Probabilistic Optimization of Laminated Composites Considering Both Ply Failure and Delamination Based on PSO and FEM

TSINGHUA SCIENCE AND TECHNOLOGY ISSNll1007-0214ll18/21llpp89-93 Volume 14, Number S2, December 2009

Probabilistic Optimization of Laminated Composites Considering Both Ply Failure and Delamination Based on PSO and FEM* TANG Yuanfu (൹ၙ‫)ؖ‬, CHEN Jianqiao (чߙ஢)**, PENG Wenjie (଎ำࠍ) Department of Mechanics, Huazhong University of Science and Technology, Wuhan 430074, China Abstract: Reliability-based design is one of major strategies for cost-effective designs under uncertainties in either engineering designs or manufacturing processes. This paper presents an approach integrating particle swarm optimization algorithm (PSO) and finite element method (FEM) into the reliability-based design optimization (RBDO). The total weight of the laminate was taken as the objective function, the fiber directional angle and thickness ratio as the design variables. Both in-plane damage and delamination were taken into account. The result shows that the value of object function using the proposed method is much greater than those without considering delamination in literature and demonstrate that the proposed method produces more conservative result. Key words: reliability-based design optimization; interlaminar stresses; particle swarm optimization; finite element method; failure criterion

Introduction In recent decades, laminated composites have been applied in many fields, i.e., aerospace, aviation, construction, automobile, machinery, and other fields for the excellent performance. Composite laminate is a multi-layered structure with low interlaminar intensity, whose bearing capacity often decreases with interlaminar failure. Therefore, the potential performance of composite laminate can not be fully exploited. Pipes and Pagno[1] used the finite-difference calculation of free-edge stress fields to give the results of stress distribution, which indicated that free edge effect occurs everywhere along free edges. Due to the singularity of interlaminar stresses in practice[2,3], delamination failure near the free edges of laminate often occurs before the strength limit predicted by the classic laminate

theory (CLT) is reached. In order to obtain reliable design in engineering practice, the three-dimensional stresses near the free edges should be considered[4,5]. Moreover, reliability-based design is one of major strategies for cost-effective designs under uncertainties in either engineering designs or manufacturing processes[6,7]. Recently, several theories under uncertainties were developed for engineering[8-11]. In this paper, we propose a novel method that integrates particle swarm optimization algorithm (PSO) and finite element method (FEM) into the reliabilitybased design optimization (RBDO), which is applied to laminated composites. Both in-plane failure criterion and interlaminar failure criterion are taken into account when calculating the reliability of the laminates. PSO can solve continuous and discrete global optimization problem efficiently and accurately for a high-dimensional and non-smooth objective function[12].

Received: 2009-05-08; revised: 2009-06-20

* Supported by the National Natural Science Foundation of China

1

Reliability-Based Design Optimization

A

general

(No. 10772070) and Ph.D. Programs Foundation of Ministry of Education of China (No. 20070487064)

** To whom correspondence should be addressed. E-mail: [email protected]; Tel: 86-13507194966

reliability-based

design

optimization

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90

problem with only random variables and deterministic design variables can be stated as Minimize f (d , X ),

1.1

d

s.t. g i (d , X ) . bi , i 1, 2, ..., p;

accurate. Three modules will be illustrated in detail in the following sections. PSO algorithm

(1)

Prz j (d , X ) . 0) . R j , j 1, 2, ..., q where f denotes the design objective function, d is the vector of deterministic design variables or control factors, X is a vector of random variables, gi (X) is the i-th general constraint function, p is the number of the general constraints, Pr is the reliability constraint function, q is the number of the reliability constraints, Rj is the j-th prescribed safe probability limit. This paper combines PSO and FEM and divides the RBDO of composite laminates into three nested modules: optimization module, the module of reliability analysis, and the module of stress analysis. The whole process of the method is shown in Fig. 1. The optimization module is performed by PSO, while the module of stress analysis by FEM. The reliability in the module of reliability analysis is evaluated by the iterative method. Three modules are nested each other by transferring corresponding variables and parameters. In each optimization cycle, reliability analysis and stress analysis are required. The stresses obtained by stress analysis are substituted into the performance function in the module of reliability analysis. The module reliability of reliability analysis behaves as a constraint of the optimization module. Since the stresses are obtained by FEM, the reliability evaluation will be more

The PSO is a heuristic global optimization search technique based on community intelligence. Each particle represents a potential solution for the optimization design problem and has a position represented by a position vector. Initially, particle swarms are initialized with random position values and random initial velocities and then they are accelerated towards their previously best positions (the solutions attained by each particle) and their best global position (the solution attained by the whole swarm). Each particle updates its own velocity and position according to the following two formulas. vi( k+1) wvi( k )  c1 rand1 (pbest i  xi( k ) )  c2 rand 2 (gbest i  xi( k ) )

xi( k )

