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Bending problem of a finite composite laminated plate weakened by multiple elliptical holes
Acta Mechanica Solida Sinica, Vol. 26, No. 4, August, 2013 Published by AMSS Press, Wuhan, China
ISSN 0894-9166
BENDING PROBLEM OF A FINITE COMPOSITE LAMINATED PLATE WEAKENED BY MULTIPLE ELLIPTICAL HOLES Chunjian Mao
Xiwu Xu
(State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China)
Received 20 November 2011, revision received 22 May 2012
ABSTRACT Based on the classical composite laminate theory, the bending problem of a finite composite plate weakened by multiple elliptical holes is studied by means of the complex variable method. The present work is intended to express the complex potentials in the form of Faber series aided by the use of the least squares boundary collocation techniques on the finite boundaries. As a result, concise and high accuracy solutions are presented for the stress distribution around the holes. Finally, numerical examples are presented to discuss the effects of some parameters on the stress concentration around the holes.
KEY WORDS anisotropic plate bending, finite plate, elliptical holes, Faber series, least squares method
I. INTRODUCTION Composite materials are widely used in many engineering fields, such as in aerospace, automobile and chemical engineering. As is well known, holes often exist in engineering materials or structures. The effects arising therefrom may lead to local stress concentration and finally result in fracture and failure of the materials or structures. Hence, much attention has been given by many researchers to the problems of stress concentration in composite materials and/or structures. Since most composite structures used in engineering are symmetric and free from bending-extension coupling, they can usually be treated as anisotropic plates. The complex potential method is an effective method of analyzing the stress concentration of plates with holes. For the plane problem of an anisotropic body, Lekhnitskii[1, 2] and Savin[3] first gave the closed-form solutions of stress fields around a circular hole in an infinite plane under uniform stress. Gao and Yue[4] , and Gao and Long[5] gave the closed form solutions for the complex potentials in an infinite and anisotropic plane under a concentrated force and moment, respectively, which can be used as basic solutions to solve the problem of a finite plate with the boundary element method. Xu et al.[6–8] developed a new method by using the Faber series expansion and the least squares boundary collocation techniques to solve the plane problem of a finite plate with elliptical holes. For the bending problem of an anisotropic plate, Lekhnitskii[1, 2] and Savin[3] presented the closed form solution for the bending of an infinite anisotropic plate under a uniform moment. Chen and Shen[9] , and Chen and Nie[10, 11] gave the Green’s function of an anisotropic plate with an elliptical hole subject to a center moment and a normal point force. Qu[12] studied the stress concentration of
Corresponding author. E-mail: [email protected] Project supported by the National Natural Science Foundation of China (No. 11271146).
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bending of infinite symmetric composite laminates with an elliptical hole. Mao[13] studied the stress of a finite anisotropic plate with an elliptical hole. However, to the authors’ knowledge, no report on the case of a finite composite laminated plate bending weakened by multiple elliptical holes is available. Facing the fact that most composite structures have quite a few holes, we have to study the stress in a finite composite laminated plate with defects. It is the purpose of the present work to analyze the bending problem of a finite composite laminated plate with multiple elliptical holes. This work is organized as follows: following the brief introduction, basic equations are outlined in §II for later use. In §III, complex potentials are derived for a finite plate with multiple elliptic holes, and then numerical results of the stress distributions around holes are presented in §IV. Finally, §V concludes the present work.
