Jifan Zhong·Yaochen Zheng·Jianqiao Chen·Zhao Jing

Abstract The large design freedom of variable-stiffness(VS)composite material presupposes its potential for wide engineering application.Previous research indicates that the design of VS cylindrical structures helps to increase the buckling load as compared to quasi-isotropic(QI)cylindrical structures.This paper focuses on the anti-buckling performance of VS cylindrical structures under combined loads and the efficient optimization design method.Two kinds of conditions,bending moment and internal pressure, and bending moment and torque are considered. Influences of the geometrical defects, ovality, on the cylinder's performances are also investigated. To increase the computational efficiency, an adaptive Kriging meta-model is proposed to approximate the structural response of the cylinders. In this improved Kriging model, a mixed updating rule is used in constructing the meta-model.A genetic algorithm(GA)is implemented in the optimization design.The optimal results show that the buckling load of VS cylinders in all cases is greatly increased as compared with a QI cylinder.

Keywords Variable-stiffness composite·Optimal anti-buckling design·Combined loading·Ovality·Kriging meta-model

1 Introduction

Due to the excellent mechanical properties and lightweight feature, composites are widely used in different engineering fields,such as offshore platforms,pipelines in deep seas,pressure vessels in industry,rocket launchers and space vehicles, and aircraft cabins and wings. In order to take full advantage of composite structures,they can be continuously tailored and with variable thickness or fiber angles,forming a variable-stiffness (VS) composite structure. Recently, the design of VS composite structures has attracted much attention: a framework for design and optimization of tapered composite structure [1] is proposed with a tailoring technique-global shared-layer blending(GSLB)method[2]and thickness distribution[3,4].Manufacturing of VS composite materials is an important issue.Research has showed that automated fiber placement (AFP) technology can not only reduce the labor costs and scrap materials,but also improve the quality and performance of the VS structures[5-10].The resulting VS laminates from AFP machines provide large designability for designers to fully exploit the directional properties of compositematerials[11].As aresult,adesigned VS material can offer significantly better performances as compared with constant-stiffness (CS) counterparts at the same weight[12-16].

In 1994,Hyer et al.[17]optimized the fiber path of a flat plate by reducing the angle variation between adjacent elements.The laminated plate was fabricated and the buckling load was verified by experiments.In engineering,composite cylinders are widely used and, hence, are of importance to study its designability and mechanical properties.The design optimization of VS composite cylinders for buckling was first studied by Tatting[18].The optimization of maximum buckling load of VS composite cylinders subject to bending moment or axial compression was studied by Rouhi et al.[19,20].Although VS design of composites can greatly enlarge the design space,the exponentially increased complexity and computational costs should be faced at the same time. To reduce the calculation burden,meta-modeling techniques are often adopted.Active meta-models have been integrated into evolutionary algorithms for conducting optimal design of composites more efficiently,as per Chen et al.[21-24].For the optimization of a VS composite cylinder, a multi-step design optimization approach has been used for bendinginduced buckling design[25].In the multi-step optimization method,the bound(side constraints)of design variables are narrowed down around the optimum point found in the previous step.This method requires high accuracy of the model.Among different meta-model techniques,one of the popular meta-models is the Kriging model[26].Some updating criteria were listed by Han et al. [27] and applied in building the Kriging meta-model.

In this paper, an actively updating Kriging meta-modal method is proposed for designing VS composite cylinders.The expected improvement(EI)criterion and the minimum of surrogate prediction(MSP)criterion are mixed to update the Kriging meta-model.This way,the Kriging meta-model doesn't require good accuracy at the beginning. The metamodeling and optimization process is repeated iteratively until a stop criterion is satisfied.By utilizing the meta-model,the bending-induced buckling design of VS composite cylinders is carried out by a genetic algorithm(GA).It is known that the ovality of a composite cylinder is a defect and its effect on the mechanical performance is important [28].Accordingly,the effect of different composite cylinder ovalities is studied.On the other hand,as combined loadings are common in engineering, the bending-internal pressure,and the bending-torsion are considered respectively for the optimal design of VS composite cylinders.

