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On the buckling of cracked composite cylindrical shells under axial compression
Composite Structures 80 (2007) 152–158 www.elsevier.com/locate/compstruct
Technical Note
On the buckling of cracked composite cylindrical shells under axial compression Ashkan Vaziri
*
Division of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138, United States Available online 14 June 2006
Abstract Potential sensitivity of the buckling behavior of cracked composite cylindrical shells to service life cracking is explored by carrying out linear buckling analysis. Computational models of cracked composite cylindrical shells are developed by exploiting a special meshing scheme in which the element size is reduced incrementally from the element size employed in the uncracked region by approaching the crack tip. The effect of crack size and orientation, as well as the composite ply angle on the buckling behavior of cylindrical shells under axial compression is investigated. The results provide some insight into designing a composite laminate, which enhances the load capacity of cylindrical shells and minimizes their potential sensitivity to the presence of defects. Ó 2006 Elsevier Ltd. All rights reserved. Keywords: Cracked cylindrical shell; Laminated composite; Buckling behavior; Finite element model
1. Introduction Fiber-reinforced composites have been used extensively in various fields of modern engineering due to distinct structural advantages they offer. One such a field is aerospace engineering where structures are mostly assemblies of shell structures. Comprehensive understanding of the mechanical behavior of composite shells is vital to assure the integrity of these structures during their service life. Several studies have focused on predicting optimum laminate configurations for enhancing the load capacity of composite cylindrical shells under various loading conditions such as pure axial compression [1–6], combined axial compression and torsion [7–9] and transverse load [10]. Here, we focus on another important aspect associated with buckling behavior of composite cylindrical shells; its potential sensitivity to the presence of a crack. Presence of defects such as cracks, which may develop during manufacturing or service life of composite cylindrical shells, could severely affect the buckling behavior of structures not only by reducing their *
Tel.: +1 617 496 5167. E-mail address: [email protected]
0263-8223/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2006.05.007
load carrying capacity but also by introducing local buckling at the crack region [11,12]. In addition, many of the aerospace shell structures comprise cutouts due to functionality requirements. Considering the effect of these cutouts on buckling and post-buckling behavior of composite cylindrical shells is vital for assuring safe operating conditions during the service life of these structures [14]. In general, reinforcements around cutouts have been used to suppress the local buckling in composite shells. However, a recent study by Hilburger and Starnes [15] revealed that certain reinforcement configurations could lead to an unexpected increase in the magnitude of local deformations and stresses in the shell and a subsequent reduction in the buckling load. These recent results further emphasize the importance of understanding the mechanisms driving the buckling and post-buckling behavior of composite cylindrical shells comprising defects and cutouts. In this study, the potential sensitivity of buckling behavior of axially-compressed cylindrical shells to the presence of a through crack is investigated, emphasizing on the role of composite ply angle. Linear eigenvalue analyses are carried out for composite cylindrical shells under axial compression, which is the most significant and common type of loading considered
A. Vaziri / Composite Structures 80 (2007) 152–158
for theoretical buckling studies on shells and plates, using finite element methods. It is noteworthy that the potential sensitivity of buckling behavior of cylindrical shells on the presence of defects highly depends on loading condition [11,12] and it is conceivable that the results presented here would not be applicable under other loading conditions. The computational models of cracked cylindrical shells are generated by employing the meshing scheme proposed by Estekanchi and Vafai [11] in which the element size reduces incrementally from the constant element size employed in the uncracked region by approaching the crack tip, Fig. 1. One of the main advantages of this meshing scheme is the simplicity it offers for generating the computational models, which is indeed crucial for studies entailing large number of computational models. The validity of this approach for studying the buckling behavior of cracked thin plates and shells are established in [11,12]. Eight-node shell element, which has six degree of freedom at each node and quadratic deformation shape in both in-plane directions, is employed in the computational models. In the employed meshing scheme, the orientation of the elements is preserved at different levels of mesh zooming as shown in Fig. 1. An alternative approach is based on gradual change of the element orientation at each zoom level. The latter approach is shown to be more effective and accurate for fracture mechanics studies, while the former approach is capable of capturing the main features of local buckling and deformation of cracked cylindrical shells with a remarkable fidelity [11]. Through cracks are modeled by allowing the relative displacement and rotation (in all six degrees of freedom) of the neighbor nodes located at two edges of the crack. A mesh sensitivity
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study is carried out to ensure the independence of the results on the computational mesh. The calculations are performed using ANSYS (ANSYS, Inc., Canonsburg, PA), a commercial finite element package. The nodes located at the boundary are allowed to move in the cylinder axial direction while other degrees of freedom are constrained to zero. The computational model of the cylindrical shell has the length of L = 6.0 m and the radius of R = 1.0 m. The composite is taken as linear elastic material with the elastic moduli of E1 = 53 GPa in the fiber direction and E2 = 17.75 GPa in transverse to the fiber direction with the Poisson ratio of m12 = 0.25 (representing typical Fiberglass composites). The ply thickness of the composite is 0.125 mm with the laminate stacking of [h/ h]3 (h is measured from the cylinder longitudinal direction), which is antisymmetric about the middle surface, corresponding to the total thickness of t = 0.75 mm. The potential failure mechanisms of cracked composite cylindrical shells are (i) Euler buckling with a wavelength related to the cylinder length, (ii) surface buckling with a wavelength smaller than the cylinder length, (iii) local buckling in the crack region, and (iv) material failure, such as plasticity and delamination. The cylindrical shell under study clearly falls in the range of intermediate length for which the Euler buckling is not the dominant mechanism of failure. This is validated using the developed finite element model. Buckling of the cylindrical shell with a wavelength smaller than the cylinder length of the composite cylindrical shell (surface buckling) is studied in Section 2, while the local buckling behavior of the cracked cylindrical shell is explored in Section 3. The material failure is not considered in this study. It is noteworthy that the current
Fig. 1. Computational models of a cylindrical shell with (a) a circumferential crack and (b) an axial crack, developed by employing a special meshing scheme at the crack region proposed by Estekanchi and Vafai [11]. The computational model comprises 38 elements in circumferential and longitudinal directions. In the crack region, the element size is reduced to 1/2 of its original size with five level of zooming which results in crack tip element size of 1/32 of those used in the uncracked region.
