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AD-AI05 598 MARYLAND UNIV COLLEGE PARK SCHOOL OF ENGINEERING B2/iSTRESS DISTRIBUTION AROUND A CIRCULAR HOLE IN SQUARE PLATES. F/6 T0/11SEP Ri M ERICKSON, A UEL, AAA OR-l-Rl4 (Uoi
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OFFICE OF NAVAL RESEARCH x TDEPARTMENT OF THE NAVY
M. /Erickson A. J. /DurelljSI NTDC205K. IRajaiah CNRC ~~~-
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4,LLS run P. 27L
14
OPlMIZA ION OF EIH SHAPE OF A HOLE
IN A SOUARE PLATE AND IN A CIRCULAR SHELL
REPORTS 55, 56 AND 57 D T C\ OCT 1 41981 .
The following three reports are being distributed together and reprints
of the papers previously published with their texts, in scientific journals,
are used for the purpose. Circumstantial reasons, primarly cost, made this
procedure advisable. One of the reports (56) deals with the optimization
of the shape of an origionally circular hole present in a square plate,
loaded uniformly and uniaxially in its plane. A second report (57) deals
with the optimization of the shape of the hole, when the hole is present
in a cylindrical shell subjected to axial tension. The third report (55)
deals with the necessary background for the optimization of the hole in
the square plate.
The abstracts of the previously published reports, prepared with O.N.R.
support, are presented only once, before the text of the three present reports.
The O.N.R. reports distribution list is also presented only once, at the
end. A Report Documentation Page follows each individual report.
FAccession For
N~SGRA&I$ DTIC TABU:itnnounced F33uSt~fication ..
Distribution/
Availability CodesImail and/or
'Spec lalII
II I* :... -
Previous Technical Reoorts to the Office of Naval Research
I. A. J. Durelli, "Development of Experimental Stress Analysis Methods toDetermine Stresses and Strains in Solid Propellant Grains"--June 1962.Developments in the manufacturing of grain-propellant models arereported. Two methods are given: a) cementing routed layers andb) casting.
2. A. J. Durelli and V. J. Parks, "New Method to Determine RestrainedShrinkage Stresses in Propellant Grain Models"--October 1962.The birefringence exhibited in the curing process of a partiallyrestrained polyurethane rubber is used to determine the stress associatedwith restrained shrinkage in models of solid propellant grains partiallybonded to the case.
3. A. J. Durelli, "Recent Advances in the Application of Photoelasticity inthe Missile Industry"--October 1962.Two- and three-dimensional photoelastic analysis of grains loaded bypressure and by temperature are presented. Scme applications to theoptimization of fillet contours and to the redesign of case joints arealso included.
4. A. J. Durelli and V. J. Parks, "Experimental Solution of Some MixedBoundary Value Problems"--April 1964.Means of applying known displacements and known stresses to the boundariesof models used in experimental stress analysis are given. The applica-tion of some of these methods to the analysis of stresses in the fieldof solid propellant grains is illustrated. The presence of the "pinchingeffect" is discussed.
5. A. J. Durelli, "Brief Review of the Stare of the Art and Expected Advanceip Experimental Stress and Strain Analysis of Solid Propellant Grains"--April 1964.A brief review is made of the state of the experimental stress and strainanalysis of solid propellant grains. A discussion of the prospects forthe next fifteen years is added.
6. A. J. Durelli, "Experimental Strain and Stress Analysis of Solid PropellantRocket Motors"--March 1965.A review is made of the experimental methods used to strain-analyze solidpropellant rocket motor shells and grains when subjected to differentloading conditions. Methods directed at the determination of strains inactual rockets are included.
7. L. Ferrer, V. J. Parks and A. J. Durelli, "An Experimental Method to AnalyzeGravitational Stresses in Two-Dimensional Problems"--October 1965.Photoelasticity and moir' methods are used to solve two-dimensional problemsin which gravity-stresses are present.
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8. A. J. Dureli, VI. T. Parks and C. J. del Rio, "Stresses in a Square SlabBonded on One Face co a Rigid Plate and Shrunk"--November 1965.A square epoxy slab was bonded to a rigid plate on one of its faces inthe process of curing. In the same process the photoelastic effectsassociated with a state of restrained shrinkage were "frozen-in."Three-dimensional photoelasticity was used in the analysis.
9. A. J. Durelli, V. J. Parks and C. J. del Rio, "Experimental Determinationof Stresses and Displacements in Thick-Wall Cylinders of ComplicatedShape"--April 1966.Photoelasticity and moir4 are used to analyze a three-dimensional rocketshape with a star shaped core subjected to internal pressure.
10. V. J. Parks, A. J. Durelli and L. Ferrer, "Gravitational StressesDetermined Using Immersion Techniques"--July 1966.The methods presented in Technical Report No. 7 above are extended tothree-dimensions. Immersion is used to increase response.
11. A. J. Durelli and V. J. Parks. "Experimental Stress Analysis of LoadedBoundaries in Two-Dimensional Second Boundary Value Problems"--February 1967.The pinching effect that occurs in two-dimensional bonding problems,noted in Reports 2 and 4 above, is analyzed in some detail.
12. A. J. Durelli, V. J. Parks, H. C. Feng and F. Chiang, "Strains andStresses in Matrices with Inserts,"-- May 1967.Stresses and strains along the interfaces, and near the fiber ends, fordifferent fiber end configurations, are studied in detail.
13. A. J. Durelli, V. J. Parks and S. Uribe, "Optimization of a Slot EndConfiguration in a F-inite Plate Subjected to Uniformly. DistributedLoad,"--June 1967.Two-dimensional photoelasticity was used to study various elliptical endsto a slot, and determine which would give the lowest stress concentrationfor a load normal to the slot length.
14. A. J. Durelli, V. J. Parks and Han-Chow Lee, "Stresses in a SplitCylinder Bonded to a Case and Subjected to Restrained Shrinkage,"--January 1968.A three-dimensional photoelastic study that describes a method andshows results for the stresses on the free boundaries and at thebonded interface of a solid propellant rocket.
15. A. J. Durelli, "Experimental Stress Analysis Activities in SelectedEuropean Laboratories"--August 1968.through several European countries. A list is given of many of the
laboratories doing important experimental stress analysis work and ofthe people interested in this kind of work. An attempt has been madeto abstract the main characteristics of the methods used in some ofthe countries visited.
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16. V. J. Parks, A. J. Durelli and L. Ferrer, "Constant Acceleration Stressesin a Composite Body"--October 1968.Use of the immersion analogy to determine gravitational stresses intwo-dimensional bodies made of materials with different properties.
17. A. J. Durelli, J. A. Clark and A. Kochev, "Experimental Analysis of HighFrequency Stress Waves in a Ring"--October 1968.A method for the complete experimental determination of dynamic stressdistributions in a ring is demonstrated. Photoelastic data is supple-mented by measurements with a capacitance gage used as a dynamic lateralextensometer.
18. J. A. Clark and A. J. Durelli, "A Modified Method of Holographic Inter-ferometry for Static and Dynamic Photoelasticity"--April 1968.A simplified absolute retardation approach to photoelastic analysis isdescribed. Dynamic isopachics are presented.
19. J. A. Clark and A. J. Durelli, "Photoelastic Analysis of Flexural Wavesin a Bar"--May 1969.A complete direct, full-field optical determination of dynamic stressdistribution is illustrated. The method is applied to the study offlexural waves propagating in a urethane rubber bar. Results arecompared with approximate theories of flexural waves.
20. J. A. Clark and A. J. Durelli, "Optical Analysis of Vibrations inContinuous Media"--June 1969.Optical methods of vibration analysis are described which are independentof assumptions associated with theories of wave propagation. Methods areillustrated with studies of transverse waves in prestressed bars, snaploading of bars and motion of a fluid surrounding a vibrating bar.
21. V. J. Parks, A. J. Durelli, K. Chandrashekhara and T. L. Chen, "StressDistribution Around a Circular Bar, with Flat and Spherical Ends,Embedded in a Matrix in a Triaxial Stress Field"--July 1969.A Three-dimensional photoelastic method to determine stresses in compositematerials is applied to this basic shape. The analyses of models withdifferent loads are combined to obtain stresses for the triaxial cases.
22. A. J. Durelli, V. J. Parks and L. Ferrer, "Stresses in Solid and HollowSpheres Subjected to Gravity or to Normal Surface Tractions"--October 1969.The method described in Report No. 10 above is applied to two specificproblems. An approach is suggested to extend the solutions to a class
of surface traction problems.
23. j. A. Clark and A. J. Durelli, "Separation of Additive and SubtractiveMoiri Patterns"--December 1969.A spatial filtering technique for adding and subtracting images of severalgratings is described and employed to determine the whole field ofCartesian shears and rigid rotations.
Ii iv "'
2u. R..;. Sanford and A. 1. Durelli, "Interoretation of Fringes in Stress-iolo-Interferomerry"--,Tul7 137').
