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Buckling of symmetrical cross-ply composite rectangular plates under a linearly varying in-plane load
Composite Structures 80 (2007) 42–48 www.elsevier.com/locate/compstruct
Buckling of symmetrical cross-ply composite rectangular plates under a linearly varying in-plane load Hongzhi Zhong a
a,*
, Chao Gu
b
Department of Civil Engineering, Tsinghua University, Beijing 100084, PR China Shanghai Municipal Engineering Design Institute, Shanghai 200092, PR China
b
Available online 3 April 2006
Abstract An exact solution for buckling of simply supported symmetrical cross-ply composite rectangular plates under a linearly varying edge load is presented. It is developed based on the first-order shear deformation theory for moderately thick laminated plates. Buckling loads of cross-ply rectangular plates with various aspect ratios are obtained and the effects of load intensity variation and layup configuration on the buckling load are investigated. The results are verified using the computer code ABAQUS. Ó 2006 Elsevier Ltd. All rights reserved. Keywords: Cross-ply; Buckling; Linearly varying load; Rectangular plate; Exact solution
1. Introduction Laminated composite structures are being increasingly used in aerospace, automotive, marine, and other areas. This is primarily due to their large values of specific strength and stiffness and the advantage that their properties can be tailored to meet practical requirements. The buckling load is one of the most important design considerations for laminated composite plates. Over the years, the buckling behavior of laminated composite plates has been studied extensively by many researchers. The parametric dependence of buckling load on the layup configuration and fiber orientation, etc. can be found in some handbooks and even texts, [1–3]. However, these available curves and data are restricted to idealized loading, namely, uniaxial or biaxial uniform compression. In contrast, the information about buckling of laminated composite plates under non-uniform load has been rather limited. Undoubtedly, various numerical techniques such as finite element method can be employed to perform buckling analysis of laminated plates under non-uniform load. But the analyti*
Corresponding author. Fax: +86 10 62771132. E-mail address: [email protected] (H. Zhong).
0263-8223/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2006.02.030
cal solution has been rare due to the complexity of the problem. Early research efforts for buckling analysis of anisotropic plates under in-plane bending load date back to the work of Lekhnitskii [4]. Papazoglou et al. [5] investigated the buckling of asymmetric laminates under linearly varying biaxial in-plane loads combined with shear. Their analysis was performed using the Rayleigh-Ritz method on the basis of classical lamination theory. Using finite strip method, Chai and Khong studied the optimization of laminated composite rectangular plates under a linearly varying in-plane load [6,7]. The optimum ply angle for buckling of antisymmetric laminates was acquired on the basis of classical lamination theory. Kam and Chu studied the buckling of laminated composite plates subjected to non-uniform in-plane edge loads using a shear-deformable finite element method [8]. They also carried out tests to verify their numerical results. Under classic lamination theory, Badir and Hu employed the Rayleigh-Ritz method to study the buckling of simply supported composite rectangular plates subjected to a parabolically varying axial load [9]. To the best knowledge of the authors, there is no exact solution available in the open literature for buckling analysis of laminated composite plates under non-uniform
H. Zhong, C. Gu / Composite Structures 80 (2007) 42–48
load. It should be mentioned that there have been significant advances in this aspect for buckling analysis of isotropic rectangular plates subject to non-uniform load [10–14]. However, as reported by Kang and Leissa [11], the development of exact solutions for plates under nonuniform load is rather formidable. The difficulty arises in the solution of the likely bivariant in-plane forces in the governing differential equations for plates under nonuniform load. The present work aims to develop an exact buckling solution for symmetric cross-ply laminated composite rectangular plates under a linearly varying in-plane load. All edges of the plates are assumed to be simply supported. Under such circumstances, the plate is devoid of stress diffusion and accordingly an exact solution is rendered likely. Trigonometric series are used to represent the displacement mode of the plate. Although the present solution technique is the same as that in [14], where buckling of isotropic moderately thick plates under linearly varying loads was studied, the present investigation is complicated due to the composite material behavior. The results of the solution compare fairly well with those of computer code ABAQUS. Buckling loads of symmetric cross-ply rectangular plates with various aspect ratios are obtained and depicted graphically. In addition, the effects of load intensity variation and layup configuration on the buckling load are also investigated.
43
h = 90°. The Cartesian coordinate system xyz is attached to the mid-plane of the plate. The plate is subjected to linearly varying unidirectional in-plane axial loads Nx in the x direction, which is assumed to be positive for compression. The governing differential equations for buckling of the plate [3] are given as D11
o2 wy o2 wx o2 wx ow w þ F þ D þ ð D þ D Þ ¼0 66 12 66 55 x ox ox2 oy 2 oxoy ð1Þ
ðD12 þ D66 Þ
o2 w y o2 w y o2 wx ow wy þ D66 2 þ D22 2 þ F 44 oy oxoy ox oy
¼0 ð2Þ
F 55
2 2 o w owx o w owy o2 w ¼0 þ F N 44 x ox2 oy 2 ox2 ox oy
ð3Þ
where w, wx, and wy represent the deflection and rotations about the y and x axes, respectively. The bending stiffness constants and the transverse shear stiffness constants are computed from Dij ¼
N 3 1X zk z3k1 ðQ0ij Þk 3 k¼1
F ij ¼ j
N X
ð4Þ
ðzk zk1 ÞðC 0ij Þk
ð5Þ
k¼1
2. Formulation Consider a symmetrical cross-ply composite rectangular plate of length a with width b (see Fig. 1), having all four edges simply supported. The plate is comprised of equal thickness cross-ply laminates with fiber angle h = 0° or
b
where ðQ0ij Þk and ðC 0ij Þk stand for the transformed reduced stiffness constants of each layer, which depend on the mechanical properties and the orientation angle [3]; j is a shear correction coefficient; zk zk1 is the thickness of layer k.
h z y x a Z
Layer Number N k
ZN
ZN-1 Z k
Midplane
Zk-1
Z2 2 1
Fig. 1. Geometry of a cross-ply composite plate.
