Roark’s Formulas
for Stress and Strain
WARREN C. YOUNG
RICHARD G. BUDYNAS
Seventh Edition
McGraw-Hill
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Roark’s Formulas
for Stress and Strain
WARREN C. YOUNG
RICHARD G. BUDYNAS
Seventh Edition
McGraw-Hill
New York Chicago San Francisco Lisbon London Madrid Mexico City Milan New Delhi San Juan Seoul Singapore Sydney Toronto
Cataloging-in-Publication Data is on file with the Library of Congress.
Copyright # 2002, 1989 by the McGraw-Hill Companies, Inc. All rights reserved. Printed in the United States of America. Except as permitted under the United States Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a data base or retrieval system, without the prior written permission of the publisher.
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ISBN 0-07-072542-X
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Chapter 5 Numerical Methods 73
The Finite-Difference Method. The Finite-Element Method. The Boundary- Element Method. References.
Chapter 6 Experimental Methods 81
Measurement Techniques. Electrical Resistance Strain Gages. Detection of Plastic Yielding. Analogies. Tables. References.
Part 3 Formulas and Examples
Chapter 7 Tension, Compression, Shear, and Combined Stress 109 Bar under Axial Tension (or Compression); Common Case. Bar under Axial Tension (or Compression); Special Cases. Composite Members. Trusses. Body under Pure Shear Stress. Cases of Direct Shear Loading. Combined Stress.
Chapter 8 Beams; Flexure of Straight Bars 125
Straight Beams (Common Case) Elastically Stressed. Composite Beams and Bimetallic Strips. Three-Moment Equation. Rigid Frames. Beams on Elastic Foundations. Deformation due to the Elasticity of Fixed Supports. Beams under Simultaneous Axial and Transverse Loading. Beams of Variable Section. Slotted Beams. Beams of Relatively Great Depth. Beams of Relatively Great Width. Beams with Wide Flanges; Shear Lag. Beams with Very Thin Webs. Beams Not Loaded in Plane of Symmetry. Flexural Center. Straight Uniform Beams (Common Case). Ultimate Strength. Plastic, or Ultimate Strength. Design. Tables. References.
Chapter 9 Bending of Curved Beams 267
Bending in the Plane of the Curve. Deflection of Curved Beams. Circular Rings and Arches. Elliptical Rings. Curved Beams Loaded Normal to Plane of Curvature. Tables. References.
Chapter 10 Torsion 381
Straight Bars of Uniform Circular Section under Pure Torsion. Bars of Noncircular Uniform Section under Pure Torsion. Effect of End Constraint. Effect of Longitudinal Stresses. Ultimate Strength of Bars in Torsion. Torsion of Curved Bars. Helical Springs. Tables. References.
Chapter 11 Flat Plates 427
Common Case. Bending of Uniform-Thickness Plates with Circular Boundaries. Circular-Plate Deflection due to Shear. Bimetallic Plates. Nonuniform Loading of Circular Plates. Circular Plates on Elastic Foundations. Circular Plates of Variable Thickness. Disk Springs. Narrow Ring under Distributed Torque about Its Axis. Bending of Uniform- Thickness Plates with Straight Boundaries. Effect of Large Deflection. Diaphragm Stresses. Plastic Analysis of Plates. Ultimate Strength. Tables. References.
iv Contents
Chapter 12 Columns and Other Compression Members 525
Columns. Common Case. Local Buckling. Strength of Latticed Columns. Eccentric Loading: Initial Curvature. Columns under Combined Compression and Bending. Thin Plates with Stiffeners. Short Prisms under Eccentric Loading. Table. References.
Chapter 13 Shells of Revolution; Pressure Vessels; Pipes 553
Circumstances and General State of Stress. Thin Shells of Revolution under Distributed Loadings Producing Membrane Stresses Only. Thin Shells of Revolution under Concentrated or Discontinuous Loadings Producing Bending and Membrane Stresses. Thin Multielement Shells of Revolution. Thin Shells of Revolution under External Pressure. Thick Shells of Revolution. Tables. References.
Chapter 14 Bodies in Contact Undergoing Direct Bearing and Shear Stress 689 Stress due to Pressure between Elastic Bodies. Rivets and Riveted Joints. Miscellaneous Cases. Tables. References.
Chapter 15 Elastic Stability 709
General Considerations. Buckling of Bars. Buckling of Flat and Curved Plates. Buckling of Shells. Tables. References.
