Katlanmış plaklar için sonlu eleman formülasyonu

Abstract

Kalınlıkları diğer boyutlarına göre küçük düzlem elemanlarla oluşturulan katlanmış plaklar, genellikle çatılarda kullanılmalarına rağmen hangarlarda, ambarlarda, malzeme depolama tanklarında, döşemelerde ve hatta temellerde ekonomik, pratik ve estetik oldukları için kullanılan yapı elemanlarıdır. Reissner teorisine dayalı plak ve levha fonksiyonelleri geometrik ve dinamik sınır koşullarını da içerecek şekilde elde edilmiştir. İfadelerin benzerlerine literatürde rastlanmamıştır. Elde edilen fonksiyoneller klasik enerji ifadesine dönüştürülebilmektedirler. Bu özelliklerinden dolayı iki fonksiyonelin toplamından katlanmış plak fonksiyoneli elde edilmiştir. İzoparametrik sonlu eleman formülasyonu kullanılarak plak ve levhaların, bunların toplamından da katlanmış plakların eleman matrisleri elde edilmiştir. Ayrıca üçgen plakların çözümü için de üçgen sonlu eleman formülasyonu geliştirilmiştir. Reissner plakları incelenmiş, sınır koşullan üzerinde detaylı olarak durulmuş ve kalın plak için elde edilen formülasyon kullanılarak ince plak çözümü yapılmıştır. Basit ve ankastre mesnetli plakların çözümlerinde bazı büyüklükler için alt ve üst limitler olduğu belirlenmiştir. Literatürde az rastlanan kesme kuvvetlerinin dağılımı incelenmiştir. Değişik sınır koşullarına sahip plak çözümleri, literatürdekilerle birlikte tablolar halinde verilmiştir. Delikli plaklarla ilgili deneyler yapılmıştır. Basit mesnetli kare plakların serbest titreşim modlarına karşı gelen açısal frekans değerleri hesaplanarak literatürdeki değerlerle karşılaştırılmış ve delikli plak ile elastik zemine oturan plakların serbest titreşimleri incelenmiştir. Reissner plaklarının detaylı olarak incelenmesi ve levhalarla ilgili çalışmaların da dahil edilmesiyle elde edilen sonuçlar katlanmış plaklara uygulanmıştır. Katlanmış plakların çözümünde, katlanmış plak global eksen takımı ile plak ve levhanın eksen takımı çakışmadığı için, eleman matrisi ya dönüşüm matrisiyle çarpılarak, yada çarpılmadan eksen dönüşümünden kaynaklanan farklılıklar Lagrange denklemleriyle hesaba katılmak suretiyle kullanılmıştır. Elde edilen sonuçların nitelik olarak karşılaştırılabilmesi için deney yapılmıştır.

