The theory and design of photoelastic load gauges incorporating glass element transducers

Int. J. Rock Alech. Min. Sci. Vol. 9, pp. 363-401. Pergamon Press 1972. Printed in Great Britain

THE THEORY A N D DESIGN OF PHOTOELAST1C LOAD GAUGES INCORPORATING GLASS ELEMENT TRANSDUCERS J. A. HOOPER Ove Arup & Partners, Consulting Engineers, London (Received 15 AII,~IISI 1971)

Abstract-- A glass cylinder compressed across a diameter can be used as an effective load transducer by utilizing the birefringent properties of the glass. Calibration of applied load in terms of fringe order can 1eadily be obtained by using a suitable polariscope. Incorporating the basic cylindrical transducer into a steel body greatly extends the load-measuring range, enabling a flexible load-gauge system to be developed. Stresses applied to the glass and steel components are kept to well below their respective elastic limits which, combined with tile absence of moving parts, gives rise to gauges of excellent long-term stability. The various factors affecting the design and performance of this type of load gauge are discussed herein. The stress dishibution and photoelastic effects within the glass cylinder are considered, together with its mode of deformation under diametral compression. Attention is also directed towards determining the stresses and displacements in various gauge components ; in this connexion, an analytical solution is given for the deformation of a circular hole in a uniaxially loaded strip of finite width. A range of theoretical calibration curves for column-type and ring-type gauges is given, and further aspects such as temperature susceptibility and longterm stability are examined. An example of the design and application of a photoelastic load gauge which has been used in practice is also included. 1. INTRODUCTION IN A REMARKABLEp a p e r published by the R o y a l Society o f L o n d o n in 1816, BREWSrER [[] gave the tirst detailed scientific account* o f the artificial double refraction p r o d u c e d by the a p p l i c a t i o n o f stress to i s o t r o p i c t r a n s p a r e n t solids. Glass was the principal test material used in the experiments, a n d the p a p e r even includes several practical suggestions for a p p l y ing the photoelastic principle to the design o f various measuring instruments. One o f these instruments is no less t h a n a l o a d gauge or d y n a m o m e t e r based u p o n the p h o t o e l a s t i c effect in glass beams loaded in bending. A n original d r a w i n g o f the instrument is r e p r o d u c e d in Fig. l, together with the relevant description given by Brewster, who evidently recognized the potential o f this form o f device and anticipated the use o f glass as a t r a n s d u c e r material. The account which follows here relates to an extension o f Brewster's original conception o f a ' c h r o m a t i c d y n a m o m e t e r ' , and considers the design o f photoelastic load gauges a p p l i c a b l e to a wide range o f usage. Active d e v e l o p m e n t o f this f o r m o f i n s t r u m e n t a t i o n has taken place over the p a s t 10 years [2-5], and the intention o f this p a p e r is to present the current situation with regard to the theory a n d related technical details o f p h o t o e l a s t i c load gauges. C o n s i d e r a b l e experience has now been gained in the design, use a n d a p p l i c a t i o n o f these gauges, a n d no further changes in basic design are envisaged. * The discovery of artificial double refraction induced by stressing transparent materials was made by Brewster in the previous year. In a short letter to the Royal Society in 1815, the birefringent effects in compressed animal jellies are described. 363

364

J. A. HOOPER

.5."

C

PROPOSITION XIV.

To construct a chromatic dynamometerfor measuring the intensily (forces. In almost every dynamometer, which has hitherto been constructed, it is assumed that a steel spring recovers its original shape after repeated bendings, and upon this assumption the scale of the instrument is formed.* The perfect elasticity of glass, however, renders it, in this respect, a much fitter substance than steel, and though it does not admit-of such a great diange of shape, yet the slightest variations in its structure can be rendered visible. If a number of narrow and thick plates of glass AB, Fig. 14, (P1. IX.) are firn{Iy fixed at each end in brass caps A, B ; then if any force is applied to a ring at C in the middle of the plates, when the ends A and B are fixed, or if C is fixed, and the three applied at the points A, B, the plates of glass will be bent in the middle, and the force by which this is produced, will be measured by the tints that appear on each side of the black space ran. By dimi0ishing the length of the plates, or increasing their number, they may be made to lZesistand to measure any degree of force. When the force to be ascertained is small, a single plate of glass will enable us to measure its intensity with great exactness. FIG. I. Part of paper [1] by David Brewster published by Royal Society of London in 1816 illustrating original conception of chromatic dynamometer. [Reproduced by permission of the Royal Society of London] I n all the p h o t o e l a s t i c gauges described herein, the transducer element is a d i a m e t r i c a l l y l o a d e d solid glass cylinder t h r o u g h which p o l a r i z e d light is passed. By using a n a n a l y s e r in the f o r m o f a p o r t a b l e hand-viewer, the l o a d a p p l i e d to the cylinder can readily be c a l i b r a t e d in terms o f the resulting fringe o r d e r a l o n g the cylinder axis. In m o s t p r a c t i c a l a p p l i c a t i o n s , the glass cylinder is i n c o r p o r a t e d into a transverse hole passing t h r o u g h a steel b o d y . T h i s greatly extends the l o a d r a n g e o f the system; l o a d a p p l i e d to the steel b o d y causes the

(a)

Before compensation.

(b) After compensation. FIG. 2. Isochromatic

RM f.p. 3641

fringe pattern

produced in diametrically compressed goniometric compensation.

cylinder and Tardy method of

FIG. 8 (a). Circular hole.

FIG. 8 (b). Slotted circular hole.

FIG. 8 (c). Glass transducer positioned in slotted hole. FIG. 8. Isochromatic

fringe patterns in perforated acrylic resin slabs under same applied uniaxial load.

FIG. 15. Laborator!

arrangement

showing 10 tonf ring-type gauge under sustained load in creep apparatus.

.^._“_..l_,.”

“.-_..-I-.

FIG. 17. 50 tonf tunnel load gauge (prior to application

of external protective coatingl

THEORY AND DESIGN OF PHOTOELASTIC LOAD GAUGES

365

transverse hole to deform, which in turn gives rise to diametral deformation of the glass cylinder located within the hole. Hence the design of a load gauge for any particular application involves consideration of the stresses and displacements in both the glass cylinder and steel body, with the latter normally in the form of either a thick ring or a cylindrical column containing a transverse hole at mid-height. In the following section, consideration is given to the photoelastic effect in glass cylinders, together with the physical and mechanical properties of the glass itself. The third section deals in some detail with stresses and displacements in the principal load-gauge components, and is followed by an account of the general calibration and stability characteristics of photoelastic load gauges. The final section contains a brief description of a practical application of these gauges, namely the long-term measurement of hoop-load in a tunnel lining. 2. GLASS PROPERTIES IN RELATION TO TRANSDUCER APPLICATION 2.1 Photoelastic effects in glass cylinder

For a concentrated line load P acting across the diameter of a cylinder, radius R, length I, the two-dimensional solution [6] for the principal stress difference at any point in the x - y plane may be written in the form 4PR (~1 -

~2) = - -

~l

(R 2 __ X2 __ y2)

(x 2 + R 2 + y2)2 _ 4RZ y2

tl)

where the co-ordinate origin is taken at the centre of the bounding circle, and the loading direction is coincident with the y-axis. In any dark-field circular polariscope, the relative retardation can be expressed in terms of fringe order, n, by (2) where C denotes the stress-optic coefficient, t the length of the light path and A the wavelength of light. Hence n)t = Ct (~1 - - ~2)

HT/A

P :

4-C

( x 2 _~ R 2 _~ y2)2 _

R(R 2

--

X

2

--

4R 2 y2 yZ)

(3)

Thus for a given light source and birefringent material, the fringe order is directly related to the total applied load, and is independent of the cylinder length. Furthermore, the locus of the zero fringe, obtained by putting n -: 0 in equation (3), is described by R 2 = X2 -~- y 2 which also defines the circular boundary of the cylinder. It therefore follows that, except at the loading points, shear stresses vanish at this boundary and the zero fringe is always present irrespective of the magnitude of the applied load. The type of fringe pattern produced in a diametrically compressed cylinder using a monochromatic light source is illustrated in Fig. 2. Fringes generated at the loading points move inwards towards the centre, and then spread laterally to stack along the x-axis. Isoclinics are removed from the field of view by incorporating quarter-wave plates in both polarizer and analyser, and the polariscope is set to give a dark field with an unloaded cylinder. There are many ways in which the axes of the four filters can be orientated with respect to each other and to the transducer, but arrangements which have been standardized for illuminating and viewing the isochromatics in diametrically loaded cylinders are as shown in Fig. 3. Figure 3(a) shows the standard filter orientations relative to the loading direction for transmitted light viewing, and Figs 3(b) and (c) show the corresponding filter

366

J. A. HOOPER ?-! .

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i

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I

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.... I

i

/

~ "-"-'N.j

p,t 4ri:'e;

,

I

~ r:m~e

cmcfiyzer

(a) Transmitted light viewing. Fc s r ~ e r

i

.... [-/

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~

-----_

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--7'/

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ttttt

crml~,zer

(b) Reflected light viewing (separate polarizer and analyser).

)~i pkt~e

po{arlzer end analyzer

cylinder

4

(c) Reflectedlight viewing (combined polarizer and analyser). F[6.3. Polariscope arrangements for use with glass transducers. (Arrows indicate optical transmission axes of filters.) orientations for reflected light viewing. In the latter two examples, one of the cylinder faces is provided with a coating of reflective cement or a mirror backing. In each case, however, the relative orientation of the linear analyser and its adjacent quarter-wave plate remains the same. Thus by producing an incident polarized beam of the correct rotation, only one analyser is required for viewing cylinders in both transmitted and reflected light, The most convenient point at which to measure fringe order is the centre o f the cylinder, in which case equation (3) simplifies to P-

~nhD 8C

(4)

THEORY AND DESIGN OF PHOTOELASTIC LOAD GAUGES

367

where D denotes the cylinder diameter. Fractional fringe orders are normally measured using the Tardy method of goniometric compensation [7], in which the analyser is rotated in order to optically advance or retard the fringe order at the centre of the cylinder. This method is illustrated in Fig. 2, in which the central fringe order of the original pattern in (a) is retarded to t~, = 2, in (b) to give an overall reading of 2.25. This particular case corresponds to Fig. 3(a) or Fig. 3(b) in ',~hich a 180 ° clockwise rotation of the analyser produces a complete wavelength retardation of the fringe order; in Fig. 3(c), the same effect is produced by a 90 ~ clockwise rotation of the combined polarizer and analyser. With the cylinder properly illuminated, the central fringe order can be measured to L 0 "02 using the normal field hand-viewer, which is adequate for most purposes. However, there are several techniques by which the accuracy can be improved, although these are normally confined to laboratory work. By aligning the central fringes with an oblique cross marked on the cylinder face, for example, HmTSCHER et al. [8] have obtained accuracies of between ~ 0.005 0-01 fringe. Using a more sophisticated system, BRowN and H~cKsoy [9] have obtained repeatable readings of fringe order to -- 0-002 by using a single photoelectric cell in conjunction with a stabilized light source. A further technique for improving accuracy is by way of the fringe sharpening and multiplication techniques developed by Post [10]. By inserting parallel partial mirrors at each end of the cylinder, the emergent light rays combine to produce sharp peaks in the intensity-birefringence curve at integral and halffringe orders. The entire fringe pattern is thus sharpened, which enables the centre of a fringe to be located more accurately. If the partial mirrors are inclined to each other at a small angle, the length of the light path is effectively increased and the number of fringes multiplied. With both techniques, however, the loss in light intensity is substantial; the peak intensity for a sharpened pattern is only about 5 per cent of that of normal peak intensity, whilst in fringe multiplication, the maximum light intensity is less than 1 per cent of the normal value. In practice, fringe sharpening is readily effected by coating the faces vf a cylinder cored from polished plate glass to provide the reflective surfaces, and using a brighter light source. For fringe multiplication, on the other hand, provision of inclined mirrors is more difficult and, together with the necessity of specialized optical equipment to receive the emergent rays, clearly limits the practical application of the technique. For the purpose of controlling tensile stresses and for ease of reading, the operating range of a glass transducer is normally limited to five fringes under transmitted light. In practice, white light gives the most distinctive fringe pattern up to about three fringes, with fringe values accurately determined by reference to the 'tint of passage'. Beyond this point, however, the higher relative retardations give rise to excessive extinction of light of certain wavelengths, which in turn leads to a 'washing out' of the colours. For this reason, monochromatic light is normally used to illuminate the cylinder. In the laboratory, it is often convenient to use gas discharge lamps to provide this illumination; in the field, it is more convenient to use coloured filters. Such filters, of course, are not 'truly monochromatic' in the sense that they pass light within a specified band width. With a high-quality interference filter this band width can be reduced to well below 100 ~, whereas normal instrument filters have a band width of around 500 ~. This is still large compared with that produced by a sodium vapour lamp, for example, where the visible energy is concentrated in the region 5890-5896 A. In order to supplement data provided by the manufacturers, the transmission properties of a number of commercially available instrument filters were measured using a Cary-14 recording spectrophotometer. Each trace was found to be slightly asymmetrical, but initial

