Numerical Validation of a Finite Element

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1 Numerical Validation of a Finite Element GABRIEL JIGA, ANTON HADĂR, ŞTEFAN PASTRAMĂ, IOAN N. CONSTANTINESCU Department of Strength of Materials University POLITEHNICA of Bucharest Splaiul Independenţei, 313, Sector 6 Bucharest, RO ROMANIA Abstract : The mechanical behavior of laminated composite structures can be estimated both numerically and experimentally. One of the phenomena that are widely studied is delamination. Starting from an initial deterioration due to the shear stresses that appear between the laminae, this phenomenon extends in different modes, leading to a loss of strength and finally to the failure of the composite material. Together with the analytical approaches, the numerical methods and especially the finite method are widely used today for the determination of the stress and strain state. In order to validate the numerical model and to prove the reliability of the new finite, experimental tests were performed. This paper presents different numerical analysis, where a new finite type has been developed by the authors to be used for the analysis of the stresses and strains in structures made from laminated fiber reinforced composites. Key-words : - finite, layer, laminated composite, structure, stress analysis, 1. Introduction The testing of a new finite type, of its software and its calculus methodology represent compulsory steps in order to validate and to use it for an established purpose. These tests are done generally on simple structures, from the point of view of the geometry and loadings, where results (stresses and strains) are well known, further to analytical or numerical calculus or by experimental determinations. The results obtained through this tested finite are often compared with results obtained by simple analytical calculus or by a specialized software, in whose library exist finite s proximal as destination and performances. If after comparative analysis small differences between results (5 10%) are obtained, one can affirm that the proposed finite is adequate and could be used for the purpose for which it has been proposed. More difficult is the testing of a finite for which the specialized litterature does not offer comparative possibilities. In these conditions experimental analysis are required [1]. With respect to the proposed finite subjected to validation, destined to local and global analysis of fiber reinforced laminated composite structures, the authors established several possibilities of its testing as well by analytical, numerical or experimental analysis [2], [3], [4]. The first tests have been achieved on plate structures, with different thicknesses, made from an isotropic and homogeneous material, loaded in tension and flexion. T The results have been compared with those obtained by analytical and numerical calculus. Similar tests have been achieved on same structures, except that these last ones were made from fiber reinforced laminated composites. The comparative study was performed on the results given by software in whose library one can found a finite called «LAMINATE» very close to the one proposed by the authors, especially for the global stress analysis. Finally, in this paper are presented comparative analysis of plate structures, made from fiber reinforced composites, loaded by an uniaxial tension, the results necessary to be compared being obtained also by code. Further to comparative numerical analysis achieved information linked to the precision of the proposed finite have been obtained. In the same time, the authors formulate conclusions and recommendations regarding to the use procedure of this finite. 2. Finite Element Used for Laminated Fiber Reinforced Composites In order to pursue the initiation and propagation of damages in laminated fiber reinforced composite structures (matrix or fiber breakings, delaminations etc.), one could utilize the numerical analysis, by using a specialized finite code, conceived by the authors (fig.1). This has eight nodes, being a hexahedron with basis an any quadrilater, with thickness equal to the thickness of a lamina [5], [6]. ISSN: ISBN:

2 It is an isoparametric, so the same interpolation functions are used for the approximation of the displacement field and for the description of its geometry. The applied loadings are considered to be concentrated forces in nodes, oriented on the three directions of the global system of axis Oxyz. 3. Analytical and Numerical Tests for the Validation of the Proposed Finite Element There are studied three types of plates, as presented in the next paragraphs. 3.1 Homogeneous and isotropic plate The first tests have been achieved on a very simple structure (a plate) built-in at one end and free at the other one, loaded in traction (fig.3). The plate material has been considered homogeneous and isotropic, with different thicknesses : 0,5 mm, 1 mm, 1,5 mm, 2 mm and 2,5 mm. The plate was meshed through 56 nodes and 18 s. The obtained results were compared to the analytical ones. For the analytical calculus of displacements and stresses, well known relations were used : Fig.1 The proposed finite u x P = ; EA P σ =. (1) A The displacements are calculated in each node of the model proposed and the stresses are calculated as well in each lamina as at the interface between two adjacent laminae (interlaminar stresses). The whole structure will be meshed through finite layers, the number of layers being equal to the number of laminae included in the plate (fig.2). where P = 1600 N, l = 30 mm and E = MPa. The errors obtained were below 1% for stresses and below 5% for displacements. A second case analyzed was a homogeneous and isotropic plate, but modeled with a variable number of finite layers. This testing was necessary to check the way how the finite works on the direction of the plate thickness. That s why homogeneous and isotropic plates with different thicknesses loaded in traction have been taken into consideration. Fig.2 Type of meshing on layers Fig.3 Finite modeling of an homogeneous and isotropic plate, loaded in traction ISSN: ISBN:

3 As one can see, the small differences between those obtained by code and those obtained by the finite proposed show that it has a very good behavior on the thickness direction when the plate is subjected to traction. Fig.4 Modeling with a variable number of finite layers of a homogeneous and isotropic plate, loaded in traction The layers thickness has been maintained constant (equal to 0,5 mm), the number of layers being varied. In figure 4 is presented the structure when the number of finite layers was considered to be equal to six, with 196 nodes and 108 s. The loadings were concentrated in each of the nodes situated at the extremity of the plate. The results obtained further to a finite calculus were compared to the analytical ones, obtained with relation (1), but also with those obtained by code. In Table 1 are presented for comparison the displacements of the nodes situated at the end of the plate (nodes 7 and 28) and the stresses in the s situated in the vicinity of the embedded end (s 1 and 13). Table 1 Displacements layer node u x7 u x ,0204 0, ,0182 0, ,0171 0, ,0159 0,0156 Stresses layer node ,7 134, ,76 119, ,27 112, ,79 105,07 Finally, a homogeneous and isotropic material, loaded in flexion has been also modelled with variable number of layers. As one can expect, with this finite type it is difficult to analyze with accurate results every type of plate subjected to flexion, due to the small number of degrees of freedom wherewith the is provided (the proposed has no rotations). That s why it s very important to state the conditions where the finite becomes performant in flexion. For this purpose, plates with variable thicknesses have been analyzed, the number of layers being also variable. In the same analysis, the plate was meshed maintaining a constant thickness, but varying the number of layers, in order to establish the optimum number of plies, corresponding to good performances of this finite. In Table 2 is specified the minimum number of layers having constant thickness whereby could be modelled a structure subjected to flexion, in order to obtain results with minimum errors. For the plate presented in figure 5, loaded in flexion, with a variable number of layers (4, 5, 6, 7 and 8), but with a constant thickness (equal to 1 mm), the results are presented in Table 3. Thickness Minimum no. of layers Thickness Minimum no. of layers Table 2 0,2 0,3 0,4 0,5 0,6 0, ,8 0,9 1,0 1,2 1,4 1,6 1, Table 3 Displacements layers nodes P u z7 u z ,0709-0, ,0549-0, ,0535-0, ,0478-0, ,0371-0,0366 ISSN: ISBN:

4 Table 3 - continued Stresses layer node ,65 61, ,47 67, ,09 85, ,12 88, ,57 81,98 Another analysis has been done on a plate made from the same material (glass-epoxy) but with a variable number of layers (2, 3, 4 and 6), with constant thickness (equal to 0,5 mm). The fiber orientations in the matrix were considered 0 o and 90 o, in different stacking sequences, as one can see in Table 4. Figure 6 represents a three-layered plate subjected to traction, with fibers oriented at 0 o (for external laminae) and 90 o for the central lamina. Table 4 u Orientation x7 u x7 plies angle 2 0/ ,1186 0, /90/ ,1378 0, /0/0/ ,1547 0, /90/0/0/90/ ,1205 0,1212 Fig.5 Modeling with a variable number of finite layers of a homogeneous and isotropic plate subjected to flexion As one can see, the results obtained in the analysis of homogeneous and isotropic plates could be estimated as good ones, especially if there are taken into account the recommendations for the use of this finite in the analysis of plates subjected to flexion. plies u y7 u Orientation y7 angle 2 0/ ,0138 0, /90/ , , /0/0/ , , /90/0/0/90/ , , Plate Made of Composite Material As a first case a plate with different thicknesses (0,5 mm, 1 mm, 1,5 mm, 2 mm, 2,5 mm) subjected to traction was analyzed. The plate consisted in one glass-epoxy lamina, with the following elastic characteristics : E l = N/mm 2 ; E t = 8600 N/mm 2 ; G lt = 3900 N/mm 2 ; ν lt = 0,28; ν tz = 0,45 and with a fiber oriented angle of 0 o, The displacements u x of nodes 7 and 28 situated in the free extremity of the plate and the normal stresses σ l in s situated in the vicinity of the fixed part of the plate (s 1 and 11) were compared. The errors are practically null, which shows that the software as well as the proposed methodology are correct. Fig.6 Modeling of a fiber glass reinforced plate (0/90/0) subjected to traction ISSN: ISBN:

5 Table 5 plies σ l1-0 σ l1-0 σ t1-0 σ t ,97 133,92 3,66 4, ,80 159,40 7,45 8, ,93 179,03 8,87 9, ,70 140,69 6,58 7,18 Table 5 - continued plies τ lt6-0 τ lt ,15-11, ,24-12, ,46-12, ,59-11,81 Fig.7 Three layered composite plate (0/0/0) from glass-epoxy provided with concentrator. As regarding the stresses, in Table 5 have been presented some of the stresses which appear in the finite s of laminae oriented at 0 o : normal stresses σ l (on the fiber direction), σ t (on a direction normal to the fibers) and shear stresses τ lt, from s 1 and 13 (the normal ones), from s 6 and 18 (the shear ones), these s having the highest stresses. Finally it has been analyzed the behavior of the finite at flexion. In this case a laminated composite plate (0/90/0) S has been taken into account. Further to a numerical analysis achieved with the proposed finite and after a comparison of the results one concludes that, if the number of laminae is superior to six, the results are accurate. 3.3 Plate with Concentrator, Made of Composite Material The tests achieved with the finite proposed continued with the study of a plane plate provided with a concentrator (type hole), the plate being subjected to traction (fig.7). The plate material was considered a laminated glass-fiber reinforced composite, with similar elastic and mechanical characteristics in comparison with laminae studied in the previous tests. The thickness of each lamina was 0,4 mm, the fibers being oriented at 0 o by respect of Ox axis. The plate material was considered a laminated glass-fiber reinforced composite, with similar elastic and mechanical characteristics in comparison with laminae studied in the previous tests. The thickness of each lamina was 0,4 mm, the fibers being oriented at 0 o by respect of Ox axis Fig.8 Modeling of a quarter of structure Due to symmetry (geometric and of loading), only a quarter of the plate has been analyzed. The uniform distributed load was considered p = 50 N/mm and divided in nodes under concentrated loads (fig.8). The normal and shear stresses, respectively σ l, σ t, τ lt in the center of some of the most solicited s have been compared in Table 6. Table 6 elemnt. number σ ll -0 σ ll -0 E.F. σ tl -0 σ tl ,76 113,88 10,88 10, ,47-1,36 25,85 24, ,33 20,78 41,11 38, ,33-44,98-1,09-0,98 ISSN: ISBN:

6 Table 6 - continued τ Nr. ltl -0 τ ltl -0 E.F. elem. 1-5,01-5, ,90 28, ,38 6, ,52 7,10 4. Conclusions and Recommendations on the Proposed Finite Element After several tests have achieved, by comparising the results presented above, one could specify the following remarks : - the proposed finite is correct from the mathematical point of view ; - the software conceived with this type of is also correct ; - the calculus methodolgy is also accurate for the proposed purpose, although it leads to an increase of number of nodes and s ; it s in fact the only way to penetrate in the interior of the laminated composite ; - better results will be obtained with a refined mesh, with a greater number of nodes and s ; - for the study of finite s of a structure subjected to flexion it is necessary to respect the conditions related to number of laminae and their thickness. References [1] Hadǎr, A., Bordeaşu, I., Mitelea, I., Vlăsceanu, D., Validarea experimentală a unui model teoretic folosit în calculul de rezistenţă al structurilor realizate din materiale compozite, Mat. Plast., vol. 43 (1), 2006, pp [2] Hadăr, A., Constantin, N., Jiga, G., Element finit pentru analiza stărilor locale de tensiuni din compozite stratificate unidirecţionale, Revista Construcţia de maşini, Bucureşti, nr. special 1, 1998, p [3] Hadăr, A., Constantin, N., Goga, N., Jiga, G., Program cu e finite specializate pentru analiza locală şi globală a compozitelor stratificate unidirecţionale, Revista Construcţia de maşini, Bucureşti, nr. special 1, 1998, p , Bucureşti [4] Zgîrian, G., Demetrescu, I., Gheorghiu, H., Iovu, H., Hadǎr, A., Atanasiu, C., Modelarea unor compozite polimerice: de la sintezǎ la proprietǎţi mecanice şi calcule cu e finite, Revista de Chimie, Bucureşti, nr. 7/ 2005, vol. 56 (7), p [5] Wei, J., Zhao, J. H., Three-Dimensional Finite Element Analysis on Interlaminar Stresses of Symmetric Laminates, Computers and Structures, 1991 [6] Lee, D., Trehmernîi konecino îi analiz nacoplenia povrejdenii v sloistom compozite, Procinosti i razrusenie compozitnîh materialov, Ed. Riga, ZinaTIE, 1983 [7] Adams, D. F., Crane, D. A., Finite Element Micromechanical Analysis of an Unidimensional Composite Including Longitudinal Shear Loading, Computers and Structures, vol. 18, 1984, p [8] Bathe, K. J., Ho, L. W., A Simple and Effective Element for Analysis of General Shell Structures, Computers and Structures., Vol. 13, 1981 [9] Flanagan, G., A sublaminate analysis method for predicting disbond and delamination loads in composite structures, Journal of Reinforced Plastics and Composites, Vol. 12, 1993, p [10] **** code ISSN: ISBN: