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Orientation and size effect of a rectangle cutout on the buckling of composite cylinders
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Orientation and size effect of a rectangle cutout on the buckling of composite cylinders
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School of Mechanical Engineering, Iran University of Science and Technology, P.O.B. 1684613114, Tehran, Iran
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Article history: Received 19 May 2018 Received in revised form 7 November 2018 Accepted 28 February 2019 Available online xxxx
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Abolfazl Shirkavand, Fathollah Taheri-Behrooz ∗ , Milad Omidi
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Keywords: Composite shells Nonlinear buckling Initial imperfections LBMI Square shaped cutouts Rectangular cutouts
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In this article the effect of a rectangular cutout on the buckling behavior of a thin composite cylinder was investigated using numerical and experimental methods. To verify the finite element results, a limited number of tests was carried out on perforated and non-perforated glass/epoxy cylinders with [90/−23/23/90] layups. In the numerical analysis, linear and nonlinear approaches were employed to study the effect of initial imperfections on the buckling of the cylinders. Several key findings including the effects of cutout size and orientation, and the mutual effects of the cutout and initial imperfections on the buckling behavior were investigated in detail. In the presence of cutouts, the effect of initial imperfections on the buckling load is a function of the cutout size. In cylinders with rectangular cutouts, buckling analysis revealed that a rectangular cutout in the circumferential direction causes around 8% more reduction in the buckling load than the same cutout in the axial direction. Also, numerical findings illustrated that elastic stress concentration factors for the circumferential cutouts are much greater than those for the axial cutouts; thus premature failure around the cutout will trigger earlier buckling in the cylinder with circumferential cutouts. © 2019 Elsevier Masson SAS. All rights reserved.
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1. Introduction
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Inclusion of cutouts in the construction of cylindrical shelltype structures is required to enable access to the inner parts of the structure for maintenance or installation of internal systems and other such usages. Based on the function of such cylinders in industry, cutouts can be made in various geometries like circle, square, rectangle, etc. It is of crucial importance to examine the buckling phenomenon because of the axial loading to optimize the usage of such cylinders. In the late 19th century, a great number of numerical, analytical and experimental researches on composite cylindrical shells were conducted, and attention to the differences between theoretical predictions and the experiments was paid constantly. In fact, experimental results always predict the buckling load of cylinders lower than the numerical and analytical results. In 1945, Koiter et al. [1] realized that the difference between such results is due to the presence of imperfections in structure. Imperfections like non-uniform loading, initial defects in property and imperfect boundary conditions. In 1968, NASA released a report named NASA SP-8007, which presented design guidelines by many experiments on metallic cylinders [2]. To this end, this report is
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Corresponding author. E-mail address: [email protected] (F. Taheri-Behrooz).
https://doi.org/10.1016/j.ast.2019.02.042 1270-9638/© 2019 Elsevier Masson SAS. All rights reserved.
still being used, but this design is very conservative and predicts the buckling load much lower than the experiments also makes the structure heavier which is not ideal especially in aerospace industries. Cutouts on cylinders play a role as an imperfection and cause a large drop in the buckling load of the structure. Hu and Wang [3] investigated fiber composite laminate shells with and without cutouts to optimize the buckling resistance of cylinders. They studied three important optimizing problems as: 1) optimization of fiber orientations to maximize the buckling resistance of composite shells without cutouts; 2) optimization of fiber orientations to maximize the buckling resistance of composite shells with circular cutouts; and 3) optimization of cutout geometry for maximizing the buckling resistance of a composite shell. The effect of imperfections on the buckling of such cylinders without cutouts under axial compression was investigated by Hilburger and Starnes [4] using analytical and experimental methods. The results identify the effects of traditional and non-traditional initial imperfections on the nonlinear response and buckling load of shells. Hilburger [5] presented a design criterion for composite shells buckling based on manufacturing imperfection signatures by measuring initial geometric imperfections in some graphite/epoxy shells. A numerical and an experimental study was performed by Hilburger and Wass [6] to demonstrate the laminate orthotropy on the buckling of composite cylindrical shells with a square cutout under axial
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compressive loading. Tafreshi [7] carried out a numerical study to investigate the buckling behavior of composite cylinders subjected to internal pressure and axial compression. She studied the effects of different sizes of rectangular and square shaped cutouts on the buckling behavior of such cylinders, but she did not mention the relation between imperfections and cutout size on the buckling load of cylinders. Orifici and Bisgani [8] have conducted a numerical research on both monolithic composite laminate and sandwich constructions with and without different sizes for square shaped cutouts. In their research, imperfections were applied to the cylinder by single perturbation load approach (SPLA). The results illustrated that the effect of small and large cutouts is analogous to the effect of small and large perturbation loads, respectively. In 2014 Arbelo et al. [9] investigated the effect of circular cutouts and imperfections on the buckling of cylinders was studied. The results extracted from the numerical analysis predicted the buckling load for cylinders with circular cutout under axial compression in three zones. It was concluded that the effect of imperfections is a function of cutout diameter to the cylinder radius. Taheri-Behrooz et al. [10,11] used a stochastic procedure to modify linear buckling mode-shape imperfection (LBMI) and SPLA methods for the buckling analysis of the composite cylinders with and without circular cutouts. They carried out buckling tests on glass/epoxy cylinders to verify their numerical findings. Yan et al. [12] investigated the axial compression behavior of a composite cylindrical shell with an opening by experiment and the finite element method. Stability and post-buckling response of sandwich pipes under external pressure was studied by Arjomandi and Taheri [13]. Taheri-Behrooz et al. [14] studied the response of perforated composite tubes subjected to axial compressive loading using numerical and experimental methods. Shokrieh and Nouri [15] investigated post buckling analysis of shallow composite shells based on the third order shear deformation theory. They derived a nonlinear mathematical model using Green-Lagrange type geometric nonlinearity which may not require the shear correction factors. Hoang et al. [16] studied stochastic buckling behavior of laminated composite structures with uncertain material properties and represented a spectral representation method based on Monte Carlo simulation and Iso Geometric Analysis method to evaluate effects of such uncertain material property on the buckling response. Cestino et al. [17] employed numerical/experimental methods to evaluate buckling behavior and residual tensile strength of composite aerospace structures. Although a lot of researches have studied the effect of cutouts on the buckling behavior of composite cylinders, there is a lack of information regarding the effect of cutout direction on the buckling of cylinders. In other words, it has been shown that the presence of a rectangular cutout on a cylinder body would reduce its compressive strength, regardless of the cutout direction. However, if the cutout is aligned in the circumferential direction (θ ) the amount of the cylinder’s strength reduction will be 2–3 times more than that of a cutout in the axial direction (Z). In the current manuscript, the buckling behavior of composite cylindrical shells with and without rectangular and square shaped cutouts under axial compression loading is investigated using numerical and experimental procedures. The effects of different cutout shapes and sizes on the buckling behavior of such cylinders are examined. Initial imperfections are applied to the models using LBMI method, and their effects on the buckling loads are also discussed. Validation is conducted by tests on composite cylinders made from glass/epoxy composites using filament winding method with and without cutouts. The effect of cutout size and orientation on the buckling loads and the static stress concentration factors is investigated numerically, and results are presented in suitable design charts. It is the first time the effect of rectangular cutouts size
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Table 1 Mechanical properties of glass/epoxy composites [10].
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Property
Symbol
Value
Elastic modulus in x-direction Elastic modulus in y-direction In-plane shear stress Poisson’s ratio Fiber volume fraction
Ex Ey E xy
35.5 GPa 5.4 GPa 4.085 GPa 0.28 56%
νxy Vf
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Table 2 Dimensions of the specimens (values in mm).
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Code
Cutout length
Shell length
Internal diameter
Thickness
Number of specimens
A D50 D80
– 50 80
700 ± 2 700 ± 2 700 ± 2
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1.5 1.5 1.5
3 2 2
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Fig. 1. Schematic illustration of the specimen geometry.
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and orientation on the buckling behavior of cylinders has been addressed.
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2. Specimen definition
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All cylindrical shells are made using E-glass fiber (1200 Tex direct roving), Araldite LY 556 resin and HY917 hardener using filament winding method. The composite plies are laid up to form four-ply laminates having [90/−23/23/90] stacking sequences. The mechanical properties of the composite layers are given in Table 1 [10], and the nominal dimensions of the models are presented in Table 2. The cylindrical shells had a free length of 700 mm and an internal diameter of 378 mm. Each layer has a thickness of 0.375 mm and overall a 4-layer composite is obtained which has a thickness of 1.5 mm. Cylindrical shells with and without cutout are tested in the current research. Three groups of specimens with a square shaped cutout on their surface and one group of specimens without cutout were fabricated and tested during this research. The cutout dimensions are presented in Table 2. A schematic illustration of the geometry of the specimens is shown in Fig. 1.
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Fig. 2. A schematic illustration of the fixture (left) and reinforced edge of the shell before cutting (right).
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Fig. 3. a) Strain gauges on the shell with the cutout. b) Strain gauges on the shell without the cutout.
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3. Testing procedure
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3.1. Bucking test set up
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For the buckling test, an STM-150 testing machine provided by SANTAM Co. with 15 tons of loading capacity was employed. The test is conducted in displacement control mode, and speed of the axial loading to the top of the shell is 0.01 mm/s. A metallic fixture is designed to provide nearly ideal boundary conditions and prevent the imperfections to affect the test procedure and buckling loads. As shown in Fig. 2, there is a 3 mm deep groove in the surface of both the top and bottom parts of the loading fixture. Both sides of the cylinder were placed in the mentioned groove to develop a clamped boundary condition. A schematic of different parts of the fixture is shown in Fig. 2a. Also, two glass/epoxy strips with 5 cm of width and 1.5 mm of thickness are used to prevent premature damage in cylinder edges (Fig. 2b). Electrical strain gauges are used to survey the exact behavior of the shell near the buckling point also to verify the uniformity of the loading. For specimens with a cutout, two bidirectional strain gauges are bonded near the cutout to extract the strains value in the axial and the circumferential directions in the top and beside of the cutout (Fig. 3a). For specimens without a cutout, two unidirectional strain gauges are bonded to the front and the back of the shell as shown in Fig. 3b. To extract strains from the strain gauges, a TMR-211 data logger manufactured by TML Co. is employed. Also to ensure the uniformity of the introduced loading on the shell, two dial gauges are used on top of the loading plate as depicted in Fig. 4. The gauges show the value of displacement in 2 points on the top part of the fixture. The vertical displacement values measured by these gauges at the top of the fixture verified the uniformity of the loading during the buckling test.
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Fig. 4. Dial gauges to measure the vertical displacement of the top part of the fixture.
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3.2. Buckling test results
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Buckling tests are conducted in 4 steps; first three specimens without cutouts named A1, A2, and A3, are tested, and their load-displacement response curves are recorded. Then three other groups of specimens with square shaped cutouts of 50, 80 and 100 mm sizes named D50, D80 and D100 are tested, and their corresponding response curves are recorded too. A representative response curve of each group is presented in Fig. 5. The nonlinearities illustrated in the first part of the load-end shortening response curves (Fig. 5) could be due to the initial dislocations or the friction between different parts of the test fixture and test samples. After applying about 10 kN load on the samples, the slope of the diagrams became constant, and diagrams are nearly linear up to the buckling point. Reduction in the buckling load can be observed in the response curves by increasing the ge-
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Fig. 7. Specimen D50 a) before and b) after buckling. Table 3 Experimental values of buckling load. Specimen
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Fig. 5. Representative load-end shortening curves of all specimens.
A1 D50 D80 D100
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Cutout length (mm)
Buckling load (kN)
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Fig. 8. Recorded axial strains of specimen A1.
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Fig. 6. Specimen A1 a) before and b) after buckling.
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ometrical imperfections. It means that growing the cutout length leads to a decrease in the buckling load. For the specimen without a cutout (A1), the buckling load is about 69 kN. By creating a 50 mm square shaped cutout on the surface of the shell, a big reduction of 38% happened in the buckling load compared to the specimen without the cutout. The buckling load values for D50, D80, and D100 specimens are about 39, 32 and 27 kN. Recorded buckling loads of various cylinders are tabulated in Table 3. Distortion in the surface of the shell can be observed at the buckling moment of the cylinder without cutout as shown in Fig. 6. In the cylinders with a cutout, initially, a local buckling was observed around the cutout which was spread to the entire wall of the shell by increasing the axial compressive load as shown in Fig. 7. After unloading the shells, the structure of specimens turns back to their initial state approximately. Bonded strain gauges on the surface of the specimen A1, show the value of strains in the axial direction. The recorded values of
strain by the back, gauge 1 and front, gauge 2 are nearly identical and the maximum difference between them is about 2%. The negative strain shows compressive loading on the structure. Fig. 8 presents a polynomial trend line for the data extracted from strain gauges 1 and 2 up to the buckling point of the cylinder without cutout which confirms the accuracy and uniformity of the axial loading. The buckling behavior of imperfect shells can be discussed considering the values of strain obtained from attached gauges on the surface of the shells near the cutout as shown in Figs. 9 and 10. There are 2 strain gauges along the axial direction and 2 strain gauges along the circumferential direction of the shell near the cutout. Strain gauges 1 and 3 show strain values along the axial direction and strain gauges 2 and 4 show strain values along the circumferential direction. In Fig. 10, a sudden shift happens near the 0.6 point on the vertical axis in the values obtained from strain gauge 2 due to the developed curvature in the surface of the shell. Strain gauge 1 shows a nearly linear behavior for axial loading up to the buckling point. Also in Fig. 11, the local structural behavior of a point next to the cutout can be inspected. By careful examination of the two diagrams, an approximately symmetric behavior can be observed which expresses that a symmetric curvature is created near the cutout in the structure. The positive value of strain gauge 4 is due to the outward motion of the surface of the shell near cutout by increasing the axial compressive loading. A more accurate picture of the buckling moment before unloading specimen D50 is shown in Fig. 11. A buckling mode with different half-waves can be observed on the surface of the shell and near the cutout due to local buckling.
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Table 4 Dimensions of the specimens (values in mm).
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Cutout length
Height
Internal radius
Thickness
Without cutout With cutout
–
–
700
189
1.5
Square Rectangle
D = 0 to 150 a, b = 0 to 150
700 700
189 189
1.5 1.5
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Fig. 9. Axial and circumferential strains at the top of the cutout for specimen D50.
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Fig. 12. Boundary conditions [10].
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Fig. 10. Axial and circumferential strains near the cutout of specimen D50.
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given in Table 2. Three groups of numerical models are created and investigated. The first group is cylinders without cutouts to find their original buckling load. The second group consists of cylinders with different sizes of square shaped cutouts from 10 ∗ 10 mm to 150 ∗ 150 mm. The last group is made from cylinders with different sizes of rectangular cutouts in the axial and circumferential directions to evaluate the effect of cutout width to length ratio on the buckling behavior of such composite cylinders. For this kind of cutout, at first, the length of the cutout in the axial direction was increased while its width in the circumferential direction was kept constant. Then its length along the circumferential direction was increased while its width in the axial direction was kept constant. Parameters ‘a’ and ‘b’ are considered for the rectangle length in axial and circumferential directions, respectively, as shown in Fig. 1. A summary of shells and cutouts dimensions is presented in Table 4. 4.2. Finite element analysis and results
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Fig. 11. The buckling moment of specimen D50.
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4. Numerical simulations
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The buckling responses of shells with and without cutouts are numerically investigated by finite element method using ABAQUS version 6.13. At the first step cylinders used in the experimental section are simulated numerically to verify the accuracy of the FE analysis. In the second step, the effects of various parameters such as cutout size and direction on the buckling response are investigated numerically.
