Table Of ContentCOMPUTATIONAL
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.
Nopartofthispublicationmaybereproducedortransmittedinanyformorbyanymeans,
electronicormechanical,includingphotocopying,recording,oranyinformationstorageand
retrievalsystem,withoutpermissioninwritingfromthepublisher.Detailsonhowtoseek
permission,furtherinformationaboutthePublisher’spermissionspoliciesandourarrangements
withorganizationssuchastheCopyrightClearanceCenterandtheCopyrightLicensingAgency,
canbefoundatourwebsite:www.elsevier.com/permissions.
Thisbookandtheindividualcontributionscontainedinitareprotectedundercopyrightbythe
Publisher(otherthanasmaybenotedherein).
Notices
Knowledgeandbestpracticeinthisfieldareconstantlychanging.Asnewresearchandexperience
broadenourunderstanding,changesinresearchmethods,professionalpractices,ormedical
treatmentmaybecomenecessary.
Practitionersandresearchersmustalwaysrelyontheirownexperienceandknowledgeinevaluating
andusinganyinformation,methods,compounds,orexperimentsdescribedherein.Inusingsuch
informationormethodstheyshouldbemindfuloftheirownsafetyandthesafetyofothers,
includingpartiesforwhomtheyhaveaprofessionalresponsibility.
Tothefullestextentofthelaw,neitherthePublishernortheauthors,contributors,oreditors,
assumeanyliabilityforanyinjuryand/ordamagetopersonsorpropertyasamatterofproducts
liability,negligenceorotherwise,orfromanyuseoroperationofanymethods,products,
instructions,orideascontainedinthematerialherein.
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