II. BASIC EQUATION Consider a composite laminate which can be treated as an anisotropic plate. The constitutive equation for the plate under bending can be expressed as[1] ⎧ ⎫ ∂ 2w ⎪ ⎪ ⎧ ⎫ ⎡ ⎤⎪ − 2 ⎪ ⎪ ⎪ Mx ⎪ D11 D12 D13 ⎪ ⎪ ⎪ ⎪ ∂x ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎨ ⎬ ⎢ ⎬ 2 ⎥ ∂ w ⎢ ⎥ = ⎣ D12 D22 D26 ⎦ (1) My − 2 ⎪ ⎪ ⎪ ∂y ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎪ Hxy D16 D26 D66 ⎪ ⎪ ⎪ ∂2w ⎪ ⎪ ⎪ ⎩ −2 ⎭ ∂x∂y where Dij (i, j = 1, 2, 6) is the equivalent compliance coefficients depending on the fiber orientation, the stacking sequence and the property of each lamina. Mx , My are the moment in the direction of x and y, Hxy is torsion of plate, and w is the deflection of the plate. According to the bending theory of anisotropic plate[1] , we have w = 2Re
2
wj (zj ),
zj = x + μj y
(2)
j=1
Mx = −2Re
2
pj wj (zj )
j=1
My = −2Re
2
qj wj (zj )
(3)
j=1
Hxy = −2Re
2
rj wj (zj )
j=1
where wj (zj ) (j = 1, 2) is an analytic function in the generalized region Sj by the affine transformation zj = x + μj y from the physical region S, and μj (j = 1, 2) are the complex parameters representing the anisotropic extent of the laminated plate that can be obtained from the following characteristic equation: D22 μ4 + 4D26 μ3 + 2(D12 + 2D66 )μ2 + 4D16 μ + D11 = 0 (4) where: pj = D11 + D12 μ2j + 2D16 μj ,
qj = D12 + D22 μ2j + 2D26 μj ,
rj = D16 + D26 μj2 + 2D66 μj
Along the boundary of the plate, the following boundary conditions should be satisfied: 2 kj wj + lj wj = f (t) j=1
(5)
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When the bending moment m(s) and the forces p(s) normal to the mid-plane are applied to the per unit arc length s, the boundary conditions are of the form pj pj + iqj , lj = + iqj μj μj s f (t) = − [(m − vi)dy + (v + mi)dx] − C(x − yi) + C1 + C2 i
kj =
0
where C, C1 , C2 are real constants, v =
s
pds, and the integration is done from the start point to 0
the integration point. On the other hand, when the deflection and the angle w∗ (s), α∗ (s) are prescribed on the boundary, the boundary conditions are of the form kj = 1 + iμj , lj = 1 + iμj dw∗ dw∗ cos(n, y) + α∗ cos(n, x) − cos(n, x)i + α∗ cos(n, y)i f (t) = − ds ds where n is the outward normal on the boundary. Up to now, the bending problem of a thin plate is translated to finding two analytical complex functions wj (zj ) which satisfy the boundary equation (5).
III. ANALYSIS Consider the case of a finite composite laminated plate weakened by multiple elliptical holes with contours L0 , L1 ,...,Ll as shown in Fig.1(a) where L0 , L1 ,...,Ll are the exterior contours of the finite plate, am , bm (m = 1, 2....l) stand for the two semi-axes of the holes, and zm are the centre of the holes. By affine transformation zj = x + μj y (j = 1, 2) the region S and the point zm are transformed onto the region Sj and the points zjm in the region Sj , as shown in Fig.1(b).
Fig. 1. Mapping of a finite plate with multiple elliptical holes.
Let ϕj (zj ) = wj (z) and the principal vector of the resultant forces acting on the contour of holes be equal to zero. Then, the complex potential function ϕj (zj ) can be expressed as[6] ϕj (zj ) = ϕj0 (zj ) +
∞
cjk pk (zj ) (j = 1, 2)
(6)
k=0
where ϕj0 (zj ) is an holomorphic function in the finite region with elliptical holes; pk (zj ) is the Faber polynomial of the region limited by contour Lj0 , cjk are Faber series coefficients. Introduce the following mapping function as
tjm zj − zjm = Rjm ξjm + (7) ξjm where Rjm =
am − iμj bm , 2
tjm =
am + iμj bm am − iμj bm
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This mapping function transforms the exterior of hole in the complex plane zjm into the exterior of a unit circle, ξjm = exp(iθ) in the plane ξjm . By using the Faber polynomial in the general region, the complex potential function can be given as l ∞ ∞ −k ϕj (zj ) = bjmk ξjm + ajk zjk (8) m=1 k=1
k=0
Obviously, the complex potential function equation (8) is analytic in the region Sj . Once the unknown coefficients, bjmk and ajk , are determined by using the boundary conditions, and the stress fields can be uniquely obtained according to the uniqueness theorem in the theory of elasticity. From the mapping function equation (7), it can be seen that the function zjk is holomorphic in the complex plane zj weakened by the hole m. Therefore, the function zjk is holomorphic in the interior of the p-th hole and continuous to its boundary. Thus the functions can be expanded into a Faber series ∞ j zjn = Hn,k Pkp (zj ) (9) k=0
Similarly, −n ξjm =
∞
Ajm n,k Pkp (zj )
(10)
k=0 j [14] where the coefficients Hn,k , Ajm n,k in the Faber series can be determined by Fourier expansion method and Pkp (zj ) is the k-th Faber polynomial for the ellipse Ljp of the complex zj plane and k + Pkp (zj ) = ξjp
tkjp k ξjp
,
P0p (zj ) = 1
Substituting Eq.(9) into Eq.(8) and using ξjp = exp(iθ) = σ in the elliptical contour Ljp , we obtain the boundary values of ϕj (zj )(j = 1, 2) in power series of σ. The right side of Eq.(5) can be expanded into the complex Fourier series: ∞ fp = (Fn σ n + F−n σ −n ) + C (11) n=1