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A. Vaziri / Composite Structures 80 (2007) 152–158
approach is proper for modeling the cylindrical shells with through or thumbnail cracks, while it fails to model delamination. The thumbnail crack model can be assessed by constraining the relative linear displacement of the neighbor nodes at two edges of the crack, while no constrain is imposed in regard to their relative rotational displacement as described in [11]. This model mimics a plastic hinge with negligible bending resistance at the crack edge. A computer program is developed to automatically generate finite element models of cracked cylindrical shells based on the proposed meshing scheme. In addition, a Matlab code is developed to calculate the anisotropic elastic properties of composite laminates. The calculated material properties are the inputs to the finite element model of composite cylindrical shells. Maximum length of the crack is taken to be a/R = 0.25, where a denotes the crack length. The numerical studies are carried out for the crack angle of a = 0–90°, measured from the cylinder circumferential line (i.e. a = 0° corresponds to the circumferential direction).
2. Buckling of composite cylindrical shells Fig. 2(a) displays the buckling shapes of the composite cylindrical shell, which may occur as the first or second buckling mode depending on the composite ply angle. Buckling shape A is a surface buckling mode having one long wavelength equal to the cylindrical shell length. Buckling shapes B and C are chessboard shapes with different wavelengths, while the buckling shape D is the ring buckling shape. We have checked that the presence of a crack does not considerably alter the surface buckling shapes of the shell for the geometries and properties consider here. It is noteworthy that the anisotropy factor defined as qffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 d ¼ max D16 = D311 D22 ; D26 = 4 D322 D11 is equal to zero for the composite laminate under study, which is antisymmetric about the middle surface. The flexural/twist coupling in composite laminates which leads to spiral buckling shapes in composite cylindrical shells is related
Fig. 2. (a) Buckling shapes of a composite cylindrical shell with ply sequence of [h/h]3, which appear as the first and second buckling modes depending on the composite ply angle. Variation of (b) the first normalized buckling load, c(1), and (c) the second normalized buckling load, c(2), of a composite cylindrical shell vs the composite ply angle. The buckling loads are normalized by the first buckling load of the composite shell with ply angle of h = 0° under axial compression. The associated buckling modes are identified for each buckling load and composite ply angle.
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to this anisotropy factor for compression buckling of composite plates and cylindrical shells [16,17] and is negligible here. In the aforementioned relation, Dijs are the bending stiffness terms of the laminate defined in the usual way [18]. The dependence of the first and second buckling loads of the composite cylindrical shell on the composite ply angle is displayed in Fig. 2(b) and (c), respectively. The buckling loads are normalized by the first buckling load of the composite shell with ply angle of h = 0° under axial compression and denoted by c(n) for the nth buckling load. For the ply angle in the range of 0° < h < 68°, the first buckling load is associated with buckling shape A, while increasing the ply angle causes buckling shape B to precede. On the other hand, all the buckling shapes presented in Fig. 2(a) could appear as the second buckling mode of the composite cylindrical shell depending on the composite ply angle. The composite laminate of [25°/25°]3 appears to have the maximum instability load under axial load, having the first buckling load about 12% higher than that of the cylindrical shell with [0°]6 stacking. We further studied the effect of composite ply angle on the first seven buckling loads of the composite cylindrical shell (data not shown for the sake of brevity). The results indicate that the sensitivity of composite buckling loads to the composite ply angle increases slightly for higher buckling modes. On the other hand, having all the fibers in the circumferential direction, [90°]6 results in the lowest load capacity, with a 20% reduction in the first buckling load as compared to the maximum load capacity of the composite cylindrical shell (associated with the ply angle of h = 25°). At composite ply angle of 65° while different buckling curves intersect, Fig. 2(a) exhibits another maximum. It is noteworthy that the interaction between Euler buckling and surface buckling could lead to a significant reduction in the load capacity of cylindrical shells. It is speculated that this effect is related to in-plane displacements in the buckling modes, which is neglected in the Donnel’s theory for shell buckling. Weaver and Dickenson [19] proposed that the influence of this interaction on composite cylindrical shells buckling behavior is related to a dimensionless parameter, X = kl2t/R3, where k accounts for material anisotropy (k = 1 for isotropic cylindrical shells, see [19] for details). It was observed that the interaction between the two aforementioned modes of buckling is negligible for X < 3, which is indeed the case for the cylindrical shells under study.
