Buckling Analysis of Ring-Stiffened Laminated Composite Cylindrical Shells by Fourier-Expansion Based Differential Quadrature Method

Transcription

1 Applied Mechanics and Materials Vol. 225 (2012) pp (2012) Trans Tech Publications, Switzerland doi: / Buckling Analysis of Ring-Stiffened Laminated Composite Cylindrical Shells by Fourier-Expansion Based Differential Quadrature Method Saeed Barani 1,a, Davood Poorveis 2,b and Shapoor Moradi 3,c 1,2 Department of Civil Engineering, Shahid Chamran University, Ahvaz, Iran 3 Department of Mechanical Engineering, Shahid Chamran University, Ahvaz, Iran a seebb86@yahoo.com, b dpoorveis@scu.ac.ir, c moradi@scu.ac.ir Keywords: differential quadrature, buckling, ring stiffened, smeared model, virtual work Abstract. This article focuses on the application of the Fourier-expansion based differential quadrature method (FDQM) for the buckling analysis of ring-stiffened composite laminated cylindrical shells. Displacements and rotations are expressed in terms of Fourier series expansions in longitudinal direction and their first order derivatives are approximated with FDQM in circumferential direction. The 'smeared stiffener' approach is adopted for the stiffeners modeling. Two FORTRA programs prepared for linear and nonlinear analysis and results were compared by ABAQUS finite element software. Buckling loads of stiffened and unstiffened shells considering the effects of changes in shell and stiffener geometric and material properties and also shell lay-ups are investigated. Introduction Today the development of new techniques from the standpoint of computational efficiency and numerical accuracy is of primal interest. The differential quadrature method (DQM) was introduced by Bellman et al. [1] in the early seventies. DQM has been shown to be a powerful contender in solving initial and boundary value problems and thus has become an alternative to the existing methods such as the finite elements or finite differences. One of the areas among which one can find extensive applications of DQM is structural mechanics. Stiffened thin cylindrical shells subject to axial compression are widely used in aerospace structures and the offshore-oil industry. Researches on the buckling and post-buckling of ringstiffened cylindrical shells in the past three decades have increased. Sheinman, Shaw and Simitses [2] used a smeared stiffener technique for orthogonally stiffened laminated cylinders. In applying the DQM to analysis of shells, Mirfakhraei and Redekop [3] used one-dimentional DQM for Buckling of circular cylindrical shells. Jiang, Wang and Wang [4] applied DQM for Buckling analysis of stiffened circular cylindrical panels. To the authors' best knowledge, to date, there has been no reference in applying the FDQ to solve the problem of buckling of ring-stiffened laminated composite cylindrical shells subject to axial compression. Various ring-stiffened composite cylinders, differing in the number of stiffeners and laminate architectures, are analyzed to estimate the range of applicability of the smeared stiffener model. Differential Quadrature Method As mentioned earlier, the differential quadrature method (DQM) was presented for the first time by Bellman et al. [1], for solving differential equations. The DQM uses the basis of the Gauss method in deriving the derivative of a function. It follows that the partial derivative of a function with respect to a space variable can be approximated by a weighted linear combination of function values at some intermediate points in that variable. In order to show the mathematical representation of DQM, consider a function f=f(x) on the domain a x b then the nth-order differential of the function f at an intermediate point x i can be written as: All rights reserved. o part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of TTP, (ID: /11/12,10:05:15)

2 208 AEROTECH IV n d f(x i) = C n f x i=1,,, n=1,,-1 dx n k1 ik k where n C ik are the weighting coefficients of the nth derivative in the domain that is divided by discrete points [5]. Fourier Expansion-based Differential Quadrature (FDQ). ow, it is supposed that f(x) is approximated by a Fourier series expansion, we can use the following equations to compute the non-diagonal weighting coefficients of the first, second and third order derivatives: (1) q(x i) Cij i, j 1,...,, j i xi xj 2sin q(x j) 2 (2) (1) x (1) (1) i xj Cij Cij 2Cii Cij cot i, j 1,...,, j i 2 (3) (1) 1 (2) x (1) (1) i xj Cij Cij 3Cii 2Cii Cij cot i, j 1,...,, j i. 2 2 It is interesting to realize that Eq. 2, Eq. 3 and Eq. 4 can be applied to periodic and non-periodic problems. For non-periodic problems, the range of x in the computational domain should be 0 x while for periodic problems, the range of x in the computational domain must be 0 x 2. Using linear vector space properties, we can compute dimensional weighting coefficients as follow ij (n) (n) (n) ij ii ij ij, ji C 0 C C i 1,...,. (5) The FDQ method uses the following formulation to determine the coordinates of the grid points i1 xi 2 i 1,...,. (6) When Eq. 6 is applied, the periodic condition is automatically satisfied by the FDQ method. As for the PDQ method, the periodic condition is implemented in such a way that the continuity of the function and its first order derivative is enforced at x=0 and x = 2π [6]. (1) (2) (3) (4) Shell Theory Sanders s nonlinear shell theory is used and the formulation accounts for the effects of shear deformation and bending-extension coupling.the vitual work principle is used to derive equations for both linear and nonlinear methods. The linearized equations can be solved using iterative ewton-raphson and arclength methods. In the smeared stiffener approach, shell rigidities are modifed based on material and geometrical properties and the spacing of stringers. Then, analysis, such as that of an unstiffened shell, is carried out. Displacements and Rotations The displacement of an arbitrary point of the cylindrical shell is expressed as: U x,,z U x, z x,, V x,,z V x, z x,, W x,,z W x, (7) x

