Buckling load analysis of grid stiffened composite cylinders

Composites: Part B 34 (2003) 1–9 www.elsevier.com/locate/compositesb

Buckling load analysis of grid stiffened composite cylinders Samuel Kidanea,*, Guoqiang Lia, Jack Helmsa, Su-Seng Panga, Eyassu Woldesenbetb a

Department of Mechanical Engineering, Louisiana State University, Baton Rouge, LA 70803, USA b Department of Mechanical Engineering, Southern University, Baton Rouge, LA 70813, USA Received 26 June 2002; accepted 10 August 2002

Abstract Stiffened cylindrical shells are the major components of aerospace structures. In this study global buckling load for a generally cross and horizontal grid stiffened composite cylinder was determined. This was accomplished by developing an analytical model for determination of the equivalent stiffness parameters of a grid stiffened composite cylindrical shell. This was performed by taking out a unit cell and smearing the forces and moments due to the stiffeners onto the shell. Based on this analysis the extensional, coupling and bending matrices (A, B and D matrices, respectively) associated with the stiffeners were determined. This stiffness contribution of the stiffeners was superimposed with the stiffness contribution of the shell to obtain the equivalent stiffness parameters of the whole panel. Making use of the energy method the buckling load was solved for a particular stiffener configuration. Buckling test was also performed on a stiffened composite cylinder and compared with analytical results. Finally, using the analytical model developed, parametric analysis of some of the important design variables was performed and based on these results conclusions were drawn. q 2002 Elsevier Science Ltd. All rights reserved. Keywords: A. Polymer-matrix composites; B. Buckling; C. Analytical modeling; Grid stiffened cylinder

1. Introduction Excellent mechanical properties such as high-strength, low-weight, and tailorability make composite materials ideal for aircraft and aerospace application. Stiffened cylindrical shells play a big role in these industries in fuselage and fuel tank applications. This has resulted in an extensive research work in the field of composite cylinders with stiffening structures [1 –8]. Advanced foam-tooling materials like MARCOREe have also contributed to a cost effective means of making composite cylinders with integral stiffeners [1]. The buckling failure modes of a stiffened cylindrical shell can generally be categorized as global buckling, skin buckling, and stiffener crippling. Several methods have so far been developed to predict these failure modes. The different approaches can be broadly classified as (1) the discrete method, (2) the branched plate and shell approach, and (3) The smeared stiffener approach. In the discrete approach, stiffeners are modeled as lines of axial bending and torsional stiffness on the skin. This approach is difficult to use when the panel is stiffened in more than two * Corresponding author. E-mail address: [email protected] (S. Kidane).

directions and when the stiffener is not symmetric about the skin mid-surface. The branched plate and shell approach is more flexible and more accurate and usually involves the use of finite element analysis. However, the detailed spatial discretization of the finite element model is tedious, and the solution is computationally expensive [2]. The smeared stiffener approach uses a mathematical model to smear the stiffeners into an equivalent ply and determine the equivalent orthotropic stiffness of the ply. The smeared stiffener approach is applicable in general to stiffened panel where the local buckling load is equal or greater than the global buckling load. This approach, for preliminary design is consistent with the aeronautical design philosophy where a buckling-resistant design is the design goal [2]. Hence, it should be emphasized that the smeared model developed cannot be used to predict local failure modes like ‘local akin buckling’ or ‘stiffener crippling’. The application of the smeared technique on stiffened isotropic cylinders goes back to 1960s. Using the smeared method, Baruch and Singer discussed a problem of instability of a stiffened metallic cylinder under hydrostatic pressure [9]. A smeared stiffener theory that accounts for the skin – stiffener interaction was developed by Navin et al. [2]. In that work, Navin et al. presented a method for the derivation of

