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Steel Structures 7 (2007) 263-276 www.kssc.or.kr
Different Approaches for Remaining Fatigue Life Estimation
of Critical Members in Railway Bridges
Siriwardane Sudath Chaminda1, Mitao Ohga2,*, Ranjith Dissanayake3 and Kazuhiro Taniwaki4
1Graduate Student, Department of Civil and Environmental Engineering, Ehime University, Bunkyo-cho 3, Matsuyama 790-8577, Japan2Professor, Department of Civil and Environmental Engineering, Ehime University, Bunkyo-cho 3, Matsuyama 790-8577, Japan
3Senior Lecturer, Department of Civil Engineering, University of Peradeniya, Peradeniya 20400, Sri Lanka4Lecturer, Department of Civil and Environmental Engineering, Ehime University, Bunkyo-cho 3, Matsuyama 790-8577, Japan
Abstract
Rail authorities all over the world are paying attractive attentions to evaluate the remaining fatigue lives of riveted railwaybridges since most of these bridges are currently reaching the ends of their theoretical fatigue lives. This paper proposes threedifferent approaches to evaluate remaining fatigue life of an existing riveted railway bridge. First proposed approach is basedon combination of measured stress histories, Miner’s rule and partially known Wöhler curve. The second approach mainlyconsists of measured stress histories, recently developed sequential law and fully known Wöhler curve. The both mentionedapproaches are specially based on evaluation of primary stresses and code provided fatigue curve. Therefore, proposed thirdapproach is anchored in secondary (local) stresses, sequential law and experimental Wöhler curve. Finally the obtained fatiguelives are compared. Thus, it has been concluded that the second approach is more advisable for general use and third approachhas been recommended for detail studies.
Keywords: Fatigue life, Railway bridges, Structural appraisal, Stress concentration, Sequential law
1. Introduction
In the decades of 1960’s and 1970’s, civil engineering
was dominated by design and construction of new civil
infrastructures. Considering the management of structures
in terms of maintenance, member replacement had a wide
acceptance during this period (Junk et al., 2004). At
present, rail authorities all over the world are paying special
attention to evaluate the remaining fatigue life of riveted
railway bridges, since most of these bridges are nearing
the end of their theoretical fatigue lives. Furthermore, the
fatigue behaviour of wrought-iron and older steels, which
were mainly used for the construction of these bridges, is
not well known. These observations coupled with the lack
of information on loading history of these bridges raise
question about their fatigue performance (Imam et al.,
2005). As a result, the assessment of remaining fatigue
life of riveted railway bridges for continuing services has
become more important than ever, especially when
decision making regarding structure replacement, deck
replacement or other major retrofits.
Even though considerable amount of the past studies
have been done on this area, experiences from engineering
practices have indicated that fatigue analysis based on
specification loads and distribution factors usually
underestimates the remaining fatigue life of the existing
bridges by overestimating the live load stress ranges. In
this context fatigue evaluation based on field measured
stress range histograms under actual traffic load proves to
be a more accurate and efficient method for existing
bridges (Köröndi et al., 1998; Constantine et al., 2004).
Most of the present day used fatigue assessment approaches
of railway bridges are generally based on combination of
measured stress histories, Miner’s rule (Miner, 1945) and
railway code provided fatigue curve (also referred to as S-
N or Wöhler curve). However, the Miner’s rule does not
properly take account of loading sequence effect (Suresh,
1998; Dattoma et al., 2006; Mesmacque et al., 2005). As
a result, real fatigue life due to same loading pattern is
higher than the Miner’s expectation for increasing of
loads and it is lower than the Miner’s expectation for
decreasing of loads. Recently, a new damage indicator-
based sequential law (Mesmacque et al., 2005) was
originated to overcome this shortcoming of Miner’s rule
and it has been proved that sequential law gives more
realistic results than Miner’s rule when material is subjected
to variable amplitude loading. As for the authors view,
application of the sequential law to estimate the remaining
fatigue life of existing railway bridges is not yet published.
Main objective of this study is to apply above mentioned
sequential law to evaluate the remaining fatigue life of an
existing railway bridge based on both field measured
*Corresponding authorTel: +81-89-9279816, Fax: +81-89-9279816E-mail: [email protected]
264 Siriwardane Sudath Chaminda et al.
stresses and its current condition. To achieve this objective,
this paper presents methodologies to estimate remaining
fatigue life of a riveted railway bridge based on three
different approaches. The first approach is based on
combination of measured stress histories, Miner’s rule
and code provided Wöhler curve. This approach shows
proximity to present day available approach. The considered
fatigue curve only describes stress ranges, which are
corresponding to more than ten thousands of failure
cycles (usually called as partially known Wöhler curve).
The proposed second approach mainly consists of
measured stress histories, recently developed sequential
law and fully known Wöhler curve. In this approach it is
essential to use the fully known Wöhler curve as the
related fatigue curve. Therefore, the technique, which
utilizes to transfer the partially known Wöhler curve to
fully known curve, is also discussed. This paper also
describes the reasonably accurate procedure to obtain the
past stress histories from present day measured stress
histograms. This is of extreme importance because most
of the bridges do not have the past strain measurements.
