- Email: [email protected]
circular cutout using FEM
Advances in Engineering Software 41 (2010) 161–164
Contents lists available at ScienceDirect
Advances in Engineering Software journal homepage: www.elsevier.com/locate/advengsoft
Buckling analysis of laminated composite plates with an elliptical/circular cutout using FEM M. Aydin Komur a, Faruk Sen b,*, Akın Atasß c, Nurettin Arslan d a
Aksaray University, Department of Civil Engineering, Aksaray, Turkey Aksaray University, Department of Mechanical Engineering, 68100 Aksaray, Turkey c The University of Sheffield, Department of Mechanical Engineering, Sheffield, UK d Balikesir University, Department of Mechanical Engineering, Balikesir, Turkey b
a r t i c l e
i n f o
Article history: Received 12 May 2009 Received in revised form 4 August 2009 Accepted 14 September 2009 Available online 22 October 2009 Keywords: Buckling Laminated composites Elliptical hole FEM
a b s t r a c t In this study, a buckling analysis was carried out of a woven–glass–polyester laminated composite plate with an circular/elliptical hole, numerically. In the analysis, finite element method (FEM) was applied to perform parametric studies on various plates based on the shape and position of the elliptical hole. This study addressed the effects of an elliptical/circular cutout on the buckling load of square composite plates. The laminated composite plates were arranged as symmetric cross-ply [(0°/90°)2]s and angleply [(15°/ 75°)2]s, [(30°/ 60°)2]s, [(45°/ 45°)2]s. The results show that buckling loads are decreased by increasing both c/a and b/a ratios. The increasing of hole positioned angle cause to decrease of buckling loads. Additionally, the cross-ply composite plate is stronger than all other analyzed angle-ply laminated plates. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction Application fields of composite materials are continuously expanding, from traditional application areas such as military aircraft to various engineering fields including commercial aircrafts, automobiles, robotic arms and even architecture [1]. The correct understanding of their structural behavior is necessary, such as the deflections, buckling loads and modal characteristics, the through-thickness distributions of stresses and strains, the large deflection behavior and, of extreme significance for obtaining strong, reliable multi-layered structures, the failure characteristics [2]. Throughout operation the composite laminate plates are generally subjected to compression loads that may basis buckling if overloaded. Consequently their buckling behaviors are significant factors in safe and reliable design of these structures [3]. For predicting the buckling load and buckling mode form of a structure in the finite element program, the linear (or eigenvalue) buckling analysis is an existing technique for estimation [4]. In general, the analysis of composite laminated plates is more complicated than the analysis of homogeneous isotropic ones [5]. In the literature, there are a range of published studies on the buckling of composite plates. Akbulut and Sayman [6] carried out a buckling analysis of a rectangular composite laminates with a central square hole. Using the first order shear deformation theory, * Corresponding author. Tel.: +90 382 2150953; fax: +90 382 2150592. E-mail address: [email protected] (F. Sen). 0965-9978/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.advengsoft.2009.09.005
the critical buckling loads of composite plates which were designed as symmetric angle-ply, antisymmetric cross-ply or angleply under the inplane loads were found for constant or various thicknesses, simple or clamped boundary conditions, various modules ratios, simple or biaxial loading versus hole sizes. Kundu and Sinha [7] performed the geometrically nonlinear postbuckling analysis of laminated composite doubly curved shells by finite element method. The principle of virtual work forms the origin to derive the nonlinear finite element equations. Kong et al. [8] analyzed buckling and postbuckling behaviors both numerically and experimentally for composite plates with a hole. In the finite element analysis, the updated Lagrangian formulation and the eight-node degenerated shell element were used. The effect of hole sizes and stacking sequences was examined on the compression behavior of the plate. Experiments showed fine agreement with the finite element results in the buckling load and the postbuckling strength. Ghannadpour et al. [9] studied the influences of a cutout on the buckling performance of rectangular plates made of polymer matrix composites (PMC). The study was concentrated on the behavior of rectangular symmetric cross-ply laminates. Finite element analysis was also carried out to obtain the effects of cutout on the buckling behavior of these plates. Jain and Kumar [10] carried out the finite element method for the postbuckling response of symmetric square laminates with a central cutout under uniaxial compression. The governing finite element equations were solved using the Newton–Raphson method. For the purpose of analysis, laminates with circular and elliptical cutouts were considered with
