Genel biçimli kabuklar için bir sonlu eleman formülasyonu
Başlık çevirisi mevcut değil.
- Tez No: 39453
- Danışmanlar: PROF. DR. NAHİT KUMBASAR
- Tez Türü: Doktora
- Konular: İnşaat Mühendisliği, Civil Engineering
- Anahtar Kelimeler: Kabuklar, Sonlu elemanlar yöntemi, Shells, Finite element method
- Yıl: 1993
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Belirtilmemiş.
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: Belirtilmemiş.
Özet
Kabuklar hem matematik formülasgonu hem de geometrisi nedeni ile karmaşık ya.pi. sistemleri olarak bilinirler. Pek çok mühendislik yapısında karşılaşılan, eğilme tesir lerinin önemli olduğu genel biçimli kalın sayılabilecek kabukların kullanımının artması sonucu Ki rchof f -Love hipotezi yerine daha doğru bir teori geliştirmek zorunlu olmuştur. Kalın sayılabilecek kabuklar için geliştiril miş bir çözüm yolu olarak yer değiştirmelerin kalınlık doğrultusundaki değişimi için bir seri alınması ve bu serinin yeterli sayıda teriminin hesaba katılması söylenebilir. Bilgisayara uygulanması açısından elveriş li olmayan bu yöntem yerine daha yaklaşık bir yöntem ola rak kalınlık doğrultusundaki kayma şekil değiştirmeleri gözönüne alınmakta veya <z/R) terimi l'in yanında ihmal edilmeyerek hesaba katılmaktadır. Bu çalışmada, kalınlık doğrultusundaki kayma şekil değiştirmelerini ve yarıçap yanında kalınlığın etkisini de hesaba katarak genel biçimli kabuklar için bir şekil değiştirme enerjisi ifadesi elde edilmiştir. Elde edilen bu fonksiyonel yardımı ile yer değiştirmelerin bilinmeyen olarak tanımlandığı eğrisel izoparametrik bir dörtgen eleman geliştirilmiştir. Sonraki aşamada, kabuk geometrik büyüklüklerini daha iyi tanımlamak amacı ile eğrilik değişiminin yeter yakla şıklıkta ifade edilemediği doğrultuda eleman kenarların daki nokta sayısı artırılarak ve Lagrange interpolasyon fonksiyonlarından yararlanarak yüksek mertebeden terimler içeren biçim fonksiyonları elde edilmiştir. Problemlerin çözümü için Fortran kodlama dilinde bir bilgisayar programı hazırlanmıştır. Geliştirilen sonlu elemanın yaklaşıklık derecesi değişik yükleme ve sınır koşulları altında genel biçimli kabuklarda belirlenmeye çalışılmış ve sonuçlar mühendislik açısından yeterli düzeyde bulunmuştur.
Özet (Çeviri)
Shells are known as complex, structural systems due to 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. With the development of computer systems, numerical analysis has become an essential tool in engineering mechanics and finite element method came into definition as an extension to matrix structural analysis. The finite element method has made possible the development of computer programs which may be used for analyzing complete arbitrary structural systems. Different methods of analysis are used for analysis of shells to obtain adequate solutions, with increasing thickness-radius ratio. On the other hands, thickness shear deformations and the ratio z/R near unity is neglected for thin shells. They have to be considered for moderate ones, while asymptotic expansions or basic equations of elasticity must be used for thicker shells. One of the refined solutions for small deflections theory of shells is to express the displacements in power series taking into account of sufficient number of terms, instead of using Ki rchof f -Love hypotesis. There are various solution methods developed especially for the computer application, where thickness shear deformation is taken into account and the first three terms of power series of l/(l+z/R) are considered for integration, as (l-z/R+z2/R2). Novozhilov, stated that additional terms of refinements of order <h/R) will not yield any significant improvement, since due to the Ki rchof f -Love hypothesis, there is an admissible error of the order h/R in the expression of the potential energy. He has deduced in this regard that, additional terms in potential energy originated by including the terms z/R, instead of neglecting in comparison with unity, will be in order of h/R or (h/R), assuming the bending and membrane stresses are of the same order. XII iThe most important part of the error originated from the Kirchof f -Love hypothesis is due to the neglect of thickness shear deformation. The ratio of maximum shear to normal stresses in a cylindrical shell is of order h/R for axially symmetrical loads. This order may be accepted or several types of shells considering the Geckeler approximation. Inclusion of the shear deformation will thus add terms of error of the same order, the inclusion of these terms may be considered as consistent and necessary. In this thesis, an expression for the potential energy of a shell of general shape, including thickness shear deformation and without neglecting z/R in comparison with unity, is derived. Then a finite element formulation based on this functional is obtained. The following assumptions are considered; -The transverse normal stress is small compared with the other normal stress components and may be neglected. -Strains and displacements are