368

J . A . HOOPER

and final cut-offpositions in the spectrum were clearly defined. Except for red tilters, all band widths were within the 500-600 A range. No abrupt final cut-off levels are obtained for red filters due to their capacity to transmit infra-red radiation, so that where these are to be included in the transducer polariscope system, initial calibration must be carried out in order to determine tile predominant wavelength of the particular filters used. 2.2 General and birefringent properties of glass Glass is an extremely variable material with mechanical and physical properties which differ widely according to its chemical composition. For general usage as the transducer element in photoelastic load gauges, soda-lime plate glass has been found most suitable. It is relatively cheap and easily obtainable, and can be cored without difficulty using thin-walled diamond drills. It also has suitable stress-optic qualities, together with adequate tensile strength and good dimensional stability [l 1]. Glass used in the present investigations had the following approximate percentage composition by weight: Si02 73.3

Na20 12.6

CaO 9.2

MgO 3.2

A1203 0.8

K20 0-3

Rein. 0.6

Young's modulus and Poisson's ratio were measured a s / 0 . 7 × 106 lb/in 2 and 0-24 respectively. The thermal softening temperature and coefficient of linear expansion were given by the manufacturer as 739°C and 8.7 ~ 10-6/vC respectively. The stress-optic coefficients of different glasses are known to vary considerably, COKER and FtLON [12] having quoted values ranging from -- 2.0 to + 4.5 brewsters. The coefficient is fairly sensitive to chemical composition, and so a 2-0 in. diameter, 1.0 in. long, cylinder was calibrated under sodium light (A -- 5893 A) in order to establish a value for the above mentioned soda-lime glass. From equation (4), the measured sensitivity of 970 lbf/ fringe gives a stress-optic coefficient of + 2.72 brewsters, which is similar to the values of + 2.62 brewsters determined by WAXLER [13] and 4-2"68 brewsters measured by VAN ZEE and NORITAKE [14] for a similar plate glass. It is sometimes more convenient to express birefringent properties in terms of a material fringe value, f, rather than a stressoptic coefficient. Defining f-

t

( ~ -- <,~).-

(SJ

n

the equivalent fringe value becomes 1230 lb/inZ/fringe/in thickness under sodium light. Equations (4) and (5) imply an invariance between stress sense and birefringence; tests by SAVUR [15] on a number of different glasses confirmed that the relationship between relative retardation and applied stress was linear, irrespective of whether the stress was tensile or compressive. The effect of moderate intervals of time on the stress-optic coefficient of soda-lime glass was checked by subjecting two cylinders to constant diametral compression in a lever-type loading rig for a period of 2 years. The applied loads were sufficiently high to give approximately 5 fringes at the centre of the cylinder under transmitted light, but during this time no discernible change in fringe order was observed. For time intervals in the region of 20 years there is the possibility of some variation in the stress-optic coefficient associated with internal changes in the glass structure. HARRIS [16] has reported such variations over this length of time for various technical glasses, but no corresponding measurements appear to have been made on soda-lime glass. Most of the variations which do occur probably take place during

THEORY AND DESIGN OF PHOTOELASTIC LOAD GAUGES

369

the first 2 or 3 years after casting, and as the plate glass used for producing the cylinders is kept in stock for about 2 years prior to coring, it is reasonable to assume that the effect of time on birefringent properties is negligible and that it should not in any way affect the performance of the glass cylinder as a practical load transducer. The effect of temperature on the stress-optic coefficient of technical glasses was also studied by Harris, who found that, except for glasses with high lead oxide content, the coefficient increased with rise in temperature. For a soda-lime glass, Van Zee and Noritake (loc. cit.) found that the stress-optic coefficient increased approximately linearly by 5 per cent as the temperature was raised from 20 to 540°C, but increased very rapidly with further rise in temperature. Hence thermal effects can be neglected in relation to birefringent properties for transducers operating within their normal temperature range. When glass cylinders are incorporated in steel bodies, however, temperature effects assume greater significance due to the differential thermal expansion of the two materials, and this aspect is considered later in the text. 2.3 Strength of glass cylinders The load-carrying capacity of glass cylinders in diametral compression is clearly an important factor in the overall design of load gauges. Glass is relatively weak in tension, and diametrically compressed cylinders are required to withstand substantial tensile stresses; in fact, the way in which the transducer is loaded is directly analogous to that used in the well-known Brazilian or indirect tensile test [17] for brittle materials. The tensile strength of glass depends upon so many factors that it is difficult to define or specify with any degree of precision. Failure is normally considered to initiate at pre-existing cracks or flaws, usually at the surface, which act as stress raisers and may propagate under applied stress. If such cracks are virtually absent, as in freshly drawn glass fibres, the measured strength can be of the order of 10 6 lb/in 2, which is the approximate theoretical strength calculated on the basis of intermolecular forces. In contrast, HOLLAND and TURNER [18] tested over 2000 soda-lime glass specimens in bending and found that mean tensile strengths varied from around 7000-16,000 lb/in 2, depending upon test conditions and surface preparation of the specimens. Measured strengths also depend upon the area subjected to tensile stress; hence bending strengths are usually higher than uniaxial tensile strengths, and when very small areas are loaded, as in spherical indentation tests of the type performed by ARGON et al. [19], the measured strengths may be in the region of a hundred times greater than the modulus of rupture. In order to ascertain suitable working loads for glass transducers, a considerable number of soda-lime glass cylinders were diametrically loaded to failure between flat ground-steel platens at a rate of 4000 lbf/min. Cylinders of various sizes were tested in batches of ten, and values of mean failure load, w, (with standard deviation, s, and coefficient of variation, v) and ultimate central tensile stress, ~,~, (from ~c -- 2PITt lD) are listed in Tables 1 and 2. It is interesting to note the wide variation in values of c~c, which are a measure of tensile strength according to conventional two-dimensional theory. This suggests the possibility of an alternative mode of failure, and close examination of the fracture surfaces established that failure always initiates at some point on the loaded surface, and not at the centre of the cylinder. This problem has been discussed in detail elsewhere [20], in which the additional factors contributing to failure were considered to be the modification of contact stress distribution due to the finite length of cylinder and the stresses produced by interfacial friction forces in the contact areas between cylinder and platens. The load capacity of the

370

.I.A. HOOPER

cylindrical transducers, however, is more than adequate for practical purposes. With ~. 1 "25 in. diameter, 0.74 in. long cylinder, for example, the mean failure load from Table 2 corresponds to central fringe order of 12.0 under transmitted light (t - 6450/k), giving a safety factor of 2.4 based upon a maximum working value of 5 fringes. Assuming a normal frequency distribution for failure loads, the chance of a cylinder failing at 5 fringes is remote ( < 0.002 per cent). In producing glass cylinders for use as transducers, it is normal practice to bevel the circular edges at both ends; this prevents subsequent spalling and increases cylinder strength. Additional increase in strength may be obtained by providing ~he cylinder with barrel-shaped ends which serve to reduce the stress concentrations in the glass still further. These improvements in cylinder strength are particularly useful i~ cases where transducers are to be subjected to high sustained loads, in which case the nomina! strength of the cylinder is reduced due to the effect of static fatigue in the glass. TABLE 1. MEAN EAILURE LOADS AND CENTRAL EENSILE STRESSES FOR l - 2 5 2 - i n . DIAMETER SODA-[.E~IE (;/_ASS (YI.INDERS COMPRESSED BETWEEN FLAT STEEL PLATENS

l (in.) 0"497 0' 738 1 "000

% (lb/in 2)

w (lbf)

s (lbf)

v

5320 5460 5630

5200 7940 11,070

530 1110 1800

0" 10 O' 14 0" 16

TABLE2. MEANFAILURELOADSANDCENTRALTENSILESTRESSES FOR 0.738-in. LONGSODA-LIMEGLASSCYLINDERSCOMPRESSED BETWEENFLATSTEELPLATENS D (in.) 0-507 0-743 1'000 1.252 1'498 1'996

% (lb/in 2)

w (lbf)

s (Ibf)

c

8130 6350 5620 5460 4610 4920

4780 5470 6510 7940 8000 11,390

900 1120 620 1110 570 l(~)0

0'19 0-20 0"10 0"14 0"07 0'14

In point of fact, glass is one of the few engineering materials whose strength is markedly dependent upon the duration of loading. This fatigue effect exhibited by glass under static load is in some ways analogous to the dynamic fatigue experienced by metals under cyclic loading. In both cases, it is the gradual spread of flaws which eventually causes failure of the specimen. In contrast to metal fatigue however, GURNEY and PEARSON [21] measured the bending strength of soda-lime glass rods under static and cyclic loading, and found little difference in the times to fracture for these two cases when specimens were subjected to applied stresses of equal magnitude. Relatively few observations have been made on the long-term strength of glass, but experimental work by HOLLAND and TURNER [22] and later by BAKER and PRESTON [23] indicated a fatigue limit for glass, as opposed to a breaking stress diminishing indefinitely over long periods of time. Experimental results obtained by SHAND [241 support this concept; soda-time glass rods were subjected to constant bending stresses for a 10-year test period, and a high proportion of the rods which did not fail during the first few weeks remained intact for the entire duration of the test.

THEORY AND DESIGN OF PHOTOELASTIC LOAD GAUGES

371

It is well established that fatigue in glass is closely associated with the penetration of surface cracks by atmospheric constituents. Experiments by GtJRNEYand PEARSON[25], in which bending tests on soda-lime glass rods were carried out in air and under partial vacuum, have indicated that the main cause of delayed fracture is due to the attack by water vapour, and possibly also by carbon dioxide. The temperature of the surrounding atmosphere also influences the process of delayed fracture, VOYNEGUr and GLATHARV [26] having found that at both very low and very high temperatures, there is a marked reduction in the fatigue effect. It follows that in applications where glass transducers are to be subjected to high loads for long duration, particularly when in damp surroundings at normal ambient temperatures, the maximum allowable working load acting upon the cylinder must be reduced to an acceptable level. In practice, there are several ways in which cylinder stresses can be effectively lowered without reducing the overall range or sensitivity of the load gauge. One further aspect concerning strength relates to the chemical durability of the glass, which is high compared with most other engineering materials due mainly to the inert nature of the silica. Chemical attack by water or acids differs from that by alkaline solutions in that the former group limit their action to constituents other than silica, whereas the lat~er attack all constituents. For glass exposed to the atmosphere, some fogging and occasionally pitting may occur when ambient conditions are severe, this weathering becoming more pronounced with increasing alkali content. Tests by BAKERand PRESTON [27] have indicated some decrease in strength with increasing relative humidity, saturated glass having the lowest strength. This effect was checked by submerging ten soda-lime glass cylinders under water (pH = 6.6) for a period of 3 months, and then loading them to failure between flat steel platens. The mean strength obtained was 20 per cent lower than the corresponding value for a batch of cylinders tested under normal laboratory conditions. 2.4 Non-elastic effects and plastic flow & glass Glass is usually considered not to exhibit the property of plastic flow which is common to metals, and is often regarded as an ideal brittle solid which has no yield point. Yet evidence of plastic flow can be simply demonstrated by drawing a hard point over a glass surface and observing the distinct plastic furrows produced. BRIDGMANand SIMON [28] have observed plastic flow and permanent deformation by loading thin glass disks between hardened steel dies; from density measurements obtained before and after compression, most glasses were found to show a marked threshold pressure below which no permanent effect was evident. For soda-silicate glasses, this pressure was approximately 400,000 lb/in 2. In a similar context, early experiments by Pml~LI~'S [29] on the stretching of glass fibres demonstrated the property of time-dependent strain bebaviour. In addition, JEssoP [30] has detected small changes in elastic modulus for specimens in bending, while experiments by MURGATROVD and S¥~:~s [31 ] on the torsion of soda-lime glass rods have indicated the presence of delayed elastic strain. Having thus established that some creep and plastic flow in glass can definitely occur, it must be examined in relation to the performance of cylindrical glass transducers, especially as high stresses are generated in the platen contact regions at working loads. Consideration of glass hardness forms a useful approach to the problem. For an ideal elastic-plastic material, TABOR [32] gives the approximate theoretical relationship between the Vickers diamond pyramid hardness, V, and the yield stress in compression, Y, as V ~ 3 Y.

