Issue 
Matériaux & Techniques
Volume 105, Number 3, 2017



Article Number  301  
Number of page(s)  13  
Section  Matériaux désordonnés : verres, vitrocéramiques... / Disordered materials: glasses, clays, vitroceramics...  
DOI  https://doi.org/10.1051/mattech/2017033  
Published online  06 October 2017 
Regular Article
Stress analysis of symmetric laminates with rectangular or square holes subjected to inplane loading
^{1}
ISAECnam Liban, Department of mechanical engineering, Lebanese university,
Beirut, Lebanon
^{2}
Faculty of engineering 1, Department of mechanical engineering, Lebanese university,
Tripoli, Lebanon
^{*} email: tjabbour@cnam.fr
Received:
10
May
2017
Accepted:
13
June
2017
In this paper we propose an analytical method for determining the behavior of symmetric laminates containing a rectangular or a square hole. This method is based on the theory developed by Savin [G.N. Savin, Stress concentration around holes, Pergamon Press, New York, 1961], for an elliptical hole, and uses the parametric representation of a given rectangle. The results obtained by this method are compared with those obtained by finite elements. This method allows the better understanding of the behavior of composite plates under inplane loading for different types of loading, different side ratios, different fillet radii, and different types of materials.
Key words: composite materials / symmetric laminate / stress concentration factor / rectangular hole / square hole
© EDP Sciences, 2017
1 Introduction
Virtually, every structure contains rivets and connections of some sort. The necessity of holes in plates, for instance, stems from the need to gain access to both sides of the plate, or the need to use that plate as a connector. The presence of cutouts in structural components is, therefore, a requirement. It is well known that cuts and flaws in stressed members induce stress concentrations, which can greatly weaken the strength of those members and, as consequence, a reliable procedure to calculate those stresses is of interest.
The problem of determining the stress concentration around a hole in an isotropic plate has been solved and there is an abundance of work published in the literature on this subject. In particular, the problem of calculating the stress concentration around a circular opening in an infinite isotropic plate has been completely solved and a closed form solution exists [1]. However, for anisotropic plates, the procedures used to arrive at such a solution are much more complex, especially if the cut takes on forms other than circular, for example, in the shape of a polygon.
Rectangular holes in anisotropic plates find their applications in the design of aircraft components and in the study of various noncircular openings in composite fuselages, such as windows, doors, and access holes in modern planes.
Investigations of plates containing rectangular holes have been performed by Savin [2] and Lekhnitskii [3]. However, the work proposed by Savin [2] treats mainly isotropic materials via conformal mapping [4,5]. This method from complex analysis has proven to be an effective tool in solving problems of this type and has been originally introduced by Mushkhelishvili [6]. The works by Lekhnitskii [3] and by Zuxing and Yuansheng [4] which are also based on Mushkhelishvili's work, provide a solution to rectangular holes in orthotropic plates, without any extension to generally anisotropic materials. An analytical method is proposed by Nageswara et al. [5] for the case of symmetric laminates and by Anil et al. [6] for the case of composite laminate with rectangular cutout under biaxial loading. A calculation procedure based on the finite element method is proposed by Özben and Arslan [7] to determine the stresses in the plates with rectangular holes. Ukadgaonker [8] used the Stroh formalism to calculate the stress field in plates containing rectangular and square holes under arbitrary biaxial loading condition while Dave et al. [9] studied the case of unsymmetric composite plates with variance of ovalshaped cutout. An interesting work is proposed by Hwu [10] for the analysis of anisotropic plates containing polygonal holes of various shapes and subjected to uniform loading (in the x and y directions) or pure bending. This work is based on Stroh's formalism but it is not valid for the calculation of the plates in the presence of shear loading.
In this paper, we propose a method to calculate the stress field around a rectangular hole in a symmetric laminate. This method is based on Savin's formulation [1], for an elliptical hole, and uses the parametric representation of a given rectangle [3]. The results are validated analytically by finite element method. The originality of this work is that it allows a better understanding of the behavior of composite plates subjected to inplane loading (xaxis, yaxis, and shear loading) for different types of discontinuity and for different types of materials and to determine the critical case to which the stress field around a discontinuity of this plate grows infinitely.
