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Abstract

Experimental studies were made on isotropic cylindrical skew panels made of Aluminum 7075-T6 and laminated composite cylindrical skew panels under uniaxial compression. The experimental values of the critical buckling load (P _cr) were determined using five different methods. The values of P _cr were also determined using MSC/Nastran and CQUAD8 finite element. The experimental values of the P _cr obtained by different methods were compared with the finite element solution. The effects of the skew angle and aspect ratio on the critical buckling load of isotropic cylindrical skew panels made of Aluminum 7075-T6 were studied. The effects of the skew angle, aspect ratio, and the laminate stacking sequence on the critical buckling load of laminated composite cylindrical skew panels were also studied. It is found that the method IV (based on a plot of applied load (P) vs. average axial strain) yields the highest value for P _cr and method III (based on a plot of P vs. square of out-of-plane-deflection) the lowest value for P _cr. The experimental values given by method IV are seen to be closest to the finite element solution, the discrepancy being in the range of 5–23% for laminated composite cylindrical skew panels. For isotropic panels, it is found that the value P _cr initially increases with an increase in the skew angle and later decreases as the skew angle increases beyond 15°. For laminated composite panels, the P _cr value decreases as the aspect ratio increases for all laminate stacking sequences.

Keywords

Cylindrical skew panel laminate stacking sequence skew angle aspect ratio finite element analysis critical buckling load

Introduction

The skew or oblique cylindrical panels find wide application in the aircraft and spaceship industries. Experience shows that such structures fail frequently on account of instability arising from the slenderness of the members. The use of fiber-reinforced composite materials has increased multifold in recent years due to their light weight, high strength, and stiffness. The application areas of composite materials are continuously expanding from the traditional areas such as military aircraft to various other areas such as automobiles, robotics, day-to-day appliances, building industry, and so on. As the components and structures composed of laminated composite materials are usually very thin and hence more prone to buckling, their design requires accurate assessment of the critical buckling load (P _cr). There are few studies made on experimental determination of P _cr value of isotropic and laminated composite cylindrical panels, and most of them have been discussed in detail by Singer et al.¹ Becker et al.² studied the instability behavior of composite cylindrical panels. Hahn et al.³ investigated the post-buckling strength of simply supported corrugated board panels subjected to edge compressive loading using a specially developed test fixture experimentally. Rao and Gopalkrishna⁴ dealt with the optimization of the orientation of plies in panels made of composite materials for maximum buckling strength. Krishna Reddy and R. Palaninathan⁵ extended the general high precision triangular plate bending finite element to the buckling analysis of laminated skew plates. They used a transformation matrix between global and local degrees of freedom for nodes lying on the skew edges and performed suitable transformation of the element matrices. The accuracy of the formulation was verified against literature values. The P _cr values for antisymmetric angle-ply and cross-ply–laminated skew plates with different skew angles, different boundary (simply supported, clamped) and loading (uniaxial, biaxial) conditions were obtained and presented in graphical form. A sandwich curved beam subjected to a uniform loading was experimentally investigated by Bozhevolnaya and Kildegaard.⁶ The explicit through-thickness integration schemes for geometric nonlinear analysis of laminated composite shells by finite element method (FEM) have been discussed by Prema Kumar and Palaninathan.⁷ Liang et al.⁸ used hybrid genetic algorithm to optimize the design of filament-wound multilayer-sandwich submersible pressure hulls taking into consideration the shell buckling strength constraint, the angle-ply–laminated facing failure strength constraint and the low-density isotropic core yielding strength constraint under hydrostatic pressure. Arman et al.⁹ studied the effect of a single circular delamination around the circular hole on the critical buckling load of woven fabric laminated composite plates both experimentally and numerically. Han et al.¹⁰5 investigated the response of aluminum cylinders with a cutout subject to axial compression using the experimental method and the results were compared with the finite element solution. Li and Batra¹¹ investigated the buckling of axially compressed thin cylindrical shells with functionally graded middle layers using experimental techniques. Anil et al.¹² have made an attempt to incorporate the effect of prebuckled stress on the stability analysis of moderately thick/very thick composite laminated plates with cutouts under in-plane compressive loading using a FEM that incorporates simple higher order shear deformation theory. Guduru and Xia¹³ investigated shell buckling of imperfect multiwalled carbon nanotubes subjected to uniaxial compression. Mittelstedt¹⁴ investigated the initial buckling loads and the corresponding buckling modes of symmetric rectangular laminated plates. Young and Zhou^15,16 made an extensive study on aluminum tubing sections and proposed design equations. The exact solutions for the buckling analysis of rectangular Mindlin plates subjected to uniformly and linearly distributed in-plane loading on two opposite edges simply supported resting on elastic foundation were investigated by Akhavan et al.¹⁷ El-Sawy et al.¹⁸ used FEM to investigate the major-axis buckling characteristics and associated buckling capacity of axially loaded I-shaped steel columns. Extensive numerical analyses were conducted to evaluate the reduction in buckling capacity of castellated columns due to shear and flexural deformations. Havasi et al.¹⁹ investigated the buckling behavior of laminated composite shells with circular cutouts and initial geometric imperfections. The effects of cutout geometry and size, material properties, fiber angle, laminate stacking sequences, and initial geometric imperfections are discussed. Maalawi²⁰ presented an exact method for obtaining column designs with maximum possible critical buckling load while maintaining the total structural mass at prescribed value equal to that of a known baseline design.

