Roarks formulas for stress and strain

549fm 1 12 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. Roarks formulas for stress and strain Roarks formulas for stress and strain Roarks formulas for stress and strain Roarks formulas for stress and strain Roarks formulas for stress and strain Roarks formulas for stress and strain Roarks formulas for stress and strain Roarks formulas for stress and strain Roarks formulas for stress and strain Roarks formulas for stress and strain Roarks formulas for stress and strain Roarks formulas for stress and strain

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 123456789 DOC=DOC 07654321 ISBN 0-07-072542-X The sponsoring editor for this book was Larry Hager and the production supervisor was Pamela A Pelton It was set in Century Schoolbook by Techset Composition Limited Printed and bound by R R Donnelley & Sons Company McGraw-Hill books are available at special quantity discounts to use as premiums and sales promotions, or for use in corporate training programs For more information, please write to the Director of Special Sales, Professional Publishing, McGraw-Hill, Two Penn Plaza, New York, NY 10121-2298 Or contact your local bookstore This book is printed on recycled, acid-free paper containing a minimum of 50% recycled, de-inked fiber Information contained in this work has been obtained by The McGraw-Hill Companies, Inc (‘‘McGraw-Hill’’) from sources believed to be reliable However, neither McGraw-Hill nor its authors guarantee the accuracy or completeness of any information published herein and neither McGraw-Hill nor its authors shall be responsible for any errors, omissions, or damages arising out of use of this information This work is published with the understanding that McGrawHill and its authors are supplying information but are not attempting to render engineering or other professional services If such services are required, the assistance of an appropriate professional should be sought Contents List of Tables vii Preface to the Seventh Edition ix Preface to the First Edition xi Part Introduction Chapter Introduction Terminology State Properties, Units, and Conversions Contents Part Facts; Principles; Methods Chapter Stress and Strain: Important Relationships Stress Strain and the Stress–Strain Relations Stress Transformations Strain Transformations Tables References Chapter The Behavior of Bodies under Stress 35 Methods of Loading Elasticity; Proportionality of Stress and Strain Factors Affecting Elastic Properties Load–Deformation Relation for a Body Plasticity Creep and Rupture under Long-Time Loading Criteria of Elastic Failure and of Rupture Fatigue Brittle Fracture Stress Concentration Effect of Form and Scale on Strength; Rupture Factor Prestressing Elastic Stability References Chapter Principles and Analytical Methods 63 Equations of Motion and of Equilibrium Principle of Superposition Principle of Reciprocal Deflections Method of Consistent Deformations (Strain Compatibility) Principles and Methods Involving Strain Energy Dimensional Analysis Remarks on the Use of Formulas References iii iv Contents Chapter Numerical Methods 73 The Finite-Difference Method The Finite-Element Method The BoundaryElement Method References Chapter Experimental Methods 81 Measurement Techniques Electrical Resistance Strain Gages Detection of Plastic Yielding Analogies Tables References Part Formulas and Examples Chapter 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 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 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 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 UniformThickness Plates with Straight Boundaries Effect of Large Deflection Diaphragm Stresses Plastic Analysis of Plates Ultimate Strength Tables References 427 Contents Chapter 12 Columns and Other Compression Members v 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 Appendix B Glossary: Definitions 813 Appendix C Composite Materials 827 Table Composite Materials Laminated Composite Materials Laminated Composite Structures Index 841 List of Tables 1.1 1.2 1.3 2.1 2.2 2.3 5.1 6.1 6.2 8.1 8.2 8.3 8.4 8.5 8.6 8.7a 8.7b 8.7c 8.7d 8.8 8.9 8.10 8.11a 8.11b 8.11c 8.11d 8.12 8.13 9.1 9.2 9.3 Units Appropriate to Structural Analysis Common Prefixes Multiplication Factors to Convert from USCU Units to SI Units Material Properties Transformation Matrices for Positive Rotations about an Axis Transformation Equations Sample Finite Element Library Strain Gage Rosette Equations Applied to a Specimen of a Linear, Isotropic Material Corrections for the Transverse Sensitivity of Electrical