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Buckling response of advanced grid stiffened carbon–fiber composite cylindrical shells with reinforced cutouts
Composites: Part B 44 (2013) 26–33
Contents lists available at SciVerse ScienceDirect
Composites: Part B journal homepage: www.elsevier.com/locate/compositesb
Buckling response of advanced grid stiffened carbon–fiber composite cylindrical shells with reinforced cutouts Shanshan Shi a, Zhi Sun a,b, Mingfa Ren a,⇑, Haoran Chen a, Xiaozhi Hu b a b
State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian 116023, PR China School of Mechanical and Chemical Engineering, University of Western Australia, Perth, WA 6009, Australia
a r t i c l e
i n f o
Article history: Received 8 January 2012 Received in revised form 5 May 2012 Accepted 17 July 2012 Available online 16 August 2012 Keywords: A. Carbon fiber B. Buckling C. Finite Element Analysis (FEA) Reinforcing design
a b s t r a c t In this paper, we present the initial buckling and post-buckling responses of axial loaded advanced grid stiffened (AGS) composite cylindrical shells with reinforced rectangular or circular cutouts. The AGS cylindrical shells were reinforced by various local grid configurations near the cutout areas. The effects of different reinforcing grid configurations on critical loads were then examined and compared to those of different skin-reinforcing designs using Finite Element Analysis (FEA) simulations. A high-fidelity nonlinear analysis procedure was proposed to predict the non-linear buckling response of the shell structures. The simulation results indicated that the grid reinforcements can reduce or eliminate the risk of local buckling response near the cutout areas and increase the critical load of the shell more effectively than the skin reinforcements. Furthermore, those results showed that an optimum grid reinforcement configuration exists, which significantly improved the initial buckling and post-buckling resistance of the cylindrical shells under axial loading. The above findings can potentially be useful to the analysis and optimum design of AGS composite cylindrical shells with cutouts. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction Advanced gird stiffened (AGS) composite structures have been increasingly used in the field of aerospace, as they offer superior specific stiffness and strength. Compared to the traditional metallic structures, AGS counterparts are capable of providing much superior mechanical properties for the same structural weight. The structural weight has been considered as one of the key factors for development of AGS structures, since the launching weight and cost is a major consideration in the aerospace engineering. The AGS cylindrical shells studied in this paper are widely used in the aircraft and aerospace industries, for instance, the main fuselage structures of commercial aircrafts and the launch vehicle sections of rockets, which all mainly experience axial loading. In many cases, cutouts need to be introduced on the AGS cylindrical shells for the installations of doors, windows or access ports. Inevitably, those cutouts interrupt the continuous distribution of stress and strain within the AGS cylindrical shells, leading to the stress concentration at the cutout edge, and thus significantly reduce the buckling resistance of the AGS cylindrical shells. To avoid such potential buckling failure, reinforcements need to be placed around the cutout regions to reduce the stress and strain concentrations near the cutouts and thus the load capacity of AGS ⇑ Corresponding author. Tel.: +86 411 84708393; fax: +86 411 84709161. E-mail address: [email protected] (M. Ren). 1359-8368/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compositesb.2012.07.044
cylindrical shells with the reinforced cutouts is increased. This study deals with different reinforced cutout designs of the AGS cylindrical shells, and it is expected that the results will be useful to AGS shell design, manufacturing and structural integrity analysis. For decades, many studies have been conducted on the buckling and post-buckling characteristics of composite plates with cutouts under different loading and boundary conditions. Komur et al. [1] analyzed the buckling behavior of laminated composite plates with an elliptical cutout. Using Finite Element Analysis (FEA) simulations, the critical buckling loads of different composite plates with various elliptical holes of different shapes and at the different positions were determined. Jain and Kumar [2] investigated the post-buckling response of square composite plates with a central circular or elliptical cutout. The substantial factors such as cutout shape and size were considered and the post-buckling loads were obtained. Kumar and Singh [3,4] examined the buckling responses, particularly shear load and failure characteristics, of composite laminate plates with various shaped cutouts including circular, square, diamond, elliptical-vertical and elliptical-horizontal. Xie et al. [5] presented a thorough discussion on the tailoring concept which was proved to be effective in improving buckling loads and ultimate loads for plates with central circular cutouts. Hilburger et al. [6] used a geometrically non-linear FEA method to study the effects of plate curvature and initial geometric imperfection on the buckling response of compression-loaded curved composite
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S. Shi et al. / Composites: Part B 44 (2013) 26–33
Fig. 1. Geometry and FE model of the cylindrical shells with single cutout: (a) square cutout and (b) circular cutout.
plates with a circular cutout. Oh et al. [7] put forward an efficient post-buckling analysis technique for stiffened curved plates, and the computational time required was dramatically reduced with improved accuracy. Tercan and Aktas [8] investigated the effects of cutout shape on buckling of 1 1 rib knitting glass/epoxy laminated plates in three different knitting tightness levels, adding more to the vast literature on the cutout-affected buckling properties of the AGS shells. The buckling responses of cylindrical shells with various types of openings of different sizes were studied as well. Tafreshi [9] focused on the buckling and post-buckling responses of a series of cylindrical shells with different cutout sizes subjected to internal pressure and axial compression. Buckling modes were suggested, and the results agreed well with relevant published results. Numerical simulation and analysis were used by Han et al. [10] for the study of the influence of cutout size, cutout location and shell aspect radio on the buckling and post-buckling responses of the cylindrical shells. In addition, an experimental investigation on moderately thickwalled shells was also carried out [10]. The results predicted by general commercial FEA platform ANSYS were compared with the experimental results. Yeh et al. [11] investigated the buckling behavior of an elastoplastic cylindrical shell with a cutout under pure bending using both analytical and experimental methods. Schenk and Schueller [12,13] surveyed the individual and combined effects of random geometric imperfections and boundaries of isotropic thin-walled cylindrical shells under axial compression with rectangular cutouts using non-linear static FEA. Stress concentrations near cutouts and decreases in load capacity were observed in all the aforementioned plate and shell structures with cutouts. An effective approach to reduce the stress concentration near cutouts and increase the load capacity of composite shells with cutouts is to apply reinforcements around the cutout areas. A significant reduction in the stress and strain concentration was observed for appropriate cutout shapes and edge reinforcements in a composite C-section beam with diamond and circular cutouts under static shear load [14]. Guo [15,16] designed four different types of cutout reinforcements to evaluate the stress concentration and buckling behavior of composite plates under shear loading. Hilburger and Starnes [17] implemented a numerical study technique on the buckling behavior and non-linear response of compression-loaded composite cylindrical shells, in which the effects of cutout reinforcements, size and thickness were consid-
Table 1 Mechanical properties of skin and grid materials. Material property
E1, E2 (GPa)
m
G12 (GPa)
Skin and ribs
127.557, 11.308
0.3
5.998
ered. All those previous studies showed that reinforcements were capable to increase the load capacity of composite structures with cutout. However, only limited studies were conducted on the buckling behavior of stiffened composite cylindrical shells with cutouts. Cervantes and Palazotto [18] studied the optimum placement and volume of a reinforcing frame around a cutout for a stiffened metal cylindrical shell. The effects of skin reinforcements with different size, thickness or ply angle on AGS cylindrical shells with cutouts were considered and the buckling responses were compared with non-reinforced shells [19]. In this paper, seven grid configurations of reinforcements were adopted and analyzed around the cutout area for axial loaded AGS cylindrical shells with single square or circular cutout. The effects of different grid reinforcements on the initial buckling and post-buckling responses of these AGS shells were calculated by FE models. In addition, the predicted load capacities with the grid reinforcements were compared with the results obtained for AGS shells with skin reinforcements so that grid and skin reinforcements were evaluated.