Z1
Z0
44
H. Zhong, C. Gu / Composite Structures 80 (2007) 42–48
For moderately thick laminated rectangular plates with all edges simply supported, the following boundary conditions should be satisfied: at x = 0, a w ¼ M x ¼ wy ¼ 0
g11
ð6Þ g12
and at y = 0, b w ¼ M y ¼ wx ¼ 0
ð7Þ
in which Mx and My are the bending moments acting on the relevant edge. The relationships between the rotations and the moments are given by owy owx þ D12 ox oy owy owx M y ¼ D21 þ D22 ox oy
ð8Þ
M x ¼ D11
ð9Þ
The displacement components of the plate are expressed as 1 mpx X npy wðx; y Þ ¼ sin W n sin a n¼1 b 1 X mpx npy wx ðx; yÞ ¼ cos X n sin a n¼1 b 1 mpx X npy wy ðx; yÞ ¼ sin Y n cos a n¼1 b
ð10Þ ð11Þ ð12Þ
ð13Þ
mp2 mnp X n þ D66 ðD12 þ D66 Þ a ab n¼1 np2 mp mpx npy W n sin cos ¼0 þ D22 þ F 44 Y n F 44 b a a b 2
g13 g23
np2 ¼ D11 þ D66 þ F 55 ; a b mnp2 ¼ g21 ¼ ðD12 þ D66 Þ ; ab mp np2 mp 2 ¼ F 55 þ D22 þ F 44 ; ; g22 ¼ D66 a a b np ¼ F 44 b ð17Þ
ð14Þ
mp2
Substitution of Eqs. (10)–(12) into Eq. (3), combined with Eqs. (15) and (16), yields 1 X
TW n sin
n¼1
where
in which Wn, Xn, and Yn are unknown coefficients to be determined. It is easy to verify that the simply supported boundary conditions in (6) and (7) are satisfied by the displacement components. Substitution of Eqs. (10)–(12) into Eqs. (1) and (2) yields 1 mp2 np2 X D11 þ D66 þ F 55 X n a b n¼1 mnp2 mp mpx npy W n cos sin ¼0 Y n F 55 þ ðD12 þ D66 Þ a a b ab 1
X
where
1 mp2 X npy npy þ Nx ¼0 W n sin b a b n¼1
mp g13 g22 g12 g23 mp2 T ¼ F 55 a g11 g22 g12 g21 a np g11 g23 g13 g21 np2 þ F 44 b g11 g22 g12 g21 b
ð18Þ
ð19Þ
The in-plane axial load distribution is expanded in terms of the Fourier series, i.e. N x ¼ kb0 þ k
1 X j¼1
bj cos
jpy b
ð20Þ
in which k denotes the multiple of the load to be determined. When the distribution of load is given as f(y), y 2 [0, b], the coefficients in (20) are determined as follows: Z Z 1 b 2 b jpy dy; j P 1 b0 ¼ f ðyÞdy; bj ¼ f ðyÞ cos b 0 b 0 b ð21Þ Introducing Eq. (20) into Eq. (18) and performing some simple trigonometric substitutions, one arrives at the following equation 1 mp2 X npy Tþ kb0 W n sin a b n¼1 þ
1 X 1 1 mp2 X npy ¼0 k bj ðW nj þ W nþj W jn Þ sin 2 a b n¼1 j¼1
Obviously, each of the above two equations is then satisfied only if it is satisfied for each individual term of the sine and cosine series. Cancelling out the cos mpx sin npy a b npy mpx and sin a cos b and solving the resulting equations simultaneously for Xn and Yn, one finds
W j ¼ 0;
g g g12 g23 X n ¼ 13 22 Wn g11 g22 g12 g21 g g g13 g21 Wn Y n ¼ 11 23 g11 g22 g12 g21
In a similar manner, Eq. (22) only holds when each indiis identically equal to zero. Canvidual term of series sin npy b celling out sin npy results in the following homogeneous b linear system of Wn
ð15Þ ð16Þ
ð22Þ It is understood herein that j ¼ 0; 1; 2; 3; . . . .
ð23Þ
H. Zhong, C. Gu / Composite Structures 80 (2007) 42–48
1 1 1 X X X 2b0 1 bj bj bj þ Wnþ W nj þ W nþj W jn ¼ 0; k S S S S j¼1 j¼1 j¼1 n ¼ 1; 2; . . .
ð24Þ
where 2
2
2
S ¼ 2Ta =ðp m Þ
ð25Þ
It is noticeable that bj/S diminishes with more terms included in the coefficient matrix so that the convergence of the infinite determinant is assured. Using the double QR algorithm [15], the eigenvalues 1/ki (i = 1, 2, . . .) of the coefficient matrix are extracted. Obviously, the maximum eigenvalue of the coefficient matrix is the desired buckling load. In the case of uniaxial uniform compression (bj = 0, j = 1, 2, 3, . . .), the eigenvalue is obtained as kcr ¼ S=2b0 ¼ Ta2 =ðp2 m2 b0 Þ.
