- Email: [email protected]
−θ)] containing an elliptical notch
Composites: Part B 55 (2013) 575–579
Contents lists available at SciVerse ScienceDirect
Composites: Part B journal homepage: www.elsevier.com/locate/compositesb
Buckling analysis of laminated composite plates [(h/h)] containing an elliptical notch Djamel Ouinas a,⇑Belkacem Achour b a Laboratoire de modélisation numérique et expérimentale des phénomènes mécaniques, Department of Mechanical Engineering, University Abdelhamid Ibn Badis, Mostaganem 27000, Algeria b Department of Civil Engineering, University of H’ail, Saudi Arabia
a r t i c l e
i n f o
Article history: Received 14 April 2012 Accepted 10 July 2013 Available online 20 July 2013 Keywords: Elliptical notch B. Buckling A. Laminated composite materials C. Finite element analysis
a b s t r a c t In this study, a buckling analysis was performed on square plates made of composite material with and without elliptical notch using the finite element method. The boron/epoxy laminated plates were asymmetrically orderly arranged into the following way [(h/h)]. The resistance to buckling of the laminated plates subjected to mono-axial compression is highlighted according to the fiber orientations, the degree of anisotropy, size and the orientation of the notch. Results show that the minimum buckling load obtained corresponds to the largest ratio a/b = 1 (circular notch), while maximum buckling loads correspond to the smallest ratio of the geometric defect a/b. More the size of the geometric defect is important less the buckling load will be. Thus, the amplification factor N⁄ grows with the increase of the ply thickness. When u = 0° and the angle of orientation of fibers is h = 0°, N⁄ is of the order of 5 and doubles when h = 90°. While for u = 90°, the maximum amplification is of the order of 4% for h = 45° corresponding to the minimum value of the first case where u = 0° and h = 0°. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction As part of the massive introduction of the composite materials within aircraft manufacturing, a particular attention is turned on the buckling behaviour and energy dissipation capacities of basic structures. In industry, composite laminated materials have proved their efficiency in the manufacture of primary structures parts, due to their performance, lightness and form versatility [1]. The design of this type of structures requires more and more sophisticated mechanical modeling tools for taking into account the particularities of these materials. Numerical methods and notably the finite element method are necessary for dimensioning complex composite structures. Due to their behaviour complexity, the analysis of laminated plates remains an open research problem [2]. In fact, besides their generally anisotropic behaviour and the presence of significant transverse shear deformations, flexionextension parameters are to be taken into account. Most of the researches carried out on laminated plates, were devoted in the determination of stresses, strains or displacements of flexural origin. Structural instability becomes an important challenge regarding reliable and feasible design of composite plates. Several studies of stability of the laminated plates were carried out mainly on rectan⇑ Corresponding author. E-mail address: [email protected] (D. Ouinas). 1359-8368/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compositesb.2013.07.011
gular plates [3–6]. It is known that the resistance to buckling of rectangular plates depends on boundary conditions [4], orientations of plies [4,5,7] and geometric ratios [4,6–8]. Thin composite structures, which are broadly used, become instable when subjected to loads of mechanical or thermal nature leading therefore to buckling. Consequently, their buckling behaviours are therefore significant factors of reliable design [9]. To predict the buckling load and the mode of deformation of a structure, the linear analysis was used as an evaluation technique [10]. In general the analysis of laminated plates is more complicated than the analysis of a supposed homogeneous and isotropic material [11]. The buckling response of a woven/glass/polyester composite laminated square plate with elliptical hole is investigated by Komur et al. [12] using ANSYS code. They indicated that the designer must avoided the big elliptical holes in laminated composite plates, if it is wanted to prevent buckling loads at lower pressures. Hamani et al. [13] have determined the effect of fiber orientation on the critical buckling load of symmetrical laminated composite plates having a crack emanating from a circular notch. They pointed out that the critical buckling loads attain important maximum values when the fibers are oriented in the range of 50°–90°, whereas the minimum values are obtained when the fibers are perpendicular to the applied pressure. In the present paper, which is a contribution to the analysis of thin laminated composite plates using the finite element method,
576
D. Ouinas, B. Achour / Composites: Part B 55 (2013) 575–579
different geometries of laminated plates in the presence of elliptical notch and subjected to compression are studied. The effects of the notch size and its position, the orientation of the plies, ratio of elastic moduli E2/E1 of the composite and the ply thickness on the buckling load were highlighted.
