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I" 1 "v. I ‘ '_' , ...!" "37 V.“ 5,} "l‘ #0:. j’y‘g ‘1 ‘A .'-. P vi ’ 1 ‘5 r2." ‘4. . ' . -‘.,‘ . .... «Hint. .‘ V. . _ “I lhl'f‘ A" ‘{ v.' ‘ ~ . .:. . .54 J , 4 A I d ‘ ‘4 I, . . ‘ “ ‘ I , ' 4 nnnnnn . . : 4444444 4‘ " I . - r ‘ '. ' ‘ 4 I ,4 . 4' '¢ ‘ ~‘ f" ' I , . A *4: 2-4;! ;. ‘1: ' ' v. ' y . ' k4;" :fi'. ..'. J ' ‘. I?‘ 0. .‘ fl". .4 . -.“> . ' I b I .g, ‘9' 4.1, ‘,~' 6 :-' . ' . . 4.. ‘l'iJ' ' '1 I ‘ ' | I A A ' ' " 4 - \ ' a. “ ~ ,,,,,,,,, s. V {S- r .‘ “'6“ V . . . 7' . . . I. ‘- ..‘- . ’- _ r» 2 ' A . "-33-..5214'3 it ‘ . . . "" .":.' :. W}: "'4 A : A REVIEW OF PHOTOELASTIC THEORY AND METHODS, WITH AN ACCOUNT OF THE RESULTS FROM AN APPLICATION TO SCREW THREADS by Russell G. Lloyd A THESIS Submitted to the Graduate School of Michigan State College of Agriculture and Applied Science in partial fulfilment of the requirements for the degree of Master of Science in Mechanical Engineering Department of Mechanical Engineering June, 1941 I" 'l '5 J‘4 3-3 5A.: i“ PREFACE The first discoveries and subsequent developments in photoelasticity were regarded as a branch of physics. However, with the recent improvements in photoelastic equipment and technique, it has become the ideal method for the commercial engineer in the eXperimental determi- nation of stresses. Hence, the large majority of papers dealing with photoelastic theory were written in this period by physicists who did not always appreciate the desire of the engineer for practical information in a con- cise form. Little was said regarding actual laboratory pro- cedure and technique, and treatment of the theory was largely mathematical, being unduly long and abstract. However, in the last few years there have appeared several papers written by investigators who were commer- cial engineers, and these papers have well covered actual laboratory procedures. The authors have largely confined themselves in these papers to the results of the particular investigations undertaken, assuming that the reader has studied previous accounts of the theory. It is therefore the purpose of Part I of this thesis to review, in as brief and simple a manner as possible, the fundamentals of photo- elastic theory as well as to give a unified account, in review form, of the laboratory procedure and technique as given in the current literature. As stated above, it was decided to plan this thesis in two parts. The first part is an account of the theory involved, while the second part deals with a particular in- vestigation performed by the writer. In giving an account of photoelastic theory and prac- tice, an effort was made to keep this as brief and simple as possible, yet giving a comprehensive review of the field to the present time. It is realized by the writer that this can- not be too easy or too short, but maximum.use was made of pictorial methods of presentation, thereby clarifying and reducing the amount of text material needed. Given in resume form using a large number of sources, it is believed that this method will be of great value in assisting those to whom this thesis is submitted in evaluating the second part, as well as being of help to those who may have occasion to use this paper as a reference in the future. It is assumed that the reader will be familiar with the subject matter of strength of materials, but Chapter III is included only as a review of those elements which apply to photoelastic work. It is also imperative that a clear con- ception of principal stresses be had before proceeding with the work. Chapter V includes a discussion of the new polarizing material Polaroid and its action upon ordinary white and monochromatic light. Polaroid has found wide favor in photo- elastic work and has many advantages over the formerly used Nicol prisms, hence this polarizing medium is given a com- plete discussion. An eXplanation of the polariscope and its essential elements is also given in Chapter V, as well as drawings and a discussion of the construction of the polariscope built by the writer. Ordinarily, sufficient information may be obtained from the stress pattern without separating the principal stresses p and q. Occasionally this is required, and methods for this procedure are reviewed in Chapter VII, these in- cluding the graphical integration method, the extensometer method, the interferometer method, the membrane method and others. The results of the writer's work on stresses in screw threads with undercut shanks is given in regular engineering report form in Part II. The author wishes to express his appreciation for the many helpful suggestions received from faculty members of the Engineering Department. Especially does he wish to ac- knowledge his gratitude to Mr. Seble and Mr. Pearson, lab- oratory technicians, for the valuable assistance which they have rendered in the construction of apparatus. CONTENTS CHAPTER I. INTRODUCTION 1. 2. Definition History and Advantages of Photoelasticity CHAPTER II. GENERAL METHODS OF INVESTIGATION 1. General Procedure CHAPTER III. ELEMENTARY THEORY OF STRESS l. 2. 5. 4. 5. AS APPLIED TO PHOTOELASTICITY Definitions Two Cases of Tension (a) Simple, or Pure Tension (b) Combined Tensions Equality of Shear Stresses General Case of Plane Stress Summary CHAPTER IV. GENERAL OPTICAL THEORY 1. 2. 5. 4. Theories of Light Nature of Light Double Refraction and Interference Appearance of Pattern with White Light and with Monochromatic Light CHAPTER V. THE POLARISCOPE l. 2. 5. 4. 5. Theory and Elements of the Polariscope Plane and Circular Polariscopes Light Sources Polarizing Methods Construction of Polariscope used for this Investigation Page 1 12 12 17 19 22 25 26 55 57 59 4O 4O 42 44 CONTENTS CHAPTER VI. THE MODEL; MATERIALS, PREPARATION, 1. 2. 5. 4. 5. LOADING. Materials Methods of Preparation Annealing The Loading Frame Constructed at M.S.C. Methods of Loading CHAPTER VII. PHOTOELASTIC DETERMINATION 1. 2. 5. 4. 5. 6. 7. 8. 9. OF STRESSES General Interpretation of Fringe Value and Stress Pattern Points of Concentrated Stress Calibration and Adjustment of the Polariscope Calibration by Bending Isochromatics Boundary Stresses Isoclinics; Principal Stress Directions Shear Lines 10. Principal Stresses 11. Stress Concentration Factors 12. Recent Developments 15. Stress Pattern Photographs 14. Bibliography Page 49 50 55 55 54 56' 57 59 59 62 65 64 65 68 68 75 74 74 ii PART I PHOTOELASTIC THEORY AND PRINCIPLES CHAPTER I INTRODUCTION 1. Definition. Photoelasticity is an eXperimental Optical method of finding stresses and stress distri- butions in machine parts and structural members, using polarized light and tranSparent models made of plastics, such as Bakelite. Use is made of the effect of a stressed transparent model on the passage of polarized light through the model. Double refraction and interference phenomena combine to produce a stress pattern which is thrown upon a screen or recorded by means of a camera, from.which the various stresses may be evaluated. The Optical equipment used in this method is termed a polari- scope, which consists of a light source, polarizing equip- ment, a suitable lens system, a means of supporting and loading the model, and a screen or camera for viewing or recording the image produced. The application of the photoelastic method in engi- neering work consists of finding the stresses present in the model, using the information given by the stress pat- tern. The model results may then be easily transferred to the prototype if there exists a difference in scale of size or loading. 2. History and Advantages of Photoelasticity. In 1816, Sir David Brewster discovered that glass became doubly refracting in polarized light when placed under load. All photoelastic stress investigations are based on this fundamental phenomena of double refraction of trans- parent materials when stressed. This property of double refraction not only accounts for what is happening within the stressed model, but is the basis for the explanation of how polarized light is produced. The first part of Sir David's paper, read before the Royal Society on Feb. 19, 1861, may be summarized in the following statement: When light passes through a plate of glass which is stressed transversely to the direction of propagation, the axes of polarization in the glass are along, and perpendicular to, the direction of stress. Thus we see that stressed glass becomes double refracting and splits the entering polarized ray into two components, the speed of each component being influenced differently while traversing the glass due to the stressed condition at that point. In this way the relative retardation of each ray was seen to have a definite relationship with the stress. This phenomena is basic in photoelastic work, and will be discussed at greater length in a later chapter. Other 19th century physicists contributed greatly to the fundamental ideas of photoelasticity, among these being Neumann, Maxwell, and many others. Among the first to make an engineering application of the method was Mesnager, a French engineer, who construc- ted a glass model of a proposed bridge over the Rhone river in order to check results of the designers. Coker and Filon did excellent work in applying the method to engineering problems using celluloid, which was a great improvement over glass because of its greater optical sensitivity and ease of cutting to shape. However, it has been only in the last five to ten years that the greatest advances have been made in the sub- ject, these recent improvements being largely in procedure, equipment and materials. Among these are the use of a mono- chromatic light source and the use of Bakelite as a material. These recent refinements have aided greatly in making the photoelastic method a practical one for the engineer. In designing a structural member or machine part, one may find stresses and stress distributions in two ways. The first method is to compute mathematically the stresses at various points or sections, using the conventional for- mulas of stress analysis. These formulas, however, are based on a number of assumptions, these holding only in special cases. Some of these assumptions are: that longitudinal stresses follow a linear law in bending; that axial loads produce a uniform stress distribution; that horizontal and vertical shear stresses follow a parabolic law of distri- bution in beams in bending. This latter assumption has been shown to be greatly in error in most cases by M. M. Frocht.l From his investigations it was found that the maximum ver- tical shears are much higher than the maximum computed by the parabolic formula, and furthermore this maximum does not occur at the neutral axis but close to the point of load 1"The Place of Photoelasticity in Engineering Instruction", M.M.Frocht. Carnegie Institute of Technology. 