A Photoelastic Investigation of the Distribution of Shearing Stresses in a Stiffened Flat Panel Citation Bell, Willard Newton and Bussey, John Kenneth (1938) A Photoelastic Investigation of the Distribution of Shearing Stresses in a Stiffened Flat Panel. Master's thesis, California Institute of Technology. doi:10.7907/9JFS-SF60. https://resolver.caltech.edu/CaltechETD:etd-01102006-131047 Abstract No abstract. Item Type: Thesis (Master's thesis) Subject Keywords: Aeronautical Engineering Degree Grantor: California Institute of Technology Division: Engineering and Applied Science Major Option: Aeronautics Thesis Availability: Public (worldwide access) Research Advisor(s): Sechler, Ernest Edwin (advisor) von Kármán, Theodore (advisor) Group: GALCIT Thesis Committee: Unknown, Unknown Defense Date: 1 January 1938 Record Number: CaltechETD:etd-01102006-131047 Persistent URL: https://resolver.caltech.edu/CaltechETD:etd-01102006-131047 DOI: 10.7907/9JFS-SF60 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 100 Collection: CaltechTHESIS Deposited By: Imported from ETD-db Deposited On: 12 Jan 2006 Last Modified: 03 Oct 2019 22:56 Thesis Files Preview PDF - Final Version See Usage Policy. 16MB Repository Staff Only: item control page

A PHOTOELASTIC INVESTIGATION OF THE DISTRIBUTION OF SHEARING STRESSES IN A STIFFENED FLAT PANEL Thesis by In Partial Fulfillment of the Requirements for the Degree of Master of Science in Aeronautical Engineering. California Institute of Technology Pasadena, California 1938 ACKNOWLEDGMENTS The authors wish to express their apprecia- tion to Dr. The von Karman, who suggested the research problem of this thesis, and to Dr. E. Ee Sechler for his helpful criticisms and suggestions during the year. Grateful acknowledgment is also made to Arthur Gross, senior in mechanical engineering, for his work as assistant during the final term. TABLE OF CONTENTS SUMMaArye 0 © © © © © © «© © © © ew ee ew Notation 2 & e 6 e ¢ € e e e s e € se e s Part I. Introduction . . « « « » «© «© » « ae Design considerations. .. be. Statement of thesis problem Ce Previous investigations .. Part II. Methods of investigation. « « e ae Theory of elasticity ... 1. Airy's stress function solutio for an unstiffened panel . 2e A simple extension to the case of a stiffened panel. * b. The photoelastic method. . 1. General remarks . 6 .« ee Working fundamentals . Part III. Laboratory procedure . .. . . ae Apparatus De Model @ e e e s e e e e e ® ce Procedure . .« e@ 8® @ © 8® @ 6 2s e @ @ @ e ° @ @ Part IV. Results ...e.+e eos e «© e Ge Data. e ® @ e e 6 e e e e @ be Calculations and graphical work Ge Gonclusion .. .« « « e e e References . « « « « © ee «© «© s © ec we Appendices 2. « « © © «© «© «© «© © e © © © A. Photographs « « « « « « « « « « Be Fundamental stress relations . C. Graphical method for separation pand Qq .« . « « « e e e e e D. Comparison tests on additional mo del Pm VER 10 10 11 14 16 20 23 23 25 28 50 od exelil xEV XXXVI B- NOTATION Vertical coordinate, positive upward. Horizontal coordinate, positive to the right. Total applied load. Moment. Thickness of model. Normal stress (tension or compression). Principal normal stresses, mutually perpendicular; P= Omaxs = Opin» OF vice-versa. Tangential stress (shear). max = & (Pp 7 4) Isoclinic angle, the inclination of one principal stress (p, say) as measured clockwise from the X-axise Angle of inclination of a principal stress trajectory to an isoclinic line. rv, - Polar coordinates; pax" 4 ye, tant =Z PART I. INTRODUCTION ae Design considerations. In the construction of modern metal airplanes it is common practice to employ panels of thin sheet reinforced by longitudinal stiffeners; and, since it is necessary to design to close strength limits in order to obtain an efficient structure, it is important to know the allowable load which the individual stiffened panel may carrye Compression loads are transferred in shear through the sheet, and it has been found that difficulties in the prediction of strength character- istics arise because of the non-uniform stress distribu- tion between stiffeners in such an assembly. This is the so-Galled "shear lag" phenomenon. be Statement of thesis problem. The final attainment of a completely valid design criterion in the case of the stiffened thin sheet panel should be facilitated, it seems reasonable to believe, by an investigation of the shear properties of the sheet inthis type of structure. With this idea in mind, the research problem of this thesis has been undertaken. It was proposed to examine by photoelastic means a model typifying the very simplest case of a stiffened panel, namely, a flat rectangular panel with one centrally located stiffener. By loading this model in compression through the stiffener and recording photographically the resulting stress patterns, a study has been made of the distribution of shearing stresses through the sheet. The photoelastic method, it may be remarked, appears to be of considerable advantage in the investigation of this particular problem, since its most direct result is the determination of the maximum shearing stresses in magnitude and direction. Ce Previous investigations. Numerous tests have been made in various laboratories to determine the compressive strengths of flat and curved panels, both stiffened and unstiffened. in the Guggenheim Aeronautical Laboratory of the California Institute of Technology, in specific, investigations have been made by Sechler*, White and Ante”, Lovett and Rodee” , and others, into the particular problems of stiffened panel construction, including the effects of various stiffener-sheet combinations, the transfer of stress from stiffener to stiffener, and the calculation of "effective widths" of sheet for use in design. Aside from a photoelastic test with celluloid sheets, supported at the edges by V-blocks (to simulate stiffeners) and subjected to compression, proposed by August Zinnser, Ir.