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case study fatigue rivets
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Residual Lifetime Assessment of an AncientRiveted Steel Road Bridge
A. M. P. de Jesus*†, M. A. V. Figueiredo‡†, A. S. Ribeiro*†, P. M. S. T. de Castro‡†
and A. A. Fernandes‡†
*Engineering Department, University of Tras-os-Montes and Alto Douro, 5001-801 Vila Real, Portugal†IDMEC-FEUP, Campus FEUP, Porto, Portugal‡Mechanical Engineering Department, Faculty of Engineering, University of Porto, Porto, Portugal
ABSTRACT: The present paper reports research work carried out to characterise the fatigue
behaviour of the Portuguese Pinhao riveted road bridge, built in 1903 over the Douro river. The
present traffic conditions are completely different from those foreseen by the bridge designer, rising
new concerns, with respect to the bridge integrity, namely its fatigue behaviour. An experimental
programme was performed using original material removed from the bridge. The chemical com-
position and microstructures of the removed materials were characterised. Also, the notch
toughness, at room temperature, was evaluated using both notch impact and Crack Opening
Displacement (COD) tests. Fatigue crack growth tests were also used to evaluate the fatigue crack
growth behaviour. Finally, fatigue tests of riveted joints were conducted in order to define an
appropriate S-N curve. The experimental results were used to evaluate the residual fatigue strength
of the bridge, adopting both S-N and Fracture Mechanics approaches. The analysis revealed a good
tolerance to fatigue cracking, even in the presence of small fatigue cracks, detected in the joints.
KEY WORDS: ancient road bridge, fatigue behaviour, fracture mechanics, riveted joints, S-N
approach
Introduction
The maintenance and safety of existing bridges is a
major concern of governmental agencies. In particu-
lar, the safety of old riveted highway bridges fabricated
and placed into service at the end of the 19th/begin-
ning of 20th centuries deserve a particular attention,
since they were designed taking into account traffic
conditions, both in terms of vehicle gross weight and
frequency, completely different from those observed
currently. Also, the current design procedures were
not yet fully developed or even did not exist in the 19th
century and designer engineers were not aware of
some important phenomena such as fatigue. Fatigue
was only intensively studied in the 20th century. In
order to assure high safety levels in old riveted steel
bridges, highway authorities have to invest heavily in
their maintenance and retrofitting. In this context,
knowledge of the fatigue behaviour of riveted joints is
of paramount importance.
The present paper reports research work carried out
to characterise the fatigue behaviour of the Portuguese
Pinhao riveted road bridge, designed by Eiffel at the
end of 19th century and built between 1903 and 1906.
A similar study was carried out by authors with the
Luiz I bridge of Porto [1]. The Pinhao bridge, illustrated
in Figure 1, crosses the Douro river and links Pinhao to
Sao Joao da Pesqueira and Peso da Regua. The bridge
has three spans of 68.8 meters each and one span of 10
meters; there is only one deck with 6 meters width,
divided in one traffic lane with 4.60 meters width and
two sidewalks with 0.675 meters width each. The
paper reports studies concerning the assessment of the
residual fatigue life of the bridge. Both traditional S-N
approaches and Fracture Mechanics approaches were
used in the analysis. The study is supported by an
experimental programme aimed the evaluation of
material properties such as tensile strength, toughness
and crack growth properties. Also, fatigue tests of riv-
eted joints were carried out. The material and riveted
joints were extracted from bridge members, which
were replaced by new material, according to a previ-
ously approved procedure.
Experimental Programme
The experimental programme was carried out with
material extracted from the bridge. One piece
1500 mm in length was extracted from a diagonal
e402 � 2009 The Authors. Journal compilation � 2009 Blackwell Publishing Ltd j Strain (2011) 47, e402–e415doi: 10.1111/j.1475-1305.2008.00596.x
An International Journal for Experimental Mechanics
member. Another piece 1400 mm in length was
removed from a bracing member. Both cuts were
located in the first span, from Pinhao side, as illus-
trated in Figure 2(A). The members were in service
since the construction of the bridge. Figure 2(B)
illustrates the cross sections of the diagonal and
bracing members. The bracing cross section is com-
posed by two equal-leg angles, riveted to each other.
These two cross sections are representative of the
cross sections existing in the bridge.
Several types of specimens were prepared using
the material samples removed from the bridge.
These specimens were used in chemical and
metallographic analyses, hardness measurements,
tensile tests, notch toughness tests, fatigue crack
propagation tests and fatigue tests of riveted lap
joints. The riveted lap joints were prepared only
from the bracing member. All other specimens were
machined from both members extracted from the
bridge. This paper only gives a brief description of
Figure 1: Riveted road Pinhao bridge
Figure 2: Locations of the extracted diagonal and bracing members (A) and respective cross sections (B)
� 2009 The Authors. Journal compilation � 2009 Blackwell Publishing Ltd j Strain (2011) 47, e402–e415 e403doi: 10.1111/j.1475-1305.2008.00596.x
A. M. P. de Jesus et al. : Residual Lifetime Assessment of a Road Bridge
the experimental programme. More details can be
found in reference [2].
