Fatigue in steel structures under random loading

Journal of Constructional Steel Research 53 (2000) 283–305
www.elsevier.com/locate/jcsr

Fatigue in steel structures under random loading
Henning Agerskov

*

Department of Structural Engineering and Materials, Technical University of Denmark, Lyngby,
Denmark
Received 16 July 1998; received in revised form 5 July 1999; accepted 5 July 1999

Abstract
Fatigue damage accumulation in steel structures under random loading is studied. The
fatigue life of welded joints has been determined both experimentally and from a fracture
mechanics analysis. In the experimental part of the investigation, fatigue test series have been
carried through on various types of welded plate test specimens and full-scale offshore tubular
joints. The materials that have been used are either conventional structural steel with a yield
stress of fyෂ360–410 MPa or high-strength steel with a yield stress of fyෂ810–1010 MPa.
The fatigue tests and the fracture mechanics analyses have been carried out using load
histories, which are realistic in relation to the types of structures studied, i.e. primarily bridges,
offshore structures and chimneys. In general, the test series carried through show a significant
difference between constant amplitude and variable amplitude fatigue test results. Both the
fracture mechanics analysis and the fatigue test results indicate that Miner’s rule, which is
normally used in the design against fatigue in steel structures, may give results, which are
unconservative, and that the validity of the results obtained from Miner’s rule will depend on
the distribution of the load history in tension and compression.  2000 Elsevier Science Ltd.
All rights reserved.
Keywords: Steel structures; Fatigue; Random loading; Variable amplitude fatigue

1. Introduction

In the design of steel structures against fatigue, one of the problems that has
attracted increasing attention in recent years is the problem of fatigue damage

* Tel.: +45 45251706; fax: +45 45883282.
E-mail address: (H. Agerskov)
0143-974X/00/$ - see front matter  2000 Elsevier Science Ltd. All rights reserved.
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Nomenclature
A
fu
fy
I
M
m
N
n
R
⌬s
⌬se
r

constant
ultimate tensile strength
yield stress

irregularity factor
Miner sum
slope of S–N line
total number of cycles
number of cycles
stress ratio
stress range
equivalent constant amplitude stress range
correlation coefficient

accumulation. Codes and specifications normally give simple rules, using a Miner
summation and based on the results of constant amplitude fatigue tests.
Over the years, fatigue test series have been carried through using different types
of block loadings, and for these types of loading, Miner’s rule has in many cases been
found to give reasonable results, see e.g. [1–5]. However, in a real steel structure the
loading normally does not consist of loading blocks, but the structure is subjected
to a stochastic loading, due to traffic, wind, waves, etc. Thus, the need for a better
understanding of the fatigue behaviour under more realistic fatigue loading conditions is obvious.
The question of the validity of Miner’s rule is the background for a series of
research projects on fatigue in steel structures, carried out at the Department of Structural Engineering and Materials of the Technical University of Denmark over the
last eight years. The main purpose of these projects is to study the fatigue life of
steel structures, primarily bridges, offshore structures and chimneys, under various
types of stochastic loading that are realistic in relation to these types of structures.
The fatigue tests in these investigations have been carried out on various types of
welded plate test specimens and full-scale offshore tubular joints. The test specimens
have been fabricated either in conventional structural steel with a yield stress of
fyෂ360–410 MPa or in high-strength steel with a yield stress of fyෂ810–1010 MPa.
Besides the fatigue tests, these projects also include analytical determination of the
fatigue life under the actual types of random loading by use of fracture mechanics, to
be able to compare experimentally and theoretically determined fatigue lives.

The present paper gives an overview of the experimental and analytical investigations carried out, the types of loading used in the fatigue tests, and in the fracture
mechanics analysis, and the main results obtained in the fatigue tests and in the
analytical investigations.


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285

2. Experimental investigations
2.1. Welded plate test specimens
The majority of the fatigue test series have been carried through using welded
plate test specimens with transverse attachments. These test specimens consist of a
40 or 90 mm wide main plate with two transverse secondary plates welded to the
main plate by means of full penetration butt welds. Two different dimensions have
been used to study size effects. The applied loading is a central normal force in the
main plate. The test specimens are shown in Fig. 1.
For the plate test specimens with transverse attachments, both conventional structural steel with a yield stress of fy=400–409 MPa and an ultimate tensile strength,

Fig. 1.

Welded plate test specimens with transverse attachments.


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fu=537–575 MPa, and high-strength steel with fy=810–840 MPa and fu=845–875 MPa
have been used [6,7].

