Stacking sequence optimization in composite tubes under internalpressure based on genetic algorithm accounting for progressivedamageAlmeida Jr, J. H. S., Ribeiro, M. L., Tita, V., & Amico, S. C. (2017). Stacking sequence optimization in compositetubes under internal pressure based on genetic algorithm accounting for progressive damage. CompositeStructures, 178, 20-26. https://doi.org/10.1016/j.compstruct.2017.07.054

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Download date:01. Oct. 2021

1

STACKING SEQUENCE OPTIMIZATION IN COMPOSITE TUBES UNDER INTERNAL

PRESSURE BASED ON GENETIC ALGORITHM ACCOUNTING FOR PROGRESSIVE

DAMAGE

José Humberto S. Almeida Jr.a*, Marcelo L. Ribeirob, Volnei Titab, Sandro C. Amicoc

aLeibniz-Institut für Polymerforschung Dresden e.V., Department of Composite Materials,

Hohestraße 6, 01067 Dresden, Germany

bDepartment of Aeronautical Engineering, São Carlos School of Engineering, University of

São Paulo, São Carlos/SP, Brazil

cPPGE3M, Federal University of Rio Grande do Sul, Av. Bento Gonçalves, 9500. 91501-970

Porto Alegre/RS, Brazil

___________________________________________________________________________

Abstract

Due to the large number of design variables for laminate composite structures, the use of an

optimum stacking sequence is a key step in the design of a structure with the most suitable

mechanical properties. This work presents a genetic algorithm (GA) for the optimization of

the stacking sequence to improve strength of a cylindrical shell under internal pressure. The

GA, which is associated to a meso-scale damage model, was written in Fortran and later

linked to a Finite Element (FE) package to simulate composite damage and failure. Two

scenarios were considered: i) without restriction, where an ideal situation is simulated; and ii)

with manufacturing restrictions, accounting for limitations on feasible winding angles. The

results show that progressive failure analysis generates asymmetric and unbalanced laminates

*Corresponding author: humberto@ipfdd.de; jhsajunior@globomail.com.

Phone: +49 351 4658 1423; Fax: +49 351 4658 362

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in both cases. Furthermore, the simulations with manufacturing restrictions present internal

pressure strengths lower than the idealized case, providing more realistic results.

Keywords: optimization; genetic algorithm; composite shell; progressive failure; finite

element method; filament winding.

___________________________________________________________________________

1. Introduction

The advent of high-performance carbon fibers has enabled the evolution of composite

tubes for structural applications. These structures are typically manufactured by filament

winding (FW) and have advantages compared to metallic-based conventional materials, such

as high stiffness- and strength-to-weight ratios, allied to high corrosion strength [1]. For an

efficient design of composite tubes for a particular application, avoiding extra costs with re-

manufacturing and testing, appropriate knowledge of the effect of laminate layup on the

required mechanical properties is essential.

Even though numerical analyses are a useful tool in the design of filament wound

structures, most decisions are based on processing experience and experiments. That is, most

parts are not optimized for optimum ply-orientation considering progressive damage/failure

phenomenon [2]. Indeed, efficient stacking sequence optimization algorithm associated with

progressive failure of the laminas is not yet well established and most contributions in the

literature have used first ply failure criterion to optimize filament wound structures.

Optimization of the winding sequence for a composite tube is a challenge because the

optimal composite structure depends on the physical and mechanical performance of the

materials, shape, and manufacturing constraints, therefore generally requires computer-

assisted design tools [3]. The possibility of easily winding the continuous fibers to match the

particular stress needs and boundary conditions in a composite structure for optimized

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performance gives an extra advantage in the use of composite materials compared to

traditional ones, such as steel or aluminum [4].

Popular algorithms for the optimization of filament wound composite tubes and

pressure vessels include genetic algorithm (GA) [5], simulated annealing [6] and artificial

immune system [7]. Optimization usually focuses on the improvement of strength/weight

ratio, reliability and lifetime of a composite shell by designing optimum winding angles and

thicknesses, as in the case of the current study. A GA is a search algorithm based on

mechanics of natural selection and genetics which simulates natural evolution so that multiple

design points evolve to converge to a global optimum solution [8]. In every new generation, a

new set of artificial strings is created using pieces from the fittest of the old generation, based

on a non-deterministic scheme. A key advantage of this method is that it uses discrete design

variables by nature [9].