1957. application. The fact that these formulas generally yield in- <:c>rrect results in the case of the second assumption may toea illustrated by a simple prismatic bar of rectangular czlross section subjected to pure tension as shown in the figure below. The stress will closely approximate that 1% - m“ Figure l Egiven by the formula 3 - 2. However, if a small hole be Iaut in the bar as shown in the following figure, the stress A y a 141 A Figure 2 :found from the formula s a E', where A' is the area of the (:ross section at A-A after the hole has been placed in the lDar, the stress will be approximately the same as in the IDrevious case. But experiments have shown that the stress (Dbtained by formula in the second instance is greatly in Earror, since the effect of a small circular hole is known 130 increase the stress to about thggg_times this value. 3Enstead of being uniform as assumed, the actual stress dis- 13ribution contains definite stress concentrations, these fiitresses increasing greatly at the edges of the hole as Shown in the figure on the following page. By formula Actual Figure 5 Thus it may be seen that the elementary formulas may be used only with caution in simple cases, and that they are not applicable in the case of members of com- plicated or irregular shape containing holes, notches, grooves, reentrant corners etc. Results obtained from the photoe;astic method have had their accuracy well established by precise agreement with results from the theory of elasticity for the more simple cases. Photoelastic results are not only exact and reliable, but the scope of the method goes far beyond available mathematical solutions. It includes statically indeterminate cases, all problems having equally simple ‘photoelastic determinations. The photoelastic method pre- eludes the need of questionable assumptions as in the case of theoretical solutions, which are generally long and difficult. In those cases where analytical methods fail to give information on stress distributions, recourse may be had to the experimental method, using models of similar shape and subjected to the same type of loading as the prototype. The most valuable of these eXperimental methods is that of photoelasticity. This has been of inestimable value to de- signers in supplying data on stress concentrations, not only making it possible to increase strength by adding material at these points but also to conserve material and weight at points of low stress. The commercial ap- plications of photoelastic work in design are in- numerable. In recent years it has also proven an effective means in the educational field of bringing to the stu- dent a way to actually see stress distributions in various structural members and machine shapes. It has also served as a means of checking the accuracy of test Specimens used in determining the strength of various materials. For example, in the standard cement testing briquet, the usual tensile strength given for the section of minimum area is around 40% less than the actual stress existing over the outside part of this section where failure begins. Most photoelastic investigations have been perform- ed using two-dimensional stress problems. Although the two-dimensional case covers a wide variety of problems met in practice, it remained until recently to develop a satis- factory method of applying photoelasticity to three-dimens- ional stress problems. However, there have been developed two methods suitable for three-dimensional work. It is interesting to note that at the present time, photoelastic- ity is the only experimental method of finding the true stress conditions within a solid body. The first of these is the fixation methodl, wherein a three-dimensional model 1 M. Hetenyi, "Fundamentals of Three-dimensional Photo- elasticity , Jour. App. Mech., 5, A149 (1938). is loaded in the manner of its prototype and heated while under load. Upon cooling, "slices" are cut from the model along suitable cross sections and examined in the polari- sCOpe. It has been shown that the patterns are correctly preserved by this "freezing“ method, and that sawing of the slices does not disturb the pattern. The second method is the scattering methodl. This is a very recent method, and the reader is referred to the Bibliography for refer- ences to these methods. In this thesis, two-dimensional photoelasticity only will be discussed. 1 Fried & Weller, "Photoelastic Analysis of Two- and Three-dimensional Stress Systems", Eng. Exp. Station Bulletin No. 106, Ohio State University. CHAPTER II GENERAL METHODS OF INVESTIGATION 1. General procedure. So that the reader may more easily follow the treatment of each individual topic, a brief indication of the stress determinations made from the photoelastic pattern will be given here. The first step is to make a scale model out of Bake- lite or other suitable transparent material, and record the pattern produced by the stressed model when placed in the polariscope. By methods to be described in Chapter VII, the following more important determinations may be made: (a) Shear stresses. A stressed model projects an image on the screen, this image consisting of a number of brightly colored bands. (If a monochromatic source of light is used instead of white light, these colored bands appear as plain black and white fringes.) These colored bands, called iso- chromatics, are actual representations of the maximum shear distribution in the model. Thus by a glance at the image on the screen, the lines of maximum shear stress may be clearly seen. In a straight beam loaded by couples at the extremities, these shear lines will be straight, parallel, and horizontal. By a simple calibration experiment the actual value in p.s.i. of these stresses may be found at any point in the model. (b) Localized points of high stress. The spacing or degree of concentration of -9- 10 lines in any region is indicative of the amount of stress present at that point. That is, in a region where the lines are far apart indicates the presence of little stress. A region having a large number of closely packed lines will be one of high stress. This can be used in an analytical manner, as will be shown later. (c) Principal stresses. Superimposed upon the isochromatic, or shear line network is a second set of blggk lines, called i39- clinics. These lines have the property of shifting their position while the model is under a.£22§£§2£.l£§§ when the polarizer and analyzer are rotated together, the axis of polarization of one being kept at 900 to the axis of the other. The eXplanation of this will be given in a later chapter. By sketching the positions of these isoclinics for various angular positions of the polarizer and analyzer, a network of lines is obtained from which the directions of the principal stresses (tension and compression) may be com- pletely determined. By subsequent methods, these principal stresses may be evaluated for any 23313 3.13.1 the m. This procedure is more involved than those previously described, although this determination is not necessary for many problems. ‘(d) Boundary stresses. The fundamental photoelastic equation is p - q - 2nF where p and q are the principal stresses, n is the fringe 11 order, and F is a constant which may be determined by cali- bration. Since the boundary is free from shear, the stresses acting upon it are either purely tensile or compressive. Hence in the formula given, one principal stress reduces to zero, making it extremely easy to plot the boundary stresses. When q is zero, the equation reduces to p = 2nF. Methods of determining F and n will be given later. The results obtainable from a photoelastic investi- gation have been indicated in only a very brief manner here. The methods used to obtain these results, and their practi- cal application will receive full attention in Chapter VII. CHAPTER III ELEMENTARY THEORY OF STRESS AS APPLIED TO PHOTOELASTICITY 1. Definitions. Before proceeding, it will be well to define the symbols ordinarily used in discussions of elastic theory: d........Angle from the X-axis to the direc- tion of P measured counterclockwise. P,Q......Principal stresses. (P-Q)....Principa1 stress difference. SX,S ....Normal components of stress in X and y Y directions respectively. vxy,vyx..Shearing components of stress. Stresses normal to a plane will be denoted by S, the subscript showing the direction in which the stress is acting; e.g., Sx would be a normal stress acting in the X-direction. Stresses acting normal to an inclined plane will be denoted by Sn° Stresses acting parallel to a plane (shear stresses) will be denoted by v, the first letter in the double sub- script showing which plane the shearing stress is acting upon, and the second showing its direction (parallel to the X or Y axis.) 2. Two cases of tension. (a) Simple, or pure tension. To obtain a clear understanding of how the various stresses on an element change in magnitude when -12- CHAPTER III ELEMENTARY THEORY OF STRESS AS APPLIED TO PHOTOELASTICITY 1. Definitions. Before proceeding, it will be well to define the symbols ordinarily used in discussions of elastic theory: d........Angle from the X-axis to the direc- tion of P measured counterclockwise. P,Q......Principal stresses. (P-Q)....Principa1 stress difference. Sx,Sy....Normal components of stress in X and Y directions respectively. vxy,vyx..Shearing components of stress. Stresses normal to a plane will be denoted by S, the subscript showing the direction in which the stress is acting; e.g., Sx would be a normal stress acting in the X-direction. Stresses acting normal to an inclined plane will be denoted by Sn. Stresses acting parallel to a plane (shear stresses) will be denoted by v, the first letter in the double sub- script showing which plane the shearing stress is acting upon, and the second showing its direction (parallel to the X or Y axis.) 2. Two cases of tension. (a) Simple, or pure tension. To obtain a clear understanding of how the various stresses on an element change in magnitude when -12- 15 the element is rotated, a case of simple tension in a prismatic bar may be considered, Figure 4. The bar is loaded axially in the one direction only, so that it is subjected to pure tension. It will be evedent that the normal stress on a perpendicular section abcd (Area A), is S = E. Now examine a section bcef (Area A') which is inclined at an angle d to the normal section. Taking half the bar as a free body in Figure 5, it will be Figure 5 seen that the stress Sx may now be resolved into two com- ponents. One, Sn, is normal to the particular inclined sec- tiOn that is being considered, the other, vs, is a shear- ing stress parallel to this plane. Hence, on planes other than a perpendicular section, even in pure tension, there appears a shear stress on those planes taken at various angles. It may next be asked when this shearing stress is a maximum, and at what angle. First, an expression may be written for the stress Sx over the section ab as it is rotated to any angle d: 14 P Sx = ------------------------ (For g = 0°) Area, section abcd = A P ' ....... I E 003 g (For any 5) A cos d A series of sketches is shown for variations of SX with the angle a in Figure 6. “_“—‘—‘E§—__"—" -———u ———~—>- .u——_- __. 5;, I I I _.