MEASUREMENTS FOR STRESSES IN MACHINE COMPONENTS V. F. YAKOVLEV AND L S. I N Y U T IN Translated from the Rwsian by J. J. CORNISH Translation edited by M. L. MEYER Senior Lecturer Postgraditate Department of Applied Mechanics Sheffield University PERGAMON PRESS OXFORD · LONDON · EDINBURGH · NEW YORK PARIS · FRANKFURT 1964 PERGAMON PRESS LTD. Headington Hill Hall, Oxford 4 de δ Fitzroy Square, London W. 1 PERGAMON PRESS (SCOTLAND) LTD. 2 ά 3 Teriot Place, Edinburgh 1 PERGAMON PRESS INC. 122 East 55th Street, New York 22, N, F. GAUTHIER-VILLARS ED. 55 Quai des Grands-Äugustins, Paris 6 PERGAMON PRESS G.m.b.H. Kaiserstrasse 75, Frankfurt am Main Distributed in the Western Hemisphere by THE MACMILLAN COMPANY · NEW YORK pursuant to a special arrangement with Pergamon Press Limited Copyright © 1964 PEBGAMON PRESS LTD Library of Congress Catalog Card Number 63-20583 This book is a translation of the original Russian HaMepemn HanpHOtceHUü öemaAeü Mauim (Izmereniya napryazhenii detalei mashin), published in 1961 by Mashgiz, Moscow PREFACE IN PRACTICE, the experimental work carried out in works, laboratories, design offices and scientific research institutes re­ quires measurement of the most diverse quantities, such as accelerations, velocities, displacements, forces, deformations, stresses and so forth. In the design and experimental examination for the strength of machines and structures, the state of stress and strain of components and members has to be investigated frequently. A basic and complex problem here is the study of the stresses at points inside a component, since the magnitude of these stresses may be particularly decisive in many cases, determining the service life of structures, for example contact stresses in runners, bearings, railway lines, etc. Strength may be checked both theoretically and experimen­ tally. Due to the complexity of individual phenomena and the consequent approximate character of the calculation schemes, theoretical calculations do not always give satisfactory results. Many problems have no theoretical solutions at all. In recent years, therefore, experimental methods of studying the state of stress have been appHed widely, alongside the develop­ ment of the theory of strength calculations. A number of measuring methods are used in the experimenta work: strain gauges, X-rays, brittle coatings, grids and photo- elasticity. INTRODUCTION THE DESIGN and operation of machines frequently requires an investigation of their state of stress. Similar work is carried out when new designs are made for machines and mechanisms, when their operating conditions are changed, when their strength in service is checked, and so on. The state of stress can be studied theoretically and experimen- taUy. Attempts to calculate theoretically the strength of structures were made as long ago as the fourteenth and sixteenth centuries by Leonardo da Vinci and Galileo [1, 2]. In the seventeenth and eighteenth centuries, the fundamentals of the theory were established by the work of Hooke, Mariotte, Bernoulli, Euler, Lomonosov, Young and others. Their work prepared the necessary basis of the classical theory for calculating the strength of members, machines and structures, which was finally formulated in the work of Cauchy, Poisson, Zhukovskii, Yasinskii, Kirpichev, Lame, Clapeyron, Saint-Venant and others. The development of modern experimental methods for measur­ ing stresses started considerably later. Wire strain gauges were first used in 1938 [3], brittle coatings in 1932 [3]. Photoelasticity and X-ray diffraction have been in use since the beginning of the twentieth century [4]. Thus, the principal methods of experi­ mental research have been developed mainly in the last 30 to 40 years. In the initial period of mechanical engineering the dimensions of individual machine components were determined from geo­ metrical conditions. Subsequently, the formulae of strength of materials for plane sections were applied. These calculations, however, did not indicate the real character of the stress distri­ bution in components of complex configuration or with complex χ INTRODUCTION application of loads. They did not allow the determination of stress concentrations and contact or other local stresses. The surfaces of most components of machines and mechanisms have complex shapes. The transition from one part of a compo­ nent to another generally involves various forms of fillets or notches (shallow or deep, external or internal, single or multiple, circular or angular). Practical experience in the operation of machines and mecha­ nisms indicates that in the majority of cases their components or members fail where the shape of the body surface shows a sharp change. For example, when the average stresses in a concentration zone are 3000-3500 kg/cm^ (42,500-50,000 Ib/in^), the maximum stresses may amount to 9000—10,000 kg/cm^ (128,000-142,000 Ib/in^), and the strength naturally is determin­ ed by the maximum stress. Therefore, in order to determine the actual conditions for sufficient strength, the service Hfe of members stressed in fatigue and the optimum shape of components from the point of view of stress distribution, the stresses in regions of stress concentration must be investigated theoretically and experimentally. The study of stress concentrations has become particularly necessary because of the increase in the operating speed of machines and the consequent increase in the dyiiamic action on their components It is frequently impossible to calculate stresses theoretically. Theoretical calculations are sometimes too inaccurate because