2 Mapping function for the rectangle
Consider an infinitely large anisotropic plate with a small rectangular opening, subjected to inplane tensile, compressive and/or shear loading at infinity. It is required to determine the stress field around the contour of the hole. The infinite plate with a cutout, in the xyplane is mapped onto the outside of the unit circle, in the (ξ, η) plane (Fig. 1).
The following quantities are defined (1) where i is the unit pure imaginary number, ρ and θ denote curvilinear coordinates and σ is the value of on ζ the unit circle.
The contour of the rectangle can be represented in parametric form by the following equations [3] (2) where 0 < c ≤ 1, ε is the parameter which controls the curvature of the angles of the rectangle. When c = 1, equations (2) produce a square, and when ε = 0, we obtain an ellipse of semi major axes a and ac.
On the contour of the unit circle, equations (2) can be rewritten (3)
Thus, the mapping function which represents an infinite plate with a rectangular opening onto the outside of the unit circle is given by (4)
When is substituted for ζ in equation (4), we obtain the mapping function which represents the infinite plate onto the inside of the unit circle, which is (5)
Fig. 1 Mapping of an infinite plate with a rectangular opening onto the outside of the unit circle. 
3 Fundamental relations for an anisotropic material
The stressstrain relations for a symmetric laminate (Jones [12]) are given by (6) where A_{ij} are the elements of the symmetric stiffness matrix. By inversion of the stiffness matrix we obtain the compliance matrix whose elements are denoted by a_{ij}.
For twodimensional problems, in the absence of body forces, the equilibrium equations will be satisfied if we set (7) where U(x,y) is the Airy's stress function which must satisfy (8)
For an especially orthotropic material a_{16} = a_{26} = 0, and equation (8) reduces to (9)
The general solution of equation (8) depends on the roots of the characteristic equation (10)
Lekhnitski [3] has proved that equation (10) cannot have real roots. Therefore, assuming unequal complex roots s_{1}, s_{2}, s_{3}, s_{4}, we set (11) with β_{1} > 0, β_{2} > 0 and β_{1} ≠ β_{2}
U(x,y) can be written as (12) where z_{1} and z_{2} are defined by the affine transformations (13)
F_{1}, F_{2} are two analytic functions and and are their respective conjugates. (14) (15)
From equations (7), (12), (14) and (15) the stress components in rectangular coordinates are obtained as (16) where Re denotes the real part of a complex number and the primes (′) denote differentiation of the functions with respect to their corresponding arguments.
4 Determination of stresses around a rectangular opening
It is assumed that at infinity the stress state is given by (17)
For a stressfree hole, the derived boundary conditions are given by the equations of Savin [2] by (18) where B*, B'*, and C'* are defined as follows (19)
From equations (5) and (13) we get (20) (21)
Differentiating equation (20) with respect to z_{1} and equation (21) with respect to z_{2}, and solving for and , we get (22) (23)
Substituting z_{1} and z_{2} from equations (20) and (21) into equations (18), we obtain, on the contour of the unit circle (24) where coefficients K_{1} through K_{8} are given as follows (25)
For a stressfree hole and ψ(z_{2}) in equations (16) are given by Savin [1] by (26) where and ψ_{0}(z_{2}) are obtained by substituting z_{1} and z_{2} for ζ in and Ψ_{0}(ζ), respectively. The latter functions are given by Savin [1] as follow (27)where the integration is performed around the unit circle γ.
In order to carry out the integration in equation (30), use is made of the Schwartz formula (28) where U(θ) is the value of the real part of F(ζ) on the contour of the unit circle and α_{0} is some constant.
Substituting and from equations (24) into equations (27) and applying the Schwartz formula, we determine and Ψ_{0}(ζ) as (29) (30)
Differentiating equations (29) and (30) with respect to ζ we get (31) (32) where in equations (29–32) is the complex conjugate of K_{i}.
At this point, we need to determine ζ as a function of z_{1} from equation (20) to put in equation (29) and ζ as a function of z_{2} from equation (21) to put in equation (30), to obtain ϕ_{0}(z_{1}) and ψ_{0}(z_{2}). Putting these functions in equations (26) we determine ∅(z_{1}) and ψ(z_{2}). Upon substitution of these latter functions in equations (16), we obtain the stress field around the rectangular hole, which is given by (33)