2. The Finite element models of cylindrical shells with reinforced cutouts 2.1. Structure geometry and Finite Element models Cylindrical shell models designed for the FEA with a square and circle cutout are shown in Fig. 1a and b respectively. Each cylindrical shell has a radius R 800 mm, length L 1600 mm and thickness of skin t 2 mm. In Fig. 1 the geometric shapes and details of the structures are fully described. The width of square cutouts and the diameter of circle cutouts are defined by c and d respectively. A 250 mm 250 mm square cutout with 10-mm radius chamfer
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S. Shi et al. / Composites: Part B 44 (2013) 26–33
Table 2 Structure identification codes and corresponding reinforcements. Identification codes
Description
G0
No reinforcement
G1
Diamond grids reinforcement combined short transversal and axial ribs
G2
Diamond grids reinforcement combined two short transversal ribs
G3
Diamond grids reinforcement
G4
Hexagon grids reinforcement combined two short axial ribs
G5
Hexagon grids reinforcement
Schematic of configuration
S. Shi et al. / Composites: Part B 44 (2013) 26–33
29
Table 2 (continued) Identification codes
Description
G6
Irregular grids reinforcement
Schematic of configuration
G7
Square grids reinforcement
S1–S5
Skin reinforcement
Table 3 Codes and corresponding geometrical parameters for skin reinforcements. Structure identification code
Reinforcement size, a (mm)
Reinforcement thickness (mm) (plies)
S1 S2 S3 S4 S5
502 502 502 502 502 502 408 408 314 314
2.5 (10) 5 (20) 8 (32) 2.5 (10) 2.5 (10)
are supported. Additionally, meticulous mesh process of triangular elements was carried out for the areas near cutouts, since the adjacent regions of cutouts were critical during FEA analysis. Furthermore, the boundary conditions and loads of the FE models are also illustrated in Fig. 1. The top and bottom edges of both models were simply supported. Meanwhile, a uniform displacement D was loaded by Multi-Point Constraint (MPC) technique [22], which applied the load on a central node of the top edge and constraint to other nodes at the top edges. 2.2. Configurations of reinforcement
corners is located at the mid-span of the AGS shell, as shown in Fig. 1a, while a circle cutout with a diameter of 250 mm locates at the same position in Fig. 1b. In other words, c and d both equal to 250 mm. In addition, since the cutout area suffers from stress concentration, the area of the around cutouts is meshed using smaller elements and parameters a and b are exploited to mark the size of the area with finer mesh. Except for the cutout area, there are 40 longitudinal ribs and 13 circumferential grids on the shells. In this study, each grid is 10 mm in height and 5 mm in width. The skins of cylindrical shells were modeled as geometrically perfect, and made by T700 carbon fiber/epoxy composites as mentioned in references [20,21]. Additionally, the thickness of each ply was 0.25 mm and shells were fabricated by a symmetric lay-up [0/45/45/90]s to a final skin thickness of 2 mm (8 plies). The mechanical properties of the skin and grid are given in Table 1. As can be seen in Fig. 1a and b, general skin areas and all grids were divided into quadrilateral elements, the reinforced area were divided into triangle elements. The elements were flat face-type based on the Kirchoff–Love shell theory and the non-linear Lagrangian strain tensor. Each element node has six degrees of freedom and the element was well suited for large deformation non-linear applications. Both full and reduced integration schemes
The analyzed configurations in this study included eight reinforcing configurations and each reinforcing configuration was used for structures with either square or circular cutouts. Meanwhile, the AGS shells with unreinforced cutouts were studied as well for comparison. These structures were referred as G0 (no reinforcement), G1–G7 (grid reinforcements) and S1–S5 (skin reinforcements). Each grid reinforcing configuration possessed different shapes, while the skin reinforcements were reinforced by axial composite laminates with different sizes and thicknesses. Tables 2 and 3 summarized the detailed descriptions for all the configurations. As shown in Table 2, solid lines, dashed lines and shade areas represented reinforcing grid, edges of cutout area and skin reinforcements respectively. The width b of grid reinforced area was 500 mm. And geometrical parameters of skin reinforcements were shown in Table 3. 3. Numerical results and discussions 3.1. Analysis approach PATRAN/NASTRAN was employed for the FEA. Eigenvalue buckling analysis was carried out to calculate the buckling resistance of
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S. Shi et al. / Composites: Part B 44 (2013) 26–33
AGS cylindrical shells without cutouts. Considering the influence of unstable post-buckling response on critical axial load, modified Newton–Raphson method was used to perform post-buckling analysis of structures, and large deformation equation was taken into account. The modified Newton–Raphson method works well in snap-through problems, and can also be used to solve problems both with stable and unstable post-buckling behaviors. Skin and rib local buckling, global buckling and other forms of buckling possibly occur during the loading process on AGS cylindrical shell structures. Meanwhile, there will possibly be large deformation and large rotation in the structures during the post-buckling process. Therefore, considering large deformation and large rotation, the author analyzed buckling and post-buckling responses for axial loaded AGS cylinders with cutout, employing the modified Newton–Raphson method. 3.2. Cylindrical shells with unreinforced rectangular or circular cutouts The results for the cylindrical shells with single rectangular or circular cutout referred to herein as shell G0 are shown in Fig. 2, where the load is plotted against the end-shortening displacement. The solid curve refers to the cylindrical shell with a circular cutout and the dashed curve refers to the cylindrical shell with a rectangular cutout. To illustrate a clear description, the compression load P was normalized to Pcr = 3.91 106 N (the simulated critical buckling load of the present AGS cylindrical shell without cutout), while the end-shortening displacement D was normalized to the structure length L. The normalized load–displacement curves indicate an overall tendency that an AGS shell with a circular cutout had a higher load capacity than an AGS shell with a rectangular cutout at the same displacement. It is thus suggested that the higher stress concentration behavior caused by quasi-vertical corners of the rectangular cutout will further reduce the load capacity of the structures. Both shells with rectangular and circle cutouts experience linear
