Partial differential equations of classical structural members

96 Pages·2020·2.009 MB·English

by  Ochsner A

SPRINGER BRIEFS IN APPLIED SCIENCES
AND TECHNOLOGY  CONTINUUM MECHANICS
Andreas Öchsner
Partial Differential
Equations of
Classical Structural
Members
A Consistent
Approach
SpringerBriefs in Applied Sciences
and Technology
Continuum Mechanics
Series Editors
Holm Altenbach, Institut für Mechanik, Lehrstuhl für Technische Mechanik,
Otto von Guericke University Magdeburg, Magdeburg, Sachsen-Anhalt, Germany
Andreas Öchsner, Faculty of Mechanical Engineering, Esslingen University of
Applied Sciences, Esslingen am Neckar, Germany
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 subseries at http://www.springer.com/series/10528
Ö
Andreas chsner
Partial Differential Equations
of Classical Structural
Members
A Consistent Approach
123
Andreas Öchsner
Faculty of MechanicalEngineering
Esslingen University of Applied Sciences
Esslingen, Baden-Württemberg, Germany
ISSN 2191-530X ISSN 2191-5318 (electronic)
SpringerBriefs inApplied SciencesandTechnology
ISSN 2625-1329 ISSN 2625-1337 (electronic)
SpringerBriefs inContinuum Mechanics
ISBN978-3-030-35310-0 ISBN978-3-030-35311-7 (eBook)
https://doi.org/10.1007/978-3-030-35311-7
©TheAuthor(s),underexclusivelicensetoSpringerNatureSwitzerlandAG2020
Thisworkissubjecttocopyright.AllrightsaresolelyandexclusivelylicensedbythePublisher,whether
thewholeorpartofthematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseof
illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and
transmissionorinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilar
ordissimilarmethodologynowknownorhereafterdeveloped.
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
authors or the editors give a warranty, expressed or implied, with respect to the material contained
hereinorforanyerrorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregard
tojurisdictionalclaimsinpublishedmapsandinstitutionalaffiliations.
ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG
Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland
Preface
Partial differential equations (PDEs) form the basis to mathematically describe the
mechanical behavior of all classical structural members known in engineering
mechanics. Nevertheless, there are concerning trends in some places of tertiary
education to no more derive and work with these equations. This may lead to a
serious lack of knowledge and may affect the reliable design of engineering
structures and processes.
The derivation and understanding of PDEs relies heavily on the fundamental
knowledge of the first years of engineering education, i.e., higher mathematics,
physics, materials science, applied mechanics, design, and programming skills.
Thus, it is definitely a challenging topic for prospective engineers.
ThisvolumeintheSpringerBriefsseriesshouldprovideacompactoverviewon
theclassicalPDEsofstructuralmembersandtoprovideaformalwaytouniformly
describetheseequationsinasimilarway.Allderivationsinthefollowingchapters
follow a common approach: the three fundamental equations of continuum
mechanics, i.e., the kinematics equation, the constitutive equation, and the equi-
librium equation, are combined to construct the partial differential equations.
The structural members covered in this book are rods, thin and thick beams,
plane elasticity members, thin and thick plates, and three-dimensional solids. The
last chapter gives a brief introduction to the topic of transient analysis.
Esslingen, Germany Andreas Öchsner
October 2019
v
Contents
1 Introduction to Structural Modeling. . . . . . . . . . . . . . . . . . . . . . . . . 1
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Rods or Bars. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Kinematics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 Constitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.4 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.5 Differential Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3 Euler–Bernoulli Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.2 Kinematics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.3 Constitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.4 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.5 Differential Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4 Timoshenko Beams. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.2 Kinematics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.3 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.4 Constitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.5 Differential Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5 Plane Members. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5.2 Kinematics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5.3 Constitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
vii
viii Contents
5.3.1 Plane Stress Case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5.3.2 Plane Strain Case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