ð26Þ
3. Results and discussion The mechanical properties assumed in the analysis are: EL ¼ 40; ET
GLT GLZ ¼ ¼ 0:6; ET ET
GZT ¼ 0:5; ET
vLT ¼ 0:25
where EL, ET, GLT, GLZ, GTZ, and vLT are the Young’s modulus along the fiber, Young’s modulus across the fiber, inplane shear modulus, transverse shear moduli, and major Poisson’s ratio, respectively. The commonly used shear correction coefficient 5/6 is taken and a symmetrical cross-ply plate [0°/90°/0°] is considered in all computations unless stated otherwise. Sixty terms are used in the Fourier expansion of the linearly varying load (see Eq. (20)) although it is usually sufficient to take 20–30 terms. The reason that more terms are used in the Fourier expansion is to assure the accuracy and convergence of results for any linearly varying loads and plate properties. The infinite linear system (see Eq. (24)) is truncated to the first 60 equations which are found adequate to acquire the eigenvalue of satisfactory accuracy. The compressive force is expressed as y Nx ¼ N0 1 g ð27Þ b where parameter g determines the linear variation of the inplane load. The present discussion is confined to the case for 0 6 g 6 2 since similar scenario arises for the value of g beyond this range. For g = 0, it corresponds to the case of uniform compression. The pure in-plane bending comes about for g = 2. For 0 < g < 2, an eccentric bending is procured, which is a combination of pure bending and uniform compression. The commonly adopted non-dimensional critical load is introduced, i.e. N 0 b2 kcr b2 k¼ ¼ E T h3 E T h3
ð28Þ
In order to verify the present solution, cross-ply laminate plates subjected to the uni-axial uniform loads (g = 0) are studied first. In Table 1, comparison of buckling
45
Table 1 Comparison of non-dimensional buckling load factors for symmetric cross-ply square plates subjected to uni-axial uniform loads (g = 0), h/b = 0.1 Source
Layup configuration
EL/ET 20
30
40
Khdeir [16] Present ABAQUS
[0°/90°/0°]
14.985 14.987 14.836
19.027 19.029 18.820
22.315 22.317 22.048
Khdeir [16] Present ABAQUS
[0°/90°/0°]S
15.736 15.750 15.642
20.485 20.497 20.375
24.547 24.558 24.416
Khdeir [16] Present ABAQUS
[(0°/90°)2/0°]S
16.068 16.088 15.997
21.117 21.135 21.050
25.495 25.511 25.427
loads for plates subjected to the uni-axial uniform loads (g = 0) is made with the results of computer code ABAQUS and those given by Khdeir [16]. It is noted that his results were obtained based on the same first-order shear deformation theory using a state space approach. Good to excellent agreement is reached. Further verification is made in Table 2 where the results of cross-ply plates subjected to linearly varying loads by the present method are compared with the results of ABAQUS. It is shown that they are in fairly good agreement. In the present exact solution, large m is usually required to obtain the accurate buckling load for moderately thick plates under severe uneven load. An extreme case is shown in Fig. 2, where m reaches some 10,000 before a virtually invariant buckling load for a square plate under pure bending is procured. The shell element S8R5 in ABAQUS element library is utilized in the buckling analysis. The least qualified mesh density varies with the load distribution parameter g. The smaller g, the coarser the mesh that suffices. In the case of uniform compression (g = 0), mere 200 elements are needed in the ABAQUS analysis for either thin or moderately thick plates. For the moderately thick plates under severe uneven load, say g P 1.5, 3200 elements are used and appreciable discrepancies still exist between ABAQUS results and the present solution. This is ascribed to the complexity of the buckling modes Table 2 Comparison of non-dimensional buckling load factors for symmetric cross-ply square plates [0°/90°/0°] subjected to various linearly varying loads h/b = 0.1
h/b = 0.15
0.5
Present ABAQUS
h/b = 0.01 47.267 47.185
41.075 40.849
29.432 29.121
20.364 20.037
1.0
Present ABAQUS
64.982 64.847
56.705 56.392
40.999 40.599
21.131 21.559
1.5
Present ABAQUS
91.374 91.142
80.336 79.905
47.708 49.210
21.204 21.872
2.0
Present ABAQUS
129.785 129.450
114.837 114.296
47.872 49.955
21.277 22.214
g
h/b = 0.05
46
H. Zhong, C. Gu / Composite Structures 80 (2007) 42–48
Fig. 3. Variation of the buckling factor k versus the aspect ratio a/b of [0°/90°/0°] plates for g = 0.5. Fig. 2. Selection of number of axial half-waves in determination of buckling load.
for the plates in question. An even finer ABAQUS mesh is employed and it takes exorbitantly long running time before a result with limited improvement is gained. In Table 3, the effect of layup configuration on the nondimensional buckling factor is shown. It is seen that the number of layers has less significant effect on the buckling factor except for the thin plates under pure bending where the buckling factor increases when more layers are used in the symmetric cross-ply laminated plates. The largest discrepancy between the buckling factors for plates with three layers and plates with 15 layers is only of 13% of the buckling factor for plates with three layers. Figs. 3–6 display the variation of the non-dimensional buckling load versus the aspect ratio (a/b) of [0°/90°/0°] plates subjected to linearly varying in-plane loads. With the increase of aspect ratio, the non-dimensional buckling load decreases for short plates and the effect of aspect ratio on the buckling load is negligible for long plates especially for relatively thin plates. For plates with large thicknessto-width ratio under severe uneven load (g P 1.5), it is found that large m is required in the exact solution and the aspect ratio has little effects on the non-dimensional buckling factor. Fig. 7 shows the effect of the thickness-to-width ratio on the non-dimensional buckling factor for a cross-ply laminated square plate. For plates under load distribution close
Fig. 4. Variation of the buckling factor k versus the aspect ratio a/b of [0°/90°/0°] plates for g = 1.