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neutral surface profile of the grid/shell assembly by using minimum potential energy principle and static conditions. However, their analysis was developed for a symmetric shell laminate and assumes a semi-infinite stiffened flat panel. Another work using the smeared approach was done by Phillips and Gurdal [3]. They analyzed the forces on a unit cell that represented the whole grid network and came up with equivalent stiffness parameters of the panel. The model developed was limited in the sense that it was restricted to symmetric panels, i.e. panels stiffened on both faces. This paper, on the other hand, develops a more general smeared model for determining the global buckling load of a stiffened composite cylindrical shell. This is done by considering the moment effect of the stiffeners in addition to force analysis performed on a unit cell. The model developed can be used not only for symmetrically stiffened panels, but also for unsymmetrically stiffened panels, i.e. panels stiffened only on one side. Furthermore, stiffened cylinders having either symmetrical or unsymmetrical shell laminates can be analyzed with equal ease.

2. Equivalent stiffness It is first required to determine the equivalent stiffness parameters of the overall structure in order to calculate the global buckling load of a composite cylinder with inner stiffening structure. This involves determination of stiffness contribution of the grid (stiffeners) as well as that of the shell. In developing the equivalent stiffness model, a unit cell is taken out of the stiffener structure and the stiffness contribution of the stiffeners to the total structure is determined. The unit cell is a representative unit such that the whole stiffener arrangement can be constructed by repetition of this unit cell (Fig. 1). In the process of determining the stiffness parameter contribution of the stiffeners to the total structure, the force and moment reactions of the stiffeners on the shell are analyzed as

a function of the shell mid-plane strains and curvatures ð10x ; 10u ; 10xu ; kx ; ku ; kxu Þ: The following assumptions are relevant in this analysis: 1. Since the transverse modulus of the unidirectional stiffeners is much lower than the longitudinal modulus, and as the cross-section dimensions are very small compared to the length, the stiffeners are assumed to support axial load only. 2. The strain is uniform across the cross-sectional area of the stiffeners. Hence, a uniform stress distribution is assumed across the cross-sectional area of the stiffeners. 3. The load on the stiffener/shell is transferred through shear forces between the stiffeners and shell. 2.1. Force analysis The mid-plane strains and curvatures of the shell are given by 10x ; 10u ; 10xu ; and kx ; ku ; kxu ; respectively. Based on laminated plate theory, the strains on the inner surface of the shell (the interface of the stiffener and the shell) are given by Eq. (1) [10]. 1x ¼ 10x þ kx ðt=2Þ;

1u ¼ 10u þ ku ðt=2Þ;

ð1Þ

1xu ¼ 10xu þ kxu ðt=2Þ where t is the thickness of the shell. The transformation matrix [11] is used to resolve these strains along the stiffener direction (1l), and normal to the stiffener directions (1t) (Fig. 1). 32 2 3 2 2 3 1x 1l c s2 sc 6 7 6 7 6 2 7 76 6 1t 7 ¼ 6 s 61 7 ð2Þ c2 2sc 7 4 5 4 54 u 5 1lt

22sc

2sc

c2 2 s2

1 xu

where c ¼ cosðfÞ; s ¼ sinðfÞ; and f is the orientation of the cross-stiffener with respect to the axial direction.

Fig. 1. Unit cell and coordinate system.

S. Kidane et al. / Composites: Part B 34 (2003) 1–9

3

Fig. 2. Force distribution.

The longitudinal strain (1l) expression given below by Eq. (3) is obtained from the transformation relation given by Eq. (2). This is the only strain of interest as transverse forces on the stiffener are neglected. 2

2

1l ¼ c 1x þ s 1u þ sc1xu

ð3Þ

The appropriate angle is substituted in Eq. (3) to obtain the strains along all the stiffener directions. Then applying assumptions (1) and (2), the forces corresponding to these strains are calculated according to Eq. (4). Fig. 2 shows the forces on the unit cell. F1 ¼ AEl 1l1 ¼ AEl ðc2 1x þ s2 1u 2 sc1xu Þ 2

2

F2 ¼ AEl 1l2 ¼ AEl ðc 1x þ s 1u þ sc1xu Þ

ð4Þ

F3 ¼ AEl 1l3 ¼ AEl ð1u Þ where El is the longitudinal modulus of the stiffeners. Summing up the circumferential (u ) and axial (x ) forces on the sides of the unit cell, one obtains: on the axial direction Fx ¼ F1 cosðfÞ þ F2 cosðfÞ