The both mentioned approaches are specially based on
evaluation of primary stresses and code provided fatigue
curve. In reality secondary stress (local stress concentration)
effect in riveted connection between the primary members
of bridges was found to be one of main reasons for
fatigue damage (Fisher et al., 1984). Further it has been
identified that the rotational fixity of riveted connection
and the variation in the clamping force of rivets (Akesson,
1994) are the major causes leading to fatigue cracking in
riveted connection (Imam et al., 2005). However, in the
cases of both discussed approaches, some safety factors,
which are included in code provided fatigue curve, help
in some extent to capture the effect of fatigue damage due
to secondary stresses. Since these safety factors are
corresponding to the assessment code, they are mostly
applicable within the design life of the bridge. But the
applicability of these factors for old structures, which are
already exceeded the design age, has not yet been confirmed.
Therefore, third approach of the study is proposed to
evaluate the remaining lives of railway bridges based on
secondary stresses evaluation, multiaxial formulation of
sequential law and experimental Wöhler curve.
Initially paper describes the details of the considered
railway bridge and the appraisals related to condition
evaluation, finite element analysis, material testing,
experimental static and dynamic load testing. Secondly,
the remaining fatigue life of each critical component of
the bridge is discussed separately based on the above
mentioned approaches. Finally comparisons of the results
are made and the validity of obtained fatigue lives is
confirmed. Hence the applicability of the proposed
approaches is discussed.
2. Bridge Description and Present Condition
The selected bridge is one of the longest railway bridge
in Sri Lanka spanning 160 m (Fig. 1). It is a six span
riveted bridge with double lane rail tracks having warren
type semi through trusses, supported on cylindrical piers.
The bridge deck is made of wrought iron and the piers are
made of cast iron casings with infilled concrete. The
bridge was constructed in 1885. Details of trains carried
by the bridge at present and their frequencies illustrate
that the bridge is subjected to variable amplitude loading.
A condition survey was carried out to assess the present
condition of the bridge with the contribution of expert
practicing engineers. Basically it consisted of detailed visual
Figure 1. General views of the riveted bridge.
Figure 2. Some identified corroded locations of the bridge.
Different Approaches for Remaining Fatigue Life Estimation of Critical Members in Railway Bridges 265
examination and in situ measurements of each component
of the bridge. Further, the identified critical connections
of the bridge were subjected to X-Ray examination.
Although the identified critical members with respective
connections should be subjected to various non-destructive
field tests such as ultrasonic examination, magnetic particle
examination, radiographic examination and etc, the lack of
facilities with experiences hindered such detail investigation
of the bridge.
The visual examination revealed that some places of
the bridge have been subjected to mild corrosion due to
the absence of anti corrosive coating (see Fig. 2). Some
rivets connecting the diagonal bracings of the bridge deck
were found to be loose. No visual cracks were observed
in any component of the super structure. However, the
results of X- ray examination showed very small internal
gaps between rivet-plate contact surfaces at a few
connections. In situ measurements of member sizes,
connections and support bearings verified the fact that the
existing drawings were applicable and only few significant
variations were observed. Further the bridge components
have been categorized to several groups entitled “member
set” by considering similar cross sectional properties as
shown in Fig. 3. Finally it was revealed that comparative
maintenance work carried out on the bridge thus far is
satisfactory.
3. Structural Appraisal
Since fatigue evaluation based on field-measured stress
range histograms under actual traffic loads of the bridge is
a more accurate and efficient method for the existing
bridges (Köröndi et al., 1998; Constantine et al., 2004),
this section describes the evaluation methodology of real
performance in the bridge according to the current state.
Extensive laboratory and field-testing, as well as analytical
work were performed to assess the condition of the
superstructure and hence determine the remaining fatigue
life of each critical component of the bridge.
3.1. Testing of bridge materials
Laboratory tests were carried out to determine the
mechanical properties and chemical composition of the
materials that were used for the construction of the
bridge. As-built drawings of the bridge show that only
one type of material has been used for the superstructure
(main truss girder, secondary cross girders, stringers,
bracings and bearings). The sampling of materials, specimen
preparation and testing were carried out according to the
ASTM standards. The chemical analyses as well as
microscopic examinations lead to the conclusion that the
bridge super structure material is wrought iron. The
fatigue strength of the bridge material was obtained by
the rotating bend test. The rotating bending smooth
specimens were prepared from extracted members. From
the results of primary analysis, it was confirmed that one
of the extracted members is subjected to compressive
stress and the other one is operating well bellow the
fatigue strength of bridge constructed material (wrought
iron). Therefore, the test specimens are free of fatigue
damage. As a result, the obtained fatigue curve (from
rotating bend test) exhibits the full fatigue life of the
bridge-constructed material (wrought iron). Further, it
describes only the fatigue behavior of wrought iron
material while closely representing the detail class B
structure (BS5400, 1980). The obtained values for elastic
modulus, yield strength and fatigue strength are 195 GPa,
240 MPa and 155 MPa, respectively.
3.2. Field load testing
Static and dynamic load testing was performed to study
the real behavior of the bridge under various load
combinations. The obtained results were used to develop
a validated analytical model. The test results further
assisted in evaluating actual dynamic factors of each
structural component. The in situ measurements were
performed using two M8 engines, each weighting 1120
kN, which is the heaviest rail traffic in current operation.