Table 3 Effect of number of layers on the non-dimensional buckling factor for symmetric cross-ply square plates, a/b = 1 Layup configuration
g=1
g=2
h/b = 0.1
h/b = 0.01
h/b = 0.1
h/b = 0.01
[0°/90°/0°] [0°/90°/0°]S [0°/90°/0°/90°]S [(0°/90°)2/0°]S [(0°/90°)30°/90°]S
40.999 46.985 46.746 46.613 46.426
64.982 69.396 70.033 70.295 70.599
47.872 47.309 47.067 46.933 46.745
129.785 200.847 226.042 239.171 256.626
Fig. 5. Variation of the buckling factor k versus the aspect ratio a/b of [0°/90°/0°] plates for g = 1.5.
H. Zhong, C. Gu / Composite Structures 80 (2007) 42–48
Fig. 6. Variation of the buckling factor k versus the aspect ratio a/b of [0°/90°/0°] plates under pure in-plane bending (g = 2).
47
Fig. 8. Effect of modulus ratio EL/ET on the buckling load of a square plate.
dimensional buckling factor begin to level off for plates with large EL/ET under severe non-uniform loads. For instance, the buckling load of the square plate under pure bending (g = 2.0) remains unchanged when EL/ET is larger than 12.0. The phenomenon is also checked and verified by the result of the computer code ABAQUS. This peculiar observation implies that there exists an ultimate buckling resistance capacity for moderately thick symmetric crossply laminated plates under a severe non-uniform in-plane load with respect to the increase of EL/ET. 4. Conclusion
Fig. 7. Effect of thickness-to-width ratio on the buckling load of a laminated square plate.
to uniform load distribution (g 6 1), the non-dimensional buckling load decreases slowly with the increase of thickness-to width ratio. For plates under severe non-uniform load distribution (g > 1), more pronounced decrease of the non-dimensional buckling factor occurs when the thickness-to-width ratio is sufficiently large. In the case of pure bending, a dramatic decrease occurs when thicknessto-width ratio is larger than 0.068, indicating the transition of buckling modes. Meanwhile, large m is needed to obtain the accurate buckling load in the exact solution while many iterations are needed in the ABAQUS runs. Fig. 8 shows the effect of the Young’s modulus ratio EL/ ET on the non-dimensional buckling factor of a square plate with thickness-to-width ratio h/b = 0.1 while GLT/ ET, GLZ/ET and GTZ/ET remain constant. It is seen that the buckling factor increases with EL/ET for plates under moderately non-uniform load (g < 1.0). A similar scenario arises for plates under severe non-uniform load when EL/ ET is not quite large. However, the curves of the non-
An exact solution has been developed to investigate the buckling behavior of simply supported symmetrical crossply rectangular plates subjected to unidirectional linearly varying in-plane loads. The present exact solution is developed on the basis of the first-order shear deformation lamination theory. The results of the present solution are verified by those of computer code ABAQUS. A parametric study is conducted to investigate the effects of aspect ratio, thickness-to-width ratio and the modulus ratio on the buckling load factor. The present solution confirms the commonly recognized conclusion that the buckling factor estimate based on uniform load is non-conservative for plates under non-uniform load. A novel discovery in the present investigation is that there exists ultimate buckling resistance capacity for moderately thick symmetric crossply laminated plates under a severe non-uniform in-plane load with respect to the increase of modulus ratio. References [1] Leissa AW. Buckling of laminated composite plates and shells panels. Air Force Wright-Patterson Aeronautical Laboratories, Final report, No. AFWAL-TR-85-3069; 1985. [2] Herakovich CT, Tarnopolskii YM. Handbook of composites. Structures and design, vol. 2. Amsterdam: North-Holland; 1989.
48
H. Zhong, C. Gu / Composite Structures 80 (2007) 42–48
[3] Berthelot JM. Composite materials mechanical behavior and structural analysis. New York: Springer; 1999. [4] Lekhnitskii SG. Anisotropic plates. New York: Gordon and Beach Science Publishers; 1968. [5] Papazoglou VJ, Tsouvalis NG, Kyriakopoulos GD. Buckling of unsymmetric laminates under linearly varying, biaxial in-plane loads, combined with shear. Compos Struct 1992;20:155–63. [6] Chai GB, Khong PW. The effect of varying the support conditions on the buckling of laminated composite plates. Compos Struct 1993;24: 99–106. [7] Chai GB, Ooi KT, Khong PW. Buckling strength optimization of laminated composite plates. Comput Struct 1993;46:77–82. [8] Kam TY, Chu KH. Buckling of laminated composite plates subject to nonuniform in-plane edge loads. Mater Des Technol, ASME 1995;71:207–15. [9] Badir A, Hu H. Elastic buckling of laminated plates under varying axial stresses. In: Collection of technical papers – AIAA/ASME/ ASCE/AHS/ASC structures, structural dynamics and materials conference, vol. 1. AIAA-98-1774; 1998. p. 635–40.
[10] Timoshenko SP, Gere JM. Theory of elastic stability. 2nd ed. New York: McGraw-Hill; 1961. [11] Kang JH, Leissa AW. Vibration and buckling of SS-F-SS-F rectangular plates loaded by in-plane moments. Int J Struct Stabil Dynam 2001;1:527–43. [12] Leissa AW, Kang JH. Exact solutions for vibration and buckling of an SS-C-SS-C rectangular plate loaded by linearly varying in-plane stresses. Int J Mech Sci 2002;44:1925–45. [13] Bert CW, Devarakonda KK. Buckling of rectangular plates subjected to nonlinearly distributed in-plane loading. Int J Solids Struct 2003;40:4097–106. [14] Zhong H, Gu C. Buckling of simply supported rectangular ReissnerMindlin plates subjected to linearly varying in-plane loading. J Eng Mech, ASCE 2006. May. [15] Press WH, Flannery BP, Teukolsky SA, Vetterling WT. Numerical recipes – the art of scientific computing. Cambridge: Cambridge University Press; 1986. [16] Khdeir AA. Free vibration and buckling of symmetric cross-ply laminated plates by an exact method. J Sound Vib 1988;126:447–61.
