Technical Note

Variable material property method in the analysis ofcold-worked fastener holes

H Jahed1�, S B Lambert2 and R N Dubey2

1School of Mechanical Engineering, Iran University of Science and Technology, Tehran, Iran2Department of Mechanical Engineering, University of Waterloo, Ontario, Canada

Abstract: Based on a general axisymmetric method of elastic±plastic analysis presented by Jahed and

Dubey, elastic±plastic boundaries and residual stress fields induced by cold expansion of fastener holes is

predicted. The method uses a linear elastic solution to construct an elastic±plastic solution. The material

parameters are treated as field variables and their spatial distributions are obtained as part of the solution.

This method uses the actual loading±unloading behaviour of the material and therefore is capable of

predicting an accurate residual stress field. Results obtained here are compared with available experimental

and finite element results. The agreement of the results with experimental measurements is very good. It is

shown that employment of the actual unloading material curve can make a significant change in residual

field prediction.

Keywords: variable material property method, cold working, fastener holes

1 INTRODUCTION

The variable material property method of axisymmetric

elastic±plastic analysis [1] is capable of predicting the

residual stress field induced by fastener hole expansion.

This method is able to consider:

(a) elastic±perfectly plastic, multilinear plastic, the Ram-

berg±Osgood formula or the actual material behaviour

during loading and unloading,

(b) the von Mises, Tresca or any other yield criterion,

(c) plastic unloading with kinematic or isotropic hard-

ening rules,

(d) the Bauschinger effect and its changes as a function of

plastic strain induced during loading and

(e) material compressibility.

As described in reference [1], the total strain is related to

current stress through

åij � 1� íeff

Eeff

ó ij ÿ íeff

Eeff

ó kkäij (1)

where

Eeff � 3E

3� 2Eö

íeff � 3í� Eö

3� 2Eö(2)

Here the function ö is a scalar valued function defined

from Hencky's total deformation theory:

ö � åpij

Sij

(3)

where Sij denotes the deviatoric stress components and åpij

is the equivalent plastic strain defined by

åpeq �

�����������23åp

ijåpij

q(4)

This method is applied here for the analysis of the

unloading behaviour of fastener holes and some numerical

results are presented here. These results include (a) the

residual stress predictions based on consideration of the

actual loading±unloading behaviour and (b) the prediction

of the elastic±plastic boundary. Results are compared with

available experimental measurements and finite element

calculations.

137

The MS was received on 5 May 1998 and was accepted after revision forpublication on 13 July 1999.�Corresponding author: School of Mechanical Engineering, Iran Uni-versity of Science and Technology, Narmak, Tehran 16844, Iran.

S02698 # IMechE 2000 JOURNAL OF STRAIN ANALYSIS VOL 35 NO 2

2 ACTUAL STRESS±STRAIN UNLOADING

CURVE

There have been many experimental measurements made

to find the residual stress in fastener holes. With the

exception of a few, the majority did not record the actual

loading±unloading behaviour of the material used. In many

of these investigations the uniaxial stress±strain curve (for

monotonic loading) was obtained to find the values of the

modulus of elasticity, Poisson's ratio and yield strength.

However, the actual curves were not used in the analysis.

Poussard et al. [2] provided the actual unloading curve for

aluminium alloy 2024-T351. Figure 1 is based on Fig. 1

from Poussard et al. [2] showing this curve. Two commonly

used models of the behaviour are also shown in the same

figure. Neither of these models represents the unloading

behaviour precisely. There exists a pronounced Bauschin-

ger effect in this aluminium alloy. While the monotonic

loading curve represents a linear hardening behaviour, with

a slope of 0.022E, the reversed yield stress (based on 0:1per cent proof stress) remains at a constant value of

ÿ110 MPa for unloading from different plastic strain

states. This suggests that the Bauschinger effect is a

function of plastic strain.

The experimental determination of residual stresses in

the same aluminium alloy 2024-T351 was recorded by

Priest et al. [3]. The results are rather scattered. For

example, the measured hoop residual stress in a 4 per cent

cold-worked plate is compressive even far away from the

hole edge. Priest et al. [3] suggested that the measured

values are highly influenced by the stresses existing in the

plate prior to cold working. However, they did not record

the as-received residual stress field.

In earlier independent work, Mann and Jost [4] also

recorded the results of experimental measurements on

aluminium alloy 2024. The experimental results, based on

the work of Lowak [5], are for a 4:5 per cent cold-worked

fastener hole with an initial hole radius of 8 mm.