ð5Þ

on the circumferential direction: Fu ¼ F1 sinðfÞ þ F2 sinðfÞ þ 2F3

ð6Þ

Expression for the shear force1 ðFxu Þ is obtained by adding the forces along any directions of the unit cell. Performing this on the axial direction Fxu ¼ F2 cosðfÞ 2 F1 cosðfÞ

Substituting Eqs. (4) into Eqs. (5) – (7): Fx ¼ AEl cðc2 1x þ s2 1u 2 sc1xu Þ þ AEl cðc2 1x þ s2 1u þ sc1xu Þ ¼ AEl ð2c3 1x þ 2s2 c1u Þ

ð8Þ

Fu ¼ AEl sðc2 1x þ s2 1u 2 sc1xu Þ þ AEl sðc2 1x þ s2 1u þ sc1xu Þ þ AEl ð1u Þ ¼ AEl ðsc2 1x þ ð2s3 þ 2Þ1u Þ Fxu ¼ AEl cðc2 1x þ s2 1u þ sc1xu Þ 2 AEl cðc2 1x þ s2 1u 2 sc1xu Þ ¼ AEl ð2sc2 1xu Þ The smeared forces, i.e. the forces per unit length Nx ; Nu ; and Nux ; are obtained by dividing the above force expressions by the corresponding edge width of the unit cell. After performing this and substituting for the strain terms from Eq. (1), the expressions for the resultant forces on the unit cell are obtained.      AEl t t Nx ¼ 2c3 10x þ 2c3 kx þ 2s2 c10u þ 2s2 cku 2 2 a    AEl t 2sc2 10x þ 2sc2 kx Nu ¼ 2 b ð9Þ   t 3 0 3 þ ð2s þ 2Þ1u þ ð2s þ 2Þku 2    AEl t 2 0 2 Nux ¼ 2sc 1xu þ 2sc kxu 2 b

ð7Þ 2.2. Moment analysis

1

The same shear force expression will result even if the circumferential direction is used instead of the axial direction because of the geometrical relationship between a, b, cos(f ), and sin(f ).

The reaction moment due to the stiffeners is caused by the shear forces on the surface of the shell. Fig. 3(a) shows

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the different moments created by these forces. Only Msh is of main interest since it is the moment effect of the shear force on the shell. As observed from the free body diagram (Fig. 3(b)), a net moment M is produced on the shell/ stiffener assembly. This moment represents the coupling of moment and force resulting from the non-symmetric structure of the shell/stiffener arrangement. Following the same procedure as the force analysis on a unit cell (refer to Fig. 3(a) for moment free body diagram), the resultant moments on the circumferential and axial directions of the unit cell will be: Mx ¼ M1 cosðfÞ þ M2 cosðfÞ

ð10aÞ

Mu ¼ M1 sinðfÞ þ M2 sinðfÞ þ 2M3

ð10bÞ

Mxu ¼ M2 cosðfÞ 2 M1 sinðfÞ

ð10cÞ

Substituting F1 ðt=2Þ; F2 ðt=2Þ and F3 ðt=2Þ for M1 ; M2 ; and M3 ; respectively, and then simplifying results in expressions for the resultant moments on the unit cell.    AEl t t 3 0 3 2c 1x þ 2c kx Mx ¼ 2a 2   t þ 2s2 c10u þ 2s2 cku 2    AEl t t 2sc2 10x þ 2sc2 kx Mu ¼ 2b 2 ð11Þ   t þ ð2s3 þ 2Þ10u þ ð2s3 þ 2Þku 2    AEl t t 2sc2 10xu þ 2sc2 kxu M xu ¼ 2b 2

Fig. 3. Moment distribution.

