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Dietmar Gross · Wolfgang Ehlers

Ralf MüllerEngineering Mechanics 1

Statics ... Formulas and Problems

123

Statics-Formulas and Problems

Dietmar Gross

Wolfgang Ehlers

Peter Wriggers

Ralf Müller

Statics-Formulas

and Problems

Engineering Mechanics 1

123

Dietmar Gross

Division of Solid Mechanics

TU Darmstadt

Darmstadt

Germany

Wolfgang Ehlers

Institute of Applied Mechanics

Stuttgart

Germany

Peter Wriggers

Institute of Continuum Mechanics

Hannover

Institute of Mechanics

Essen

Germany

Ralf Müller

Institute of Applied Mechanics

TU Kaiserslautern

Kaiserslautern

Germany

ISBN 978-3-662-53853-1 ISBN 978-3-662-53854-8 (eBook)

DOI 10.1007/978-3-662-53854-8

Library of Congress Control Number: 2016956635

©Springer-Verlag GmbH Germany 2017

This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microlms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specic statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. 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, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made.

Printed on acid-free paper

This Springer imprint is published by Springer Nature The registered company is Springer-Verlag GmbH Germany The registered company address is: Heidelberger Platz 3, 14197 Berlin, Germany

Preface

This collection of problems results from the demand of students for sup- plementary problems and support in the preparation for examinations. With the present collection Engineering Mechanics 1 - Formulas and Problems, StaticsŽ we provide more additional exercise material. The subject StaticsŽ is commonly taught in the basic course of Engineering

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. We would like to make the reader aware that pure reading and trying to comprehend the presented solutions will not provide a deeper under- standing of mechanics. Neither does it improve the problem solving skills. Using this collection wisely, one has to try to solve the problems 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 not familiar with the formulae, explanations, or technical terms the rea- 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, Summer 2016W. Ehlers

P. Wriggers

J. Schr¨oder

R. M¨uller

Table of Contents

Bibliography, Nomenclature....................................IX

2Center of Gravity, Center of Mass, Centroids...............29

3Support Reactions................................................45

5Beams, Frames, Arches..........................................97

7Work and Potential Energy.....................................169

8Static and Kinetic Friction......................................193

9Moments of Inertia...............................................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., Vector Mechanics for Engineers:

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

Collection of Problems

Hibbeler, R. C., Engineering Mechanics, Statics, Prentice Hall, 2012 Schaums Outlines of Engineering Mechanics: Statics, 6th Edition, McGraw-

Hill Education 2010

Beer, F., Johnston, E.R., Mazurek, D., Vector Mechanics for Engineers:

Statics, 11th edition, McGraw-Hill Education 2015

Notation

For the solutions of the problems we used the following symbols: α: Short notation forsum of all forces in direction of the arrow equal to zero. A: Short notation forsum of all moments with respect to reference pointA(with predetermined direction of rotation) equal to zero. ?Short notation forfrom this follows that. 1

Chapter 1

Equilibrium

2Equilibrium

Forces with a common point of application in a

plane

A system of forces with a common point

of application can be replaced by a statically equivalent force R= Fi.

The system is inequilibrium, if

Fi=0 or in cartesian components

Fix=0?

Fiy=0?

Here we used the notation

Fi=Fixex+Fiyey?

Fix=Ficos?i?

Fiy=Fisin?i?

|Fi|=Fi= F 2 ix +F 2 iy i F iy F 1 F i F ix y x x y F i In agraphical solution, the equilibrium condition is expressed by aclo- sed force polygon. lines of actionforce polygon f i f F 1 F i

Forces with a common point of application in

space

Equilibriumexists, if the resultantR=

Fivanishes, i.e. if

Fi=0 or in cartesian components

Fix=0?

Fiy=0?

Fiz=0?

Equilibrium3

Here, the following notation is used

Fi=Fixex+Fiyey+Fizez,

Fix=Ficosαi,

Fiy=Ficosβi,

Fiz=Ficosγi,

cos 2

αi+ cos

2

βi+ cos

2

γi=1,

|Fi|=Fi= F 2 ix +F 2 iy +F 2 iz i i F ix i F iy F i F iz x z y

General systems of forces in a plane

A general system of forces can be re-

placed by a resultantR= Fiand a resulting momentM (A) R with re- spect to an arbitrary reference point

A. Equilibrium exists, if

Fix=0,

Fiy=0,

M (A) i =0. F 1 F i A y x Instead of using the two force conditions, two alternative moment con- ditions with dierent reference points (e.g.BandC) may be applied. Here the pointsA,BandCmust notlie on astraightline. Graphical solutions for the resultant force are obtained with the help of the link polygon and the force polygon. link polygon in layout diagramforce polygon f f f f 4 s ?s s s 4s r S S S S 4 S II Pol F 1 F 2 R F 3 F 4

4Equilibrium

€The link linessiare parallel to the linesSiin the force polygon. €The line of actionrof the resultantR(amplitude and direction follow from the force polygon) goes through the intersection of the outer link liness1ands5of the link polygon.quotesdbs_dbs4.pdfusesText_7

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