Resistance Strain Gages Shear, Moment, Slope, and Deflection Formulas for Elastic Straight Beams Reaction and Deflection Formulas for In-Plane Loading of Elastic Frames Numerical Values for Functions Used in Table 8.2 Numerical Values for Denominators Used in Table 8.2 Shear, Moment, Slope, and Deflection Formulas for Finite-Length Beams on Elastic Foundations Shear, Moment, Slope, and Deflection Formulas for Semi-Infinite Beams on Elastic Foundations Reaction and Deflection Coefficients for Beams under Simultaneous Axial and Transverse Loading: Cantilever End Support Reaction and Deflection Coefficients for Beams under Simultaneous Axial and Transverse Loading: Simply Supported Ends Reaction and Deflection Coefficients for Beams under Simultaneous Axial and Transverse Loading: Left End Simply Supported, Right End Fixed Reaction and Deflection Coefficients for Beams under Simultaneous Axial and Transverse Loading: Fixed Ends Shear, Moment, Slope, and Deflection Formulas for Beams under Simultaneous Axial Compression and Transverse Loading Shear, Moment, Slope, and Deflection Formulas for Beams under Simultaneous Axial Tension and Transverse Loading Beams Restrained against Horizontal Displacement at the Ends Reaction and Deflection Coefficients for Tapered Beams; Moments of Inertia Vary as ỵ Kx =lịn , where n ¼ 1:0 Reaction and Deflection Coefficients for Tapered Beams; Moments of Inertia Vary as ỵ Kx =lịn , where n ẳ 2:0 Reaction and Deection Coefcients for Tapered Beams; Moments of Inertia Vary as ỵ Kx =lịn , where n ẳ 3:0 Reaction and Deection Coefficients for Tapered Beams; Moments of Inertia Vary as ð1 ỵ Kx =lịn , where n ẳ 4:0 Position of Flexural Center Q for Different Sections Collapse Loads with Plastic Hinge Locations for Straight Beams Formulas for Curved Beams Subjected to Bending in the Plane of the Curve Formulas for Circular Rings Reaction and Deformation Formulas for Circular Arches 5 33 33 34 76 102 104 189 202 211 212 213 221 225 226 227 228 229 242 245 246 249 252 255 258 260 304 313 333 vii viii 9.4 10.1 10.2 10.3 11.1 11.2 11.3 11.4 12.1 13.1 13.2 13.3 13.4 13.5 14.1 15.1 15.2 16.1 17.1 A.1 C.1 List of Tables Formulas for Curved Beams of Compact Cross-Section Loaded Normal to the Plane of Curvature Formulas for Torsional Deformation and Stress Formulas for Torsional Properties and Stresses in Thin-Walled Open Cross-Sections Formulas for the Elastic Deformations of Uniform Thin-Walled Open Members under Torsional Loading Numerical Values for Functions Used in Table 11.2 Formulas for Flat Circular Plates of Constant Thickness Shear Deflections for Flat Circular Plates of Constant Thickness Formulas for Flat Plates with Straight Boundaries and Constant Thickness Formulas for Short Prisms Loaded Eccentrically; Stress Reversal Impossible Formulas for Membrane Stresses and Deformations in Thin-Walled Pressure Vessels Shear, Moment, Slope, and Deflection Formulas for Long and Short Thin-Walled Cylindrical Shells under Axisymmetric Loading Formulas for Bending and Membrane Stresses and Deformations in Thin- Walled Pressure Vessels Formulas for Discontinuity Stresses and Deformations at the Junctions of Shells and Plates Formulas for Thick-Walled Vessels Under Internal and External Loading Formulas for Stress and Strain Due to Pressure on or between Elastic Bodies Formulas for Elastic Stability of Bars, Rings, and Beams Formulas for Elastic Stability of Plates and Shells Natural Frequencies of Vibration for Continuous Members Stress Concentration Factors for Elastic Stress (K t) Properties of Sections Composite Material Systems 350 401 413 417 455 457 500 502 548 592 601 608 638 683 702 718 730 765 781 802 830 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 implementing problem solutions on a programmable calculator, or especially, 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 convenience 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 particular 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 is an introduction, where Chapter provides terminology such as state properties, units and conversions, and a description of the contents of the remaining chapters and appendices The definiix x Preface to the Seventh Edition tions incorporated in Part of the previous editions are retained in the seventh edition, and are found in Appendix B as a glossary j Properties of plane areas are located in Appendix A j Composite material coverage is expanded, where an introductory discussion is provided in Appendix C, which presents the nomenclature associated with composite materials and how available computer software can be employed