The bridge was instrumented with strain gauges placed
at selected locations to measure normal strains. In addition,
the triaxial vibrations were recorded at several locations
using accelerometers. In order to measure free vibration,
accelerations were recorded after the M8 engines had
crossed the bridge. Displacement transducers were used
to measures the vertical deflection at three places around
the mid span area of the bridge. The measured locations
Figure 3. Member set categorization (a) Main truss girder (b) Horizontal bridge deck.
266 Siriwardane Sudath Chaminda et al.
are as shown in Fig. 4. The static and dynamic responses
of strains, displacements and accelerations were acquired
using the sophisticated static and dynamic data loggers.
The different type of load combinations, which are
critical to the bridge were obtained by placing as well as
moving the two test-engines under different speeds. The
considered three static load combinations are defined as
static load case (SLC) 1, 2 and 3 by considering criteria
of maximum shear effect, maximum bending effect
(maximum deflection) and maximum torsion effect to the
bridge deck respectively. The loading positions corresponding
to the mentioned three load cases are shown in Fig. 5.
The criteria, which were considered for dynamic load
combinations, basically illustrate that impact effect to the
bridge with different levels of speed and traction force
effect. Apart from the above mentioned formal field load
testing, the bridge was subjected to a two days continuous
field measurement program under present day actual
traffic. Even though under this investigation many types
of load combinations were considered, only the combinations,
which were used to evaluate the fatigue life of the bridge,
are only specify in the paper. When the bridge is affected
by maximum load due to the present day heaviest train
passage, the obtained sample measurements are shown in
Fig. 6. Finally the dynamic factors were obtained as 1.3,
1.4 and 1.4 for main truss girders, secondary cross girders
and stringers respectively by using the curves illustrated
in Fig. 7.
Figure 4. Locations of the strain gauges and displacement gauges (a) Main truss girder (b) Horizontal bridge deck.
Figure 5. Loading positions corresponding to three static load cases (a) SLC 1 (b) SLC 2 (c) SLC 3.
Different Approaches for Remaining Fatigue Life Estimation of Critical Members in Railway Bridges 267
3.3. Development of the validated analytical model
The Bridge girder was analyzed using the finite element
(FE) method employing the general-purpose package
SAP 2000. A three-dimensional (3D) model (Fig. 8) of
one complete middle span of the bridge was analyzed
under test loadings and actual loadings to determine
stresses in members and deflections, as well as variations
of stresses under moving loads. The bridge deck was
modeled with 3D frame elements and the riveted
connections are assumed to be fully-fixed (Imam et al.,
2005). Even though the cross girders are ideally supported
on bottom chord of the main truss girder, the assumption
of rotational stiffness behavior with magnitude 18200
kNm/rad about second local axis for representative
connection of cross girder to truss were found to be in
better agreement with field measurements than the pinned
connection assumption. Every riveted connections of
cross girders with both stringers and bracings were
assumed to be fixed.
The validation of FE model was done by comparing the
results from analysis with those from field-tests as shown
in Table 1. From the results of static load cases it was
seen that there is good agreement among analytical
results of FE model and the measurements of the actual
bridge. Therefore, the considered 3D frame element
model was termed as “validated analytical model”. Then
the validated analytical model was used to analyze the
bridge for loading combinations involving the train types
Figure 6. Field measurements of the bridge due to heaviest load (a) Stresses at bottom chord of the main girder (b)Stresses at top chord of the main girder (c) Stresses at diagonal members which are usually subjecetd to tensile stress (d)Stresses at diagonal members which are usually subjecetd to compression stress (e) Stresses at stringers (f) Stresses atsecondry cross girders (g) Vertical displacemnt at midspan (h) Vertical acceleration at midspan.
Figure 7. Dynamic factor determination curves (Maximum responds variation with speed) (a) Main truss girder (b)Secondary cross girders (c) Stringers.
268 Siriwardane Sudath Chaminda et al.
specified by the owner, in order to assess the strength,
stability and remaining fatigue life of the bridge.
4. Remaining Fatigue Life Estimation -Proposed Approach 1
Remaining fatigue life evaluation of the critical members
in each member set is discussed in this section. Evaluations
are specially based on primary stresses, which are
determined by the global analysis of whole structure.
Previously mentioned validated analytical model was used
to evaluate primary stresses and Miner’s rule (Miner,
1945) was employed to obtain remaining fatigue life for
the bridge.
4.1. Primary stress evaluation
To apply the Miner’s rule, it is essential to determine
the primary stress ranges generated by the passage of
trains over the bridge. Therefore, it is required to know
the stress cycles (stress histories) distributions of all the
critical members of each member sets for trains that are
included in the timetables obtained from bridge owner for
present and past rail traffic over the bridge. Since the used
types of trains are changed with age of the bridge, the age
had to be divided in to eight periods. According to the
past and present time tables of the bridge, it could be
decided that the traffic sequence is almost constant during
a single week of each period of age. Then the validated
analytical model was used to obtain the static stress
histories of each critical member during a single week of
each period. Due to the dynamic effect of moving trains,
the actual working stresses should be higher than the
analytical static stress. Therefore, the dynamic factor of
each member, which was found experimentally in sub
section 3.2, was used to multiply the static stress to get
the service stresses. Then the stress histories were
converted into stress ranges by using the reservoir
counting method (BS 5400 part 10,1980) and hence the
stress range histogram can be obtained for all the
members. The stress range histograms for one critical
member (critical member of member set DT 3) are drawn
in Fig. 9 for all the periods of its age.