2. Finite elements modeling In this study, a thin square plate of length 2H (see Fig. 1) and width 2w and having an elliptical notch, is considered. The ratio between the length 2h ¼ 100 mm and the width 2w ¼ 100 mm of the plate is h=w ¼ 1 and the thickness e ¼ 1:016 mm. The material has properties E1 = 118 GPa, E2 = 10 GPa, G12 = 6.2 GPa, G13 = 6.2 GPa, G23 = 4.1 GPa, m ¼ 0:237; and is laid up in 8 plies. The considered plate is subjected to mono-axial compression in the vertical direction under an applied stress per unit length r ¼ 1N= mm. The finite element commercial code ABAQUS 6.11 has been used for the analysis. 43471 quadrilateral type S8R elements and 1352 of type STRI65 were used with a mesh size refined in the neighborhood of the notch as shown in Fig. 1. The analysis has been undertaken in plane stress. The plate is constituted of eight plies, each having a thickness of 0.127 mm. The symmetrically cross-layered were alternately arranged in the following pattern h and h respectively (Fig. 2). In real applications of composite plates, different forms of the notch can be used for design purposes. The form of the notch was assumed to be an elliptical hole centered in the plate. So, the effect of different positions of the elliptical notch on the buckling load is highlighted. These positions are represented by an angle between the major axis of the notch and the abscissa axis by 0°, 15°, 30°, 45°, 60°, 75°, and 90°. When the angle (u = 0°), the major axis coincides with the xx axis and when the angle equals u = 90°, it coincides with the ordinate axis. The dimensions of the principal diameters of the ellipse are represented by ‘a’ and ‘b’ respectively. The parameters ‘a’ and ‘b’ vary according to several chosen ratios. Consequently, the effect of circular notch was also analyzed as a particular case when a = b. The buckling analysis was carried out for various elliptical and circular holes in terms of different created models. Besides, a particular attention is focused on the number of alternated crossed plies of the laminated plate.
Fig. 2. The fiber orientation in the laminated plate.
3. Results and discussion 3.1. Influence of the plies orientation on the buckling load In Fig. 3, the influence of the angle of the fibers orientation on the variation of the buckling load is shown. In this case comparison is made with and without the presence of the elliptical notch. The ratio of the principal axes of the notch is taken equal to a=b ¼ 1=3: Thus the variation of the angle of inclination of the elliptical with regard to the abscissa axis is highlighted. Fig. 1, shows an exponential growth of the buckling load with the increase of the inclination angle of the composite material fibers. This growth is much more important and rapid when h P 45°. For cases where h 6 45°, the variation of the buckling load is quasi–linear. The lowest values of the buckling load are obtained when the fibers are orientated perpendicularly to the applied load (h = 0°) whereas the maximum values are obtained when the fibers are set parallel to the applied load (h = 90°). Moreover, the effect of the orientation of the elliptical notch was also considered. It can be noted in this case that whatever is the inclination angle of the
Fig. 1. Geometric modeling and meshing of the plate having an elliptical notch.
577
D. Ouinas, B. Achour / Composites: Part B 55 (2013) 575–579
45
15,0
40
Without ellipltical notch
30 25 20
Bucklingload (N/mm)
35
Bucklingload (N/mm)
14,8
With ellipltical notch ϕ=0° ϕ=15° ϕ=30° ϕ=45° ϕ=60° ϕ=75° ϕ=90°
15
14,6 ϕ=0° ϕ=15° ϕ=30° ϕ=45° ϕ=60° ϕ=75° ϕ=90°
14,4
14,2
10 5 0
Elliptical notch orientation
14,0 0
10
20
30
40
50
60
70
80
90
Angle (θ ) Fig. 3. Effect of the fiber orientation on the variation of the buckling load.
notch about the abscissa axis; the difference between the buckling loads is insignificant for the highest values of the ply orientation angles (h > 45°), while it is negligible even zero for smaller angles (h 6 45°). The difference between the curves becomes noticeable for angles varying from 45° to 90°. The largest values are calculated for the case of a plate without a notch. In fact, it is very clear that the buckling load is highly affected by the piling up order of the laminated composite materials. Therefore to have a laminated plate which resists better to buckling, the ply orientations must be closer to the ordinate axis.
5
15
25
35
45
55
65
75
85
95
Height h (mm) Fig. 5. Variation of the buckling load with regard to the notch position.
biggest ratio (a/b = 1), that is to say for a circular notch, and it is identical for all cases, while the maximum values are obtained for the smallest geometrical defect ratio. This means that the increase of the notch radii causes the smallest buckling loads in plates subjected to mono-axial pressure. The slope of the curves increases with the increasing values of the angles u. Therefore more the size of geometric defect is important less is the buckling load, in other words the buckling load decreases with the increasing size of the notch which shows a reduced resistance of the composite laminated plate.