162
M. Aydin Komur et al. / Advances in Engineering Software 41 (2010) 161–164
a view to investigate the effect of cutout shape, size and the alignment of the elliptical cutout on the buckling and the first-ply failure loads of laminates. In this study, a buckling analysis was carried out for laminated composite plates with an elliptical/circular hole centered in the plate using FEM. The buckling loads were computed for different sizes of cutout placed with an angle. 2. Materials and methods In this study, the effects of elliptical/circular hole on the buckling load of laminated composite plates have been investigated numerically. The laminated plates were taken into account woven–glass fibers as reinforcement material and from polyester as
Table 1 Mechanical properties of the woven–glass–polyester composite material [14]. E1 = E2 (MPa)
G12 (MPa)
m12
Xt = Yt (MPa)
Xc = Y c (MPa)
S (MPa)
Vf (%)
31,610
3220
0.206
344
359
82
76
c
y α
a b
a Fig. 1. Geometry of the model.
x
matrix material by IZOREL company in Izmir, based on previous study [14]. The mechanical properties of the woven–glass–polyester composite material are listed in Table 1 [14]. All of these properties were determined experimentally using standard test methods (ASTM 3039-76, ASTM 3518-76, ASTM D 5379, ASTM 3410-75) and test procedures [11–13]. The detailed information about analyzed composite material during this work can be found in a previous study [14]. The geometry of the model is shown in Fig. 1. In real composite applications different plate and cutout form may be used owing to design necessities. Therefore, as seen in Fig. 1, the composite plate was considered as a square with dimensions of 120 mm 120 mm. The thickness of plate was 1.6 mm. Nonetheless, the cutout shape was assumed an elliptical hole centered in the square plate in this work. The hole was also positioned according to a angle rotated according to x-axis as 0°, 15°, 30°, 45°, 60°, 75° and 90°. The diameters of the major and minor axis dimensions of ellipse were presented by b and c respectively. In other words, the width was b and height was c. The parameters b and c were changed according to selected ratios, hence the elliptical hole was also positioned as circular hole when b/a = 0.5. As a result, the effect of circular hole was also analyzed at these same conditions. Briefly, buckling analysis was performed for both various elliptical holes and circular holes in terms of created different models. Meanwhile, the laminated plates were also analyzed without a hole when c/a = 0 to compare the influences having a hole and without a hole conditions on buckling loads. Furthermore, the laminated plates were assumed to stack eight laminas onto together namely cross-ply [(0°/90°)2]s and angle-ply [(15°/ 75°)2]s, [(30°/ 60°)2]s, [(45°/ 45°)2]s symmetrically. Consequently, four different composite plates based on stacking sequences were analyzed. In this manner, the effects of orientations of laminated composite plates on the buckling loads were also analyzed. During the analysis, ANSYS which is known general purpose finite element software was preferred as numerical tool. Because of the plane stress analysis of the composite plate, shell element type was selected. SHELL91 [15] element type illustrated in Fig. 2 was used to produce for mesh structure. Since, SHELL91 can be used for layered applications of a structural shell model. Up to 100 different layers are permitted for applications with the sandwich option turned off. Additionally, the element has six degrees of freedom at each node: translations in the nodal x, y, and z directions and rotations about the nodal x, y, and z-axes. Because of the different hole dimensions and angles, the different models and mesh structures were made. For an example, a sample mesh structure is shown in Fig. 3. Additionally, the element
Fig. 2. SHELL91 element geometry [15].