Errors associated with interpretin; stres-holo-interferomety patternsas the supertosition of isopachics (wiTh half order fringe shifts) andisochromatics are analy:ed theoretically and illustrated with computergenerated holographic interference patterns.
25. J. A. Clark, A. J. Durelli and P. A. Laura, "On the -ffec- of TnitialStress on the Propagation of Flexural Waves in Elastic RectangularBars"--December 1370.Experimental analysis of the propagation of flexural waves in prismatic,elastic bars with and without prestressing. Tne effects of prestressingby axial tension, axial compression and pure bending are illustrated.
2S. A. J. Durelli and J. A. Clark, "Ex.perimental Analysis of Stresses in aBuoy-Cable System Using a Birefringent Fluid"--February 1971.An extension of the method pf photoviscous analysis is presented whichpermits nuantitative studies of strains associated with steady statevibrations of immersed structures. The method is applied in aninvestigation of one forn of behavior of buoy-cable systems loaded bythe action of surface waves.
27. A. J. Durelli and T. L. Chen, "Isplacements and Finite-Strain Fields ina Sphere Subjected to Large Defor.ations"--February 1972.Displacements and ztrains (ranging from 0.001 to 0.50) are determined ina polyurethane sphere subjected to several levels of diametral compression.A 500 lines-per-inch grating was embedded in a meridian plane of theaphere and moir4 effect produced with a non-deformed master. The maximumapplied vertical displacement reduced the diameter of the sphere by 27per cent.
28. A. .1. Durelli and S. Machida, "Lcresses and Strain in a Disk with VariableModulus of Elasticity"--March 1972A transparent material with variable modulus of elasticity has beenmanufactured that exhibits good photoelastic properties and can also bestrain analyzed by moiri. The results obtained suggests that the stressdistribution In the disk of variable E Is practically the same as thestress distribution in the homopeneous disk. It also indicates that thestrain fields in both cases are very different, but that it is possible,approximately, to obtain the stress field from the strain field using thevalue of E at every point, and Ilooke's law.
29. A. J. Durelli and .1. Bultrapo, "State of Stress and Strain In a RectangularBelt Pulled Over a Cylindrical Pulley"--June 1972.Two- and three-dimensional pliotoelasticitv as well as electrical strainpages, dial pages and micrometers are used to determine the stress distri-bution in a belt-pulley system. Contact and tangential stress for variouscontact angles and friction coefficients are piven.
VV
30. T. L. Chen and A. J. Durelli, "Stress Field in a Sphere Subjected tolarge Oeformationw"--June 1972.Strain fields obtained in a sphere subjected to large diametral compressionsfrom a previous paper were converted into stress fields using two approaches.First, the concept of strain-energy function for an isotropic elasticbody was used. Then the stress field was determined with the Hookean
prpe natural sress-natuhal strain relation. The results so obtainediwere also compared.
31. A.L. Cue a . J. Parks and H. M. Hasseem, "Helices Under Load"--July 1973.Previous solutions for the case of close coiled helical springs and forhelices made of thin bars are extended. The complete solution ispresented in gtaphs for the use of designes. The theoretical developmentis corelated with exper b pnts.
32. T.L. Chen and A. J. Durelli, "Displacements and Finite Strain Fields ina Hollow Sphere Subjected to Large Elastic Deformations"--September 1973.The same methods described in No. 27, were applied to a hollow spherewith an inner diameter one half the outer diameter. The hollow spherewas loaded up to a strain of 30 per cent on the meridian plane and areduction of the diamete by 20 per cent.
33. A.J. Durelli, H. H. Hasseem and V. J. Parks, "New Experimental Methodin Three-Dimensional Elastostatics"--December 1973.A new material is reported which is unique among three-dimensionalstress-freezing materials, in that, in its heated (or rubbery) state
it has a Poisson's ratio which is appreciably lower than 0.5. For aloaded model, made of this material, the unique property allows thedirect determination of stresses from strain measurementsc-taken atinterior points in the model.
3. J. Wolak and V. J. Parks, "Evaluation of Large Strains in IndustrialApplications"--April 1975.It was shown that Mohr's circle permits the ransformation of strain fromon* axis of reference to another, irrespective of the magnitude of thestrain. and leads to the evaluation of the principal strain componentsfrom the measurement of direct strain in three directions.
35. A. J. Durelli, "Experimental Stress Analysis Activities in SelectedEuropean Laboratories"--April 197S.Continuation of Report No. 15 after a visit to Belgium, Holland, Germany,France, Turkey, England and Scotland.
36. A. J. DurelLi, V. J. Parks and J. 0. Bihler-Vidal, "Linear and Non-linearZlastic and Plastic Strains in a Plate with a Big Hole Loaded Axially inits Plane"--July 1975.Strain analysis of the ligament of a plate with a big hole indicate&.thatboth geometric and material non-linearity may take place. The strainconcentration factor was found to vary from 1 to 2 depending on the levelof deformation.
vi
0• S .
37. A. J. Durelli, V. ?avlin, J. 0. Bihler-Vidal and G. Ome, "Elastostaticsof a Cubic Box S;,bjected to Concentrated Loads"--August 1975.Analysis of experimental strain, stess and deflection of a cubic boxsubjected to concentrated loads applied at the center of two oppositefaces. The ratio between the inside span and the wall thickness wasvaried between approximately 5 and 121.
38. A. J. Durelli, V. J. Parks and J. 0. BUhler-Vidal, "Elastostatics ofCubic Boxes Subjected to Pressure"--March 1976.Experimental analysis of strain, stress and deflections in a cubic boxsubjected to either internal or external pressure. Inside span-to-wallthickness ratio varied from 5 to 14.
39. Y. Y. Hung, J. D. Hovanesian and A. J. Duelli, "New Optical Method toDetermine Vibration-Induced Strains with Variable Sensitivity AfterRecording"--November 1976.A steady state vibrating object is illuminated with coherent light andits image slightly misfocused. The resulting specklegram is "time-integrated" as when Fourier filtered gives derivatives of the vibrationalamplitude.
40. Y. Y. Hung, C. Y. Liang, J. D. Hovanesian and A. J. Durelli, "CyclicStress Studies by Time-Averaged Photoelasticity"--November 1976."Time-averaged isochromatics" are formed when the photographic film isexposed for more than one period. Fringes represent amplitudes of theoscillating stress according to the zeroth order Bessel function.
41. Y. Y. Hung, C. Y. Liang, J. D. Hovanesian and A. J. Durelli, "Time-Averaged Shadow Moir Method for Studying Vibrations"--November 1976.Time-averaged shadow moir permits the determination of the amplitudedistribution of the deflection of a steady vibrating plate'.
42. J. Buitrago and A. J. Duelli, "On the Interpretation of Shadow-Moir4Fringes"--April 1977.Possible rotations and translations of the grating are considered Ln ageneral expression to interpret shadow-moird fringes and on thesensitivity of the method. Application to an inverted perforated tube.
43. J. der Hovanesian, "18th Polish Solid Mechanics Conference." Published inEuropean Scientific Notes of the Office of Naval Research, in London,England, Dec. 31, 1976.Comments on the planning and organization of, and scientific content ofpaper presented at the 18th Polish Solid Mechanics Conference held inWisla-Jawornik from September 7-14, 1976.
Lt4. A. J. Durelli, "The Difficult Choice,"--May 1977.The advantages and limitations of methods available for the analysesof displacements, strain, and stresses are considered. Comments aremade on several theoretical approaches, in particular approximatemethods, and attention is concentrated on experimental methods: photo-elasticity, moiri, brittle and photoelastic coatings, gages, grids,holography and speckle to solve two- and three-dimensional problems inelasticity, plasticity, dynamics and anisotropy.
vi i
45. C. Y. Liang, Y. Y. Hung, A. J. Durelli and J. D. Hovanesian,"Direct Determination of Flexural Strains in Plates Using ProjectedGratings, "--June 1977.The method requires the rotation of one photograph of the deformedgrating over a copy of itself. The moird produced yields strains byoptical double differentiation of deflections. Applied to projectedgratings the idea permits the study of plates subjected to much largerdeflections than the ones that can be studied with holograms.
46. A. J. Durelli, K. Brown and P. Yee, "Optimization of GeometricDiscontinuities in Stress Fields"-March 1978.The concept of "coefficient of efficiency" is introduced to evaluatethe degree of optimization. An iddal design of the inside boundary ofa tube subjected to diametral compression is developed which decreasesits maximum stress by 25Z, at the time it also decreases its weight by10%. The efficiency coefficient is increased from 0.59 to 0.95.Tests with 4 brittle material show an increase in strength of 20%. Anideal design of the boundary of the hole in a plate subjected to axialload reduces the maximum stresses by 26% and increases the coefficientof efficiency from 0.54 to 0.90.
