Structural Vibration: A Uniform Accurate Solution for Laminated Beams, Plates and Shells with General Boundary Conditions

318 Pages·2015·10.5 MB·English

by  Guoyong Jin

Guoyong Jin · Tiangui Ye
Zhu Su
Structural
Vibration
A Uniform Accurate Solution for
Laminated Beams, Plates and Shells
with General Boundary Conditions
Structural Vibration
Guoyong Jin Tiangui Ye
(cid:129) (cid:129)
Zhu Su
Structural Vibration
A Uniform Accurate Solution for Laminated
Beams, Plates and Shells with General
Boundary Conditions
123
Guoyong Jin ZhuSu
College ofPower andEnergy Engineering College ofPower andEnergy Engineering
Harbin EngineeringUniversity Harbin EngineeringUniversity
Harbin Harbin
China China
Tiangui Ye
College ofPower andEnergy Engineering
Harbin EngineeringUniversity
Harbin
China
ISBN 978-3-662-46363-5 ISBN 978-3-662-46364-2 (eBook)
DOI 10.1007/978-3-662-46364-2
Jointlypublished withScience Press, Beijing
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Preface
Inpracticalapplicationsthatrangefromouterspacetothedeepoceans,engineering
structuressuchasaircraft,rockets,automobiles,turbines,architectures,vessels,and
submarines often work in complex environments and can be subjected to various
dynamic loads, which can lead to the vibratory behaviors of the structures. In all
these applications, the engineering structures may fail and collapse because of
material fatigue resulting from vibrations. Many calamitous incidents have shown
the destructive nature of vibrations. For instance, the main span of the famous
Tacoma Narrows Bridge suffered severe forced resonance and collapsed in 1940
due to the fact that the wind provided an external periodic frequency that matched
oneofthenaturalstructuralfrequenciesofthebridge.Furthermore,noisegenerated
byvibrationsalwayscausesannoyance,discomfort,andlossofefficiencytohuman
beings. Therefore, it is of particular importance to understand the structural
vibrations and reduce them through proper design to ensure a reliable, safe, and
lasting structural performance. An important step in the vibration design of an
engineeringstructureistheevaluationofitsvibrationmodalcharacteristics,suchas
natural frequencies and mode shapes. This modal information plays a key role in
the design and vibration suppression of the structure when subjected to dynamics
excitations. In engineering applications, a variety of possible boundary restraining
casesmaybeencounteredforastructure.Inrecentdecades,theabilityofpredicting
the vibration characteristics of structures with general boundary conditions is of
primeinteresttoengineersanddesignersandisthemutualconcernofresearchersin
this field as well.
Beams, plates, and shells are basic structural elements of most engineering
structuresandmachines.Athoroughunderstandingoftheirvibrationcharacteristics
isofgreatsignificanceforengineerstopredictthevibrationsofthewholestructures
anddesignsuitablestructureswithlowvibrationandnoiseradiationcharacteristics.
There exists many books, papers, and research reports on the vibration analysis of
beams, plates, and shells. In 1969, Prof. A.W. Leissa published the excellent
monograph Vibration of Plates, in which theoretical and experimental results of
approximately 500 research papers and reports were presented. And in 1973, he
organized and summarized approximately 1,000 references in the field of shell
v
vi Preface
vibrations and published another famous monograph entitled Vibration of Shells.
New survey shows that the literature on the vibrations of beams, plates, and shells
has expanded rapidly since then. Based on the Google Scholar search tool, the
numbers of article related to the following keywords from 1973 up to 2014 are:
315,000itemsfor“vibration&beam,”416,000itemsfor“vibration&plates,”and
101,000 items for “vibration & shell.” This clearly reveals the importance of the
vibration analysis of beams, plates, and shells.
Undeniably, significant advances in the vibration analysis of beams, plates, and
shells have been achieved over the past four decades. Many accurate and efficient
computational methods have also been developed, such as the Ritz method, dif-
ferentialquadraturemethod(DQM),Galerkinmethod,wavepropagationapproach,
multiquadric radial basis function method (MRBFM), meshless method, finite
element method (FEM), discrete singular convolution approach (DSC), etc.
Furthermore, a large variety of classical and modern theories have been proposed
by researchers, such as the classical structure theories (CSTs), the first-order shear
deformation theories (FSDTs), and the higher order shear deformation theories
(HSDTs).
