Ortotrop plak sistemlerinin hesabı için yeni bir sonlu eleman formülasyonu

Abstract

ÖZET Bu çalışmada, Ortotrop plaklardan meydana gelen çeşitli yapı sistemlerinin hesabı için bir me t od geliştirilmiştir. Yaklaşık Çözüm tekniklerinden son zamanlarda bilgisayarların yaygınlaşması ile sıkça kullanılan metodl arından biri olan `SONLU ELEMANLAR YÖNTEMİ` ile ortotrop plaklar için çok kullanışlı olacak bir mixed sonlu eleman formulasyonu çıkarılmıştır. Bulunan bu eleman yardımıyla ortotrop yapı sistemlerinin statik yükler altında hesabı yapılabilmektedir. Ayrıca bu elemanla; Çelik karkas binalarda ve Gemi İnşaatlarında sıkça kullanılan iki yönde farklı eğilme rijitliğine haiz takviyeli çelik saç döşemeler, Kompozit Döşemeler, Nervürlü betonarme taşıyıcılar ve Asmolen döşeme sistemleri ç öz ülebilmek t edir. Bulunan sonlu elemanın kullanılırlığı diğer çözüm teknikleri ile karşılaştırmalı olarak verilmiştir. Altı bölümden oluşan bu çalışmanın birinci bölümü, çalışmanın amacına ayrılmıştır. İkinci bölümde, ortotrop plak teorisi ve plak denklemleri özet halinde verilmiştir. Üçüncü bölümde, Ortotrop plak sistemine sonlu elemanlar metodunun uygulanması anlatılmıştır. Dördüncü bölümde, üçgen Ortotrop plak elemanının fonksiyoneli elde edilerek varyasyon prensibi uygulanmış Eleman davranış denklemleri, Alan koordinatları kullanılarak denklemlerin integrasyonu genel halde verilmiş. Eleman davranış ve Eleman etki denklemleri matris formunda elde edilmişlerdir. Beşinci bölümde, Nümerik uygulamalar üzerinde sonlu elemanın kullanılabilirliği diğer çözüm teknikleri ile karşılaş t ırılmıştır. Altıncı bölümde. Sayısal uygulamalar için hazırlanan ve QBASI C dilinde kodlanmış E. H. M. programının çalışma düzeni ve kullanma esasları anlatılmıştır. Ayrıca sistem denklem takımının çözümünde hafıza kaybını en aza indirmek için sağlanmış bir kolaylık bulunarak bir algoritma verilmiştir. VII

SUMMARY A NEW FINITE ELEMENT FORMULATION FOR ORTHOTROPIG PLATE SYSTEMS In this study, it has been developed a method of calculation for structural systems which consist of Orthotropic Plates. A finite element formulation, practical for the calculation of Orthotropic Plates, is found by `FINITE ELEMENTS METHOD` using computers as an approximate solution technics. By the aid of this element discovered, Orthotropic Structural Systems could be designed statical loadings. Moreover, the practice of this finite element used for reinforced sheet iron slabs, composite slabs, ribbed slab systems and hollow core slab systems having different bending stiffness in both directions used frequently in steel structures & boat construction, is given in order to compare with other solution technics. In the first part of this study which consists of six parts, the purpose is outlined. In the second part, Orthotropic Plates theory and equations of plate are summarized. In the third part, plate functional is obtained, using functional analysis method on Orthotropic Plates systems. In the fourth part, a general integration of equations is given by using finite elements equations of triangular Orthotropic Plate element and area coordinates, by the way element stiffness matrice and loading matrices are obtained. In the fifth part, the practice of finite elements with numerical applications is compared with other solution technichs. In the sixth part, it has been explained the working order and using principles of computer programme coded in `QUICKBASIC` computer