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M. Aydin Komur et al. / Advances in Engineering Software 41 (2010) 161–164
Buckling Load, N/mm
a
30 b/a=0.5, c/a=0.1
[(0/90)2]s [(15/-75)2]s [(30/-60)2]s
25
[(45/-45)2]s
20
15
10 0
b
15
30
45 α
60
75
90
30 b/a=0.4, c/a=0.2
[(0/90)2]s
Table 2 The element number of created some models. b/a
c/a
a
Element number
0.5
0.1
0 15 30 45 60 75 90
2930 2844 3029 3007 3013 3037 2943
0 15 30 45 60 75 90
3126 3142 3019 3115 2939 3127 3213
0.3
0.2
number of created some models are presented in Table 2 related to b/a, c/a and a angle. As seen from this figure, the model was divided with quadrilateral elements for a good mesh generation. Additionally, a refine mesh process was performed for surrounding of hole, since the close areas of the hole were very critical for FEM solutions. The refine mesh provided a sensitive solution. Therefore, the obtained results could be calculated as good with refine mesh than normal mesh. Furthermore, the boundary conditions and loadings of the model are also illustrated in Fig. 3, clearly. 3. Results and discussion The effects of a angle on buckling loads are shown in Fig. 4 for various b/a and c/a ratios. As seen from this figure, firstly, the buckling loads are decreased by increasing a. Therefore, the maximum values of it are calculated when a = 0°, whereas the lowest values are computed if a = 90°. This case is also valid for selected parameters. Additionally, the decreasing of buckling loads is very fast between a = 0° and 45°, while this falling is continue a = 45° and 90° with smaller levels. Besides, the differences of buckling loads of each analyzed plates are very higher, when a = 0°. The differences are decreasing after from a = 45° to 90°. Second important point in Fig. 4, the buckling loads are increased by increasing b/a ratio, so the highest values of it are obtained when b/a = 0.5. Furthermore, according to Fig. 4, the magnitudes of buckling loads are affected
[(30/-60)2]s
25
[(45/-45)2]s
20
15
10 0
c
15
30
45 α
60
75
90
30 [(0/90)2]s
b/a=0.3, c/a=0.2
[(15/-75)2]s
Buckling Load, N/mm
Fig. 3. Sample mesh structure and boundary conditions.
Buckling Load, N/mm
[(15/-75)2]s
[(30/-60)2]
25
[(45/-45)2]s
20
15
10 0
15
30
45 α
60
75
90
Fig. 4. The effect of parameter a on buckling load.
from the stacking sequences of laminated composite plates. The uppermost values of it are calculated for cross-ply [(0°/90°)2]s plates, while the lower values are computed for angle-ply [(15°/ 75°)2]s, [(30°/ 60°)2]s, [(45°/ 45°)2]s laminates. Nonetheless, the lowest values of buckling loads are obtained for [(45°/ 45°)2]s laminates. In other words, the cross-ply laminated plate is stronger than angle-ply laminated plates to resist for buckling. The calculated result of [(30°/ 60°)2]s, and [(45°/ 45°)2]s laminates are done very close if a = 45° or 60°. The effects of c/a ratio on buckling load are illustrated for a = 0° and 90° in Fig. 5a and b, respectively, when b/a = 0.5. According to Fig. 5a, the buckling loads are decreased by increasing of c/a ratios. The reducing is continued from c/a = 0 to 0.5 for a = 0°, significantly. Therefore the differences between c/a = 0 and 0.5 are very high. However, the decreasing is occurred with high values
164
a
M. Aydin Komur et al. / Advances in Engineering Software 41 (2010) 161–164
[(0/90)2]s
b/a=0.5 and α=0
[(15/-75)2]s
30
Buckling Load, N/mm
The comparison of buckling loads related to a is illustrated in Fig. 6. But the curves are only plotted for a = 0–90°. It is clearly seen from this figure that the magnitudes of buckling loads are equal to each other when c/a = 0 and 0.5. However, the buckling loads for a = 0° are higher than a = 90° for other c/a ratios. The difference of buckling loads between a = 0° and 90° for small c/a ratios is also bigger than increasing values of c/a ratios. In other words, their values are close to each other. The buckling loads are decreased by increasing of c/a ratio for a = 0°, especially. Although, a sharp decreasing is observed between c/a = 0 and 0.05, a small increasing is seen between c/a = 0.05 and 0.1. After the c/a = 0.1, the buckling loads are seen almost linear. The minor changes are negligible levels for a = 90°.
