COMPUTATIONAL STRUCTURAL MECHANICS - Static and Dynamic Behaviors PDF
Preview COMPUTATIONAL STRUCTURAL MECHANICS - Static and Dynamic Behaviors
COMPUTATIONAL STRUCTURAL MECHANICS COMPUTATIONAL STRUCTURAL MECHANICS Static and Dynamic Behaviors KARANKUMARPRADHAN ParalaMaharajaEngineeringCollege,Berhampur Berhampur,India SNEHASHISHCHAKRAVERTY DepartmentofMathematics NationalInstituteofTechnologyRourkela Rourkela,India AcademicPressisanimprintofElsevier 125LondonWall,LondonEC2Y5AS,UnitedKingdom 525BStreet,Suite1650,SanDiego,CA92101,UnitedStates 50HampshireStreet,5thFloor,Cambridge,MA02139,UnitedStates TheBoulevard,LangfordLane,Kidlington,OxfordOX51GB,UnitedKingdom ©2019ElsevierInc.Allrightsreserved. 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LibraryofCongressCataloging-in-PublicationData AcatalogrecordforthisbookisavailablefromtheLibraryofCongress BritishLibraryCataloguing-in-PublicationData AcataloguerecordforthisbookisavailablefromtheBritishLibrary ISBN:978-0-12-815492-2 ForinformationonallAcademicPresspublications visitourwebsiteathttps://www.elsevier.com/books-and-journals Publisher:MatthewDeans AcquisitionEditor:DennisMcGonagle EditorialProjectManager:CharlotteKent ProductionProjectManager:VijayarajPurushothaman Designer:MatthewLimbert TypesetbyVTeX PREFACE Presentbookiswrittentosatisfytheneedoftheteachersandresearchersto understand the static and dynamic (or vibration) problems of Functionally Graded(FG)structuralbeamsandplates.TheEuler–Bernoullibeamtheory and classical plate theory of linear vibration are well established. The com- putational algorithms of Laplace transform in static bending of FG beams and three well-known numerical methods, Rayleigh–Ritz, Finite Element and Differential Quadrature (DQ), in finding vibration characteristics of FGbeamarealsoclearlyoutlined.Also,thisbookinvolvesnewlyproposed alternate forms of deformation plate theories in the study of vibration of isotropic thick rectangular plates. In addition, the effects of complicating environments on structural vibration of FG plates are also addressed. Re- centresearchonthesubjectsofstaticsandvibrationofFGbeamsandplates in the form of referencesto books and papers are incorporated. It is worth mentioning that static and vibration analysis of complex- shaped structures is commonly encountered in various engineering and architectural practices. In aeronautical, marine, mechanical and civil struc- tural designs, regular-, irregular- and complex-shaped members are some- timesincorporatedtoreducecostlymaterial,lightentheloads,provideven- tilation and alter the resonant frequencies of the structures. Moreover, the concept of Functionally Graded Materials (FGMs) was first introduced in 1984 by a group of material scientists in Japan during a space plane project intheformofthermalbarriermaterialwhichcanwithstandahugetemper- ature fluctuation across a very thin cross-section. Since, FGMs have taken majorattentionasheat-shieldingadvancedcompositesinaerospace,nuclear reactor, automobile, aircraft and space vehicle, biomedical and steel indus- tries.Thesecompositesgenerallyconsistofceramicandmetalconstituents, in which material properties vary continuously in thickness direction from one interface to another in a specific mathematical pattern. In this respect, static and dynamic characteristics of FG beams and plates are of consid- erable importance in both research and industrial sectors. The effects of WinklerandPasternakelasticfoundationsontheirvibrationcharacteristics are also a major concern in this book. Accordingly, structural members of variousshapesmadeofFGmaterialsneedtobeanalyzedfromanengineer- ingpointofviewwithgoodaccuracyalongwithcomputationallyefficient methods. In particular, beams, plates and other structural members are an vii viii Preface integralpartofmostengineeringstructuresandtheirvibrationanalysesare needed for safe design of structures. Analysis and design of such structures call for efficient computational tools. The Finite Element Method (FEM), Finite DifferenceMethod (FDM), Boundary ElementMethod (BEM) etc. are the standard industrial approach to deal with such situations. But with irregular (complex) shapes of structural components, design is based on numerousapproximations. Theselead sometimesto inaccuracies and more computing time. VibrationanalysisofFGbeamsandplatesofvariousshapesandconfig- urations have been studied extensively in the past, whereas the studies of static problems of FG plates are very limited in the available literature. The correspondingbehaviorofthesestructuresisstronglydependentonbound- ary conditions, geometrical shapes, material properties, different theories and various complicating effects. In the initial stages, results were available for some simple cases, viz. a limited set of boundary conditions and ge- ometries, in which the analytical solution could easily be obtained. The lack of good computational facilities made it almost impossible to get ac- curate results even in these cases. With the advent of fast computers and various efficient numerical methods, there has been a tremendous increase in the amount of research done for getting better accuracy in the results. Although the discretization methods in term of FEM, FDM and BEM provideageneralframeworkforgeneralstructures,theyinvariablyresultin problems with a large number of degrees of freedom. This deficiency may be overcome by using the Rayleigh–Ritz method. Recently, a tremen- dous amount of work has been done throughout the world by using the Rayleigh–Ritz method with suitable selection of shape functions in terms of different geometries. This method provides better accuracy of results, is more efficient and simple and is easier for computer implementation. While investigating the static and vibration problems of FG structural members, we could not find books that systematically address the basics to start with the subject. As such, we thought to write a book on the mentioned title so that readers understand the topic easily and work on