum cost for applying pressure water supply. For this purpose, all design requirements dictated by normative standards and manufacturing limitations are considered as the constraints, and the design architecture of composite layers are treated as design variables. Thus, design constraints are first identified. Secondly, proper computational tools for estimating the required structural properties of GFRP pipes are developed and validated. Finally, an optimization process is developed and implemented through developing a computer code. The workflow of the study is shown in Fig.1, illustrating the involved steps.

3 Computational methods

The structural design of the GFRP pipes in this research aims to be optimized to convey pressurized water/waste water. Hence, design constraints are identified following AWWA C950 [28]. In this section, computational methods used for estimating design constraints are presented. Simple tools are developed for estimating the structural properties of the GFRP pipes. These tools are required to be computationally quick and accurate enough at the same time since the core of the optimization procedure is constructed on them. Before developing necessary computational tools, fundamental parameters must be calculated. Consisting of geometrical specifications and initial mechanical properties of the layers, fundamental tools are input data of the computational tools. Consequently, these fundamental parameters are first explained, and then required computational tools is outlined.

3.1 Fundamental parameters

The structural layers of the GFRP pipes produced through the reciprocal filament winding technique consists of hoop and helical layers. In the hoop layers, the fibers are oriented along circumferential directions. Helical layers are angle plies where the fiber orientation is measured from the axial direction of the pipe. A pair of angle plies in the form of balanced angle plies (i.e., ±θ) forms a cross-layer. The thickness of each hoop layer is calculated using below formula:

(1)t_{\mathrm{h}\mathrm{o}\mathrm{o}\mathrm{p}}=\frac{{\rho }_{\mathrm{A}}^{\mathrm{H}}}{{\rho }_{\mathrm{F}\mathrm{R}\mathrm{P}}\times W_{\mathrm{f}}},

where W_f , {\rho }_{\mathrm{A}}^{\mathrm{H}}, and {\rho }_{\mathrm{F}\mathrm{R}\mathrm{P}} are fiber weight fraction, areal density of the hoop layer, and density of composites, respectively. These parameters are calculated as below:

(2){\rho }_{\mathrm{A}}^{\mathrm{H}}=\frac{N_{\mathrm{S}}T}{B},

(3){\rho }_{\mathrm{F}\mathrm{R}\mathrm{P}}=(V_{\mathrm{f}}{\rho }_{\mathrm{f}}+V_{\mathrm{m}}{\rho }_{\mathrm{m}}),

(4)W_{\mathrm{f}}=\frac{V_{\mathrm{f}}{\rho }_{\mathrm{f}}}{V_{\mathrm{f}}{\rho }_{\mathrm{f}}+V_{\mathrm{m}}{\rho }_{\mathrm{m}}},

where N_S, T, and B stand for number of fiber strands in the fiber bundle, fiber Tex (i.e., mass of fiber per on km of fiber length) and band width of the fiber bundle, respectively. V_f, V_m, ρ_f, and ρ_m denote fiber volume fraction, matrix volume fraction, fiber density, and matrix density, respectively.

The thickness of cross layers is calculated as below:

(5)t_{\mathrm{c}\mathrm{r}\mathrm{o}\mathrm{s}\mathrm{s}}=\frac{2\times {\rho }_{\mathrm{A}}^{\mathrm{C}}}{{\rho }_{\mathrm{F}\mathrm{R}\mathrm{P}}\times W_{\mathrm{f}}},

(6){\rho }_{\mathrm{A}}^{\mathrm{C}}=\frac{N_{\mathrm{S}}T}{B\mathrm{s}\mathrm{i}\mathrm{n}\theta },

where {\rho }_{\mathrm{A}}^{\mathrm{C}} is areal density of the cross layer and θ is the winding angle in cross layer.

It is widespread to use a core layer made of sand/resin in-between GFRP layers to increase the thickness of the pipe with an economical method. These types of pipes are referred to as GFRP mortar pipes, and they are mostly used as buried pipes. The thickness of this layer is calculated using the below formula:

(7)t_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}=\frac{{\rho }_{\mathrm{A}}^{\mathrm{S}}}{{\rho }_{\mathrm{S}}\times V_{\mathrm{S}}^{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}},

where

(8){\rho }_{\mathrm{A}}^{\mathrm{S}}=\frac{m_{\mathrm{S}}}{\text{π}DL}.

