Free Access
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

© 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 non-circular 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 oval-shaped 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 (x-axis, y-axis, 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 cut-out, in the xy-plane 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)

thumbnail 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 stress-strain relations for a symmetric laminate (Jones [12]) are given by (6) where Aij are the elements of the symmetric stiffness matrix. By inversion of the stiffness matrix we obtain the compliance matrix whose elements are denoted by aij.

For two-dimensional 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 a16 = a26 = 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 s1, s2, s3, s4, we set (11) with β1 > 0, β2 > 0 and β1 ≠ β2

U(x,y) can be written as (12) where z1 and z2 are defined by the affine transformations (13)

F1, F2 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 stress-free 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 z1 and equation (21) with respect to z2, and solving for and , we get (22) (23)

Substituting z1 and z2 from equations (20) and (21) into equations (18), we obtain, on the contour of the unit circle (24) where coefficients K1 through K8 are given as follows (25)

For a stress-free hole  and ψ(z2) in equations (16) are given by Savin [1] by (26) where  and ψ0(z2) are obtained by substituting z1 and z2 for ζ in  and Ψ0(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 α0 is some constant.

Substituting and from equations (24) into equations (27) and applying the Schwartz formula, we determine  and Ψ0(ζ) as (29) (30)

Differentiating equations (29) and (30) with respect to ζ we get (31) (32) where in equations (2932) is the complex conjugate of Ki.

At this point, we need to determine ζ as a function of z1 from equation (20) to put in equation (29) and ζ as a function of z2 from equation (21) to put in equation (30), to obtain ϕ0(z1) and ψ0(z2). Putting these functions in equations (26) we determine ∅(z1) and ψ(z2). 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 z1 and z2 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(z1) and ψ′0(z2) 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) 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 x-axis, 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-, y-directions, equi-biaxial tension and shear loading. Also, for calculating the complex parameters s1 and s2, 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.

Table 1

Properties of the unidirectional lamina.

Table 2

Complex parameters s1 and s2 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 57, for the case of square hole, and in Figures 911 for the rectangular hole. These figures show the variation of the stress concentration factor on the edge of each hole for the cases of x-axis loading, equi-biaxial 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 57, 911 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 σx and σy on the contour of the hole is symmetrical relative to the axes x and y while the distribution of τxy is antisymmetric with respect to these axes. In Figures 7 and 11 the distribution of the stresses σx and σy is antisymmetric relative to the axes x and y while the distribution of τxy is symmetrical with respect to these axes.

In Figures 8 and 12, we can notice that the stresses σρ and τρθ 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 equi-biaxial loading, the distribution of σθ 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 512, 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.

thumbnail Fig. 2

Shapes of square and rectangular holes considered in the calculations.

thumbnail Fig. 3

Meshing of the plate and contour plot of the stress around the rectangular hole.

thumbnail Fig. 4

Meshing of the plate and contour plot of the stress around the square hole.

thumbnail Fig. 5

Variation of σx/σ, σy/σ, and τxy/σ on the contour of the square hole in Graphite/epoxy for the case of x-axis loading.

thumbnail Fig. 6

Variation of σx/σ, σy/σ, and τxy/σ on the contour of the square hole in Graphite/epoxy for the case of equi-biaxial loading.

thumbnail Fig. 7

Variation of σx/σ, σy/σ, and τxy/σ on the contour of the square hole in Graphite/epoxy for the case of shear loading.

thumbnail Fig. 8

Variation of σρ/σ, σθ/σ, and τρθ/σ on the contour of the square hole in Graphite/epoxy for different types of loading.

thumbnail Fig. 9

Variation of σx/σ, σy/σ, and τxy/σ on the contour of the rectangular hole in Glass/epoxy for the case of x-axis loading.

thumbnail Fig. 10

Variation of σx/σ, σy/σ, and τxy/σ on the contour of rectangular hole in Glass/epoxy for the case of equi-biaxial loading.

thumbnail Fig. 11

Variation of σx/σ, σy/σ, and τxy/σ on the contour of the rectangular hole in Glass/epoxy for the case of shear loading.