Ozben²¹ obtained the critical buckling load of fiber-reinforced composite plate using analytical and FEMs. Topal and Uzman²² performed frequency optimization of laminated composite skew sandwich plates using finite element solution. The first-order shear deformation theory was used in the finite element formulation and modified feasible direction method was used for the frequency optimization. Shariati et al.²³ carried out experimental studies of buckling and post-buckling of cylindrical panels subjected to axial compressive load and determined the effects of variation of panel length, panel angle, and boundary conditions on the critical buckling load. Shariati and Rokhi²⁴ examined the influence of the cutout size, cutout angle, and the shell aspect ratios L/D and D/t on the pre-buckling, buckling, and post-buckling responses of the cylindrical shells. Zabihollah and Ganesan²⁵ studied the buckling behavior of laminated tapered composite beams using a higher order finite element formulation. Morovat et al.²⁶ have proposed a preliminary methodology to study the phenomenon of creep buckling in steel columns subjected to fire. Preliminary analytical solutions were presented and compared with computational predictions for creep buckling. The analytical and computational results indicated that accurate knowledge of material creep is essential in studying creep buckling phenomenon at elevated temperatures. Srinivasa et al.²⁷ evaluated experimentally the physical, flexural, and impact properties of composites made of randomly distributed areca fibers. Tsuji and Meshii²⁸ have proposed an image processing strain measurement system to evaluate fracture behavior of thin-walled pipes. Prabu et al.²⁹ studied the neighborhood effect of two circumferential short dents on the buckling behavior of thin short stainless steel cylindrical shell using finite element analysis. Prabu et al.³⁰ investigated the individual and combined effects of distributed and local geometrical imperfections on the limit load of an isotropic, thin-walled cylindrical shell under axial compression using nonlinear static finite element analysis.

Tahir and Mandal³¹ have presented a numerical study on buckling and post-buckling behavior of laminated carbon fiber-reinforced plastic (CFRP) thin-walled cylindrical shells under axial compression using asymmetric meshing technique. Upadhyay and Shukla³² have presented the large deformation flexural response of composite laminated skew plates subjected to uniform transverse pressure. Third-order shear deformation theory and von-Karman’s nonlinearity are used for the analysis. Zahurul and Young³³ conducted experiment studies on high-strength aluminum tubular structural members strengthened with CFRP. Laudiero et al.³⁴ investigated the post-buckling behavior of pultruded fiber-reinforced plastic (PFRP) beams in uniform major-axis bending using a nonlinear finite element analysis that accounts the initial out-of-straightness. Umbarkar et al.³⁵ investigated the effects of various geometrical parameters of circular single perforation on the critical/ultimate buckling load of a circular lean duplex stainless steel stub column loaded axially using ABAQUS (Version 6.9) finite element software. Zhao et al.³⁶ used digital image correlation method to predict the buckling load in shells and concluded that the theoretical value of buckling strength is much higher than the experimental value. Laudiero et al.³⁷ have presented buckling and post-buckling behavior of commercial PFRP I-section profiles subjected to pure compression using nonlinear finite element analyses. The imperfection sensitivity was investigated with reference to different imperfection shapes. They concluded that at equal cross section area, the narrow flange profiles may exhibit higher ultimate loads with respect to the wide flange profiles in a broad range of column slenderness. Najafov et al.³⁸ studied the vibration and stability behavior of axially compressed three-layer truncated conical shells with a functionally graded middle layer surrounded by elastic media. A great need exists for an extensive study of the buckling behavior of skew cylindrical panels. The present investigation deals with the buckling studies on isotropic and laminated composite cylindrical skew panels using experimental and FEMs. The experimental results are compared with the finite element solution obtained using CQUAD8 finite element of MSC/Nastran. In this study, the effects of skew angle, fiber orientation angle, laminate stacking sequence, and aspect ratio on the critical buckling load of cylindrical skew panels are investigated keeping the panel angle constant at 60° and total number of layers constant at 20°.

Determination of the critical buckling load using finite element analysis

FEM was employed to obtain the P _cr values and natural frequencies using MSC/Nastran software.

In linear static analysis, a structure is assumed to be in a state of stable equilibrium. As the applied load is removed, the structure is assumed to return to its original, undeformed position. Under certain combinations of loadings, however, the structure continues to deform without an increase in the magnitude of loading. In this case, the structure has become unstable; it has buckled. For elastic, or linear, buckling analysis, it is assumed that there is no yielding of the structure and that the direction of applied forces does not change.

