The 4th International Conference Advanced Composite Materials Engineering COMAT 2012 18-20 October 2012, Brasov, Romania ANALYSIS OF COMPOSITE RECTANGULAR PLATES BASED ON THE CLASSICAL LAMINATED PLATE THEORY I Sprinţu Military Technical Academy, Bucharest, Romania, sprintui@yahoocom ABSTRACT: In this paper are developed some analytical solutions for static analysis of thin orthotropic plates, based on the Classical Laminated Plate Theory (CLPT The considered reasons for the solutions were to exactly satisfy the boundary conditions and to verify as close as possible the differential equation of the plate The weighted residue method was considered to optimise the chosen analytical solutions Interesting evaluations were performed for different types of functions, especially with respect to the orthotropic answer of the plate Article contains a comparative analysis of analytical results with the numerical (FEM and those obtained experimentally KEY WORDS: composite plates, orthotropic, static analysis, simply supported edge, clamped edge 1 Assumptions To establish the constitutive equation of flat plates, is considering Kirchhoff Love hypothesis, "weak" as follows: We accept as working hypotheses to determine the constitutive equation of composite flat plate, assumptions of Kirchhoff-Love, extended as follows [5]: a During deformations, the normal to the median of the plate remain straight and normal to the deformed middle surface ( xz yz 0 wich implies xz yz 0 Although efforts unit xz, yz are very small compared xz yz with the normal, we extend the assumption considering their gradient high enough 0, 0 z z b During deformation of the plate its thickness is constant, equivalent with z 0 c We accept the hypothesis of small strains, wich approximates: 2 2 2 w w w x z,, 2, 2 y z 2 xy z where z is the distance from the point where displacements x y x y are been computed and the midplane 2 ANALYSIS FOR COMPOSITE PLATE EQUATIONS 21 Case of clamped edge For a rectangular plate, on which pressure p is evenly distributed, using the assumptions from the first chapter, we obtain a generalization of Sophie-Germain equation in particular case, when ox-axis is oriented along the fiber direction, [5],[8]: 4 4 4 w w w 12 p Q11 Q22 2Q12 2 G12, (1 4 4 2 2 3 x y x y h where E1, E2 are elasticity moduli in the longitudinal and transversal directions, respectively, G12 is the shear modulus in the plane of the ply, and 12 is the Poisson coefficient, 233
E1 E2 21 E1 12 E Q 2 11, Q22, Q12 112 21 112 21 112 21 112 21 For example, we consider a flat plate with size a 300 mm, b 200mm, thickness h 15 mm, plate made by incorporating the 8 layers of glass fiber fabric, type EWR 300 g / mp, Polylite 440-M888 resin, with technology VARTM (Vacuum Assisted Resin Transfer Molding, at SC STRAERO SA For each layer of fabric fibers have been removed on one direction (2 of 4 fibers (Fig 1(a Sails were arranged to have the same warp and weft direction, laminate being treated as orthotropic material The main directions of elasticity are the warp and weft To determine the elasticity modules were made standard flat specimens, the size 25 300 Following tensile testing (SC STRAERO SA, were obtained values E1 22263MPa, 12 018, and E2 19035 MPa In the same time, the orthotropic plate with clamped edge is required to stretching and bending Using weights of 12 kg and flat specimens size ( 24 100, were obtained values E1 22065 MPa, E2 17079MPa For this, we used the relations: 3 3 3 3 F c h F M g v, I, which implies E 48 E I 12 3 3 4v c h 4v c h The orthotropic plate was driven by a pressure p 000465 MPa, uniformly distributed over the entire surface To ensure better clamped conditions, contour plate was reinforced (fig 1(b In table 1 are presents the comparative results between the experimental measurements and the numerical results (using - shell Elastic 4 node 63, mesh 10, taking into account the elasticity modules in stretching In table 2 are presents the comparative results between the experimental measurements and the numerical results, taking into account the elasticity modules in bending For a better approximation, we consider an average for modules of elasticity, respectively E1 22164 MPa, E2 18057 MPa (table3 Mathematic, clamped edge for x 0, x a and y 0, y b follows: w w w0, y wa, y 0, 0, y a, y x x w w w x,0 w x, b 0, x,0 x, b 0 y y In [8], Reddy proposed solution for equation (1, as: m n i1 j1 2 2 x y x y w x, y cij 1 1, and solved it for m n 1 i1 j1 a b a b In this paper, using Ritz method, we look for a solution as n m i1 i1 j1 j1 w x, y cij x a x y b y i1 j1 c The coefficients ij i, j will be determined using the weighted residue method For p 000465 MPa, E1 22164 MPa, E2 18057 MPa, we get coefficients (Maple: 16 c11 12937810, 21 c12 3215910, 26 c13 20920410, 21 c21 3347110, 26 c31 1574610, 25 c22 1246610, 35 c33 4944410, for m n 3 a b In this case, the maximum value of the arrow is w, 2 2 3053 mm Using finite element method (shell Elastic 4 node 63, mesh 10, we obtain a maximum displacement a b wmax w, 2 2 3055 mm In table 4 are presented results comparison between the analytical (Maple and numerical solution (2 (3 234