thumbnail Fig. 12

Variation of σρ/σ, σθ/σ, and τρθ/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 x-axis, y-axis, equi-biaxial tension and shear loading on Glass/epoxy and Graphite/epoxy [0/−45/]s and [0/45/0/45/0] laminates, the maximum normalized stresses σθ/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 σθ/σ is shown because its distribution is symmetric with respect to x and y axes in the case of x-axis loading, y-axis loading and equi-biaxial 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 x-axis loading, the value of σθ/σ decreases with the increase of the side ratio of the rectangular hole. In the case of y-axis loading, equi-biaxial loading and shear loading, the value of σθ/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 equi-biaxial 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 σθ/σ is the same in the case of x-axis and y-axis 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 equi-biaxial 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 x-axis, the maximum stress has occurred at 48°, 132°, 228°, 312°, for y-axis tension the maximum stress has occurred at 36°, 144°, 216°, 324°, for equi-biaxial 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 x-axis loading and towards the start of the fillet for y-axis 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 x-axis, the shorter edge is under compression including the corner region (see Fig. 12) while for y-axis 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 x-axis loading, the maximum stress is at 42°, 138°, 224°, 318°, for y-axis tension, the maximum stress is at 30°, 150°, 220°, 330° while for equi-biaxial and shear loading, the maximum stress is at 36°, 144°, 216°, 324°. For R6, the corner is at 29°. For x-axis loading, the maximum σθ/σ is located at 33°, 147°, 213°, 327°, for y-axis loading, the maximum stress is at 24°, 156°, 204°, 336° while for equi-biaxial and shear loading, the maximum stress is at 27°, 153°, 207°, 333°.

Table 3

Normalized maximum stress σθ/σ and their locations around rectangular hole for different side ratio and different loading in two Graphite/epoxy laminates.

Table 4

Normalized maximum stress σθ/σ and their locations around rectangular for different side ratio and for different loading in two Glass/epoxy laminates.

Table 5

Normalized maximum stress σθ/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 (rc/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 x-axis loading, the maximum value of σθ/σ has increased from 2.92 for  = −1/11 to 5.25 for  = −1/6. In the case of y-axis loading, it has increased from 3.10 for  = −1/11 to 5.11 for  = −1/6. For equi-biaxial 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 σθ/σ for the above corner radii. The maximum σθ/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 (rc/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, rc 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 x-axis loading and y-axis loading, the maximum value of σθ/σ has increased from 3.19 for  = −1/11 to 5.78 for  = −1/5. For equi-biaxial 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 σθ/σ for the above corner radii. The maximum σθ/σ is ± 6.00 for  = −1/9, ± 6.99 for  = −1/7, ± 7.99 for  = −1/6 and ± 9.81  = −1/5 .

Table 6

Normalized maximum stress σθ/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 σθ/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 σθ/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 σθ/σ in Figure 14 is higher than the value σθ/σ obtained in Figure 8 in the case of x-axis and y-axis 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 σθ/σ is on the fillet for θ = 0 for y-axis and equi-biaxial loading and the fillet for θ = 90° for x-axis loading, which is different from the locations of maximum σθ/σ for square hole with normal orientation.

Note also that extremely large values of σθ/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 σθ/σ is governed by the orientation of the hole with respect to angle of loading.

thumbnail Fig. 13

Rotated holes by 45°.

thumbnail Fig. 14

Variation of σθ/σ on the contour of the square hole rotated by 45° for different types of loading.

thumbnail Fig. 15

Variation of σθ/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 x-axis, the maximum values of maximum σθ/σ is 2.84; for tension along y-axis, the maximum σθ/σ is 4.62; for equi-biaxial loading, it is 4.70 and for shear stress, the maximum σθ/σ is 9.90. In case of Glass/epoxy laminate of [0/45/0/45/0] for tension along x-axis, the maximum σθ/σ is 3.12; for y-axis loading, it is 4.5; for equi-biaxial loading, it is 5.17 while for shear loading, the maximum σθ/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 x-axis loading to 9.01 for the shear loading. We can conclude that the values of σθ/σ have gradually increased for uni-axial tension to shear loading.

Another parameter which affects the value of σθ/σ is the angle of loading λ between the applied stress σ and the x-axis. In this case, the stresses at infinity become equal to

Figure 16 shows the variation of the maximum value of σθ/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 σθ/σ transiting of x-axis loading to y-axis loading. In the case of the rectangular hole, the maximum value of σθ/σ is obtained for λ = 45°. In the case of the square hole, the symmetry of the problem imposed two maximum values of σθ/σ obtained for λ = 27° and λ = 63°. These results highlight the difference of behavior between square and rectangular holes.

thumbnail Fig. 16

Variation of the maximum value of σθ/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 σθ/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 σθ/σ are 3.11 for x-axis loading, 5.34 for y-axis loading, 6.39 for equi-biaxial 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 σθ/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 x-axis to 7.61 for shear loading. For plate with R3.5 the value of maximum σθ/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 σθ/σ is at 3.6 for x-axis and y-axis tension, it is 4.02 for equi-biaxial loading and 6.93 for shear loading.