prebuckling responses during the initial loading process. There is an obvious local buckling (marked as A) for shell with rectangular cutout at the load level of P/Pcr = 0.31. The displacement and stress nearby the cutout area varied dramatically while the overall stiffness of shell degenerate slightly during the following local buckling period. The complex displacement and stress fields caused by local buckling respond to fluctuation of the load–displacement curve between the local buckling point and global buckling point. Typical shell with circular cutout in present study do not comport an evident local buckling. The load capacities for shells with rectangular and circular cutouts achieves the peak at P/Pcr = 0.43 and P/Pcr = 0.48 respectively. The displacement cloud pictures for both shells exhibit that buckling behaviors extends from grid cells nearby the cutout to a broader area at the post-buckling section. The snap-through phenomenon following global buckling responses results in a significant reduction in the axial load capacity of the shells. In the post-buckling section, load transmission switches from axial to bending which thereby reduces the effective loading-carrying cross-section. The post-buckling displacement mode gradually varies from single buckling waves around the cutouts to several waves flooding along the whole cylindrical shells. Overall, the axial load-shortening response for the typical AGS cylindrical shells with rectangular and circle cutouts exhibit complete local buckling, global buckling and post-buckling responses. 3.3. Cylindrical shells with grid or skin reinforced rectangular and circular cutouts In the previous section, typically buckling responses for the cylindrical shells with single rectangular or circular cutout referred to herein as G0 were discussed. Moreover, the larger displacements and local buckling behaviors could be found around the cutout areas for AGS cylindrical shells, especially for shells with rectangular cutouts. Based on those results, it is suggested that an optimum
Fig. 2. Compression load-shortening response curve of AGS cylindrical shell with unreinforced cutout (G0).
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0.6
0.5
0.4
(a)
G0 G1 G2 G3 G4 G5 G6 G7
0.5
0.3
0.2
0.4
0.3
0.2
0.1
0.1
0.0 0.000
G0 G1 G2 G3 G4 G5 G6 G7
0.6
Axial load, P/ Pcr
Axial load, P/Pcr
(a)
0.001
0.002
0.003
0.0 0.000
0.004
0.001
0.002
(b) 0.6
(b)
s1 s2 s3 s4 s5
0.4
0.3
0.2
0.004
0.005
0.004
0.005
s1 s2 s3 s4 s5
0.4
0.3
0.2
0.1
0.1
0.0 0.000
0.6
0.5
Axial load, P/ Pcr
Axial load, P/Pcr
0.5
0.003
End-shortening, Δ/L
End-shortening, Δ/L
0.001
0.002
0.003
0.004
0.0 0.000
0.005
0.001
0.002
reinforcement can be identified to enhance the stiffness and strength in this area, which can reduce or eliminate the risk of local buckling response near the cutout areas and increase the load capacity of the shells. Therefore, a range of buckling and post-buckling simulation studies were processed to demonstrate the diversity between normal skin reinforcement and certain grid reinforcing configurations selected in Section 2.2. The reinforcing configurations consisted of special grid reinforcing frames near the cutout area, which were added to inner side of the skin laminates. Similar to the shells with unreinforced cutouts, the compression load P was normalized to Pcr = 3.91 106 N, while the end-shortening displacement D was normalized to the structure length L. The normalized load–displacement curves for reinforced shells with rectangular cutouts and circle cutouts were indicated respectively. For shells with rectangular cutouts, the load–displacement curves for grid-reinforcements and skin-reinforcements are indicated in Fig. 3a and b. Clearly, G5 performs a significant rise in the global collapse load compared with unreinforced shell. The result indicates that grid configuration G5 expressed to be smoother in load transmission, because its geometric composition is favorable in the improvement process of AGS cylindrical shells with rectangular cutout. Structure reinforced by configuration G6, however, is even weaker than original shell and lost load capacity at lower compression loading displacement. This suggests that reinforcement without smooth geometry might increase the effect of local buckling and reduce collapse load. The others structures reinforced by grid or skin configurations experience relatively close results, while the structures with grid reinforcements exhibit little
Fig. 4. Compression load-shortening response curve of AGS cylindrical shell with circle cutout: (a) with grids reinforcement and (b) with skin reinforcement.
higher collapse loads than structures with skin reinforcements on average. In addition, there are no obviously local buckling responses for reinforced structures. Furthermore, the load–displacement curves for grid-reinforcements and skin-reinforcements of AGS shells with circle cutouts are indicated in Fig. 4a and b. The AGS shell reinforced by configuration G4, which is similar to the grid configuration G5 – the best configuration for shells with rectangular cutouts in present study, exhibits to own the highest global collapse load. The results further
rectangular circular 0.6
Axial load, P/ Pcr
Fig. 3. Compression load-shortening response curve of AGS cylindrical shell with rectangular cutout: (a) with grids reinforcement and (b) with skin reinforcement.
0.003
End-shortening, Δ/L
End-shortening, Δ/L
0.4
0.2 g1
g2
g3
g4
g5
g6
g7
s1
s2
s3
s4
s5
Configuration Fig. 5. Normalized global collapse loads for AGS cylinders with rectangular or circular cutout reinforced by various configurations.
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S. Shi et al. / Composites: Part B 44 (2013) 26–33
4. Conclusions 6000
rectangular circular
Efficiency
3000
0
-3000
-6000
g1
g2
g3
g4
g5
g6
g7
s1
s2
s3
s4
s5
Configuration Fig. 6. Reinforcing effectiveness normalized by weight increase for AGS cylinders with rectangular or circular cutout reinforced by various configurations.
indicate that configurations which could transmit load smoothly are competitive. The overall buckling responses illustrate that shells with circle cutouts could undertake more load than shells with rectangular cutouts. In addition, the results of AGS shells with grid and skin reinforced circle cutouts less scatters, so it is suggested that the circle-shape cutout to be a more desirable form of cutouts for the AGS cylindrical shells. Moreover, grid reinforcements are more effective than skin reinforcements.
