SPRINGER BRIEFS IN APPLIED SCIENCES AND TECHNOLOGY  CONTINUUM MECHANICS Alexander Ya. Grigorenko Wolfgang H. Müller Yaroslav M. Grigorenko Georgii G. Vlaikov Recent Developments in Anisotropic Heterogeneous Shell Theory Applications of Refined and Three-dimensional Theory—Volume IIB 123 SpringerBriefs in Applied Sciences and Technology Continuum Mechanics Series editors Holm Altenbach, Magdeburg, Germany Andreas Öchsner, Southport Queensland, Australia These SpringerBriefs publish concise summaries of cutting-edge research and practical applications on any subject of Continuum Mechanics and Generalized Continua, including the theory of elasticity, heat conduction, thermodynamics, electromagnetic continua, as well as applied mathematics. SpringerBriefs in Continuum Mechanics are devoted to the publication of fundamentals and applications, presenting concise summaries of cutting-edge research and practical applications across a wide spectrum of fields. Featuring compact volumes of 50 to 125 pages, the series covers a range of content from professional to academic. More information about this series at http://www.springer.com/series/10528 ü Alexander Ya. Grigorenko Wolfgang H. M ller (cid:129) Yaroslav M. Grigorenko Georgii G. Vlaikov (cid:129) Recent Developments in Anisotropic Heterogeneous Shell Theory fi Applications of Re ned and — Three-dimensional Theory Volume IIB 123 Alexander Ya.Grigorenko Yaroslav M.Grigorenko S.P.Timoshenko Institute of Mechanics S.P.Timoshenko Institute of Mechanics National Academy of Sciences ofUkraine National Academy of Sciences ofUkraine Kiev Kiev Ukraine Ukraine WolfgangH.Müller GeorgiiG.Vlaikov Institut für Mechanik Technical Center Technische UniversitätBerlin National Academy of Sciences ofUkraine Berlin Kiev Germany Ukraine ISSN 2191-530X ISSN 2191-5318 (electronic) SpringerBriefs inApplied SciencesandTechnology ISBN978-981-10-1595-3 ISBN978-981-10-1596-0 (eBook) DOI 10.1007/978-981-10-1596-0 LibraryofCongressControlNumber:2015958914 ©TheAuthor(s)2016 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authorsortheeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinor foranyerrorsoromissionsthatmayhavebeenmade. Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerScience+BusinessMediaSingaporePteLtd. From the Preface of Book 1 The theory of shells is an independent and highly developed science, logically based onthe theory ofelasticity. Constructionsconsisting ofthin-walled elements, have found widespread application in mechanical engineering, civil and industrial construction, ships, planes and rockets building, as well as transport systems. The developmentofdifferent shellmodels requirestheapplicationofhypotheses based on elasticity theory leading to a reduction in terms of two-dimensional equations that describe the deformation of the shell’s middle surface. The solution of shell problems requires use of various numerical methods and involves great difficulties ofcomputationalnature.Theauthorspresentdiscrete-continuumapproacheswhich theydevelopedforsolvingproblemsofelasticitytheoryandwhichallowtoreduce the initial problem to systems of ordinary differential equations. These are then solved by the stable numerical method of discrete orthogonalization and will be presentedinthisbook.Onthebasisoftheseapproachesasolutionforawideclass of problems of stationary deformation of anisotropic heterogeneous shells is obtained. Themonographconsistsoftwobooks,eachofwhichconsistsofthreechapters. A summary of the chapters is as follows. Chapter1(VolumeIIA):Thesolutionsofstress-strainproblemsforawideclass of anisotropic inhomogeneous shells obtained by the refined model are presented. Studying these problems results in difficult calculations due to partial differential equationswithvariablecoefficients.Forsolvingtheproblem,spline-collocationand discrete-orthogonalization methods are used. The influence of geometrical and mechanicalparameters,oftheboundaryconditions,andoftheloadingcharacteron thedistributionsofstressanddisplacementfieldsinshallow,spherical,conical,and noncircular cylindrical shells is analyzed. The dependence of the stress-strain pat- ternonshellthicknessvariationisstudied.Theproblemwassolvedalsointhecase ofthethickness varyingintwodirections.It isstudied howtheruleofvariationin thethicknessoftheshellsinfluencestheirstress-strainstate.Noncircularcylindrical shells with elliptical and corrugated sections are considered. v vi FromthePrefaceofBook1 The results obtained in numerous calculations support the efficiency of the discrete-orthogonalization approach proposed in the monograph for solving static problems for anisotropic inhomogeneous shells when using the refined model. Chapter 1 (Volume IIB): A wide class of problems of natural vibrations of anisotropic inhomogeneous shells is solved by using a refined model. Shells with constructional (variable thickness) and structural inhomogeneity (made of func- tionally gradient materials) are considered. The initial boundary-value eigenvalue partial derivative problems with variable coefficients are solved by spline-collocation, discrete-orthogonalization, and incremental search methods. In the case of hinged shells, the results obtained by making use of analytical and proposed numerical methods are compared and analyzed. It is studied how the geometrical and mechanical parameters as well as the type of boundary conditions influence the distribution of dynamical characteristics of the shells under consid- eration. The frequencies and modes of natural vibrations of an orthotropic shallow shell of double curvature with variable thickness and various values of curvature radiusaredetermined.Fortheexampleofcylindricalshells madeofafunctionally gradient material, the dynamical characteristics have been calculated with the thicknessbeingdifferentlyvariedincircumferentialdirection.Thevaluesofnatural frequencies obtained for this class of shells under some boundary conditions are compared with the data calculated by the three-dimensional theory of elasticity. Chapter2(VolumeIIB):Themodelofthethree-dimensionaltheoryofelasticity is employed in order to study stationary deformation of hollow anisotropic inho- mogeneouscylindersoffinitelength.Solutionsofproblemsofthestress-strainstate and natural vibrations of hollow inhomogeneous