In above equations, {\rho }_{\mathrm{A}}^{\mathrm{S}}, {\rho }_{\mathrm{S}}, and V_{\mathrm{s}}^{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}} are areal density of sand/resin layer, density of sand, and sand volume fraction in core layer, respectively. m_S, D, and L stand for mass of sand in the full-length of pipe, pipe diameter, and pipe length, respectively.

After calculating the thickness of each layer, it is also required to obtain the mechanical properties of each layer using the following micromechanical rules [29]:

(9)E_X=E_{\mathrm{f}}{V}_{\mathrm{f}}+E_{\mathrm{m}}{V}_{\mathrm{m}},

(10){\vartheta }_X={\vartheta }_{\mathrm{f}}{V}_{\mathrm{f}}+{\vartheta }_{\mathrm{m}}{V}_{\mathrm{m}},

(11)E_Y=E_{\mathrm{m}}\frac{\left(1+2{\eta }_{\mathrm{T}}{V}_{\mathrm{f}}\right)}{\left(1-{\eta }_{\mathrm{T}}{V}_{\mathrm{f}}\right)};{\eta }_{\mathrm{T}}=\frac{{\displaystyle \frac{E_{\mathrm{f}}}{E_{\mathrm{m}}}}-1}{{\displaystyle \frac{E_{\mathrm{f}}}{E_{\mathrm{m}}}}+2},

(12)G_{XY}=E_{\mathrm{S}}=G_{\mathrm{m}}\frac{(1+{\eta }_{\mathrm{T}}{V}_{\mathrm{f}})}{(1-{\eta }_{\mathrm{T}}{V}_{\mathrm{f}})};{\eta }_{\mathrm{T}}=\frac{{\displaystyle \frac{G_{\mathrm{f}}}{G_{\mathrm{m}}}}-1}{{\displaystyle \frac{G_{\mathrm{f}}}{G_{\mathrm{m}}}}+1},

where E, G, and ν are Young’s modulus, shear modulus, and Poisson’s ratio, respectively. Subscripts of ‘f’ and ‘m’ stand for fiber and matrix, respectively.

Treated as an isotropic material and categorized as particulate composites, mechanical properties of the core layer are also calculated using the below formulas [30]:

(13)E_{\text{core}}=\frac{{\left(V_{\text{s}}^{\text{core}}\right)}^{0.67}{E}_{\text{m}}}{1-{\left(V_{\text{s}}^{\text{core}}\right)}^{0.33}\left(1-{\displaystyle \frac{E_{\text{m}}}{E_{\text{s}}}}\right)}+\left(1-{\left(V_{\text{s}}^{\text{core}}\right)}^{0.67}\right)E_{\text{m}},

(14)G_{\text{core}}=\frac{{\left(V_{\text{s}}^{\text{core}}\right)}^{\text{0}\text{.67}}{G}_m}{1-{\left(V_{\text{s}}^{\text{core}}\right)}^{0.33}\left(1-{\displaystyle \frac{G_{\text{m}}}{G_{\text{s}}}}\right)}+\left(1-{\left(V_{\text{s}}^{\text{core}}\right)}^{0.67}\right)G_{\text{m}},

(15){\nu }_{\text{core}}=\frac{E_{\text{core}}}{2G_{\text{core}}}-1,

where V_{\mathrm{s}}^{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}} is sand volume fraction in core layer.