in conjunction with the tables contained within this book j Stress concentrations are presented in Chapter 17 j Part 2, Chapter 2, is completely revised, providing a more comprehensive and modern presentation of stress and strain transformations j Experimental Methods Chapter 6, is expanded, presenting more coverage on electrical strain gages and providing tables of equations for commonly used strain gage rosettes j Correction terms for multielement shells of revolution were presented in the sixth edition Additional information is provided in Chapter 13 of this edition to assist users in the application of these corrections The authors wish to acknowledge and convey their appreciation to those individuals, publishers, institutions, and corporations who have generously given permission to use material in this and previous editions Special recognition goes to Barry J Berenberg and Universal Technical Systems, Inc who provided the presentation on composite materials in Appendix C, and Dr Marietta Scanlon for her review of this work Finally, the authors would especially like to thank the many dedicated readers and users of Roark’s Formulas for Stress & Strain It is an honor and quite gratifying to correspond with the many individuals who call attention to errors and=or convey useful and practical suggestions to incorporate in future editions Warren C Young Richard G Budynas 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 are defined all terms whose exact meaning might otherwise not be clear In Part certain useful general principles 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 the behavior of structural elements under various conditions of loading is discussed, and extensive 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 appropriate in a textbook, are omitted, but a sufficient number of examples xi SEC 16.6] Dynamic and Temperature Stresses 761 mum stress is the tangential stress at the ends of the ellipse and is st ẳ DT gE=ẵ1 ỵ b=aị, where a is the major and b the minor semiaxis of the ellipse (Ref 7) 12 If the disk of case 10 is heated symmetrically about its center and uniformly throughout its thickness so that the temperature is a function of the distance r from the center only, the radial and tangential stresses at any point a distance r1 from the center are ! ð ð R r1 sr1 ¼ gE Tr dr À Tr dr R r1 st1 ẳ gE T ỵ R R Tr dr ỵ r1 ! r1 Tr dr where R is the radius of the disk and T is the temperature at any point a distance r from the center minus the temperature of the coldest part of the disk [In the preceding expressions, the negative sign denotes compressive stress (Ref 7).] 13 A rectangular plate or strip ABCD (Fig 16.5) is heated along a transverse line FG uniformly throughout the thickness and across the width so that the temperature varies only along the length with x At FG the temperature is T1 ; the minimum temperature in the plate is T0 At any point along the edges of the strip where the temperature is T, a tensile stress sx ẳ EgT T0 ị is developed; this stress has its maximum value at F and G, where it becomes EgðT1 À T0 Þ Halfway between F and G, a compressive stress sy of equal intensity is developed along line FG (Ref 7) 14 The plate of case 13 is heated as described except that the lower face of the plate is cooler than the upper face, the maximum temperature there being T2 and the temperature gradient through the thickness being linear The maximum tensile stress at F and G is (see Ref 7)   1n T1 T2 ị sx ẳ Eg T1 þ T2 À 2T0 þ 3þn 15 A long hollow cylinder with thin walls has the outer surface at the uniform temperature T and the inner surface at the uniform Figure 16.5 762 Formulas for Stress and Strain [CHAP 16 temperature T ỵ DT The temperature gradient through the thickness is linear At points remote from the ends, the maximum circumferential stress is 12 DT gE=ð1 À nÞ (compression at the inner surface; tension at the outer surface) and the longitudinal stress is DT gE=ð1 À nÞ (compression at the inside; tension at the outside) (These formulas apply to a thin tube of any cross section.) At the ends, if these are free, the maximum tensile stress in a tube of circular section is about 25% greater than the value given by the formula (Ref 7) 16 A hollow cylinder with thick walls of inner radius b and outer radius c has the outer surface at the uniform temperature T and the inner surface at the uniform temperature T ỵ DT After steady-state heat flow is established the temperature decreases logarithmically with r and then the maximum stresses, which are circumferential and which occur at the inner and outer surfaces, are (Outer