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4.1. Finite element models definition
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Mechanical and geometrical properties of the cylinders simulated for numerical analysis are similar to the specimens tested in the experimental section. A 4-layer glass/epoxy composite laminate with the same stacking sequence and properties as the test specimen is assigned to the cylinder numerical model. Cylinder free length and internal radius are 700 mm and 189 mm, respectively. The thickness of each layer is 0.375 mm which results in a total shell thickness of 1.5 mm. The layer elastic properties are
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To calculate the buckling load of different models, linear and nonlinear approaches are employed. The linear solver is the buckle step in ABAQUS and predicts the linear buckling loads and corresponding mode shapes. The nonlinear method is a dynamic/implicit step in ABAQUS that provides a condition more similar to the test procedure and considers the nonlinear geometry effects. The solvers for buckling analysis in the current research are chosen according to many studies such as [6,10]. The boundary conditions are clamped in the bottom and top parts of the shells except for axial displacement at the top which is free as shown in Fig. 12. For the present investigation, the standard large-strain shell elements are appropriate. A 4-node doubly curved thin shell, reduced integration element is used for meshing the structure which is named S4R in ABAQUS. For models without a cutout, a convergence test is conducted through the entirety of the structure, and a mesh size of 13 mm and a shell elements count of 5278 are used for the perfect model. Convergence of results with different mesh sizes is shown in Fig. 13. For models with a cutout, in addition to total convergence analysis, an analysis should be conducted near the cutout due to the local buckling which affects the results. For this purpose, partition-
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Fig. 13. Appropriate mesh size of the perfect cylinder.
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the structures by the LBMI method is investigated for the shells, and the relation between cutout dimensions and imperfections and their effect on the buckling load is presented. The LBMI applies a scaled factor of the linear buckling modes to the cylinders as initial imperfections. The two main questions of this approach which should be carefully addressed are a) which mode shape is appropriate for the cylinder and b) what is the most acceptable scaling factor for different cylinders. The effect of the linear buckling mode shape and scaling factor has been fully investigated in many studies, and the results have been presented clearly [10]. The appropriate mode shape and the acceptable scaling factor are presented as mode number 24 and 18% of shell thickness for perfect cylinders. For perforated cylinders, these quantities are mode number 14 and 18% of shell thickness, as comprehensively investigated in [10].
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Fig. 14. Mesh details around cutout.
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ing should be done around the cutout to increase the mesh density by a bias ratio. Also, a convergence analysis should be conducted for bias ratio and element size near the cutout. The appropriate element number on the edge of cutout and bias ratio are 30 and 5, respectively, and in total 5289 shell elements are used to analyze the imperfect models [10]. Cylinders with cutout are shown in Fig. 14 after the meshing procedure. At first, a linear solution is obtained for shells to extract the different buckling eigenvalues and mode shapes. These buckling modes are used for applying the imperfections by the LBMI method. In many investigations, the LBMI approach is used to put the initial imperfections on the shells [6,10]. In the present study, the sensitivity of buckling load to initial imperfections applied to
4.2.1. FE analysis of cylinders with a square shaped cutout on the surface In the presence of various cutouts on the cylinder body the effect of geometrical imperfections on the buckling load is unclear and needs to be investigated. Composite cylinders with a square shaped cutout are studied by increasing the cutout length from 0 to 150 mm. The associated buckling loads are extracted using nonlinear methods with and without imperfections. As mentioned, cylinders have a 700 mm free length and 189 mm of internal radius and a stacking sequence of [90/+23/−23/90]. Many linear and nonlinear analyses with and without imperfections are conducted for D/R ratios ranging from 0 to 0.8, where D is the length of the square shaped cutout and R is the internal radius of the cylinder. Fig. 15 shows the reaction force-end shortening response curves of the cylinder for different values of D/R ratio which are obtained by nonlinear methods without applying imperfections. Depending on Fig. 15, the global stiffness of the structure was reduced by increasing the cutout size. Also, for D/R ratios more than 0.4 there was no clear sudden drop in the values of the reaction forces. Also, the buckling load was decreased by raising the cutout dimension up to D/R = 0.16, called C1 , after which the loads do not change much and remain nearly constant. This fact is illustrated in Fig. 16 more clearly for wider ranges of D/R ratios. All nonlinear analyses of the perfect cylinder are repeated for the imperfect cylinder using the LBMI method; results are presented in Fig. 16. In fact, this figure shows the relation between imperfections and cutout dimensions and their effect on P/Pcr ratio for different cylinders, where P is the buckling load of the perforated cylinder, and Pcr is the buckling load of the cylinder without cutout and imperfections.
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Fig. 15. Reaction load end shortening response curves for various D/R ratios.
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Fig. 18. Cylinders with a rectangular cutout in (a) circumferential and (b) axial directions.
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Fig. 17. The buckling mode shape of cylinder D50 predicted by a) test and b) the LBMI method.
axial and circumferential directions as shown in Fig. 18. Parameters ‘a’ and ‘b’ are considered as rectangle dimensions in the axial and circumferential directions, respectively. One dimension of the cutout was kept constant at 50 mm, and the other one was increased from 0 up to 150 mm. The effect of cutout size growth in the axial and transverse directions on the buckling behavior of the cylinder is investigated in detail numerically.
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Depending on Fig. 16, the effect of initial imperfections applied by the LBMI method on the buckling load was decreased by increasing the length of the cutout. On the other hand, for D/R ratios between 0 and 0.16, there was on average a 23% drop in the reaction force due to initial imperfections. Also, for 0.16 < D/R < 0.46 (D/R = 0.46 called C2 ) and for 0.46 < D/R < 0.8 the reduction in reaction force was about 7% and 1% on average, respectively. In fact, for D/R < 0.16, predicted buckling loads using the LBMI method are close to the experimental results. Conversely, predicted results without considering the imperfections deviate greatly from the experiment. So in this range, both the cut out, and the geometrical imperfections are responsible for the buckling of the cylinders. For D/R > 0.16, the cutout effect is dominant, which means that the initial imperfections are not impactful on the buckling loads. Also, geometrical imperfections effect is negligible for D/R ratios more than 0.46. Fig. 17 presents a comparative illustration of cylinder D50 buckling patterns recorded from the experiment and nonlinear finite element analysis.
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4.2.2. FE analysis of cylinders with a rectangular cutout on the surface To verify the effect of a rectangular cutout orientation on the buckling load of the composite cylinders two groups of the FE models are created using ABAQUS 6-13 and investigated in this section. A primitive hole of 50 ∗ 50 mm dimensions is considered in the middle of the shell. The cutout length is increased along
4.2.2.1. Results of the cylinder with a rectangular cutout in the axial direction In this section, the cutout length was increased in the axial direction, and its effect on the buckling load was investigated. Nonlinear approaches are employed, and the imperfections are applied to the structure using the LBMI method with an appropriate scaling factor and mode shape which were explained in the first part of the paper. The P/Pcr versus a/R ratios are illustrated in Fig. 19, where ‘a’ is the length of the cutout in the axial direction and ‘R’ is the internal radius of the cylinder. As shown in Fig. 19, for a/R ratios less than 0.07, reduction in the reaction forces of the perfect and imperfect cylinder is about 20% on average, and for 0.07 < a/R < 0.37, the difference between reaction forces is about 8%. Also, for a/R ratios more than 0.37 the drop in the buckling load caused by imperfections is about 3% which is negligible (a/R = 0.07 is denoted by C1 , while a/R = 0.37 is denoted by C2 ).
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4.2.2.2. Results of the cylinder with a rectangular cutout in the circumferential direction In this section, the rectangular cutout was aligned in the circumferential direction, and the effect of its length growth on the buckling load was investigated using the LBMI method. The P/Pcr versus b/R ratios are depicted in Fig. 20, where ‘b’ is the length of the cutout in the circumferential direction and ‘R’ is the internal radius of the cylinder. As shown in Fig. 20, there are three zones to explain the effect of the imperfections on the buckling load due to the different values of ‘b’. Where the b/R ratio
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Fig. 19. P/Pcr versus a/R ratios of cylinder with and without imperfection.