(6)

372

J.A.

HOOPER

This theory is adequate for certain work-hardened metals, but its application to glass predicts flow stresses well below those associated with so-called brittle fracture. An alter'native interpretation of the indentation test has been given by MARSH [33], who assumed that material is displaced radially outwards from the indenter instead of towards ihc free surface as in the rigid die theory. A similar mode of hemispherical symmetry in the strata regime has been observed in brass specimens by SAMUELSand MULHEARN[34]. This alternative mode is analogous to the expansion of a spherical cavity in an elastic material (Young'~ modulus/2. Poisson's ratio v) by an internal pressure, V, the relevant expression fox"which is* V _ 2( 1 Y 3

3 3-

1,1 ~

3 ~_~) a-{ 3~--

(7)

where a = 6 (1 -- 2 v) Y/E and/3 = (1 -5 ~) Y/E. From indentation tests carried out on a wide range of materials, Marsh established the semi-empirical relationship V g

- -=0.28

+0.60QlnZ

(8)

where Q and Z are the appropriate functions of c~and/~. For the soda-lime glass used to make transducers, the mean Vickers hardness number was measured as 554, giving a yield stress of 260,000 lb/in 2 based upon the rigid die theory, and 440,000 lb/in 2 based upon the spherical cavity theory. From physical considerations, this latter value is by far the more reasonable of the two in that it more closely approximates to the breaking stress of thin glass fibres. As indicated in a subsequent section, diametral compression of glass cylinders does generate high stresses in the platen contact zone, but these are still well below flow stress levels and do not give rise to any stability problems with transducers of this type. 2.5 Selected properties of various types of glass Although soda-lime glass is the standard material for photoelastic transducers, other types of glass may be used provided that they satisfy the necessary requirements of strength and stability. In order to investigate this possibility, various specimens of optical glass were obtained from Chance-Pilkington Ltd., as well as one sample of Pyrex glass manufactured at the Corning Glass Works, New York. Where possible, three 1.25 in. diameter, 0-75 in. long, cylinders were cored from each blank and loaded to failure between fiat steel platens. A fourth cylinder was also cored and its end faces polished to enable the relevant birefringent properties to be determined. Test results for the various glasses are summarized in Table 3. with the values for soda-lime glass included for reference. The relatively low strength of the optical glass cylinders is immediately apparent, and it is this property which normally precludes their use as transducers. This becomes self-evident when comparing their failure loads with the corresponding loads required to produce 5 fringes at the centre of each cylinder under transmitted sodium light. Measured values of the stress-optic coefficient ranged from 4- 3.95 brewsters for the low expansion borosilicate glass (Pyrex) to 1.23 brewsters for the double extra-dense flint, a glass containing a high percentage of lead oxide. Vickers hardness tests also showed this latter glass to be comparatively soft in relation to the other glasses. Material fringe values defined by equation (5) and corresponding to sodium light (A -- 5893 A) are included in the Table. It is of interest to note that cylinders of the * N o t e the typographical error in the a term given by MARSH [33].

THEORY AND DESIGN OF PHOTOELASTIC LOAD GAUGES

373

same diameter but of different glass types can be loaded together to form a composite transducer as the total effect is to integrate the birefringence throughout the entire light path. In this technique, however, differences in elastic moduli have to be taken into account, and cylinders have to be cored together in order to ensure accurate alignment during subsequent loading. TABLE 3. SELECTED PROPERTIES OF A RANGE OF DIFFERENT GLASSES ( L o a d values relate to 1.25 in. diameter, 0.75-in. long cylinders; fringe values relate to s o d i u m light)

Glass type Soda-lime Borosilicate (Pyrex) Borosilicate Crown Zinc Crown Dense Barium Crown Barium Light Flint Dense Flint Double Extra Dense Flint Irtran 5

Ref. no.

Mean failure load (Ibf)

Load for 5 fringes (lbf)

-7740 510644 508612 612585 574520 620362 927210 --

8060 4630 2440 2890 2310 2290 2500 ---

3040 2090 2870 2230 4420 3060 2940 6720 4130

C f (brewsters)(lb/in2/f/in.) + 2.72 + 3" 95 ~--2-88 +3-70 +1.87 + 2.70 +2.81 --1.23 +2.00

+ 1230 + 850 ~-1160 +910 4-1780 + 1240 +1190 --2710 +1680

V 554 498 536 572 566 508 525 270 421

With regard to the general problem of load measurement at elevated temperatures, the use of a cylindrical photoelastic transducer of some suitable material could well provide an acceptable solution in certain circumstances. The maximum operating temperature of a Pyrex glass transducer is approximately 350°C, and higher temperatures would necessitate the use of a material other than glass. One such material which satisfies the dual requirements of high transmittance and temperature resistance is a polycrystalline magnesium oxide ceramic known as 'Irtran 5' manufactured by the Eastman K o d a k Co. A small sample was borrowed for testing, and the results are included in Table 3. Although expensive, the high melting point (2800°C) of this material suggests the possible use and application of photoelastic transducers in the field of high-temperature load measurement. It is interesting to note, however, that with few exceptions, ordinary soda-lime plate glass is by far the most suitable material for photoelastic transducers; the fact that it is also the cheapest and most readily available type of quality glass simply reveals a situation which is not altogether c o m m o n in the field of engineering materials and design. 3. STRESSES AND DISPLACEMENTS 1N GAUGE COMPONENTS 3.1 Stresses in diametrically compressed cylinder

The solution to the two-dimensional problem of determining the stress distribution in a solid disk or cylinder compressed across a diameter by concentrated loads was first given by HERTZ [35]. Taking the co-ordinate origin at the centre of the bounding circle, the principal stresses at any point y : R ~ along the axis of loading (y-axis) are crx - -

2P 7rlD

(9) -

Cry-ROCK 9 / 3 - - D

+

7riD \ 1 - - ~ 2 ]

374

J. A. HOOPER

in which tensile stresses are taken as positive. Thus the classical theory predicts a constant tensile stress along the entire loading axis, and implies infinite compressive stresses at the loading points. If the applied load is considered to be spread uniformly over strips of finite width, the stress distribution within the cylinder is modified to the form given by HONDROS [36]. For a uniform pressure, p, applied over the arcs - .~i < 0 < aandTr --- ,,,~ • ~ < ~-~ Q where ct denotes the angular half-width, the principal stresses along the loading axis are c~x .

2P[ ( t - - ~ 2 ) sin2 a . 2a. -~- ~:4. ~r . 1 - . 2~ 2 cos

. tan-1

(1÷~:2) (I

-

~2)

tan a

i

(10)

%

--

--2p[ (1--~2)sin2a (1 + ~ Z ) t a n a ] g 1 -~- 2-~"2-C--OOS"2a "~ ~:4 -~- tan-1 (2 -- ~:2-----~ '

From equations (10), both principal stresses become compressive in regions close to the loaded surfaces, and attain a maximum value equal to the applied pressure at the loaded surface. By St Venant's principle, however, the stresses at the centre of the cylinder would be expected to be substantially the same irrespective of whether the load was considered to act along a line or over a small but finite area. By assuming concentrated loads, for example, the calculated principal stress difference at the centre of the cylinder would be in error by an amount 2(z

With ~ = tan- 11/ 12, for example, J = -- 0.3 per cent, and with 5 fringes at the centre of the cylinder, the corresponding reduction in fringe order is 0.015, which is less than the normal reading accuracy. In practice, glass transducers are loaded between fiat steel platens which results in even smaller loading widths and which also produces a non-uniform load distribution at the contact surfaces. The stress distribution within a cylinder compressed by parabolically distributed loads has been investigated by KAKUZEN et al. [37], but for small loading arcs, the results will be similar to those for the case of uniformly distributed loads in regions away from the immediate vicinity of the contact surfaces. A further consideration in the loading of glass transducers relates to the situation that, in practice, not all the applied load is normal to the surface. Tangential surface tractions are generated at the loaded surface due to the dissimilar elastic properties of the cylinder and platens, and the distribution of such tractions depends upon both the friction coefficient and the compliance between the contacting surfaces [20]. An indication of the effect of tangential tractions on the stresses within the cylinder can be obtained from equations given by ADDINALLand HACKETT[38]. For the case of tangential tractions in the form q sin 0, 1~1 ~< a, acting over opposite arcs, the resulting stresses along the y-axis are* (c~x -t- ~y) = __2qTr[2 sin a

(1 --~:2) tan -a sesina ]

(l

* Note the typographical errors in the corresponding equations given in Reference [38].

THEORY AND DESIGN OF PHOTOELASTIC LOAD GAUGES

(O'x -- O'Y) ~- q~ [ ~ 2T sin ~r "@ 2 sin ~ ~ [.(1 (1-- -~2-~ -- ~2)2.~ (l4 ~"~-S~2 ~2) C~}

)(

(1

~2 33

tan-

375 (12)

, 2~ sin ~*] {-i ----}T)]"

The limiting stresses at the centre and surface of the cylinder are readily obtained by expanding equations (12) in series form; at the surface (~: ==- 1): 4q .

%=----sma, 77"

%~0

(13)

and at the centre (e = 0): q , % =-- % = - sm a.

7r

(14)

It follows that although tangential surface tractions modify the stress distribution within the cylinder, they produce no change of shear stress at the centre and hence do not affect the fringe order in the transducer at this point. In normal use, glass transducers are diametrically compressed between flat steel platens, and thus the contact area increases as the cylinder is pressed against the steel. This gives rise to a non-uniform pressure distribution across the contact zone, and the two-dimensional solution to the problem of establishing this distribution is once again due to HERTZ [39]. For smooth contacting surfaces, the so-called Hertzian pressure distribution over the contact width is 2P

p ._ ~a_/(l _ ~2)~ = p o ( 1 __ ~2)½

(]5)

where Po denotes the maximum contact pressure, and the distance x = ~a is measured along the interface from the loading axis. The contact half-width, a, is given by a2 --_ 4 P R ( 1 - - v , 2 + I .- p2 2) 7rl \ E, E;

(16)

where the subscripts 1, 2 refer to the cylinder and platens. The stresses within the contacting bodies reach a maximum at the contact surface, where both principal stresses are compressive and equal to the applied pressure distribution defined in equation (15). Thus for transducers loaded between flat steel platens, the maximum working stress in the glass is in the region of 130,000 lb/in 2, which is less than a third of the stress required to produce flow and permanent deformation of the material, according to the spherical cavity theory referred to earlier. 3.2 Displacements in cylinder and loading platens As the sensitivity of a load gauge of given geometry is largely dependent upon the deformation of the glass transducer located within the steel body, the displacement characteristics of diametrically loaded cylinders must be accurately known. Considering a uniform pressure, p, applied over opposite arcs of angular half-width, a, the two-dimensional elastic solution

376

J. A. HOOPER

by JAEGER [40] gives the diametral deformation, A, along the loading axis of" the cylinder as ATrE 2pR(1 -?v)

1 a(X-- 1) + .

.

n.