3.2 Estimating HTS and LTS

HTS and LTS stand for the ultimate strength of GFRP pipe in circumferential and axial directions, respectively. The netting method is employed for predicting HTS and LTS as a quick tool for the composite structures undergoing tensile loading [30]. In netting analysis, only the contribution of fiber is taken into account, and the role of the matrix is ignored. The tensile strength of hoop plies is computed as below:

(16)X_{\mathrm{H}\mathrm{H}}=2\frac{N_{\mathrm{s}}{S}_{\mathrm{f}}T}{B},

where S_f is tensile strength of filament, X_HH is the tensile strength of hoop plies along hoop direction per unit-width. The reflected coefficient of 2 in Eq. (16) is originated from this fact that each hoop layer in the pipe can be viewed as two Uni-Directional plies around the pipe: one is placed from 12 to 6 o’clock, and another is positioned from 6 to 12 o’clock. The hoop strength of cross plies is also calculated using the below formula:

(17)X_{\mathrm{H}\mathrm{C}}=2\left(2\frac{N_{\mathrm{s}}{S}_{\mathrm{f}}T}{B}\right){\mathrm{s}\mathrm{i}\mathrm{n}}^2\theta ,

where X_HC is the tensile strength of cross plies along hoop direction per unit-width. Since each cross ply consists of a pair of helical layers, an additional coefficient of 2 in Eq. (17) is considered compared to Eq. (16). Finally, the hoop tensile strength of the pipes is calculated using the below formula considering the contributions of the hoop and helical layers:

(18){HTS}_{\mathrm{n}\mathrm{e}\mathrm{t}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{g}}=V_{\mathrm{f}}\left(\sum _1^p{X}_{\mathrm{H}\mathrm{H}}+\sum _1^q{X}_{\mathrm{H}\mathrm{C}}\right),

where p and q stand for the total number of hoop plies and the total number of cross plies in the pipe structures, respectively.

The strength of helical plies along the axial direction is calculated using the below formula:

(19)X_{\mathrm{L}\mathrm{C}}=2\left(\frac{N_{\mathrm{s}}{S}_{\mathrm{f}}T}{B}\right){\mathrm{c}\mathrm{o}\mathrm{s}}^2\theta ,

where X_LC is the tensile strength of cross plies along axial direction per unit-width. For calculating the LTS of the pipes, only the contribution of cross layers is taken into account. Since netting analysis is employed, the contribution of hoop plies in LTS is ignored. Thus, we have:

(20){LTS}_{\mathrm{n}\mathrm{e}\mathrm{t}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{g}}=V_{\mathrm{f}}\left(\sum _1^q{X}_{\mathrm{L}\mathrm{C}}\right).

3.3 Estimating failure pressure

The maximum allowable internal pressure is known as failure pressure. For thin-walled GFRP pipes (i.e., t_r/D < 0.1), the induced hoop stress in a pipe subjected to internal pressure can be estimated using the below formula:

(21){\sigma }_{\mathrm{H}}\approx \frac{P_{\mathrm{N}}{D}_{\mathrm{i}}}{2t_{\mathrm{r}}},

where P_N, D_i, and t_r stand for nominal pressure, internal diameter, and thickness of the structural layers without liner, respectively. Substituting {\sigma }_{\mathrm{H}} with HTS in Eq. (21), the failure pressure can be indirectly estimated based on netting analysis:

(22)P_{\mathrm{F}}\approx \frac{2t_{\mathrm{r}}{\sigma }_{\mathrm{H}}^{\mathrm{f}\mathrm{a}\mathrm{i}\mathrm{l}\mathrm{u}\mathrm{r}\mathrm{e}}}{D_{\mathrm{i}}}=\frac{2{HTS}_{\mathrm{n}\mathrm{e}\mathrm{t}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{g}}}{D_{\mathrm{i}}},

where {\sigma }_{\mathrm{H}}^{\mathrm{f}\mathrm{a}\mathrm{i}\mathrm{l}\mathrm{u}\mathrm{r}\mathrm{e}} is failure hoop stress.

3.4 Pipe stiffness

Since pipes are subjected to various source of loadings and internal pressure, they should also withstand them. Pipe stiffness is defined as the resistance of the buried pipe against service loads during installation and/or operation. The circular cross-section of pipes is intended to experience ovality under external transverse loadings. The pipe stiffness governs the vertical deflection of the pipe as a structural property. Pipe stiffness plays a key factor in avoiding the negative side effect of pipe cross-section deformation from both hydraulic and structural integrity viewpoints.

The pipe stiffness is generally expressed using the below formula [20]:

(23)\frac{E\overline{I}}{D^3}={PS}_{}.