where σ = exp(iθ), and Fn , F−n are the coefficients in Fourier series. Taking the partial sum of ϕj (zj ) up to the N th power and substituting them into the boundary condition of the elliptical holes, and then equating the coefficients of the same power σ k (k = +1, +2, ..., +N ) on both sides of the equation, we obtain 2N l linear equations about the coefficients bjmk and ajk . It is obvious that the equations obtained from the inner boundary alone are not enough to determine all coefficients. Therefore it is necessary to use outer boundary conditions. The least squares boundary collocation technique is used in this paper. Taking collocation points zcn (n = 1, 2, ..., M, M > 2N ) along the outer L0 and substituting zcn into the boundary condition equation (5), we can obtain linear equations about the unknown coefficients bjk and ajk that satisfy the outer boundary conditions. These equations together with the equations that satisfy inner boundary conditions are used to determine the complex potential functions ϕj (zj ). Then the stress field in the laminated plate can be calculated by the following equations: 12Mx 12My 12Hxy σx = z, σy = z, τxy = τyx = z (12) 3 3 h h h3 where h is the thickness of plate. Obviously, the complex potential function ϕj (zj ) is an analytic function in the region Sj . Therefore the accuracy of the solution can be justified according to whether all boundary conditions are satisfied accurately (absolutely error less than 10−5 ), and the outer boundary conditions can be well satisfied to ensure the relative error within 1% by increasing the number of collocation points. According to the Saint-Venant principle, more accurate results of the stress distribution around the hole, the main concern of many researchers, is obtained by using the present method. In this paper all results are obtained by taking a partial sum up to the 10th power and 32 collocation points on the boundary. It is shown that numerical results are exact enough to indicate that to satisfy boundary conditions and the expected accuracy.
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IV. STRESS ANALYSIS OF COMPOSITE LAMINATE WITH MULTIPLE HOLES Consider a finite laminated plate weakened by a free elliptical hole in the center, subjected to Mx = 1, along the outer boundary, the bending moments and normal force acting on the contour of holes are zero, as shown in Fig.2. The semi-major, semi-minor axes and elliptical orientation are a, b, ϕ respectively. Define the stress concentration as Mθ /Mx then the effect of various parameters on the stress concentration is discussed below. Figure 3 show the stress distribution with circular hole of Fig. 2 Bending of a composite plate with an diameter D. The material properties of the laminated plate are elliptical hole. E1 = 16.9 GPa,
E2 = 1.4 GPa,
G12 = 0.7 GPa,
ν12 = 0.31
Results are compared with the exact solutions of the infinite laminated plates[1] . It is found that the relative size W/D has significant effects on the stress concentrations. The stress concentration increases rapidly with the decrease of W/D. When W/D ≥ 10, the results are closed to the exact solutions of the corresponding infinite laminated plates in reference[1] . Therefore it is rational to treat a finite plate as an infinite plate in engineering analysis when W/D ≥ 10. Figure 4 show the effect of the ellipticity on distribution of Mθ around hole. The laminated plate is composed by T300/QY8911. The material properties are E1 = 135.0 GPa,
E2 = 8.8 GPa,
Fig. 3. The effect of composite plate size on the stress distribution around hole.
G12 = 4.47 GPa,
ν12 = 0.33
Fig. 4. The effect of ellipticity on the stress distribution around hole.
It is found that as a/b increases, the stress concentrations become smaller. Especially, when a/b = 0, the hole degenerates into a crack in a finite plate, and the stress will approach infinity, but the solution of the present work is still valid in this case. Figure 5 shows the effect of the elliptical orientation on distribution of Mθ around hole, where the laminated plate is composed by T300/QY8911. The results indicate that when 0◦ ≤ ϕ ≤ 90◦ the stress concentration increases with the increase of ϕ. The location of maximum stress Mθ changes from elliptical orientation angle to elliptical angle. Consider a composite laminated plate weakened by two free circular holes in the center, subjected to Mx = 1 along the outer boundary, the bending moments and normal force acting on the contour of holes are zero, as shown in Fig.6. The laminated plate is composed by T300/QY8911. The diameters of the circular holes are D. Figures 7 and 8 show the effect of the relative center-to-center distance and the layups of laminate on the stress distribution around the holes.
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ACTA MECHANICA SOLIDA SINICA
Fig. 5. The effect of the elliptical orientation angle on the stress distribution around hole.
Fig. 7. The effect of the relative center-to-center distance on the stress distribution around the holes.
2013
Fig. 6. Bending of a composite plate with two elliptical holes.
Fig. 8. The effect of layups of laminate on the stress distribution around hole.