3 Applied Mechanics and Materials Vol where Ux,, Vx, and circumferential and radial directions, respectively, and and x, normal vector to the surface. W x, are displacements of the mid-surface of the shell in axial, x x, are the rotations of the Approximations. For classical simply supported end conditions, displacements and rotations are expressed in terms of Fourier series expansions in longitudinal direction and their firs order derivatives are approximated with FDQM in circumferential direction: M M M U(x, ) U cos( x), V(x, ) V sin( x), W(x, ) W sin( x), mi m mi m mi m m1 m1 m1 M (x, ) cos( x), (x, ) sin( x) x xmi m mi m m1 m1 M in which m m / L. U x0, U mn, V mn, W mn, xmn and mn are the unknown coefficients. Subscript m is for the longitudinal harmonics and i is for the circumferential points. Then we can obtain the unknown coefficients for all of the points and harmonics. ow we obtain the first derivative of the U x, with respect to, we have M U (x, ) C U com( x) (9), in mn m m1 n1 in Eq. 13 is the number of circumferential points and and will be obtained by Eq. 2, Eq. 3 and Eq. 4. (8) Cin are the weighting coefficients of FDQM Strain-Displacement Relationships. The strains at an arbitrary point of the shell thickness can be expressed as follows: e z, e z, e z (10) xx xx xx x x x where strains of mid-surface and curvatures, based on Sanders and the first order shear deformation theories are as follows (for C=0, we have Donnell s shell theory): 2 2 u 1 w 1 v w 1 w xx, Cv 2 x 2 x R R 2R 1 u v 1 w w v w x C R x R x R x 1 x w 1 w v xx,, x, xz x, x C. x R x x R R Virtual Work The equilibrium equations of cylinder are obtained by combining the internal virtual work and external virtual work of shell due to applied loads: W W W 0. (12) int ext The internal virtual work of the cylindrical shell in terms of generalized stresses and strains is [7]: T (13) W dv ( M int v s x x x x x x M M Q Q )ds x x x xz x and external virtual work of shell due to axial compression will be: 2 Wext P[ U(L, ) U(0, )]Rd. (14) 0 The non-linear equations are solved by the ewton-raphson method. (11)

4 210 AEROTECH IV Ring Smeared Model Shell rigidities are modified based on material properties, cross-sectional dimensions and spacing of rings. The ring-stiffened shell rigidities, according to Fig. 1 and Ref. [2] are: A11 A12 A B11 B12 B A A A A 0 E A / S 0, B B B B 0 E A e / S p p A16 A26 A B16 B26 B D11 D12 D16 2 D D12 D22 D 26 0 E1 I1 A1e 1 / Sp 0 D16 D26 D G 1J 1 / S p / 4 where E 1 A 1 and G 1 J 1 are the extensional and torsional rigidities of the ring in the circumferential direction, respectively, I 1 is the moment of inertia of the ring stiffener cross-section about its centroidal axis and e 1 is the ring eccentricity (positive outside). Table 1: Axial buckling load of isotropic cylindrical shell [K/mm 2 ] ABAQUS EVFDQ (Sanders theory) EVFDQ (Donnell s theory) LFDQ (Donnell s theory) 9.85 umerical Result and Discussion Fig. 1 Longitudinal section of the stiffened shell The described analysis is used as a base for the development of two computer program, EVFDQ and LFDQ, for both linear and nonlinear procedures. The numerical study is designed to compare the results of miscellaneous ring-stiffened and unstiffened cylindrical shells, different in shell layups, ring geometry with those of ABAQUS finite element software. Unstiffened Isotropic Cylindrical Shell. Fig. 2 shows the effect of number of grid points on the linear buckling load of two unstiffened isotropic cylindrical shells, respectively by Sanders s and Donnel s shell theories, that M is number of nonzero terms of Furier series expansion in longitudinal direction. This investigation demonstrates the capability of the selected numerical method. Material properties are E=2000 K/mm 2, ν=0.3, L=R=100 mm and h=1mm. (15) (a) (b) Fig. 2 Buckling load for isotropic cylindrical shell by: (a) Donnell s shell theory, (b) Sanders s shell theory

