Document Type : Research Article
Authors
^{1} Shahrood university of technology
^{2} Department of Mechanical Engineering, University of shahrood
Abstract
Keywords

Mechanics of Advanced Composite Structures 5 (2018) 115 – 130


Semnan University 
Mechanics of Advanced Composite Structures journal homepage: http://MACS.journals.semnan.ac.ir 
Optimization of infinite composite plates with quasitriangular holes under inplane loading
S. A. Mahmodzade Hoseyni, M. Jafari^{*}
Department of Mechanical Engineering, Shahrood University of Technology, Shahrood, Iran, 3619995161, Shahrood, Iran
Paper INFO 

ABSTRACT 
Paper history: Received 20161120 Received in revised form 20170528 Accepted 20180130 
This study used particle swarm optimization (PSO) to determine the optimal values of effective design variables acting on the stress distribution around a quasitriangular hole in an infinite orthotropic plate. These parameters were load angle, hole orientation, bluntness, fiber angle, and material properties, which were ascertained on the basis of an analytical method used by Lekhnitskii [3]. The cost function was regarded as the maximum stress created around the hole and was calculated using the aforementioned analytical approach. The finite element method was then employed to verify the results of the analytical calculation. The overlap in the analytical and FEM calculations confirmed the validity of the solution proposed in this research. The findings further indicated that the design variables significantly affect the stress distribution around quasitriangular holes and structural loadbearing capacity. The performance of the PSO algorithm was also investigated. 



Keywords: Infinite orthotropic plate Quasitriangular hole Particle swarm optimization Analytical solution Complex variable method 


