İnce silindirik kabukların karışık sonlu eleman ve ritz metoduyla çözümü

Abstract

Kabuklar hem matematiksel formülasyonu hem de geometrisi nedeniyle karmaşık yapı sistemleri olarak bilinirler. Bu çalışma da ele alınan silindirik kabuklar; ortalama yüzeyi tek eğrilikli olan yüzeysel taşıyıcı sistemlerdir. Silindirik kabuklar pratikte baca, tonoz, boru, silindirik yapılar, sıvı hazneleri gibi yapı elemanları olarak karşımıza çıkarlar. Bu çalışmada silindirik kabuklar teoriden çözüm yöntemleri ne kadar adımlar halinde incelenmiştir. Birinci bölümde kabukla rın çözümü için nümerik yöntemler kullanıldığı anlatılmıştır. Bu yöntemlerin tarihçeleri ve bunlarla ilgili araştırmalar ve makale lerden sözedilmiştir. îkinci bölümde kabuk teorisinden bahsedilmiş, birim şekil değiştirme ve birim kayma açılarının genel ifadeleri ele alınarak silindirik kabuklar için kesit zorları bulunmuştur. Denge denk lemlerinin hesaplanmasıyla alan denklemleri elde edilmiştir. Üçüncü bölümde alan denklemlerinden yola çıkılmış, Gateaux türevinden yararlanarak operatörün potansiyel olduğu bulunmuş ve silindirik kabuklar için fonksiyonel elde edilmiştir. Dördüncü bölümde çözüm yöntemleri hakkında bilgiler veril miş, rijitlik matrisleri oluşturularak çözüme gidilmiştir. Beşinci bölümde silindirik kabuklarla ilgili uygulamalar yapılmıştır. Altıncı bölümde sonuçlar incelenmiştir.

Shells are known as complex, structural systems due to their complexity in mathematical formulation and geometric shape. For that reason, both in theoretical and experimental analysis, certain problems were met and only systems with severely idealized situations under certain conditions were solvable. The theory of shells forms a part of the theory of elasticity concerned with the study of the deformation of elastic bodies under the influence of given loads. For this purpose, it will be assumed that the material of the shell is isotropic and obeys Hooke's Law and that the displacements at a point are small in comparison with the thickness of the shell. A shell will be called thin, if the maximum value of the ratio h/R can be neglected. The theory of thin shells was first considered by H. Aron who derived an expression for the potential energy of a shell and the equations of equilibrium and of deformations in the curvilinear coordinates of the middle surface. A. Love gave a derivation of the equations of equilibrium and motion for shells that was free from some assumptions made by Aron. The Love theory of shells was built on analogy with Kirchoff s theory of plates, and based on the assumptions usual in that theory Numerical methods are preferred for general shell problems. One of the widely used numerical method is the finite element method. Cylindrical shells are superficial transporting systems of which middle surfaces are single curvature. Funnel, cylindrical structures, vault are examples of cylindrical shells. A survey of finite element formulation of cylindrical shells is given by Ashwell and Gallagher [2]. The mixed formulation of finite element method was first studied by Hermann [3], [4] for plate bending problems. Prager [5], [6] established the theoretical basis for the mixed formulation using variational principles. Another widely used numerical method is the Ritz method. The main difference between the finite elements and the Ritz vu method is that Ritz method is valid on the complete field whereas finite element method is valid only on the element inside the field. In this thesis functional for thin cylindrical shells with geometrical and dynamic boundary conditions are presented. This functional was also been proved to be potential and is trans formable to the classical energy equations. In this study finite elements and Ritz method are used. For cylindrical shells the following assumptions are con sidered; - The displacement components are small compared to the shell thickness. - Kirchoff s - Love hypothesis is valid. - Hooke's Law is valid. - The shell thickness is one degree smaller than the other dimensions. The field equations are written as; The equilibrium equations; 3Q dP m+şq+h 0 as & r Mz *+Ş£-2İ 0 (!) 1 r+mx = U Sx 3s as ax s The constituve equations; v as ax rJ {da ax KJ V1U F-c(f+nx)=o H-c(f+^-i>0 <2> k-d[^-+9^.]=o I ax ds ) m-d(- +»- V<> k ds dx. ) -<¥)( ^X, ^S )-Q ds dx J where B=j^-,; D=,**'.; O^^r- (3) (l-a2) 12(l-d2) 6(l+d) V ' The dynamic boundary conditions; -x+x=0 l-İ = 0 (4) Geometric boundary conditions; -Q+Q=0 8-5 = 0 (5) written in symbolic form, x, £, CI, 8 are force, moment, rotation and displacement. Quantities with hat have known values on the boundary. Field equations can be written in operator form as; Q=£y-f (6) Using Gateaux differential if the operator for thin cylindrical shells satisfy the equation (7), we can say that the operator is potential. «*Q(y, y), />= (7) IX The inner products on the boundary are ; [x, 5]=[Q, u] + [P, u]+[Q, v]+[N, v]+[F, w]+[H, w] [ç, n]=[K, nx]+[T, nx]+[M, nj+[T, ns] (8) The functional is obtained as; 1 i(y)=jQ(sy, y)ds Ky)=-[Q,f]-[P,§]-[N,^]-[Q,^]+i-[v,H]-[^,F]-[^,H] OS OK OS OX. R OX OS -^[N,w]+[^,nx]+[^,nx]-[F,Qx]+[^,ns]+[g,ns]- [H,ns]+[qx,u]+[qs,v]+[qz,w]+[mx>nx]+[ms,ns]+^{[F,F] + [H,H]}+2B(lU2){[P'P]+[N,N3"2^[P,N]} + B(ri)[Q>Q] + D(I^&)[T'T]+ 2d(iU2) { [K»K1+[M>M]-2atK>M3 >+< C". QJ +[u,P]+[v,Q]+[v,N]+[w,F]+[w,H]-[nx,(K-K)]-[nx,(T-T)]- [ns,(M-M)]-[ns,(T-T)]}a+{-[K,nx]-[T,nx]-[M,ns]-[T,ns] +[P,(u-û)]+[N,(v-v)]+[Q,(u-Û)]+[Q,(v-v)]+[F,(w-w)] + [H,(w-w)]}s In this thesis a functional for thin cylindrical shells with geometric and dynamic boundary conditions are presented. For the solution of the problem, finite element method and Ritz method are used. In finite element method, an isoparametric rectangular element is used by applying variational principles to the functional for thin cylindrical shells. Displacements, rota tions, in plane axial and shear forces, transverse shear forces, bending and torsional moments are the unknowns. An element rigidity matrice is obtained for thin cylindrical shells of constant thickness and then the element rigidity matrice is put in order for the shells of variable thickness. A program written in Fortran language is used for the solution of the problem. The second method is the Ritz method. The rigidity matrice is obtained by deriving the functional with respect to the unknown constants. Another program written in Fortran language helps us to solve the problem. The comparison of the finite element method's results with the examples given in the literature is in good agreement. But the results of the Ritz method for the chosen functions is not in good agreement.