3.3. Influence of the notch position 3.2. Influence of the notch geometric ratio on the buckling load The influence of the geometric ratio a/b of an elliptical notch on the variation of the buckling load is illustrated in Fig. 4 for several notch inclinations with regard to the abscissa axis. The laminated composite material studied in this case is symmetrical of type [(0/90)2]S. The major axis of the ellipse is maintained constant (b = 10 mm) and the minor axis (a) variable. The figure shows that the buckling load decreases progressively and quasi-linearly with the increasing ratio of the elliptical notch. This reduction is noticeable whatever the inclination angle u of the geometrical defect is. The minimum value is obtained for the
15,0 Elliptical notch orientation ϕ=0° ϕ=15° ϕ=30° ϕ=45° ϕ=60° ϕ=75° ϕ=90°
Bucklingload (N/mm)
14,8 14,6 14,4
3.4. Influence of the rigidity ratio
14,2 14,0 13,8 13,6
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
Fig. 5 shows the buckling load curves when the position of the centre of the notch changes with regard to the applied load. The effect of the notch inclination with the abscissa axis is highlighted for symmetrical [(0/90)2]S laminated composite ply orientations. The size of the considered elliptical notch is a = b/8. It is noted that the largest values of the buckling load are obtained when the inclination angle of the notch is superior to 45° regardless its position h. For angles smaller than 45°, sine curves are obtained in which their maximum values are in the range 15°–25° and 75°–85° while the minimum values are between 45° and 55°. The resistance of the plate is better when the minor axis of the notch is perpendicular to the applied load (u = 90°) while it is weak when it is parallel (u = 0°). Extremum values have the tendency to disappear when the angle approaches (90°). In this case the buckling load is practically constant regardless the height h.
0,9
Ratio (a/b) Fig. 4. Effect of a/b ratio on the variation of the buckling load.
1,0
The effect of the modulus of rigidity ratio was also studied for different orientations of the laminated structure with and without the presence of an elliptical notch. With the presence of the notch, two cases (u = 0° and u = 90°) were studied. As shown in Fig. 6, the buckling load increases linearly with the increasing Young modulus ratio when the orientations of fibers are less than 45°. Beyond this angle, the curves have the tendency to be more and more constant. In other words, the buckling load is constant for angles h > 45°. It could be concluded that the reduction of the load is faster for smaller ratios E2/E1 and smaller fiber orientations of the laminated structure. The maximum value is obtained for h = 90° with and without the presence of the notch. When the inclination angle of the elliptical notch is h = 90°, the values of the buckling loads are
578
D. Ouinas, B. Achour / Composites: Part B 55 (2013) 575–579
45
11
40
10
Bucklingload (N/mm)
35
Layer tickness e=0.127 mm e=0.175 mm e=0.225 mm e=0.250 mm e=0.300 mm ϕ =0°
9
30 8
25
N
20 Without notch
15
θ=0° θ=10° θ=20° θ=30° θ=40° θ=45°
10 5 0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,9
7 6
θ=50° θ=60° θ=70° θ=80° θ=90°
0,8
*
5 4 1,0
0
10
20
30
40
50
60
70
80
90
Angle ( θ )
E2/E1
Fig. 7. Effect of the laminated plate fiber orientation on the buckling amplification factor (u = 0°).
40
Bucklingload (N/mm)
35 30
4,5
25
4,0
20
3,5
15
θ=0° θ=10° θ=20° θ=30° θ=40° θ=45°
10 5 0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
ϕ=90°
3,0
With notch ϕ=90° θ=50° θ=60° θ=70° θ=80° θ=90°
Layer tickness
N
* 2,5
2,0
1,0
1,5 1,0
E2/E1
0,5
40
0
10
20
30
40
50
60
70
80
90
Angle ( θ )
35
Bucklingload (N/mm)
e=0.127 mm e=0.175 mm e=0.225 mm e=0.250 mm e=0.300 mm
Fig. 8. Effect of the laminated plate fiber orientation on the buckling amplification factor (u = 90°).
30 25
x-direction. In the second case, the minor axis of the notch is taken perpendicular to the y-direction. The obtained results are represented in Figs. 7 and 8. On these last two figures, the variation of the dimensionless amplification factor of the buckling load versus the fiber orientation of composite material is represented. The amplification parameter is defined by:
20 15
θ=0° θ=10° θ=20° θ=30° θ=40° θ=45°
10 5 0
With nocth ϕ=0° θ=50° θ=60° θ=70° θ=80° θ=90°
N ¼ 1 0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1,0
E2/E1 Fig. 6. Buckling load for orientation angle versus Young modulus ratios.