Contribution to this paper is threefold:First,an improved Kriging model is proposed in optimizing VS cylinders in which a mixed-sample adding criterion is used. Then,influences of ovality on the performance of a cylinder are investigated.After that,optimization of VS cylinders under two kinds of combined loadings is studied.The structure of this work is as follows.Improved Kriging metal-modeling is described in Sect. 2. In Sect. 3, design of a VS cylinder is studied and different ovalities are considered.In Sect.4,two kinds of combined loadings are analyzed and the optimized ply angles are obtained by using the GA. Conclusions are provided in Sect.5.

2 Kriging meta-modeling and the updating criteria

To reduce the computation cost in performing a structural optimization, meta-models can be utilized. The Kriging model with a mixed updating rule is used in this work.Below is a description of this model.

2.1 Kriging model

Krige, a South African engineer, constructed Kriging for geostatistics,and Matheron improved this meta-model later.In this section, the Kriging model is briefly introduced in the framework of structural reliability problems.It assumes that the performance function of a structure consists of two parts,

where FT(β,x)is the deterministic part,holding the information of the whole trend of F(x),and h(x),a realization of stationary Gaussian process,is the random part.As the basis function, fT(x) is a scalar or multi-variable polynomial. β is the coefficient vector of F(x)estimated with generalized least squares. h(x) flows the Gaussian process with zeromean and constant variance. And the covariance between h(xi)and h(xj)is defined as:

where σ2and R(xi,xj)denote the variance and the correlation function of h(x),respectively.The Gaussian correlation function is adopted in this paper.For an n-dimensional problem,

where xi,lis the l-th component of xi, θlis the correlation parameter which is determined through the maximum likelihood estimation method,i.e.,

After transformation,θ can be determined by solving the following optimization problem:

Parameters of the Kriging model can be obtained by the least squares method,i.e.,

2.2 Updating the Kriging model

A perfect Kriging model construction is not attempted at the beginning since it requires a lot of samples to ensure its accuracy.Instead,strategies to update the model sequentially by adding samples are used [27]. A small number of samples are needed to establish the initial meta-model.Then,multiple new samples in terms of certain criteria can be utilized to update the meta-model. As new samples are added, the meta-model becomes more accurate.For the Kriging model optimization algorithm, a variety of updating criteria have been developed[27]which include MSP criterion,EI criterion,the probability of improvement(PI)criterion,the lower confidence bounding(LCB)criterion,the mean square error(MSE)criterion and so on.To improve the efficiency of the meta-model,the MSP criterion is coupled with the EI criterion in this work and named as the MSP+EI criterion.

For MSP, the target point is determined through the following computation:

The optimal solution x*and the true response value are added into the database to update the meta-model. However, the overall approximate accuracy is relatively poor and it is liable to fall into the local optimum.The EI criterion is also used to improve the Kriging model.

Each response predicted by the Kriging meta-model follows a normal distribution:Assume that yminis the minimum response among the current samples.A target function is defined as:

Its expectation can be expressed as:

φ(·) and φ(·) present respectively the cumulative distribution function and probability density function of the standard normal distribution.The target point is found through the following optimization:

Fig.1 Schematic of the 2D Rosenbrock function

By using the MSP criterion, the local improvement is obvious, but it is not suitable for complex functions. EI in efficient global optimization(EGO)is another efficient criterion to update the meta-model[29].This study combines the advantages of the two methods.By constructing the Kriging meta-model in each iteration, samples based on the MSP+EI criterion are obtained and added to the database to update the meta-model.

2.3 Verification of the mixed MSP+EI criterion

The benchmark Rosenbrock function and Rastrigin function are adopted to verify the effectiveness of the mixed MSP+EI criterion.The two-dimensional(2D)Rosenbrock function is shown in Fig.1.It can be expressed as:

The global optimal solution for this problem is fmin=f(1.0,1.0)=0,the optimum solution with the mixed MSP+EI criterion is fmin= f(1.000,1.000)=9.850 × 10-8.

Figure 2 shows the results of the minimum function value versus the number of iterations by using the Kriging model and the three criteria.The mixed MSP+EI criterion is obviously better than the other two.

