Jtam-A4.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 56, 1, pp. 297-311, Warsaw 2018 DOI: 10.15632/jtam-pl.56.1.297 OPTIMUM DESIGN OF FIBER ANGLE AND HOLE ORIENTATION OF AN ORTHOTROPIC PLATE Xiaoli Zhang, Aizhong Lu, Shaojie Wang, Ning Zhang North China Electric Power University, Institute of Hydroelectric and Geotechnical Engineering, Beijing, China e-mail: zning1125@ncepu.edu.cn (N. Zhang) With the goal of decreasing the stress concentration along the hole boundary in an ortho- tropic plate under inequi-biaxial loadings, an optimum design of the fiber angle and hole orientation is presented. The maximum absolute tangential stress along the hole boundary is taken as the objective function, and the fiber orientation angle and the hole orientation angle are considered as design variables.The conformal transformationmethod of a complex function and the Differential Evolution (DE) algorithm are used. Two non-circular shapes, ellipse and hexagon are taken as examples to analyze the problem. Based on the results, we can conclude that themajor axis of elliptical holes should be designed in the direction of the maximum external loading for a perforated structure in an orthotropic plate. However, the principal direction that has the larger Young’s modulus should be inclined to the direction of the minimum loading, especially for a significantly orthotropic plate. Keywords: orthotropic plate, fiber orientation angle, hole orientation angle, conformal trans- formationmethod, differential evolution algorithm 1. Introduction Natural materials, such as wood and rock, and composite materials, such as fiberboard, epoxy resin bonded fiber and fiber reinforced polymer, all display anisotropic properties. Specifically, orthotropic plates with different holes have foundwidespread applications in various fields such as aerospace, marine, automobile and mechanics because of high specific stiffness and specific strength as well as the designability of the properties (Jain, 2009; Romeo, 2001; Li and Zheng, 2007). The stress concentration along the boundary of the hole becomes an important research problem for this structure (Toubal et al., 2005; Sharma, 2011; Engels et al., 2001). Generally, tangential stresses at different points along the hole boundary in orthotropic plates are different and, usually, it is the maximum value that depends the stability on the plate (Savin, 1961). Thus, it is important to decrease the tangential stress concentration along the hole boundary in the design of orthotropic plates. Some researchers have studied the shape optimization of holes in isotropic plates based on different optimization criteria or methods. Bjorkman and Richards (1976, 1979) proposed the concept of a harmonic hole and obtained the optimal shape of the hole under different loads. Taking theminimum integration of the square of tangential stress as the objective, Dhir (1981) solved a series of shape optimization problems by the complex variables function, the same method as Bjorkman. Aiming at minimizing the maximum value of the tangential stress along the hole boundary, Lu et al. (2014a,b) obtained the optimum shape of the support section of a tunnelat greatdepthsusingthecomplexvariables functionmethod.Thehole-shapeoptimization problem has also been solved using the Evolutionary Structural Optimization (ESO) procedure (Ren et al., 2005) and the Simulated Annealing (SA) procedure (Sobótka et al., 2013). So far, the existing studies havemainly focused on the shapeoptimization of hole in isotropic plates to decrease stress concentration. For many practical applications, however, the shape 298 X. Zhang et al. of the hole is always determined by some specific demands. Hence, the optimization results sometimes cannotmeet the actual requirement. Furthermore,many plates in actual applications are orthotropic. For orthotropic plates, however, the determination of stress distribution ismore complex than that for isotropic plates. Lekhnitskii (1968, 1981) used the complex function method, developed by Muskhelishvili (1963), to determine the stress in an anisotropic plate, and gave an accurate solution of stress around an elliptic hole under in-plane loading at infinity. Some researchers have obtained analytical solutions for other non-elliptic