Elastic buckling incorporates the effect of the differential stiffness, which includes higher order strain displacement relationships that are functions of the geometry, element type, and applied loads. From a physical standpoint, the differential stiffness represents a linear approximation of softening (reducing) the stiffness matrix for a compressive axial load and stiffening (increasing) the stiffness matrix for a tensile axial load.

In buckling analysis, the equations are solved for the eigenvalues that are scale factors that multiply the applied load in order to produce the critical buckling load. In general, only the lowest buckling load is of interest, since the structure will fail before reaching any of the higher order buckling loads. Therefore, usually only the lowest eigenvalue needs to be computed.

The buckling eigenvalue problem reduces to:

[K] + λ_{i} [K_{d}] = 0,

where, K is the system stiffness matrix and K _d is the differential stiffness matrix (generated automatically by MSC/Nastran, based on the geometry, properties, and applied load), and are the eigenvalues to be computed. Once the eigenvalues are found, the critical buckling load is calculated using the equation:

P_{cr} = λ_{i} P,

where, P _cr is the critical buckling load and P is the applied load.

The Lanczos method was used in the present study as it combines the best features of the other methods and computes accurate eigenvalues and eigenvectors.

Figure 1 shows the geometries of regular and skew cylindrical panel. A linear buckling analysis was performed using MSC/Nastran software. CQUAD8 and CQUAD4 finite elements were validated in the present study. The CQUAD4 element is a four-node plate element having six degrees of freedom/node (translational (u, v, w) and rotational (θ _x, θ _y, θ _z)). The CQUAD8 element is an eight-node isoparametric shell element having six degrees of freedom/node (translational (u, v, w) and rotational (θ _x, θ _y, θ _z)). Both the elements take into account the shear deformations. Table 1 shows the results of validation. It is clear from Table 1 that the CQUAD8 element is more accurate than the CQUAD4 element of MSC/Nastran. Hence, it was decided to employ CQUAD8 element for further computation in the present work. To arrive at the size of elements to be used in the finite element mesh for reliable results, a convergence study was undertaken. The entire panel was taken for discretization. It was performed on simply supported isotropic cylindrical panels subjected to axial compression. The convergence details are presented in Table 2.

Figure 1.

Geometry of the regular cylindrical and cylindrical skew panel.

Table 1.

Comparison of critical buckling load for isotropic cylindrical panels subjected to uniform in-plane load (υ = 0.3, t = 0.096 in, and E = 10 × 10⁶ psi).

Cylindrical panel dimensions (in)			P _cr
a	b	R	Rao and Gopalkrishna⁴	Timoshenko and Gere³⁹	Present
a	b	R	Rao and Gopalkrishna⁴	Timoshenko and Gere³⁹	CQUAD4	CQUAD8
100	10	1000	323	323	336.6	323.7
100	20	1000	89	90	93.9	90.6
50	10	500	329	329	334.7	330.6
25	5	100	1340	1340	1349.3	1340.8

a: length of panel; b: curved width of panel; R: panel radius; t: panel thickness; E: modulus of elasticity of the material of isotropic panel; υ: Poisson’s ratio; P _cr: critical buckling load.

Table 2.

Convergence study for Aluminum 7075-T6 cylindrical panels subjected to uniform in-plane load (L = 100mm, R = 40° mm t = 2 mm, E = 71.7 GPa, and μ = 0.33).

Mesh size	P _cr (kN)
Mesh size	CQUAD4	CQUAD8
10 × 25	16.20	10.02
16 × 30	15.35	10.82
22 × 35	14.96	11.10
28 × 40	14.10	11.45
34 × 45	13.58	11.52
40 × 50	13.46	11.53

P _cr: critical buckling load.

For the finite element study in the present work, the entire skew panel was discretized with a finite element mesh of size 40 × 50 CQUAD8 elements. The straight edges were completely free. One loaded curved edge was translationally restrained in all three directions and the other loaded curved edge translationally restrained except in the direction of loading. Figure 2 shows the finite element mesh of skew cylindrical panel with global and local coordinate systems. u and v are the displacement components in the global x and y directions, respectively. Since u and v are inclined to the skew edges, the displacement boundary conditions cannot be applied directly. In order to overcome this, a local coordinate system (x′, y′) normal and tangential to the skew edges is chosen and the software performs the required transformation. The axial load on the specimen was applied as a pressure loading on the end section.

Figure 2.

Global and local coordinate systems for finite element mesh of cylindrical skew panels.