Chapter 16 Dynamic and Temperature Stresses 743
Dynamic Loading. General Conditions. Body in a Known State of Motion. Impact and Sudden Loading. Approximate Formulas. Remarks on Stress due to Impact. Temperature Stresses. Table. References.
Chapter 17 Stress Concentration Factors 771
Static Stress and Strain Concentration Factors. Stress Concentration Reduction Methods. Table. References.
Appendix A Properties of a Plane Area 799
Table.
Appendix B Glossary: Definitions 813
Appendix C Composite Materials 827
Composite Materials. Laminated Composite Materials. Laminated Composite Structures.
Index 841
Contents v
9.4 Formulas for Curved Beams of Compact Cross-Section Loaded Normal to the Plane of Curvature 350 10.1 Formulas for Torsional Deformation and Stress 401 10.2 Formulas for Torsional Properties and Stresses in Thin-Walled Open Cross-Sections 413 10.3 Formulas for the Elastic Deformations of Uniform Thin-Walled Open Members under Torsional Loading 417 11.1 Numerical Values for Functions Used in Table 11.2 455 11.2 Formulas for Flat Circular Plates of Constant Thickness 457 11.3 Shear Deflections for Flat Circular Plates of Constant Thickness 500 11.4 Formulas for Flat Plates with Straight Boundaries and Constant Thickness 502 12.1 Formulas for Short Prisms Loaded Eccentrically; Stress Reversal Impossible 548 13.1 Formulas for Membrane Stresses and Deformations in Thin-Walled Pressure Vessels 592 13.2 Shear, Moment, Slope, and Deflection Formulas for Long and Short Thin-Walled Cylindrical Shells under Axisymmetric Loading 601 13.3 Formulas for Bending and Membrane Stresses and Deformations in Thin- Walled Pressure Vessels 608 13.4 Formulas for Discontinuity Stresses and Deformations at the Junctions of Shells and Plates 638 13.5 Formulas for Thick-Walled Vessels Under Internal and External Loading 683 14.1 Formulas for Stress and Strain Due to Pressure on or between Elastic Bodies 702 15.1 Formulas for Elastic Stability of Bars, Rings, and Beams 718 15.2 Formulas for Elastic Stability of Plates and Shells 730 16.1 Natural Frequencies of Vibration for Continuous Members 765 17.1 Stress Concentration Factors for Elastic Stress (K (^) t ) 781 A.1 Properties of Sections 802 C.1 Composite Material Systems 830
viii List of Tables
ix
Preface to the
Seventh Edition
The tabular format used in the fifth and sixth editions is continued in this edition. This format has been particularly successful when imple- menting problem solutions on a programmable calculator, or espe- cially, a personal computer. In addition, though not required in utilizing this book, user-friendly computer software designed to employ the format of the tabulations contained herein are available. The seventh edition intermixes International System of Units (SI) and United States Customary Units (USCU) in presenting example problems. Tabulated coefficients are in dimensionless form for conve- nience in using either system of units. Design formulas drawn from works published in the past remain in the system of units originally published or quoted. Much of the changes of the seventh edition are organizational, such as:
j (^) Numbering of equations, figures and tables is linked to the parti- cular chapter where they appear. In the case of equations, the section number is also indicated, making it convenient to locate the equation, since section numbers are indicated at the top of each odd-numbered page.
j (^) In prior editions, tables were interspersed within the text of each chapter. This made it difficult to locate a particular table and disturbed the flow of the text presentation. In this edition, all numbered tables are listed at the end of each chapter before the references.