holes (Romeo, 2001; Rao et al., 2010; Daoust and Hoa, 1991; Ukadgaonker and Kakhandki, 2005; Rezaeepazhand and Jafari, 2008). Lu et al. (2015) found an accurate analytic solution of stress for plates with an arbitrarily shaped hole using the power-series method. It can be seen from the stress analysis of an orthotropic plate that the stress distribution along the hole boundarymainly depends on the fiber orientation angle and the hole orientation angle for certain loadings and hole shapes. Therefore, how to decide the two angles of an or- thotropic plate in minimization of the stress concentration is important. Sharma et al. (2014) researched the optimization of fiber orientation angle for single lamina of composite materials (graphite/epoxy and glass/epoxy) with circular and elliptical holes; however, the hole orienta- tion angle was not taken into consideration. In this paper, not only the fiber orientation angle but also the hole orientation angle are taken into consideration, and other more complicated shapes of holes are considered, in which the maximum absolute value of the tangential stress reaches its minimum value. Herein, the Differential Evolution (DE) algorithm (Storn and Price, 1997) is used, and the following assumptions are made: the orthotropic plate is infinite and elastic under the in-plane loading at infinity (see Fig. 1); the problem can be treated as a plane stress problem. 2. Fundamental theories Although the optimization of fiber orientation angle and hole orientation of an orthotropic plate is an inverse problem, the process of obtaining the optimal results requires solving a series of forward problems. In every optimization process, the solution for the tangential stress along the hole boundary shouldbebased on thematerial properties, external loads, shapes andorientation angles of the hole. In this paper, the conformal transformation method of the complex function is adopted to firstmap the outer regions of the hole in the physical plane to the outer regions of the unit circle in the image plane.Then, the analytical solution of stress along the hole boundary can be obtained using the power-series method. 2.1. Transformation of the mapping function under different coordinates Figure 1 illustrates an orthotropic plate with an arbitrarily shaped hole under an in-plane loading. xoy is the global Cartesian coordinate system, where x and y are along the directions of σ∞x and σ ∞ y , respectively. x1oy1 and x ′oy′ are two local Cartesian coordinates. x1 is the symmetric axis of the hole, and the angle α between the positive direction of the x-axis and x1-axis denotes the hole orientation. x ′ and y′ are along the principal direction of the elastic materials, and the angleϕ denotes the fiber orientation. The outer region of the hole in the local coordinates x1oy1 (z-plane) is mapped to the outer region of the unit circle (ζ-plane) using the following mapping function z∗ =x1+iy1 =R1 ( ζ+ n∑ k=1 Ckζ −k ) (2.1) where i = √ −1, and R1 is a real constant and Ck are complex constants denoting the size and shape of the hole, respectively. ζ = ρeiθ (θ is the polar angle in the ζ-plane, ρ= 1 on the hole Optimum design of fiber angle and hole orientation... 299 boundary).When n is large enough, Eq. (2.1) can describe a variety of shapes of the hole. The mapping function in the global coordinates should be obtained using the following method due to the change of the hole orientation angle. Fig. 1. Orthotropic plate with an arbitrarily shaped hole under an in-plane loading As shown in Fig. 1, the local coordinates x1oy1 overlap the global coordinates xoy after rotatingα degrees clockwise. The vector z (z=x+iy) with a length of r in the coordinates xoy makes an angle β with the x-axis. In the global coordinates xoy z= reiβ (2.2) In the local coordinates x1oy1 z∗ = re i(β−α) (2.3) Then, Eq. (2.4) can be obtained because Eqs. (2.2) and (2.3) share the same vector z z∗ = re i(β−α) =e−iαreiβ =e−iαz (2.4) Then z=ω(ζ)= eiαz∗ =e iαR1 ( ζ+ n∑ k=1 Ckζ −k ) =R ( ζ+ n∑ k=1 Ckζ −k ) (2.5) It can be seen that themapping function in the coordinates xoy can be easily determined by Eq. (2.5) as long as the correspondingmapping function in the coordinates x1oy1 is given even if the hole orientation angleα changes. Changing the real