Considering GFRP pipes as layered structures, a very simple and efficient method in the context of solid mechanics is employed here to estimate the stiffness of GFRP using the reformed shape of Eq. (23) as below [20]:

(24)PS=\frac{{\displaystyle \sum _{i=1}^{lastply}{E}_{\mathrm{H}}^i{\overline{I}}^i}}{{(D_{\mathrm{i}}+2NA)}^3},

where \overline{I} is centroidal moment of inertia of the cross-sectional area of the wall per unit length. E_H and NA stand for hoop modulus and location of the neutral axis and calculated using the following relations [20]:

(25)E_{\mathrm{H}}=\frac{1}{{\displaystyle \frac{m^4}{E_{\mathrm{x}}}}+{\displaystyle \frac{n^4}{E_{\mathrm{y}}}}+{\displaystyle \frac{m^2{n}^2}{E_{\mathrm{s}}}}-{\displaystyle \frac{{2{\nu }_{\mathrm{x}}m}^2{n}^2}{E_{\mathrm{x}}}}},m=\mathrm{c}\mathrm{o}\mathrm{s}\theta ,n=\mathrm{s}\mathrm{i}\mathrm{n}\theta ,

(26)A=\frac{{\displaystyle \sum _{i=1}^{lastply}{E}_{\mathrm{H}}^i{t}^i{\overline{Y}}^i}}{{\displaystyle \sum _{i=1}^{lastply}{E}_{\mathrm{H}}^i{t}^i}},

(27){\overline{Y}}^i=\frac{t^i}{2}+\sum _{m=1}^{i-1}{t}^m,

where {\overline{Y}}^i denotes centroid of each layer. It can be realized from Eqs. (24)–(27) that the pipe stiffness also depends on the lay-up sequence of constructing layers.

4 Validation of computational tools

To validate abovementioned computational tools, various GFRP pipes are manufactured using reciprocal filament winding techniques. Then HTS, LTS, failure pressure, and pipe stiffness are measured using tensile test [31], split-disk experiment [32], hydrostatic test [33], and parallel plate loading [34], respectively. All four testing procedures are shown in Fig.2.

Mechanical properties of the utilized fiber and resin are inserted in Tab.1 based on the technical data sheet of utilized materials in this study. The volume fraction of constituents is also measured through procedure G of ASTM D3171 [35], and obtained values are used as input data in corresponding equations (i.e., Eqs. (3), (9)–(14), (18), (20)). The bandwidth of the fiber bundle was adjusted at 180 mm, containing 42 strands.

It is formidable to measure the thickness of cross, hoop, and core plies separately. Thus, the whole thicknesses of all pipes are measured and compared with the summation of all thicknesses, and a perfect agreement was observed for the thickness comparison. Finally, the computational tools are executed, and the results of estimations are compared with experimentally measured values in Tab.2–Tab.4.

As shown in Tab.2, the results of estimating LTS are more accurate than those of HTS values, since the measurement of LTS is done on a flat specimen rather than ring specimen used for HTS. Generally, the error percentage falls below 15% for all cases that are acceptable for this study. In all cases, computational modeling underestimates both HTS and LTS, and thus the proposed method is categorized as a conservative method.

The results of Tab.3 imply the proper accuracy of the solid mechanic method for predicting stiffness. In all cases, estimated stiffness are lower than that of experimental measurement. This can be considered an advantage of the proposed method, avoiding unreliable design schemes through the optimization process.

It is evident from the comparison of the results in Tab.4 that the proposed method for calculating failure pressure has good accuracy. But the estimated failure pressure is overestimated. The experimental failure pressure is associated with leakage pressure, while the proposed indirect netting analysis predicts the complete rupture. Because the proposed method is constructed based on estimating HTS, and the HTS is also associated with the last-ply-failure. Thus, the overestimation in the result is rational. Since the method is very fast in obtaining the results, it can be accepted as a reasonable compromise for optimization. Moreover, for optimization, the input pressure entered by a user as nominal pressure would be automatically enhanced, accounting for long-term behavior considerations, and thus this shortcoming would be overcome. This strategy is explained in the preceding section when long-term behaviors are also considered design constraints.