Folded plates are surface structures made from individual plane surfaces in which the thicknesses are smaller than other dimensions. The folded plates which are constructed from rectangular plates are called prismatic folded plates, so their cross - section is constant throughout its length. If the cross - section varies throughout its length it is called non - prismatic folded plate. The plates may individually be either prismatic or non - prismatic, the whole system is termed non - prismatic. Folded plates are formed as multiple bay or multispan construction systems. The intersection lines between individual plates are usually termed fold lines. The folded plates are principally used for roof structures but also adapted to bins, floors and even foundations, resulting in an economic, rational and pleasing design. First systems were developed for reinforced concrete as it can be adapted to different forms. Then it has been extended to timber construction, wherein plywood is shown to be readily adaptable to the large flat areas and to metal construction. To construct economical structures folded trusses may be used to suit architectural forms. In folded plate systems it is possible to construct simpler and less expensive formwork than curved shells, also concreting and screeding is easier. More complex structures such as arches and frames will have same advantages too, with folded plates. Longer spans may be obtained with folded plates without an increase in material requirement because of their inherent stiffness. In this study, before the solution of folded plates, the plates and in-plane elements are separately studied. New functionals for Reissner plates and in-plane elements with geometric and dynamic boundary conditions are obtained. In the literature survey the same functionals were not found. These functionals are transformable to the classical energy equations. Below assumptions are made to prepare the necessary equations to obtain the functional for Reissner plates: VII . Material is linearly elastic.. Lateral stresses are taken.(az, %xz, xvz ). In equilibrium equations the volume forces are neglected.. Bernoulli - Navier hypothesis is valid. Depending on these assumptions the plate equations are obtained as follows, dM dM dx ? + ? dy ** dM dM dx dy y dQx 6Q + - *- +q-kw=0 dx dy **x = 12 dx Ehz x ^ y jo \i(q-kw) dD. y _ 12 Qy Eh3 W J \ M -\M - y x jo \i(q-kw) (2.23) sn sq n dy dx Gh3 xy n + dw x dx 5Gh x Q=0 a +^.-J-q =o y dy SGh^y where, D = Eh > /1 2(1 - \x2 ), G= E 1 2(1 + \i). Dynamic boundary conditions, M-M = 0 Q-Q=o (2.24) VI» geometric boundary conditions, -n+h=o -W+W=0 (2.25) written in symbolic form. Quantities with hat are known values on the boundary. For the condition which elastic foundations are not considered, elastic foundation coefficient is taken as k=0. Reissner plate equations are written in operator form as, Q = L y - f and to find the functional it is necessary to show that Q is potential as in equation 2.28. dQ(y ; y ) and dQ(y ; y ) show the Gateaux derivative of operator Q in direction y and y After showing that the operator is potential, using equation 2.29 functional is obtained by equation 2.32 as below. i(y) = l[Q(sy,y),y]ds o (2.32) Here, s is a scalar. Following the calculations the functional is obtained as, fr(y)=[Qxwx+wiX)]+ o.(a +w ) -rx y y Mxy>ax, y\ ^kw,w]-[(w-w),Q]e-[(çi-h),M] -[w.e]a-[M,n]CT In-plane element equations are given as follows, (2.33) dN dN dx dy (2.39.a) IX = 0 = 0 = 0 (2.39.b) (2.39.c) (2.39.d) Dynamic boundary conditions, -N + N = 0 and geometric boundary conditions, w-« = 0 (2.40) (2.41) are written in symbolic form. After showing that the operator Q = L y - f is potential, using Gateaux differential, equation 2.39, 2.40 and 2.41 yields to the following functional after a few manipulations, -[^'M]CT-[^rv]CT-[^'wL"[^'vl~[(M~")'^ -[fv-v;,/^] -{(u-UXN^j -[(v-vlNxy] (2.43) The new functional which is obtained by summation of the above two fiinctionals can be used to solve folded plates. ıAy) = {[Qx,(nx+wx)]+ e/VV T*'% + W W ^'^I'^H'^l'K^J-^'^]"2^^^'^]} ' îİÂİex-eJ4e>"eJ}+^{^MJ+hMJ-h "*]-[*"' My]}-M [kw,w]-[(w-$),Q]e-[{n-cı),M] -[w,ğ] -[û,n] }J +{ Nx'U, l + N,v y > >J N,v N,u xy, y\ 2E N,N 1 x x\ **-i -[ vL -[ vi -[ vL -Mkl -[? fv-vj,# (2.44) Depending on the development in computer systems, the finite element method is widely used, as it becomes easier to calculate matrixes with many unknowns. As the functionals have derivative of first degree only, lineer shape functions for Reissner plates and in-plane elements would be necessary and sufficient. In isoparametric finite elements, the relationship between the unknowns on any place of the element and the unknowns on the nodes are achieved using shape functionals. So, the element matrices are obtained easily. Reissner plates are investigated, boundary conditions are studied in detail and using the formulation of Reissner plates, thin plate solutions are done. For some unknown values, the top and bottom limits for simple and cantilever supported plates are defined. The distribution of shear forces which can be rarely met in literature is researched. The solutions of plates which have different boundary conditions are given in tables with the solutions in literature. The tests of the plates with holes are also performed. The angular frequency values corresponding to free vibration modes xi of simply supported square plate are calculated and compared with the ones in literature. The free vibrations of the plates with hole and the plate on an elastic foundation are also researched. In-plane elements are studied, results are compared with exact results and given as tables. In the solution of folded plates, as the global axis of folded plate and the axis of plate and in-plane element are not met, the differences are used by considering the Lagrange equations in the calculations or. by multiplying the element matrix by transformation matrix. To check the results, tests are done. A computer program written in Fortran programming language is developed for the analysis of plates, in-plane element and folded plates. All results are also compared with the examples given in the literature and found that they are in good agreement.