Four test series have been carried through on welded plate test specimens with
longitudinal attachments. For these test specimens, an 80 mm wide main plate is
used, and two longitudinal attachments are welded to the main plate by means of fillet
welds. Also in this case, two different dimensions are used to study size effects [7].
The material used for the plate test specimens with longitudinal attachments is
high-strength steel with a yield stress, fy=965–1010 MPa and an ultimate tensile
strength, fu=995–1045 MPa. These test specimens are shown in Fig. 2.
A third type of plate test specimen has been used in two test series. This test
specimen is a 40×8 mm plate with a transverse partial penetration butt weld. A
penetration of about 2/3 of the plate thickness has been used. The material has a
yield stress of fy=405 MPa and an ultimate tensile strength, fu=566 MPa [8].
2.2. Tubular joint test specimens
The dimensions that have been chosen for the tubular joints, see Fig. 3, correspond
to a large number of the joints in the platforms of the Tyra Field in the North
Sea. Compared with the largest joints in these platforms, the actual test joints are
approximately half size.
The test specimens are carried out as double T–joints. The test joints are loaded
in in-plane bending. Each test series comprises three test specimens with two tubular
joints in each of them.
In the investigation on joints in conventional offshore structural steel, the material
used has a yield stress of fy=363–381 MPa and an ultimate tensile strength of fu=506–
548 MPa [6].
In the test series on joints in high-strength steel, the material used in the fabrication
of the test specimens is a quenched and tempered high-strength steel. The yield stress
of the material used is fy=823–830 MPa, and the ultimate tensile strength is fu=853–
863 MPa [9].
After the first series of fatigue tests on the tubular joints in conventional offshore
structural steel, the fatigue cracks in the joints were repair-welded, and the fatigue
tests were repeated. In the repair-welding, it has been emphasised that the welding
procedures correspond as precisely as possible to procedures used presently in the

North Sea in repair-welding of fatigue cracks in tubular structures. After both series
of tests had been carried through, the fatigue life of the repair-welded tubular joints
could be compared with the fatigue life of the original joints [10].
2.3. Test equipment and test procedure
2.3.1. Plate test specimens
The tests on the plate specimens have been carried out in two fixed test frames,
one with a capacity of ±100 kN and the other with a capacity of ±500 kN. The
equipment — actuators, computers, valves, etc. — have been chosen so that a high
frequency is possible in the tests of these series.


H. Agerskov / Journal of Constructional Steel Research 53 (2000) 283–305

Fig. 2. Welded plate test specimens with longitudinal attachments.

287


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Fig. 3.

Tubular joint test specimen.

Small eccentricities due to the welding of the test specimens are inevitable in these
test series. This results in additional secondary bending stresses at the joint. Strain
gages are used on all test specimens in these series to determine the resulting stresses
from normal force and eccentricity moment.

2.3.2. Tubular joint test specimens
In the test equipment used in the investigation on the tubular joints, the test specimen has a fixed support in the central plane, whereas the rest of the test specimen
is free to move. The test joint is loaded in in-plane bending, using a 125 kN servocontrolled hydraulic actuator between the two secondary tubes.
Strain gages are used to determine the stresses in the test specimens. Furthermore,
the stresses in the most critical areas with respect to fatigue have been determined
from finite element analysis and by use of the thermoelastic technique (SPATE).
This is an experimental stress analysis technique based on the measurement of infrared radiant flux emitting from the surface of a body under cyclic stress [11–13].
Fatigue crack propagation during the test is determined by use of the AC-potential
drop technique. A computer is used to store sequences of recorded maxima and
minima of the load history.


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289

3. Variable amplitude loading
Various types of random loading have been used in the fatigue tests and in the
fracture mechanics determination of the fatigue life. The load histories applied correspond to offshore structures, highway bridges and chimneys.
3.1. Offshore structures
In the investigations on offshore structures, five different types of load histories
have been used. These load histories are generated by a computer program,
developed at the Department of Structural Engineering and Materials of the Technical
University of Denmark [14]. The program simulates a stationary Gaussian stochastic
process in real time. Only the extremes of the process are needed, since the load
course between consecutive extremes is considered unimportant. In the load simulation, a one-step Markov model is used. In this load model, the next extreme to be
generated will depend only on the present extreme, and not on the preceding load
history, i.e. it has a one-step memory. Each column and each row in the Markov
matrix contains cumulated transition probabilities. The matrix element to be chosen,