© 2018 Published by Semnan University Press. All rights reserved. 
Composite plates are characterized by superior mechanical properties, such as high strengthtoweight ratios, which is why these materials are increasingly used in building industries and many other important sectors. The wide application of composite plates and their complicated behavioral nature compared with that of metals necessitate a detailed study of their failure strength. The numerous applications of composite plates also generate a variety of geometric discontinuities with different sizes and shapes. Examples of such discontinuities are holes on the doors and windows of submarines and airplanes, points for mounting manometers and barometers on a furnace wall, points of junction between plates and rivets and screws, holes that enable the easy flow of cooling fluids across turbine blades, and holes created to enable the passing of electrical cables and hydraulic hoses or the facilitation of fuel flow on a wing. In all these cases, geometric changes in different structures create highly localized stresses around discontinuous areas, where structural failure usually occurs. Analyzing stress concentration is therefore essential for designers of engineering structures because the fracture strength of these constructions depends strongly on the stress concentration caused by holes. Stress concentration is also critical in evaluating the reliability of engineering structures. The stress surrounding fastened joints, for example, causes 80% of fatigue failures in aircraft structures [1].
Savin [2] was the first to study an infinite plate containing a triangular hole with rounded edges—a task that he carried out using conformal mapping and the Schwarz relation. The author examined stress concentration around different holes in isotropic plates and around circular and elliptical holes in anisotropic plates. He also calculated the stress distribution around an equilateral triangular hole with rounded corners in infinite plates. Lekhnitskii [3] explored anisotropic plates containing circular and elliptical holes, while Theocaris and Petro [4] investigated the stress distribution around an equilateral triangular hole and the effects of bluntness. With the expansion of Savin’s method, Daoust and Hoa [5] analyzed equilateral triangular holes in infinite isotropic and anisotropic plates subjected to uniaxial loading. Apart from scrutinizing these holes, they investigated other triangular holes with different aspect ratios and the effects of curvature bluntness on the stress distribution around the holes. Tsutsumi et al. [6] explored the solution of a semiinfinite plane with one circular hole. The authors arrived at their solution by repeatedly superposing the solution of an infinite plane with one circular hole and that of a semiinfinite plane without holes to cancel out the stresses arising on both boundaries.
Rezaeepazh and Jafari [7–8] illuminated the stress distribution around several noncircular holes in isotropic and composite plates using Lekhnitskii’s [3] solution. Asmar and Jabbour [9] calculated the stress distribution around a rectangular hole in an infinite anisotropic plate under tensile loading. They examined the effects of bluntness and load angle on the stress analysis of perforated plates. On the basis of Savin’s [2] potential function, Rao et al. [10] presented relationships for the stress analysis of isotropic and anisotropic plates containing a rectangular hole. Yang et al. [11] used the Airy stress function to look into exact solutions for the stresses, strains, and displacements in a perforated rectangular plate—problems that stem from an arbitrarily located circular hole subjected to both linearly varying inplane normal stresses on two opposite edges and inplane shear stresses.
Sivakumara et al. [12] probed into the optimization of laminate composites containing an elliptical hole by using a genetic algorithm. The design variables used by the authors were the stacking sequence of laminates, the thickness of each composite layer, the relative size of a hole, hole orientation, and ellipse diameters. They also regarded first and second natural frequencies as cost functions. Liu et al. [13] presented a newly developed fixed grid evolutionary structural optimization method to obtain the optimum shape of multiple holes in composite structures on the basis of the Tsai–Hill failure criterion. The authors first considered the effects of the number of holes and the spacing relative to each hole, then they used the Tsai–Hill failure criterion for a composite plate to evaluate the cost function. Cho and Rowlands [14] demonstrated the effectiveness of a genetic algorithm in minimizing the tensile stress concentration in composite laminates containing holes. Along with the algorithm, a specially developed finite element program was used by the authors. Hudson et al. [15] employed ant colony optimization for the multiple objective optimization of composite sandwich structures for rail vehicle floor panels. Almeida and Awruch [16] delved into the optimal design of laminate composites using a genetic algorithm and finite element approaches. The authors regarded fiber angle and the thickness of each composite layer as design variables and indicated the cost function as achieving the lowest deflection and weight.
Alonso et al. [17] used particle swarm optimization (PSO) on the basis of moving bird groups to optimized composite structures. Also using PSO and a finite element method (FEM), Chen et al. [18] presented an optimization approach to ensuring the reliability of composite structures. The authors used ANSYS data in MATLAB and presented numerical examples of laminates, composite cylindrical shells, and composite pressure vessels to verify the effectiveness of their method. Sharma et al. [19] optimally designed symmetrical laminate composites containing an elliptical hole by using a genetic algorithm. They obtained the optimum fiber angle in the composites under inplane loading conditions and calculated the cost function using the Tsai–Hill failure criterion. The design variable that they used was the stacking sequence of laminates. Zhu et al. [20] considered the optimization of composite struts using a genetic algorithm and the Tsai–Hill failure criterion, paying attention to minimizing the weight of a structure and increasing buckling load. The authors adopted fiber volume fraction and the stacking sequence of laminates as design variables. Jafari et al. [21] calculated the optimum values of the effective design variables acting on the stress distribution around different holes by using the genetic algorithm.
In this study, a PSO algorithm was used to determine the optimal values of effective parameters that act on the stress distribution around a quasitriangular hole in an orthotropic plate. Previous studies minimally investigated the effects of these parameters on the stress distribution around holes. Specifically, the research relied on Lekhnitskii’s [3] analytical solution and expanded it for application to the quasitriangular hole for the purpose of determining the optimal values of the design variables for uniaxial, equibiaxial, and shear loading and generating the minimum normalized stress. Because of the tractionfree boundary conditions imposed on a hole’s edge, stresses s_{r} and t_{r}_{q} at the hole’s edge are zero, and circumferential stress is the only remaining stress. In this research, minimizing normalized stress around the hole is considered the cost function. Normalized stress is the ratio of the maximum value of circumferential stress at the edge of the hole (s_{q}) to nominal or applied stress.
PSO was presented by Kennedy and Eberhart [22]. In an problem, a particle includes a row vector with elements. This arrangement is defined as Eq. (1). To initiate the algorithm, a number of these particles (as the number of the primary particle algorithm) must be created. The total particle matrix is then formed and expressed as Eq. (2).
(1) 

