Ortotrop Zemine Oturan Süper Eliptik Kesitli Akışkan Tankı İnce Taban Plağının Serbest Titreşim Analizi

Konferans Bildirisi-- İstanbul Teknik Üniversitesi, Teorik ve Uygulamalı Mekanik Türk Milli Komitesi, 2017Conference Paper -- İstanbul Technical University, Theoretical and Applied Mechanical Turkish National Committee, 2017Bu çalışmada, süper eliptik kesitli, rijit akışkan tanklarının ince taban plaklarının serbest titreşim analizi gerçekleştirilmiştir. Akışkan tankının ortotrop Pasternak elastik zeminine oturduğu düşünülmüş ve içerideki akışkan ideal olarak alınmıştır. Akışkanın plak davranışı üzerindeki eylemsizlik etkileri sınır eleman yöntemi ile tanımlanırken, zeminle temas eden süper eliptik geometrili taban plağı Hellinger Reissner prensibi kullanılarak karışık sonlu eleman formülasyonu ile modellenmiştir. Taban plağı doğal frekans değerlerinin farklı tank geometrisi, akışkan yüksekliği ve zemin parametrelerine bağlı değişimi ayrıntılı olarak incelenmiştir.In this study, the free vibration analysis of thin plates forming the bottom of rigid fluid storage tanks with super-elliptic section is performed. The fluid storage tank is considered to rest on orthotropic Pasternak foundation while storing ideal quiescent fluid. The added mass terms included from the fluid domain into plate equation of motion are computed by a boundary element method and the super-elliptic bottom plate interacting with foundation is simulated by a mixed finite element formulation using the Hellinger-Reissner principle. The variation of the natural angular frequency values of the bottom plate with respect to various tank geometries, fluid depth and foundation parameters are investigated intensively

The static and free vibration analyses of axially functionally graded elliptical beams via mixed FEM

The objective of this study is to investigate the behavior of the static and free vibration analyses of axially functionally graded elliptical planar curved beams using a mixed finite element method (MFEM) based on the Timoshenko beam theory. A two-noded curved mixed finite element has 12 field variables at each node. These variables denote three displacements, three cross-sectional rotations, three forces, two bending moments, and torque, respectively. The functionally graded material is composed of ceramic-particle material and metal-matrix material. The volume fraction of ceramic and metal materials varies along the beam axis. The effective material properties (modulus of elasticity, Poisson's ratio, and density) of the functionally graded material are determined according to the rule of mixture. It is aimed in the benchmark examples to present the influence of ceramic-particle material and non-homogeneity index of material gradation, the minimum radius of the elliptical beam, and boundary condition to the results of static and free vibration analysis in detail

Mixed finite element method formulation for stiffened cylindrical shells

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

Buckling of rectangular FSDT plates resting on orthotropic foundation by mixed FEM

This study presents a mixed type finite element procedure for the linear buckling analysis of moderately thick plates lying on orthotropic elastic foundation. Kinematical expressions are due to the Mindlin plate theory and von Karman strains. The force intensity exerted by orthotropic foundation on the plate is reflected according to the Pasternak model. Material directions of the foundation coincides with the global axes of the plate. The first variation of the systems nonlinear functional is obtained by following the Hellinger-Reissner principle. This expression is linearized according to the incremental formulation, thus the system and geometric matrices of the problem are obtained. Finite element equations are constructed by discretizing the plate domain with four noded isoparametric quadrilateral elements. After a static condensation procedure, force and couple type field variables are removed from the equations in order to reduce the problem into the solution of a standard Eigen-value system. Firstly, a convergence and comparison study is presented to verify the formulation and numerical procedure. The effects of foundation and plate parameters on the critical buckling loads are investigated

Contact problem between a rigid punch and a functionally graded orthotropic layer resting on a Pasternak foundation

The present work is a pioneering study on the contact mechanics including Pasternak foundation model. The context of this research is frictionless plane contact problem between a rigid punch and a functionally graded orthotropic layer lying on a Pasternak foundation in the limits of the linear elasticity theory. The layer is pressed by rigid cylindrical or flat punches that apply a concentrated force in the normal direction. The orthotropic material parameters are assumed to vary exponentially in the in-depth direction. Applying the Fourier integral transform technique and the boundary conditions of the problem, a singular integral equation is obtained, in which the contact stress and the contact width are unknowns. Using the Gauss-Chebyshev integration formula the singular integral equation is solved numerically. Effects of the Pasternak foundation parameters, material inhomogeneity, external load, punch radius or punch length on the contact stress, the contact width, the vertical displacements on the top and bottom surfaces of the layer, the subsurface and in-plane stresses are given

Zemin Ve Akışkan İle Etkileşen Plakların Serbest Titreşim Problemi İçin Bir Sınır Eleman-Karışık Sonlu Eleman Çözümü

