STRESS CONCENTRATION FACTORS FOR SHOULDERED PLATES

by

James Arthur VassB.Sc.(Sng*), CoEngo, M0I0HecheE., MoInstoM.C.

Submitted for the Degree of Doctor of Philosophy

Department of Mechanical Engineering,

University of Surrey,, September, 1971.

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SUMMARY

The need for rational and reliable design methods to meet the continual demand for higher standards of performance and lower unit costs is the incentive for providing the designer with comprehensive and accurate information on the magnitude of stress concentrations associated with the geometry of a component. Theoretical stress concentration factors are used indirectly for the calculation of strength reduction factors from which the life of components suffering fluctuating loading can be predicted and are also of direct relevance in cases of failure by brittle fracture under conditions of steady loading. Fairly comprehensive and accurate information is available for the simpler loading cases such as torsion, direct load or uniform bending of many basic engineering components but, in practice, bending is generally accompanied by shear and for this condition the existing . information is both scanty and unreliable*

A comprehensive series of tests using two-dimensional photoelastic methods avejbeen carried out on shouldered plate models covering a . wide range of useful parameters and subjected to various loading configurations of bending with shear* By means of a consistent extrapolation procedure based on curve fitting by orthogonal polynomials and the use of other statistical methods it has been demonstrated that boundary Stresses can be described by linear expressions involving two basic factors associated, independently, with the respective effects of bending and shear. Values of the resulting basic factors are presented in the form of curves plotted against the model parameters. For design purposes, methods are examined whereby the basic information can be used, either in the form of empirical expressions from which peak stresses can be calculated for any given set of parameters, or as

input data to a digital computer programmed to control an incremental

plotter to give, automatically, curves of stress variation for any selected configuration of bending v/ith shear.

ACKNOWLEDGEMENTS

The author wishes to express his gratitude to Dr. I.M. Allison, Ph.D.(London), C.Eng., A.F.R.Ae.S., Reader in the Department of Mechanical Engineering, University of Surrey, who supervised this

work, for his helpful criticism and advice, constant encouragement and unfailing patience;

to his friend and former colleague Mr. R.H. Blakemore, B.Sc.(Eng.)rC.Eng., M.I.Mech.E., for his kind help and advice on computing matters;

to the staff of University College, London, for their helpful advice on the techniques of preparing photoelastic models;

to the County Borough of Luton Education Authority for their

financial support;to his wife Jean, for her patience and forbearance.

September, 1971*

CONTENTSFAGE

CHAPTER 1 INTRODUCTION AND REVIEW OF EXISTING INFORMATION1*1 Introduction 71*2 Review of Existing Information 8

CHAPTER 2 GENERAL TEST PROCEDURE

2*1 Notation 122*2 Dimensions used for Shouldered Plate Models2*3 Production of Models 152*^ The Loading Systems and Rigs 20

2*5 The Optical System and Photoelastic Readings 25CHAPTER 3 REDUCTION OF TEST RESULTS AND DETERMINATION OF

THE BASIC STRESS CONCENTRATION FACTORS 3*1 Notes on the Stress Concentration Factor 293*2 Definition of Basic Stress Concentration Factors 303*3 Determination of Boundary Values of Stress

or Fringe Order 3T3*4 Justification of the Linear Relationship between the

Basic Stress Concentration Factors 3*5 Determination of the Basic Stress Concentration Factors 373*6 Accuracy of the Results ^9

CHAPTER k DISCUSSION OF RESULTS .

^*1 The Carpet Plotting Technique 55^*2 The Basic S.C.F. for the Bending Effect, 55

A

*f«3 The Peak Basic S.C.F. for the Bending Effect, l( 58

h*k The Basic S.C*F* for the Shear Effect, Kc, 65A

4*5 The Peak Basic S.C.F. for the Shear Effect, K s 67* A/*i*6 The Stress Concentration Factor and its Peak Value 69

CHAPTER 5 PRESENTATION OF CRITICAL STRESS DATA FORDESIGN PURPOSES

5*1 Introductory Comments A8?

5.2 The Stress Concentration Factor and its Peak Value K T 88

5-3 A wEmpirical Representations of Peak Basic S.C.F.s T 98

5 A Uses of the Empirical Relationships 110

CHAPTER 6 GENERAL CONCLUSIONS CONCERNING SHOULDERED PLATES SUBJECTED TO BENDING WITH SHEAR

6*1 Summary of the Experimental Results 115

6 .2 Use of the Information for Design Purposes 118

6.3 Suggestions for Further Work 120

APPENDIX A GENERATION AND USE OF ORTHOGONAL POLYNOMIALS

A. 1 Introduction 122

A.2 Forsythe’s Method 122

APPENDIX B COMPUTER PROGRAM FOR CALCULATION OF BASIC STRESS CONCENTRATION FACTORS

B.1 Introductory Comments 126

B«2 Brief Description of Program Structure 126

B*3 Layout of Input Data 135Bc*f Output Data 1^1

APPENDIX C LINEAR CORRELATION OF DATA USING LEAST-SQUARE REGRESSION 163

APPENDIX D AUTOMATIC PLOTTING PROCEDURES FOR THE S.C.F.

D. 1 Introductory Comments 16.5

D«2 Brief Description of Program 165

D*3 Layout of the Input Data 170

D»^ Output Data 171

D.5 Some Alternative Methods of Specifying and Using the Data 172REFERENCES 178

CHAPTER 1INTRODUCTION AND REVIEW OF EXISTING INFORMATION

IntroductionIn the field of Engineering Design the importance and relevance

of local stress concentration associated with the geometry of a component is now widely accepted. The continual striving for higher standards of performance in modern aircraft and the consequent rapid

increase in their cost and complexity are incentives for the attainment of a rational and reliable design procedure. Even in simpler products such as the automobile, the achievement of a more efficient design can give increased reliability and, possibly, savings in unit costs of material, these being important factors in the highly competitive areas of production for a mass market.

Unfortunately, the relationship between stresses and failure of an actual component is not generally straightforward and the direct use of the theoretical stress concentration factor for conditions of steady stress is confined to materials in which failure would be by brittle fracture. For ductile materials under steady stress, failure is unaffected by local stress concentration because of the redistribution of stress which occurs as a result of local yielding. However, under conditions of fluctuating stress, failure of a ductile material can be predicted by the use of a fatigue strength reduction factor and it has been suggested that this in turn can be related to the theoretical stress concentration factor by means of a notch sensitivity factor.This latter, experimentally determined factor is influenced primarily by the gradient of stress and the relative size of grain structure, the effect of a theoretical stress concentration being most severe in a fine-grain, heat-treated steel. A prerequisite, therefore, in the process of producing an efficient and reliable design, is the necessity

for the engineer to have at his disposal comprehensive and accurate information about the theoretical characteristics of regions of stress concentration.

The behaviour of a commonly used part, the shouldered shaft and its close relative, the shouldered plate, subjected to the simpler loading conditions, i.e. torsion, direct load or uniform bending moment, has been studied fairly extensively and reliable information is available

on the stress concentration factors relevant to a large range of useful parameters. In practice, however, bending is generally accompanied by shear and for this condition the existing information is scanty and, where available, is largely inaccurate, being based mainly on semi- empirical formulae which do not represent the behaviour satisfactorily except, perhaps, over a very limited range of parameters. For this reason the following work on shouldered plates was carried out with the object of obtaining information 011 the stress concentrations produced by various configurations of bending with shear, the results also being useful as a first approximation to similar loading configurations on shouldered shafts and projections such as gear teeth. The problem studied is essentially two-dimensional and the investigations were carried out by means of the well known photoelastic techniques.

1.2 Review of Existing Information1.2.1 For Uniform Bending Moment

The earliest results of any relevance were published in 1928 by E.G. COKER^^\‘ who carried out experiments on two plate models of stepped beams, each having a discontinuity, on one edge only, in the form of a shoulder and fillet. Using the photoelastic method, the two isolated values of stress concentration factor obtained for fillet radius to minor width ratios of 0«22 and 0 *06^ are, respectively, 15% above and 11% below the corresponding values obtained in this present

Papers published by M.M.-FROCHT^*^, in 1935 .and 1936, give results for models of shouldered plates having parameters ranging between width ratios of 0*6 to 0*9 2.at a fillet radius to minor width ratio of 0®08, to width' ratios of 0*11 to 0*5 for a radius to width ratio of 1. The photoelastic method used in conjunction with the "photographic” and "semi-photographic” techniques for estimating boundary values, gave results in good agreement with those of the author for the extremes of the radius to width parameter and show a maximum

'jdiscrepancy of 7*5% high for a radius to width ratio of ^ . Similarresults to those of Frocht are included in a paper by J.B. HARTMAN and

(7)M.M. LEVEN , 1951s using the same techniques on models of plates withcentrally enlarged sections*.

Investigations by I.M. ALLISON , 1958, dealing with a variety ofloading conditions and component geometries, include results forshouldered shafts subjected to uniform bending moment* A suggestion byR.E. PETERSON^ that a relationship exists between stress concentration.factors for shouldered shafts and plates having corresponding profiles,

(17)which is the same as that between the factors given by the NStfBER theory for grooved shafts and notched plates, is also discussed by Allison in connection with shouldered shafts and plates having tension loading* Values from the author’s results for shouldered plates in uniform bending have been used in conjunction with the Peterson relationship to calculate "predicted” values of stress concentration factors for shouldered shafts and these show some similarity with the corresponding results of Allison, the maximum difference v/ithin the range of parameters available for comparison being 13%«

1*2*2 For Bending With Shear

An early example of the use of the photoelastic method is to be

C 1 ^found in a paper published in 1925 by S. TIMOSHENKO and W. DIETZ inwhich an isolated value of stress concentration factor is given for a celluloid, shouldered plate model. Although the information is presented as being relevant to bending, the actual loading configuration used produced bending with shear and the rather inaccurate value obtained for the stress concentration factor, using the colour matching technique, points to the inadequacies of the methods then in use.

(2)The results of work by R.E. PETERSON and A.M. WAHLV , published in193o? include values of stress concentration factors obtained fromlarge diameter, shouldered shafts of steel, with a diameter ratio of 2— , using a precision extensometer of 0*1" gauge length. The three isolated values quoted are for a loading configuration which is predominantly bending but with some shear and the most significant difference, compared with the author's values, is for the shaft of smallest radius of fillet, for which the result is 12% low.

Included in a survey of existing information on stress concentration

factors published in 19^3 by G.H. NEUG2BAUER , is information quotedspecifically for shouldered plates subjected to bending v/ith shear.The stress concentration factors are presented as curves for a major tominor width ratio of at least 4 and covering fillet radius to minor

v/idth ratios of 0*1 to 0*5 for various load positions. The method usedis unspecified but is probably empirically based and the limited rangeof results obtained agree reasonably v/ith those of the author onlyfor a radius to minor width ratio of approximately 0*2.

(14)R.H. HEYU'OOD , in his book "Designing by Photoelasticity", published in 1952, gives some discussion on the problem of a projection having a load applied close to the fillet. Details of some investigations of tapered projections subjected to bending with shear and analysed by the photoelastic method are given and a semi-empirical formula,

purporting to give the peak stresses in the fillets, is quoted. Unfortunately, apart from deficiencies in the experimental technique, the formula is derived from a small range of model parameters and a single load configuration for each model and has a form which clearly does not represent conditions satisfactorily for loads in close proximity to the fillet.

CHAPTER 2 ■GENERAL TEST PROCEDURE

Notation'B Major width of shouldered plate,b Minor width of shouldered plate.b Width of plate at position of peak stress in fillet.f Material fringe value for a photoelastic material.!vg> Basic Stress Concentration Factor for uniform bending effect.K$ Basic Stress Concentration Factor for shear effect®Kq- Stress Concentration Factor defined by < / .k; Stress Concentration Factor defined by 6/, •A onom

Peak Basic Stress Concentration Factor for bending effect.A

Peak Basic Stress Concentration Factor for shear effect.A A,KTjKt Critical Stress Concentration Factors, i.e. peak values

of respectively.L Distance from shoulder to effective line of action of V>- .M Uniform bending moment.N Isochromatic fringe order. •T Radius of shoulder fillet.t Thickness of model.

W In-plane shear load.X Distance from a given boundary point to effective line of

action of W *'j Normal distance of internal point from boundary.H Distance measured along boundary from some datum.0 Angular coordinate on shoulder fillet measured from normal

td-minor width boundary.6 Non-zero principal stress at any given boundary point.dnorf) Maximum stress due to moment M on minor section, as given

by .

Mean shear stress on minor section, i.e.

The shouldered plate model geometry, specified by the basic parameters B i b and P , is illustrated in Figure 2.1.

B

FIGURE 2.1 Basic Parameters for Models

FIGURE 2.2 Details of Grid for Main Measurements*

D is tance aW\q \>oundar

10 divisions feacVi Q-05" W r > 0*3and 0*025 for r< 0*2)

Figure 2.2 shows the method by which positions on the boundary are specified by the measurement 2 , or, alternatively, for points on the fillet boundary, by the angular quantity © •

2.2 Dimensions used for Shouldered Plate Models

FIGUHEi 2.3 Details of Basic Dimensions for Models

Details of the dimensions of the shouldered plate models are given in

Figure 2.3i the values of $ , b and r shown providing for models having\? 'width ratios of 0«2, 0»6, 0»8 and 0»9 for each of the radius to minor

1 1 1 1 width ratios of 1, - , g and-^ .

2.3 Production of Models2.3*1 Manufacture and Initial Treatment of Moulds

3MFor each mould to be manufactured, two pieces of ^ thick duralumin

plate were selected, each l4fr square and as flat as possible. Aftercycling these several times to 150°C in an oven to minimise any subsequent

tendency to bowing, the plates were surface ground, approximately 0*01"being removed from each face. Spacers for three sides of each mouldwere fabricated from 1" wide mild steel strip and surface ground to the

-| ** 3"required thicknesses of g and . The respective plates and spacers were clamped together for drilling suitably arranged holes for clamping bolts, after which the inner faces of the plates were polished by hand, using progressively finer carborundum papers down to ^20 grade and finishing with metal polish. All plates and spacers were cleaned thoroughly, then covered with a film of RELSASIL 7 silicone grease and returned to the oven for several cycles of temperature up to 150°C. Immediately prior to the use of moulds in a casting operation the inner face of each plate was coated with a fresh film of RELEASIL 7 grease, wiped with a pad of cotton wool soaked in carbon tetrachloride to remove surplus grease and polished with a pad of dry cotton wool to a mirror- like finish. The moulds were then assembled using a thin smear of a putty made from Kaolin and RELSASIL 7 on each mating face of the spacers in order to give a good seal, the bolts, washers and nuts having previously been soaked in a solution of 1 part RELSASIL Ik oil to 10 parts of chloroform to facilitate their later removal.