X

1

2n . + 1 ' 2n --

sin2m~

(17) 7r

=--~(X-- 1)(1 - - c o s s ) + ( X +

1) s i n a l n c o t

where X denotes the factor (3 -- 4 ~,) for plane-strain conditions and (3 - v)/(l ~- v) for the case of plane stress. The series summation is readily determined using the results of JAEGER and COOK [41], and selected numerical values for the right-hand side of equation (17) are listed in Table 4. Displacements in the loading platens can be obtained from the standard plane elasticity solutions for the case of uniform load acting upon the surface of an elastic half-space or layer. TABLE4. VALUESOF A~rE/2pR (1 + v) IN EQUATION(17) Loaded half-angle 4° Plane strmn v=0.2 v = 0.3 Plane stress ~=0.2 v =~0.3

I

2

3

4

5

0.2651 0.2319

0.4532 0.3963

0.6126 0-5355

0'7534 0-6583

0.8805 0.7689

0.2761 0.2549

0.4722 0.4357

0.6383 0.5889

0'7851 0.7242

0-9177 0"8462

In practice, the glass transducer is loaded between flat steel platens, and the required surface displacements in both cylinder and platens are obtained by summating the strains produced by the Hertzian loading along the length under consideration. The necessary integration has been performed by JOrINSON [42], from which the mutual approach, At, of two points on the loading axis, each a distance (R + z) from the cylinder centre, may be shown to be Az = ~

E1

a

0+' 'nez E2

a

jilt

1 -- v

for plane-strain conditions. The corresponding expression for plane stress is obtained by replacing (1 -- v2)/E by 1 / E and remaining terms in v by v/(1 + v). A set of theoretical (plane-strain) displacement curves for a 1.25 in. diameter, 1 in. long, glass cylinder compressed between flat steel platens is shown in Fig, 4, taking El ---- 10.7 × 106 lb/in ~, vl -0.24 for glass, and E z = 30 x 106 lb/in 2, v2 = 0.30 for steel. In order to provide a check on computed displacements, measurements of overall deformation were made on cylinders of various sizes. Each glass cylinder and set of flat steel platens (V = 220) were assembled in a jig to ensure accurate alignment and held in position with silicone rubber. A displacement transducer of the differential inductance type was clamped across the loaded diameter in the manner illustrated in Fig. 5, and used in conjunction with a frequency bridge to give direct readings of displacement. Initial calibration of

THEORY AND DESIGN OF PHOTOELASTIC LOAD GAUGES

,000-

-T

-

i

W

-

i

377

-

% a~

P~ 1000

......

O

0.001

0002

--~

0 003

0 00~,

disp[ocement Z~z - inch

FIG. 4. Plane-strain deformation for 1.25 in. diameter, 1.0 in. long, glass cylinder diametrically compressed between flat steel platens, computed from equation (18). the displacement transducer was carried out using a micrometer frame. The measured l o a d displacement curve shown in Fig. 6 for a 1.25 in. diameter, 1 in. long, glass cylinder is typical of the results obtained. The slight shift in the curves between first and third loading cycles is due mainly to the flattening of surface asperities in the contact area of the platens, and serves to emphasize the need for proof loading all assembled gauges at the time of

,ooo

r

i

,5o0 .

.

.

.

.

.

.

.

! c~

2000 . . . . . o

1st ~Oodiog 1500 L

//•/•

,

3 rd and subsequent ~oading

1008 ~ . . . . i i

500

//

0

0 5

i 10

15

20

2.5

disp~acmment ~'z - inch × 10 3

FIG. 6. Measured and calculated displacements for 1.25 in. diameter, 1.0 in. long, glass cylinder loaded in diametral compression between flat steel platens.

378

J. A. HOOPER

manufacture. Also included in Fig. 6 is the plane-strain displacement curve computed from equation (18), taking z = 0.1 in., and the agreement between theoretical and measured values is more than adequate for the purposes of load-gauge design. 3.3 Elastic and shakedown limits ,/'or steel loading platens The behaviour of glass under the high contact stresses encountered in diametrically compressed cylinders has already been mentioned, but of equal importance to the general stability of photoelastic load gauges is the response of the steel loading platens to the same high stresses. When the load on the cylinder is small, the stresses in the steel will fall within the elastic range. As the load is increased, the steel begins to yield at a point beneath the centre of the loaded surface, and the applied load corresponding to this onset of yielding is termed the elastic limit, Pc. Further increase in load will cause additional plastic flow of the material, and if this load is subsequently removed, residual stresses will remain in the contact zones. If the same load is re-applied, the combination of elastic and residual stresses in the contact region may not reach the yield point of the material. For such a case, the regime is considered to 'shake-down' to an entirely elastic system, and the maximum load at which this takes place is referred to as the shakedown limit, P~. For loads greater than the shake-down limit, cumulative plastic flow occurs. With reference to the elastic limit, the solution for the stress field within normally loaded circular cylinders [43] may be combined with the yon Mises yield criterion to define the onset of yielding. From MERWlN and JOHNSON [44], the elastic limit is given by

['~, == 2-81 al Y.

~!9)

To determine the shakedown limit, Merwin and Johnson considered possible distributions of residual stress in conjunction with elastic stresses in the x - - y plane for the case of rolling contact, and concluded that provided (rxy).... ~ Y/~/3, a residual stress distribution can be found which will eliminate yield. They then introduced Melan's theorem [45] which states that: "if any residual stress can be found which, together with the stress due to the applied load, constitute a system within the yield limit then, under repeated loading, the system will shake down to some system of purely elastic deformation". For the loading in question. (rxy),,~x -----po/4, and hence the shakedown limit in rolling contact is P~- : 3.63 al Y~

(20)

These latter values relate specifically to the Tresca yield criterion, but approaching the problem on the basis of the von Mises criterion gives the same shakedown limit. For the case of repeated static loading to beyond the elastic limit, there is no definitive or conclusive account of the behaviour of the material beneath the contact surface of the indentor. TABOR (loc. cit.) has argued that in the process of cyclic indentation of a metal surface, the deformation remains elastic after the first loading. This conclusion has been questioned by JOHNSON [46] on theoretical grounds, but in subsequent experiments on the plastic deformation of copper spheres and cylinders [47], no evidence of cyclic plastic flow was found. On the other hand, there is some experimental evidence [48, 49] of cumulative plastic deformation after a large number of repeated loads. Hence the question is not entirely settled, but it can be stated that the shakedown limit in static loading is undoubtedly high and approaches the indentation hardness. Values of load corresponding to elastic and shakedown limits for the steel normally used for loading platens (En 57) are listed in Table 5. The onset of flow is assumed to be governed

THEORY AND DESIGN OF PHOTOELASTIC LOAD GAUGES

379

by the primitive yield point of the steel which is taken as 100,000 lb/in 2 in accordance with the minimum values for uniaxial tension quoted in BS970: 1955.* The shakedown limit, on the other hand, represents a condition where plastic deformation will have occurred, and it is more appropriate to use a yield stress associated with the strain-hardened state of the steel. Vickers hardness tests were therefore carried out and gave a mean hardness number of 320 for the En 57 steel, corresponding to an approximate yield stress of 152,000 lb/in 2, and the shakedown limits listed in the Table have been calculated using this value in equation (20). It can easily be shown that in the limiting condition for the static case, i.e. shakedown limit equal to indentation hardness, the ratio of static to rolling shakedown loads is 1.65 ajar, where the subscripts s, r refer to the static and rolling cases respectively. The contact halfwidth during a fully plastic static indentation, as, is much larger than that at the shakedown load in rolling, at, so that the shakedown limits for repeated static loading of the steel platens will be appreciably higher than those listed in the Table. TABLE 5. ELASTIC AND SHAKEDOWN LIMITS FOR En 57 STEEL PLATENS

Glass cylinder dia. (in.)

Load for 5 fringes (Ibf)

Elastic limit (lbf/in)

Shakedown limit (lbf/in)

0"5 1"0 1"25 1"5 2-0

1330 2660 3330 3990 5320

3000 6000 7400 8900 11,900

11,500 23,000 28,500 34,000 45,500

Table 5 thus serves as a guide for determining the maximum working loads which may be transferred by En 57 steel platens to glass transducers of various diameters. Although theory admits the use of loads up to the shakedown limit to retain subsequent elastic behaviour, it is clearly reasonable to consider the elastic limit as the more useful design criterion. In fact the platen loads encountered in practice fall well below the elastic limit, and to illustrate this point the loads required to produce 5 fringes at the centre of soda-lime glass cylinders under transmitted red light (2, = 6450/~) are included in Table 5. Thus the safety factor is approximately 2-2 with respect to the elastic limit, and the inherent reserve against cumulative plastic deformation is more than adequate. 3.4 Stresses in perforated strip of finite width One of the most common types of load-gauge body takes the form of a solid-steel circular cylinder or column containing a transverse hole at mid-height. Pressure applied over the ends of the cylinder causes distortion of the transverse hole, which in turn leads to deformation of the glass transducer located within the hole. The steel body must clearly be able to accommodate the stress concentrations involved, and the deformation of the transverse hole must be known with sufficient accuracy to enable the gauge-body geometry to be determined for any specified gauge sensitivity. * BS 970:1955 is currently under revision; in BS 970: Part 4: 1970, En 57 steel is leplaced by 431S29 steel, but the specified minimum yield stress remains the same.

380

J.A. HOOPER ..............

i

Ks1 (W} K n (W)

2 21b

- - t

.......

1 -r

i .... Ks~ (M) Ks= (M) Ks= ON)

0.1

0.2

0.3

O.t,

0.5

2b

(a) In-plane loading. % 2.0

1-0 0-5

o

u

2

R

1

o

I

o.i

0.2

!

J

i

0,3

0,4

0-5

2b

(b) Transverse bending. FIG. 7. Stress and strain concentration factors for perforated strip of finite width. The presence of a hole in a loaded body gives rise to considerable stress gradients, with maximum stresses occurring at the hole boundary. For a uniaxially loaded plate of infinite width containing a circular hole, the stress concentration factor, Ks, equals _s_ 3.0, the critical points being on the diameter transverse to the loading axis [50]. For column-type load gauges, however, the ratio of hole diameter to column diameter is usually in the O. 30.4 range, which results in stress concentration factors of the order of + 3- 5, based upon the unpierced section. Provided that the transverse hole is sufficiently f a r removed from the cylinder ends, the stresses in the central section of a uniaxially loaded column may be estimated by reference to the plane elasticity solution of HOWLAND [51]; this solution relates specifically to the stresses around a circular hole located at the centre o f a uniaxially loaded strip o f finite width, and the results are summarized graphically in Fig. 7(a). Also included in the figure are theoretical stress concentration values (based upon unpierced sections) obtained by HOWLAND and STEVENSON[52] for the case where a bending moment is applied in the plane of the strip. The corresponding problem for a thick perforated strip