Buckling of symmetrical cross-ply composite rectangular plates under a linearly varying in-plane load Hongzhi Zhong a
a,*
, Chao Gu
b
Department of Civil Engineering, Tsinghua University, Beijing 100084, PR China Shanghai Municipal Engineering Design Institute, Shanghai 200092, PR China
b
Available online 3 April 2006
Abstract An exact solution for buckling of simply supported symmetrical cross-ply composite rectangular plates under a linearly varying edge load is presented. It is developed based on the first-order shear deformation theory for moderately thick laminated plates. Buckling loads of cross-ply rectangular plates with various aspect ratios are obtained and the effects of load intensity variation and layup configuration on the buckling load are investigated. The results are verified using the computer code ABAQUS. Ó 2006 Elsevier Ltd. All rights reserved. Keywords: Cross-ply; Buckling; Linearly varying load; Rectangular plate; Exact solution
1. Introduction Laminated composite structures are being increasingly used in aerospace, automotive, marine, and other areas. This is primarily due to their large values of specific strength and stiffness and the advantage that their properties can be tailored to meet practical requirements. The buckling load is one of the most important design considerations for laminated composite plates. Over the years, the buckling behavior of laminated composite plates has been studied extensively by many researchers. The parametric dependence of buckling load on the layup configuration and fiber orientation, etc. can be found in some handbooks and even texts, [1–3]. However, these available curves and data are restricted to idealized loading, namely, uniaxial or biaxial uniform compression. In contrast, the information about buckling of laminated composite plates under non-uniform load has been rather limited. Undoubtedly, various numerical techniques such as finite element method can be employed to perform buckling analysis of laminated plates under non-uniform load. But the analyti*
Corresponding author. Fax: +86 10 62771132. E-mail address: [email protected] (H. Zhong).
0263-8223/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2006.02.030
cal solution has been rare due to the complexity of the problem. Early research efforts for buckling analysis of anisotropic plates under in-plane bending load date back to the work of Lekhnitskii [4]. Papazoglou et al. [5] investigated the buckling of asymmetric laminates under linearly varying biaxial in-plane loads combined with shear. Their analysis was performed using the Rayleigh-Ritz method on the basis of classical lamination theory. Using finite strip method, Chai and Khong studied the optimization of laminated composite rectangular plates under a linearly varying in-plane load [6,7]. The optimum ply angle for buckling of antisymmetric laminates was acquired on the basis of classical lamination theory. Kam and Chu studied the buckling of laminated composite plates subjected to non-uniform in-plane edge loads using a shear-deformable finite element method [8]. They also carried out tests to verify their numerical results. Under classic lamination theory, Badir and Hu employed the Rayleigh-Ritz method to study the buckling of simply supported composite rectangular plates subjected to a parabolically varying axial load [9]. To the best knowledge of the authors, there is no exact solution available in the open literature for buckling analysis of laminated composite plates under non-uniform
H. Zhong, C. Gu / Composite Structures 80 (2007) 42–48
load. It should be mentioned that there have been significant advances in this aspect for buckling analysis of isotropic rectangular plates subject to non-uniform load [10–14]. However, as reported by Kang and Leissa [11], the development of exact solutions for plates under nonuniform load is rather formidable. The difficulty arises in the solution of the likely bivariant in-plane forces in the governing differential equations for plates under nonuniform load. The present work aims to develop an exact buckling solution for symmetric cross-ply laminated composite rectangular plates under a linearly varying in-plane load. All edges of the plates are assumed to be simply supported. Under such circumstances, the plate is devoid of stress diffusion and accordingly an exact solution is rendered likely. Trigonometric series are used to represent the displacement mode of the plate. Although the present solution technique is the same as that in [14], where buckling of isotropic moderately thick plates under linearly varying loads was studied, the present investigation is complicated due to the composite material behavior. The results of the solution compare fairly well with those of computer code ABAQUS. Buckling loads of symmetric cross-ply rectangular plates with various aspect ratios are obtained and depicted graphically. In addition, the effects of load intensity variation and layup configuration on the buckling load are also investigated.
43
h = 90°. The Cartesian coordinate system xyz is attached to the mid-plane of the plate. The plate is subjected to linearly varying unidirectional in-plane axial loads Nx in the x direction, which is assumed to be positive for compression. The governing differential equations for buckling of the plate [3] are given as D11
o2 wy o2 wx o2 wx ow w þ F þ D þ ð D þ D Þ ¼0 66 12 66 55 x ox ox2 oy 2 oxoy ð1Þ
ðD12 þ D66 Þ
o2 w y o2 w y o2 wx ow wy þ D66 2 þ D22 2 þ F 44 oy oxoy ox oy
¼0 ð2Þ
F 55
2 2 o w owx o w owy o2 w ¼0 þ F N 44 x ox2 oy 2 ox2 ox oy
ð3Þ
where w, wx, and wy represent the deflection and rotations about the y and x axes, respectively. The bending stiffness constants and the transverse shear stiffness constants are computed from Dij ¼
N 3 1X zk z3k1 ðQ0ij Þk 3 k¼1
F ij ¼ j
N X
ð4Þ
ðzk zk1 ÞðC 0ij Þk
ð5Þ
k¼1
2. Formulation Consider a symmetrical cross-ply composite rectangular plate of length a with width b (see Fig. 1), having all four edges simply supported. The plate is comprised of equal thickness cross-ply laminates with fiber angle h = 0° or
b
where ðQ0ij Þk and ðC 0ij Þk stand for the transformed reduced stiffness constants of each layer, which depend on the mechanical properties and the orientation angle [3]; j is a shear correction coefficient; zk zk1 is the thickness of layer k.
h z y x a Z
Layer Number N k
ZN
ZN-1 Z k
Midplane
Zk-1
Z2 2 1
Fig. 1. Geometry of a cross-ply composite plate.