*, of Massachusetts Institute of Technology, however, the authors have knowledge of no other photoelastic in- vestigations of stiffened panel construction. foam comp cS core tia tem ney ORT GEMS GRD RS OH GNP OST ES MN a GS OT ee HD le, E, Sechler, Stress Distribution in Stiffened Panels under Compression. Journal of the Aeronautical Sciences, June, 1937. 2 White and Antz, Tests on the Stress Distribution in Reinforced Panels. Journal of the Aeronautical Sciences, April, 19356. ° Lovett and Rodee, Transfer of Stress from Main Beams to Intermediate Stiffeners in Metal Sheet - covered Box Beams. Journal of the Aeronautical Sciences, October, 1936. August Zinnser, Jr., Letter to the Journal of the Aeronautical Sciences, May 15, 1954. PART II. METHODS OF INVESTIGATION. ae Theory of elasticity. 1. The problem of this research project is fundamentally of an experimental nature. In general, such a laboratory test inthe field of structural design and analysis is intended toaid inthe deter- mination of an analytical expression for stress distri- bution. Besides the experimental method, however, analysis is also possible by the mathematical theory of elasticity. In order toa void toog reat mathematical complexities inthis latter method, either fairly specialized cases must usually be considered or assump- tions must be made which will simplify the analytical solution of the problenis. Because of the physical limitations wpon engineering materials, therefore, these approximate theories of elasticity do not always give an exact picture of the distribution of stresses through- out a structure. These methods, nevertheless, give a valuable clue to the stress field in any loaded structure, and in the simpler cases may be considered to give, within practical limits, the exact solution. For the problem under consideration, the simplest type of analytical solution will be obtained if the sheete-stiffener combination is assumed toc omprise a completely two-dimensional stress field. This simpli- fication points to the consideration of an unstiffened panel of thickness t, loaded by means of a uniform pressure w exerted over a distance 2a along the edge of the panel. If it is assumed further that the dimen- sions of the panel are largein comparison with the distance 2a, then the solution obtained for a semi-infinite panel will be valid in the region near the point of application of the load. For this case, the following results may be obtained’: X 4 EP BH GAD Re ad OD GND Ime oD cay coe Smt nw Ue Gu oD oD oD 1 References (2), (4), & (6). (a) Airy's stress function: Fa oe (4 6 - %7°¢,) (>) Stresses at any point A, in directions parallel to the axes OX and OY : 0 = Spl 2($.-G) - ar 2b, + 4-28] Oy = Flz(b,-b) tain 24, -4 2g] Cy = KE (eos 24, - Cov 292) (c) Maximum shear at any point A: Cra. x = nin (9, - 2) » Hence, the lines of constant maximum shear (isochromatics) are circles with centers on OX and passing through O- and Oo The greatest value of the maximum shear is Cex = a which occurs when (¢ -¢,) = % (Lee, on a semicircle with center at O and radius equal to a). (d) The isoclinic lines (see Part II,b(2) for definition) are a set of rectangular hyperbolas passing through 07 and. Ons and whose asymptotes make an angle ($.+ Pe) /y with OX and OY respectively. (e) The trajectories of the principal normal stresses are systems of confocal ellipses and hyperbolas with foci at 07 and, One (f) The trajectories of the maximum shearing stresses (the so-called Luders', or ~G= Slip, lines) form a second set of orthogonal systems of curves, which everywhere cross the principal stress trajectories at an angle of ™, ° Two factors prevent an exact. experimental reproduction of this case. The first is the presence of the boundaries of the finite panel which causes the stress conditions todeviate from those of the ideal case. Secondly, a uniformly distributed load is diffi- cult to obtain since the intensity of the pressure becomes ~ infinite at the edges of a rigid contact. As shown by Timoshenko (reference 5), the distribution of pressure under a rigid contact is given by the expression: _ P where q is the intensity of pressure, P is the total applied load, and the coordinates are the same as above. Notwithstanding these practical limitations, the theoretical s olution is approximately true at a sufficient distance from the sides and bottom of the panel, as has been shown by Coker and Filon (reference Z)e 2. If now we consider the case of a stiffened panel, assuming for our purposes that the stiffener is integral with the panel and of the form shown below, we gee immediately that the exact stress function solution will be considerably more. complex than for the — eect rt Te HTH Gia tt Gee red —e| ae — 20. NN Unstiffened Panel Stiffened Panel case above. Hence, let us attempt to obtain a less involved, though approximate, solution by making such simplifying assumptions as will permit us to deduce the stress field in the stiffened panel from that inthe unstiffened panel. In the first place, since we define unit stress as the stress per