2.3*2 The Casting, Curing and Annealing ProceduresThe material chosen for the photoelastic models, epoxy resin GT 200

(formerly known as Araldite B), is a hot setting casting resin and is used extensively for this purpose because of its good photoelastic properties and the relative ease with which it can be machined. The resin

was supplied in the form of small, yellow-brown platelets and the hardener (phthalic anhydride) as a white powder. The two components must be mixed in the liquid state (i.e. when hot), in the ratio of three parts of resin to one part of hardener by weight, the cured resin having a specific weight of, approximately, 0*045 lbf/in3. To ensure uniformity throughout the test series sufficient sheets of CT 200 were produced at one time to provide for all models, including spares, the following procedure being adopted.

An oven with circulating fan was set to a temperature of 140°C and in it were placed the prepared moulds for pre-heating, spare containers and a No. 70 wire gauze filter and the requisite amount of resin in a clean container. After allowing sufficient time for the resin to melt, the appropriate quantity of hardener was heated in a covered container until melted, poured and stirred well into the resin, the mixture then being filtered into a clean, pre-heated container and returned to the oven for five minutes in order to bring the temperature of the mixture up to the casting level of 140°C. Each mould was then removed from the oven, supported vertically and filled with the resin mixture via the open edge of the mould and returned to the oven. The oven temperature'was retained at 140°C for 30 min to allow the release from the mixture of any included air bubbles and then reduced to 105°C, at which level the resin was cured over a period of 15 hours. Following this the oven temperature was reduced at a rate of 5°C/hr until it reached 70°C, the moulds then being removed from the oven and allowed to cool to room temperature. The complete sequence of temperature changes v/as controlled by means of an anticipatory controller fitted with a suitably prepared programme cam.

After removal of the cured sheets from the moulds, traces of the releasing agent were removed from their surfaces with pads of cotton

wool soaked in carbon tetrachloride. Except for strips extending up to•zuft in from the edges which had been in contact with the spacers, the surface finish of the cast plates was such as to require no subsequent polishing operation, the maximum variation in thickness did not exceed 0«002M and the birefringence due to initial stresses was reasonably low. Careful removal of these strips by means of a fine toothed, slov; running

-|tl "jf!band saw left usable rectangular plates approximately 12— by 10^ 1 fromeach of which could be cut two models. Each plate was laid flat on asheet of clean paper placed on plate glass and replaced in the oven toundergo one annealing cycle controlled automatically by the temperaturecontroller and suitable programme cam. The annealing cycle comprised aperiod in which the temperature was raised at a rate of 5°C/hr from roomtemperature to 130°C, a period of six hours at this value, followed byone in which it was reduced at the same rate to a temperature of 70°C.All plates so produced were checked carefully on the photoelastic benchat selected points and in no case was the initial birefringence found toexceed 0*04 fringes for a nominal thickness of g . All plates werestored flat as in the annealing process, but at a constant temperature

(8)of 70°C in order to avoid time-edge stress effects , until required for cutting and testing.

2.3*3 Marking out and Machining of ModelsTo facilitate the production of models to the required accuracy

a set of templates was prepared from 12 gauge duralumin plate. That for the basic blank consisted of a rectangular plate 10M by 6U with the two

■Z11rows of equi-spaced holes corresponding to the loading holes in the models. The other templates had the same overall dimensions of 10" by 6” but the longer edges of each were cut to correspond to the profiles of models of various shoulder r&dii and the same major and minor widths of 6n and 1*2n, respectively. Each of these templates was provided with

holes so arranged that the other major widths could be

obtained by relocating the appropriate template on the model using four

of the existing loading holes, A photograph showing a representative

selection of templates is found below (Figure 2.*0.

FIGURE 2.^ A Representative Selection of Templates

All machining operations on the templates were performed on either a vertical spindle milling machine or a horizontal spindle jig borer, ensuring that the hole and radius centres and the profiles were located

Using the basic template the epoxy resin plates were marked outwith 1011 by 6" rectangles which were then cut out on a fine toothed,slow running band saw, leaving a machining allowance of about 0 »03M oneach edge* Each plate was clamped in turn to the template and theloading holes drilled and reamed. Locating each plate on the template

1" .by means of four tapered pins the basic shape was machined with a pr dia spiral fluted, high speed steel cutter in a routing head, taking two

to within an accuracy of - 0-002".

roughing cuts followed by a finishing cut of 0 -005". The depth of cut

and the profile were controlled by locating the template against a pin

of suitable diameter projecting from the supporting bed, the pin for

the finishing cut having a diameter equal to that of the cutter. With the

model blank located on the appropriate template, the main measuring grid

shown in Figure 2.2 and a second grid at intervals of 0*05" across the

minor width for calibration measurements were marked out by means of

a vernier height gauge and dividing head. Again leaving an allowance for

machining, the shapes were cut out on the band saw, after which the

router was used, as before, to complete the raa.chini.ng of the shouldered

profiles for the various shoulder radii required, all models so produced

having a major width of 6". Figure 2.5 is a photograph showing the

router being used to machine the profile of a model located on its

template.

FIGURE 2.5 Model and Template set up on Router for the Machining

Operation on the Profile.

Particular care was taken during the machining operations on the shouldered profiles to maintain a steady, continuous relative motion between cutter and model, in order to avoid the significant edge stresses which would result from an excessive local temperature rise. After the completion of all photoelastic tests for this set, the models were re-utilized for each succeeding set simply by reducing the major

width of each to the new value by the machining procedure just described, thus obviating the necessity of' marking out and machining a completely new set of models.

2.3*^ Inspection and Storage of ModelsThe dimensional accuracy of each model.,both with respect to the

profile and the relative location of the measuring grids, was checkedby means of a travelling microscope and found to be satisfactory, exceptfor the first model produced. This was found to be undersize along oneedge of the minor width due to lack of expertise in -the machining

technique and it was therefore discarded and later replaced.Examination of the models at selected points in the photoelastic

bench showed relative retardations of up to 0*1 fringes in regions upto about 0*01" in from the edges, due to stresses resulting from themachining process, when measured immediately after machining. Afterstorage at 70°C for one week, re-examination shov/ed significantimprovement to a value not exceeding 0*05 fringes on a model thickness

1”of ^ , this being considered satisfactory.

2.*+ The Loading Systems and Rigs 2.^*1 Measurement of the Applied Loads

In the photoelastic tests two different systems of loading were used on the models and in each case the total load acting on the loading rig was indicated by a spring balance having a circular scale reading

to 100 lbf with sub-divisions of 0*5 lbf, the indicating pointer having an

adjustment for the zero position. Preliminary tests showed that loads

between 15 lbf and J>0 lbf would be needed and, accordingly, the calibration

of the balance was checked from zero to *+0 lbf using dead weights. In

various tests using initial loads of 1, 3 and 5 lbf and with the pointer

zero adjusted to suit, the indicated values did not vary by more than•j*an estimated -0*1 lbf over the test range. When used in the photoelastic

tests, loads were adjusted by the tensioning screw to a convenient

scale marking and it was considered that it is possible to achieve this

to an accuracy of better than -0«1 lbf.

2.^.2 Rig and Loading Procedure for Uniform Bending

In order to apply a uniform bending moment to each model in its

own plane a suitable loading rig was constructed and some modifications

made to the standard photoelastic straining frame to accommodate, in

turn, both this and the other loading rig used. A photograph, giving a

general view of the arrangement, is shown in Figure 2.6

FIGURE 2.6 General View of Rig for Uniform Bending

Each end of the model under test is retained in the slotted end of the

1T'-shaped loading lever by eight pins which are a push fit in the lever and the loading holes in the model. The vertical load applied centrally to the horizontal beam above the model is split equally between the two inner vertical links and reacted through the two outer links, the system of forces and the resulting bending moments being shown in the diagrams,

Figure 2 .7 ? below.

I(1

\\ .

FIGURE 2.7 System of Forces and Bending Moments

After removal of the model to be tested from the storage oven, approximately 20 minutes was allowed for it to cool to ambient temperature

before assembly in the loading rig. With the outer vertical links free at their lower ends‘small counterbalancing masses were added to them until balance of the loading links was indicated by the presence of zero fringe order across the minor width of the model remote from the

changes of section. The spring balance was then adjusted to read zero,

the anchorage pins inserted and the tensioning screw adjusted to give

a load producing a maximum fringe order at the shoulder fillet of

y ro (d j meanwhile carefully checking that the model did not

show signs of any local buckling. Allowing a period of 3G minutes, after(18)which there would be no further significant optical creep , the

photoelastic readings were taken and the model was then returned to

the storage oven until required again.

2.^.3 Hig and Loading Procedure for Bending with Shear

To enable systems of in-plane forces corresponding to various

combinations of bending and shear to be applied to the models, a

loading rig using the same modified straining frame was constructed,

a photograph showing the arrangement being given in Figure 2.8*

FIGURE 2.8 General View of Rig for Bending with Shear

Each loading lever consists of a pair of ’L'-shaped plates clamped one

on each side of the model by eight fitted bolts through the loading

holes, with short extension strips fitted over these to act as guides

to the adjacent end of the other lever. Equal and opposite vertical forces, one to each lever, can be applied through loading links, there

-ZH -p»being a series of holes at a pitch of — m each lever providing various positions for the common line of action of these forces. Extra arms may be fitted to the levers, when required, to facilitate the

addition of balancing masses and the levers and arms are arranged in a different way when the load line is to the left of the shoulder on the model. The system of forces and the resulting actions are shown

in the diagrams, Figure 2*9, below.

a

W is vta effcc.ViMn force.

W

f)—<■& .... i. gp—

w

Bending Moment Diagram

— ^ ^

\\ \ \ JWShearing Force D iagram

FIGURE 2.9 System of Forces and Resulting Actions

Removing the model from storage and allowing it to cool, as before, it was then assembled with the lower lever, loading link and associated parts. With' this part assembly suspended by the loading link, sufficient mass was added to balance the system and the total weight determined. The upper lever and link were then added and the complete assembly was suspended in the straining frame and balanced by the addition of further mass to the upper lever. The spring balance was adjusted to the value of weight previously determined for the lower part assembly, the anchorage pin inserted and the tensioning screw adjusted to give a load producing 3 to 6 fringes at the shoulder fillet.

The levers v/ere eased away from their guides, lubricating oil intro­duced and the load indication checked for significant changes due to friction. After a further period of 30 minutes the photoelastic readings were taken and the model returned to storage.

2.5 The Optical System and Photoelastic ReadingsA photograph showing the polariscope set up for taking measurements

by the Senarmont method is given in Figure 2.10 and a diagram showing the disposition of the optical elements in Figure 2.11. Illumination is supplied by a high pressure, mercury vapour lamp of power 250 watts, shielded by a die-cast housing and cooled by convection. The light

rays emerge through a U.V. filter and are brought to a focus at an iris diaphragm by a small lens. A green filter may be interposed at this point giving a reasonably monochromatic green light output of mean wavelength 5^61 A°. A lens of 8M focal length and diameter produces a beam of collimated light and a sheet of translucent paper converts this to diffused light illumination. This passes in turn through the polariser, the model, a quarter wave plate when required, and the

analyser. The model is viewed through a travelling microscope fitted

with a 9" focal length objective and for each observation the crosswire

FIGURE 2.10 View of Polariscope set up for

Senarmont Method

is brought into a position of no parallax with the point considered.

With the green filter and the quarter wave plate removed and the polariser and analyser crossed and coupled, the isoclinic parameter was noted at each grid point. Then with the polariscope set up for the Senarmont method of measuring fractional fringe orders, observations were made, taking points in blocks in which the principal inclinations lie within a 5° band^*10 The value of total fringe order at each point observed was determined by reference to the fringe pattern

produced without compensation, i.e. before rotation of the analyser

from its datum position. Typical fringe patterns for both uniform

S LIGHT SOURCE D TjIFFUSELR Q QUftV-TeR WftNF- TLME

I LENS MO W ft P TOLMUSE'R ft KNftLISE.'R

L COLUHKima LENS n HODEL T TRMELIIUG. ttlC*OSC.OTt

FIGURE 2.11 Disposition of Optical Elements

bending and bending with shear loading systems are shown in Figures 2*12 and 2«1J, respectively.

■ - •••

FIGURE 2.12 Typical Fringe Pattern for Uniform Bending

FIGURE 2.13 Typical Fringe Pattern for Bending with Shear

CHAPTER 3REDUCTION OF TEST RESULTS AND DETERMINATION OF

THE BASIC STRESS CONCENTRATION FACTORS

3.1 Notes on the Stress Concentration Factor

In general, the stress concentration factor for a particular geometrical feature is expressed as the ratio of a relevant peak stress in a component to some arbitrarily chosen nominal stress. For any given component geometry and loading case, a stress concentration factor may have differing values dependent upon the basis on which the nominal stress is chosen. However, provided this basis is fully defined for any values of stress concentration factor quoted, the choice can be made solely from the point of view of the convenience with which the designer can make subsequent use of the information. Conversely, it is incumbent upon the designer using such information, to be aware of and take due account of this basis of definition when making stress calculations.

In this investigation all stress concentration factors quoted are based on nominal stresses calculated for the minor width of the shouldered plate. A linear relationship, showing that the stress at any point on the boundary is dependent upon two basic types of factor, is proposed and justified in this chapter and curves of the resulting values for these factors are given. This information is later re-presented in alternative ways, either for greater convenience of usage or for the purposes of comparison with the results of other investigators.

Since it will be found that these basic factors have values which vary from point to point on the boundary, the peak values for the two types not occurring at the same position, it is necessary, in order to

calculate the stress at any boundary point, to provide the information regarding the variation of the basic factors with position. In this investigation the basic factors will be termed "Basic Stress Concentration Factors" and their peak values, "Peak Basic Stress Concentration Factors"J~ A factor representing the ratio of the boundary stress at any point to some nominal stress will be called a "Stress Concentration Factor" and a peak value of such a factor, a "Critical Stress Concentration Factor".

3o2 Definition of Basic Stress Concentration FactorsThe Basic Stress Concentration Factors relate to the non-zero

principal stress at the boundary of the shouldered plate subjected to in-plane forces producing, in general, bending with shear and as a special case, uniform bending. A diagram showing the quantities involved

is given in Figure 3«1*

a WL

■Wa 6

b —

d is VW noft-y.CA' 0 principal sl'cess at any given poinV on \\e. Wmdauj.VI is tie. sit ca r Wt<U t is tW [VucVnc-SS plate..

■nom “ w /tt is tW mean nominal4ic,ar stress bn minor section.

X is VW distance.of V\\is pomffrom \ifi*. cA action o \ W.b is minor Vvi\cLVV\ of plate.

~ foVlx/tf?- is maximum nominal bcndino stress calculate simpte-bearn VV<cor foe ntamc<\V VIX on minor sccbtcm.