Experimental determination of the critical buckling load

Test specimens

Isotropic cylindrical skew panel specimens made of Aluminum 7075-T6 were used in the studies. The material was supplied by Rio-Tinto Alcon (Canada). The material properties of the isotropic cylindrical panels made of Aluminum 7075-T6 are as follows: E = 71.7 GPa, µ = 0.33, and ρ = 2800 kg/m³ and these data were supplied by the manufacturer. The laminated composite cylindrical specimens were fabricated by hand layup technique using unidirectional glass fibers, epoxy-556 resin, and the hardener (HY951) supplied by Hindustan Ciba-Geigy Ltd, Mumbai (India). The cylindrical skew panel specimens were prepared using a mandrel of 600 mm length and 76.2 mm diameter. The surface of the cylindrical mandrel was thoroughly cleaned using acetone to remove any dust, dirt, or rust. Then a layer of thin releasing film was smeared over the surface of the mandrel before wrapping the layers of prepreg around it. The laminate was fabricated using hand layup technique. After fabrication, the entire surface was covered with a thin layer of releasing film, whose main purpose was to provide a smooth external surface and to protect the fibers from direct exposure to the environment. At a time one cylindrical panel of 500 mm length and 76.2 mm inner diameter was cast and it was later cut into required specimen lengths. The percentage of fiber and matrix was taken as 50:50 in weight for fabrication of the cylindrical panels. The test specimens were prepared in accordance with the relevant American Society for Testing and Materials (ASTM) standards. For laminated glass/epoxy composite cylindrical panels, the material constants E ₁ and E ₂ were evaluated experimentally using INSTRON 1195 universal testing machine (Norwood, Massachusetts, USA) as per ASTM Standard D 3039/D 3039M.⁴⁰ The average of three experimental determinations was adopted. For the determination of Poisson’s ratio (υ ₁₂), two strain gages were bonded to the specimen, one in the direction of the loading and the other at right angles to it. The strains were measured in longitudinal and transverse directions using strain indicator. The ratio of transverse to longitudinal strain gives the Poisson’s ratio within the elastic range. The average of three experimental determinations was adopted. The shear modulus (G ₁₂) was computed using standard expression available in Jones.⁴¹ The adopted material properties are as follows: E ₁ = 38.07 GPa, E ₂ = 8.1 GPa, G ₁₂ = 3.05 GPa, υ ₁₂ = 0.22, ρ = 2200 kg/m³. In this study, the skew angle is varied from 0° to 45° and the panel angle is maintained constant at 60°. The panel lengths considered are 100, 150, and 200 mm. Extreme precaution was taken to ensure that the compressive load was applied axially and no geometric imperfections were present in the fabricated test specimens.

Experimental procedure

The fixture for holding the test specimen is shown in Figure 3(a) and (b). The test specimen was inserted between the plates at the ends and the screws were tightened properly so that no slippage of the test specimen occurs. The tests were conducted on a computerized universal testing machine after positioning properly the test specimen using universal vice as shown in Figure 4. The measuring instrumentation consists of back-to-back strain gages and three linear variable differential transformers (LVDTs). The strain gages were placed at the center of the test panels on each side. Two LVDTs were positioned equidistant along the horizontal center line of the test specimen to measure the out-of-plane displacement; the third one was fixed to the moving jaw of UTM to measure the in-plane displacement. The testing was carried out with unloaded straight edges completely free. One loaded curved edge was restrained completely and the other loaded curved edge restrained except translationally in the direction of loading.

Figure 3.

Fixture for holding the test specimen.

Figure 4.

Experimental setup for cylindrical skew panel.

Methods for determining the experimental value of critical buckling load

Several procedures have been used by different investigators to evaluate the critical buckling load of regular cylindrical panels (Singer et al.¹). These are depicted in Figure 5. In the present study, five different methods or procedures are used, which are designated as method I, method II, and so on. Method I employs a plot of P versus out-of-plane deflection (W) at midspan. Method II employs a plot of P versus end shortening (Δ) in the direction of applied load. Method III employs a plot of P versus square of out-of-plane deflection (W ²). In method IV, the P is plotted against the algebraic mean strain, $ε_{A} = (ε_{1} + ε_{2}) / 2$ , where ε ₁ and ε ₂ are strains at the two surfaces of the specimen at midspan in the direction of loading. Method IV utilizes the fact that the surface strain on one side of the specimen becomes tensile when the specimen has buckled. Method V employs a plot of P versus strain difference $(ε_{D} = (ε_{1} - ε_{2}))$ .

Figure 5.

Methods used to determine critical buckling load.

Experimental work

Isotropic cylindrical skew panels:

Isotropic cylindrical panels were tested in uniaxial compression, the skew angle varying from 0° to 45°, panel length varying from 100 mm to 200 mm (a/b = 2.50–5.00), and the panel angle (Ø) being kept constant at 60°.The experimental values of the P _cr were determined in accordance with the methods I through V.