surface)   DT gE 2b2 c st ¼ 1À ln 2ð1 À nÞ lnðc=bÞ c À b2 b tension (Inner surface)   DT gE 2c2 c 1À ln st ¼ 2ð1 À nÞ lnðc=bÞ c À b2 b compression At the inner and outer surfaces, the longitudinal stresses are equal to the tangential stresses (Ref 7) 17 If the thick tube of case 16 has the temperature of the outer surface raised at the uniform rate of m =s then, after a steady rate of temperature rise has been reached throughout, the maximum tangential stresses are (Outer surface)   Egm 4b4 c 2 3b À c À st ẳ ln 8A1 nị c b2 b compression (Inner surface) st ¼   Egm 4b2 c2 c b2 ỵ c2 ln 8A1 nÞ c À b2 b tension where A is the coefficient of thermal diffusivity equal to the coefficient of thermal conductivity divided by the product of density of the material and its specific heat (For steel, A may be SEC 16.6] Dynamic and Temperature Stresses 763 taken as 0:027 in2 =s at moderate temperatures.) [At the inner and outer surfaces, the longitudinal stresses are equal to the tangential stresses (Ref 9).] The stated conditions in this case 17 as well as those in cases 19 to 21 are difficult to create in a short time except for small parts heated or cooled in liquids 18 A solid rod of circular section is heated or cooled symmetrically with respect to its axis, the condition being uniform along the length, so that the temperature is a function of r (the distance from the axis) only The stresses are equal to those given by the formulas for case 12 divided by À n (Ref 7) 19 If the solid rod of case 18 has the temperature of its convex surface raised at the uniform rate of m =s, then, after a steady rate of temperature rise has been reached throughout, the radial, tangential, and longitudinal stresses at any point a distance r from the center are sr ¼ Egm c2 À r2 À n 16A st ¼ Egm c2 À 3r2 À n 16A sx ¼ Egm c2 À 2r2 À n 8A Here A has the same meaning as in case 17 and c is the radius of the shaft, [A negative result indicates compression, a positive result tension (Ref 9).] 20 A solid sphere of radius c is considered instead of a solid cylinder but with all other conditions kept the same as in case 19 The radial and tangential stresses produced at any point a distance r from the center are sr ¼ Egm ðc2 À r2 ị 15A1 nị st ẳ Egm c2 2r2 Þ 15Að1 À nÞ [A negative result indicates compression, a positive result tension (Ref 9).] 21 If the sphere is hollow, with outer radius c and inner radius b, and with all other conditions kept as stated in case 17, the stresses at 764 Formulas for Stress and Strain [CHAP 16 any point are   Egm 5b3 r ỵfc sr ẳ 15A1 nị r   Egm 5b3 c st ẳ 2r ỵfỵ 15A1 nị 2r where c5 ỵ 5c2 b3 À 6b5 c À b3 c b 6c3 b5 ỵ 5c2 b6 cẳ r3 c3 b3 ị fẳ [A negative result indicates compression, a positive result tension (Ref 9).] Other problems involving thermal stress, the solutions of which cannot be expressed by simple formulas, are considered in the references cited above and in Refs 3, 10 and 25 to 29; charts for the solution of thermal stresses in tubes are given in Ref 11 Derivations for many of the thermal loadings shown above along with thermal loadings on many other examples of bars, rings, plates, and cylindrical and spherical shells are given in Ref 28 16.7 Tables Natural frequencies of vibration for continuous members Case no and description Uniform beam; both ends simply supported Natural frequencies 1a Center load W , beam weight negligible 1b Uniform load w per unit length including beam weight rffiffiffiffiffiffiffiffiffi K EIg fn ¼ n 2p wl4 Mode Kn 9:87 39:5 88:8 158 247 2b Uniform load w per unit length including beam weight fn ¼ 3a Right end load W , beam weight negligible 3b Uniform load w per unit length including beam weight f1 ¼ Kn 2p rffiffiffiffiffiffiffiffiffi EIg wl4 Nodal position=l 0:0 0:0 0:0 0:0 0:0 1:00 0:50 0:33 0:25 0:20 1:00 0:67 0:50 0:40 1:00 0:75 1:00 0:60 0:80 Ref: 22 1:00 approximately Mode Kn 22:4 61:7 121 200 299 Nodal position=l 0:0 0:0 0:0 0:0 0:0 1:00 0:50 0:36 0:28 0:23 1:00 0:64 0:50 0:41 1:00 0:72 0:59 1:00 0:77 1:00 f1 ¼ Ref: 22 765 r 13:86 EIg approximately 2p Wl3 ỵ 0:383wl4 rffiffiffiffiffiffiffiffiffi 1:732 EIg f1 ¼ 2p Wl3 rffiffiffiffiffiffiffiffiffi K EIg Mode Kn Nodal position=l fn ¼ n 2p wl4 3:52 0:0 22:0 0:0 0:783 61:7 0:0 0:504 0:868 121 0:0 0:358 0:644 0:905 200 0:0 0:279 0:500 0:723 0:926 Ref: 22 Dynamic and Temperature Stresses 2a Center load W , beam weight negligible 2c Uniform load w per unit length plus a center load W Uniform beam; left end fixed, right end free (cantilever) rffiffiffiffiffiffiffiffiffi 6:93 EIg f1 ¼ 2p Wl3 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6:93 EIg 2p Wl3 ỵ 0:486wl4 r 13:86 