Statics - Formulas and Problems: Engineering Mechanics 1

240 Pages·2017·3.75 MB·English

by  Dietmar Gross

Dietmar Gross · Wolfgang Ehlers
Peter Wriggers · Jörg Schröder
Ralf Müller
Statics –
Formulas and
Problems
Engineering Mechanics 1
123
–
Statics Formulas and Problems
Dietmar Gross Wolfgang Ehlers
(cid:129)
ö
Peter Wriggers Jörg Schr der
(cid:129)
ü
Ralf M ller
–
Statics Formulas
and Problems
Engineering Mechanics 1
123
Dietmar Gross Jörg Schröder
Division of Solid Mechanics Institute of Mechanics
TU Darmstadt Universität Duisburg-Essen
Darmstadt Essen
Germany Germany
WolfgangEhlers Ralf Müller
Institute of Applied Mechanics Institute of Applied Mechanics
Universität Stuttgart TU Kaiserslautern
Stuttgart Kaiserslautern
Germany Germany
PeterWriggers
Institute of Continuum Mechanics
LeibnizUniversität Hannover
Hannover
Germany
ISBN978-3-662-53853-1 ISBN978-3-662-53854-8 (eBook)
DOI 10.1007/978-3-662-53854-8
LibraryofCongressControlNumber:2016956635
©Springer-VerlagGmbHGermany2017
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Preface
Thiscollectionofproblemsresultsfromthedemandofstudentsforsup-
plementaryproblemsandsupportinthepreparationforexaminations.
With the present collection “Engineering Mechanics 1 - Formulas and
Problems, Statics” we provide more additional exercise material. The
subject“Statics”iscommonlytaughtinthebasiccourseofEngineering
Mechanics classes at universities.
The problems analyzed within these courses use equilibrium condi-
tions and the principle of virtual work to analyze static problems and
to compute reaction forces and stress resultants. These concepts are
the basic of many structural analyses of components used in civil and
mechanical engineering.
Wewouldliketomakethereaderawarethatpurereadingandtrying
tocomprehendthepresentedsolutionswillnotprovideadeeperunder-
standing of mechanics. Neither does it improve the problem solving
skills. Usingthiscollection wisely,onehastotrytosolvetheproblems
independently.The proposed solution should only be considered when
experiencing major problems in solving an exercise.
Obviously this collection cannot substitute a full-scale textbook. If
notfamiliarwiththeformulae,explanations,ortechnicaltermstherea-
der has to consider his or her course material or additional textbooks
on mechanics of materials. An incomplete list is provided on page IX.
Darmstadt, Stuttgart, Hannover, D.Gross
Essen and Kaiserslautern, Summer2016 W.Ehlers
P.Wriggers
J.Schr¨oder
R.M¨uller
Table of Contents
Bibliography,Nomenclature.................................... IX
1 Equilibrium ......................................................... 1
2 CenterofGravity,CenterofMass,Centroids............... 29
3 SupportReactions................................................ 45
4 Trusses............................................................... 65
5 Beams,Frames,Arches.......................................... 97
6 Cables................................................................ 157
7 WorkandPotentialEnergy..................................... 169
8 StaticandKineticFriction...................................... 193
9 MomentsofInertia............................................... 217
Literature
Textbooks
Gross,D.,Hauger,W.,Schr¨oder,J.,Wall,W.,Rajapakse,N.,Enginee-
ring Mechanics 1: Statics, 2nd edition, Springer 2013
Hibbeler, R.C., Engineering Mechanics, Statics, Prentice Hall, 2012
Beer,F.,Johnston,E.R.,Mazurek,D.,VectorMechanicsforEngineers:
Statics, 11th edition, McGraw-Hill Education 2015
Beer, F., Johnston, E.R., Eisenberg, E., Cornwell, P., Mazurek, D.,
Vector Mechanics for Engineers: Statics and Dynamics, 11th edition,
McGraw-Hill Education 2015
CollectionofProblems
Hibbeler, R.C., Engineering Mechanics, Statics, Prentice Hall, 2012
Schaum’sOutlinesofEngineeringMechanics:Statics,6thEdition,McGraw-
Hill Education 2010
Beer,F.,Johnston,E.R.,Mazurek,D.,VectorMechanicsforEngineers:
Statics, 11th edition, McGraw-Hill Education 2015
Notation
For thesolutions of the problems we used the following symbols:
↑: Short notation for sum of all forces in direction of the arrow
equal to zero.
(cid:2)
A: Short notation for sum of all moments with respect to reference
point A (with predetermined direction of rotation) equal to zero.
(cid:2) Short notation for from this follows that.
1
Chapter 1
Equilibrium
2 Equilibrium
Forces with a common point of application in a
plane
Asystemofforceswithacommonpoint
of application can be replaced by a
staticallyequivalentforce
(cid:2)
F
i R= Fi.
The system is in equilibrium,if
y (cid:2)
Fi =0
x F1 or in cartesian components
(cid:2) (cid:2)
F =0, F =0.
ix iy
Fiy F Here we used the notation
i
y αi Fi =Fixex+Fiyey,
Fix F =F cosα ,
ix i i
x
F =F sinα ,
iy i i
(cid:3)
|F |=F = F2 +F2 .
i i ix iy
Inagraphical solution,theequilibriumconditionisexpressedbyaclo-
sed force polygon.
lines of action force polygon
fi
F
1
F
i
f1
Forces with a common point of application in
space
(cid:4) (cid:4)
Equilibriumexists,iftheresultantR= Fi vanishes,i.e.if Fi =0
or in cartesian components
(cid:2) (cid:2) (cid:2)
F =0, F =0, F =0.
ix iy iz
Equilibrium 3
Here, thefollowing notation is used
z Fi =Fixex+Fiyey+Fizez ,
F =F cosα ,
ix i i
Fiz Fi Fiy =Ficosβi ,
γi Fiz =Ficosγi,
αi βi Fiy cos2αi+cos2βi+cos2γi =1,
(cid:3)
Fix y |F |=F = F2 +F2 +F2 .
i i ix iy iz
x
General systems of forces in a plane
Ageneralsystemofforcesc(cid:4)anbere-
placed by a resultant R= Fi and A
a resulting moment MR(A) with re- y
spect to an arbitrary reference point F Fi
1
A. Equilibrium exists, if
x
(cid:2) (cid:2) (cid:2)
F =0, F =0, M(A) =0.
ix iy i
Insteadofusingthetwoforceconditions,twoalternativemomentcon-
ditions with different reference points (e.g. B and C) may be applied.
Here thepoints A,B and C must not lie on a straight line.
Graphicalsolutionsfortheresultantforceareobtainedwiththehelp
of the link polygon and theforce polygon.
link polygon in layout diagram force polygon
F
1
S1
r
f1 f4 F2 S2 Pol
s1 s2f2 s3f3 s4 s5 R SS34 II
F
3
S5
F
4