constantR1 in Eq. (2.1) to a complex constant R (R=R1e iα) is what we need to do. 2.2. The affine transformation and mapping The three complex variables involved in solving the anisotropic plane problems are z, z1 and z2, and the affinemathematical relationships between the three complex variables are z1 =x+µ1y= γ1z+ δ1z z2 =x+µ2y= γ2z+ δ2z (2.6) 300 X. Zhang et al. where γ1 = (1− iµ1)/2, γ2 = (1− iµ2)/2, δ1 = (1+ iµ1)/2, δ2 = (1+ iµ2)/2, µ1 = α1 + iβ1, µ2 =α2+iβ2, αk and βk (k=1,2) are real constants related to the material properties. If three polar coordinates are introduced, and by transforming the outer regions of the hole in the z-, z1-, and z2-planes into the outer regions of the unit circle in the ζ-, ζ1-, and ζ2-planes, respectively, the three mapping functions which satisfy Eqs. (2.6) can then be obtained. The three mapping functions are expressed by z = ω(ζ), z1 = ω1(ζ1) and z2 = ω2(ζ2), respectively (Lu et al., 2015) z1 =ω1(ζ1)= γ1R ( ζ1+ n∑ k=1 Ckζ −k 1 ) + δ1R ( 1 ζ1 + n∑ k=1 Ckζ k 1 ) z2 =ω2(ζ2)= γ2R ( ζ2+ n∑ k=1 Ckζ −k 2 ) + δ2R ( 1 ζ2 + n∑ k=1 Ckζ k 2 ) (2.7) where ζ1 = ρ1e iθ1 and ζ2 = ρ2e iθ2. The angles θ1 and θ2 are the polar angles in the ζ1- and ζ2-planes, respectively. ρ1 = ρ2 =1 on the hole boundary. Assuming that ζ1 = ζ2 = ζ = σ = e iθ along the unit circle, the relationships between ζ, ζ1 and ζ2 can also be established γ1R ( ζ1+ n∑ k=1 Ckζ −k 1 ) + δ1R ( 1 ζ1 + n∑ k=1 Ckζ k 1 ) = γ1R ( ζ+ n∑ k=1 Ckζ −k ) + δ1R ( ζ+ n∑ k=1 Ckζ −k ) γ2R ( ζ2+ n∑ k=1 Ckζ −k 2 ) + δ2R ( 1 ζ2 + n∑ k=1 Ckζ k 2 ) = γ2R ( ζ+ n∑ k=1 Ckζ −k ) + δ2R ( ζ+ n∑ k=1 Ckζ −k ) (2.8) Equations (2.8) are suitable for any point in the region |ζ| ­ 1. Given a point ζ, then points ζ1 and ζ2 can be determined by Eqs. (2.8). z, z1, and z2 can be calculated by Eqs. (2.5) and (2.7), respectively. 2.3. Calculation of the analytical solution of stress For the plane stress problemof an orthotropic plane,when the body forces are not taken into consideration, the compatibility equation for Airy’s stress function F = F(x′,y′) in the local coordinates x′oy′ can be given as (Lekhnitskii, 1968, 1981) a′22 ∂4F ∂x′4 −2a′26 ∂4F ∂x′3∂y′ +(2a′12+a ′ 66) ∂4F ∂x′2∂y′2 −2a′16 ∂4F ∂x′∂y′3 +a′11 ∂4F ∂y′4 =0 (2.9) where a′ij are constants related to the material properties. The solution to Eq. (2.9) is related to the roots of the following characteristic equation a′11µ ′4−2a′16µ′ 3 +(2a′12+a ′ 66)µ ′2−2a′26µ′+a′22 =0 (2.10) The four conjugate complex roots inEq. (2.10) areµ′1,µ ′ 1,µ ′ 2 andµ ′ 2,whichcanbecalculated by the principal elastic constants. However, the complex roots in the global coordinates xoy should be calculated due to the change of the fiber orientation.When the local coordinatesx′oy′ overlap the global coordinates xoy after rotating ϕ degrees counter clockwise, µ1 and µ2 in the global coordinates can be given as (Lekhnitskii, 1968) µ1 = µ′1cosϕ− sinϕ cosϕ+µ′1 sinϕ µ2 = µ′2cosϕ− sinϕ cosϕ+µ′2 sinϕ (2.11) Optimum design of fiber angle and hole orientation... 301 In this paper, only the situation of µ1 6= µ2 is discussed. The solution to Eq. (2.9) can be expressed by the two analytical functions F1(z1) and F2(z2) as F =2Re[F1(z1)+F2(z2)] (2.12) where z1 and z2 are exactly the same as given in the previous Sections. Let Φ1(z1) = dF1(z1)/dz1 and Φ2(z2) = dF2(z2)/dz2, respectively. The stress boundary conditions on the edge of the hole expressed byΦ1(z1) andΦ2(z2) can be given as 2Re[Φ1(z1)+Φ2(z2)] = f1 2Re[µ1Φ1(z1)+µ2Φ2(z2)]= f2 (2.13) where z1 and z2 are two points on the boundary. The problem discussed here is an infinite field with holes, and no loads exist along the edge of the hole. Therefore, f1 = f2 =0, and Φ1(z1) andΦ2(z2) can be given in the following form Φ1(z1)=B ∗z1+Φ 0 1(z1) Φ2(z2)= (B ′∗+iC′ ∗ )z2+Φ 0 2(z2) (2.14) whereB∗,B′ ∗ , andC′ ∗ can be determined according to the stress components acting at infinity (i.e., σ∞x , σ ∞ y and τ ∞ xy) as B∗ = σ∞x +(α 2 2+β 2 2)σ ∞ y +2α2τ ∞ xy 2[(α2−α1)2+(β22 −β21)] B′ ∗ = (α21−β21 −2α1α2)σ∞y −σ∞x −2α2τ∞xy 2[(α2−α1)2+(β22 −β21)] C′ ∗ = (α1−α2)σ∞x +[α2(α21−β21)−α1(α22−β22)]σ∞y +[(α21−β21)− (α22−β22)]τ∞xy 2β2[(α2−α1)2+(β22 −β21)] Here, Φ01(z1) and Φ 0 2(z2) in Eqs. (2.14) should have the following form after substituting z1 =ω1(ζ1) and z2 =ω2(ζ2) into them Φ01(z1)=Φ 0 1[ω1(ζ1)] = ∞∑ k=0 akζ −k 