5.4 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5.5 Differential Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
6 Classical Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
6.2 Kinematics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
6.3 Constitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
6.4 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
6.5 Differential Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
7 Shear Deformable Plates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
7.2 Kinematics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
7.3 Constitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
7.4 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
7.5 Differential Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
8 Three-Dimensional Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
8.2 Kinematics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
8.3 Constitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
8.4 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
8.5 Differential Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
9 Introduction to Transient Problems: Rods or Bars. . . . . . . . . . . . . . 89
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
9.2 Kinematics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
9.3 Constitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
9.4 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
9.5 Differential Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
Chapter 1
Introduction to Structural Modeling
Abstract The first chapter classifies the content as well as the focus of this text-
book.Inengineeringpractice,thedescriptionofprocessesiscenteredaroundpartial
differential equations, and all the classical approximation methods such as the fi-
niteelementmethod,thefinitedifferencemethod,thefinitevolumemethod,andthe
boundaryelementmethodofferdifferentwaysofsolvingtheseequations.
Engineersdescribephysicalphenomenaandprocessestypicallybyequations,par-
ticularlybypartialdifferentialequations[3, 4, 9].Inthiscontext,thederivationand
the solution of these differential equations (see Fig.1.1) is the task of engineers,
obviously requiring fundamental knowledge from physics and engineering mathe-
matics. The classical engineering methods for providing approximate solutions of
partialdifferentialequationsarethefiniteelementmethod(FEM),thefinitediffer-
ence method (FDM), the finite volume method (FVM), and the boundary element
method(BEM)[1, 2, 8].
Theimportanceofpartialdifferentialequationsisclearlyrepresentedinthefol-
lowingquote:‘Formorethan250yearspartialdifferentialequationshavebeenclearly
themostimportanttoolavailabletomankindinordertounderstandalargevariety
of phenomena, natural at first and then those originating from human activity and
technological development. Mechanics, physics and their engineering applications
werethefirsttobenefitfromtheimpactofpartialdifferentialequationsonmodeling
anddesign,...’[5].
Intheone-dimensionalcase,aphysicalproblemcanbegenerallydescribedina
spatialdomainΩ bythedifferentialequation
L{y(x)}=b (x ∈Ω) (1.1)
and by the conditions which are prescribed on the boundary Γ. The differential
equationisalsocalledthestrongformortheoriginalstatementoftheproblem.The
expression‘strongform’comesfromthefactthatthedifferentialequationdescribes
exactlyeachpointx inthedomainoftheproblem.TheoperatorL{...}inEq.(1.1)
isanarbitrarydifferentialoperatorwhichcantake,forexample,thefollowingforms:
©TheAuthor(s),underexclusivelicensetoSpringerNatureSwitzerlandAG2020 1
A.Öchsner,PartialDifferentialEquationsofClassicalStructuralMembers,
SpringerBriefsinContinuumMechanics,
https://doi.org/10.1007/978-3-030-35311-7_1
2 1 IntroductiontoStructuralModeling
Fig.1.1 Modelingbasedon
partialdifferentialequations
d2
L{...}= {...}, (1.2)
dx2
d4
L{...}= {...}, (1.3)
dx4
d4 d
L{...}= {...}+ {...}+{...}. (1.4)
dx4 dx
Furthermore, variable b in Eq. (1.1) is a given function, and in the case of b=0,
theequationreducestothehomogeneousdifferentialequation:L{y(x)}=0.More
specificexpressionsofEqs.(1.3)till(1.4)cantakethefollowingform[6]:
d2y(x)
a =b, (1.5)
dx2
d4y(x)
a =b, (1.6)
dx4
andwillbeusedtodescribethebehaviorofrodsandbeamsinthefollowingsections.
Let us highlight at the end of this section that the derivations in the following
chaptersfollowacommonapproach,seeFig.1.2[7].
Acombinationofthekinematicsequation(i.e.,therelationbetweenthestrainsand
displacements)withtheconstitutiveequation(i.e.,therelationbetweenthestresses
andstrains)andtheequilibriumequation(i.e.,theequilibriumbetweentheinternal
reactions and the external loads) results in a partial differential equation. Limited
to simple cases, analytical solutions can be derived and are exact under the given
assumptions.Formorecomplexproblems,engineersrelyontheclassicalnumerical
methodsasalreadymentionedinthischapter.