However, solving for ζ in terms of z_{1} and z_{2} requires finding the roots of two sixth degree complex polynomials for which no formula is possible. Instead, we employ the chain rule for differentiation to express the functions ∅′_{0}(z_{1}) and ψ′_{0}(z_{2}) in equations (33) as (34)
In accordance with equations (34), the stress field around the hole (Eq. (33)) becomes (35) where , , and are determined, respectively, from equations (22), (23), (31) and (32).
For transforming the given hole in Cartesian coordinate plane xoy to the orthogonal curvilinear coordinate axes given by ρ = const., θ = const., conformal transformation is applied using the transformation function z = ω(ζ) = ω(ρe_{iθ}) as given in equation (5). The purpose of applying conformal transformation is to ensure that for an infinite area weakened by the hole of any shape, one of the coordinate lines, ρ = const. should concide with the contour of the hole. The stresses in Cartesian coordinates given by equation (35) are transformed into orthogonal curvilinear coordinate system (ρ, θ) using the following relations: (36)
The method proposed by Timoshenko [13], allows to determine the relationship between the angle α, between the normal to the hole boundary and xaxis, and the angle θ. In this study, the angle θ can be expressed as (37)
5 Results and discussions
To validate the proposed method, we considered the case of Graphite/epoxy and Glass/epoxy [0,90]_{s}, [60/0/−60]_{s}, [0/45/0/45/0] and [0/±45/90]_{s} laminates with square and rectangular holes. The laminates are subjected to tension in x, ydirections, equibiaxial tension and shear loading. Also, for calculating the complex parameters s_{1} and s_{2}, we considered for the unidirectional lamina two materials, whose properties [13] are shown in Table 1. The complex parameters, for each laminate, are listed in Table 2.
Properties of the unidirectional lamina.
Complex parameters s_{1} and s_{2} for different laminates.
5.1 Validation of the proposed method
To begin our study, we first validated the proposed calculation method by considering the laminate [0,90]_{s} in Graphite/epoxy for the square hole shown in Figure 2a (obtained for c = 1 and ε = −1/7) and the laminate [0/±45/90]_{s} in Glass/epoxy for the rectangular hole of side ratio R1.5 shown in Figure 2b (obtained for c = 0.7 and ε = −1/9). We compared the results with those obtained by finite element (FE) by using the Abaqus® simulation software for the calculation of the stresses on the contour of each hole. The meshing of each plate is made with quadrangles. Figures 3 and 4 show a part of the finite element models of the plates. Note in this figure that a very fine mesh size of 0.1 mm, is considered near the contour of each hole because of the high rate of change of stress.
The results obtained analytically (generated at a rate of variation of the rotation angle of one degree) and those obtained by finite elements (generated at a rate of variation of the rotation angle of 2.5 degrees) are shown in Figures 5–7, for the case of square hole, and in Figures 9–11 for the rectangular hole. These figures show the variation of the stress concentration factor on the edge of each hole for the cases of xaxis loading, equibiaxial loading and shear loading. This factor is defined as the ratio between the stress at a given point of the plate, corresponding to an angle of rotation, and the stress applied to the plate. In Figures 8 and 12 we showed, for each type of loading, the variation of the curvilinear stresses resulting from the application of equation (37).
Figures 5–7, 9–11 show that there is good agreement between the values of the stress concentration factor obtained by the proposed method and the finite element method which prove the validity of the proposed method.
In Figures 5, 6, 9 and 10 the distribution of the stresses σ_{x} and σ_{y} on the contour of the hole is symmetrical relative to the axes x and y while the distribution of τ_{xy} is antisymmetric with respect to these axes. In Figures 7 and 11 the distribution of the stresses σ_{x} and σ_{y} is antisymmetric relative to the axes x and y while the distribution of τ_{xy} is symmetrical with respect to these axes.
In Figures 8 and 12, we can notice that the stresses σ_{ρ} and τ_{ρθ} are almost zero on the contour of the hole, which is normal because the hole edge is free of stresses. Moreover, in the case of axial and equibiaxial loading, the distribution of σ_{θ} is symmetrical relative to the axes x and y, while it is antisymmetric with respect to these axes in the case of shear loading.