more important than when u = 0° which shows in this second case that the notch presents an acute risk of damage. 3.5. Influence of the plate thickness To highlight the effect of the fiber orientation of the composite material of the plate with regard to the elliptical notch position, two cases were considered. The first case consists in taking the major axis of the notch in the direction of abscissas i.e. parallel to the
Nwn Nwtn
where Nwn and N wtn are respectively the buckling load of the plate with and without a notch. It can be pointed out that the presence of an elliptical notch in two positions in the same composite material introduces different results of the amplification factor of the buckling load. When the major axis of the notch is taken parallel to the x-axis (Fig. 7), three stages could be considered. In the first stage, the amplification factor is found practically stable with the increasing fiber orientation angle up to 20°. Beyond this value and up to 60° (stage II), the factor N increases quadratically. The third stage is characterized by a speedy increase of the buckling load which shows the growth of the plate resistance to buckling when the orientation angle gets larger. When h = 0° the amplification factor N is of the order of 5%, this value doubles when h = 90°. It is to be noted that for the
D. Ouinas, B. Achour / Composites: Part B 55 (2013) 575–579
same value of the orientation angle, the amplification factor becomes more and more significant with the increasing ply thickness. Fig. 8 shows the variation of the buckling load amplification factor N⁄ with regards to the fiber inclination angle. It increases proportionally with the inclination angle and reaches its maximum for h = 45°. In this case, the major axis of the notch is parallel to the applied load and it could be noticed that the amplification factor curves are symmetrical with regard to h = 45°. The minimum values are obtained for h = 0°. and h = 90°. For this case (u = 90°), the maximum amplification is of the order of 4% for h = 45° corresponding to the minimum value in the first case ((u = 0°)) for h = 0°. This show that the presence of elliptical notch in the first case has a considerable influence on the resistance of the plate in comparison with the second case. 4. Conclusion The design and application parameters have a great influence on the failure mode of a structure. In this study, the buckling response of a composite laminated material square plate having an elliptical notch is studied. The effects of the orientation angle of the elliptical notch, the geometry and position of the latter, the fiber orientation of the composite material, the ratio of rigidity of the material and the effect of the plate thickness on the variation of the buckling load were highlighted leading to the following findings: – The buckling load increases exponentially with regard to the increase of the fiber orientation angle of the composite material. This growth is much quicker and more important for h P 45°. – When the fibers are oriented perpendicularly to the applied stress, minimum buckling loads are obtained and vice versa. – The minimum value of N obtained corresponds to the largest ratio a/b = 1 (circular notch), while the maximum values of N are obtained for the weakest ratio of the geometric defect a/b. Therefore more the size of geometric defect is important less the buckling load is. – The maximum values of the buckling load N are obtained when the orientation angle of the elliptical notch is greater than 45° regardless its position h.
579
– Higher degrees of anisotropy increase considerably the buckling loads. – The amplification factor N⁄ increases significantly with the importance of the ply thickness. When u = 0° and the fiber orientation angle is h = 0°, the factor N is of the order of 5%, this value is doubled when h = 90°. While for u = 90°, the maximum amplification is of the order of 4% for h = 45° corresponding to the minimum value in the first case u = 0° and h = 0°.
References [1] Kweon JH, Jung JW, Kim TH, Choi JH, Kim DH. Failure of carbon composite – toaluminum reached with combined mechanical fastening and adhesive bonding. Compos Struct 2006;75:192–8. [2] Komur Aydin, Sen Faruk, Atas Akın, Arslan Nurettin. Composite buckling analysis of laminated flat with elliptical/circular cutout using year FEM. Adv Eng Software 2010;41:161–4. [3] Rhodes MD, MIKULAS MM, McGowan PE. Effects of orthotropy and width one the compression strength of graphite–epoxy with holes panels. AIAA Newspaper 1984;22:1283–92. [4] Leissa AW. Composite buckling of laminated flat and shell panels. AFWAL-TR85-3069, Flight dynamics laboratory, Air Force Wright Aeronautical Laboratories Wright-Patterson Air Forces Foundation: Ohio; 1985. [5] Muc A. Optimum grain orientation for simply-supported acute angle-ply flat under biaxial compression. Compos Struct 1988;9:161–72. [6] Nemeth MP. Buckling behavior of compression-loaded symmetrically laminated acute angle-ply flat with holes. AIAA Newspaper 1988;26:330–6. [7] Hu H-T, Flax B-H. Buckling optimization of symmetrically laminated rectangular flat with various geometry and end conditions. Compos Sci Technol 1995;55:277–85. [8] Vellaichamy S, Prakash BG, Brown S. Optimum design of cutouts in laminated composite structures. Comput Struct 1990;37:241–6. [9] Baba BO. Composite buckling behavior of laminated flat. J Reinf Plast Compos 2007;26:1637–55. [10] Zor M, Sen F, TOYGAR MISS. Year investigation of small public garden delamination effects one the buckling behavior of laminated composite flat with has small public garden hole by using three-dimensional FEM analysis. J Reinf Plast Compos 2005;24:1119–30. [11] Shufrin I, Rabinovitch O, Eisenberger M. Buckling of laminated flat with general boundary conditions under combined compression, tension, and in shear-A semi-analytical resolution. Thin-Wall Struct 2008;46:925–38. [12] Aydin Komur M, Faruk S, Akın A, Nurettin A. Buckling analysis of laminated composite plates with an elliptical/circular cutout using FEM. Adv Eng Software 2010;41:161–4. [13] Hamani N, Ouinas D, Benderdouche N, Sahnoun M. Buckling analyses of the antisymetrical composite laminate plate with a crack from circular notch. Adv Mater Res, vol. 365. Switzerland: Trans Tech Publications; 2012. ISSN: 1022– 6680, p. 56–1.