Laminated composite cylindrical skew panels:

Laminated composite cylindrical skew panels were tested in uniaxial compression, varying the skew angle from 0° to 45°, the panel length from 100 mm to 200 mm (curved width of the panel remaining constant), and panel angle (Ø) being kept constant at 60°. Four laminate stacking sequences, namely, antisymmetric angle-ply [+0°/−0°/…/−0°], antisymmetric angle-ply [+45°/−45°/…/−45°], antisymmetric angle-ply [+90°/−90°/…/−90°], and antisymmetric cross-ply [0°/90°/…/90°] were considered. The number of layers were held constant at 20°. The total thickness of the laminate was kept at 2.0 mm. The experimental values of the P _cr were determined according to the methods I through V.

Results and discussion

Isotropic cylindrical skew panels

Figure 6 shows a typical buckled shape of the test specimen. The test results are presented in Table 3 and Figure 7. The following observations are made:

Figure 6.

A typical buckled shape of the isotropic cylindrical skew specimen (α = 15°, Ø = 60°).

Table 3.

Critical buckling load for isotropic cylindrical skew panels.^a

Skew angle (α)	Aspect ratio (a/b)	P _cr (kN)
		Experimental values					FEM (CQUAD8)
		Method I	Method II	Method III	Method IV	Method V	FEM (CQUAD8)
0°	2.50	10.731 (1.22)	10.500 (1.20)	10.385 (1.23)	10.962 (0.90)	10.847 (0.91)	11.539
	3.75	6.870 (0.32)	6.722 (0.30)	6.648 (0.33)	7.018 (0.32)	6.944 (0.25)	7.387
	5.00	5.451 (0.25)	5.334 (0.24)	5.275 (0.30)	5.568 (0.33)	5.509 (0.26)	5.861
15°	2.50	14.505 (0.75)	14.189 (0.80)	14.032 (0.77)	14.820 (0.80)	14.662 (0.82)	15.766
	3.75	13.120 (0.85)	12.835 (0.90)	12.692 (0.90)	13.405 (077)	13.263 (0.80)	14.261
	5.00	6.736 (0.45)	6.590 (0.50)	6.517 (0.44)	6.883 (0.43)	6.809 (0.40)	7.322
30°	2.50	14.200 (0.88)	13.888 (0.90)	13.732 (0.78)	14.512 (0.95)	14.356 (0.85)	15.604
	3.75	9.592 (0.55)	9.381 (0.50)	9.276 (0.56)	9.803 (0.60)	9.698 (0.56)	10.541
	5.00	6.400 (0.35)	6.259 (0.34)	6.189 (0.34)	6.541 (0.33)	6.470 (0.30)	7.033
45°	2.50	13.047 (0.44)	12.747 (0.60)	12.747 (0.39)	13.497 (0.40)	13.347 (0.41)	14.997
	3.75	8.544 (0.30)	8.348 (0.32)	8.348 (0.33)	8.839 (0.30)	8.741 (0.32)	9.821
	5.00	5.727 (0.24)	5.596 (0.23)	5.596 (0.25)	5.925 (0.26)	5.859 (0.26)	6.583

FEM: finite element method; P _cr: critical buckling load.

^aThe numbers in parentheses represent the standard deviation.

Figure 7.

Critical buckling load for isotropic cylindrical skew panels (Ø = 60°).

Method IV yields the highest experimental value for the P _cr and method III yields the lowest value. The experimental values given by method IV are closest to the finite element solution. The percentage of discrepancy between the finite element solution and method IV is less than about 10%. For any given cylindrical skew panel, the experimental values of P _cr are less than the value given by the finite element analysis. The discrepancy may be attributed to the higher stiffness of the finite element model arising out of the finite degrees of freedom chosen, differences between actual boundary conditions in the experiment and idealized conditions considered in the finite element analysis, and inaccuracies in the geometry and load application during experiment among others. For any given panel, the discrepancies among the experimental values given by methods I through V are not much.

For any particular aspect ratio of the panel, the P _cr value initially increases and becomes a maximum for a skew angle of about 15° and later decreases.

The P _cr value is observed to decrease as the aspect ratio increases for all skew angles.

The experimental values are in good agreement with the finite element solution, the maximum discrepancy being about 10% (for method IV).

For a particular skew angle, the P _cr value decreases as the aspect ratio increases. The rate of decrease is initially large and becomes smaller for higher values of aspect ratio.

For a particular aspect ratio, the P _cr value is observed to increase with the skew angle, the increase being not substantial.

Laminated composite cylindrical skew panels

Figure 8 shows a typical buckled shape of the test specimen. The values of the P _cr for various values of skew angle, laminate stacking sequence, and aspect ratios are tabulated in Tables 4 to 7 and the same is presented in a graphical manner in Figures 9 to 12 for skew angles of 0°, 15°, 30°, and 45°, respectively. The following are observed from Tables 4 to 7 and Figures 9 to 12.

Figure 8.

A typical buckled shape of the laminated composite cylindrical skew specimen (α = 15°, a/b = 3.75, Ø = 60° cross-ply [0°/90°/…/90°]).

Table 4.