THEORY AND DESIGN OF PHOTOELASTIC LOAD GAUGES

381

in transverse bending has not been solved analytically, but useful results have been obtained from photoelastic studies by GOODIERand LEE [53], DRUCKER [54] and others, and these are represented in Fig. 7(b). In this connexion, FESSLERand ROBERTS [55] used the frozen stress photoelastic technique to determine the bending stresses in a circular shaft containing a transverse hole, and found that the maximum stresses occurred at the hole boundary close to the ends of the hole. However, the resulting stress concentration factor was only slightly greater than the corresponding value determined from Fig. 7(b), and it is reasonable to use the data contained in this figure for the design of column-type load gauges. in practice, the transverse hole containing the glass transducer is provided with two diametrically opposite slots in order to accommodate the internal loading platens, and these slots cause some modification to the basic stress distribution relating to the idealized case of a perfectly circular hole. The extent of this modification was checked by uniaxial loading tests on a perforated slab of acrylic resin (8 in. square, 1 •5 in. thick), accurately machined for squareness. Isochromatic fringes in the vicinity of a centrally placed 1 •625 in. diameter circular hole are shown in Fig. 8(a), and gave a stress concentration factor slightly greater than 3.0. The slab was again loaded to the same value after 0.5 in. wide slots (1.875 in. between flats) had been formed, and the resulting fringe patterns are illustrated in Fig. 8(b). The corners of each slot are seen to lie in zones of relatively low stress, and thus the possibility of these corners acting as significant stress raisers can be disregarded for the purposes of load gauge-design. The introduction of a glass transducer (1.25 in. diameter, 1.0 in. long) relieves the compressive stresses on the diameter transverse to the loading axis and induces compressive stresses in the slot-loading zones, as shown in Fig. 8(c). The magnitude of these compressive stresses depends upon the ratio of the stiffness of the combined cylinder and platen assembly to the stiffness of the material it replaces, and the influence of transducer rigidity upon the calibration characteristics of steel-body load gauges is discussed in a later section. In connexion with the behaviour of gauges loaded beyond their elastic range, THEOCARIS and MARKETOS[56] have provided a useful insight into the elastic-plastic response of a thin perforated strip under uniaxial load. For small values of R/b (hole radius, R; strip width, 2b), plastic deformation commences at the hole boundary on the transverse diameter and continues along approximately 45 ° lines until reaching the edges of the strip. For higher values of R/b, plastic zones develop at both the rim of the hole and the straight boundaries, and converge with increasing applied load. Ultimate collapse will be due to the simultaneous formation of four plastic hinges at the critical sections, and so approximate ultimate loads for column-type gauges can be estimated for any specified yield stress of the material. As photoelastic load gauges may be used under dynamic as well as static loading, the work on dynamic stress concentrations by DALLY and HALBLEIB [57] is relevant. Resin struts containing a centrally placed hole were dynamically loaded, and the photoelastic stress patterns recorded. It was found that for R/b ratios between 0.15 and 0-35, the dynamic stress concentrations at the hole boundary on the transverse diameter were about equal to the static stress concentrations, but that on the longitudinal axis, the dynamic stresses were appreciably higher than those corresponding to static loading. The maximum differences occurred for a strip with R/b = 0.15, where the dynamic stress concentration factor was -- 1.9, compared to the static value of --1.1. For Rib ratios greater than about 0.35, dynamic stresses at the hole boundary along both principal axes were found to be lower than corresponding static stresses. Hence for practical purposes, any modification of the stress field within load-gauge bodies caused by dynamic loading may safely be neglected.

382

3.5

J. A. HOOPER

Deformation of hole in strip of finite width

As mentioned earlier, an accurate assessment of hole deformation is one of the chief requisites in predicting load-gauge sensitivity. Derivation of an expression for the deformation of a circular hole in an infinite plate is relatively straightforward, and has been given by MUSKHELISHV~LI [58]. For uniaxial loading the strain concentration factor, K~, defined as the ratio of the hole deformation along the loading axis to the corresponding deformation with no hole, is equal to 3.0. But this factor will clearly increase for sections of finite width, and a solution for this case is given in Appendix A. In the analysis, the equilibrium equations and Hooke's law are used in conjunction with Howland's stress equations to give an expression for relative displacement along the hole boundary. In polar co-ordinates, ~he change in hole diameter, U, along the loading axis of a strip, width 2b, is given by n/2

E

o

eo cos 0 -- sin 0 --~z/2

~r

(reo) dO

dO

(21)

7t/2

for the case of plane strain, where 0 is measured from the transverse diameter. The relevant equation for tangential stresses is

do

~ . ~n(2n +

% = crz ~- (1 q- cos 20) + 2mo + ~'7 4- 2 ~

L

1) d2, (n -- 1) (2n -- 1) e2, ~o%~-~ 4a,2,

n=l

4- n(2n --

1)/2, w2"-24- (n 4- 1)(2n 4-

1)m2, w 2"} cos 2nO]

(22)

where ~1 denotes the applied uniaxial stress, co - r/b, and d2,, e2,, 12,, m2, are coefficients in the original stress function. Performing the necessary integration and assembling coefficients in the form P2,,/~2,, the hole deformation may be finally written as

w

d~l.(1--v

2)

(

E

l) "+'

(2n -- 1) (2n + 1) (P2, -- P2,)

(23)

rl~0

where d denotes the hole diameter. The coefficients P2,,/32, are listed in Table A1 for various values of ~b (-- R/b), and corresponding values of K,, are included in Table A2 and Fig. 7. In order to provide a check on theoretical displacements, the diametral deformation of transverse holes in column-type steel bodies of various sizes was measured using an inductance-type linear transducer. Arrangements for housing and locating the transducer are shown in Fig. 9. The armature and coil are mounted in a spring-loaded jig which can be positioned at any point along the transverse hole. Rods holding the unit together are then unscrewed to allow free movement of each component. Most measurements were then taken with the transducer located at the centre of the hole i.e. at the intersection of hole and cylinder axes, and in every case experimental results were within 5 per cent o f those predicted by equation (23) for plane-strain conditions. It is interesting to note, however, that a uniaxial compressive load applied at the cylinder ends leads to a slight barrelling of the transverse hole due to the variation in lateral restraint along the length of the hole. Conditions resembling those of plane strain will apply at the central section of the hole. whilst conditions at

THEORY AND DESIGN OF PHOTOELASTIC LOAD GAUGES

383

",.C

1 I

"

i

k,

" \ "

"4

\\

'i

L : ,o:

screwed red m ,a

I

~

c
c~Qmp

625 in dlometer

. ~37g

slotted hote

\" ,

\

"

\,l

steer body

i

1:

.J Ik~I d FnenSiL ?i~ in h~ches Elev~a t ion

Seclion A - A

FK;. 9. Arrangements for measuring deformation of transverse hole in steel body using an inductance-type displacement transducer. the ends will tend to approach those of plane stress. The differential displacement along the length of the transverse hole was measured during the tests, and found to be only slightly less than the theoretical difference between plane-strain and plane-stress displacements. In practice, glass transducers are located at the hole centre, and the small differential deformations along the length of the cylinder (approximately 10 -4 in. at 5 fringes) are readily accommodated and do not materially affect the behaviour of the transducer. 3.6 Stresses and displacements in diametrically compressed ring Whereas column-type gauges are commonly used for lneasurement of relatively high loads, it is generally more convenient to use a gauge body in the form of a thick steel ring or holloxx cylinder for measurement of lower loads. Hence the stresses and displacements in such a body are required for design purposes. A solution for the stress distribution within a ring compressed by concentrated loads acting across an external diameter has been obtained by RIPPERGER and DAVIDS [59], and supplemented by Poeov [60]. The maximum stresses occur at points where the loading and transverse axes intersect the hole boundary, the absolute maximum depending upon the ratio of inner to outer radius, o ( = Ri/Ro). Tangential stresses at r -- Ri may be expressed in the form

K,W % -- 7rLRo

(24)

where the applied load, W, acts across a ring of length L, and the ring stress factors, K,, for the critical points are plotted in Fig. 10. As ring-type load gauges are normally loaded between steel platens, the applied load will be spread over a contact zone of finite width, but the results will be very similar to those given by equation (24). For example, stress factors relating to the case where the applied load is uniformly spread over opposite 15 ° arcs have

384

J.A. HOOPER

been computed by JAEGERand HOSKINS[61 ], and for point 2 in Fig. 10, the factors are up to about 5 per cent lower than those derived on the basis of concentrated loads applied to rings of moderate thickness. The contact widths encountered in practice arc much smailer, so that the relevant stress factors will more nearly approach those pertaining t¢~concentrated applied loads. 60

!

/ 40

.

TW ;

R0

~ /

/

20

~ ~

0

/

/

O~

0.2

[ lcornpre~ive/

0-~

03

0.5

06

0.7

P

FIG. I0. Displacementsand stress factors for circular ring in diarnetral compression(plane elasticityvalues). When a ring is loaded beyond the elastic limit, yield commences at the critical points on the hole boundary and plastic deformation progressively spreads throughout the thickness of the ring. Collapse occurs when the material across the four critical sections becomes fully plastic, and the smallest load corresponding to this condition is referred to as the collapse load, Wf. Using elementary methods of analysis, applied to a perfectly elastoplastic material, HWAN6 [62] found the collapse load of a diametrically compressed ring to be given by 4LMp Wf ~

Rm

(1 -- p)2 -- 2 R o L Y ~

1 -I-- p

(25)

where Rm denotes the mean radius, Mp the full plastic bending moment across the section, and Y the uniaxial yield stress of the material. As the derivation of equation (25) is based only upon bending stresses, the results should normally be confined to problems dealing with thin rings, yet experimental results of SOWERBYet aL [63] on rings in the range 0.5 ~< p ~< 0.75 have shown good correlation between values given by equation (25) and measured collapse loads. Their work also includes details of a more complicated analytical method in which numerical iteration is combined with slip-line field theory to predict collapse loads; an etching technique was used to reveal the shapes of plastic hinges formed at the quarter points of steel rings, and good agreement was obtained between theoretical and measured collapse loads. However, for the purposes of designing ring-type load gauges, where usually p w, 0.4-0-6, values given by the simplified equation (25) are sufficiently accurate: On this basis, the ratio of collapse load to maximum working load varies from about 4 for the thinner rings to about 8 for the thicker ones.

THEORY AND DESIGN OF PHOTOELASTIC LOAD GAUGES

385

Ring displacements can also be estimated using standard 'strength of materials' theory but this is most inaccurate even for rings of moderate thickness [64]. Modified expressions derived by MORLEY [65], in which account is taken of both bending and normal stresses, give far more accurate results and compare well with displacements obtained from the plane elasticity solution of SILVERMANand MOODY [66]. The deduced expressions for displacement are rather cumbersome, but relevant details are given in Appendix B, including a list of numerical values for reference. Elastic solution values are also included in Fig. 10 with denoting the change in hole diameter along the loading axis. As for the hole deformation problems mentioned in the previous section, ring displacements measured using an inductance-type transducer agreed well with computed values over a wide range of ring geometry.

4. C A L I B R A T I O N C H A R A C T E R I S T I C S O F S T E E L - B O D Y L O A D G A U G E S

4.1 Calibration cur~es for OTical column-type and ring-t)Te load gauges Sufficient information on stresses and displacements has been given in the two previous sections to enable the theoretical fringe-order response of photoelastic load gauges to be calculated. In view of the large number of variables involved, computed calibration curves will be confined here to gauges with a 1.25 in. diameter, 1.0 in. long, soda-lime glass cylinder located in a 1 •625 in. diameter slotted hole. The distance between flats of the 0.5 in. wide slots is 1-875 in. and the transducer is taken to be viewed under transmitted light (;~ -- 6450/~). 21.0

! 200

~

r t

q

i

! o ×

160

i

I 120

----

t .........

i o

'/

80

computed

40

1

3 fringe

order

i

!

4

5

1

{ X = 64,50~)

FIG. 11. C o m p u t e d and measured calibration curves for 100 tonf column-type gauge under concentric loading. (Steel c o l u m n : 4.5 in. diameter, 12.0 in. long, 1.625 in. diameter hole; glass cylinder: 1.25 in. diameter, 1.0 in. long; transmitted light.)