4.2. Determination of fatigue curve
In order to capture the fatigue damage due to the
secondary stresses near the riveted connection or
discontinuities, detail class (BS 5400 part 10, 1980) of
riveted connection based S-N curves are considered for
life estimation. The detail class is determined by considering
the quality of the workmanship and current condition of
the riveted connection. Since field investigations reveal
that the connections of the bridge represent the lapped or
spliced connection behavior with normal clamping force,
riveted connections were classified as class Wrought-iron
(WI) which is proposed by the UK railway assessment
code (Network Rail, 2001). Hence the S-N curve, which
Figure 8. 3D frame element model for single span (a) Deflected shape for SLC 2 (b) Axial force diagram at SLC 2 (c)Bending moment diagram at SLC 2.
Table 1. Comparison of FE analytical results with load test results
Static load caseDisplacement (mm) Stress (MPa)
Location of measurement Load test FEM Location of measurement Load test FEM
SLC 1 Main girder mid span 19.4 21.0 Critical member of DC3 -40.2 -40.6
Critical member of DT3 51.4 57.3
Critical member of MT3 47.3 48.2
SLC 2 Main girder mid span 21.3 22.5 Critical member of DC3 -37.8 -37.7
Critical member of DT3 44.5 43.6
Critical member of MT3 53.5 53.9
SLC 3 Main girder mid span - 19.1 Critical member of DC3 -39.5 -39.9
Critical member of DT3 35.2 41.5
Critical member of MT3 39.0 44.7
Different Approaches for Remaining Fatigue Life Estimation of Critical Members in Railway Bridges 269
is mentioned under the UK railway assessment code for
WI detail class connection (Network Rail, 2001), was
considered as the suitable fatigue curve for this evaluation.
4.3. Fatigue life estimation
With help of stress cycle distributions and the daily rail
frequency, the specified number of yearly repetition of
various stress ranges in the stress histories (n) for each
time period can be found. For the particular ith period of
age, Miner’s summation of fatigue damage, αi = (Σn/N)i x
(Number of years for period where sequence of operations
was constant throughout). Miner’s summation of the total
cumulative fatigue damage for all periods α = Σαi where
i = 1 to k which is equal to number of divided periods of
age. For this bridge, value of k was 8. Therefore remaining
Miner’s summation of fatigue damage at present = (1 −
α). Hence the remaining life of each member, assuming
that the future sequence of passage is the same as that for
Figure 9. Sample stress range histograms for critical member of set DT3 for (a) Period from 1995 to date (b) Period from1985 to 1994 (c) Period from 1975 to 1984 (d) Period from 1970 to 1974 (e) Period from 1946 to 1969 (f) Period from1928 to 1945 (g) Period from 1910 to 1927 (h) Period from 1885 to 1909.
270 Siriwardane Sudath Chaminda et al.
the present period = [(1 − α)/αpresent )] years, where αpresent
= (Σn/N)i.
The remaining fatigue lives for each critical member of
corresponding member set are calculated by following the
above methodology and summarized as shown in Table 2.
Since the Miner’s rule does not properly take account of
loading sequence effect (Suresh, 1998; Dattoma et al.,
2006; Mesmacque et al., 2005), the validity of the obtained
remaining lives cannot be assured. Therefore, the recently
developed new damage indicator based on sequential law
(Mesmacque et al., 2005), is utilized as second approach
for remaining fatigue life estimation of this bridge.
5. Remaining Fatigue Life Estimation - Proposed Approach 2
This section also discuses remaining fatigue life evaluation
of the critical members in each member set. Evaluations
are specially based on primary stresses, which are determined
by the global analysis of whole structure. Previously
mentioned validated analytical model was used to
evaluate primary stresses as above section and uniaxail
sequential law (Mesmacque et al., 2005) was employed
to obtain a more realistic fatigue life for the bridge.
5.1. Primary stress evaluation
In order to apply the uniaxil sequential law, it is
essential to determine the primary stress ranges generated
by the passage of trains over the bridge. Primary stress
evaluation is as same as previous approach and the age of
the bridge also divided in to same number of periods as
before. The stress ranges, which correspond to each
period of age, were obtained by using reservoir counting
method as similar to proposed approach 1 (see subsection
4.1).
5.2. Determination of fatigue curve
The chosen fatigue curve in previous approach only
describes stress ranges, which are corresponding to more
than ten thousands of failure cycles (usually called as
partially known Wöhler curve). In the case of sequential
law it is essential to know the Wöhler curve for full range
of number of cycles. Therefore, the chosen partially
known Wöhler curve in proposed approach 1, which is
mentioned under the UK railway assessment code
(Network Rail, 2001), was transferred to fully known
Wöhler curve by using Kohout and Vechet Wöhler curve
modeling technique (Kohout et al., 2001). The obtained
function and the geometrical shape of new fatigue curve,
which corresponds to class WI riveted connection, are
illustrated in Fig. 10 (a).