EIg f1 ẳ 2p Wl3 1c Uniform load w per unit length plus a center load W Uniform beam; both ends fixed 16.7] f ¼ natural frequency (cycles per second); Kn ¼ constant where n refers to the mode of vibration; g ¼ gravitational acceleration (units consistent with length dimension); E ¼ modulus of elasticity; I ¼ area moment of inertia; D ¼ Et3 =12ð1 À n2 Þ NOTATION: SEC TABLE 16.1 Natural frequencies of vibration for continuous members (Continued) Case no and description Natural frequencies Uniform beam; left end fixed, right end hinged Uniform beam; left end hinged, right end free Uniform bar or spring vibrating along its longitudinal axis; upper end fixed, lower end free 4a Uniform load w per unit length including beam weight 5a Uniform load w per unit length including beam weight 6a Uniform load w per unit length including beam weight 7a Weight W at lower end, bar weight negligible rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1:732 EIg f1 ẳ approximately 2p Wl3 ỵ 0:236wl4 r K EIg Mode Kn Nodal position=l fn ¼ n 2p wl4 22:4 0:224 0:776 61:7 0:132 0:500 0:868 121 0:095 0:356 0:644 0:905 200 0:074 0:277 0:500 0:723 0:926 299 0:060 0:226 0:409 0:591 0:774 0:940 rffiffiffiffiffiffiffiffiffi K EIg fn ¼ n 2p wl4 rffiffiffiffiffiffiffiffiffi K EIg fn ¼ n 2p wl4 rffiffiffiffiffiffi kg f1 ¼ 2p W rffiffiffiffiffiffiffiffiffiffi AEg f1 ¼ 2p Wl 7c Uniform load w per unit length plus a load W at the lower end f1 ¼ Kn 15:4 50:0 104 178 272 Mode Kn 15:4 50:0 104 178 272 Nodal position=l 0:0 0:0 0:0 0:0 0:0 1:000 0:557 0:386 0:295 0:239 0:0 0:0 0:0 0:0 0:0 0:736 0:446 0:308 0:235 0:190 1:000 0:692 1:000 0:529 0:765 1:000 0:428 0:619 0:810 1:000 Ref: 22 Nodal position=l 0:853 0:617 0:898 0:471 0:707 0:922 0:381 0:571 0:763 0:937 Ref: 22 for a spring where k is the spring constant for a bar where A is the area; l the length; and E the modulus where K1 ¼ 1:57 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kg 2p W þ wl=3 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi AEg f1 ¼ 2p Wl þ wl2 =3 K2 ¼ 4:71 K3 ¼ 7:85 approximately for a spring where k is the spring constant approximately for a bar where A is the area [CHAP 16 7b Uniform load w per unit length including bar weight rffiffiffiffiffiffiffiffiffiffi K AEg fn ¼ n 2p wl2 Mode Ref: 22 Formulas for Stress and Strain 3c Uniform load w per unit length plus an end load W Uniform beam; both ends free 766 TABLE 16.1 SEC Natural frequencies of vibration for continuous members (Continued) Uniform shaft or bar in torsional vibration; one end fixed, the other end free 8a Concentrated end mass of J mass moment of inertia, shaft weight negligible 8b Uniform distribution of mass moment of inertia along shaft; Js ¼ total distributed mass moment of inertia fn ¼ rffiffiffiffiffiffiffiffi GK 2p Jl Kn 2p sffiffiffiffiffiffiffiffi GK Js l G is the shear modulus of elasticity and K is the torsional stiffness constant (see Chap 10) where K1 ¼ 1:57 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi GK 2p ðJ ỵ Js =3ịl K2 ẳ 4:71 K3 ẳ 7:85 8c Uniformly distributed inertia plus a concentrated end mass f1 ¼ 9a Uniform load w per unit length including own weight f¼ Kn 2p rffiffiffiffiffiffiffiffi Tg wl2 f¼ Kn 2p rffiffiffiffiffiffiffiffiffi Dg wr4 where K1 K2 K3 K4 ¼ 10:2 ¼ 21:3 ¼ 34:9 ¼ 39:8 fundamental one nodal diameter two nodal diameters one nodal circle approximately where K1 ¼ p K2 ¼ 2p K3 ¼ 3p 10 Circular flat plate of uniform thickness t and radius r; edge fixed 10a Uniform load w per unit area including own weight 11 Circular flat plate of uniform thickness t and radius r; edge simply supported 11a Uniform load w per unit area including own weight; n ¼ 0:3 rffiffiffiffiffiffiffiffiffi K Dg f¼ n 2p wr4 where K1 K2 K3 K4 ¼ 4:99 ¼ 13:9 ¼ 25:7 ¼ 29:8 fundamental one nodal diameter two nodal diameters one nodal circle 12a Uniform load w per unit area including own weight; n ¼ 0:33 rffiffiffiffiffiffiffiffiffi K Dg f¼ n 2p wr4 where K1 K2 K3 K4 ¼ 5:25 ¼ 9:08 ¼ 12:2 ¼ 20:5 two nodal diameters one nodal circle three nodal diameters one nodal diameter and one nodal circle 12 Circular flat plate of uniform thickness t and radius r; edge free Ref: 20 Ref: 20 Ref: 20 Dynamic and Temperature Stresses String vibrating laterally under a tension T with both ends fixed f1 ¼ 16.7] TABLE 16.1 767 TABLE 16.1 Natural frequencies of vibration for continuous members (Continued) 14 Elliptical flat plate of major radius, a, minor radius b, and thickness t; edge fixed Natural frequencies 13a Uniform load w per unit area including own weight; n ¼ 0:3 rffiffiffiffiffiffiffiffiffi K Dg f¼ n 2p wr4 where Kn is tabulated for various degrees of