(2) 
As previously stated, the cost function is to minimize normalized stress, which is the ratio of the maximum value of circumferential stress at the edge of the hole (s_{q}) to nominal or applied stress. The cost of each particle is obtained by evaluating the cost function for variables in accordance with Eq. (3).
(3) 
In the PSO algorithm, the particles update their velocities and positions on the basis of the best absolute and local solution, as in Eqs. (4) and (5) [23].
(4) 

(5) 
In the equations above, and are the current velocity and position of a particle, respectively. Let be the position of particle i in an search space at iteration t. The updated velocity and position of the particle are and , respectively. Item is the inertia weight coefficient, and and denote two positive constants called cognitive and social coefficients, respectively. Items and _{ }are two random vectors, whose elements are , and and are the best positions of individual and group experiences, respectively. Finally, each particle is updated on the basis of the best performance of its relationship with condition (6). Moreover, the velocity and position of a particle determined on the basis of the best position among particles are updated according to Eq. (7) [24].
(6) 
if f( ) < f( ) then 
(7) 
if f( ) < f( ) then 
3.1. Convergence performance
PSO has become very popular because of the simplicity with which it is implemented and its effectiveness in quickly converging to a reasonably good solution. The convergence performance of the algorithm depends on the conditions presented in Eq. (8) [25].
(8) 
We know that As a result, ensuring the convergence of the PSO algorithm requires expressing the abovementioned inequalities with innovative parameters as Eq. (9). The upper and lower bounds of design variables are defined as Eq. (10).
(9) 
(10) 
As reflected in Eqs. (4) and (5), some of the particles may leave the feasible region of a search space. To handle these infeasible solutions, the “absorb technique” is applied. In this technique, invalid particles are moved to the nearest boundary by setting all variables outside the feasible region to their nearest bound (Eq. (11)). The affected velocity components are set to zero (Eq. (2)). In Eqs. (11) and (12), and are the lower and upper bounds of the jth component of the search space.
(11) 

(12) 
The values of the effective parameters for the PSO algorithm are listed in Table 1. Changing and optimizing these parameters produce a highperformance algorithm. To illustrate, the effects of the parameters related to the algorithms used in this work were investigated. Note that the results represent boron/epoxy composites and uniaxial tensile loading at w = 0.1.
4.1. Effects of c1 and c2 in the PSO algorithm
Fig. 1 illustrates the effects of the number of iterations on the cost function at different values of c_{1} and c_{2}. Table 2 presents the values of the cost function and the optimum design variables at different values of c_{1} and c_{2}.
To emphasize the general search of particles in the search space at the beginning of algorithm operation and local search at the end of operation, is reduced and is increased. Correspondingly, the particles are attracted to the best position of the swarm. The parameters are updated as Eqs. (13) and (14) [26].
(13) 

(14) 
where , , , are constant, t denotes the number of current repetitions of a particle, and represents the maximum number of allowable iterations.
Table 1. Values of effective parameters for the PSO algorithm 


4.2. Effects of ω in the PSO algorithm
The inertia weight coefficient (w) directly affects the convergence rate of PSO. This coefficient can be used to control the effects of previous velocities on current velocities, and its value can be modified to improve balance between global and local searches. Shi and Eberhart [22] stated that the optimum value of w can be improved for most problems by changing the weight coefficient from 0.9 at the beginning of a search to 0.4 at the end of the search.
Figure 1. Convergence of cost function at different , values
Table 2. Optimal values and corresponding cost function at different , values 


Fig. 2 depicts the effects of the number of iterations on the cost function at different methods of calculating w. Table 3 presents the values of the cost function and the optimum parameters under different methods of calculating w. In the first approach using [26], inertia weight is defined as follows (linearly varying):
(15) 
in which and are the primary and final values of inertia weight, respectively. In the second approach (Convergence factor), a PSO algorithm with a convergence factor was defined in [24], and the following formula is used to update the position and speed of particles:

(16) 
where , (c should be smaller than 4.). The third approach enables a dynamic decrease in the inertia weight value if a swarm does not improve the solution after a certain number of iterations [25]. An update is implemented from an initial weight ( ) value based on a fraction multiplier ; that is,
(17) 
Figure 2. Convergence of the cost function at different approaches to ω
Table 3. Optimal values and corresponding cost function for different approaches to ω

Population size 
Number of iterations 
C.F. 

