SHELL STRUCTURES. THEORY AND APPLICATIONS

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Maude Glenn
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1 The 5th Conference SHELL STRUCTURES. THEORY AND APPLICATIONS Janowice * October MODERATE ROTATION THEORY FEM ANALYSIS OF LAMINATED ANISOTROPIC COMPOSITE PLATES AND SHELLS I. Kreja 1 ) and R. Schmidt Fachgebiet Baumechanik Bergische Universitat 5600 Wuppertal Germany 1) On leave from Technical University of Gdansk Poland.

6 Hinged cylindrical roof under center load p u v ~1 0.9 present analysis (2x2 9-SRI) Sablr &: Lock Horrigmoe &: Bergan h s:::. L&J... '-'" c.. h- "' II) c -II) (.) 0.1 R in L = 10 rn f1 = 0.1 rad E ksi in II ~~~~~~~~~~~~~~~tnll~"nlTn~IM" Cenier deflection (in)

7 Hinged orthotropic cylindrical roof under point load p R - L p - h E1 100 in loin 0.1 red 1 in = 40 msi E2 = 1 msi G12 = G13 = 0.6 msi G23 = 0.5 msi V12 = SRI elements mesh 4x4 - I 2 layers 0/90 I loyers 9%0 I - - loyer fo - ~ layer 90 90).; 0.; 0 0.; /.- / 60.0 '-" /.; / / II).;.; 0... / / 40 / II) / c::./ II) 0././ / center deflection (in) /

8 Short cylinder under internal pressure R = L h 1 in 20 in Boundary conditions: fixed at 9 1 = + / - O.SL 12 ~ATERIAL DATA: GLASS-EPOXY E=7500 ksl. E:=2000 ksr G 12 = 1250 ksi v12=o.25 G23=G1~= 625 ksi 10 /' /' -"iii /' /' ~ 8 /'..... /'/' ::J '- /' /' ' a.. g SRI mesh 1x SRI mesh 2x SRI mesh 4x1 c Chong & Sawamiphakdi axisymmetric analysis -' O~~~~~~TMTn~nTrrnTMTnTITMTrrn~TnTITPT~~T"~~ radial displacement (in)

9 Deformation profile p=o.4 h=o.l linear analysis -..r:.; 0)..r: NASH 9~0 00[JCJ0 SHELL 0/ SHELL 0/ _... : :! : : :. : I :. :.. NASH radius radius ; L.---~~~r+"~TT"rr~'-TT~~TT"~TT"rr~'-rr~" Deformation profile p=o.4 h=o.l nonlinear analysis -..r:.; 0)..r: " " II radius ~-...""...t. : I. NASH 0 90 : : SHELL % SHELL NASH 9~0 L ' radius

10 Deformation profile p=15.0 h=1.0 linear analysis -.s:: 0) CD.s:: _ _.._-_..._ : :. I I r"-"-"l : : i : : i ~ :. '. ' : ~ '.'.. I I ' I.' ' I. NASH 9~0 '. NASH I I ' : ' DDDCIJ SHELL 010 I I '! SHELL 0/90! : ~ m...r_+..._..._.._..._r_t...t""1r_t..._...r_t""t"" radius radius Deformation profile p= 15.0 h= 1.0 nonlinear analysis -.s:: 0) CD.s:: 10.0 :.._ r DO. : f I I ' J t t I J I : I : :! : : I i ' i : : 8.0 I I t I I : : f iii i t I I i : : : : I I I ~. I It I. I I ' I I : : : II II I I I I I I I I I I I I I I ' I I I ' I I I I f : : I i I i i NASH 9~O i : i --I-r--r--1--i--lIlJ-r.!-...O...OOOO SHEll _lih-r-r-r...-t-r r-t"t"t...-rr-r-r...-r radius radius