5 Optimization framework

Three nominal parameters classify GFRP pipes at the design stages as diameter (DN) in millimeter, pressure (PN) in bar and stiffness (SN) in Pascal. Therefore, the objective is to find an optimal structural design for a GFRP pipe with arbitrary {DN-PN-SN} provided by client. The minimum requirements of HTS and LTS are reflected in normative standards based on the nominal DN and PN of the pipe [28]. The pipe stiffness should also be equal to or greater than nominal SN.

Recalling from Section 3, simple micromechanical rules are used to calculate the pipe layers’ mechanical properties. To estimate design constraints, netting analysis is utilized to estimate HTS, LTS, and failure pressure, and simple solid mechanic theory is utilized for estimating pipe stiffness.

Each GFRP pipe should withstand the hydrostatic test with twice the nominal pressure for a duration of 30 seconds as a quality control test [28]. Therefore, the failure pressure should be more than twice the nominal pressure by default. On the other hand, GFRP pipes for application in water and waste water are required to remain in operation for 50 years as the long-term consideration according to the international rules and regulations [28]. Both remaining pipe stiffness and failure pressure after 50 years should not fall below the nominal values. For this reason, a series of long-term experiments are required to be conducted for each type of pipe for a duration of 10000 hours. The obtained results are extrapolated to 50 years, and the remaining pressure/stiffness is estimated [36,37]. In this research, another constraint is proposed to cover both short-term and long-term considerations for the failure pressure. Namely, the nominal pressure entered by the user is enhanced by some correction factors. This strategy can also overcome the shortcoming of indirect netting analysis, where failure pressure was overestimated in Section 5. For long-term stiffness, a similar strategy is considered for the sake of simplicity. All the constraints considered in the optimization process of this research are presented in Tab.5, accompanied by objective and design variables. Thus, the optimization problem is outlined in Tab.5.

Reflected manufacturing constraints in Tab.5 resemble the limitations of the reciprocal filament winding technology utilized in this study, and they are obtained through the practical experience of production. Aforementioned values are subjected to change for other machinery, and thus, these values can be customized. Given the minimum wall thickness as 6 mm originated from the practical experience of manufacturing, GFRP pipes with a thickness of less than 6 mm are highly susceptible to experience buckling either during the hydrostatic test or during extracting the mandrel. Although the occurrence of buckling during mandrel extraction can be avoided using collapsible mandrels, the former persists.

Rafiee and Ghorbanhosseini have done experimental and theoretical studies on long-term stiffness and found that initial stiffness is reduced by about 30% after 50 years [38–40]. Thus, C_SL is considered as 1.25 in this research. C_PL is also assumed as 1.8 based on the simulation performed for the long-term hydrostatic test [41,42]. It is worth mentioning that C_SL and C_PL are required to be obtained through experimental tests for each pipe, and suggested values in this research are provided to conduct an optimization process without loss of generality. Explicitly stated in normative standards, long-term experiments as qualification tests should be conducted to approve the structural design of GFRP pipes.

An in-house computer code is written on the Visual Basic platform, where all computational procedures are included. The code requires entering {DN-PN-SN} by the user, and then it starts generating possible structural designs fulfilling the requirements of all constraints. Then direct search method is utilized to find the optimal design(s) based on the defined objective. The direct search method is chosen as a simple tool among various gradient-free optimization methods [43–46]. Therefore, the efficiency of the optimization method is not required to be discussed, since all possible alternatives are generated and evaluated.

Users can adjust all input data, including mechanical properties and the ranges of winding angle, fiber volume fraction in GFRP layers, and sand volume fraction in the core layer. The code has also powered with this feature to design and optimize GFRP pipes or GFRP mortar pipes (containing core layer). The optimization framework is limited to the case of composite pipes for water or waste-water application according to the identified design constraints. As the main assumptions of the modeling, thin-walled structures, and linear behaviors are assumed.

In this research, topology optimization [47,48] is not utilized, since topology optimization results might be challenging to manufacture. Thus, topology optimization is not used in this research, assuring the manufacturability of the optimal design as the output of this research.