By comparing Figures 5–12, it is clear that, for the same geometry and for the same material, the stress distribution varies depending on the nature of the applied load. Similarly, the change in geometry, of the material and of the stacking sequence also have the effect of changing the value and the position of the maximum stress concentration factor. We can also conclude from Figures 8 and 12 that the maximum stress concentration factor is obtained in the case of shear loading. For this reason, the effect of geometry, stacking sequence, the type of loading, material variation and angle of loading are discussed in the following.
Fig. 2 Shapes of square and rectangular holes considered in the calculations. 
Fig. 3 Meshing of the plate and contour plot of the stress around the rectangular hole. 
Fig. 4 Meshing of the plate and contour plot of the stress around the square hole. 
Fig. 5 Variation of σ_{x}/σ, σ_{y}/σ, and τ_{xy}/σ on the contour of the square hole in Graphite/epoxy for the case of xaxis loading. 
Fig. 6 Variation of σ_{x}/σ, σ_{y}/σ, and τ_{xy}/σ on the contour of the square hole in Graphite/epoxy for the case of equibiaxial loading. 
Fig. 7 Variation of σ_{x}/σ, σ_{y}/σ, and τ_{xy}/σ on the contour of the square hole in Graphite/epoxy for the case of shear loading. 
Fig. 8 Variation of σ_{ρ}/σ, σ_{θ}/σ, and τ_{ρθ}/σ on the contour of the square hole in Graphite/epoxy for different types of loading. 
Fig. 9 Variation of σ_{x}/σ, σ_{y}/σ, and τ_{xy}/σ on the contour of the rectangular hole in Glass/epoxy for the case of xaxis loading. 
Fig. 10 Variation of σ_{x}/σ, σ_{y}/σ, and τ_{xy}/σ on the contour of rectangular hole in Glass/epoxy for the case of equibiaxial loading. 
Fig. 11 Variation of σ_{x}/σ, σ_{y}/σ, and τ_{xy}/σ on the contour of the rectangular hole in Glass/epoxy for the case of shear loading. 
Fig. 12 Variation of σ_{ρ}/σ, σ_{θ}/σ, and τ_{ρθ}/σ on the rectangular hole in Glass/epoxy for different types of loading. 
5.2 Effect of the geometry of the hole
5.2.1 Effect of side ratio
In this study we will consider rectangular holes of side ratios R1.5 (c = 0.7 and ε = −1/9), R3.5 (c = 0.28 and ε = −1/12), and R6 (c = 0.16 and ε = −1/16) respectively.
For remotely applied xaxis, yaxis, equibiaxial tension and shear loading on Glass/epoxy and Graphite/epoxy [0/−45/]_{s} and [0/45/0/45/0] laminates, the maximum normalized stresses σ_{θ}/σ and their locations are given in Tables 3 and 4 respectively. In these tables, the first value of the position of the maximum value of σ_{θ}/σ is shown because its distribution is symmetric with respect to x and y axes in the case of xaxis loading, yaxis loading and equibiaxial loading while it is antisymmetric relative to the axes x and y in the case of shear loading. We can deduct from these tables that, in the case of xaxis loading, the value of σ_{θ}/σ decreases with the increase of the side ratio of the rectangular hole. In the case of yaxis loading, equibiaxial loading and shear loading, the value of σ_{θ}/σ increases with the increase of the side ratio for both laminates. For R1.5, the maximum stress is occurring at 36° and 42° and 48°. For R3.5, the maximum stress values are at 30°, 36° and 41° while for R6 the maximum stress values are 24°, 27° and 33°. It may be noted that, for the case of equibiaxial and shear loading, there is a shift in the location of maximum stress towards the corner (see Fig. 12).
For square hole, the maximum value of σ_{θ}/σ is the same in the case of xaxis and yaxis loading and is located at 36° and 54° respectively, as shown in Figure 8 while the maximum stress is located at two symmetrical positions relative to the fillet of the hole for equibiaxial and shear loading.
For rectangular hole, the location of maximum stress is also influenced by the type of loading. For R1.5, from Tables 4 and 5 it can be seen that for tension along xaxis, the maximum stress has occurred at 48°, 132°, 228°, 312°, for yaxis tension the maximum stress has occurred at 36°, 144°, 216°, 324°, for equibiaxial tension and shear loading, the maximum stress has occurred at 42°, 138°, 222°, 318°. A drift in the location of the maximum stress towards the end of the fillet for xaxis loading and towards the start of the fillet for yaxis loading is observed in the results shown in Tables 3 and 4. Further, the stress is tensile on the edges parallel to the direction of loading and compressive on the edges normal to the direction of loading. For example, due to tension along xaxis, the shorter edge is under compression including the corner region (see Fig. 12) while for yaxis loading, the middle region of the longer edges is under compression while the shorter edges are under tension. Further, there is a transition from compression to tension over some region along the longer edge near the corner.
For R3.5, the corner is at 35°. For r xaxis loading, the maximum stress is at 42°, 138°, 224°, 318°, for yaxis tension, the maximum stress is at 30°, 150°, 220°, 330° while for equibiaxial and shear loading, the maximum stress is at 36°, 144°, 216°, 324°. For R6, the corner is at 29°. For xaxis loading, the maximum σ_{θ}/σ is located at 33°, 147°, 213°, 327°, for yaxis loading, the maximum stress is at 24°, 156°, 204°, 336° while for equibiaxial and shear loading, the maximum stress is at 27°, 153°, 207°, 333°.