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Fig. 20. P/Pcr versus b/R ratios of cylinder with and without imperfection.
is less than 0.07, reaction force reduction is about 20% on average and where 0.05 < b/R < 0.37, the difference between reaction force of the perfect and imperfect cylinder is about 9%. Finally for the b/R ratios more than 0.37 the buckling load reduction due to imperfections is about 4% which is negligible (b/R = 0.07 is labeled C1 and b/R = 0.37 is labeled C2 ). Depending on Figs. 19 and 20, one may conclude, in a cylinder with a rectangular cutout, for a/R and b/R ratios less than 0.07, the effect of both cutout and initial imperfections is responsible for the buckling load reduction, regardless of the cutout orientation. While for a/R and b/R ratios more than 0.37, cutout effect is dominant and the effect of imperfections on the buckling load reduction could be ignored. To discuss the mutual effects of cutout size and orientation on the buckling load of composite cylinders with details, more numerical analyses are conducted. At first, a square shaped cutout of different sizes is created on the surface of the shell, and then axial and circumferential length of the cutout is increased up to 150 mm, separately. Different nonlinear analyses on perfect cylinders are conducted with and without considering the initial imperfections and results are presented in Fig. 21. This figure shows the effect of rectangular cutouts along axial and circumferential directions on the buckling load of such cylinders. Curves are labeled as a10, a30, a50, and a70 for axial cutouts and b10, b30, b50 and b70 for circumferential cutouts. As an example, a10 curve means that
at first a cutout of 10 ∗ 10 mm is created on the surface and then the axial dimension, a, is increased from 10 to 150 mm. Also for b10, we can express that a square shaped cutout of 10 ∗ 10 mm is created at first, and then the circumferential dimension, b, is increased up to 150 mm. As Fig. 21 shows, cutouts oriented in the circumferential direction have a more profound effect on the buckling loads than the cutouts in the axial direction. For instance at Ra = Rb = 0.58 the buckling load of cylinders with a cutout of 50 ∗ 110 mm at circumferential direction shows 7% more reduction compared to the same cutout in the axial direction. It is worth mentioning for a/R and b/R ratios less than 0.26 the effect of the cutout orientation on the buckling load can be ignored.
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4.2.2.3. Stress concentrations around cutouts In the previous section, it was shown that cutout in the circumferential direction has a greater effect on the buckling load reduction of the cylinder than the cutout in the axial direction. The effect of the cutout size growth on the theoretical stress concentration factor under compressive axial loading is reported in this section. Parameters Ka and Kb are considered as stress concentration factors near the cutout for the rectangular cutouts aligned in the axial and the circumferential directions, respectively. The theoretical stress concentration factor is calculated using the av-
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Fig. 21. Buckling loads versus b/R or a/R ratios of cylinders with a rectangular cutout.
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Fig. 22. Stress concentrations versus b/R or a/R ratios of cylinders with a rectangular cutout.
erage stress of all layers at point “x” as shown in Fig. 22, for more details readers are referred to [18,19]. For this purpose, cylinders with different sizes of cutouts are modeled and investigated as explained in section 4.2.2. It should be mentioned that all stresses are extracted from critical points which are shown in Fig. 22. Any growth in the cutout length in both directions will increase the stress concentration factor near the cutout, but the value of this increase is different for the axial and circumferential cutouts. The growth of dimension ‘b’ is more impactful on the stress concentration near the cutout than the growth of dimension ‘a’. As shown in Fig. 22, by increasing the parameter ‘a’ or ‘b’, the stress concentration factors of the axial and circumferential cutouts, Ka and Kb , were raised to around 2.6 and 5.6, respectively, at a dimensional ratio of 0.58. In other words, at Ra = Rb = 0.58 the stress concentration factor of a cylinder with a 50 ∗ 110 mm cutout in the circumferential direction is 51% more than the same cutout in the axial direction. One may conclude that circumferential cutouts are more critical than the axial cutouts from a damage point of view. Also, during the buckling analysis of cylinders with circumferential cutouts, earlier damage initiation around the cutout would cause premature buckling of the corresponding cylinders.
4.2.2.4. Comparison of axial and circumferential oriented cutouts In sections 4.2.2.1 and 4.2.2.2, the effect of a rectangular cutout in the axial and circumferential directions on the buckling load with and without imperfections is examined. But it is not determined which type of cutout is more effective on the reaction force of the structures. For the perfect cylinders (without applying the imperfections) the effect of increasing the length of cutout along axial and circumferential directions is similar, and there is no significant difference between the values of the reaction forces. However, for the imperfect cylinders, it was shown that the circumferential cutout had 8% more effect on the reduction of the buckling load compared to the axial cutout (in the largest case). Although it was noticed that there is a remarkable difference between the stress concentration factors stemming from the two different cutouts. Based on Fig. 21, one may conclude that for b/R and a/R ratios less than 0.26, there is a negligible difference between the buckling load of the cylinders regardless of the cutout direction. While, for the b/R and a/R ratios more than 0.26, the effect of cutouts directions on the buckling load was increased up to 8%. In other words, cutouts in the circumferential direction are more critical than those in the axial direction. The same trend can be seen for the stress concentration factors in Fig. 22. Stress concentration factors for the b/R and a/R ratios more than 0.26 seriously depend on the cutout direction. As the value of stress concentration factors for the circumferential cutouts are much more than the
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axial cutouts, under compressive loading premature failure around the cutout will trigger an earlier buckling of the cylinder with the circumferential cutout. In the current research, the effect of earlier damage around the cutout was not considered in the buckling analysis for due to the fact that at larger D/R ratios (more than 0.4), there was a large difference between experimental and numerical results (Fig. 16). 5. Results in brief and conclusions
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In the current article, many linear and nonlinear FE analyses along with limited buckling tests were conducted to verify the effect of initial geometrical imperfections and cutout parameters on the critical buckling load of the composite cylinder. The extracted results are summarized in the following paragraphs.
• Considering the initial geometrical imperfections in the buckling analysis of the perfect composite cylinders resulted in an acceptable agreement between the experiment and the numerical predictions. • In the cylinders with cutout, the role of initial imperfections on the buckling load depends on the size of the cutout. For D/R ratios (ratio of cutout size to the cylinder radius) less than 0.16 the reduction of reaction forces caused by the initial imperfections is as large as about 23% and 20% for square and rectangular cutouts, respectively. For D/R ratios between 0.16 and 0.46, the difference between the reaction forces of the two perfect and imperfect cases are about 7% for cylinders with a square cutout, 8% for the cylinders with a rectangular cutout in the axial direction and 9% for the cylinders with a rectangular cutout in the circumferential direction. For D/R ratios more than 0.46, the effect of initial imperfections on the buckling load is negligible regardless of the cutout direction. In other words, the effect of cutout size is dominant in ratios more than 0.46 compared to the effect of initial imperfections. • In the cylinders with rectangular cutout, buckling analysis revealed that; for a/R (ratio of cutout length to the cylinder radius) or b/R (ratio of cutout width to the cylinder radius) ratios less than 0.26, the predicted buckling loads are almost the same, no matter of the cutout direction. However, for a/R or b/R ratios more than 0.26, the difference between the axial and circumferential cutouts increased to around 8%. Which means that a rectangular cutout along the circumferential direction causes severe reduction in the cylinder’s buckling load than the same cutout in the axial direction. • Variations of the cylinders stress concentration factors, Ka and Kb , are nearly equal for both cutout directions as long as a/R and b/R ratios are less than 0.26. But, by increasing the dimensional ratio, Ka parameter was increased almost linearly up to around 2.5, while Kb parameter was increased nonlinearly to around 6. It means that circumferential cutouts are more critical than the axial cutouts. Higher stress concentration factors around the circumferential cutouts would result in
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Conflict of interest statement
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None declared.
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References
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premature failure around the cutouts and subsequently causes earlier buckling of the cylinder.