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is a function of loading condition and cylindrical shell radius and thickness. (For example for circumferentially cracked isotopic shells, the critical crack length increases remarkably on increasing the shell internal pressure as the internal pressure stabilizes the shell against buckling by suppressing the first possible local buckling shape and causing higher local buckling shapes to become critical.) The dependence of the first buckling load associated with local buckling of a circumferentially cracked composite shell on the composite ply angle is depicted in Fig. 3(a) for various normalized crack lengths, a/R. As expected, the buckling load associated with local buckling of cracked composite shell decreases on increasing the crack length. The variation of the first buckling load of an identical uncracked cylindrical shell vs the composite ply angle is also depicted in the same figure. The influence of crack length on this buckling load decreases on increasing the ply angle from 0°. For ply angle of 20° < h < 75°, local buckling does not precede the surface buckling for the range of crack length studied here and the overall linear buckling behavior of the cracked shell is not altered by the presence of a crack. Maximum reduction in the first buckling load of the cracked cylindrical shell is revealed for the ply angle of 0°, where a circumferential crack with
3. Local buckling of cracked composite cylindrical shells In this section, the effect of crack length and orientation on the buckling behavior of cracked cylindrical shells with different composite ply angles is investigated. For each composite laminate and crack orientation, a critical crack length can be defined which is the shortest crack causing the local buckling to precede the surface buckling of the cylindrical shell. This concept is studied in a companion paper [12]. It is noteworthy that the critical crack length
Fig. 3. (a) Variation of the first normalized buckling load associated with local buckling of a circumferentially cracked cylindrical shell vs the composite ply angle for various crack lengths. Inset: schematic of a circumferentially cracked cylindrical shell. (b) Typical first local buckling shape of a cracked cylindrical shell comprising a circumferential through crack. The composite cylindrical shell has the ply sequence of [h/h]3.
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a normalized length of a/R = 0.25 results in 20% reduction in the first buckling load of the composite cylindrical shell. Fig. 3(b) displays the first local buckling shape of an axially-compressed cracked shell with a circumferential through crack. The results indicate that the composite ply angle and the crack length do not affect the first local buckling shape of a circumferentially cracked composite shell significantly. The buckling mode shape resembles that of Fig. 3(b) with minor changes in surface curvatures. The influence of composite ply angle and crack orientation on the linear buckling behavior of a cracked composite cylindrical shell is examined by systematically varying the composite ply angle for a composite cylindrical shell comprising a through crack of normalized length, a/R = 0.25. Fig. 4 displays the variation of the first buckling load associated with local buckling of the composite shell vs the composite ply angle for various crack orientations, revealing the considerable role of crack orientation on buckling behavior of cracked composite shells. The variation of
the first buckling load of an identical uncracked cylindrical shell vs the composite ply angle is also depicted in the same figure for comparison. Interestingly for 30° < h < 75°, local buckling does not precede the surface buckling of the cylindrical shell for all crack orientations. Note also that for a crack oriented near axial direction, local buckling does not alter the buckling behavior of the composite shell significantly for any composite ply angle. Fig. 5 shows the dependence of the first buckling load associated with local buckling of the cracked cylindrical shell on the crack orientation for various composite ply angles. The results emphasize the considerable sensitivity of the composite buckling behavior on the crack orientation. Clearly the crack orientated close to a 45° results in maximum reduction in the buckling load (40% for a/R = 0.25 and h = 0°). For cylindrical shells with crack oriented near the circumferential direction, the buckling load associated with the local buckling of a composite shell highly depends on the composite ply angle where the composite ply angle of h = 45° leads
Fig. 4. Variation of the first normalized buckling load associated with local buckling of a cracked cylindrical shell vs the composite ply angle for various crack orientations (a) 0° 6 a 6 45° (b) 45° 6 a 6 90°. The normalized crack length is a/R = 0.25 and the composite cylindrical shell has the ply sequence of [h/h]3.
Fig. 5. Variation of the first normalized buckling load associated with local buckling of a cracked cylindrical shell vs the crack orientation for various composite ply angles (a) 0° 6 h 6 45° (b) 45° 6 h 6 90°. The normalized crack length is a/R = 0.25 and the composite cylindrical shell has the ply sequence of [h/h]3.