edge stiffness in the form of br=D: Kn br=D Fundamental nodal diameter 0:1 0:01 10:2 10:2 10:0 8:76 21:2 21:2 20:9 18:6 34:8 34:8 34:2 30:8 39:7 39:7 39:1 35:2 0:001 6:05 4:93 15:0 13:9 26:7 25:6 30:8 29:7 14a Uniform load w per unit area including own weight K f¼ 2p rffiffiffiffiffiffiffiffiffi Dg wa4 a=b 1:0 nodal diameters nodal circle where K1 is tabulated for various ratios of a=b 1:1 1:2 1:5 2:0 3:0 K1 10:2 11:3 12:6 17:0 27:8 57:0 rffiffiffiffiffiffiffiffiffi K Dg where K1 is tabulated for various ratios of a=b f¼ 2p wa2 a=b 0:9 0:8 0:6 0:4 0:2 15a Uniform load w per unit area including own weight 16 Rectangular flat plate with short edge a, long edge b, and thickness t; all edges simply supported 16a Uniform load w per unit area including own weight K f¼ n 2p 17a Uniform load w per unit area including own weight a=b 1:0 0:8 0:6 K1 19:7 16:2 13:4 ẳ 1; mb ẳ 1ị K2 49:3 35:1 24:1 ẳ 1; mb ẳ 2ị K3 49:3 45:8 ẳ 2; mb ẳ 1ị 41:9 K3 ẳ 1; mb ẳ 3ị r K1 Dg where K1 is tabulated for f¼ 2p wa4 17 Rectangular flat plate with two edges a fixed, one edge b fixed, and one edge b simply supported a=b K1 Ref: 20 36:0 32:7 29:9 25:9 23:6 22:6 22:4 rffiffiffiffiffiffiffiffiffi  a2  Dg where Kn ẳ p2 m2a ỵ m2b wa4 b ma ðma ðma ðma 3:0 213 2:0 99 1:6 67 1:2 42:4 1:0 0:8 33:1 25:9 Ref: 20 0:4 0:2 11:5 10:3 16:2 11:5 0:0 9:87 24:1 13:4 Ref: 20 [CHAP 16 15 Rectangular flat plate with short edge a, long edge, b, and thickness, t; all edges fixed K1 Ref: 20 Formulas for Stress and Strain 13 Circular flat plate of uniform thickness t and radius r; edge simply supported with an additional edge constraining moment M ¼ bc per unit circumference where c is the edge rotation 768 Case no and description a various ratios of b 0:6 20:8 0:4 :2 17:8 16:2 15:8 Ref: 22 SEC 16.8] 16.8 Dynamic and Temperature Stresses 769 References Hodkinson, B.: Rotating Discs of Conical Profile, Engineering, vol 115, p 1, 1923 Rushing, F C.: Determination of Stresses in Rotating Disks of Conical Profile, Trans ASME, vol 53, p 91, 1931 Stodola, A.: ‘‘Steam and Gas Turbines,’’ 6th ed., McGraw-Hill, 1927 (transl by L C Loewenstein) Loewenstein, L C,: ‘‘Marks’ Mechanical Engineers’ Handbook,’’ McGraw-Hill, 1930 Donnell, L H.: Longitudinal Wave Transmission and Impact, Trans ASME, vol 52, no 1, p 153, 1930 Love, A E H.: ‘‘Mathematical Theory of Elasticity,’’ 2nd ed., Cambridge University Press, 1906 Goodier, J N.: Thermal Stress, ASME J Appl Mech., vol 4, no 1, 1937 Maulbetsch, J L: Thermal Stresses in Plates, ASME J Appl Mech., vol 2, no 4, 1935 Kent, C H.: Thermal Stresses in Spheres and Cylinders Produced by Temperatures Varying with Time, Trans ASME, vol 54, no 18, p 185, 1932 10 Timoshenko, S.: ‘‘Theory of Elasticity,’’ McGraw-Hill, 1934 11 Barker, L H.: The Calculation of Temperature Stresses in Tubes, Engineering, vol 124, p 443, 1927 12 Robinson, E L.: Bursting Tests of Steam-turbine Disk Wheels, Trans ASME, vol 66, no 5, p 373, 1944 13 Holms, A G., and J E Jenkins: Effect of Strength and Ductility on Burst Characteristics of Rotating Disks, Natl Adv Comm Aeron., Tech Note 1667, 1948 14 Goodier, J N.: Thermal Stress and Deformation, ASME J Appl Mech., vol 24, no 3, 1957 15 Eichmann, E S.: Note on the Maximum Effect of a Moving Force on a Simple Beam, ASME J Appl Mech., vol 20, no 4, 1953 16 Ayre, R S., L S Jacobsen, and C S Hsu: Transverse Vibration of and 2-span Beams under Moving Mass-Load, Proc 1st U.S Nail Congr Appl Mech., 1952 17 Burr, Arthur H.: Longitudinal and Torsional Impact in a Uniform Bar with a Rigid Body at One End, ASME J Appl Mech., vol 17, no 2, 1950 18 Schwieger, Horst: A Simple Calculation of the Transverse Impact on Beams and Its Experimental Verification, J Soc Exp Mech., vol 5, no 11, 1965 19 Thomson, W T., and M D Dahleh: ‘‘Theory of Vibrations with Applications,’’ 5th ed., Prentice-Hall, 1998 20 Leissa, A W.: Vibration of Plates, NASA SP-160, National Aeronautics and Space Administration, 1969 21 Leissa, A W.: Vibration of Shells, NASA SP-288, National Aeronautics and Space Administration, 1973 22 Huang, T C.: Eigenvalues and Modifying Quotients of Vibration of Beams, and Eigenfunctions of Vibration of Beams, Univ Wis Eng Exp Sta Repts Nos 25 and 26, 1964 23 Jones, R.: An Approximate Expression for the Fundamental Frequency of Vibration of Elastic Plates, J Sound Vib., vol 38, no 4, 1975 24 Blevins, R D.: ‘‘Formulas for Natural Frequency and Mode Shape,’’ Van Nostrand Reinhold, 1979 25 Fridman, Y B (ed.): ‘‘Strength and Deformation in Nonuniform Temperature Fields,’’ transl from the Russian, Consultants Bureau, 1964 26 Johns, D J.: ‘‘Thermal Stress Analyses,’’ Pergamon Press, 