Advanced Fatigue Assessment of Riveted Railway Bridges on Existing Masonry Abutments: An Italian Case Study

Abstract

Riveted railway bridges with already long structural lives can still be commonly found in service in Europe. In light of their peculiarities, they are often prone to fatigue damage; nevertheless, very few prescriptions regarding fatigue assessment of these structures can be found in current European provisions. Within this framework, the present paper illustrates the advanced fatigue assessment of an Italian riveted railway bridge selected as a case study. For this purpose, multi-scale finite element modelling of the bridge was developed, and the most critical details were assessed through application of the advanced strain energy density (SED) method. The obtained outcomes were compared both with other studies in the literature and prescriptions from the current and upcoming versions of EN1993-1-9.

Full text

Citation: Milone, A.; D’Aniello, M.;
Landolfo, R. Advanced Fatigue
Assessment of Riveted Railway
Bridges on Existing Masonry
Abutments: An Italian Case Study.
Buildings 2024,14, 2271. https://
doi.org/10.3390/buildings14082271
Academic Editor: Weixin Ren
Received: 16 June 2024
Revised: 19 July 2024
Accepted: 20 July 2024
Published: 23 July 2024
Copyright: © 2024 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
buildings
Article
Advanced Fatigue Assessment of Riveted Railway Bridges on
Existing Masonry Abutments: An Italian Case Study
Aldo Milone * , Mario D’Aniello and Raffaele Landolfo
Department of Structures for Engineering and Architecture, Via Forno Vecchio 36, 80134 Naples, Italy;
mdaniel@unina.it (M.D.); landolfo@unina.it (R.L.)
*Correspondence: aldo.milone@unina.it
Abstract: Riveted railway bridges with already long structural lives can still be commonly found in
service in Europe. In light of their peculiarities, they are often prone to fatigue damage; nevertheless,
very few prescriptions regarding fatigue assessment of these structures can be found in current
European provisions. Within this framework, the present paper illustrates the advanced fatigue
assessment of an Italian riveted railway bridge selected as a case study. For this purpose, multi-scale
finite element modelling of the bridge was developed, and the most critical details were assessed
through application of the advanced strain energy density (SED) method. The obtained outcomes
were compared both with other studies in the literature and prescriptions from the current and
upcoming versions of EN1993-1-9.
Keywords: railway bridges; riveted connections; fatigue assessment; strain energy density method;
finite element analyses
1. Introduction
Riveted connections represent the most common structural detail implemented in steel
bridges erected in the XIXth century and the first half of the XXth century [
1
–
3
]. Hot-driven
rivets (HDRs) were usually adopted to either couple hot-rolled members into built-up
sections—that is, by means of riveted battens—or to assemble truss nodes with a high
density of rivets resisting to shear. These recurring practices had their roots in previous,
empirical knowledge gained from timber structures [4–6].
In particular, riveted connections are commonly found in older railway bridges; no-
tably, about 30% of them have already endured more than 100 years of service life [
7
,
8
].
This condition—in conjunction with (i) structural shortages deriving from the lack of ade-
quate provisions at the time of erection, (ii) peculiar constructional imperfections deriving
from the hot-driving process (HDP), and (iii) the diffuse increase of traffic loads in recent
times—leaves riveted details exposed to severe fatigue damage. This further proves to be a
critical issue if one considers the limited redundancy of truss bridge structures [9–12].
Nevertheless, in current European provisions (EN1993-1-9 [
13
]), very limited indica-
tions are given to designers for fatigue assessment of such details. Namely, while two detail
classes (DCs) were proposed in the Eurocode 3 background document [
14
]—i.e., DC71 for
one-sided riveted joints and DC90 for double-covered riveted joints—the current version
of EN1993-1-9 does not include any DC related to riveted connections. Therefore, it should
not be a surprise that the proper fatigue assessment of such details still represents an open
field of research [5,7,15].
To this end, it is worth noting that some remarkable research efforts devoted to the
fatigue performance of riveted connections can be found in the scientific literature.
Bertolesi et al. [
7
] investigated the fatigue performance of riveted details belonging
to a 110-year-old, 170 m long truss bridge located in Spain. To this end, both numerical
and experimental tests were performed, suggesting that a variable DC between 63 and
Buildings 2024,14, 2271. https://doi.org/10.3390/buildings14082271 https://www.mdpi.com/journal/buildings

Buildings 2024,14, 2271 2 of 17
71 MPa
should be selected for riveted details belonging to bridge crossbeams. The authors
also reported that the fatigue life of full-scale assemblies had been almost reached prior to
testing, i.e., highlighting the potential risk of fatigue failures in older riveted bridges still
in service.
In the same fashion, Pedrosa et al. [
2
] investigated the fatigue response of single- and
double-splice riveted shear connections assembled with rivets drawn from the well-known
Luiz I bridge (Porto, Portugal). According to the authors, the fatigue resistance of single-
splice details (DC = 55 MPa) was significantly lower than that of double-splice ones
−
49%)
and not in line with reported recommendations [
14
]. Estimated slopes of S-N curves
(m= 6 and 10, respectively) also appreciably differed from each other and from normative
prescriptions.
Useful data regarding strain-life and crack propagation properties in old Portuguese
riveted bridges can be also found in De Jesus et al. [1].
With reference to comprehensive fatigue assessment of existing riveted bridges, it is
worth mentioning the contributions of Kuzawa et al. [
11
], Landolfo et al. [
16
], and De Jesus
et al. [
17
], which assessed the remaining service life of still-operational old infrastructures
(having endured service life in the range 65–120 years), accounting for accurate geometry
of members and details, traffic load evolution up to the present time, and ongoing material
degradation.
Within this framework, the present paper illustrates the advanced fatigue assessment
of a case-study riveted railway bridge located in Italy. For this purpose, a multi-scale
refined modelling of the structure was performed, and critical details were investigated
through the advanced strain energy density (SED) method [18–20].
The derived results are hence compared with the literature recommendations [
2
,
21
]
and normative prescriptions from the upcoming version of prEN1993-1-9:2023 [
22
], with
the aim of providing some useful insights about key factors affecting the fatigue life of
existing riveted infrastructures.
In light of the above, this work is mainly divided into four parts, as follows: (i) in
the first section, main features of the selected case study are presented; (ii) modelling
assumptions for multi-scale finite element models (FEMs) of the bridge are discussed
in the second part, with a particular focus on the developed numerical application of
the SED method; (iii) the third section illustrates relevant results from advanced fatigue
analysis; (iv) finally, derived outcomes are compared with the literature and normative
recommendations.
2. Main Features of the Selected Case Study
2.1. Generality
The selected case study (“Gesso Railway Bridge” in Montecalvo Irpino, Italy—see
Figure 1a) was erected in 1963–1966 to replace a former five-span masonry bridge which
was severely damaged by the 1962 Irpinia earthquake, as massive existing arches partially
collapsed under the combined effect of self-weight and seismic loads (size effect [23]).
For this purpose, both original masonry abutments and the central masonry pier
were preserved, while the new deck was realized via three identical, simply supported 3D
Warren trusses (L = 29 m). Remarkably, a low degree of continuity among the spans was
provided by maple sleepers and rail tracks (UIC 60 type, unit weight gr= 60 daN/m).
Sleepers were placed on built-up steel stringers (430
×
170 mm, spacing s= 1.5 m) made
of a core plate and four angle profiles. Stringers were in turn connected to H-shaped upper
transverse beams (480
×
200 mm), interrupting the longitudinal trusses’ continuity at
≈
5.7 m
intervals. Built-up X-bracings (2L 80
×
10 mm) and lower beams (2L 100
×
10 mm) complete
the transverse stiffening system.
The main longitudinal trusses (spacing s= 3.3 m, see Figure 1b) have a
Λ
-shaped
configuration with consecutive tensile and compressive diagonals being interspersed with
built-up H-shaped vertical struts (120
×
100 mm). Diagonals have a ] [-shaped section
connected by 10 mm riveted battens.