Z1
Z0
44
H. Zhong, C. Gu / Composite Structures 80 (2007) 42–48
For moderately thick laminated rectangular plates with all edges simply supported, the following boundary conditions should be satisfied: at x = 0, a w ¼ M x ¼ wy ¼ 0
g11
ð6Þ g12
and at y = 0, b w ¼ M y ¼ wx ¼ 0
ð7Þ
in which Mx and My are the bending moments acting on the relevant edge. The relationships between the rotations and the moments are given by owy owx þ D12 ox oy owy owx M y ¼ D21 þ D22 ox oy
ð8Þ
M x ¼ D11
ð9Þ
The displacement components of the plate are expressed as 1 mpx X npy wðx; y Þ ¼ sin W n sin a n¼1 b 1 X mpx npy wx ðx; yÞ ¼ cos X n sin a n¼1 b 1 mpx X npy wy ðx; yÞ ¼ sin Y n cos a n¼1 b
ð10Þ ð11Þ ð12Þ
ð13Þ
mp2 mnp X n þ D66 ðD12 þ D66 Þ a ab n¼1 np2 mp mpx npy W n sin cos ¼0 þ D22 þ F 44 Y n F 44 b a a b 2
g13 g23
np2 ¼ D11 þ D66 þ F 55 ; a b mnp2 ¼ g21 ¼ ðD12 þ D66 Þ ; ab mp np2 mp 2 ¼ F 55 þ D22 þ F 44 ; ; g22 ¼ D66 a a b np ¼ F 44 b ð17Þ
ð14Þ
mp2
Substitution of Eqs. (10)–(12) into Eq. (3), combined with Eqs. (15) and (16), yields 1 X
TW n sin
n¼1
where
in which Wn, Xn, and Yn are unknown coefficients to be determined. It is easy to verify that the simply supported boundary conditions in (6) and (7) are satisfied by the displacement components. Substitution of Eqs. (10)–(12) into Eqs. (1) and (2) yields 1 mp2 np2 X D11 þ D66 þ F 55 X n a b n¼1 mnp2 mp mpx npy W n cos sin ¼0 Y n F 55 þ ðD12 þ D66 Þ a a b ab 1
X
where
1 mp2 X npy npy þ Nx ¼0 W n sin b a b n¼1
mp g13 g22 g12 g23 mp2 T ¼ F 55 a g11 g22 g12 g21 a np g11 g23 g13 g21 np2 þ F 44 b g11 g22 g12 g21 b
ð18Þ
ð19Þ
The in-plane axial load distribution is expanded in terms of the Fourier series, i.e. N x ¼ kb0 þ k
1 X j¼1
bj cos
jpy b
ð20Þ
in which k denotes the multiple of the load to be determined. When the distribution of load is given as f(y), y 2 [0, b], the coefficients in (20) are determined as follows: Z Z 1 b 2 b jpy dy; j P 1 b0 ¼ f ðyÞdy; bj ¼ f ðyÞ cos b 0 b 0 b ð21Þ Introducing Eq. (20) into Eq. (18) and performing some simple trigonometric substitutions, one arrives at the following equation 1 mp2 X npy Tþ kb0 W n sin a b n¼1 þ
1 X 1 1 mp2 X npy ¼0 k bj ðW nj þ W nþj W jn Þ sin 2 a b n¼1 j¼1
Obviously, each of the above two equations is then satisfied only if it is satisfied for each individual term of the sine and cosine series. Cancelling out the cos mpx sin npy a b npy mpx and sin a cos b and solving the resulting equations simultaneously for Xn and Yn, one finds
W j ¼ 0;
g g g12 g23 X n ¼ 13 22 Wn g11 g22 g12 g21 g g g13 g21 Wn Y n ¼ 11 23 g11 g22 g12 g21
In a similar manner, Eq. (22) only holds when each indiis identically equal to zero. Canvidual term of series sin npy b celling out sin npy results in the following homogeneous b linear system of Wn
ð15Þ ð16Þ
ð22Þ It is understood herein that j ¼ 0; 1; 2; 3; . . . .
ð23Þ
H. Zhong, C. Gu / Composite Structures 80 (2007) 42–48
1 1 1 X X X 2b0 1 bj bj bj þ Wnþ W nj þ W nþj W jn ¼ 0; k S S S S j¼1 j¼1 j¼1 n ¼ 1; 2; . . .
ð24Þ
where 2
2
2
S ¼ 2Ta =ðp m Þ
ð25Þ
It is noticeable that bj/S diminishes with more terms included in the coefficient matrix so that the convergence of the infinite determinant is assured. Using the double QR algorithm [15], the eigenvalues 1/ki (i = 1, 2, . . .) of the coefficient matrix are extracted. Obviously, the maximum eigenvalue of the coefficient matrix is the desired buckling load. In the case of uniaxial uniform compression (bj = 0, j = 1, 2, 3, . . .), the eigenvalue is obtained as kcr ¼ S=2b0 ¼ Ta2 =ðp2 m2 b0 Þ.