47. J. D. Hovanesian, Y. Y. Hung and A. J. Durelli, "New Optical Methodto Determine Vibration-Induced Strains With Variable Sensitivity AfterRecording"--May 1978.A steady-state vibrating object is illuminated with coherent light andits image is slightly misfocused in the film plane of a camera. Theresulting processed film is called a "time-integrated specklegram."When the specklegram is Fourier filtered, it exhibits fringes depictingderivatives of the vibrational amplitude. The direction of the spatialderivative, as well as the fringe sensitivity may be easily and continu-ously varied during the Fourier filtering process. This new method isalso much less demanding than holographic interferometry with respect tovibration isolation, optical set-up time, illuminating source coherence,required film resolution. etc.
48. Y. Y. I1unp and A. J. Durelli, ": Imultaneous Determination of ThreeStrain Components in Speckle Interferometry Using a Multiple ImageShearing Camera,"--September 1978This paper describes a multiple image-shearing camera. Incorporatingcoherent light illumination, the camera serves as a multiple shearingspeckle interferometer which measures the derivatives of surfacedisplacements with respect to three directions simultaneously. Theapplication of the camera to the study of flexural strains in bentplates is shown, and the determination of the complete state of two-dimensional strains is also considered. The multiple image-shearingcamera uses an interference phenomena, but is less demanding thanholographic interferometry with respect to vibration isolation and thecoherence of the light source. It is superior to other speckletechniques in that the obtained frinpes are of much better quality.
vii
I.I
49. A. J. Durelli and K. Rajaiah, "Quasi-square Hole With Optimum Shapein an Infinite Plate Subjected to In-plane Loading"-January 1979.This paper deals with the optimization of the shape of the cornersand sides of a square hole, located in a large plate and subjectedto in-plane loads. Appreciable disagreement has been found betweenthe results obtained previously by other investigators. Using anoptimization technique, the authors have developed a quasi-squareshape which introduces a stress concentration of only 2.54 in auniaxial field, the comparable value for the circular hole being 3.The efficiency factor of the proposed optimum shape is 0.90. whereasthe one of the best shape developed previously was 0.71. The shapealso is developed that minimizes the stress concentration in thecase of biaxial loading when the ratio of biaxiality is 1:-i.
50. A. J. Durelli and K. Rajaiah, "Optimum Hole Shapes in Finite PlatesUnder Uniaxial Load,"-February 1979.This paper presents optimized hole shapes in plates of finite widthsubjected to uniaxial load for a large range of hole to plate widths(D/W) ratios. The stress concentration factor for the optimizedholes decreased by as much as 44% when compared to circular holes.Simultaneously, the area covered by the optimized hole increasedby as much as 26% compared to the circular hole. Coefficients ofefficiency between 0.91 and 0.96 are achieved. The geometries ofthe optimized holes for the D/W ratios considered are presented ina form suitable for use by designers. It is also suggested thatthe developed geometries may be applicable to cases of rectangularholes and to the tip of a crack. This information may be ofinterest in fracture mechanics.
51. A. J. Durelli and K. Rajaiah, "Determination of Strains inPhotoelastic Coatings,'--May 1979Photoelastic coatings can be cemented directly to actual structuralcomponents and tested under field conditions. This important advantagehas made them relatively popular in industry. The information obtained,however, may be misinterpreted and lead to serious errors. A correctinterpretation requires the separation of the principal strains and sofar, this operation has been found very difficult. Following a previouspaper by one of the authors, it is proposed to drill small holes in thecoating and record the birefringence at points removed from the edge of
the holes. The theoretical background of the method is reviewed; the
technique necessary to use it is explained and two applications aredescribed. The precision of the method is evaluated and found satisfactoryin contradiction to information previously published in the literature.
ix
52. A. J. Durelli and K. Rajaiah, "Optimized Inner Boundary Shapes in
Circular Rings Under Diametral Compression,"--June 1979.
Using a method developed by the authors, the configuration oi the inside
boundary of circular rings, subjected to diametral compression, has been
optimized, keeping cleared the space enclosed by the original circular
inside boundary. The range of jiameters studied was 0.33 4 ID/OD 4 0.7.
In comparison with circular rings of the same ID/OD, the stress concentra-
tions have been reduced by about 30%, the weight has been reduced by about
10% and coefficients of efficiency of about 0.96 have been attained. The
maximum values of compressive and tensile stresses on the edge of the hole,
are approximately equal, there are practically no gradients of stress
along the edge of the hole, and sharp corners exhibit zero stress. The
geometries for each ID/OD design are given in detail.
53. A. J. Durelli and K. Rajaiah, "Lighter and Stronger,"--February 1980.A new method has been developed that permits the direction design of shapes
of two-dimensional structures and structural components, loaded in their
plane, within specified design constrains and exhibiting optimum distributionof stresses. The method uses photoelasticity and requires a large field
diffused light polariscope. Several problems of optimization related tothe presence of holes in finite and infinite plates, subjected to uniaxialand biaxial loadings, are solved parametrically.
Some unexpected results have been found: 1) the optimum shape of a largehole in a bar of finite width, subjected to uniaxial load, is "quasi" square,but the transverse boundary has the configuration of a "hat"; 2) for the
small hole in the large plate, a "barrel" shape has a lower s.c.f. than
the circular hole and appreciably higher coefficient of efficiency; 3) the
optimum shape of a tube, subjected to diametral compression, has small"hinges" and is much lighter and stronger than the circular tube. Applicationsare also shown to the design of dove-tails and slots in turbine blades and
rotors, and to the design of star-shaped solid propellant grains for rockets.
54. C. Brdmond and A. J. Durelli, "Experimental Analysis of Displaceme.nts and
Shears at the Surface of Contact BEtween two Loaded Bodies,"--July 1980.
The displacements which exist at the contact between two loaded 'cdies depend
on the geometry of the surface of contact, the type of the loading and the
propzrty of the materials. A method has been developed to deterrine these
displacements experimentally. A grid has been photographically printed on
an interior plane of a transparent model of low modulus of elasticity. The
displacements were recorded photographically and the analysis was conducted on
the photbgraphs of the deformed grids. Shears were determined from the changein angles. The precision of the measurements at the interface is estimated
to be plus or minus 0.05mm. Examples of application are given for the casesof loads applied normally and tangentially to a rigid cylindrical punchresting on a semi-infinite solft plate. Important observations can be made
on the zones of friction and of slip. The proposed method is three-dimensionaland the distributions can be obtained at several interior planes by changing
the position of the plane of the grid. The limitations of the method are
pointed out. The possibility of using gratings (12 to 40 Ipmm) is considered
as well as the advantages of using moir6 to analyze the displacements.
x
5. M. Erickson and 2 . )tirel1 "St r .Fs i ;t ribut itn Around a CircularHole in SqcIi rt, 1l.,tes, I(.idid Uniform] ' ii , 'nt * or, two Oppc. iteSidLS of the Souare"--'r1,t o ,],.u strts, :tr int iu., around - ( 'rI. larhole, located in the center of ,1 square plate, ha.; becn dotcmr- i(.photot-l;!-;tical1v fpr the (as- of the plate load'cIl ntfurnly n two
0'-U i I. side . Th, study WA; conduCt., d pararotrifally for a large rangeof tht r.tif, ',: tbi, side of Lhe 'qu;,r( to the dit,-,ter of th( E , 1(-. eresults tbta ncd p. rl it the determinatLLw of t tbe 1.t IeEses for 1,.: I 'axialcondition z.nd ,erifv , rt.',[ Lis sft ion r .tain,. fur the cai:-i . tire
press1urized ho . ' :ija ,.- t I , t al p roc.dt ir i.s ),r eC y d , cri ed56. A. J. Durelli, M. Erickson and K. Rajaiah, " "This paper presents the
shapes that will optimize the stress distribution about central holes insquare plates subjected to uniform load on two opposite sides of the plate.The study is conducted for a large range of bole to plate widths ratios (D/W).The stress concentration factor for the optimized holes decreased by asmuch as 21% when compared to the one associated with a circular hole.Simultaneously, the weight of the plate with optimized hole is reduced byas much as 36% as compared to the circular hole. Coefficient of efficiecyof around 0.92 is achieved for all D/W ratios. The geometry of theoptimized holes are presented in a form suitable for use by designers.
xi
STRESS DISTRIBUTION AROUND A CIRCULAR HOLE IN SQUARE
PLATES. LOADE) UNIFORMLY IN THE PLANE, ON TWO
OPPOSITE SIDES OF THE SQUARE
by
M. Erickson and A. J. Durelli
Sponsored by
Office of Naval ResearchDepartment of the NavyWashington, D.C. 20025
on
Contract No. N00014-81-K-0186U.N. Project No. SF-CARS
Report No. 55
School of EngineeringUniversity of MarylandCollege Park, Md. 20742
STRESS DISTRIBUTION AROUND A CIRCULAR HOLE IN SQUARE PLATES, LOADED
UNIFORMLY IN THE PLANE, ON TWO OPPOSITE SIDES OF THE SQUARE
by M. Erickson and A. J. Durelli
TABLE OF CONTENTS
ABSTRACT...................................