Buildings 2024,14, 2271 3 of 17
Buildings 2024, 14, x FOR PEER REVIEW 3 of 18
built-up H-shaped vertical struts (120 × 100 mm). Diagonals have a ] [-shaped section con-
nected by 10 mm riveted baens.
(a)
(b)
Figure 1. (a) View of the investigated riveted truss bridge at launch and present time; (b) main geo-
metrical features as retrieved from the original design drawings.
Remarkably, while the C-shaped parts of the outer (i.e., most stressed) diagonals
were made via assembling plates and angles (484 × 380 mm), progressively smaller UPN
proles were used for the inner ones (UPN300—UPN260—UPN220, spacing s = 324 mm).
In order to prevent buckling phenomena due to high compressive forces, large built-
up box proles (550 × 520 mm, composed of two 12 mm webs, four 100 × 10 mm + two 80
× 10 mm Ls, and 10 mm backing plates on both sides) were adopted for the upper chords.
On the other hand, lower chords feature an inverted Π section (532 × 510 mm) realized via
two 12 mm webs, four 100 × 10 mm Ls, and two 10 mm backing plates on the lower side.
1966
Present
Figure 1. (a) View of the investigated riveted truss bridge at launch and present time; (b) main
geometrical features as retrieved from the original design drawings.
Remarkably, while the C-shaped parts of the outer (i.e., most stressed) diagonals were
made via assembling plates and angles (484
×
380 mm), progressively smaller UPN profiles
were used for the inner ones (UPN300—UPN260—UPN220, spacing s= 324 mm).
In order to prevent buckling phenomena due to high compressive forces, large built-up
box profiles (550
×
520 mm, composed of two 12 mm webs, four 100
×
10 mm + two 80
×
10 mm Ls, and 10 mm backing plates on both sides) were adopted for the upper chords.
On the other hand, lower chords feature an inverted
Π
section (532
×
510 mm) realized via
two 12 mm webs, four 100 ×10 mm Ls, and two 10 mm backing plates on the lower side.
The 3D truss structure is completed by both lower and upper floor bracings, i.e., made
of back-to-back built-up 80
×
10 mm Ls. Notably, the upper bracings have been placed at
double spacing (5.66 m) compared with the lower ones. Nevertheless, the upper side of the
truss is further stiffened by single 80 ×10 mm Ls connecting adjacent sleepers.

Buildings 2024,14, 2271 4 of 17
The connection between the steel structure and existing masonry piers and abutments
uses casted-steel rollers and pinned bearings placed at each span ends.
Steel members and plates are all made of Fe 50.2 steel (yield strength f
y
= 340 N/mm
2
,
ultimate tensile strength f
u
= 500 N/mm
2
), which is no longer produced in Europe, al-
though it can be compared to modern S355 steel grade. “High-quality” Aq 34 mild steel
(fyr = 185 N/mm2,fur = 340 N/mm2) was instead used for Ø22 hot-driven rivets.
2.2. Description of Investigated Connections
As discerned from global-scale analyses, the most critical elements from a fatigue
perspective are represented by the midspan segment of the lower chord (LC) and by
the tensile diagonals (TDs) being closer to the end supports (see Section 3.1 for further
details). Therefore, the geometrical features of the relevant connections were thoroughly
investigated by comparing original design drawings and outcomes from on-site surveys.
Figure 2summarizes the key features of these connections.
Buildings 2024, 14, x FOR PEER REVIEW 4 of 18
The 3D truss structure is completed by both lower and upper oor bracings, i.e.,
made of back-to-back built-up 80 × 10 mm Ls. Notably, the upper bracings have been
placed at double spacing (5.66 m) compared with the lower ones. Nevertheless, the upper
side of the truss is further stiened by single 80 × 10 mm Ls connecting adjacent sleepers.
The connection between the steel structure and existing masonry piers and abut-
ments uses casted-steel rollers and pinned bearings placed at each span ends.
Steel members and plates are all made of Fe 50.2 steel (yield strength fy = 340 N/mm2,
ultimate tensile strength fu = 500 N/mm2), which is no longer produced in Europe, alt-
hough it can be compared to modern S355 steel grade. “High-quality” Aq 34 mild steel (fyr
= 185 N/mm2, fur = 340 N/mm2) was instead used for Ø22 hot-driven rivets.
2.2. Description of Investigated Connections
As discerned from global-scale analyses, the most critical elements from a fatigue
perspective are represented by the midspan segment of the lower chord (LC) and by the
tensile diagonals (TDs) being closer to the end supports (see Section 3.1 for further details).
Therefore, the geometrical features of the relevant connections were thoroughly investi-
gated by comparing original design drawings and outcomes from on-site surveys.
Figure 2 summarizes the key features of these connections.
(a)
(b)
SYM
100
50
90 45
4 × 6 Ø 22 HDRs
2450 ×400 Plates
10 mm GP
Ø 22 HDRs/118Ø 22 HDRs/118
Built-up
section
Π
510
532
2 L 100 × 10
Web 500 ×10
Plate 220 × 10
2 UPN 300
10 mm GP
SYM
Figure 2. Key features of the most critical connections: (a) LC—midspan and (b) TD—most stressed.
With reference to LC, each 5.6 m-long segment is interrupted in correspondence of a
couple of 10 mm trapezoidal gusset plates (GPs) being the centrepiece of the KT joints. LC

Buildings 2024,14, 2271 5 of 17
webs are juxtaposed to the GP ends and connected through a couple of riveted backing
plates (450
×
400 mm) featuring 4
×
6 Ø22 HDRs (in-line configuration, see Figure 2a).
Moreover, 100
×
10 mm Ls constituting the LC lower flanges are riveted to the GP as well
(Ø22 HDRs, spacing s= 118 mm).
Conversely, UPN 300 profiles belonging to most stressed TDs are directly connected
to the relevant GPs by means of 3 ×7 Ø22 HDRs (staggered configuration, see Figure 2b).
In order to further accommodate the stress transmission, a couple of 100
×
10 mm angle
profiles are connected to both UPN flanges and the GP via 2 + 2 Ø22 HDRs.
3. Modelling Assumptions
3.1. Global FEMs
Global FEMs were developed in SAP2000 v.23 (see Figure 3a) [
24
]. All steel members
were modelled via one-dimensional frame elements. In order to capture the structural
response of battened members without considerably increasing the computational effort,
equivalent cross-sectional properties (i.e., derived with a fiber model) were introduced
for built-up sections. The variation of shear stiffness due to intermittent battening was
properly accounted for according to provisions in force in Italy [25,26].
The rather low degree of flexural restraint provided by the riveted connections was
modelled via hinge releases located at member ends. Moreover, pinned and roller restraints
were introduced at the spans’ ends to reproduce the intended structural scheme.
Buildings 2024, 14, x FOR PEER REVIEW 6 of 18
Table 1. Main features of “real” train loads and relevant daily passages according to RFI indications.
Train
Type
[-]
Total
Load
[kN]
Total
Length
[m]
Number
of Axes
[-]
Nominal
Speed
[m/s]
Dynamic
Factor φ
[-]
Average Daily
Passages
[1/d]
3
9400
385.5
60
70
1.31
7
4
5100
237.6
30
70
1.31
6
8
10,350
212.5
46
28
1.18
1
9
2960
134.8
24
33
1.16
1
(a)
(b)
Roller support
Pinned support
Rail tracks
TD connection
LC connection
Fix X
Y
Z
Δσ
ΩSED
Figure 3. Cont.