‘_‘—- _+__..._- -_+_:';r:2_-:R;—.HF> _. Figure 6 It will be remembered that this shows the stress Sx only, it being the stress over the inclined section in the direction of load. The components of Sx may now be considered, these being the normal stress Sn and the shear stress vs. A study of the shear stress is important for there are materials which are much weaker in shear than in tension. Writing the expressions for the normal and tangential (shear) stresses: Normal: P Sn 3 -E-Eg§_é- I - 0032 d - Sx cos2 d..(2-l) 15 Shear: P vS - ---A--— a 8x cos s sin d 653'; a _§§-EEE-Eé_ oeoeooooooeeeoeeoeoee(2-2) From this it is evident that Max. value of S - Sx when d I 0° Min. value of s - 0 when d - 90° Max. value of v = 23 when d - 45° Min. value of v = 0 when d = 0° or 90° Letting d1 = 90°—-¢ and referring to the formula for shear stress, it will be evident that values of v for d - #1 and d = 90° - #1 are equal and that the values .22 z for d = d1 and d = 90° + H1 are gqual_§u§dgpposite in .3359. Also from equation (2-1), 31 3 Sx cos2 d, (See Fig. 7 below) and 32 = Sx cos2 (90° + 51) = Sx sing d from which 31 + 82 - Sx(cosg d1 + sin2 #1) - sx. . f N D / J + — a - - .- rig—— 9o°+¢' ‘ Figure'7 16 From the preceding statements two important conclusions may be stated: 1. The sum of the normal stresses acting on two perpendicular sides of an element within a bar under pure tension is ggnstgnt and is equal to Sx max. 2. The tangential (shearing) stresses on two per- pendicular sides of an element within a bar under pure tension are of equal magnitude. A graphic representation of how the two stresses Sn and vs vary as the plane is turned through various angles in shown in Figure 8. In this example the area was taken as 1 sq. in. and the load as 2000 lbs. This appears on the following page. Shear stress v3\ Normal stress Sn 5 4L32. \_\\//__ -t - __ I Plane Angle Sn, p.s.i. vs, p.s.i. 0 0° 2000 0 1 20° 1766 645 2 400 1174 985 3 45° 1000 1000 4 50° 826 985 5 700 234 645 6 90° 0 0 Figure 8 17 2. Two cases of tension (cont). (b) Combined tensions. Consider now the stress condition in a thin prismatic bar subjected to two tensions at right angles to each other and acting along the X and Y axes, Figure 9. Figure 9 The conditions of static equilibrium may be applied to the above sketch. It must be remembered that the vectors shown represent unit stresses and not total forces. Hence these unit stresses must be multiplied by the areas over which they act to obtain the total forces used in the equa- tions of equilibrium. Let A' be the area of face bf; then the area of face ab is A' cos d and the area of face af is A' sin 5. Normal stress: Considering all forces acting on face bf as projected in a direction normal to bf, the equation of equilibrium may be written for Sn: 18 SnA' = (SXA' cos d)(cos d) + (SVA' sin d)( sin 5) The areas A' cancel, leaving: Sn I Sx c032 d - Sy sin2 5 . ------- b- " ----:--""’ COS 2¢oeeoeooooeee(2-5) 2 Shear stress: In a similar way an equation of equilibrium may be written for the forces acting parallel to the plane bf: vsA' - (SXA' cos ¢)(sin d) - (SyA' sin d)(cos 5) Again the areas A' cancel, leaving: v3 - (Sx - Sy) sin 0 cos d . -§z-:~§y- Sin 2d...ooeeoooooeoeeoeooooeo(2-4) 2 Maximum and minimum stresses. As in the case of pure tension, it is evident upon examining the variation of the stresses on a section as it is rotated through 900 that the normal stresses act- ing upon the section reach a maximum or minimum value when d ' 0° or 900. Under these circumstances the shear stress disappears. These particular directions are Vagin- cipal Stress Directions", and the normal stresses in this case become the "Principal Stresses“. In this particular case of two axial tensions it should be noted that the directions of principal stress happen to coincide with the directions of the external forces applied along the axes of the bar. This is not the case for more complicated shapes and systems of loading. The case just discussed helps to introduce the reader to 19 the idea Of principal stresses; the_general case will be discussed in Article 4. The general case considers the effect Of shear stresses applied. 5. Equality Of shear stresses. Before studying the general case of plane stress, the relation between shearing stresses on planes at right angles may be briefly shown. This relation has been stated previously in Article 2a, but is usually proved as given here. Let Figure 10 represent a small prismatical element of unit thickness cut out of a Qfi‘ug 5 '-~ «e stressed member. If a shearing ‘51 ~:, , ‘IABLvs stress vS acts on the right hand ‘ 1 tdfi ffft face, the shearing force on this T“ 5 face will be vsdy. There must be Figure 10 a shearing force equal but oppo- site in direction acting on the left hand face in order that the sum Of the vertical forces be zero to satisfy the conditions of equilibrium. However, these two forces form a couple, and so to prevent rotation, there must be a second couple made up Of shearing forces vgdx which act on the top and bottom faces. These two couples must be numeri- callyequal and must act in Opposite directions. The first couple may be seen to have a moment arm dx, and the second couple a moment arm of dy. Then (vsdy)dx = (védx)dy from which VS 8 Véoeoeeeoeeoeeeeeeoo(2"5) 20 Thus it may be seen that unit shearing stresses acting on planes at right angles to each other are numeri- cally equal. Hence in the following discussion v . v xy yr and either may be written in any case. 4. General case of plane stress. The general case of an element subjected to plane stress may now be considered, such as when an irregularly shaped member is under any type of external loading or combination of loads. Referring to Figure 11, a small elementary prism may be imagined to be taken from a stressed member and isolated 1 .v t' ' ' "A a 3“ ‘( -'7":’4'5T“.’a§"._ ii (b) as a free body in Figure llb. Representing the face ab as area A, then the area of face ac is A sin d and the area of face cb is A cos d. Using the method of Article 2b of applying the con- 21 ditions of static equilibrium to the prism: SnA = SXA cos d cos d + SyA sin 5 sin e - vxyA cos ¢ sin d - vxyA sin 0 cos d The areas cancel, and this reduces to: Sn - Sx cos2 d + Sy sin2 e - 2va sin 0 cos d 3.3 SJ-S = --§-é--E- - --}-§--X- cos 2d - vxy sin 2d..(2-6) In a similar manner expressions for v8 may be obtained. These are: vS - (Sx - Sy) sin e cos d + vxy(cosg d - sing a) - -§§-é-§z- sin Zd 4 ny cos 2d..............(2-7) The two equations developed above, (2-6 and 2-7), give the normal and shear stresses at any point and for any direction for the general case of plane stress. In photoelastic work the principal stresses, maximum shear stresses, and the directions of these planes are of prime importance. These will now be discussed for the general case. Taking the equation of normal stress (2-6) and dif- ferentiating it with respect to d, one obtains: tan 2¢ fl ' ”-gyzy'-- 0.000000000000000(2-8) sx-sv Differentiating the equation for vs, gives: ‘s-s tan 2% = --§----y‘ oeeoeeoo...eoeooooe(2-9) vay Considering equation (2-8), it is evident that there are two values of 2d between 0° and 560°, these differing by 22 180° because they have the same value for the tangent. From this we infer a statement important in photoelastic work: Since the values of d differ by 90°, the planes on which principal stresses occur are 900 apart, or, more simply, the principal stresses are at right angles to each other. Considering equation (2-9), a similar reasoning shows that the planes on which the maximum shearing stress- es occur are at right angles to each other. It is now desired to find the planes on which the shear stress is zero. To do this, equation (2-7) may be rewritten after setting vs = 0. Solving for tan 2% gives: 2v tan 2% = e —----3Y-- Since by this procedure an expression is obtained which is the same as equation (2-8), a further important conclusion may be made: The shear stress is zero on planes of maximum and minimum normal stress (principal stresses). If it be noted that the right hand member of equa- tion (2-8) is the negative reciprocal of the right hand member of equation (2-9), and that since the values of 2d differ by 90° then the values of d differ by 45°, a last conclusion may be drawn that will be of interest photo- elastically: The planes of maximum shear stress are at 45° to the planes Of principal stress. 5. Summary. It has been shown that the normal and shear stresses on a section of a stressed member vary with the 25 rotation of the section. The maxima and minima of these stresses were investigated mathematically as well as the directions of the planes on which they occurred. Quantitative statements of the amounts of the stress- es were given by the numbered equations of the preceding article. Directions of the planes and relations between planes of principal stress and planes of maximum shear will be given again in this summary. The following conclusions develOped in this chapter are important in evaluating stress patterns and under- standing photoelastic theory. They are given here in sum- mary form: 1. When the normal stresses reach a maximum or a mini- mum they become the principal stresses, and the shear stress than is zero. 2. The principal stresses are on planes which are at right angles to each other. 5. The planes on which the maximum shearing stresses occur are at right angles to each other. 4. The shear stress is zero on planes of maximum and minimum normal stress (principal stresses). 5. The planes of maximum shear stress are at 45° to the planes of principal stress. 6. The maximum shear stress is given by the formula v I ------- ...............(2-10) where P and Q are principal stresses. Hereafter in the text, the two maximum and minimum 24 normal stresses (principal stresses) denoted formerly by maximum or minimum Sx and Sy, will be denoted by P and Q. The formula (2-10) is basic in interpreting stress patterns, since the isochromatic lines of a pattern are interpreted as loci of points of maximum shear stress, or or loci of points of equal (P - 0) stress. The formula for maximum shear (2-10) is a Special case of the plain shear formula, this being P - Q v I ------- sin 2 s 2 d The fraction is a maximum when sin 2% - 1. This occurs when ad is 900, 5 then being 45°. CHAPTER IV GENERAL OPTICAL THEORY l. Theories of light. The nature of light is a difficult subject, and nO theory exists yet which is completely satisfactory. A brief statement Of the theories which have been proposed are listed here. (a). Corpuscular theory of Newton. About 1866 New- ton prOposed that light consisted of a stream of very small particles or "corpuscles", radiating from a source in straight lines. This explained reflection, but could not explain later phenomena and was rejected. Physicists recently returned to this theory but only for certain un- usual phenomena in conjunction with modern theory. (b). Wave theory Of Huygens. Huygens suggested, about 1678, that light consisted of a longitudinal motion, such as that of sound. This explained reflection, refrac- tion, and double refraction. (0). Modern wave theory Of Fresnel. This theory stated that light was of a wave form, but the waves were of a trans- verse nature such as the wave motion Of a vibrating wire. The motion was considered to be oscillatory in a direction perpendicular to the direction of propagation. (d). Electro-magnetic theory of Maxwell. This retains the transverse wave motion theory but in addition states that there are electrical oscillations which give rise to mag- netic fields, each being perpendicular to the other and to -25- 26 the path Of the ray. The exact nature of light is still an unsettled one, and one theory is used to explain certain phenomena, while another theory is used to eXplain other phenomena. As yet, no one theory is entirely adequate. However, it is defi- nitely establiShed that for ordinary phenomena such as po- larization and double refraction which are dealt with in photoelasticity, the light may be regarded simply as a trans- verse wave motion. Hence in the following discussion this will be the theory assumed as basis for the explanations. This is theory (c) given above. 