Contents lists available at SciVerse ScienceDirect
Composites: Part B journal homepage: www.elsevier.com/locate/compositesb
Buckling analysis of laminated composite plates [(h/h)] containing an elliptical notch Djamel Ouinas a,⇑Belkacem Achour b a Laboratoire de modélisation numérique et expérimentale des phénomènes mécaniques, Department of Mechanical Engineering, University Abdelhamid Ibn Badis, Mostaganem 27000, Algeria b Department of Civil Engineering, University of H’ail, Saudi Arabia
a r t i c l e
i n f o
Article history: Received 14 April 2012 Accepted 10 July 2013 Available online 20 July 2013 Keywords: Elliptical notch B. Buckling A. Laminated composite materials C. Finite element analysis
a b s t r a c t In this study, a buckling analysis was performed on square plates made of composite material with and without elliptical notch using the finite element method. The boron/epoxy laminated plates were asymmetrically orderly arranged into the following way [(h/h)]. The resistance to buckling of the laminated plates subjected to mono-axial compression is highlighted according to the fiber orientations, the degree of anisotropy, size and the orientation of the notch. Results show that the minimum buckling load obtained corresponds to the largest ratio a/b = 1 (circular notch), while maximum buckling loads correspond to the smallest ratio of the geometric defect a/b. More the size of the geometric defect is important less the buckling load will be. Thus, the amplification factor N⁄ grows with the increase of the ply thickness. When u = 0° and the angle of orientation of fibers is h = 0°, N⁄ is of the order of 5 and doubles when h = 90°. While for u = 90°, the maximum amplification is of the order of 4% for h = 45° corresponding to the minimum value of the first case where u = 0° and h = 0°. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction As part of the massive introduction of the composite materials within aircraft manufacturing, a particular attention is turned on the buckling behaviour and energy dissipation capacities of basic structures. In industry, composite laminated materials have proved their efficiency in the manufacture of primary structures parts, due to their performance, lightness and form versatility [1]. The design of this type of structures requires more and more sophisticated mechanical modeling tools for taking into account the particularities of these materials. Numerical methods and notably the finite element method are necessary for dimensioning complex composite structures. Due to their behaviour complexity, the analysis of laminated plates remains an open research problem [2]. In fact, besides their generally anisotropic behaviour and the presence of significant transverse shear deformations, flexionextension parameters are to be taken into account. Most of the researches carried out on laminated plates, were devoted in the determination of stresses, strains or displacements of flexural origin. Structural instability becomes an important challenge regarding reliable and feasible design of composite plates. Several studies of stability of the laminated plates were carried out mainly on rectan⇑ Corresponding author. E-mail address: [email protected] (D. Ouinas). 1359-8368/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compositesb.2013.07.011
gular plates [3–6]. It is known that the resistance to buckling of rectangular plates depends on boundary conditions [4], orientations of plies [4,5,7] and geometric ratios [4,6–8]. Thin composite structures, which are broadly used, become instable when subjected to loads of mechanical or thermal nature leading therefore to buckling. Consequently, their buckling behaviours are therefore significant factors of reliable design [9]. To predict the buckling load and the mode of deformation of a structure, the linear analysis was used as an evaluation technique [10]. In general the analysis of laminated plates is more complicated than the analysis of a supposed homogeneous and isotropic material [11]. The buckling response of a woven/glass/polyester composite laminated square plate with elliptical hole is investigated by Komur et al. [12] using ANSYS code. They indicated that the designer must avoided the big elliptical holes in laminated composite plates, if it is wanted to prevent buckling loads at lower pressures. Hamani et al. [13] have determined the effect of fiber orientation on the critical buckling load of symmetrical laminated composite plates having a crack emanating from a circular notch. They pointed out that the critical buckling loads attain important maximum values when the fibers are oriented in the range of 50°–90°, whereas the minimum values are obtained when the fibers are perpendicular to the applied pressure. In the present paper, which is a contribution to the analysis of thin laminated composite plates using the finite element method,
576
D. Ouinas, B. Achour / Composites: Part B 55 (2013) 575–579
different geometries of laminated plates in the presence of elliptical notch and subjected to compression are studied. The effects of the notch size and its position, the orientation of the plies, ratio of elastic moduli E2/E1 of the composite and the ply thickness on the buckling load were highlighted.
