Thin Plates and Shells: Theory, Analysis, and - NumGeom

651 Pages·2005·3.28 MB·English

Thin Plates and Shells
Theory, Analysis, and Applications
Eduard Ventsel
Theodor Krauthammer
The Pennsylvania State University
University Park,Pennsylvania
MarcelDekker,Inc. NewYork Basel
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Copyright 2001 by Marcel Dekker, Inc. All Rights Reserved.
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Preface
Thin-walled structures in the form of plates and shells are encountered in many
branches of technology, such as civil, mechanical, aeronautical, marine, and chemi-
calengineering.Suchawidespreaduseofplateandshellstructuresarisesfromtheir
intrinsic properties. When suitably designed, even very thin plates, and especially
shells,cansupportlargeloads.Thus,theyareutilizedinstructuressuchasaerospace
vehicles in which light weight is essential.
In preparing this book, we had three main objectives: first, to offer a compre-
hensive and methodical presentation of the fundamentals of thin plate and shell
theories,basedonastrongfoundationofmathematicsandmechanicswithemphasis
on engineeringaspects. Second,we wantedtoacquaintreaders withthemostuseful
and contemporary analytical and numerical methods for solving linear and non-
linearplateandshellproblems.Ourthirdgoalwastoapplythetheoriesandmethods
developed in the book to the analysis and design of thin plate-shell structures in
engineering. This bookis intended asa text for graduate and postgraduate students
in civil, architectural, mechanical, chemical, aeronautical, aerospace, and ocean
engineering, and engineering mechanics. It can also serve as a reference book for
practicing engineers, designers, and stress analysts who are involved in the analysis
and design of thin-walled structures.
As a textbook, it contains enough materal for a two-semester senior or grad-
uate course on the theory and applications of thin plates and shells. Also, a special
effort has been made to have the chapters as independent from one another as
possible, so that a course can be taught in one semester by selecting appropriate
chapters, or through equivalent self-study.
The textbook is divided into two parts. Part I (Chapters 1–9) presents plate
bending theory and its application and Part II (Chapters 10–20) covers the theory,
analysis, and principles of shell structures.
Copyright 2001 by Marcel Dekker, Inc. All Rights Reserved.
The book is organized in the following manner. First, the general linear the-
oriesofthinelasticplatesandshellsofanarbitrarygeometryaredevelopedbyusing
the basic classical assumptions. Deriving the general relationships and equations of
the linear shell theory requires some familiarity with topics of advanced mathe-
matics, including vector calculus, theory of differential equations, and theory of
surfaces. We tried to keep a necessary rigorous treatment of shell theory and its
principles and, at the same time, to make the book more readable for graduate
students and engineers. Therefore, we presented the fundamental kinematic and
static relationships, and elements of the theory of surfaces, which are necessary
for constructing the shell theory, without proof and verification. The detailed deri-
vation and proof of the above relationships and equations are given in Appendices
A–E so that the interested reader can refer to them.
Later on, governing differential equations of the linear general theory are
applied to plates and shells of particular geometrical forms. In doing so, various
approximate engineering shell theories are presented by introducing some supple-
mentary assumptions to the general shell theory. The mathematical formulation of
theaboveshelltheoriesleads,asarule,toasystemofpartialdifferentialequations.
Asolutionoftheseequationsisthefocusofattentionofthebook.Emphasisisgiven
tocomputer-orientedmethods,suchasthefinitedifference andfiniteelementmeth-
ods, boundary element and boundary collocation methods, and to their application
toplateandshellproblems.Nevertheless,theemphasisplacedonnumericalmethods
is not intended to deny the merit of classical analytical methods that are also pre-
sented in the book, for example, the Galerkin and Ritz methods.
A great attempt has been made to emphasize the physical meanings of engi-
neering shell theories, mathematical relationships, and adapted basic and supple-
mentaryassumptions.Theaccuracyofnumericalresultsobtainedwiththeuseofthe
abovetheories,andpossibleareasoftheirapplication,arediscussed.Themaingoal
is to help the reader to understand how plate and shell structures resist the applied
loads and to express this understanding in the language of physical rather than
purely mathematical aspects. To this end, the basic ideas of the considered plate
andshellmodelsaredemonstratedbycomparisonswithmoresimplemodelssuchas
beams and arches, for which the main ideas are understandable for readers familiar
with strength of materials. We believe that understanding the behavior of plate and
shell structures enables designers or stress analysts to verify the accuracy of numer-
ical structural analysis results for such structures obtained by available computer
code, and to interpret these results correctly.