( k 1) i

x

v

( k 1) i

(2) (3)

where i denotes the i-th particle, k denotes the k-th iteration, vi is the fly velocity of the i-th particle, xi is the current position of the i-th particle, c1 and c2 are the acceleration constants (c1=c2=2.05), rand1 and rand2 are uniformly distributed random numbers in the interval [0, 1], w is the inertia weight, pbesti is the the individual’s i-th best position found so far and gbest is the global best position attained by the swarm found so far. This process is iterated until a minimum error is achieved or a set number of times are achieved. 1.2

Reliability analysis

When only random variables and design variables are involved, the structural reliability is defined as R =Pr[z (d ,X ) . 0}= ³ f X (X )dX (4) z (d , X ).0

where z is a performance function (also called limitstate function) and fX is the joint probability density function (pdf) of the random variables. Once the original random vector X is transformed into a set of random vector U whose elements follow a standard normal distribution, then the reliability is rewritten as R =Pr[z (d , X ) . 0}= ³ fU (U ) dU =) (E ) (5) z (d, U ).0

Fig. 1

The process of the proposed method

where fU is the joint pdf of U, () is the cumulative distribution function (cdf) of the standard normal

91

TANG Yuanfu (൹ၙ‫ )ؖ‬et al.ġProbabilistic Optimization of Laminated Composites Considering …

distribution, and  is the reliability index. In the first order reliability method (FORM), reliability index  is geometrically the minimum distance from the origin to the limit-state surface with z(d, U) 0 in U space. In this method, the limit-state is approximated by the tangent plane at the most probable point (MPP), which is on the surface of z(d, U) 0[7]. The vector from MPP to origin is normal to the surface of z(d, U) 0. The distance between the MPP and the origin is used to quantify the reliability index and could be obtained by the iterative method[13]. The iterative method is not only simple in iteration and converges as fast as JC method, but also can guarantee converge in general. 1.3

Interlaminar stress analysis

Traditionally, stress analysis is performed based on classical lamination theory, which is not precise enough, especially for complex structures. However, ANSYS can employ the various types of layer elements in the thickness direction to simulate the laminated structure, compensate for the inadequacies of the classical theory, and obtain stresses accurately. Since there is stress singularity near the free-edges and the compressive interlaminar normal stress are capable of delaying the delamination, Lecuyer and Engrand propose a criterion to predict the stress level related to delamination occurrence[14]. Lecuyer’s criterion[15] is expressed as 2

2

§ V xz · § V yz · V zz ¨ D ¸  ¨¨ D ¸¸  D 1, © S xz ¹ © S yz ¹ Z T 1 b V ij V ij dy d 0 ³ b  d0

V ij is evaluated by the following compound trapezoid formula: 1 b V ij dy d 0 ³ bd0

n 1 1 ª º V ij (1)  2¦ V ij (k )  V ij (n) » (7) « 2(n  1) ¬ k 2 ¼

where n denotes the number of the nodes within the characteristic length near the free-edge interface, V ij (1) and V ij (n) are respectively the stresses of the starting node and terminal node.

2 Numerical Examples and Discussion Considering a laminated rectangular plate[15] with simply supported boundary conditions under a transverse uniform bending load q 4.8×104 N/m2, an x-axial compressive load Nx 3.7×105 N/m, and a y-axial compressive load Ny 2.4×105 N/m, the laminate has length a 20 cm, width b 12.5 cm, and a symmetric [0/+//90°]s-layup. Both the thicknesses of the angle 0o and 90o plies are 0.2 mm, and the thicknesses of the angle + and – lamina respectively 0.1h1and 0.1h2. Strength parameters are assumed to follow standard normal distribution as shown in Table 1, each parameter being non-correlative with the others. The material properties are given as: EX 181 GPa, EY 10.7 GPa, EZ 40 MPa, GXY 7.17 GPa, GXZ 7.17 GPa, GYZ 5.26 GPa, xy 0.28, xz 0.28, yz 0.3. The task is to determine the thicknesses h1 and h2 and the angle , to minimize the total weight or total thickness of the laminate with required reliability index of bucking constraint E . 2.5 (or reliability R . 0.9938 ). Table 1

(6)

where y b represents the free edge, d0 is a material’s interface characteristic length representing the zone width wherein the stress field is disturbed by the freeedge singularity. The distance d0 from the edge must be experimentally determined. In this study, the critical length d0=1.5t (t=ply thickness) is taken in order to minimize the effect of high stress gradients at the free edge. Z TD is the tensile strength of lamina in principal material direction, while S xzD and S yzD are respectively the interlaminar shear strengths of mode III and mode II for which the transverse shear strengths of the lamina Sxz and Syz are taken[16]. On account of nodes with equidistance, the average

Strength parameters

Variable

Mean

S.D.