It can be seen from Fig.7 that the relative center-to-center distance affects the stress distribution around holes. The stress concentration increases with the increase of the relative center-to-center distance. When L/D ≥ 10, the stress distribution is the same as the laminate with one hole. Therefore it is reasonable to treat a finite plate with multiple holes as a plate with one hole when L/D ≥ 10. Figure 8 shows the effect of layups (0α / ± 45β /90γ )s on the distribution of Mθ around the hole. The results show that the stress concentration strongly depends on the percentage of each layup in the whole plate. The more anisotropic the plate, the more intensesevere the stress concentration. The increase of number of ±45◦ lamina is beneficial to the decrease of stress concentrations, because it reduces the extent of anisotropy of laminates. Consider a composite laminated plate weakened by three free circle holes in the center, subjected to Mx = 1 along the outer boundary, the bending moments and normal force acting on the contour of holes are zero, as shown in Fig.9. The laminated plate is composed by T300/QY8911. Figure 10 shows the effect of the position of the holes on the stress concentration of a laminated plate with three holes. Figure 10 shows the effect of the position of holes on the stress concentration of a laminated plate with three holes. For hole 1*, the plate shown in Fig.9(c) has the most stress concentration around holes
Fig. 9. Bending of a composite plate with three elliptical holes.
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Fig. 10. The effect of position of laminate on the stress concentration around holes.
and the plate shown in Fig.9(a) has the least stress concentration around holes. For hole 2*, the plate shown in Fig.9(b) has the most stress concentration around holes and the plate shown in Fig.9(a) has the least stress concentration around holes. For the plate with three holes, the plate shown in Fig.9(a) has the least stress concentration for both hole 1* and hole 2*.
V. CONCLUSIONS The bending problems of a finite laminated plate are studied with the complex variable method. Concise and high accuracy solutions are presented when the plate is with multiple elliptic holes. Numerical examples are also given to discuss the effects of plate sizes, defect geometries and laminate layups on the stress concentration around the holes. From the results obtained, the following conclusions can be drawn: As the relative size W/D of a laminated plate decreases, the stress concentration may increase rapidly for the case where the laminate is with a free hole. Especially, when W/D ≥ 10, it is rational to assume that the finite plate can be treated as an infinite plate in most engineering analyses. In general, the increase of ellipticity causes more stress concentration for a laminate with a free hole. In general, when 0◦ ≤ ϕ ≤ 90◦ , the stress concentration increases with the increase of elliptical orientation angle for a laminate with a free hole. The location of maximum stress differs from elliptical orientation angle to elliptical orientation angle. The stress concentration increases with the increase of the relative center-to-center distance. When L/D ≥ 10, it is rational to treat a finite plate with multiple holes as a plate with one hole. The increase of numbers ±45◦ lamina is beneficial to the decrease of stress concentration for the laminate with two free holes as it reduces the extent of anisotropy of laminates. The present method is not only very efficient for analysis of the stress distribution of finite composite laminates with an elliptical hole owing to the application of Faber series, but also highly accurate and computer time-saving.
References [1] [2] [3] [4] [5] [6] [7] [8]
Lekhnitskii,S.G., Anisotropic Plate. New York: Gordon and Brench, 1968. Lekhnitskii,S.G., Theory of Elasticity of an Anisotropic Body. Moscow: Mir Publishers, 1981. Savin,G.N., Stress Concentration Around Holes. London: Pergamon Press, 1961. Gao,C.F. and Yue,B.Q., General solutions of loaded anisotropic plane with elliptical hole. Journal of China University of Petroleum, 1992, 16(2): 54-62. Gao,C.F. and Long,L.C., Complex stress function due to concentrated loads applied in an anisotropic half plane. Chinese Journal of Applied Mechanics, 1996, 13(3): 62-66. Xu,X.W., Sun,L.X. and Fan,X.Q., Stress concentration of finite element composite laminates weakened by multiple elliptical holes. International Journal of Solids and Structures, 1995, 32: 3001-3014. Xu,X.W., Sun,L.X. and Fan,X.Q., Stress concentration of finite composite laminates with elliptical hole. Computer Structure, 1995, 57(1): 29-34. Xu,X.W., Yue,T.M. and Man,H.C., Stress analysis of finite composite laminate with multiple loaded holes. International Journal of Solids and Structures, 1999, 36: 919-931.