Postgraduate students, stress analysts, and engineers will be interested in the
advanced topics on plate and shell structures, including the refined theory of thin
plates, orthotropic and multilayered plates and shells, sandwich plate and shell
structures, geometrically nonlinear plate, and shell theories. Much attention is also
giventoorthotropicandstiffenedplatesandshells,aswellastomultishellstructures
that are commonlyencountered in engineering applications.The peculiarities of the
behavior and states of stress of the above thin-walled structures are analyzed in
detail.
Sincethefailureofthin-walledstructuresismoreoftencausedbybuckling,the
issueofthelinearandnonlinearbucklinganalysisofplatesandshellsisgivenmuch
attention in the book. Particular emphasis is placed on the formulation of elastic
stability criteria and on the analysis of peculiarities of the buckling process for thin
Copyright 2001 by Marcel Dekker, Inc. All Rights Reserved.
shells. Buckling analysis of orthtropic, stiffened, and sandwich plates and shells is
presented. The important issues of postbuckling behavior of plates and shells—in
paticular, the load-carrying capacity of stiffened plates and shells—are discussed in
detail.Someconsiderationsofdesignstabilityanalysisforthinshellstructuresisalso
provided in the book.
Anintroductiontothevibrationofplatesandshellsisgivenincondensedform
and the fundamental concepts of dynamic analysis for free and forced vibrations of
unstiffened and stiffened plate and shell structures are discussed. The book empha-
sizes the understanding of basic phenomena in shell and plate vibrations. We hope
that this materal will be useful for engineers in preventing failures and for acousti-
cians in controlling noise.
Each chapter contains fully worked out examples and homework problems
that are primarily drawn from engineering practice. The sample problems serve a
doublepurpose:tohelpreadersunderstandthebasicprinciplesandmethodsusedin
plate and shell theories and to show application of the above theories and methods
to engineering design.
The selection, arrangement, and presentation of the material have been made
with the greatest care, based on lecture notes for a course taught by the first author
atThePennsylvaniaStateUniversityformanyyearsandalsoearlierattheKharkov
Technical University of Civil Engineering, Ukraine. The research, practical design,
and consulting experiences of both authors have also contributed to the presented
material.
The first author wishes to express his gratitude to Dr. R. McNitt for his
encouragement, unwavering support, and valuable advice in bringing this book to
its final form. Thanks are also due to the many graduate students who offered
constructive suggestions when drafts of this book were used as a text. A special
thanks is extended to Dr. I. Ginsburg for spending long hours reviewing and criti-
quingthemanuscript.WethankMs. J.Fennemaforherexcellenceinsketchingthe
numerous figures. Finally, we thank Marcel Dekker, Inc., and especially, Mr. B. J.
Clark, for extraordinary dedication and assistance in the preparation of this book.
Eduard Ventsel
Theodor Krauthammer
Copyright 2001 by Marcel Dekker, Inc. All Rights Reserved.
Contents
Preface
PART I. THIN PLATES
1 Introduction
1.1 General
1.2 History of Plate Theory Development
1.3 General Behavior of Plates
1.4 Survey of Elasticity Theory
References
2 The Fundamentals of the Small-Deflection Plate Bending Theory
2.1 Introduction
2.2 Strain–Curvature Relations (Kinematic Equations)
2.3 Stresses, Stress Resultants, and Stress Couples
2.4 The Governing Equation for Deflections of Plates in
Cartesian Coordinates
2.5 Boundary Conditions
2.6 Variational Formulation of Plate Bending Problems
Problems
References
3 Rectangular Plates
3.1 Introduction
3.2 The Elementary Cases of Plate Bending
3.3 Navier’s Method (Double Series Solution)
Copyright 2001 by Marcel Dekker, Inc. All Rights Reserved.