Several studies have dealt with optimization in composite laminates. For instance,

Rahul et al. [10] used GA to optimize the weight of graphite/aramid/epoxy laminates. Jing et

al. [11] optimized the stacking sequence for composite laminates using a multicriteria

objective function with respect to the critical buckling. Irisarri et al. [12] applied a

multiobjective stacking sequence optimization for composite plates and concluded that non-

conventional ply orientations may lead to improved optimal designs compared to classical

angles (0°, ±45° and 90°). Zu et al. [13] applied GA to find the best non-geodesic trajectory

for filament wound toroidal pressure vessels. Francescato et al. [14] used GA to study

composite pressure vessels overwrapping a metallic liner (type III COPV) under internal

pressure and concluded that optimal design is mainly dependent on the calculation strategy,

particularly considering first-ply or last-ply failure design objective. However, these authors

did not focus on optimum stacking sequence, which is the main motivation and contribution

of the current study.

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Studies dealing with optimization procedures for filament wound structures evaluating

fiber path predictions, structural analyses, weight reduction and design procedures are not

easily found, especially accounting for progressive damage/failure evolution. Thus, the scope

of this study is to optimize the stacking sequence of filament wound composite cylindrical

shell under internal pressure using a genetic algorithm connected to a mesoscale damage

finite element model accounting for progressive failure. Simulations with and without

manufacturing restrictions of feasible winding angles were also carried out.

2. Finite element model

Structural modeling was performed based on finite element (FE), using Abaqus™ 6.14

commercial software platform. Non-linear geometry was considered in all cases since large

displacements and strains can take place during the simulations. The composite tubes/

cylinders (381 mm length and 136 mm diameter) were modeled using an equivalent single

layer (ESL) four node reduced integration shell element (S4R) with hourglass control (Figure

1). This element was chosen in order to reduce simulation time and avoid numerical issues.

The composite layers were modeled as conventional shell with Simpson’s thickness

integration rule and three integrations points in each of the ten layers.

Internal pressure was applied on the inner surface of the shell and on the inner surface

of the flanges located at the ends of the structure (Figure 1) and axial displacements of both

flanges were restricted. As the damage model applied is very step-size-dependent, a

convergence study was performed and, after mesh sensibility study, a typical mesh with 1,672

elements and 1,716 nodes was selected. A maximum increment size of 5% for applied internal

pressure was used throughout the simulation.

In order to predict material failure, the proposed damage model was compiled as a

UMAT (User Material Subroutine) and linked to Abaqus™. The material properties of the

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carbon/epoxy laminates were experimentally measured and reported in a previous study [15]

(Table 1). The damage model parameters used may be found in [16,17].

Figure 1. Applied loading and boundary conditions, typical mesh and FE element used.

Table 1. Representative material properties used as input in the numerical models [15].

Symbol Description Value

Ela

stic

pro

per

ties

E1 (GPa) Longitudinal elastic modulus 129.3

E2 = E3 (GPa) Transversal elastic modulus 9.11

= Poisson’s ratio in plane 1-2 0.32

Poisson’s ratio in plane 2-3 0.35

G12 = G13 (GPa) In-plane shear modulus 5.44

G23 (GPa) Transverse shear modulus in plane 2-3 2.10

Str

ength

s

Xt (MPa) Longitudinal tensile strength 1409.9

Yt (MPa) Transverse tensile strength 42.5

Xc (MPa) Longitudinal compressive strength −740.0

Yc (MPa) Transverse compressive strength −140.3

Spl (MPa) In-plane shear strength 68.9

2.1 Damage model

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In order to evaluate progressive failure of the tubes, a damage model based on the

work of Ribeiro et al. [18] was applied. The model regards the composite lamina as under

plane stress state and damage is considered uniform throughout the laminate thickness [19].

Regarding fiber failure modeling, a unidirectional carbon/epoxy laminate under tensile

loading in the fiber direction (𝜎11) is considered linear elastic with brittle fracture. The model

assumes that fiber behavior is not influenced by the damage state of the matrix and, for tensile

loading in the fiber direction, the maximum stress criterion is used to identify fiber failure. On

the other hand, under compressive longitudinal loading, fiber behavior is considered linear

elastic until a specified value is reached (𝑋𝐶0), and non-linear elastic after that. To model the

damage process in the fiber, an internal damage variable 𝑑1 is used.