1965 27 Boley, B A., and J H Weiner: ‘‘Theory of Thermal Stresses,’’ John Wiley & Sons, 1960 28 Burgreen, D.: ‘‘Elements of Thermal Stress Analysis,’’ C P Press, 1971 29 Nowacki, W.: ‘‘Thermoelasticity,’’ 2nd ed., English transl by H Zorski, Pergamon Press, 1986 30 Timoshenko, S.: ‘‘Vibration Problems in Engineering,’’ Van Nostrand, 1955 Chapter 17 Stress Concentration When a large stress gradient occurs in a small, localized area of a structure, the high stress is referred to as a stress concentration Near changes in geometry of a loaded structure, the flow of stress is interfered with, causing high stress gradients where the maximum stress and strain may greatly exceed the average or nominal values based on simple calculations Contact stresses, as discussed in Chapter 14, also exhibit high stress gradients near the point of contact, which subside quickly as one moves away from the contact area Thus, the two most common occurrences of stress concentrations are due to (1) discontinuities in continuum and (2) contact forces Discontinuities in continuum include changes in geometry and material properties This chapter is devoted to geometric changes Rapid geometry changes disrupt the smooth flow of stresses through the structure between load application areas Plates in tension or bending with holes, notches, steps, etc are simple examples involving direct normal stresses Shafts in tension, bending, and torsion, with holes, notches, steps, keyways, etc., are simple examples involving direct and bending normal stresses and torsional shear stresses More complicated geometries must be analyzed either by experimental or numerical techniques such as the finite element method Other, less obvious, geometry changes include rough surface finishes and external and internal cracks Changes in material properties are discussed in Chap 7, and demonstrated in an example where a change in modulus of elasticity drastically changed the stress distribution Changes in material properties can occur both at macroscopic and microscopic levels which include alloy formulation, grain size and orientation, foreign materials, etc 771 772 Formulas for Stress and Strain 17.1 Static Stress and Strain Concentration Factors [CHAP 17 Consider the plate shown in Fig 17.1, loaded in tension by a force per unit area, s Although not drawn to scale, consider that the outer dimensions of the plate are infinite compared with the diameter of the hole, 2a It can be shown, from linear elasticity, that the tangential stress throughout the plate is given by (see Ref 60) sy ¼     s a2 a4 ỵ ỵ cos 2y r r ð17:1-1Þ The maximum stress is sy ¼ 3s at r ¼ a and y ẳ ặ90 Figure 17.2 shows how the tangential stress varies along the x and y axes of the plate For the top (and bottom) of the hole, we see the stress gradient is extremely large compared with the nominal stress, and hence the term stress concentratiom applies Along the surface of the hole, the tangential stress is Às at y ¼ 0 and 180 , and increases, as y increases, to 3s at y ¼ 90 and 270 Figure 17.1 Circular hole in a plate loaded in tension Figure 17.2 Tangential stress distribution for y ¼ 0 and 90 SEC 17.1] Stress Concentration 773 The static stress concentration factor in the elastic range, Kt , is defined as the ratio of the maximum stress, smax , to the nominal stress, snom That is, s ð17:1-2Þ Kt ¼ max snom For the infinite plate containing a hole and loaded in tension, snom ¼ s, smax ¼ 3s, and thus Kt ¼ 3.* The analysis of the plate in tension with a hole just given is for a very wide plate (infinite in the limit) As the width of the plate decreases, the maximum stress becomes less than three times the nominal stress at the zone containing the hole Figure 17.3(a) shows a plate of thickness t ¼ 0:125 in, width D ¼ 1:50 in, with a hole of diameter 2r ¼ 0:50 in, and an applied uniform stress of s0 ¼ 320 psi Figure 17.3 Stress distribution for a plate in tension containing a centrally located hole *See Case 7a of Table 17.1 As 2a=D ! 0, Kt ! 