Buildings 2024,14, 2271 6 of 17
Buildings 2024, 14, x FOR PEER REVIEW 7 of 18
(c)
Figure 3. Main features of FE models: (a) global FEM of the bridge, (b,c) local FEMs of critical details.
3.2. Principles of the Strain Energy Density Method
The strain energy density (SED) method was rst introduced by Lazzarin and Zam-
bardi [18] to assess the fatigue and fracture behavior of notched elements.
The authors noticed that applying stress-based approaches to such components
would lead in many cases to the prediction of a higher fatigue strength compared with
the value obtained by simply dividing the fatigue limit of a plain specimen by the theo-
retical value of the stress concentration factor Kt [29]. This suggested that fatigue failure
of notched/drilled components was not governed by the notch (hot-spot) stress but rather
by a mean stress averaged over a nite neighbourhood centred at the notch tip. This aspect
was thus addressed by means of an energetic formulation.
Accordingly, an energetic and yet stress-related parameter was introduced to de-
scribe fatigue and fracture behaviour of notched components, i.e., the averaged strain en-
ergy density (ASED, also referred to as W



) over a control volume ΩSED centred at (or lo-
cated near to) the notch tip (Equation (1)):
W



= 1
ΩSED ∫σij dεij = 1
2
σij εij
ΩSED
= 1
2
εij
0
(σθθ εθθ + σrr εrr + σzz εzz + τrθ γrθ)
ΩSED
(1)
where εij and σij are the notch strain and stress components in a cylindrical (r, θ, z) coor-
dinate system, respectively.
The shape and size of the control volume ΩSED are determined based on theoretical
considerations related to the stress distribution around the notch tip. Accordingly, a single
material parameter (control volume radius R0) is required to address ASED calculations.
With reference to round (U-)notches, ΩSED is described by a moon-shaped curve re-
volving around the point where the maximum principal stress is aained (see Figure 4a
[18–20]). As shown in a previous publication by the authors [5] this insight can be used to
address the fatigue performance of riveted connections by regarding rivet holes as equiv-
alent U-notches (see Figure 4b).
To this end, it is worth highlighting that the usual values of R0 for mild steels (10−1–
100 mm [20]) often result in ΩSED being included in the HDP-aected zone of drilled plates.
Therefore, the selected value of R0 should be properly reduced to account for localized
material damage. According to [5], R0 = 0.9–1.0 mm can be used to account for the
Fix
X
Y
ZΔσ
ΩSED
Figure 3. Main features of FE models: (a) global FEM of the bridge, (b,c) local FEMs of critical details.
Stress histories for connections were estimated through moving load analysis, i.e.,
based on “real” train loads reported in Italian Railway Network (RFI) instructions (DTC
INC PO SP IFS 003 A [
27
]). For this purpose, rail tracks were explicitly modelled to serve
as load paths and to account for the small degree of continuity among consecutive spans.
Dynamic load amplification was considered depending on the train speed and the
span length as suggested in [27], i.e., via the dynamic factor φ.
Load spectra were derived by (i) accounting for trains’ daily passages as declared
by the infrastructure’s owner (see Table 1) and then (ii) processing oscillograms via the
rainflow method. For this purpose, equivalent stress reversals (half-cycles n/2) were
identified according to the following criteria [13,28]:
•Ideal clockwise rotation of 90 degrees of the stress history;
•
Identification of “water sources” in correspondence with each peak of the stress
history;
•
Counting the number of half-cycles by detecting flow terminations according to three
stop criteria, i.e., (i) intersection with a flow started at an earlier peak, (ii) intersection
with an opposite tensile peak of equal or greater magnitude, (iii) end of the stress
history;
•Repetition of previous steps for compressive valleys;
•
Definition of equivalent stress ranges for each half-cycle, i.e., equal to the stress
difference between its start and termination.
Table 1. Main features of “real” train loads and relevant daily passages according to RFI indications.
Train
Type
[-]
Total
Load
[kN]
Total
Length
[m]
Number
of Axes
[-]
Nominal
Speed
[m/s]
Dynamic
Factor φ
[-]
Average Daily
Passages
[1/d]
3 9400 385.5 60 70 1.31 7
4 5100 237.6 30 70 1.31 6
8 10,350 212.5 46 28 1.18 1
9 2960 134.8 24 33 1.16 1

Buildings 2024,14, 2271 7 of 17
Reversals sharing the same magnitude were then combined into full cycles n. The
collaboration among global and local FEM was established by adopting the processed stress
histories as inputs for the energy calculations (see Section 3.3 for more details).
3.2. Principles of the Strain Energy Density Method
The strain energy density (SED) method was first introduced by Lazzarin and Zam-
bardi [18] to assess the fatigue and fracture behavior of notched elements.
The authors noticed that applying stress-based approaches to such components would
lead in many cases to the prediction of a higher fatigue strength compared with the value
obtained by simply dividing the fatigue limit of a plain specimen by the theoretical value of
the stress concentration factor K
t
[
29
]. This suggested that fatigue failure of notched/drilled
components was not governed by the notch (hot-spot) stress but rather by a mean stress
averaged over a finite neighbourhood centred at the notch tip. This aspect was thus
addressed by means of an energetic formulation.
Accordingly, an energetic and yet stress-related parameter was introduced to describe
fatigue and fracture behaviour of notched components, i.e., the averaged strain energy
density (ASED, also referred to as
W
) over a control volume
ΩSED
centred at (or located
near to) the notch tip (Equation (1)):
W=1
ΩSED Zεij
0
σijdεij =1
2
σij εij
ΩSED
=1
2
(σθθ εθθ +σrr εrr +σzz εzz +τrθγrθ)
ΩSED
(1)
where
εij
and
σij
are the notch strain and stress components in a cylindrical (r,
θ
,z) coordi-
nate system, respectively.
The shape and size of the control volume
ΩSED
are determined based on theoretical
considerations related to the stress distribution around the notch tip. Accordingly, a single
material parameter (control volume radius R0) is required to address ASED calculations.
With reference to round (U-)notches,
ΩSED
is described by a moon-shaped curve re-
volving around the point where the maximum principal stress is attained
(see Figure 4a [
18
–
20
]). As shown in a previous publication by the authors [
5
] this in-
sight can be used to address the fatigue performance of riveted connections by regarding
rivet holes as equivalent U-notches (see Figure 4b).
Buildings 2024, 14, x FOR PEER REVIEW 8 of 18
reduction of fracture toughness in Fe 50.2 steel due to drilling and hammering (–32% com-
pared with pristine material).
(a)
(b)
Figure 4. Applications of the SED method: (a) control volume for ASED calculations in U-notches;
(b) extension to hot-driven riveted connections.
Fatigue collapse can therefore be addressed by assuming the ASED range ΔW