unit area, we may say that having given two unstiffened panels of thicknesses t, and to respec- tively (t1 > to), but with other dimensions identical and having equal loads applied in the manner of the first case, the distribution of stresses through the panels will be the same, but the intensity of stress at corresponding points in the two panels will be in the ratio of t/t, « Now, if we assume, for a stiffened panel having a thickness t, at the stiffener, and a panel thickness to, that the stress field remains essentially two-dimensional throughout (i.6., neglect the effects of the discontinuity in thickness due to the presence of the stiffener, as well as the method of transfer of stress from the stiffener into the panel), we may conclude that,(1) the stress distribution in the stiffened oF oe panel will be fundamentally the same as that in the unstiffened panel; (2) the stress intensities at the stiffener will be the same as for the unstiffened panel of thickness t,, and (3) the intensities in the panel will be greater and inthe ratio of t/t, to those in the unstiffened panel. Hence, for a uniformly distributed load w over a stiffener of width 2a and thickness t,, the maximum shearing stress at any point A in the panel of thickness to becomes, according to our deduction: - a ‘ Tmax = t G ~(¢, - $2) In practice, the stiffened sheet must necessarily deviate considerably from the case of the semi-infinite panels; as would be expected, the stress patterns ina finite panel of relatively small dimensions are greatly altered from those of the more ideal case. It is this case of a small panel with a comparatively large stiffener which nas been investigated inthis project by photo- elastic means, to a discussion of which method we now proceed. The assumptions made above, being of a general nature, would indicate that although the stress distri- bution corresponding to a givenapplied load will vary for panels of different dimensions, the only variation for panels of equaldimensions throughout (except for different sheet thicknesses) will, to a first approxima- tion, be in the intensity of the stresses. The tests er made will perhaps help to substantiate this assumption. be The photoelastic method. le General remarks. Photoelasticity is an experimental method which uses polarized light for determining the stresses in transparent models of structural parts loaded either statically or dynamically. It provides a convenient tool for checking the results obtained by means of the mathematical theory of elasticity and for obtaining in- dependent solutions to problems too complex for a reasonably simple mathematical treatment. The great advantage of the photoelastic method is the possibility of observing directly the formation, growth, and in- tensity of internals tresses. Any problem of two-dimen- Sional stress analysis may be investigated solely by ne optical method to completely determine the distribution, magnitude, and direction of the internal f orces. The usefulness of the photoelastic method lies in its adaptability to practical investigation, i.e., in the validity of transfering the results obtained fron the model to the actual structure. According to Alex- ander (reference 1), an accuracy of within 2 per cent is possible, for the stresses at corresponding points are rigorously proportional, provided that, (1) the elastic imit is exceeded in neither the structure nor its model: (2) thee xternal forces are every wherein strict -10= proportion, and (3) the ratio of elastic constants is the same in both materials if the plate contains any holes. Since the mechanism of polarization is actuated by stress alone (i.e., the polarizing effect of the model is independent of strain), the results are fully independent of the stress-strain relations of the materials. ee Working Fundamentals. The optical theory of photoelasticity may be found in a number of texts, among which are references (1), (2), and (3). In undertaking such an investigation as this, one is primarily interested inthe results applicable for direct use; the working fundamentals of the method will, therefore, be briefly summarized here. A list of the necessary laboratory apparatus is given in Part III. If an understanding of the polariza- tion phenomenon may be assumed, we will proceed directly to list the data obtained from the image of the loaded model and the use which is made of that data. isochromatic lines. Referring to Appendix A, photographs 94 - 135A, 1t is seen that there appears on the loaded model a series of alternating bright and dark lines. These so-called "“isochromatics” are shear contour lines; i.¢., lines along which the maximum shearing stress is a constant. The increment of stress value between lines is constant for a given thickness of the models; and by means of a proper calibration -ll- method these lines may be readily evaluated and © max determined at every point. 