5.3. Fatigue life estimation
A new damage indicator based sequential law in
multiaxial fatigue, (Mesmacque et al., 2005), was used to
obtain a more realistic fatigue life for the bridge. A
detailed description of the damage stress model and the
definition of damage indicator, Di is available in the
corresponding paper (Mesmacque et al., 2005). In
appendix A, only the concept is summarized with an
algorithm for understanding.
The new damage indicator (present Di value) was
calculated from the date of bridge construction to the
present by considering the sequence of stress ranges of
each critical member. Assuming that future sequence of
passage is similar to that of the present day, the time
taken to reach the present day’s Di values to one (when
Di = 1 is considered as fatigue failure) was considered as
the remaining fatigue life for each critical member. The
calculated remaining fatigue lives for critical members of
each member set are shown in Table 2. The critical
members of which the stress range is entirely in
compression zone, the effect of fatigue damage were
ignored (BS 5400 part 10,1980).
6. Remaining Fatigue Life Estimation - Proposed Approach 3
Remaining fatigue life evaluation of a critical member
of one set is discussed in this section and evaluations are
Figure 10. W?hler curve for wrought iron material (a) Predicted from UK railway assessment code (b) Predicted fromfatigue test data.
Different Approaches for Remaining Fatigue Life Estimation of Critical Members in Railway Bridges 271
especially based on secondary stresses, which are
generated around the riveted connection due to stress
concentration effect of primary stress. The considered
secondary stresses are usually subjected to multiaxial
state of stress. To evaluate the so-called secondary
stresses, the riveted area of the member was subjected to
further fine mesh FEM analysis. The Wöhler curve,
which was determined from the rotating bend tests of the
material, was utilized for the remaining fatigue life
calculation. Since this evaluation considers more realistic
critical stress histories (secondary stress) and real fatigue
damage behavior of the construction material (experimental
Wöhler curve), it can be said that the fatigue life which
are calculated using this method gives more realistic
predictions to fatigue damage. Since this secondary stress
evaluation approach is also a macroscopic scale and the
highly stressed area has been subjected to various types
of micro structural changes with age, actual fatigue life
may somewhat deviate from this estimation. Therefore,
main objective of this section is not to give an exact result
for fatigue life but to apply the sequential law to estimate
remaining life due to multiaxial fatigue.
6.1. Secondary stress evaluation
When a hot rivet is inserted into the hole of plates in
order to connect them and when the second head is
formed from the protruding shank, the rivet gets shortened
in length due to cooling. However most of shrinkage of
the free rivets is restricted by the connected plates, which
consequently are compressed through the thickness. The
residual tensile force in the rivet and the compressive
force in the plates get balanced each other; i.e. called as
clamping force. Therefore the clamping force from the
rivet generates a complex tri-axial stress state in the
connected plate in the vicinity of the rivet hole (Akesson.,
1994). Finally the clamping force seems to be affected by
the mechanism of distribution of stresses along the
connection. The experience from the field practice reveals
that resulting clamping force could vary substantially due
to normal conditions. Therefore, it could consequently
not be given a reliable value. Furthermore, one can
assume a certain relaxation of the rivet clamping force
due to creep, fretting of the interfacing plate surfaces,
overloading (due to residual plastic deformation) and etc
with the time (when bridge is getting older). However, the
secondary stress analysis, which corresponds to normal
clamping force, becomes more difficult. Because the
geometry of the problem consists of all the rivets with all
members and the connected ply. But considered criterion
of this section corresponds to the low or high clamping
force at the rivets. Therefore, to obtain reasonable
accurate results only critical member without rivets can
be considered as relevant geometry for secondary stress
analysis in this section.
Figure 11. (a) Critical riveted connection of the main truss girder (b) Close view of the critical connection and the criticalmember (c) Schematic representation of the critical member and related areas for primary and local stresses.
Table 2. Summary of remaining fatigue lives for critical members in member sets
Bridge component Member setRemaining Fatigue life from today (years)
Approach 1 Approach 2
Main girder bottom chord MT1 305 323
Main girder bottom chord MT2 156 165
Main girder bottom chord MT3 157 169
Cross girders CG 20 12
Stringers ST 24 13
Truss diagonal (tension member) DT1 179 191
Truss diagonal (tension member) DT2 168 171
Truss diagonal (tension member) DT3 131 138
Truss diagonal (tension member) DT4 152 162
272 Siriwardane Sudath Chaminda et al.
The most critical member of the truss girder (see Fig.
11), which belongs to DT 3, was subjected to further
analysis of secondary stress evaluation. Usually, applied
load transfers to selected member through six rivets. The
nine-node isoperimetric shell elements were used for the
FE mesh as shown in Fig. 12. To represent the effect of
no clamping forces in the rivets, the actual air gap restraint
conditions were applied to represent the unilateral contact
between rivet and plate. Similarly, to simulate the effect
of friction forces due to clamping of the rivet (rotational
fixity of riveted connection), the fully bonding unilateral
contact behavior between rivet and plate was implemented.