ð26Þ
3. Results and discussion The mechanical properties assumed in the analysis are: EL ¼ 40; ET
GLT GLZ ¼ ¼ 0:6; ET ET
GZT ¼ 0:5; ET
vLT ¼ 0:25
where EL, ET, GLT, GLZ, GTZ, and vLT are the Young’s modulus along the fiber, Young’s modulus across the fiber, inplane shear modulus, transverse shear moduli, and major Poisson’s ratio, respectively. The commonly used shear correction coefficient 5/6 is taken and a symmetrical cross-ply plate [0°/90°/0°] is considered in all computations unless stated otherwise. Sixty terms are used in the Fourier expansion of the linearly varying load (see Eq. (20)) although it is usually sufficient to take 20–30 terms. The reason that more terms are used in the Fourier expansion is to assure the accuracy and convergence of results for any linearly varying loads and plate properties. The infinite linear system (see Eq. (24)) is truncated to the first 60 equations which are found adequate to acquire the eigenvalue of satisfactory accuracy. The compressive force is expressed as y Nx ¼ N0 1 g ð27Þ b where parameter g determines the linear variation of the inplane load. The present discussion is confined to the case for 0 6 g 6 2 since similar scenario arises for the value of g beyond this range. For g = 0, it corresponds to the case of uniform compression. The pure in-plane bending comes about for g = 2. For 0 < g < 2, an eccentric bending is procured, which is a combination of pure bending and uniform compression. The commonly adopted non-dimensional critical load is introduced, i.e. N 0 b2 kcr b2 k¼ ¼ E T h3 E T h3
ð28Þ
In order to verify the present solution, cross-ply laminate plates subjected to the uni-axial uniform loads (g = 0) are studied first. In Table 1, comparison of buckling
45
Table 1 Comparison of non-dimensional buckling load factors for symmetric cross-ply square plates subjected to uni-axial uniform loads (g = 0), h/b = 0.1 Source
Layup configuration
EL/ET 20
30
40
Khdeir [16] Present ABAQUS
[0°/90°/0°]
14.985 14.987 14.836
19.027 19.029 18.820
22.315 22.317 22.048
Khdeir [16] Present ABAQUS
[0°/90°/0°]S
15.736 15.750 15.642
20.485 20.497 20.375
24.547 24.558 24.416
Khdeir [16] Present ABAQUS
[(0°/90°)2/0°]S
16.068 16.088 15.997
21.117 21.135 21.050
25.495 25.511 25.427
loads for plates subjected to the uni-axial uniform loads (g = 0) is made with the results of computer code ABAQUS and those given by Khdeir [16]. It is noted that his results were obtained based on the same first-order shear deformation theory using a state space approach. Good to excellent agreement is reached. Further verification is made in Table 2 where the results of cross-ply plates subjected to linearly varying loads by the present method are compared with the results of ABAQUS. It is shown that they are in fairly good agreement. In the present exact solution, large m is usually required to obtain the accurate buckling load for moderately thick plates under severe uneven load. An extreme case is shown in Fig. 2, where m reaches some 10,000 before a virtually invariant buckling load for a square plate under pure bending is procured. The shell element S8R5 in ABAQUS element library is utilized in the buckling analysis. The least qualified mesh density varies with the load distribution parameter g. The smaller g, the coarser the mesh that suffices. In the case of uniform compression (g = 0), mere 200 elements are needed in the ABAQUS analysis for either thin or moderately thick plates. For the moderately thick plates under severe uneven load, say g P 1.5, 3200 elements are used and appreciable discrepancies still exist between ABAQUS results and the present solution. This is ascribed to the complexity of the buckling modes Table 2 Comparison of non-dimensional buckling load factors for symmetric cross-ply square plates [0°/90°/0°] subjected to various linearly varying loads h/b = 0.1
h/b = 0.15
0.5
Present ABAQUS
h/b = 0.01 47.267 47.185
41.075 40.849
29.432 29.121
20.364 20.037
1.0
Present ABAQUS
64.982 64.847
56.705 56.392
40.999 40.599
21.131 21.559
1.5
Present ABAQUS
91.374 91.142
80.336 79.905
47.708 49.210
21.204 21.872
2.0
Present ABAQUS
129.785 129.450
114.837 114.296
47.872 49.955
21.277 22.214
g
h/b = 0.05
46
H. Zhong, C. Gu / Composite Structures 80 (2007) 42–48
Fig. 3. Variation of the buckling factor k versus the aspect ratio a/b of [0°/90°/0°] plates for g = 0.5. Fig. 2. Selection of number of axial half-waves in determination of buckling load.
for the plates in question. An even finer ABAQUS mesh is employed and it takes exorbitantly long running time before a result with limited improvement is gained. In Table 3, the effect of layup configuration on the nondimensional buckling factor is shown. It is seen that the number of layers has less significant effect on the buckling factor except for the thin plates under pure bending where the buckling factor increases when more layers are used in the symmetric cross-ply laminated plates. The largest discrepancy between the buckling factors for plates with three layers and plates with 15 layers is only of 13% of the buckling factor for plates with three layers. Figs. 3–6 display the variation of the non-dimensional buckling load versus the aspect ratio (a/b) of [0°/90°/0°] plates subjected to linearly varying in-plane loads. With the increase of aspect ratio, the non-dimensional buckling load decreases for short plates and the effect of aspect ratio on the buckling load is negligible for long plates especially for relatively thin plates. For plates with large thicknessto-width ratio under severe uneven load (g P 1.5), it is found that large m is required in the exact solution and the aspect ratio has little effects on the non-dimensional buckling factor. Fig. 7 shows the effect of the thickness-to-width ratio on the non-dimensional buckling factor for a cross-ply laminated square plate. For plates under load distribution close
Fig. 4. Variation of the buckling factor k versus the aspect ratio a/b of [0°/90°/0°] plates for g = 1.
Table 3 Effect of number of layers on the non-dimensional buckling factor for symmetric cross-ply square plates, a/b = 1 Layup configuration
g=1
g=2
h/b = 0.1
h/b = 0.01
h/b = 0.1
h/b = 0.01
[0°/90°/0°] [0°/90°/0°]S [0°/90°/0°/90°]S [(0°/90°)2/0°]S [(0°/90°)30°/90°]S
40.999 46.985 46.746 46.613 46.426
64.982 69.396 70.033 70.295 70.599
47.872 47.309 47.067 46.933 46.745
129.785 200.847 226.042 239.171 256.626
Fig. 5. Variation of the buckling factor k versus the aspect ratio a/b of [0°/90°/0°] plates for g = 1.5.