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[1] W.T. Koiter, A Translation of the Stability of Elastic Equilibrium, Technische Hooge School at Delft, Department of Mechanics, Shipbuilding and Airplane Building, 1945. [2] V.I. Weingarten, P. Seide, J.P. Peterson, Buckling of Thin-Walled Circular Cylinders, NASA SP-8007, NASA Space Vehicle Design Criteria - Structures, 1965 (revised 1968). [3] H.T. Hu, S.S. Wang, Optimization for buckling resistance of fiber-composite laminate shells with and without cutouts, Compos. Struct. 22 (1) (1992) 3–13. [4] M.W. Hilburger, J.H. Starnes, Effects of imperfections on the buckling response of compression-loaded composite shells, Int. J. Non-Linear Mech. 37 (4–5) (2002) 623–643. [5] M.W. Hilburger, M.P. Nemeth, J.H. Starnes, Shell buckling design criteria based on manufacturing imperfection signatures, AIAA J. 44 (3) (2006) 654–663. [6] M.W. Hilburger, A.M. Waas, J.H. Starnes, The response of composite shells with cutouts to internal pressure and compression loads, AIAA J. 37 (2) (1999) 232–237. [7] A. Tafreshi, Buckling and post-buckling analysis of composite cylindrical shells with cutouts subjected to internal pressure and axial compression loads, Int. J. Press. Vessels Piping 79 (5) (2002) 351–359. [8] A.C. Orifici, C. Bisagni, Perturbation-based imperfection analysis for composite cylindrical shells buckling in compression, Compos. Struct. 106 (2013) 520–528. [9] M.A. Arbelo, A. Herrmann, S.G. Castro, R. Khakimova, R. Zimmermann, R. Degenhardt, Investigation of buckling behavior of composite shell structures with cutouts, Appl. Compos. Mater. 22 (6) (2015) 623–636. [10] F. Taheri-Behrooz, M. Omidi, M.M. Shokrieh, Experimental and numerical investigation of buckling behavior of composite cylinders with cutout, Thin-Walled Struct. 116 (2017) 136–144. [11] F. Taheri-Behrooz, M. Omidi, Buckling of axially compressed composite cylinders with geometric imperfections, Steel Compos. Struct. (2018), in press. [12] C. Yan, Z. Fan, X. Cheng, Z. Li, Compressive behavior of composite cylindrical shell with an open hole, J. Beijing Univ. Aeronaut. Astronaut. 38 (7) (2012) 920–924. [13] K. Arjomandi, F. Taheri, Stability and post-buckling response of sandwich pipes under hydrostatic external pressure, Int. J. Press. Vessels Piping 88 (4) (2011) 138–148. [14] F. Taheri-Behrooz, R.A. Esmaeel, F. Taheri, Response of perforated composite tubes subjected to axial compressive loading, Thin-Walled Struct. 50 (2012) 174–181. [15] M.M. Shokrieh, A. Nouri Parkestani, Post buckling analysis of shallow composite shells based on the third order shear deformation theory, Aerosp. Sci. Technol. 66 (2017) 332–341. [16] Hoang X. Nguyen, Ta Duy Hien, Jaehong Lee, H. Nguyen-Xuan, Stochastic buckling behavior of laminated composite structures with uncertain material properties, Aerosp. Sci. Technol. 66 (2017) 274–283. [17] E. Cestino, G. Romeo, P. Piana, F. Danzi, Numerical/experimental evaluation of buckling behavior and residual tensile strength of composite aerospace structures after low velocity impact, Aerosp. Sci. Technol. 54 (2016) 1–9. [18] F. Taheri-Behrooz, N. Bakhshi, Neuber’s rule accounting for the material nonlinearity influence on the stress concentration of the laminated composites, J. Reinf. Plast. Compos. 36 (3) (2017) 214–225. [19] Zixing Su, Cihang Xie, Yang Tang, Stress distribution analysis and optimization for composite laminate containing hole of different shapes, Aerosp. Sci. Technol. 76 (2018) 466–470.
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Orientation and size effect of a rectangle cutout on the buckling of composite cylinders
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School of Mechanical Engineering, Iran University of Science and Technology, P.O.B. 1684613114, Tehran, Iran
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Article history: Received 19 May 2018 Received in revised form 7 November 2018 Accepted 28 February 2019 Available online xxxx
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Abolfazl Shirkavand, Fathollah Taheri-Behrooz ∗ , Milad Omidi
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Keywords: Composite shells Nonlinear buckling Initial imperfections LBMI Square shaped cutouts Rectangular cutouts
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In this article the effect of a rectangular cutout on the buckling behavior of a thin composite cylinder was investigated using numerical and experimental methods. To verify the finite element results, a limited number of tests was carried out on perforated and non-perforated glass/epoxy cylinders with [90/−23/23/90] layups. In the numerical analysis, linear and nonlinear approaches were employed to study the effect of initial imperfections on the buckling of the cylinders. Several key findings including the effects of cutout size and orientation, and the mutual effects of the cutout and initial imperfections on the buckling behavior were investigated in detail. In the presence of cutouts, the effect of initial imperfections on the buckling load is a function of the cutout size. In cylinders with rectangular cutouts, buckling analysis revealed that a rectangular cutout in the circumferential direction causes around 8% more reduction in the buckling load than the same cutout in the axial direction. Also, numerical findings illustrated that elastic stress concentration factors for the circumferential cutouts are much greater than those for the axial cutouts; thus premature failure around the cutout will trigger earlier buckling in the cylinder with circumferential cutouts. © 2019 Elsevier Masson SAS. All rights reserved.
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1. Introduction
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Inclusion of cutouts in the construction of cylindrical shelltype structures is required to enable access to the inner parts of the structure for maintenance or installation of internal systems and other such usages. Based on the function of such cylinders in industry, cutouts can be made in various geometries like circle, square, rectangle, etc. It is of crucial importance to examine the buckling phenomenon because of the axial loading to optimize the usage of such cylinders. In the late 19th century, a great number of numerical, analytical and experimental researches on composite cylindrical shells were conducted, and attention to the differences between theoretical predictions and the experiments was paid constantly. In fact, experimental results always predict the buckling load of cylinders lower than the numerical and analytical results. In 1945, Koiter et al. [1] realized that the difference between such results is due to the presence of imperfections in structure. Imperfections like non-uniform loading, initial defects in property and imperfect boundary conditions. In 1968, NASA released a report named NASA SP-8007, which presented design guidelines by many experiments on metallic cylinders [2]. To this end, this report is
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Corresponding author. E-mail address: [email protected] (F. Taheri-Behrooz).
https://doi.org/10.1016/j.ast.2019.02.042 1270-9638/© 2019 Elsevier Masson SAS. All rights reserved.
still being used, but this design is very conservative and predicts the buckling load much lower than the experiments also makes the structure heavier which is not ideal especially in aerospace industries. Cutouts on cylinders play a role as an imperfection and cause a large drop in the buckling load of the structure. Hu and Wang [3] investigated fiber composite laminate shells with and without cutouts to optimize the buckling resistance of cylinders. They studied three important optimizing problems as: 1) optimization of fiber orientations to maximize the buckling resistance of composite shells without cutouts; 2) optimization of fiber orientations to maximize the buckling resistance of composite shells with circular cutouts; and 3) optimization of cutout geometry for maximizing the buckling resistance of a composite shell. The effect of imperfections on the buckling of such cylinders without cutouts under axial compression was investigated by Hilburger and Starnes [4] using analytical and experimental methods. The results identify the effects of traditional and non-traditional initial imperfections on the nonlinear response and buckling load of shells. Hilburger [5] presented a design criterion for composite shells buckling based on manufacturing imperfection signatures by measuring initial geometric imperfections in some graphite/epoxy shells. A numerical and an experimental study was performed by Hilburger and Wass [6] to demonstrate the laminate orthotropy on the buckling of composite cylindrical shells with a square cutout under axial
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compressive loading. Tafreshi [7] carried out a numerical study to investigate the buckling behavior of composite cylinders subjected to internal pressure and axial compression. She studied the effects of different sizes of rectangular and square shaped cutouts on the buckling behavior of such cylinders, but she did not mention the relation between imperfections and cutout size on the buckling load of cylinders. Orifici and Bisgani [8] have conducted a numerical research on both monolithic composite laminate and sandwich constructions with and without different sizes for square shaped cutouts. In their research, imperfections were applied to the cylinder by single perturbation load approach (SPLA). The results illustrated that the effect of small and large cutouts is analogous to the effect of small and large perturbation loads, respectively. In 2014 Arbelo et al. [9] investigated the effect of circular cutouts and imperfections on the buckling of cylinders was studied. The results extracted from the numerical analysis predicted the buckling load for cylinders with circular cutout under axial compression in three zones. It was concluded that the effect of imperfections is a function of cutout diameter to the cylinder radius. Taheri-Behrooz et al. [10,11] used a stochastic procedure to modify linear buckling mode-shape imperfection (LBMI) and SPLA methods for the buckling analysis of the composite cylinders with and without circular cutouts. They carried out buckling tests on glass/epoxy cylinders to verify their numerical findings. Yan et al. [12] investigated the axial compression behavior of a composite cylindrical shell with an opening by experiment and the finite element method. Stability and post-buckling response of sandwich pipes under external pressure was studied by Arjomandi and Taheri [13]. Taheri-Behrooz et al. [14] studied the response of perforated composite tubes subjected to axial compressive loading using numerical and experimental methods. Shokrieh and Nouri [15] investigated post buckling analysis of shallow composite shells based on the third order shear deformation theory. They derived a nonlinear mathematical model using Green-Lagrange type geometric nonlinearity which may not require the shear correction factors. Hoang et al. [16] studied stochastic buckling behavior of laminated composite structures with uncertain material properties and represented a spectral representation method based on Monte Carlo simulation and Iso Geometric Analysis method to evaluate effects of such uncertain material property on the buckling response. Cestino et al. [17] employed numerical/experimental methods to evaluate buckling behavior and residual tensile strength of composite aerospace structures. Although a lot of researches have studied the effect of cutouts on the buckling behavior of composite cylinders, there is a lack of information regarding the effect of cutout direction on the buckling of cylinders. In other words, it has been shown that the presence of a rectangular cutout on a cylinder body would reduce its compressive strength, regardless of the cutout direction. However, if the cutout is aligned in the circumferential direction (θ ) the amount of the cylinder’s strength reduction will be 2–3 times more than that of a cutout in the axial direction (Z). In the current manuscript, the buckling behavior of composite cylindrical shells with and without rectangular and square shaped cutouts under axial compression loading is investigated using numerical and experimental procedures. The effects of different cutout shapes and sizes on the buckling behavior of such cylinders are examined. Initial imperfections are applied to the models using LBMI method, and their effects on the buckling loads are also discussed. Validation is conducted by tests on composite cylinders made from glass/epoxy composites using filament winding method with and without cutouts. The effect of cutout size and orientation on the buckling loads and the static stress concentration factors is investigated numerically, and results are presented in suitable design charts. It is the first time the effect of rectangular cutouts size
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Table 1 Mechanical properties of glass/epoxy composites [10].