Variable

Mean

S.D.

XT / MPa

1500

150.0

XC / MPa

1500

150.0

YT / MPa

40

4.0

YC / MPa

246

24.6

Sxy / MPa

68

6.8

Sxz / MPa

68

6.8

Syz / MPa

68

6.8

ZT Z / MPa

40

4.0

S.D: Standard deviation

The optimization design problem is modeled as Minimize f (d ) h1  h2 , s.t. E . 2.5;

(8)

h1 , h2 . 0, 90q . T . 90q In this problem, first ply failure criterion (FPF) is applied to predict the failure of a laminate. In order to compare the proposed method with other methods, four cases are presented, wherein the optimization

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processes are carried out by PSO algorithm. In case 1, the 2-dimensional Tsai-Wu tensor criterion is considered as the limit-state function and classic laminate theory is adopted to obtain stresses. Only the in-plane stresses are considered. The stresses of each ply can be represented by the stresses of any point of the ply. In this case, the center point of each ply is selected. Once the stresses are obtained, the stresses are substituted into the limit-state function in reliability analysis module, and then the reliability of each ply can be evaluated. Thusˈthe reliability of the laminate is the minimum among the reliabilities of the eight plies. In case 2, the stress analysis is perfumed by FEM and 2-dimensional Tsai-Wu tensor criterion is applied to predict the in-plane failure in each ply. The failure occurs first in the element with largest failure index whose stresses are treated as the stresses of the ply. In case 3, 2-dimensional Tsai-Wu tensor theory is replaced by 3-dimensional Tsai-Wu tensor theory and the rest is the same as case 2. In case 4, the laminate is divided into two parts: the general region and the region where the free-edge exerts a vital influence. The finite element mesh near free edges is shown in Fig. 2. Lecuyer’s criterion and Tsai-Wu tensor theory are used to predict the delamination failure and in-plane failure respectively in the related regions. The reliability of each ply is determined by the smaller reliability between two regions.

compared with those in case 1, case 2, and case 3, where interlaminar failure criterion is not taken into account. There is little difference among the values of object function in the former three cases. The results presented are in agreement with expectations. Consider the same reliability-based design problem as above but the thicknesses h1 and h2 are discrete limited to a certain set of integer. The optimum results are shown in Table 3. It can be found that the reliability constraint in each case is satisfied and the minimum total thickness is equal to each other in the former three cases. Table 2

Optimization results for four cases

4

h1/(10 m) h2/(104m)

/(o)



f/(104m)

Case 1

1.5701

1.5832

30.6308 2.5000

3.1533

Case 2

1.7569

1.2267

65.7248 2.5000

2.9836

Case 3

1.2724

1.9213

66.1157 2.5000

3.1937

Case 4

3.5862

3.5586

20.7798 2.5000

7.1447

Table 3

Discrete results for four cases

h1 / (104m) h2 / (104m)

 / (o)



f / (104m)

Case 1

2

2

30.6308 3.1225

4

Case 2

2

2

65.7248 4.5062

4

Case 3

2

2

66.1157 4.1126

4

Case 4

4

4

20.7798 3.3747

8

From Fig. 3, it can be found that the minimum total thickness f decreases with decreasing of the transverse uniform bending load q and they behave an approximate linear relation.

Fig. 2 Finite element mesh near free edges

All the four cases are under the same loads and have the same material properties. In cases 2, 3, and 4, the stress analysis is performed by commercial software ANSYS. The element type “solid 64” is chosen from which interlaminar stresses can be obtained precisely. The optimization results for four cases are listed in Table 2. It is seen that the value of object function in case 4 is much greater, which indicates that the proposed method produces the most conservative result as

Fig. 3 The relation between the transverse uniform bending load q and minimum total thickness

3

Conclusions

This paper suggests a new RBDO model incorporating PSO and FEM. A laminated composites example is used to demonstrate the effectiveness of the proposed

TANG Yuanfu (൹ၙ‫ )ؖ‬et al.ġProbabilistic Optimization of Laminated Composites Considering …

method. Since laminated composites’ strength and life may be influenced by interlaminar stresses on account of free edges, both in-plane damage and delamination are considered. The optimization results indicate that the methodology provides a much more conservative and preferable solution as compared with those which ignore the delamination. Thus, the proposed method is a more feasible and effective approach for the probabilistic optimization of complex composite structures. References

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