given a certain load level, is determined by use of a random number generator. The
elements in the matrix have been determined numerically from the spectral density
function of the wave elevation spectrum [15,16].
The load histories used in the investigations on offshore structures are equally in
tension and compression and with irregularity factors, I, varying from 0.745 to 0.987.
The irregularity factor is defined as the number of positive-going mean-value crossings divided by the number of maxima of the load history. For narrow band loading,
the irregularity factor will be close to unity. Typical load histories for fixed offshore
structures will be more broad banded, with irregularity factors in the range from
ෂ0.6 to 0.8.
Details of the load simulation procedures and the main characteristics of the various load histories used may be found in [6,17]. Fig. 4 shows examples of typical
load histories, generated by use of the matrices BROAD64 and PMMOD64.
BROAD64 was evaluated from a truncated white noise spectral density function,
and PMMOD64 from a modified Pierson–Moscowitz wave elevation spectrum.
3.2. Highway bridges
The variable amplitude loading that was used in the investigation on highway
bridges has been determined from strain gage measurements on the orthotropic steel
deck structure of the Farø Bridges in Denmark. The measurements were carried out
during the months of May and November [18,19]. The load histories correspond to
one week’s traffic loading. Strain gage measurements were taken at 10 different
locations in the orthotropic deck. The load histories that have been used in the present
investigation were measured by two strain gages, both placed on the bottom of one
of the trapezoidal longitudinal stiffeners of the deck plate. The stiffener chosen is
located under the most heavily loaded lane of the motorway. Strain gage No. 1 is


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Fig. 4. Examples of load histories from investigations on offshore structures. 150 extremes generated

by use of the matrices BROAD64 and PMMOD64, respectively.

placed in the middle of the longitudinal stiffener span, which has a length of 4 m.
Strain gage No. 5 is placed at a distance of 0.5 m from one of the transverse diaphragms. This means that the stresses measured by strain gage No. 1 are primarily
tensile stresses, whereas the stresses registered by strain gage No. 5 are almost equal
in tension and compression. For strain gage No. 1, the stress measurements were
taken during the month of May, whereas for strain gage No. 5, load histories were
measured in both May and November.
Figs. 5 and 6 show examples of typical load histories based on the measurements
from strain gages No. 1 and 5, respectively. In both cases, 200 extremes are included
in the load history shown. The load history based on strain gage No. 1 has an irregularity factor, I=0.617, while the load histories from strain gage No. 5 have I=0.793–
0.834. Further details of these load histories may be found in [19].
Furthermore, one fatigue test series has been carried through, in which a cantilever
bridge girder of a cable-stayed bridge during construction is studied. The bridge
girder is subjected to vertical oscillations due to transverse wind. The stress history
has been generated from the bridge girder response, simulating a 50-year storm with


H. Agerskov / Journal of Constructional Steel Research 53 (2000) 283–305

Fig. 5.

Fig. 6.

291

Example of load history. 200 extremes based on the measurements from strain gage No. 1.

Example of load history. 200 extremes based on the measurements from strain gage No. 5, May.

five days duration. This stress history is equally in tension and compression, and has
an irregularity factor of I=0.888 [8].
3.3. Chimneys
The variable amplitude loading that was used in the investigation on chimneys
has been determined from wind tunnel tests. An undamped chimney model, subject
to transverse oscillations due to vortex shedding was studied. Two load histories
have been used, both determined from strain gage measurements on the model. These
load histories are equal in tension and compression and very narrow-banded, with
an irregularity factor of I=0.998 [20].


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4. Fatigue test results
In the following is given an overview of the main results that have been obtained
in the various fatigue test series on the plate test specimens and the tubular joints.
A total of 520 fatigue tests on welded plate specimens have been carried out in these
investigations, and 18 full-scale tubular joints have been tested.
In all investigations, initial test series with constant amplitude loading were carried
out as a reference, and also — for the plate test specimens — to obtain the actual
value of the exponent m for calculation of the equivalent stress ranges of the tests
with variable amplitude loading, cf. Eq. (1).
In the results from the variable amplitude tests, the stress parameter used is the
equivalent constant amplitude stress range, ⌬se, defined as:
⌬seϭ

΄

͸

i

N

1
m

΅

(ni·⌬smi )

(1)

in which ni=number of cycles of stress range ⌬si; ⌬si=variable amplitude stress
range; N=total number of cycles (=⌺ini); and m=slope of corresponding constant
amplitude S–N line.
The cycle counting method that has been chosen for the analysis of the stress
history is in all investigations “rainflow counting”. This method is usually recommended for the analysis of random loading histories in steel structures [21].
Examples of the results that have been obtained in the various test series are shown
in the S–N diagrams in Figs. 7–9. Both the linear regression S–N line from the

Fig. 7. Results obtained from variable amplitude tests with BROAD64 spectrum. Small plate specimens
with transverse attachments. Conventional structural steel.