theircomplicatedpracticalproblems.Wedohopethatthisbookwillbean important benchmark for the teachers, for future researchers and also for the industry. Chapter 1 deals with the origin and salient features of FG materials. Subsequently, the governing equations corresponding to different classical beam (or plate) theories associated with static and dynamic characteristics Preface ix ofFGbeams(orplates)areoutlined.Furthermore,therecentdevelopments on the corresponding studies have been addressed. Next, the historical bases of the Rayleigh–Ritz method are mentioned in Chapter 2, followed by the emerging trends claimed by different re- searchers on static and dynamic characteristics of different structural mem- bers. This book particularly assumes this method in solving varieties of staticanddynamicproblems(thosearereportedinrespectivechapters),but the computational algorithms related to vibration of FG beams are subse- quently incorporated. Chapter 3 involves another efficient computational technique, referred toasthemethodofDQ.Initially,thehistoryandoriginofthismethodare reported, followed by its recent developments towards static and dynamic problems. Furthermore, the computational procedure of the DQ method in handling vibration of FG beams is provided. In reference to the numerical approach, the FEM is also one of the major inclusions in this book; it is particularly applied in finding natu- ral frequencies of FG beams. As such, Chapter 4 initiates the origin of this well-known technique first, followed by different studies proposed by various researchers towards static and dynamic characteristics. At last, the corresponding numerical procedure in estimating the vibration character- istics of FG beams is given. Afterwards, static bending of Euler–Bernoulli FG beams subjected to uniformly distributed load is analyzed in Chapter 5, using the analytical approach of the Laplace transform. The aim here is to provide the expres- sionofbendingdeflectionofFGbeamsunderdifferentsetsofclassicaledge supports. Again related to static bending, Chapters 6, 7 and 8 estimate bending behaviors of FG rectangular, elliptic and triangular plates respectively sub- jected to different external mechanical loads (uniformly distributed load andhydrostaticpressure).Usualthepower-lawgradationpatternoftheFG materialpropertiesisconsideredtovaryspatiallyinthethicknessdirection. The numerical modeling of these problems is based on the Rayleigh–Ritz methodtoobtainthecorrespondingsystemoflinearequations.Specifically, Chapter6estimatesnumericalfactorsassociatedwithmaximumdeflection, bendingmomentsandnormalstressesbasedontheeffectofaspectratioand volume fraction of the constituents. Chapter 7 and 8 include maximum plate deflections in terms of various physical and geometric parameters. FurtherchaptersassumethedynamicbehaviorsofFGbeamsandplates. Accordingly, Chapter 9 presents free vibration of Euler–Bernoulli FG x Preface beams subject to various classical boundary supports. As usual, the beam material properties are assumed to vary continuously along the thickness direction in power-law form. In particular, the computational algorithms given in Chapters 2, 3 and 4 are considered here to generate the gener- alized eigenvalue problems. In this regard, natural frequencies of different FG beams under four sets of classical edge supports have been evaluated along with 2-D mode shapes after finding the convergence in terms of the concerned numerical methods and validation with available literature. In spite of taking only FG plates, Chapter 10 develops four new in- versetrigonometric shear deformation plate theories to estimate transverse vibration of thick isotropic rectangular plates. The proposed theories ex- actly satisfy the transverse stress boundary conditions on the bottom and top surfaces of the plate, which were also true in earlier shear deformation theories. Numerical formulation is based on the Rayleigh–Ritz method because it can very well handle all types of classical boundary conditions. The primary objective here is to estimate the effect of geometric config- urations and various deformation theories on the natural frequencies after performing a test of convergence and comparison in special cases. In addi- tion, 3-D mode shapes of the plate with a few specific edge supports are also depicted. Chapters 11, 12 and 13 deal with vibration problems of thin FG skew andannularandFG(ellipticandskew)platesrestingonelasticfoundations. TheusualgradationformofFGmaterialconstituentsseemstoholdtruein thesecasesandsolutionproceduresfollowtheRayleigh–Ritzmethod.Free vibration of FG skew plates subjected to different classical edge supports is consideredinChapter11withvariousskewangles.Chapter12isbasedon vibration of FG annular plates (defined as a ring by two concentric elliptic regions), whereas vibration characteristics of FG elliptic and skew plates resting on elastic foundations are reported in Chapter 13. The primary objectives in these chapters are to evaluate the effects of various physical and geometric parameters on the natural frequencies in view of the test of convergence along with the validation with existing results. The purpose of the present book is to have a systematic understanding ofthestaticanddynamicbehaviorsofFGbeamsandplates,whichinvolvea descriptionofalgorithmsrelatedtoefficientanalyticalandnumericaltech- niques,differentplategeometries(rectangular,elliptic,triangular,skewand annular), a proposition of alternate forms of plate theories and the effect of elastic foundations in reference to various analyses and results. This may prove to be a handy book for graduate and postgraduate students, teachers Preface xi and researchers throughout the world in the mentioned subject area. The book provides comprehensive results and an up-to-date and self-contained reviewofthetopic,alongwithanapplication-orientedtreatmentoftheuse of analytical as well as numerical methods in different static and vibration problems. K.K. Pradhan and S. Chakraverty
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