2. Nature of light. (a). White light. Ordinary white light is composed (of the different monochromatic colors. Blended together, their combined effect is the appearance of white light. When white light is passed through a prism, it is bent by refraction and separated into its colored components, thus giving the spectrum of colors. White light may be thought of as being composed Of vibrations lying in planes passed at £11 angles through a center line whose direction is in the direction of the ray. Each vibration, or wave, is a sine curve, and may be repre- sented as shown in Figure 12 by a rotating vector. The length of the vector is equal to the amplitude,and the po- sition of a point (at the end of the vector) from the X-sX1s at any time is plotted against time. This figure also gives the notation which will be used. It should be remembered 27 aeo DISPLACEMENT IEF\\5 I I‘\ . ' n—AMF’LITUDEI 17o ‘ ’ 90 O 90, ‘ I80 2.70 ‘ sec .1. \ , , PHASE. AAIGLLE I80 ' I ~<———————JM#NJE.LEmm311fi ————- Figure 12 ' that white light is composed of several different colors, and hence will have vibrations of several different wave- lengths. Figure 15a shows the nature of ordinary light for Figure 15 one wave length only, while Figure 15b gives a vector re- presentation Of the vibrations in the different planes at maximum amplitude. Figure 14 shows, on one plane only, the Figure 14 different wave-lengths which would exist in white light. Hence it is evident that ordinary white light con- 28 sists Of random, chaotic vibrations which are not ordered *with respect to any one direction or plane. Polarized light, it will be shown, has definite directional prOperties. The intensity or brightness of light is proportional to the square of the amplitude. Color is a function of the frequency. This does not change when the light is reflected, refracted, or transmitted through various media. Although the frequency does not change, the velocity varies when light is transmitted and depends on the material. The ve- locity is equal to the product of the wave-length and the frequency; hence the wave-length also changes. The velocity is given by the familiar equation: V a n}....................(5-1) In this discussion the velocity of light through air, then Bakelite, then emergence into air again, will be of greatest interest. It will be evident that the light will regain its original speed upon emergence from.the Bakelite to the air again. In this case, no effect would be apparent but as will be seen later, the Bakelite will have a differ- ent effect on two certain ray components, which gives a per- manent retardatiOn of one behind the other upon emergence. (b). Monochromatic light. Monochromatic light is light of a single color and hence of acertain frequency. In other words, it is simply a Special case of white light. Noting this limitation, the discussion of the preceding article applies to monochromatic light as a special case of white 29 (c). Polarized light. There are several methods of polarizing light, one of which is by means of discs of Polaroid. These Various methods will be discussed in Chap- ter V, but Polaroid is mentioned here since it serves ad- mirably as a means of visualizing how polarized light is produced. Polarized light may be Obtained then, by passing ordinary light through a disc of Polaroid. This material has Optical prOperties that are directional, which may be described by saying that the Polaroid has an optical axis. In the direction Of this axis, the Polaroid may be thought Of as having a number of very small slots which stop all of the vibrations except those in a single plane. Thus, the light emerging from the Polaroid will be polarized, the vi- brations all being in a single plane. The total effect over the surface of the Polaroid will be to transmit po- larized light, the vibrations of which will be in parallel planes. This polarized condition of the light may be detected by the use of a second disc of Polaroid. This too may be thought of as having a large number of tiny slots oriented in the same direction. Thus, if the axis of the second Po- laroid is set parallel to that of the first, the polarized light will be transmitted, while if the axis of the second disc is set at 90° to that of the first, the vibrations will be stopped and no light will be transmitted. The first of a pair of Polaroid discs used in the 5O manner described above is termed the polarizer, while the second disc is termed the analyzer, since it detects or analyses the polarized condition of the light. A pair of Polaroids used in this manner comprises the fundamental parts of a polariscope. The beam of parallel, polarized light between the polarizer and analyzer is termed the Ilgld Of the polarisc0pe, and it is in this field that the stressed transparent model is placed. The action of Polaroids upon light is shown in Fig- ure 15, which appears on the following page. (d). Circularly polarized light. For reasons to be discussed later, it is sometimes desirable tO examine the model in circularly polarized light. This condition is ob- tained by the use Of quarter-wave plates which are made of mica or quartz. These materials may be cut so that they po- ssess a principal crystalline axis called the Optical axis, and will have different Optical properties in two perpen- dicular directions. Let it be assumed that a disc of mica or quartz be placed in the path of a ray of polarized light so that the Optical axis of the crystal is at 45° to the plane of po- larized light. The polarized light, it will be remembered, is vibrating in one direction only, and may be represented by a single vector lying in the plane Of polarization. When the polarized ray strikes the mica disc, the single com- ponent splits into 339 components which then proceed through the mica at 90° to each other and at 45° to the original plane Of polarization. Due to the directional properties of 51 the mica, one ray will travel faster than the other. This is due to the fact that the molecules are closer together in one direction; thus the vibrating electrons have more difficulty in traversing the mice in this direction than do the electrons penetrating the mica at 90°, on which plane the molecules are farther apart. Thus one component Of the original wave vector travels through the mica at a greater speed than does the other component. The mica is set at 45° to the plane Of polarization of the incident light in order that the two components formed will be equal. Now if the thickness of the mica be such that the two components reach the far side when they are i of a wave out of phase, they will retain this retardation upon entry into the air again where the speed of each component will resume a constant velocity. In other words, upon leaving the mica they will have a permanent relative retardation of i of a wave length. A crystal whose thickness is pre- pared to produce this effect is termed a i-wave plate. Figure 16, on the following page, shows pictorially the effect of a %—wave plate on polarized light. After leaving the %-wave plate, the two components will be vibrating at right angles to each other and } wave out Of phase. Each component will have a simple harmonic vibration in its plane, and will be in the form of a sine curve. If these two components be combined at each instant to form a resultant vector, it will be apparent that the terminus of this resultant vector will trace out a helix. The projection of this motion on a plane perpendicular to 52 the ray would be a circle. Hence the name circularly polar- ized light. Circularly polarized light may be converted back to plane polarized light by means of a second %-wave plate with its principal Optic axis at 90° to that of the first. This second i-wave plate will retard the ray component ad- vanced by the first and restore the light to the plane po- larized condition. The use of circularly polarized light is to remove the isoclinic lines from the pattern. This will be discussed in a later chapter. (e). Vector representation and summary. A vector re- presentation of each type of light will help to fix them in mind. They are given in Figure 17, below. I (a) Ordinary (b) Plane (c) Circularly Light Polarized Polarized Figure 17 White light is composed of random vibrations in all / \- planes, and is composed of the different colors having different wave lengths. Plane-polarized white light consists of vibrations in one direction only, but still contains vibrations of diff- erent wave-lengths. Ordinary monochromatic light consists of vibrations in all planes, but has only one wave length. Plane-polarized monochromatic light has vibrations in one direction only, and has but a single wave length. 