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2.3. The equivalent stiffness matrix

5

Eq. (14) results. #" 0 # " # " N 1 As þ Ash l Bs þ Bsh ¼ s þ Bsh l Ds þ Dsh B M k

ð14Þ Eqs. (9) and (11) are, respectively, the force and moment contribution of the stiffener, henceforth denoted by the superscript s. These equations are written in a matrix form in where A sh, B sh, and D sh are the shell stiffness parameters Eq. (12): obtained from laminate theory. 3 2 2c3 2s2 c c3 t s2 ct 6 0 0 7 7 6 a a a a 7 6 72 6 3 2 3 2 3 2 s 3 7 6 2sc ð2s þ 2Þ sc t ð2s þ 2Þt 7 10x 6 Nx 0 0 76 6 b b b 2b 76 0 7 6 6 s 7 7 6 1u 7 6 6 Nu 7 7 7 6 6 7 2sc2 sc2 t 7 76 7 6 6 76 6 0 6 Ns 7 0 0 0 0 7 6 1u x 7 6 6 xu 7 b 7 b 7 76 7 6 6 ð12Þ 7 6 7 6 s 7 ¼ AEl 6 3 6 2 3 2 2 2 76 kx 7 6 c t 6 Mx 7 s ct c t s ct 7 7 7 6 6 7 0 0 76 6 a 6 s 7 7 a 2a 2a 76 6 6M 7 7 k 76 6 4 u5 u 5 4 7 6 sc2 t 3 2 2 3 2 ð2s þ 2Þt sc t ð2s þ 2Þt 7 6 Mxsu 7 kxu 6 0 0 7 6 b 2b 2b 4b 7 6 7 6 4 sc2 t sc2 t2 5 0 0 0 0 b 2b

The above matrix elements are the functions of the midplane strain and curvature parameters of the shell. We denote these stiffness parameters by Asij ; Bsij ; Cijs ; which corresponds to the extensional, coupling and bending matrices, respectively. Here, the superscript s stands for stiffener. At first glance the stiffness matrix given by Eq. (12) might seem non-symmetrical (i.e. Aij – Aji and Dij – Dji ; but due to the geometric relationship between the parameters a, b, cos(f ) and sin(f ), these stiffness quantities can be shown to be equal. It can also be observed that the same Bij elements result from the summation of forces and summation of moments analysis of the unit cell. This is in good agreement with laminate theory, hence further validating the initial assumptions. The total force and moment on the panel is the superposition of the force and moment due to the stiffener and the shell. As the stiffener force and moment contributions have been developed based on the mid-plane strains and curvatures, these quantities can be directly superimposed (Eq. (13)). "

N M

#

" ¼

N s þ N sh M s þ M sh

The resultant stiffness parameters obtained from Eq. (14) are thus the equivalent stiffness parameters of the whole panel. 3. Buckling load analysis The Ritz energy method is used to calculate the buckling load of the cylinder [10]. The total potential energy of the cylinder P, is composed of the strain energy U and the work done by the external force VðP ¼ U þ VÞ: This total energy is a function of the unknown parameters w, u, and v which are the displacements in the z (radial), x (axial) and u (hoop) directions, respectively. These parameters are represented by a double Fourier series that satisfies the boundary condition requirement. For a simply supported end condition they are given as follows [4], u¼ v¼

# ð13Þ

N sh and M sh are the force and moment contribution of the shell, respectively. Laminate theory is used to obtain these resultant forces and moments. Substituting the force and moment expressions for the stiffener network from Eq. (12), and the force and moment expressions for the shell from the laminate theory, the panel constitutive equation given by

w¼

1 X 1 X m¼1 n¼1 1 X 1 X

Amn cosðmxÞ  sinðnsÞ Bmn sinðmxÞ  cosðnsÞ

m¼1 n¼1 1 X 1 X

ð15Þ

Cmn sinðmxÞ  sinðnsÞ

m¼1 n¼1

where m  ¼ mp=L; n ¼ n=r; s ¼ r u; L is the height of cylinder and m; n ¼ 1; 2; 3; …: While for a clamped boundary condition, the expression for u, v, and w are given by Eq. (16) u¼