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to maximum resistance against local buckling. In contrast, the ply angle of h = 60° provides the maximum resistance against local buckling if the crack orientation is a 45°. In addition, the ply angle does not significantly alter the local buckling behavior of the composite shells with a crack oriented close to the axial direction. It is also noteworthy that for each composite ply angle, the first buckling load associated with local buckling of the composite shell is almost constant for a > 60°, signifying the minimal sensitivity to the crack orientation in this range. 4. Discussion and concluding remarks In this study, eigenvalue analysis is carried out to explore the linear buckling behavior of uncracked and cracked composite cylindrical shells subject to axial compression. Linear buckling analysis, apart from its theoretical importance, is the basis for most of the available design guides and provides design engineers with an initial estimate of the buckling load for various configurations in early stages of design. This issue is particularly important in design of composite shells, where the number of design alternatives can make the design procedure very extensive. The designer has several options at his disposal for enhancing the load capacity of the composite shell and for minimizing its potential sensitivity to the presence of defects, including optimizing the composite laminate and employing structural reinforcements. In this study, we focused on the former option by keeping the ply sequence constant and varying the associated ply angle. For the case of uncracked shell, the cylindrical shell with the composite ply angle of h 25° leads to maximum load capacity against buckling for the ply sequence under study, while the ply angle of 90° (composite fibers oriented in the circumferential direction) exhibits the lowest load capacity. In addition, the effect of crack length and orientation on the buckling behavior of composite cylindrical shells comprising a through crack is studied, using a special meshing scheme demonstrated in Section 1. The results indicate that the composite ply angle can be chosen to minimize the potential sensitivity of composite cylindrical shell to the presence of a crack. It should be emphasized that the current numerical approach is capable of analyzing the buckling behavior of cylindrical shells comprising through or partiallythrough cracks. Another type of common defect in the composite structures is delamination. The influence of multiple delaminations on the buckling behavior of laminated composite plates under uniaxial compression is studied by Hwang and Liu [20] using nonlinear buckling finite element analysis. The results indicated that near-surface delaminations could significantly influence the load capacity of the composite plates. Kim and Kedward [13] presented an analytical method for predicting the buckling initiation of delaminated composite plates, concluding that the local buckling can severely decrease the critical buckling load of plates. In addition, the bifurcation buckling analysis car-
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ried out here does not provide any information regarding the post-buckling behavior of composite cylindrical shells and also does not account for the effect of pre-buckling deformations. Hilburger and Starnes [21] show that laminate orthotropy not only has a significant influence on the buckling load capacity of composite shells but also affects the post-buckling behavior of these structures. Interestingly, in their study the laminate stacking, which has the lowest buckling load (load capacity before instability) leads to the highest post-buckling load of all three composite laminates studied. Therefore, although in most cases linear eigenvalue analysis may be sufficient for the preliminary design evaluation, but if there is a concern about material non-linearity or post-buckling response, a nonlinear postbuckling analysis should also be performed [22,23]. Acknowledgement The author is thankful to Drs. Homayoon E. Estekanchi, Hamid Nayeb-Hashemi and John W. Hutchinson for many insightful discussions. References [1] Jaunky N, Knight Jr NF. An assessment of shell theories for buckling of circular cylindrical laminated composite panels loaded in axial compression. Int J Solids Struct 1999;36:3799–820. [2] Ferreira AJM, Barbosa JT. Buckling behavior of composite shells. Compos Struct 2000;50:93–8. [3] Weaver PM, Driesen JR, Roberts P. The effects of flexural/twist anisotropy on compression buckling of laminated cylindrical shells. Compos Struct 2002;55:195–204. [4] Onoda J. Optimal laminate configurations of cylindrical shells for axial buckling. AIAA J 1985;23:1093–8. [5] Weaver PM. Design of laminated composite cylindrical shells under axial compression. Composites Part B 2000;31:669–79. [6] Geier B, Meyer-Peiening HR, Zimmermann R. On the influence of laminated stacking on buckling of composite cylindrical shells subjected to axial compression. Compos Struct 2002;55:467–74. [7] Sun G, Hansan JS. Optimal design of laminated composite circular– cylindrical shells subjected to combined loads. J Appl Mech 1998;55:136–42. [8] Diaconu CG, Masaki S, Sekine H. Buckling characteristics and layup optimization of long laminated composite cylindrical shells subjected to combined loads using lamination parameters. Compos Struct 2002;58:423–33. [9] Meyer-Peiening HR, Farshad M, Geier B, Zimmermann R. Buckling loads of CFRP composite cylinders under combined axial and torsion loading-experiments and computations. Compos Struct 2001;53: 427–35. [10] Sai Ram KS, Sreedhar Babu T. Buckling of laminated composite shells under transverse load. Compos Struct 2002;55:157–68. [11] Estekanchi HE, Vafai A. On the buckling of cylindrical shells with through cracks under axial load. Thin Wall Struct 1999;35:255–74. [12] Vaziri A, Estekanchi HE. Buckling of cracked cylindrical thin shells under combined internal pressure and axial compression. Thin Wall Struct 2006;44:141–51. [13] Kim H, Kedward KT. A method for modeling the local and global buckling of delaminated composite panels. Compos Struct 1999;44:43–53. [14] Tafreshi A. Buckling and post-buckling analysis of composite cylindrical shells with cutouts subjected to internal pressure and axial compression loads. Int J Pres Ves Pip 2002;79:351–9.
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[15] Hilburger MW, Starnes Jr JH. Buckling behavior of compressionloaded composite cylindrical shells with reinforced cutouts. Int J Non-Linear Mech 2005;7:1005–21. [16] Nemeth MP. Importance of anisotropy on buckling of compositeloaded symmetric composites plates. AIAA J 1986;24:1831. [17] Weaver PM. Anisotropy induced spiral buckling in compressionloaded composite shells. AIAA J 2002;40:101–7. [18] Jones RM. Mechanics of composite materials. Taylor and Francis, Inc.; 1998. [19] Weaver PM, Dickenson R. Interactive local/Euler buckling of composite cylindrical shells. Comput Struct 2003;81:2767–73.