The present method of analysis was used to predict the

residual stress field for a 4:5 per cent cold-worked plate.

The actual unloading behaviour of aluminium alloy 2024

shown in Fig. 1 was used. The changes in the Bauschinger

Fig. 1 Uniaxial loading±unloading response of aluminium alloy 2024-T351. (Based on Fig. 1 of reference [2])

JOURNAL OF STRAIN ANALYSIS VOL 35 NO 2 S02698 # IMechE 2000

138 H JAHED, S B LAMBERT AND R N DUBEY

effect were accounted for in the analysis, by setting the

compressive yield at ÿ110 MPa as observed from the

uniaxial curve.

An infinite plate with a hole of 8 mm initial radius was

considered. The residual stress field due to 4:5 per cent

expansion of the hole was obtained and the results are

shown in Fig. 2. The shape of the predicted curve near and

at the hole edge indicates the possibility of reversed yield.

However, the experimental results do not show any reversed

yielding and show only elastic unloading. This may be due

to the method of measurement (X-rays). The general

agreement between the present method and the experimen-

tal values is good. This agreement suggests the importance

of actual unloading curve employment in the analysis. The

prediction by Rich and Impelizzerri [6], which is one of the

few methods that accounts for reversed yielding is also

shown in this figure.

Poussard et al. [2] have employed the actual loading

behaviour of aluminium alloy 2024 with the ABAQUS

finite element package to predict the residual stress field.

Different hardening models provided by ABAQUS (iso-

tropic and kinematic hardening rules) were used. Their

finite element analysis was on a plate with an initial hole

radius of 3:175 mm and width of 200 mm. The hole was

assumed to be 4 per cent cold expanded. The same

dimensions were employed to obtain the residual stress

field using the variable material property method. The

results are shown in Fig. 3. While the kinematic hardening

model underestimates the compressive stress at the hole

edge by 20 per cent in comparison with the actual loading±

Fig. 2 Comparison of calculated and measured residual stress distribution for 4.5 per cent cold expanded hole in

aluminium alloy 2024 plate. (Experimental results from reference [4])

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VARIABLE MATERIAL PROPERTY METHOD IN THE ANALYSIS OF COLD-WORKED FASTENER HOLES 139

unloading curve, the isotropic model shows a uniform

compressive field near the hole. Both model predictions

away from the hole are very close to the present analysis

based on the actual unloading behaviour. The isotropic

hardening model stays closer to the present analysis away

from the hole. The difference in the radial residual stresses

is not significant.

3 ELASTIC±PLASTIC BOUNDARY PREDICTION

Knowledge of the size of the plastic zone in a fastener hole

is important in design and spacing of hole locations. Some

theoretical studies have been based entirely on predicting

the size of the plastic zone [7]. A comparison of the

prediction from different theories with the prediction from

the present method for the plastic zone radius is given

herein.

Poolsuk and Sharpe [8] conducted a series of experi-

ments to measure the exact size of the plastic zone and to

examine the validity of different theories in this regard.

They argued that it was easier to measure the location of

the elastic±plastic boundary, rather than the complete

residual stress field. Since most of the experimental results

for the stress field have been found to be in poor agreement

with the values predicted by different theories, Poolsuk and

Sharpe [8] suggested that the theories could be evaluated

on the basis of their capabilities of predicting the elastic±

plastic boundaries. The experiment was conducted on a

plate with a central hole of 3:3 mm radius made of

aluminium alloy 7075-T6. They used four different levels

of cold work and compared their measurements with

different theories.

Endo and Morrow [9] and Landgraf et al. [10] gave the

loading±unloading behaviour of aluminium alloy 7075-T6.

The data points for the monotonic loading curve are given

in Table 1. The yield stress and modulus of elasticity for

this curve match the specifications of one of the samples

used in the experiments of Poolsuk and Sharpe [8]. The

behaviour of this aluminium alloy cannot be modelled

precisely by any of the available hardening models.

However, consideration of the actual unloading curve have

more influence on the prediction of residual stresses closed

by the hole, as suggested by Fig. 2.

Fig. 3 Comparison of residual stress distribution for 4 per cent cold expanded hole in aluminium alloy 2024 plate.

(FEM results from reference [2])

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140 H JAHED, S B LAMBERT AND R N DUBEY

The elastic±plastic radius results obtained from the

present method are shown in Table 2. The maximum elastic

displacement at the hole, uaE, in this table is given by

uaE � (1� í)ó 0a

E���3p (5)

where í is Poisson's ratio, E is Young's modulus and ó0 is

the yield stress.