Concerning matrix modeling, the damage process in the matrix of a unidirectional

quasi-flat filament wound laminate is essentially dominated by transverse loading (𝜎22) and

shear loading (𝜏12), and non-linear behavior is reported due to inelastic strains and damage

[20]. Thus, two internal matrix damage variables were used, 𝑑2 (related to 𝜎22) and 𝑑6

(related to 𝜏12). Based on Continuous Damage Mechanics (CDM), the hypothesis of effective

stress links the damage variables to the stresses [19]. Table 2 summarizes the constitutive law

for each type of failure in the damage model herein presented.

A full description of this model can be found in [18], where the authors used

traditional flat composite coupons to identify the required damage parameters through well-

known mechanical tests. This model has also been successfully applied to flat filament wound

laminates under 3-point bending [17] and to carbon/epoxy filament wound composite tubes

under hydrostatic external pressure [21] or radial compression [22].

The damage model is very effective in predicting progressive failure when loading

involves essentially in-plane stresses, since it intrinsically considers these stresses. Also, it

requires low computational cost and simple tests to identify the parameters, making the model

7

very attractive. As a drawback, if out-of-plane shear stresses are significant and the dominant

failure is driven by delaminations, the model might not predict failure well.

Table 2. Summary of the constitutive laws for each type of failure in the damage model used.

Failure type Failure criteria Degradation law

Fiber tension 𝜎11𝑋𝑡

≤ 1 𝐸11 = 0

Fiber compression |𝜎11|

𝑋𝐶0≤ 1 𝐸11 =

𝑋𝐶0|𝜀11|

(1 − ℎ(𝜀11)) + ℎ(𝜀11)𝐸110

Matrix tension 𝑓 > 0 𝑑2 = 𝐴(𝜃)𝑌2 + 𝐵(𝜃)

Matrix compression 𝑓 > 0 𝐸22 =𝜎22𝑦|𝜀22|

(1 − 𝑓(𝜀22)) + 𝑓(𝜀22)𝐸220

Shear 𝑓 > 0 𝑑6 = 𝐶(𝜃)𝑌6 + 𝐷(𝜃)

In Table 2, ℎ(𝜀11) is obtained from the fitting of stress-strain plots for 0° specimens

under compressive loading, and 𝑓 is defined as shown in Eq. (1).

𝑓 = √𝜎222 + 𝜏12

2 − (−𝑆12𝑦 +2𝑆12𝑦

1+(|𝜎22|

𝜎220⁄ )

3) (1)

3. Optimization process via genetic algorithm

Genetic algorithms use the evolution theory concept to search the global optimal [9].

Each possible solution of a problem is considered an “individual” with an encoded

“chromosome”. This codification is obtained by a “genes” group that keeps individual’s

characteristic and there is a particular value for each “Individual” associated with its problem

solution potentiality, which is usually known as “fitness”. A method based on binary numbers

allows crossing individuals represented by a number that saves its genes characteristics.

8

To start the optimization process, an initial “Population” of individuals is randomly

generated. This generation can decode its chromosome and apply its fitness function after new

individuals are generated by combining individuals of the previous population. This process

can be divided into “Individual” selection, “Crossover” and “Mutation”. The first one is

related with method and criteria, which are used to choose individuals. The crossover creates

new individuals by changing genes with selected individuals, whereas mutation ensures that,

with low probability, a few genes are modified and a new search space can be explored,

thereby increasing the chance of achieving a global optimum. The process is repeated until a

new complete population is established, finishing a generation. The algorithm is further

iterated if a termination criterion is not satisfied. The identification process is based on the

conventional GA as described by Goldberg [9] and implemented in MatLab software®. The

population is randomly generated and, at each generation, reproduction continues until the

size of this population is doubled. By sorting the new doubled population, the scheme

discards the half with the worst fitness values.

3.1 Angle layer optimization

The GA works creating one angle population at random, based on the angle range of

interest. These angles need to be qualified according to how it may be more able than others,

to achieve the project objective. When this is carried out by the FE model, population

crossing can produce a new generation, which would be again qualified by the FE model, and

this process repeats until the best generation is found, as shows the flowchart in Figure 2.

After each crossing, the algorithm makes elitism pre-definition, comparing the new

generation with the previous one, and selecting the best members to compose the next

generation to be crossed. For the genetic algorithm, mutation probability is 1% and the

crossover probability is 100%.