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[9] Chen,P. and Shen,Z., Green’s functions for an unsymmetrical laminated plate with an elliptic hole. Mechanics Research Communications, 2001, 28: 519-524. [10] Chen,P. and Nie,H., Green’s function for bending problem of a thin anisotropic plate with an elliptic hole. Mechanics Research Communications, 2003, 31: 423-428. [11] Chen,P. and Nie,H., Green’s function for bending problem of an unsymmetrical laminated plate with bending-extension coupling containing an elliptic hole. Archive of Applied Mechanics, 2004, 73: 846-856. [12] Qu,Y.Z. and Gai,B.Z., Stress concentration of bending symmetric composite laminates with elliptical holes. Journal of Astronautics, 2007, 28(4): 1065-1069. [13] Mao,C.J., Xu,X.W. and Guo,S.X., Stress analysisi of a finite anisotropic thin plate with an elliptical hole. Chiese Journal of Solid Mechanics, 2010, 31(1): 80-85. [14] Xu,X.W., The Strength Analysis of Mechanically Multi-fastened Momposite Maminate Joints [Ph.D Thesis]. Nanjing: Nanjing Aeronautical Institute, 1992 (in Chinese).
ISSN 0894-9166
BENDING PROBLEM OF A FINITE COMPOSITE LAMINATED PLATE WEAKENED BY MULTIPLE ELLIPTICAL HOLES Chunjian Mao
Xiwu Xu
(State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China)
Received 20 November 2011, revision received 22 May 2012
ABSTRACT Based on the classical composite laminate theory, the bending problem of a finite composite plate weakened by multiple elliptical holes is studied by means of the complex variable method. The present work is intended to express the complex potentials in the form of Faber series aided by the use of the least squares boundary collocation techniques on the finite boundaries. As a result, concise and high accuracy solutions are presented for the stress distribution around the holes. Finally, numerical examples are presented to discuss the effects of some parameters on the stress concentration around the holes.
KEY WORDS anisotropic plate bending, finite plate, elliptical holes, Faber series, least squares method
I. INTRODUCTION Composite materials are widely used in many engineering fields, such as in aerospace, automobile and chemical engineering. As is well known, holes often exist in engineering materials or structures. The effects arising therefrom may lead to local stress concentration and finally result in fracture and failure of the materials or structures. Hence, much attention has been given by many researchers to the problems of stress concentration in composite materials and/or structures. Since most composite structures used in engineering are symmetric and free from bending-extension coupling, they can usually be treated as anisotropic plates. The complex potential method is an effective method of analyzing the stress concentration of plates with holes. For the plane problem of an anisotropic body, Lekhnitskii[1, 2] and Savin[3] first gave the closed-form solutions of stress fields around a circular hole in an infinite plane under uniform stress. Gao and Yue[4] , and Gao and Long[5] gave the closed form solutions for the complex potentials in an infinite and anisotropic plane under a concentrated force and moment, respectively, which can be used as basic solutions to solve the problem of a finite plate with the boundary element method. Xu et al.[6–8] developed a new method by using the Faber series expansion and the least squares boundary collocation techniques to solve the plane problem of a finite plate with elliptical holes. For the bending problem of an anisotropic plate, Lekhnitskii[1, 2] and Savin[3] presented the closed form solution for the bending of an infinite anisotropic plate under a uniform moment. Chen and Shen[9] , and Chen and Nie[10, 11] gave the Green’s function of an anisotropic plate with an elliptical hole subject to a center moment and a normal point force. Qu[12] studied the stress concentration of
Corresponding author. E-mail: [email protected] Project supported by the National Natural Science Foundation of China (No. 11271146).
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2013
bending of infinite symmetric composite laminates with an elliptical hole. Mao[13] studied the stress of a finite anisotropic plate with an elliptical hole. However, to the authors’ knowledge, no report on the case of a finite composite laminated plate bending weakened by multiple elliptical holes is available. Facing the fact that most composite structures have quite a few holes, we have to study the stress in a finite composite laminated plate with defects. It is the purpose of the present work to analyze the bending problem of a finite composite laminated plate with multiple elliptical holes. This work is organized as follows: following the brief introduction, basic equations are outlined in §II for later use. In §III, complex potentials are derived for a finite plate with multiple elliptic holes, and then numerical results of the stress distributions around holes are presented in §IV. Finally, §V concludes the present work.