Linearly varying 
50 
16 
90 
22.42 
157.5 
3.0484 
Convergence factor 
50 
26 
75.15 
7.59 
82.70 
3.0484 
Dynamic decrease 
50 ( = 0.975) 
31 
19.38 
86.94 
11.83 
3.0484 
The problem disentangled in this study is an infinite plate containing a quasitriangular hole. As shown in Fig. 3, the plate is subjected to biaxial loading at angle (load angle) with respect to the xaxis. A triangular hole has arbitrary orientations such that its major axis is directed at angle (rotation angle) with respect to the xaxis. The fiber angle is .
Replacing stress–strain relations in a compatibility relation and rewriting the resultant equation in terms of the stress function yield [3]
(18) 
Eq. (18) is the compatibility equation for anisotropic materials, where represents the compliance coefficients of perforated plates. The general solution of this equation depends on the roots of the following characteristic equation:
(19) 
In general, Eq. (19) can be proved as having four conjugate roots, after which the general expression of the stress function can be indicated thus:
(20) 
The equation ( ) and S_{k} are the roots of the characteristic equation of anisotropic materials. Finally, stress components in terms of two potential functions of and are expressed according to Eq. (21) [7].
(21) 
where
(22) 
In Eq. (22), adopting appropriate values of and for stress applied at infinity enables the consideration of uniaxial loading, equibiaxial loading, and shear loading. The following values of and may be incorporated into Eqs. (22) to obtain various cases of inplane loading:
Items and are the derivatives of functions and with respect to and These analytic functions can be determined by applying boundary conditions.
Eqs. (23) and (24) are used to calculate stress components in the polar coordinate system. In these equations, W is the angle between the positive xaxis and r (Fig. 4).
(23) 

(24) 
As shown in Fig. 5, the infinite orthotropic plate with an arbitrary hole subjected to inplane loading was examined. No external forces are acting on the hole’s edge. The superposition method was used to calculate the stress distribution around the hole. For the nohole plate subjected to inplane loading, stress functions and were calculated (Fig. 5a).
Figure 3. Perforated orthotropic plate subjected to biaxial loading
Figure 4. Curvilinear coordinates
(a) (b) (c)
Figure 5. Scheme for the superposition of a solution for a perforated plate under biaxial loading; (a) plate without a hole, loading at infinity; (b) plate with negative loading on the edge of the hole; (c) plate with an arbitrary hole, loading at infinity [27]
In this stage, using the aforementioned stress functions enabled the calculation of the boundary conditions (f_{1} and f_{2}( located on the edge of a fictitious hole. Then, the plate with a hole and subjected to negative distributed loads –f_{1}and –f_{2}at the hole’s edge was analyzed (Fig. 5b). In this case, stress functions and were obtained on the basis of the Ukadgaonker solution [27].
The stress functions for a perforated plate with a tractionfree hole and loading at infinity are obtained via the superposition of the stress functions in the stages discussed above; that is,
(25) 

(26) 
Finally, stress components are obtained by replacing the stress functions in Eq. (21) with the abovementioned stress functions.
5.1. Conformal mapping
With the analytical solution proposed by Lekhnitskii [3], the stress distribution around a circular or an elliptical hole in anisotropic plates was obtained. To use this method as a means of examining plates with triangular holes, an essential requirement is to apply the mapping function, similar to what is presented in Eq. (27). According to this conformal mapping, the infinite area external to a hole can be transformed by the area outside a unit circle.
(27) 
where
(28) 