The individual deformations of rivets due to loading were
not captured in this model. To make the continuity of
stress field between global model (shown in Fig. 8) and
the sub-model (shown in Fig. 12), it is required to use any
interface between the two models at every iterative step.
In this model, the tensile stress history of the critical
member of member set DT3, which has been obtained
from global model (shown in Fig. 8) in section 3, is
applied on the bottom face (ab of Fig. 12 (a)) as a uniform
pressure P (Imam et al., Article in press, Kiss et al., 2000,
Liu et al., 2006). The position of the ab boundary of the
sub-model (shown in Fig. 12) was determined based on
far field primary stress of the member. In house FEM
code was employed to perform a two surface plasticity
theory based nonlinear kinematic hardening elasto-plastic
analysis. Corresponding increment of far field stress histories
(P) of the member was imposed for the elasto-plastic
incremental analysis. The obtained maximum stress
contours are shown in Fig. 12 for two considered features
of riveted connection. Since related fatigue theory (new
damage indicator based sequential law) describes the
stress field at critical locations in terms of equivalent von
Mises stress (Mesmacque et al., 2005), from this analysis
the von Mises stress histories at critical location due to
daily passage of trains was obtained for fatigue life
evaluation. The von Mises stress histories are converted
in to stress ranges as similar to previous case by using the
reservoir counting method (BS 5400 part 10,1980).
6.2. Determination of fatigue curve
The Wöhler curve was obtained from the rotating bend
test results of bridge material (subsection 3.1). The
rotating bending smooth specimens were prepared from
extracted members. From the results of structural analysis
and field load tests (section 3.2 and 3.3), it was realized
that one of the extracted member is subjected to
compressive stress and the other one is operating well
bellow the fatigue strength of bridge constructed material
(wrought iron). Therefore it could be concluded that the
test specimens are still free of fatigue damage. As a
result, the obtained fatigue curve (from rotating bend test)
exhibits the full fatigue life of the bridge-constructed
material (wrought iron). Further, it describes only the
material behavior of wrought iron and does not represent
any riveted feature of Wrought iron connection. The
mathematical expression for the test curve was obtained
using Kohout and Vechet fatigue curve modeling technique
(Kohout et al., 2001) and the obtained expression with
geometrical shape of the curve is illustrated in Fig. 10 (b).
This new function is proposed to describe the fatigue
curves in both low and high cycle fatigue regions i.e for
the whole region of cycles from tensile strength to
permanent fatigue limit.
6.3. Fatigue life estimation
A new damage indicator based on sequential law in
multiaxial fatigue (Mesmacque et al., 2005) was utilized
in this section to obtain a more realistic service life for the
Figure 12. (a) Fine FE mesh (b) Maximum von Mises stress contour when all six rivets are active (c) Maximum vonMises stress contour when five rivets are active (d) Maximum von Mises stress contour when four rivets are active (e)Maximum von Mises stress contour when three rivets are active (f) Maximum von Mises stress contour when two rivetsare active.
Different Approaches for Remaining Fatigue Life Estimation of Critical Members in Railway Bridges 273
bridge. Although the hypothesis behind the fatigue model
is the same as the previous case, only difference is that
the active stress field is considered as the equivalent von
Misses stress for multiaxial state of stress. Therefore, it is
possible to use the previous fatigue concept with algorithms,
which has been described under Appendix A, for fatigue
life estimation by replacing active stress by equivalent
von Misses stress. The critical member of member set
DT3, which is the most critical member of main girder,
was subjected to fatigue evaluation by considering three
major steps. These steps describe the variation of
remaining fatigue life with condition of riveted connection,
such as level of clamping force effect, degree of surface
contact-ness of rivets and etc.
Step 1: In the first step the significance of remaining
life with effect of clamping force on all the rivets was
considered at once. When a rivet is subjected to plastic
loading, the effect of clamping force begins to deviate
from initial value. But considered connection is operating
in a elastic state of stress when it has a high clamping
force effect. (The maximum von Mises stress value is
132.2 MPa). When only the low clamping forces are
presents at all six rivets, the maximum stresses are
subjected to plastic state (The maximum von Mises stress
value is 256.7 MPa). In this case, there is no clamping
force as described before. Therefore it is possible to
assume that the clamping force of the considered connection
is not significantly changing with the time. As a result,
fatigue life was obtained by considering that the deemed
feature of clamping force remains the same from bridge
construction to the date of failure. The calculated results
are shown in Table 3.
Step 2: In this step, the fatigue life is obtained
considering that the effect of clamping force disappears
from one rivet to the next and the calculated results are
shown in Table 4. Maximum stresses for first five features
of the Table 4 (until five rivets have low clamping force,
remaining one has high clamping force) are operating
bellow the yield limits. (The maximum von Mises stress
values are 194.3, 196.1, 198.2, 208.4, 217 MPa respectively
from top feature of the Table 4). Only the last case (all six
rivets have low clamping force) is subjected to plastic
state of stress (The maximum von Mises stress value is
256.7 MPa). Therefore, here also it is possible to assume
that there is no significant change of clamping force with
the time and it is remained constant (at high clamping
force rivets) or disappeared (at low clamping force rivets)
in the elasto-plastic analysis of this step. The defined
fatigue life in this section describes the time duration
from the date when considered feature of riveted connection
appeared, to the date of fatigue failure. Further it was
considered that the sequence and density of rail passage
were similar to present period of operation.