INTRODUCTION........................... .. .
TEST PROCEDURE............................ ..
ACKNOWLEDGMENTS ........................
REFERENCES. . . . . . . . . . . . . . . . . . . . . . .
Stress Distribution Around aCircular Hole in Square Plates,Loaded Uniformly in the Plane,on Two Opposite Sides of theSquare
M. Erickson' and A. J. Durell 2
The complete stress distribution around a circular hole. located in BRIEF NOTESthe center of a square plate. has been determined photoelasticallyfor the case of the plate loaded uniformly on two opposite sides Thestudy was conducted parametrically for a range of the ratio DIW ofthe diameter of the hole to the side of the square from 0.20 to 0.83. .. ..........The results obtained permit the determination of the stresses for . $ /any biaxial condition and verify a previous solution obtained for the * . *case of the pressurized hole. The experimental procedure is brieflydescribed. ,
IntroductionThe classical problem of the stress distribution around a circular T
hole in an infinite plate subjected to a uniaxial uniform loading in the "plane of the plate was solved by Kirsch (11 in closed form. The ap-preciably more complicated case of the finite plate with the circularhole was solved by Howland [21 using an infinite series solution, but * 0 .results were evaluated only for DIW < 0.5, D being the diameter of v _-the hole and W the width of the plate. The distribution of stress for - A .... A A
Fig. Losing devi used to apply udorm pressure to two oppoeie sidesof a eluae plate
the cases when DIW > 0.5 was obtained experimentally by Wahl andBeeuwkes 131. The stress-concentration factors referred to both thegross area and the net area, for the total range of D/W values are givenin 141. The case of the very large hole in the plate, when D/W ap- 3.7 ., 1.sproaches one presented some problems of interpretation, which have
been dealt with in 151.The stress distribution for the case of a square plate with a circular
hole was solved experimentally [61 when a uniform pressure is appliedinside the hole, or what is equivalent 171, when the four sides of thesquare are subjected to uniform pressure. The problem of the squareplate with a circular hole, subjected to in-plane uniform pressureapplied to two opposite sides of the plate, has not been solved. Theproblem is important and if the solution were available, the solution 3-S
of the previously mentioned problem for any ratio of biaxiality couldbe obtained as a special case by superposition. That is the problemdealt with in this Note. The solution is obtained photoelasticallv fora range of D/W values from 0.20 to 0.83.
Test Procedure 0-44
The analysis was conducted in a 3-in-sq. 1-in-thick (Homalite 100) W
FI. 2 Typical lseochromal pattern aound a circular hole In 8a quare pla0e* subJe)Wtd to wilorm pere on two opposite sdis
I Adjunct Professr, School of Engineering, Oakland University, Rochester,Mich. 48W63.
P Jrofeor, Deprtment of Mechanical Engineering. University of Marylend,Colsite Park. Md 20742, Fellow ASME.
=pt received by ASME Applied Mechanics Divisio. April, 1980. finalrevision, October. 190
Journal of Applied MechanicsMARCH 1981, VOL. 48 / 203
k ' ' 4 ' "
wwawi itl 1*%s , to a
kI i
It!I
60-
CCto
tet dooforlt
Fii3 4 t ds dmtu on oute bouyndarya nd hobinaeas platwtharutensubeed t unifrm praswo on two oppieo W (aveirae stress on tho /0not ectlued for c~atalr)
Filg. 4 idreas dbrtions onouler boundary of a&opuro plate with a roundlhole subjected to uniorm pressure applied to two opposite skies ol the plate(averge stre" on the net section used tor comparsion)
specimen (Fig. 1). The uniform pressure is applied by means of aspecially built device as described in 181. Two rubber hoses, one placedon each of the opposite aides of the plate are used. The deformationof the pressurized hose is restrained by Plexiglass sheets. The loading O K
frame had to be calibrated to determine the amount of pressure ac-Dtually applied to the specimen. For this purpose, a strain gaged loadcell was specially designed. The average fringe order was computedusing the applied pressure and the fringe value of the material anda check obtained by algebraically averaging the areas above and below 40"the zero axis for those specimens with high D/W. -- . ...
Seven specimens were used with the inner hole diameter varying ne ,w r e -,from 0.6 in. to 2.5 in. giving DIW values from 0.2 to 0.83, where D isthe hole diameter and W the width of the specimen. Dark field andlight field photographs were taken in a diffused light polariscope ofthe seven specimens, subjected to pressure sufficient to produce a Umaximum of about 5 fringes (Fig. 2). Fractional fringe orders were ai 0l os °recorded using Tardy's method of compensation at every 100 at theedge of the hole from 0° (horizontal) to 900 (vertical). Readings were sibeced to a uila.lal uillrm pressur and compulations to the blaxloalso taken on the outer edge of the plate at the 0° and 90° points. A ceecalibration test on a 2.5-in-dia round disk of the material gave a ma-terial constant of 156 lb/in./fringe.
The results obtained are given as stress distributions along theinside and the outside boundaries (Figs. 3 and 4), and as stress con- Referencescentrations at the intersection of the longitudinal and transverse axes IKiech G.. "Die Theorie der Elstizitact and die Beduerfnise der Fes-with the boundary of the hole. All values are given parametrically as tigkeitlehre." Z Ver. deut Ing W. Vol. 32. 1898. pp 797-807.functions of DIW. These results permit, by superposition, the de- 2 Howland C. J., "On the Stresses in the Neighborhood of a Circular Holetermination of stresses for any ratio of biaxial loading of the plate. The in s Strip Under Tension," Transactions of the Royal Societ, London. Seriescase of equal biaxiality was computed and is shown in Fig. 5. The A, Vol. 229, 1929-1930, pp. 49-8EWvalues obtained verify those previously published for the case of the 3 Wahl, M,, and Besuwkes, R.. "Stress Concentration Produced by Holes
and Notches," ASME TRANS., Vol. 56. Aug 1934, pp 617-623hydrostatically loaded hole 16], using the transformation explained 4 Durelli A. J., Phillips, E A., and Tso, C. J., Introduction to the Theo.
in 17]. retacol and Experimental Analysis of Stress and Strain. McGrsw-Hill. NewIt may be noted that, in the present problem, K, increases as DIW York, 1958.
5 Durelli A. J., Parks. V. J.. and Buhler-Vidal. J O., "Linear and Nonlinearincreases while for the case of circular holes in long rectangular plates, Elastic and Plastic Strsins in s Plate With s Large Hole Loaded Axially in ItsKt decreases as DIW increases. Plane," International Journal of Nonlinear Mechanws, Vol. 1I, pp. 207-
211.Acknowledgments 6 Riley, W. F., Durelli. A. J., and Theocaris. P. S., "Further Stresm Studieson a Square Plate With a Pressurized Central Circular Hole," Proceedings of
The research program from which this Note was developed was the 4th Midwestern Conference on Solid Mechanics. Austin, Texas, Septsupported, in part, by the Office of Naval Research (Contract No. 1959.N00014-76-C-0487). The authors are grateful to N. Perrone and N. 7 Parks. V. J.. and Durelli. A. J., "Transfer of Hydrostatic Loading From.
Basdekas of ONR for their support. The photoelastic specimens hsve One Boundary to Another." Experimental Mecha.iics. Vol 11, No. 4,' ppbeen prepared by S. Nygren and the manuscript reproduction by P. 148-149, Apr. 1975.
8 Dureli A. J.. Applied Stress Analysits. Prentice-Hill, Englewood Cliffs,Baxter. N.J., 1967, pp. 78-79
204 / VOL. 48, MARCH 1981 Transactions of the ASME
-. '" _.. , - i --i- '' ' ' " " 'a ''' *' " . ii --
SECURITY CLASSIFICATION OF THIS PAGE (Mmw Data Entered)
READ INSTRUCTIONSREPORT DOCUMENTATION PAGE BEFORE COMPLETIRG FORMI. REPORT NUMBER 12. GOVT ACCESSION NO. 3. RECIPIENT*S CATALOG NUMBER5 54. TITLE (ad Subtitle) STRESS DISTRIBUTION AROUND A S. TYPE OF REPORT A PERIOD COVERED
CIRCULAR HOLE IN SQUARE PLATES, LOADED UNIFORMLYIN THE PLANE, ON TWO OPPOSITE SIDES OF THE SQUARE
6. PERFORMING ORG. REPORT NUMBER
7. AUTHOR(.) S. CONTRACT OR GRANT NUMBER(s)
M. Erickson and A. J. DurelliN00014-81-K-0186
9. PERFORMING ORGANIZATION NAME AND ADDRESS tO. PROGRAM ELEMENT. PROJECT, TASK
AREA & WORK UNIT NUMBERS
University of MarylandCollege Park, Md.