[20] Hwang ShF, Liu GH. Buckling behavior of composite laminates with multiple delaminations under uniaxial compression. Compos Struct 2001;53:235–43. [21] Hilburger MW, Starnes Jr JH. Effects of imperfections on the buckling response of compression-loaded composite shells. Int J NonLinear Mech 2002;37:623–43. [22] Hutchinson JW. Post-bifurcation behavior in the plastic range. J Mech Phys Solids 1973;21:163–90. [23] Kunda CK, Sinha PK. Postbuckling analysis of laminated composite shells. Compos Struct, in press.
Technical Note
On the buckling of cracked composite cylindrical shells under axial compression Ashkan Vaziri
*
Division of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138, United States Available online 14 June 2006
Abstract Potential sensitivity of the buckling behavior of cracked composite cylindrical shells to service life cracking is explored by carrying out linear buckling analysis. Computational models of cracked composite cylindrical shells are developed by exploiting a special meshing scheme in which the element size is reduced incrementally from the element size employed in the uncracked region by approaching the crack tip. The effect of crack size and orientation, as well as the composite ply angle on the buckling behavior of cylindrical shells under axial compression is investigated. The results provide some insight into designing a composite laminate, which enhances the load capacity of cylindrical shells and minimizes their potential sensitivity to the presence of defects. Ó 2006 Elsevier Ltd. All rights reserved. Keywords: Cracked cylindrical shell; Laminated composite; Buckling behavior; Finite element model
1. Introduction Fiber-reinforced composites have been used extensively in various fields of modern engineering due to distinct structural advantages they offer. One such a field is aerospace engineering where structures are mostly assemblies of shell structures. Comprehensive understanding of the mechanical behavior of composite shells is vital to assure the integrity of these structures during their service life. Several studies have focused on predicting optimum laminate configurations for enhancing the load capacity of composite cylindrical shells under various loading conditions such as pure axial compression [1–6], combined axial compression and torsion [7–9] and transverse load [10]. Here, we focus on another important aspect associated with buckling behavior of composite cylindrical shells; its potential sensitivity to the presence of a crack. Presence of defects such as cracks, which may develop during manufacturing or service life of composite cylindrical shells, could severely affect the buckling behavior of structures not only by reducing their *
Tel.: +1 617 496 5167. E-mail address: [email protected]
0263-8223/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2006.05.007
load carrying capacity but also by introducing local buckling at the crack region [11,12]. In addition, many of the aerospace shell structures comprise cutouts due to functionality requirements. Considering the effect of these cutouts on buckling and post-buckling behavior of composite cylindrical shells is vital for assuring safe operating conditions during the service life of these structures [14]. In general, reinforcements around cutouts have been used to suppress the local buckling in composite shells. However, a recent study by Hilburger and Starnes [15] revealed that certain reinforcement configurations could lead to an unexpected increase in the magnitude of local deformations and stresses in the shell and a subsequent reduction in the buckling load. These recent results further emphasize the importance of understanding the mechanisms driving the buckling and post-buckling behavior of composite cylindrical shells comprising defects and cutouts. In this study, the potential sensitivity of buckling behavior of axially-compressed cylindrical shells to the presence of a through crack is investigated, emphasizing on the role of composite ply angle. Linear eigenvalue analyses are carried out for composite cylindrical shells under axial compression, which is the most significant and common type of loading considered
A. Vaziri / Composite Structures 80 (2007) 152–158
for theoretical buckling studies on shells and plates, using finite element methods. It is noteworthy that the potential sensitivity of buckling behavior of cylindrical shells on the presence of defects highly depends on loading condition [11,12] and it is conceivable that the results presented here would not be applicable under other loading conditions. The computational models of cracked cylindrical shells are generated by employing the meshing scheme proposed by Estekanchi and Vafai [11] in which the element size reduces incrementally from the constant element size employed in the uncracked region by approaching the crack tip, Fig. 1. One of the main advantages of this meshing scheme is the simplicity it offers for generating the computational models, which is indeed crucial for studies entailing large number of computational models. The validity of this approach for studying the buckling behavior of cracked thin plates and shells are established in [11,12]. Eight-node shell element, which has six degree of freedom at each node and quadratic deformation shape in both in-plane directions, is employed in the computational models. In the employed meshing scheme, the orientation of the elements is preserved at different levels of mesh zooming as shown in Fig. 1. An alternative approach is based on gradual change of the element orientation at each zoom level. The latter approach is shown to be more effective and accurate for fracture mechanics studies, while the former approach is capable of capturing the main features of local buckling and deformation of cracked cylindrical shells with a remarkable fidelity [11]. Through cracks are modeled by allowing the relative displacement and rotation (in all six degrees of freedom) of the neighbor nodes located at two edges of the crack. A mesh sensitivity
153