In this paper, a FEA model has been established for the investigation of pre-buckling, local buckling and global buckling responses of the advanced composite grid stiffened cylindrical shells with unreinforced or reinforced cutouts under axial compressive loading. The resultant local buckling and post-buckling responses indicate that the initial buckling load underestimates the global bucking load of the AGS shells with cutouts. Therefore, post-buckling analysis and design is necessary. For unreinforced structures, the simulated results illustrate that an AGS shell with a circular cutout possesses a higher global collapse load than that of an AGS shell with a rectangular cutout due to the obvious difference in the cutout geometry and thus associated stress concentration. More specifically, the stress concentration behavior induced by quasi-vertical corners leads to premature local buckling response around the cutout area, resulting in a noticeable reduction in the global load capacity of the AGS shells. Different grid and skin reinforced configurations have also been investigated and compared in the present study. The simulation results indicate that a grid reinforced configuration which can transmit loading smoothly is more desirable and more effective as the reinforcement. Clearly, results from the present study and the analytical process adopted can be useful to relevant structural analysis and design of reinforced cutouts in the AGS cylindrical shells. Acknowledgments
3.4. Response trends Reinforced ribs and skins with higher thickness, height or width generally lead to a better performance on stability of the AGS cylinders with cutouts. However, the increase in loading capacity for unit weight is complex. Thus, the efficiency of reinforcements versus weight-increasing could be used to exhibit the effects of each reinforcing configuration, as the structural weight is a key factor for this sort of structures. In Fig. 5 the normalized global collapse loads for AGS cylinders with rectangular or circle cutout reinforced by various configurations is described. As illustrated in Fig. 6, the normalized effects (either growth or drop) of reinforcing configurations on collapse loads are presented as normalized collapse load divided by normalized weight increase. The weight of reinforcements is normalized to the structural weight of complete AGS cylindrical shell without cutout. The results indicate that the grid reinforcements are marginally favored in comparison with the skin reinforcements as in Fig. 5. However, the advantage of grid reinforcements is more obvious when weight is considered (Fig. 6). The grid reinforcements can provide a higher bending stiffness, which is crucial to the structural stability. For the same weight, the skin reinforcements also reinforce less important area. Furthermore, shells with a circular cutout are a much referred choice than shells with a rectangular cutout if a cutout is unavoidable. As aforementioned, the global collapse loads for rectangularcut shells are affected by the stress concentration behavior caused by quasi-vertical corners. It is interesting to see that the efficiency of skin reinforcements for a circle-cutout is comparable to that of a comparable rectangular-cutout. While the efficiency of grid reinforcements for circle-cut shells is much higher than that of a comparable rectangular-cutout. In other words, grid reinforcements are desirable for the AGS cylinders with circle cutouts. Overall, G4 and G5 are the best reinforced configuration for the AGS cylinders with circle and rectangular cutout respectively, based on the present study.
The authors are grateful to the supports of Key National Science Foundation of China (Grant No. 90816025), National Basic Research Program of China (Grant No. 2011CB610304), Doctor Startup Research Foundation of Liaoning Province (Grant No. S08207) and the China Scholarship Council. References [1] Komur MA, Sen F, Akin, Atas, Arslan N. Buckling analysis of laminated composite plates with an elliptical/circular cutout using FEM. Adv Eng Softw 2010;41:161–4. [2] Jain P, Kumar A. Postbuckling response of square laminates with a central circular/elliptical cutout. Compos Struct 2004;65:179–85. [3] Kumar D, Singh SB. Postbuckling strengths of composite laminate with various shaped cutouts under in-plane shear. Compos Struct 2010;92:2966–78. [4] Kumar D, Singh SB. Stability and failure of composite laminates with various shaped cutouts under combined in-plane loads. Compos B: Eng 2012;43:142–9. [5] Xie D, Biggers SB. Postbuckling analysis with progressive damage modeling. Compos Struct 2003;59:199–216. [6] Hilburger MW, Britt VO, Nemeth MP. Buckling behavior of compression-loaded quasi-isotropic curved panels with a circular cutout. Int J Solids Struct 2001;38:1495–522. [7] Oh SH, Kim KS, Kim CG. An efficient postbuckling analysis technique for composite stiffened curved panels. Compos Struct 2006;74:361–9. [8] Tercan M, Aktas M. Buckling behavior of 1 1 rib knitting laminated plates with cutouts. Compos Struct 2009;89:245–52. [9] Tafreshi A. Buckling and post-buckling analysis of composite cylindrical shells with cutouts subjected to internal pressure and axial compression loads. Int J Press Vessels Pip 2002;79:351–9. [10] Han HP, Cheng JQ, Taheri F, Pegg N. Numerical and experimental investigations of the response of aluminum. Thin-Wall Struct 2006;44:254–70. [11] Yeh MK, Lin MC, Wu WT. Bending buckling of an elastoplastic cylindrical shell with a cutout. Eng Struct 1999;21:996–1005. [12] Schenk CA, Schueller GI. Buckling analysis of cylindrical shells with random geometric imperfections. Int J Non-Lin Mech 2003;38(7):1119–32. [13] Schenk CA, Schueller GI. Buckling analysis of cylindrical shells with cutouts including random boundary and geometric imperfections. Comput Methods Appl Mech Eng 2007;196(35–36):3424–34. [14] Guo S, Morishima R, Zhang X, Mills A. Cutout shape and reinforcement design for composite C-section beams under shear load. Compos Struct 2009;88:179–87. [15] Guo SJ. Stress concentration and buckling behaviour of shear loaded composite panels with reinforced cutouts. Compos Struct 2007;80:1–9.
S. Shi et al. / Composites: Part B 44 (2013) 26–33 [16] Guo S, Zhou L, Cheung CW. Cutout reinforcements for shear loaded laminate and sandwich composite panels. Int J Mech Mater Des 2008;4(2):157–71. [17] Hilburger MW, Starnes JH. Buckling behavior of compression-loaded composite cylindrical Shells with reinforced cutouts. Int J Non-Lin Mech 2005;40:1005–21. [18] Cervantes JA, Palazotto AN. Cutout reinforcement of stiffened cylindrical shells. J Aircraft 1979;16:203–8. [19] He JX, Ren MF, Huang QZ, Shi SS, Chen HR. Buckling behavior of compressionloaded advanced grid stiffened composite cylindrical shells with reinforced cutouts. Polym Polym Compos 2011;19(4&5):357–62.