Step 3: In the third step the fatigue damage is evaluated
based on a criterion called critical state of stress due to
release of contact-ness of rivet while all the rivets have
low clamping force. When the rivet is not properly in
contact with the plate, particular rivets tend to transfer
less or zero amount of total load, and this leads to
unexpected stress redistribution around the riveted area.
In this stage other rivets, which are carrying the load, are
called as active rivets. The fatigue lives were evaluated
stepwise by reducing the contribution of active number of
rivets in the connection. The calculated lives are shown in
Table 5 and the defined fatigue life in this section
describes the time duration from date when considered
feature of riveted connections appeared, to the date of
fatigue failure. Further it was considered that the sequence
and density of rail passage was similar to present period
of operation as similar to previous step.
7. Comparisons and Discussions of Results
Obtained remaining fatigue lives of approach 1 and 2
for critical members of each member sets were compared
each other as shown in Table 2. Even though the
Table 3. Comparison of fatigue lives obtained from three approaches
Member set
Remaining fatigue life (years)
Approach 1 Approach 2Approach 3
High clamping force Low clamping force
DT3 131 138Infinite
(Upper bound)Failed 26 years before
(Lower bound)
Table 4. Fatigue life variation when the effect of clamping force is gradually decreasing
Considered features of the riveted connection Fatigue life (years)
One rivet has low clamping force, remaining five have high clamping force Infinite fatigue life
Two rivets have low clamping force, remaining four have high clamping force Infinite fatigue life
Three rivets have low clamping force, remaining three have high clamping force Infinite fatigue life
Four rivets have low clamping force, remaining two have high clamping force Infinite fatigue life
Five rivets have low clamping force, remaining one has high clamping force Infinite fatigue life
All six rivets have low clamping force 18
274 Siriwardane Sudath Chaminda et al.
predicted lives of two approaches illustrate some amount
of deviation from each other, it can be said that both
approaches have highlighted cross girders become the
most critical members to fatigue failure. Further it reveals
that in case of main girder consisted truss member, the
sequential law-based approach 2 gives higher values than
approach 1 values. But it is the opposite for bridge deck
members. Since the Miner’s rule estimation produces
pessimistic results with increasing of loads and optimistic
results with decreasing loads (Mesmacque et al., 2005), it
can be said that in case of truss members, the global
increment of live load of trains with each period of age
has greater effect on fatigue damage than local variation
(increase and decrease of loading during a week) of
loading in each week. Similarly it can be seen that in the
case of bridge deck members (cross girders, stringers and
bracings), the local variation of loading has a greater
effect on fatigue damage than global increment of loading.
Although these types of conclusions are particular to this
bridge and fatigue criticality of structure varies from
bridge to bridge.
Considered critical member DT3 has been subjected to
three types of approaches to find out remaining fatigue
life. (section 4, section 5 and section 6).The obtained
results are summarized as shown in Table 3 and it shows
that lives, which are obtained through approach 1 and 2,
lie in between upper and lower bound of approach 3.
Therefore, it is possible to confirm that the UK railway
code provided S-N curve, (Network rail, 2001), represents
the normal or intermediate effect of clamping force for
wrought iron riveted connections of existing bridges.
Further, the particular results which are related to the
second step of approach 3 (see Table 4), illustrated that
the effect of clamping force in riveted connection tends to
deviate the fatigue life considerably. Likewise, fatigue life
results, which are shown in Table 5, reveal that the active
number of rivets, which are able to transfer the load, also
changes the fatigue life significantly.
8. Conclusions
Remaining fatigue life estimation of a riveted railway
bridge has been presented based on structural appraisal.
Handling of sequential law in both uniaxail and maltiaxial
fatigue of a railway bridge were described. Hence the
fatigue damages due to both the primary stresses and
secondary stresses of a riveted railway bridge have been
considered and the corresponding approaches were clearly
indicated. Reasonably accurate procedure, which is based
on structural appraisal, was presented to obtain the past
and present stress histories. Finally the study flows to
highlight the major conclusions as follows.
Condition evaluation of the bridge exhibits that the
overall maintenance of the considered bridge is satisfactory,
but there are localized mild corrosion at few places, and
these need immediate attention. Due to fatigue, under
current loadings, speeds and frequencies of operation, the
lowest remaining life found for a member is 12 years.
Thus it may be concluded that the bridge deck can be
used for another 12 years provided that the speed,
frequency, and weight of the trains are not increased. If
proper maintenance work is carried out and the critical
members are replaced with new members with longer
life, the bridge will be able to provide further service.
There is a 10 to 15 years variation among the estimated
remaining lives when comparing of approach 1 and
approach 2. Obviously the effect of members where the
lives are very low (eg. cross girders, stringers) becomes
more significant in percentage terms. These observation
and the phenomenological validity of the new damage
indicator-based sequential law tend to conclude that the
application of sequential law is much advisable for the
evaluation of remaining fatigue life of riveted railway
bridges where the detailed stress histories are known.