Figure 4 displays the results obtained by the present

method and the experimental measurements of Poolsak and

Sharpe [8] for a similar plate with 3:18 mm thickness

which shows the best agreement. The prediction from some

other theories are also shown in the same figure. As can be

seen from Fig. 4, some of these predictions are far from the

experimental results [7, 11]. The work of Nadai [12] and

Hsu and Forman [13] showed the same trend as the

experimental results. However, the plastic radius predicted

by these two methods is not accurate. The solution given by

Table 1 Data for loading behaviour of aluminium alloy

7075-T6. (From reference [9])

StrainElasticstrain

Plasticstrain

Stress(lbf=in2)

Stress(MPa)

0.0057 0.0057 0.0000 60.92 4200.0063 0.0063 0.0000 65.41 4510.0079 0.0079 0.0000 72.95 5030.0121 0.0114 0.0007 77.89 5370.0185 0.0145 0.0040 87.89 6060.0290 0.0090 0.0200 95.87 6610.0450 0.0095 0.0355 99.93 689

Fig. 4 Comparison of predicted elastic±plastic boundaries with measurements for aluminium alloy 7075-T6.

(Experimental results from reference [8])

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VARIABLE MATERIAL PROPERTY METHOD IN THE ANALYSIS OF COLD-WORKED FASTENER HOLES 141

Rich and Impelizzerri [6] is the best of the different

theories compared here. Nevertheless, the trend shown by

Rich and Impelizzerri [6] is different from that shown by

the experiment. It appears that this solution will deviate

very much from the experimental results for higher values

of ua=uaE. Chang's [14] solution is very close to the

solution obtained by Rich and Impelizzerri [6].

4 CONCLUSIONS

The general axisymmetric method of elastic±plastic analy-

sis [1] has been employed to predict the elastic±plastic

boundary and the residual stress field induced by the cold-

work expansion of the fastener holes. Results obtained by

this method were compared with experimental and finite

element results. From the results presented here it can be

concluded that the present method offers flexibility and

accuracy that may be useful in axisymmetric analysis of a

fastener hole.

REFERENCES

1 Jahed, H. and Dubey, R. N. An axisymmetric method of

elastic±plastic analysis capable of predicting residual stress

field. Trans. ASME, J. Pressure Vessel Technol., 1997, 119,

264±273.

2 Poussard, C., Pavier, M. and Smith, D. J. Analytical and

finite element predictions of residual stresses in cold worked

fastener holes. J. Strain Analysis, 1995, 30(4), 291±304.

3 Priest, M., Poussard, C. G., Pavier, M. J. and Smith, D. J.

An assessment of residual-stress measurements around cold-

worked holes in Al 2024. In Proceedings of the Fourth

International Conference on Residual Stresses, Baltimore,

Maryland, 1995, pp. 324±332.

4 Mann, J. Y. and Jost, G. S. Stress fields associated with

interference fitted and cold-expanded holes. Metal Forum,

1983, 6(1), 43±53.

5 Lowak, H. Berichte FB-157, Fraunhofer-Institut Betriebs.,

1981.

6 Rich, D. L. and Impelizzerri, L. F. Fatigue analysis of cold-

worked and interference fit fastener holes. In Cyclic Stress±

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ASTM STP 637, 1977, pp. 153±175 (American Society for

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plastic boundary around cold worked fastener holes. Trans.

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9 Endo, T. and Morrow, J. Cyclic stress±strain and fatigue

behavior of representative aircraft metals. J. Mater., 1969,

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the cyclic stress±strain curve. J. Mater., 1969, 4(1), 176±188.

11 Potter, R. M., Ting, T. W. and Grandt, A. F. An analysis of

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12 Nadai, A. Theory of the expanding of boiler and condenser

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13 Hsu, Y. C. and Forman, R. G. Elastic±plastic analysis of an

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14 Chang, J. B. Analytical prediction of fatigue crack growth at

cold worked fastener holes. AIAA paper 75-805.

Table 2 Results of the present method on the elastic±

plastic boundary location

Expansion (%) ua (mm) ua=uaE rp=a

1.15 0.038 2 1.5501.73 0.057 3 1.8162.3 0.076 4 1.9703.0 0.102 5.368 2.1803.85 0.127 6.68 2.2884.60 0.152 8 2.394

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142 H JAHED, S B LAMBERT AND R N DUBEY