The 2D Rastrigin function is a multi-peak function.Candidate solutions converge to local optimum easily. The 2D Rastrigin function is shown in Fig.3.It can be expressed as:

The global minimum for this problem is fmin=f(0.0,0.0)=0,the optimum solution with the mixed MSP+EI criterion is fmin= f(-0.487×10-6,-0.011×10-6)=4.710×10-11.

Fig.2 Comparison of the convergence process for the 2D Rosenbrock function based on the three criteria

Fig.3 Schematic of the 2D Rastrigin function

Fig.4 Comparison of the convergence process for the 2D Rastrigin function based on different criteria

Figure 4 indicates that Kriging model with MSP provided thelocalminimum.TheobtainedresultsusingtheEIcriterion are better than the MSP criterion and with faster convergence.When the mixed MSP+EI criterion is used,the convergence is much faster and the result is the best among the three criteria.Hence,the coupled MSP+EI criterion will be used in this work.

Fig.5 A composite plate with curved fiber path

Fig.6 Description of the monofilament curved fiber path

3 Optimization of composite structures with variable stiffness

3.1 Composite cylinders with variable stiffness

A common composite material cylinder is the filament winding composite cylinder in which fiber is wrapped in a certain winding angle, producing the uniform stiffness along the cylindrical surface. A technique called AFP can be used for manufacturing VS composite structures,leading to an increase of interest on their optimal design.The AFP technique changes the fiber laying of composite structures through the curve lay, thus greatly influencing the material's properties.The following pictures demonstrate how the curves lay in a lamina.

As shown in Fig.5,along the x direction,the fiber angle changes.And along the y direction,the ply orientation angle does not change. A description of one fiber path shown in Fig. 6 is simplified as Fig. 7, i.e., a linearly varying angle[30]is used to discretize the curves in each segment in Fig.6,so that the ply orientation angle at a specific position x is identified.

Fig.7 Simplified description of the ply angle for a curved fiber path

According to Figs. 6 and 7, in the Cartesian coordinate system,fiber ply angle variation can be expressed as:

Here,φi(x)is the fiber angle at the location x；θiis the fiber angle at xi； N is the number of fiber segments； Li+1is the characteristic length of the segments(Li+1= xi+1-xi).

For a VS cylinder structure, in Fig. 8a, the longitudinal axis of the cylinder is similar to the y direction of a plate,and the loop direction corresponds to the x direction of a plate.The cylinder structure is formed by finding the fiber angle in each narrow band as shown in Fig. 8b. The orientation angle in a single ply is defined to vary in four steps from the keel to the crown of the cylinder. Each segment is divided into M narrow bands.The symmetry about the vertical axis is assumed so that there are five design variables per VS ply:T1,T2,...,T5should be considered,as shown in Fig.9.In each segment,the orientation angle of the fiber varies linearly along the circumferential direction[31]:

Fig.9 Circular cylindrical design nodes

where [αi,αi+1] denotes the lower boundary and the upper boundaryofthei-thsegment,θi,kisthefiberorientationangle in the k-th narrow band,αkis the circumferential angle of the k-th band,Tiis the fiber orientation angle of the i-th segment lower boundary,and Ti+1is the upper boundary.

A composite cylinder is studied in this section whose diameter is 0.457 m and length is 0.457 m.Sixteen layers are arrangedintheformof[0°/+θ/90°/-θ/-θ/90°/+θ/0°]s,where θ ∈[0°,90°]is the ply angle that varies in the circumferential direction.The thickness of each layer is 0.127 mm.

Fig.8 VS composite cylinder:a fiber path,b the cylinder is divided into narrow strips

Table 1 AS4D/9310 material properties of carbon fiber-epoxy material

Fig.10 Finite element model of a composite cylinder

The plies are made of AS4D/9310 carbon/epoxy materials for which the material properties are given in Table 1.

Buckling analysis of the cylinder subjected to bending moments is performed.When θ =45°,the buckling moment is maximal for the baseline CS case. This corresponds to the quasi-isotropic(QI)case.For VS cases,in each VS ply,θ =θ(α).Analysis based on ANSYS software using element SHELL181 is conducted in which the cylinder is divided into 138 narrow bands,and T1-T5as shown in Fig.9 are chosen to be the design variables.The finite element model is shown in Fig.10.