Normalized maximum stress σ_{θ}/σ and their locations around rectangular hole for different side ratio and different loading in two Graphite/epoxy laminates.
Normalized maximum stress σ_{θ}/σ and their locations around rectangular for different side ratio and for different loading in two Glass/epoxy laminates.
Normalized maximum stress σ_{θ}/σ and their locations around rectangular hole with side ratio R1.5 for different corner radii and different loading in [0/−45/]_{s} Graphite/epoxy laminate.
5.2.2 Effect of corner radius
In this section, the hole with R1.5 is considered with the values of ∊ of −1/11, 1/9, −1/7, and −1/6. The normalized corner radius (r_{c}/a) is 0.18 for ∊ = −1/11, for ∊ = −1/9, it is 0.12, for ∊ = −1/7 it is 0.099 and for ∊ = −1/6 it is 0.073. The rectangular hole is considered in [0/−45/]_{s} laminate of Graphite/epoxy and in [0/45/0/45/0] laminate of Glass/epoxy. The obtained results are shown in Table 5. Concerning the [0/−45/]_{s} laminate, for xaxis loading, the maximum value of σ_{θ}/σ has increased from 2.92 for ∊ = −1/11 to 5.25 for ∊ = −1/6. In the case of yaxis loading, it has increased from 3.10 for ∊ = −1/11 to 5.11 for ∊ = −1/6. For equibiaxial loading, it has increased from 3.67 for ∊ = −1/11 to 7.22 for ∊ = −1/6. For shear loading, this increase is from 6.60 for ∊ = −1/11 to 10.83 for ∊ = −1/6. However, for [0/45/0/45/0] laminate under shear stress, there is a gradual increase in maximum σ_{θ}/σ for the above corner radii. The maximum σ_{θ}/σ is ± 8.97 for ∊ = −1/11, ± 10.45 for ∊ = −1/9, ± 11.94 for ∊ = −1/7 and ± 14.2 for ∊ = −1/6.
For square hole, we considered the values of ∊ of −1/9, −1/7, −1/6 and −1/5 in the mapping function for [0/90]s in Glass/epoxy laminate and for [0/±45/90]_{s} in Graphite/epoxy laminate. For ∊ = −1/9, the normalized corner radius (r_{c}/a) is 0.12, for ∊ = −1/7, it is 0.10, for ∊ = −1/6, it is equal to 0.075, and for ∊ = −1/5, it is 0.06 for where, r_{c} is the corner radius and a is the side of the square. The obtained results are shown in Table 6. It can be noted that the corner radius has decreased for decreasing ∊ in the mapping function. For [0/−45/]_{s} laminate, for xaxis loading and yaxis loading, the maximum value of σ_{θ}/σ has increased from 3.19 for ∊ = −1/11 to 5.78 for ∊ = −1/5. For equibiaxial loading, it has increased from 3.21 for ∊ = −1/11 to 6.21 for ∊ = −1/6. For shear loading, this increase is from 4.97 for ∊ = −1/11 to 8.25 for ∊ = −1/6. However, for [0/±45/90]_{s} laminate under shear stress, there is a gradual increase in maximum σ_{θ}/σ for the above corner radii. The maximum σ_{θ}/σ is ± 6.00 for ∊ = −1/9, ± 6.99 for ∊ = −1/7, ± 7.99 for ∊ = −1/6 and ± 9.81 ∊ = −1/5 .
Normalized maximum stress σ_{θ}/σ and their locations around square hole for different corner radii and different loading in [0/90]s Glass/epoxy laminate.
5.2.3 Particular case of hole rotated by 45°
The case of rotated holes by 45° is a special case where the value of σ_{θ}/σ value may increase to high values. This is highlighted by considering the case of the square hole rotated 45° (obtained for c = 1 and ε = 1/9) shown in Figure 13a and the case of the hole rotated 45° with a sharp fillet for θ = 0 (obtained for c = 1 and ε = 1/7) shown in Figure 13b. The studied laminate is [0/±45/90]_{s} in Graphite/epoxy. The variation of σ_{θ}/σ as a function of the angle of rotation and for different types of loads are shown in Figure 14, in the case of the hole in Figure 13a, and in Figure 15 for the hole shown in Figure 13b.
Note that in the case of the square hole shown in Figure 13a, the maximum value of σ_{θ}/σ in Figure 14 is higher than the value σ_{θ}/σ obtained in Figure 8 in the case of xaxis and yaxis loading for the square hole shown in Figure 2a. The two holes have the same fillet radius. From Figures 14 and 15, the location of maximum σ_{θ}/σ is on the fillet for θ = 0 for yaxis and equibiaxial loading and the fillet for θ = 90° for xaxis loading, which is different from the locations of maximum σ_{θ}/σ for square hole with normal orientation.
Note also that extremely large values of σ_{θ}/σ equal to 23.7 and 23.2 are obtained in the case of the hole shown in Figure 13b. We can then deduce that the maximum value of σ_{θ}/σ is governed by the orientation of the hole with respect to angle of loading.
Fig. 13 Rotated holes by 45°. 
Fig. 14 Variation of σ_{θ}/σ on the contour of the square hole rotated by 45° for different types of loading. 
Fig. 15 Variation of σ_{θ}/σ on the contour of the hole shown in Figure 13b for different types of loading. 
5.3 Effect of type and angle of loading