1 X 1 X m¼1 n¼1

Amn cosðmxÞ  sinðnsÞ

6

v¼

S. Kidane et al. / Composites: Part B 34 (2003) 1–9 1 X 1 X

Bmn sinðmxÞ  cosðnsÞ

ð16Þ

m¼1 n¼1

w¼

1 X 1 X

Cmn ð1 2 cosðmxÞÞ sinðnsÞ 

m¼1 n¼1

where m  ¼ mp=L; n ¼ n=r; s ¼ r u; and m; n ¼ 1; 2; 3; …: Because u, v and w are kinematically admissible functions, i.e. displacement satisfying boundary conditions, the total potential energy P becomes a function of Amn ; Bmn ; and Cmn : For the equilibrium to be stable, the total potential energy must be a minimum. This can be satisfied by finding the first derivative of the total potential energy P with respect to the unknown constants Amn; Bmn ; and Cmn and equating to zero. This results in an eigenvalue equation. A Matlab code was developed to determine the buckling loads satisfying this condition. Infinite number of eigenvalues satisfy the expression for the buckling load. The minimum eigenvalue is the buckling load of the structure.

4. Experimentation A buckling test was performed to assess the reliability of the analytical model developed. The specimen tested was an iso-grid stiffened composite cylinder made by filament winding process. The mechanical properties and other significant parameters of the tested specimen are presented in Table 1. The test was performed on an Instron MTS machine. The specimen was placed between two rigid steel plates with cushioning material between the plates and the cylinder to avoid crushing of the rims. It is noted that even though the sample was simply supported the transverse frictional force between the plates and the cylinder cannot be avoided, particularly due to the introduction of the cushion material. Hence, the end conditions simulated in the experiment are considered to be somewhere between the clamped and simply supported end conditions. The test was conducted in a displacementcontrolled mode with loading rate of 0.26 mm/s. The displacement vs. load data was obtained from a data

Table 1 Physical property of specimen tested Composite system Cylinder height Cylinder diameter Shell-winding angle Stiffeners orientation Horizontal stiffener spacing Cross-stiffeners spacing Shell thickness Stiffener cross-section

IM7/977-2 180 mm 146 mm ^308 08, þ 608, 2608 38.5 mm 42.5 mm 0.3 mm 6 £ 2.8 mm2

acquisition interface. Strains on the surface of the cylinder were also measured at two positions, one at mid-height and the other closer to the rim of the cylinder. The cylinder under discussion failed by a global buckling mode at 88.0 kN, at which point the highest strains were measured by the strain gauges. Pictures of the specimen before testing and after buckling are shown in Fig. 4(a) and (b). The experimental load and strain results obtained have been plotted in Fig. 5.

5. Results and discussion Using the models developed in Sections 2 and 3, analytical buckling load calculation was carried out for the specimen tested. The calculation was performed for clamped and simply supported end conditions. The buckling load for the clamped model was found to be 107.9 kN, while the buckling load for the simply supported model was 82.8 kN. The experimental buckling load (88 kN) is observed to be between the calculated values of clamped and simply supported end conditions. Hence, given the complexity of buckling problems, the fact that this buckling load occurred between the relatively close analytical values of the two end conditions indicates the reliability of the analytical model. Based on the clamped end model, a parametric study was performed on some important design variables. The parameters analyzed are stiffener orientation and cross-sectional area, stiffener spacing, and shell-winding angle and thickness. Stiffener orientation angle and cross-sectional area. While maintaining the circumferential stiffeners, the angle of the cross-stiffeners was varied from completely axial (08) to completely circumferential (908). The result shows (Fig. 6(a)) the load smoothly increases with the increase in stiffener orientation angle up to approximately 428 and starts to drop after that. Hence, we can conclude that 428 is the optimum stiffener orientation angle for this particular sample. With the increase of the stiffener cross-sectional area, the buckling load also increases. This is in agreement with the expectation. In Fig. 6(a) it can also be observed that the buckling load is higher at 08 than at 908 stiffener orientation. This is in agreement with physical intuition, as all axial stiffeners would result in a more axially stiff cylinder compared to all circumferential stiffener orientation. Stiffener spacing. From Fig. 6(b) the parameters a and b of the unit cell were varied continuously to simulate the circumferential and axial spacing variation of the stiffeners. The result shows the expected load drop as the stiffeners are spaced wider. It can also be noted that as the stiffener spacing is increased further the decrease in the buckling load becomes less and less, which means the contribution of the stiffener becomes small and the skin becomes the main load bearing component.