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293

Fig. 8. Results obtained from variable amplitude tests with BROAD64 spectrum and constant amplitude
tests. Small plate specimens with transverse attachments. High-strength steel.

Fig. 9. Results obtained from variable amplitude tests with load history from strain gage No. 5, May.
Small plate specimens with transverse attachments. Conventional structural steel.


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variable amplitude test series and the S–N line obtained in the corresponding constant
amplitude test series are shown in Figs. 7–9.
Figs. 7 and 8 show the results of the test series on the small plate specimens
with transverse attachments and the offshore broad-band spectrum, BROAD64, for
conventional structural steel and high-strength steel, respectively. In the S–N diagrams, points marked with a star have not been included in the regression analysis.
These points correspond to a number of cycles to failure of 107 or more, and they
are thus expected to be situated on the transition curve from the linear relationship
up to about 107 cycles to a possible horizontal line at a high number of cycles. Points
marked with an arrow correspond to a test with a non-broken test specimen.
For the test results obtained in each test series on the plate specimens, the data
are fitted to an expression:
log Nϭ log AϪm· log ⌬s

(2)

by the method of least squares. In Eq. (2), m and A are constants, N is the number
of cycles to failure, and ⌬s is the stress range.
Fig. 9 shows the results obtained in the test series on the small plate specimens

with transverse attachments in conventional structural steel, and with highway bridge
loading, strain gage No. 5, May.
Details of the results obtained in the individual test series of the various investigations may be found in [7,17,19].
When comparing the results obtained in the various test series, it appears that
there is in general a significant difference between constant amplitude and variable
amplitude fatigue test results. This was specially pronounced in the investigations
with stochastic loading corresponding to offshore structures, but the same tendency
was observed in the other investigations.
The difference in fatigue life between constant amplitude and variable amplitude
test results has in these investigations been quantified by the Miner sum, M, determined as the number of cycles to failure at variable amplitude loading, Nva, divided
by the number of cycles to failure at constant amplitude loading, Nca, at the same
equivalent stress range level. When the slope of the linear regression S–N lines from
variable amplitude and constant amplitude tests are not identical, M will be a function
of the stress range level.
Miner’s rule, which is the cumulative damage rule generally used today in the
design of steel structures against fatigue, assumes that fracture occurs for a Miner
sum of M equal to 1.
The main observations with respect to the values of the Miner sum, M, obtained
in the fatigue test series of the various investigations are given in the following.
4.1. Offshore structures. Conventional structural steel
For the investigations on variable amplitude fatigue in offshore structures in conventional structural steel, the values of the Miner sum obtained are given in [22,23],
for the test series with the various load histories investigated. These values are based
on approximately 160 fatigue tests.


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295

In Table 1, values of the Miner sum calculated at different stress range levels

from the regression S–N lines that have been obtained, are given for the test series
on the plate specimens with transverse attachments. Four different offshore load
histories, PM32, BROAD64, PMMOD64, and NARROW64 were investigated.
As may be seen from Table 1, the Miner sum corresponding to failure in the
variable amplitude test series, varies in the range ෂ0.40–0.85, in all but one case.
The value for the small plate specimens and the load history PMMOD64 at ⌬se=100
MPa is given in brackets and should be taken with some precaution. A value of
M=1.35 is obtained, when the linear regression S–N lines are used directly. However,
the test results indicate that the transition curve for the constant amplitude test series
starts already at 5–6×106 cycles and continues down to a fatigue limit of approximately 100 MPa at a high number of cycles. Taking this into consideration, it seems
appropriate to estimate the value of M to ෂ1.0 at ⌬se=100 MPa for this test series.
This is in good agreement with the value obtained in the fracture mechanics analysis.
For the tubular joints, the number of test results in the present investigation is too
limited to make possible any significant statistical analysis. If best fit S–N lines are
determined, assuming a slope of m=3 for both constant amplitude and variable amplitude tests, a Miner sum of Mෂ0.6–0.8 is found for the variable amplitude tests on
the tubular joints in conventional offshore structural steel. With an irregularity factor,
I=0.82–0.84 for the load histories used in the variable amplitude tests on the tubular
joints, these results are in good agreement with the observations from the test series
on the welded plate test specimens.
The fatigue cracks that developed during this first series of tests on the tubular
joints were repair-welded according to specifications used presently in the North Sea
in repair-welding of fatigue cracks in tubular structures. After this, the fatigue tests
were repeated. It was expected beforehand that the repair-welded joints would have
fatigue lives that were shorter than those of the original joints. However, a comparison of the results obtained showed that the repair-welded joints had fatigue lives of
1.9–5.0 times the life of the original joints. Furthermore, the fatigue crack initiation
Table 1
Values of Miner sum, M, at different equivalent stress range levels, ⌬se. Test series on plate specimens
with transverse attachments. Conventional structural steel
Load history