Critical buckling load for laminated composite cylindrical skew panels (α = 0°, Ø = 60°).^a

Aspect ratio (a/b)	Laminate stacking sequence	P _cr (kN)
		Experimental values					FEM (CQUAD8)
		Method I	Method II	Method III	Method IV	Method V	FEM (CQUAD8)
2.50	Angle-ply [+0°/−0°/…/−0°]	2.117 (0.22)	2.093 (0.25)	2.069 (0.21)	2.165 (0.20)	2.141 (0.19)	2.406
	Angle-ply [+45°/−45°/…/−45°]	3.574 (0.21)	3.533 (0.23)	3.492 (0.18)	3.655 (0.24)	3.614 (0.22)	4.061
	Angle-ply [+90°/−90°/…/−90°]	4.416 (0.18)	4.366 (0.15)	4.315 (0.20)	4.516 (0.21)	4.466 (0.19)	5.018
	Cross-ply [0°/90°/…/90°]	3.311 (0.15)	3.274 (0.19)	3.236 (0.18)	3.387 (0.20)	3.349 (0.21)	3.763
3.75	Angle-ply [+0° /−0°/…/−0°]	1.221 (0.11)	1.207 (0.10)	1.193 (0.12)	1.250 (0.10)	1.236 (0.11)	1.404
	Angle-ply [+45°/−45°/…/−45°]	1.756 (0.12)	1.735 (0.11)	1.715 (0.15)	1.796 (0.13)	1.776 (0.12)	2.018
	Angle-ply [+90°/−90°/…/−90°]	2.807 (0.15)	2.774 (0.16)	2.742 (0.12)	2.871 (0.12)	2.839 (0.12)	3.226
	Cross-ply [0°/90°/…/90°]	2.275 (0.10)	2.249 (0.11)	2.223 (0.12)	2.327 (0.11)	2.301 (0.10)	2.615
5.00	Angle-ply [+0°/−0°/…/−0°]	0.740 (0.08)	0.732 (0.08)	0.723 (0.07)	0.758 (0.08)	0.749 (0.07)	0.861
	Angle-ply [+45°/−45°/…/−45°]	1.031 (0.08)	1.019 (0.07)	1.007 (0.07)	1.055 (0.07)	1.043 (0.06)	1.199
	Angle-ply [+90°/−90°/…/−90°]	2.189 (0.07)	2.163 (0.06)	2.138 (0.06)	2.240 (0.07)	2.214 (0.06)	2.545
	Cross-ply [0°/90°/…/90°]	1.798 (0.07)	1.777 (0.08)	1.756 (0.07)	1.840 (0.07)	1.819 (0.06)	2.091

FEM: finite element method; P _cr: critical buckling load.

^aThe numbers in parentheses represent the standard deviation.

Table 5.

Critical buckling load for laminated composite cylindrical skew panels (α = 15°, Ø = 60°).^a

Aspect ratio (a/b)	Laminate stacking sequence	P _cr (kN)
		Experimental values					FEM (CQUAD8)
		Method I	Method II	Method III	Method IV	Method V	FEM (CQUAD8)
2.50	Angle-ply [+0°/−0°/…/−0°]	2.394 (0.21)	2.366 (0.20)	2.339 (0.22)	2.450 (0.20)	2.422 (0.21)	2.784
	Angle-ply [+45°/−45°/…/−45°]	3.701 (0.22)	3.658 (0.23)	3.615 (0.23)	3.787 (0.21)	3.744 (0.22)	4.303
	Angle-ply [+90°/−90°/…/−90°]	4.180 (0.32)	4.131 (0.33)	4.082 (0.33)	4.277 (0.30)	4.228 (0.31)	4.860
	Cross-ply [0°/90°/…/90°]	3.485 (0.27)	3.444 (0.28)	3.404 (0.28)	3.566 (0.25)	3.525 (0.24)	4.052
3.75	Angle-ply [+0°/−0°/…/−0°]	2.366 (0.18)	2.339 (0.20)	2.311 (0.21)	2.422 (0.20)	2.394 (0.19)	2.684
	Angle-ply [+45°/−45°/…/−45°]	2.816 (0.21)	2.783 (0.22)	2.750 (0.22)	2.882 (0.19)	2.849 (0.18)	3.313
	Angle-ply [+90°/−90°/…/−90°]	3.360 (0.23)	3.321 (0.22)	3.281 (0.25)	3.439 (0.20)	3.400 (022)	3.953
	Cross-ply [0°/90°/…/90°]	3.363 (0.26)	3.324 (0.24)	3.284 (0.25)	3.443 (0.23)	3.403 (0.24)	3.957
5.00	Angle-ply [+0°/−0°/…/−0°]	0.715 (0.11)	0.706 (0.10)	0.698 (0.10)	0.732 (0.05)	0.723 (0.06)	0.851
	Angle-ply [+45°/−45°/…/−45°]	1.094 (0.08)	1.081 (0.07)	1.068 (0.06)	1.120 (0.06)	1.107 (0.06)	1.302
	Angle-ply [+90°/−90°/…/−90°]	1.992 (0.11)	1.968 (0.10)	1.944 (0.09)	2.039 (0.08)	2.015 (0.07)	2.371
	Cross-ply [0°/90°/…/90°]	1.530 (0.11)	1.512 (0.12)	1.494 (0.06)	1.567 (0.05)	1.549 (0.06)	1.822

FEM: finite element method; P _cr: critical buckling load.