1 Φ 0 2(z2)=Φ 0 2[ω2(ζ2)] = ∞∑ k=0 bkζ −k 2 (2.15) where ak = a1k + ia2k, bk = b1k + ib2k, a1k, a2k, b1k and b2k are undetermined real constants (k=0, . . . ,∞). Along the edge of the hole, ζ1 = σ = e iθ and ζ2 = σ = e iθ. The undetermined coefficients ak and bk can be determined from Eqs. (2.13)-(2.15) by using the power-series method. Then, the stress components σx, σy and τxy in the Cartesian coordinates can be determined by the following equations (Lu et al., 2015) σx =2Re[µ 2 1Φ ′ 1(z1)+µ 2 2Φ ′ 2(z2)] σy =2Re[Φ ′ 1(z1)+Φ ′ 2(z2)] τxy =−2Re[µ1Φ′1(z1)+µ2Φ′2(z2)] (2.16) where Φ′1(z1)=B ∗− n∑ k=1 kakζ −k−1 1 /[ γ1R ( 1− n∑ k=1 kCkζ −k−1 1 ) + δ1R ( − 1 ζ21 + n∑ k=1 kCkζ k−1 1 )] Φ′2(z2)=B ′∗+iC′ ∗− n∑ k=1 kbkζ −k−1 2 /[ γ2R ( 1− n∑ k=1 kCkζ −k−1 2 ) + δ2R ( − 1 ζ22 + n∑ k=1 kCkζ k−1 2 )] (2.17) 302 X. Zhang et al. Thestress componentsσρ,σθ and τρθ in orthogonal curvilinear coordinates canbedetermined by the following equations σρ+σθ =σx+σy σθ−σρ+2iτρθ = ζ2 ρ2 ω′(ζ) ω′(ζ) (σy −σx+2iτxy) (2.18) Along the hole boundary where ζ1 = ζ2 = ζ = σ = e iθ and ρ = 1, the analytical solution of stress can be solved easily by Eqs. (2.16)-(2.18). In this way, when the mapping functions are known and the material parameters and external loads are given, no matter how the fiber orientation angle and hole orientation angle change, the analytical solution of the stress along and near the hole boundary can be calculated. In addition, this analytical solution of the stress is suitable for an arbitrarily shaped hole. 2.4. Optimization Tangential stress concentration along the boundary of the hole will cause damage if its magnitude exceeds the material strength. Consequently, the optimization criterion we take is that the absolute maximum value of the tangential stress along the hole boundary should be kept at theminimum. In the optimization, the fiber orientation angleϕ and the hole orientation angle α are the unknown variables. A set of the initial values of ϕ and α are given; then, the tangential stress with the maximum absolute value max |σθ| could be obtained through the complex variable method. By revising the values of design variables ϕ and α, the final values of ϕ̂ and α̂ that lead to the minimal value of max |σθ| should be the optimized results. The mathematical model could be expressed as follows F(X) = max θ∈[0,2π] |σθ| minF(X) X ∈D= {ϕ,α} s.t. 0¬ϕ¬ 2π 0¬α¬ 2π (2.19) The original DE algorithm (Storn and Price, 1997) is used in the computations. It can be written as DE/rand/1/bin. As defined by Storn and Price, the DE algorithm is characterized by three main parameters NP , F andCR as xi,G i=1,2, . . . ,NP vi,G+1 =xr1,G+F(xr2,G−xr3,G) uji,G+1 = { vji,G+1 if (randb(j)¬CR) ∨ j= rnbr(i) xji,G if (randb(j)>CR) ∧ j 6= rnbr(i) j=1,2, . . . ,D (2.20) In this paper, the number of population vectorsNP equals 200. The real and constant factor F equals 0.8, and the crossover constant CR is equal to 0.5. 3. Examples As shown in Fig. 1, the external loads are chosen as σ∞x : σ ∞ y = 2 : 7 and τ ∞ xy = 0. Sign convention is defined as positive for tension and negative for compression. It can be seen from the results of the following examples that all of the maximum tangential stresses are tensile stresses. The values of the independent principal elastic constants in the local coordinates x′oy′ are:E1 =1.4 ·105Pa, υ12 =0.46 andG12 =1.2 ·104Pa, where the subscripts 1 and 2 represent the directions along the x′- and y′-axis, respectively. To analyze the effect of Young’s modulus on the optimization results, we take different values of E2/E1, i.e., 0.2, 0.5, 2.0 and 5.0. The Optimum design of fiber angle and hole orientation... 