386

J.A. HOOPER

Computed calibration curves for a 4.5 in. diameter, 12 in. long, concentrically loaded 100 tonf column-type gauge are shown in Fig. 11. It is normal practice to preload the glass cylinder to about 0.5 fringe in order to fix it in position within the transverse hole, but this preload, which is applied using an adjustable wedge mechanism, may be set above or below this level. In Fig. 1 I, computed calibration curves relating to preloads of zero. 0-5 and ! .0 fringe are shown for comparison. All curves are non-linear because of the corresponding displacement mode of the glass cylinder in diametral compression; in addition, they are not quite parallel due to the relatively high rate of displacement in the contact zones at low applied loads. Also included in Fig. 11 is the measured calibration curve for a load gauge in En 24 steel with a 0.5 fringe preload. The measured curve is in reasonable agreement with the corresponding computed curve, calculated for plane-strain conditions, and cont\~rms to the small differences between measured and computed cylinder displacements shown in Fig. 6. However, calibration curves for a number of similar gauges will always be slightly different due to the effect of normal dimensional tolerances and surface tlatness of the various components. In the present case, for example, the variation for a batch of six gauges was ~ 0.25 fringe at maximum working load. Calibration tests were also carried out on column-type gauges of different shapes in order to examine the effect of various end-loading conditions. Gauges were loaded through flat steel disks having diameters varying between one quarter and three quarters of the column diameter. In addition, tests were performed in which the disks were positioned away fi'om the longitudinal axis of the column. Detailed results are too numerous to inch~de here, but for concentric loadings, the effect of platen size on gauge calibration was negligible provided that the length of the column was greater than about 2-5 times its diameter. For these longer columns, the effect of eccentric loading was to cause a slight shift ,:~f the original calibration curve. The centroid of the applied load was positioned at various points within a concentric circle of radius R/4, which defines the boundary of the kernel c>f the section..At m a x i m u m eccentricity, the effect was to increase the fringe order m the {ransducer by approximately 4 per cent with the centroid of the applied load in the plane containing t!~e axis of the transverse hole, this reducing to about 2 per cent with the load applied in the plane perpendicular to the hole axis. For shorter columns, both eccentric loading and size of the loaded area have a more marked effect upon gauge calibration, and in cases where the use of short gauges is unavoidable, considerable care must be exercised in the design of the steel bodies to ensure that the additional stress concentrations can be adequalely accommodated and are not such as to adversely affect the performance of the gauge. Where measurements of relatively low Loads are required, it is usual practice ~o incorporate the glass transducer in a ring-type steel body. Computation of calibration cmve~ for such gauges follows the procedure for column-type gauges except that the rigiditT, of the transducer itself has to be taken into account. In the case of a 100 tonf gauge, for example, the load taken by the glass cylinder is a small enough proportion of the applied load to be neglected for the purposes of determining gauge body geometry. But for ring-type gauges up to about 20 tonf capacity, the glass transducer offers a considerable proportionate resistance to hole deformation, which in turn has a marked effect upon gauge calibration. Thus an iterative process is used to calculate calibration curves for this type of gauge. Values of fringe order are first calculated on the assumption that no resistance is offered by the transducer, and the cylinder loads corresponding to these fringe orders are then deducted f r o m the applied load to give a net diametral load acting across the ring. A new set of fringe orders is therefore obtained, and the procedure is repeated until the variation between

THEORY A N D DESIGN OF PHOTOELASTIC LOAD GAUGES

387

25

20

? x

==

1

2

3

/,

5

6

fringe order [ ;k = 6Z,50~,)

FIG. 12. Computed and measured calibration curves for 10 tonf ring-type load gauge. (Steel ring: i.d. 1.625 in., o.d. 3.0 in., length 3.5 in. ; glass cylinder: l. 25 in. diameter, 1.0 in. long; transmitted light.)

2oo I

. . . . . . J

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160

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externai ring diameter (in)

35

{Length= 35 in )

3D

2

3

4

5

,

6

fringe order ( k = 6/~50~)

FIG. 13. Computed calibration curves for steel-body load gauges of various shapes. (Constant internal bore ]. 625 in. diameter; constant glass cylinder shape, ]. 25 in. diameter, ! "0 in. long; transmitted light.)

388

J. A. HOOPER

successive values of fringe order is acceptably small. In fact, convergence is quite rapid and sufficiently accurate values are normally obtained after three iterations. Computed and measured calibration curves for a 10 tonf ring-type gauge are shown in Fig. 12, and the agreement is adequate. Five iterations were carried out in this particular case, the successive computed fringe orders at an applied load of 22500 lbf being 6.07, 5.07, 526, 5.20 and 5-21. The extent to which load-gauge capacity is related to steel-body geometry is demonstrated in Fig. 13 for the particular case of constant internal bore (1.625 in.) and constant transducer dimensions (1 -25 in. diameter, 1.0 in. long). Even with these restrictions, it is evidently possible to produce a wide range of load gauges, and by varying some of the other available parameters, this range can be increased. The diameter of both glass cylinder and transverse hole may be varied, and the stiffness of the internal assembly reduced by using platens of curved section. But the simplest and most effective way of modifying gauge sensitivity is to vary the length of the glass cylinder. Equation (4) indicates that for a given diametral load applied to a cylinder, the observed fringe order is independent of cylinder length. When incorporated in a load gauge, however, the cylinder is subjected to a load directly proportional to its length due to the essentially uniform deformation along the length of the transverse hole. Thus by combining this technique with the data presented in Fig. 13, photoelastic gauges can be designed to operate within virtually any specified load range. 4.2 Effect of temperature on load-gauge readings Photoelastic load gauges are relatively insensitive to temperature changes compared with most other forms of load measuring device, but as the coefficient of expansion of soda-lime glass is less than that for steel, change in temperature will produce small but significant 03

computed 2 - 5 fringes 0 - 2 fringes 0.2 # e~

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-10

F ......

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temperature in use

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fringe temperature correction

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FIG. 14. Tempcrature-con:ection curves for 100 tonf column-type load gauge.

(Steel column: 4"5 in. diameter, 12.0 in. long, 1 "625 in. diameter hole; glass cylinder: 1-25 in, diameter, 1 "O in. long; transmitted light.)

THEORY AND DESIGN OF PHOTOELASTIC LOAD GAUGES

389

variations in fringe order which have to be accounted for in practice. Temperature-correction curves for a specific load gauge are shown in Fig. 14, but these are typical of the curves applicable to photoelastic gauges in general. As the stiffness of any cylinder and wedge assembly is non-linear, the precise change in fringe order for a given temperature change depends upon the position of the gauge reading to be corrected in the fringe-order scale. But for the purposes of temperature correction, calibration curves are flat enough to be considered in the form of two linear sections with the break at 2 fringes. Nevertheless, computed values of fringe order are sensitive to values assumed for the linear coefficients of expansion, taken in this case to be 8.7 × 10-6/°C for soda-lime glass, and 12.4 and 10.4 x 10-6/°C for En 24 and En 57 steels respectively [67]. In laboratory tests carried out in order to check on temperature effects, the measured change in fringe order for a given temperature variation was found to be rather less than the computed change, but the overall response of the gauges agreed with that predicted by theory. Hence by using correction curves of the type shown in Fig. 14, fringe-order corrections can be immediately read off for a given difference between initial calibration temperature and the temperature of the gauge in use. 4.3 Long-term stability of photoelastie gauges Perhaps the most important attribute of photoelastic load gauges is their excellent longterm stability. In the majority of applications, these gauges are used in field construction of one sort or another where load changes occur over lengthy periods of time. This is particularly true in civil and structural engineering, for example, where significant variations in applied load commonly occur over a period of years rather than weeks or months. It would clearly be unwise in such cases as these to spend time and effort installing instrumentation of doubtful long-term stability. With photoelastic load gauges, however, laboratory and field experience gained over a period of several years has shown them to be exceptionally stable. One of the principal reasons for this state of affairs is that the gauges contain no 'moving parts' in the conventional sense. All the gauges consist essentially of components of glass and steel which, if properly annealed and not overstressed, are two very stable materials. The basic principle to be applied in aiming for long-term stability is to ensure that the stresses applied to these two materials are kept well within their respective elastic ranges. It also follows that the degree of success achieved in this respect depends to a great extent upon performing a detailed and reliable stress analysis on each gauge component. But provided that these stresses are accurately known, there is every reason to expect good stability. The auxiliary equipment used to read the gauges is, of course, inherently stable as it simply consists of a light source and filters in the form of a standard circular polariscope. In addition, the overall reliability of this equipment is assured as the light source and polarizer are portable and in no way fixed to the gauge, and can therefore be easily replaced if necessary. The selection of soda-lime glass as the most suitable transducer material has already been mentioned, but the choice of gauge body material also has an important bearing upon overall gauge stability. For both column-type and ring-type bodies, the material commonly used is En 24 (1- 5 per cent nickel--chromium-molybdenum) steel,* hardened and tempered to condition T. The minimum specified yield of stress of this steel is 100,000 lb/in 2, which is * BS 970:1955 is currently under revision; in BS 970 : Part 2: 1970, En 24 steel is replaced by 817M40 steel, but the specifiedminimum yield stress remains the same. ROCK 9/3--E

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5.2 Design of 50 tonf tunnel load gauge In designing photoelastic load gauges, it is normally preferable to arrange for the glass transducer to be viewed under transmitted polarized light, but from considerations of access and ease of reading, a reflected light system was adopted for the tunnel load gauges. The gauge is drawn in section in Fig. 16, and an assembled gauge is illustrated in Fig. 17, prior to application of its external protective coating. The gauge is read using a handviewer incorporating a separate polarizer and analyser in which polarized light is bounced off the back

392

J. A. HOOPER

face oftheglass cylinder before entering the analyser. The preload on each glass transducc~ was fixed at about 1.0 fringe, and the gauges read approximately 5.0 fringes at 50 tonf applied load. The arrangements for sealing the internal gauge components from dirt and moisture are also shown in Fig. 16. The cavity at the back of the glass cylinder is filled with silicone rubber, and the opposite end is sealed with O-rings against both the front face of the glass and the side of the transverse hole. Where direct access to gauges is always available, the sealing tube and retaining ring can be replaced simply by a removable screw-cap. The sealing pieces are in cadmium plated mild steel; the gauge bodies are of En 24 steel with chromium-plated counterbores, and the internal platens and wedge mechanisms are of En 57 steel. All external gauge surfaces are coated with a thick layer of an epoxy-type paint for protection against corrosion. In order to accommodate the non-parallel surfaces of the tunnel segments, each gauge is provided with tilting end platens. Applied load is transferred from platens to gauge body through a steel ball located in spherical seatings, with the gaps between column and platens filled with silicone rubber. The anticipated non-parallelism of the adjacent concrete segments was approximately 1°, but a series of laboratory tests in which gauges were loaded through various end platens having tapers of up to 2.5 ° showed that load applied in this way resulted in no discernible departure from the basic calibration curve obtained under concentric loading. 5.3 Location of gauges and measured loads in tunnel lining The positioning of a pair of load gauges relative to the adjoining concrete segments is shown diagrammatically in Fig. 18. These segments were specially cast to give plane instead of convex contacting surfaces, and their lengths also reduced to make room for the gauges. Prior to installation, the gauges were matched in pairs to give an equal overall length, any . . . . . . 2.4 . in.. . . . . . . . .

•4 t i

12in. J

i

i

,

½ Etevation

Section

A-A

FXo. 18. Positioning of 50 tonf load gauges in concrete tunnel lining.

THEORY A N D DESIGN OF PHOTOELASTIC LOAD G A U G E S

393

fine adjustments being made by grinding the end platens. The gauges were installed in four positions, namely at the crown, near the invert, and at axis level on either side. Due to the absence of elaborate auxiliary equipment, the gauges could be installed during normal erection of the rings in such a way as to cause virtually no delay to the construction process. Gauges located at axis level were placed by hand, but gauges at the crown and invert were installed using the normal erection equipment. Each pair of gauges were temporarily bolted to frames designed to be jacked into position along with adjacent concrete segments, the frames being removed on completion of the tunnel ring. The measured variations in hoop-load during the 18-month period following construction of the ring are shown in Fig. 19. The gauges were designed so that their overall stiffness matched that of concrete section to be replaced, thereby ensuring that representative hooploads were measured. As expected, most of the applied load established itself during and shortly after construction of the ring, but subsequent measurements effectively demonstrate the gradual development of pressure on to the tunnel lining over an extended period of time. '

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F]o. 19. Recorded hoop-loads in tunnel lining. Acknowledgements--Part of the work described above was carried out at the former Postgraduate School of Mining, Sheffield University, with the aid of a grant from the Science Research Council. The helpful suggestions of Dr B. POWDaILL on the solution to the hole deformation problem considered in Appendix A are gratefully acknowledged. Ove Arup and Partners have generously supported much of the work and provided computer facilities. The tunnel load-gauge components were manufactured by Messrs Horstman Ltd, Bath, Somerset.

REFERENCES 1. BREWSTERD. On the communication of the structure of doubly refracting crystals to glass, muriate of soda, fluor spar, and other substances, by mechanical compression and dilatation. Phil. Trans. R. Soc. I06, 156-178 (1816).