II. BASIC EQUATION Consider a composite laminate which can be treated as an anisotropic plate. The constitutive equation for the plate under bending can be expressed as[1] ⎧ ⎫ ∂ 2w ⎪ ⎪ ⎧ ⎫ ⎡ ⎤⎪ − 2 ⎪ ⎪ ⎪ Mx ⎪ D11 D12 D13 ⎪ ⎪ ⎪ ⎪ ∂x ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎨ ⎬ ⎢ ⎬ 2 ⎥ ∂ w ⎢ ⎥ = ⎣ D12 D22 D26 ⎦ (1) My − 2 ⎪ ⎪ ⎪ ∂y ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎪ Hxy D16 D26 D66 ⎪ ⎪ ⎪ ∂2w ⎪ ⎪ ⎪ ⎩ −2 ⎭ ∂x∂y where Dij (i, j = 1, 2, 6) is the equivalent compliance coefficients depending on the fiber orientation, the stacking sequence and the property of each lamina. Mx , My are the moment in the direction of x and y, Hxy is torsion of plate, and w is the deflection of the plate. According to the bending theory of anisotropic plate[1] , we have w = 2Re
2
wj (zj ),
zj = x + μj y
(2)
j=1
Mx = −2Re
2
pj wj (zj )
j=1
My = −2Re
2
qj wj (zj )
(3)
j=1
Hxy = −2Re
2
rj wj (zj )
j=1
where wj (zj ) (j = 1, 2) is an analytic function in the generalized region Sj by the affine transformation zj = x + μj y from the physical region S, and μj (j = 1, 2) are the complex parameters representing the anisotropic extent of the laminated plate that can be obtained from the following characteristic equation: D22 μ4 + 4D26 μ3 + 2(D12 + 2D66 )μ2 + 4D16 μ + D11 = 0 (4) where: pj = D11 + D12 μ2j + 2D16 μj ,
qj = D12 + D22 μ2j + 2D26 μj ,
rj = D16 + D26 μj2 + 2D66 μj
Along the boundary of the plate, the following boundary conditions should be satisfied: 2 kj wj + lj wj = f (t) j=1
(5)
Vol. 26, No. 4
Chunjian Mao et al.: Bending Problem of a Finite Composite Laminated Plate
· 421 ·
When the bending moment m(s) and the forces p(s) normal to the mid-plane are applied to the per unit arc length s, the boundary conditions are of the form pj pj + iqj , lj = + iqj μj μj s f (t) = − [(m − vi)dy + (v + mi)dx] − C(x − yi) + C1 + C2 i
kj =
0
where C, C1 , C2 are real constants, v =
s
pds, and the integration is done from the start point to 0
the integration point. On the other hand, when the deflection and the angle w∗ (s), α∗ (s) are prescribed on the boundary, the boundary conditions are of the form kj = 1 + iμj , lj = 1 + iμj dw∗ dw∗ cos(n, y) + α∗ cos(n, x) − cos(n, x)i + α∗ cos(n, y)i f (t) = − ds ds where n is the outward normal on the boundary. Up to now, the bending problem of a thin plate is translated to finding two analytical complex functions wj (zj ) which satisfy the boundary equation (5).
III. ANALYSIS Consider the case of a finite composite laminated plate weakened by multiple elliptical holes with contours L0 , L1 ,...,Ll as shown in Fig.1(a) where L0 , L1 ,...,Ll are the exterior contours of the finite plate, am , bm (m = 1, 2....l) stand for the two semi-axes of the holes, and zm are the centre of the holes. By affine transformation zj = x + μj y (j = 1, 2) the region S and the point zm are transformed onto the region Sj and the points zjm in the region Sj , as shown in Fig.1(b).
Fig. 1. Mapping of a finite plate with multiple elliptical holes.
Let ϕj (zj ) = wj (z) and the principal vector of the resultant forces acting on the contour of holes be equal to zero. Then, the complex potential function ϕj (zj ) can be expressed as[6] ϕj (zj ) = ϕj0 (zj ) +
∞
cjk pk (zj ) (j = 1, 2)
(6)
k=0
where ϕj0 (zj ) is an holomorphic function in the finite region with elliptical holes; pk (zj ) is the Faber polynomial of the region limited by contour Lj0 , cjk are Faber series coefficients. Introduce the following mapping function as
tjm zj − zjm = Rjm ξjm + (7) ξjm where Rjm =
am − iμj bm , 2
tjm =
am + iμj bm am − iμj bm
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This mapping function transforms the exterior of hole in the complex plane zjm into the exterior of a unit circle, ξjm = exp(iθ) in the plane ξjm . By using the Faber polynomial in the general region, the complex potential function can be given as l ∞ ∞ −k ϕj (zj ) = bjmk ξjm + ajk zjk (8) m=1 k=1
k=0
Obviously, the complex potential function equation (8) is analytic in the region Sj . Once the unknown coefficients, bjmk and ajk , are determined by using the boundary conditions, and the stress fields can be uniquely obtained according to the uniqueness theorem in the theory of elasticity. From the mapping function equation (7), it can be seen that the function zjk is holomorphic in the complex plane zj weakened by the hole m. Therefore, the function zjk is holomorphic in the interior of the p-th hole and continuous to its boundary. Thus the functions can be expanded into a Faber series ∞ j zjn = Hn,k Pkp (zj ) (9) k=0
Similarly, −n ξjm =
∞
Ajm n,k Pkp (zj )
(10)
k=0 j [14] where the coefficients Hn,k , Ajm n,k in the Faber series can be determined by Fourier expansion method and Pkp (zj ) is the k-th Faber polynomial for the ellipse Ljp of the complex zj plane and k + Pkp (zj ) = ξjp
tkjp k ξjp
,
P0p (zj ) = 1