study is carried out to ensure the independence of the results on the computational mesh. The calculations are performed using ANSYS (ANSYS, Inc., Canonsburg, PA), a commercial finite element package. The nodes located at the boundary are allowed to move in the cylinder axial direction while other degrees of freedom are constrained to zero. The computational model of the cylindrical shell has the length of L = 6.0 m and the radius of R = 1.0 m. The composite is taken as linear elastic material with the elastic moduli of E1 = 53 GPa in the fiber direction and E2 = 17.75 GPa in transverse to the fiber direction with the Poisson ratio of m12 = 0.25 (representing typical Fiberglass composites). The ply thickness of the composite is 0.125 mm with the laminate stacking of [h/ h]3 (h is measured from the cylinder longitudinal direction), which is antisymmetric about the middle surface, corresponding to the total thickness of t = 0.75 mm. The potential failure mechanisms of cracked composite cylindrical shells are (i) Euler buckling with a wavelength related to the cylinder length, (ii) surface buckling with a wavelength smaller than the cylinder length, (iii) local buckling in the crack region, and (iv) material failure, such as plasticity and delamination. The cylindrical shell under study clearly falls in the range of intermediate length for which the Euler buckling is not the dominant mechanism of failure. This is validated using the developed finite element model. Buckling of the cylindrical shell with a wavelength smaller than the cylinder length of the composite cylindrical shell (surface buckling) is studied in Section 2, while the local buckling behavior of the cracked cylindrical shell is explored in Section 3. The material failure is not considered in this study. It is noteworthy that the current
Fig. 1. Computational models of a cylindrical shell with (a) a circumferential crack and (b) an axial crack, developed by employing a special meshing scheme at the crack region proposed by Estekanchi and Vafai [11]. The computational model comprises 38 elements in circumferential and longitudinal directions. In the crack region, the element size is reduced to 1/2 of its original size with five level of zooming which results in crack tip element size of 1/32 of those used in the uncracked region.
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approach is proper for modeling the cylindrical shells with through or thumbnail cracks, while it fails to model delamination. The thumbnail crack model can be assessed by constraining the relative linear displacement of the neighbor nodes at two edges of the crack, while no constrain is imposed in regard to their relative rotational displacement as described in [11]. This model mimics a plastic hinge with negligible bending resistance at the crack edge. A computer program is developed to automatically generate finite element models of cracked cylindrical shells based on the proposed meshing scheme. In addition, a Matlab code is developed to calculate the anisotropic elastic properties of composite laminates. The calculated material properties are the inputs to the finite element model of composite cylindrical shells. Maximum length of the crack is taken to be a/R = 0.25, where a denotes the crack length. The numerical studies are carried out for the crack angle of a = 0–90°, measured from the cylinder circumferential line (i.e. a = 0° corresponds to the circumferential direction).
2. Buckling of composite cylindrical shells Fig. 2(a) displays the buckling shapes of the composite cylindrical shell, which may occur as the first or second buckling mode depending on the composite ply angle. Buckling shape A is a surface buckling mode having one long wavelength equal to the cylindrical shell length. Buckling shapes B and C are chessboard shapes with different wavelengths, while the buckling shape D is the ring buckling shape. We have checked that the presence of a crack does not considerably alter the surface buckling shapes of the shell for the geometries and properties consider here. It is noteworthy that the anisotropy factor defined as qffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 d ¼ max D16 = D311 D22 ; D26 = 4 D322 D11 is equal to zero for the composite laminate under study, which is antisymmetric about the middle surface. The flexural/twist coupling in composite laminates which leads to spiral buckling shapes in composite cylindrical shells is related
Fig. 2. (a) Buckling shapes of a composite cylindrical shell with ply sequence of [h/h]3, which appear as the first and second buckling modes depending on the composite ply angle. Variation of (b) the first normalized buckling load, c(1), and (c) the second normalized buckling load, c(2), of a composite cylindrical shell vs the composite ply angle. The buckling loads are normalized by the first buckling load of the composite shell with ply angle of h = 0° under axial compression. The associated buckling modes are identified for each buckling load and composite ply angle.
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to this anisotropy factor for compression buckling of composite plates and cylindrical shells [16,17] and is negligible here. In the aforementioned relation, Dijs are the bending stiffness terms of the laminate defined in the usual way [18]. The dependence of the first and second buckling loads of the composite cylindrical shell on the composite ply angle is displayed in Fig. 2(b) and (c), respectively. The buckling loads are normalized by the first buckling load of the composite shell with ply angle of h = 0° under axial compression and denoted by c(n) for the nth buckling load. For the ply angle in the range of 0° < h < 68°, the first buckling load is associated with buckling shape A, while increasing the ply angle causes buckling shape B to precede. On the other hand, all the buckling shapes presented in Fig. 2(a) could appear as the second buckling mode of the composite cylindrical shell depending on the composite ply angle. The composite laminate of [25°/25°]3 appears to have the maximum instability load under axial load, having the first buckling load about 12% higher than that of the cylindrical shell with [0°]6 stacking. We further studied the effect of composite ply angle on the first seven buckling loads of the composite cylindrical shell (data not shown for the sake of brevity). The results indicate that the sensitivity of composite buckling loads to the composite ply angle increases slightly for higher buckling modes. On the other hand, having all the fibers in the circumferential direction, [90°]6 results in the lowest load capacity, with a 20% reduction in the first buckling load as compared to the maximum load capacity of the composite cylindrical shell (associated with the ply angle of h = 25°). At composite ply angle of 65° while different buckling curves intersect, Fig. 2(a) exhibits another maximum. It is noteworthy that the interaction between Euler buckling and surface buckling could lead to a significant reduction in the load capacity of cylindrical shells. It is speculated that this effect is related to in-plane displacements in the buckling modes, which is neglected in the Donnel’s theory for shell buckling. Weaver and Dickenson [19] proposed that the influence of this interaction on composite cylindrical shells buckling behavior is related to a dimensionless parameter, X = kl2t/R3, where k accounts for material anisotropy (k = 1 for isotropic cylindrical shells, see [19] for details). It was observed that the interaction between the two aforementioned modes of buckling is negligible for X < 3, which is indeed the case for the cylindrical shells under study.
