AXIALITY OF LOADING IN THE TENSILE TEST

C . GRANT

Department of Mechanical Engineering, University of Newcastle upon Tyne

Axiality of loading is very important in the tensile test, particularly under creep conditions. An experimental evaluation of bending of the test section using resistance strain gauges has been devised. This test showed that excessive bending was present in spite of the adoption of a coupling design recommended in the literature. The cause of bending was found to be solely due to a friction moment exerted by the pin joints which formed the essential feature of the couplings. A crossed knife-edge design was substituted for them with satisfactory results. Analysis of the pin joint showed it to be unsuitable for specimen alignment in the tensile test.

1 INTRODUCTION

BY FAR THE MAJORITY of experimental data regarding the strength of materials continues to be accumulated by means of the uniaxial tensile test. Such data are usually expressed in terms of parameters related specifically to the tensile test, indeed, design codes of practice in this country and abroad are often expressed in these terms. Imperfections in practical tensile-test machines have previously been subjected to close scrutiny by many experimenters to ensure that material data would be largely independent of the test machine. During the past two or three decades, a great deal of attention has been focused on materials operating at high temperatures where time-dependent deformation or creep is the primary consideration. Standard test machines are often employed to study such behaviour, with little modification except for the possible addition of a furnace to surround the testpiece. The additional parameters of time and temperature in the creep test inevitably cause the constitutive relations for creep to be extremely complex. It is not surprising that no satisfactory general theory of creep behaviour yet exists. The result is that both the designer and materials scientist require very many data collected over very long periods of time. The practical answer has all too often been to provide banks of tensile-creep machines operating simultaneously. The situation is well illustrated by plate 12 in Fenner's book 011 mechanical testing (I)*. Kennedy (2) observed that 'a single test machine of high sensitivity would provide a great deal of illuminating information more quickly than a large number of identical, expensive machines often employed over very extended times.' Design of the creep-test machine may influence both temperature and stress distributions in the testpiece, both of which are very important parameters affecting creep behaviour. These considerations have been more fully discussed by Smith, Grant, and Booth (3) in a paper dealing with an experimental project at Newcastle University involving creep testing under variable load and at variable temperature, of which the present work forms a part. Axiality of loading is of particular importance in such tests, which are designed to determine the transient reThe M S . of this paper was received at the Institution of Mechanical Engineers on 3rd August 1971 and accepted for publication on 28th January 1972. 33 * References are given in Appendix 2. JOURNAL OF STRAIN ANALYSIS VOL

7

NO

4 1972

sponse of the material under test. Load is transmitted to the testpiece by a precision-built universal coupling, and eccentricity of the coupling relative to the testpiece may introduce flexure into the test section. Some of the geometric factors affecting the eccentricity and their effects upon tensile-test results have been discussed by Penny, Ellison, and Webster (4), although the results claimed for the recommended universal block system were far from realized in the Newcastle tests. As a result of these discrepancies, bending of the test section has been investigated experimentally by use of resistance strain gauges. Results are compared with predictions based upon a simple coulombic-friction model of the behaviour of pin joints under load. 1.1 Notation a Pin radius. d Test-section diameter. e Eccentricity of load. ex x component of e. eg y component of e. F Tensile force acting on specimen. M Limiting friction moment on pin under load. p ( 0 ) Distributed pressure on pin. p o Modulus of pressure on pin. u Strain measured by gauge 1. v Strain measured by gauge 2. wStrain measured by gauge 3. x In-plane co-ordinate direction associated with gauge 1. In-plane co-ordinate direction normal to x. y cc In-plane angle where bending is a maximum. Surface strain at angle 8. e0 eB Bending strain. eu Direct strain. 0 General in-plane angle. p Coefficient of coulombic friction. 2 T H R E E - G A U G E EVALUATION O F BENDING IN T H E T E N S I L E - T E S T S P E C I M E N

T o ensure that testpiece couplings perform in accordance with design criteria, a room-temperature experimental determination of bending in the test section has been devised. Three electrical-resistance strain gauges are attached to the centre of the test section, spaced at 120" intervals around the circumference as shown in Fig. 1, with their sensitive axes parallel to the testpiece axis. 26 1