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Property
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Value
Elastic modulus in x-direction Elastic modulus in y-direction In-plane shear stress Poisson’s ratio Fiber volume fraction
Ex Ey E xy
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νxy Vf
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Table 2 Dimensions of the specimens (values in mm).
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Fig. 1. Schematic illustration of the specimen geometry.
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and orientation on the buckling behavior of cylinders has been addressed.
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2. Specimen definition
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All cylindrical shells are made using E-glass fiber (1200 Tex direct roving), Araldite LY 556 resin and HY917 hardener using filament winding method. The composite plies are laid up to form four-ply laminates having [90/−23/23/90] stacking sequences. The mechanical properties of the composite layers are given in Table 1 [10], and the nominal dimensions of the models are presented in Table 2. The cylindrical shells had a free length of 700 mm and an internal diameter of 378 mm. Each layer has a thickness of 0.375 mm and overall a 4-layer composite is obtained which has a thickness of 1.5 mm. Cylindrical shells with and without cutout are tested in the current research. Three groups of specimens with a square shaped cutout on their surface and one group of specimens without cutout were fabricated and tested during this research. The cutout dimensions are presented in Table 2. A schematic illustration of the geometry of the specimens is shown in Fig. 1.
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Fig. 2. A schematic illustration of the fixture (left) and reinforced edge of the shell before cutting (right).
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Fig. 3. a) Strain gauges on the shell with the cutout. b) Strain gauges on the shell without the cutout.
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For the buckling test, an STM-150 testing machine provided by SANTAM Co. with 15 tons of loading capacity was employed. The test is conducted in displacement control mode, and speed of the axial loading to the top of the shell is 0.01 mm/s. A metallic fixture is designed to provide nearly ideal boundary conditions and prevent the imperfections to affect the test procedure and buckling loads. As shown in Fig. 2, there is a 3 mm deep groove in the surface of both the top and bottom parts of the loading fixture. Both sides of the cylinder were placed in the mentioned groove to develop a clamped boundary condition. A schematic of different parts of the fixture is shown in Fig. 2a. Also, two glass/epoxy strips with 5 cm of width and 1.5 mm of thickness are used to prevent premature damage in cylinder edges (Fig. 2b). Electrical strain gauges are used to survey the exact behavior of the shell near the buckling point also to verify the uniformity of the loading. For specimens with a cutout, two bidirectional strain gauges are bonded near the cutout to extract the strains value in the axial and the circumferential directions in the top and beside of the cutout (Fig. 3a). For specimens without a cutout, two unidirectional strain gauges are bonded to the front and the back of the shell as shown in Fig. 3b. To extract strains from the strain gauges, a TMR-211 data logger manufactured by TML Co. is employed. Also to ensure the uniformity of the introduced loading on the shell, two dial gauges are used on top of the loading plate as depicted in Fig. 4. The gauges show the value of displacement in 2 points on the top part of the fixture. The vertical displacement values measured by these gauges at the top of the fixture verified the uniformity of the loading during the buckling test.
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Fig. 4. Dial gauges to measure the vertical displacement of the top part of the fixture.
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3.2. Buckling test results
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Buckling tests are conducted in 4 steps; first three specimens without cutouts named A1, A2, and A3, are tested, and their load-displacement response curves are recorded. Then three other groups of specimens with square shaped cutouts of 50, 80 and 100 mm sizes named D50, D80 and D100 are tested, and their corresponding response curves are recorded too. A representative response curve of each group is presented in Fig. 5. The nonlinearities illustrated in the first part of the load-end shortening response curves (Fig. 5) could be due to the initial dislocations or the friction between different parts of the test fixture and test samples. After applying about 10 kN load on the samples, the slope of the diagrams became constant, and diagrams are nearly linear up to the buckling point. Reduction in the buckling load can be observed in the response curves by increasing the ge-
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Fig. 6. Specimen A1 a) before and b) after buckling.
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ometrical imperfections. It means that growing the cutout length leads to a decrease in the buckling load. For the specimen without a cutout (A1), the buckling load is about 69 kN. By creating a 50 mm square shaped cutout on the surface of the shell, a big reduction of 38% happened in the buckling load compared to the specimen without the cutout. The buckling load values for D50, D80, and D100 specimens are about 39, 32 and 27 kN. Recorded buckling loads of various cylinders are tabulated in Table 3. Distortion in the surface of the shell can be observed at the buckling moment of the cylinder without cutout as shown in Fig. 6. In the cylinders with a cutout, initially, a local buckling was observed around the cutout which was spread to the entire wall of the shell by increasing the axial compressive load as shown in Fig. 7. After unloading the shells, the structure of specimens turns back to their initial state approximately. Bonded strain gauges on the surface of the specimen A1, show the value of strains in the axial direction. The recorded values of
strain by the back, gauge 1 and front, gauge 2 are nearly identical and the maximum difference between them is about 2%. The negative strain shows compressive loading on the structure. Fig. 8 presents a polynomial trend line for the data extracted from strain gauges 1 and 2 up to the buckling point of the cylinder without cutout which confirms the accuracy and uniformity of the axial loading. The buckling behavior of imperfect shells can be discussed considering the values of strain obtained from attached gauges on the surface of the shells near the cutout as shown in Figs. 9 and 10. There are 2 strain gauges along the axial direction and 2 strain gauges along the circumferential direction of the shell near the cutout. Strain gauges 1 and 3 show strain values along the axial direction and strain gauges 2 and 4 show strain values along the circumferential direction. In Fig. 10, a sudden shift happens near the 0.6 point on the vertical axis in the values obtained from strain gauge 2 due to the developed curvature in the surface of the shell. Strain gauge 1 shows a nearly linear behavior for axial loading up to the buckling point. Also in Fig. 11, the local structural behavior of a point next to the cutout can be inspected. By careful examination of the two diagrams, an approximately symmetric behavior can be observed which expresses that a symmetric curvature is created near the cutout in the structure. The positive value of strain gauge 4 is due to the outward motion of the surface of the shell near cutout by increasing the axial compressive loading. A more accurate picture of the buckling moment before unloading specimen D50 is shown in Fig. 11. A buckling mode with different half-waves can be observed on the surface of the shell and near the cutout due to local buckling.
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Table 4 Dimensions of the specimens (values in mm).
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Fig. 10. Axial and circumferential strains near the cutout of specimen D50.
