Optimization of infinite composite plates with quasi-triangular holes under in-plane loading

S. A. Mahmodzade Hoseyni, M. Jafari^*

Department of Mechanical Engineering, Shahrood University of Technology, Shahrood, Iran, 36199-95161, Shahrood, Iran

1. Introduction

Composite plates are characterized by superior mechanical properties, such as high strength-to-weight 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].

1. Literature review

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 semi-infinite 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 semi-infinite plane without holes to cancel out the stresses arising on both boundaries.

          Rezaeepazh and Jafari [7–8] illuminated the stress distribution around several non-circular 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 in-plane normal stresses on two opposite edges and in-plane 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 in-plane 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 quasi-triangular 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 quasi-triangular hole for the purpose of determining the optimal values of the design variables for uniaxial, equi-biaxial, and shear loading and generating the minimum normalized stress. Because of the traction-free boundary conditions imposed on a hole’s edge, stresses s_r and t_rq 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.

3. Particle Swarm Optimization

          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).

          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).

          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].

          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].

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].

          We know that  As a result, ensuring the convergence of the PSO algorithm requires expressing the above-mentioned inequalities with innovative parameters as Eq. (9). The upper and lower bounds of design variables are defined as Eq. (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.

1. Effects of algorithm parameters

          The values of the effective parameters for the PSO algorithm are listed in Table 1. Changing and optimizing these parameters produce a high-performance 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 c₁ and c₂. Table 2 presents the values of the cost function and the optimum design variables at different values of c₁ and c₂.

          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].

where  ,  ,  ,  are constant, t denotes the number of current repetitions of a particle, and  represents the maximum number of allowable iterations.

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

          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):

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:

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,

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

5. Problem description

          The problem disentangled in this study is an infinite plate containing a quasi-triangular hole. As shown in Fig. 3, the plate is subjected to biaxial loading at angle  (load angle) with respect to the x-axis. A triangular hole has arbitrary orientations such that its major axis is directed at angle  (rotation angle) with respect to the x-axis. 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]

          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:

          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:

          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].

where

          In Eq. (22), adopting appropriate values of  and  for stress applied at infinity  enables the consideration of uniaxial loading, equi-biaxial loading, and shear loading. The following values of  and  may be incorporated into Eqs. (22) to obtain various cases of in-plane 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 x-axis and r (Fig. 4).

As shown in Fig. 5, the infinite orthotropic plate with an arbitrary hole subjected to in-plane 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 no-hole plate subjected to in-plane 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 (f₁ and f₂( located on the edge of a fictitious hole. Then, the plate with a hole and subjected to negative distributed loads –f₁and –f₂at 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 traction-free hole and loading at infinity are obtained via the superposition of the stress functions in the stages discussed above; that is,

Finally, stress components are obtained by replacing the stress functions in Eq. (21) with the above-mentioned 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.

where

          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.

6. Verification of results

          FEM (ABAQUS) was used to verify the accuracy of the analytical results. First, for the orthotropic plate with a quasi-triangular 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.

7. Results

          As mentioned earlier, this research ascertained the optimal values of design variables that affect the stress distribution around a quasi-triangular 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 in-plane 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.

Table 4. Properties of the plates

          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 θ₁=0 (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°.

          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 equi-biaxial 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 equi-biaxial and shear loading and w = 0.1.

Figure 14. Effects of bluntness on the cost function for different materials

8. Conclusion

This research inquired into the optimum values of effective design variables for the stress analysis of an infinite orthotropic plate with a quasi-triangular hole. The plate was subjected to uniaxial, equi-biaxial, 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.

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