C. GRANT

equation (6) and in terms of the components in equations (7): d €B . . . . . . e=-(6)

yl

ED

ell = -823~;

I

Equations (7)plotted in Cartesian co-ordinates gives the trajectory of the centroid of the test section as the load is varied. The limiting free position of the specimen centroid is given when the load tends to zero. As the load increases, the flexibility of the specimen string causes the centroid to traverse towards the zero offset position as the specimen string tends to ‘straighten out’. Fig. I . Disposition of strain gauges around the circumference of the test section For a given load, let the gauges read u, w and w . Provided that plane cross-sections of the test section remain plane during application of the load, the three gauge readings completely determine the strain distribution in the cross-section. Let the principal bending axis be inclined at 8 = a+r/2. (8 is measured from gauge 1 and is positive in the direction of gauge 2 as shown in Fig. 1.) By definition the surface strain at this position is zD, whilst at 8 = a, the surface strain is ED+EB. In general, the strain will vary around the circumference in accordance with equation (1). €0

. . .

= ED+EBcos(~-~)

3 INVESTIGATION OF THE C A U S E S OF BENDING IN THE TENSILE T E S T

A tensile-creep machine has been constructed at Newcastle University for the purpose of conducting tests at

El

,Pull

rod

O u t e r yoke

(1)

/r

Insertion of the appropriate values of z0 and 0 into equation (1) allows eD and eBto be identified in terms of u, o, and W:

u = ED+EB cos CL ZJ

EB

= q,--

w = cD--

2

EB

2

I

(cos a - d 3 sin a) (cos a+ 4 3 sin a)

Equations (3) and (4) follow immediately from a rearrangement of equations (2) to give zD and cB explicitly in terms of the strain-gauge readings : u+v+w

ED

=3

EB

=

. . . . . . .

4 2 [(u-v)2+(o-w)2+(w-#)2]1’2

*

(3)

.

(4)

The angular position a of the maximum bending strain may then be obtained from equations (2), (3), and (4) as @-w> sin CL = d3rB

cos a =

}

(224- w -w )

. . .

(5)

3EB

Let e be defined as eccentricity of loading, with components e,, ey in the x and y directions. Then, if the test section remains elastic, the total eccentricity is given by

262

H U

Specimen

Fig. 2. Original arrangement of specimen and couplings in the tensile-test machine JOURNAL OP S T R A l N ANALYSIS VOL

7

NO

4 I972

AXIALITY OF LOADING I N THE TENSILE TEST

I

Test 4

+y

t\ 1 ,

0.7 Test 2

? Free hanging

, (normal) I

Test 3

t* + x

1:

-0.7 -0.8

Test 6

-Y

F&. 3. Trajectories of the specimen centroid during loading variable load and temperature (3). A universal-block type of coupling was used to transmit load to the testpiece. This is essentially a hybrid coupling consisting of a pinjoint and crossed knife-edge as shown in Fig. 2. Specimen construction is such that the upper pin-joint axis lies parallel with the lower knife-edge, whilst the lower pinjoint axis lies parallel with the upper knife-edge. Components in the load string were manufactured to close tolerances and selectively assembled to give an anticipated axial alignment of load within 0-05 mm (0.002 in), equivalent to 5 per cent bending on a testsection diameter of 8 mm. Bending of the test section under load was determined by the three-gauge test described in the previous section. The strain gauges used in this test were i n x i in Saunders-Roe foil gauges of 120fl Q resistance, gauge factor 2.16f0.1 per cent. Each gauge formed a single active arm of a Wheatstone-bridge network. Phasesensitive Sanborn amplifiers (Model 8805A) operating with a carrier frequency of 2.4 kHz were used to amplify and demodulate the signals which were all automatically recorded on punched paper tape by a Hewlett-Packard Model 2010D data-acquisition system. Results of the tests were analysed directly by digital computer. Fig. 3 shows the trajectories of the test section during loading, referred to the axes defined in Fig. 1. Loads of up to 200 MN/m2 were applied in approximately equal increments of 10 MN/m2. The first two test results show the trajectories with the