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is a function of loading condition and cylindrical shell radius and thickness. (For example for circumferentially cracked isotopic shells, the critical crack length increases remarkably on increasing the shell internal pressure as the internal pressure stabilizes the shell against buckling by suppressing the first possible local buckling shape and causing higher local buckling shapes to become critical.) The dependence of the first buckling load associated with local buckling of a circumferentially cracked composite shell on the composite ply angle is depicted in Fig. 3(a) for various normalized crack lengths, a/R. As expected, the buckling load associated with local buckling of cracked composite shell decreases on increasing the crack length. The variation of the first buckling load of an identical uncracked cylindrical shell vs the composite ply angle is also depicted in the same figure. The influence of crack length on this buckling load decreases on increasing the ply angle from 0°. For ply angle of 20° < h < 75°, local buckling does not precede the surface buckling for the range of crack length studied here and the overall linear buckling behavior of the cracked shell is not altered by the presence of a crack. Maximum reduction in the first buckling load of the cracked cylindrical shell is revealed for the ply angle of 0°, where a circumferential crack with
3. Local buckling of cracked composite cylindrical shells In this section, the effect of crack length and orientation on the buckling behavior of cracked cylindrical shells with different composite ply angles is investigated. For each composite laminate and crack orientation, a critical crack length can be defined which is the shortest crack causing the local buckling to precede the surface buckling of the cylindrical shell. This concept is studied in a companion paper [12]. It is noteworthy that the critical crack length
Fig. 3. (a) Variation of the first normalized buckling load associated with local buckling of a circumferentially cracked cylindrical shell vs the composite ply angle for various crack lengths. Inset: schematic of a circumferentially cracked cylindrical shell. (b) Typical first local buckling shape of a cracked cylindrical shell comprising a circumferential through crack. The composite cylindrical shell has the ply sequence of [h/h]3.
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a normalized length of a/R = 0.25 results in 20% reduction in the first buckling load of the composite cylindrical shell. Fig. 3(b) displays the first local buckling shape of an axially-compressed cracked shell with a circumferential through crack. The results indicate that the composite ply angle and the crack length do not affect the first local buckling shape of a circumferentially cracked composite shell significantly. The buckling mode shape resembles that of Fig. 3(b) with minor changes in surface curvatures. The influence of composite ply angle and crack orientation on the linear buckling behavior of a cracked composite cylindrical shell is examined by systematically varying the composite ply angle for a composite cylindrical shell comprising a through crack of normalized length, a/R = 0.25. Fig. 4 displays the variation of the first buckling load associated with local buckling of the composite shell vs the composite ply angle for various crack orientations, revealing the considerable role of crack orientation on buckling behavior of cracked composite shells. The variation of
the first buckling load of an identical uncracked cylindrical shell vs the composite ply angle is also depicted in the same figure for comparison. Interestingly for 30° < h < 75°, local buckling does not precede the surface buckling of the cylindrical shell for all crack orientations. Note also that for a crack oriented near axial direction, local buckling does not alter the buckling behavior of the composite shell significantly for any composite ply angle. Fig. 5 shows the dependence of the first buckling load associated with local buckling of the cracked cylindrical shell on the crack orientation for various composite ply angles. The results emphasize the considerable sensitivity of the composite buckling behavior on the crack orientation. Clearly the crack orientated close to a 45° results in maximum reduction in the buckling load (40% for a/R = 0.25 and h = 0°). For cylindrical shells with crack oriented near the circumferential direction, the buckling load associated with the local buckling of a composite shell highly depends on the composite ply angle where the composite ply angle of h = 45° leads
Fig. 4. Variation of the first normalized buckling load associated with local buckling of a cracked cylindrical shell vs the composite ply angle for various crack orientations (a) 0° 6 a 6 45° (b) 45° 6 a 6 90°. The normalized crack length is a/R = 0.25 and the composite cylindrical shell has the ply sequence of [h/h]3.
Fig. 5. Variation of the first normalized buckling load associated with local buckling of a cracked cylindrical shell vs the crack orientation for various composite ply angles (a) 0° 6 h 6 45° (b) 45° 6 h 6 90°. The normalized crack length is a/R = 0.25 and the composite cylindrical shell has the ply sequence of [h/h]3.