finite-length cylinders are pre- sented, which were obtained by making use of spline-collocation and discrete- orthogonalization methods. The influence of geometrical and mechanical parame- ters, ofboundaryconditions, andof theloading characteron distributionsof stress and displacement fields, as well as of dynamical characteristics in the above cylindersisanalyzed.Forsomecasestheresultsobtainedbythree-dimensionaland shell theories are compared. When solving dynamical problems for orthotropic hollow cylinders with different boundary conditions at the ends, the method of straight-line methods in combination with the discrete-orthogonalization method was also applied. Computations for solid anisotropic finite-length cylinders with differentendconditionswerecarriedoutbyusingthesemi-analyticalfiniteelement method. In the case offree ends the results of calculations the natural frequencies werecomparedwiththosedeterminedexperimentally.Theresultsofcalculationsof mechanical behavior of anisotropic inhomogeneous circular cylinders demonstrate the efficiency of the discrete-continual approaches proposed in the monograph for solvingshellproblemsusingthethree-dimensionalmodelofthetheoryofelasticity. Contents 1 Solutions of Dynamic Problems Based on the Refined Model. . . . . . 1 1.1 Free Vibrations of Rectangular Plates. . . . . . . . . . . . . . . . . . . . . 1 1.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.2 Problem Formulation. Governing Equations . . . . . . . . . . . 2 1.1.3 Solution Technique. . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.1.4 Problem Solving. Analysis of the Results. . . . . . . . . . . . . 9 1.2 Free Vibrations of Shallow Shells . . . . . . . . . . . . . . . . . . . . . . . 12 1.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.2.2 Problem Formulation. Governing Equations . . . . . . . . . . . 12 1.2.3 Problem Solving Method. . . . . . . . . . . . . . . . . . . . . . . . 17 1.2.4 Numerical Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.3 Free Vibrations of Closed and Open Cylindrical Shells . . . . . . . . 29 1.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 1.3.2 Main Relations. Governing Equations . . . . . . . . . . . . . . . 29 1.3.3 Solution Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 1.3.4 Analysis of the Results . . . . . . . . . . . . . . . . . . . . . . . . . 36 1.4 Free Vibrations of Cylindrical Shells Made of Functionally Gradient Materials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 1.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 1.4.2 Problem Statement. Basic Relations. . . . . . . . . . . . . . . . . 47 1.4.3 Three-Dimensional Elasticity Theory. . . . . . . . . . . . . . . . 51 1.4.4 Problem Solving Method. . . . . . . . . . . . . . . . . . . . . . . . 53 1.4.5 Numerical Results and Their Analysis. . . . . . . . . . . . . . . 55 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 2 Some Solutions of Stationary Problems Based on 3D Theory. . . . . . 61 2.1 Stress–Strain State of Anisotropic Cylinders of Finite Length . . . . 61 2.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 2.1.2 Transversally-Isotropic and Orthotropic Cylinders . . . . . . . 63 vii viii Contents 2.2 Stress–Strain State of Heterogeneous Cylinders of Finite Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 2.2.1 Basic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 2.2.2 Solution Technique. . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 2.2.3 Numerical Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 2.3 Free Vibrations of Anisotropic Heterogeneous Cylinders of Finite Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 2.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 2.3.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 2.3.3 Approach Based on Using the Straight-Line Method. . . . . 80 2.3.4 Approach Based on the Application of the Semi-analytical Finite-Element Method. . . . . . . . . . 87 2.3.5 Spline-Approximation Approach. . . . . . . . . . . . . . . . . . . 96 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 Chapter 1 Solutions of Dynamic Problems Based fi on the Re ned Model Abstract A wide class of problems on natural vibrations of anisotropic inhomo- geneous shells is solved by using the refined model. Shells with constructional (with variable thickness) and structural inhomogeneity (made offunctionally gra- dient materials) are considered. Initial boundary-value, eigenvalue, and partial derivative problems with variable coefficients are solved by spline-collocation, discrete-orthogonalization, and incremental search methods. In the case of hinged shells,theresultsobtainedbymeansofanalyticalandproposednumericalmethods are compared and analyzed. It is studied how the geometrical and mechanical parameters as well as the type of boundary conditions influence the distribution of dynamical characteristics of the shells under consideration. The frequencies and modesofnaturalvibrationsofanorthotropicshallowshellofdoublecurvaturewith variable thickness and various values of a radius of curvature are determined. The dynamicalcharacteristicshavebeencalculatedfortheexampleofcylindricalshells made of a functionally gradient material with thickness varying differently in cir- cumferential direction. The values of natural frequencies obtained for this class of shells under some boundary conditions are compared with the data calculated by means of three-dimensional theory of elasticity. 1.1 Free Vibrations of Rectangular Plates 1.1.1 Introduction Natural vibrations of orthotropic plates subjected to various boundary conditions havebeenstudiedquiteactively.Thisisreflectedinanumberofpublications(e.g., Leissa[13,14]).Whencomparedtosimilarinvestigationswithintheframeworkof the classical theory of plates natural vibrations of rectangular plates with varying thickness have been studied less actively by using Mindlin theory. Of particular importance in this context isthe work by Mindlin, Mizusava and Condo [22]. The collocation method based on the orthogonal polynomials was used in Mikami and Yoshimura [20] in order to analyze vibrations of a plate with linearly varying ©TheAuthor(s)2016 1 A.Y.Grigorenkoetal.,RecentDevelopmentsinAnisotropicHeterogeneous ShellTheory,SpringerBriefsinContinuumMechanics, DOI10.1007/978-981-10-1596-0_1