303 material coefficients in Eq. (2.9) can be obtained by the three elastic constants as a′11 =1/E1, a′22 =1/E2, a ′ 16 = a ′ 26 =0, a ′ 12 =−υ12/E1, and a′66 =1/G12 (Lekhnitskii, 1968; Chen, 1994). The hexagonal hole and the elliptical hole are selected as examples. The size of the hole has no influence on the calculation of the stress field, because the domain occupied by the plate is infinite. In order not to lose the generality, take R1 = 1.0. The mapping function of Eq. (2.5) can be described as z=ω(ζ)= 1.0eiα ( ζ+ n∑ k=1 Ckζ −k ) (3.1) The imaginary part of the coefficientCk is equal to zero, because the hole shapes of the examples are all symmetric about the x1-axis. For the elliptical hole: a/b=1.5, n=1,C1 =(a− b)/(a+b)= 0.2. For the hexagonal hole: n=29,C5 =0.0667, C11 =0.0101, C17 =0.0036, C23 =0.0018 and C29 =0.0010 (Savin, 1961). The other Ck are all equal to zero. 3.1. Optimization of the fiber orientation angle ϕ In this Section, only the fiber orientation angle ϕ is taken as the design variable, while the external loads and hole orientation angle α are given. The hole orientation angle α shown in Fig. 1 is set to 0◦ (see Fig. 2), meaning that the coordinates x1oy1 coincide with xoy. The local coordinatesx′oy′ still follow the fiber orientation, coincidingwithxoy after rotation byϕdegrees counterclockwise. The scope of the optimized variable isϕ∈ 0◦-90◦ because of the symmetry of the external loads and the hole shape. Fig. 2. The location of holes with two coordinates: (a) elliptical hole, (b) hexagonal hole 3.1.1. Elliptical hole The optimal fiber orientation and its corresponding tangential stress along the boundary of the elliptical hole are illustrated in Figs. 3 and 4, respectively. The optimized fiber orientation angles and the maximum tangential stresses are listed in Table 1. The global coordinates xoy in Fig. 3 are in accordance with that in Fig. 2a, and the oblique lines in Fig. 3 represent the fiber directions in the x′-axis (Fig. 2a) for different values ofE2/E1. It should be noted that the 304 X. Zhang et al. Fig. 3. The optimal fiber orientation of the orthotropic plate with the elliptical hole Fig. 4. Tangential stresses along the boundary of the elliptical hole at the optimal fiber orientation Table 1.The optimal fiber orientation angles and themax |σθ| of an orthotropic plate with the elliptical hole E2/E1 0.2 0.5 2.0 5.0 ϕ [◦] 30.46 39.89 49.96 56.49 max |σθ| [MPa] 21.84 23.34 28.50 32.04 angle θ in Fig. 4 is the polar angle in the ζ-plane and θ=0◦ corresponds to the intersection of the positive x-axis and the excavation boundary. It can be seen fromFigs. 3 and 4 andTable 1 that both the optimal fiber orientation angleϕ and themaximum tangential stress increase with the increasing value ofE2/E1. Themaximum tangential stress concentration is evident in approximately 30◦, 150◦, 210◦ and 330◦, not similar to the isotropic plate, where themaximumtangential stress usually occurs at 0◦ and180◦.Under the given loading condition, the compressive stress can be found around the intersection of the y-axis and the hole boundary with a smaller magnitude than the tensile stress. Considering that there is only one variable in this case, the optimal fiber orientation angleϕ can be obtainedwithout any optimization algorithm. Figure 5 illustrates themaximumabsolute tangential stress max |σθ| for different fiber orientation angles, showing that theminimumvalue of max |σθ| and the corresponding fiber orientation angles are identical with the results listed in Table 1, which verifies the results obtained by the DE algorithm. For an elliptical hole, the stresses have also been analyzed byUkadgaonker and Rao (2000). To verify the fundamental theories of Section 2, we compared the normalized stresses of σx, σy and τxy along the hole boundary with the results obtained by Ukadgaonker and Rao (2000, Optimum design of fiber angle and hole orientation... 305 Fig. 5. The max |σθ| for different fiber orientation angles Table 2. Normalized stresses of σx, σy and τxy for equi-biaxial tension on a graphite/epoxy plate, 60◦ fibers containing an elliptical hole (a/b=2) θ [deg] Ukadgaonker and Rao (2000) Zhang et al. σx σy τxy σx σy τxy 0 0 3.44 0 0 3.4393 0 20 0.82 1.54 −1.12 0.8172 1.5421 −1.1226 40 1.46 0.52 −0.87 1.4589 0.518 −0.8693 60 1.54 0.13 −0.45 1.5423 0.1285 −0.4452 80 1.41 0.01 −0.12 1.4076 0.0109 −0.1241 90 1.3 0 0 1.3033 0 0 100 1.18 0.01 0.1 1.1812 0.0092 0.1041 120 0.89 0.07 0.26 0.8867 0.0739 0.256 140 0.57 0.2 0.34 0.5651 0.2007 0.3367 160 1.23 2.32 1.69 1.2272 2.316 1.6859 180 0 3.44 0 0 3.4393 0 page 348, Table 3) in Table 2. Taking the same values of material parameters and external loadings,Table 2 shows that the results obtainedby the twopapers are in avery close agreement. 