Remaining life is 130-140 years for critical member in
set DT3 when all six rivets have normal clamping force
and it has failed before 26 years when it does not have
clamping force at all six rivets. Further, it increases to
infinity when it has high clamping force (see Table 3).
Therefore, it is possible to give an assurance to some
extent that the UK railway code provided design S-N
curve (Fig. 10-(a)) captures the stress concentration effect
of riveted connection when it has a normal or intermediate
effect of clamping force. Therefore obtained function and
the geometrical shape of this fully known design S-N
curve can be employed to assess the fatigue damages of
other wrought iron bridges, which have riveted connections
with normal clamping force.
The fatigue life evaluation based on secondary stress
analysis revealed that the clamping force and the activeness
of rivets play a big role in fatigue damage. Hence, it can
be concluded that it is great important to investigate
accurately the condition of places where the stress
concentration effect is severe such as notch, crack or
Table 5. Fatigue life variation when active numbers of rivets are gradually decreasing while low clamping force
Considered features of the riveted connection Maximum von Mises stress (MPa) Fatigue life (years)
All six rivets are active 256.7 18
Five rivets are active 257.2 17
Four rivets are active 258.4 15
Three rivets are active 260.5 11
Two rivets are active 263.1 9
Different Approaches for Remaining Fatigue Life Estimation of Critical Members in Railway Bridges 275
connection area especially in old bridges for good
judgment of fatigue life. Finally it can be concluded that
the second approach is more advisable for general use
and approach three has been recommended for detail
studies.
Since this investigation has not been captured the effect
of various type of micro structural changes and the effect
of mesoscopic damage variables of particular material at
highly stressed locations, comparisons of above approach
with microsopic level fatigue theories are currently on the
way.
Acknowledgments
The authors wish to express their sincere gratitude to
Senior Professor M.P Ranaweera and other team of
experts who works in the Sri Lankan Railway Bridge
project, for their great advices to carry out this research.
The kind support given by the Sri Lanka Railways (SLR)
is also appreciated.
Appendix A
Here only the concept of the new damage indicator
based sequential law in multiaxial fatigue (Mesmacque et
al., 2005), is summarized with an algorithm for understanding
(see flow chart described in Fig. A-1).
The hypothesis behind the model is that if the physical
state of damage is the same, then fatigue life depends
only on loading condition. Therefore, the life can be
assessed using the Wöhler curve for new structures,
which are still free of damage. At load level i, a certain
stress amplitude σi is applied for a number of cycles ni.
Here the number of cycles to failure from the Wöhler
curve for σi is Ni. Thus, after ni applied cycles, the
residual life is considered as (Ni-ni) for load level i. From
the Wöhler curve, σ(i)ed is said to be ith level damage
stress (otherwise can be introduced as stress relevant to
the residual life) which corresponds to the failure life (Ni-
ni), (see Fig. A-2). Hence, the damage stress, Di is defined
as,
(A-1)
Where σu is the magnitude of ultimate stress. The stress
field can be considered in terms of equivalent von Mises
stress and in this way the model can be applied to the
multiaxial fatigue. In the case of uniaxial loading
condition, the stress field can be considered in terms of
corresponding stress values. The σ(i)ed is equal to σ1 at
first cycle when damage indicator Di = 0 and σ(i)ed is
equal to σu at the last cycle when Di = 1. Therefore, the
damage indicator is normalized to 1 at the failure of
material.
Same damage is then transformed to load level i + 1
and hence damage equivalent stress at level i + 1 is
calculated with the relation,
(A-2)
Further simplification of Eq. (A-2),
(A-3)
where is damage equivalent stress at the level
i + 1. Thus the corresponding equivalent number of
cycles to failure, can be obtained from the Wöhler
Di
σi( )ed σ
i–
σu
σi
–--------------------=
Di
σi( )ed σ
i–
σu
σi
–--------------------
σ'i 1+( )ed σ
i 1+–
σu
σi 1+
–-------------------------------= =
σ'i 1+( )ed D
iσu
σi 1+
–( ) σi 1+
+=
σ'i 1+( )ed
N' i 1+( )R
Figure A-1. Flow chart for damage stress based sequentiallaw.
Figure A-2. Schematic representation of parameters inWöhler curve.
276 Siriwardane Sudath Chaminda et al.
curve as shown in Fig. A-2. The is the magnitude of
applied stress and it is subjected to number of
cycles at the level i + 1. Then the corresponding residual
life at the load level i + 1, is calculated as,
(A-4)
Hence the damage stress σ(i+1)ed, which corresponds to
N(i+1)R at loading level i + 1, can be obtained from the
Wöhler curve as shown in Fig. A-2. Then the cumulative
damage at loading level i + 1 is defined as,
(A-5)
The same procedure is followed until the failure of
material, that is, when damage indicator Di = 1.
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σi 1+
ni 1+( )
Ni 1+( )R
Ni 1+( )R N' i 1+( )R n
i 1+( )–=
Di 1+( )
σi 1+( )ed σ
i 1+–
σuσi 1+
–
------------------------------=