6 Results and discussion

In this section, the code is implemented for various case studies, and the results are presented. For each case, the user can define the starting values for the hoop number and cross layers (i.e., p and q) as the starting point based on the design background referred to as the knowledge of expert (KOE). Usually, this information is practically available based on either previous experience of a designer or old design configurations. If such information is not accessible, the user can adjust starting values for the number of hoop and cross layers to 0 and 1, respectively, as the minimum possible number of layers. In other words, the code can find the optimal design even without starting point of search (i.e., KOE), but the runtime significantly decreases employing KOE. To reduce the runtime without KOE, a meta-heuristic optimization technique must be utilized, which is the subject of another research on composite pipes by the first author and his co-workers.

The optimization procedure is first executed for simple cases where some design parameters are unchanged, resembling restricted optimizations. Then, the optimization problem smoothly switches to more complicated cases where no restriction is pertinent to the design variables. This strategy has been taken to trace the outputs of the optimization for verification. Different implemented optimization scenarios are shown in Fig.3. As is shown, scenarios “A” to “D” belong to GFRP pipes, and scenarios “E” and “F” are related to GFRP mortar pipes. Moreover, all scenarios except scenarios “D” and “F” are carried out based on KOE. Consequently, the restrictions are removed in a step-by-step process through the scenarios “B” and “C” and finally, scenario “D” is implemented without KOE for GFRP pipes.

Scenarios “A” to “D” are implemented on a {600-16-1000} GFRP pipe as a case study. As an additional case study, scenario “D” is also implemented on a {800-6-2500}. Scenarios “E” and “F” are executed for the GFRP mortar pipe with {600-16-4000} specification. The detailed outputs of all scenarios accompanied by initial settings are presented in Tab.6. Scenario “A” is implemented based on available KOE, and the obtained results are extensively discussed in the proceeding parts. The required runtimes for the accomplishment of each optimization scenario are also reported in Tab.6. All cases were executed on a computer equipped with 16-GB memory and a Core i7-1065G7 processor running at 1.5 GHz frequency.

6.1 Scenarios “A” and “B”

For implementing scenario “A”, the total number of layers is chosen as 7 layers, and the weight fraction of fiber is assumed as 73%. Winding angle for cross layers is also selected as 60° based on available KOE. For scenario “A”, just 23 outputs among the 128 alternatives of design patterns fulfill the design and manufacturing constraints requirements. The generated design patterns fall into three main categories with three distinct overall thicknesses based on the combinations of the number of hoop and cross layers. As mentioned earlier, the lay-up sequence does not change the HTS, LTS, and failure pressure. Therefore, generally, three distinct values are presented for the aforementioned categories.

In contrast, pipe stiffness (SN) depends on the lay-up sequence, and thus three different ranges of estimated stiffness are presented in Fig.4. As expected, increasing the pipe thickness, the pipe’s stiffness also increases, and the increasing the number of hoop and/or cross layers, HTS also increases. For scenario “A”, the optimal designs with the thickness of 6.9 mm belong to the combination of two hoops and five cross layers. Among these combinations, the highest stiffness is achieved by the lay-up sequence of [90/±60₅/90] as 1919 Pa.

For scenario “B”, the winding angle is also added to the list of design variables and the other design variables assumed in scenario “A”. Different values for winding angles are reflected in Tab.5. For scenario “B”, 261 outputs among the 1152 alternatives of design patterns satisfy the design and manufacturing constraint requirements.

The green ovals in Fig.5 imply the optimal design for scenario “B”. Recalling from scenario “A”, the minimum thickness is obtained as 6.9 mm, while the minimum thickness in scenario “B” is 6.5 mm. Unlike scenario “A”, the combination of four cross and three hoop layers is resulted in the minimum thickness in scenario “B”. The optimal design of the B-scenario is associated with a winding angle of 57.5°. In B-scenario, the best winding angle in the range of 50° to 70° is associated with 57.5° based on all design constraints for the fiber weight fraction of 73%. It is very well known that by increasing the winding angle, the thickness of cross layers decreases according to the Eqs. (5) and (6).