5. Double refraction and interference. When a transparent isotropic material such as Bakelite is stressed and placed in the path of plggg p9- _1arized light, it behaves as a temporary doubly-refract- ing crystal much as did the %-wave plate discussed in the preceding article. However, the difference is that the 4- ‘wave plate possessed directional properties which were parallel and perpendicular at all points of the material. In the case of the Bakelite model, these directional pro- perties vary in direction and amount at different points throughout the model. The distribution, or amount and di- rection of these properties, are dependent at any parti- cular point upon the amount and direction of the internal stresses at that point. Let a particular point be examined in a stressed transparent model, as in Figure 18 below. As the plane- polarized ray Original plane Of polarization Principal stress directions nent splits into two components at right angles to each other, these two components taking the as at that point. The principal Figure 18 enters the model, the single compo- directions of the principal stress- stresses, it will be remembered, are always at right angles to each other. The model may be thought of as having two tiny slots at the point in question, 54 each slot being along a principal stress direction. The two ray components entering these slots travel at differ- ent velocities just as they did in the case of the %-wave plate, and for the same reason. However, in this case the orientation and length of these slots is not the same for all points in the model, but will vary according to the (lirections and amounts of the principal stresses at each point. As stated previously, the action of the stressed Bakelite on plane polarized light is very similar to that Of the %-wave plate. In the case of the i—wave plate, its axis is always set at 45° to the plane of polarization to give equal length to each vector produced. In other words, the fi-wave plate is simply a Special condition Of what is now being considered. The length of each imaginary slot in the stressed model is prOportional to the stress in that direction, and it is only in special cases where the P and Q stresses are equal (or the difference between them is zero), that each slot will be equal. This is a special case which will be discussed later. Figure 19 on the following page shows the action of a stressed model on polarized light when placed in the field of a polariscOpe. The plane polarized light is in a vertical plane. Upon reaching the stressed model (consider- ing a single point as shown on the drawing) the single vertical vector Splits into two components at right angles to each other. These two components take the directions of the principal stresses at that particular point, and 55 start through the model. Each component will travel with a different velocity while moving through the loaded model. Thus they will be out of phase with each other when they leave the model. The difference in speed existing while the rays are within the model is caused by a stretching of the molecules in the direction of tension, leaving more Space for the electrons vibrating in the direction of a tensile stress. Similarly compression (at right angles tO tension) will crowd the molecules closer together, and the speed of the two components in the direction Of the ten- sile and compressive stresses will Obviously be differ- ent. In other words, the change in velocity of each com- ponent will be proportional to the stress in that direct- ion. Upon leaving the model, the two component.rays are vibrating at right angles to each other and are Out of phase with each other an amount which is prOportional to the difference between the principal stresses. Since the two components regain a constant velocity upon reentering the air, this difference in phase, or retardation, will be maintained. All that remains is to resolve the two rays into two components in the same plane so that interference will be obtained. This is done by passing the two rays through a second Polaroid disc, the analyzer. Figure 19a, on the following page, gives a vector representation of plane-polarized light. Figure 19b indi— cates how the single vertical vector is split into two vectors at right angles and in the direction of the prin- 56 cipal stresses. Figure 19c shows how the two rays are re- i . A "X \ / U .(a) - (b) Figure 19 solved into cOplanar horizontal components in order to secure interference. A succession Of these rays coming from related points in the model whose principal stress differences are the same will form a band, or fringe, on the screen. This band will be a locus of points whose principal stress differences are the same. A word may be said here concerning the thickness of the model. If the model is of constant thickness through- out, all rays will have traversed the same distance in going through the model, and hence all relative retar- dations will be proportional. Hence the thickness of the model does not matter. However, the thickness should be small enough compared with the dimensions of the model so that conditions of 21223 stress Obtain. A good thick- ness for models of average size is usually taken as i". Considering the total effect of several points along a section of the model having various stress-differences, it will be seen that with monochromatic light a series of dark and light bands will appear on the screen. This is the result of interference of the light waves which either reinforce or cancel each other, thus producing bands of 57 varying intensity on the screen. It has now been shown that the difference in the principal stresses produced a retardation which was pro- portional to this difference, and that the analyzer re- duced these out-Of-phase vibrations to components which were cOplanar. It has also been shown that by interfer- ence of the final vibrations, bands are produced on the screen due to varying intensities. These bands form the image on the screen which will give the desired infor- mation pertaining to the stresses producing them. Inter- pretation and evaluation of these bands is discussed in Chapter VII. 4. Appearance of pattern with white light and with monochromatic light. The eXplanation of the preceding articles was based upon monochromatic light as a source. This forms the more simple case for discussion, since it is a special case of white light. Monochromatic light. When a monochromatic source of light is used, the image thrown upon the screen will con- sist of black and white fringes. White light. When a white source of light is used, the image thrown upon the screen will consist of colored bands, these bands being termed isochromatics, meaning lines of equal color. In the colored image, the pattern (distribution, shape and position of the bands) is the same as the monochromatic pattern. However, in place of 58 each black fringe, there will be a complete spectrum of colors. This Spectrum of colors from yellow through orange, red, violet, green etc. and back to yellow, is repeated once on the white light source pattern for every corresponding black fringe appearing on the monochromatic pattern. Thus the two are actually the same, but each has different advantages in evaluating stresses. These will be discussed later. Monochromatic light is a special case of white light, and interference gives only the black and white fringes. White light, however, is composed of the spectrum Of colors and forms a more complex case to eXplain. The discussion of the preceding article based on monochromatic light applies the same in this case, the process simply being repeated for each color at a point. CHAPTER V THE POLARISCOPE 1. Theory and elements of the polariscope. The essential elements of a polariscope would in- clluie a light source, a suitable lens system, a means for supporting and loading the model, and a screen or camera for*viewing or recording the image produced. Two sketches of polariscOpes are given on the following pages, the first with.a white light source and a screen, the second with a monochromatic light source and a camera. The lens system shown is designed to give a beam Of ppgrallel rays for passage through the polarizer, model, and analyzer. If the light is a point source, two collimating lenses may be used. The first will give a parallel beam of rays for passage through the model, while the second will converge these parallel rays. The projection lens is used for focusing the image on the screen. The various lenses and other parts are usually mounted in suitable holders, these having graduations in 5 or 10 degree increments for the Polaroids and i-wave plates. These holders are suppor- ted by rods mounted on some type of saddle, these in turn being supported on an optical bench of the proper rigidity. The relative positions of these elements may now be discussed. Since the use of Polaroid has become such a popu- lar means of Obtaining polarized light, it will be assumed that the polariscOpe being discussed is equipped with it. The first collimating lens after the light source will -39.. \(NNWUO u I WUW\Q Q\0W.¢7VOO. I Q mks.» e>\\o\\a - O mzws 02R {23.400 - O NNQONK QMwoMmRm. - k $2.“: 02R 0.9300 . Q musk. 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N x \\ Xx .. \Iu «.4 . k. ... 5...... ... . f \n .\ VI. ll‘ t—-i.__.__ ”—1 - . -7 ' -_,___.._. —— \J l 7 ... 3.4...» x- x... ..., . 'II. I‘ ‘l ah. .I.‘ p\ \ \\ lo 1 . _ _ 1 / . m. m are a, I a) I. 2. \\.....\<\»\o 054 35...}... we \.. \C 5. tr...- NW- _ 7'... Y _ ~61". . 1311) -r .. ‘ __-. I l I .\\ r\\.l\\4« IJKI A \\ 4. .-\ 9x; - .. n d. r .- Q- '1 ts C. H \i .50 _ .49.. .4 not \ Wm. H “N m 9 - .C A..\ .... \v ' (III . .... I -//- v. If], I./. \. ...\ -z. .. / I. NUUIVQ .0\- .\. . , . , // 2 \ \ , , ,/ ../ .,l 4‘ 1.6 1&4 ... 1, k _ .. . @055. p.651 5845 UN +7 m g. ..w.\ ..W. 3 .«\ k \ .. .. or. - \\\ In. .x |\ \\ 1:], (I; , \\.- \ / / .k. ._ \ Isl \.. \ .. x. . / 5 .. /, . ., 2 . .‘. ,. .1... .. / ... . ., l. 11 a , 1 . l 11 . x V -- .«W \ 1 . . V .... . .. . N._/\ r . x \v \- \ a .\ .O .. a N / \ — I x \ /fl .... \ / a \ / , . . w\ . l \\ /. 7 - A /./.- I .\., \ . ”hills!“ .\ .4 {xx - 1...... . .. - ...... 5 a ...l A r . A , k I I o- _ I I...» I1. _ _ g . _ A A t"lllll-.'l . . |. a O, 'I‘I' . AI ' ,. y a 1M -:. ‘ \ , . | u \ I v . x 4 - m III1II. J - III. . . , V I. I‘lu 4. . 1 — . I ‘ _. . 5 fl . v J .. II» I - ll- VIII! III"! . u I I ‘V .( A I .\ IVI‘II’IO II) F}.Il.. _ w -4. _. . . _ ,. ... n ._ .. _ _ I1.-.6....I.I-. ., _ _ v . . a _ fl . I; . _ _. . 7.. I v . y . w .. . _ ~ . / . ... _ _ . _ q p > ; g . a . , / x \ x o u ‘ / \ \ . . L . . . I \ u a. RI/ \ . ‘ r. . .r / ill. ., - \\« x \. RX r . \ \\ Lw't ” .1 // I I \\W \ \ H .... 'r/ ., / llll‘l‘\\ \ 1:. \U; 48 Light source. Both white and monochromatic light sources were used. For the white light source, a SOC-watt pro- Jection bulb with a pre-focus socket was used. The lamp housing is shown in the accompanying print. The monochromatic light source used for taking pic- tures was kindly loaned by the Physics Department. The single unit consists of a mercury light, parabolic re- flector, and other necessary parts. The unit produces a field of parallel rays. Hence no collimating lens is nece- ssary, the unit being placed just before the first Polar- oid disc. Suitable Wratten and Wainwright filters were used to obtain green light of 5461 A0. Details of the parts described are shown in the prints and photographs included in this section. III... I. III ...! I (1“! Ct- I I‘ll-Ill!" . £cl”.E tyi‘t.tru. Ekg$nw' 2.4 ' ’- l\'.:(tf’g1f.u mm“ 4» . *3. 1'1 .. I..I. -lpllINuvaLI flIi ...... - CHAPTER VI THE MODEL; MATERIALS, PREPARATION, LOADING 1. Materials. Several materials have been used for photoelastic analeis work, each having its advantages and disadvan- tanges. Materials used by early investigators included glass and celluloid, but these have very poor Optical sen- sitivity. Recent materials include pyralin, marblette, and Bakelite. Bakelite has been used most widely in recent years, and the type BT-6l-893 has been accepted as having the best general prOperties. A list of desirable properties to consider has been listed by Filon (Ref. 3), Solakian (Ref. 44), and other investigators. Some of these properties may be listed here: (1) High Optical sensitivity, (2) Good machinability, (5) Linear stress-strain relation, (4) Absence or easy re- moval of initial double refraction, (5) Little or no creep, (6) Little or no "edge effect", (7) High transparency. Mindlin (Ref. 33) has given a list of materials arranged in order according to their desirability from the view- point Of Optical sensitivity only: Phenolite Bakelite Celluloid Pollopas Charmoid Vinylite Glass A newly developed plastic, Lucite, also ranges near the tOp of this list according to the producers. Marblette -49- 50 also has a high optical sensitivity, but has a lower elastic limit. A material having a higher coefficient of optical sensitivity will give a pattern having more isochromatics, and hence gives more accurate data. Figure 21 shows a com- parison between the optical sensitivity of Bakelite and celluloid, using two beams loaded under identical con- ditions. ELAA6£1LI7ZT (ZEZHLLJLJD/C7 Figure 21 2. Methods of preparation. Bakelite may be obtained in sheets about 10" x 18" in unpolished form, %" being the usual thickness used. It may also be obtained, at a higher price, ready polished and annealed. In the latter case it is only necessary to mark out the model desired and carefully form it to shape. 