=
W


max W



min as a fracture indicator [18–20]. To this end, the well-known Basquin’s for-
mula [30] can be conveniently expressed in terms of ΔW



as follows (Equation (2) [18–20]):
N*= C ∆W



m
(2)
with N* being the expected number of cycles at failure and C and m being material pa-
rameters to be properly calibrated (intercept and slope of the fatigue curve, respectively).
When constant tensile stresses σm > 0 are superimposed onto uctuating actions (i.e.,
typical conditions for bridge details due to permanent loads [31–33]), the mean stress ef-
fect can be explicitly accounted for by means of a non-dimensional prestress coecient cw
depending on the stress ratio R = σmin/σmax as follows (Equations (3) and (4)):
∆W



(R≠0) =cw∆W



(R = 0)
(3)
cw = 1sign󰇛R󰇜 R2
󰇛1R󰇜2
(4)
cw can be graphically regarded as the ratio among the areas underlying the σ–ε curves
(that is, ∆W



), bounded by the same stress range Δσ but for R ≠ 0 and R = 0, respectively.
As shown in multiple contributions in the literature, the SED method has been
proven to be highly eective in predicting the fatigue performance of structural compo-
nents, e.g., through validation against 900+ experimental tests on welded joints [19,20] and
150+ experimental tests on notched specimens subjected to high stress ratios [32].
Nevertheless, it should be remarked that potential limitations of the method lie in (i)
the requirement of dedicated ΔW



-N curves (Equation (2)) for details of interest in order
to assess the fatigue performance of real structures, and (ii) some issues in the denition
of the control volume shape for rather complex geometries [20].
Also for the above reasons, the SED-based results reported in this work were com-
pared against the established literature and normative formulations for fatigue analysis
of riveted connections (see Section 5 for further details).
3.3. Local FEMs for SED Calculations
Local FEMs for SED calculations were developed in ABAQUS v6.14 [34] (see Figure
3b,c). As anticipated, LC and TD connections were selected owing to (i) non-negligible
mean tensile stresses due to gravity loads and (ii) signicant stress histories due to train
P
σP= σI,max
ρ
rR0
Δσ
Plate portion affected by HDP
Control volume for ASED calculations
U-notch
R0
Figure 4. Applications of the SED method: (a) control volume for ASED calculations in U-notches;
(b) extension to hot-driven riveted connections.
To this end, it is worth highlighting that the usual values of R
0
for mild steels
(10
−1
–10
0
mm [
20
]) often result in
ΩSED
being included in the HDP-affected zone of
drilled plates. Therefore, the selected value of R
0
should be properly reduced to account for
localized material damage. According to [
5
], R
0
= 0.9–1.0 mm can be used to account for

Buildings 2024,14, 2271 8 of 17
the reduction of fracture toughness in Fe 50.2 steel due to drilling and hammering (
−
32%
compared with pristine material).
Fatigue collapse can therefore be addressed by assuming the ASED range
∆W
=
Wmax −Wmin
as a fracture indicator [
18
–
20
]. To this end, the well-known Basquin’s
formula [
30
] can be conveniently expressed in terms of
∆W
as follows (Equation (2) [
18
–
20
]):
N∗=C(∆W)−m(2)
with N* being the expected number of cycles at failure and C and mbeing material parame-
ters to be properly calibrated (intercept and slope of the fatigue curve, respectively).
When constant tensile stresses
σm
> 0 are superimposed onto fluctuating actions (i.e.,
typical conditions for bridge details due to permanent loads [
31
–
33
]), the mean stress effect
can be explicitly accounted for by means of a non-dimensional prestress coefficient c
w
depending on the stress ratio R = σmin/σmax as follows (Equations (3) and (4)):
∆W(R=0)=cw∆W(R=0)(3)
cw=1−sign(R)R2
(1−R)2(4)
c
w
can be graphically regarded as the ratio among the areas underlying the
σ
–
ε
curves
(that is, ∆W), bounded by the same stress range ∆σbut for R =0 and R = 0, respectively.
As shown in multiple contributions in the literature, the SED method has been proven
to be highly effective in predicting the fatigue performance of structural components,
e.g., through validation against 900+ experimental tests on welded joints [
19
,
20
] and 150+
experimental tests on notched specimens subjected to high stress ratios [32].
Nevertheless, it should be remarked that potential limitations of the method lie in
(i) the requirement of dedicated
∆W
-N curves (Equation (2)) for details of interest in order
to assess the fatigue performance of real structures, and (ii) some issues in the definition of
the control volume shape for rather complex geometries [20].
Also for the above reasons, the SED-based results reported in this work were compared
against the established literature and normative formulations for fatigue analysis of riveted
connections (see Section 5for further details).
3.3. Local FEMs for SED Calculations
Local FEMs for SED calculations were developed in ABAQUS v6.14 [
34
]
(see Figure 3b,c). As anticipated, LC and TD connections were selected owing to (i) non-
negligible mean tensile stresses due to gravity loads and (ii) significant stress histories due
to train loads. In order to enable SED calculations, all parts were discretized by means
of C3D20R elements (20-node solid brick, reduced integration, quadratic geometry as
suggested in [19,35,36]).
To ensure an accurate estimation of ASED range, a minimum mesh size equal to R
0
/4
was adopted in the neighborhood of the most stressed hole tip. A coarser mesh with average
size of 1 mm was instead used for rivets, plates, and other portions of connected members.
Indeed, as reported in [
20
,
35
–
37
], energetic calculations are rather insensitive to the overall
mesh sizing of the model, provided that the control volume has been
properly partitioned
.
Based on preliminary stress analyses, the outermost holes were monitored in both
cases, and control volumes were accordingly partitioned. In favor of safety, the minimum
suggested value of R
0
= 0.9 mm (i.e., resulting in the highest fatigue demand) was assumed
for calculations, in compliance with outcomes from [5].
In order to balance computational effort and accuracy of analyses, geometrical and
mechanical symmetry was explicitly accounted for by means of proper boundary conditions
(BCs). The fatigue behavior of connections was investigated by applying surface pressures
with magnitude
∆σ
at members’ ends. Equivalent fixed restraints were also introduced to
model the structural continuity of KT joints.

