3.8. RESIDUAL STRESS COMPUTATIONS 59
Front Bush
Gundrill
Electrode
Air Probe
Air Probe
Back Bush
Stage 1 Stage 2
Stage 3 Stage 4
Figure 3.13: Deep-hole measurement in a large metal component. (1) Attach front and back
bushes and gun-drill a pilot hole, (2) measure the diameter along the length of the pilot hole,
(3) overcore the pilot hole, and (4) re-measure the diameter along the length of the pilot hole
(diagram courtesy of Veqter Ltd.).
A fundamental distinguishing feature of the Deep-Hole Method is that the deformation
measurements are made within the interior of the hole rather than at the surface. is is done
to move the location of stress sensitivity into the deep interior. In contrast, hole-drilling and
ring-coring measurements focus on near-surface stresses because their measurements are at the
surface. For metal components, deep-hole measurements can be made to about 750 mm deep,
while for geological applications, measurement depths over 1 km are routinely achieved.
3.8 RESIDUAL STRESS COMPUTATIONS
For the Ring-Core Method, calculation of the residual stresses corresponding to the measured
strains is fairly straightforward because complete strain relief occurs within the central core and
so the standard equations of linear elasticity may be directly applied. However, for the Hole-
Drilling Method, only a fraction of the strains within the material surrounding the drilled hole
are relieved, so the residual stress calculation also requires knowledge of the specific numerical
value of that fraction. In practice, the needed data take the form of calibration constants provided
either in numerical or graphical form.
Historically, two approaches have been taken to determine hole-drilling calibrations, the-
oretical and experimental. In his pioneering 1934 work, Mathar initiated the use of both ap-
60 3. HOLE-DRILLING METHOD CONCEPT AND DEVELOPMENT
proaches. On the theoretical side, he used the classical Kirsch solution for the stress distribution
around a through hole in a thin stressed plate to determine the relationship between hole di-
ameter change and the in-plane stresses. On the experimental side, he also measured diameter
change vs. hole depth for known in-plane stresses for the case of blind hole drilling in a thick
material. He noted that the experimental results asymptotically approach the theoretical solution
when the hole depth reaches approximately one hole diameter.
Both the theoretical and experimental approaches were taken up by subsequent re-
searchers. e greater accuracy provided by the advent of strain gauge use in the 1940s and
1950s made it possible to go beyond the simple evaluation of the bulk residual stresses at the
hole location to the determination of the profile of the residual stresses through the hole depth.
Since the Kirsch analytical solution only applies to stresses that are uniform within a through-
hole in a thin plate, calibrations for non-uniform stresses within a blind hole in thick plate could
at that time only be done experimentally. us, computational procedures had to be chosen to
use only experimental calibration data.
Kelsey (1956) described a stress profile computation method that later become known as
the Differential Strain Method. e method is based on the assumption that the change in strain
measured at a given hole depth depends on the stresses existing at that depth. us, the residual
stress profile can be determined by comparing point-by-point the slopes of the strain vs. depth
profiles of the test measurements with experimental calibration data for a known uniform stress
state. is method works reasonably well for residual stress profiles that do not greatly differ
from uniform, however its accuracy significantly deteriorates when used with significantly non-
uniform residual stress profiles. e reason is that the evolution of the measured strain response
additionally depends on the hole geometry change caused by its increasing depth. us, the
change in relieved strain at any stage combines the effects of hole enlargement on the previously
exposed near-surface stresses as well as those on the newly exposed interior stresses.
e Average Stress Method introduced by Nickola in 1986 sought to address the concerns
with the Differential Strain Method by recognizing that the strain response at a given hole
depth combines the effects of all the stresses at the various depths within the hole depth. A
running average stress at each hole depth is computed using the calibration data for the uniform
stress case. e stress within each hole depth increment is then evaluated by determining the
stress values that combine to give the computed sequence of running averages. In practice, the
strain response is highly nonlinear, with much greater sensitivity to near-surface stresses than
to interior stresses. us, the use of a simple average stress value underweights the effects of
the near-surface stresses and overweights those of the interior stresses. us, the Average Stress
Method also has limited usefulness and is also suitable only for near-uniform residual stress
profiles.
e growth in the practical use of the finite element method in the 1970s enabled the
development of residual stress computation methods that could break through the limitations
3.8. RESIDUAL STRESS COMPUTATIONS 61
imposed by dependence on experimental calibrations. During the 1970s and 1980s, advances
were made in three important areas:
1. accuracy improvement of hole-drilling calibration data,
2. formulation of calibration data not measurable experimentally, and
3. creation of calibration data for additional applications.
e early works of Bijak-Zochowski (1978), Beaney and Procter (1974), Schajer (1981)
were aimed at improving calibration accuracy and consistency and established the finite element
method as a practical method for evaluating hole-drilling calibration data. An important devel-
opment was the introduction of the Integral Method for calculating the residual stress profile
through the depth of the drilled hole. Several researchers contributed to this initial development,
notably Bijak-Zochowski, Flavenot and Lu, and Schajer (1988). e significant feature of the
Integral Method is that it correctly accounts for the contributions of all stresses within the hole
depth to the measured strain response, thereby avoiding the limitations of the Differential Strain
and Average Stress methods. e use of finite element calculations was an essential prerequisite
to the development and use of the Integral Method because the needed calibration data are not
measureable experimentally. Since 1999, the ASTM Standard Test Method E837 has specified
the use of the Integral Method for residual stress profile evaluations.
e flexibility of the finite element method has enabled substantial computational de-
velopments to be made in many further areas, notably for corrections for various experimental
artifacts that can occur during the course of practical measurements. One such artifact occurs
when the drilled hole is not exactly at the geometric center of the strain gauge rosette. e
resulting eccentricity causes a systematic shift in the measured strains, thereby distorting the
corresponding computed residual stresses. In the late 1970s, Sandifer and Bowie (1978) and
Ajovalasit (1979) introduced computational approaches for correcting the effect providing that
the size and direction of the hole eccentricity are accurately known. However, the opportunity
to correct for hole eccentricity should not be allowed to grant a tolerance of such errors. Cer-
tainly it is always the best strategy is to seek to refine the experimental technique used so as to
minimize the occurrence and size of all preventable errors.
Another significant artifact occurs when the Hole-Drilling Method is used to measure
large residual stresses close to the material yield stress. e drilling of the hole creates a stress
concentration in the adjacent material, which causes local yielding. e resulting material plas-
ticity adds to the relieved strains, causing them to be larger than they would be if only elastic
deformations had occurred. Consequently, the residual stresses evaluated from the measured
strain reliefs are overestimated, often suggesting residual stresses larger than the material yield
stress. is artifact can also be ameliorated using finite-element based compensation methods.
Starting in the 1990s, Beghini and coworkers have led the initiative to develop approaches to
compensate for material plasticity and to allow hole-drilling measurements to be made for resid-
ual stresses close to yield stress. Without such corrections, the Hole-Drilling Method can be
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