35
[(30/-60)2]s [(45/-45)2]s
25
20
15
10 0
0.1
0.2
0.3
0.4
0.5
4. Conclusions
c/a
b
35
b/a=0.5 and α=90
[(0/90)2]s
Buckling Load, N/mm
30
[(15/-75)2]s [(30/-60)2]s
25
[(45/-45)2]s
20 15 10 5 0 0
0.1
0.2
0.3
0.4
0.5
c/a Fig. 5. The effect of c/a ratio on buckling load.
In this study, the buckling response of a woven–glass–polyester composite laminated square plate with centered elliptical hole is investigated. The elliptical hole is positioned according to various angles from a = 0° to 90°. Additionally, the effect of c/a and b/a ratios on buckling loads are calculated. During the modeling process and solutions were done with FEM using ANSYS finite element software. From the present study, the following conclusions can be made. Firstly, the magnitudes of buckling loads are decreased by increasing c/a ratio, whereas it is increased by increasing b/a ratio. This means that big elliptical holes cause the weakest plates under the pressure. Secondly, the increasing of hole positioned angle cause to decrease of buckling loads. Lastly, the cross-ply [(0°/ 90°)2]s composite plates is stronger than other analyzed angleply [(15°/ 75°)2]s, [(30°/ 60°)2]s, [(45°/ 45°)2]s laminated plates. Meanwhile, the [(45°/ 45°)2]s laminated plate is observed as the weakest angle-ply plate. All in all, the designer must avoided the big elliptical holes in laminated composite plates, if it is wanted to prevent buckling loads at lower pressures. Additionally, crossply laminated plates can be preferred to provide higher buckling loads. References
35
Buckling Load. N/mm
α=0
30
α=90
25 20 15 10 5 0 0
0.1
0.2
0.3
0.4
0.5
c/a Fig. 6. The comparison of buckling loads related to a.
between c/a = 0 and 0.05 for a = 90°, suddenly as seen from Fig. 5a. But after the c/a = 0.005 the decreasing is very lower, so it can be neglected for all laminates. It can be said that the values of buckling loads are very close between from c/a = 0.1 to 0.5 for each laminates if a = 90°. But an unexpected increasing is observed between c/a = 0.05 and 0.1 for [(0°/90°)2]s and [(15°/ 75°)2]s. Furthermore, the highest values of buckling loads are calculated [(0°/90°)2]s plates, while the lower values are computed angle-ply [(45°/ 45°)2]s plates like Fig. 4.