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to maximum resistance against local buckling. In contrast, the ply angle of h = 60° provides the maximum resistance against local buckling if the crack orientation is a 45°. In addition, the ply angle does not significantly alter the local buckling behavior of the composite shells with a crack oriented close to the axial direction. It is also noteworthy that for each composite ply angle, the first buckling load associated with local buckling of the composite shell is almost constant for a > 60°, signifying the minimal sensitivity to the crack orientation in this range. 4. Discussion and concluding remarks In this study, eigenvalue analysis is carried out to explore the linear buckling behavior of uncracked and cracked composite cylindrical shells subject to axial compression. Linear buckling analysis, apart from its theoretical importance, is the basis for most of the available design guides and provides design engineers with an initial estimate of the buckling load for various configurations in early stages of design. This issue is particularly important in design of composite shells, where the number of design alternatives can make the design procedure very extensive. The designer has several options at his disposal for enhancing the load capacity of the composite shell and for minimizing its potential sensitivity to the presence of defects, including optimizing the composite laminate and employing structural reinforcements. In this study, we focused on the former option by keeping the ply sequence constant and varying the associated ply angle. For the case of uncracked shell, the cylindrical shell with the composite ply angle of h 25° leads to maximum load capacity against buckling for the ply sequence under study, while the ply angle of 90° (composite fibers oriented in the circumferential direction) exhibits the lowest load capacity. In addition, the effect of crack length and orientation on the buckling behavior of composite cylindrical shells comprising a through crack is studied, using a special meshing scheme demonstrated in Section 1. The results indicate that the composite ply angle can be chosen to minimize the potential sensitivity of composite cylindrical shell to the presence of a crack. It should be emphasized that the current numerical approach is capable of analyzing the buckling behavior of cylindrical shells comprising through or partiallythrough cracks. Another type of common defect in the composite structures is delamination. The influence of multiple delaminations on the buckling behavior of laminated composite plates under uniaxial compression is studied by Hwang and Liu [20] using nonlinear buckling finite element analysis. The results indicated that near-surface delaminations could significantly influence the load capacity of the composite plates. Kim and Kedward [13] presented an analytical method for predicting the buckling initiation of delaminated composite plates, concluding that the local buckling can severely decrease the critical buckling load of plates. In addition, the bifurcation buckling analysis car-
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ried out here does not provide any information regarding the post-buckling behavior of composite cylindrical shells and also does not account for the effect of pre-buckling deformations. Hilburger and Starnes [21] show that laminate orthotropy not only has a significant influence on the buckling load capacity of composite shells but also affects the post-buckling behavior of these structures. Interestingly, in their study the laminate stacking, which has the lowest buckling load (load capacity before instability) leads to the highest post-buckling load of all three composite laminates studied. Therefore, although in most cases linear eigenvalue analysis may be sufficient for the preliminary design evaluation, but if there is a concern about material non-linearity or post-buckling response, a nonlinear postbuckling analysis should also be performed [22,23]. Acknowledgement The author is thankful to Drs. Homayoon E. Estekanchi, Hamid Nayeb-Hashemi and John W. Hutchinson for many insightful discussions. References [1] Jaunky N, Knight Jr NF. An assessment of shell theories for buckling of circular cylindrical laminated composite panels loaded in axial compression. Int J Solids Struct 1999;36:3799–820. [2] Ferreira AJM, Barbosa JT. Buckling behavior of composite shells. Compos Struct 2000;50:93–8. [3] Weaver PM, Driesen JR, Roberts P. The effects of flexural/twist anisotropy on compression buckling of laminated cylindrical shells. Compos Struct 2002;55:195–204. [4] Onoda J. Optimal laminate configurations of cylindrical shells for axial buckling. AIAA J 1985;23:1093–8. [5] Weaver PM. Design of laminated composite cylindrical shells under axial compression. Composites Part B 2000;31:669–79. [6] Geier B, Meyer-Peiening HR, Zimmermann R. On the influence of laminated stacking on buckling of composite cylindrical shells subjected to axial compression. Compos Struct 2002;55:467–74. [7] Sun G, Hansan JS. Optimal design of laminated composite circular– cylindrical shells subjected to combined loads. J Appl Mech 1998;55:136–42. [8] Diaconu CG, Masaki S, Sekine H. Buckling characteristics and layup optimization of long laminated composite cylindrical shells subjected to combined loads using lamination parameters. Compos Struct 2002;58:423–33. [9] Meyer-Peiening HR, Farshad M, Geier B, Zimmermann R. Buckling loads of CFRP composite cylinders under combined axial and torsion loading-experiments and computations. Compos Struct 2001;53: 427–35. [10] Sai Ram KS, Sreedhar Babu T. Buckling of laminated composite shells under transverse load. Compos Struct 2002;55:157–68. [11] Estekanchi HE, Vafai A. On the buckling of cylindrical shells with through cracks under axial load. Thin Wall Struct 1999;35:255–74. [12] Vaziri A, Estekanchi HE. Buckling of cracked cylindrical thin shells under combined internal pressure and axial compression. Thin Wall Struct 2006;44:141–51. [13] Kim H, Kedward KT. A method for modeling the local and global buckling of delaminated composite panels. Compos Struct 1999;44:43–53. [14] Tafreshi A. Buckling and post-buckling analysis of composite cylindrical shells with cutouts subjected to internal pressure and axial compression loads. Int J Pres Ves Pip 2002;79:351–9.
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