3.1.2. Hexagonal hole The optimized fiber orientation angles and their corresponding tangential stresses along the boundary of the hexagonal hole are illustrated in Figs. 6 and 7, respectively. The maximum tangential stress occurs at the points marked with an asterisk. When the ratios of the elastic moduli are E2/E1 = 0.2, 0.5, 2.0 and 5.0, the corresponding optimal fiber orientation angles are ϕ= 32.15◦, 40.47◦, 42.74◦ and 57.90◦, respectively (shown in Fig. 6), which shows a great difference for the different ratios ofE2/E1. Figure 7 illustrates that the tangential stress reaches the corresponding extreme values in/near the corner points. The maximum tangential stresses for different E2/E1 are found in different positions but are still around the corner points, and the values of themaximum tangential stress range from 42.89MPa (E2/E1 =0.5) to 70.77MPa (E2/E1 = 5.0), meaning that the location and magnitude of the maximum tangential stress are highly dependent on Young’s modulus for a given hole shape and loading condition. We can also see that the tangential stresses along the boundary are mainly tensile stresses with the exception of the two sides BC and EF, which are parallel to the direction of the minimum loading. Moreover, the magnitude of the compressive stress is much smaller than that of the tensile stress. 306 X. Zhang et al. Comparing with the results in Fig. 4, the stress concentration along the boundary of the hexagonal hole is much larger than that along the boundary of the elliptical hole because there are sharp corners on the hexagonal hole. Fig. 6. The optimal fiber orientation of the orthotropic plate with the hexagonal hole Fig. 7. Tangential stresses along the boundary of the hexagonal hole at the optimal fiber orientation 3.2. Optimization of the fiber and hole orientation angles Considering that the hole orientation angle also influences the tangential stress along the boundary of the hole, both the fiber orientation angle and the hole orientation angle are chosen as the design variables in this Section with the external loads given (refer Fig. 1). Optimum design of fiber angle and hole orientation... 307 3.2.1. Elliptical hole Because of the symmetry of the external loads and the elliptical hole, the scopes of optimized variables are given asα∈ 0◦-180◦ andϕ∈ 0◦-90◦. The optimal fiber and hole orientation angles and the corresponding tangential stress along the boundary of the hole are illustrated in Figs. 8 and 9, and Table 3, respectively. Fig. 8. The optimal fiber and hole orientation angles of the orthotropic plate with the elliptical hole Fig. 9. Tangential stresses along the boundary of the elliptical hole at the optimal fiber and hole orientation Table 3.The optimal fiber and hole orientation and themax |σθ| of the orthotropic plate with the elliptical hole E2/E1 0.2 0.5 2.0 5.0 ϕ [◦] 33.64 39.25 53.98 66.13 α [◦] 90.95 89.29 92.43 94.84 max |σθ| [MPa] 12.85 14.33 18.12 20.11 It can be seen from Fig. 8 and Table 3 that the optimal fiber orientation angle increases greatly with an increase in the ratio of E1/E2. Nevertheless, the hole orientation angles are all near the same degree of 90◦, that is to say, the major axis of the elliptical hole should be set in the direction of themaximum external load. It can also be verified by comparing Figs. 4 and 9. Under the given loadings, vertical elliptical holes (Fig. 9) always produce smaller tangential stress concentration compared to the horizontal ones (Fig. 4). In addition, the similar laws of the two cases are that the maximum tangential stress increases with the ratio of E2/E1 and compressive stress is found in Fig. 9 as well. 308 X. Zhang et al. 3.2.2. Hexagonal hole Because a hexagonal hole hasmore symmetry axes, the scopes of the optimized variables are reduced to α∈ 0◦-60◦ and ϕ∈ 0◦-90◦. Figures 10-13 show the optimal placements of the fiber and hole orientation angles and the corresponding tangential stress along the hole boundary. It should be noted that the angle θ=0◦ corresponds to the intersection of the positive x1-axis and the hole in this case. Fig. 10. The optimal placement of fiber and hole orientation angles of orthotropic plates and the tangential stresses along the boundary of the hexagonal hole whenE2/E1 =0.2 Fig. 11. The optimal placement of fiber and hole orientation angles of orthotropic plates and the tangential stresses along the boundary of the hexagonal hole whenE2/E1 =0.5 Fig. 12. The optimal placement of fiber and hole orientation angles of orthotropic plates and the tangential stresses along the boundary of the hexagonal hole whenE2/E1 =2.0 It can be seen fromFigs. 