language and prepared for numerical applications. VTIIAlso an algorithme is given, in order to discover a facility to extremum the memory loss in the solution of equation system. For any triangular plate sample, practical for Orthotropic Plates; Plate settlements w, and M, M, M, x y xy M moments are chosen as independent variables and yx converted to a `Variation Principle` using `Weigted Residuals Method` on Linear Self-Adjoint differantial equation systems and a n function is obtained. np, function is discovered then by a partial integration of n function t 3] The element equation, which causes some problems in Classical finite element methods, is clearly obtained in order to choose linear Interpolation functions for the variables because only the first derivatives of the variables is applied in this function. Also by the some reason, making the choice of linear interpolation functions carries out appropriateness and exactness conditions which guarantees the convergence when the elements get smaller C20], 1 22 J, C361.. THEORY OF ORTHOTROPIC PLATES AND EQUATION OF PLATES The plate will be defined in x, y, z coordinate system x and y axises, being in the plane of the plate, are parallel to the main directions of the plate unisothropy. < FIGURE 3.1). ASSUMPTIONS The main assumptions are as follows; 1) The material is linear and homogeneous and there exist orthogonal unisothropy. 2) The deformations are smalls. The superposition is valid. 3) The mid surface remain plane after deformation. 4) Plate thickness is small with respect to other dimensions of the plate. IXDEFORMATION RELATIONS (FIGURE 2.2) 1/p curvature which takes place after the application of - Mx/(EI)x C2. 1) 1/p curvature which takes place after the application of =-Ai. Mx/(EI)x (2.2) x 1/p curvature which takes place after the application of ¦ My/(EI)y <2. 3) 1/p curvature which takes place after the application of - v.My/(EI>y <2.4> y (FIGURE 2.3) 1/p - Mx/(EI)x - v. My/(EI)y (2.5) x y 1/p - My/(EI)y - p. Mx/(EI)x (2.6) y x i/p - - d/«x.(aw/dx) « - dw2/ax2 (2.7) x 1/p = - a/dy.(dw/dy) - - dw2/dy2 (2.8) 1/p - - d/Öx. (dto/öy) - - Öu>Z/dxây (2.9)(FIGURE 2.4) u> =w ; u =- Z. aw/ax ; v »- Z. aw/ay (2.10) z z z (FIGURE 2. 5) DEFORMATIONS e = 0u / «x - z. a2w/ax2 (2.11) x z £ » 9v S &y - z. »2w/0y2 (2.12) y - au / ay + av /ax -2. z. a^w/dxay (2.13) 'xy z ' z ELASTICmr RELATIONS ft ELEMENT STRESS COMPENENT a « (E.z /(l- (J.^)). (i /p +/i. 1 /p ) (2.14) x x y x y y a =-(E.z /u- /x./j >). (a2» /ax2 + /i.a2» /ay2) (2.i4.a> x x y y o - (E.z /(l- *i.^i >). ( 1 / p + /i. 1 / p ) (2.15) Y x y y x x a «-(E.z /a- fj.n >>. (a2^ /ay2 + u. a^w /ax2) (2. is. a) y x y x t = G. r ¦ -2.G. z. aw2/ axdy (2. 16) xv * xy ' t - G. y - -2.G. z. aw2/ ayax (2. 17) YX * yx -h/Z ¦h/2 z. dz. dy ¦ M. dy (2.18) M - -C(EI) /b. (1-ju./j >]. (a^w/ax2 + m.a^w/dy2) (2. 18. a) X x x rx y y XI-h/2 J a. z. dz. dx » M. dx <2.19> y y +h/2 M - -C(EI> /b. <l-*i./u >]. Ca^w/dy2 + /j. a^ux/dx2) (2. 19. a) y y y x y x -İV2 J t. z. dx. dz = M. dx (2.20) xy xy ¦h/2 M » - 2. G. < Jw / b ). < a2* / «xdy ) (2.20. a) xy bx x -Ytsz J t. z. dy. dz » M. dy (2. 21 > yx * yx ¦h/2 Myx » - 2. G. < J^ / b ). < d2w. «xdy ) (2.2i.a> by y. PLATE MAIN EQUATIONS ft BOUNDARY CONDITIONS <FIGURE 2. 6) J Mjj - 0 J *y * ° J Z ` ° <2.22) Neglect oMx aMyx 1 aqx J M a dxdy + dydx - q dydx. < )dxdxdy *. x ax ay x 2 ax =0 (2.23) aMx aMyx >M -0 - + = q (2.24) *. x ax ay x XIIdMy dMxy V m - 0 - + - q (2.25) L y ay ax y y Z - 0 - + + ( P- K. w > (2.26) * ax ay z a^M a^M a^M a^M x yx y xy + + «. + (p _fc #fc)) (2.27) 2 2 2 ax ay ax ay ax ay Plate general equation Is put in order in the case of Orthotropic Plates. 