However, after the review of the literature in this subject, it appears that most
of the books deal with a technique that is only suitable for a particular type of
classicalboundaryconditions(i.e.,simplysupportedsupports,clampedboundaries,
free edges, shear-diaphragm restrains and their combinations), which typically
requires constant modifications of the solution procedures and corresponding
computation codes to adapt to different boundary cases. This will result in very
tedious calculations and be easily inundated with various boundary conditions in
practical applications since the boundary conditions of a beam, plate, or shell may
not always be classical in nature, a variety of possible boundary restraining cases,
including classical boundary conditions, elastic restraints, and their combinations
may be encountered. In addition, with the development of new industries and
modern processes, laminated beams, plates, and shells composed of composite
laminasareextensivelyusedinmanyfieldsofmodernengineeringpracticessuchas
space vehicles, civil constructions, and deep-sea engineering equipments to satisfy
special functional requirements due to their outstanding bending rigidity, high
strength-weight and stiffness-weight ratios, excellent vibration characteristics, and
goodfatigueproperties.Thevibrationresultsoflaminatedbeams,plates,andshells
arefarfromcomplete.Itisnecessaryandofgreatsignificancetodevelopaunified,
efficient, and accurate method which is capable of universally dealing with lami-
nated beams, plates, and shells with general boundary conditions. Furthermore, a
systematic, comprehensive, and up-to-date monograph which contains vibration
resultsofisotropicandlaminatedbeams,plates,andshellswithvariouslamination
schemesandgeneralboundaryconditionswouldbehighlydesirableandusefulfor
the senior undergraduate and postgraduate students, teachers, engineers, and indi-
vidual researchers in this field.
Inviewoftheseapparentvoids,thepresentmonographpresentsanendeavorto
complementthevibrationanalysisoflaminatedbeams,plates,andshells.Thetitle,
Preface vii
Structural Vibration: A Uniform Accurate Solution for Laminated Beams, Plates
andShellswithGeneralBoundaryConditions,illustratesthemainaimofthisbook,
namely:
(1) Todevelopanaccuratesemi-analyticalmethodwhichiscapableofdealingwith
thevibrationsoflaminatedbeams,plates,andshellswitharbitrarylamination
schemesandgeneralboundaryconditionsincludingclassicalboundaries,elastic
supports and their combinations, aiming to provide a unified and reasonable
accuratealternativetootheranalyticalandnumericaltechniques.
(2) Toprovideasummaryofknownresultsoflaminatedbeams,plates,andshells
withvariouslaminationschemesandgeneralboundaryconditions,whichmay
serve as benchmark solutions for the future research in this field.
The book is organized into eight chapters. Fundamental equations of laminated
shellsintheframeworkofclassicalshelltheoryandsheardeformationshelltheory
are derived in detail, including the kinematic relations, stress–strain relations and
stress resultants, energy functions, governing equations, and boundary conditions.
The corresponding fundamental equations of laminated beams and plates are spe-
cialized from the shell ones. Following the fundamental equations, a unified
modified Fourier series method is developed. Then both strong and weak form
solution procedures are realized and established by combining the fundamental
equations and the modified Fourier series method. Finally, numerous vibration
results are presented for isotropic, orthotropic, and laminated beams, plates, and
shellswithvariousgeometryandmaterialparameters,differentlaminationschemes
and different boundary conditions including the classical boundaries, elastic ones,
and their combinations. Summarizing, the work is arranged as follows:
The theories of linear vibration of laminated beams, plates, and shells are well
established. In this regard, Chap. 1 introduces the fundamental equations of lami-
nated beams, plates, and shells in the framework of classical shell theory and the
first-order shear deformation shell theory without proofs.
Chapter2presentsamodifiedFourierseriesmethodwhichiscapableofdealing
with vibrations of laminated beams, plates, and shells with general boundary
conditions.InthemodifiedFourierseriesmethod,eachdisplacementofalaminated
beam,plate,orshell,regardlessofboundaryconditions,isinvariantlyexpressedas
a new form of trigonometric series expansions in which several supplementary
terms are introduced to ensure and accelerate the convergence of the series
expansion.Thenonecanseekthesolutionseitherinstrongformsolutionprocedure
or the weak form one. These two solution procedures are fully illustrated in this
chapter.
Chapters 3–8 deal with laminated beams, plates, and cylindrical, conical,
spherical and shallow shells, respectively. In each chapter, corresponding funda-
mental equations in the framework of classical and shear deformation theories for
the general dynamic analysis are developed first, which can be useful for potential
readers. Following the fundamental equations, numerous free vibration results are
presented for various configurations including different boundary conditions, lam-
inated sequences, and geometry and material properties.
viii Preface
Finally, the authors would like to record their appreciation to the National
Natural Science Foundation of China (Grant Nos. 51175098, 51279035, and
10802024) and the Fundamental Research Funds for the Central Universities of
China (No. HEUCFQ1401) for partially funding this work.