394

J . A . HOOPER

2. ROBERTS A., HAWKES i. and GILL V. L. Optical control methods applied to prestressing. ,Uiv. /:n.~m~~ publ. Wks Rev. 58, 1111-1115 (1963). 3. HAWKES I., HOOPLa J. A. and ROSE H. Photoelastic Instrumentation in Civil Engineering Structures, Proceedings o f the Conference on Experimental Methods of Investigating Stress and Strai~t h~ S~ruetures, Prague. Vol. II, p. 642 (1965). 4. HOOVER J. A. The Theory and Development of Load Cells incorporating Photoelastie Glass Disc 7)'ans. ducers, Ph.D. Thesis, University of Sheffield (1968). 5. HAWKES I. and HOOPER J. A. Photoelastie load cells for civil engineering structures. ~7~'. /~)tgng pith/, Wks Rev. 64, 660-664 (1969). 6. FROCHT M. M. Photoelasticity, Vol. II, p. 134, Wiley, New York (1948). 7. TARDY H. L. M&hode pratique d'examen et de tonsure de la bir6fringence des verres d'optique. Revue Opt. thdor, instrum. 8, 59-69 (1929). 8. HILTSCHER R., FLORIN G. and STRtNDELL L. Arbeitsanleitung zur spannungsoptishen Messung ebener Spannungszustande mit Polariskop und Lateralextensometer. Bautechnik 43, 3-12 (1966). 9. BROWN A. F. C. and HICKSON V. M. Improvements in photoelastie technique obtained by the use of a photometric method. Br. J. appl. Phys. 1, 39-44 (1950). 10. POST D. Isochromatic fringe sharpening and fringe multiplication in photoelasticity. Proc~ Soc. exp. Stress Analysis 12, 143-156 (1955). l 1. ANDERTON P. and COTTER J. The Dimensional Stability of Plate Glass, Metrotogy Centre Report No. 3, N.P.L. (1969). 12. COKER E. G. and FILON L. N. G. A Treatise on Photoelasticity, 2nd edn, p. 215, Cambridge University Press (1957). 13. WAXLER R. M. The stress-optical coefficient of plate glass. Glass Ind. 34, 258-259 (1953). 14. VAN ZEE A. F. and NORITAKE H. M. Measurement of stress-optical coefficient and rate of stress release in commercial soda-lime glasses. J. Am. Ceram. Soc. 41, t64-175 (1958). 15. SAVUR S. R. On the stress-optical coefficients for direct tension and pressure measured in the case of glass. Phil. Mag. 50, 453-463 (1925). 16. HARRIS F. C. The photoelastic constants of glass as affected by high temperature and by lapse of time. Proc. R. Soc. A106, 718-723 (1924). 17. WRIGHT P. J. F. Comments on an indirect tensile test on concrete cylinders. Mag. Concr. Res. 7, 87-96 (1955). 18. HOLLAND A. J. and TURNER W. E. S. A study of the breaking strength of glass. J. Soc. Glass Technol. 22, 225-251 (1934). 19. ARGON A. S., HORI Y. and OROWAN E. Indentation strength of glass. J. Ant. Ceram. Soc. 43, 86-96 (1960). 20. HOOVERJ. A. The failure of glass cylinders in diametral compression. J. Mech. Phys. SolMs 19, 179-200 (1971). 21. GURNEY C. and PEARSON S. Fatigue of mineral glass under static and cyclic loading. Proc. R. Soc. A192, 537-544 (1948). 22. HOLLAND A. J. and TURNER W. E. S. The effect of sustained loading on the breaking strength of sheet glass. J. Soc. Glass Technol. 24, 46-57 (1940). 23. BAKER T. C. and PRESTON F. W. Fatigue of glass under static load. J. appL Phys. 17, 170.-178 (1946). 24. SHAND E. B. Experimental study of fracture of glass. 1--The fracture process. J~ Am. Ceram. Soc. 37, 52-60 (1954). 25. GURNEY C. and PEARSON S. The effect of the surrounding atmosphere on the delayed fracture of glass. Proc. phys. Soc. B62, 469-476 (1949). 26. VONNEGUT B. and GLATHART J. L, The effect of temperature on the strength and fatigue of glass. J. appl. Phys. 17, 1082-1085 (1946). 27. BAKERT. C. and PRESTON F. W. The effect of water on the strength of glass. J. appl. Phys. 17, 179-188 (1946). 28. BRmGMAN P. W. and SIMON i. Effects of very high pressures on glass. J. appl. Phys. 24, 405-413 (1953). 29. PhiLLIPS P. The slow stretch in indiarubber, glass, and metal wires when subjected to a constant pulll Phil. Mag. 9, 513-531 (1905). 30. JESSOP H. T. On C o m u ' s method of determining the elastic constants of glass. Phil. Mag. 42, 551-568 (1921). 31. MURGATROYDJ. B. and SYKES R. F. R. The delayed elastic effect in silicate glasses at room temperature. J. Soc. Glass Technol. 31, 17-35 (1947). 32. TABOR D. The Hardness of Metals, p. I01, Clarendon Press, Oxford (1951). 33. MARSH D. M. Plastic flow in glass. Proc. R. Soc. A279, 420-435 (1964). 34. SAMUELSL. E. and MULHEARN T. O. An experimental investigation of the deformed zone associated with indentation hardness impressions. J. Mech. Phys. Solids 5, 125-134 (1957). 35. HERTZ H. On the Distribution of Stress in an Elastic Right Circular Cylinder, Miscellaneous Papers (D. E. Jones and G. A. Schott, Translators), p. 261, Macmillan, London (1896).

THEORY A N D DESIGN OF PHOTOELASTIC LOAD G A U G E S

395

36. HONDROSG. The evaluation of Poisson's ratio and the modulus of materials of a low tensile resistance by the Brazilian (indirect tensile) test with particular reference to concrete. Aust. J. appl. Sei. 10, 243-268 (1959). 37. KAKUZEN M., NAKAJIMA N. and HAYASHI S. Mathematical stress analysis in a circular disc under parabolically distributed loads on its circumference. Doshisha Univ. Sci. Engng Rev. 9, 69-83 (1968). 38. ADDINALLE. and HACKETT P. Tensile Failure in Rock-like Materials, Proceedings o f the Sixth Symposium on Rock Mechanics, Rolla, p. 515 (1964). 39. HERTZ H. On the Contact of Rigid Elastic Solids and on Hardness, Miscellaneous Papers (D. E. Jones and G. A. Schott, Translators), p. 163, Macmillan, London (1896). 40. JAEGER J. C. Elasticity, Fracture and Flow, 3rd edn., p. 206, Methuen, London (1969). 41. JAEGER J. C. and COOK N. G. W. Fundamentals o f Rock Mechanics, p. 245, Methuen, London (1969). 42. JOHNSON K. L. Private communication. 43. RADZ~MOVSKYE. I. Stress distribution and strength condition of two rolling cylinders pressed together. Bull. Ill. Univ. Engng Exp. Stn No. 408 (1953). 44. MERWIN J. E. and JOHNSON K. L. An analysis of plastic deformation in rolling contact. Proe. Instn mech. Engrs 177, 676-685 (1963). 45. SYMONDS P. S. Shakedown in continuous media. J. appl. Mech. 18, 85-89 (1951). 46. JOHNSON K. L. Reversed plastic flow during the unloading of a spherical indenter. Nature, Lond. 199, 1282 (1963). 47. JOHNSON K. L. An experimental determination of the contact stresses between plastically deformed cylinders and spheres. Engineering Plasticity (J. Heyman and F. A. Leckie, Eds), p. 341, Cambridge University Press (1968). 48. TYLER J. C., BURTON R. A. and K u P, M. Contact fatigue under oscillatory normal load. Trans. Am. Soc. Lubric. Engrs 6, 255-266 (1963). 49. PINEGINS. W., ORI OVA. W. and GOODCHENKOV. M. Failure of Material under Pulsating Contact Load, Proceedings o f the Second Conference on Dimensioning and Strength Calculations, p. 411, Hungarian Academy of Sciences, Budapest (1965). 50. KIRSC~I G. Die Theorie der Elasticit~it und die Bedfirfnisse der Festigkeitlehre. Z. Ver. dt. Ing. 42, 797 807 (1898). 51. HOWLAND R. C. J. On the stresses in the neighbourhood of a circular hole in a strip under tension. Phil. Trans. R. Soc. A229, 49-86 (1930). 52. HOWLAND R. C. J. and STEVENSON A. C. Biharmonic analysis in a perforated strip. Phil. Trans. R. Soe. A232, 155 222 (1933). 53. GOODIER J. N. and LEE G. H. An extension of the photoelastic method of stress measurement to plates in transverse bending. J. appl. Mech. 8, 27-29 (1941). 54. DRU('KER D. C. The photoelastic analysis of transverse bending of plates in the standard transmission polariscope. J. appl. Mech. 9, 161-164 (1942). 55. FESSLER H. and ROBERTS E. A. Bending Stresses in a Shaft with a Transverse Hole, Proceedings o f the Conference o f the Institute o f Physics on Stress Analysis, Delft, pp. 45-49 (1959). 56. THEOCAR~SP. S. and MARKE'rOS E. Elastic plastic analysis of perforated thin strips of a strain-hardening material. J. ~Iech. Phys. Solids 12, 377-390 (1964). 57. DALLY J. W. and HAL~LE~B W. F. Dynamic stress concentrations at circular holes in struts. J. mech. Engng Sci. 7, 23-27 (1965). 58. MUSKHEI~ISIlVIHN. I. Some Basic Problems o f the A~Iathematieal Theory o f Elasticity (J. R. M. Radok, Translator), 3rd edn, p. 205, Noordhoff, Groningen (1963). 59. Rn'P~RGER E. A. and DAVIDS N. Critical stresses in a ci~cular ring. Proc. Am. Soc. cir. Engrs 72, 159-168 (1946). 60. Popov E. P. Discussion on paper by E. A. Ripperger and N. Davids. Proc. Am. Soe. cir. Engrs 72, 1291-1294 (I 946). 61. JAEGER J. C. and HOSKINS E. R. Stresses and failure in rings of rock loaded in diametral tension or compression. Br. J. appl. Phys. 17, 685-692 (1966). 62. HWANG C. Plastic collapse of thin rings. J. aeronaut. Sci. 20, 819-826 (1953). 63. SOWER~Y R., JOHr~SON W. and SAMANTAS. K. The diametral compression of circular rings by 'point' loads, hlt. J. mech. Sci. 10, 369-383 (1968). 64. BLAKEA. Deflection of a thick ring in diametral compression by test and by strength of materials theory. .L appl. Mech. 26, 294-295 (1959). 65. MORLEY A. Strength o f Materials, l l t h edn, p. 421, Longmans (1954). 66. SILVERMAN I. K. and MOODY W. T. Displacements in closed circular rings subject to concentrated diametral loads. J. Franklin Inst. 279, 374-386 (1965). 67. WOOLMAN J. and MOTTRAM R. A. The Mechanical and Physical Properties o f the British Standard En Steels, Vol. II, p. 72, Pergamon Press (1966); Vol. II1, p. 257 (1969). 68. EVANS U. R. and RANCE V. E. Corrosion and its Prevention at Bimetallic Contacts, 3rd edn, H.M.S.O. (1963).

396

J. A. HOOPER

69. HOOPER J. A. Apparatus for applying sustained loads to large specimens, lnt. J. Rock: Mech. N[#~. ?.~i 4, 353-361 (1967). 70. MUIR WOOD A. M. Contribution to Main Session 4; Deep Excavations and Tunnelling in Soft Ground, Proceedings of the Seventh International Conference on Soil Mechanics and Foundation Engineering, Mexico, Vol. III, p. 363 (1969). 71. SMYTH-OsBOLIRNEK. R. Discussion on paper by A. M. Muir Wood and F. R. G i b b Proc. lnstn ci~,. Engrs 50, 190-196 (1971). APPENDIX

A

Deformation of a Circular Hole in a Uniaxially Loaded Strip of Finite Width The problem considered here is that of determining the deformation of a circular hole located mid-way between the parallel edges of a long strip under uniaxial load. In this analysis, attention is given solely to deriving an expression for the hole deformation along the loading axis of the strip. A complete analytical solution for the elastic stress distribution in the vicinity of a circular hole in a strip under tension has been obtained by HowLayo [51]. Stress equations are given in the form of infinite series, which may be considered as rapidly convergent when the hole diameter is not greater than half the width of the strip. The problem therefore reduces to one of formulating a method by which these stress equations can be integrated to give the required hole deformation. Consider a strip of uniform thickness bounded in the (x, y) plane by the lines y = ± b, and containing a hole of radius R (diameter d) with its centre located at the origin. Using polar co-ordinates (r, 0), it is convenient to take the initial line along the x-axis. At any point in the strip, the deformation ux in the x-direction is related to the radial and tangential displacements u and v by ux = : U C O S 0 -

vsin

0.