11. CONTROLLING OFFICE NAME AND ADDRESS 12. REPORT DATEOffice of Naval Research September1981Dept. of the Navy I,. NUMBER OF PAGES
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19. KEY WORDS (Continue on revre side It neeeorsa aw Identity by block number)
Holes
Plates, squareStress ConcentrationsPhotoelasticity
20. ABSTRACT (Continue a t-..r..e, ide it. coeo a Idetiy by bo.. m s )The complete stress distribution around a circular hole, located in the
center of a square plate, has been determined photoelastically for the case ofthe plate loaded uniformly on two opposite sides. The study was conductedparametrically for a large range of the ratio of the side of the square to thediameter of the hole. The results obtained permit the determination of theStresses for any biaxial condition and verify a previous solution obtained forthe case of the pressurized hole. The experimental procedure is brieflydescribed.
DO , J 1473 EDITION OF I NOV 6S IS OBSOLETED N7 / 0102"014-60 1| CURITY CLA*SIFICATION Of THIS PAG1 (Maen D&te RAOSwe
OPTIMUM SHAPES OF CENTRAL HOLES IN SQUARE PLATES
SUBJECTED TO UNIAXIAL UNIFORM LOAD
by
A. J. Durelli, M. Erickson and K. Rajaiah
Sponsored by
Office of Naval ResearchDepartment of the NavyWashington, D.C. 20025
on
Contract No. N00014-81-K-0186
U.M. Project No. SF-CARS
Report No. 56
School of EngineeringUniversity of MarylandCollege Park, Md. 20742
w- .-
OPTIMUM SHAPES OF CENTRAL HOLES IN SQUARE PLATES
SUBJECTED TO UNIAXIAL UNIFORM LOAD
by
A. .J. Durelli, M. Etickson and K. Rajaiah
TABLE OF CONTENTS
ABSTRACT........................ ..
INTRODUCTION.................. . . ..... .. .. .. ..
OPTIMIZATION PROCEDURE........... . .. .. .. .. .. .
EXPERIMENTAL DETAILS ..................
RESULTS . . . . . . . . . . . . . . . . . . . . . . . . .
DISCUSSION. .. ................. .. .. ..
ACKNOWLEDGMENTS .....................
REFERENCES. . . . . . . . . . . . . . . . . . . . .
ji,, J Solid, Strwrtjs Vol. 17, lip, 797-703. 1981 Q520-7(A3t1V507-02.0WlPrinted in Grcal Britain Prgamon Press Lid
OPTIMUM SHAPES OF CENTRAL HOLES INSQUARE PLATES SUBJECTED TO UNIAXIAL
UNIFORM LOAD
A. J. DURELLI, M. ERICKSON and K. RAJAIAH
Department of Mechanical Engineering, University of Maryland, College Park, MD 20742, U.S.A.
(Received 30 June 1980)
Abstract-This paper presents the shapes that will optimize the stress distribution about central holes insquare plates subjected to uniform load on two opposite sides of the plate. The study is conducted for alarge range of hole to plate widths ratios (0 W). The stress concentration factor for the optimized holesdecreased by as much as 21% when compared to the one associated with a circular hole. Simultaneously,the weight of the plate with optimized hole is reduced by as much as 36% as compared to the circular hole.Coefficient of efficiency of around 0.92 is achieved for all DI W ratios. The geometry of the optimized holesare presented in a form suitable for use by designers.
INTRODUCTIONThis paper is one in a series of papers dealing with the optimization of discontinuities in twodimensional stress fields. The optimum shape of a hole in an infinite plate subjected to uniaxialuniformly distributed load was presented in[l]. The shapes to be given to an optimized centralhole in finite plates subjected to uniaxial uniform loading have been given in[2] for differentratios of the diameter of the hole to the width of the plate. The optimized inner boundaryshapes of rings with circular outer boundaries subjected to diametral compression have beengiven in [3] for different ratios of outer to inner diameters. The description of the basic featuresof the method have been presented in(4, 51. References to other contributions in the literaturecan be found in [5] among which the most important one is due to Heywood [6).
OPTIMIZATION PROCEDURE
The method consists in using photoelasticity, in a systematic way, to idealize a configurationso that its boundaries do not have gradients of stress along the length of the boundaries. Inother words, the structure will have stresses uniformly, or almost uniformly distributed alongthe boundary. The procedure permits the direct design of the geometry of the structure ratherthan the conventional step by step design and analysis, satisfying the requirement that themaximum stress should be lower than an allowable stress and at the same time, the distributionshould be as efficient as possible, The geometric constrains for the problem are stripulatedinitially. A transparent model of the structure is placed in a diffused light circular polariscope.(The material of the model should exhibit birefringence when under load and should besufficiently sensitive to produce several fringes of interference.) The operator should be able towork on the model with a hand file or portable router at the same time that he looks at themodel through the analyzer. The method requires that the operator file away material from theboundaries starting at the points where the stress (and therefore the fringe order) is at aminimum. The filing operation redistributes the fringes. The operator continues filing awaymaterial from the low stressed zones of the boundary until as much as possible of the length ofthe boundary shows the same order of birefringence. This is easy to detect because at thatmoment, the same fringe falls all along the length of the boundary. If the body has more thanone boundary, it may be necessary to operate by steps back and forth from one boundary to theother. In some cases, white light may be more practical than monochromatic light, the objectivebeing then to have the same color along the boundary.
The degree of optimization can be evaluated quantitatively by a coefficient of efficiency:
I futds ,- dsk =ff + -
787
788 A. . DU RaL el aL.
where a, is the tangential stress, o, represents the maximum allowable stress (the positive andnegative superscripts referring to tensile and compressive stresses, respectively), So and Si arethe limiting points of the segment of boundary subjected to tensile stresses and S' and S2 arethe limiting points of the segment of boundary with compressive stresses. The significance ofthe coefficient of efficiency was discussed in Refs. [2, 5]. The above criterion will be used in thepresent work to evaluate the optimized hole shapes. The design procedure will be particularlyuseful for components made with brittle materials, or components made with ductile materialssubjected to fatigue.
Ple I Fa
441
( .t~omelite tO I00 Nmhodel -
1P 0 Clamping
- A - I -
section IA - A
Fig. 1. Loading device used to apply uniform pressure to two opposite sides of a square plate.
0-4
4.1. . S 0 0,
2.S- - I.5
4.5 I
WW
Fig. 2. Typical isochromatic pattern around an optimized hole in a square plate subjected to a uniaxialuniform pressure.
Optimum shapes of central holes in square plates 789
In this paper, using the technique described above, optimized hole shapes are presented insquare plates subjected to uniaxial uniform compression.
EXPERIMENTAL DETAILS
Experiments were conducted with 0.23 in. (5.8 mm) thick Homalite-100 plates (fringe con-stant of 156 lb/in-fr (27.0 kNm-fr)). The plate size was chosen as 3 x 3 in. (76.2 x 76.2 mm) forall DI W ratios. Optimization was carried out for DI W = 0.20; 0.41; 0.56; 0.65; 0.71; 0.79 and0.89, with the models subjected to uniaxial uniformly distributed compression. Material wasremoved from the low stress regions by careful hand filing. To improve the precision, inparticular at the corner zones, the operator used a binocular magnifier with a set of polarizerand quarter wave plates attached to each of its lenses, during the filing process. The uniformcompression on the two parallel boundaries was applied following the methods developedpreviously[7,8] and used recently in [9]. The position of a pressurized rubber tube located inthe special device is shown in Fig. I. This loading frame had to be calibrated to determine the
10 --%
88
I-_ _IW
CicuarD/W- 8
6 Op timized
6 bo7 .;
V ICI ,"A-I
-Degrees
0 10 20 30 40 50 60 70 90 90
Fig. 3. Stress ratios for points on the boundary of optimized circular holes in a square plate subjected to uniaxialuniform pressure.
K PPP
1.- o I/ D K ,
W-D
0 - CIRCULAR
&- 7- 250
o0. o2 0.4 0' O ,D/W
Fig. 4. Comparison of stress concentration factors for an optimized hole vs a circular hole in a square platesubjected to uniaxial uniform pressure.
I-.,!
790 A. J. DURELLI et al.
amount of pressure actually applied to the specimen. For this purpose, a strain gaged load cellwas specially designed.
RESULTS
The isochromatic pattern for a typical hole shape is shown in Fig. 2. The stress distributionsaround circular and optimized holes for the DI W ratios considered are presented in Fig. 3. Thes.c.f. for the tensile and compressive regions of the circular and optimized holes for differentDIW ratios are plotted in Fig. 4. The information on coefficient of efficiency and percentageweight reduction achieved are given in Fig. 5. The stress distributions on the outer edges areshown in Fig. 6.
P
4030 - "'D- /0
ECoefficient of
20 090
0 oWeigh retion 0 080
01 0 70 01 02 03 04 05 06 07 08
D/W
Fig. 5. Percentage weight reduction and coefficient of efficiency for an optimized hole in a square platesubjected to a uniaxial uniform pressure.