S. Kidane et al. / Composites: Part B 34 (2003) 1–9

Fig. 4. Test specimen: iso-grid stiffened composite cylinder.

Fig. 5. Experimental result.

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S. Kidane et al. / Composites: Part B 34 (2003) 1–9

Fig. 6. Effect of variation of stiffener parameters.

Fig. 7. Effect of variation of shell-winding angle and thickness.

S. Kidane et al. / Composites: Part B 34 (2003) 1–9

Shell-winding angle and thickness. The shell-winding angle and thickness were also varied continuously and the effects on buckling load are plotted in Fig. 7. The shellwinding angle was varied from completely circumferential fibers (08) to completely axial fibers (908). The result shows the buckling load continuously increases with the increase of winding angle and shell thickness. It is noted that for a non-stiffened cylinder, the optimum winding angle is 458. The optimum angle of 908 in this study is due to the stiffness contribution of the stiffeners to the skin. The buckling load resistance is also noted to continuously increase with increase in shell thickness.

6. Conclusion An analytical model was developed using the smeared method. The model developed was shown to be reliable by performing a buckling load experiment. A parametric study was also performed based on the analytical model developed. The results obtained from the parametric study show reasonable and expected trends, hence further validating the analytical model developed. The calculated results show that stiffener orientation has a significant effect on the buckling load. For the specific specimen analyzed this angle was found out to be approximately 428. Increasing the stiffener spacing results in a decrease in buckling load. However, this decrease becomes insignificant with large enough stiffener spacing. In addition, buckling load varies as the skin-winding angle and thickness changes. Increasing the winding angle and thickness of the skin enhances buckling resistance.

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Acknowledgements Support for this research was provided by a grant from the NASA/Louisiana Space Consortium and the Louisiana Board of Regents under LaSPACE (BOR12662) and LEQSF (2001-04)-RD-B-03. The authors are also very grateful to Mr Brett H. Smith (NASA/MSFC) for providing technical support and test specimens.

References [1] Helms JE, Li G, Smith BH. Analysis of grid stiffened cylinders. ASME/ETCE 2001;. [2] Navin J, Norman FK, Damodar R. Formulation of an improved smeared stiffener theory of buckling analysis of grid-stiffened composite panels. NASA Technical Memorandum 110162; June 1995. [3] Phillips JL, Gurdal Z. Structural analysis and optimum design of geodesically stiffened composite panels. Report NASA CCMS-90-50 (VPI-E-90-08), Grant NAG-1-643; July 1990. [4] Brush DO, Almroth BO. Buckling of bars, plates, and shells. New York, NY: McGraw-Hill; 1975. [5] Bruhn EF. Analysis and design of flight vehicle structures. Carmel, IN: Jacobs Publishing; 1973. [6] Ramm E. Buckling of shells. Berlin: Springer; 1982. [7] Gerdon G, Gurdal Z. Optimal design of geodesically stiffened composite cylindrical shells. AIAA J November 1985;23(11): 1753– 61. [8] Troisky MS. Stiffened plates, bending, stability and vibration. Amsterdam: Elsevier; 1976. [9] Baruch M, Singer J. Effect of eccentricity of stiffeners on the general instability of stiffened cylindrical shell under hydrostatic pressure. J Mech Sci 1963;6(1). [10] Whitney JM. Structural analysis of laminated anisotropic plates. Lancaster: Technomic; 1987. [11] Agarwal BD, Broutman LJ. Analysis and performance of fiber composites. New York: Wiley; 1990.