By using the discrete linear interpolation Eq. (15) to fit the curved fiber path,the finite element model is simplified,and the computational cost is reduced in optimizing the VS cylinder.The active learning meta-model method introduced in Sect.2 is used to approximate the responses,and the GA is applied to get the optimum design of VS cylinders. One hundred samples are used to construct the initial Kriging meta-model. After 100 iterations, the optimal result can be obtained.It has 300 finite element models in total.

3.2 Structural optimization design of variable-stiffness composite cylinders

A VS composite cylinder optimization model is expressed as:

where Mcr is the critical buckling moment of the cylinder and Tiis the ply angle at the circumferential node.is the optimal VS design based on the Kriging meta-model using the MSP+EI criterion.

The first buckling modes of the QI cylinder and the optimized VS cylinder are shown in Fig.11a,b.It can be seen that buckling area appears in the compression zone.The stress is distributed more extensively in the VS cylinder under the critical buckling moment. When θ = 45°in the structure[0°/+θ/90°/-θ/-θ/90°/+θ/0°]s,a baseline laminate,a QI cylinder,is obtained,and the maximum buckling moment of the QI cylinder is 0.9742×105N·m.From Table 2,it is seen that the buckling moment of the VS cylinder is increased by 23.9%.Similar results can be found in Ref.[25].Figure 12 shows the optimal angles of the VS cylinder versus the QI one.The ply angle of the optimized VS cylinder at the cylinder bottom is close to 0°, for resisting the stretching in the area.The ply angle at the cylinder top is close to 55°where compression is exerted.

3.3 Design of a variable-stiffness cylinder with ovality

The ovality defect of a cylinder is inevitable in engineering,and it often leads to a decrease in performance[28].Ovality is defined as (a-b)/a where a is the long axis and b is the short axis of the section. A bending load is applied about the major axis of the VS cylinders,i.e.,the most dangerous case is investigated. The stacking sequences are the same as in Sect. 3.2. Cylinders with different ovalities are taken into account. Table 3 and Fig. 13 show the first buckling moments of QI and VS cylinders under different ovalities.It can be found that the buckling moment decreases with the increase of ovality,both for QI and VS cylinders.However, the structure performance is greatly improved by the VS design.Figure 14a,b shows the first buckling modes of a QI cylinder and the optimized VS cylinder with 0.1 ovality.Similar to Fig.11, it can be found that the buckling area is larger in the VS structure,and the deformation distribution is relatively uniform.That is to say,an optimized VS cylinder takes full advantage of the material.Figure 15 shows the optimal fiber angles under different ovalities.It can be seen that the fiber angle at the cylinder bottom takes values near zero.

Fig.11 The first buckling modes:a QI cylinder,b VS cylinder

Table 2 Buckling load of QI and VS composite cylinder(D=0.457 m,L=0.457 m)

Fig.12 Optimized QI ply angle and VS ply angle

This is because the bottom is pulled and it requires strong tensile strength.At the top,compression occurs and the fiber angles are approximately 55°. The overall structural resistance to buckling has been enhanced with a set of suitable ply fiber angles.

From the above discussion, the structural buckling area becomes larger and the critical load increases when a VScylinder is considered and optimized as compared to its QI counterpart.In the tension area,the fiber angle at the cylinder bottom is close to 0°,while in the compression zone,the fiber angle is approximately 55°.

Table 3 Critical buckling moment of QI and VS cylinders under different ovalities

Fig.13 Critical buckling loads under different ovalities (QI and VS cylinders)

Fig.14 First buckling modes with an ovality of 0.1:a QI cylinder,b VS cylinder

Fig.15 Optimization results of VS cylinders with different ovalities

4 Designs of composite cylinders under combined loadings

In engineering,a common situation is a cylinder under combined loadings. Previous studies [32, 33] found that the following combined loadings often occur for the cylindrical structure: axial force and external pressure, axial force and internal pressure, bending moment and axial force, bending moment and internal pressure,and bending moment and torque. In literature, most emphasis is placed on the axial compressive force and the bending force.Buckling and critical strength are very important considerations in the design of a cylinder under a variety of load conditions,thereby ensuring better performance.In order to enhance the strength and buckling properties of cylinders simultaneously, the design of VS cylinders under combined loads has to be considered.