51 The general procedure for this will now be described. In the event that unpolished pieces are obtained, a piece should be cut from the sheet so that there is about 1/8" of material in excess Of the finished dimen— sions. This is to obtain a perpendicular edge on the fi- nished model, since polishing wears down the edge sur- faces slightly. If the model blank is warped or has un- even surfaces, these should be made plane and parallel by surfacing on a milling machine, turning in a lathe, or if the condition is not excessive, by hand surfacing with emery cloth. Usually the unpolished sheets are suf- ficiently plane and parallel so that they may be polished directly. The author has obtained good results by beginning the polishing process with a NO. 240 metallographic polishing wheel, then working through No. 520, levi- gated alumina of two grades, and finishing with a rouge wheel. The 240 and 520 wheels were canvas covered, the remainder being velvet covered. Scratches may be elimi- nated by holding the model each time at right angles to the position on the preceding wheel. The model should be washed in water at room temperature when changing wheels. A wheel speed of about 250 R.P.M. is recommended. The models should always be polished wet, using the abrasive in suSpension. In Figure 22 is shown a series of metallo- graphic polishing wheels used by the author, which serve very well in polishing models. gFigure 22 A.) 52 A hand or power jig saw may be used for cutting the blank roughly to shape before polishing,but the final shaping must be done much more carefully. The excess material may be removed slowly on a jig saw fairly close to the finish line, the final out being made on a milling machine using sharp cutters and lard Oil lubrication. Holes should not be drilled, but either reamed, bored, or turned on a lathe. On models of irregular shape, good results may be obtained by careful filing. Mindlin (Ref. 53) gives three points to observe in finishing models: (1) Avoid heating the material, (2) Form sharp edges, (5) Have the boundary surfaces accurately normal to the faces of the model. The model will be useless if fast or heavy cuts are taken. In some materials, notably Bakelite, there occurs a phenomenon known as "edge effect". This is due to a "drying out" of the edges which have just been cut. This effect causes a change in the Optical properties of the material near the edges, thus causing an error. Figure 23 shows a portion of a model exhibiting edge effect. It ___._ . l- ., _ -.l -..-.e is characterized by a hooking back of the fringes near the boundary. In some materials this effect appears within a few hours; in others not Figure 23 until several hours have elapsed. Hence it is best to use the model directly after it has been finished. 53 5. Annealing. When the rough, unannealed sheets are used, the piece should be inserted between the crossed Polaroids and examined for initial stresses. There is usually some initial stress evident at the boundaries of the pieces. Ordinarily, the clear areas may be marked and the models carefully cut from these areas. If initial stresses are present in a piece to be used as a model, it is necessary to anneal the piece. The writer has found that a small electric oven of the type used for household cooking serves admirably for this purpose. However, to insure a uniform temperature and slow cooling rate, several fairly large pieces of metal were placed inside, and the oven covered with blankets. The oven may be brought up to heat fairly rapidly, using about three hours for this. A soak- ing temperature of about 200° F. may be used, the model being kept at this temperature for about three hours. The heat may then be turned off, the model being left tocool overnight. Before removing the model, it is well to leave the oven door Open for about an hour before bringing the model out into the room. While in the Oven, the model should be fully supported on a piece of asbestos board to prevent warping. 4. The loading frame constructed at M.S.C. A desirable loading device should possess the following characteristics: (1) Capability of loading models with a wide variety of loading systems, (2) Accurate means of measuring the applied load, (3) Large, 54 clear opening for passage Of the optical field, (4) Pro- vision for moving the loading frame so that different parts of a large model may be brought into the field. The design of the loading frame constructed by the author is very similar to that of H. E. Wessman (Ref. 50), with the exception of being somewhat smaller. It consists of a frame sliding within a larger frame, thus making pro- vision for bringing different parts of a large model into view. The inner frame is adjusted vertically by means of a handwheel and screw. Loading is by means Of a lever supporting weights at its extremity. Models may be loaded in tension, compression, and under various types of bend- ing. Materials used in constructing the frame were largely angle iron and strap iron. Details are given in the accom- panying print. A photograph of the frame is shown in Fig- ure 24. 5. Methods of loading. From an inspection of the drawing and photograph, it may be seen that models may be tested in tension, com- pression, or under various types of bending. For models to be tested in tension, holes should be reamed to take a bushing and pin. Great care must be taken to see that a truly axial load is applied with no eccentricity. Various types of compression loading may be easily arranged. Pre- caution should be taken to see that the applied load or loads are exactly vertical and in good alignment. When testing beams and other models in bending, distances should be carefully and accurately determined. For com- 55 pression and bending tests, the model should be checked to see that it remains vertical when under a heavy load. In many cases where it is desired to load a model at several points, loading may be accomplished by loop- ing music wire Over the model at the point or points de- sired and loading directly with known weights. For side loading, music wire may be used with turnbuckles and small spring balances. CHAPTER VII PHOTOELASTIC DETERMINATION OF STRESSES 1. General. In making photoelastic stress determinations, .the first step is to make a suitable model Of the struc- ture or part to be studied, the model being made of Bakelite or other similar material. This is then placed in the loading frame and loaded in the same manner as is the prototype. Examination with white light in a circular polariscope will give a brightly colored pattern con- sisting of colored bands, called isochromatics. If a monochromatic light source is used, the pattern will con- sist of black and white fringes, as eXplained in Chapter IV. Evaluation of this pattern will give the shear and boundary stresses. If the i-wave plates be removed from the circular polariscope, a new set of lines will appear, these being the isoclinic lines. These lines are always black, re- gardless of the light source, and will be superimposed on the isochromatic pattern. However, the isoclinics may be unmistakably recognized, and are easily discerned from the isochromatics. The direction of the isoclinics are inde- pendent Of the load, and will have the same direction at all loads for any particular setting Of the Polaroids. The directions of the isoclinics will vary as the Polaroids are rotated angularly, this being for a constant load. Directions of the principal stresses are determined from -55- 57 the isoclinic lines. The above material has been given in Chapter IV, but is included here briefly as a review before discuss— ing evaluation of the patterns. 2. Interpretation of fringe value and stress pattern. The general shear formula is It was explained in Chapter III that this shear stress be- comes a maximum when sin 2a is I. This occurs when 2e = 90°, or d = 45°, when the above formula becomes Photoelastic analysis is based on the fundamental stress- Optic law, which states that in a model of constant thick- ness, the fringe order is constant at all points where the principal stress difference is constant. This is expressed mathematically as n = ct(P - Q)....................(7-l) where n is the fringe order, t is the thickness of the model, P and Q are the principal stresses, and c is a con- stant to be determined. Application of this law to the interpretation Of stress patterns may now be discussed. Equation (7-1) may be rewritten: - -9- P ' Q ' ct Along a particular fringe n is constant, and hence along 58 a locus of such points, P - Q - K From.this, it may now be said that a fringe represents the locus of points at which there is a constant differ- ence between the principal stresses. It will be remembered that the maximum shear stress is given by or, rewriting this: P - Q = 2vmax If this last expression for P - Q is substituted in the stress-optic equation (7-1), there results: Vmax ' égt or vmax = Fn where F is given by F‘éts F is known as the fringe value of the model, and is a constant for any one material or sheet. Methods of de- termining F will be given later. It may now be seen that where n is constant, as along a fringe, vmax is also constant; hence a fringe may now be defined as the locus Of points of equal maxi- mum shear stress. It may also be seen from this equation that the maximum shear stress is directly proportional to the fringe order. Hence for a point in a model at which sev- eral fringes have passed by during loading, the stress 59 will be greater than for a point at which only a few fringes have passed, the stresses at the two points being proportional to the number Of fringes passing each point. For example, if 12 fringes have passed a certain point, the stress for that point will be 3 times as great as that for a point where only 4 fringes have passed. 