Engineering Mechanics 2: Strength of Materials. An introduction with many examples

328 Pages·2023·4.351 MB·English

by  Christian Mittelstedt

Christian Mittelstedt
Engineering
Mechanics
2: Strength
of Materials
An introduction with many examples
Engineering Mechanics 2: Strength of
Materials
Christian Mittelstedt
Engineering Mechanics 2:
Strength of Materials
An introduction with many examples
ChristianMittelstedt
TechnicalUniversityofDarmstadt
Darmstadt,Germany
ISBN978-3-662-66589-3 ISBN978-3-662-66590-9(eBook)
https://doi.org/10.1007/978-3-662-66590-9
SpringerVieweg
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Preface
Thisbookoriginatedfrommylecturenotesforthecourse“EngineeringMechanics
2”, which I hold for students of mechanical engineering in the second semester,
butalsoforstudentsofotherdisciplinesattheTechnicalUniversityofDarmstadt.
The book follows the classical division of engineering mechanics as it is taught
at universities in Germany and is dedicated to the determination of stresses and
deformationsinelasticbodies.
Feedbackofanykindiswelcomeanytime.
Darmstadt ChristianMittelstedt
Autumn2022
v
Contents
1 Introductiontolinearelasticity . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Stateofstress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.1 Baruntertension . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.2 Stressvectorandstresstensor . . . . . . . . . . . . . . . . . . 4
1.2.3 Equilibriumconditions . . . . . . . . . . . . . . . . . . . . . . 7
1.3 Deformationandstrainstate. . . . . . . . . . . . . . . . . . . . . . . . 9
1.3.1 Barundertension . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3.2 Infinitesimalstraintensor . . . . . . . . . . . . . . . . . . . . . 10
1.4 ThegeneralizedHooke’slaw . . . . . . . . . . . . . . . . . . . . . . . 14
1.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.4.2 Three-dimensionalmateriallaw . . . . . . . . . . . . . . . . . 16
1.4.3 Temperatureinfluence . . . . . . . . . . . . . . . . . . . . . . . 17
2 Planestressstate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2 Stresstransformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.3 Principalstresses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.4 Mohr’sstresscircle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3 Bars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.1 Bars. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.1.1 Barstresses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.1.2 Bardeformations . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.1.3 Staticallyindeterminatebars . . . . . . . . . . . . . . . . . . . 63
3.2 Barsystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.2.1 Staticallydeterminatebarsystems. . . . . . . . . . . . . . . . 65
3.2.2 Staticallyindeterminatebarsystems . . . . . . . . . . . . . . 68
vii
viii Contents
4 Beams. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.2 Basicequationsforanarbitraryreferenceframe . . . . . . . . . . . 75
4.3 Firstcross-sectionalnormalization:CenterofgravityC . . . . . . . 88
4.3.1 Steiner’stheorem . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.3.2 Selectedelementarycases. . . . . . . . . . . . . . . . . . . . . 91
4.3.3 Compositecross-sections . . . . . . . . . . . . . . . . . . . . . 97
4.3.4 Stressanalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
4.4 Secondcross-sectionalnormalization:Principalaxes . . . . . . . . 115
5 Beamdeflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
5.1 Basicequationsofbeambending. . . . . . . . . . . . . . . . . . . . . 127
5.2 Staticallydeterminatesingle-spanbeams . . . . . . . . . . . . . . . . 129
5.3 Staticallyindeterminatesingle-spanbeams. . . . . . . . . . . . . . . 137
5.4 Multi-spanbeams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
5.5 Standardbendingcases . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
5.5.1 Simplysupportedbeams . . . . . . . . . . . . . . . . . . . . . 153
5.5.2 Cantileverbeams . . . . . . . . . . . . . . . . . . . . . . . . . . 155
5.5.3 Staticallyindeterminatesystems . . . . . . . . . . . . . . . . . 158
5.5.4 Multi-spanbeamsandangledsystems . . . . . . . . . . . . . 165
5.6 Biaxialbending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
6 Shearstressesinbeams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
6.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
6.2 Thick-walledcross-sections . . . . . . . . . . . . . . . . . . . . . . . . 176
6.3 Thin-walledcross-sections. . . . . . . . . . . . . . . . . . . . . . . . . 185
6.4 Shearcenter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
7 Torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
7.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
7.2 Solidbarwithcircularcylindricalcross-section . . . . . . . . . . . . 209
7.3 Thin-walledbarwithcircularcylindricalcross-section . . . . . . . 211
7.4 Barswitharbitrarythin-walledcylindricalcross-sections . . . . . . 213
7.5 Barswithopenthin-walledcross-sections . . . . . . . . . . . . . . . 219
7.6 Determinationofinternalmoments . . . . . . . . . . . . . . . . . . . 224
8 Energymethods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
8.1 Workandenergy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
8.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
8.1.2 Internalandexternalwork . . . . . . . . . . . . . . . . . . . . 236
8.1.3 Principleofworkandenergy . . . . . . . . . . . . . . . . . . . 237
8.2 Strainenergyandcomplementarystrainenergy . . . . . . . . . . . . 237
8.2.1 Thebar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
8.2.2 TheEuler–Bernoullibeam . . . . . . . . . . . . . . . . . . . . 244
Contents ix