sufficiently small so that the quantities of second and higher order magnitude in the strain-displacement relations may be neglected in comparison with the first-order terms. -In moderately thick shells (z/E) is included instead of neglecting in comparison with unity. -Normals to the undeformed middle surface remain straight but not perpendicular to the deformed surface. -Hooke's Law is valid. s, e, y, y, y strain- displacement relations 1 2*12 12*22 obtained in three dimensional elasticity for small displacements can be expressed in terms of shell coordinates <xi,cxz,z and the displacements U,V,W along these coordinate directions. These expressions are the same as those used by Novozhilov except the thickness shear deformations. Considering the strains obtained, together with Hooke's Law and inserting in the expression. 1 r V=- -- I (.a £ +o £ +o £ +t y +t y +t y 2J 11 22 Z Z 12 12 İZ lz 2z )dv 2z“ 2 z various types of integrals, depending on z, are encountered. It is possible to integrate them with respect to z and find exact expressions instead of expanding 1/(1 +z/R) in power series. They may be used for solutions having the desired degree of accuracy. xivSome typical integrations are given below: + h/2 I, 1 +h/2R. TTT3”= ln TTTT“ = ~*~ (1 V w l-h/2R. I -h/2 Where, 1+- h R. 2RT ZL=-Fr ln t_ hv -1 2R. or V -M-ar] +^Mc) +^Mt) +^NH a +. These integrals man be obtained in terms of Zt. The strain energy may be formed after performing all of these integrals and a field integral is obtained in the result. In order to obtain an expression similar to those obtained in previous works, one may write Eh ç Eh p Eh (. V= Q dA+ Q dA+ Q dA 2(l-v2) J ° 24<l-v2> J 2 2(l-v2) J 3 A A A in which Q3 represents the effects of thickness shear stresses. Where Q o I1 2J which is the same expression given by Novozhilov. In order to make the expression Q2 similar to Q2 given by Novozhilov and retain the terms consistent with the accuracy of the present analysis, one may write. h2 h2 Z £, Z =S 1 12R2 2 12R2 1 2 XVThus the followings are obtained. Q2=(^+^]2-2(1-v: *±*2 4 £ £ 1 2 R R 1 2 ']? -a 'i i *j r -i rı i i r ^ o2 OZ 1 '2 R R 1 2 -(l-v> 1 11 ?r”~~~r“~ 1 2 (J 1 1 ' 2 lj Q3 = -(1. f o2^ 02^ [' İ2 2zJ The reason for the difference of Q2 from that given bg Novozhilov is the independence of the functions a. and ft from the displacements u,v,w, due to the inclusion of shear deformations. An additional strain energy term Qa is also obtained due to the same reason. This formulation of the functional were not met in the literature survey. A curved isoparametric trapezoidal fini formulation based on this expression for the energy of a shell of general shape, including shear deformation and without neglecting comparison with unity, is derived. The shell eight nodes and at each node three displacemen directions of the local coordinates and two ro defined i.e. a total of five parameters, element stiffness matrices, shape func derivatives of shape functions of this element t e element potential thickness z/R in element has ts at the tations are To obtain t i ons and are used. Shape functions of the shell finite element with eight nodes are used to describe the surface goemetry similar to the calculation of element stiffness matrices. But in existing obtained for the insufficient. For function presented terms to describe approach in the geometry shape functions, that are study, the level of approximation geometrical values are indicated that reason, a finite element shape in this study contains high degree the surface geometry better. The calculation is improved with found by increasing the number of points on two sides of the element and making use of the Lagrange interpolation functions. XVIThese shape functions are; N. =-4.(l+ÇÇ. ) (.1+7)7). ) ı 4 t ı çç. +(1-7777. > (--^”--g^i"1) 1=1,2,3,4 at the corner nodes, and Ç.=0, FT. =-İ-(l-Ç2Xl+7)7?i) i »6,8 7) =0, M. =2<1+ÇÇ. )<7)2-D<7)2-l/4) i =5,7 77. =+1/2, FT. =- İ_<l+Ç£. )7?(7?2-l)<7?+r?. ) 1=9,10.11.12 V 1. w 1> L at the middle nodes. The level of approximation obtained for the geometrical values are indicated in the numerical examples. Shape functions with high degree terms are used only to describe the surface geometry. In the calculation of element stiffness matrices shape functions with quadratic terms are preferred. In the solution of a problem, data enter in respect of the finite element with twelve nodes, that contains high degree terms. When displacements which belong to last four nodes enter as zero, using different elements doesn't form ana difficulty during data input. A computer program, written in Fortran programming language is developed for the analysis of moderately thick shells of general shape. Accuracy of finite element solutions of moderately thick shells of general shape is shown on examples. Looking from the view of necessary engineering precision, the comparison of the results with the examples given in the literature has shown in good agreement.
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