,- 0 2 3 4 5 6
8 --
0 0
unfom rssreonto opostesies
#/ -
W/2
0
2 -
AVWj I W-
6- 84 - --- ocirhe
--- 0- Optimized hole
Fig. 6. Stress distribution along the outer boundaries of a square plate, with an optimized hole, subjected touniform pressure on two opposite sides.
Optimum shapes of central hole% in square plates 791
D
p /1
R 0-253
3 R3
073 1 hbt.7 51
71
Fig. 7. Optimum shape of a central hole in a square plate subjected to uniform pressure on two oppositesides (as function of DI W).
R225 w
J.2- 2
/
01w
Fig. S. Radii of (fhe elements of holes producing optimum distribution of sittvt in square plates subjected touniaxial uniform pressure.
792 A. J. DtUI i i et al.
The experimentally developed optimum hole geometries have been fitted with a combinationof circles of different diameters and common tangents at the points of intersection. The holegeometries for the different DIW ratios are given in Fig. 7. The information given in Fig. 7 isconsolidated in Fig. 8 in a graphical form to make easy its application.
DISCUSSION
The isochromatic pattern in Fig. 2 shows that the newly developed hole shape exhibit', .1high degree of optimization with the stresses remaining uniform along major portion' of thetensile as well as compressive segments of the boundary. The information in Figs. 3 and 4shows that, as compared to the circular holes, the optimum shapes have lead to significantreduction in s.c.f., the reduction ranging from 16% for DIW = 0.14 to about 21%',- for D/0.84.
A coefficient of efficiency of about 0.92 has been achieved for all DI ratios, as seen fromFig, 5. The optimum shapes have also lead to significant reduction in weight as compared to thecircular holes, in the range of 1% for DIW = 0.14 to 36% for Dll = 0.84.
Comparison of the optimized shapes and the s.c.f. presented here for square plates withthose given in Ref. [2] for long rectangular plates under identical loading conditions (except forthe reversal of sign) brings out certain interesting features.
For DIW <0.3, the optimized shapes are found to be identical, whereas for DIl' 0.3. thetwo shapes tend to become quite different especially on the hole edges perpendicular to the loadaxis. As DIW becomes larger, the hole shapes in square plates are predominantly influenced b.the bending effect of the horizontal segments. No such bending effect can be observed in longrectangular plates.
For a very large value of DIW, the problem can be considered as that of a portal framesubjected to uniform load on top. The optimum shape presented can be taken as the shape to begiven to a stress optimized portal frame.
While the s.c.f. for square plates decreases initially slightly and then increases monotonic-ally with increase in DIW (Fig. 4). it was found to decrease monotically for long rectangularplates as DIW increases. For a given DIW ratio beyond D1W = 0.5. the s.c.f. for square platesis found to be very much higher than that for long rectangular plates. The reason for thisincrease can be traced to the behavior of the square plate as a portal frame, which increasesappreciably the bending stresses in both the vertical and horizontal members.
For large DW. the vertical elements of the plate becorite thinner and the fringes are notonly parallel to the inner, but also to a large extent, to the outer boundaries indicating a linearvariation of stress across the section. Ii also shows that, for large Di1l', as the inner edge getsoptimized, the outer edge also tends to become optimum. The distribution of ;tresses on theouter edges shown in Fig. 6 confirms this. It is also seen from Fig. 6 that the stress distributionbecomes favorable also on the loaded edges with the tangential stresses remaining constantover a considerable length.
It is also seen from Fig. 4 that the location of maximum stress shifts from the vertical edgeto the horizontal edge for DI W > 0.81.
Acknowledgements-The research program from which this paper Awas developed %sas supported. in part. h%. the Otlice ofNaval Research (Contact No. N00014-76-C-04871. The authors ;ire grateful to N. Perrone and N. Basdckas of ONR for thi,support. The photoelastic specimens have been prepared by S. Nygren and the manuscript reproduction lh' I' Baxter.
REFERENCESI. A. J. Durelli and K. Rajaiah. Quasi-square hole with optimum shape in an infinite plate subjec'ed to in-plane loading
ONR Rep. No. 49. Oakland University (Jan. 19791. To appear in the J. Mech. ksign (.4.,S..M.F.l2. A. J. Durelli and K. Rajaiah. Optimum hole shapes in finite plates under uniaxial load. J. 4ppl. Mech 46. 6ql--6q5 (Sept
1979).3. A. J. Durelli and K. Rajaiah. Optimized inner boundary shapes in circular rings under diametral compression Strain
127-130 (Oct. 1979).4. A. J. Durelli, K. Rajaiah, J. D. Hovanesian and Y. Y. Hung. General method of directly design stres-,Aise optimum
two-dimensional structures. Mech. Res. Comm. 63). 159-165 (19791.5. A. J. Durelti. K. Brown and P. Yee. Optimization of geometric discontinuities in stress fields. Fop. Mech. t8(g). 103-(
(Aug. 1978).
~ .2
Optimum shapes of central holes in square plates 793
6. R. B. Heywood, Designing by Photoelasticity. Chapman & Hall, London 01958).7. A. J. Durelli. Distribution of stresses in partial compression and a new method of determining the isostatics in
photoclasticity. Proc. 13th Semi-Annual Eastern Photoelasticity Conf. pp. 25-50 (1941).8. A. J. Durelli, Experimental means of analyzing stresses and strains in rocket propellant grains. Exp. Mech. 2(4), 102-110
(1962).9. M. Erickson and A. J. Durelli. Stress distribution around a circular hole in square plates loaded uniformly in the plane
on two opposite sides of the square. ONR Rep. No. 55. Oakland University (1980).
SS Vol I'. No 9--F
0 . -. V i
.LLUICITY CLASSIFICATION OF THIS PAGE(Whe, Dal Zteomg)
circular hole. Coefficient of efficiency of around 0.92 is achievedfor all D/W ratios. The geometry of the optimized holes are presentedin a form suitable for use by designers.
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4. TITLE (and Subtitle) OPTIMUM SHAPES OF CENTRAL HOLES IN S TYPE OF REPORT . PERIOD COVERED
SQUARE PLATES SUBJECTED TO UNIAXIAL LOAD
s. PERFORMING ORG. REPORT NUMBER
7. AuTOR(,) S. CONTRACT OR GRANT NUMBER()N00014-81-K-0186
A. J. Durelli, M. Erickson and K. Rajaiah I,
I PERFORMING ORGANIZATION NAME ANO ADDRESS 10. PROGRAM ELEMENT. PROJECT. TASKAREA A WORK UNIT NUMBERS
University of MarylandCollege park, Md.
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Office of Naval Research September 1981Dept. of the Navy ,, NUMOER OF PAGES 12Washtn tnn. n 2OO25 -_____________
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16. DISTRIBUTION STATEMENT ( l thi. Report)
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17. DISTRIBUTION STATEMENT (of the abot,act entered In Block 20, II different from Report)
1S. SUPPLEMENTARY NOTES
IS. KEY WORDS (Continue on reveors side if nIcoearn md IdeitifI by block nhmber)
OptimizationHolesSquare PlatesPhotoelasticity
SO. ABSTRACT (Conifinu0 an 0e 0~e id* It ng~ca ,ad idemIlt? by block maeSbou)
This paper presents the shapes that will optimize the stress distributionabout central holes in square plates subjected to uniform load on twoopposite sides of the plate. The study is conducted for a large range ofhole to plate widths ratios (D/W). The stress concentration factor forthe optimized holes decreased by as much as 21% when compared to the oneassociated with a circular hole. Simultaneously, the weight of the platewith optimized hole is reduced by as much as 36% as compared to the
Do, s % 1473 EoITION OF I NOV 65 1S OBSOLETES/N O|O- 0 | 4" SSO | I I#¢SECURITY CLASSIFICATION OF THIS PA@S (Wheat tnlerud)
OPTIMIZATION OF HOLE SHAPES IN CIRCULAR
CYLINDRICAL SHELLS UNDER AXIAL TENSION
by
K. Rajaiah and A. J. Durelli
Sponsored by
Office of Naval Research
Department of the NavyWashington, D.C. 20025
on
Contract No. N00014-81-K-0186
U.M. Project No. SF-CARS
Report No. 57
School of EngineeringUniversity of MarylandCollege Park, Md. 20742
-A
OPTTMIZATION OF HOLE SHAPES IN CIRCULAR
CYLINDRICAL SHELLS UNDER AXIAL TENSION
by
K. Rajaiah and A. J. Durelli
TABLE OF CONTENTS
ABSTRACT. .................... .. ..
INTRODUCTION. .................. .. ...
MODEL MAKING. ................... .. ...
LOADING SYSTEM . .................. .. ..
f TYPE OF POLARISCOPE USED....... .... .. .. .. .. ...
OPTIMIZATION PROCESS. ............... .. ..
RESULTS ADDISCUSSION....... .... .. .. .. .. .. ...