H. Zhong, C. Gu / Composite Structures 80 (2007) 42–48
Fig. 6. Variation of the buckling factor k versus the aspect ratio a/b of [0°/90°/0°] plates under pure in-plane bending (g = 2).
47
Fig. 8. Effect of modulus ratio EL/ET on the buckling load of a square plate.
dimensional buckling factor begin to level off for plates with large EL/ET under severe non-uniform loads. For instance, the buckling load of the square plate under pure bending (g = 2.0) remains unchanged when EL/ET is larger than 12.0. The phenomenon is also checked and verified by the result of the computer code ABAQUS. This peculiar observation implies that there exists an ultimate buckling resistance capacity for moderately thick symmetric crossply laminated plates under a severe non-uniform in-plane load with respect to the increase of EL/ET. 4. Conclusion
Fig. 7. Effect of thickness-to-width ratio on the buckling load of a laminated square plate.
to uniform load distribution (g 6 1), the non-dimensional buckling load decreases slowly with the increase of thickness-to width ratio. For plates under severe non-uniform load distribution (g > 1), more pronounced decrease of the non-dimensional buckling factor occurs when the thickness-to-width ratio is sufficiently large. In the case of pure bending, a dramatic decrease occurs when thicknessto-width ratio is larger than 0.068, indicating the transition of buckling modes. Meanwhile, large m is needed to obtain the accurate buckling load in the exact solution while many iterations are needed in the ABAQUS runs. Fig. 8 shows the effect of the Young’s modulus ratio EL/ ET on the non-dimensional buckling factor of a square plate with thickness-to-width ratio h/b = 0.1 while GLT/ ET, GLZ/ET and GTZ/ET remain constant. It is seen that the buckling factor increases with EL/ET for plates under moderately non-uniform load (g < 1.0). A similar scenario arises for plates under severe non-uniform load when EL/ ET is not quite large. However, the curves of the non-
An exact solution has been developed to investigate the buckling behavior of simply supported symmetrical crossply rectangular plates subjected to unidirectional linearly varying in-plane loads. The present exact solution is developed on the basis of the first-order shear deformation lamination theory. The results of the present solution are verified by those of computer code ABAQUS. A parametric study is conducted to investigate the effects of aspect ratio, thickness-to-width ratio and the modulus ratio on the buckling load factor. The present solution confirms the commonly recognized conclusion that the buckling factor estimate based on uniform load is non-conservative for plates under non-uniform load. A novel discovery in the present investigation is that there exists ultimate buckling resistance capacity for moderately thick symmetric crossply laminated plates under a severe non-uniform in-plane load with respect to the increase of modulus ratio. References [1] Leissa AW. Buckling of laminated composite plates and shells panels. Air Force Wright-Patterson Aeronautical Laboratories, Final report, No. AFWAL-TR-85-3069; 1985. [2] Herakovich CT, Tarnopolskii YM. Handbook of composites. Structures and design, vol. 2. Amsterdam: North-Holland; 1989.
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H. Zhong, C. Gu / Composite Structures 80 (2007) 42–48
[3] Berthelot JM. Composite materials mechanical behavior and structural analysis. New York: Springer; 1999. [4] Lekhnitskii SG. Anisotropic plates. New York: Gordon and Beach Science Publishers; 1968. [5] Papazoglou VJ, Tsouvalis NG, Kyriakopoulos GD. Buckling of unsymmetric laminates under linearly varying, biaxial in-plane loads, combined with shear. Compos Struct 1992;20:155–63. [6] Chai GB, Khong PW. The effect of varying the support conditions on the buckling of laminated composite plates. Compos Struct 1993;24: 99–106. [7] Chai GB, Ooi KT, Khong PW. Buckling strength optimization of laminated composite plates. Comput Struct 1993;46:77–82. [8] Kam TY, Chu KH. Buckling of laminated composite plates subject to nonuniform in-plane edge loads. Mater Des Technol, ASME 1995;71:207–15. [9] Badir A, Hu H. Elastic buckling of laminated plates under varying axial stresses. In: Collection of technical papers – AIAA/ASME/ ASCE/AHS/ASC structures, structural dynamics and materials conference, vol. 1. AIAA-98-1774; 1998. p. 635–40.
[10] Timoshenko SP, Gere JM. Theory of elastic stability. 2nd ed. New York: McGraw-Hill; 1961. [11] Kang JH, Leissa AW. Vibration and buckling of SS-F-SS-F rectangular plates loaded by in-plane moments. Int J Struct Stabil Dynam 2001;1:527–43. [12] Leissa AW, Kang JH. Exact solutions for vibration and buckling of an SS-C-SS-C rectangular plate loaded by linearly varying in-plane stresses. Int J Mech Sci 2002;44:1925–45. [13] Bert CW, Devarakonda KK. Buckling of rectangular plates subjected to nonlinearly distributed in-plane loading. Int J Solids Struct 2003;40:4097–106. [14] Zhong H, Gu C. Buckling of simply supported rectangular ReissnerMindlin plates subjected to linearly varying in-plane loading. J Eng Mech, ASCE 2006. May. [15] Press WH, Flannery BP, Teukolsky SA, Vetterling WT. Numerical recipes – the art of scientific computing. Cambridge: Cambridge University Press; 1986. [16] Khdeir AA. Free vibration and buckling of symmetric cross-ply laminated plates by an exact method. J Sound Vib 1988;126:447–61.