3:00 774 Formulas for Stress and Strain [CHAP 17 A photoelastic* model is shown in Fig 17.3(b) From a photoelastic analysis, the stresses at points a, b, and c are found to be zone A A: zone B B: sa ¼ 320 psi sb ¼ 280 psi; sc ¼ 1130 psi The nominal stress in zone B B is snom ¼ D 1:50 s ¼ 320 ¼ 480 psi D À 2r 1:50 À 0:5 If the stress was uniform from b to c, the stress would be 480 psi However, the photoelastic analysis shows the stress to be nonuniform, ranging from 280 psi at b to a maximum stress at c of 1130 psi Thus, for this example, the stress concentration factor is found to be Kt ¼ smax 1130 ¼ 2:35 ¼ 480 snom The static stress concentration factor for a plate containing a centrally located hole in which the plate is loaded in tension depends on the ratio 2r=D as given for case 7a of Table 17.1 For our example here, 2r=D ¼ 0:5=1:5 ¼ 13 The equation for Kt from Table 17.1 gives Kt ẳ 3:00 3:1313ị ỵ 3:6613ị2 1:5313ị3 ẳ 2:31 which is within 2% of the results from the photoelastic model Table 17.1 provides the means to evaluate the static stress concentration factors in the elastic range for many cases that apply to fundamental forms of geometry and loading conditions If the load on a structure exceeds the value for which the maximum stress at a stress concentration equals the elastic limit of the material, the stress distribution changes from that within the elastic range Neuber (Ref 61) presented a formula which includes stress and strain Defining an effective stress concentration factor, Ks ¼ smax = snom , and an effective strain concentration factor, Ke ¼ emax =enom , Neuber established that Kt is the geometric means of the stress and strain factors That is, Kt ¼ ðKs Ke Þ1=2 , or Neuber’s Formula for Nonlinear Material Behavior Ks ¼ Kt2 Ke *Photoelasticity is discussed at some length in Ref 60 ð17:1-3Þ SEC 17.1] Stress Concentration 775 In terms of the stresses and strains, Eq (17.1-3) can be written as smax emax ẳ Kt2 snom enom 17:1-4ị Kt and snom are obtained exactly the same as when the max stress is within the elastic range The determination of enom is found from the material’s elastic stress-strain curve using the nominal stress EXAMPLE A circular shaft with a square shoulder and fillet is undergoing bending (case 17b of Table 17.1) A bending moment of 500 N-m is being transmitted at the fillet section For the shaft, D ¼ 50 mm, h ¼ mm, and r ¼ mm The stress– strain data for the shaft material is tabulated below and plotted in Fig 17.4 Determine the maximum stress in the shaft e; 10À5 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 s; ðMPaÞ 50 100 150 200 235 252 263 267 272 276 279 282 285 287 289 290 Solution From the given dimensions, h=r ¼ 9=3 ¼ From case 17b of Table 17.1, pffiffiffi C1 ẳ 1:225 ỵ 0:831 0:0103ị ẳ 2:634 p C2 ẳ 3:790 ỵ 0:958 0:2573ị ẳ 2:902 p C3 ẳ 7:374 4:834 ỵ 0:8623ị ẳ 1:587 C4 ẳ 3:809 ỵ 3:046 0:5953ị ẳ 0:3182 With 2h=D ¼ 18=50 ¼ 0:36, Kt ¼ 2:634 À 2:9020:36ị ỵ 1:5870:36ị2 0:31820:36ị3 ẳ 1:780 Figure 17.4 776 Formulas for Stress and Strain [CHAP 17 The nominal stress at the minor radius of the step shaft is snom ẳ 32M pD 2hị3 ẳ 32500ị pẵ50 29ị3 103 ị3 ẳ 155:4106 ị N=m2 ẳ 155:4 MPa If smax is in the elastic range, then smax ¼ Kt snom ẳ 1:780155:4ị ẳ 276:6 MPa However, as one can see from the stress–strain plot that this exceeds the elastic limit of 200 MPa Thus, smax must be determined from Neuber’s equation The modulus of elasticity in the elastic range of the material is E ¼ 20 GPa Thus, the nominal strain is found to be enom ¼ snom =E ¼ 155:4 ð106 Þ=20ð109 Þ ¼ 77:7 ð10À5 Þ Thus, Kt2 snom enom ẳ 1:780ị2 155:4ị77:7ị105 ị ẳ 0:3826 MPa From the tabulated data, the product s e can be tabulated as a function of s This results in the following: s ðMPaÞ 50 100 150 200 235 252 263 267 s e ðMPaÞ 0.0125 0.05 0.1125 0.2 0.29375 0.378 0.46025 0.534 s ðMPaÞ 272 276 279 282 285 287 289 290 se ðMPaÞ 0.612 0.69 0.76725 0.846 0.92625 1.0045 1.08375 1.16 Since, based on Eq (17.4), we are looking for the value of smax emax ¼ 0:3826, we will interpolate s e between 0.378 and 0.46025 Thus, smax À 252 263 À 252 ¼ 0:3826 À 0:378 0:46025 À 0:378 This yields smax ¼ 252:6 MPa For dynamic problems where loading is cycling, the fatigue stress concentration factor is more appropriate to use See Sec 3.20 for a discussion of this 17.2 Stress Concentration Reduction Methods Intuitive methods such as the flow analogy are sometimes helpful to the analyst faced with the task of reducing stress concentrations When dealing with a situation where it is necessary to reduce the cross section abruptly, the resulting stress concentration can often be mini- ... Loading Formulas for Bending and Membrane Stresses and Deformations in Thin- Walled Pressure Vessels Formulas for Discontinuity Stresses and Deformations at the Junctions of Shells and Plates Formulas. .. of Tables Formulas for Curved Beams of Compact Cross-Section Loaded Normal to the Plane of Curvature Formulas for Torsional Deformation and Stress Formulas for Torsional Properties and Stresses... Units, and Conversions Contents Part Facts; Principles; Methods Chapter Stress and Strain: Important Relationships Stress Strain and the Stress? ? ?Strain Relations Stress Transformations Strain Transformations