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[20] Wang B, Wu LZ, Ma L, Feng JC. Low-velocity impact characteristics and residual tensile strength of carbon fiber composite lattice core sandwich structures. Compos B: Eng 2011;42:891–7. [21] Hong C et al. Novel phenolic impregnated 3-D Fine-woven pierced carbon fabric composites: microstructure and ablation behavior. Composites: Part B 2012;43:2389–94. [22] MSC. Patran PCL handbook, MSC. Patran PCL Workshop. USA; 2006.
Contents lists available at SciVerse ScienceDirect
Composites: Part B journal homepage: www.elsevier.com/locate/compositesb
Buckling response of advanced grid stiffened carbon–fiber composite cylindrical shells with reinforced cutouts Shanshan Shi a, Zhi Sun a,b, Mingfa Ren a,⇑, Haoran Chen a, Xiaozhi Hu b a b
State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian 116023, PR China School of Mechanical and Chemical Engineering, University of Western Australia, Perth, WA 6009, Australia
a r t i c l e
i n f o
Article history: Received 8 January 2012 Received in revised form 5 May 2012 Accepted 17 July 2012 Available online 16 August 2012 Keywords: A. Carbon fiber B. Buckling C. Finite Element Analysis (FEA) Reinforcing design
a b s t r a c t In this paper, we present the initial buckling and post-buckling responses of axial loaded advanced grid stiffened (AGS) composite cylindrical shells with reinforced rectangular or circular cutouts. The AGS cylindrical shells were reinforced by various local grid configurations near the cutout areas. The effects of different reinforcing grid configurations on critical loads were then examined and compared to those of different skin-reinforcing designs using Finite Element Analysis (FEA) simulations. A high-fidelity nonlinear analysis procedure was proposed to predict the non-linear buckling response of the shell structures. The simulation results indicated that the grid reinforcements can reduce or eliminate the risk of local buckling response near the cutout areas and increase the critical load of the shell more effectively than the skin reinforcements. Furthermore, those results showed that an optimum grid reinforcement configuration exists, which significantly improved the initial buckling and post-buckling resistance of the cylindrical shells under axial loading. The above findings can potentially be useful to the analysis and optimum design of AGS composite cylindrical shells with cutouts. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction Advanced gird stiffened (AGS) composite structures have been increasingly used in the field of aerospace, as they offer superior specific stiffness and strength. Compared to the traditional metallic structures, AGS counterparts are capable of providing much superior mechanical properties for the same structural weight. The structural weight has been considered as one of the key factors for development of AGS structures, since the launching weight and cost is a major consideration in the aerospace engineering. The AGS cylindrical shells studied in this paper are widely used in the aircraft and aerospace industries, for instance, the main fuselage structures of commercial aircrafts and the launch vehicle sections of rockets, which all mainly experience axial loading. In many cases, cutouts need to be introduced on the AGS cylindrical shells for the installations of doors, windows or access ports. Inevitably, those cutouts interrupt the continuous distribution of stress and strain within the AGS cylindrical shells, leading to the stress concentration at the cutout edge, and thus significantly reduce the buckling resistance of the AGS cylindrical shells. To avoid such potential buckling failure, reinforcements need to be placed around the cutout regions to reduce the stress and strain concentrations near the cutouts and thus the load capacity of AGS ⇑ Corresponding author. Tel.: +86 411 84708393; fax: +86 411 84709161. E-mail address: [email protected] (M. Ren). 1359-8368/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compositesb.2012.07.044
cylindrical shells with the reinforced cutouts is increased. This study deals with different reinforced cutout designs of the AGS cylindrical shells, and it is expected that the results will be useful to AGS shell design, manufacturing and structural integrity analysis. For decades, many studies have been conducted on the buckling and post-buckling characteristics of composite plates with cutouts under different loading and boundary conditions. Komur et al. [1] analyzed the buckling behavior of laminated composite plates with an elliptical cutout. Using Finite Element Analysis (FEA) simulations, the critical buckling loads of different composite plates with various elliptical holes of different shapes and at the different positions were determined. Jain and Kumar [2] investigated the post-buckling response of square composite plates with a central circular or elliptical cutout. The substantial factors such as cutout shape and size were considered and the post-buckling loads were obtained. Kumar and Singh [3,4] examined the buckling responses, particularly shear load and failure characteristics, of composite laminate plates with various shaped cutouts including circular, square, diamond, elliptical-vertical and elliptical-horizontal. Xie et al. [5] presented a thorough discussion on the tailoring concept which was proved to be effective in improving buckling loads and ultimate loads for plates with central circular cutouts. Hilburger et al. [6] used a geometrically non-linear FEA method to study the effects of plate curvature and initial geometric imperfection on the buckling response of compression-loaded curved composite
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S. Shi et al. / Composites: Part B 44 (2013) 26–33
Fig. 1. Geometry and FE model of the cylindrical shells with single cutout: (a) square cutout and (b) circular cutout.