Other changes=additions included in the seventh addition are as follows:
j (^) Part 1 is an introduction, where Chapter 1 provides terminology such as state properties, units and conversions, and a description of the contents of the remaining chapters and appendices. The defini-
xi
Preface to
the First Edition
This book was written for the purpose of making available a compact, adequate summary of the formulas, facts, and principles pertaining to strength of materials. It is intended primarily as a reference book and represents an attempt to meet what is believed to be a present need of the designing engineer. This need results from the necessity for more accurate methods of stress analysis imposed by the trend of engineering practice. That trend is toward greater speed and complexity of machinery, greater size and diversity of structures, and greater economy and refinement of design. In consequence of such developments, familiar problems, for which approximate solutions were formerly considered adequate, are now frequently found to require more precise treatment, and many less familiar problems, once of academic interest only, have become of great practical importance. The solutions and data desired are often to be found only in advanced treatises or scattered through an extensive literature, and the results are not always presented in such form as to be suited to the requirements of the engineer. To bring together as much of this material as is likely to prove generally useful and to present it in convenient form has been the author’s aim. The scope and management of the book are indicated by the Contents. In Part 1 are defined all terms whose exact meaning might otherwise not be clear. In Part 2 certain useful general princi- ples are stated; analytical and experimental methods of stress analysis are briefly described, and information concerning the behavior of material under stress is given. In Part 3 the behavior of structural elements under various conditions of loading is discussed, and exten- sive tables of formulas for the calculation of stress, strain, and strength are given. Because they are not believed to serve the purpose of this book, derivations of formulas and detailed explanations, such as are appro- priate in a textbook, are omitted, but a sufficient number of examples
are included to illustrate the application of the various formulas and methods. Numerous references to more detailed discussions are given, but for the most part these are limited to sources that are generally available and no attempt has been made to compile an exhaustive bibliography. That such a book as this derives almost wholly from the work of others is self-evident, and it is the author’s hope that due acknowl- edgment has been made of the immediate sources of all material here presented. To the publishers and others who have generously permitted the use of material, he wishes to express his thanks. The helpful criticisms and suggestions of his colleagues, Professors E. R. Maurer, M. O. Withey, J. B. Kommers, and K. F. Wendt, are gratefully acknowledged. A considerable number of the tables of formulas have been published from time to time in Product Engineering, and the opportunity thus afforded for criticism and study of arrangement has been of great advantage. Finally, it should be said that, although every care has been taken to avoid errors, it would be oversanguine to hope that none had escaped detection; for any suggestions that readers may make concerning needed corrections the author will be grateful.
Raymond J. Roark
xii Preface to the First Edition
3
Chapter
1
Introduction
The widespread use of personal computers, which have the power to solve problems solvable in the past only on mainframe computers, has influenced the tabulated format of this book. Computer programs for structural analysis, employing techniques such as the finite element method, are also available for general use. These programs are very powerful; however, in many cases, elements of structural systems can be analyzed quite effectively independently without the need for an elaborate finite element model. In some instances, finite element models or programs are verified by comparing their solutions with the results given in a book such as this. Contained within this book are simple, accurate, and thorough tabulated formulations that can be applied to the stress analysis of a comprehensive range of structural components. This chapter serves to introduce the reader to the terminology, state property units and conversions, and contents of the book.
1.1 Terminology
Definitions of terms used throughout the book can be found in the glossary in Appendix B.
1.2 State Properties, Units, and Conversions
The basic state properties associated with stress analysis include the following: geometrical properties such as length, area, volume, centroid, center of gravity, and second-area moment (area moment of inertia); material properties such as mass density, modulus of elasti- city, Poisson’s ratio, and thermal expansion coefficient; loading proper- ties such as force, moment, and force distributions (e.g., force per unit length, force per unit area, and force per unit volume); other proper-
ties associated with loading, including energy, work, and power; and stress analysis properties such as deformation, strain, and stress. Two basic systems of units are employed in the field of stress analysis: SI units and USCU units.y^ SI units are mass-based units using the kilogram (kg), meter (m), second (s), and Kelvin (K) or degree Celsius (C) as the fundamental units of mass, length, time, and temperature, respectively. Other SI units, such as that used for force, the Newton (kg-m=s 2 ), are derived quantities. USCU units are force-based units using the pound force (lbf), inch (in) or foot (ft), second (s), and degree Fahrenheit (F) as the fundamental units of force, length, time, and temperature, respectively. Other USCU units, such as that used for mass, the slug (lbf-s^2 =ft) or the nameless lbf- s^2 =in, are derived quantities. Table 1.1 gives a listing of the primary SI and USCU units used for structural analysis. Certain prefixes may be appropriate, depending on the size of the quantity. Common prefixes are given in Table 1.2. For example, the modulus of elasticity of carbon steel is approximately 207 GPa ¼ 207 10 9 Pa ¼ 207 109 N=m^2. Pre- fixes are normally used with SI units. However, there are cases where prefixes are also used with USCU units. Some examples are the kpsi (1 kpsi ¼ 10 3 psi ¼ 10 3 lbf =in^2 ), kip (1 kip ¼ 1 kilopound ¼ 1000 lbf ), and Mpsi (1 Mpsi ¼ 10 6 psi). Depending on the application, different units may be specified. It is important that the analyst be aware of all the implications of the units and make consistent use of them. For example, if you are building a model from a CAD file in which the design dimensional units are given in mm, it is unnecessary to change the system of units or to scale the model to units of m. However, if in this example the input forces are in
TABLE 1.1 Units appropriate to structural analysis
Property SI unit, symbol (derived units)
USCU unit,y^ symbol (derived units)
Length meter, m inch, in Area square meter (m^2 ) square inch (in^2 ) Volume cubic meter (m^3 ) cubic inch (in^3 ) Second-area moment (m^4 ) (in^4 ) Mass kilogram, kg (lbf-s^2 =in) Force Newton, N (kg-m=s^2 ) pound, lbf Stress, pressure Pascal, Pa (N=m^2 ) psi (lbf=in^2 ) Work, energy Joule, J (N-m) (lbf-in) Temperature Kelvin, K degrees Fahrenheit, F
y (^) In stress analysis, the unit of length used most often is the inch.