As shown in Fig.5, for the minimum thickness of 6.5 mm, 16 different classes of pipe stiffness are obtained because of various admissible sequences of three hoops and four cross layers. The optimal design of scenario “B” consists of three hoops and four cross plies, while the optimal design of scenario “A” comprises two hoop and five cross layers. This stems from utilizing the optimal winding angle of 57.5° in scenario “B” compared to the winding angle of 60° in scenario “A”. The pipe stiffness for the optimal designs of the B-scenario varies from 1251 to 1775 Pa. Considering the maximum stiffness as the supplementary objective than the main objective of minimizing thickness, the best lay-up configuration would be [90/±57.5₄/90₂].

6.2 Scenarios “C” and “D”

For the case of C-scenario, just the total number of layers is kept fixed and other design variables are varied according to Tab.5. The influence of fiber weight fraction on the optimum winding angle is elaborated through this scenario for a GFRP pipe with 7 layers. The requirements of defined design constraints are met by 575 alternatives out of 5760 possible permutations. The thickness of the optimal designs as the outputs of C-scenario optimization is 6.1 mm. Different lay-up sequences are permissible for this minimum thickness, and thus nine different values for stiffness are identified. In all of these optimal designs circled in Fig.6, both HTS and LTS have the same values as highlighted in green, while stiffness values are different. In Fig.6, a part of outputs is shown avoiding messy illustration. The same as the outputs of the B-scenario, the combinations of four cross layers with a winding angle of 57.5° and three hoop layers lead to the optimal designs of the C-scenario. The pipe stiffness for the optimal designs of the C-scenario varies from 1309 to 1503 Pa. The latter is associated with the lay-up pattern of [90/±57.5₄/90₂], like the output of the scenario “B”.

In D-scenario, all restrictions are lifted, and the optimal design among all possible design patterns is intended to be found. Considering all reflected constraints in Tab.5, 2230 design patterns are acceptable out of generated 22960 cases. In this scenario, the optimization procedure starts from the total layers of 2 and continues up to 8 layers. The optimal design for the GFRP pipe with {600-10-1000} specification through scenario “D” converges to the same optimal design reported by Scenario “C”. Thus, the optimal design for this specific pipe {600-16-1000} is a 7-layer GFRP pipe (three hoop and four cross layers with winding angle of 57.5°) with fiber weight fraction of 77%. Some design patterns are acceptable for 6-layer GFRP pipes, but the thickness is higher than that of 7-layer pipes. The minimum thickness is associated with lay-up [±62.5₅/90] and fiber weight fraction of 75%. Considering the main objective of the current optimization scenarios, i.e., minimizing the thickness, that 7-layer pipe is the optimal design. But, the 6-layer design configuration is also appealing because of less required cycle time for the production process of a 6-layer pipe than a 7-layer pipe.

The thickness variation trend in terms of fiber weight fraction and winding angle are shown in Fig.7 as the outputs of optimization scenarios “C” and “D” for both optimum groups associated with 6- and 7-layer GFRP pipes with {600-16-1000} specification. It is worth mentioning that the presented data in Fig.7 are associated with the minimum thickness avoiding messy figure, since different lay-up sequences are permissible for each combination of winding angle and fiber weight fraction. As can be seen in Fig.7, for the fiber weight fraction of 73%–75%, the minimum thickness is associated with the winding angle of 62.5° for a 6-layer GFRP pipe, but for the fiber weight fraction of 76% and 77%, the optimal winding angle is adjusted to 57.5° and 55°, respectively. In contrast, for the 7-layer pipes, the minimum thickness occurs at the winding angle of 57.5° for all investigated fiber weight fractions, and similar trends are observed for them. This verifies the effect of winding angle and fiber weight fraction on the results.

Since the outputs of the optimization scenarios “C” and “D” coincide, the D-scenario is implemented on another GFRP pipe with {800-6-2500} specification to show the proficiency of the developed method without any KOE. It is interesting to observe that for this specific pipe, the obtained minimum thickness is the same for all fiber weight fractions, and five categories of the optimal design are found accordingly. Among the optimal designs of each category, those with maximum SN are inserted in Tab.6. The optimal designs for the GFRP pipe with {800-6-2500} contain 13 layers, and the different optimal winding angle is obtained in cross plies for each fiber weight fraction.