[1] Kweon JH, Jung JW, Kim TH, Choi JH, Kim DH. Failure of carbon composite-toaluminum joints with combined mechanical fastening and adhesive bonding. Compos Struct 2006;75:192–8. [2] Zhang YX, Yang CH. Recent developments in finite element analysis for laminated composite plates. Compos Struct 2009;88:147–57. [3] Baba BO. Buckling behavior of laminated composite plates. J Reinf Plast Compos 2007;26:1637–55. [4] Zor M, Sen F, Toygar ME. An investigation of square delamination effects on the buckling behavior of laminated composite plates with a square hole by using three-dimensional FEM analysis. J Reinf Plast Compos 2005;24:1119–30. [5] Shufrin I, Rabinovitch O, Eisenberger M. Buckling of laminated plates with general boundary conditions under combined compression, tension, and shear-A semi-analytical solution. Thin-Wall Struct 2008;46:925–38. [6] Akbulut H, Sayman O. An investigation on buckling of laminated plates with central square hole. J Reinf Plast Compos 2001;20:1112–24. [7] Kundu CK, Sinha PK. Post buckling analysis of laminated composite shells. Compos Struct 2007;78:316–24. [8] Kong CW, Hong CS, Kim CG. Postbuckling strength of composite plate with a hole. J Reinf Plast Compos 2001;20:466–81. [9] Ghannadpour SAM, Najafi A, Mohammadi B. On the buckling behavior of crossply laminated composite plates due to circular/elliptical cutouts. Compos Struct 2006;75:3–6. [10] Jain P, Kumar A. Postbuckling response of square laminates with a central circular/elliptical cutout. Compos Struct 2004;65:179–85. [11] Jones RM. Mechanics of composite material. Taylor and Francis; 1999. [12] Gibson RF. Principals of composite material mechanics. Mc Graw-Hill; 1994. [13] Reddy JN. Mechanics of laminated composite plates; theory and analysis. CRC Press; 1997. [14] Atasß A, Sen F, Arslan N. Failure analysis of laminated composite plates with two parallel pin-loaded holes. J Reinf Plast Compos 2009;28:1265–76. [15] ANSYS Procedures. Engineering analysis system verification manual, vol. 1. Houston, PA, USA: Swanson Analysis System Inc.; 1993.
Contents lists available at ScienceDirect
Advances in Engineering Software journal homepage: www.elsevier.com/locate/advengsoft
Buckling analysis of laminated composite plates with an elliptical/circular cutout using FEM M. Aydin Komur a, Faruk Sen b,*, Akın Atasß c, Nurettin Arslan d a
Aksaray University, Department of Civil Engineering, Aksaray, Turkey Aksaray University, Department of Mechanical Engineering, 68100 Aksaray, Turkey c The University of Sheffield, Department of Mechanical Engineering, Sheffield, UK d Balikesir University, Department of Mechanical Engineering, Balikesir, Turkey b
a r t i c l e
i n f o
Article history: Received 12 May 2009 Received in revised form 4 August 2009 Accepted 14 September 2009 Available online 22 October 2009 Keywords: Buckling Laminated composites Elliptical hole FEM
a b s t r a c t In this study, a buckling analysis was carried out of a woven–glass–polyester laminated composite plate with an circular/elliptical hole, numerically. In the analysis, finite element method (FEM) was applied to perform parametric studies on various plates based on the shape and position of the elliptical hole. This study addressed the effects of an elliptical/circular cutout on the buckling load of square composite plates. The laminated composite plates were arranged as symmetric cross-ply [(0°/90°)2]s and angleply [(15°/ 75°)2]s, [(30°/ 60°)2]s, [(45°/ 45°)2]s. The results show that buckling loads are decreased by increasing both c/a and b/a ratios. The increasing of hole positioned angle cause to decrease of buckling loads. Additionally, the cross-ply composite plate is stronger than all other analyzed angle-ply laminated plates. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction Application fields of composite materials are continuously expanding, from traditional application areas such as military aircraft to various engineering fields including commercial aircrafts, automobiles, robotic arms and even architecture [1]. The correct understanding of their structural behavior is necessary, such as the deflections, buckling loads and modal characteristics, the through-thickness distributions of stresses and strains, the large deflection behavior and, of extreme significance for obtaining strong, reliable multi-layered structures, the failure characteristics [2]. Throughout operation the composite laminate plates are generally subjected to compression loads that may basis buckling if overloaded. Consequently their buckling behaviors are significant factors in safe and reliable design of these structures [3]. For predicting the buckling load and buckling mode form of a structure in the finite element program, the linear (or eigenvalue) buckling analysis is an existing technique for estimation [4]. In general, the analysis of composite laminated plates is more complicated than the analysis of homogeneous isotropic ones [5]. In the literature, there are a range of published studies on the buckling of composite plates. Akbulut and Sayman [6] carried out a buckling analysis of a rectangular composite laminates with a central square hole. Using the first order shear deformation theory, * Corresponding author. Tel.: +90 382 2150953; fax: +90 382 2150592. E-mail address: [email protected] (F. Sen). 0965-9978/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.advengsoft.2009.09.005