2 Tsoclinic Lines. The isochromatics have no bearing onthe directions of the stresses inthe material. By a suitable manipulation of the apparatus (described under Procedure), a second series of dark lines, called isoclinies, is obtained. Along each of these the directions of the principal stresses (and, hence, also that of 7 ) are constant. mar Isotropic points. A point at which t he value of (p ~ q) is zero is called an isotropic point. This means that at such a point there is radial tension, radial compression, or zero stress. A locus of such points is termed a neutral line. Since at an isotropic point the directions of the stresses are indeterminate, an infinite number of isoclinics pass through the point. An example of this appears onthe model investigated. Principal stress trajectorics. The directional distribution of each of the principal stresses may be plotted graphically after the isoclinic lines have been determined; the two systems form a network of orthogonal curves. It should be noted that p and q are not necessarily constant along these curves, but vary according to the relations: st = (~-f) cot /F an. Ge-g/) Lan {i Maximum shearing stress trajectories. Since ~12= the maximum shearing stress at any pointacts at an angle Ce. to the directions of p and q, the directional distribution of may be similarly © max plotted. This determines a second network of orthogonal curves which coincide approximately with the lines of slip (the so-called Luders! lines) which appear ina test piece of mild-steel, or other ductile material, when the limit of plasticity is reached, neparation of the principal stresses. The isolation of the individual values of p and g are not so simple, nor so accurate as the (p - q) determination. There are available such analytical methods as separation by graphical integration processes and by plotting the so-called isopachics, lines along which the value of (0 + q) is constant. There are also such mechanical means for determining (p + q) as the membrane method and the lateral strain measurement method. In view of the necessity for placing limits upon the scope of this investigation, no attempt has been made to isolate the values of the principal stresses. From a practical standpoint, in fact, the main value of the photoelastic method lies not in the possibility of obtaining the exact values of the principal stresses, but in giving directly a physical picture of the stress distributions and concentrations. For purposes of reference, however, an analytical method of separating p and q is outlined in Appendix ¢,. PART IIL LABORATORY PROCEDURE &. Apparatus. A complete diagram of the photoelasticity equipment is shown in Fige le Ingeneral, it is the usual polariscope employing: a light source, a polarizer, and an analyser, with their respective quarter - wave plates, a projection screen, and a loading frame. However, the polariscope shown has advantages over the usual type. Two spherical mirrors and condensing lenses are employed to give a large beam of polarized light through the model. The polarizer and analyser are placed at the focal points of their res-~- pective mirrors as shown, with the result that a beam of 20 centimeters in diameter is obtained from polaroids 5 centimeters in diameter. Since the light beam is broken by the use of these mirrors, the light path can be long in actual length yet fit into a small room. A polariscope of this type will operate without distortion to the image if the angle, of , is less than 8 degrees, The light source is a monochromatic 60-watt Sodium Lab-Arc manufactured by the Vapor Lamp Company of the General Electric Company. The monochromatic light together with the quarter - wave plates for the light, produces circularly polarized light and allows the isochromatic lines of the model to bs cbserved ~[4@ without the presence of the isoclinic lines. This light also produces definite light and dark fringes on the screen which are necessary for a clear picture. A set of 5.0 centimeter polaroids are used as polarizer and analyser instead of the usual nicol prismse The polaroid is a transparent material that produces plane polarized light by simple transmission. The degree of polarization of light produced by commercial 20026 inch polaroid is 99.8% effective throughout the visible spectrum. The two polaroids are mounted in individual stands and are free to rotate 180 degrees as required in determining the isoclinic lines. Attached directly to each polaroid is the quarter ~- wave plate inclined. at 45 degrees to the axis of the polaroid. These quarter — wave plates are quickly detachable as the procedure demands. The loading frames are shown in Fig. 2, and Fig. 3 The first is used toa pply pure moment to a simple calibration beam, and hence, the stress per fringe order is determined. The second is the frame in which the model is loaded to 2000 pounds. It consists of a machined flat as a base, a method of applying the load, and a means of determining the load. The loading beam shown is calibrated, giving dial reading against load, and the calibration is shown in Fig. 7. =-15= The sereen of the polariscope is a five by seven inch camera. This gives a ground-glass screen for focusing,as well as a plate holder for pictures. As the optical system of the polariscope is complete within itself, the lens of the camera must be removed, but the iris may be used to good advantage in obtaining a clear image. be Models. A summary of the desired requirements for a material tobe used in making the model is given as follows:+ 1. High stress-optical coefficient. ee Facility of machining into intricate shapes. Se Linear relation of stress to strain within limit of working stresse 4. Uigh elastic constant. 5e Perfect homogeneity (physical and optical). 6. Absence of initial stresses. 7. j<Absence of creep while under constant Stress. 8. No permanent set or residual double refraction upon removal of the applied force. 9. Dependence of double refraction on stress rather than on strain. 