Contents
1 Fundamental Equations of Laminated Beams,
Plates and Shells. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Three-Dimensional Elasticity Theory in Curvilinear
Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Fundamental Equations of Thin Laminated Shells. . . . . . . . . . 3
1.2.1 Kinematic Relations . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.2 Stress-Strain Relations and Stress Resultants . . . . . . . 5
1.2.3 Energy Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2.4 Governing Equations and Boundary Conditions . . . . . 11
1.3 Fundamental Equations of Thick Laminated Shells . . . . . . . . . 16
1.3.1 Kinematic Relations . . . . . . . . . . . . . . . . . . . . . . . . 17
1.3.2 Stress-Strain Relations and Stress Resultants . . . . . . . 18
1.3.3 Energy Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.3.4 Governing Equations and Boundary Conditions . . . . . 26
1.4 Lamé Parameters for Plates and Shells. . . . . . . . . . . . . . . . . . 29
2 Modified Fourier Series and Rayleigh-Ritz Method . . . . . . . . . . . 37
2.1 Modified Fourier Series. . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.1.1 Traditional Fourier Series Solutions. . . . . . . . . . . . . . 39
2.1.2 One-Dimensional Modified Fourier
Series Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.1.3 Two-Dimensional Modified Fourier
Series Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.2 Strong Form Solution Procedure. . . . . . . . . . . . . . . . . . . . . . 53
2.3 Rayleigh-Ritz Method (Weak Form Solution Procedure) . . . . . 58
3 Straight and Curved Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.1 Fundamental Equations of Thin Laminated Beams . . . . . . . . . 64
3.1.1 Kinematic Relations . . . . . . . . . . . . . . . . . . . . . . . . 64
3.1.2 Stress-Strain Relations and Stress Resultants . . . . . . . 65
3.1.3 Energy Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 66
ix
x Contents
3.1.4 Governing Equations and Boundary Conditions . . . . . 68
3.2 Fundamental Equations of Thick Laminated Beams. . . . . . . . . 71
3.2.1 Kinematic Relations . . . . . . . . . . . . . . . . . . . . . . . . 71
3.2.2 Stress-Strain Relations and Stress Resultants . . . . . . . 72
3.2.3 Energy Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.2.4 Governing Equations and Boundary Conditions . . . . . 74
3.3 Solution Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.3.1 Strong Form Solution Procedure. . . . . . . . . . . . . . . . 77
3.3.2 Weak Form Solution Procedure
(Rayleigh-Ritz Procedure) . . . . . . . . . . . . . . . . . . . . 80
3.4 Laminated Beams with General Boundary Conditions . . . . . . . 83
3.4.1 Convergence Studies and Result Verification . . . . . . . 83
3.4.2 Effects of Shear Deformation and Rotary Inertia. . . . . 85
3.4.3 Effects of the Deepness Term (1 + z/R). . . . . . . . . . . 87
3.4.4 Isotropic and Laminated Beams with General
Boundary Conditions. . . . . . . . . . . . . . . . . . . . . . . . 90
4 Plates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.1 Fundamental Equations of Thin Laminated
Rectangular Plates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.1.1 Kinematic Relations . . . . . . . . . . . . . . . . . . . . . . . . 100
4.1.2 Stress-Strain Relations and Stress Resultants . . . . . . . 102
4.1.3 Energy Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.1.4 Governing Equations and Boundary Conditions . . . . . 105
4.2 Fundamental Equations of Thick Laminated
Rectangular Plates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
4.2.1 Kinematic Relations . . . . . . . . . . . . . . . . . . . . . . . . 108
4.2.2 Stress-Strain Relations and Stress Resultants . . . . . . . 108
4.2.3 Energy Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 109
4.2.4 Governing Equations and Boundary Conditions . . . . . 111
4.3 Vibration of Laminated Rectangular Plates. . . . . . . . . . . . . . . 113
4.3.1 Convergence Studies and Result Verification . . . . . . . 116
4.3.2 Laminated Rectangular Plates with Arbitrary
Classical Boundary Conditions. . . . . . . . . . . . . . . . . 117
4.3.3 Laminated Rectangular Plates with Elastic
Boundary Conditions. . . . . . . . . . . . . . . . . . . . . . . . 122
4.3.4 Laminated Rectangular Plates with Internal
Line Supports. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
4.4 Fundamental Equations of Laminated Sectorial, Annular
and Circular Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
4.4.1 Fundamental Equations of Thin Laminated
Sectorial, Annular and Circular Plates . . . . . . . . . . . . 134
4.4.2 Fundamental Equations of Thick Laminated
Sectorial, Annular and Circular Plates . . . . . . . . . . . . 137