the critical buckling loads of composite plates which were designed as symmetric angle-ply, antisymmetric cross-ply or angleply under the inplane loads were found for constant or various thicknesses, simple or clamped boundary conditions, various modules ratios, simple or biaxial loading versus hole sizes. Kundu and Sinha [7] performed the geometrically nonlinear postbuckling analysis of laminated composite doubly curved shells by finite element method. The principle of virtual work forms the origin to derive the nonlinear finite element equations. Kong et al. [8] analyzed buckling and postbuckling behaviors both numerically and experimentally for composite plates with a hole. In the finite element analysis, the updated Lagrangian formulation and the eight-node degenerated shell element were used. The effect of hole sizes and stacking sequences was examined on the compression behavior of the plate. Experiments showed fine agreement with the finite element results in the buckling load and the postbuckling strength. Ghannadpour et al. [9] studied the influences of a cutout on the buckling performance of rectangular plates made of polymer matrix composites (PMC). The study was concentrated on the behavior of rectangular symmetric cross-ply laminates. Finite element analysis was also carried out to obtain the effects of cutout on the buckling behavior of these plates. Jain and Kumar [10] carried out the finite element method for the postbuckling response of symmetric square laminates with a central cutout under uniaxial compression. The governing finite element equations were solved using the Newton–Raphson method. For the purpose of analysis, laminates with circular and elliptical cutouts were considered with
162
M. Aydin Komur et al. / Advances in Engineering Software 41 (2010) 161–164
a view to investigate the effect of cutout shape, size and the alignment of the elliptical cutout on the buckling and the first-ply failure loads of laminates. In this study, a buckling analysis was carried out for laminated composite plates with an elliptical/circular hole centered in the plate using FEM. The buckling loads were computed for different sizes of cutout placed with an angle. 2. Materials and methods In this study, the effects of elliptical/circular hole on the buckling load of laminated composite plates have been investigated numerically. The laminated plates were taken into account woven–glass fibers as reinforcement material and from polyester as
Table 1 Mechanical properties of the woven–glass–polyester composite material [14]. E1 = E2 (MPa)
G12 (MPa)
m12
Xt = Yt (MPa)
Xc = Y c (MPa)
S (MPa)
Vf (%)
31,610
3220
0.206
344
359
82
76
c
y α
a b
a Fig. 1. Geometry of the model.
x
matrix material by IZOREL company in Izmir, based on previous study [14]. The mechanical properties of the woven–glass–polyester composite material are listed in Table 1 [14]. All of these properties were determined experimentally using standard test methods (ASTM 3039-76, ASTM 3518-76, ASTM D 5379, ASTM 3410-75) and test procedures [11–13]. The detailed information about analyzed composite material during this work can be found in a previous study [14]. The geometry of the model is shown in Fig. 1. In real composite applications different plate and cutout form may be used owing to design necessities. Therefore, as seen in Fig. 1, the composite plate was considered as a square with dimensions of 120 mm 120 mm. The thickness of plate was 1.6 mm. Nonetheless, the cutout shape was assumed an elliptical hole centered in the square plate in this work. The hole was also positioned according to a angle rotated according to x-axis as 0°, 15°, 30°, 45°, 60°, 75° and 90°. The diameters of the major and minor axis dimensions of ellipse were presented by b and c respectively. In other words, the width was b and height was c. The parameters b and c were changed according to selected ratios, hence the elliptical hole was also positioned as circular hole when b/a = 0.5. As a result, the effect of circular hole was also analyzed at these same conditions. Briefly, buckling analysis was performed for both various elliptical holes and circular holes in terms of created different models. Meanwhile, the laminated plates were also analyzed without a hole when c/a = 0 to compare the influences having a hole and without a hole conditions on buckling loads. Furthermore, the laminated plates were assumed to stack eight laminas onto together namely cross-ply [(0°/90°)2]s and angle-ply [(15°/ 75°)2]s, [(30°/ 60°)2]s, [(45°/ 45°)2]s symmetrically. Consequently, four different composite plates based on stacking sequences were analyzed. In this manner, the effects of orientations of laminated composite plates on the buckling loads were also analyzed. During the analysis, ANSYS which is known general purpose finite element software was preferred as numerical tool. Because of the plane stress analysis of the composite plate, shell element type was selected. SHELL91 [15] element type illustrated in Fig. 2 was used to produce for mesh structure. Since, SHELL91 can be used for layered applications of a structural shell model. Up to 100 different layers are permitted for applications with the sandwich option turned off. Additionally, the element has six degrees of freedom at each node: translations in the nodal x, y, and z directions and rotations about the nodal x, y, and z-axes. Because of the different hole dimensions and angles, the different models and mesh structures were made. For an example, a sample mesh structure is shown in Fig. 3. Additionally, the element
Fig. 2. SHELL91 element geometry [15].