9

Regarding the optimization scheme presented in Figure 2, two routines were

developed separately: (i) the FE routine is represented by the steps described within the green

dashed square box; and ii) the optimization Routine, represented by the steps within the blue

dashed square box. Regarding FE Routine (i), the first step is to develop a .CAE FE model,

which is generated by a Python script. After that, processing of the FE model is carried out

by solver of Abaqus, exporting an initial stacking sequence as a .TXT file. Then, the

optimization process (ii) begins with the definition of the fitness function, which establishes

the basis for selection of pairs of individuals that will be mated during reproduction. GA

optimization is then defined by maximizing the fitness function defined, where a new set of

approximation by selecting individuals respecting the level of the fitness at each new

generation. This process leads to the evolution of population of individuals, and then the

results exported as a .CSV file.

Figure 2. Flowchart of the optimization process.

Optimization has been performed with and without manufacturing restrictions. In the

FW process, the machine needs a minimum length to contour the turnaround zones, in which

10

both winding angle and thickness vary to allow the pay-out eye to turn to the opposite

longitudinal direction in the programmed symmetric angle (+φ or −φ). In the current study,

the mandrel is 420 mm long and, for the friction factors used, the equipment is not able to

wind within −20° and +20° due to the limited turnaround length available.

In an idealized scenario, if the FW machine was capable of winding any angle, the

simulation may run without restriction. Although this scenario is not possible for the actual

mandrel, it was also studied since alternative strategies, such as extended mandrel and/or

insert pins at its extremities, might make lower angles possible. The first scenario is expected

to produce structures with higher internal pressure strength, as an upper bound, whereas the

second scenario should produce more realistic results in terms of internal pressure strengths.

The first 20 layups were randomly chosen for both scenarios. After several

simulations, the results for 30 generations with 20 individuals each converged, with no further

increase in maximum internal pressure with the increase in generation number. In all, 600

finite element simulations were carried out in each case.

4. Results and discussion

Figure 3(a) shows the results of the optimization process without manufacturing

restrictions. After 30 generations with 20 individuals each, the improvement in maximum

internal pressure strength is of 14.4 MPa (from 31.46 MPa to 45.85 MPa). In fact, this value

was reached after 8 generations only. Also, average and minimum fitness display a significant

improvement, converging to the maximum pressure at higher number of generations.

Moreover, even though strength does not increase further, there are many layups able to

withstand that maximum applied internal pressure.

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Figure 3. Fitness evolution for simulations without layup restriction with a randomly chosen

initial angle (a), and with a pre-defined initial angle of 0° (b) or 45° (c).

Table 3 shows the first generation layup (randomly chosen) and the 30th generation.

The optimization process yields an asymmetric layup, and several of the winding angles

shown are not feasible due to the mandrel restrictions. Furthermore, if a particular layer is

forced into the initial population, the generation procedure takes less time to reach the

maximum internal pressure supported by the laminate, as shown in Figure 3(b) for an initial

angle of 0o. Nevertheless, similar pressure levels were reached compared to the previous

random simulation (Figure 3(a)). In Figure 3(c), the maximum internal pressure is achieved in

the 16th generation, later than the other two cases (as shown in figures 3(a) and (b)), but the

pressure level achieved is similar to the other cases.

Table 3. Optimum stacking sequence for the 1st and 30th generations for the simulations

without layup restriction and randomly chosen initial angle.

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Simulations with a higher number of generations with the initial angle randomly

chosen have also been performed for comparison (Figure 4). Although the obtained stacking

sequence is quite similar to the one shown in Table 3, the results from the simulations with

more generations provided better results, with higher pressure levels. This optimum stacking

sequence is shown in Table 3.

It is clear that an asymmetric winding sequence yielded the best results considering

progressive failure analyses, that is, a symmetric laminate is not the best option for composite

tubes under internal pressure based on last ply failure approach. Nevertheless, most

publications report that the optimum winding angle for composite shells under hydrostatic

internal pressure is symmetric ±55° (considering first ply failure approach). In such case, the

current results are quite relevant, and aid in a paradigm break for the design of composite

tubes under internal pressure, where unusual angles and non-symmetric laminates are found.

Figure 4. Fitness evolution for simulations without layup restriction with a randomly chosen

initial angle (50 generations, 20 individuals each).