PM 32
BROAD64
PMMOD64
NARROW64

Plate test specimen M

Small
Large
Small
Large
Small
Large
Small
Large

⌬se=100 MPa

⌬se=200 MPa

⌬se=300 MPa

0.46
0.62
0.51
0.38
(1.35)
0.38
0.84
0.53

0.49
0.66
0.44
0.41
0.67
0.56
0.82
0.62

0.51
0.69
0.41
0.43
0.45
0.71
0.80
0.68


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in general moved from the chord wall in the first series of tests to the branch wall
in the test series on the repair-welded joints. The main reason for these observations
is assumed to be the differences in weld shape and thus also in stress concentrations
in the two cases. Details of the results obtained in the investigation on repair-welded
tubular joints may be found in [10].
4.2. Offshore structures. High-strength steel

In the investigations on offshore structures in high-strength steel, the values of the
Miner sum obtained may be found in [7,24], for the various test series investigated. A
total of 170 fatigue tests were carried out in these test series.
Table 2 gives values of the Miner sum calculated at different stress range levels
from the regression S–N lines obtained, for the test series on the plate specimens
with longitudinal and transverse attachments. In these investigations, two different
offshore load histories, BROAD64 and PMMOD64 were applied.
For the test series on both small and large plate specimens with longitudinal attachments, little difference in fatigue life was found between the series with constant
amplitude loading and with the broad-band offshore load history, BROAD64, at an
equivalent stress range of about 160–180 MPa. At higher stress range levels, longer
fatigue lives were obtained for the stochastic loading, while the constant amplitude
loading at lower stress ranges resulted in longer fatigue lives, for both small and
large test specimens. In Table 2, one of the values of the Miner sum — at ⌬se=300
MPa — is given in brackets. The reason for this is that the regression S–N line for
the corresponding test series (BROAD64, small plate specimens) is based on results
obtained in the stress range interval ⌬se=100–200 MPa.
For the test series on plate specimens with transverse attachments, there is in all
series a significant difference between constant amplitude and variable amplitude
fatigue test results. For the test series on both small and large plate specimens, and
Table 2
Values of Miner sum, M, at different equivalent stress range levels, ⌬se. Test series on plate specimens.
High-strength steel
Type of test
specimens

Longitudinal
attachments
Transverse
attachments

Load history

Size of test
specimens

M
⌬se=100 MPa

⌬se=200 MPa

⌬se=300 MPa

Small

0.49

1.24

(2.13)

Large

0.48

1.22

2.12

BROAD64

Small

0.50

0.52

0.54

PMMOD64

Large
Small
Large

0.63
0.55
0.56

0.66
0.61
0.81

0.68
0.65
1.01

BROAD64


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297

with the two different load histories investigated, there is a clear indication, that the
fatigue life is shorter with variable amplitude loading than with constant amplitude
loading for the same stress range level. It appears from Table 2 that the Miner sum
corresponding to failure in the variable amplitude test series, varies in the range
ෂ0.50–0.80, in all but one case.
With respect to the tubular joints, the number of tests carried out is not sufficient
to make possible any significant statistical analysis. However, if best fit S–N lines
are determined, assuming a slope of m=3 for both constant and variable amplitude
tests, a Miner sum of Mෂ0.75 is obtained for the variable amplitude tests on the
tubular joints in high-strength steel. The irregularity factor is I=0.82 for the load
history used in the variable amplitude tests on the tubular joints, and thus this result
agrees well with the observations from the test series on the plate specimens.
4.3. Highway bridges
For the investigations with bridge traffic loading, determined from strain gage
measurements on the Farø bridges, the main results obtained may be found in [19,25].
The test series on the small and the large plate specimens gave Miner sums of 0.5–
1.0, corresponding to failure in the variable amplitude tests with the load history
based on strain gage No. 5. These values of M correspond to stress range levels of
approximately 90–200 MPa.
For the test series on the small plate specimens with the load history based on
strain gage No. 1, the interval of the equivalent constant amplitude stress range
covered by the fatigue tests carried out is approximately 80–150 MPa. In the corresponding constant amplitude test series with stress ratio, R=smin/smax=Ϫ1/5, fatigue
failure occurred for stress ranges in the interval 100–275 MPa. For the stress range
area covered by both the test series with constant amplitude loading, R=Ϫ1/5, and
the series with the load history based on strain gage No. 1, values of the Miner sum
of Mෂ1.2–1.8 were obtained. Thus, in this case Miner’s rule apparently was found
to give conservative predictions of the fatigue life.