163
M. Aydin Komur et al. / Advances in Engineering Software 41 (2010) 161–164
Buckling Load, N/mm
a
30 b/a=0.5, c/a=0.1
[(0/90)2]s [(15/-75)2]s [(30/-60)2]s
25
[(45/-45)2]s
20
15
10 0
b
15
30
45 α
60
75
90
30 b/a=0.4, c/a=0.2
[(0/90)2]s
Table 2 The element number of created some models. b/a
c/a
a
Element number
0.5
0.1
0 15 30 45 60 75 90
2930 2844 3029 3007 3013 3037 2943
0 15 30 45 60 75 90
3126 3142 3019 3115 2939 3127 3213
0.3
0.2
number of created some models are presented in Table 2 related to b/a, c/a and a angle. As seen from this figure, the model was divided with quadrilateral elements for a good mesh generation. Additionally, a refine mesh process was performed for surrounding of hole, since the close areas of the hole were very critical for FEM solutions. The refine mesh provided a sensitive solution. Therefore, the obtained results could be calculated as good with refine mesh than normal mesh. Furthermore, the boundary conditions and loadings of the model are also illustrated in Fig. 3, clearly. 3. Results and discussion The effects of a angle on buckling loads are shown in Fig. 4 for various b/a and c/a ratios. As seen from this figure, firstly, the buckling loads are decreased by increasing a. Therefore, the maximum values of it are calculated when a = 0°, whereas the lowest values are computed if a = 90°. This case is also valid for selected parameters. Additionally, the decreasing of buckling loads is very fast between a = 0° and 45°, while this falling is continue a = 45° and 90° with smaller levels. Besides, the differences of buckling loads of each analyzed plates are very higher, when a = 0°. The differences are decreasing after from a = 45° to 90°. Second important point in Fig. 4, the buckling loads are increased by increasing b/a ratio, so the highest values of it are obtained when b/a = 0.5. Furthermore, according to Fig. 4, the magnitudes of buckling loads are affected
[(30/-60)2]s
25
[(45/-45)2]s
20
15
10 0
c
15
30
45 α
60
75
90
30 [(0/90)2]s
b/a=0.3, c/a=0.2
[(15/-75)2]s
Buckling Load, N/mm
Fig. 3. Sample mesh structure and boundary conditions.
Buckling Load, N/mm
[(15/-75)2]s
[(30/-60)2]
25
[(45/-45)2]s
20
15
10 0
15
30
45 α
60
75
90
Fig. 4. The effect of parameter a on buckling load.
from the stacking sequences of laminated composite plates. The uppermost values of it are calculated for cross-ply [(0°/90°)2]s plates, while the lower values are computed for angle-ply [(15°/ 75°)2]s, [(30°/ 60°)2]s, [(45°/ 45°)2]s laminates. Nonetheless, the lowest values of buckling loads are obtained for [(45°/ 45°)2]s laminates. In other words, the cross-ply laminated plate is stronger than angle-ply laminated plates to resist for buckling. The calculated result of [(30°/ 60°)2]s, and [(45°/ 45°)2]s laminates are done very close if a = 45° or 60°. The effects of c/a ratio on buckling load are illustrated for a = 0° and 90° in Fig. 5a and b, respectively, when b/a = 0.5. According to Fig. 5a, the buckling loads are decreased by increasing of c/a ratios. The reducing is continued from c/a = 0 to 0.5 for a = 0°, significantly. Therefore the differences between c/a = 0 and 0.5 are very high. However, the decreasing is occurred with high values
164
a
M. Aydin Komur et al. / Advances in Engineering Software 41 (2010) 161–164
[(0/90)2]s
b/a=0.5 and α=0
[(15/-75)2]s
30
Buckling Load, N/mm
The comparison of buckling loads related to a is illustrated in Fig. 6. But the curves are only plotted for a = 0–90°. It is clearly seen from this figure that the magnitudes of buckling loads are equal to each other when c/a = 0 and 0.5. However, the buckling loads for a = 0° are higher than a = 90° for other c/a ratios. The difference of buckling loads between a = 0° and 90° for small c/a ratios is also bigger than increasing values of c/a ratios. In other words, their values are close to each other. The buckling loads are decreased by increasing of c/a ratio for a = 0°, especially. Although, a sharp decreasing is observed between c/a = 0 and 0.05, a small increasing is seen between c/a = 0.05 and 0.1. After the c/a = 0.1, the buckling loads are seen almost linear. The minor changes are negligible levels for a = 90°.
35
[(30/-60)2]s [(45/-45)2]s
25
20
15
10 0
0.1
0.2
0.3
0.4
0.5
4. Conclusions
c/a
b
35
b/a=0.5 and α=90
[(0/90)2]s
Buckling Load, N/mm
30
[(15/-75)2]s [(30/-60)2]s
25
[(45/-45)2]s
20 15 10 5 0 0
0.1
0.2
0.3
0.4
0.5
c/a Fig. 5. The effect of c/a ratio on buckling load.
In this study, the buckling response of a woven–glass–polyester composite laminated square plate with centered elliptical hole is investigated. The elliptical hole is positioned according to various angles from a = 0° to 90°. Additionally, the effect of c/a and b/a ratios on buckling loads are calculated. During the modeling process and solutions were done with FEM using ANSYS finite element software. From the present study, the following conclusions can be made. Firstly, the magnitudes of buckling loads are decreased by increasing c/a ratio, whereas it is increased by increasing b/a ratio. This means that big elliptical holes cause the weakest plates under the pressure. Secondly, the increasing of hole positioned angle cause to decrease of buckling loads. Lastly, the cross-ply [(0°/ 90°)2]s composite plates is stronger than other analyzed angleply [(15°/ 75°)2]s, [(30°/ 60°)2]s, [(45°/ 45°)2]s laminated plates. Meanwhile, the [(45°/ 45°)2]s laminated plate is observed as the weakest angle-ply plate. All in all, the designer must avoided the big elliptical holes in laminated composite plates, if it is wanted to prevent buckling loads at lower pressures. Additionally, crossply laminated plates can be preferred to provide higher buckling loads. References
35
Buckling Load. N/mm
α=0
30
α=90
25 20 15 10 5 0 0
0.1
0.2
0.3
0.4
0.5
c/a Fig. 6. The comparison of buckling loads related to a.
between c/a = 0 and 0.05 for a = 90°, suddenly as seen from Fig. 5a. But after the c/a = 0.005 the decreasing is very lower, so it can be neglected for all laminates. It can be said that the values of buckling loads are very close between from c/a = 0.1 to 0.5 for each laminates if a = 90°. But an unexpected increasing is observed between c/a = 0.05 and 0.1 for [(0°/90°)2]s and [(15°/ 75°)2]s. Furthermore, the highest values of buckling loads are calculated [(0°/90°)2]s plates, while the lower values are computed angle-ply [(45°/ 45°)2]s plates like Fig. 4.
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