10-13 that both the optimized fiber orientation angle and the hole orientation angle are apparently different for different ratios of E2/E1. Under the given inequi- Optimum design of fiber angle and hole orientation... 309 Fig. 13. The optimal placement of fiber and hole orientation angles of orthotropic plates and the tangential stresses along the boundary of the hexagonal hole whenE2/E1 =5.0 -biaxial tensile loading, compressive tangential stress may occur and is more likely to be found in the sides that are parallel (BC andEF in Figs. 6 and 7), or inclined (BC andEF in Figs. 11, AB andDE in Figs. 12 and 13) to the direction of the minimum external loading. Combiningwith the above example on the hexagonal hole (see Section 3.1.2), wefind that for holes with sharp corners, the tangential stress concentration is more likely to occur around the corner points but not always exactly in the corner points. The pointD in Fig. 11 gives themost obvious evidence. In addition, the tangential stress concentration is smaller when both the fiber and hole orientation angles are optimized compared to the case in Section 3.1. Thus, when we design a perforated structure in an orthotropic plate, the two parameters should be considered jointly to decrease the tangential stress concentration along the boundary of the hole. From all of the examples given in Sections 3.1 and 3.2, we obtain that for a significantly orthotropic plate (E2/E1 = 0.2 or 5.0) with a central hole, setting the principal direction that has the larger Young’s modulus inclined to the direction of theminimum loading may decrease the tangential stress concentration along the hole boundary. 3.2.3. Optimality verification of fiber and hole orientation angles The optimality verification should be carried out to guarantee that the optimized fiber and hole orientation angles, ϕ and α, are the global optimal solutions. Taking the hexagonal hole as an example, only one condition is analyzed when the ratio of the elastic modulus is given as E2/E1 =2.0.The external loads andother parameters are the sameas in the previous examples. The variation of the maximum tangential stress max |σθ| with respect to ϕ, α and the contour map is illustrated in Fig. 14. The degree intervals of ϕ and α are all set as 0.01◦. It can be seen fromFig. 14 that the value ofmax |σθ| differs for differentϕ andα. According to the calculation results,max |σθ| reaches theminimumvalue of 47.30MPaonlywhenϕ=55.53◦ andα=54.60◦, which verifies the results shown in Fig. 12. 4. Conclusions Todecrease the tangential stress concentration aroundan arbitrarily shapedhole in an orthotro- pic plate, the optimization design on the fiber orientation angle and the hole orientation angle are conducted. The tangential stress distributions around non-circular shapes are given. The results show that the tangential stress concentration around a hole with corner points in an or- thotropic plate under an inequi-biaxial loading ismore likely to occur in/near the corner points. Furthermore, the tangential stress concentration around a hole with corner points is found to 310 X. Zhang et al. Fig. 14. The three-dimensional map and contourmap of max |σθ| for all possible placements of the fiber and hole orientation angles be more significant compared to that around a smooth convex hole. Under an inequi-biaxial loading, compressive stress can usually be found and is mainly located at the sides that are parallel or inclined to the direction of the minimum external loading. Based on the results, the following several treatments can be referred in the designing of orthotropic plates in order to decrease the tangential stress concentration: (1) avoid holes with corner points and choose smooth convex holes instead, such as an ellipse hole; (2) set themajor axis of an ellipse in the direction of the maximum external loading and (3) set the principal direction that has the larger Young’s modulus inclined to the direction of theminimum loading, especially for a significantly orthotropic plate. Funding This researchworkwas supported by theNatural Science Foundation of China (GrantNo. 