3. Points Of concentrated stress. In some cases sufficient information regarding points of high stress concentration may be obtained by merely examining the pattern and making no quantitative analysis. From the preceding article it is evident that areas of high stress concentration may be Observed as areas containing fringes which are closely Spaced, as at sharp corners, fillets, etc. Likewise areas of low stress are those containing only a few widely spaced lines. Weight may be saved by merely cutting these areas down. 4. Calibration and adjustment of the polariscope. It is recommended that a test model be made similar to that suggested by Orton (Ref. 39) and given in Figure 25 on the following page. The holes should be fitted with steel bushings, which are smooth and a snug fit. The model is loaded vertically in tension in the center of the field. Care should be taken to see that there is little or no eccen- tricity Of loading. As load is applied slowly on gay model, each point will alternate in brightness and darkness as the black 2M~>§UMQW ZOPbSWTx q VUI .J.m_ '1' fl -@ - , f o II III .95 If .IAII I. \ _ IJW T Iii». ..II 00w. .3me \m a I. ’L *W’ 60 and white fringes pass the point. A diagramatic repre- ( 0F WOAGE ID (0 U 2 5- I g a m hw.... - -- ._ ,,_ H. -4..._._~___.--.._,- ..., Figure 2b sentation of this variation in intensity is given in Figure 26 above. The same will now occur when the specimen used for adjusting is loaded. As load is applied slowly, the model will brighten at its midpoint, then darken again. When maximum darkness occurs again, the model will at this time be stressed to the first order. By dividing the load by the area of the mid-section (width x thickness), the material will be calibrated and the fringe value de- termined. This first value obtained should not be used, but the model should be loaded until several fringes have passed, and an average value computed. This completes the calibration of the material. In the next article, a second method of calibrating will be described. To adjust the polariscope, the same specimen may 61 be used. If Polaroids are used, the directions of the polarizing axes will be known when purchased. The polar- izer and analyzer should be set at the 45° position (po- larizer and analyzer are always set at right angles to each other), which will give maximum light at the mid- point of the specimen. If the Specimen now be loaded so that the second order fringe is centered about the center- line of the model and then backed off to one-fourth the distance from the first to the second order, the effect of a i-wave plate will be produced, since the vertical component emerges from the Specimen a %-wave length ahead of the horizontal component. This temporary fi-wave plate may now be used in checking the plates of the polariscope. One %ewave plate may be placed in position and rotated until maximum dark- ness 1; obtained at the center of the Specimen. In order to cancel the %~wave advance occurring in the Specimen, the retarding axis of this %-wave plate must therefore be vertical. By loosening the clamping ring, the graduated ring Should now be set so that the zero is horizontal. This will be the advancing axis of the plate. The zero of the adjusted i-wave plate may now be moved to the vertical position, the second plate placed in the polariscope and rotated until the field on the screen is at maximum darkness. (intensity of the speci- men is disregarded for this). The retarding axis of this second plate will then be vertical, and the graduated 62 ring should now be set with.the zero horizontal, as before. Adjustment is now complete. To prepare the polariscope for use, the %-wave plates should be removed, the polarizer set vertical and the analyzer horizontal, thus giving maximum field darkness. The first %-wave plate may then be placed in position with the axis set at 45°. The second i-wave plate, when placed in position, Should be rotated until maximum.darkness is again obtained. The polariscOpe is then ready for use. In using the tension strip, bushings are used as shown in Figure 27, these bushings having slots Figure 27 for accurately centering the load. 5. Calibration by bending. It is usually difficult to accurately center the tension strip just described, and a small eccentricity will cause a large calibration error. In the beam method to be described, all that is needed is a reasonably ac- curate determination of the load positions with assurance of their symmetrical diSposition about the center of the beam. In calibrating by the beam method, a piece of the shape shown in Figure 28 on the following page is first made, with four bushings placed tangent to the center line. When loaded as shown, the middle portion of the beam will be subjected to a constant bending moment and Calibration Beam in Bending zero vertical shear. The isochromatics, or fringes, will be straight, horizontal, and parallel. When load- ing begins, the fringes start at the beam boundaries CALIBRATION $PEC|MEN §HEAE QMGRAM- §ENDnaG MOMENT‘ omeszmv)~ Figure 28 and progress toward the neutral axis of the beam. By counting the number of fringes which pass the boundary and relating this to the stress obtained from the for- mula s = %g, fringe represents, is obtained. the fringe value, or stress which each 6. Isochromatics. Isochromatics are lines of constant color and 64 are generally referred to as fringes when monochromatic light is used. In any stressed body, there exist principal stress- es at every point. In photoelastic analysis using trans- parent materials, interference fringes are produced by the polarized light, these fringes being proportional to the difference between the principal stresses, this being denoted by (P-Q)- To determine the maximum shear stress at any point, the fringe order is first obtained. This is simply the number of fringes which have passed that point during loading. This fringe order is then multiplied by the fringe value, or stress per fringe as determined from the calibration beam. The value obtained is the differ- ence between the principal stresses, (P-Q). The maximum shear stress is ------- and hence can be obtained immedi- ately by dividing the value found above by 2. Thus the maximum shear stress at any point or along any section may be rapidly and easily determined. 7. Boundary stresses. Consider a small element in the boundary of a body, the boundary considered being free from external loads at that point. Since there can be no stress normal to the boundary and the shear stress on one face of the element is zero, it follows that one of the principal stresses is zero. Hence at a free boundary there is only one principal stress, its direction being tangent to the boundary. 65 This may be Shown as follows: Writing the funda- mental equation: P - Q = 5E and remembering that F ' ---: it follows that P - Q = 2nF. For a free boundary condition, Q may be set equal to zero. This gives: P = 2nF. Hence tensile or compressive boundary stresses may be easily and directly evaluated. 8. Isoclinics; principal stress directions. If the %-wave plates are removed from the polari— scOpe, a few black bands will appear on the screen. These lines are termed isoclinics. Although the isochromatics change position with a change in load, the isoclinics are independent of the load (excepting zero load) for any particular setting of the polarizer and analyzer. If the polarizer and analyzer be rotated, the directions of these lines will be seen to change. AS eXplained in Chap- ter IV, these lines are loci of points whose principal stress directions coincide with the axes of polarizer and analyzer.for each setting. The first step in obtaining the directions of the principal stresses is to set polarizer and analyzer to their zero positions; i.e., polarizer axis vertical, 66 analyzer axis horizontal. The center lines of the iso- clinics for this setting are then traced on a sheet of paper which may be attached to the screen. Rotating polarizer and analyzer 10° will give rise to new posi- tions of the isoclinics. These may then be traced on the paper. Each set of lines obtained should be identified by marking the setting for which they were obtained. After the isoclinic pattern is obtained, the di- rections of the principal stresses are drawn in. The isoclinics themselves are not principal stress direc- tions, but each isoclinic gives the angle which a princi- pal stress direction forms with the X-axis for a parti- cular point on the isoclinic. An easy manner in which to perform the work is to place a series of small crosses inclined at the proper angle along each iso- MET_Hgo or Mizmwgfioctwtcs R.L.. Figure 29 67 clinic as Shown in Figure 29, on the preceding page. For example, all the crosses along a 10° isoclinic would be made inclined at 10° to the axes. A pro- tractor-triangle has been found to be extremely help- ful in this work. Lines may now be drawn through the inclined crosses, forming a series of lines crossing each iso- clinic at the prOper angle, as shown in Figure 50. DETEE_MINAIIQN__O_F t PEINC|PAL 5Trae_5§_c>ig_ecno:\_s_s_ fEOM ISOCLlNICS Figure 30 The resulting network of lines will be the directions of the principal stresses. It should be remembered that the complete network consists of two sets of lines, each set always intersecting the other set at right angles at all 68 points. These lines will also be always parallel and perpendicular to the boundaries. The above illustrations are for a cantilever beam. Figure 31 shows a tension strip, the lower half showing ‘DIQECTCON 5 OF PE|NC|PAL STRESSES 5° 10’ 5° 20‘ 15' 30° Isocumucs Figure 51 the isoclinics, the upper half the principal stress direc- tions. 9. Shear lines. Since the maximum shear stresses are at 45° to the principal stresses, the directions of the shear stresses may be obtained from the network of principal stress directions by simply drawing a set of lines which intersect the principal stress directions at 45° at all points. 10. Principal stresses. For a great majority of investigations, the 69 information obtained from the procedures already described are sufficient. These include shear stresses and tensile and compressive boundary stresses. Principal stress direc- tions are also easily obtained and are sometimes of inter- est. However, there are occasional problems in which the amounts of the principal stresses at interior points are desired. This involves a more extensive procedure, and calls for additional equipment. The process is also time- consuming. A brief summary of the more important methods will be given here. These methods have been well covered in recent literature. (a). Lateral extensometer method. To find the individual principal stresses P and Q, it is first necessary to determine the principal stress difference P-Q for those regions in which investigation is desired. To separate P and Q, it is necessary to obtain a quantity (P + Q), so that this may be combined with the (P - Q) values found for a point. It was first discovered by Mesnager that the strain normal to the model is pro- portional to the sum of the principal stresses. Hence (P + Q) is constant along lines of constant thickness, these lines being determined by use of a lateral exten- someter. For a Bakelite sheet %" in thickness, an increase of one fringe produces roughly a displacement of about 0.00001 in. It will be seen that this is a delicate and tedious method. 70 (b). Interferometer method. 'These methods are based on interference phenom- ena of light reflected from the surface of a steel model which is Shaped and loaded in the same manner as the Bake- lite model. . Figure 52 shows the equipment used by Max M. Frocht, Ref. 6. Figure 32 In Figure 35 on the following page is shown the interferometer as used by Brahtz and Soehrens, (Ref. 2). 