10. No development of edge stress aging of finished model. mw aus GED OE UAE Ort OEP ene aad Om ant em ect ee eer ee a ee 7 ad s rt TT s * 2 ~ Ae Ge Solakian, A New Photoeclastic Material, Mechanical Lngineering, December 1955. ll. High transparancy. lg. Adaptability to built-up and reinforced models. 18. Low cost. Bakelite BT61895, manufactured by the Bakelite Corporation, 247 Park Avenue, New York City, has been found to most nearly fit the above require- ments, and it is used for all models in this paper. This bakelite is a clear, white material and lends to machining very readily. It usually is sufficiently free from initial stresses to be used without annealing, and no stresses will be produced during the machining process if the material is properly supported and if small cuts are made with a very sharp, well shaped tool. Actually an 1/8 inch cut was made around the six by twelve inch stock as received from the Bakelite Corporatione This area did contain initial stress, but removing it left the remainder of the sheet free so that no annealing was necessary. Had annealing been necessary, a large hot water bath with circulation provided, was constructed with a gas thermostat control to keep the temperature at 80 degrees C. for four hours and then allow gradual cooling at a rate of not more than 4 degrees per hour. Experiments were carried out on old 3/8 inch stock, and initial stresses were removed satisfactorily. The models used for study are shown in Fig. 5. They are the actual stiffened panel for study and the calibration beam. It should be noted that the comparison strip must be machined from the same sheet as the model in order to eliminate the two factors in the photo- elastic formula; namely, the optical constant and the thickness. Machining was accomplished with a fly cutter. The specimens were mounted on flat boards by cementing, and, hence, ample support was obtained without clamping loads being introduced on the specimen. Machining left the surface covered with circular arcs, the height of the ridges being comparable to a surface prepared with 400 Alundum finishing papere The machined models allowed a very faint picture to be observed on the screen when loaded. ‘This called for finishing of the surfaces through which the light was projected. Polishing of surfaces with a ridge down the middle would be a lengthy job, and rather than do this, a simple means to prepare the surface was sought. Experiment showed that the machined surfaces could be built up with oil or lacquer to a smooth surface, and a high polish and transparency obtained. To verify this statement, a collection of pictures, Fig. 6, is exhibited. These pictures are of the same Set -18-= beam under a constant load, but finished in four different wayse The four pictures in order are: 1) machined surface, 2) machined surface covered with a light oil film, 3) machined surface coated with clear lacquer, and 4) polished surface. The order of thickness of the lacquer film is .001 inch per side, as compared with the specimen thickness of .350 inches. ‘The polishing of the surface in 4) was accomplished in four steps: grinding with 400 Alundum paper, wet, grinding with 500 and 600 Alundum powder respectively on canvas laps, and finishing with Levigated Alumina on a felt lap. The optical constant of the material is not in any way changed,as the distance between fringes, the number of fringes, and the fringe location is the same for each type of finish. The pictures also illustrate the transparency of the different finishes, since the photographic negatives were e xposed the same length of time for allfour pictures. Finishing may be accomplished by coating the model with oil or lacquer, and hours of grinding and polishing time may be saved. ‘The transpar- eney in either case is as good as that of the polished specimen, but oil may prove a little inconvenient to work with. All of the results given in this paper are based upon machined models polished with clear lacquer. -19= Ge Procedure. Calibration beam. The fringe order of the photoelastic material is determined by applying a pure moment to a calibration of comparison beam. This is accomplished through the use of the loading frame, Pige Se By using the flexure f ormule, the stress at any distance from the neutral axis is calculated, and hence the stress determination per fringe is known. ~It is best toalso observe the beam in white (poly- chromatic) light to determine the position of the neutral axise Tests show that it is not necessarily inthe center of the beam (see calculation, calibration beam). A recommended procedure for this calibration is to increase the moment in about five steps to the neighbor- hood of the elastic limit of the material, as a check in determining the stress per fringe. Model loading. the model, Dig. 5, is loaded in three different wayse They are as follows: Type I. Load is applied to the stiffener at the top and absorbed as a distributed load across the bottom. (See Fig. 13.) Type Ii. Load is applied to the stiffener at the top and absorbed as a distributed load across the webs only. (See Fig. 18). Type IIL. Load is applied at thetop of the stiffener and absorbed at the bottom of the stiffener onlye (See Fig. 21). Determination of isochromatic lines. The quarter wave plates are inserted, and isochromatic lines are recorded in all three types of loading. A progressive lead is shown for Type I in five steps from zero to 2000 pounds. See pictures 7A to 13A and 2Ge Pictures 7A and 8A are included to show the model in the unstressed condition before and after loading. For Type II and III, one load of 2000 pounds is applied to eache Determination of isoclinic lines. Removing the quarter-wave plates will give isoclinic lines superimposed on the isochromatic lines. For plotting isoclinics of a model, it is a good plan to record pictures from zero through 90 degrees in 10 degree steps, and make necessary additional pictures where the lines change rapidly. If difficulty is encountered in the interpretation of the isoclinic pictures, it is -advised to view the model in white light, as the dark lines stand out more clearly than they do in mono- chromatic light. The isoclinic pictures are differentiated from the isochromatic pictures by the notation of the angle @Q through which the polaroids have been rotated in each case. Complete series of isoclinic photographs are given in Appendix A for Type I and Type II loadings. It was found advisable to take additional pictures in the range from O=70° to O:80°, since here the isoclinic lines were changing rapidly =2]= in the lower portion of the models elsewhere, the soclinic pictures are given for steps of 10° rotation We of the polaroids. PART IV. R3SULTS. ae Datae le Calibration of loading frame springe P = applied load in Lbe. D = dial gage reading in 5 ye Run No. le 2. Bo Le Reading P 5s) P D P D |’ P D Noe le ll 39 tl 59 196 51 201 Lg Qa 199 85 20% 87 has 99 509 95 Ae 207 = 126 40 127 611-138 60R = 429 Me 590 165 508 166 787 176 | 796 169 5 79T —-204 794 204 O98 21% 997 206 Se | YOOO 243 998 240 | 1199 247 | 1197 24% 7+ | 1196 280) 1211 281 | 1402 288 | Laor 278 Be | 1394 316 | 1397 414 | 1606 3025 | 1599 89416 9. | 1590 352} 1500 349 | 1802 361 | 1794 453 10. | 1793 387 | 1800 385 | 2002 Aor | 1996 390 Tle | 2000 «421 | 2000. 420 | 21907) «4356 | 2190 AO Calibration runs made on "tension machine", Galeit Structures Laboratoryve See Fige 7e for calibration curvee 2e Photographs (see Apvendix A). Series A - Calibration beam loaded in pure bendinge (see ppe ii — iii) Picture Noe Load, Lbs Moment, ine-Lbe. 3A i) fe) MA TL.16 16.9 5A 21 AS 32.2 6A 28.81 4B 62 7- 47261 5604 8- 48.10 72. ae Be 2. Photographs (cont inued). Series ©@ - Model under Type I loading (pp. iv - xiv). Picture No. Dial rdg. Load, lbs. Isoclinic ansle, 8 TA 6) 6) (isochromat ic) 8A ¢) 0 " 9A 4O ALT " 1OA 175 873 " 11a 245 1200 " Lea 417 1600 " L3A 585 2020 " 1c 0 0 " 20 390 2065 u 30 " ne 10 degrees Ag HT ft 20 i} 50 i t 30 tt 60 il 1 40 it 7c nt it} aie) Hy &o " n 6560 it 9 fe) f " 70 i} l rere! "i ft 75 " 1 1 o tt tt 8&6 n Series B - Model under Type II loading (pp. xv - xxi). Picture No. Dial rdge Load. los. Isoclinic angle, § LB 486 2020 (isoehromatic) 23 " u O degrees 3B " " 10 " LB i" it /20 0 5B i} 20 "i 6B " fi Te) " 7B it W 50 tt 8B a " 60 " OB tt u 79 Wi Ll B ff it TTs tt 123 " 1 80 “ i t 4B tt # 90 if Series & - Model under Type TII loading (p. xxii). Picture Noe Dial rdg.- Load, Ibs. 1E 384 2000 (isochromet ie) = Ohm be Calculations and graphical work. 1. The evaluation of 4? y.,, from the calibration beam. It was found that the neutral axis of the calibration beam (i.e., the line along which p- q = 0) varied in location with changes in load and that it moved from the compression edge of the beam, as load was firstapplied, toward the center line of the beam as the load was increased. At the same time, the change in stress per fringe, A? maxs increased. No explanation is advanced for these effects, since they occurred for every type of application of pure bending moment which was tried, irrespective of the position of the beame In order to evaluate the isochromatic lines on the model, a value for 4 One was determined by the following extropolation process. Taking ¢ equal to the distance of the neutral axis from the center line of the beam, a curve of AC mex VSe @€ was plotted from the e xperimentaldata; five points were obtained to give the straight line variation shown in Fige 8. It was assumed that, in order todetermine the fringe scale for the model, this curve should be carried to the point where e = O and AC ax taken there. This is admittedly an arbitrary assumption; for the purposes of this particular investigation, it seems a satisfactory ‘solution. However, for the sake of a logical explanation of the effects noted, a more thorough study of this individual problem might well be made. Since the beam is loaded with a pure moment, tension and compression are found on opposite sides of the neutral axis and may be calculated from the flexure formula: O~ 3 My I Since of = Q% =f » and Yd >» vhe expression for maximum shearing stress becomes: where stress relations). o= oO, Cm a x = Of % (See Appendix B for fundamental Therefore, the increment of shearing stress per fringe is the shearing stress at a given point divided by the fringe order n That is, M e rT o Ne. Yt SAMPLE CALCULATIONS PicrurRe WNo. , In.- lb. . inches 5 (771. >t » db. Jin® , aftine , in. {line Onve. + /b.fin line Tmax > |b. fin. line 4A 76.9 O./QA30 0O.0/Z1 377° 9./326 O.1326 18502 972.6 SA B2.2 0.0577 O. O104 Blt(Oec oO. OT48 O. 0748 Z32.6 116.3 -26- 6A F3.R O.0406 oO. COFFG 4345C O.0F5FG 0.05 73 243.3 124.0 i ed 56. 0.03766 0.0098 S7ZSe 0.0346 0.0445 Z2F3.2 126.6 0.0097 at the same point. ao Vad oO. OFF 74+ O0Fe Q034D 0.0356 257.5 4128,8 2. Evaluation of isochromatics. (Figs. 9-15, 18, 21). Since the increment in maximum shearing stress per fringe has been determined by means of the bakelite calibration beam, the value of @ max &@b any point in the loaced model may now be determined by the simple process of counting the isochromatics which have passed over that point during the loading. Thus, beginning with the fringe of order zero, which maybe readily located by observing under white light the growth of the stress pattern during application of the load, the parameter of the fringe of the nth order, say, is given by the relation: (2 max) on J43R5 rn £485 PER 1" i 3. Isoclinie lines. (Fig. 15, 20). or both Series C and Series B, isoclinic diagram I (see Figs. 14 and 19) were drawn by tracing directly from the corresponding sets of photographs the isoclinic areas shown. Then diagram II for each series was drawn by estimating the position of the center line of each area in diagram I. 4, Trajectories of the principal normal ty stresses. (Fig. 16). The first set of these two families of curves was obtained by drawing between successive isoclinics of parameters OF, @, O;, , ete., 3 the sides of a series of polygons, such that the sides of each polygon made angles of ZC, +0.) : 4(2.+2), ete., vespectively, with OX, the successive vertices lying on the successive isoclinics. These vertices then were points on a principal stress trajectory which could be easily faired in. Ina similar manner, the second system of curves was drawn in orthogonal to the first family of principal stress trajectories. 5. Trajectories of maximum shearing stress. (Fig. 17). In a manner similar to that described under (4), these curves were also drawn for Type I loading. In this case, however, the polygons for the first system of shearing stress trajectories drawn were constructed with sides at angles of [4 (+ @co1/) + VA with the axis OX. Ge Gonclusion. The model used in this project has been in= vestigated sufficiently to give a fairly complete picture of the shear characteristics of the sheet in the type of panel considered, namely an unbuckled sheet reinforced with a light stiffener. The tests made tend to confirm the fact that the stress conditions in the panel are influenced largely by the edge conditions (iee., the dimensions of the panel, the nearness of the edge boundaries either free or loaded, etc.). In = 28-0 addition it was noted that for a load applied in compression through the stiffener, the stress distri- bution in the panel was closely analogous to that in an unstiffened panel with a uniformly distributed edge load over a distance equal to the width of the stiffener. Appendix D contains the data and photographs obtained with one additional model. Due to unavoidable conflicts in the schedule of the Institute shop, this model was not obtained early enough to include the data in the main body of the thesis. ‘These later tests, however, provide an interesting comparison with the first experiments made and serve as confirmation for the above remarks. The experiments conducted indicate that a rurtner and much more complete investigation of stiffened sheet characteristics by the photoelastic method would be of considerable value. The inherent advantages of make it highly adaptable to a study of the stress ns in thin sheet structures below buckling. p: 169) cr a) fle oO = cr foe o Le REFERENCES Alexander, Nicholas. Photoélasticity. Rhode Island State College. Kingston, Rhode Island. 1936. Coker, E- G-, and Filon, L- N- G- A Treatise on Photo} Elasticity. Cambridge University Press. London. 1941. Filon, L. N. G A Manual of Photo-Zlasticity for Engineers. Cambridge University Press. London. 1936. Nadai, Ae Plasticity. MeGraw - Hill Book Company. New York. 1941. Timoshenko, S- Theory of Elasticity. MeGraw - Hill Book Company. New York. 1934. APPENDICES APPENDIX A Photographs. Pertinent data is given on each picture. The following photoelastic photographs are shown: Pages ii - iii. The bakelite calibration beam in pure bending{ isochromatics). Pages iv - viie The model under progressive stages of Type I loading (isochromstics). : Pages viii - xive Isochromatic and isoclinics of the model under Type I loading. Pages xv - xxi-e Isochromatic and isoclinics of the model under Type IT Toadin ge Page xxii. Isochromatie of the model under Type III loading. // 3A t=.350 M=0 AA +-350 M-I6.9 5A t=350 M=32.2 ‘tt c=. 350 M=- 56.4 lV — . NA i — Be ae A P=837 f) ) a 4 < 2) 9A t=.355> P=A4l7 Vi BEES, Vil Sag ONO x O ns al wn) Ln ro) ISA t= Vill come J55 P=2000 W G % 1X OOO0Z=A GGE=? DS oP O00Z=*AS GGE=2 BE P?=L000 6= 370° =F O00C*AS GGE 27 DS = € OD0Z =A GGE =F? OO02= AS GGE=? Fé 9L-°0 002 +a SSE =? IO/ XLV XV XV/ XVI w 55 -O0O 95 i {lt ‘eo o ~ rv S S . X 0 ; us S O-60° P OOO & L£2-QO 000% =f SGE-7? PY | a a : | | YL =O 00OO8%=oS GOE°=?2? Goa oa XX1 - - =? GY | oP OOOC-S GSE | 0228 0OOX=A. GE =? PA XXI1 o00e@ =d GGE APPENDIX B From the (1) Consider a square element of small dimensions with sides reavectively parallel to the reference axes « Ox + OF = const. x Pra t ox Ty = Tr | oxy a . Aw ; G Poo? BO + g sin* a a aa _y - ao roy % oy = (P+ g) y= £(P- 9) sin 26 7 ww ui SF NX J angle of AS degrees. x ~ . bed mak ~ 2 v) as (xx n for gt a Ft + ns Bye O rd © Soot Gey PS aD ra 5 vd Ee OOo & fer ard FD) of] > 7 - & a oe e and at w Ry Ao 8 By $y fa oO ee eget zo *< S a ° S 8 @ mast ht 2. ow ed 72D © ’ g a va opel ue 3 ; yo eft Sue PS ot ot dy i $2 go} © a ae PR we Ge * © 2 47 2 @ ve o 2 o 7 goa Bat fay Song fa 0 3 ay Py ba) MO 43 et eer gt = rol Be Ted a pe? et sed 4d CG oO. yo Et at Oo o. ise god Sei pod o rs a of eh © rch fed Gl go gir 2 © SS 4S a © & . o & . a So = “2 i) Ew 4? So vt “ a eet 2 2 ot os ; aed OY ha a oO Cima x -= elere of cas Gly - G is * rt Bs io et pt ei ve f =] m O © 2 © .w ao ct got oP So © 3° it a i) on oe Om GC & ord & & FO ey i a : Q : . 