FIGURE 3.1 Quantities Relating to Basic S.C.F.s.

It was postulated that the stress d at any given point on the boundary would, in general, depend upon dnomand Tncw), this dependence being expressed by the linear relationship

where are constants for a given boundary point on a model of aparticular geometry and are the Basic Stress Concentration Factors

referred to. Furthermore, it was suggested that this linear relationship, with the same constants, would hold for the case of uniform bending moment, the value of T notn then, of course, being zero.

3.3 Determination of Boundary Values of Stress or Fringe Order3.3.1 Notes on Some Existing Methods

The obvious pre-requisite for obtaining reliable information regarding stress concentration factors is the determination of boundary stresses to a sufficient degree of accuracy and when the photoelastic method is used this devolves, in so far as the optical measurements are concerned, upon the accuracy with which the isochromatic fringe

numbers at the boundary can be determined. This requirement is of particular relevance in this investigation if the separated values of the Basic S.C.F.s and Kc, ) are to show significant trends with any reliability.

(z0Some earlier investigations use the ’photographic’ and ’semi- photographic' methods suggested by Frocht but, although improvements in photoelastic techniques have minimised some of the difficulties associated with boundary vagueness, the accuracy of these methods

rely in part on the maximum fringe order attained. Thus, a figure of at least ten fringes is quoted as being necessary for errors to be less than -2«5%$ this being a condition which, for reasons of stability for example, is not always appropriate. Again, the technique of measuring fractional fringe orders by rotation of the analyser, i.e. the Tardy or

Senarmont methods, permitting observations at internal points to anH"* '* _ i—accuracy of -0»02,of a fringe , become less reliable when an attempt

is made to set a fringe on the boundary of a model, the errors involvedtending to increase as the stress gradient becomes greater.

(1 1)A more recent paper by Allison advocates a method of obtaining boundary values of isochromatic fringe number by extrapolation of a curve given by a power series of the form

N = K K 3 + f + K y 1

fitted to values of fringe order measured at discrete points distant VJj. from the boundary along a line normal to the boundary. Factors affecting the accuracy of the results are discussed, notably the model

thickness in relation to the fillet radius, the order of the fringes observed and. the effects of time-edge stress and particular reference is made to the importance of accuracy of boundary location. The polynomial curve is fitted on the basis of the ’least squares' criterion using a digital computer and the program includes a subroutine for evaluating the root mean square deviation as a check on the acceptability of the extrapolation. Practical tests indicate that the technique permits values at points of significant stress concentration to be determined consistently accurate to -3 <»

(13)A paper by Clenshaw and hayes on curve and surface fitting \gives a survey of the various methods available and discusses theirrelative merits. Some criticism is levelled at the classical method ofusing power series in that the normal equations rapidly become ill-conditioned as the order of polynomial is increased. A method due to

(12)Forsythe , using orthogonal polynomials generated by recurrence, is recommended, the principal advantages apart from the improved accuracy being that no prior knowledge of the degree is required and the time

needed for computation is relatively small. The method provides a simple' check, based on the least squares criterion, on the closeness of fit to the given data.

3.3»2 Reduction of Data in the Region of the Shoulder FilletFor the reasons mentioned in section 3»3«1 it was decided, for

this present investigation, to use the technique of extrapolation to the boundary from measured internal values of isochromatic fringe

(11)number, observing the experimental methods as described by Allison , but adopting Forsythe's method of orthogonal polynomials for the curve fitting procedure. A summary of the theory of Forsythe's method is given in.APPENDIX A. However, it must be acknowledged that, in view of the degree of polynomial finally chosen for the:fit, it is possible that there would have been no significant differences in results given by the two methods. ' *

To facilitate the performance of the necessary calculations of ^the orthogonal polynomials from each set of measured data, a program j

iwas developed for use on a digital computer. Using selected sets of tdata the program was run to give the fitted and extrapolated boundary values of fringe order corresponding in turn to polynomials of degree three, four and five. Checks for each line of readings showed acceptably low values of root mean square deviation in every case and also ’demonstrated the expected improvement in closeness of fit as the degree of the polynomial v/as increased. Graphs plotted of extrapolated values against distance along the boundary showed, in general, less scatter for the lowest degree of polynomial considered. It was therefore considered apx>ropriate to use the orthogonal polynomials of third degree as the basis for the determination of extrapolated boundary values of fringe order in the region of the shoulder fillet, this representing a satisfactory compromise between the attainment of an

exact fit to the measured values and the tendency to give undue

prominence to the effect of natural scatter in the measured values when polynomials of higher degree are used. The pilot program v/as modified to a form suitable as a subroutine for inclusion in the main program, details of which are given in APPENDIX B.

3*3*3 Reduction of Data from Calibration TestsMeasurements of isochromatic fringe number for calibration

purposes were taken at interior points along a line across the minor width, sufficiently removed from changes in section so that their disturbing influence was negligible, the model being subjected to uniform bending moment. Graphs plotted of the measured values showed good correlation with the expected linear variation across the section.

The extrapolated boundary value of fringe order used for calibration of each model was derived from a linear fit based on the least squares criterion, the necessary calculations being performed with a simple program on a digital computer.

3.A Justification of the Linear Relationship between theBasic Stress Concentration FactorsThe linear relationship between the Basic S.G.F.s, quoted in

section 3*21 niay be written in the form

+ Kc,Horn

which, on substitution of the expressions defining dnol andT , reduces to the form

= Kr = 5, + Ks'•nom P

Now, from the usual relationship between the relative retardation in fringes, the principal stress difference and the model thickness, we

may writed - i/^

where is the boundary value of fringe order corresponding to the principal stress 6 and j" is the material fringe value.Again, from the calibration test we have

where N is the boundary value of fringe order on the minor width, corresponding to a uniform bending moment of M i.

Then, eliminating |) ,

d - b M

and substituting Shi/ for TftomC v

K t * feNo HW b Nckt

Since is known from the calibration test for uniform bending moment, the value of Kp can be determined at any boundary point for a particular combination of bending with shear.It follows from the resulting linear relationship that if Kp can be determined for various values of X , the result of plotting it against DC should be a straight line of slope K and intercept K« •

P *Using sets of data obtainedfrom one model for the loading case

of bending with shear with the effective line of action of the load set successively at distances L of V', y\ 2", 1" and --111 from the shoulder on the model, (see Figure 3*1)» the pilot program for orthogonal polynomials was used to determine all boundary values of fringe order. Then, after calibration of the model by the uniform bending moment test, values of K-p v/ere determined for each boundary point and plotted against X , a selection of these graphs being shown in Figure 3*2*

R t- V

w4v,

F \6 0 K E 3 .2 . Selg.t.V<L<l p\ofc> iteVroirm^ \\t\ecv.v r&WhowaWy UWigAA Kg, AOcl Ks

Since, in general, each graph of K-j- against OC. was seen to be a good approximation to a straight line, the use of the linear relationship between the Basic Stress Concentration Factors was considered to be justifiable. In addition, using the uniform bending moment test, values of were calculated independently, from the ratios of the boundary values of fringe order in the region of the shoulder fillet to that determined at the section used for the calibration of the model. These independently determined values of Kg showed close agreement with values for the corresponding points derived previously

from the slopes of the graphs of K-j* against X «

3*5 Determination of the Basic Stress Concentration FactorsThe calculations for values of the Basic S.C.F.s and )

for all models were performed on a digital computer using the results from the photoelastic tests. The master segment for the program written for this accepts data from the uniform bending moment test and the successive loading cases of bending with shear and calls two subroutines. The first of these gives tabulated values of measured fringe order, the corresponding values from the orthogonal polynomial fit together with extrapolated boundary values, checks on the closeness of fit and the values of Kg, determined from the uniform bending moment test. The second gives tabulated values of K3 and Ks derived from the slopes and intercepts of the straight line fits to the values of against X , using the least squares criterion. Details of the program together with a typical set of output data are given in APPENDIX B.

A total of fifteen models with a minor width of 1*2" were tested, covering the range of parameters indicated in the following table, these including two models for the smallest fillet radius attempted (i.e. 0*1") as a check 011 the reliability of the results.

T A B L E OF P H 0 T 0 E L A S T I C MODELS

\ v/I

o-s. X X X X X X

CH X X...

X X

0-8 .X X

OS u j X X

.Graphs showing the variation of Wg, and for the various models tested are shown in Figures % 3 to inclusive*

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3*6 Accuracy of the ResultsIn making an assessment of the overall accuracy of the results,

a number of potential sources of error must be considered, these being

conveniently categorised as either systematic or random errors. Errors in the former category are usually of a repeatable character, such as those arising from deficiencies in the measuring systems and can usually be determined from independent tests. On the other hand, the random errors which can arise, for example, from the many subjective processes involved when deciding upon the values of readings, can only be dealt with satisfactorily on a statistical basis and in this case the level of confidence in the reliability of a result will usually rise as the number of measurements showing reasonable correlation is increased. However, accuracy in general cannot be increased without limit and usually there has to be a compromise made between several conflicting requirements. Bearing this in mind, the following notes list the various precautions taken to minimise the effects of both systematic and random errors and give individual and overall assessments of the maximum errors to be expected.

3-6.1 Dimensional Accuracy of Models and Grids

The method of machining using accurate templates ensured that model profiles were reproduced correct to “0®002n, this apparently

•f-giving a maximum possible error of -2% on the 0*1" radius fillet. Inpractice this error was considerably less and in any case this orderof error in profile dimensions would have little effect on the stressconcentration factor.

Of greater significaurice is the positioning of the grids relative.+to the boundary and an acceptable value for this was set at “0-001”

and checked on every model. With typical gradients, for a model 0*1”

thick, of 15 fringes/ in at the position of peak stress and 3 fringes/in

at the calibration section across the minor width, the resulting maximum errors in extrapolated boundary values would be of the order

of and -0.2°/o% respectively, the error being substantially constantaround any one fillet<>

3.6.2 Accuracy of Determination of Effective LoadsA check on the calibration of the spring balance used for load

indication showed it to be correct to within -0.1 lbf over the range of loads measured# This represents a maximum variation of the order of ~0.5% for both the uniform bending moment calibration test and the bending with shear tests# The loading linkages were balanced and their loading effects allowed for and a careful check was made to ensure that there was no significant hysteresis due to friction. The weights of models were within the range 0*12 lbf to 0«231bf and an approximate

allowance for each was made v/hen adjusting the spring balance zero, but in any case, this effect is insignificant over the test region in each model.

3.6 .3 Time h'dge Stresses and Machining StressesIn the models used in the tests the maximum isochromatic fringe

number due to the combined effects of time edge stresses and machining stresses was.estimated to be less than 0*05 wavelengths for a model thickness of 0.12”# Furthermore, the technique of extrapolating to the boundary from measurements taken at interior points tended to minimise the effect of this and, in verification, readings taken with the sense of the loadings reversed showed good agreement.

3.60*1 The Sdnarmont Method(9)A paper by Jessop on the measurement of fractional fringe

orders by the Senarmont method and one by A l l i s o n o n the technique

of selecting successive blocks of points having isoclinic parameters within a 5° band, indicate that the expected accuracy of determination should be better than -0*0/1 (of a fringe. The block technique undoubtedly reduces operator fatigue and enables more readings to be taken in a given time. Using these methods, the mean value of isochromatic fringe number, using at least five readings at each point, was determined for each grid point, a careful check being made to ensure that individual readings fell within an acceptable scatter band. Working on maximum fringe orders of 3 &t the fillet and 2 at the calibration position, the corresponding maximum errors would be -0*7% and -1%, respectively*

3*6*5 Other Factors Influencing AccuracyIn general, the accuracy of measurement of the isochromatic

fringe number is dependent upon the number of fringes which can be observed and this, for a model of given scale and subjected to a particular strain, is related to the thickness. Against this must be set the optical requirements for optimum resolution of the boundary and a reasonable compromise is reached by using a model of thickness not exceeding the fillet radius. For this investigation a nominal

xuthickness of was used for the models of 0*1" and 0 »15” fillet radiiDd.'jitand of g for the models with larger radii. An increase in applied load would give higher fringe numbers but with the possibility of creating instability of the system. For this reason a careful check was kept on the model as loading was applied, for signs of local buckling both by inspection of the flatness of the model and by observation of the clarity of the isoclinic pattern. Some preliminary tests were also made, taking values of fringe order at selected points for a number of different loads as a check on the load range for linear behaviour of the model.

Improved resolution of the boundary would be obtained with models

of larger size but against this must be set the practical problems ofmanufacturing larger models and accommodating these in a loading frameand the increasing likelihood of model instability and this againrepresents an area of compromise.

Care must be taken in the design of the model and loadingarrangements to ensure that there is sufficient distance on eitherside of the shoulder in order for uniform stress distributions to bere-established. For the models used in this investigation the length

to width ratio was two for the minor width and varied between 0*75 and3.4 for the major width. Measurements taken on models subjected touniform bending moment showed that linearly varying stress distributions

were re-established at distance to width ratios of approximately 0*7on either side of the shoulder fillet and this figure is in agreement

(15)with investigations by Fessler, Rogers and Stanley on shouldered(?)plates in tension and by Hartman and Leven on plates in bending,

3*6.6 Statistical Methods and the Overall AccuracyAt each stage in the determination of results from the experimental

data there is the advisability, wherever appropriate, of using methods of reduction which tend to increase the reliability and provide checks on the order of accuracy and which avoid the introduction of spurious effects due to subjective inconsistencies or mathematical ju'ocedures which are potentially ill-conditioned. Thus, for the determination of boundary values of isochromatic fringe number by extrapolation from interior values the technique of curve fitting using orthogonal polynomials was adopted, this including a subroutine for calculating the root mean square deviations as checks on the closeness of fit.After preliminary tests on the scatter of the resulting boundary values, polynomials of third degree were adopted, these producing root mean

square deviations not greater than 0*005 of a fringe on all but a very

small proportion of the lines and for these the figure was of the order of 0«02jof a fringe. These exceptions occurring on the model of smallest fillet radius, further tests were made on a new model as a check on the reliability of these results. The root mean square deviation is, of course, a check on the closeness of fit of the polynomial curve to the measured values but, in view of the low degree of polynomial used, the generally low figure for the deviation is an indication of the reliability of the extrapolated values.

Initially the separation of values of the Basic S.C.F.s K- and for each model geometry was attempted using the data from the uniform

bending moment test and only one loading case for the bending with shear test. However, realising that this increased the possibility of scatter, the technique was adopted of using a least squares, straight line fit to the data from several loading cases for bending with shear. Apart from the increase in reliability resulting from the much larger population of readings, this gave the added advantages of verification of the linear relationship between the boundary stress and the Basic Stress Concentration Factors and provided an independent check on the values of from the uniform bending moment test. Using statistical methods based on probability theory, (summarised in APPENDIX C), it

was predicted in general, that for a probability of 93% the scatter bands for the precision of values of and given by the straight line fits are approximately -2% and respectively.