2. Finite elements modeling In this study, a thin square plate of length 2H (see Fig. 1) and width 2w and having an elliptical notch, is considered. The ratio between the length 2h ¼ 100 mm and the width 2w ¼ 100 mm of the plate is h=w ¼ 1 and the thickness e ¼ 1:016 mm. The material has properties E1 = 118 GPa, E2 = 10 GPa, G12 = 6.2 GPa, G13 = 6.2 GPa, G23 = 4.1 GPa, m ¼ 0:237; and is laid up in 8 plies. The considered plate is subjected to mono-axial compression in the vertical direction under an applied stress per unit length r ¼ 1N= mm. The finite element commercial code ABAQUS 6.11 has been used for the analysis. 43471 quadrilateral type S8R elements and 1352 of type STRI65 were used with a mesh size refined in the neighborhood of the notch as shown in Fig. 1. The analysis has been undertaken in plane stress. The plate is constituted of eight plies, each having a thickness of 0.127 mm. The symmetrically cross-layered were alternately arranged in the following pattern h and h respectively (Fig. 2). In real applications of composite plates, different forms of the notch can be used for design purposes. The form of the notch was assumed to be an elliptical hole centered in the plate. So, the effect of different positions of the elliptical notch on the buckling load is highlighted. These positions are represented by an angle between the major axis of the notch and the abscissa axis by 0°, 15°, 30°, 45°, 60°, 75°, and 90°. When the angle (u = 0°), the major axis coincides with the xx axis and when the angle equals u = 90°, it coincides with the ordinate axis. The dimensions of the principal diameters of the ellipse are represented by ‘a’ and ‘b’ respectively. The parameters ‘a’ and ‘b’ vary according to several chosen ratios. Consequently, the effect of circular notch was also analyzed as a particular case when a = b. The buckling analysis was carried out for various elliptical and circular holes in terms of different created models. Besides, a particular attention is focused on the number of alternated crossed plies of the laminated plate.
Fig. 2. The fiber orientation in the laminated plate.
3. Results and discussion 3.1. Influence of the plies orientation on the buckling load In Fig. 3, the influence of the angle of the fibers orientation on the variation of the buckling load is shown. In this case comparison is made with and without the presence of the elliptical notch. The ratio of the principal axes of the notch is taken equal to a=b ¼ 1=3: Thus the variation of the angle of inclination of the elliptical with regard to the abscissa axis is highlighted. Fig. 1, shows an exponential growth of the buckling load with the increase of the inclination angle of the composite material fibers. This growth is much more important and rapid when h P 45°. For cases where h 6 45°, the variation of the buckling load is quasi–linear. The lowest values of the buckling load are obtained when the fibers are orientated perpendicularly to the applied load (h = 0°) whereas the maximum values are obtained when the fibers are set parallel to the applied load (h = 90°). Moreover, the effect of the orientation of the elliptical notch was also considered. It can be noted in this case that whatever is the inclination angle of the
Fig. 1. Geometric modeling and meshing of the plate having an elliptical notch.
577
D. Ouinas, B. Achour / Composites: Part B 55 (2013) 575–579
45
15,0
40
Without ellipltical notch
30 25 20
Bucklingload (N/mm)
35
Bucklingload (N/mm)
14,8
With ellipltical notch ϕ=0° ϕ=15° ϕ=30° ϕ=45° ϕ=60° ϕ=75° ϕ=90°
15
14,6 ϕ=0° ϕ=15° ϕ=30° ϕ=45° ϕ=60° ϕ=75° ϕ=90°
14,4
14,2
10 5 0
Elliptical notch orientation
14,0 0
10
20
30
40
50
60
70
80
90
Angle (θ ) Fig. 3. Effect of the fiber orientation on the variation of the buckling load.
notch about the abscissa axis; the difference between the buckling loads is insignificant for the highest values of the ply orientation angles (h > 45°), while it is negligible even zero for smaller angles (h 6 45°). The difference between the curves becomes noticeable for angles varying from 45° to 90°. The largest values are calculated for the case of a plate without a notch. In fact, it is very clear that the buckling load is highly affected by the piling up order of the laminated composite materials. Therefore to have a laminated plate which resists better to buckling, the ply orientations must be closer to the ordinate axis.
5
15
25
35
45
55
65
75
85
95
Height h (mm) Fig. 5. Variation of the buckling load with regard to the notch position.
biggest ratio (a/b = 1), that is to say for a circular notch, and it is identical for all cases, while the maximum values are obtained for the smallest geometrical defect ratio. This means that the increase of the notch radii causes the smallest buckling loads in plates subjected to mono-axial pressure. The slope of the curves increases with the increasing values of the angles u. Therefore more the size of geometric defect is important less is the buckling load, in other words the buckling load decreases with the increasing size of the notch which shows a reduced resistance of the composite laminated plate.