Buildings 2024,14, 2271 9 of 17
According to [
9
,
10
,
15
], a rather low level of clamping stress
σclamp
was considered
for existing rivets owing to peculiarities of HDP and due to the already long service life
(
σclamp
= 0.5 f
yr
= 92 N/mm
2
). For this purpose, the “bolt-load” command was suitably used.
The von Mises criterion was adopted to model steel yielding.
Materials’ strength values were assumed in compliance with indications from the
original design report (see Section 2.1,f
y
= 340 MPa and 185 MPa for plates and HDRs,
respectively). Related stress–strain constitutive laws were modelled based on parameters
reported in [9].
Contact among adjacent parts was modelled through “surface-to-surface” interactions.
Namely, both normal and tangential behaviour were accounted for by means of “hard
contact” and “penalty” formulations (with a friction coefficient
µ
= 0.3 being selected in the
latter case, according to [9]).
4. Results and Discussion
Results related to stress and energetic assessment of the case study are summarized in
the following sub-sections. To this end, it is worth to mentioning first that the computational
cost for the numerical application of the SED method was effectively controlled.
This goal was indeed achieved thanks to (i) the collaboration between local and global
FEM—with the latter requiring only a negligible amount of time to perform moving load
analysis, (ii) the exploitation of structural symmetries in local models, and (iii) the introduc-
tion of variable mesh size in the proximity of control volumes for ASED calculations.
4.1. Global Scale FE Analyses
The results of global-scale FE analyses are summarized in Figure 5a,b and Table 2in
terms of (i) stress histories deriving from the passage of “real” train loads and (ii) outcomes
from the application of the rainflow method. For the sake of brevity, only results related to
the most critical details are depicted in Figure 5a (LC) and Figure 5b (TD).
Table 2. Application of the rainflow method to derive stress histories for critical details.
Lower Chord—Midspan (LC)
Train
Type
[-]
Daily
Passages
[1/d]
Rainflow Method
n
[-]
∆σeq
[N/mm2]
σm,eq
[N/mm2]
R
[-]
377×1 20.6 20.4 0.33
7×13 3.0 23.8 0.88
466×1 21.0 20.6 0.32
6×12 6.3 21.2 0.74
811×1 26.0 28.1 0.37
1×22 4.0 35.2 0.89
911×1 15.0 17.6 0.40
1×6 7.4 21.4 0.71
Tensile Diagonal—Support (TD)
Train
Type
[-]
Daily
Passages
[1/d]
Rainflow Method
n
[-]
∆σeq
[N/mm2]
σm,eq
[N/mm2]
R
[-]
377×1 5.5 10.0 0.57
7×12 3.2 11.2 0.75
466×1 4.9 9.7 0.60
6×11 3.0 10.7 0.75
811×2 9.8 12.2 0.43
1×32 1.1 14.1 0.93
911×1 4.4 9.5 0.62
1×6 2.2 10.6 0.81

Buildings 2024,14, 2271 10 of 17
Buildings 2024, 14, x FOR PEER REVIEW 10 of 18
(a)
(b)
Figure 5. Moving load analysis results for the most critical details: stress histories for (a) lower chord
− midspan (LC); (b) tensile diagonal − support (TD).
This insight was conrmed by the results of the rainow method. Indeed, while mod-
erately lower equivalent stress ranges Δσeq were derived for both LC (Δσeq,LC,T3,max = 20.6
N/mm2 against Δσeq,LC,T8,max = 26.0 N/mm2, –21%) and TD (Δσeq,TD,T3,max = 5.5 N/mm2 against
Δσeq,TD,T8,max = 9.8 N/mm2, –44%) in the case of Train 3, the number of cycles n endured was
signicantly higher (+326% and +167% for LC and TD, respectively).
With reference to mean stress eect, it is worth reporting that the highest superim-
posed tensile stresses were aained for LC (σm,LC = 10.1 N/mm2, +42% compared with σm,TD
= 7.1 N/mm2), that is, as a clear consequence of the structural scheme of the adopted
trusses.
Nevertheless, while equivalent mean stresses σm,eq—estimated by adding σm and the
average stress σm,i associated with the i-th rainow rehearsal—were higher in LC com-
pared with TD, the highest values of stress ratio R are actually reached in the laer mem-
ber (Rmax = 0.91, Train 8), due to the larger (relative) impact of mean stresses with respect
to the endured stress ranges.
The derived values of Δσeq, i.e., superimposed with clamping stresses and equivalent
mean stresses, were introduced into the local FEMs for ASED calculations. Further details
are reported in the next section.
Table 2. Application of the rainow method to derive stress histories for critical details.
Lower Chord—Midspan (LC)
Train
Type
[-]
Daily Passages
[1/d]
Rainflow Method
n
[-]
Δσeq
[N/mm2]
σm,eq
[N/mm2]
R
[-]
3
7
7 × 1
20.6
20.4
0.33
7 × 13
3.0
23.8
0.88
4
6
6 × 1
21.0
20.6
0.32
6 × 12
6.3
21.2
0.74
8
1
1 × 1
26.0
28.1
0.37
1 × 22
4.0
35.2
0.89
9
1
1 × 1
15.0
17.6
0.40
1 × 6
7.4
21.4
0.71
Tensile Diagonal—Support (TD)
Train
Type
[-]
Daily Passages
[1/d]
Rainflow Method
n
[-]
Δσeq
[N/mm2]
σm,eq
[N/mm2]
R
[-]
3
7
7 × 1
5.5
10.0
0.57
7 × 12
3.2
11.2
0.75
4
6
6 × 1
4.9
9.7
0.60
0
5
10
15
20
25
30
0 2 4 6 8 10 12
Stress σ[N/mm2]
Time [s]
Train 3
Train 4
Train 8
Train 9
LC - σm= 10.1 N/mm2
0
5
10
15
20
25
30
35
40
0 2 4 6 8 10 12
Stress σ[N/mm2]
Time [s]
Train 3
Train 4
Train 8
Train 9
TD -σm= 7.1 N/mm2
Figure 5. Moving load analysis results for the most critical details: stress histories for (a) lower chord
−midspan (LC); (b) tensile diagonal −support (TD).
In line with typical outcomes for railway bridges [
30
,
31
], higher equivalent stress
ranges were in most cases associated with a smaller number of cycles, and vice versa. It
was readily apparent that, in both cases, the highest stress variations were induced by Train
8 due to it having the highest total weight (Q
tot
= 10,350 kN). Nevertheless, the highest
fatigue damage was expected to be associated with Train 3 due to it having the highest
number of average daily passages (7/day).
This insight was confirmed by the results of the rainflow method. Indeed, while
moderately lower equivalent stress ranges
∆σeq
were derived for both LC (
∆σeq,LC,T3,max
=
20.6 N/mm
2
against
∆σeq,LC,T8,max
= 26.0 N/mm
2
,
−
21%) and TD (
∆σeq,TD,T3,max
=
5.5 N/mm
2
against
∆σeq,TD,T8,max
= 9.8 N/mm
2
,
−
44%) in the case of Train 3, the number of
cycles nendured was significantly higher (+326% and +167% for LC and TD, respectively).
With reference to mean stress effect, it is worth reporting that the highest superimposed
tensile stresses were attained for LC (
σm,LC
= 10.1 N/mm
2
, +42% compared with
σm,TD
=
7.1 N/mm
2
), that is, as a clear consequence of the structural scheme of the adopted trusses.
Nevertheless, while equivalent mean stresses
σm,eq
—estimated by adding
σm
and
the average stress
σm,i
associated with the i-th rainflow rehearsal—were higher in LC
compared with TD, the highest values of stress ratio R are actually reached in the latter
member (
Rmax = 0.91
, Train 8), due to the larger (relative) impact of mean stresses with
respect to the endured stress ranges.
The derived values of
∆σeq
, i.e., superimposed with clamping stresses and equivalent
mean stresses, were introduced into the local FEMs for ASED calculations. Further details
are reported in the next section.
4.2. Local FE Analyses for ASED Calculations
Results of local-scale FE analyses on LC and TD connections are summarized in
Figure 6a,b and Table 3, in terms of (i) von Mises stress distributions associated with the
highest equivalent stress range
∆σeq,max
and (ii) ASED calculations for each equivalent stress
range value
∆σeq,i
. Relevant mesh sensitivity analyses (average mesh size s= 1–5 mm) for
such quantities are also reported.
As expected, reducing the mesh size sresulted in a moderate increase of predicted
highest stresses (+15% and +9% when sranged from 5 mm up to 1 mm for LC and TD,
respectively).
It was also observed that the highest stresses were achieved at the tips of the outermost
rivet holes in both cases (
σMises,MAX,LC
= 43.1 N/mm
2
and
σMises,MAX,TD
= 28.7 N/mm
2
,
×1.66 and ×2.87 with respect to the applied stress ranges, respectively).
In light of this outcome, energy calculations were performed by monitoring the rele-
vant control volumes as described in Sections 3.1 and 3.2.