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given in Table 2. Three groups of numerical models are created and investigated. The first group is cylinders without cutouts to find their original buckling load. The second group consists of cylinders with different sizes of square shaped cutouts from 10 ∗ 10 mm to 150 ∗ 150 mm. The last group is made from cylinders with different sizes of rectangular cutouts in the axial and circumferential directions to evaluate the effect of cutout width to length ratio on the buckling behavior of such composite cylinders. For this kind of cutout, at first, the length of the cutout in the axial direction was increased while its width in the circumferential direction was kept constant. Then its length along the circumferential direction was increased while its width in the axial direction was kept constant. Parameters ‘a’ and ‘b’ are considered for the rectangle length in axial and circumferential directions, respectively, as shown in Fig. 1. A summary of shells and cutouts dimensions is presented in Table 4. 4.2. Finite element analysis and results
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Fig. 11. The buckling moment of specimen D50.
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4. Numerical simulations
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The buckling responses of shells with and without cutouts are numerically investigated by finite element method using ABAQUS version 6.13. At the first step cylinders used in the experimental section are simulated numerically to verify the accuracy of the FE analysis. In the second step, the effects of various parameters such as cutout size and direction on the buckling response are investigated numerically.
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4.1. Finite element models definition
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Mechanical and geometrical properties of the cylinders simulated for numerical analysis are similar to the specimens tested in the experimental section. A 4-layer glass/epoxy composite laminate with the same stacking sequence and properties as the test specimen is assigned to the cylinder numerical model. Cylinder free length and internal radius are 700 mm and 189 mm, respectively. The thickness of each layer is 0.375 mm which results in a total shell thickness of 1.5 mm. The layer elastic properties are
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To calculate the buckling load of different models, linear and nonlinear approaches are employed. The linear solver is the buckle step in ABAQUS and predicts the linear buckling loads and corresponding mode shapes. The nonlinear method is a dynamic/implicit step in ABAQUS that provides a condition more similar to the test procedure and considers the nonlinear geometry effects. The solvers for buckling analysis in the current research are chosen according to many studies such as [6,10]. The boundary conditions are clamped in the bottom and top parts of the shells except for axial displacement at the top which is free as shown in Fig. 12. For the present investigation, the standard large-strain shell elements are appropriate. A 4-node doubly curved thin shell, reduced integration element is used for meshing the structure which is named S4R in ABAQUS. For models without a cutout, a convergence test is conducted through the entirety of the structure, and a mesh size of 13 mm and a shell elements count of 5278 are used for the perfect model. Convergence of results with different mesh sizes is shown in Fig. 13. For models with a cutout, in addition to total convergence analysis, an analysis should be conducted near the cutout due to the local buckling which affects the results. For this purpose, partition-
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Fig. 13. Appropriate mesh size of the perfect cylinder.
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the structures by the LBMI method is investigated for the shells, and the relation between cutout dimensions and imperfections and their effect on the buckling load is presented. The LBMI applies a scaled factor of the linear buckling modes to the cylinders as initial imperfections. The two main questions of this approach which should be carefully addressed are a) which mode shape is appropriate for the cylinder and b) what is the most acceptable scaling factor for different cylinders. The effect of the linear buckling mode shape and scaling factor has been fully investigated in many studies, and the results have been presented clearly [10]. The appropriate mode shape and the acceptable scaling factor are presented as mode number 24 and 18% of shell thickness for perfect cylinders. For perforated cylinders, these quantities are mode number 14 and 18% of shell thickness, as comprehensively investigated in [10].
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Fig. 14. Mesh details around cutout.
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ing should be done around the cutout to increase the mesh density by a bias ratio. Also, a convergence analysis should be conducted for bias ratio and element size near the cutout. The appropriate element number on the edge of cutout and bias ratio are 30 and 5, respectively, and in total 5289 shell elements are used to analyze the imperfect models [10]. Cylinders with cutout are shown in Fig. 14 after the meshing procedure. At first, a linear solution is obtained for shells to extract the different buckling eigenvalues and mode shapes. These buckling modes are used for applying the imperfections by the LBMI method. In many investigations, the LBMI approach is used to put the initial imperfections on the shells [6,10]. In the present study, the sensitivity of buckling load to initial imperfections applied to
4.2.1. FE analysis of cylinders with a square shaped cutout on the surface In the presence of various cutouts on the cylinder body the effect of geometrical imperfections on the buckling load is unclear and needs to be investigated. Composite cylinders with a square shaped cutout are studied by increasing the cutout length from 0 to 150 mm. The associated buckling loads are extracted using nonlinear methods with and without imperfections. As mentioned, cylinders have a 700 mm free length and 189 mm of internal radius and a stacking sequence of [90/+23/−23/90]. Many linear and nonlinear analyses with and without imperfections are conducted for D/R ratios ranging from 0 to 0.8, where D is the length of the square shaped cutout and R is the internal radius of the cylinder. Fig. 15 shows the reaction force-end shortening response curves of the cylinder for different values of D/R ratio which are obtained by nonlinear methods without applying imperfections. Depending on Fig. 15, the global stiffness of the structure was reduced by increasing the cutout size. Also, for D/R ratios more than 0.4 there was no clear sudden drop in the values of the reaction forces. Also, the buckling load was decreased by raising the cutout dimension up to D/R = 0.16, called C1 , after which the loads do not change much and remain nearly constant. This fact is illustrated in Fig. 16 more clearly for wider ranges of D/R ratios. All nonlinear analyses of the perfect cylinder are repeated for the imperfect cylinder using the LBMI method; results are presented in Fig. 16. In fact, this figure shows the relation between imperfections and cutout dimensions and their effect on P/Pcr ratio for different cylinders, where P is the buckling load of the perforated cylinder, and Pcr is the buckling load of the cylinder without cutout and imperfections.
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Fig. 17. The buckling mode shape of cylinder D50 predicted by a) test and b) the LBMI method.
axial and circumferential directions as shown in Fig. 18. Parameters ‘a’ and ‘b’ are considered as rectangle dimensions in the axial and circumferential directions, respectively. One dimension of the cutout was kept constant at 50 mm, and the other one was increased from 0 up to 150 mm. The effect of cutout size growth in the axial and transverse directions on the buckling behavior of the cylinder is investigated in detail numerically.
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Depending on Fig. 16, the effect of initial imperfections applied by the LBMI method on the buckling load was decreased by increasing the length of the cutout. On the other hand, for D/R ratios between 0 and 0.16, there was on average a 23% drop in the reaction force due to initial imperfections. Also, for 0.16 < D/R < 0.46 (D/R = 0.46 called C2 ) and for 0.46 < D/R < 0.8 the reduction in reaction force was about 7% and 1% on average, respectively. In fact, for D/R < 0.16, predicted buckling loads using the LBMI method are close to the experimental results. Conversely, predicted results without considering the imperfections deviate greatly from the experiment. So in this range, both the cut out, and the geometrical imperfections are responsible for the buckling of the cylinders. For D/R > 0.16, the cutout effect is dominant, which means that the initial imperfections are not impactful on the buckling loads. Also, geometrical imperfections effect is negligible for D/R ratios more than 0.46. Fig. 17 presents a comparative illustration of cylinder D50 buckling patterns recorded from the experiment and nonlinear finite element analysis.
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4.2.2. FE analysis of cylinders with a rectangular cutout on the surface To verify the effect of a rectangular cutout orientation on the buckling load of the composite cylinders two groups of the FE models are created using ABAQUS 6-13 and investigated in this section. A primitive hole of 50 ∗ 50 mm dimensions is considered in the middle of the shell. The cutout length is increased along
4.2.2.1. Results of the cylinder with a rectangular cutout in the axial direction In this section, the cutout length was increased in the axial direction, and its effect on the buckling load was investigated. Nonlinear approaches are employed, and the imperfections are applied to the structure using the LBMI method with an appropriate scaling factor and mode shape which were explained in the first part of the paper. The P/Pcr versus a/R ratios are illustrated in Fig. 19, where ‘a’ is the length of the cutout in the axial direction and ‘R’ is the internal radius of the cylinder. As shown in Fig. 19, for a/R ratios less than 0.07, reduction in the reaction forces of the perfect and imperfect cylinder is about 20% on average, and for 0.07 < a/R < 0.37, the difference between reaction forces is about 8%. Also, for a/R ratios more than 0.37 the drop in the buckling load caused by imperfections is about 3% which is negligible (a/R = 0.07 is denoted by C1 , while a/R = 0.37 is denoted by C2 ).