5 Applied Mechanics and Materials Vol Fig. 3 Axial buckling load for stiffened isotropic cylindrical shell Table 2: Stiffeners properties for isotropic cylindrical shell (Fig. 1) o. of Case ts d Sp stiffeners For both theories, we obtain the buckling load by first 2 nonzero Harmonics and 15 point that is considerably less than degrees of freedom of finite element method, which means less calculation time respectively. Table 1 compares the answers calculated by different methods. Stiffened Isotropic Cylindrical Shell. ow we investigate the effect of internal ring stiffeners on the axial buckling load of an isotropic cylindrical shell with E=75000 K/mm 2 ν=.3, L=R=100mm and h=1mm. equally spaced stiffeners with same material properties of shell, total area of 10 mm 2 and h s /t s =1 (for all cases), added to the internal face of shell. Table 2 shows ring dimensions for 8 cases. Fig. 3 shows the buckling loads for all cases, calculated by three different programs. From Fig. 3, increasing the number of stiffeners, the buckling load decreases (total area of stiffeners is 10mm 2 ), and also for the case 1, the buckling load from ABAQUS software, has a considerable difference with that of EVFDQ, that shows we have local buckling of shell skin between stiffeners, and the smeared model cannot capture it. Laminated Composite Cylindrical Shell. The influence of fiber orientation on the buckling strength of laminated composite cylindrical shell is presented in Fig. 4. This study considers symmetric and anti-symmetric laminates of (θ,-θ) s and (θ,-θ) 2T with E 11 =E 22 =23.45 GPa, G 12 =1.52 GPa, ν 12 =.2, R = mm, L= mm and h=1 mm. (a) Fig. 4 Total buckling load for different angles of fiber orientations: (a) symmetric lay-ups (θ/-θ) s, (b) anti-symmetric lay-ups (θ/-θ) 2T (b)

6 212 AEROTECH IV Summary The differential quadrature method was applied to the stability of laminated composite circular cylindrical shell subjected to axial compression considering first order shear deformation theory along with sanders nonlinear shell theory. In some context parametric studies were carried out including comparison of linear and nonlinear buckling analysis, the effect of number of sampling points in the circumferential direction on convergence of buckling load, geometric properties of ring stiffeners and the effect of shell lay-ups on axial buckling load. The results show that the FDQ method due to using total sampling points to approximate displacements derivatives has higher convergence rates than that of FEM method. The results also confirm that axial compression buckling load obtains by geometrical nonlinear analysis differs substantially with that of linear buckling analysis. In another analysis axial buckling load for two symmetric, (θ/-θ) s,and antisymmetric, (θ/-θ) 2t shell lay-ups were tested for various angles which in both cases maximum load was obtained for 20 θ 25 degrees. In all cases the results of ABAQUS finite element software were in good agreement with the present study. References [1] R.E. Bellman, B.G. Kashef, J. Casti, Differential quadrature: a technique for rapid solution of nonlinear partial differential equations, J. Comput Phys. 10 (1972) [2] I. Sheinman, D. Shaw, G.J. Simitses, onlinear analysis of axially loaded laminated cylindrical shells, J. Comput. & Struct. 16 (1983) [3] P. Mirfakhraei, D. Redekop, Buckling of circular cylindrical shells by the Differential Quadrature Method, J. Pressure Vessels and Piping 75 (1998) [4] L. Jiang, Y. Wang, X. Wang, Buckling analysis of stiffened circular cylindrical panels using differential quadrature element method, Thin-Walled Structures 46 (2008) [5] S. Moradi, F. Taheri, Application of differential quadrature method to the delamination buckling of composite plates, J. Comput. & Struct. 70 (1999) [6] C. Shu, Differential Quadrature and It s Application in Engineering, Springer, London [7] D. Poorveis, M.Z. Kabir, Buckling of Discretely Stringer-Stiffened Composite Cylindrical Shells under Combined Axial Compression and External Pressure, J. Scientia Iranica 13(2006)