11 Cross-ply (0/90) simply supported (BCl) spherical cap R = 1000 in a = 50 in h 1 in E1 = 25 msi E2 = 1 msi Gl2 - G 15 = 0.5 msi G:z:s msi V12 = 0.25 Boundary conditions - Bel: v - w at x = +/- Q U = W = +1 = 0 at y = +/- (J 3 MRT analysis 9-node elements / SRI mesh 2x2 / -- 9-SRI mesh 4x4 o Q a a 0 9-URI mesh 2x2 o::::x:x:x) Liao &: Reddy / / O~~~~~~~~~~~~~~~~~~~rT~rr~~1TI o Center deflection (in)

12 Cross-ply (0/90) simply supported (BC3) spherical cap R 1000 in a 50 in h = 1 in E1 = 25 msi E2 = 1 msi G I2 G u = 0.5 msi G msi 1112 = 0.25 Boundary conditions - BC3: u w = it2 0 at x = +/- a v = w = it1 = 0 at y = +/- a 4 WRT analysis 9-node elements 9-SRI. mesh 2x2 9-SRI mesh 4x4 3 a a a a a 9-URI. mesh 2x2 cx:x:xx:> Lioo &: Reddy lilt Q. 1 o~~~~~~~~~~~~~~~~~~~~~~~~~ Center deflection (in)

13 Square sandwich clamped plate under pressure I Ị : ~ I ~+.-!-.- P SO" I I p Facing: 1 E=10S-106 lblin2 I V= Core: E:: 0 Gxz = Gyz = SO'103lb/in2' - J S" 1 1" 001S" f' '-'"... '"' 0-0." J. - 9-SRI mesh 2x URI mesh 2x2 9-SRI mesh 4x URI mesh 4x4 <XX)QQ Chang &: Sowamiphakdi [12) Load foct~: Q = 12po (1 _2)/(E t t!) JlRT5 a.nallln -- _9 --- lineat solution O~~~~~~~~~rn~~~~~~nTrnllnl~~rrn Central deflection (in)

16 I'd like to present here a short report on the Moderate Rotation Theory FEM analysis of laminated anisotropic composite plates and shells. This report has been prepared by me and dr. Schmidt at the Bergishe Universitat in Wuppertal its contents refers to the last word in the title of this conference - I mean APPLICATIONS - so I'm not going to present here the whole theoretical background. Let me just say that our numerical algorithm based on the Moderate Rotation Shell Theory proposed by Schmidt & Reddy in the Journal of Applied Mechanics in he basic assumptions of this theory are as follows: - Large displacement are considered only in range of small elastic strains. - Additionally the in-surface rotations are assumed to be small - while the rotations of the normal can be moderately large - it means - they are larger compared to the strains but of such magnitude that their squares and products are small compared to unity. - The first order shear deformation theory with the assumption of the inextensible director have been adopted. - According to the order-of-magnitude assumption on strains the terms of relative order e square compared to strains are neglected. Our numerical algorithm can be characterized in the following way: - The Total Lagrangian formulation together with the virtual work principle have been used to obtain incremental equilibrium equations. - An explicit pre-integration in the thickness direction has been applied to obtain the constitutive relation expressed in stress and strain resultants. - The nine node isoparametric element with selective reduced integration has been employed in the FEM discretization of the problem. - The standard Newton-Raphson iterations with Riks-Wempner-Ramm arc-length control method have been adopted to solve nonlinear 1