However, the values of the Miner sum, MϾ1, obtained in the test series with the
load history from strain gage No. 1 should be taken with some precaution due to
the following reasons: The correlation coefficient for the linear regression in this test
series is r=Ϫ0.69, which is a quite bad value, compared to the values of r obtained
in the other test series (Ϫ0.82 to Ϫ0.98). Furthermore, the stress range interval
covered by both test series (constant amplitude loading, R=Ϫ1/5, and load history
from strain gage No. 1) is small, from 100 to 150 MPa. Finally, the test results
indicate that the transition curve for the constant amplitude test series starts already
at 2–3×106 cycles and continues down to a fatigue limit of approximately 105 MPa
at a high number of cycles. Taking the above into consideration, it seems appropriate
to estimate the values of the Miner sum for the test series with the load history from
strain gage No. 1 to ෂ1.0. This is in good agreement with the corresponding values
of Mෂ0.8–0.9, obtained in the fracture mechanics analysis.
In the investigation, in which a cantilever bridge girder during construction is
studied, and where the load history is a simulated 50-year storm of five days duration,


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values of the Miner sum of Mෂ0.32–0.51 have been obtained for a stress range
interval of ⌬se=100–400 MPa [8].
4.4. Chimneys
In the investigation on chimneys, two test series were carried through with very
narrow-banded loading. In these test series, values of the Miner sum of Mෂ0.85–
1.16 were obtained. The stress range interval covered was ⌬seෂ130–270 MPa.
Further details of these test series and the results obtained may be found in [20].

5. Fracture mechanics prediction of fatigue life

Besides the fatigue tests, the present investigations also include analytical determination of the fatigue life under the actual types of random loading by use of fracture
mechanics, to be able to compare experimentally and theoretically determined
fatigue lives.
Of special importance for the validity of the results that are obtained from the
fracture mechanics analysis is the consideration of crack closure in the analytical
model. The crack growth analysis model used in the present investigations is based
on the Dugdale–Barenblatt strip yielding assumption, with modifications to allow
plastically deformed material to be left along the crack surfaces as the crack grows.
The crack closure model accounts for load interaction effects, such as retardation
and acceleration, under variable amplitude loading. The model may be used to simulate fatigue crack growth under both constant amplitude and variable amplitude loading, taking into account the influence of crack closure upon fatigue crack growth.
Furthermore, in the determination of the crack growth life the effects of stress concentrations and welding residual stresses are included. More details of the crack
growth model used may be found in [26–28].
The fatigue lives have been calculated by use of fracture mechanics in two investigations: 1. Offshore structures in conventional structural steel, and 2. Highway
bridges with load histories from Farø bridges. S–N curves have been determined for
both constant amplitude loading and variable amplitude loading.
In the investigation on offshore structures, a fracture mechanics determination of
the fatigue life was carried out for both the small and the large plate test specimens
with transverse attachments and with the three different load histories, BROAD64,
PMMOD64, and NARROW64. Fig. 10 shows the analytical results obtained for the
small plate test specimens with these three load histories, together with the analytical
results for constant amplitude loading [26].
In the investigation on highway bridges, the fatigue lives have been calculated for
the small plate test specimens with transverse attachments in conventional structural
steel for constant amplitude loading and for variable amplitude loading using the
load histories measured by strain gages No. 1 and No. 5, May. Fig. 11 shows a
comparison between the results obtained at constant amplitude loading and with the
load history from strain gage No. 5, May, for the small plate test specimens [28].


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299

Fig. 10. Comparison of analytical results for small plate test specimens with transverse attachments.
Conventional structural steel. Offshore load histories and constant amplitude loading [26].

Fig. 11. Comparison of analytical results for small plate test specimens with transverse attachments.
Conventional structural steel. Highway bridge load history and constant amplitude loading [28].


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Table 3
Values of Miner sum, M, at different equivalent stress range levels, ⌬se, obtained from fracture mechanics
analysis. Small plate test specimens with transverse attachments. Conventional structural steel [27]
Load history