ACKNOWLEDGMENTS S. ................. ....
REFERENCES. ................... .. ..
Optimization of Hole Shapes in Circular CylindricalShells under Axial Tension
Optimization of hole shapes in cylindrical shells underaxial tension is obtained using two-dimensional photoelasticand birefringent-coating techniques, resulting in uniform membrane stressesalong the tensile and compressive segments of the hole boundary
by K. Rajaiah and A.J. Durelli
ABSTRAC-Hole shapes are optimized in circular cylindrical As a percentage of the total stress, the bending stressessnels suojected to axiai load considering only the predominantly constitute less than 16 percent for t/R > 0.004 (Fig. 2).large membrane stresses present around the holes Two-d,mensional photoelastic isochromatics obtained with aspecial-purpose polariscope are utilized for the optimizationprocess The process leads to a significant decrease in themembrane stress.concentraion factor and a modest decreasein weight. thus yielding a considerable increase in strenglh-to- -oweightl ratio This paper presents results for certain typicalratios of hole diameter to shell diameter Previous theoretical is 9and experimental studies for the circular hole have also been 0 -
verified it
Introduction 'q "
Optimization of hole shapes in engineering structures W PLsAE S PLUS "sAlike plates and shells is important from fatigue and Y BIENDING ,0
minimum-weight considerations. In a series of papers,Durelli and his associates- have recently presented a .S,1)practical way of arriving at optimum hole shapes in uni- to A .0 01r,axially loaded plates and rings from simple two-dimen- -, liP .00osional photoelastic experiments by removal of material t In *0 0from lot-stress regions around the hole and making an 3isochromatic fringe coincide separately with the tensile 0 2and compressive segments of the boundary. This process Fig. 1-Membrane and bending stress-concentrationleads to a significant reduction in the stress-concentration factors for circular holes in circular cylindrical shellfactor as well. The present paper is an extension of the under axial tensionsame concepts to the optimization of hole shapes incircular cylindrical shells under uniaxial tension.
The stress pattern around holes in cylindrical shells iscomplicated by the fact that both membrane and bendingstresses are present. Fortunately, for circular holes incircular cylindrical shells under axial tension, Peterson's' - too 0 000,handbook shows that membrane stresses (computed by " 20Van Dyke) are predominant on the hole edge while -000bending stresses are of a much smaller magnitude (Fig. I). 1,0 111. 0 0K. Rujeaih ISESA Meniberi is Prwfessr. Indian Instiute of Tho-lorto. [Poo.i, Busmha 40(0 076. India A.J. Durellh (SESA lunurart Menberj asPrtojesu. .eparin tn of eg A-'hal niLnneenrrng. Unffvers n of Marv.oland. C'ullexoe Patt. MD 20742. i 0 0-S 1 0
Aapir wal presnied at Fourth SESA International Ckonrgress on Lrper. CuRiVru PAurTE Rimenial Xlethunixs held in hostun. MA on Mu 23-30.Oririnal nwnuuo'ropt submitd. August I. 19 0. Authors notified oj Fig. 2-Percentage of maximum bending stress presentaeaplunce: September 19, 1960. Final version reteed: (.Auher 9. lo. around the hole for various curvatures d of the hole
Experimental Mechanes * 201
In an ideal situation, one would attempt to optimize the circle of diameter 2a, and the square of side 2a.hole shape considering both the membrane and thebending stresses. Obsiousl), such an exercise would be Model Makingquite insohed. From a ptactical-deign point of view, itshould be enough if the holt shapes are optimized by Three different types of shells were made. Preliminar%taking into account onls the predominantly large mern- experiment.% used commerciall) available Perspes tuhes ofbrane stresses. The method proposed in this paper is 3-in. (76-mm) nominal diameter with 0.125-in. 3-mn)based on this idea. nominal wall thickness (identified as Type I shell). %% hle
Defining an optimized hole as one in which unilorm the thickness distribution was within 5 percent of themembrane stresses are present in both the tensile and nominal value, the cylinder was not sufficientl) straight.compressie segments of the hole boundar), it is found Due to this, certain bending effects were obsersed. Thethat two-dimensional photoelasticity (either by trans- adsantage of this material was the negligible time-edgpmission or by reflection) can be used to obtain optimized stresses around the hole, but the major problem Aas thehole shapes using shell models made of birefringent large load required to produce a sufficient number otmaterials. In the preent insestigation, this technique is fringes.utilized for the optimization of hole shapes in circular The second set of experiments used shells made ofcylindrical shells under uniaxial loading. Optimized hole epoxy sheets, 1.3 mm ±0.05 mm thick cast between twoshapes, the corresponding stress-concentration factor and substantiall) thick flat Perspex sheets (identified as Typestress distributions around the holes are presented for II shell). Each sheet in a semipolymerized state was cut tosome typical ratios of hole diameter/shell diameter. size and wrapped around a 50-mm-diani ground-metalThe geometric constrain specified for the optimized hole cylinder coated with a releasing agent. The edges of theis that the optimized boundary should be in-between the epoxy sheet touching each other were joined together wih
a 5-mm-wide. very-thin strip providing a bonded butt joint.To provide symmetry to the shell, an identical strip wasbonded on the diagonally opposite side. The open endsof the cylindrical shell were reinforced over a distance of
sO..CL SEATh, SLOT $RO POLARIZER/ 1.0 in. (25 mm) with bonded epoxy strips to prosideA.1 "L W QUARTER-WAVE PLAT[,- ,W F,, / 1SEATION stiffness near the end fixture. After complete curing, the
t CYLI.NRCAL cylinder was removed and tested. For a no-hole condition,
\- it was verified that there were no residual stresses exceptvery close to the joints, and that the reinforcement affects
OUE €0 the stress distribution over the uniform section onl. to arR Idistance of about twice the shell thickness. Even with the
introduction of holes, there were no measurable effectsof the reinforcing strips over the hole stress distribution.
Pox, The advantage of this technique was the fine transmissionIRCULAR
sHLL properties of the shell and absence of machining process.The last set of experiments were carried out with
integrally cast circular cylindrical shells (identified as"Type Ill shell) using a plaster-of-paris mold with a centrally
Fig. 3-Loading fixture located, slightly tapered metal cylinder coated with asilicone-grease-releasing agent. After complete curing, theinner metal cylinder was first pushed out using a hydraulicpress and then the outer plaster-of-paris shell was remosed.
CIRCULAR cVL,,,,. Both the inner and outer surfaces of the epox. shell*-o~k LoCArOo, were turned on a lathe to the required dimensions (50-mm
diameter and 0.4 mm thick). The surfaces needed a thin!coating of oil to make them transparent. Dimensionall),these cylinders were the best. In the last two experiments,
, cAMR in order to overcome the time-edge effect around the" &holes, the experiments were carried out immediatel% after
the introduction of the holes. Optimization was also com-pleted on the same day.
Fig. 4-Transmission-polariscope arrangement The introduction of the circular hole was carried out indifferent stages starting from a small-diameter hole andprogressively increasing it to the required diameter andthen finishing with a reamer. The holes were stress free
.- .eErLECT[ S for all practical purposes. The length of all the shells usedHOLE LOCATION was sufficient for the ends to have negligible effect on the
LS hole. No creep effects were noticed in any of the shells.
- - - Loading System
" A - '7 Uniform axial loading of both tension and compressiontypes was applied on the models. The compression loadC€:RCU Aft CVLINMOR
4tf~erial Dobeckot resin SOC 4 hardener 738 (IV 11 mnutsfuttured op
Fig. 5-Reflection-polariscope arrangement Indi f - 44.4 Iblin. .fi (7.8 AN r.n-fr.
2 aMy 10s1
was given through rubber sheet and wooden blocks kept Jessop et at.' have determined the stress-concentrationon the model with a 12-mm ball located at the center. factors in circular tubes with transverse circular holesThe shell could be axially loaded and the uniformity of under tension for some typical values of 13 and 1/R, usingthe stress at some distance away from the load checked. the stress-freeving technique. Comparison of their results
The tensile load was applied through shear with tightly with the present values, as well as with the analyticalgripped end rings and a disk with a spherical-ended rod at results quoted in Ref. 5, shows close agreement. Fig. 9,the center ensuring the alignment of the shell (Fig. 3). In but are about 5 percent lower.these tests, compression loading was more difficult to The condition under which the tests were conductedcontrol than tensile loading and all the reported results required thai the diameter of the holes used be small. Thiscorrespond to tensile loading, increased the difficulties of the analysis. It is estimated
that the level of precision obtained is 5 percent. The
Type of Polariscope Used
Conventional transmission polariscopes are clearlyunsuitable for the study of the present problem. Apolariscope, based on Houghton's experiments., with thepolarizer and first quarter-wave plate having their planesparallel to the plane of the hole and located at the center oof the shell, was used. A diffused 130-W sodium-vaporlamp was used as light source on one side of the shell sC . 3.SS
while the second quarter-wave plate and the analyzer were .rA, P
located on the other side. The fringes were recorded using '-.5
a bellows-type camera with a 25-in. (62.5-cm) focal-lengthlens (Fig. 4).