From Table 4, it can be seen that for Glass/epoxy laminate of [0/−45/]_{s} with R3.5 subjected to tension along xaxis, the maximum values of maximum σ_{θ}/σ is 2.84; for tension along yaxis, the maximum σ_{θ}/σ is 4.62; for equibiaxial loading, it is 4.70 and for shear stress, the maximum σ_{θ}/σ is 9.90. In case of Glass/epoxy laminate of [0/45/0/45/0] for tension along xaxis, the maximum σ_{θ}/σ is 3.12; for yaxis loading, it is 4.5; for equibiaxial loading, it is 5.17 while for shear loading, the maximum σ_{θ}/σ is 9.67. It is observed for the [0/−45/]_{s} Graphite/epoxy laminate, the values by analytical solution have increased from 2.93 for xaxis loading to 9.01 for the shear loading. We can conclude that the values of σ_{θ}/σ have gradually increased for uniaxial tension to shear loading.
Another parameter which affects the value of σ_{θ}/σ is the angle of loading λ between the applied stress σ and the xaxis. In this case, the stresses at infinity become equal to
Figure 16 shows the variation of the maximum value of σ_{θ}/σ in [0/90]_{s} Graphite /epoxy laminate in function of the angle λ for the rectangular hole of side ratio R1.5 and for the square hole. This is equivalent to show the variation of the maximum value of σ_{θ}/σ transiting of xaxis loading to yaxis loading. In the case of the rectangular hole, the maximum value of σ_{θ}/σ is obtained for λ = 45°. In the case of the square hole, the symmetry of the problem imposed two maximum values of σ_{θ}/σ obtained for λ = 27° and λ = 63°. These results highlight the difference of behavior between square and rectangular holes.
Fig. 16 Variation of the maximum value of σ_{θ}/σ in [0/90]_{s} Graphite /epoxy laminate in function of the angle λ for the rectangular hole of side ratio R1.5 and for the square hole. 
5.4 Effect of material
From Tables 3 and 4 it can be seen that for the same laminate and the same side ratio the maximum value of the normalized σ_{θ}/σ depend on the properties of the material. For example, for the hole with R6 and the laminate [0/45/0/45/0] in Graphite/epoxy, the values of maximum σ_{θ}/σ are 3.11 for xaxis loading, 5.34 for yaxis loading, 6.39 for equibiaxial loading and 14.5 for shear loading. These values are 2.7, 4.82, 5.27 and 10.13 for the same laminate in Glass/epoxy. By comparing the values obtained in Tables 3 and 4 with the values obtained for an isotropic material such as steel (see Tab. 7) for R1 (square hole), R1.5, R3.5 and R6, it can be noted that the values and the location of the maximum σ_{θ}/σ obtained for isotropic case are closer to those of the anisotropic case listed in Tables 3 and 4. The locations of the maximum stress for each case indicated as post script to respective value and are explicitly indicated in the foot note of Table 7. For plate with R1.5, the maximum stress concentration factor varies from 3.03, for tension along xaxis to 7.61 for shear loading. For plate with R3.5 the value of maximum σ_{θ}/σ varies from 2.65 to 8.15. For plate with R6, it varies from 2.19 to 8.46. For different cases of loading on an isotropic plate with square hole, the maximum σ_{θ}/σ is at 3.6 for xaxis and yaxis tension, it is 4.02 for equibiaxial loading and 6.93 for shear loading.
Maximum values of normalized stress σ_{θ}/σ and their locations around R1.5, R3.5, and R6 and square holes in isotropic plate for different cases of inplane loading.
6 Conclusion
In this article we have proposed a method to determine the distribution of the stress concentration factor around holes in laminated plates of composite materials under remote inplane loading. This method is based on the parametric equation of a rectangular hole, hence the ability to study different types of discontinuities. The only data to be provided is the constants for mapping function for the shape of hole and the complex parameters for the laminate.
The results are in agreement with those obtained by finite element method regarding the value and shape of the distribution of the stress concentration factor. This allows us to confirm the validity of the analytical method proposed hence its effectiveness and interest in the calculation of cases where the growth of the stress field around a hole can reach critical levels.
The results showed that there are several parameters that affect the maximum value and the location of the stress concentration factor. These parameters relate to the geometry of the hole, the angle and the type of loading, and the material properties.