BROAD64
PMMOD64
NARROW64

M
⌬se=120 MPa

⌬se=200 MPa

⌬se=250 MPa

0.75
0.88
0.95

0.38
0.47
0.76

0.34
0.41
0.65

Also in the analytical investigations, the values of the Miner sum, M, corresponding to failure in the variable amplitude series, have been determined. The main observations with respect to the values of the Miner sum obtained in these investigations
are given in the following.
5.1. Offshore structures. Conventional structural steel
In the investigation on fracture mechanics determination of the fatigue life of
offshore structures in conventional structural steel, the values of the Miner sum that
have been obtained may be found in [26,27], for the various load histories investigated.
In Tables 3 and 4, values of the Miner sum calculated at different stress range
levels are given for three of the offshore load histories used in the test series on the
small and the large plate specimens, respectively.
As may be seen from Tables 3 and 4, the Miner sums corresponding to failure
for the various offshore load histories studied varies in the range ෂ0.35–0.95.
5.2. Highway bridges. Load histories from Farø bridges
The values of the Miner sum that were obtained in the investigation on fracture
mechanics determination of the fatigue life of highway bridges, may be found in
[19,28], for the load histories investigated.
Table 4
Values of Miner sum, M, at different equivalent stress range levels, ⌬se, obtained from fracture mechanics
analysis. Large plate test specimens with transverse attachments. Conventional structural steel [27]

Load history

BROAD64
PMMOD64
NARROW64

M
⌬se=150 MPa

⌬se=200 MPa

⌬se=250 MPa

0.74
0.83
0.93

0.55
0.63
0.81

0.42
0.49
0.69


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301

Table 5 gives the values of the Miner sum that have been calculated at different
stress range levels for two of the load histories, determined from strain gage measurements on the Farø bridges.
It appears from Table 5 that the Miner sum corresponding to failure for the highway bridge load histories investigated varies in the range ෂ0.30–0.95.
6. Observations
When comparing the results obtained in the experimental investigations, it appears
that there is in general a significant difference between constant amplitude and variable amplitude fatigue test results. This observation was confirmed by the results
obtained in the fracture mechanics analyses.
The main reason for these differences in fatigue behaviour is crack growth acceleration and/or retardation due to the high tensile and compressive loads of the variable
amplitude load histories. Important factors in this connection are crack closure mechanisms, the size of the welding residual stresses, the size of the stress concentrations,
and the yield stress of the material. Both the experimental and analytical investigations carried through show that acceleration effects dominate the fatigue crack
growth for the load histories studied.
For the test series on both plate specimens and tubular joints, and with the different
load histories investigated, it was found that the fatigue life in general is shorter
with variable amplitude loading than with constant amplitude loading at the same
equivalent stress range level, for load histories which are by and large equally in
tension and compression. This was observed both in the experimental investigations
and in the fracture mechanics determination of the fatigue life.
In all the fracture mechanics analyses carried out, higher values of the Miner sum,
M, were obtained at the lower equivalent stress range levels. The reason for this is
that the crack growth acceleration effects at the variable amplitude loading in general
are less important at the lower stress levels. This observation was not similarly clear
for the test results. In about 1/3 of the test series higher values of M were obtained
at lower stress range levels, ⌬se; in about 1/3 of the test series, generally the same
values of M were obtained for all values of ⌬se considered; and in about 1/3 of the
test series, higher values of M were obtained at higher stress range levels.
Table 5
Values of Miner sum, M, at different equivalent stress range levels, ⌬se, obtained from fracture mechanics
analysis. Small plate test specimens with transverse attachments. Conventional structural steel [19]
Load history

M

Strain gage No. 1

⌬se=100 MPa
0.94

⌬se=150 MPa
0.81

⌬se=200 MPa
0.68

⌬se=250 MPa
0.61

Strain gage No. 5,
⌬se=100 MPa
May
0.77

⌬se=200 MPa

⌬se=300 MPa

⌬se=400 MPa

0.38

0.30

0.33


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H. Agerskov / Journal of Constructional Steel Research 53 (2000) 283–305

In two cases, the test series with the bridge load history based on strain gage No.
1 and with the stresses primarily in tension, and the test series on plate specimens
with longitudinal attachments in high-strength steel, the results showed — generally — the same fatigue life at variable and constant amplitude loading. For the plate
specimens with longitudinal attachments, values of MϽ1 were obtained at lower
stress range levels, while at higher stress range levels, MϾ1 was found. Thus, in
this case the combination of a rather irregular load history (BROAD64), a high stress
level, high stress concentrations, and a high-strength material resulted in the crack
growth retardation effects being dominating, compared with the corresponding constant amplitude tests.
The results obtained in the various investigations carried through show that there
is a clear tendency that the value of the Miner sum, M, corresponding to failure,
decreases with the irregularity factor of the load history. For the offshore load histories studied, which are equally in tension and compression and with irregularity factors ranging from Iෂ0.7 to 1.0, the results indicate that, taking the uncertainties into
consideration, the use of a value of M=2•IϪ1, corresponding to failure, might be
appropriate [9].
The results obtained — both experimental and theoretical — show that the distribution of the load history in tension and compression has a significant influence on
the validity of the results, which are obtained by use of Miner’s rule.