y (^) SI and USCU are abbreviations for the International System of Units (from the French Syste´me International d’Unite´s) and the United States Customary Units, respectively.
4 Formulas for Stress and Strain [ CHAP. 1
1.3 Contents
The remaining parts of this book are as follows.
Part 2: Facts; Principles; Methods. This part describes important relationships associated with stress and strain, basic material behavior, principles and analytical methods of the mechanics of structural elements, and numerical and experimental techniques in stress analysis.
Part 3: Formulas and Examples. This part contains the many applica- tions associated with the stress analysis of structural components. Topics include the following: direct tension, compression, shear, and combined stresses; bending of straight and curved beams; torsion; bending of flat plates; columns and other compression members; shells of revolution, pressure vessels, and pipes; direct bearing and shear stress; elastic stability; stress concentrations; and dynamic and temperature stresses. Each chapter contains many tables associated with most conditions of geometry, loading, and boundary conditions for a given element type. The definition of each term used in a table is completely described in the introduction of the table.
Appendices. The first appendix deals with the properties of a plane area. The second appendix provides a glossary of the terminology employed in the field of stress analysis. The references given in a particular chapter are always referred to by number, and are listed at the end of each chapter.
6 Formulas for Stress and Strain [ CHAP. 1
Part
Facts; Principles; Methods
surface containing Q. In the general case, this distribution will not be uniform along the surface, and will be neither normal nor tangential to the surface at Q. However, the force distribution at Q will have components in the normal and tangential directions. These compo- nents will be tensile or compressive and shear stresses, respectively. Following a right-handed rectangular coordinate system, the y and z axes are defined perpendicular to x, and tangential to the surface. Examine an infinitesimal area DAx ¼ DyDz surrounding Q, as shown in Fig. 2.2(a). The equivalent concentrated force due to the force distribution across this area is DF (^) x, which in general is neither normal nor tangential to the surface (the subscript x is used to designate the normal to the area). The force DFx has components in the x, y, and z directions, which are labeled DF (^) xx, DFxy, and DF (^) xz, respectively, as shown in Fig. 2.2(b). Note that the first subscript
Figure 2.
10 Formulas for Stress and Strain [ CHAP. 2
denotes the direction normal to the surface and the second gives the actual direction of the force component. The average distributed force per unit area (average stress) in the x direction isy
ss xx ¼
DFxx DAx
Recalling that stress is actually a point function, we obtain the exact stress in the x direction at point Q by allowing DA (^) x to approach zero. Thus,
sxx ¼ lim DAx! 0
DFxx DA (^) x
or,
sxx ¼
dF (^) xx dA (^) x
ð 2 :1-1Þ
Stresses arise from the tangential forces DFxy and DF (^) xz as well, and since these forces are tangential, the stresses are shear stresses. Similar to Eq. (2.1-1),
txy ¼
dF (^) xy dA (^) x
ð 2 :1-2Þ
txz ¼
dF (^) xz dA (^) x ð 2 :1-3Þ
Figure 2.
y (^) Standard engineering practice is to use the Greek symbols s and t for normal (tensile or compressive) and shear stresses, respectively.
SEC. 2.1] Stress and Strain: Important Relationships 11