17 equilibrium equations. - This numerical algorithm has taken a final form of the computer code prepared in Fortran and implemented on a personal computer. A presentation of the numerical results I'd like to begin with the standard test example for nonlinear analysis of isotropic shells. This hinged cylindrical roof under center load has been studied first by Sabir & Lock. The calculations performed for three values of the thickness have resulted in the three different forms of the structure response: stiffening behaviour for the thickness equal one inch snap-through for h equal half inch and snap-through with snap-back for the thickness equal.25 inch. Our results obtained with the four element discretization for the one quarter of the shell illustrated with the solid lines agree quite well with the solutions of Sabir & Lock (the circles) and Horrigmoe & Bergan (the squares) except the last case h equal.25. For this last case a substantial improvement of the results has been obtained when the number of elements increased to 16. In the next example we have considered the same geometry of the cylindrical roof but this time the shell was built of the highly orthotropic material. Four models have been analysed: First two there were 2-layer models with lamination schemes (0/90) and (90/0) respectively. Let me explain herethat the lamination scheme described as (0/90) means that in the bottom layer the material axes coincide with the axes of the shell coordinate system ( ) whereas in the upper layer they are rotated by 90 degrees. For comparison reasons two 1-layer models are also included in this graph.. As one can see only the first model response is characterized by the snap-through behaviour. A short glass-epoxy cylinder clamped at both ends and subjected to the uniformly distributed internal pressure has been analysed before by Chang & Sawamiphakdi. Due to the axi-symmetry one band of shell elements with proper boundary conditions has been used in the discretization. Two elements in mesh were enough to obtain the convergent solution which is in a good agreement with the results of Chang & Sawamiphakdi. However please notice small difference between these solutions and the results of the axisymmetric analysis performed 2

18 with the degenerated finite elements as it will be shown in the next presentation by Mr. Bodefeld. This disagreement between results of our shell element model and the degenerated axi-symmetric shell element is even more evident in the case of laminated cylinder. Here we have deformation profiles for the two layers model with lamination schemes (90/0) and (0/90) lines represent results obtained with the degenerate elements symbols stand for results of our shell element. On the left hand side the real scale deformation profiles are presented on the right we have enlarged deformations to make the differences more visible. One can observe some differences in nonlinear as well as in linear solutions. At this stage we would like to remind that the thickness of the analysed shell was equal 1 inch and the radius 20 inches. We repeated above analysis for the thinner shell thickness equal.1 inch and then the results were in much better agreement as we can see on this transparent. In the next example a laminated spherical shell has been analysed. The cap shown in the picture is formed by four vertical cutting planes acting on the sphere. Two layers of the composite shell are made of the same orthotropic material but a different ply orientation has been used in each layer. The lamination scheme is described as (0/90). Two types of simply support boundary conditions have been considered after Liao & Reddy. The first type BCl can be characterized by the remark that at the supported edges only the translations perpendicular to the edge are free. In such case we can observe the stiffening type of behaviour of the structure. In the second type of boundary conditions - BC3 only tangential translations are free at the supported edges. The response of the structure demonstrates now the snap-through behaviour. We would like to point out that in those both analyses of spherical shells it was necessary to use 16 9-SRI elements in mesh to obtain the convergent solution. On the other hand an application of only 4 elements with uniformly reduced integration can produce almost the same solution. 3

19 In the last example we would like to illustrate the possibility of the proposed numerical algorithm to handle problems of sandwich structures with a core possessing only the transverse shear stiffness. The square clamped plate is loaded by the uniformly distributed load. The aluminum honeycomb is bonded between two very thin aluminum facings. The results obtained with 4 elements are in quite good agreement with the reference solution of Chang & Sawamiphakdi. Almost no difference between results for selective and uniformly reduced integration has been detected in this example. This report could be ended with the following conclusions: The FEM algorithm presented here appeared to be an efficient tool for the large deformation numerical analysis of shell structures also composites with the use of 9-SRI (or 9-URI) elements. However since our factor of self-satisfaction is not so high we would like to add some remarks. If it's really an efficient tool? Yes but we are aware of the existing limits. We can analyse large deformation but with moderate rotations only. The best results can be obtained for shallow shells. Analysed composites can not be to thick. Using 9-SRI or 9-URI elements one can expect the locking and/or hour-glass mechanisms. One can ask the question: How are the authors going to extend the possibilities of the presented algorithm? In the answer we can say that to overcome the limits in geometry and to reduce the influence of the locking and/or hour-glass mechanisms we started to implement other types of finite elements: 8-URI or 9-AS. To improve the results for large rotations we are going to change the rotation updating procedure in the numerical algorithm simultaneously with the critical review of the MRT equations. 4

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