In the relation above, parameter w determines the bluntness factor and changes the radius of curvature at the corner of the hole. Changing the value of w from zero to 0.3 reduces the curvature of the corners of the triangular hole (Fig. 6). As can be concluded from Eq. (28), w = 0 represents a circular hole. Bluntness (w) and hole orientation ( ) are important design variables that influence the stress distribution around different holes.
FEM (ABAQUS) was used to verify the accuracy of the analytical results. First, for the orthotropic plate with a quasitriangular hole (w = 0.15), the optimum values were obtained using the optimization code written in MATLAB. Second, considering the optimum values of the effective design variables ( = 90°, = 30°, = 22°) enabled the modeling of the problem described in the previous section in ABAQUS. Mesh sensitivity was scrutinized, and an acceptable mesh was chosen. Fig. 7 shows the comparison of the stress distributions obtained via the analytical solution and FEM for a glass/epoxy composite.
Fig. 8 shows the comparison of the results obtained via the analytical method and FEM for a boron/epoxy composite; the comparison involved considering the optimum values calculated through the PSO optimization code written in MATLAB ( = 90°, = 157°, = 20.40°). The findings derived via the analytical solution and FEM exhibit good agreement, indicating the accuracy and precision of the proposed analytical solution.
w = 0.3 w = 0.2 w = 0.1 w = 0 
Figure 6. Effects of w on the shape of the quasitriangular hole 
Figure 7. Comparison of the FEM and analytical results for glass/epoxy composites (w = 0.15) 
Figure 8. Comparison of FEM and analytical results for boron/epoxy (w = 0.15) 
7. Results
As mentioned earlier, this research ascertained the optimal values of design variables that affect the stress distribution around a quasitriangular hole in an infinite orthotropic plate by using PSO. To achieve this aim, for uniaxial loading and determining the specific value of each design variable, the optimum values of other design variables and their corresponding cost function were determined. Then, taking into account all the design variables, the value of the cost function was incorporated into various cases of inplane loading. The mechanical properties of the materials examined are presented in Table 4. The effects of hole orientation on the value of the cost function at a range of load angles (uniaxial loading), at a w = 0.05, and for a glass/epoxy composite are shown in Fig. 9. These results are presented in the optimum fiber angles obtained using the PSO algorithm. As indicated in the figure, at different load angles, the value of the cost function varies with hole orientation. These variations with rotation angle occurs at 60°. Although these results are applicable only to glass/epoxy composites, such periodic behavior can also be observed in other materials. Fig. 10 presents the effects of fiber angle on the value of the cost function at different load angles for a glass/epoxy composite. These findings are presented in the optimum hole orientation obtained using the PSO algorithm. As the figure reflects, the maximum and minimum values of the cost function at different load angles occur identically at different fiber angles.

Figure 9. Variations in the cost function with respect to hole orientation at an optimum fiber angle (glass/epoxy)


Table 4. Properties of the plates

(GPa) 
(GPa) 
(GPa) 