Considering the individual errors noted in the previous sections, the worst possible error in the values for Ky would be butadopting the more usual root mean square criterion for combining the

4-individual errors, a more realistic figure would be Acceptingthat the corresponding errors in the separated values of and

would be larger than this, the errors predicted by the two analyses

show reasonable agreement .and the great majority of points for the graphs of and fall well within a -5% scatter band about the mean curves drawn. In the following chapter the accuracy of the peak values obtained for the Basic S.C.F.s will be further justified by demonstrating their correlation with variations in the ratios of major width and fillet radius to the minor width.

CHAPTER k DISCUSSION OF RESULTS

^.1 The Carpet Plotting TechniqueIn displaying some of the information pertaining to the various

types of stress concentration factor the technique of carpet plotting has been used. The advantages of this technique are immediately apparent when studying the correlation of some quantity against two parameters since, apart from the more concise way in which the information is presented, any attempts at smoothing the data can be made systematically taking due account of the two governing parameters.

This in turn creates more favourable conditions when attempting extrapolations to limiting values of the parameters* For these reasons

the carpet plotting technique has been used wherever relevant* The most suitable parameters for this particular geometrical profile have been found to be ^ and . These basic parameters describe completely

the model geometry but, of course, they may be used in alternative combinations to emphasise some particular characteristic of the same information.

The Basic S.C.F, for the Bending Effect (Referring to Figures 3*3 to 3*7* inclusive, in which curves are

given showing the variation of , the Basic Stress Concentration Factor for the bending effect, for the various shouldered plate models tested, a number of common characteristics are evident.

(a) has a value of unity for the minor section for boundaryVrpoints at distances greater than approximately 0*2’', i.e. 6 %

from the junction of the fillet and minor width. Alternatively, if the distance required for uniform conditions to be reachedis expressed as a proportion of the fillet radius, this ratio

varies inversely as the fillet radius, having a value of g

for ^ = 1 and of 2 for •(b) At the end of the fillet where it either blends into the

shoulder or, in the case of incomplete fillets, intersects the major width, the value of tends to zero. In either case, at the junction of the shoulder or fillet with the major width, whichever is relevant, the conditions for equilibrium at a free boundary require that the boundary stresses and hence the value of must be zero.

(c) Kj, reaches a peak in the fillets, the angular coordinate ofthese peaks having only a small range of variation, i.e. 5° to9°» approximately. The values of 8 estimated for the models

r, V,tested have been correlated against ^ and 3 ratios by means of the carpet plot in Figure *+.2 and this shows that the B

values for the positions of the peaks have only a minimalL

increase as decreases from 0»9 to zero and that, withinp

this range, 0 increases as ^ decreases.(d) The magnitudes of the peak values of Kj, in general increase

with decreases m both and 3 and the correlation betweenthese parameters will be studied in a later section.

Considering particular results, it can be seen from Figure 3*7rthat, for the models having a ratio of unity, there is no significant

change in the variation of rvg, along the boundary for width ratios 3

betv/een 0«*2 and 0*9, even although at the latter ratio of 0*9 the fillet remaining subtends an angle of only 15°.

O'fe

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FiguRE4,2 Carpi' p U : ay l o f postKftte g>| pe.<x,k. voices on M d .

*tv3 The Peak Basic S.C.F. for the Bending Effect ( )0 r hThe carpet plot of against the parameters ^ and ^ is given

in Figure ^«3» the curves being drawn using the peak values estimatedfrom the curves of , Figures 3*3 to 3*7» for the fourteen differentmodel geometries. Negligible smoothing of the values was necessaryand the curves were extrapolated to the limits of zero and unity for. A

jjg , the value of for the latter limit (i.e. corresponding to nochange in section) being taken as unity. The extrapolations to the zero

b/limit for '32, were reasonably easy to achieve since the curves werewell-behaved in this region. (It should be pointed out that the originalplots were carried out to a scale four times that of the curves shown

in Figure ^.3)*A number of comments on the characteristics of these curves can

be made, as follows :-A Ip

(a) The value of increases significantly for decreasing Kfor any value of <0 , apart from unity.

A ^(b) The rate of change of with respect to ^ also increases

significantly with decreasing ^ , the curves for a given ^ apparently being asymptotic to ^ = 0.

(c) The rate of change of with respect to ^ is substantiallyr,constant at any given value of ^ over the whole range of

k values.u k(d) The curves of against ‘J, show a critical change at approx-

k,imately the value of ^ corresponding to the transition between complete and incomplete fillets. For incomplete fillets

\) fe,the rate of change of ^ with respect to ^ is relativelyrapid, particularly for the smaller ^ values. For complete fillets the rate of change is considerably less, the decrease

1/ p1 1in for = 1 and ~ being negligible. Thus, for example,

<?!

<

o-o-

o

6-J—

oi

r 1 ufor the smallest ^ tested, i.e. , the increase in t\^\>between = 0*2 and zero is only approximately 2%.

Bearing in mind that the curves shown in Figures 3*3 to 3*7 represent both as determined from values obtained from the various loading cases for bending with shear and the values given by the uniform bending moment tests, there being no significant differences betv/een corresponding values, the results for ^ may be compared with those obtained by other investigators for uniform bending moment tests0 Some results of a paper by Coker on "Stresses in the Hulls of Stranded Vessels", (Trans. Inst. Naval Arch., 1928) are summarised in "Treatise on Photoelasticity" by Coker and Filon^^, pages 4?1-2* Two plate models of beams, each having a discontinuity on one edge only, were tested under uniform bending moment, details of the models being given

in Figure The model geometry does not correspond entirely with that

Model

FIGURE ■ 4 Models used by Coker

used in this.investigation but some similarity in-results is apparent.

For both models Coker observed that the peak stress occurred on the

fillet boundary at a point given by 0 =10° approximately, (c.f.Author’s values of 7°-8°). Comparing his figures for the ratios of peak stress in the fillet.to the boundary stress in the minor width remote

Afrom the fillet, with values of estimated from Figure ^.3» model A r

gives the result of 1*^1 (i.e. 1 5% high) while that for model B is 1*69 (i.e. 11% low). Apart from discrepancies which are. to be expected because of the slightly different model geometry and the lower sensitivity materials then available for photoelastic models, the result for model B would be suspect in view of the large ratio of thickness to fillet radius (i.e. ^ = 3*2).

(7 MTwo papers by Frocht 1 , published in 1933 and 193&, include

the results of photoelastic investigations carried out on shoulderedplates subjected to uniform bending moment. The range of models covered

r bvalues of from 0«08 to unity and values of which cover rangesp I p

differing according to the /\p values (e.g. = 0*6 to 0*92 for^ = 0*08,r» 0*11 to 0*3 fon /jj = 1). As a general comment on the characteristics

the paper remarks on the extreme sensitivity of the rate of change ofi/ ^|\£ with respect to ^ for values of ^ greater than 0*1. Valuestaken from Frocht are superimposed on the carpet plot in Figureand these show very close agreement with the author’s curves for = 1

1sind , while the apparent errors for the remaining curves are of the

order of +7*5% for ^ , +5% for ^ = - and from -3% to +2*5% forP 1~ -g . No information is given in the paper regarding the thickness of the models used or, more important, the ratio of thickness to radius of fillet, although reference is made briefly to the clear resolution of boundaries. However, it should be noted that the ’’photographic” and "semi-photographic” techniques were used for the determination of the

peak and reference stresses, these methods using only integral andhalf-order fringes and relying for their accuracy on the attainment ofpeak values of isochromatic fringe number of at least 10, there beingno attempt at the use of a consistent extrapolation technique. If thisis coupled with the fact that the models were manufactured from BakeliteBT-6I-893 which has only 70% of the sensitivity of epoxy resin CT200,it follows that the applied loading, particularly for the models oflarger radius, would be greater than that used in this investigationby a factor of about four times and this could have some effect on the

accuracy of Frocht*s results. -(2)Peterson and Wahl performed tests on shouldered steel shafts

subjected to bending, their values for stress concentration factor being determined from strains measured by means of an extensometer of 0*1” gauge length. The three test shafts had a ratio of minor to major

pdiameters of ~ and ratios of fillet radius to minor diameter of 0*5»P

0«281 and 0*167, respectively. Their paper compares the results obtained(3) L 1 2with those of Frocht for shouldered plates for ^ ratios of ^ snd

and for this reason values interpolated from the results of Petersonand Wahl are shown superimposed on the curves of Figure ^.3 for reference.Although the results of Feterson and V/ahl apparently show good agreementwith the carpet plot curves, the greatest difference being only 2*3%»this is only fortuitous, owing to the effect of a number of factors.The apparent peak strain indicated by the extensometer will becomeprogressively lower than the correct value as decreases because ofthe relative size of the gauge length to that of the fillet radius.Also, their loading system as described in the paper would producebending with shear, although it will be shown in a later section thatthe error due to this effect is expected to be small for the ratio ofmoment arm to minor diameter used (i.e.^ = 3|#2). Finally, the comparison

is between factors for two-dimensionul and three-dimensional geometries and it is usually predicted that for the same ratios of fillet radius and major width to minor width the stress concentration factors for the three-dimensional case are less.

A suggestion has been made by Peterson^^ that a relationship exists between the S.C.F.s for thin plates and the shafts having the same profile, which will be the same for shoulder fillets and for grooved members. The relationship proposed, i.e.

Kgs, - 1 : = w - ' « -r- I - I

where » S.C.F. for a thin shouldered plateas S.C.F. for a shouldered shaft of the same profile

K^n = S.C.F. for a grooved plate, calculated by Neuber*s(17)theory

Ko>q = S.C.F. for a grooved shaft of the same profile, also(17)calculated by Neuber*s theory

is used by Peterson to predict values for by calculation using

values obtained by experimental methods for K^s and Neuber*s theoretical values for and , i.e.

K * = ' + ( K ^ - O C ^ q - O( Ktg - 0

(6)Allison has criticised this procedure, pointing out that the Neuber solutions, which are for notches or grooves in infinite plates or solids of revolution, respectively, do not satisfy the boundary conditions for practical systems having finite dimensions. He demonstrates, (pp. 81, et.seq.), that values of *R calculated from his results for shouldered plates and shouldered shafts in tension differ significantly from the values calculated from the Neuber equations, although as ^ tends to zero it is likely that the S.C.F.s for corresponding two-dimensional

and three-dimensional geometries will tend to the same limit and that

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"R will.tend to unity. It was considered that it could be instructive

to follow a similar procedure using values of *R given by the NeuberA

theory and values of K^Ci.e. ) from Figure 4.3 to predictcorresponding values of for a shouldered shaft subjected to uniform

bending moment. Experimental values obtained by Allison^^ for theshouldered shaft are carpet plotted against ^ and in Figure 4.5 andcurves, calculated by the above procedure using values of i(g fromFigure 4.3, are superimposed for comparison. Over the range of parameters

r 1 1shown the maximum errors are approximately 7% for % ~ an( <5 * anc* p 113% for ^ ~ IJ" • The characteristic rise of the stress concentration

rfactor to a maximum for a given value of ^ and the subsequent fall fork, .smaller values of as shown by the results of Allison and for which

there is no obvious explanation, is not apparent in the curves for the "predicted" values.

The Basic S.C.F. for the Shear Effect (The curves showing the variation of , the Basic Stress

Concentration Factor for the shear effect, are given in Figures 3*8 to 3*12, inclusive, for the various shouldered plate models tested. These curves also show a number of common characteristics, as follows :-

(a) In general, appears to approach a value of zero in the minor width at varying distances from the junction with the fillet. Unlike the condition discussed in section 4.2(a), there is no apparent consistency in these distances even when the possible scatter band for K$, is considered. As a possible explanation for this it is suggested that will have a non-zero value at a boundary point only when the normal to the boundary at that point is inclined to the direction of the applied loading. Apart, therefore, from the fillet, for which this inclination is significant, the boundary

oJ

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>\oV o? aftQoUr poVtlribRs <£ peak K$ \iaW s on Mlel-

of the minor width, although ideally straight and normal to

the line of action of the applied load, will in practice depart from this condition due to small discrepancies in the initial setting up and distortions due to the loading.

(b) At the end. of the fillet where it either blends 5_nto the shoulder or, in the case of incomplete fillets, intersects

the major width, the value of Kc, tends towards zero. Again, this agrees with the theoretical condition mentioned in section k02(h) in connection with the equilibrium of the free boundary and would also be expected if the explanation offered in section ^.Ma) is applied for the boundary of the

major ;width.(c) Ks reaches reasonably well-defined peaks in the fillets, the

angular coordinate 6 giving the positions of these peaksshowing a consistent and significant variation, in general*

r. k,with both < and -y • These values of o estimated for theR. ^models tested have been correlated against 3 and 3 by

means of the carpet plot in Figure ^.1 and some extrapolationto limiting values attempted. The carpet plot indicates thatthe 0 values for the positions of the peaks of Ks increaser> h,for decreases in both y and 3 , the variation for any giveni rvalue being approximately linear with respect to ^ •Within the limits of accuracy possible in the estimation of these 8 values there is no significant change in the general trend which can be associated with the transition fromcomplete to incomplete fillets. Extrapolation indicates that

ty If'the peak value of for the limiting case of = 0 , 3 =would occur at a position given by 0 = 55°» approximately.

(d) The magnitudes of the peak values of K5 in general increase

with decreases in both y and “g> and the correlation with

these parameters will be studied in the following section.A

405 The Peak Basic S.C.F. for the Shear Effect ( K<- )Sc,The carpet plot of against the parameters ^ and is given

in Figure 4C6, the curves being drawn using the peak values estimated from the curves of , Figures 3*8 to 3»12, inclusive, for the fourteen

A

different model geometries. As in the case of , only negligible smoothing of the values was necessary and the curves were extrapolated to the limits of zero and unity for ^ , the value of Kg for the latter limit being taken as zero, this being in accordance with the explanation

Loffered in section 4.4(a). The extrapolations to the zero limit for ^

were reasonably easy to achieve since the curves were well-behaved inA

this region. (It should be pointed out that the original plots for Kg were carried out to a scale twice that of the curves shown in Figure 406).

A number of characteristic features of these curves are apparent, . as follows

i? r(a) The value of increases significantly for decreasing ^for any value of \ , apart from unity.

A k(b) The rate of change of with respect to also increases

Psignificantly with decreasing , the curves for a givenb Papparently being asymptotic to ^ = 0.A ^

(c) The rate of change of with respect to decreases uniformlyas ^ decreases, there being no critical change corresponding to the transition from complete to incomplete fillets.