^aThe numbers in parentheses represent the standard deviation.

Table 6.

Critical buckling load for laminated composite cylindrical skew panels (α = 30°, Ø = 60°).^a

Aspect ratio (a/b)	Laminate stacking sequence	P _cr (kN)
		Experimental values					FEM (CQUAD8)
		Method I	Method II	Method III	Method IV	Method V	FEM (CQUAD8)
2.50	Angle-ply [+0°/−0°/…/−0°]	2.291 (0.10)	2.263 (0.11)	2.236 (0.09)	2.345 (0.08)	2.318 (0.09)	2.727
	Angle-ply [+45°/−45°/…/−45°]	3.955 (0.12)	3.908 (0.13)	3.861 (0.09)	4.049 (0.10)	4.002 (0.11)	4.708
	Angle-ply [+90°/−90°/…/−90°]	4.079 (0.13)	4.030 (0.14)	3.982 (0.13)	4.176 (0.12)	4.128 (0.12)	4.856
	Cross-ply [0°/90°/…/90°]	3.796 (0.10)	3.751 (0.11)	3.706 (0.10)	3.886 (0.10)	3.841 (0.11)	4.519
3.75	Angle-ply [+0°/−0°/…/−0°]	1.165 (0.05)	1.151 (0.06)	1.137 (0.05)	1.194 ((0.05)	1.179 (0.04)	1.421
	Angle-ply [+45°/−45°/…/−45°]	2.145 (0.06)	2.119 (0.05)	2.093 (0.05)	2.197 (0.06)	2.171 (0.05)	2.616
	Angle-ply [+90°/−90°/…/−90°]	2.270 (0.07)	2.242 (0.05)	2.214 (0.04)	2.325 (0.04)	2.297 (0.04)	2.768
	Cross-ply [0°/90°/…/90°]	2.030 (0.06)	2.005 (0.07)	1.980 (0.04)	2.079 (0.050	2.054 (0.05)	2.475
5.00	Angle-ply [+0°/−0°/…/−0°]	0.665 (0.02)	0.657 (0.03)	0.648 (0.02)	0.682 (0.02)	0.674 (0.02)	0.842
	Angle-ply [+45°/−45°/…/−45°]	1.270 (0.01)	1.254 (0.02)	1.238 (0.01)	1.302 (0.01)	1.286 (0.01)	1.608
	Angle-ply [+90°/−90°/…/−90°]	1.309 (0.02)	1.292 (0.30)	1.276 (0.10)	1.342 (0.10)	1.326 (0.11)	1.657
	Cross-ply [0°/90°/…/90°]	1.149 (0.05)	1.134 (0.06)	1.120 (0.04)	1.178 (0.05)	1.163 (0.04)	1.454

FEM: finite element method; P _cr: critical buckling load.

^aThe numbers in parentheses represent the standard deviation.

Table 7.

Critical buckling load for laminated composite cylindrical skew panels (α = 45°, Ø = 60°).^a

Aspect ratio (a/b)	Laminate stacking sequence	P _cr (kN)
		Experimental values					FEM (CQUAD8)
		Method I	Method II	Method III	Method IV	Method V	FEM (CQUAD8)
2.50	Angle-ply [+0°/−0°/…/−0°]	2.558 (0.20)	2.526 (0.21)	2.494 (0.20)	2.622 (0.18)	2.590 (0.19)	3.197
	Angle-ply [+45°/−45°/…/−45°]	3.717 (0.22)	3.670 (0.21)	3.624 (0.20)	3.810 (0.20)	3.763 (0.18)	4.646
	Angle-ply [+90°/−90°/…/−90°]	3.377 (0.22)	3.335 (0.21)	3.292 (0.22)	3.461 (0.21)	3.419 (0.20)	4.221
	Cross-ply [0°/90°/…/90°]	3.500 (0.30)	3.456 (0.31)	3.413 (0.20)	3.588 (0.24)	3.544 (0.25)	4.375
3.75	Angle-ply [+0°/−0°/…/−0°]	1.180 (0.10)	1.164 (0.11)	1.149 (0.10)	1.210 (0.10)	1.195 (0.10)	1.532
	Angle-ply [+45°/−45°/…/−45°]	2.205 (0.11)	2.177 (0.11)	2.148 (0.10)	2.263 (0.18)	2.234 (0.15)	2.864
	Angle-ply [+90°/−90°/…/−90°]	1.595 (0.11)	1.574 (0.10)	1.553 (0.15)	1.636 (0.12)	1.615 (0.12)	2.071
	Cross-ply [0°/90°/…/90°]	1.658 (0.09)	1.636 (0.08)	1.615 (0.09)	1.701 (0.15)	1.679 (0.12)	2.153
5.00	Angle-ply [+0°/−0°/…/−0°]	0.637 (0.05)	0.628 (0.06)	0.619 (0.04)	0.680 (0.05)	0.671 (0.05)	0.872
	Angle-ply [+45°/−45°/…/−45°]	1.405 (0.07)	1.385 (0.08)	1.366 (0.09)	1.501 (0.12)	1.481 (0.13)	1.924
	Angle-ply [+90°/−90°/…/−90°]	0.831 (0.07)	0.819 (0.08)	0.808 (0.07)	0.888 (0.07)	0.876 (0.07)	1.138
	Cross-ply [0°/90°/…/90°]	0.864 (0.06)	0.852 (0.06)	0.841 (0.05)	0.924 (0.07)	0.912 (0.06)	1.184