Chemical composition
The chemical composition of the materials of the
diagonal and bracing were measured using spark
emission spectrometry. Six samples, three from the
diagonal and three from the bracing, were analyzed.
The chemical analysis revealed good homogeneity in
the chemical composition. The mean values of the
chemical composition of the materials are summar-
ised in Table 1. The phosphorus and sulphur con-
tents are low and are within the acceptable values for
modern steels. According to the chemical composi-
tion, the analyzed steels are carbon steels with small
amounts of Mn, Si and C.
Figure 3 illustrates the typical microstructure of the
materials from the diagonal and bracing members.
The material from the diagonal exhibits a micro-
structure almost composed by ferrite as is expected by
the low carbon content. The material from the
bracing member shows a microstructure of ferrite
with low content of perlite.
Tensile strength properties
The tensile strength properties were evaluated using
14 specimens, seven from the diagonal and seven
from the bracing. The specimens were prepared and
tested according to the NP 10002-1 standard [3]. The
mean values of the properties are summarised in
Table 2, where Rm is the ultimate tensile strength,
ReH is the higher yield stress, A is the elongation at
fracture and Z is the reduction in cross section area,
at breaking point. The materials exhibit a high duc-
tility and elastoplastic behaviour with almost null
strain hardening. This behaviour is compatible with
the observed microstructure of ferrite with low vol-
umetric fraction of perlite.
Hardness measurements
Vickers hardness, HV40, were measured accordingly
the procedures of the NP711-1 standard [4]. Six
samples of material, three from the diagonal and
three from the bracing, were subjected to hardness
measurements. A mean hardness of 108 HV40 was
found for the material of the diagonal; a value of 116
HV40 was found for the material of the bracing. The
measured values presented small scatter.
Notch toughness testing
The notch toughness of the materials was measured
using both Charpy V-notch impact and COD tests.
The Charpy V-notch impact tests were conducted
according to the NP10045-1 standard [5]. A total of
16 specimens (thickness of 7.5 mm) were tested
namely, eight extracted from the diagonal and eight
extracted from the bracing members. One half of the
specimens were cut according to the longitudinal
direction and the others according to the transverse
direction. Tests were conducted at room temperature
(19 �C). The material from the diagonal exhibits a
mean Charpy V-notch strength of 89 J, in the longi-
tudinal direction, and 20 J in the transverse direc-
tion. The material from the bracing exhibits a mean
Charpy V-notch strength of 89 J, in the longitudinal
direction, and 26 J in the transverse direction. A sig-
nificant difference in the Charpy V-notch strength
Table 1: Chemical composition of the materials
%C %Si %Mn %P %S
Diagonal 0.06 <0.01 0.04 0.04 0.03
Bracing 0.05 <0.01 0.34 0.04 0.04
Figure 3: Microstructure of the diagonal (left) and bracing (right) materials
e404 � 2009 The Authors. Journal compilation � 2009 Blackwell Publishing Ltd j Strain (2011) 47, e402–e415doi: 10.1111/j.1475-1305.2008.00596.x
Residual Lifetime Assessment of a Road Bridge : A. M. P. de Jesus et al.
between the longitudinal and transversal directions is
observed, which can be justified by the oriented
microstructure (oriented grains, lined-up inclusions)
induced by the rolling process, used to produce the
bridge members. According to the Eurocode [6], the
minimum allowable Charpy V notch strength, for a
material classified according the EN 10025 Class B
(same type of the investigated material), should be
27 J at the temperature of 25 �C. Thus, the material
of the bridge presents very acceptable toughness
properties, even for modern design requirements.
Crack opening displacement tests were also carried
out according to the BS 5762 standard [7]. The
thicknesses of the specimens were limited by the
thickness of the bridge members from which they
were extracted. A total of six specimens were tested,
three from the material of the diagonal (5 mm thick)
and three from the material from the bracing (9 mm
thick). All the specimens were machined in the
material longitudinal direction and they were tested
at the room temperature. The two materials exhibit a
good toughness at the room temperature as is con-
firmed by the average CTOD values of 0.949 mm and
0.765 mm, for the bracing and diagonal members
respectively, and measured at maximum load. The
required minimum CTOD values, at working tem-
perature, should be 0.25 mm.
Crack propagation tests
Crack growth studies were also undertaken. Tests
were conducted according to the ASTM E647 stan-
dard [8] using the compact tension (CT) geometry.
The geometry of the specimens, with the respective
dimensions, is illustrated in Figure 4. The specimens
were machined from the bracing and diagonal
members, and were oriented according to the longi-
tudinal (rolling) direction of the members (T-L crack
plane orientation).
The critical elements of the bridge are the vertical
members connecting the arch to the deck. These
members are subjected to tensile loading. They sup-
port the structure self weight plus the variable load-
ing induced by vehicle crossing. Taking into account
the simply support configuration of the bridge spans,
the vertical elements experience a minimum tensile
stress corresponding to the structure self weight and a
maximum tensile stress corresponding to the vehicle
crossing. Thus, positive stress ratios are expected for
these members, being the actual stress ratio depen-
dent on vehicle weight/bridge self weight ratio.
R = 0.0 and R = 0.5 are two representative values. The
evaluation of the crack growth rates for these two
distinct values allows the assessment of the stress
ratio influence.
A total of ten specimens were tested, four from the
material of the diagonal (D) and six from the material
of the bracing (B), for the two referred stress ratios.
The experimental results were correlated using a
power relation between the crack growth rate and the
stress intensity factor range, as proposed by Paris and
Erdogan [9]:
Table 2: Tensile strength properties
Rm (MPa) ReH (MPa) A (%) Z (%)
Diagonal 367 284 33 70
Bracing 355 328 33 72
Figure 4: Geometry and dimensions of the Compact Tension specimen (dimensions in mm)
� 2009 The Authors. Journal compilation � 2009 Blackwell Publishing Ltd j Strain (2011) 47, e402–e415 e405doi: 10.1111/j.1475-1305.2008.00596.x
A. M. P. de Jesus et al. : Residual Lifetime Assessment of a Road Bridge
da=dN ¼ C:DKm (1)
where da/dN is the crack propagation rate, DK is the
stress intensity factor range and C and m are material
constants. Table 3 summarises the constants of the
Paris’s law, where R2 is the determination coefficient
from the linear regression analysis.
Figure 5 represents, for each tested specimen, the
crack length, a, as a function of the number of cycles,
N. The figure also includes, for each test, the maxi-
mum applied load, Fmax, and the stress ratio, R. From
these curves and using the incremental polynomial
method with seven data points, as described in the
ASTM E647 standard, the crack growth rates were
computed [8]. Figure 6 compares the crack growth
data obtained for the bracing and diagonal materials
for the stress ratios, R = 0.0 and R = 0.5. It was veri-
fied that the stress ratio influences the crack growth
data for the bracing material, mainly for lower stress
intensity factor ranges. The increase of the stress ratio
leads to higher crack growth rates. For the material
from the diagonal, the stress ratio does not influence
the respective crack growth rate. The material from
the diagonal exhibits lower crack growth rates than
Table 3: Crack propagation constants
Material
R = 0.0 R = 0.5 R = 0.0 + R = 0.5
C* m R2 C* m R2 C* m R2
Diagonal 1.9900e-17 4.3410 0.9838 2.9374e-14 3.2833 0.9874 1.8697e-15 3.6793 0.9291
Bracing 1.3128e-15 3.7482 0.9841 2.2866e-14 3.3208 0.9910 4.8966e-15 3.5548 0.9838
Diag.+Brac. 2.7874e-16 3.9684 0.9696 2.4849e-14 3.3085 0.9896 3.1961e-15 3.6117 0.9618
*da/dN expressed in mm/cycle and DK in N.mm)1.5.
(A)
(B)
Figure 5: Crack growth versus cycles curves: A) diagonal member; B) bracing member
e406 � 2009 The Authors. Journal compilation � 2009 Blackwell Publishing Ltd j Strain (2011) 47, e402–e415doi: 10.1111/j.1475-1305.2008.00596.x
Residual Lifetime Assessment of a Road Bridge : A. M. P. de Jesus et al.
the bracing material, essentially for lower stress
intensity ranges and for R = 0.0. For R = 0.5 it can be
concluded that both materials show the same crack
growth rates. Figure 6 also illustrates the correlation
of the crack growth data using the Paris’s law. In spite
of the simplicity of the Paris’s law, it gives a very
satisfactory description of the experimental data. As a
final conclusion, the influence of the stress ratio on
crack growth is small and the materials from the two
members present very similar fatigue crack propaga-
tion behaviours.
Fatigue tests of riveted joints
Finally, fatigue tests of riveted joints were performed.
The specimens were machined from the bracing
member according to the nominal dimensions of
Figure 7. Original riveted assemblies were considered
in these tests. Figure 7 also presents a macrograph of
the rivet. The observation of the macrograph of the
rivet allowed the estimation of the hole diameter
(/21 mm) and the rivet diameter (/20 mm). A total
of seven specimens were tested under stress control,
with stress ratio, R = 0.1. The number of specimens
was limited by the amount of available material. The
results of the fatigue tests are summarised in Table 4.
The fracture surfaces of specimens CF1, CF4 and CF5
showed that fatigue cracks initiated at existing flaws.
These initial flaws were developed during the opera-
tion of the bridge. Figure 8 illustrates the initial flaws.
A linear regression analysis was applied using the
data points, previously transformed with logarithms,
resulting the following expression for the S-N curve:
log Dr ¼ 3:3108� 0:2226 log Nf (2)
where Dr is the remote stress range in MPa and Nf is
the number of cycles to failure.
Figure 9 illustrates the fatigue strength data
obtained for the riveted connection. This figure also
points out the S-N curve. A determination coefficient,
R2, equal to 0.801 was found which is within the
usual values obtained in correlation of fatigue
strength data.
The proposed S-N curve includes the damaging
effects of the previous loading history, since it
resulted from fatigue tests of original riveted assem-
blies, which experienced the referred loading history.
Therefore, the proposed S-N curve can be applied to
perform residual life calculations of the bridge under
investigation, if the future loading history is foreseen,
without taking into account the previous loading
history. The complete loading history of ancient
riveted bridges is generally unknown, since it can
Figure 6: Comparison between the crack growth data for the bracing and diagonal materials: (A) diagonal; (B) bracing; (C) R = 0.0
and (D) R = 0.5
� 2009 The Authors. Journal compilation � 2009 Blackwell Publishing Ltd j Strain (2011) 47, e402–e415 e407doi: 10.1111/j.1475-1305.2008.00596.x
A. M. P. de Jesus et al. : Residual Lifetime Assessment of a Road Bridge
represent very long operation periods, in many cases
more than 100 years. If the available S-N curves
resulted from tests of undamaged riveted connec-
tions, any residual life calculation must take into
account the whole loading history – the previous and
future ones.
It is worthwhile to refer that during the fatigue
tests and after the very first cycles, a relative slip
between the riveted members is observed, being the
rivets subjected to shear loading. This a clear indica-
tion of low clamping forces of the rivets.
Residual Fatigue Life of Riveted JointsUsing S-N Approach
Studies conducted by Fisher et al. [10], DiBattista
et al. [11] and others showed that the AASHTO [12]
class D S-N curve, for riveted joints, leads to conser-
vative predictions of the fatigue strength of riveted
structural details from bridges. Figure 10 presents
fatigue results for riveted joints gathered by
DiBattista et al. [11] as well as results obtained with
fatigue tests of riveted joints from Luiz I bridge [1]
Figure 7: Nominal geometry of the riveted joint (left) and macrograph of the rivet zone (right) (dimensions in mm)
Table 4: Results of the fatigue tests of riveted joints
Specimens
Stress range Fatigue life
MPa cycles
CF2 168.3 86140
CF3 124.1 635172
CF4 103.6 574452*
CF1 83.5 1922024*
CF6 83.5 2243676†
CF5 61.8 1450789*
CF7 61.8 ‡
*Fracture surface presents initial cracks.
†Rupture occurred outside the riveted connection.
‡Run out (test interrupted at 107 cycles).
5
2.8
11
1.4
22
2
10 10.5
4
3
41,4
2,8
Figure 8: Initial crack-like flaws observed in some riveted joints (dimensions in mm)
e408 � 2009 The Authors. Journal compilation � 2009 Blackwell Publishing Ltd j Strain (2011) 47, e402–e415doi: 10.1111/j.1475-1305.2008.00596.x
Residual Lifetime Assessment of a Road Bridge : A. M. P. de Jesus et al.
and the results from this study [2]. The class D S-N
curve, from AASHTO code, is a lower bound of the
experimental data, with some exceptions, namely
some points from riveted joints of Luiz I bridge and
one point from the Pinhao bridge. The observation of
the fracture surface for these riveted joints showed
the presence of initial cracks, which were hided by
the rivet head. Figure 10, right, shows that the
Eurocode class 71 S-N curve is coincident with the
AASHTO class D S-N curve until about 5 · 106 cycles.
Above this value, the two curves diverge. In the
assessment of the residual life of the Pinhao bridge,
the class D S-N curve, from AASHTO code, will be
used. This S-N curve presents the following form:
N � Dr3 ¼ 7:21� 1011 (3)
where Dr is stress range in MPa and N is the cycles to
failure.
For a fatigue limit of 2 · 106 cycles, Equation (3)
gives a stress range of 71.17 MPa which is lower than
80.95 MPa obtained with the Equation (2). Thus, the
AASHTO class D S-N curve is more conservative than
the S-N curve obtained with the test results of the
riveted joints from Pinhao bridge.
For the evaluation of the bridge residual lifetime, it
is necessary to know the history of the load spectra
imposed by the vehicles crossing the bridge. It is
assumed that only the vertical effects of trucks with a
gross weights greater than 30 kN can induce fatigue
damage [13, 14]. Usually the calculations are based
on a standard vehicle. Several suggestions can be
found in different codes of practice. For example, the
Portuguese RSAEP code [15] suggests a standard
vehicle with a total weight of 300 kN and three axles
for a static calculation; the BS5400 [13] suggests a
standard vehicle with a total weight of 320 kN and
four axles; the AASHTO [12] suggests a standard
vehicle with a total weight of 325 kN and two axles.
The calculation was performed using the standard
vehicle proposed in the RSAEP. Generally, the cross-
ing of one truck can induce more than one fatigue
cycle in a given detail. However, the bridge is com-
posed by simply supported spans, which leads to
Figure 9: Fatigue results of a single rivet joint from the Pinhao
bridge
1000
Helmerich et al., 1997Akesson and Edlund, 1996Adamson and Kulak, 1995DiBattista and Kulak, 1995 (BD)DiBattista and Kulak, 1995 (TD)ATLSS, 1993Bruhwiler et al., 1990Fisher et al., 1987Out et al., 1984Baker and Kulak, 1982Reemsnyder, 1975AASHTO
Helmerich et al., 1997Akesson and Edlund, 1996Adamson and Kulak, 1995DiBattista and Kulak, 1995 (BD)DiBattista and Kulak, 1995 (TD)ATLSS, 1993Bruhwiler et al., 1990Fisher et al., 1987Out et al., 1984Baker and Kulak, 1982Reemsnyder, 1975AASHTO
Pinhao
Luis I
PinhaoEurocode 3 – Class 71
Luis I
100
Str
ess
rang
e (M
Pa)
10
1000
100
Str
ess
rang
e (M
Pa)
101.00E+05 1.00E+06 1.00E+07 1.00E+08
Fatigue life (cycles)1.00E+05 1.00E+06 1.00E+07 1.00E+08
Fatigue life (cycles)
Figure 10: Fatigue results of riveted joints: comparison with the AASHTO class D S-N curve (left) and with Eurocode class 71 S-N
curve (right) [11]
� 2009 The Authors. Journal compilation � 2009 Blackwell Publishing Ltd j Strain (2011) 47, e402–e415 e409doi: 10.1111/j.1475-1305.2008.00596.x
A. M. P. de Jesus et al. : Residual Lifetime Assessment of a Road Bridge
single cycles for each truck crossing [14]. The stress
cycles can be evaluated using the influence line cal-
culation technique associated to a cycle counting
technique, such as the ‘reservoir method’ [16]. The
crossing of a truck induces bridge vibrations and
consequently additional stresses. However, the stress
ranges induced by vibrations are relatively low [14]
and will be disregarded in this study. Also, it was
considered that the bridge is crossed by a truck at a
time, neglecting the possibility of superposition of
trucks on the bridge. Experimental evidence illus-
trated that the effect of trucks superposition results
on an increase of the stress range lower than 15%
[14]. The calculated equivalent stress range for the
critical element under analysis, on the basis of the
load influence lines, was Dr = 64.7 MPa [2]. This
stress range accounts for the crossing of the vehicle
type (300 kN) but also for other variable actions,
specified by the designer [2].
The total number of stress cycles can be estimated
using the following expression:
NT ¼ i� YD�DATF�NSC (4)
where i is the number of years of service, YD is the
number of days per year, DATF is the daily average
traffic flow and NSC is the number of stress cycles
induced by each truck. Using data supplied by the
Portuguese Directorate-General for Traffic on traffic
flow nearby the Pinhao bridge, a daily average traffic
flow of trucks crossing the Pinhao bridge was esti-
mated equal to 91 trucks per day. Considering a study
period of 30 years, results a total number of cycles
equal to NT = 30 · 365 · 91 · 1 = 996450 cycles.
Using the estimated stress range of Dr = 64.7 MPa
and the class D S-N curve results a number of cycles
to failure equal to 2.66 · 106 cycles which is about
2.7 times the number of predicted cycles for the
studied period. The number of cycles to failure of
2.66 · 106 cycles corresponds to an unlikely crossing
of trucks with 300 kN even taking into account that
traffic flow will rise in the future. The value of the
DATF used above represents the totality of trucks
crossing the bridge, with gross weights varying
between 30 kN and the maximum legal of 400 kN.
We assume there is no traffic of illegal weight vehi-
cles. Since the gross weights distribution of the
vehicles crossing the bridge is not available, an
approximation can be obtained using the national
distribution of gross weights of registered trucks,
made available by the Portuguese Directorate-Gen-
eral for Traffic (see Figure 11) [2]. Since the actions
are now produced by trucks of different gross
weights, critical details are submitted to variable
amplitude stress spectra. The assessment of the fati-
gue strength is performed by calculating the accu-
mulated fatigue damaged. The Palmgren-Miner rule
[17] was used, which states that rupture occurs when
damage reaches the unit value. Damage can be cal-
culated using the following formula:
D ¼X ni
Ni� 1 (5)
where ni is the number of cycles with stress range Dri,
observed during the studied period, Ni is the number
of cycles to failure for the stress range Dri evaluated
using the class D S-N curve from AASHTO code. The
stresses Dri were proportionally scaled from the value
of 64.7 MPa corresponding to a truck gross weight of
300 kN. The calculated damage for the 30 years per-
iod (NT = 996450 cycles) was D = 0.026, which is very
small. An equivalent stress range, Dre, was evaluated
taking into account the gross weights distribution
until 300 kN. This equivalent stress range yields the
same damage as the variable amplitude stress spectra
when applied for the same number of cycles. The
following expression is used:
Figure 11: Weights distribution for trucks by the Portuguese Directorate-General for Traffic
e410 � 2009 The Authors. Journal compilation � 2009 Blackwell Publishing Ltd j Strain (2011) 47, e402–e415doi: 10.1111/j.1475-1305.2008.00596.x
Residual Lifetime Assessment of a Road Bridge : A. M. P. de Jesus et al.
Dre ¼X
ni Drið Þm=X
ni
h i1=m(6)
where m is the slope of the S-N curve. Using the
previous equation results an equivalent stress range
equal to 27.9 MPa. This stress range yields a number
of cycles to failure of 3.32 · 107 cycles, which is
much higher than the calculated value, using the
standard vehicle of 300 kN.
Residual Life of Riveted Joints UsingFracture Mechanics Approach
The authors performed fatigue tests of riveted joints
that were extracted from the bridge, which details
and results were reported in previous sections. The
analysis of the fracture surfaces of some riveted joints
revealed the existence of initial cracks, prior to the
fatigue tests, which nucleated and developed during
the previous bridge operation. These cracks are usu-
ally hidden by the rivet head or due to the overlap of
the members. Figure 8 illustrates some examples of
initial cracks. It is observed semi-elliptical or circular
surface cracks, developed in the interface of the two
riveted members; also quarter-elliptical corner cracks,
at the rivet hole, can be visualised. Based on these
observations, the geometry specified in Figure 12 was
adopted, in this paper, as the base geometry to be
used in the assessment of the residual fatigue life of a
typical riveted joint. A semi-elliptical surface crack
with a crack width, c, and a crack depth, a, located at
the interface between the two riveted members, in a
plane perpendicular to the longitudinal axis of the
member was considered. This geometry leads to a
circular crack if one consider c = 2a. Linear Elastic
Fracture Mechanics was applied to derive the number
of cycles to propagate the crack until it reaches crit-
ical dimensions, responsible for the failure of the
riveted joint, or dimensions that allows the crack
detection during the inspection routines. The critical
dimensions of the cracks generally correspond to
crack sizes at which the unstable crack propagation
will occur. The critical dimensions depend on the
material toughness. Higher toughness values corre-
spond to higher critical dimensions before unstable
propagation can occur. The material from the Pinhao
bridge presents a Charpy V notch energy of 89 J at
room temperature which tolerates a defect of 10 mm
depth and 48 mm width at the maximum stress of
169.6 MPa [2]. Thus, the occultation of small cracks
by the head of the rivets is not a handicap because
these cracks are far from the critical dimensions.
The simulation of the propagation of the semi-
elliptical surface crack was performed integrating the
Paris’s law in the crack depth direction, between
the initial depth, ai, and a final depth, af, resulting
the required number of cycles, Nf (residual lifetime):
Nf ¼Zaf
ai
1
CDKmda (7)
The stress intensity range, DK, at the maximum depth
point of the semi-elliptic crack of Figure 12, can be
defined using the following generic equation:
DK ¼ Kmax � Kmin ¼ F a;Yð Þ Drffiffiffiffiffiffipap
(8)
where Dr is the tress range, F(a,Y) is a function of the
geometry that takes into account the possible stress
concentration and Y is a vector of geometrical
parameters, such as the dimensions of the crack and
component under consideration. Substituting Equa-
tion (8) into Equation (7) results the following Frac-
ture Mechanics based S-N curve:
NfDrm ¼ 1
C
Zaf
ai
da
F a;Yð Þffiffiffiffiffiffipap
½ �mda (9)
The geometry function F(a,Y) can be defined using
the formulation proposed by Cheung and Li [18]:
F a;Yð Þ ¼ Fe � Fs � Fw:Fg (10)
where Fe, Fs, Fw and Fg are correction factors,
respectively, the crack shape correction, the free
surface correction, the finite plate width correction
and the non uniform stress correction. The crack
shape correction Fe is defined as follows:
Fe ¼1
E kð Þ (11)
where E(k) is the elliptic integral of second kind,
defined in the following way:Figure 12: Semi-elliptical crack located in the interface
between riveted members
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A. M. P. de Jesus et al. : Residual Lifetime Assessment of a Road Bridge
E kð Þ ¼Zp=2
0
1� k2 sin2 h� �0:5
dh (12)
with k2 ¼ c2�a2
c2 .
The following approximation was used to calculate
the elliptic integral [19]:
E kð Þ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 1:464
a
c
� �1:65r
(13)
The correction factor for the effect of free surface Fs is
defined as follows:
Fs ¼ 1:211� 0:186
ffiffiffia
c
r(14)
The correction factor for a central crack in a plate of
finite thickness Fw is given by:
Fw ¼ffiffiffiffiffiffiffiffiffiffiffiffiffisec
pa
2t
r(15)
where t is the member thickness.
The correction factor for non uniform stress, i.e.
the effect of stress concentration factor, Fg, is as-
sumed to be equal to:
Fg ¼ 1:36 (16)
The integral of Equation (7) does not admit an exact
solution. Thus, the numerical integration scheme,
illustrated on Figure 13, was adopted. The initial
surface crack depth, ai, is inputted; the initial crack
width is estimated using a given relation between the
crack width and the crack depth. Two relations are
analysed in this paper, namely, the empirical relation
proposed by Cheung and Li [18]:
c ¼ 3:549a1:133 (17)
and a simplification proposed by authors:
c ¼ 2a (18)
The latter relation consists on admitting a crack with
circular crack front. The crack is forced to propagate
in depth direction admitting constant crack incre-
ments, Da (to be inputted). In this paper, crack
increments,Da, equal to 0.01 mm were assumed. It
was verified that crack increments lower than this
value does not produce significant changes on
results. For an actual crack configuration, its stress
intensity factor range is evaluated. This stress
intensity factor range is assumed to be constant
during the crack increment, which allows the com-
putation of the increment in the number of cycles.
Finally, the dimensions of the crack are compared
with inputted final crack depth and/or crack width.
Several scenarios for crack propagation, namely,
two initial crack depths (ai = 2.0 mm and
ai = 2.8 mm), which were observed on fracture sur-
faces (see Figure 8) were simulated. Two stress ran-
ges were also considered, namely Dr = 64.7 MPa and
Dr = 27.9 MPa. The stress range of 64.7 MPa were
obtained for the most stressed member using the
influence lines method and considering the crossing
of a single standard truck with a total weight of
300 kN and three axles. Again, a daily average traffic
flow of 91 trucks per day was assumed resulting, for
a study period of 30 years, in n = 996450 cycles.
A more accurate analysis can be performed if one
assumes that the daily average traffic flow
corresponds to the total number of trucks from a
distribution ranging from 30 kN and the maximum
legal of 400 kN as considered in previous section
(see Figure 11). An equivalent stress range equal to
27.9 MPa was computed, for the most stressed
Figure 13: Integration procedure adopted in crack propaga-
tion simulations
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Residual Lifetime Assessment of a Road Bridge : A. M. P. de Jesus et al.
member, assuming a linear damage summation
hypothesis, which will be used in these Fracture
Mechanics computations.
Figures 14 and 15 present crack propagation curves
for initial crack depths of 2 and 2.8 mm, respectively,
and for a stress range of 64.7 MPa. The crack depth, a,
is expressed as a function of the number of cycles:
a ¼ aðNÞ. Curves i) and iii) were obtained assuming
that the crack width, c, is given by Equation (18).
Curves ii) and iv) assumes that the crack width, c, is
given by Equation (17). The crack propagation curves
were interrupted as soon as the crack depth reached
the thickness of the member (a = 11 mm), becoming
a part through crack. The crack configuration, at
which the crack width reaches the maximum of
14 mm is also point out on Figures 14 and 15. After
Figure 14: Crack propagation curves for an initial crack depth of 2 mm
Figure 15: Crack propagation curves for an initial crack depth of 2.8 mm
Table 5: Residual fatigue lives for several scenarios
Dr (MPa) R
ai = 2.0 mm
af = 11.0 mm
cf = 53.73 mm
ai = 2.0 mm
af = 11.0 mm
cf = 22.0 mm
ai = 2.0 mm
af = 3.36 mm
cf = 14.0 mm
ai = 2.0 mm
af = 7.0 mm
cf = 14.0 mm
27.9 0.0 4.706 · 107 8.939 · 107 3.037 · 107 8.723 · 107
0.5 1.353 · 107 2.318 · 107 7.819 · 106 2.222 · 107
64.7 0.0 1.671 · 106 3.174 · 106 1.078 · 106 0.310 · 106
0.5 0.837 · 106 1.434 · 106 0.484 · 106 1.734 · 106
ai = 2.8 mm
af = 11.0 mm
cf = 53.73 mm
ai = 2.8 mm
af = 11.0 mm
cf = 22.0 mm
ai = 2.8 mm
af = 3.36 mm
cf = 14.0 mm
ai = 2.8 mm
af = 7.0 mm
cf = 14.0 mm
27.9 0.0 2.501 · 107 4.842 · 107 8.323 · 106 4.625 · 107
0.5 8.018 · 106 1.394 · 107 2.310 · 106 1.298 · 107
64.7 0.0 0.888 · 106 1.717 · 106 0.296 · 106 1.642 · 106
0.5 0.496 · 106 0.863 · 106 0.143 · 106 0.691 · 106
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A. M. P. de Jesus et al. : Residual Lifetime Assessment of a Road Bridge
reaching a width of 14 mm, the crack propagates
with constant width, approaching a constant depth
crack. However, simulations were run even for crack
widths greater than 14 mm, which constitutes ‘ficti-
tious’ cracks. The resulted simulations should be
applied with care.
Figures 14 and 15 also illustrate the effects on
results of the stress ratio. Global fatigue crack
growth data, derived for R = 0.0 and R = 0.5 (see
Table 3), was used in the Paris’s equation. The
highest stress ratio leads to lower failure lives. The
stress ratio has a very significant influence on
the crack propagation curves in spite of the da/dN
versus DK curves being not so distinct. The relation
between the crack depth and the crack width also
has a noticeable influence on the crack propagation
curves. A circular crack front leads to failure lives
much higher than using the relation proposed by
Cheung.
Table 5 summarises the residual fatigue lives
resulted from the application of Fracture Mechanics
for several simulation scenarios. The lowest residual
fatigue lives were obtained for Dr = 64.7 MPa with
R = 0.5. Inspection routines must be performed
periodically to detect growing cracks since some
scenarios predict residual fatigue lives below the
expected number of cycles, for the study period.
Some simulation scenarios reported on Table 5 gave
predicted lives consistent with the S-N curve based
predictions.
Conclusions
The main conclusions of this study can be summar-
ised as follows:
• The material of the Pinhao bridge presents
mechanical strength properties similar to values
obtained with materials of other European bridges
built at same time.
• The toughness values of the material are much
higher than values demanded by current design
codes of practice, which allows a high tolerance to
the presence of cracks. Cracks hidden by rivet
heads are not critical unless if they become visible.
Inspection routines for crack detection are
required.
• The crack growth behaviour of the bridge steels
were assessed through crack propagation tests. Two
stress ratios and two distinct materials of the bridge
were tested (materials from a bracing and a diago-
nal). The stress ratio influenced the crack growth
data only for the material from the diagonal
member. Both materials presented very similar
crack growth behaviours for R = 0.5. However, for
R = 0.0 the material from the diagonal presents
lower crack growth rates. The crack growth data
was well correlated using the Paris’s law.
• The fatigue resistances obtained with the fatigue
tests of the riveted joints are compatible with the
recommendations of actual international codes of
practice such as the AASHTO. Some exceptions
found were justified by the presence of initial cracks
developed during the previous bridge operation.
• The residual fatigue lives of the riveted joints were
evaluated using Linear Elastic Fracture Mechanics.
Several simulation scenarios were tested. For the
worst cases (higher stress range and ratio) fatigue
lives of the same order of magnitude of those
expected for the study period of 30 years were
observed. These results enforce the need for peri-
odic inspections to detect fatigue cracks.
• The present study demonstrates the bridge safety
against fatigue, after rehabilitation, for a period of
30 years. This analysis was supported by important
assumptions related with the stress spectra at the
critical locations. More accurate analysis can be
achieved if the stresses/loads are experimentally
monitored, during a representative period of time.
ACKNOWLEDGEMENTS
Authors gratefully acknowledge the GEG (Gabinete de
Estruturas e Geotecnia, Lda) for their co-operation in
this study. This work was partially supported by the
Portuguese Scientific Foundation (FCT) through the
project PTDC/EME-PME/78833/2006, which is also
acknowledged.
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� 2009 The Authors. Journal compilation � 2009 Blackwell Publishing Ltd j Strain (2011) 47, e402–e415 e415doi: 10.1111/j.1475-1305.2008.00596.x
A. M. P. de Jesus et al. : Residual Lifetime Assessment of a Road Bridge