Table 7

Maximum values of normalized stress σθ/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 x-axis loading and toward the start of the fillet in the case of y-axis loading. The maximum stress points are located at the proximity of the fillet in the case of equi-biaxial 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 y-axis, equi-biaxial 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

sj (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

zj: anisotropic complex coordinate, zj = x + sjy

λ: angle of loading

εx, εy, εxy: longitudinal and shear strains

ζ: mapped coordinate of the complex variable, z

ζ1, ζ2: mapped coordinates of the two complex variables, z1, z2

ρ, θ: orthogonal curvilinear coordinates

: stresses applied at infinity

σx, σy, τxy: stresses in Cartesian coordinates

σρ, σθ, σρθ: stresses in curvilinear coordinates

σθ/σ: normalized tangential stress

∅(z1), ψ(z2): stress functions for given plate problem

0(z1), ψ0(z2): stress functions of second stage solution

∅'0(z1), ψ'0(z2): first derivatives of the stress functions

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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

Table 1

Properties of the unidirectional lamina.

Table 2

Complex parameters s1 and s2 for different laminates.

Table 3

Normalized maximum stress σθ/σ and their locations around rectangular hole for different side ratio and different loading in two Graphite/epoxy laminates.

Table 4

Normalized maximum stress σθ/σ and their locations around rectangular for different side ratio and for different loading in two Glass/epoxy laminates.

Table 5

Normalized maximum stress σθ/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.

Table 6

Normalized maximum stress σθ/σ and their locations around square hole for different corner radii and different loading in [0/90]s Glass/epoxy laminate.

Table 7

Maximum values of normalized stress σθ/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

thumbnail Fig. 1

Mapping of an infinite plate with a rectangular opening onto the outside of the unit circle.

In the text
thumbnail Fig. 2

Shapes of square and rectangular holes considered in the calculations.

In the text
thumbnail Fig. 3

Meshing of the plate and contour plot of the stress around the rectangular hole.

In the text
thumbnail Fig. 4

Meshing of the plate and contour plot of the stress around the square hole.

In the text
thumbnail Fig. 5

Variation of σx/σ, σy/σ, and τxy/σ on the contour of the square hole in Graphite/epoxy for the case of x-axis loading.

In the text
thumbnail Fig. 6

Variation of σx/σ, σy/σ, and τxy/σ on the contour of the square hole in Graphite/epoxy for the case of equi-biaxial loading.

In the text
thumbnail Fig. 7

Variation of σx/σ, σy/σ, and τxy/σ on the contour of the square hole in Graphite/epoxy for the case of shear loading.

In the text
thumbnail Fig. 8

Variation of σρ/σ, σθ/σ, and τρθ/σ on the contour of the square hole in Graphite/epoxy for different types of loading.

In the text
thumbnail Fig. 9

Variation of σx/σ, σy/σ, and τxy/σ on the contour of the rectangular hole in Glass/epoxy for the case of x-axis loading.

In the text
thumbnail Fig. 10

Variation of σx/σ, σy/σ, and τxy/σ on the contour of rectangular hole in Glass/epoxy for the case of equi-biaxial loading.

In the text
thumbnail Fig. 11

Variation of σx/σ, σy/σ, and τxy/σ on the contour of the rectangular hole in Glass/epoxy for the case of shear loading.

In the text
thumbnail Fig. 12

Variation of σρ/σ, σθ/σ, and τρθ/σ on the rectangular hole in Glass/epoxy for different types of loading.

In the text
thumbnail Fig. 13

Rotated holes by 45°.

In the text
thumbnail Fig. 14

Variation of σθ/σ on the contour of the square hole rotated by 45° for different types of loading.

In the text
thumbnail Fig. 15

Variation of σθ/σ on the contour of the hole shown in Figure 13b for different types of loading.

In the text
thumbnail Fig. 16

Variation of the maximum value of σθ/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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