To validate the solutions we performed measurements on the same plate, driven uniformly with different pressures ( p 000305MPa, table 5, and p 0004MPa, table 6 In fig 2 (a,b,c are experimental graphics solutions for requests made Given the need for several sets of measurements of orthotropic plate arrows, depending on the conditions imposed on the shape and position in plan, I chose the option to apply a pressure plate perpendicular, evenly distributed over the entire surface with a pressurized air chamber Solution readily allows repeated measurements, and application of pressure change Measurement of air pressure in the chamber and so the pressure plate application is made with a liquid manometer To measure arrows of deformed plate, we use inductive displacement transducer In these conditions, the entire system covers the following modules: - Application module orthotropic plate - Displacement transducer module in the horizontal plane xoy - Arrow measuring module - Acquisition and storage module measurements (steper electric motor EM-257, 17PM-K212-P1T for moving the transducer 12 on the ox and EM-464, 2Y28AD2 for moving the transducer 12 on the oy Air chamber pressure was made from a thick 1 mm rubber sheet 340 240 mm and, on where were bonded two valves, one for charging and one for connection to the manometer Deformable wall of the chamber is a plastic foil (Fig 1(c 22 Tables Table 1: ( p 000465MPa, E1 22263MPa, E2 19035MPa (% 00208 0303-10 0401 0439-38 1004 1056-51 1619 17-81 2145 2248-103 2539 2651-112 2788 2861-73 2894 2893 01 2859 2878-19 2682 2807-125 236 248-120 1898 2001-103 1319 1383-64 0691 0701-10 0164 0166-02 Table 2: ( p 000465MPa, E1 22065MPa, E2 17079MPa (% 00222 00303-08 0431 0439-08 1066 1055 31 1764 17 64 2348 2248 100 2788 2651 137 307 2861 209 319 2893 297 315 2878 272 2949 2807 142 2587 248 107 2071 2001 70 1431 1383 48 235
0746 0701 44 0176 0166 10 Table 3: ( p 000465MPa, E1 22164MPa, E2 18057MPa (% 00215 00303-09 0416 0439-24 1044 1055-11 1689 17-11 2242 2248-06 2658 2651 07 2871 2861 10 2923 2893 30 2871 2878-07 2809 2807 02 2469 248-11 1981 2001-2 1373 1383-1 0717 0701 16 017 0166 04 Table 4: ( p 000465MPa, E1 22164MPa, E2 18057MPa 0022 0415 1689 2242 2658 2923 2998 2809 2469 198 0717 017 Maple 0022 0416 1688 2241 2657 2921 2996 2808 2467 198 0717 017 Table 5: ( p 000305MPa, E1 22164MPa, E2 18057MPa (% 0014 0015 011 0272 031 38 1109 1105-03 1471 1508 37 1743 1723-2 1917 1916-01 1966 1982 16 1843 1845 02 1619 1647 28 1299 1267-12 047 0508 38 0111 0137 26 Table 6: ( p 0004MPa, E1 22164MPa, E2 18057MPa (% 0019 0000 18 0617 0619-02 1453 1423 30 2123 2119 04 2514 2501 13 2615 2607 08 2417 244-23 1929 1921 08 1181 1198-17 0357 0328 29 236
23 Figures Figure1(a Figure1(b Figure1(c a b c Figure 2 237
3 CONCLUSION This paper is an analysis of orthotropic plates, based on CLPT, characterizing thin plates, assuming small deformations I proposed an analytical solution for case of clamped edge After comparing the results with those treated in the literature, as well as the results from finite element, and experimentally analysis, we conclude that the solution presented in this paper approximates well plate subjected to compressive deformation Mechanical device allows measurements for other conditions imposed on the outline of the rectangular plate, simply supported, articulated and free REFERENCES [1] Alămoreanu, E, Negruţ, C, Jiga, G, Calculul structurilor din materiale compozite, UPB, 1993 [2] Ambartsumyam SA, Theory of Anisotropic Plates, Tehnomic Publishing, 1969 [3] Araujo, AL, Mota Soares, CM, Moreira de Freitas, MJ, Pedersen, P, Herskovits, J, Combined numerical-experimental model for the identification of mechanical properties oflaminated structures Composite Structures, 2000 [4] Buzdugan, Gh, Rezistenţa materialelor, Ed Academiei, Bucureşti, 1987 [5] Cristescu, N, Crăciun, M, Soos, E, Mechanics of Elastic Composites, Chapman&Hall/CRC, 2004 [6] Jones, RM, Mechanics of Composite Materials, Scripta Book, Washington DC, 1975 [7] Lekhnitskii, S G, Anisotropic Plates Gordon and Breach Science Publishers, New York, 1968 [8] Reddy, JN, Mechanics of Laminated Composite Plates- Theory and Analysis, CRC Press, 1997 [9] Springer, GS, Kollar, LP, Mechanics of Composite Structures, Cambridge University Press, 2003 [10] Vasiliev, VV, Morozov, EV, Mechanics and Analysis of Composite Materials, Elsevier, 2001 [11] Voyiadjis, GZ, Kattan, PI, Mechanics of Composite Materials with MATLAB, Springer, 2005 238