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ENGINEER -Vol. XXXXI, No. 02, pp. 7-16, 2008
© The Institution of Engineers, Sri Lanka
An Accurate Life Estimation Method for
Existing Railway Bridges
P. B. R. Dissanayake and S.C. Siriwardane
Abstract: The paper proposes an accurate methodology to estimate remaining fatigue lives of riveted
railway bridges. The proposed method mainly consists of measured stress histories, recently developed
sequential law and fully known Wbhler curve. Here, it is essential to use the fully known Wdhler curve
as the related fatigue curve. Therefore the technique, which utilizes to transfer the partially known
Wbhler curve to fully known curve, is also discussed under this paper. Since most of the bridges do not
have past strain measurements, this method describes reasonably accurate procedure to obtain the past
stress histories from present day measured stress histograms. Initially paper describes the proposed
method for remaining fatigue life estimation. Secondly remaining fatigue life of an existing railway
bridge is estimated by performing a case study. Case study describes the details of the considered
railway bridge and the appraisals related to condition evaluation, FE analysis, material testing,
experimental static and dynamic load testing. Then the remaining fatigue lives of each critical
components of the bridge are obtained. Hence validity and merits of proposed method is confirmed by
comparing the results with previous method-based fatigue lives.
Keywords: Remaining life, Railway bridge, Sequential law, Health monitoring
1.
Introduction
In past two decades, a significant amount of
efforts have been directed towards the
development of structural health monitoring
and non-destructive assessment methods to
manage civil structures more efficiently [1]. 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 wroughtiron and older
steels,
which were chiefly 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
[2].
As a result, the assessment of remaining
fatigue life of a riveted railway bridge for
continuing services has become more important
than ever, especially when decision making
regarding structure replacement, deck
replacement or other major retrofits.
Experiences from engineering practices have
indicated that fatigue analysis based on
specification loads and distribution factors
usually underestimates the remaining fatigue
life of 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
[3,4].
Most of the present day
fatigue assessment approaches used for railway
bridges are generally based on combination of
measured stress histories, Miner's rule [5] and
railway code provided fatigue curve (also
referred to as S-N or Wohler curve). However,
the Miner's rule does not properly take account
of loading sequence effect
[6-8].
As a result, real
fatigue life due to same loading pattern is higher
than the Miner's expectation for increasing type
loads and it is lower than the Miner's expectation
for decreasing type loads. Recently, a new
damage indicator-based sequential law [8] 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.
Generally, railway bridges are subjected to
changes of traffic load and frequency of
operations with rapid development of
transportation facilities which is encountered
with in the period of age. Therefore, most of the
Eng. (Dr.) P.B. R. Dissanayake, B.Sc. Eng. (Hons) (Peradeniya), C.
Eng., MlE(Sri
Lanka),
M.Eng., Dr. Eng. (Ehime), Senior Lecturer in
Civil Engineering, Department of Civil Engineering, the University
of
Peradeniya.
S. C. Siriwardane,
B.Sc.
Eng. (Hons)
(Peradeniya),
M.Phil
(Peradeniya).
7 ENGINEER
SSlg
railway bridges subjected to variable amplitude
loadings including both increment and
decrement of live load. Since the Miner's rule
produces inaccurate predictions to the real
failure in variable amplitude loading
[6-8],
it is
doubtful to use the Miner's rule for remaining
fatigue life estimation of railway bridges. But as
for the authors view, related investigations to
remove the usage of Miner's rule against the
sequential law for remaining fatigue life
estimation of existing railway bridges have not
been so far attended.
Therefore major objective of this paper is to
check the significance and applicability of the
sequential law to estimate the remaining fatigue
life of a riveted railway bridge by proposing a
new method based on measured stress histories,
recently developed sequential law [8], and
Wohler curve. Railway code provided Wohler
curve only describes stress ranges, which are
corresponding to more than ten thousands of
failure cycles (usually called as partially known
Wohler curve). But for the application of
sequential law to estimate the fatigue life, it is
compulsory to have fully known Wohler curve.
Therefore the technique, which utilizes to
transfer the partially known Wohler curve to
fully known curve, is also discussed under this
paper. Further this paper 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.
Initially paper describes the proposed method
for remaining fatigue life estimation. The details
of case study railway bridge and the appraisals
related to condition evaluation, material testing,
field static and dynamic load testing, structural
analysis are mentioned. Then, the remaining
fatigue life of each critical component of the
bridge is discussed. Finally comparisons of the
results are made with Miner's rule-based
previous estimation. Hence, validity and
applicability of the proposed approach is
discussed.
2. Proposed method for remaining
fatigue life estimation
Proposed method for remaining fatigue life
estimation of an existing riveted railway bridge
is discussed in this section. Particular
methodology follows three major steps such as
stress evaluation, determination of Wohler
curve and application of sequential law.
2.1 Structural appraisal and stress evaluation
In order to apply the uniaxial sequential law, 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 for trains that are included
in present and past timetables. 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 existing bridges
[3,4],
this section describes
the evaluation methodology of real stresses in
the bridge related to the current state.
Initially a condition survey has to be carried out
to assess the present geometric condition and
damages. Generally, it consists of detailed visual
examination, in-situ measurements of each
component of the bridge and non-destructive
field examinations. Then laboratory tests will
need to be carried out to determine the current
state of mechanical properties and chemical
composition of the bridge materials. The static
and dynamic load testing can be recommended
as next major step to study the real behavior of
the bridge under various load combinations.
The obtained results is used to develop a proper
analytical model and further assists in
evaluating actual dynamic factors of each
structural component. Finally the bridge will be
subjected to finite element (FE) analysis under
test and actual loadings to determine stresses
and deflections, as well as variations of stresses
under moving loads. Material properties which
are obtained through laboratory tests and
current geometric properties obtained from
condition assessment are applied to the FE
model for more realistic outputs. The validation
of the FE model has to be carried out by
comparing the results from analysis with those
from field-tests. The FE model, which gives
better comparison to load test results can be
nominated as "validated analytical model".
Hence, a validated analytical model is used to
obtain past and present static stress histories
due to passage of trains specified by the owner.
|88j8
ENGINEER 8
Due to the dynamic effect of moving trains, the
actual working stresses should be higher than
the analytical static stress. Therefore, the
dynamic factors of each member, which are
found experimentally, is used to multiply the
static stress to get the service stresses. Finally,
the stress histories have to be converted into
stress ranges by using the reservoir counting
method (BS 5400 part 10,1980). The described
stress evaluation procedure is briefly
summarized as shown in
pig-1.
Condition Survey
Material Testing
1
Field Load Testing
1
Development of validated analytical model
i
Stress Evaluation (Past and Present)
Figure
1.
Flow of stress evaluation procedure
2.2 Determination of Wbhler curve
To capture the fatigue damage due to the
secondary stresses near the riveted connection
or discontinuities, detail class [9] of riveted
connection based Wohler 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.
Generally, the S-N curve, which is mentioned
under the UK railway assessment code [10], is
considered as the suitable fatigue curve for this
evaluation. But, chosen fatigue curve only
describes stress ranges, which are corresponding
to more than ten thousands of failure cycles
(usually called as partially known Wbhler
curve). In the case of sequential law it is
essential to know the Wbhler curve for full
range of the number of cycles. Therefore, the
chosen partially known Wbhler curve, which is
mentioned under the UK railway assessment
code [10], has to be transferred to fully known
Wbhler curve by using Kohout and Vechet
Wbhler curve modeling technique [11].
2.3 Application of sequential law
A new damage indicator based sequential law in
multiaxial fatigue8), is used to obtain a more
realistic fatigue life for the bridge. A detailed
description of the sequential law is available in
the corresponding paper [8]. Here only the
concept is summarized with an algorithm for
understanding (see Fig. 2).
A new damage indicator based sequential
r»i
cycles
at
a
i
stress
Ni failure cycles number
at oi
stress level (Wohler)
1
Nw=Ni-nf.
Residual
life
a
(ijfrf:
Damage stress
for Nm
cycles (Wohler curve)
I
A
=
n/
cycles
at oi
stress level
Fatigue
failure
Damage
transformation
from
previous
step
to
next step
a
l(i)ed
associated number of cycles
N^R
(Wohler curve)
N(,)R
=N/(i)R-ni:
I
a
:
Damage stress for
N
^ cycles (Wohler)
I
i=>i+l
Figure
2.
Flow chart for damage stress
based sequential law
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
Wbhler curve for new structures, which are still
free of damage. At load level i, a certain stress
amplitude a is applied for a number of cycles n.
Here the number of cycles to failure from the
Wbhler curve for a is N.. Thus, after n. applied
cycles, the residual life is considered as {N.-n)
for load level i. From the Wbhler curve, a,.. , is
ENGINEER
Figure
3.
Schematic representation of parameters in
Wohler curve
said to be zth level damage stress (otherwise can
be introduced as stress relevant to the residual
life) which corresponds to the failure life (Ni-ni),
(see Fig. 3). Hence, the damage stress, Di is
defined as,
D,
= (1)
where ou 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 a is equal to
o, at first cycle when damage indicator D=0 and
a(j)ed
is equal to au at the last cycle when D=l.
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+l is calculated with the relation,
D,
= '(i+\)ed 'i+l (2)
'i+l
Further simplification of Eq. (2),
'{:+\)ed (3)
where a'iH)edis damage equivalent stress at the
level i+l. Thus, the corresponding equivalent
number of cycles to failure, N'. can be
obtained from the Wohler curve as shown in
Fig. 3. The o'.+I is the magnitude of applied
stress and it is subjected to n.+] number of cycles
at the level i+l. Then the corresponding residual
life at the load level i+l, N' is calculated as,
(i+l) (4)
Hence the damage stress o ed, which
corresponds to N . R at loading level i+l, can
be obtained from the Wohler curve as shown in
Fig. 3. Then the cumulative damage at loading
level i+l is defined as,
a(i+\)ed aM (5)
cr„
- a
i+l
The same procedure is followed until the failure
of material, that is, when damage indicator D.=l.
3.
Case study: Remaining life
estimation of a riveted railway
bridge
The selected bridge to evaluate the remaining
fatigue life is one of the longest railway bridges
in Sri Lanka having a length of 160 m (Fig.4). 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 has 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.
il
f vy
• i
Figure
4.
General vieivs of the riveted bridge
ENGINEER 10
Figure 5. Some of photos to show the corrosion of the bridge
Ml 1 MCI MCI MC2 MC2 MC2 MC3 MC3
\ DT2 J v DT4 J ^ ITTS ,/\
/uci\ /n 4\
/bcs\
/
MT1 MT1 MTl MT2 MT2 MT2 MT3 MT3
(a)
If!
S'l
/ST
ST\
ST
K ST J
1 ST M |
^ ST /
1 ST
t(N/
CG
ST
CG
ST
CG
ST
CG
ST
k
ST
ST
/ \ ST
\ V
1
\
EB2 EB1 EB4 BR3 _ x
,!i<;
(b)
Figure
6.
Member set categorization (a) Main truss girder
(b)
Horizontal bridge deck
3.1 Structural appraisal and stress evaluation
Extensive condition survey, laboratory testing,
field-testing, and analytical work were
performed to obtain more realistic stresses of
critical components of the bridge [12,13].
3.1.1 Condition survey
The condition survey revealed that some places
of the bridge have been subjected to mild
corrosion due to the absence of anti corrosive
coating (see Fig.5). No visual cracks were
observed in any component of the super
structure. 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.6. Finally it
can be said that comparative maintenance work
carried out on the bridge thus far is satisfactory.
3.1.2 Material testing
The sampling of materials, specimen
preparation and testing were carried out
according to the ASTM standards. The chemical
analyses as well as microscopic examinations
led to conclusion that the bridge super structure
material is wrought iron. A summary of the
Table 1: Mechanical properties of super structure
material
Property Values
Density
Yield Strength in tension
Ultimate tensile strength
Elastic modulus
Fatigue limit
7600 kg/m3
240 MPa
383 MPa
195 GPa
155 MPa
obtained mechanical properties of the wrought
iron is shown in Table 1.
3.1.3 Field load testing
Static and dynamic load testings were
performed to study the real behavior of the
bridge under various load combinations. 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 vertical
deflection at three places around the mid span
area of the bridge.
11 ENGINEER
Time (sec) Time (sec) Time (sec)
(d) (e) (f)
Figure
7.
Field measurements of the bridge due to heaviest load (a) Stresses at bottom chord of the main girder (b)
Stresses at diagonal members which are usually subjecetd to tensile stress
(c)
Stresses at stringers (d) Stresses at
secondry cross girders (e) Vertical displacemnt at midspan (f) Vertical acceleration at midspan.
To acquire static and dynamic responses of
strains, displacements and accelerations,
sophisticated static and dynamic data loggers
were used. To obtain the different type of load
combinations, which are critical to the bridge,
the two test-engines were placed as well as
moved 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 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 shown in this 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. 7. Maximum
values of responses which were measured while
moving the test-engine with different speeds,
are plotted against the speed of the train (Some
of plots are illustrated in pig- 8). Finally the
dynamic factors were obtained as
1.3,1.4
and 1.4
for main truss girders, secondary cross girders
by the ratio of maximum dynamic response to
static response.
3.1.4 Development of 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- 9) 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
ENGINEER 12
0 10 20 30 40 50
Train speed (km/h)
10 20 30 40 50
Train speed (km/h)
(a) (b)
0 10 20 30 40 50
Train speed (km/h)
(c)
Fig. 8. Dynamic factor determination curves (Maxitnutn responds variation with speed)
(a) Main truss girder
(b)
Secondary cross girders
(c)
Stringers
Figure
9.
3D frame element model for single span
(a) Deflected shape for SLC 2
(b)
Axial force diagram atSLC2
(c)
Bending moment diagram at SLC 2
bridge deck was modeled with 3D frame
elements and the riveted connections are
assumed to be fully-fixed [2]. Even though the
cross girders are ideally supported on bottom
chord of the main truss girder, the assumption
of rotational stiffness behavior with small
magnitude 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 carried out by
comparing the results from analysis with those
from field-tests as shown in Table 2. From the
results of static load cases it was seen that there
is a good agreement among analytical results of
the FE model and the measurement of the actual
bridge. Therefore, the considered 3D frame
element model was defined as ivalidated
analytical model.
3.1.5 Stress evaluation
Since the types of trains used are changed with
age of the bridge, the age had to be divided in to
several periods. According to the past and
present timetables of the bridge, it could be
assumed 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, 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 [9].
3.2 Determination of Wohler curve
Field investigations reveal that the riveted
connections of the bridge represent lapped or
spliced connection behavior with normal
clamping force. Therefore, riveted connections
were classified as class Wrought-iron (WI),
which is proposed by the UK railway
assessment code [10]. Hence the S-N curve,
which is mentioned under the UK railway
13 ENGINEER
Table 2. Comparison of FE analytical results with load test results
Static load case Displacement (mm) Stress (MPa)
Static load case
Location of measurement Load test FEM Location of measurement Load test FEM
SLC1 Main girder mid span 19.4 21.0 Critical member of DC3 -40.2 -40.6
Main girder mid span Critical member of DT3 51.4 57.3
Critical member of MT3 47.3 48.2
SLC2 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 Main girder mid span Critical member of DT3 35.2 41.5
Critical member of MT3 39.0 44.7
assessment code for WI detail class connection
[10],
was transferred to fully known Wohler
curve by using Kohout and Vechet Wohler
curve modeling technique [11]. The obtained
function and the geometrical shape of new
fatigue curve, which corresponds to class WI
riveted connection, are illustrated in Fig.10.
3.3 Application of sequential law
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 values
to one (when D.=l 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 3. The
critical members of which the stress range is
entirely in compression zone, the effect of
fatigue damage were ignored [9].
I.E-HW
l£*G2
l.E+04 l£405 I.EWK l.E+07
1.E+A8
1E^» 1£*10
Number of Cycln N
Figure
10.
Predicted Wohler curve for wrought iron
material
4.
Comparisons of remaining fatigue
lives
Obtained remaining fatigue lives for the critical
members of each member sets were compared
with the Mineris rule-based estimations
(previously used method) as shown in Table 3.
Even though the predicted lives from the two
approaches illustrates some amount of
deviation from each other, it can be said that
both approaches have highlighted that the cross
girders becomes the most critical members to
fatigue failure. Further it reveals that in the case
of truss members of main girder, the sequential
law-based remaining fatigue life gives higher
values than Mineris rule-based values.
However, it is the opposite for bridge deck
members. Since Mineris rule estimation
produces pessimistic results with increase of
loads and optimistic results with decreases
loads [8], 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 type of a bridge and fatigue
criticality of structure could vary from bridge to
bridge.
5.
Conclusions
Condition evaluation of the bridge exhibits that
the overall maintenance of the considered
bridge is satisfactory. However, there are
localized mild corrosion at few places, and these
1«|S ENGINEER 14
Table 3. Summary
of
remaining fatigue lives
for
critical members
in
member sets
Bridge component Member
set
Remaining Fatigue life from today (years)
Bridge component
Miner's Rule Sequential
Law
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) DTI 179 191
Truss diagonal (tension member) DT2 168 171
Truss diagonal (tension member) DT3 131 138
Truss diagonal (tension member) DT4 152 162
need immediate attention.
It is
found that,
the
lowest remaining life
due to
fatigue
for a
member
is 12
years, under current loadings,
speeds
and
frequencies. 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.
Comparison
of
Miner's estimations
and
sequential
law
predictions
(the
proposed
method) illustrates considerable amount
of
deviation
of
remaining lives. This deviation
and
the phenomelogical validity
of the new
damage
indicator-based sequential
law
tend
to
conclude
that
the
application
of
Miner's rule-based
previous approach
is not
much advisable
for the
evaluation
of
remaining fatigue life
of
riveted
railway bridges
in
future
and
proposed method
is recommended
for
general application.
The
obtained function
and the
geometrical shape
of
the fully known Wohler curve, which
corresponds
to
normal
or
intermediate effect
of
clamping force
at
riveted connections,
can be
employed
to
assess
the
fatigue damages
of
other
wrought iron riveted bridges.
Since this investigation
has not
been captured
the effect
of
various types
of
micro structural
changes
at
highly stressed locations,
comparisons
of
above approach with microsopic
level fatigue theories
are
currently underway.
Acknowledgement
The authors wish
to
express their sincere
gratitude
to
Senior Professor
M.P
Ranaweera
and other team
of
experts
who
work
in the Sri
Lankan Railway Bridge project.
The
kind
support given
by the Sri
Lanka Railways
(SLR)
is also appreciated.
References
1.
Sherif,
B.,
Shuichi,
M. and
Toshiyuki,
O.,
Nondestructive damage detection scheme
for
steel bridges,
Journal
of
Applied
Mechanics,
JSCE,
Vol.9,
pp.
63-74, 2006.
2.
Imam,
B.,
Righiniotis,
T.D. and
Chryssanthopoulos, M.
K.,
Fatigue Assessment
of
Riveted Railway Bridges.
International Journal
of
Steel
Structures,
KSSC,
Vol.5 (5), pp. 485-494,2005.
3.
Kbrondi, L., Szittner,
A.,
Kall6, M.
and
Krisr6f,
L.,
Determination
of
fatigue safety
and
remaining
fatigue life
on a
riveted railway bridge
by
measurement.
Journal
of
Constructional Steel
Research, Elsevier,
paper number 327, Vol.
46
(1-3),
pp.
430,1998.
4.
Constantine,
C.S.,
Ioannis,
G.R. and
John,
Ch. E.,
Condition assessment
and
retrofit
of
historic
steel-truss railway bridges.
Journal
of
Constructional
Steel
Research,
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