3 o <, os e coo iS oo 8 aeoarl BL 4o eed ot oO eto Ooo 7 ~ 50 Sod oOo ® & ord geod eid ect wd Seo eg f oe & 2 oe , & Be << edy & pir OM h © 2 fa) ot Oy ,r © ,@ G4 © Sa 5 Sot gg D 4 oO @ Mo EF & 2 D qo er} go erd 1S) gd a a. @ 4 Gaeci on “3 B Ep enerally occurs & Llure Pal 2 i creed g rt fe} i © O° ner * Ga 4 o we © o Sa oO can oS Oo & 3 BD ew Smt 0m o oh > Oo » be a © n cat b a ow 3 eet ou *~% ceo} Oo © oer gc ®;P «+ fe) So 42 os! a. . ow | os ® oH wis ® ed & © @Q aon @ 1 ‘ os oP rat) Qo « ® jo S ' o oO mt © wee IF} : Sa scl OB ord and co ca) a oO C$ yo wn x aD Gi oO fan) ae jd w oN aN o Oo eet Os os £2 ert ae ron << mo > cay a in o 9° } b 47 as cl % re) 3 Q 4 mG et 3 S ce oO ae a or ord = erd ro) © oO ots sed yp yg Ss OM fe ah ooo o oO & ae © gx ° QO 4 Sa oO sed et a ed © DB noe @ oo 49 0 o a x aor 3 qe ba ot ard o Goo 4 = EO @ ed a O ei qo hd Be sy © So a C7 2 2 2 2 s e rd al NS =< iO ‘oO hm Qs pan Py 2. 2. & on o © 0 © o © 3) +3 2 49 42 2 re 43 wm fea mn nm wo com) on £3 This set of pletures, series F, consiste of isochromatic photo- graphs of the second model under each of the three types of loadings for the Ly considered and of calibration beam photographs previous no ‘ee oS Beery 5 fea NS © is @ iS af Om OO EIR OCs Ga @ OO OO % oP gt OD CQ ND GD NO <f OO © oN e¢ @ 8 od BN 8 @ & @ ahh pol rmvyer O O OS NN fe, @ CO Pe Pe £2 6 gy SOA a ON ef OO © e& > 8 BONY god e © tego ond ® Se ey ae a ed Be © A ® HR a Zry & ~~ ss ge ¢ bw Bs inerement is taken to bee Lbee/ine /Line 20606 4a Gna x The compressive Load on the model im each cage was 2000 lbs. The and 6 2 rea phe OOF Y oh ; en on the giv 6) hrowetie is YO ¥ t Picture I fe ft oy bhi g i for evaluatin FEC Sines the type of ca accurate oe es stress patterns in t Quantitative anal ines) does not ¥ i o the me g- or OF) Avery simple m contours are ideritical. lar loa Or Sanit g atives £ insofar as stress, wf? OL aa is) the 4 Thie fact indicates the Ehe thick- nd ¢@ ion a it pos oe) Ga .e) a oe 0 fet £ “gy wel s 9 44 ot LECuss vd given applied load increases for a decrease in the thickness of the panel. Because of the inaccuracy of the stress evaluation ¥ dus to the necessity for eporoximating the values of DTC inax 2 it is impossible unfortunately to determine the exact relation for this varilatione XX1K SF M=FH18 C230 XXX FL N= FP €=. FO BF M=5E5 C= 230 XXX! XXXII =aS eS “oe ce XXXII 7 ma eh anv oxtyon cg = ME) SHOURTE sarava) Sdose in TRAOWD % OATI & € (MELE T (SULLEN 2 bf SHVld HAVA waaEvAd (m BOUIN ZO UNIO IvOOd LV GALVOOT j B Say (TW HOMMIN GNVOT GNET 30 GNIQd TvOOd ZY asdvooT) tw oy SNe any Ut ST JO Ao Troy — apes Fy LLSVM - GZS 25 LNANdUINOS ALIO 2 |v = “iS ; ~T | - BALL BEARING | Le LEAF ; = : SPRING ok a \pertection ARM ° J NOBEL mtn bl Zz cp) Not: To Scae COMPRESSION FRAME FIG. 2 ] oo N COUNTER. BALLANCE LITTTTT ST TTT WEIGHT PAN VIT77 ORT 77 CALIBRATION BEAM a Ln ‘ 4 ro ‘ S ee e a v m ny FIC. €. QEPARPAITUS PCVIOTO-ELASTICIT ¥ SFE CINIEN S roe CIVEW AS oR PER PICTURE { Ie ail | | tm ara — Le ; eae 74 ee _ LH 55> / Ig ba gp po es? “1 pp x { So ~ 4 od mt rom / 3 a | ro MODEL. (—— GIVE / AS PER FICTORE f { ] 1 | CAL/BEAATIOM BEANT Z1IC. 3 FIC 2 MACHINE - O1L M= 56.4% t=.350 FI6.F POLISHED SURFACE | M-G6.F €- 350 A-689 FIG. 6 ee ee arenes | fe de ef fe 0. WE. BEAM READING 1 ee eee we pte ry PEMOQING ALVBFRRATION CURVE “ ABO. | b f ! fo. J ae ep ce fee be eenenneee _ [: cS Os ie Gn One iS tt + 43 “3 SINE +3} | opNE Sul: SERIES C EVALUATION OF /SOCAHROMATIC LINES P22 F/P POUNDS - Caan = fFLS ML | LBSLi = | PICTURE, GA ee, SSS (NCA SCALE 24 OBO SIZE F/G. 7 a ae \/ Cutax 2@SIS Sy LBS //M* eS ‘ \ \ /s aT PICTURE 104 Lo fo a EVALUATION OF [SOCHRONMATIC LINES Cmax 2 JFS ES mM LBS /IN® [{--—- | PICTURE SSA Ce FSIS /NCA SOCAL EF LOSO Sl E FILE. Sf SERIES C LVALUOA TION OF ISOCHROMATIC LINES TT) f= f 6OO POUNDS Ny Cmax = SFIS. S vr, LES Jin Freel Pra : 2 po pete ff hg Fx PFS few fo mg fof f eo EP es he I MES ACE EK cA CAT ¢ A VISTA ER LFEPA M3 COOC POUNDS ar nea lFZ 5S A5 LBSSIN ~ Co SALLAKNAL A LALLA AA LLLAELLE LUCTUPE 2C C2 GSE INGA SOALE ® LOBOS Peal 7 e LYMM L STE SS IAM DIRECTIONS O&O + ™% | MAXIMUM SHEARS 1M DIRECTIONS O+% & O+3hy, on a Joe. fee ERIES c. SSOCLINIC LIAG RAN IT | f? = AGO? LBS, °° | } RR ee ad I SING ro ae OS ae 0 Hy op >> | . ee af NASA 20? S << _ PRINCIPAL STRESS Iv DIRECTIONS 2 6 Er TE. MM AKAM SHEAR (NM DIRECTIONS 84788 Or 204 FIC IS SERIES C-~ TRASECTORIES OF THE LIM CEA L MOR MMIAL STAE SSES PE Q I = C000 28s. = = | Ze | Ss z AGUSUORUOS DODUROCUTUOUCTRURRERORET CRAP HI CAL COMETRUCZIONM FROM (SOCLIN I LIACRALY awe “ Ape C19. £@ SERIES C— TRAJECTORIES OF TH APAX/MUA SHEAPING STRESSES ) Pp = 2000 L485. ‘ , Oot ae od ae Se ON q Cae gg EN LP mms ,m@ x os ar a oh - € aa t W.. s t i i} CETTE TA SSR GRGCHRRGCRNSTT BAA HCA L COMMS Ti BY CTIOM FROM Co ae Ok Khe 7 a LAA" SS AG fet SERIES LE EVALUATION OF ISOCHROMATIC LIMES bay 0° = OOD POUNDS i é Cursx = JIS AD LES/SIN M, PICT UE Ef &8 2. ISS IMC SE1 RIE S a. SIO’, LIMTE LIA CRAM LZ P=2O00LBS. a \ \ \ \ \ \ \ \ \ PRIVEE £ STRESS 1M DIRECTION | a £ Er % MAKI MU ML SHEAR (MM OIREL Tiete PL Ge FOO e. LVEVF — SEAIES B : mt PRS : ; “gs0" ) tA a! = | a pee ae en LE. 3 Y. aN | PRINCIPAL STRESS 1 OURECTION @ £ OF Fe "MAXIMUM SHEAR IN DIRECTION 8+ Ta 0+ 27H LV. 20 ae fa Nava Cia re f Fay ee SLOVO FA iy ly Si) arma an re tran Swat ffi on ge B 4A at LG. oe 4 LO } : | 4 i i ! af Lod | a i" ?. : | 1 coe anna ee an if i + | i : t t wa a | EACBRATION, 2 cee ae wt