Finite element studies on buckling of laminated cylindrical skew panels

DOI 10.1515/secm-2013-0204 Sci Eng Compos Mater 2014; 21(4): 551–558

Chikkol Venkateshappa Srinivasa*, Yalaburgi Jayadevappa Suresh and Wooday Puttiah Prema Kumar

Finite element studies on buckling of laminated cylindrical skew panels

Abstract: The present paper presents the finite element studies made on critical buckling load of isotropic and laminated composite cylindrical skew panels. Analysis is performed using CQUAD4 and CQUAD8 elements of MSC/NASTRAN. It is found that the CQUAD8 element yields bet-ter results compared to the CQUAD4 element in terms of accuracy and convergence. Using the CQUAD8 element, the effects of the panel angle, skew angle, aspect ratio, and length-to-thickness ratio on the critical buckling load of isotropic cylindrical skew panels have been studied. The effects of additional parameters such as fiber orienta-tion angle, numbers of layers (NL), and stacking sequence on the critical buckling load of laminated composite cylin-drical skew panels have also been studied. The critical buckling loads are found to increase with the increase in panel angle and skew angle. When the NL in the laminate is large, the variation of the critical buckling load with the NL is not appreciable. The boundary conditions are found to have significant influence on the critical buckling load.

Keywords: critical buckling load parameter; cylindrical skew panel; finite element analysis; laminated compos-ites; panel angle; skew angle.

*Corresponding author: Chikkol Venkateshappa Srinivasa, Department of Mechanical Engineering, GM Institute of Technology, Davangere, Karnataka, 577006 India, e-mail: [email protected] Jayadevappa Suresh: Department of Mechanical Engineering, J.N.N. College of Engineering, Shivamogga, Karnataka, 577204 IndiaWooday Puttiah Prema Kumar: Department of Civil Engineering, Reva Institute of Technology and Management, Member, Academic Council, Reva University, Bangalore, Karnataka, 560064 India

List of symbols

a curved length of panelb longitudinal dimension of panelt panel thicknessNL number of layers in the laminateE modulus of elasticity of the material of isotropic

panel

El Young’s modulus of the lamina in the longitudinal direction

Et Young’s modulus of the lamina in the transverse direction

Glt in-plane shear modulus of the laminaD bending rigidity, Et3/12(1-μ2)σ applied compressive stress in the longitudinal

directionα skew angle of the panelØ panel angleθ fiber orientation angle of the laminaμ Poisson’s ratio of isotropic panelvlt major Poisson’s ratio of the laminaKcr critical buckling load parameterλ eigenvalue

1 IntroductionThe skew cylindrical panels find wide application in the aircraft and spaceship industry. Such structures are usually very thin and fail frequently due to buckling or instability. The use of fiber-reinforced composite materi-als has increased multifold in recent years due to their high strength-to-weight and stiffness-to-weight ratios. The practical application of composite materials has widened from the traditional application in aircraft industry to other applications in automobiles, robotics, day-to-day appliances, building industry, etc. The laminated com-posite cylindrical panels are usually very thin and their design requires an accurate assessment of the critical buckling load.

An excellent review of the plate/shell finite elements developed during the past 20 years is presented by Zhang and Yang [1]. Studies on buckling of laminated compos-ite plates/shells were made by Aydogu [2]. Studies on buckling behavior of laminated composite cylindrical panels under axial compression were made by Jun and Hong [3]. Rao and Gopalkrishna [4] have discussed the optimization of the orientation of plies in panels made of composite materials for maximum buckling strength. The buckling behavior of laminated cylindrical panels subject

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552 C.V. Srinivasa et al.: Finite element studies on buckling of laminated cylindrical skew panels

Z

a

θ α

y

x b

Figure 1 Geometry of the cylindrical skew panels with dimensions.

x

x′

y′

α

y

Figure 2 Global and local coordinate systems for cylindrical skew panel used in finite element analysis.

A great need exists for an elaborate study on the buckling behavior of laminated composite skew cylindrical panels and the present work is an attempt in this direction. Using CQUAD8 finite element of MSC/NASTRAN, the effect of aspect ratio, skew angle, panel angle, fiber orientation angle, number of layers (NL) in the laminate, and lami-nate sequence on the critical buckling load parameter of cylindrical skew panels has been studied in the present work.

