# Mixed finite element method formulation for stiffened cylindrical shells

## Abstract

Tez (Doktora) -- İstanbul Teknik Üniversitesi, Fen Bilimleri Enstitüsü, 1990Thesis (Ph.D.) -- İstanbul Technical University, Institute of Science and Technology, 1990Ortalama yüzeyi tek eğrili kli yüzeysel taşıyıcı sistemler olan silindirik kabuklar; tonozlarda, baca larda, borularda, sıvı haznelerinde, silindirik yapı larda ve kiriş takviyeli kabuklar ise uçak, gemi, deniz altı, geniş açıklıklı silindirik tonoz gibi yapı sistem lerinde sıkça karşılaşılan yapı elemanlarıdır. Bu çalışmada silindirik kabuklarla, daire ve doğru eksenli uzay çubuklar için yeni fonksiyoneller dinamik ve geometrik sınır koşullarımda içerecek şekilde elde edilmiştir. İfadelerin literatürde orijinal ol dukları görülmüştür. Elde edilen kabuk fonksiyoneli klasik enerji ifadesine dönüştürülebilmektedir. Varyasyon tekniği kullanılarak karışık izoparametrik sonlu elemanlar, yapı elemanlarının değişken kesit özel liklerini rijitlik matrisleri içinde bulunduracak biçim de türetilmiştir. Kabuk global eksen takımı ile çubuk asal eksen takımlarının çakışmama durumunu da gözönüne alacak şekilde eksenleri döndürülmüş çubuklar için de rij itlik matrisi elde edilmiştir. Kabuk elemanda yer değiştirmeler, kuvvetler, momentler, uzay çubuk eleman da bunlara ek olarak dönmelerde bilinmeyen olarak he- sapl anmak tadır. Problemlerin çözümü için Fortran kodlama dilinde bir bilgisayar programı hazırlanmış ve yardımcı disk üniteleri ile geçici dosyalama imkanlarından faydala- nılmıştır. Bu çalışmada önce silindirik kabuklar, sonra uzay çubuklar literatürdeki örneklerle karşılaştırılmış ve sonuçlar mühendislik açısından gerekli yeter yakınsak lıkta bulunmuştur. Bilinmeyen sayısına göre sonuçlara yaklaşım ise hızlıdır. Daha sonra değişken kesitli ka buklar ile, kiriş takviyeli kabuklar parametrik olarak incelenmiş elde edilen neticeler sonuç bölümünde sunul muştur.Shells are known as complex, structural systems due to complexity in mathematical formulation and geo metric 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 became an essential tool in engineering mechanics and finite element method came into picture as an extension to matrix structural analysis. Prob ably the greatest advantage of finite element method that it has made possible the development of general purpose computer programs which may be used for analys ing complete arbitrary structural systems. Numerious different finite element models are possible by selecting different displacement or stress fields or both defined in terms of variety of generalized coordinates and introducing different equilibrium conditions at the nodes or compatibility conditions along inter element boundaries or both in classical finite element formulation. Approaches in the finite element analysis, are associated with application of variational principles in solid mechanics. The improvements in finite elements first began by application of matrix displacement method to plane stress problems using triangular and rectangular elements. In the assemblage of discrete elements and / or stress field are assumed in the element for represen tation of a solid continuum. The application of variational principles gives simultaneous algebraic equations which are in terms of generalized stresses or generalized displacement or both or generalized forces and generalized moments and generalized displacements at the nodal points. xiii In this thesis some new functional s are obtained for cylindrical shells, and curved space bars by functional analysis method. Parameters in this func tional can be choosen as required with respect to ne cessity. These functional s reduce to the classical potential energy function as a special case. These functional s gain an advantage over classical potential energy function when finite element and variational methods are used. A new finite element formulation for cylindrical shells, stiffened by curved and straight space bars are given. The following assumptions are considered, - Transverse stresses are omitted, - Kirchof f-Love hypothesis is valid. That is the straight fibres of the shell which are perpen dicular to the middle surface before deforma tion remain straight and perpendicular to the deformed surface, - The displacement components are small compared to the shell thickness, - Hook * s Law i s val i d, in derivation of the field equatins of the shell as, 3S" + 3x" + qi " ° i \au. srr\ E 13s 3xj § + 5§ + ff[£ + £| + % = ° as2 dx2 ^^ * 9 p - -B £ - B £ - -I w - ° "-"fi-"**?