strain

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


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)



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



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



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)



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]



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



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



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



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



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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strain

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residual fatigue strength