For the reflection studies, a conventional polariscope ofthe type shown in Fig. 5 was tried. With this polariscope,the shadow, around the hole edge is very pronounced. The ,soc-'.zs, A&:u,,D L CItjA Z .Zresults obtained are not very reliable. Due to this difficulty,the reflection method was only used in the present studyas an occasional check. A special reflection polariscope ofthe type used by Slot' and, more recently, by Durelli eial.' would appear to be ideal for this problem. With onepolariscope as with the other, only membrane stresses 05 C
are recorded. j LOD 4
Optimization Process 1 I
For optimization, the constraint was stipuhied that theboundary of the hole lie between the circle of diameter 4 O.,* K2a and the square side 2a. The process followed is versysimilar to the one mentioned in Refs. 1-4. It starts with Fig. 6--sochromatic-fringe patternsthe viewing of the isochromatics around the circular hole around circular and optimized holes in ain the cylinder loaded axially, using the polariscope des- circular cylindrical shell under axial
cribed above. From a study of the isochromatics, the load (a R = 0.24, tIR= 0.056)
tensile and compressive segments and the low and highstress regions around the hole boundary are established.The operator starts filing away material from the boundaryat the points where the stress is minimum, say, on thetensile segment of the boundary, and with the hole regionin view through the analyzer. This operation redistributesthe stresses bringing down the maximum stress along thatsegment. The operation is continued until an isochromatic -fringe coincides with the tensile boundary. The same 7operation is conducted on the other segments.
CIRCULAR NOLC
Results and Discussion . Plieco .The isochromalic patterns for a typical circular and .
corresponding optimized holes are shown in Fig. 6. The . 's
stress distributions at the boundaries of the holes for three -----. ' Xdifferent ratios of hole diameter to shell diameter are I - -- 0 _ ,presented in Fig. 7. For the sake of comparison, Fig. 7 - 0 o0also includes the distributions corresponding to thecircular holes as determined in the present experiments. 0 i 0 30 &o W_ To 010 opThe empirically developed geometry has been fitted with 0a combination of circles of different diameters and Fig 7-Stress distribution at the boundary of optimizedcommon tangents at the points of intersections. The and circular holes in circular cylindrical shells under axialgeometry of a typical optimized shape is shown in Fig. 8. load (f/R = 0.056)
Experimental Mechanics * 203
IThe stress-concentration factor increases with increasinghole diameter for optimized holes also. The optimum holc
a/Ruo 24 shapes obtained for the cases investigated are close to adouble barrel.
Comparison of the results obtained using shells with(Type 11) and without (Type 11) reinforcements indicatethat, for the range of hole diameters considered, the
d-. stress-concentration-factor values in both cases are inclose agreement. The optimization process, apart fromthe reduction in stress-concentration factor has alsoresulted in a decrease in weight as evidenced b) theincrease in the hole area in the optimized case. The 4
increase in hole area is found to be of the order of 5d/18040 percent to I I percent. One can conclude that the optimiza.
tion process has thus led to a significant increase in thestrength/weight ratio of the shell.
As mentioned earlier, the hole optimization has beencarried out considering only the effect of membranestresses. Whether this process leads to an increase ordecrease of the bending stresses around the hole is notknown yet. For a complete understanding of the problem,estimation of the bending stresses for both the circularand optimized shapes would be required, particularl) forvery thin shells.
Fig. B-Optimized geometries of holes for As a matter of interest, the shape shown in Fig, 9 forminimum stress-concentration factor in a circular the small hole could be compared to the one developed incylindrical shell under axial load Ref. 2, for the case of the large plate under uniaxial
loading, and the one developed in Ref. 4 for the case ofthe large plate subjected to equal biaxial loading ofopposite signs. It can be noted that the transverse boun-dary of the optimized hole in the case of the plate underuniaxial load is flat, rather than curved as in the othertwo cases. it is possible that further work on the shellcase will shoA that an improvement in optimization canbe obtained by using a flatter transverse boundary for
D -. ".n oO, the hole.
a MEWMANE-t*1RA E pus- , Acknowledgments
,/ ""~'CRCULAR LE The research program from which this paper was- - , ,developed was supported, in part, by the Office of Naval
-/ .Research (Contract No. N()014-76-C-048"). The authorsa JESSOP ETAL9 are grateful to N. Perrone and N. Basdekas of ONR for
(100I.-0o their support. The photoelastic specimens have been.-"" ,prepared by S. Nygren and the manuscript reproduction
-- \~ P 2: i (j..) by P. Baxter.
___ReferencesIi S I. Durelli, A.J.. Brown. K. and kee. P., "Optamization of Geometric
D/sconunules in Stress Fids." tAPt /tILNTAI 51&LHAIS. I (18),Fig. 9-Stressconcentration factors for a sir gle 303-308 Aug. 1978).circular and optimized hole in circular cylindrical 2. Durelli, A.J. and Rajoih. A.. "Optimum Hole Shapes in Finiteshells subjected to axial loading Plates Under Unnuxl Load, " J. of Appl. Mech., 4, 691-695 (Sepi.
(979),J. Durelh. AJ and Ratio. Ah.. "'Oplmard Inner Boundary Shapes
in Circular Rings Under Diametral Compression. " Strain. iS. 127.130(Oct. 19791.
optimization process is also difficult on small holes.' 4. Durellt, A.J. and Rajaiah. k., "Lighter and Stronger." I\PIIMiLNTAI 6IM. HANI(S. 20 (111, 369-3,0INov. 1980.However, the optimized hole shapes and the percentage . Peterson, R.E., "Stress Concentration Factors.' 4dey Iniencierre
reduction in stress-concentration factor can be considered (1974).to be reliable. For better accuracy, larger hole sizes in 6. Houghton, D.S., "Stres Concttraion Around Cut-outs in alarge-diameter shells should be used. Cylhnder. "" J. of the RoVl Aervn. Soc., U. M0160 J rl9l.large-ime.er 9 l c sonld oe stress-con. Slat, T., "Reflection Polanrmope fur Photograph of photoeliwti
In Fig. 9, comparison of the stres-oncent ration - Coatings"LXPLuIMINIAI MtkLHANKs. 2( 2). 41-47(19021factor values for circular and optimized holes shows that 8. Dureli. A.J and Rajaiah. A.. "Detrmination of Strains inthe optimization process has brought down the stress- Photoelatc Coatings." i.,PLIMLNIAL MlHAii.lts, 20 (2). 57-64concentration factor by as much as 25 percent. The .€a n " S c
P. Jew. H. T.. Snell. C. and Alisuon, .t,"'The Sirem Concentr-reduction in stress-concentration factor is larger for the ion F ctou in Circular Trutes with Tronsvere Circular Holes.large hole sizes. Larger gains can be anticipated for > 1. Aeronautical Quarterly, 1. J26-11 (1959j.
204 II May 1961
. . . . . . . . .. . . ... . .,. . .,.-
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1. REPORT NUMBER 2. GOVT ACCESSION NO. 3. RECIPIENT'S CATALOG NUMER57
4. TITLE (nd Subtite) OPTIMIZATION OF HOLE SHAPES IN s TYPE oF REPORT & PERIOD COVEREDCIRCULAR CYLINDRICAL SHELLS UNDER AXIAL TENSION
S. PERFORMING ORG. REPORT NUMBER
7. AUTHOR(e) 6. CONTRACT OR GRANT NUMBER(e)
K. Rajaiah and A. J. Durelli N00014-81-K-0186
s. PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT. PROJECT, TASKAREA A WORK UNIT NUMBERS
Ufliversity of MarylandCollege Park, Md.
11. CONTROLLING OFFICE NAME AND ADDRESS 12. REPORT DATEOffice of Naval Research September 1981
Dept. of the Navy, IS NUMBEROF PAGES 2'go,; ____D_._______ 1
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Distribution of this report is unlimited
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OptimizationShellsHolesStress Concentration
20. ABSTRACT (Conftnue t .#dorl ait nfecessary, rant fdontst by Weoek nuibw)
Hole shapes are optimized in circular cylindrical shells subjected to axialload considering only the predominantly large membrane stresses present aroundthe holes. Two dimensional photoelastic isochromatics obtained with a specialpurpose polariscope are utilized for the optimization process. The processleads to a significant decrease in the membrane s.c.f. and a modest decreasein weight, thus yielding a considerable increase in strength to weight ratio.This paper presents results for certain typical ratios of hole diameter to
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LU'IVTV CLASSIFICATION OF THI4S PAGE(Man Dot& 5Ahem'.)
circular hole. Coefficient of efficiency of around 0.92 is achievedfor all D/W ratios. The geometry of the optimized holes are presentedin a form suitable for use by designers.
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