3.4 Rectangular Plates Subjected to a Concentrated Lateral Force P
3.5 Levy’s Solution (Single Series Solution)
3.6 Continuous Plates
3.7 Plates on an Elastic Foundation
3.8 Plates with Variable Stiffness
3.9 Rectangular Plates Under Combined Lateral and Direct Loads
3.10 Bending of Plates with Small Initial Curvature
Problems
References
4 Circular Plates
4.1 Introduction
4.2 Basic Relations in Polar Coordinates
4.3 Axisymmetric Bending of Circular Plates
4.4 The Use of Superposition for the Axisymmetric Analysis of
Circular Plates
4.5 Circular Plates on Elastic Foundation
4.6 Asymmetric Bending of Circular Plates
4.7 Circular Plates Loaded by an Eccentric Lateral Concentrated Forc
4.8 Circular Plates of Variable Thickness
Problems
References
5 Bending of Plates of Various Shapes
5.1 Introduction
5.2 Elliptical Plates
5.3 Sector-Shaped Plates
5.4 Triangular Plates
5.5 Skew Plates
Problems
References
6 Plate Bending by Approximate and Numerical Methods
6.1 Introduction
6.2 The Finite Difference Method (FDM)
6.3 The Boundary Collocation Method (BCM)
6.4 The Boundary Element Method (BEM)
6.5 The Galerkin Method
6.6 The Ritz Method
6.7 The Finite Element Method (FEM)
Problems
References
7 Advanced Topics
7.1 Thermal Stresses in Plates
7.2 Orthotropic and Stiffened Plates
7.3 The Effect of Transverse Shear Deformation on the Bending of
Elastic Plates
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7.4 Large-Deflection Theory of Thin Plates
7.5 Multilayered Plates
7.6 Sandwich Plates
Problems
References
8 Buckling of Plates
8.1 Introduction
8.2 General Postulations of the Theory of Stability of Plates
8.3 The Equilibrium Method
8.4 The Energy Method
8.5 Buckling Analysis of Orthotropic and Stiffened Plates
8.6 Postbuckling Behavior of Plates
8.7 Buckling of Sandwich Plates
Problems
References
9 Vibration of Plates
9.1 Introduction
9.2 Free Flexural Vibrations of Rectangular Plates
9.3 Approximate Methods in Vibration Analysis
9.4 Free Flexural Vibrations of Circular Plates
9.5 Forced Flexural Vibrations of Plates
Problems
References
PART II. THIN SHELLS
10 Introduction to the General Linear Shell Theory
10.1 Shells in Engineering Structures
10.2 General Definitions and Fundamentals of Shells
10.3 Brief Outline of the Linear Shell Theories
10.4 Loading-Carrying Mechanism of Shells
References
11 Geometry of the Middle Surface
11.1 Coordinate System of the Surface
11.2 Principal Directions and Lines of Curvature
11.3 The First and Second Quadratic Forms of Surfaces
11.4 Principal Curvatures
11.5 Unit Vectors
11.6 Equations of Codazzi and Gauss. Gaussian Curvature.
11.7 Classification of Shell Surfaces
11.8 Specialization of Shell Geometry
Problems
References
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12 The General Linear Theory of Shells
12.1 Basic Assumptions
12.2 Kinematics of Shells
12.3 Statics of Shells
12.4 Strain Energy of Shells
12.5 Boundary Conditions
12.6 Discussion of the Governing Equations of the General
Linear Shell Theory
12.7 Types of State of Stress for Thin Shells
Problems
References
13 The Membrane Theory of Shells
13.1 Preliminary Remarks
13.2 The Fundamental Equations of the Membrane Theory of Thin
Shells
13.3 Applicability of the Membrane Theory
13.4 The Membrane Theory of Shells of Revolution
13.5 Symmetrically Loaded Shells of Revolution
13.6 Membrane Analysis of Cylindrical and Conical Shells
13.7 The Membrane Theory of Shells of an Arbitrary Shape in
Cartesian Coordinates
Problems
References
14 Application of the Membrane Theory to the Analysis of Shell
Structures
14.1 Membrane Analysis of Roof Shell Structures
14.2 Membrane Analysis of Liquid Storage Facilities
14.3 Axisymmetric Pressure Vessels
Problems
References
15 Moment Theory of Circular Cylindrical Shells
15.1 Introduction
15.2 Circular Cylindrical Shells Under General Loads
15.3 Axisymmetrically Loaded Circular Cylindrical Shells
15.4 Circular Cylindrical Shell of Variable Thickness Under
Axisymmetric Loading
Problems
References
16 The Moment Theory of Shells of Revolution
16.1 Introduction
16.2 Governing Equations
16.3 Shells of Revolution Under Axisymmetrical Loads
16.4 Approximate Method for Solution of the Governing Equations
(16.30)
Copyright 2001 by Marcel Dekker, Inc. All Rights Reserved.