Substituting Eq.(9) into Eq.(8) and using ξjp = exp(iθ) = σ in the elliptical contour Ljp , we obtain the boundary values of ϕj (zj )(j = 1, 2) in power series of σ. The right side of Eq.(5) can be expanded into the complex Fourier series: ∞ fp = (Fn σ n + F−n σ −n ) + C (11) n=1
where σ = exp(iθ), and Fn , F−n are the coefficients in Fourier series. Taking the partial sum of ϕj (zj ) up to the N th power and substituting them into the boundary condition of the elliptical holes, and then equating the coefficients of the same power σ k (k = +1, +2, ..., +N ) on both sides of the equation, we obtain 2N l linear equations about the coefficients bjmk and ajk . It is obvious that the equations obtained from the inner boundary alone are not enough to determine all coefficients. Therefore it is necessary to use outer boundary conditions. The least squares boundary collocation technique is used in this paper. Taking collocation points zcn (n = 1, 2, ..., M, M > 2N ) along the outer L0 and substituting zcn into the boundary condition equation (5), we can obtain linear equations about the unknown coefficients bjk and ajk that satisfy the outer boundary conditions. These equations together with the equations that satisfy inner boundary conditions are used to determine the complex potential functions ϕj (zj ). Then the stress field in the laminated plate can be calculated by the following equations: 12Mx 12My 12Hxy σx = z, σy = z, τxy = τyx = z (12) 3 3 h h h3 where h is the thickness of plate. Obviously, the complex potential function ϕj (zj ) is an analytic function in the region Sj . Therefore the accuracy of the solution can be justified according to whether all boundary conditions are satisfied accurately (absolutely error less than 10−5 ), and the outer boundary conditions can be well satisfied to ensure the relative error within 1% by increasing the number of collocation points. According to the Saint-Venant principle, more accurate results of the stress distribution around the hole, the main concern of many researchers, is obtained by using the present method. In this paper all results are obtained by taking a partial sum up to the 10th power and 32 collocation points on the boundary. It is shown that numerical results are exact enough to indicate that to satisfy boundary conditions and the expected accuracy.
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IV. STRESS ANALYSIS OF COMPOSITE LAMINATE WITH MULTIPLE HOLES Consider a finite laminated plate weakened by a free elliptical hole in the center, subjected to Mx = 1, along the outer boundary, the bending moments and normal force acting on the contour of holes are zero, as shown in Fig.2. The semi-major, semi-minor axes and elliptical orientation are a, b, ϕ respectively. Define the stress concentration as Mθ /Mx then the effect of various parameters on the stress concentration is discussed below. Figure 3 show the stress distribution with circular hole of Fig. 2 Bending of a composite plate with an diameter D. The material properties of the laminated plate are elliptical hole. E1 = 16.9 GPa,
E2 = 1.4 GPa,
G12 = 0.7 GPa,
ν12 = 0.31
Results are compared with the exact solutions of the infinite laminated plates[1] . It is found that the relative size W/D has significant effects on the stress concentrations. The stress concentration increases rapidly with the decrease of W/D. When W/D ≥ 10, the results are closed to the exact solutions of the corresponding infinite laminated plates in reference[1] . Therefore it is rational to treat a finite plate as an infinite plate in engineering analysis when W/D ≥ 10. Figure 4 show the effect of the ellipticity on distribution of Mθ around hole. The laminated plate is composed by T300/QY8911. The material properties are E1 = 135.0 GPa,
E2 = 8.8 GPa,
Fig. 3. The effect of composite plate size on the stress distribution around hole.
G12 = 4.47 GPa,
ν12 = 0.33
Fig. 4. The effect of ellipticity on the stress distribution around hole.
It is found that as a/b increases, the stress concentrations become smaller. Especially, when a/b = 0, the hole degenerates into a crack in a finite plate, and the stress will approach infinity, but the solution of the present work is still valid in this case. Figure 5 shows the effect of the elliptical orientation on distribution of Mθ around hole, where the laminated plate is composed by T300/QY8911. The results indicate that when 0◦ ≤ ϕ ≤ 90◦ the stress concentration increases with the increase of ϕ. The location of maximum stress Mθ changes from elliptical orientation angle to elliptical angle. Consider a composite laminated plate weakened by two free circular holes in the center, subjected to Mx = 1 along the outer boundary, the bending moments and normal force acting on the contour of holes are zero, as shown in Fig.6. The laminated plate is composed by T300/QY8911. The diameters of the circular holes are D. Figures 7 and 8 show the effect of the relative center-to-center distance and the layups of laminate on the stress distribution around the holes.