The Peak Basic Stress Concentration Factor for the shear effect,AK , being a newly defined factor, there are no available results from 5other investigations with which it may be compared. Moreover, the Basic S.C.F.s, Ka and K* , reaching peak values at different points in the

* A Afillet for any given geometry precludes the use of Kg and in

c\J

<u

CO

Fl6Uto

4-,6

combination. However, the use of all four factors will be demonstrated in later sections.

| a / ^.6 The Stress Concentration Factor Kt and its Peak Value Kt

It was noted previously that the Basic S.C.F.s, and , varyfrom point to point along the boundary of the shouldered plate and that their peak values occur at differing positions on the fillet for any given geometry. In order, therefore, to determine the peak stress for

a given combination of bending and shear, the stress d must be computed for points along the boundary using the appropriate Kg and K<. values in the expression

d ~ \\$ +■ K sthe peak value of d then being estimated from the resulting curve. Onepossible method of defining a stress concentration factor is by theratio d>s , designated here as the factor Kp , where is the maximumnominal stress at a given point, corresponding to the bending momentat that point and calculated by the simple flexure formula using theminor section. The peak value of VCj- , designated here as the CriticalS.C.F. Kp , is a definition adopted in other papers in. which the problemof the shouldered plate subjected to bending with shear is treated.

(2)Two such papers, one by Peterson and Wahl already mentioned(1)in section and an earlier one by Timoshenko and Dietz , designate

the S.C.F.s quoted as being relevant to bending, although the loadingarrangements depicted show that bending with shear configurations were

(5)used. A paper by Neugebauer on stress concentration factors and their effect on design (Product Engineering 19^3)» gives a fairly compre­hensive summary of the results of work on S.C.F.s by various investigators Included in this is a section giving data specifically for shouldered plates subjected to bending with shear, the section being unreferenced and therefore presumably attributable to Neugebauer. He presents the data

as curves of peak values of plotted against ^ over a range of 0*1

to 0*5- for various values of a parameterX/, covering the range 0*5 tobinfinity. The nominal stress dnofliis calculated by the simple beam flexure theory on the basis of the effective bending moment at an assumed position for the peak stress and details of the parameters involved are given in Figure 40?. The positions C and C assumed for the peak

wL

B -

x

b -

Thickness t

L

TWo-Wla

Teak value ol stress, Acnax, assumed \o ke aV C cuvd C#.A

k 't = < H 4 o«,

rfna,,, =

FIGURE ^.7 Parameters Involved in the Neugebauer Data

stress are given by the points of tangency between the parabola CftC*

and the fillets, this parabola being drawn with its apex h on the line of action o f W and at the mid-point of the minor width, the dimension \? being the width Ct .

The Neugebauer data being quoted for a ratio -'L not less than 4,- "-\ A

it was decided to use as comparison values of calculated from the author’s values of and for the width ratio ^ = 0*2, (i.e.*^ = 3)*

Thus, from the expression

d - Ks inorn

ds “ K-ti, d* K c norn<4 * *'Wnorti

■i.e. Ky = Kb + K5

LUsing discrete values of and K 5 for = 0*2 from Figures 3*3 to3.12, values of Ky were calculated for points on the fillets for value

of ^ = 0*3, 0»6, 0.8, 0*9, 1*0, 2*0 and oQ and the resulting curves

plotted against 0 are shown in Figures k,$ to k.12. (This method ofcalculation means that, for a given , the load position is moved to

ockeep the same for each point).

o

©

in=> XI~TS > tjto 3

v M © <- *n£ -*A<2 _h f “ - s ^ - *? J» tf.»

<\4©

OOJ

00 oo«n

©

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o*-

0001 o£ atS>

O

in

o<po oo

lv-(D

OcS

SO

o

o

ipo oo

a

*5>iH

o

M

o

P - O iX> CO o oO O o cU

oi

uJr>c*

1/1.Estimating peak values of KT from these curves, together v/ith the positions 8 at which the peaks occur, the corresponding v measurementswere determined thus enabling curves of K-j- to be plotted against ^

£, \/f for constant values of \q . From these curves values of l\p v/ere takenat = 0*5 , 0 *6 , 0 *8, 1*0 , 2*0 and oo and plotted against ^ , these

(5)curves, together with those of Neugebauer and superimposed points

from the results of Timoshenko and Dietz^^ and Peterson and V/ahl^\being shown in Figure *4.13*

Referring to Figure *U13» & number of observations can be made.A /

(a) The values of K|- f or - syo should be equal, at correspondingV, to the values of , this condition being that due to uniform bending moment. Neugebauer's curve for — oO t

compared with the author's is 8% low at = 0 *0 8, coincidentat ^ == 0*18 and 12% high at ^ = 0*5* Values from the results

(x L) Wof Frocht 9 are available for ^ = 1, 0*5 and 0*25 andcan be estimated from the other points superimposed on the

Al i |Acarpet plot for in Figure k.J for ^ = 0*125 and 0*0 8 3,the values of Frocht and the author agreeing within 5%. It

pis significant that Neugebauer's curve at 4 =0*5 has almost

K zero slope and this would imply that his value of for

= 1 would be approximately 20% higher than both Frocht *s and the author's values.

(b) All the curves of Neugebauer are significantly higher thanthe author's at = 0*5 , the error increasing as ^ decreases,being approximately 32% for = 0*5* The respective curves show good agreement for between 0*12 and 0*22 and the maximum differences at = 0*08 are -8% for ^ and +5% for = 0*5*

(c) An isolated value is available from the investigations of

LAc- oi g 2o)era •dco

I't— r

C Oo

ro oo>

(1 ) TO «>Timoshenko and Dietz for ^ = 2 and \ - 0»375* This was1"determined using a plate model of thickness , fillet

3M }t, 'lradius and a width ratio ^ of . The model being of

celluloid, the colour matching technique using a white lightsource was adopted for taking the photoelastic measurements

and the inadequacies of this method are reflected in theresult for the stress concentration factor which is 21%higher than that of Neugebauer and 33% higher than that ofthe author’s.

(2)(d) The results of Peterson and Wahl , obtained by means ofL

extensometer readings on steel shafts, are for a ratio 2 cCof — and an ^ value of approximately 3t and the most

significant difference compared with the author’s values is

one of -12% at \ = 0*125.Referring again to the Neugebauer results it must be pointed out that no details are given in his paper of the method of determination of his values. The method he uses for deciding upon the position at which maximum stress occurs is indicated in Figure ^.7. This device of fitting a parabola against the load line and the two fillets is, in fact, a basis for the calculation of strength factors for gear teeth

(B.SoSo +36). It arises from the application of the simple bending theory, this giving the parabola so constructed as the profile of the cantilever of constant strength for the given loading. In view of Neugebauer’s use of this device it may be reasonably speculated, in the absence of any other information, that his values of stress concentration factors are based on some semi-analytical procedure. In any case, the assumed positions for the maximum stresses which result from the use of the fitted parabola are, in general, at variance with experimental evidence. Values of 0 obtained by a simple graphical

Q_

Q~> o

olet%s>

construction for the fitted parabola criterion have been carpet plotted c l_against ^ and ^ in Figure 4.14. A second carpet plot is superimposed,

this being drawn using values estimated from the curves of ,Figures 4,8 to 4.12, derived from and values. It is immediately apparent from an examination of the two sets of curves that the region of reasonable agreement is confined to ^ values of approximately 2. Further consideration of the fitted parabola method leads to limiting results of 0 = 0° for the uniform bending moment case (i.e. ^ = 00 ) and B = 90° for = 0. These do not accord with the values expected from the positions of peak values for Kg> and which would require the positions of peak stress in the fillet to lie between the limiting values for 6 of 5° and 55°• Timoshenko and Dietz quote a value, albeit very approximate, of B = 22«5° for their model, for which ^ = g ,

= |r and ^ = 4«4 and the photographs of the isochromatic fringes(7)for the models of Hartman and Leven appear to indicate the same

order of result.(14The problem of loaded projections has been considered by Heywood

and some results obtained by him are summarised in his book, '’Designing by Photoelasticity”, pages 205, et.seq. Heywood notes that for a projection having a load applied close to the fillet, the maximum stress in the fillet cannot be assessed accurately by considering the effect of the bending moment alone. Stresses so obtained will be too low and in some cases compressive stresses may be indicated where, in fact, tensile stresses are produced. Heywood used photoelastic methods to study the stress distributions in models of tapered projections, the eight models used having parameters covering the ranges 0*09 to 0*27

for with ^ = 0*47 and 0»j52 to 0*67 for with ^ = 0*27, where b is taken as the width of the projections at the junction with the

fillets. Using the results of the photoelastic tests, Heywood evolved

an empirical formula for determination of the maximum stress in the fillet, namely

d

the parameters involved being shown in Figure ^.15.

VnJT

FIGURE 4.15 Parameters for Heywood * s Formula

The positions of C and C* given by 0 « 30° for the peak stress in the fillet and suggested by Heywood agree reasonably well with the author'

values of 3$° to 40° for the range of and ^ values used but, although he notes that, in practice, 0 will be smaller for larger distances of load from the fillet* the statement that 0 *= 0° for the uniform bending moment condition does not accord with the experimental evidence* However, use of the assumed value for 0 will not give rise to significant error when applying the formula. Since the main body to which the projections are attached in Heywood's models is relatively

large, it is appropriate to make a comparison with the author's resultsLfor shouldered plates of width ratio = 0«2. Substitution of the

shouldered plate parameters in the Heywood formula leads to the expression

/ ,0*76 -

and, dividing by d ^ , the Critical S.C.F. is given by

fc , - ( , * + / * § : ]

or K't = ji + a . l 4 ! 4 ^ |

where dimension b is the width at the critical section. Taking the position of the peak stress in the fillet at © = 30°, as suggested by

I * fHeywood, (i.e. width V = V 'V-0*£t&r), K-j* was calculated for values of^ = 0-25, 0*5, 1*0, 2.0 and and ^ = 0*083, 0.125, 0-25 and 0*5.The resulting curves are shown in Figure 4*16 together with theA fcorresponding curves of KT determined in a similar manner as for

Figure ^.13, i.e. by estimation of the peak values of calculated from the author’s curves for Kg and . From an examination of the two sets of curves the following comments can be made

(a) For ^ = 0.08, Heywood’s values are below those of the author, the errors ranging from -5% at ^ = 0*25 to -12% at = 0-0 .

(b) For = 1*0, 2*0 and the errors are approximately -10%rover the range = 0.08 to 0*5.

(c) A t = 0*25 and the error is +23%*Accepting that there would be some differences expected between

the results for tapered projections and shouldered plates there are, nevertheless, certain deficiencies in the procedures used by Heywood which give rise to some doubts concerning the accuracy of his results. The difficulties caused by poor resolution of isochromatic fringes near the boundary associated with ratios of thickness to fillet radius

o

o-

r .. j f,

sjr £ =

ur»©© o

Figure 4

.lfe

greater than unity, ( ^ ratios of up to 2»5> in this case), (ofpotential instability due to relatively high loads and the possible errors due to the lack of a consistent extrapolation technique, have been discussed previously. In addition, examination of photographs of typical isochromatic fringe patterns produced in the tapered projection models indicates the presence of time edge stress equivalent to a fringe order of at least 0*5, Referring again to Figure ^.16, it is ; -

significant that reasonable correlation between the results of Heywood and the author occur only within the limited range of = 0*32 to 0*6? used by Heywood and this illustrates the dangers of extensive extrapolation

of results derived from a small range of model parameters and using only one loading configuration in each case. Indeed, the implication made by Heywood that his empirical formula is conceived from a sound theoretical basis is not entirely accurate. Putting the angle = 0, so that dimension QL becomes equal to OC , leads to the simplified expression

for the peak stress in the fillet and this can be written in the form

It is obvious from an examination of this expression that it does not satisfactorily represent the conditions for which it was intended, namely loading in close proximity to the fillet, since the fillet stress calculated from it would tend to infinity as X tends to zero and this is at variance with the experimental evidence.

In further discussion in connection with the proximity effect on page 212, Heywood mentions also the local effects produced by loading applied to the boundary. Reference is made to the difference between the

ideal case of the serai-infinite plate with loading applied to part of

the boundary and for which the mathematical analysis gives zero stress on the free boundary, and the practical case for which a free boundary stress is seen to be present. The mathematical analysis, of course,

neglects the effect of the free boundary displacements in the determination of the stresses and it is accepted that distortion of the boundary will, in practice, lead to free boundary stresses other than those produced by bending moments. However, it is important to point out that in this present investigation the various configurations of bending with shear were achieved, using arrangements in which no loads were applied to the boundary of the minor width. Bearing this in mind, it can be inferred from the characteristics of the Basic S.C.F. K s discussed in section that the increase in boundary stress dueto the proximity of the load line to the fillet is mainly due to the presence of shear at sections where the normal to the boundary is ~"inclined to the line of action of the load, this effect being most significant in the fillet but also is present where the boundary is distorted.

CHAPTER 5PRESENTATION OF CRITICAL STRESS DATA

FOR DESIGN PURPOSES

5*1 Introductory CommentsIn an earlier section it was noted that a stress concentration

factor for a particular component geometry and loading case may have differing values dependent upon the basis upon which the nominal stress is,chosen* It was noted also that, provided this basis is defined fully

for any stress concentration factor quoted, the choice can then be made solely from the standpoint of the convenience with which the designer can make subsequent use of the information. It has been demonstrated that boundary stresses for the shouldered plate subjected to bending with shear can be described by a linear function involving the two Basic S.C.FoS, and , and the nominal stresses corresponding to the bending and shear effects. Elsewhere, these Basic S.C.F.s have been used with the linear relationship to calculate values for a stress

concentration factor Kq using the expression

geometries and load positions, shown in Figures *f.8 to were then

with the results of other investigations in Figures ^.13 andNow, the existing results considered are relevant either to

uniform bending moment or to configurations of bending with shear in which the bending effect is predominant and for these situations the S.C.F. based on the nominal bending stress is acceptable. However, it is clear from the expression for Kq- that it will not produce useful results for cases where the effective line of action of the load passes

Curves giving the variation of Kq around the fillet for different model

used to determine peak values Kq- , the Critical S.C.F., for comparison

through or near to the fillet, since KT tends to infinity as 3C tends to zero, whereas the actual boundary stresses corresponding to this

situation are finite* Again, .the parameters against which the S.C.F.s are plotted should also be chosen, both for their relevance to the problem and the ease with which calculations can be made. In this connection, the parameter used by Neugebauer to describe the effective load position is not favoured by the author for two reasons, namely

(i) the use of the width p of the section at the assumed position for peak stress seems to be unnecessarily complicated and

(ii) the method of describing the effective load position relative to the actual or assumed position of peak stress, this position itself being dependent upon the loading configuration, seems to be artificial and not entirely convenient*

In the following section, therefore, presentation of the information by means of the S.C.F. K-j- , based on the nominal shear stress, is considered for general use and the parameterS^ will be adopted for describing the

effective load position relative to the shoulder*A

5*2 The Stress Concentration Factor Kt and its Peak Value KtThe S.C.F* (Op , defined as^< , is given by writing the linear

'nomrelationship in the form

Kt = K j S W + Kc,'•nom

‘i.e. Kt = Kfc

as shown previously in section 3*^i details of the various quantities

involved being given in Figure 3*1• The S.C.F. Ky for any point on theboundary of the shouldered plate may thus be calculated for a given

\component geometry, using the appropriate values of Basic S.C.F.s, K$

and , and the distance X of the effective load from the point —Lyselected. Using the parameter to describe the position of the

effective load, where L is the distance of the load from the shoulder, the corresponding distance X for a selected jjoint can be expressed as a function of

When determining values of Ky , the problems of specifying the

loading configuration, the input data for and K^ and the method of presenting the results may be dealt with in a variety of ways. To illustrate one method, a program has been written for a digital computer to read in and as values for discrete points on the fillets for any given model geometry for which results are available and hence calculate Ky values for any selected values of the parameter^, • The program produces tabulated values of Ky at the discrete points using

a line printer output facility but also contains the necessary subroutines to control automatically an incremental plotter to give graphical output in the form of scaled and labelled axes, plotted points and spline fitted curves through these points. Complete details of the program are given in APPENDIX D together with a description of some possible amendments for dealing with the specifications in alternative ways* The program has been used to produce tabular and graphical output for each of the models tested, for the width ratiotv / 1 1 i= 0*2 and for jp ratios of 0, g, ^ 1 and 2 and copies of thecurves produced are given in Figures 5*1 to 5.5* inclusive.