plates with a circular cutout. Oh et al. [7] put forward an efficient post-buckling analysis technique for stiffened curved plates, and the computational time required was dramatically reduced with improved accuracy. Tercan and Aktas [8] investigated the effects of cutout shape on buckling of 1 1 rib knitting glass/epoxy laminated plates in three different knitting tightness levels, adding more to the vast literature on the cutout-affected buckling properties of the AGS shells. The buckling responses of cylindrical shells with various types of openings of different sizes were studied as well. Tafreshi [9] focused on the buckling and post-buckling responses of a series of cylindrical shells with different cutout sizes subjected to internal pressure and axial compression. Buckling modes were suggested, and the results agreed well with relevant published results. Numerical simulation and analysis were used by Han et al. [10] for the study of the influence of cutout size, cutout location and shell aspect radio on the buckling and post-buckling responses of the cylindrical shells. In addition, an experimental investigation on moderately thickwalled shells was also carried out [10]. The results predicted by general commercial FEA platform ANSYS were compared with the experimental results. Yeh et al. [11] investigated the buckling behavior of an elastoplastic cylindrical shell with a cutout under pure bending using both analytical and experimental methods. Schenk and Schueller [12,13] surveyed the individual and combined effects of random geometric imperfections and boundaries of isotropic thin-walled cylindrical shells under axial compression with rectangular cutouts using non-linear static FEA. Stress concentrations near cutouts and decreases in load capacity were observed in all the aforementioned plate and shell structures with cutouts. An effective approach to reduce the stress concentration near cutouts and increase the load capacity of composite shells with cutouts is to apply reinforcements around the cutout areas. A significant reduction in the stress and strain concentration was observed for appropriate cutout shapes and edge reinforcements in a composite C-section beam with diamond and circular cutouts under static shear load [14]. Guo [15,16] designed four different types of cutout reinforcements to evaluate the stress concentration and buckling behavior of composite plates under shear loading. Hilburger and Starnes [17] implemented a numerical study technique on the buckling behavior and non-linear response of compression-loaded composite cylindrical shells, in which the effects of cutout reinforcements, size and thickness were consid-
Table 1 Mechanical properties of skin and grid materials. Material property
E1, E2 (GPa)
m
G12 (GPa)
Skin and ribs
127.557, 11.308
0.3
5.998
ered. All those previous studies showed that reinforcements were capable to increase the load capacity of composite structures with cutout. However, only limited studies were conducted on the buckling behavior of stiffened composite cylindrical shells with cutouts. Cervantes and Palazotto [18] studied the optimum placement and volume of a reinforcing frame around a cutout for a stiffened metal cylindrical shell. The effects of skin reinforcements with different size, thickness or ply angle on AGS cylindrical shells with cutouts were considered and the buckling responses were compared with non-reinforced shells [19]. In this paper, seven grid configurations of reinforcements were adopted and analyzed around the cutout area for axial loaded AGS cylindrical shells with single square or circular cutout. The effects of different grid reinforcements on the initial buckling and post-buckling responses of these AGS shells were calculated by FE models. In addition, the predicted load capacities with the grid reinforcements were compared with the results obtained for AGS shells with skin reinforcements so that grid and skin reinforcements were evaluated.
2. The Finite element models of cylindrical shells with reinforced cutouts 2.1. Structure geometry and Finite Element models Cylindrical shell models designed for the FEA with a square and circle cutout are shown in Fig. 1a and b respectively. Each cylindrical shell has a radius R 800 mm, length L 1600 mm and thickness of skin t 2 mm. In Fig. 1 the geometric shapes and details of the structures are fully described. The width of square cutouts and the diameter of circle cutouts are defined by c and d respectively. A 250 mm 250 mm square cutout with 10-mm radius chamfer
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Table 2 Structure identification codes and corresponding reinforcements. Identification codes
Description
G0
No reinforcement
G1
Diamond grids reinforcement combined short transversal and axial ribs
G2
Diamond grids reinforcement combined two short transversal ribs
G3
Diamond grids reinforcement
G4
Hexagon grids reinforcement combined two short axial ribs
G5
Hexagon grids reinforcement
Schematic of configuration
S. Shi et al. / Composites: Part B 44 (2013) 26–33
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Table 2 (continued) Identification codes
Description
G6
Irregular grids reinforcement
Schematic of configuration
G7
Square grids reinforcement
S1–S5
Skin reinforcement
Table 3 Codes and corresponding geometrical parameters for skin reinforcements. Structure identification code
Reinforcement size, a (mm)
Reinforcement thickness (mm) (plies)
S1 S2 S3 S4 S5
502 502 502 502 502 502 408 408 314 314
2.5 (10) 5 (20) 8 (32) 2.5 (10) 2.5 (10)
are supported. Additionally, meticulous mesh process of triangular elements was carried out for the areas near cutouts, since the adjacent regions of cutouts were critical during FEA analysis. Furthermore, the boundary conditions and loads of the FE models are also illustrated in Fig. 1. The top and bottom edges of both models were simply supported. Meanwhile, a uniform displacement D was loaded by Multi-Point Constraint (MPC) technique [22], which applied the load on a central node of the top edge and constraint to other nodes at the top edges. 2.2. Configurations of reinforcement
corners is located at the mid-span of the AGS shell, as shown in Fig. 1a, while a circle cutout with a diameter of 250 mm locates at the same position in Fig. 1b. In other words, c and d both equal to 250 mm. In addition, since the cutout area suffers from stress concentration, the area of the around cutouts is meshed using smaller elements and parameters a and b are exploited to mark the size of the area with finer mesh. Except for the cutout area, there are 40 longitudinal ribs and 13 circumferential grids on the shells. In this study, each grid is 10 mm in height and 5 mm in width. The skins of cylindrical shells were modeled as geometrically perfect, and made by T700 carbon fiber/epoxy composites as mentioned in references [20,21]. Additionally, the thickness of each ply was 0.25 mm and shells were fabricated by a symmetric lay-up [0/45/45/90]s to a final skin thickness of 2 mm (8 plies). The mechanical properties of the skin and grid are given in Table 1. As can be seen in Fig. 1a and b, general skin areas and all grids were divided into quadrilateral elements, the reinforced area were divided into triangle elements. The elements were flat face-type based on the Kirchoff–Love shell theory and the non-linear Lagrangian strain tensor. Each element node has six degrees of freedom and the element was well suited for large deformation non-linear applications. Both full and reduced integration schemes
The analyzed configurations in this study included eight reinforcing configurations and each reinforcing configuration was used for structures with either square or circular cutouts. Meanwhile, the AGS shells with unreinforced cutouts were studied as well for comparison. These structures were referred as G0 (no reinforcement), G1–G7 (grid reinforcements) and S1–S5 (skin reinforcements). Each grid reinforcing configuration possessed different shapes, while the skin reinforcements were reinforced by axial composite laminates with different sizes and thicknesses. Tables 2 and 3 summarized the detailed descriptions for all the configurations. As shown in Table 2, solid lines, dashed lines and shade areas represented reinforcing grid, edges of cutout area and skin reinforcements respectively. The width b of grid reinforced area was 500 mm. And geometrical parameters of skin reinforcements were shown in Table 3. 3. Numerical results and discussions 3.1. Analysis approach PATRAN/NASTRAN was employed for the FEA. Eigenvalue buckling analysis was carried out to calculate the buckling resistance of