For rectangular and square holes, the points of maximum stress are shifted toward the end of the fillet for the case of xaxis loading and toward the start of the fillet in the case of yaxis loading. The maximum stress points are located at the proximity of the fillet in the case of equibiaxial and shear loading. The maximum stress has increased for decreasing corner radius, it has also increased for increasing the side ratio in the case of yaxis, equibiaxial and shear loading. This study can be also applied to isotropic plates where the results have shown that the values and the location of the maximum stress concentration factor obtained for isotropic case are closer to those of the anisotropic case.
The projected objective to continue this work is to consider other types of holes such as triangular or elliptical holes. The proposed method can also be extended to study the behavior under loading of other types of laminates such as unsymmetric laminates.
Nomenclature
c: parameter which controls the side ratio of the rectangle
ε: parameter which controls the curvature of the fillet of the rectangle
b: length of the longer edge of the hole
: boundary conditions on the hole
: complex constants and their conjugates
s_{j} (j = 1, 4): complex parameters of anisotropy
R1.5, R3.5, R6: rectangular holes of side ratio 1.5, 3.5, and 6
z: complex coordinate, z = x + iy
z_{j}: anisotropic complex coordinate, z_{j} = x + s_{j}y
ε_{x}, ε_{y}, ε_{xy}: longitudinal and shear strains
ζ: mapped coordinate of the complex variable, z
ζ_{1}, ζ_{2}: mapped coordinates of the two complex variables, z_{1}, z_{2}
ρ, θ: orthogonal curvilinear coordinates
: stresses applied at infinity
σ_{x}, σ_{y}, τ_{xy}: stresses in Cartesian coordinates
σ_{ρ}, σ_{θ}, σ_{ρθ}: stresses in curvilinear coordinates
σ_{θ}/σ: normalized tangential stress
∅(z_{1}), ψ(z_{2}): stress functions for given plate problem
∅_{0}(z_{1}), ψ_{0}(z_{2}): stress functions of second stage solution
∅'_{0}(z_{1}), ψ'_{0}(z_{2}): first derivatives of the stress functions
References
 G.N. Savin, Stress concentration around holes, Pergamon Press, New York, 1961 [Google Scholar]
 N.I. Mushkhelishvili, Some basic problems of the mathematical theory of elasticity, P. Noordhoff Ltd, Groningen, The Netherlands, 1963 [Google Scholar]
 S.G. Lekhnitskii, Anisotropic plates, Second edition, Foreign Technology Division, WrightPatterson Air Force Base, Ohio, 1968 [Google Scholar]
 P.C. Zuxing, L. Yuansheng, Stress analysis of a finite plate with a rectangular hole subjected to uniaxial tension using modified stress functions, Int. J. Mech. Sci. 75, 265 (2013) [CrossRef] [Google Scholar]
 D.K. Nageswara Rao, M. Ramesh Babu, R. Raja Narender, D. Sunil, Stress around square and rectangular cutouts in symmetric laminates, Compos. Struct. 92 (12), 2845 (2010) [CrossRef] [Google Scholar]
 V. Anil, C.S. Upadhyay, N.G.R. Iyengar, Stability analysis of composite laminate with and without rectangular cutout under biaxial loading, Compos. Struct. 80 (1), 92 (2007) [CrossRef] [Google Scholar]
 T. Özben, N. Arslan, FEM analysis of laminated composite plate with rectangular hole and various elastic modulus under transverse loads, Appl. Math. Model. 34 (7), 1746 (2010) [CrossRef] [Google Scholar]
 V.G. Ukadgaonker, D.K.N. Rao, A general solution for stresses around holes in symmetric laminates under inplane loading, Compos. Struct. 49, 339 (2000) [CrossRef] [Google Scholar]
 J. Dave, D. Sharma Chauhan, Stress analysis of an unsymmetric composite plate with variance of ovalshaped cutout, Math. Mech. Solid 22(4), 692 (2017) [CrossRef] [Google Scholar]
 C. Hwu, Polygonal holes in anisotropic media, Int. J. Solids Struct. 29 (19), 2369 (1992) [CrossRef] [Google Scholar]
 R.M. Jones, Mechanics of composite materials, Second Edition, Taylor & Francis, Philadelphia, PA, 1999 [Google Scholar]
 S.P. Timoshenko, J.N. Goodier, Theory of elasticity, Singapore, McGrawHill, 1970 [Google Scholar]
 K. Kaw, Mechanics of composite materials, CRC Press, Boca Raton, Florida, 1997 [Google Scholar]
Cite this article as: Toni Jabbour, Mohamad Abdel Wahab, Stress analysis of symmetric laminates with rectangular or square holes subjected to inplane loading, Matériaux & Techniques 105, 301 (2017)
All Tables
Normalized maximum stress σ_{θ}/σ and their locations around rectangular hole for different side ratio and different loading in two Graphite/epoxy laminates.
Normalized maximum stress σ_{θ}/σ and their locations around rectangular for different side ratio and for different loading in two Glass/epoxy laminates.
Normalized maximum stress σ_{θ}/σ and their locations around rectangular hole with side ratio R1.5 for different corner radii and different loading in [0/−45/]_{s} Graphite/epoxy laminate.
Normalized maximum stress σ_{θ}/σ and their locations around square hole for different corner radii and different loading in [0/90]s Glass/epoxy laminate.
Maximum values of normalized stress σ_{θ}/σ and their locations around R1.5, R3.5, and R6 and square holes in isotropic plate for different cases of inplane loading.
All Figures
Fig. 1 Mapping of an infinite plate with a rectangular opening onto the outside of the unit circle. 