Buildings 2024,14, 2271 11 of 17
Buildings 2024, 14, x FOR PEER REVIEW 11 of 18
6 × 11
3.0
10.7
0.75
8
1
1 × 2
9.8
12.2
0.43
1 × 32
1.1
14.1
0.93
9
1
1 × 1
4.4
9.5
0.62
1 × 6
2.2
10.6
0.81
4.2. Local FE Analyses for ASED Calculations
Results of local-scale FE analyses on LC and TD connections are summarized in Fig-
ure 6a,b and Table 3, in terms of (i) von Mises stress distributions associated with the high-
est equivalent stress range Δσeq,max and (ii) ASED calculations for each equivalent stress
range value Δσeq,i. Relevant mesh sensitivity analyses (average mesh size s = 1–5 mm) for
such quantities are also reported.
As expected, reducing the mesh size s resulted in a moderate increase of predicted
highest stresses (+15% and +9% when s ranged from 5 mm up to 1 mm for LC and TD,
respectively).
It was also observed that the highest stresses were achieved at the tips of the outer-
most rivet holes in both cases (σMises,MAX,LC = 43.1 N/mm2 and σMises,MAX,TD = 28.7 N/mm2,
×1.66 and ×2.87 with respect to the applied stress ranges, respectively).
In light of this outcome, energy calculations were performed by monitoring the rele-
vant control volumes as described in Sections 3.1–3.2.
(a)
(b)
Figure 6. Von Mises stress distributions for the most critical details: (a) LC; (b) TD.
LC web
Most stressed hole
Δσ = 26 N/mm2
ASED range ∆W [mJ/mm3]
25
30
35
40
45
12345
σmax,Von Mises [N/mm2]
Mesh size s[mm]
LC - VM Stress
3.0E-03
3.2E-03
3.4E-03
3.6E-03
3.8E-03
4.0E-03
LC - ASED
45
40
35
30
25
4.0×10–3
3.8×10–3
3.6×10–3
3.4×10–3
3.2×10–3
3.0×10–3
5 4 3 2 1
TD profile
Δσ = 9.8 N/mm2
Most stressed hole
ASED range ∆W [mJ/mm3]
25
30
35
40
45
12345
σmax,Von Mises [N/mm2]
Mesh size s[mm]
TD - VM Stress
3.0E-03
3.2E-03
3.4E-03
3.6E-03
3.8E-03
4.0E-03
TD - ASED
45
40
35
30
25
4.0×10–3
3.8×10–3
3.6×10–3
3.4×10–3
3.2×10–3
3.0×10–3
5 4 3 2 1
Figure 6. Von Mises stress distributions for the most critical details: (a) LC; (b) TD.
Table 3. SED-based fatigue demand on investigated critical details.
Lower Chord—Midspan (LC) Tensile Diagonal—Support (TD)
Train Type
[-]
SED Method Train Type
[-]
SED Method
n
[-]
cw∆W
[mJ/mm3]
n
[-]
cw∆W
[mJ/mm3]
378.7 ×10–5
371.7 ×10–3
91 4.0 ×10–3 84 1.1 ×10–3
468.0 ×10–4
461.4 ×10–3
72 1.3 ×10–2 66 9.7 ×10–4
813.7 ×10–3
823.6 ×10–3
22 8.5 ×10–2 32 4.7 ×10–4
911.2 ×10–2
911.3 ×10–3
69.3 ×10–2 67.0 ×10–4
To this end, the substantial mesh-insensitivity of the ASED range can be observed in
Figure 6a,b, in compliance with remarks from [
35
,
36
]. Table 3also shows how significantly
higher values of
∆W
were attained in the LC connection compared with the TD. This
outcome plausibly resulted from (i) the presence of larger mean stresses, (ii) the influence of
secondary bending due to the moderate degree of flexural restraint provided by the riveted
backing plates, and (iii) the larger reduction of net cross-rection due to rivet holes.
The derived energetic parameters were then used to perform a SED-based fatigue
assessment of the critical details. Further insights are provided in the next section.