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S. Shi et al. / Composites: Part B 44 (2013) 26–33
AGS cylindrical shells without cutouts. Considering the influence of unstable post-buckling response on critical axial load, modified Newton–Raphson method was used to perform post-buckling analysis of structures, and large deformation equation was taken into account. The modified Newton–Raphson method works well in snap-through problems, and can also be used to solve problems both with stable and unstable post-buckling behaviors. Skin and rib local buckling, global buckling and other forms of buckling possibly occur during the loading process on AGS cylindrical shell structures. Meanwhile, there will possibly be large deformation and large rotation in the structures during the post-buckling process. Therefore, considering large deformation and large rotation, the author analyzed buckling and post-buckling responses for axial loaded AGS cylinders with cutout, employing the modified Newton–Raphson method. 3.2. Cylindrical shells with unreinforced rectangular or circular cutouts The results for the cylindrical shells with single rectangular or circular cutout referred to herein as shell G0 are shown in Fig. 2, where the load is plotted against the end-shortening displacement. The solid curve refers to the cylindrical shell with a circular cutout and the dashed curve refers to the cylindrical shell with a rectangular cutout. To illustrate a clear description, the compression load P was normalized to Pcr = 3.91 106 N (the simulated critical buckling load of the present AGS cylindrical shell without cutout), while the end-shortening displacement D was normalized to the structure length L. The normalized load–displacement curves indicate an overall tendency that an AGS shell with a circular cutout had a higher load capacity than an AGS shell with a rectangular cutout at the same displacement. It is thus suggested that the higher stress concentration behavior caused by quasi-vertical corners of the rectangular cutout will further reduce the load capacity of the structures. Both shells with rectangular and circle cutouts experience linear
prebuckling responses during the initial loading process. There is an obvious local buckling (marked as A) for shell with rectangular cutout at the load level of P/Pcr = 0.31. The displacement and stress nearby the cutout area varied dramatically while the overall stiffness of shell degenerate slightly during the following local buckling period. The complex displacement and stress fields caused by local buckling respond to fluctuation of the load–displacement curve between the local buckling point and global buckling point. Typical shell with circular cutout in present study do not comport an evident local buckling. The load capacities for shells with rectangular and circular cutouts achieves the peak at P/Pcr = 0.43 and P/Pcr = 0.48 respectively. The displacement cloud pictures for both shells exhibit that buckling behaviors extends from grid cells nearby the cutout to a broader area at the post-buckling section. The snap-through phenomenon following global buckling responses results in a significant reduction in the axial load capacity of the shells. In the post-buckling section, load transmission switches from axial to bending which thereby reduces the effective loading-carrying cross-section. The post-buckling displacement mode gradually varies from single buckling waves around the cutouts to several waves flooding along the whole cylindrical shells. Overall, the axial load-shortening response for the typical AGS cylindrical shells with rectangular and circle cutouts exhibit complete local buckling, global buckling and post-buckling responses. 3.3. Cylindrical shells with grid or skin reinforced rectangular and circular cutouts In the previous section, typically buckling responses for the cylindrical shells with single rectangular or circular cutout referred to herein as G0 were discussed. Moreover, the larger displacements and local buckling behaviors could be found around the cutout areas for AGS cylindrical shells, especially for shells with rectangular cutouts. Based on those results, it is suggested that an optimum
Fig. 2. Compression load-shortening response curve of AGS cylindrical shell with unreinforced cutout (G0).
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S. Shi et al. / Composites: Part B 44 (2013) 26–33
0.6
0.5
0.4
(a)
G0 G1 G2 G3 G4 G5 G6 G7
0.5
0.3
0.2
0.4
0.3
0.2
0.1
0.1
0.0 0.000
G0 G1 G2 G3 G4 G5 G6 G7
0.6
Axial load, P/ Pcr
Axial load, P/Pcr
(a)
0.001
0.002
0.003
0.0 0.000
0.004
0.001
0.002
(b) 0.6
(b)
s1 s2 s3 s4 s5
0.4
0.3
0.2
0.004
0.005
0.004
0.005
s1 s2 s3 s4 s5
0.4
0.3
0.2
0.1
0.1
0.0 0.000
0.6
0.5
Axial load, P/ Pcr
Axial load, P/Pcr
0.5
0.003
End-shortening, Δ/L
End-shortening, Δ/L
0.001
0.002
0.003
0.004
0.0 0.000
0.005
0.001
0.002
reinforcement can be identified to enhance the stiffness and strength in this area, which can reduce or eliminate the risk of local buckling response near the cutout areas and increase the load capacity of the shells. Therefore, a range of buckling and post-buckling simulation studies were processed to demonstrate the diversity between normal skin reinforcement and certain grid reinforcing configurations selected in Section 2.2. The reinforcing configurations consisted of special grid reinforcing frames near the cutout area, which were added to inner side of the skin laminates. Similar to the shells with unreinforced cutouts, the compression load P was normalized to Pcr = 3.91 106 N, while the end-shortening displacement D was normalized to the structure length L. The normalized load–displacement curves for reinforced shells with rectangular cutouts and circle cutouts were indicated respectively. For shells with rectangular cutouts, the load–displacement curves for grid-reinforcements and skin-reinforcements are indicated in Fig. 3a and b. Clearly, G5 performs a significant rise in the global collapse load compared with unreinforced shell. The result indicates that grid configuration G5 expressed to be smoother in load transmission, because its geometric composition is favorable in the improvement process of AGS cylindrical shells with rectangular cutout. Structure reinforced by configuration G6, however, is even weaker than original shell and lost load capacity at lower compression loading displacement. This suggests that reinforcement without smooth geometry might increase the effect of local buckling and reduce collapse load. The others structures reinforced by grid or skin configurations experience relatively close results, while the structures with grid reinforcements exhibit little
Fig. 4. Compression load-shortening response curve of AGS cylindrical shell with circle cutout: (a) with grids reinforcement and (b) with skin reinforcement.
higher collapse loads than structures with skin reinforcements on average. In addition, there are no obviously local buckling responses for reinforced structures. Furthermore, the load–displacement curves for grid-reinforcements and skin-reinforcements of AGS shells with circle cutouts are indicated in Fig. 4a and b. The AGS shell reinforced by configuration G4, which is similar to the grid configuration G5 – the best configuration for shells with rectangular cutouts in present study, exhibits to own the highest global collapse load. The results further
rectangular circular 0.6
Axial load, P/ Pcr
Fig. 3. Compression load-shortening response curve of AGS cylindrical shell with rectangular cutout: (a) with grids reinforcement and (b) with skin reinforcement.
0.003
End-shortening, Δ/L
End-shortening, Δ/L
0.4
0.2 g1
g2
g3
g4
g5
g6
g7
s1
s2
s3
s4
s5
Configuration Fig. 5. Normalized global collapse loads for AGS cylinders with rectangular or circular cutout reinforced by various configurations.