2 Validation and convergence studies

The geometry of the skew cylindrical panel with dimen-sions are shown in Figure 1. Figure 2 indicates cylin-drical skew panel with global and local coordinate systems, and u and v are the displacement components in the x and y directions, respectively. Because u and v are inclined to the skew edges, the displacement bound-ary conditions cannot be applied directly. To overcome this, a local coordinate system (x′, y′) normal and tan-gential to the skew edges is chosen.

2.1 Validation check

CQUAD4 is a four-node plate element of MSC/NASTRAN having 6 degrees of freedom/node. CQUAD8 is an eight-node shell element of MSC/NASTRAN having 6 degrees of freedom/node. Both elements take into account shear deformations. The validation of the CQUAD4 and CQUAD8 finite elements has been made by comparing the values of the critical buckling load obtained with those of the litera-ture for the case of simply supported isotropic cylindrical panel. The same is presented in Table 1. It can be seen from Table 1 that the results obtained using the aforesaid finite

Table 1 Comparison of critical buckling load for isotropic cylindrical panels subjected to uniform in-plane load (ν = 0.3, t = 0.096 in, E = 10 × 106 psi).

Cylindrical panel dimensions (in)

Critical buckling load (lb/in)

a b R Rao and Gopalkrishna [4] Timoshenko and Gere [10]

Present

CQUAD4 CQUAD8

100 10 1000 323 323 336.6 323.7100 20 1000 89 90 93.9 90.6

50 10 500 329 329 334.7 330.625 5 100 1340 1340 1349.3 1340.8

to axial compressive load using finite element method is reported by Hu and Yang [5]. Liew et al. [6] have discussed the dynamic stability of composite cylindrical shells via mesh-free kernel particle (kp) Ritz method under static and periodic axial forces. Singh et al. [7, 8] have inves-tigated the buckling response of composite cylindrical panels with random material properties. The dynamic response of functionally graded skew shell was investi-gated by GulshanTaj and Chakrabarti [9] using a C0 finite element formulation. J.N. Reddy’s higher-order theory was adopted in the analysis. Limited studies have been made on buckling of laminated composite skew panels.


C.V. Srinivasa et al.: Finite element studies on buckling of laminated cylindrical skew panels 553

Table 2 Convergence study for isotropic cylindrical panels sub-jected to uniform in-plane load (ν = 0.3, t = 0.096 in, E = 10 × 106 psi).

Cylindrical panel dimensions (in)

Critical buckling load (lb/in)

a b R CQUAD8 element size (mm)

10 5 3.33 2.5

100 10 1000 335.5 327.5 324 323.7100 20 1000 93.7 92.2 90.8 90.6

50 10 500 340.3 330.1 330.8 330.625 5 100 1353.7 1345.0 1342.1 1340.8

Table 3 Nondimensional critical buckling load parameter for isotropic cylindrical skew panels (Ø = 15°).

a/b

a/t

Nondimensional critical buckling load parameter (Kcr)

Skew angle (α)

0° 15° 30° 45°

0° 15° 30° 45°

S-S-S-S C-C-C-C

0.5 1000 17.375 18.613 23.088 34.345 34.512 36.423 45.039 67.936 500 8.786 9.403 11.659 17.559 17.366 18.420 23.196 35.450 100 2.175 2.345 2.949 4.437 3.751 4.084 5.258 7.983 50 1.025 1.123 1.490 2.515 2.316 2.546 3.429 5.833 20 0.599 0.660 0.895 1.580 1.799 1.952 2.539 4.124

1.0 1000 8.738 9.357 11.667 18.108 17.373 18.308 22.099 32.542 500 4.627 5.008 6.453 10.001 8.716 9.276 11.502 17.257 100 0.932 1.004 1.266 1.970 1.699 1.840 2.345 3.576 50 0.451 0.522 0.686 1.119 1.188 1.154 1.448 2.147 20 0.289 0.321 0.436 0.721 0.760 0.813 0.993 1.390