7. Conclusions
A series of research projects on fatigue in steel structures, primarily bridges, offshore structures and chimneys, has been carried out at the Department of Structural
Engineering and Materials of the Technical University of Denmark. The main purpose of these projects has been to study the fatigue life under various types of random
loading, which are realistic in relation to the types of steel structures investigated.
Comparisons between experimental results, results of fracture mechanics analysis,
and results obtained using current codes and specifications, i.e. Miner’s rule, are

given.
In the experimental investigations, test series with a total of approximately 540
fatigue tests on welded plate specimens and full-scale tubular joints have been carried
through. The materials used are either conventional structural steel with a yield stress
of fyෂ360–410 MPa or high-strength steel with a yield stress, fyෂ810–1010 MPa.
The experimental investigations in general show a significant difference between
constant amplitude and variable amplitude fatigue test results. For the variable amplitude tests, the stress parameter used is the equivalent constant amplitude stress range,
⌬se, and the difference in fatigue life between constant amplitude and variable amplitude test results is quantified by the Miner sum, M, determined as the number of
cycles to failure at variable amplitude loading divided by the number of cycles to
failure at constant amplitude loading, at the same equivalent stress range level.
The values of the Miner sum that were obtained in the variable amplitude test
series, generally vary in the range ෂ0.40–0.85 for the test series with the offshore


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303

load histories. In the investigations on highway bridges and chimneys, values of the
Miner sum of 0.50–ෂ1.0 were obtained.
The fatigue lives of the welded joints under both constant amplitude and variable
amplitude loading have been determined theoretically by use of fracture mechanics.
These investigations have shown that the crack closure mechanisms are of paramount
importance for the physical understanding and explanation of the fatigue crack
growth under variable amplitude loading.
The main reason for the difference between constant amplitude and variable amplitude fatigue behaviour is crack growth acceleration and/or retardation due to the high
tensile and compressive loads of the variable amplitude load histories. The most
important factors in this connection are the crack closure mechanisms, the size of
the welding residual stresses, the size of the stress concentrations, and the yield stress
of the material. Both the experimental and analytical investigations carried through

show that acceleration effects dominate the fatigue crack growth for the load histories studied.
A comparison of the experimental results and the results of the fracture mechanics
calculations in general shows good agreement, when the calculations are based on
the estimated values of the actual welding residual stresses and crack closure is
included. The results of the fracture mechanics calculations also show that Miner’s
rule may give unconservative predictions of the fatigue life, since the Miner sums,
which were obtained from the calculations in the present investigations, all are less
than 1.
In most of the investigations carried through, e.g. in all the fracture mechanics
analyses, greater values of the Miner sum, M, were obtained at the lower equivalent
stress range levels. The reason for this is believed to be that the crack growth acceleration effects at the variable amplitude loading in general are less important at the
lower stress levels.
The results obtained in the various investigations show that there is a clear tendency that the value of the Miner sum, M, corresponding to failure, decreases with
the irregularity factor of the load history.
For the test series on both plate specimens and tubular joints, and with the different
load histories investigated, it was found that the fatigue life in general is shorter
with variable amplitude loading than with constant amplitude loading at the same
equivalent stress range level, for load histories which are by and large equally in
tension and compression. This was observed both in the experimental investigations
and in the fracture mechanics determination of the fatigue life.
The results obtained demonstrate that Miner’s rule, which is normally used in the
design against fatigue in steel structures, may give quite unconservative predictions
of the fatigue life, and that the distribution of the load history in tension and compression has a significant influence on the validity of the results, which are obtained
by use of Miner’s rule.
On the basis of the results obtained in the present investigations, the following
general recommendations can be given: For rather broad-banded types of random
loading, which are by and large equal in tension and compression, a value of the
Miner sum, corresponding to failure, of Mෂ1/3–1/2 should be used. For rather nar-

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H. Agerskov / Journal of Constructional Steel Research 53 (2000) 283–305

row-banded types of random loading, which are primarily in tension, a value of M
of ෂ0.8–1.0 seems appropriate. However, it should be emphasized that in special
cases both higher and lower values of M may be obtained.
Acknowledgements
The funding for the various investigations carried out has been provided by the
Danish Technical Research Council, the Nordic Fund for Technology and Industrial
Development, the Technical University of Denmark, and SSAB Oxelo¨sund AB,
Sweden, who are gratefully acknowledged. The permission from the Road Directorate, Danish Ministry of Transport to carry out the strain gage measurements on the
Farø Bridges is greatly appreciated.
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