Buildings 2024,14, 2271 12 of 17
4.3. SED-Based Fatigue Assessment of Critical Details
Once the energetic demand parameters associated with equivalent stress ranges were
estimated, SED-based fatigue assessment of critical details was carried out through cumu-
lating fatigue damage via the well-known Miner’s rule (Equation (5) [38]):
DTOT =∑idi=∑i
nTOT,i
N∗
i
(5)
where D
TOT
is the total fatigue damage composed of the sum of i-th elementary damage
values d
i
,n
TOT,I
is the total number of (equivalent) endured cycles for a given ASED range
during the reference life, and N
i
* is the associated value of endurable cycles for the same
value of ASED range.
According to the above criterion, a given component is deemed safe against the
relevant fatigue demand if DTOT ≤1.
For assessment purposes, the master ASED curve for riveted details reported in
Milone [5] was considered (C = 14,675, m= 3.42, see Figure 7a).
Buildings 2024, 14, x FOR PEER REVIEW 13 of 18
(a)
(b)
(c)
Figure 7. SED-based fatigue assessment of critical riveted details: (a) master ASED fatigue curve [5];
(b) LC checks; (c) TD checks.
Total fatigue damage was estimated assuming a reference life Lref = 100 years, in com-
pliance with provisions in force for railway bridges in Italy [25,26]; the total number of
endured cycles nTOT,i was estimated accordingly.
As a result, both LC and TD connections can be deemed safe with respect to fatigue
issues, although with signicantly dierent margins of safety (DTOT = 1.6 × 10−2 and 4.5 ×
10−6, respectively).
In order to assess the robustness of the described SED-based procedure, and for com-
parison purposes, in the next section, the fatigue performance of critical riveted details is
reassessed with reference to both the literature [21] and normative prescriptions [13,14,22].
5. Comparison with the Literature and Normative Fatigue Assessment Techniques
As stated in the introduction, several researchers have investigated the fatigue per-
formance of hot-driven riveted connections via nominal stress-life methods (see Table 4).
Taras et al. [21] proposed seven DCs for riveted details based on the conguration of
global connections, i.e., including two- and one-sided joints, laice members, cover plates,
and built-up girders. For the relevant cases of double and single GP joints, DC 90 and DC
85 were proposed, respectively. Remarkably, (i) an invariant slope of S-N curves (m1 = m2
= 5, see Figure 8, red solid curves) was suggested as opposed to EN1993-1-9 recommen-
dations, and (ii) the gross cross-section stress range Δσgross was indicated for the estimation
of N*.
Maljaars and Euler [39] reassessed the fatigue performance of multiple congura-
tions of bolted joints to serve as a background for the prEN1993-1-9:2023 draft [22]. To this
end, although no results were reported by the authors with explicit reference to hot-driven
riveted connections, indications for ed, non-pre-tightened bolted joints can still be suit-
ably used owing to strong structural similarities.
Indeed, as discussed [5,9,10], HDP results in a lateral shank expansion which nullies
the rivet-hole gap. Moreover, due to the large variety of thermomechanical parameters
N* = 14675 (ΔW) −3.42
R² = 0.8633
0.1
1
1.00E+04 1.00E+05 1.00E+06 1.00E+07
LOG ΔW [mJ/mm3]
LOG N* [-]
MAX value
Average value
MIN value
Milone, 2023
104105106
1.0E-05
1.0E-04
1.0E-03
1.0E-02
1.0E-01
1.0E+00
1.0E+06 1.0E+10 1.0E+14 1.0E+18
Log ΔW [mJ/mm3]
Log N [-]
1.0E-05
1.0E-04
1.0E-03
1.0E-02
1.0E-01
1.0E+00
1.0E+06 1.0E+10 1.0E+14 1.0E+18
Log ΔW [mJ/mm3]
Log N [-]
LC –SED-based assessment TD –SED-based assessment
100
10−1
10−2
10−3
10−4
10−5
100
10−1
10−2
10−3
10−4
10−5
1061010 1014 1018 1061010 1014 1018
Figure 7. SED-based fatigue assessment of critical riveted details: (a) master ASED fatigue curve [
5
];
(b) LC checks; (c) TD checks.
A graphical representation of SED-based assessment is provided in Figure 7b,c, where
the expected number of cycles at failure N* is derived for both LC and TD details with
reference to each estimated ASED range.
As expected, more dispersed values of N* were attained for LC, with N*
LC
being
in the range
≈
10
7
–10
18
. This outcome plausibly resulted from the peculiar connection
configuration, in which secondary bending stresses significantly affected the stress (and
thus, strain energy) distribution near the rivet holes, while further non-linearity was
induced by friction.
Total fatigue damage was estimated assuming a reference life L
ref
= 100 years, in
compliance with provisions in force for railway bridges in Italy [
25
,
26
]; the total number of
endured cycles nTOT,i was estimated accordingly.

Buildings 2024,14, 2271 13 of 17
As a result, both LC and TD connections can be deemed safe with respect to fatigue
issues, although with significantly different margins of safety (D
TOT
= 1.6
×
10
−2
and
4.5 ×10−6, respectively).
In order to assess the robustness of the described SED-based procedure, and for com-
parison purposes, in the next section, the fatigue performance of critical riveted details is
reassessed with reference to both the literature [21] and normative prescriptions [13,14,22].
5. Comparison with the Literature and Normative Fatigue Assessment Techniques
As stated in the introduction, several researchers have investigated the fatigue perfor-
mance of hot-driven riveted connections via nominal stress-life methods (see Table 4).
Table 4. Relevant parameters for stress-life analysis of riveted details according to [13,14,21,22,39].
Reference Detail DC
[N/mm2]
Slope m1
[-]
Slope m2
[-]
Stress
Parameter
EN1993-1-9 B.D. [13,14]Two-sided joint 90 3 5 ∆σnet
One-sided joint 71 3 5 ∆σnet
prEN1993-1-9:2023 [22];
Maljaars and Euler [39]
Two-sided joint 90 5 5 ∆σ* (Equation (6))
One-sided joint 90 5 5 ∆σ* (Equation (6))
Taras et al. [21]Two-sided joint 90 5 5 ∆σgross
One-sided joint 85 5 5 ∆σgross
Taras et al. [
21
] proposed seven DCs for riveted details based on the configuration of
global connections, i.e., including two- and one-sided joints, lattice members, cover plates,
and built-up girders. For the relevant cases of double and single GP joints, DC 90 and DC 85
were proposed, respectively. Remarkably, (i) an invariant slope of S-N curves (m
1
= m
2
= 5,
see Figure 8, red solid curves) was suggested as opposed to EN1993-1-9 recommendations,
and (ii) the gross cross-section stress range ∆σgross was indicated for the estimation of N*.
Buildings 2024, 14, x FOR PEER REVIEW 14 of 18
inuencing shank contraction during cooling, the resulting rivet clamping is rather low
and unreliable, with typical values being equal to σclamp ≈ 0.5 fyr ≈ 100 N/mm2 [9,10,15].
Therefore, according to [22,39], load transfer in joints’ plates occurs via both “pin
loading” (i.e., bearing contact) at the outermost holes and “bypass loading” (i.e., net area
diusion) at subsequent holes. As a consequence, the relevant stress parameter for fatigue
analysis Δσ* should be estimated as follows (Equation (6) [22,39]):
Δσ*= ∆σnet×[a + (b − cd0
w)3]= k*∆σnet
(6)
where Δσnet is the net cross-section stress range, d0 is the hole diameter, w is the plate
width, k* is the net stress range modication factor, and a, b, c are empirical parameters
depending on the number of rows of fasteners nf.
For the relevant case of nf > 3, a = 1.0, b = 1.1, c = 1.8 may be used.
For the sake of comparison, in the present section, the results of the SED-based as-
sessment of LC and TD details are compared with the above formulations from the liter-
ature.
For thoroughness, removed DCs from EN1993-1-9 background document for HDRs
(DC90/DC71 [13,14], see Table 4 and Figure 8) are also accounted for.
According to indications reported in [13,14,21,22,39], outcomes from the rainow
method (see Section 4.1) were used to estimate stress ranges referred to the gross cross-
section Δσgross.
Hence, net (Δσnet) and modied (Δσ*) were derived based on the details’ geometrical
features. For instance, the presence of rivet holes in LC and TD connections resulted in
nominal increases of +19% and +8% in terms of stress ranges, respectively (see Table 5). In
the same fashion, fairly similar modication factors kLC* = 2.06 and kTD* = 1.91 were ob-
tained in both cases according to prEN1993-1-9:2023 prescriptions [22,39].
Table 4. Relevant parameters for stress-life analysis of riveted details according to [13,14,21,22,39].
Reference
Detail
DC
[N/mm2]
Slope
m1
[-]
Slope
m2
[-]
Stress
Parameter
EN1993-1-9 B.D. [13,14]
Two-sided joint
90
3
5
Δσnet
One-sided joint
71
3
5
Δσnet
prEN1993-1-9:2023 [22];
Maljaars and Euler [39]
Two-sided joint
90
5
5
Δσ* (Equation (6))
One-sided joint
90
5
5
Δσ* (Equation (6))
Taras et al. [21]
Two-sided joint
90
5
5
Δσgross
One-sided joint
85
5
5
Δσgross
Figure 8. Literature and normative fatigue curves for the assessment of riveted details
[13,14,21,22,39].
10
100
1000
1.0E+04 1.0E+05 1.0E+06 1.0E+07
LOG Δσ [N/mm2]
1.0E+04 1.0E+05 1.0E+06 1.0E+07
LOG N* [-]
Two-sided riveted splices One-sided riveted splices
EN1993-1-9: DC90; m1= 3, m2= 5; Δσ = Δσnet
prEN1993-1-9: DC90; m1= m2= 5; Δσ = Δσ*
Taras et al.: DC90; m1= m2= 5; Δσ = Δσgross
EN1993-1-9: DC71; m1= 3, m2= 5; Δσ = Δσnet
prEN1993-1-9: DC90; m1= m2= 5; Δσ = Δσ*
Taras et al.: DC85; m1= m2= 5; Δσ = Δσgross
1.0E+04 1.0E+05 1.0E+06 1.0E+07
LOG N* [-]
Taras et al. [21]
EN1993-1-9:2005 [12-13]
prEN1993-1-9:2023 [22]
Taras et al., 2019
EN1993-1-9
prEN1993-1-9:2023
105106107105106107
Figure 8. Literature and normative fatigue curves for the assessment of riveted details [
13
,
14
,
21
,
22
,
39
].
Maljaars and Euler [
39
] reassessed the fatigue performance of multiple configurations
of bolted joints to serve as a background for the prEN1993-1-9:2023 draft [22]. To this end,
although no results were reported by the authors with explicit reference to hot-driven
riveted connections, indications for fitted, non-pre-tightened bolted joints can still be
suitably used owing to strong structural similarities.
Indeed, as discussed [
5
,
9
,
10
], HDP results in a lateral shank expansion which nullifies
the rivet-hole gap. Moreover, due to the large variety of thermomechanical parameters
influencing shank contraction during cooling, the resulting rivet clamping is rather low
and unreliable, with typical values being equal to σclamp ≈0.5 fyr ≈100 N/mm2[9,10,15].
Therefore, according to [
22
,
39
], load transfer in joints’ plates occurs via both “pin
loading” (i.e., bearing contact) at the outermost holes and “bypass loading” (i.e., net area