8.2.3 Barundertorsion . . . . . . . . . . . . . . . . . . . . . . . . . . 248
8.2.4 Combinedload . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
8.3 Applicationoftheprincipleofworkandenergytothedetermination
ofelasticdeformations . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
8.4 Theprincipleofvirtualforces. . . . . . . . . . . . . . . . . . . . . . . 253
8.4.1 Formulationforthebeam . . . . . . . . . . . . . . . . . . . . . 253
8.4.2 Theunitloadtheorem . . . . . . . . . . . . . . . . . . . . . . . 255
8.4.3 Useofintegraltables. . . . . . . . . . . . . . . . . . . . . . . . 259
8.5 Theforcemethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
8.5.1 Determinationofdeformationsofstatically determinate
systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
8.5.2 Staticallyindeterminatesystems . . . . . . . . . . . . . . . . . 270
8.6 Reciprocitytheorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
8.6.1 Betti’stheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
8.6.2 Maxwell’stheorem . . . . . . . . . . . . . . . . . . . . . . . . . 279
8.7 Staticallyindeterminatesystems . . . . . . . . . . . . . . . . . . . . . 280
9 Bucklingofbars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
9.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
9.2 Typesofequilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290
9.3 Determinationofcriticalloads . . . . . . . . . . . . . . . . . . . . . . 292
9.4 Bucklingofbars:ThefourEulercases . . . . . . . . . . . . . . . . . 297
9.4.1 Introductoryexample:EulercaseII . . . . . . . . . . . . . . . 297
9.4.2 EulercaseI. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300
9.4.3 EulercaseIII. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302
9.4.4 EulercaseIV . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305
9.4.5 Summaryoftheresults . . . . . . . . . . . . . . . . . . . . . . 306
9.5 Bucklinglength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307
9.6 Generalformofthebucklingdifferentialequation . . . . . . . . . . 312
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321
1
Introduction to linear elasticity
Thepresentchapterdealswiththebasicsoflinearelasticity theoryandintroduces
thecorrespondingstatevariables,i.e.stresses,strainsanddisplacements.Thegov-
erning equations are the equilibrium conditions, the kinematic relations and the
materiallaw,hereintheformofthegeneralizedHooke’slaw.Studentsareenabled
toderiveandformulatethebasicequationsoflinearelasticitytheoryandtoformu-
lateandsolvesimplefundamentalproblems.
1.1 Introduction
In a solid (a so-called continuum) under load, a stress state will develop. Due to
theload,deformationsoccurinthesolid,thusdisplacementsoftheindividualbody
pointsoccur.Displacements areaccompaniedbystrains, theso-called strain state
isformed.
For a three-dimensional body, the above quantities – displacements, strains,
stresses – are described by the following set of equations under a given load and
prescribedboundaryconditions:
(cid:2) Kinematicequations:Theso-calledkinematicequationsestablisharelationship
betweenthedisplacementsandtheresultingstrains.
(cid:2) Constitutiveequations:Theso-calledconstitutiveequationsestablisharelation-
ship between the stresses and the strains. In the case of linear elasticity, the
constitutiveequationsaredescribedbyHooke’slaw.
(cid:2) Equilibriumconditions:Equilibriummustbeensuredateverypointofthesolid
underconsideration.
©TheAuthor(s),underexclusivelicensetoSpringer-VerlagGmbH,DE,partofSpringer 1
Nature2023
C.Mittelstedt,EngineeringMechanics2:StrengthofMaterials,
https://doi.org/10.1007/978-3-662-66590-9_1
2 1 Introductiontolinearelasticity
1.2 Stateofstress
1.2.1 Baruntertension
Tomotivatetheconceptofthestress state,asanintroductoryelementaryexample
we consider the straight linear elastic bar unter tension as shown in Fig. 1.1, top.
Thebarhasthelengthl andthecross-sectionalareaAandisloadedatitscenterof
gravitybythetensileforceF.
The applied tensile force F causes forces inside the bar. To investigate this in
more detail, we perform a vertical cut at an arbitrary point x of the bar (Fig. 1.1,
middle). The forces acting inside the bar must be thought of as being uniformly
distributed, and these distributed forces are called stress. It is common to use the
symbol (cid:2) for the stress. Stresses are given in a corresponding unit of a force per
unit area, e.g. in the unit [N=m2]. It is also possible to use the unit Pascal, where
1PaD1 N.
m2
The stresses occurring in the given section are assumed to be constantly dis-
tributedoverthecross-sectionalareaandalsoorientednormaltothecross-section,
so that at this pointwealso speak of the so-called normalstresses (cid:2). Thenormal
stressescanbesummeduptothenormalforceN ofthebar.Sincethenormalstress
is constant across the cross-section, the following relationship holds between the
normalforceN DF andthenormalstress(cid:2):
N F
(cid:2) D D : (1.1)
A A
Thesign of thenormalstress (cid:2) dependsonthe sign of thenormalforceN.If, as
in this case, the bar is subjected to tension, the normal force N is positive, and
thus, according to (1.1), the normal stress (cid:2) will also be positive, i.e. it will be
atensilestress. IfthenormalforceN isnegative,then(cid:2) isalso negativeandthus
acompressivestress.
We now consider the bar under tension of Fig. 1.1 again and make a cut at an
arbitrary location x under an inclination angle (cid:3) as shown in Fig. 1.2. The cross-
sectionalareatobeconsideredhereiscalledANandiscalculatedasAND A .Such
cos(cid:3)
Fig.1.1 Barundertension A
(top),normalstress(cid:2) for
F F
aperpendicularcutatan
arbitrarylocationx(middle), x
l
replacementofthenormal
stressbythenormalforceN
F F
(bottom).
σ σ
x
F N F