11572126, 51704117) and the Fundamental Research Funds for the Central Universities (NCEPU2016XS55). References 1. Bjorkman G.S., Richards R., 1976, Harmonic holes – an inverse problem in elasticity, Journal of Applied Mechanics, 43, 3, 414-418 2. Bjorkman G.S., Richards R., 1979, Harmonic holes for nonconstant fields, Journal of Applied Mechanics, 46, 3, 573-576 3. Chen Z.Y., 1994,Analytical Method of RockMechanics Analysis, ChinaCoal Industry Publishing House, Beijing 4. Daoust J., Hoa S.V., 1991, An analytical solution for anisotropic plates containing triangular holes,Composite Structures, 19, 2, 107-130 5. Dhir S.K., 1981, Optimization in a class of hole shapes in plate structures, Journal of Applied Mechanics, 48, 4, 905-908 6. Engels H., Zakharov D., Becker W., 2001, The plane problem of an elliptically reinforced circular hole in an anisotropic plate or laminate,Archive of Applied Mechanics, 71, 9, 601-612 Optimum design of fiber angle and hole orientation... 311 7. Jain N.K., 2009, Analysis of stress concentration and deflection in isotropic and orthotropic rec- tangularplateswith central circularholeunder transverse static loading,WorldAcademy of Science Engineering and Technology, 3, 12, 1513-1519 8. Lekhnitskii S.G., 1968,Anisotropic Plates, Gorden and Breach, NewYork 9. Lekhnitskii S.G., 1981,Theory of Elasticity of an Anisotropic Body, Mir Publishers,Moscow 10. Li C., Zheng Y.P., 2007, Influence of different orifice figure on hole-edge stress of composite material plate with hole (in Chinese),Engineering Mechanics, 24, 10, 19-24 11. Lu A.Z., Chen H.Y., Qin Y., Zhang N., 2014a, Shape optimisation of the support section of a tunnel at great depths,Computers and Geotechnics, 61, 3, 190-197 12. Lu A.Z., Chen H.Y., Qin Y., Zhang N., 2014b, Shape optimization of tunnel support section under contact condition of pure slip, Chinese Journal of Rock Mechanics and Engineering, 33, 8, 1563-1571 13. LuA.Z.,ZhangN.,ZhangX.L.,LuD.H.,LiW.S., 2015,Analyticmethodof stress analysis for an orthotropic rockmasswith an arbitrary-shaped tunnel, International Journal of Geomechanics, 15, 4, 04014068 14. Muskhelishvili N.I., 1963, Some Basic Problems of the Mathematical Theory of Elasticity, No- ordhoff, Groningen 15. RaoD.K.N.,BabuM.R.,ReddyK.R.N., SunilD., 2010,Stress aroundsquareand rectangular cutouts in symmetric laminates,Composite Structures, 92, 12, 2845-2859 16. Ren G., Smith J.V., Tang J.W., Xie Y.M., 2005,Underground excavation shape optimization using an evolutionary procedure,Computers and Geotechnics, 32, 2, 122-132 17. RezaeepazhandJ., JafariM., 2008,Stress analysis of compositeplateswithnon-circularcutout, Key Engineering Materials, 385-387, 365-368 18. Romeo G., 2001, Analytical and experimental behavior of laminated panels with rectangular opening under biaxial tension, compression and shear loads, Journal of Composite Materials, 35, 8, 639-664 19. Savin G.N., 1961, Stress Concentration Around Holes, PergamonPress, NewYork 20. Sharma D.S., 2011, Stress concentration around circular/elliptical/triangular cutouts in infinite composite plate,Proceedings of the World Congress on Engineering, 3, 2626-2631 21. Sharma D.S., Patel N.P., Trivedi R.R., 2014, Optimum design of laminates containing an elliptical hole, International Journal of Mechanical Sciences, 85, 8, 76-87 22. Sobótka M., Łydżba D., Różański A., 2013, Shape optimization of underground excavation by simulated annealing, Studia Geotechnica et Mechanica, 35, 1, 209-218 23. Storn R., Price K., 1997, Differential evolution – a simple and efficient heuristic for global optimization over continuous spaces, Journal of Global Optimization, 11, 4, 341-359 24. Toubal L., KaramaM., LorrainB., 2005, Stress concentration in a circular hole in composite plate,Composite Structures, 68, 1, 31-36 25. Ukadgaonker V.G., Kakhandki V., 2005, Stress analysis for an orthotropic plate with an irregular shaped hole for different in-plane loading conditions – Part 1,Composite Structures, 70, 3, 255-274 26. UkadgaonkerV.G., Rao D.K.N., 2000,A general solution for stresses around holes in symme- tric laminates under in plane loading,Composite Structures, 49, 3, 339-354 Manuscript received September 9, 2016; accepted for print September 21, 2017