+

JOURNAL OF STRAIN ANALYSIS VOL

7

NO

4 I972

specimen turned through 180" about its own axis relative to the end fittings, and in the normal position. Similar results for these tests would indicate that the specimen ends were not aligned with the gauge length and a reversal of the bending direction would indicate that the end fittings were misaligned. Fig. 3 shows that neither of these things happened in practice; since the amount of bending was very large (72 per cent bending, equivalent to 0.72 mm eccentricity for an 8 mm gauge diameter) the causes were investigated further. The excessive bending of the specimen was tentatively attributed to coulombic-friction torque at the two pinjoints (see Fig. 2). A simple analysis of friction in the pin joints (see Appendix 1) indicated quantitatively that this would most likely be the case. The limiting friction moment was found to be proportional to the applied load, under the assumptions of the analysis. In these circumstances the joint should lock into whatever orientation it has at the time the first load increment is applied, provided that the misalignment is such that the limiting friction moment is not exceeded. This hypothesis was tested experimentally by applying a lateral force during application of the first load increment, and then continuing the test normally with the lateral force removed. Several tests were performed, each with a lateral force applied in a different direction. These results are also shown in Fig. 3. When the upper end of the specimen is pushed in the + x direction, the upper pin locks in this orientstion as the load is applied. The load passes through the corresponding 263

C. GRANT

lower knife-edge causing compressive bending on the +x side of the specimen. Conversely, when the upper end of the specimen is pushed in the +y direction, the lower pin joint locks and the upper knife-edge defines the line of action of the load, causing tensile bending on the +y side of the specimen. On average, the results of the last four deliberately misorientated tests show that friction in the pin joint is capable of sustaining an eccentricity of approximately 0.8mm, similar to that predicted in Appendix 1. Furthermore, additional tests show that the eccentricity can be further increased by increasing the initial load increment. The simple coefficient of friction used as the basis for the predictions in Appendix 1 is therefore slightly dependent on the load. The results given in Fig. 3 are for an initial stress of 5 MN/m2 (0.3 ton/in2), this is the lightest load increment to give acceptable results within the limits set by the instrumentation. The trajectories of the specimen centroid (Fig. 3) are seen to be equally disposed around the origin. This shows conclusively that the excessive load misalignment that occurs with this coupling arrangement is due entirely to friction in the pin joints. It is also clear that pin joints of conventional design have no place in precision alignment couplings.

Specimen couplings for the Newcastle tests were redesigned as crossed knife-edges, which may be thought of as pin joints with a very small effective pin diameter. This causes a reduction in the friction moment. The modified couplings, shown in Fig. 4, were again manufactured to give load alignment within 0.05 mm (0.002 in). A threegauge experimental evaluation of elastic bending at room temperature confirmed that this degree of alignment had actually been achieved in practice. The overall eccentricity of the modified specimen and couplings is shown plotted to a base of direct strain in Fig. 5. Test 2 on the original couplings is also shown in this form for comparison. These results are clearly better by an order of magnitude than those for the original design. They show an estimated eccentricity of approximately 0.1 mm, equivalent to 10 per cent bending at zero load. No significant variation in these results was observed when small lateral forces were applied during the first load increment. Friction in the knife-edges is therefore insignificant in its effect on axiality of the couplings. The eccentricity has in fact been traced to an adverse accumulation of manufacturing tolerances, confirmed in both magnitude and direction. 4 ERRORS IN ECCENTRICITY ESTIMATES