Graphite/epoxy (T300/5208) 
181 
10.3 
7.17 
0.28 
Carbon/epoxy (IM6/SC1081) 
177 
10.8 
7.6 
0.27 
Boron/epoxy(B5.6/5505) 
201 
21.7 
5.4 
0.17 
Glass/epoxy 
38.6 
8.27 
4.14 
0.26 
Fig. 11 shows the effect of hole orientation on the cost function at different values of bluntness factor. These results are applicable to the optimum fiber angle and . As expected, a decreasing w and hole softening reduce the value of the cost function. The minimum value of the cost function occurs at w = 0, which is equivalent to a circular hole.
Table 5 lists the optimum values provided in Figs. 9 to 11. We endeavored to present the results for glass/epoxy (uniaxial loading) composites in detail, but for other materials, only the optimum values of design variables are provided.
Fig. 12 shows the variations in the cost function with respect to fiber angle at various rotation angles. In this case, the design variable used was load angle, which was determined using the PSO algorithm. Fig. 13 shows the variations at different values of bluntness factor and . Fiber angle was considered a design variable. As shown in the figure, except for the circular hole (w = 0), the maximum value of the cost function occurs at a load angle close to 50°. A load angle of 0° leads to the lowest possible value of the cost function.
Figure 11. Effects of rotation angle on the cost function at different bluntness values, optimum fiber angle, and θ_{1}=0 (glass/epoxy)
Table 5. Optimum hole orientation and fiber angle at various load angles and bluntness values (glass/epoxy) 
Figure 12. Variations in the cost function in terms of fibers angle at an optimum load angle (glass/epoxy, w = 0.05)
Figure 13. Effects of load angle on the cost function at an optimum fiber angle and different bluntness values (glass/epoxy, θ_3=0)
Table 6 presents the optimum values of load and fiber angles at different hole orientations. Note that at w = 0 during various runs of the program, the minimum value of the cost function occurs at different load and fiber angles; in all runs, the absolute difference between load and fiber angles is 60°.
Table 6. Optimum load and fiber angles at different hole orientations (glass/epoxy, w = 0.05) 


Fig. 14 shows the variations in the cost function with bluntness factor w (uniaxial tensile loading) for different materials. In this case, the values of other design variables ( , , ) are their optimum values obtained via the PSO algorithm.
According to the figure, for all the materials, a decreasing w reduces the value of the cost function. For w = 0, the value of the cost function is the lowest possible value. Fig. 15 indicates the stress distribution at different values of w and different materials under optimal situations.
Figs. 16 and 17 display the changes in the normalized stress (cost function) around a triangular hole in optimal conditions for glass/epoxy and boron/epoxy composites at three values of w, respectively.
Finally, Table 7 presents the optimum values of different design variables, such as load angle, fiber angle, and hole orientation, which were obtained through PSO for different w values, all the materials, and various cases of loading, such as equibiaxial loading, uniaxial tensile loading, and shear loading. Figs. 18 and Fig. 19 show the changes in the normalized stress around a triangular hole in optimal conditions for glass/epoxy and boron/epoxy composites under equibiaxial and shear loading and w = 0.1.
Figure 14. Effects of bluntness on the cost function for different materials






Figure 15. Optimum stress distribution around the hole at different bluntness values and uniaxial tensile loading. (a) Graphite/epoxy, (b) glass/epoxy, (c) boron/epoxy, (d) carbon/epoxy 

This research inquired into the optimum values of effective design variables for the stress analysis of an infinite orthotropic plate with a quasitriangular hole. The plate was subjected to uniaxial, equibiaxial, and shear loading, and the PSO algorithm was used to calculate the optimum values of design variables. As the design variables, bluntness, hole orientation, fiber angle, and load angle are the most important design variables that influence the stress distribution around triangular holes. The optimum values of these design variables were determined for different materials, minimizing normalized stress around the hole was considered the cost function. The results showed that the curvature radius of the hole’s corners is not the only parameter that effectively reduces normalized stress and that the appropriate selection of other effective design variables, such as fiber angle, load angle, and hole orientation, significantly reduces normalized stress.





w = 0.05 
w = 0.1 
w = 0.15 



Figure 16. Stress distribution around the hole at different bluntness factors and optimum conditions (glass/epoxy) 






w = 0.05 
w = 0.1 
w = 0.15 



Figure 17. Stress distribution around the hole at different bluntness factors and optimum conditions (boron/epoxy) 

Table 7. Optimum values of design variables at different bluntness factors and different loading cases 





(a) 
(b) 

Figure 18. Stress distribution around the hole under optimum conditions and w = 0.1 for glass/epoxy composites; (a) equibiaxial loading, (b) shear loading 



(a) 
(b) 


Figure 19. Stress distribution around the hole under optimum conditions and w = 0.1 for boron/epoxy composites; (a) equibiaxial loading, (b) shear loading 


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