Layer 1 Layer 2 Layer 3 Layer 4 Layer 5 Layer 6 Layer 7 Layer 8 Layer 9 Layer 10

Gen. 1 −36.8º −76.5º 9.9º −86.5º −5.7º −14.2º 41.1º −17.0º 60.9º −19.8º

Gen. 30 −86.5º 14.2º 34.0º −72.3º −68.0º −7.1º 65.2º −19.8º 65.2º 35.4º

13

Indeed, in a last-ply failure approach, after the first ply fails, unbalancing of the

laminate appears, and a symmetric laminate loses its symmetry. Hence, even though the

laminate is originally symmetric, it becomes asymmetric as the failure process initiates. These

results can be related to those of Irisarri et al [12], who found that non-conventional ply

angles for composite plates under compression and shear loadings behave better than

laminates using 0°, ±45° and 90° angles. Another consequence of the unbalancing of the

laminate is that the coupling stiffness [𝑩] matrix begins to have nonzero elements. Likewise,

the 𝐴16 and 𝐴26 elements from the [𝑨] matrix are not null anymore.

Regarding the results for simulations performed with mandrel restrictions, i.e. for a

more realistic scenario, the optimization process yields the results presented in Figure 5 for a

fitness evolution over 30 generations. This optimization uses the same conditions applied for

the first case (30 generations with 20 individuals each, and first 20 individuals randomly

chosen).

The resulting optimum layup is quite different from the previous ones. Nevertheless,

asymmetric laminates are also found, corroborating the previous results. In such asymmetric

and unbalanced laminates, even more unique couplings may arise and even result in fully

anisotropic behavior in that bending, stretching, shearing and twisting responses are fully

coupled. In the laminates herein studied, they exhibit coupling between mechanical

deformations generated by in-plane shear loads and moments, along with in-plane shear

strains and curvatures inherent to cylindrical structures [23].

As the initial layup was randomly chosen, maximum pressure (38.35 MPa) is already

achieved at the first generation (Figure 5). On the other hand, average and minimum fitness

increase over the generations.

14

Figure 5. Fitness evolution for simulations with layup restriction and randomly chosen initial

angle (30 generations, 20 individuals each).

Table 4 presents then the layup for first and last generation. As well as for the case

without restrictions, there is no single layup for the cylinder that would lead to maximum

internal pressure. Indeed, the layup may differ significantly from one generation to the next

and the cylinder would still support that pressure level. Here again, optimization lead to an

asymmetric laminate. Finally, without manufacturing limitations, the maximum internal

pressure is around 7.5 MPa higher than those with manufacturing restrictions limitations

(45.85 MPa and 38.35 MPa, respectively), which means that processing parameters (in this

case, mandrel length, can have a significant effect on the predictions.

Table 4. Optimum stacking sequence for the 1st and 30th generations for the simulations with

layup restriction and randomly chosen initial angle.

Layer 1 Layer 2 Layer 3 Layer 4 Layer 5 Layer 6 Layer 7 Layer 8 Layer 9 Layer 10

Gen. 1 −48.7º −79.5º 28.3º −87.2º −24.4º −31.0º 52.5º −33.2º 68.0º −35.4º

Gen. 30 −87.2º 35.9º 47.0º 69.1º 32.7º −57.5º −36.5º −54.2º −49.8º −44.2º

15

5. Conclusions

An optimization process using a genetic algorithm was employed to find the optimum

winding angle for filament wound composite cylinders under internal pressure. The FE model

accounted for progressive failure considering, or not, manufacturing angle restrictions

(between −20° and +20°) due to the limited mandrel length. For the first scenario, without

manufacturing restrictions, the composite cylinders reached an internal pressure of up to

45.85 MPa. The simulations taking into account angle limitations and allowing the first 20

individuals to be randomly chosen quickly converged, and one of them reached the maximum

pressure in the first generation. In addition, the progressive failure analysis allowed the

optimization process to yield asymmetric and unbalanced laminates in both scenarios.

Therefore, strain-curvature coupling between layers is verified, and bending load acts on the

structure once the cylinder is pressurized.

Finally, based on the last ply failure approach adopted, this research aids in a paradigm

break considering that it suggests asymmetric and non-conventional angles as the best

solution for internally pressurized composite tubes, contrasting to the well-known ±55°

winding angle recommendation (for first ply failure approach).

Acknowledgements

The authors are grateful to CNPq and AEB for the financial support. Marcelo L.

Ribeiro would like to thank FAPESP (project 2015/13844-8) and Volnei Tita acknowledges

the financial support from CNPq (projects 401170/2014-4 and 310094/2015-1).

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