B

Pull r o d

Knife edge

Calculations of eccentricity are based upon differences between the individual strain-gauge readings, and small relative errors between readings, such as those due to amplifier drift, greatly affect the result at small values of direct strain. Each strain gauge and associated bridge circuit and amplifier were cross-calibrated to minimize consistent relative errors between readings. If an absolute standard error on readings of f1 per cent f l o p strain is assumed, and a tenfold increase in relative accuracy results from the cross-calibration procedure, then the relative standard error would be f0.1 per centf 1p strain. If this error is further assumed to be random and normally distributed, the error in eccentricity is likely to fall within the range f~/5(0.1+10-6/cD)mm. The I .o

fi$$j;% Co-planar k n i f e quide

Knife-edqe support

0

M o d i f i e d c o u p l i n g design

v Original coupling design

support

Specimen

0

Fig. 4. Modified arrangement of specimen and couplings 264

200

400 600 D I R E C T STRAIN p

800

I( I0

Fig. 5. Comparison of eccentricities obtained with original and modified arrangements JOURNAL OF STRAIN ANALYSIS VOL

7

NO

4 I972

AXIALITY OF LOADING IN T H E TENSILE T E S T

error bands shown in Fig. 5 are based upon these assumptions. 5 ACKNOWLEDGEMENTS

Thanks are due particularly to Dr E. M. Smith and Mr J. M. Booth, both colleagues in the Department of Mechanical Engineering, University of Newcastle upon Tyne, for helpful participation in this work. Thanks are also due to Mr A. Warren of the Departmental Technical Staff for the skilful manufacture of the components in difficult materials. The work has been performed under direct grant from the Science Research Council (Grant No. B/SR/3960) whose support is gratefully acknowledged.

Fig. 6. Distribution of pressure around the pin

APPENDIX 1 F R I C T I O N I N A P I N J O I N T USED T O T R A N S M I T LOAD

Let the total load transmitted by the joint be F, and let the pin diameter be 2a. The normal interface pressure between pin and housing will in general be distributed around the circumference. For this analysis the pressure is assumed to be distributed according to the following law :

p = p , cos 8,

n

n

-2 < B < 5

p is zero around the remainder of the circumference, and p o is the peak interface pressure which is dependent upon F. Fig. 6 shows this arrangement diagrammatically. The force F is found in terms of p , by integrating the vertical component of pressure around the circumference. ni2

F = / - = , . p a cos 0 d0 =

Note that the constant 4/n is dependent upon the assumption of sinusoidal pressure distribution around the pin. The more severe assumption of a uniform pressure gives a similar result with a constant of n/2, a? increase of approximately 23 per cent in the limiting friction moment. The limiting friction moment can be expressed as an eccentricity of loading simply by dividing the moment M by the force F. For the Newcastle tests, the anticipated eccentricity of the joint would be 1.14-1.67 mm, a roomtemperature friction coefficient for the Nimonic series of alloys in the range 0 . 1 5 4 2 2 being assumed in accordance with Betteridge (5). The eccentricity measured at the specimen centre would then be 0 . 5 7 4 8 4 mm, which is equivalent to between 57 and 84 per cent bending. Under creep-test conditions the lack of lubrication at the high operating temperature could give rise to a fourfold increase in the friction coefficient accompanied by a corresponding increase in the eccentricity of loading.

7T

= ZPoa

APPENDIX 2 REFERENCES

Hence 2F nu If a simple coulombic friction coefficient p is assumed, the limiting friction moment on the joint can similarly be found. Limiting friction moment

Po

M =

=--

j'" ppa2 d6' = pp,a2 /:22 -n 2

cos 6' dB

4

= -uaF

FENNER, A. J. Mechanical testing of materials 1965 (George Newnes Ltd, London). (2) KENNEDY, A. J. Processes of creep and fatigue in metals 1962 428 (Oliver and Boyd, Edinburgh). (3) SMITH,E. M., GRANT,C., and BOOTH,J. M. 'Equipment for creep testing at variable load and temperature', J. Strain Analysis 1970 5, 145. (4) PENNY, R. K., ELLISON,E.G., and WEBSTER, G. A. 'Specimen alignment and strain measurement in axial creep tests', Mater. Res. Stands 1966 6 , 76. ( 5 ) BETTERIDGE, W. The Nimonic alloys 1961 (Edward Arnold Ltd, London). (I)

n'

J O U R N A L O F STRAIN A N A L Y S I S V O L

7

NO

4 I972

265