The curves for Ky apply to either fillet and, for example, for

the upper fillet a positive value for Ky represents a tensile boundary stress. The following comments may be made, applying them to the upper fillet for convenience

(a) For all values selected for illustration, Ky reachesmaximum positive values in the fillet for values of 6 in the

90

o00 CO

X>

o 6 5 ° ^

\<x>

%s>

91

O0CO cJ

LOo o

©

o

U5 cJ -r- / ^* v _ _

OCO

92

cJoo CJ

ino

qJ

c_C«JOo

U -c\^

oJCO

O

OCQ

in

9*f

cO OJ

t.

oJO

ir» vr»

cP

range 7° to 55° i a carpet plot showing, the variation of 6 forLy IV .these maximum values against ^ an^ P being given in

Figure The maximum positive values of Ky , estimatedfrom the curves in Figures 5*3 to 5*5 and designated the

A

Critical S.C.F. Ky f show a consistent correlation against the parameters find ^ , as demonstrated by the carpetplot in Figure $.6. ■

(b) The method adopted in this investigation for producing the bending with shear configurations of loading means that bending moments to the right of the effective line of action of the downward load will be sagging* Where this situation arises in practice it is possible for small or negative values

I .of to have the effects of the sagging moment sufficiently

large so that part or all of the fillet boundary is in compression. This effect can be seen in Figures p.2 to 5*5 and, to illustrate it more clearly, values of Ky for

= 0o2 and ^ =1, 0 and -1, plotted against B , are shownin Figure 5*7* The dotted curves superimposed represent the theoretical value of Ky for no stress concentration, i.e. putting l{ = 1 and = 0, thus giving . It can beseen that f o r ^ = -1, the Ky* values are everywhere negative and a maximum compressive stress occurs in the fillet at the point where the effect of the sagging moment is predominant.

(c) Since K-g and Kt, tend, respectively, to unity and zero onu

the minor width boundary within ^ of the start of the fillet, Ky will tend to its theoretical value within the same distance and the simple bending theory is adequate for the determination of maximum stresses in the minor width.

o

cJo

o —r?

Q

cvl «0 Jr 2 CO

Ft6U

ftE

S.b

97

/

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Figu

re-5.7

From the foregoing illustration it can be seen that complete

information regarding the variation of stress at the fillet boundarycan be obtained for any configuration of bending with shear, using thevalues available for the Basic S.C.F.s, Kg and K<- . In the following

Asections empirical representations of the Peak Basic S.C.F.s, and A

, are obtained and these may be used:-(a) to provide simpler procedures for loading cases where either

the bending effect or the shear effect is predominant and(b) in conjunction with the carpet plots of the angular positions

of the peak values of and to produce idealised, interpolated curves of Kg, and Kc, for shouldered plate geometries other than those tested, in order that the methods of this section can be implemented*

A A

5*3 Empirical Representations of Peak Basic S.C.F.s, K-% and K S A number of investigators have produced empirical formulae to.

represent stress concentration data for various types of component and loading, either for obtaining interpolated or extrapolated values from, existing numerical information. The advantage in being able to specify the information concisely, in terms of tie parameters rather than by curves or tabulated data, is obvious. However, care must be taken to ensure that the empirical expression is based on a sufficient number of results within the working range and that the form of the expression has suitable characteristics if it is to be used for extrapolated values. Examples of some suggested empirical formulae from other investigators follow.

(7)(a) Hartman and Leven used the following expression to give interpolated and extrapolated values of S.C.F. for plates with a centrally enlarged section, subjected to uniform bending moment. Expressing the formula in terms of parameters

used in this investigation with one additional parameter C for the axial length of the enlarged section, the 3.C.F* is given by ^

K = I ^ h a n l i i s . f c 1 Win ("b ” ') ) 1 6‘li +1 L 1 b iJ n 0 - y l l (5>J>

for \ < 1 > 1 > 1. | > o .Large values of correspond to the limiting condition of the shouldered plate and K then tends to infinity for =0, and to 1 for = 1*

(b) Heywood^^ suggests empirical formulae covering a wide variety of component types and loading configurations, one example for the tapered projection being discussed in section 4*6* To describe the results of a number of investigationson the shouldered plate under Uniform bending moment, Heywood gives the formula

k - i + (■ f t - o . ar (

and this produces values of K which tend to infinity forr 'a

= 0 and to 1 for ^ = 1 .(c) Fessler, Rogers and Stanley have derived a relationship

for shouldered plates in tension, namelyO l l v u a U v/X w U l /C O X I I t C l i O X U I l ^ l i d U i v X j v

C /BY) ,rs-(Q^8 + Q'0a%)

which gives values in the range 1*44 K for ^ 1and 1 e rne formula is based on a limited range of

3width ratios (i*e. "1*5 < *''£ ^ 2*5) arid fillet radius to minorp

width ratios (i«e» 0 -036<"'£> < 0*144) and the models used hadfc✓p ratios betv/een 0®3 and 4 (see discussion in section 3-3 on the importance of small -p ratios)* The expression gives values considerably higher than results obtained by other

investigators for width ratios larger than 2*5*A A

The correlation of the Peak Basic S.C.F.s, and , againstb/ j> ,the parameters ^ and b is shown by the carpet plots in Figures 4*3

and 4.6 and discussed in the sections 4.3 and 4.5. The carpet plotsshow characteristics consistent with-asymptotic values at ''C = 0 , of

a a a A

infinity for both and and values of Kg = 1 and = 0 for =1

It was decided to examine the suitability of the following empiricalforms for the Peak Basic S.C.F.s, namely

it5- If,(4)]# k« i , - i

by studying the correlation of the relevant experimental results against the respective straight lines represented by

locj(K$) = lo^F,(^ + talD] lo^)shov/n in Figure 5*8 and

loj(Kg-l) = + ^ ( ^ l o j . ( r )

in Figure 5*9* The plotted points are derived from the respective carpet plots using the values at = 0 , 0-2 , 0*6 , 0- 8 and 0*9 fo**

j*each of the values of - 1, 0-5? 0*25, 0*125 and 0*083. The straight

Lline shov/n for each value of corresponds to the least-squares fit to the appropriate points, the calculations being performed on a digital computer in accordance with the theory outlined in APPENDIX C.Assuming that the errors relative to the respective regression lines are normally distributed and using scatter bands of i 2 standard errors, the intercept and slope values for the various fitted straight lines are given,

A

with a 93% level of confidence, to a precision of £4*5% for the dataA

and ±7*2% for the data.p

101

ooo

102

<7\O O

(0 3 00 <po 6 6 6 o

Fieufte 5.9

re;

Now the intercepts and slopes of tie straight line fits represent, inactively, l°Cj[ ij and ft in Figure 1*8 and and ft in

. Figure 5»9 and the values obtained were used to plot the points forbF| » ft 9 ft 311 ft against *•£ shov/n, respectively, in Figures 3<>10

to holpt with two additional points given by (1) = ft (1) = 0. A

number of solutions were considered for the functions fi . Ft ,F, and ft and it v/as decided finally that satisfactory representations are given by

r* ' b /bN1 "F, = 1.52 - 0.128^ - 1*392^ )

ft - 0*k62 + OoOOS^ 4- Oul6 3 ^^^ VVcr— bft = 1.166 - 0*016;$ - 0.15 1$ f

ft = 0.0636 - 0.008^ bo < ^ < ( % 7r* h, ft rr -0*^1 + 1.281^ - o.867(^f VV

JC^VIN-•O

Using these functions in the empirical relationships, the resulting A ^ p

values of fft and , carpet plotted against ^ and ^ , are shov/n in

Figures 5o1^ and 5*^5 i respectively. These empirically derived curvesshow reasonable agreement with the points superimposed from the

horiginal carpet plots in Figures ^*3 and ^*6 and, in the case of ,

4)with the results of Frocht .

L asF Soo'dcos FiV consK<x\(\c3l to pass V W o o ^ \ v F t - ° | * l * o ani givitnjF. = 1-5 -OM e&^-K

0-4

105

Curve co n s tra in t Iro £iV points at k 0 o, o ^ , o-$

art gWiag .

Fe * o*4<&+o*oo3j|+o*\<&(^

10 6

o-oi

107

m

1-04

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3

Ficuue s.is

Uses of the Empirical Relationships~ " " A A

The determination of isolated values of or from the empirical relationships can be undertaken conveniently by hand calculation, while for obtaining large numbers of values a digital computer can be controlled by a trivial program. In certain special situations the peak stress in the fillet for a given geometry can be determined using one or other of the Peak Basic o.C.F.s and for certain other loading conditions this procedure can give reasonable approximations, as follows:-

(a) If the line of action of the effective load coincides with th

position of peak stress in the fillet for a particular geometry, the contribution of the bending effect at that point is zero and the peak stress is then determined by the

A

value of the Peak Basic 3.C.F. K<. . The position of the peak stress for a given geometry can be estimated from the carpet plot in Figure 4.1h.

(b) If some degree of approximation is acceptable the appropriateA

value of may also be used if the line of action of the effective load is reasonably close to the position of peak

A

stress. With this in mind, the error €. incurred in usingA .

instead of KT has been correlated against and Ap (andI .

'Y as a more appropriate parameter in this case) in thecarpet plot given in Figure 5„16. From this it can be seen,

I*' / | ^for example, that for ^ ^ ^ and 0 < s' , the error doesnot exceed --10%. This type of carpet plot of error could beused either to determine whether or no a particular loading

Aconfiguration can be dealt with satisfactorily using the value or, alternatively, could be used to correct the value given by this procedure for any value of p and y .

111

-g "2 <s£l«*a- cu o d ^

I

Fi&U'Rt

5.1b

(c) If the loading corresponds to urn form bending moment, the peak stress is determined by the value of the Peak Basic

AS.C.F. K b .

(d) If the loading configuration of bending with shear is suchthat the bending effect predominates, the peak stress can bedetermined approximately by the use of . The error incurred

v Ain using instead of KT (taking note of the fact that thesefactors are based on C>nofn and TRQm , respectively) has been

r L.cori*elated against ^ and in the carpet plot given inFigure In this case it can be seen that the use ofunderestimates the peak stress for the range of ^ valuesshown and, for example, that the error in peak stress using A ^Kb is less than -10% for values of ^ greater than 2. As in section (b), the carpet plot of error can be used either to

g gives anacceptable error, or to apply a correction for any value of ^ and ,

determine A values for which the use of K

113

o<*- er-»

UJ

oe> o

Fieimt 5.n

Returning to the determination of peak stress in the fillet byA

means of the Critical S.C.F. Kp » there remains the problem of thep

T at values of the parameters ^ and for which curves of the Basic S«C.F*s, Kj, and , are not available* For this situation it is suggested that ICg, and can be specified by idealised curves over a limited range for each fillet, say, 0 = C to 60°, by means of polynomial or other functions* The coefficients of the describing functions for a given geometry could be determined using boundary

A A -------------------------------------------------------- ---- -

conditions such as the peak values K3 and calculated by the empirical relationships, together with the positions at which they occur* This procedure would obviously be tedious for hand calculation and would best be dealt with by a suitable subroutine which could be included as a modification of the program for the automatic plotting of K-p values, this subroutine then replacing the procedure for reading in numerical values of and « Examples of describing functions suitable for this procedure are detailed in APPENDIX D.

CHAPTER 6

GSRSEAL CONClUoIuiVS CONGiiiRNING SHOULDERED PLaTES SUBJECTS!) TO BENDING WITH SHEAR

These general conclusions fall into three main sections, thefirst summarising'the-salient points arising from the experimentalwork and results detailed in Chapters 3 and k, the second dealing with the re-presentation and use of the information obtained, in forms suitable for design procedures, these being described in Chapter 5 and APPENDIX D and the third and final section giving suggestions for further investigations.

6.1 Summary, of the Experimental Results(a) The stress at any boundary point of the shouldered plates

subjected to bending with shear can be defined satisfactorily

by the linear relationship

= M n o c n + K s L ™ ,this relationship also being applicable to the condition of uniform bending moment where the mean nominal shear stress Tnon,is zero.

(b) Curves showing the variation of the Basic Stress Concentration

Factors and !(*, » with distance along the boundary have been produced for each model geometry from straight line, least-squares fits of results from various configurationsof bending with shear. The values of so determined show close agreement with those obtained from uniform bending moment

tests. At the boundary of the minor width, tends to a value of

unity and to a value of zero, i.e. the maximum stress in

the minor section la satisfactorily determined by the simple bending formula.

A(c) The peak values, , of the Basic 8.C.F. Kg in general show.,

reasonable agreement with values of B.C.F.s for uniform(E A)bending moment obtained by Frocht "* and the range of the

b parameter for which models have been tested has beenT Vextended for the smaller values. The carpet plotting

technique has been used to provide a systematic and reliable\>ymethod of extrapolation to extend the ^ parameter to the

limits of 0 and 1.A

(i) The carpet plot of the author's values of Kg shows ab/consistent correlation against the parameters ^ and

, this pointing to the advantages gained by adopting 11)the techniques recommended by Allison^ • In contrast,

carpet plotting of Frocht's results indicates someu 1 1inconsistencies for his values at = — and jj* , but

this is to be expected in view of the experimental method used and the lack of a consistent extrapolation technique.