1.5 1000 6.215 6.718 8.534 12.941 11.591 12.240 14.807 21.479 500 3.506 3.725 4.484 6.283 5.773 6.165 7.618 11.297 100 0.607 0.662 0.873 1.412 1.160 1.244 1.547 2.285 50 0.413 0.450 0.566 0.845 0.780 0.833 1.019 1.460 20 0.289 0.310 0.386 0.553 0.549 0.578 0.676 0.880

2.0 1000 4.898 5.266 6.614 9.343 8.697 9.168 11.090 16.106 500 2.294 2.408 2.853 4.152 4.318 4.607 5.677 8.394 100 0.505 0.543 0.684 1.037 0.926 0.988 1.209 1.749 50 0.324 0.352 0.452 0.689 0.662 0.702 0.842 1.178 20 0.221 0.246 0.306 0.431 0.420 0.448 0.513 0.633

2.5 1000 3.866 4.021 4.641 6.123 6.950 7.344 8.908 12.955 500 1.618 1.738 2.218 3.489 3.491 3.723 4.579 6.742 100 0.426 0.463 0.599 0.871 0.790 0.843 1.027 1.467 50 0.305 0.334 0.412 0.391 0.578 0.614 0.739 1.013 20 0.208 0.223 0.273 0.064 0.347 0.360 0.399 0.466

elements are in good agreement with the literature values. From Table 1, it can be seen that CQUAD8 element yields better results when compared to the CQUAD4 element. Hence, CQUAD8 element is employed in the present work for carrying out further study.

2.2 Convergence study

To arrive at the optimum size of elements to be used in the finite element mesh for analysis, a convergence study was made. It was performed on simply supported isotropic cylindrical panels subject to axial compression. The con-vergence details are furnished in Table 2.

3 Results and discussionThe material properties used are E = 200 GPa and μ = 0.3 for isotropic material and representative high modulus graphite epoxy material as a laminated composite with (El = 206.91 GPa, Et = 5.173 GPa, Glt = 2.586 GPa, and vlt = 0.25) El/Et = 40, Glt/Et = 0.5, and vlt = 0.25. The results of the present work are presented in the form of nondimensional critical buckling load parameter (Kcr) as



Table 4 Nondimensional critical buckling load parameter for isotropic cylindrical skew panels (Ø = 30°).

a/b

a/t

Nondimensional critical buckling load parameter (Kcr)

Skew angle (α)

0° 15° 30° 45°

0° 15° 30° 45°

S-S-S-S C-C-C-C

0.5 1000 34.526 36.020 45.663 67.108 66.750 68.205 77.062 105.021 500 17.339 18.553 22.846 33.710 34.415 35.505 41.588 58.175 100 4.222 4.180 5.343 7.896 6.865 7.406 9.169 13.518 50 1.735 2.214 2.637 3.829 3.497 3.715 4.586 6.784 20 0.701 0.724 0.931 1.466 1.293 1.392 1.741 2.557

1.0 1000 17.363 18.557 22.853 33.708 34.416 35.500 41.575 58.170 500 8.719 9.326 11.592 17.787 17.287 18.195 21.979 31.494 100 2.124 2.213 2.638 3.824 3.497 3.715 4.587 6.784 50 0.872 0.938 1.186 1.828 1.585 1.708 2.149 3.187 20 0.353 0.388 0.519 0.832 0.825 0.881 1.076 1.498

1.5 1000 11.603 12.435 15.541 23.891 23.062 23.902 28.092 39.181 500 6.192 6.684 8.446 12.711 11.502 12.133 14.575 20.749 100 1.109 1.185 1.470 2.255 2.204 2.376 2.954 4.281 50 0.545 0.596 0.786 1.237 1.042 1.113 1.361 1.950 20 0.313 0.343 0.423 0.598 0.577 0.607 0.709 0.921

2.0 1000 9.112 9.803 12.295 18.730 17.295 18.010 21.257 31.020 500 4.865 5.222 6.525 9.213 8.589 9.060 10.881 16.152 100 0.824 0.899 1.188 1.773 1.566 1.675 2.067 3.117 50 0.442 0.475 0.595 0.926 0.810 0.859 1.032 1.510 20 0.243 0.261 0.322 0.470 0.440 0.461 0.527 0.688