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4. Conclusions 6000
rectangular circular
Efficiency
3000
0
-3000
-6000
g1
g2
g3
g4
g5
g6
g7
s1
s2
s3
s4
s5
Configuration Fig. 6. Reinforcing effectiveness normalized by weight increase for AGS cylinders with rectangular or circular cutout reinforced by various configurations.
indicate that configurations which could transmit load smoothly are competitive. The overall buckling responses illustrate that shells with circle cutouts could undertake more load than shells with rectangular cutouts. In addition, the results of AGS shells with grid and skin reinforced circle cutouts less scatters, so it is suggested that the circle-shape cutout to be a more desirable form of cutouts for the AGS cylindrical shells. Moreover, grid reinforcements are more effective than skin reinforcements.
In this paper, a FEA model has been established for the investigation of pre-buckling, local buckling and global buckling responses of the advanced composite grid stiffened cylindrical shells with unreinforced or reinforced cutouts under axial compressive loading. The resultant local buckling and post-buckling responses indicate that the initial buckling load underestimates the global bucking load of the AGS shells with cutouts. Therefore, post-buckling analysis and design is necessary. For unreinforced structures, the simulated results illustrate that an AGS shell with a circular cutout possesses a higher global collapse load than that of an AGS shell with a rectangular cutout due to the obvious difference in the cutout geometry and thus associated stress concentration. More specifically, the stress concentration behavior induced by quasi-vertical corners leads to premature local buckling response around the cutout area, resulting in a noticeable reduction in the global load capacity of the AGS shells. Different grid and skin reinforced configurations have also been investigated and compared in the present study. The simulation results indicate that a grid reinforced configuration which can transmit loading smoothly is more desirable and more effective as the reinforcement. Clearly, results from the present study and the analytical process adopted can be useful to relevant structural analysis and design of reinforced cutouts in the AGS cylindrical shells. Acknowledgments
3.4. Response trends Reinforced ribs and skins with higher thickness, height or width generally lead to a better performance on stability of the AGS cylinders with cutouts. However, the increase in loading capacity for unit weight is complex. Thus, the efficiency of reinforcements versus weight-increasing could be used to exhibit the effects of each reinforcing configuration, as the structural weight is a key factor for this sort of structures. In Fig. 5 the normalized global collapse loads for AGS cylinders with rectangular or circle cutout reinforced by various configurations is described. As illustrated in Fig. 6, the normalized effects (either growth or drop) of reinforcing configurations on collapse loads are presented as normalized collapse load divided by normalized weight increase. The weight of reinforcements is normalized to the structural weight of complete AGS cylindrical shell without cutout. The results indicate that the grid reinforcements are marginally favored in comparison with the skin reinforcements as in Fig. 5. However, the advantage of grid reinforcements is more obvious when weight is considered (Fig. 6). The grid reinforcements can provide a higher bending stiffness, which is crucial to the structural stability. For the same weight, the skin reinforcements also reinforce less important area. Furthermore, shells with a circular cutout are a much referred choice than shells with a rectangular cutout if a cutout is unavoidable. As aforementioned, the global collapse loads for rectangularcut shells are affected by the stress concentration behavior caused by quasi-vertical corners. It is interesting to see that the efficiency of skin reinforcements for a circle-cutout is comparable to that of a comparable rectangular-cutout. While the efficiency of grid reinforcements for circle-cut shells is much higher than that of a comparable rectangular-cutout. In other words, grid reinforcements are desirable for the AGS cylinders with circle cutouts. Overall, G4 and G5 are the best reinforced configuration for the AGS cylinders with circle and rectangular cutout respectively, based on the present study.
The authors are grateful to the supports of Key National Science Foundation of China (Grant No. 90816025), National Basic Research Program of China (Grant No. 2011CB610304), Doctor Startup Research Foundation of Liaoning Province (Grant No. S08207) and the China Scholarship Council. References [1] Komur MA, Sen F, Akin, Atas, Arslan N. Buckling analysis of laminated composite plates with an elliptical/circular cutout using FEM. Adv Eng Softw 2010;41:161–4. [2] Jain P, Kumar A. Postbuckling response of square laminates with a central circular/elliptical cutout. Compos Struct 2004;65:179–85. [3] Kumar D, Singh SB. Postbuckling strengths of composite laminate with various shaped cutouts under in-plane shear. Compos Struct 2010;92:2966–78. [4] Kumar D, Singh SB. Stability and failure of composite laminates with various shaped cutouts under combined in-plane loads. Compos B: Eng 2012;43:142–9. [5] Xie D, Biggers SB. Postbuckling analysis with progressive damage modeling. Compos Struct 2003;59:199–216. [6] Hilburger MW, Britt VO, Nemeth MP. Buckling behavior of compression-loaded quasi-isotropic curved panels with a circular cutout. Int J Solids Struct 2001;38:1495–522. [7] Oh SH, Kim KS, Kim CG. An efficient postbuckling analysis technique for composite stiffened curved panels. Compos Struct 2006;74:361–9. [8] Tercan M, Aktas M. Buckling behavior of 1 1 rib knitting laminated plates with cutouts. Compos Struct 2009;89:245–52. [9] Tafreshi A. Buckling and post-buckling analysis of composite cylindrical shells with cutouts subjected to internal pressure and axial compression loads. Int J Press Vessels Pip 2002;79:351–9. [10] Han HP, Cheng JQ, Taheri F, Pegg N. Numerical and experimental investigations of the response of aluminum. Thin-Wall Struct 2006;44:254–70. [11] Yeh MK, Lin MC, Wu WT. Bending buckling of an elastoplastic cylindrical shell with a cutout. Eng Struct 1999;21:996–1005. [12] Schenk CA, Schueller GI. Buckling analysis of cylindrical shells with random geometric imperfections. Int J Non-Lin Mech 2003;38(7):1119–32. [13] Schenk CA, Schueller GI. Buckling analysis of cylindrical shells with cutouts including random boundary and geometric imperfections. Comput Methods Appl Mech Eng 2007;196(35–36):3424–34. [14] Guo S, Morishima R, Zhang X, Mills A. Cutout shape and reinforcement design for composite C-section beams under shear load. Compos Struct 2009;88:179–87. [15] Guo SJ. Stress concentration and buckling behaviour of shear loaded composite panels with reinforced cutouts. Compos Struct 2007;80:1–9.
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