FEM: finite element method; P _cr: critical buckling load.

^aThe numbers in parentheses represent the standard deviation.

Figure 9.

Critical buckling load for laminated composite cylindrical skew panels (α = 0°, Ø = 60°).

Figure 10.

Critical buckling load for laminated composite cylindrical skew panels (α = 15°, Ø = 60°).

Figure 11.

Critical buckling load for laminated composite cylindrical skew panels (α = 30°, Ø = 60°).

Figure 12.

Critical buckling load for laminated composite cylindrical skew panels (α = 45°, Ø = 60°).

The experimental values given by method IV are closest to the finite element solution. The percentage of discrepancy between the finite element solution and method IV is in the range of 5–23%.

For any given cylindrical skew panel, the experimental values of P _cr are less than the value given by finite element analysis. For any given panel, the discrepancies among the experimental values given by methods I through V are not much.

Method IV yields the highest experimental value for P _cr for all laminate stacking sequences and skew angles 15° and 30°.

Method III yields the lowest experimental value for P _cr for all laminate stacking sequences and all skew angles considered.

The percentage of discrepancy between the numerical or finite element solution and experimental value increases as the skew angle increases.

Conclusions

The following conclusions are made in respect of buckling of isotropic and laminated composite cylindrical skew panels under uniaxial compression:

All the experimental values are less than the corresponding FEM/numerical values.

The percentage of discrepancy between the numerical or finite element solution and experimental value increases as the skew angle increases.

The P _cr value decreases sharply as the aspect ratio increases from 1.0 to 2.5 for all the stacking sequences.

For both isotropic cylindrical skew panels and laminated composite cylindrical skew panels, method IV (based on a plot of P vs. average axial strain) yields the highest experimental value for the P _cr and method III (based on a plot of P vs. square of out-of-plane deflection) yields the lowest value.

For any given cylindrical skew panel, the experimental values of the P _cr are less than the value given by finite element analysis. For any given panel, the discrepancies among the experimental values given by methods I through V are not much.

The values given by method IV are closest to the finite element solution. The percentage of discrepancy between the finite element solution and method IV is less than about 10% for isotropic skew panels and may be neglected for all practical purposes. It is about 5–23% in case of laminated composite cylindrical skew panels. This information is of importance in the design.

For a particular skew angle, the value of P _cr decreases as the aspect ratio increases, the rate of decrease being initially high and becomes smaller for larger values of aspect ratio. This trend is observed both in isotropic and laminated composite cylindrical skew panels for all laminate stacking sequences.

In case of isotropic cylindrical skew panels, the value of P _cr initially increases and becomes a maximum for a skew angle of about 15° and later decreases for any particular value of the aspect ratio.

Footnotes

Acknowledgments

The first author would like to thank the Management and Principal Dr P Prakash of GM Institute of Technology, Davanagere, Karnataka, India, for the kind encouragement and support provided. The second author would like to thank the Management, ACS College of Engineering, Bengaluru, Karnataka, India, for the kind encouragement and support provided. The third author would like to thank the Management of Jawaharlal Nehru College of Engineering, Shivamogga, Karnataka, India, for the kind encouragement and support provided. Authors would like to appreciate and acknowledge the reviewers for their useful comments and suggestions.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

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Buckling of laminated composite cylindrical skew panels

Abstract

Keywords

Introduction

Determination of the critical buckling load using finite element analysis

Experimental determination of the critical buckling load

Test specimens

Experimental procedure

Methods for determining the experimental value of critical buckling load

Experimental work

Isotropic cylindrical skew panels:

Laminated composite cylindrical skew panels:

Results and discussion

Isotropic cylindrical skew panels

Laminated composite cylindrical skew panels

Conclusions

Footnotes

Acknowledgments

Declaration of Conflicting Interests

Funding

References