(ii) The angular coordinate 0 giving the positions at whichA

the peak values occur, shows only a small variationo b iv(3° to 9°) over the range 'g = 0 to 0*9 for all values

and this is in agreement with other experimental evidence(d) The Basic S.C.F. being a newly defined factor, there are

no results available from other investigators with which theA

Peak Basic S.C.F. can be compared.A

(i) The carpet plot of IC shows a consistent correlation

against the parameters ^ and ^ .(ii) The angular coordinate 9 giving the position at which

the peaks occur shows a significant variation (55° to. 0°) over the parameter ranges 0 to 1 for and

(e) Since the factors and peak at differing positions for a given geometry, information regarding their variation along the boundary must be used to determine the peak stress produced by a given configuration of bending with shear. For comparison with other results, values of the Critical kS.C.F.Ak; have been calculated, these being the peak values of thefactor =9jf and the following relevant comments can be made:-

(1)(i) An isolated value by Timoshenko and Diets for a loadingconfiguration predominantly due to bending is J>J>% highand merely reflects the inadequacies of the photoelasticmaterials and techniques then available.

(5)(ii) Values produced by Neugebauer ' by some unspecifiedbmethod, for the limited range^ 0«25, agree approx-

imately with those of the author for ^ ~ 0«2, but show trends which Indicate increasing errors for both smaller

pand larger values of . The Neugebauer values are based on an empirical method giving assumed positions for the peak stresses which are inconsistent at the limiting values of the parameters.

(iii) Heywood^^ specifically mentions the effect of a load in "proximity11 to the fillet of a projection and gives results of tests on tapered projections. Apart from deficiencies in the experimental method and the lack of a consistent extrapolation technique, his empirical formula quoted is developed from a small range of model parameters and a single loading configuration for each model geometry and, in any case, has a form which does

not represent satisfactorily the conditions for loading

in close proximity to the fillet.(f) In this present investigation the various configurations of

bending with shear are achieved by the application of systemsof forces at positions remote from the shoulder fillet andminor width. It may be inferred, therefore, from thecharacteristics of the factor K, , that the increase inbboundary stress produced by the proximity of the load line to the fillet is due mainly to the presence of shear. In cases where the load is actually applied to the boundary it would seem reasonable to allow for this, approximately, by super­imposing the additional stress system produced by the appropriate

edge loading on a semi-infinite plate.

6.2 Use of the Information for Design Purposes

(a) The linear relationship for the boundary stress has been rewritten to give the S.C.F. , based on the mean nominal shear stress, from the expression

Kt = ^ = Kb

where X is the distance from the boundary point considered to the effective line of action of the load. This S.C.F. is preferred to one based on the nominal bending stress since

'?it satisfactorily represents the variation of stress along

the boundary for all positions of the load.(b) Taking advantage of the growing availability of digital

computers, a program for the calculation and automatic plotting of K-p values, for any selected load positions, using discrete values of the Basic S.C.F.s, and , has been developed.

A(c) The possibility of representing the Peak Basic S.C.F.s, and

A

Kgj , by empirical relationships of the form

has been examined and least-squares fits of the experimentalresults show a good correlation with the straight lines% for

\bconstant values of the width ratio ^ , represented by

!o<j('Ks)- = Vocj(r)

lo3 (KB-l) -- \ ^ { k ) \ + \ ? M ■

Satisfactory representations of the functions Fj , Fi , F* and were achieved using power series of up to second degree

and the values given by the empirical expressions agree closely with the experimental results of both this and other investigations, where relevant.

A

(d) The Peak Basic S.C.F. can be used as a simple basis for the determination of peak stress if the effective line of action of the load is sufficiently close to the position of the peak stress. The error involved in using this procedure for any load position is presented in the form of a carpet

plot.A

(e) The Peak Basic S.C.F. can be used for the approximate determination of the peak stress if the bending effect is predominant. The error involved in using this procedure for any load position is presented in the form of a carpet plot and this shows that for loads at distances greater than two minor widths from the shoulder the error incurred is less than -10$.

(f) For the determination of stresses in the fillet by means oft/ ^the S.C.F. Ky,at values of the parameters £ and ^ for which

there are no available experimental results, a modifiedapproach is suggested in which idealised curves of and

are specified over a limited part of each fillet usingsuitable functions with coefficients derived from the

A A

empirical relationships for and .

6*3 Suggestions for Further WorkThe suggestion made by Peterson^^ that a relationship exists

j '

between the S.C.F.s for shouldered plates and shafts of correspondingprofile, which will be the same as that between notched plates andgrooved shafts derived from the Neuber theory, has been examined by

(6)Allison in connection with plates and shafts in tension. The procedure is shown for that case to be useful as a method of obtaining a first approximation to the S.C.F. for the three-dimensional geometry using the corresponding two-dimensional results, although the Neuber *9ESBtheory does not satisfy boundary conditions for practical, finite systems. In a similar manner the author’s results for the uniform bending case for the shouldered plate, when used to predict values of S.C.F. for the corresponding shouldered shaft, show approximate agreement with Allison's results for this case over a limited range of parameters, but do not exhibit the characteristic drop at small values of ^ •

Although the Peterson procedure applied to the shouldered plate results for bending with shear configurations may be justifiable for the prediction of S.C.F.s for shouldered shafts in the absence of any other information, it is dangerous to infer that similar trends occur for all types of loading and the results should be used with caution. It is clear, therefore, that information is required concerning the behaviour of the shouldered shaft under significant configurations of bending with

shear and it is suggested that this could be profitably studied, using the techniques of three-dimensional photoelasticity and the extrapolation and statistical procedures described for this investigation. Also, investigations of the behaviour, under bending with shear configurations, of plate models having other profiles, could be undertaken with the object of determining whether the device of representing boundary stress at a given point by a linear function of Basic S.C.F.s can be applied as a general principle.

APPENDIX A

GENERATION AND UoD OF ORTHOGONAL POLYNOMIALS

A*1 IntroductionCurve fitting is essentially the process of finding a -smooth curve

which passes near to each of a number of prescribed points in a plane. In a numerical, as opposed to a graphical, approach to this problem, it is customary to use a polynomial to define the curve, the degree of the

polynomial affording a convenient measure of smoothness. If the nearness is achieved by imposing the "least-squares" criterion the original vague requirement is converted to a definite and tractable problem.

The numerical solution of this problem involves a considerable amount of arithmetic except in the more trivial cases. The classical method fits a set of function values by a power series of the form

k0 + k,0C + ..... 4- knXn

using the least-squares criterion. It has the grave disadvantage ofrequiring the solution of a set of simultaneous equations (the normalequations) for the coefficients k. that may be ill-conditioned and,moreover, for each degree of polynomial tried, an essentially separatecomputation is involved. If, however, fitting is carried out in termsof a series of polynomials of increasing degree, orthogonal over the

(12)data points, these objections disappear. Forsythe has described a

method of fitting orthogonal polynomials to sets of data, the coefficients being generated by recurrence formulae.

A.2 Forsythe1s MethodLet ^ (M = I; 2./ f m) be the values of an observed or

computed real dependent variable $ at corresponding values of a real independent variable X . Suppose that it is desired to fit the data values (j},; by the truncated series

F f c f x J = S ^ p o f X o ) + S®p,(xn) + . . . + S(kk) pfc (x^,)where pifcx ) is a polynomial in X of proper degree tand k + I < m .

The m numbers eM = 4m ~ Fk (*m ) are the components of anerror vector which, since the X/t and jju. are regarded as fixed, depends

only on the parameters f sj . Fitting the data bymeans roughly that k J is small for all Ai - I; / th and oneway of defining this precisely is by the use of the euclidean norm

fielle = { ef + ...... e*}*determining the values '5^ ••'•••* 5£ so that |H|Z is a minimum,

i.e. , Sf) =• ^ R,(Xju)] is a minimum.;

Then = O' (for all L - 0, L •’**, k )^ J k) 71 7

leads to 2 S{ [ 2 DC1 ) = 2 (L 3Ci (for all t = 0, • • * • k )h=o h I «•=» "• A a =i * 7 7

and using the abbreviationsm m

0)t*y ~ (^>l) ) " 2^ YA pi (

the k+l normal equations which determine uniquely are

n ^ + + n c M - noo o + + WokSk - o

to i ..............+ to = to.ko 0 kk k kMaking the polynomials p;(DCJ orthogonal over the point set X () X m ,

m . *i.e. putting to-j = 2^ Pi ( 7*3 pj (%-/i) = 0 ( j ^ i )

will decouple the normal equations, leaving

and these can be solved without difficulty.The orthogonal polynomials are generated using a three-term

recurrence, thus

The polynomial of "best fit" may be conceived as that which most effectively compromises between smoothness as represented by the degree of the polynomial and closeness to the data values as indicated by the root mean square deviation. The R.M.5. deviation can be obtained

conveniently as a by-product of the main calculations, the particular

formula used being derived from the orthogonality relationships,

pc (*) = I

p, (x) - x p 0 (x) - cVp<>(x)

pi(x) = xp,(x) -c<£ p, (x) ~ 0 , p 6(x)

p;+i M *: x p d x ) - oCiv-,' pt(x) - ft pi-, (x)

where d and are numbers chosen to make the orthogonality relations

hold. Then

i+l

All calculations required to be carried out using the foregoing relationships were performed on a digital computer and the corresponding program segment is found as part of a subroutine in the program detailed in APPENDIX B.

APPENDIX B COMPUTER PROGRAM FOR CALCULATION OF

BASIC STRESS CONCENTRATION FACTORS

B.1 Introductory CommentsThe calculations giving the Basic S.C.F.s, Kg and , were

performed on a digital computer using punched cards to input both the source program and data, the output being produced via a high speed line printer. The particular computer installation used was an ICL 1902A configuration and the source program is written in the 1900 version of FORTRAN, this being essentially FORTRAN IV with a small number of both

extensions and restrictions. The PROGRAM DESCRIPTION segment, or other initial statements before the source program proper, are, in general, peculiar to the particular installation and the operating system used and,-therefore, these have been omitted.

B.2 Brief Description of Program Structure

The program comprises a MASTER segment (BV07) and two SUBROUTINES (SCF and LST3Q). The MASTER segment is contained in statements numbered, consecutively, 1 to 129, inclusive and for each model tested

(a) reads the model parameters and the degree of polynomial to be used,

(b) calculates distances along the boundary,(c) calls the SUBROUTINE SCF for each line of readings of

isochromatic fringe number taken in the uniform bending moment test,

(d) for each configuration of bending v/ith shear, calls the SUBROUTINE; SCF for each line of readings of isochromatic number, computes and stores the corresponding values of KT and X for each boundary point,

(e) calls the'SUBROUTINE LSTSQ.

The SUBROUTINE 3CF is contained in statements numbered, consecutively,130 to 227, inclusive and for each line of readings for which it is called

(a) calculates the coefficients of the required orthogonal polynomial, the fitted and extrapolated boundary values of isochromatic fringe number and values from which the R.H.3. deviation can be calculated,

(b) in the case of the uniform bending moment test calculates the value of for the respective boundary point.

The SUBROUTINE LSTSQ is contained in statements numbered, consecutively, 228 to 2if?J inclusive and, for each model geometry tested, calculates the values of and from a straight line, least-squares fit of the respective KT and X values stored for each boundary point.A copy of the print-out of the complete program is given in the following seven pages.

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B.3 Layout of Input DataApart from the first data card, all other input data is entered

in free format using the field specifications of 10 for integer values

and FO.O for real values* This device, peculiar to 1900 FORTRAN, allows data fields of any length to be used, provided the numbers they represent are within the maximum limits prescribed for the computer and so long as at least one blank is left between each field. For convenience each line of internal points at which readings of isochromatic fringe number are taken, together with the corresponding boundary point, is given an identifying number as shown in Figure B.1.

In the following description of the order in which the input

data is presented the field specifications are given in parentheses

and the names in upper case letters are those used in the source program for the respctive data values. All linear dimensions should be in inches. Any units, consistent throughout the data set, may be used for forces.

FIGURE B.1 Details of Identifying Numbers

Card 1 (12)

NSET The number of models of different geometries for which data is to be given.

Card 2 (5F0.0)J3MAJ The major width B .BMIN The minor width b .T The model thickness t .R The fillet radius r .N100 The fringe number on the boundary of the minor width for

a uniform bending moment of 100. (from calibration test).

Card 3 (10)NDEG Degree of orthogomil polynomial plus 1. Value can be b,

5 or 6.Card k (10)

IND Value 1 if data following is for uniform bending.Value 2 if data following is for bending with shear.

For uniform bending moment test:-

. Card 5 (2F0.0, 210)W Effective load indicated by spring balance.

ECC Moment arm Z of loading lever, (see Figure 2.7)LIFST Identifying number of first line of readings given.

(see Figure B.1).LILST Identifying number of last line of readings given.

(see Figure B.1).

Card 6.1 (10, 10F0.0) - data for line LIFST..NREAD Number of internal points on this line for which readings

are to be given, (up to 10 readings allowed).N(J) Isochromatic fringe numbers taken in order from point

adjacent to boundary.

Cards 6.2, 6.3* etc.As Card 6.1 but for lines.LIFST + 1, LIFST + 2, etc., in order, up to line LILST.

Card 7 (10).IND (See Card *t).

For bending with shear tests:- Card 8 (10)

NL Number of configurations of bending with shear for which data sets are to be given.

For first configuration of bending with shear:- Card 9«1 (2F0.0, 210)

S Effective load indicated by spring balance.L Distance L of load from shoulder.LIFST (See Card 5).LILST (See Card 5)*

Card 9*2 (10, 10F0.0) - data for line LIFST.NREAD (See Card 6.1).N(J) (See Card 6.1).

Cards 9*3« 9.^. etc.As Card 9*2 but for lines LIFST + 1, etc., in order, up to line LILST.

For each successive configuration of bending with shear repeat as for

Cards 9*1? 9*2, 9*3» etc., until the data set for the model is completed. Data sets for succeeding models are presented starting with Card 2, continuing until the data sets for a total of NSET models are completed. To illustrate the foregoing, the first three pages of a data set, beginning at Card 2, are given on the next three pages.

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Output DataFor each model geometry tested

(a) the first page of output produced by the line printer contains definitions of some names occurring in the program, togetherwith the value used for NDEJG. (i.e. degree of polynomial + 1).

(b) the last page of output is headed with the model parameters BMAJ, BMIN, T and R (i.e.3 , b , t , P ) and, for each boundary point for which data is obtained, values of the basic S.C.F.s,

and K5 1 and distance along the boundary, 2 , are tabulated. These and values are those determined by the straight line, least-squares fit of Kp against X from the bending with shear tests.