4 4 4 a o> aw a w AAw - K. +K. + <K + K ). » <P-k. « ) * & y L x V 2 a z ax* ay* axsays (2.28) Here, the symbol indication is as follows. K - (EI) / b. ( 1-/J.*i ) (2.29) x xx x y K - (EI). b. ( 1-ai.u ) (2. 30) y y y x *y K - 2. G. ( J / b > (2.31) xy bx x K - 2. G. (,1 s b > (2.32) yx by y XIIIW »K, ` Poission ratio of the plate about X,Y direction. q » z External loads in direction of Z - axis. to = z Deflections in direction of Z - axis. K,K - Plate bending stiffness about x,y direction, x y * ' K,K =» Plate torsion stiffness about x»y direction, yx xy ' E,G b Hoduls of Elasticity&shear of the plate material The dynamic and geometric boundary conditions are considered. T » To Dynamic Boundary condition. M - Mo to* » w Geometric ` _. w* » co Geometric ` In the case of mathematical operations, to make extremum the Tip function» plate equations and boundary conditions must be secured. Replacing the moments expressed in terms of derivation of w in n function» the well-known plate energy function is obtained in order to secure limit conditions. n function can be obtained as below : (2.33) n « (Tf 1/2.<u >r.< EL 3.<u > > - <u >T.<f >lds JJ t » B B BBj (3.19) XIVHere are the clear expresions of La, U., f. ; oDD. (4.3) <4.11> n<u> function became as below after the mathematical operations on equation (4.11) (4.12) T m Valid in the limit in which geometric boundary condition has come out. r = Valid in the limit in which dynamic boundary condition has come out. In I7p function, there exist 5 unknowns;», M, M,M,M x y xy yx Indeed, Informing plate settlement function w ¦ caCx.y) the other variables could be known easily. Considering »,M,M,M,M as five unknowns x y xy yx independent from each other, we can assume that they change linearly in a triangular element shown in figure (4.1) (4.5) (4.19)..(4.20) Here «, M,M » M, M, w, M, M,M i xl yi xy* yxl 9 x9 y9 xys M, M show the values of unknowns in i node. xy» yxa N (x.y) ' s are obtained by the solution of linear interpolation function N(x.y) = a + a. x +a.y Mostly the first derivations of the five unknows exist in n <u) <4. 18) function. P By this reason, selected linear functions secure the continuity and exactness conditions. So, when the element get smaller, It can beapproached to definite results. XVTo find equations, replacing 4.5 and 4. 21 expressions in rip(u)...(4.18) function an equalizing the derivations of unknowns in IKu) function, which are the nodal values of w., M., M., M., and M., So that the expressions ı xt yv xyi yxi form an minimal, dflp dw. v drip An, xi arip - 0 <4.23> dm, yi dflp dm xyi dTIp - 0 dm yxv Five equations as the number of unknowns can be obtained in each nodal point. By the way, a new triangular finite element formulation is produced for any orthotropic plate. XVI. DERIVATION OF FINITE ELEMENT EQUATIONS Replacing (4.5) and (4.20) expressions, completing all of the operarions, using area coordinates on the element and then integrating rip function given in (3.18) with the element analysis is done, totally fifteen equations could be written so that five equations in each nodal point. (4.24) < M ) (4. 38) Integrating (4.24) and (4.38) equations in their own limits. Element Stiffness and loading matrices are given in part 4.4. XVII