3.3. Influence of the notch position 3.2. Influence of the notch geometric ratio on the buckling load The influence of the geometric ratio a/b of an elliptical notch on the variation of the buckling load is illustrated in Fig. 4 for several notch inclinations with regard to the abscissa axis. The laminated composite material studied in this case is symmetrical of type [(0/90)2]S. The major axis of the ellipse is maintained constant (b = 10 mm) and the minor axis (a) variable. The figure shows that the buckling load decreases progressively and quasi-linearly with the increasing ratio of the elliptical notch. This reduction is noticeable whatever the inclination angle u of the geometrical defect is. The minimum value is obtained for the
15,0 Elliptical notch orientation ϕ=0° ϕ=15° ϕ=30° ϕ=45° ϕ=60° ϕ=75° ϕ=90°
Bucklingload (N/mm)
14,8 14,6 14,4
3.4. Influence of the rigidity ratio
14,2 14,0 13,8 13,6
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
Fig. 5 shows the buckling load curves when the position of the centre of the notch changes with regard to the applied load. The effect of the notch inclination with the abscissa axis is highlighted for symmetrical [(0/90)2]S laminated composite ply orientations. The size of the considered elliptical notch is a = b/8. It is noted that the largest values of the buckling load are obtained when the inclination angle of the notch is superior to 45° regardless its position h. For angles smaller than 45°, sine curves are obtained in which their maximum values are in the range 15°–25° and 75°–85° while the minimum values are between 45° and 55°. The resistance of the plate is better when the minor axis of the notch is perpendicular to the applied load (u = 90°) while it is weak when it is parallel (u = 0°). Extremum values have the tendency to disappear when the angle approaches (90°). In this case the buckling load is practically constant regardless the height h.
0,9
Ratio (a/b) Fig. 4. Effect of a/b ratio on the variation of the buckling load.
1,0
The effect of the modulus of rigidity ratio was also studied for different orientations of the laminated structure with and without the presence of an elliptical notch. With the presence of the notch, two cases (u = 0° and u = 90°) were studied. As shown in Fig. 6, the buckling load increases linearly with the increasing Young modulus ratio when the orientations of fibers are less than 45°. Beyond this angle, the curves have the tendency to be more and more constant. In other words, the buckling load is constant for angles h > 45°. It could be concluded that the reduction of the load is faster for smaller ratios E2/E1 and smaller fiber orientations of the laminated structure. The maximum value is obtained for h = 90° with and without the presence of the notch. When the inclination angle of the elliptical notch is h = 90°, the values of the buckling loads are
578
D. Ouinas, B. Achour / Composites: Part B 55 (2013) 575–579
45
11
40
10
Bucklingload (N/mm)
35
Layer tickness e=0.127 mm e=0.175 mm e=0.225 mm e=0.250 mm e=0.300 mm ϕ =0°
9
30 8
25
N
20 Without notch
15
θ=0° θ=10° θ=20° θ=30° θ=40° θ=45°
10 5 0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,9
7 6
θ=50° θ=60° θ=70° θ=80° θ=90°
0,8
*
5 4 1,0
0
10
20
30
40
50
60
70
80
90
Angle ( θ )
E2/E1
Fig. 7. Effect of the laminated plate fiber orientation on the buckling amplification factor (u = 0°).
40
Bucklingload (N/mm)
35 30
4,5
25
4,0
20
3,5
15
θ=0° θ=10° θ=20° θ=30° θ=40° θ=45°
10 5 0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
ϕ=90°
3,0
With notch ϕ=90° θ=50° θ=60° θ=70° θ=80° θ=90°
Layer tickness
N
* 2,5
2,0
1,0
1,5 1,0
E2/E1
0,5
40
0
10
20
30
40
50
60
70
80
90
Angle ( θ )
35
Bucklingload (N/mm)
e=0.127 mm e=0.175 mm e=0.225 mm e=0.250 mm e=0.300 mm
Fig. 8. Effect of the laminated plate fiber orientation on the buckling amplification factor (u = 90°).
30 25
x-direction. In the second case, the minor axis of the notch is taken perpendicular to the y-direction. The obtained results are represented in Figs. 7 and 8. On these last two figures, the variation of the dimensionless amplification factor of the buckling load versus the fiber orientation of composite material is represented. The amplification parameter is defined by:
20 15
θ=0° θ=10° θ=20° θ=30° θ=40° θ=45°
10 5 0
With nocth ϕ=0° θ=50° θ=60° θ=70° θ=80° θ=90°
N ¼ 1 0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1,0
E2/E1 Fig. 6. Buckling load for orientation angle versus Young modulus ratios.