6.3 Scenarios “E” and “F”

After implementing scenarios “A” to “D” on a GFRP pipe, the optimization is switched to the case of a GFRP mortar pipe. For this purpose, a GFRP mortar pipe with {600-10-4000} specification is chosen. This specific pipe requires a core layer to fulfil stiffness requirement. Because 6 or 7 GFRP layers can fulfill the HTS, LTS, and failure pressure constraints as discussed through the D-scenario. But, the thickness of the pipe is required to be increased to enhance stiffness. The industrial producers incorporate a core layer in-between the GFRP layers to increase the thickness of the pipe as an economical solution. In other words, increasing the thickness of GFRP pipes by utilizing extra redundant GFRP layers would lead to considerably high cost, and thus, required thickness is compensated by incorporating a core layer. In other words, the outputs of optimization for the D-scenario fulfills all design constraints for the {600-10-4000} GFRP mortar pipe except stiffness, and hence adding a core layer can satisfy this requirement as well.

In the E-scenario of optimization, 75 kg of sand is used to generate possible design patterns. In the F-scenario, the mass of sand is also considered a design variable. Since the amount of sand is considered as a constant value in the scenario “F”, the thickness of the core layer is the same in all outputs of this scenario. The details of obtained results for all optimization mentioned above scenarios are presented in Tab.6. The optimal designs in the E-scenario and F-scenario are totally different from optimal design in the D-scenario, despite the fact that the total numbers of the GFRP layers and fiber weight fraction are the same. Although the amount of incorporated sand has increased from 75 kg to 85 kg in the optimum design of the scenario “F”, the thickness was reduced from 9.6 to 9.2 mm. This reduction is rooted in the optimal winding angle of 60° for this pipe.

It is very well known that accurate adjustment of fiber weight fraction is a formidable task in the reciprocal filament winding process of GFRP pipes from a practical point of view. The fiber weight fraction depends on various parameters like the applied tension to the fiber during the winding process, the wet ability of the fiber, TEX of the fiber, ambient temperature, the viscosity of resin and etc. On the other hand, a higher fiber weight fraction would always be led to the thinner GFRP pipes as a trivial output of Eqs. (1) and (5). Consequently, in those scenarios where fiber weight fraction is considered as a design variable (i.e., scenarios “C”, “D”, “E”, and “F”), the optimal designs are associated with the maximum possible fiber weight fraction. It is emphasized that the code generates all possible cases for the whole range of defined fiber weight fractions, but just the optimum values are presented in Tab.6.

7 Conclusions

Intending to attain the minimum achievable price, this research aims to develop an integrated design and optimization process to obtain the optimal customized structural design for GFRP pipes and GFRP mortar pipes. The application of investigated GFRP pipes is considered for conveying water and wastewater. The design constraints regulated by normative standards from short-term and long-term service viewpoints are identified. Then, proper computational tools are developed to predict the GFRP pipes’ structural properties based on the lay-up configurations. The developed computational tools are validated through extensive experimental study, assuring the optimization foundation’s proper performance. The developed computational procedures are quick and trustworthy enough since the optimization’s runtime isinfluenced by these fundamental calculations. Then, the optimization framework is constructed by defining the minimum price as the primary objective and design requirements, and filament winding limitations as constraints. Design variables consist of the number of layers, lay-up sequence, fiber weight fraction, and winding angle for GFRP pipes, while for GFRP mortar pipe the amount of incorporated sand in the core layer is added to the list of variables as mentioned above, too. A computer code is written integrating design and optimization procedures. The optimization process is implemented through different scenarios to verify its proper performance. Firstly, pre-defined values are assumed for some design variables relying on the available KOE, and then involved design variables are added step-by-step to the optimization process untill no KOE is utilized.

The computer code needs pipe diameter, pressure class, and stiffness class as input data by the user and then optimal design(s) are presented based on the mechanical properties of fiber and resin. It can be applied for any new design even without background information on the structural pipe design in the form of KOE; however, the required runtime of analysis increases significantly. This becomes even more pronounced when the number of layers increases in large diameter GFRP pipes and GFRP mortar pipes. Therefore, it is necessary to employ a meta-heuristic optimization technique without using KOE.

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