Konferans Bildirisi -- Teorik ve Uygulamalı Mekanik Türk Milli Komitesi, 2015Conference Paper -- Theoretical and Applied Mechanical Turkish National Committee, 2015Bu çalışmada, Pasternak zeminine yaslanan ve durağan akışkanla etkileşen Mindlin plaklarının serbest titreşim analizi için bir çözüm yöntemi sunulmuştur. Plak zemin sistemi için Hellinger-Reissner prensibi uygulanmış, yapı-akışkan etkileşimi içinse bir sınır eleman çözümü sunulmuştur. Önerilen çözüm şemasında akışkan etkisi plak yer değiştirmeleri cinsinden tarif edilmiş ve plak hareket denklemlerine eksu kütlesi formunda dahil edilmiştir. Yöntemin uygulaması olarak rijit akışkan tanklarının dairesel taban plakları ele alınmış, plak kalınlık-genişlik oranı, akışkan doluluk oranı gibi sistem parametrelerinin plağın dinamik davranışı üzerindeki etkileri ayrıntılı olarak incelenmiştir.In this study a new solution strategy for the free vibration analysis of Mindlin plates lying on Pasternak foundation and interacting with a quiescent fluid domain is presented. The Hellinger-Reissner variational principle is adopted for the plate-foundation system to generate a mixed finite element formulation and a boundary element solution scheme is presented for the fluid-structure interaction. Through the suggested formulation, the added mass matrix is defined in terms of plate deflection, which is then integrated into the plate equations of motion. Circular bottom plates of rigid fluid storage tanks are investigated as application of the method. The effect of the system parameters, such as thickness to width and fluid filling ratios, on the plate dynamic performance is thoroughly studied

Free vibration of axially FG curved beam on orthotropic Pasternak foundation via mixed FEM

The main objective of this study is to present an accurate and consistent original formulation to describe the interaction between planar straight/curved beam structure and orthotropic foundation by highlighting the problematic orthotropic foundation models currently used in the literature. The proposed formulation introduces a consistent reduction of a 2D stress state of an elastic orthotropic foundation to a 1D form suitable for interaction with a beam. The formulation is also extended for curved beams resting on an arbitrarily orthotropic generalized Pasternak foundation by including a rocking effect. The beam is assumed to be made of a two-phase composite material of metal-matrix and ceramic-inclusion varying continuously through the parabolic beam axis. The effective material properties of the axially functionally graded material are predicted through a cubic local representative volume elements scheme. A two-noded curved beam element is employed based on a mixed finite element formulation based on Timoshenko beam theory by considering cross-sectional warping and the tests are performed over free vibration analyses. The new formulation of orthotropic Pasternak foundation for straight beam interaction is firstly verified by the results of a plate resting on orthotropic foundation. The flaws of existing formulations in the literature are explained, and the erroneous results produced by some of them are shown with comparison examples. The formulation is capable of modelling straight beams as well as general curved planar beams and orthotropic Pasternak foundation interaction problems. It is believed that the derived expression be safely implemented for further analyses, where beam like structures interacting with orthotropic foundation or substrate. Furthermore, this study presents free vibration results of several original benchmark examples using the mixed finite element formulation, e.g. straight and parabolic beams resting on an arbitrarily orthotropic generalized Pasternak foundation.International Technological Universit

Two-dimensional solution of functionally graded piezoelectric-layered beams

This study presents an exact elasticity solution for the functionally graded piezoelectric beams subjected to sinusoidally distributed transverse electromechanical loads. The elasticity solution can reflect the symmetrical exponential variation of materials about the interface of perfectly bonded two layers. The effects of the material gradation, beam span length to thickness ratio, intensity of external mechanical/electrical loads, and material properties on the axial normal stresses, transverse normal stress, transverse shear stress, axial and vertical displacements, electric displacements, and electrical potential are comprehensively investigated. The proposed elasticity solution can be used for the verification/comparison purposes of numerical procedures as a reference procedure

An exact elasticity solution for monoclinic functionally graded beams

In this study, an elasticity solution is presented for monoclinic functionally graded beams subject to a transverse pressure distributed sinusoidally. Monoclinic material properties are assumed to vary exponentially throughout the thickness of the beam's layers. An analytical formulation based on the classical Euler-Bernoulli beam theory is also derived for comparison purposes of simply supported monoclinic functionally graded beams. In benchmark examples, the numerical results of normal stresses, transverse shear stress, as well as axial and vertical displacements are presented. The effect of material grading, fiber angle, and beam length to thickness ratio on the stress and displacement distributions is comprehensively investigated. The proposed elasticity-based analytical solution and presented numerical results can be used for verification or comparison purposes of numerical procedures