In the text 
Fig. 2 Shapes of square and rectangular holes considered in the calculations. 

In the text 
Fig. 3 Meshing of the plate and contour plot of the stress around the rectangular hole. 

In the text 
Fig. 4 Meshing of the plate and contour plot of the stress around the square hole. 

In the text 
Fig. 5 Variation of σ_{x}/σ, σ_{y}/σ, and τ_{xy}/σ on the contour of the square hole in Graphite/epoxy for the case of xaxis loading. 

In the text 
Fig. 6 Variation of σ_{x}/σ, σ_{y}/σ, and τ_{xy}/σ on the contour of the square hole in Graphite/epoxy for the case of equibiaxial loading. 

In the text 
Fig. 7 Variation of σ_{x}/σ, σ_{y}/σ, and τ_{xy}/σ on the contour of the square hole in Graphite/epoxy for the case of shear loading. 

In the text 
Fig. 8 Variation of σ_{ρ}/σ, σ_{θ}/σ, and τ_{ρθ}/σ on the contour of the square hole in Graphite/epoxy for different types of loading. 

In the text 
Fig. 9 Variation of σ_{x}/σ, σ_{y}/σ, and τ_{xy}/σ on the contour of the rectangular hole in Glass/epoxy for the case of xaxis loading. 

In the text 
Fig. 10 Variation of σ_{x}/σ, σ_{y}/σ, and τ_{xy}/σ on the contour of rectangular hole in Glass/epoxy for the case of equibiaxial loading. 

In the text 
Fig. 11 Variation of σ_{x}/σ, σ_{y}/σ, and τ_{xy}/σ on the contour of the rectangular hole in Glass/epoxy for the case of shear loading. 

In the text 
Fig. 12 Variation of σ_{ρ}/σ, σ_{θ}/σ, and τ_{ρθ}/σ on the rectangular hole in Glass/epoxy for different types of loading. 

In the text 
Fig. 13 Rotated holes by 45°. 

In the text 
Fig. 14 Variation of σ_{θ}/σ on the contour of the square hole rotated by 45° for different types of loading. 

In the text 
Fig. 15 Variation of σ_{θ}/σ on the contour of the hole shown in Figure 13b for different types of loading. 

In the text 
Fig. 16 Variation of the maximum value of σ_{θ}/σ in [0/90]_{s} Graphite /epoxy laminate in function of the angle λ for the rectangular hole of side ratio R1.5 and for the square hole. 

In the text 
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