71 "A ,. ‘. H s '24,. u x 4.. K} Interferometer used by Brahtz and Soehrens Figure 36 These methods are based on the fact that each inter- ference fringe appearing in the interferometer will repre- sent a path of constant thickness, and hence a path of constant (P + Q). Thus the (P + Q) values of points on these lines may be combined directly with the correSpond- ing points whose (P - Q) values are known. This will yield the amount of each individual principal stress at the point desired. The disadvantage of this method is the cost of interferometer equipment. (c). Membrane method. In the application of this method, a plate or box is constructed so that an opening is formed in a flat surface, the opening having the same shape as the model 72 being investigated. A membrane is then stretched over the opening and fastened along the boundaries such that the ordinates of the membrane are proportional to the (P + Q) values at those boundary points. It may then be shown that the ordinate of any point on the interior region of the membrane is proportional to the (P + Q) value at that point. Combining this with the (P - Q) value of the point will separate the principal stresses. Weibel (Ref. 40) has obtained good results from this method by using a soap film as a membrane. McGivern and Supper (Ref. 23) used the same method making use of a thin rubber sheet as a membrane. This method is an easy one to apply, but is limited to regions near those boundaries along which (P + Q) can be previously determined. (d) Numerical and Graphical methods. These methods of separating the principal stresses for interior points require no additional photo- elastic equipment or laboratory procedure after the (P - Q) stresses have been determined and the directions of the principal stresses obtained. Numerical method. Liebmann has developed a process in which Laplace's equation is solved for certain bound- ary values by successive numerical approximations. Shortley, Fried, and Weller (Ref. 9, Bulletin) have im- proved on the process by devising means for obtaining a more rapid convergence of the successive approximations. Among others developing similar lethods are Por- 73 itsky, Snively, and Wylie, (Ref. 29), Neuber, (Ref. 26), and Frocht, (Ref. 8). 11. Stress concentration factors. It is well known that Sharp corners, fillets, small holes etc. have a marked tendency to greatly in- crease the stress concentration wherever they occur. The factor of stress concentration is defined as the ratio of the maximum stress to the average stress for a par- ticular region, and is denoted by K: Photoelastically, these may be determined by a compari- son of fringe orders, these being directly prOportional to the stresses: Smax ' 2anax 9 Save ' ZFnave' Substituting: nmax K ...... nave Thus the stress concentration factor may be obtained directly from the stress pattern, no knowledge of loads, dimensions, or material being necessary. Stress concentration factors are usually plotted against the ratio r/d, r being the radius of the fillet, groove, or hole, and d being the depth of the piece at that point. Frocht has plotted these curves for investi- gations of several types (Ref. 10 & 11). It is interesting to note that as the ratio r/d decreases, the stress concentration rises to extremely high values. 74 12. Recent develOpments. The photoelastic method is now being applied with success to three-dimensional problems (Ref. 8, 13, 15, 18, 28). Dynamic problems, such as impact stresses, stress conditions in rotating discs etc.are being studied by motion picture and stroboscopic means. 15. Stress pattern photographs. 0n the following Sheets are Shown photographs taken by the author of typical models loaded as de- scribed. All models are of Bakelite. Photographs were taken with monochromatic light (5461 AC) on Wratten and Wainwright Metallographic plates. Azo No. 4 printing paper was used in making prints. Figure S4 Tension Strip with Circular Hole Figure 35 Calibration Beam in Bending )-4 a Figure 36 Rocker Loaded Vertically Figure 37 Centrally Loaded Beam 1. 2. 3. 4. 5. 6. 8. 9. 10. 11. 12. BIBLIOGRAPHY BOOKS N. Alexander, “Photoelasticity", Rhode Island State College, 1936. Coker and Filon, "A Treatise on Photoelasticity", Cambridge University Press, Cambridge, 1931. L.N.G.Filon, "A Manual of Photoelasticity for Engineers“, Cambridge University Press, Cam- bridge a 1936 e Riggs and Frocht, "Strength of Materials", Chapter 13. Ronald Press Go. 1938. BULLETINS M.M.Frocht, "The Place of Photoelasticity in the Analysis of Statically Indeterminate Structures", Carnegie Institute of Technology. 1938. M.M.Frocht, "The Place of Photoelasticity in Engineering Instruction", Carnegie Institute of Technology. 1937. M.M.Frocht, "A Photoelastic Investigation of Shear and Bending Stresses in Centrally Loaded Simple Beams", Carnegie Institute of Technology. 1937. Fried and Weller, "Photoelastic Analysis of Two- and Three-Dimensional Stress Systems', Ohio University Engineering Experiment Station Bulletin N00 1060 1940. Shortley, Fried, and Weller, "Numerical Solution of Laplace's and Poisson's Equations with Application to Photoelasticity and Torsion", Ohio University Engineering Experiment Station Bulletin No. 107. 1940. ARTICLES R.V.Baud, "Further Developments in Photoelasticity", Journal of the Optical Society of America, Vol. 18, 1929. Brahtz and Soehrens,"Direct Optical Measurement of Individual Principal Stresses , Journal of Applied Physics, April 1939, Vol. 10, NO. 4. Edmonds and McMinn, "Celluloid as a Medium for Photo- elastic Investigations", A.S.M.E. Transactions, Vol. 54, 1932. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. ii M.M.Frocht, "The Behaviour of a Brittle Material at Failure", A.S.M.E. Transactions, Vol. 58, 1936. M.M.Frocht, "Recent Advances in Photoelasticity", A.S.M.E. Transactions, Vol. 53, 1931. M.M.Frocht, "ISOpachic Stress Patterns", Journal of Applied Physics, April 1939, Vol. 10, No. 4. M.M.Frocht, "Kinematography in Photoelasticity", A.S¥M.E. Transactions, Vol. 54, 1932. M.M.Frocht, "A Rapid Method for the Determination of Principal Stresses Across Sections of Symmetry from Photoelastic Data", A.S.M.E. Transactions, Vol. 60, 1939. M.M.Frocht, "The Behaviour of a Brittle Material at Failure", A.S.M.E. Transactions, Vol. 58, 1936. .M.M.Frocht, "Photoelastic Studies in Stress Concen- tration", Mechanical Engineering, Vol. 58, 1936. M.M.Frocht and H.N.Hi11, "Stress-Concentration Factors Around a Central Circular Hole in a Plate Loaded Through a Pin in the Hole", Journal of Applied Mechanics, March 1940, Vol. 7, No. l. Goodier and Lee, "An Extension of the Photoelastic Method of Stress Measurement to Plates in Transverse Bending", Journal of Applied Mechanics, March 1941. M. Hetenyi, "Application of Hardening Resins in Three Dimensional Photoelastic Studies", Journal of Applied Physics, May 1939. Vol. 10, No. 5. M. Hetenyi, "Stresses in Rotating Parts", Metal Progress, Feb. 1941. M. Hetenyi, "Photoelastic Studies of Three Dimensional Stress Problems", Procedings of the Fifth Inter- national Congress for Applied Mechanics, 1938. M. Hetenyi, "Some Applications of Photoelasticity in Turbine-Generator Design", Journal of Applied Mechanics, Dec. 1939, Vol. 6, No. 4. M. Hetenyi, "Stresses in Rotating Parts", Metal Progress, Feb. 1941. M. Hetenyi, "The Fundamentals of Three Dimensional Photoelasticity", A.S.M.E. Transactions, Vol. 60, 1938. 28. 29. 30. 31. 32. 33. ‘38. 40. 41. iii Horger, "Improving Engine Axles and Piston Rods", Metal Progree, Feb. 1941. Lee and Armstrong, "Effect of Temperature on Physical and Optical Properties of Photoelastic Materials", A.S.M.E. Transactions, Vol. 60, 1938. MacGregor, "A Two-Load Method of Determining the Average True Stress-Strain Curve in Tension", Journal of Applied Mechanics, Dec. 1939. H.B.Maris, "Photoelastic Investigations of the Ten- sile Test Specimen, The Notched Bar, The Ship Pro- peller Strut, and the Roller Path Ring", Journal of the Optical Society of America, Vol. 15, 1927. McGivern and Supper,”"A Membrane Analogy Supplement- ing Photoelasticity, A.S.M.E. Transactions, Vol. 56, 1934. R.D.Mindlin, "A Review of the Photoelastic Method of Stress Analysis", Journal of Applied Physics, Part I April, 1939, Part II May, 1939. W.M. Murray, "Seeing Stresses with Photoelasticity", Metal Progress, Feb. 1941. H.P.Neuber, "Exact Construction of the (P + Q) Network from Photoelastic Observations", A.S.M.E. Transactions, Vol. 56, 1934. R.E.Newton, "A Photoelastic Study of Stresses in Rotating Discs", Journal of Applied Mechanics, June 1940. Peterson and Wahl, "Two and Three Dimensional Cases of Stress Concentration and Comparison with Fatigue Tests", A.S.M.E. Transactions, Vol. 58, 1936. Poritsky, Snively, and Wylie,"Numerical and Graph- ical Method of Solving Two Dimensional Stress Prob- lems", A.S.M.E. Transactions, Vol. 61, 1939. R.E.Orton, "Photoelastic Analysis in Commercial Practice" Machine Design, March through July 1940. R.E.Orton, "Applying Theory of Elasticity in Practi- cal Design", Machine Design, April 1941. B.F.Ruffner, "The Photoelastic Method as an Aid in Stress Analysis and Structural Design", Aero Digest, April 1939. 42. 43. 44. 45. 46. 47. 48. 49. 50. iv B.J.Ruffner, "Stresses Due to Secondary Bending", Engineering Experiment Station, Oregon State College. Reprint 21 from Procedings of First Northwest Photo- elasticity Conference. Ryan and Rettaliata, "Photoelastic Analysis of Stresses in a Steam-Turbine Blade Root", A.S.M.E. Transactions, Aug. 1940. Vol. 62, No. 6. Solakian, "A New Photoelastic Material", Mechanical Engineering, Dec. 1935. Photoelastic Journal, A.Solakian, Editor. 55 W. 42nd St., New York, N.Y. Solakian and Karelitz, "Photoelastic Study of Shear- ing Stresses in Keys and Keyways", A.S.M.E. Trans- actions, Vol. 54, 1932. E.K.Timby, "Distortion of the Photoelastic Fringe Pattern in an Optically Unbalanced PolariSCOpe", A.S.M.E. Transactions, Vol. 60, 1938. E.E.Weibe1, "Thermal Stresses in Cylinders by the Photoelastic Method", Procedings of the Fifth Inter- national Congress for Applied Mechanics, 1938. R.E.Weibel, "Studies in Photoelastic Stress Deter- mination", A.S.M.E. Transactions, Vol. 56, 1934. R.E.Wessman, "New Universal Straining Frame aids Photoelastic Research", Civil Engineering, Sept. 1938. ROOM 113130,,ng "W, a Circulat‘ .- . I" I r I In: I! ‘ I r l I . I 9 III . - c I I I . {Int 2 'l - . I . D . 1 ‘., ‘I‘.I \I A I I I l t ) I I A ' I ..\ - - . l I A \ I I II - I ‘0 B I: - , o o l \ l I ‘ I l ‘1 l ‘I‘ I t“ | |V1II I . . . I. .. u l . III I D, .(I I .« -,- . III. I . ‘ I l I ' v ‘ ' f | I: A l I n i I e I I I (I, O . \ I i I . o I I I II I I‘ I I a r I Iill II 1 ‘ I \ ' ' I - III. I I I I . I J I I . . I . I — I . I . .. .I 1‘ a - ' I I I I .l I I. a \ I . I I I v I 1' a. I I I . . I . I I I J \ I . . .i i I I C . I' II I I .o I I I I 101 I . w . I . a . I I I . . I . I . . . . I I I ' v I I '4 ; I . I I I I . 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