-!--0 Q - B |-^-| ^ - B l-g-j ^ - o e-,££ + vd£İ + d£*«o * *» ds2 *x2 w D dv _,_ _, d2w ^ _ d2w _,. M " E 3s" + D + vD = ° C3. 13 N D fl-ul 2 2 as dx ** dxds where, B=Eh/C 1 -t>2D and D=Eh3/£ i 3C 1 -*>2:> 3. Dynamic boun dary conditions, Xiv -K+î=0 C3.1D R-R = O geometric boundary conditions, -d' +d' =0 * C3.1Z> d-d = O written in symbollic form. Quantities with hat have known values on the boundary. Field equations can be written in the operator form as, i?=fu- q=0 which is shown to be a potential operator givenin equation C3. 4D. Using Gateaux differential dJ£Cu,uD, equation C3. ID yields to the following func tional after a few manipulations, T - r^ ^t re #"-? rvi *V, r^ *V,,lr âM,Jr ^T- Ik=-[Q-3s-3"-[P'^3"[N»^3-tQ»^+HrV^3+RtV'^].au âw. -âE dw, TffT aw..err aw. ir.,. -[ 7£- a^1 -l ?&> 3x-3 -[ 7£> 7Z? -c SZ- 5ST3 -R [ N ' w3 + - -JcP.P]+CN,N]-av[P,N3 I + k*,vtQ.Q] 2BC1-U D I J ÖCJ.-UJ + - |[E,E]+CM,M]-£t;[E,M3 I + --i__.ET.T3 aDci-^1 J DC1-JüD +[q,u3+Cq,v3+Cq,w3 ^1 ^2 ^a Eu,Q3+Cu, P3 +[ v, Q3 +E v. N3 -İC v.CM-MD 3 -İE v, CT-T) 3 R R *t..S]+.£.C*4b]«..«J*tg-.C&«)*lw.£1 öw ~ + [XZ,CT-TD3+[w as.îh£-«-î"] a + [ Q, Cu-uD 3 + [ P, Cu-uZ) 3 +E N, C v-v2> 3 +[ Q, C v-vD 3 -i[ M, v3 -i[ T, v3 +[ g, C w-*0 3 +£ M. |^3 +[ %±. C w-wD 3 +[ |E, C w-wZ> 3 +CE, |^3 +[ |I, C w-wD 3 +C T, |^3 +[ T dw ] - * 1 C3.7Z) xv Field equations for curved space bars, dT - 1 + q = O ds C3. 83 M - D w = O T - C v = O Dynamic boundary conditions, - T + T = O ~ C3. S.dZ) - M + M = O and geometrical boundary conditions, n - o = o Z CS.S.dD u - u = O are written in symbol lie form. Using Gateaux differen tial, equation C3.83 yields to the following functional after a few manipulations, dT dH I =[u,-T^]-[txn,T]+[q,u3+[m,n]+[-A,Q] c *?* cts ** ~ ~ *?? ~ ** ds *" +İC D" 1M, M] +i[ C~ *T. T] -[ CT-TD, u] -[ C M-Sd, 03 -eG.t] -cn.Mi r~ 1P. ~ ~ £ ~ ~ £. C3. 1 tf J is obtained in vectorial form for three di mensi nal bars. To obtain element rijidity matrices, variational method is applied to the given functional s of sheels and bars. In the derivation of shell and bar* finite elements, isoparametric finite element formulation is followed. The principle idea of isoparametric finite element formulation is to achieve the relationship between the element unknowns at any point and the xvi element nodal point unknowns directly through the use of interpolation functions, and the element matrices corresponding to the required degrees of freedom are obtained directly. Since the functional s have only de rivatives of first degree, lineer shape functions for shell and bar element would be necessary and sufficient. The finite element matrices, include the uniform variation of crossectional properties both for shell and bar elements. The shell element has four nodes and at each node three displacements at the directions of the global coordinates and three inplane forces and three inplane moments are defined as a total of nine unknowns. The bar element has two nodes and at each node three displacements, three rotations, two axial one transverse force, one twisting two bending moments are defined as a total of twelwe unknowns. In this study, new fuctionals for thin cylindrical shells and curved space bars with geometrical and dyna mic boundary conditions are presented. In the litera ture survey the same functionals were not met. These functional s were also been proved to be potential and they are transformable to the classical energy equa- ti ons. A computer program written in Fortran programming language using discs and temporary files is developed for the analysis of shells, bars, stiffened shells of any shape. Looking from the view of necessary engineering precision and satisfaction, the comparision of the results with the examples given in the literature was in good agreement. When a comparision is made for reaching to the results in required precision with respect to the num ber- of unknowns, the given finite element formulation is comparable with all other fast reaching finite el ement studies.DoktoraPh.D

## Full text ### Ulusal Üniversitelerarası Açık Erişim Sistemi - İstanbul Teknik Üniversitesi

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oai:polen.itu.edu.tr:11527/16410Last time updated on 7/11/2018

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