a number of premises and assumptions have to be made. In a number of cases insuperable mathematical difficulties are en­ countered in theoretical calculations. Problems of stress deter­ mination which have no theoretical solution are encountered in many important problems in the field of mechanical and aero­ nautical engineering, fluid mechanics and so on. In these cases experimental methods of investigation play the most important role and lead to very simple or complex empirical factors. Hence, alongside theoretical investigations into the state of stress in machine components and structural members, ex- INTRODUCTION xi perimental methods are acquiring greater and greater impor­ tance. In practice, the most expedient way of studying strength problems is to use both experiment and theory and to supplement theoretical calculation by separate experimental data and coef­ ficients. In recent years, therefore, together with a considerable improvement in calculation theory, experimental methods of investigation have been greatly developed and are becoming more and more important. Experimenters have adequately improved the measuring equipment at their disposal, thus permitting the study of diverse processes under various conditions. In practice, the operation of machines and mechanisms has shown that in most cases wear and failure of components and members take place not only at sharp changes in the shape of the surface but also where components come in contact with one another. Components of machines and mechanisms some­ times fail as a result of contact stresses and local stresses, despite a considerable margin of safety with regard to the principal stresses of loading. Contact stresses are mostly less dangerous at the contact surface than at a certain depth below the surface where the greatest contact shearing stresses occur, causing pitting of the material. Such a deep-pitting phenomenon is observed in escala­ tor runners on underground railways [5, 6], in railway Unes, etc. The problem of determining stresses in the contact zone of elastic bodies, i. e. below the surface, arises whenever pressure is transmitted from one component or member to another. As has been stated, the theoretical solution of three-dimensional problems, and particularly of contact problems, presents consid­ erable difficulties and is not possible in every practical case. The stresses inside a component can be both residual and active. Residual stresses are those balanced inside the given body without the application of external forces. In addition to residual stresses, there may be preliminary or initial stresses in a component; such stresses arise when the component is mounted XU INTRODUCTION into the structure, for example stresses in tightened bolts, in prestressed reinforcements, etc. The present book deals with a method of determining active stresses occurring inside a component when it is subjected to FIG. 1. Stresses and strain at a point of a linearly stressed specimen: stresses: strains dynamic loads during operation or when it is loaded by external forces. When the external forces are removed this form of stress disappears, as do the deformations if the material is working within its elastic limit. INTRODUCTION XUl Most components and members are stressed three-dimension- ally, but with the existing methods of measurement only stres­ ses at the surface can be determined, and these do not give an overall picture of the stress distribution. The following conside­ rations show that the type of investigation to be adopted is sub­ stantially dependent upon the character of the state of stress. FIG. 2. Stresses and strains at a point of a specimen in plane stress: stresses; - - - - strains The simple relationship between strains and stresses defined by Hooke's law σ = s'E is valid only for a Hnear state of stress in one direction, i. e. the direction of the major principal strain. Figure 1 shows the picture of the stresses and strains at a point in a linearly stressed speci­ men, and it is evident that in the direction I—I perpendicular to the line of action of the forces there are no stresses but that there are strains, while in the direction II—II there are stresses XIV INTRODUCTION but no strains. Thus, even for a uni-direetional state of stress, the strain measured in an arbitrary direction fails to reflect the loading stresses. In the case of a plane state of stress, illustrated diagrammatic- ally in Fig. 2, the simple relationship between strain and stress cannot be used in any direction at all. It is well known from the theory of elasticity that Hooke's law here takes the form £χ ^ (σ·χ — f^(^y)^ 1 / X i.e. each of the strains is determined by two stresses and, corre­ spondingly, each of the stresses is determined by two strains. The relationship between strains and stresses is even more complex for the three-dimensional state of stress. Analjrfcically, the strains can be defined in terms of the stresses «y = ¿ [í^y — ((^χ + (^2)1 Correspondingly, the stresses are expressed in terms of the strains by -θ 1 — 2/x μ 1—2μ μ σ, = 20 1—2μ INTRODUCTION XV where ^ = 2äT7ö' ο = . ,+ ε, + ε,. It is seen from these expressions that the magnitudes of the principal stresses can only be obtained by determining the values of the three principal strains and that for a complete picture of the stress field at points inside a body in a three-dimensional state of stress, for example in a region of contact stresses, six strain components must be determined. Hence it is impossible to study the stresses inside a component by determining the strains at its surface.