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Fig. 5. The effect of the elliptical orientation angle on the stress distribution around hole.
Fig. 7. The effect of the relative center-to-center distance on the stress distribution around the holes.
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Fig. 6. Bending of a composite plate with two elliptical holes.
Fig. 8. The effect of layups of laminate on the stress distribution around hole.
It can be seen from Fig.7 that the relative center-to-center distance affects the stress distribution around holes. The stress concentration increases with the increase of the relative center-to-center distance. When L/D ≥ 10, the stress distribution is the same as the laminate with one hole. Therefore it is reasonable to treat a finite plate with multiple holes as a plate with one hole when L/D ≥ 10. Figure 8 shows the effect of layups (0α / ± 45β /90γ )s on the distribution of Mθ around the hole. The results show that the stress concentration strongly depends on the percentage of each layup in the whole plate. The more anisotropic the plate, the more intensesevere the stress concentration. The increase of number of ±45◦ lamina is beneficial to the decrease of stress concentrations, because it reduces the extent of anisotropy of laminates. Consider a composite laminated plate weakened by three free circle holes in the center, subjected to Mx = 1 along the outer boundary, the bending moments and normal force acting on the contour of holes are zero, as shown in Fig.9. The laminated plate is composed by T300/QY8911. Figure 10 shows the effect of the position of the holes on the stress concentration of a laminated plate with three holes. Figure 10 shows the effect of the position of holes on the stress concentration of a laminated plate with three holes. For hole 1*, the plate shown in Fig.9(c) has the most stress concentration around holes
Fig. 9. Bending of a composite plate with three elliptical holes.
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Fig. 10. The effect of position of laminate on the stress concentration around holes.
and the plate shown in Fig.9(a) has the least stress concentration around holes. For hole 2*, the plate shown in Fig.9(b) has the most stress concentration around holes and the plate shown in Fig.9(a) has the least stress concentration around holes. For the plate with three holes, the plate shown in Fig.9(a) has the least stress concentration for both hole 1* and hole 2*.
V. CONCLUSIONS The bending problems of a finite laminated plate are studied with the complex variable method. Concise and high accuracy solutions are presented when the plate is with multiple elliptic holes. Numerical examples are also given to discuss the effects of plate sizes, defect geometries and laminate layups on the stress concentration around the holes. From the results obtained, the following conclusions can be drawn: As the relative size W/D of a laminated plate decreases, the stress concentration may increase rapidly for the case where the laminate is with a free hole. Especially, when W/D ≥ 10, it is rational to assume that the finite plate can be treated as an infinite plate in most engineering analyses. In general, the increase of ellipticity causes more stress concentration for a laminate with a free hole. In general, when 0◦ ≤ ϕ ≤ 90◦ , the stress concentration increases with the increase of elliptical orientation angle for a laminate with a free hole. The location of maximum stress differs from elliptical orientation angle to elliptical orientation angle. The stress concentration increases with the increase of the relative center-to-center distance. When L/D ≥ 10, it is rational to treat a finite plate with multiple holes as a plate with one hole. The increase of numbers ±45◦ lamina is beneficial to the decrease of stress concentration for the laminate with two free holes as it reduces the extent of anisotropy of laminates. The present method is not only very efficient for analysis of the stress distribution of finite composite laminates with an elliptical hole owing to the application of Faber series, but also highly accurate and computer time-saving.
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[9] Chen,P. and Shen,Z., Green’s functions for an unsymmetrical laminated plate with an elliptic hole. Mechanics Research Communications, 2001, 28: 519-524. [10] Chen,P. and Nie,H., Green’s function for bending problem of a thin anisotropic plate with an elliptic hole. Mechanics Research Communications, 2003, 31: 423-428. [11] Chen,P. and Nie,H., Green’s function for bending problem of an unsymmetrical laminated plate with bending-extension coupling containing an elliptic hole. Archive of Applied Mechanics, 2004, 73: 846-856. [12] Qu,Y.Z. and Gai,B.Z., Stress concentration of bending symmetric composite laminates with elliptical holes. Journal of Astronautics, 2007, 28(4): 1065-1069. [13] Mao,C.J., Xu,X.W. and Guo,S.X., Stress analysisi of a finite anisotropic thin plate with an elliptical hole. Chiese Journal of Solid Mechanics, 2010, 31(1): 80-85. [14] Xu,X.W., The Strength Analysis of Mechanically Multi-fastened Momposite Maminate Joints [Ph.D Thesis]. Nanjing: Nanjing Aeronautical Institute, 1992 (in Chinese).