4.1 Design of variable-stiffness cylinders under bending moment and internal pressure

Fig.16 VS cylinder under bending moment and internal pressure

When a cylinder is subjected to both bending moment and internalpressure,bothbucklingandfailureareliabletooccur.First,under a given internal pressure,a bending moment is applied.As the applied bending moment increases,the composite cylinder structure can break or buckle.To improve the performance of the cylinder,the structure must satisfy both buckling and strength constraints.In this section,design of VS cylinders under the combined loads of bending moment and internal pressure is studied. The cylinder is shown in Fig.16.

The optimization problem is formulated as:where M is the bending moment,λcris the ratio of the bending moment-to-buckled bending moment,λ f is the Tsai-Wu criterion coefficient,Tiis the fiber angle at certain nodes,and T*iis the optimal fiber angle.

Using the material parameters in Table 1, the optimizations are performed and the results are shown in Table 4.

Table 4 and Fig. 17 show that as the internal pressure increases, the ultimate bending moment increases. Table 4 also shows that under different internal pressures, the failure modes are all buckling,but the failure index increases as the internal pressure increases.At p = 2.0 MPa,for example, M*= 1.5086 × 105N·m for the VS cylinder, and M*=1.03976×105N·m for the QI cylinder.The capacity of optimized VS cylinder is 45%higher than the QI counterpart.The failure mode is still buckling,but the failure index is 0.9682,indicating that in this case,the structure approaches its strength limit.

4.2 Design of variable-stiffness cylinders under bending moment and torque

This sub-section considers the case when bending moment and torque are loaded simultaneously.For a certain ratio of

Table 4 Optimization results of VS design under bending moment and internal pressure(the failure modes are all buckling)

Fig.17 Critical bending moment under different internal pressures

Fig.18 VS cylinder under bending moment and torque

T/M, as the applied bending moment and torque increases,the composite cylinder structure can break or buckle. The cylinder is shown in Fig.18.

In order to satisfy the requirements in engineering,design of a VS cylinder is formulated as:

where M is the bending moment,λcris the ratio of the bending moment-to-buckled bending moment,λfis the Tsai-Wu criterion coefficient,Tiis the fiber angle to be optimized,T*iis the optimal fiber angle,T is the torque,and u is the ratio of torque-to-the bending moment；

From Table 5, it is found that as the ratio increases, the ultimate bending moment decreases.That is to say the torque has a great influence on the structural stability.Table 5 shows that the failure modes are all buckling and the failure indexes are all small,indicating that buckling dominates the performances of the cylinder.

5 Conclusions

Optimal design of VS cylinders is studied by using an improved Kriging meta-model together with the GA. A mixed updating rule in constructing the Kriging model is proposed to increase its accuracy and convergence.The bending moment-induced buckling problem is studied first. Compared with the QI cylinder, the critical buckling load of an optimized VS cylinder increases more than 23.9%. After that, the influences of geometrical defects, i.e., ovality, on the performance of a cylinder are also investigated.Optimal design of VS cylinders with different ovalities is conducted.The bending moment is applied about the major axis of the cylinder with ovality,i.e.,the most disadvantageous situation is considered.The result indicates that the critical buckling moment of an optimized VS cylinder decreases as the ovality increases,but its performance is still better than that of a QI cylinder under the condition investigated.

Two kinds of combined loading cases are studied.Under the joint loading of bending moment and internal pressure,it is found that as the internal pressure increases,the critical bending moment of an optimized VS cylinder will increase.When the bending moment and torque are applied simultaneously,the ultimate bending moment decreases with the increasing of the torque-to-moment ratio. The structure is more and more prone to instability,but the failure index getssmaller and smaller.In this case, buckling always precedes failure.

Table 5 Optimum design of VS composite cylindrical structures with different u(the failure modes are all buckling)

AcknowledgementsThis work is supported by the National Natural Science Foundation of China (Grant 11572134) and the China Postdoctoral Science Foundation(Grant 2017M612443).