Statics – Formulas and Problems Dietmar Gross • Wolfgang Ehlers Peter Wriggers • Jörg Schröder Ralf Müller

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

Wolfgang Ehlers Ralf Müller Institute of Applied Mechanics Institute of Applied Mechanics Universität Stuttgart TU Kaiserslautern Stuttgart Kaiserslautern Germany Germany

Peter Wriggers Institute of Continuum Mechanics Leibniz Universität Hannover Hannover 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

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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, Statics” we provide more additional exercise material. The subject “Statics” 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 reader 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 2016 W. Ehlers P. Wriggers J. Schr¨oder R. M¨uller Table of Contents

Bibliography, Nomenclature ...... IX

1 Equilibrium ...... 1

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

3 Support Reactions ...... 45

4 Trusses...... 65

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

6 Cables...... 157

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

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

9 Moments 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: ↑ : Short notation for sum of all forces in direction of the arrow equal to zero. A : Short notation for sum of all moments with respect to reference point A (with predetermined direction of rotation) equal to zero. ; Short notation for from this follows that.