(c) all of the remaining pages are headed with the model parameters DMAJ, BMIN, T and R and either PURE BENDING or BENDING WITH SHEAR with the appropriate L value, in order to provide means of identification if any of the print-out becomes separated at the perforations. These intervening pages contain the following data for each loading configuration and each line:-(i) The identifying number for the line.

(ii)The distance's* along the boundary.(iii) The measured values of isochromatic fringe number N at

each interior point.(iv)The corresponding fitted values and the extrapolated

boundary value, designated NFIT, given by the orthogonal polynomial♦

K i(v) A quantity designated SIGMA A SGD GAMMA, (i.e. 52

M Odefined in APPENDIX A).

(vi)A quantity designated SIGMA EN SQD, (i.e. S fji defined

in APPENDIX A).

(vii)For the pages headed PURE BE!.'DING only, a quantitydesignated KBDA3H, this being the value of as determined from the uniform bending moment test.

The R.M.S. deviation can be determined from the data values (c)(v) and (c)(vi), thus:-

R.M.S. deviation = 2 ? ; -

N o . o^' r e a d i n g s

A copy of the data output by the line printer for one model geometry is given in the following twenty pages.

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APPENDIX C

LINEAR CORRELATION OF DATA USING LEAST-SQUARE REGRESSION

Let (/* = 1, 2, ........ ) be the values of an observed or computedreal dependent variable V at corresponding values X . of a real independent variable X . Suppose that is desired to correlate the data values to the straight line

Y = a0 + a,Xthen, in general, the deviations of the data points from the line are represented by the ro components of the error vector

6 = Yja. ( CL0 "V &, X|u3Determining the straight line which gives the ’’best fit’1 to the data means roughly that j^u| is small for all /X - 1, ♦ • • -, I'D and one way of defining this precisely is by the use of the euclidian norm

edetermining the values of 0lc and so thatlldll is a minimum. This

Y>a0 l"*1 ■) 'Sa, U--i

least-squares criterion of best fit requires that

L {2 ( V/t “ dD ~which leads to the normal equations

Z Y - a0m +• a . Z Y

2XY = 0.oZX + aSX*„ all summations being for/i- 1 to im

Solving these simultaneously, the intercept and slope of the "best fit" line are given, respectively, by

in r Z Y Z Y * - Z Y Z X Y ; a, = Z Y - m a 0

m Z X a - (S*)* E Xand it will be found that this line passes through the centroid X , Y ,where X = Z Y and Y = Z Y .

m vr\

Now, a measure of the scatter of the data about the regression line of Y on X is supplied by the standard error or r.m.s. deviation

Syx = k

or, for small samples, preferably the modified standard error

sv> « /sCv-ac-a^)y * 7/

Assuming that the errors relative to the regression line are normally distributed, lines drawn parallel to it at vertical distances of YX on either side should include approximately 95% of the data points. Thus, the precision of the values of the intercept and slope can be specified by

Intercept = & 0 t Z + f j

Slope r 0.| ~ 2

with a 95% level of confidence, whereK*S -j ---------- 7---- —T ; - standard error in slope,

- CZtfl .......... - -(3 »

--

and the correlation coefficientI w x f t U Z Y

All the calculations required in using the foregoing procedure to obtain straight line, least-squares fits were carried out by means of a digital computer using a simple program.

APPENDIX D

AUTOMATIC PLOTTING PROCEDURES FOR THE STRESS CONCENTRATION FACTOR KT

D.1 Introductory CommentsThe automatic plotting of curves of the S.C.F. KT against the

position on the fillet for given geometries and loading conditions was performed off-line using a COKPLOT, type DP-3* incremental plotter controlled by standard, eight hole, paper tape. The necessary procedures for producing the controlling tape were performed on an ICL 1902A digital computer using punched cards as the input medium for both the source program and input data. The source program is written in the 1900 FORTRAN version (as in APPENDIX B) and the plotting routines called by the main segment are all standard ICL software packages which are read in from magnetic tape during compilation.

D.2 Brief Description of ProgramThe Program (CV08) comprises a Program Description segment

contained in statements numbered k to 11 and a Master segment (MAIN)

in statements 13 to 89 and, when compiled into an object program, contains the required plotting subroutine segments read in from a magnetic tape. For each model geometry selected, the Master segment:-

(a) reads from punched cards the parameter values describing the model geometry,

(b) reads discrete values of the Basic S.C.F.s,Kg, and K s ,(c) prints out the input data (a) and (b) using a line printer,(d) reads the selected values of the distance from shoulder to

the effective line of action of the applied load and for each of these distances:-

(i) calculates the S.C.F. K-p at each discrete point and

tabulates the values by means of the line printer,

(ii) calls the necessary subroutines which produce the papertape output required for automatic control of the incremental plotter.

A copy of the print-out of the complete program is given in the following three pages.

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^•3 Layout of the Input DataIn the following description of the (order; in which the input

data is presented, the field specifications are given in parentheses and the names in upper case letters are those used in the source program for the respective data values. All linear dimensions should be in consistent units.

Card 1 (3F5.2)'"BMAJ. The major width *B .BMIN The minor width b .R The fillet radius f ,

Card 2 (210)IFST

ILSTFor convenience, each boundary point on the fillet at

coordinate Q is represented in the program by an integer I, this being 1 at 9 = 0 and increasing by 1 for each 3°increment in 0 .Data values of and are read in foreach 5° increment, starting at I = IFST and ending at I ss.ILST.

Card 3.1 (2F5.2)KB(I) Value of Kj, at 0 corresponding to I = IFST and taken from

the appropriate graphs in Figures 3*3 to 3*7*KS(I) Value of at © corresponding to I = IFST and taken from

the appropriate graphs in Figures 3*8 to 3* 12.Cards 3*2, 3*3. etc.

As card 3*1 but for data values of K3 and at I = IFST+1,IFST+2, ....., ILST.

Card *4.1 (F6.3, 12)L Distance from shoulder to effective line of action of load,

(see diagram in Figure D.1 for sign convention).

NEXT If more values of L are to be read in for this geometry, columns 7 and 8 are left blank.

Card .2, ^.3« etc.As Card ^.1 but for further values of L.After the last value of L for this geometry has been dealt with , add another card with 1 in column 8.

Having completed the data set for one model, those for any succeeding geometries follow, starting again with Card 1. Place a blank card after the last data set to cause the program to jump to the statement initiating subroutines for emptying the buffer storage used for the incremental plotting procedures.

D.4 Output DataFor each shouldered plate geometry for which input data has been

read in, the line printer tabulates the following information:-(a) The parameters 3 , t and r describing the geometry.(b) The and values read in together with the corresponding

values of 9 .(c) Each value of L read in, followed by the calculated values of

K t tabulated against the corresponding values of © •In addition the necessary paper tape is produced for controlling the incremental plotter. The plotting routines produce scaled and labelled axes, a plotted symbol for each calculated value of and a spline fit curve passing through the plotted points. The calculations required for determining the coefficients of the polynomials describing the spline fit curves involve considerably more time than the other plotting procedures and, if desired, this feature can be deleted by removing statements numbered 65 and 67 from the program.

1 1 1 1Copies of the graphical output for ^ = 0-2, ^ = 1, -, g and ^ andU/ 1 1 1p = 0, g, •£, 1 and 2 are shown in Figures 5#1 to 5*3*

Some Alternative Methods of Specifying and Using the Data

(a) Details of the sign convention for the method used in section D.3 which specifies the loading condition by giving the distance from the shoulder to the effective line of action of the load, are given in Figure D.1. An alternative method of

L (+ve)

ll (+ve)

111' (+ve)

E o u i 'm W iV Vo e m c l w c .

W A V I on\v j v iV V

L -- l ! + M '

FIGURE D.1 Sign Conventions and Alternative Methods of Specifying the Loading

defining the loading condition by specifying the shear and bending moment at any section is also illustrated and if this procedure is adopted, either(i) the equivalent value of L can be calculated from

L - L + and presented as input data

for the program as it is written, or

(ii) the program can be amended so that the alternative load parameters can be read in. In this case, statement number *+7 should be replaced by READ (5,130). L, EM, W, NEXT

an additional statement numbered, say, ^71 should follow,

this, to read L = L + EM/W

and the format specification in statement numbered 79

and labelled 130 should be replaced by

FORMAT (3F6.3, 12).

Referring back to the layout of input data detailed in section D.3» the following amendments are necessary

Card *f.1 (3F6.3, 12)L Distance from shoulder to position of section

for which shear and bending moment are given.EM Bending moment at that section.W Shear at that section.NEXT If more loading cases are to be read in for

this geometry, columns 19 and 20 should beleft blank.

Card *1.2, etc.As Card *t.1 but for further loading cases. After the last loading case for this geometry has been dealt v/ith, add another card with 1 in column 20.

(b) To deal with the problem of calculating and plotting KT forI?*' p' I/values of the parameters and ^ for which curves of IVg and

are not available, it was suggested in section 5*4 that Kjj, and Ks could be specified by idealised curves, over a limited part of each fillet, using describing functions with coefficients determined from the empirical relationships forA jjK 3 and , together with some method of specifying the positions of peak values of and • A set of possible specifications, based on representations of the idealised ICg and K s curves.by sinusoids, now follow, together with an example of their use, showing the curve of given by theidealised K,,K g curves and compared with that given by theoriginal method.

Idealised CurvesAn idealised curve is obtained first for the geometry corresponding

to the limiting condition between complete and incomplete fillets, i.e.

for

3 = b + U °r ^ = J T Z f y

For this condition(i) is specified as zero at B = 90°.

A(ii) K b is specified as given by the empirical formula in

Asection 5»3» at a position 0^ agreeing with the carpet plot of 0 for peak positions given in Figure 4,2. This shows

^ Athat for 0<i«<0*9, 0 is approximately constant for a3

given and can be represented satisfactorily by

\ = (* - * \ ) ° .

(iii) Taking the curve between points (i) and (ii) as a sinusoidof interval TT radians, the value of K at any position 0$can be specified by

K b = k . k h { I + -sin h o + g.e» -g.eVd .V 9 0 + % J t j

For any other value of ^ the idealised curve is assumed to be of identical profile but displaced vertically so that it passes through

Athe appropriate value of given by the empirical formula in section 5*3* This gives

where the suffix UH refers to values already calculated as above for the limiting case.

Idealised Ks. CurvesThe idealised curve for any values of the parameters in the range

0 < $ < 0 . 9 , can be obtained by specifying:-I/ h(i) K s is zero at a position distant ^ from the junction of

the minor width and the fillet.A

(ii) K s « Ks (as given by the appropriate empirical formula) at aA

position agreeing with the carpet plot of 0 for the peakA

K s positions given in Figure 4.1. From this, can be represented satisfactorily by

K -- fe* - •(iii)Taking the curve between the points (i) and (ii) as a sinusoid

of intervalIT radians, the value at any position B

can be specified by

Use of the Idealised CurvesThe foregoing expressions give idealised curves which show good

agreement with all the available experimental curves of K<g and K«> within the range O < 0 < 6 O ° . These expressions can be included in the program for automatic plotting of K-f values as a subroutine which will replace the statements required for reading in discrete values of Kj,

and K s • The only input data then required would be the parameters *£> , b and r » describing the shouldered plate geometry and the values selected for the parameter L ♦ describing the loading configuration.As a check on the procedure, values of Kj have been calculated by handfor ^ = 0*6, ^ * 0*125 and 1^=1, using both the available

from Figures 3*4 and 5*9 and the idealised values given by the foregoing expressions and these are plotted for comparison in Figure D.2.

177

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S. TIMOSHENKO and W. DIETZ. 'Stress Concentration Produced by Holes and Fillets’. Trans.A.S.M.S. Vol. 47 (1925), pp. 199-220.R.F. PETERSON and A.M. WAHL. 'Two- and Three-Dimensional Cases of Stress Concentration and Comparison with Fatigue Tests'. Trans.A.S.M.E. Vol. 58 (1936), J.App.Mech., pp. A15-A22.M.M. FROCHT. 'Factors of Stress Concentration Photoelastically Determined'. Trans.A.S.M.E. Vol. 57 (1935)« PP* A67-A68.M.M. FROCHT. 'Photoelastic Studies in Stress Concentration'. Selected Papers of M.M. Frocht on Photoelasticity. Fp. 65-75* Pergamon 1969* (Reprinted from Mechanical Engineering, 1936).G.H. NEUGEBAUER. 'Stress Concentration Factors and their Effect on Design'. Product Engineering (Feb. 1943), PP* 83-87.I.M. ALLISON. 'Stress Concentrations in Shouldered Shafts and in Tubes with Diametral Holes'. Ph.D. Thesis (Oct. 1958),University College London.J.B. HARTMAN and M.M. LEVEN. 'Factors of Stress Concentration for the Bending Case of Fillets in Flat Bars and Shafts with Central Enlarged Section'. Proc.S.E.S.A. Vol. 9 , No. 1 (1951), pp. 53-62.J.S. STRANNIGAN. 'Variation of Edge Stress during Storage of an Epoxy Resin'. N E L Report No. 127, (Jan. 1964).H.T. JESSOP. 'On the Tardy and Senarmont Methods of Measuring Fractional Relative Retardations'. British Journal of App. Phys., Vol. 4 (May 1953) 1 PP* 1.38-141.I.Mi ALLISON. 'Stress Separation by Block Integration*. British Journal of App. Phys., Vol. 16 (1965), pp* 1573-1581.I.M. ALLISON. 'Extrapolation Methods for Evaluating Boundary Stresses'. British Journal of App. Phys., Vol. 16 (1965), PP* 93-99*

G.E. FORSYTHE. 'Generation and Use of Orthogonal Polynomials for Data-Fitting with a Digital Computer'• Journal Socy. for Indust.App. Maths., Vol. 5, No. 2 (June 1957), pp. 74-88.

13. CoWo GLENSHAW and J.G. HAYES. ’Curve and Surface Fitting’. Journal Inst. Maths. Applications, (1963)» PP* 164-183.

1*+. R.Bo HEYWOOD. ’Designing by Photoelasticity’. Chapman andHall 1952.

15. Ho FESSLER, C.C. ROGERS and P. STANLEY. ’Shouldered Plates and Shafts in Tension and Torsion’. Journal of Strain Analysi Vol. *f, No. 3 (1969), PP* 169-179.

16. RoE. PETERSON. ’Stress Concentration Design Factors’.J. Wiley and Sons 1953-

17. H. NEUBER. ‘Kerbspannungslehre’. Springer, Berlin (1937). Translation: ’Theory of Notch Stresses’, J0W« Edwards Co.,Ann Arbor, Michigan (19^6).

18. E0G 0 COKER and L.N.G. FILON. ’A Treatise on Photoelasticity*. Cambridge University Press 1931 (Revised 19371 H.T. Jessop).