2.5 1000 7.515 7.968 9.943 14.440 13.852 14.438 17.111 25.080 500 3.812 3.958 4.579 6.945 6.852 7.244 8.725 12.841 100 0.676 0.720 0.893 1.314 1.215 1.294 1.585 2.224 50 0.367 0.398 0.512 0.717 0.616 0.709 0.854 1.143 20 0.216 0.231 0.281 0.360 0.102 0.367 0.408 0.471

2

2

2

2 3

for isotropic cylindrical skew panels andcos

for laminated cylindrical skew panels cos

cr

crt

a tKD

aKE t

λσα π

λσα π

=

=

3.1 Isotropic cylindrical skew panels

Finite element analysis for critical buckling load has been carried out for skew panels with different panel angles, aspect ratios, skew angles, and length-thickness ratios, simply supported and clamped boundary conditions. The results obtained are tabulated in Tables 3 and 4 in terms of nondimensional critical buckling load parameter (Kcr) for panel angles of 15° and 30°, respectively. The following conclusions are made from Tables 3 and 4:

– Kcr decreases as aspect ratio (a/b) increases for constant values of skew angle (α) and a/t ratio

– Kcr increases as the skew angle α increases for constant values of a/b and a/t.

– Kcr decreases as a/t decreases for constant values of a/b ratio and skew angle.

– The panel angle is seen to have considerable influence on Kcr value. Kcr increases monotonically as panel angle increases.

3.2 Laminated composite cylindrical skew panels

3.2.1 Simply supported boundary condition

The variations of nondimensional critical buckling load parameter (Kcr) with skew angle (α), fiber orientation angle (θ), and NL for antisymmetric angle-ply cylindri-cal skew laminates under S3 boundary conditions (refer to [11] for definition of S3 boundary conditions) are pre-sented in Figures 3–6 for panel angle = 15° and Figures 7–10 for panel angle = 30°, respectively. The increase in Kcr with an increase in NL beyond NL = 10 is not appreciable in all the cases. There is considerable difference in Kcr between



80NL=2NL=4NL=6NL=8NL=10NL=16NL=20NL=24

70

60

50Kcr

40

30

200 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90

Fiber angle (θ)

Figure 3 Critical buckling load parameter for simply supported antisymmetric cylindrical skew panel (Ø = 15°, α = 0°).

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90

Fiber angle (θ)

80

90

70

60

50

Kcr

40

30

20

NL=2NL=4NL=6NL=8NL=10NL=16NL=20NL=24


0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90Fiber angle (θ)

80

90

100

110

70

60

50

Kcr

40

30



0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90Fiber angle (θ)

80

90

100

110

120

130

140

70

60

50

Kcr

40



0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90

Fiber angle (θ)

80

90

100

110

120

130

140

70

60

50

Kcr

NL=2

NL=4

NL=6

NL=8

NL=10

NL=16

NL=20

NL=24


0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90

Fiber angle (θ)

80

90

100

110

120

130

140

150

70

60

Kcr





0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90Fiber angle (θ)

80

90

100

110

70

60

50

40

30

Kcr



0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90Fiber angle (θ)

8090

100110120130140150160170180190200210220

70

Kcr



0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90Fiber angle (θ)

80

60

40

100

120

140

160

Kcr


Figure 11 Critical buckling load parameter for clamped antisymmet-ric cylindrical skew panel (Ø = 15°, α = 0°).

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90Fiber angle (θ)

90

70

50

110

130

150

170

Kcr


Figure 12 Critical buckling load parameter for clamped antisym-metric cylindrical skew panel (Ø = 15°, α = 15°).

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90Fiber angle (θ)

90

70

50

110

130

150

170

190

Kcr



0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90

Fiber angle (θ)

90

70

110

130

150

170

190

210

230

250


Kcr




0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90

Fiber angle (θ)

100

120

140

160

180

200

220

Kcr


Figure 15 Critical buckling load parameter for clamped antisymmet-ric cylindrical skew panel (Ø = 30°, α = 0°).