Hence the relative displacement of any point on the hole boundary is given by o

tux]°°

o

: e0 d0 = 0

.)cos0- (u +

)sin0] d0.

(A.1)

o

The strain components in polar co-ordinates are ~u

~' = ~r'

1 ~v

"° = r ~

u

+,(A.2)

± (3.

or)

7,o = 2r\c~0 -- v + r ~

.

Hence 0

~.v COS O - rEo sin 0] dO. [Ux]oO = f [(2ry, o -- r-~-r)

(A.3)

0

Also, a s v = 0 w h e n 0 = 0 o

o

(A.4) O

0

In plane strain l÷v

e, = ~

[(1 - - v) or, - - uOo]

(A.5) l+v ~o = - ~

[(1 - v) % - v~,].

THEORY AND DESIGN OF PHOTOELASTIC

LOAD GAUGES

397

Therefore

(r%)-

~,

==

%-- ¢,÷r

de0

(1 + ~)

&

E

r(1 + v ) [

(%--cry) + ~

~a 0

c%,].

(1 -- v ) ~ - -

v ~rJ

(A.6)

This equation may be simplified by observing that at the hole boundary, the equilibrium equations give (A.7)

~F Hence, working only on r = R l+v[

-~r ( r e ° ) - e, = - - ~

l--v 2 ' g%k &ro ] e o - b R(I -- v) ~---~-- v% , : - ( % -J. R ~ r )

l--v2[~

"-E

]

~.r , (r%)

r=R"

(A.8) If the change in hole diameter along the loading axis is denoted by U, then with the condition ~',o = e, = 0 at the hole boundary

U=

[u:,]o '~ ==-- R . 0

¢o s i n O + ~ c o s O

i [oos i n 0 - t

SE-

dO=

~ 0

0

.t

c o s 0 " c~ 0

d0] dO (A.9)

In order to make direct use of Howland's stress equations, in which the initial line is taken along the y-axis, equation (A.9) must conform to the same notation. With 0 now measured clockwise from the y-axis, the expression for hole deformation becomes

n/2 U~ R ~ (1-

v z)

o

f [a0cos0--rU2

sinOf ~(r%) dO] dO.

(A.10)

~/2

A n equivalent expression for the hole deformation under plane stress conditions is obtained by replacing (1 -- v2)/E by 1/E in equation (A.10). In order to evaluate equation (A.10), recourse is made to Howland's equation

a0

=

Crl[½(1+ COS20)+ 2 m o - ~ -

+ n(2n--1)lz,

do

~

+ 2

2 I~'n(2nco+2"+21) d2.

n=l

1)(2n -t l)m2,,~oz"}cos2nO]

oJ2"-2 + ( n +

(n -- 1) (2n -- 1) e2, ¢x)2n

(A.II)

where al denotes the applied stress, oJ -- r/b and d2n, ezn, 12,, m2n are coefficients in the corresponding stress function. At the hole boundary, with ~b = R/b, equation (A.11) may be written in the form C~o = or1 (Po + P2 cos 20 ~ p,, cos 40 '- . . .) where

Po = ½ + 2 m o + ~ 7 6d2

do

P2 = ½ -]- " ~ + 212 "- 12m2

Sz

20d4 6e4 ~bz 4~4 p4 = 7 + ~ z 4- 1214 + 30m4 42(/6 20e6 ~b4 ~6 P6 = " 7 @ ~ - _t_ 3016 + 56m6 72ds 42es ~b6 ~bS. P8 = - ~ + - 7 + 5618 + 90m8

(A.12)

398

J.A.

HOOPER

Multiplying equation (A. 11) by co and differentiating

c5-~(°Ja°) = or1 ~(1 + cos 20) + 2too -- ~-~ + 2

(n -- 1) (2n .... ~)2 e2.

o~211+2

Oj2 ~

n=l

+ n(2n

I

1) 2

12.

co2n- 2

] + (n + 1)(2n + 1) 2 mz. co2"? COS 2nO].

This may be written as co

~-~ + 2

t

,o 2"+2

+

a,""

n= t

+ n(2n -- 1) [z,, ~o2"-2 + (n + 1) (2n + 1) m2. co

]

cos 2nO

where d2. 62. i2. m2.

=-(2n+ 1) d2. = --(2n -- 1) e2. = ( 2 n - 1) t2. = (2n + 1) m2..

Hence at the hole boundary 8 8-'-~(°Ja°) = ~l(Po +/52 cos 20 +/54 cos 40 + . . . )

(A. 13)

where the coefficients fiz. are obtained by inserting the above modified values o f stress function coefficientS in the expression forp2.. Howland's values o f p2,, together with c o m p u t e d values o f P2., are listed as a function o f ~bin Table A1.

TABLE A1.

¢=0.1 Po P2 P4 P6 ps /50 /52 /5~ /56 /ss

COEFFICIENTS IN EQUATIONS

tb = 0 . 2

1.00 2.03 0 0 0 0 -4"06 0 0 0

(A.12) and (A.13)

¢=0.3

l -01 2-12 0.01 0 0 0 -4"25 -0"03 0 0

1.02 2.30 0.04 0 0 0 .....4"65 --0"17 - 0' 01 0

,tb = 0 . 4

¢=0.5

1.04 2.56 0.13 0.01 0 0 --5"28 --0'54 --0"08 --0"01

t .06 2.9l 0.30 0.04 0'01 0 - 6'26 - 1 "39 - - 0 ' 30 --0'04

The hole deformation can now be written as n12 do" 1 . .

p2cos20+p4cos40+

...}--

sin0

0--

o o

40 +

))

f -~12

(A.14)

THEORY A N D DESIGN OF PHOTOELASTIC LOAD G A U G E S

399

Using the relationships .

1"cos mO cos nO = ½ [sin(m - n)O sin(m + n)O] L m--n+ m:+75-j [ s i n ( m - - n)O ½ t m---n-

,I sin mO sin nO =

sin(m + ,)0] m ~- -tl a

gives

d.

U=~(1

[

,

-- v2) (Po--/So) +5(Pz--/52)--

(p4--/L,) ?~-~(p6--/36) --~~(Ps --/Ss)÷

....

]

(A.15) The required hole deformation in plane strain may be finally written in the form m

dat y U = --if- (1 -- v2) ~

=

(-- 1)"+1 (2n - - l i ( ~ n + 1) (P2.

-

-

(A.16)

tO2n) •

The infinite series in equation (A.16) is rapidly convergent for ~ ~< 0.5, and taking the first five terms only gives an accuiacy which is adequate for most purposes. Numerical values corresponding to this summation are listed in Table A2. TABLE A2. STRESS A N D

STRAIN CONCENTRATION

FACTORS FOR PERFORATED THIN STRIP IN TENSION

~b=0

4,=0.1

~b - 0-2

~b = 0.3

~b = 0.4

~b-- 0.5

K,

3.00

3.03

3.14

3.36

3.74

4.32

K,

3.00

3.03

3" 13

3.32

3.61

4.01

In conclusion, it is useful to compare numerical values derived from equation (A.16) with the known results for a circular hole in a uniaxially loaded infinite plate. In this latter case, the original equations of KIRSCH [50] give a maximum stress at the hole boundary of 3al, while the hole deformation along the loading axis has been given by MOSKHELISHVILI [58] as

3dez

/-7o = - - ~ ( 1

-- v2).

(A.17)

It would therefore seem reasonable as a first approximation to assume that for a plate of finite width

UE dcrt(l -- v2)

%(max) - %

(A.18)

017

K~ ~ K~ where K~ and K, denote the stress and strain concentration factors respectively. Table A2 shows that the two factors have similar values, but diverge as ~ increases. With q~ = 0-5, for example, Ks is about 8 per cent greater than K,.

APPENDIX B Deflexion of a Thick Circular Ring in Diametral Compression Of particular interest in the present context is the determination of the change in length, 8, of the internal diameter of a closed circular ring subject to diametral compression. Concentrated line loads, W, are considered to act upon the external boundary over the length of the ring, L, and the displacement required is that along the loading axis. The following two solutions to the problem are relevant.

400

J . A . HOOPER

Strength o f materials solution

It has been well demonstrated by BLAKE [64] that ring deflexions computed on the basis of standard 'strength of materials' solutions become highly inaccurate as the geometry assumes 'thick ring' proportions. However, MORLEY [65] has modified the simple curved-beam theory, taking account of both bending arid normal stresses, and gives the expression 2 WRm 3

= ~

A

-~- - -

(]['}. 1)

8 - - 7rA'~ ~ rrEA'L

where A denotes the cross-sectional area, Rm the mean radius, 1 the second moment of area, p the ratio of inner to outer radius, and A' = R,,L In 0 for a ring of rectangular section. For a given applied load and elastic modulus, E, the value of ring deflexion is dependent only upon p, and equation (BA) can be written in the form

~ec

3 i l + p t 3[~

2W

2 \i--pl

2(1-p)

]

1

8 - ,,(--]-,+ p) In (1/0)J -t- (1/p-------~)" ,, In

(B.2)

Theory o f elasticity solution

Using the complex variable method, SILVERMANand MOODY [66] have obtained a complete elastic solution for the stresses and displacements in an annulus subject to diametral concentrated loads applied to its outer boundary. In particular, the theory shows that displacements of points on the inner (unloaded) boundary are independent of Poisson's ratio. With some rearrangement of the original expression, the required displacement is given by l n \(1 l _ -+- - ~o]! - - 2 o +' 2_ ( 1_ - - p2)

=4

~EL

2W

o" A*. n=3.5,7..,

2

n~3,5,7...

p~-," A * ~ _ .

]

(B.3)

where 1

A*, = ~-ff,, [p2"(l -- 04-2" ) + n(2 -- n) (I -- pz)2 _ n(1 -- p2)] D, = (p4-2n __ 1) (1 -- p2,) __ n(n -- 2) (1 -- p_,)2 1

A*2-~ -- (n -- 2) D 2 - , [(n -- 2) (p2 _ 1) -- (1 -- p2,)] D : - , = (p2, _ 1) (1 -- p4-2,) + n(2 -- n) (1 -- p2)2. The variations of 8EL~2 W with p represented by equations (B.2) and (B.3) are shown in Fig. BI and selected numerical values listed in Table BI for reference. Hence the values given by Morley's approximate solution deviate from the exact solution values (taking n = :t: 41) by less than 10 per cent in the range 0.25 < p < 0.65; for very thick rings, with p < 0-25, the discrepancies become somewhat larger. TABLE B I .

VALUES OF

~EL/2W IN

EQUATIONS

(B.2) AND (13.3) p

Equation (B.3)

Equation (B.2)

0"10 0-15 0"20 0"25 0-30 0"35 0"40 0-45 0-50 0-55 0'60 0-65 0"70

0'3302 0-5188 0-7383 1"0047 1-3403 1'7778 2-3685 3-1952 4.3984 6"2286 9'1596 14-1544 23'3569

0.5942 0.7052 0'8508 1.0433 1-3018 1"6568 2"1565 2"8803 3"9645 5-6535 8'4121 13'1893 22'1060

THEORY AND DESIGN OF PHOTOELASTIC LOAD GAUGES 19

i

I

l .... (

~+--

....

~w

:. . . . . . . . .

-

+

] ......

71

"i JI

. . . . . .

i

1

;

ILl

(S[LVERMAN and

+

MOODY

rG~i ~"//

+~'+

+

!.//: .....

+ //. I

i

--i--ZJ 0.1

i

i

+o+~+,°,° sotution (MORLEY:65~,)

/-.'I

S J

i

!i !

"

-i

:

-/

i

I

i

5

0-6

0.7

,

,

,

0.2

0.3

0~,

0

FIG. B1, Displacement of elastic circular ring in diametral compression.

401