more important than when u = 0° which shows in this second case that the notch presents an acute risk of damage. 3.5. Influence of the plate thickness To highlight the effect of the fiber orientation of the composite material of the plate with regard to the elliptical notch position, two cases were considered. The first case consists in taking the major axis of the notch in the direction of abscissas i.e. parallel to the
Nwn Nwtn
where Nwn and N wtn are respectively the buckling load of the plate with and without a notch. It can be pointed out that the presence of an elliptical notch in two positions in the same composite material introduces different results of the amplification factor of the buckling load. When the major axis of the notch is taken parallel to the x-axis (Fig. 7), three stages could be considered. In the first stage, the amplification factor is found practically stable with the increasing fiber orientation angle up to 20°. Beyond this value and up to 60° (stage II), the factor N increases quadratically. The third stage is characterized by a speedy increase of the buckling load which shows the growth of the plate resistance to buckling when the orientation angle gets larger. When h = 0° the amplification factor N is of the order of 5%, this value doubles when h = 90°. It is to be noted that for the
D. Ouinas, B. Achour / Composites: Part B 55 (2013) 575–579
same value of the orientation angle, the amplification factor becomes more and more significant with the increasing ply thickness. Fig. 8 shows the variation of the buckling load amplification factor N⁄ with regards to the fiber inclination angle. It increases proportionally with the inclination angle and reaches its maximum for h = 45°. In this case, the major axis of the notch is parallel to the applied load and it could be noticed that the amplification factor curves are symmetrical with regard to h = 45°. The minimum values are obtained for h = 0°. and h = 90°. For this case (u = 90°), the maximum amplification is of the order of 4% for h = 45° corresponding to the minimum value in the first case ((u = 0°)) for h = 0°. This show that the presence of elliptical notch in the first case has a considerable influence on the resistance of the plate in comparison with the second case. 4. Conclusion The design and application parameters have a great influence on the failure mode of a structure. In this study, the buckling response of a composite laminated material square plate having an elliptical notch is studied. The effects of the orientation angle of the elliptical notch, the geometry and position of the latter, the fiber orientation of the composite material, the ratio of rigidity of the material and the effect of the plate thickness on the variation of the buckling load were highlighted leading to the following findings: – The buckling load increases exponentially with regard to the increase of the fiber orientation angle of the composite material. This growth is much quicker and more important for h P 45°. – When the fibers are oriented perpendicularly to the applied stress, minimum buckling loads are obtained and vice versa. – The minimum value of N obtained corresponds to the largest ratio a/b = 1 (circular notch), while the maximum values of N are obtained for the weakest ratio of the geometric defect a/b. Therefore more the size of geometric defect is important less the buckling load is. – The maximum values of the buckling load N are obtained when the orientation angle of the elliptical notch is greater than 45° regardless its position h.
579
– Higher degrees of anisotropy increase considerably the buckling loads. – The amplification factor N⁄ increases significantly with the importance of the ply thickness. When u = 0° and the fiber orientation angle is h = 0°, the factor N is of the order of 5%, this value is doubled when h = 90°. While for u = 90°, the maximum amplification is of the order of 4% for h = 45° corresponding to the minimum value in the first case u = 0° and h = 0°.
References [1] Kweon JH, Jung JW, Kim TH, Choi JH, Kim DH. Failure of carbon composite – toaluminum reached with combined mechanical fastening and adhesive bonding. Compos Struct 2006;75:192–8. [2] Komur Aydin, Sen Faruk, Atas Akın, Arslan Nurettin. Composite buckling analysis of laminated flat with elliptical/circular cutout using year FEM. Adv Eng Software 2010;41:161–4. [3] Rhodes MD, MIKULAS MM, McGowan PE. Effects of orthotropy and width one the compression strength of graphite–epoxy with holes panels. AIAA Newspaper 1984;22:1283–92. [4] Leissa AW. Composite buckling of laminated flat and shell panels. AFWAL-TR85-3069, Flight dynamics laboratory, Air Force Wright Aeronautical Laboratories Wright-Patterson Air Forces Foundation: Ohio; 1985. [5] Muc A. Optimum grain orientation for simply-supported acute angle-ply flat under biaxial compression. Compos Struct 1988;9:161–72. [6] Nemeth MP. Buckling behavior of compression-loaded symmetrically laminated acute angle-ply flat with holes. AIAA Newspaper 1988;26:330–6. [7] Hu H-T, Flax B-H. Buckling optimization of symmetrically laminated rectangular flat with various geometry and end conditions. Compos Sci Technol 1995;55:277–85. [8] Vellaichamy S, Prakash BG, Brown S. Optimum design of cutouts in laminated composite structures. Comput Struct 1990;37:241–6. [9] Baba BO. Composite buckling behavior of laminated flat. J Reinf Plast Compos 2007;26:1637–55. [10] Zor M, Sen F, TOYGAR MISS. Year investigation of small public garden delamination effects one the buckling behavior of laminated composite flat with has small public garden hole by using three-dimensional FEM analysis. J Reinf Plast Compos 2005;24:1119–30. [11] Shufrin I, Rabinovitch O, Eisenberger M. Buckling of laminated flat with general boundary conditions under combined compression, tension, and in shear-A semi-analytical resolution. Thin-Wall Struct 2008;46:925–38. [12] Aydin Komur M, Faruk S, Akın A, Nurettin A. Buckling analysis of laminated composite plates with an elliptical/circular cutout using FEM. Adv Eng Software 2010;41:161–4. [13] Hamani N, Ouinas D, Benderdouche N, Sahnoun M. Buckling analyses of the antisymetrical composite laminate plate with a crack from circular notch. Adv Mater Res, vol. 365. Switzerland: Trans Tech Publications; 2012. ISSN: 1022– 6680, p. 56–1.