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90Fiber angle (θ)

90

70

50

110

130

150

170

190

Kcr



0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90

Fiber angle (θ)

90

110

130

150

170

190

210

230

250

Kcr



0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90Fiber angle (θ)


120

140

160

180

200

220

240

260

280

300

320

340

360

380

400

Kcr


NL = 2 and other values of NL. Kcr increases and reaches a maximum at about θ = 45°, 57.5°, 57.5°, and 52.5° for skew angles = 0°, 15°, 30°, and 45°, respectively, the panel angle being 15°. Kcr increases and reaches a maximum at about θ = 57.5°, 62.5°, 57.5°, and 57.5° for skew angles = 0°, 15°, 30°, and 45°, respectively, the panel being 30°.

3.2.2 Clamped boundary condition

For antisymmetric angle-ply cylindrical skew panels with skew angles of α = 0°, 15°, 30°, and 45°, the variations of Kcr with θ and NL are presented in Figures 11–14 for panel angle = 15° and Figures 15–18 for panel angle = 30°, respec-tively. For NL = 2, Kcr initially decreases and varies later as shown in aforementioned figures. The increase in Kcr with NL for NL > 10 is not appreciable. Kcr increases and reaches a maximum at about θ = 80°, 72.5°, 62.5°, and 57.5° for skew angles = 0°, 15°, 30°, and 45°, respectively, the panel angle being 15°. Kcr increases and reaches a maximum at about θ = 70°, 67.5°, 62.5°, and 62.5° for skew angles = 0°, 15°, 30°, and 45°, respectively, the panel angle being 30°. The boundary condition has a significant influence on the critical buckling load parameter, the clamped condition yielding higher value compared to the simply supported condition.

4 ConclusionsNew results on the buckling behavior of cylindrical skew laminated panels with various parameters and bound-ary conditions are presented in this work. The following



conclusions are made based on the above study of cylin-drical skew panels with simply supported and clamped boundary conditions.

4.1 Isotropic cylindrical skew panels

The aspect ratio, skew angle, panel angle, a/t ratio and boundary conditions have a considerable bearing on the value of the critical buckling load parameter of the panel. Kcr increases as the skew angle α increases for constant values of a/b and a/t. Kcr decreases as a/t decreases for constant values of a/b ratio and skew angle. Kcr increases monotonically as panel angle increases.

4.2 Antisymmetric cylindrical skew panels

The skew angle, panel angle, fiber orientation angle, NL, and boundary condition have a bearing on the value of critical buckling load parameter of the panel. The increase in Kcr with an increase in NL beyond NL = 10 is not appreci-able in all the cases. The critical buckling load parameter (Kcr) is minimum for NL = 2. If NL ≥ 4, then Kcr values are con-siderably different from that for NL = 2. Keeping the total thickness constant and increasing the NL, it is seen that the Kcr also increases. This is due to the reduction in the membrane-bending coupling stiffness terms B16 and B26. Eventually, for laminates with a very large NL, the behavior of laminated skew panels tends to that of an orthotropic

homogeneous one. The boundary condition has a signifi-cant influence on the critical buckling load parameter, the clamped condition yielding higher value to the simply sup-ported condition. The value of Kerr increases as fiber orien-tation angle increases, becomes a maximum at a particular value of fiber orientation angle, and then decreases for NL ≥ 4 for simply supported condition. The value of fiber orientation angle at which Kcr becomes maximum varies slightly with the panel angle for simply supported condi-tion. The variation in the case of clamped condition is dif-ferent and as depicted in the relevant figures.

Acknowledgments: The first author would like to thank the Management and Principal Dr. S.G. Hiremath of GM Institute of Technology, Davangere, Karnataka, India, for the kind encouragement and support provided. The second author would like to thank the Management of Jawaharlal Nehru College of Engineering, Shivamogga, Karnataka, India, for the kind encouragement and sup-port provided. The third author would like to thank the Management, Principal Dr. N. Ranaprathap Reddy and Head of the Department of Civil Engineering, Dr. Y. Ramalinga Reddy, Reva Institute of Technology and Management, Bangalore, Karnataka, India, for the kind encouragement and support provided.

Received August 28, 2013; accepted September 24, 2013; previously published online December 4, 2013

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Finite element studies on buckling of laminated cylindrical skew panels

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