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4.2.2.2. Results of the cylinder with a rectangular cutout in the circumferential direction In this section, the rectangular cutout was aligned in the circumferential direction, and the effect of its length growth on the buckling load was investigated using the LBMI method. The P/Pcr versus b/R ratios are depicted in Fig. 20, where ‘b’ is the length of the cutout in the circumferential direction and ‘R’ is the internal radius of the cylinder. As shown in Fig. 20, there are three zones to explain the effect of the imperfections on the buckling load due to the different values of ‘b’. Where the b/R ratio
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Fig. 20. P/Pcr versus b/R ratios of cylinder with and without imperfection.
is less than 0.07, reaction force reduction is about 20% on average and where 0.05 < b/R < 0.37, the difference between reaction force of the perfect and imperfect cylinder is about 9%. Finally for the b/R ratios more than 0.37 the buckling load reduction due to imperfections is about 4% which is negligible (b/R = 0.07 is labeled C1 and b/R = 0.37 is labeled C2 ). Depending on Figs. 19 and 20, one may conclude, in a cylinder with a rectangular cutout, for a/R and b/R ratios less than 0.07, the effect of both cutout and initial imperfections is responsible for the buckling load reduction, regardless of the cutout orientation. While for a/R and b/R ratios more than 0.37, cutout effect is dominant and the effect of imperfections on the buckling load reduction could be ignored. To discuss the mutual effects of cutout size and orientation on the buckling load of composite cylinders with details, more numerical analyses are conducted. At first, a square shaped cutout of different sizes is created on the surface of the shell, and then axial and circumferential length of the cutout is increased up to 150 mm, separately. Different nonlinear analyses on perfect cylinders are conducted with and without considering the initial imperfections and results are presented in Fig. 21. This figure shows the effect of rectangular cutouts along axial and circumferential directions on the buckling load of such cylinders. Curves are labeled as a10, a30, a50, and a70 for axial cutouts and b10, b30, b50 and b70 for circumferential cutouts. As an example, a10 curve means that
at first a cutout of 10 ∗ 10 mm is created on the surface and then the axial dimension, a, is increased from 10 to 150 mm. Also for b10, we can express that a square shaped cutout of 10 ∗ 10 mm is created at first, and then the circumferential dimension, b, is increased up to 150 mm. As Fig. 21 shows, cutouts oriented in the circumferential direction have a more profound effect on the buckling loads than the cutouts in the axial direction. For instance at Ra = Rb = 0.58 the buckling load of cylinders with a cutout of 50 ∗ 110 mm at circumferential direction shows 7% more reduction compared to the same cutout in the axial direction. It is worth mentioning for a/R and b/R ratios less than 0.26 the effect of the cutout orientation on the buckling load can be ignored.
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4.2.2.3. Stress concentrations around cutouts In the previous section, it was shown that cutout in the circumferential direction has a greater effect on the buckling load reduction of the cylinder than the cutout in the axial direction. The effect of the cutout size growth on the theoretical stress concentration factor under compressive axial loading is reported in this section. Parameters Ka and Kb are considered as stress concentration factors near the cutout for the rectangular cutouts aligned in the axial and the circumferential directions, respectively. The theoretical stress concentration factor is calculated using the av-
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Fig. 22. Stress concentrations versus b/R or a/R ratios of cylinders with a rectangular cutout.
erage stress of all layers at point “x” as shown in Fig. 22, for more details readers are referred to [18,19]. For this purpose, cylinders with different sizes of cutouts are modeled and investigated as explained in section 4.2.2. It should be mentioned that all stresses are extracted from critical points which are shown in Fig. 22. Any growth in the cutout length in both directions will increase the stress concentration factor near the cutout, but the value of this increase is different for the axial and circumferential cutouts. The growth of dimension ‘b’ is more impactful on the stress concentration near the cutout than the growth of dimension ‘a’. As shown in Fig. 22, by increasing the parameter ‘a’ or ‘b’, the stress concentration factors of the axial and circumferential cutouts, Ka and Kb , were raised to around 2.6 and 5.6, respectively, at a dimensional ratio of 0.58. In other words, at Ra = Rb = 0.58 the stress concentration factor of a cylinder with a 50 ∗ 110 mm cutout in the circumferential direction is 51% more than the same cutout in the axial direction. One may conclude that circumferential cutouts are more critical than the axial cutouts from a damage point of view. Also, during the buckling analysis of cylinders with circumferential cutouts, earlier damage initiation around the cutout would cause premature buckling of the corresponding cylinders.
4.2.2.4. Comparison of axial and circumferential oriented cutouts In sections 4.2.2.1 and 4.2.2.2, the effect of a rectangular cutout in the axial and circumferential directions on the buckling load with and without imperfections is examined. But it is not determined which type of cutout is more effective on the reaction force of the structures. For the perfect cylinders (without applying the imperfections) the effect of increasing the length of cutout along axial and circumferential directions is similar, and there is no significant difference between the values of the reaction forces. However, for the imperfect cylinders, it was shown that the circumferential cutout had 8% more effect on the reduction of the buckling load compared to the axial cutout (in the largest case). Although it was noticed that there is a remarkable difference between the stress concentration factors stemming from the two different cutouts. Based on Fig. 21, one may conclude that for b/R and a/R ratios less than 0.26, there is a negligible difference between the buckling load of the cylinders regardless of the cutout direction. While, for the b/R and a/R ratios more than 0.26, the effect of cutouts directions on the buckling load was increased up to 8%. In other words, cutouts in the circumferential direction are more critical than those in the axial direction. The same trend can be seen for the stress concentration factors in Fig. 22. Stress concentration factors for the b/R and a/R ratios more than 0.26 seriously depend on the cutout direction. As the value of stress concentration factors for the circumferential cutouts are much more than the
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axial cutouts, under compressive loading premature failure around the cutout will trigger an earlier buckling of the cylinder with the circumferential cutout. In the current research, the effect of earlier damage around the cutout was not considered in the buckling analysis for due to the fact that at larger D/R ratios (more than 0.4), there was a large difference between experimental and numerical results (Fig. 16). 5. Results in brief and conclusions
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In the current article, many linear and nonlinear FE analyses along with limited buckling tests were conducted to verify the effect of initial geometrical imperfections and cutout parameters on the critical buckling load of the composite cylinder. The extracted results are summarized in the following paragraphs.
• Considering the initial geometrical imperfections in the buckling analysis of the perfect composite cylinders resulted in an acceptable agreement between the experiment and the numerical predictions. • In the cylinders with cutout, the role of initial imperfections on the buckling load depends on the size of the cutout. For D/R ratios (ratio of cutout size to the cylinder radius) less than 0.16 the reduction of reaction forces caused by the initial imperfections is as large as about 23% and 20% for square and rectangular cutouts, respectively. For D/R ratios between 0.16 and 0.46, the difference between the reaction forces of the two perfect and imperfect cases are about 7% for cylinders with a square cutout, 8% for the cylinders with a rectangular cutout in the axial direction and 9% for the cylinders with a rectangular cutout in the circumferential direction. For D/R ratios more than 0.46, the effect of initial imperfections on the buckling load is negligible regardless of the cutout direction. In other words, the effect of cutout size is dominant in ratios more than 0.46 compared to the effect of initial imperfections. • In the cylinders with rectangular cutout, buckling analysis revealed that; for a/R (ratio of cutout length to the cylinder radius) or b/R (ratio of cutout width to the cylinder radius) ratios less than 0.26, the predicted buckling loads are almost the same, no matter of the cutout direction. However, for a/R or b/R ratios more than 0.26, the difference between the axial and circumferential cutouts increased to around 8%. Which means that a rectangular cutout along the circumferential direction causes severe reduction in the cylinder’s buckling load than the same cutout in the axial direction. • Variations of the cylinders stress concentration factors, Ka and Kb , are nearly equal for both cutout directions as long as a/R and b/R ratios are less than 0.26. But, by increasing the dimensional ratio, Ka parameter was increased almost linearly up to around 2.5, while Kb parameter was increased nonlinearly to around 6. It means that circumferential cutouts are more critical than the axial cutouts. Higher stress concentration factors around the circumferential cutouts would result in
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Conflict of interest statement
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None declared.
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References
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premature failure around the cutouts and subsequently causes earlier buckling of the cylinder.
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