Buildings 2024,14, 2271 14 of 17
diffusion) at subsequent holes. As a consequence, the relevant stress parameter for fatigue
analysis ∆σ* should be estimated as follows (Equation (6) [22,39]):
∆σ∗=∆σnet ×"a+b−cd0
w3#=k∗∆σnet (6)
where
∆σnet
is the net cross-section stress range, d
0
is the hole diameter, wis the plate width,
k* is the net stress range modification factor, and a,b,care empirical parameters depending
on the number of rows of fasteners nf.
For the relevant case of nf> 3, a= 1.0, b= 1.1, c= 1.8 may be used.
For the sake of comparison, in the present section, the results of the SED-based
assessment of LC and TD details are compared with the above formulations from the
literature.
For thoroughness, removed DCs from EN1993-1-9 background document for HDRs
(DC90/DC71 [13,14], see Table 4and Figure 8) are also accounted for.
According to indications reported in [
13
,
14
,
21
,
22
,
39
], outcomes from the rainflow
method (see Section 4.1) were used to estimate stress ranges referred to the gross cross-
section ∆σgross.
Hence, net (
∆σnet
) and modified (
∆σ
*) were derived based on the details’ geometrical
features. For instance, the presence of rivet holes in LC and TD connections resulted in
nominal increases of +19% and +8% in terms of stress ranges, respectively (see Table 5).
In the same fashion, fairly similar modification factors k
LC
* = 2.06 and k
TD
* = 1.91 were
obtained in both cases according to prEN1993-1-9:2023 prescriptions [22,39].
Fatigue assessment in terms of elementary and total damage d
i
, D
TOT
is summarized
in Table 5for both LC and TD, based on all described methodologies.
With reference to the lower chord connection, it was apparent that the SED method
and prEN1993-1-9:2023 [
22
,
39
] yielded the most severe predictions. This outcome plausibly
descends from both methods being able—although to different extents—to capture local
stress amplifications.
Conversely, indications from the EN1993-1-9 background document [
13
,
14
] proved to
be moderately unconservative (DTOT,SED = 2.37 DTOT,EN1993-1-9-BD).
This condition was even more noticeable with reference to Taras et al.’s [
21
] formula-
tion, where
∆σgross
was used (that is, stress amplification was not explicitly considered),
with DC being the same as prEN1993-1-9:2023.
Interestingly, the opposite trend was obtained for the TD connection. For instance, the
most severe result was obtained with reference to the EN1993-1-9 background document’s
predictions. This outcome plausibly resulted from the relevant DC (71 N/mm2) being the
lowest among all the formulations.
As expected, the SED method instead provided an intermediate prediction between
the prEN1993-1-9:2023 [
22
,
39
] and Taras et al. [
21
] formulations. Indeed, the former model
slightly overestimated the fatigue damage due to it being not calibrated for staggered joint
configurations, in which the destructive interference among close rivet holes reduces the
resulting stress amplification [
29
,
40
] and thus increases fatigue life. Conversely, in the same
fashion as LC, Taras et al.’s [
21
] formulation resulted in the lowest fatigue damage due to
∆σgross being used for calculations.
The observed differences in results were also dependent on the peculiar configuration
of the diagonals. Indeed, due to presence of riveted battens, the out-of-plane bending
moments that would arise for a one-sided connection are actually countered, as longitudinal
symmetry would be violated instead. As a consequence, local stress amplifications on rivet
holes were slightly smaller than the ones accounted for by nominal stress-life methods.
Conversely, the intrinsic local nature of the SED method ensured a more accurate
consideration of such an effect, although at the price of requiring multi-scale modelling of
the investigated case study.

Buildings 2024,14, 2271 15 of 17
Table 5. Fatigue assessment of critical details according to described formulations [13,14,21,22,39].
Lower Chord (LC)
nTOT,i
ki[-] 1.19 2.06 di=nTOT,i/N*i
∆σ∆σnet ∆σ*SED [13,14] [22,39] [21]
[-] [N/mm2][-]
2.6 ×10520.6 24.5 50.6 2.2 ×10–13 2.6 ×10–3 7.2 ×10–3 8.0 ×10–5
3.3 ×1063.0 3.6 7.4 1.4 ×10–6 1.0 ×10–4 6.1 ×10–6 6.8 ×10–8
2.2 ×10521.0 25.0 51.6 3.7 ×10–10 2.3 ×10–3 6.7 ×10–3 7.6 ×10–5
2.6 ×1066.3 7.5 15.5 6.2 ×10–5 7.6 ×10–4 2.0 ×10–4 2.2 ×10–6
3.7 ×10426.0 31.0 63.9 1.2 ×10–8 7.4 ×10–4 3.3 ×10–3 3.7 ×10–5
8.0 ×1054.0 4.8 9.8 1.2 ×10–2 5.9 ×10–5 6.2 ×10–6 7.0 ×10–8
3.7 ×10415.0 17.9 36.9 6.6 ×10–7 1.4 ×10–4 2.1 ×10–4 2.3 ×10–6
2.2 ×1057.4 8.8 18.2 4.4 ×10–3 1.0 ×10–4 3.7 ×10–5 4.1 ×10–7
DTOT [-] 1.6 ×10–2 6.8 ×10–3 1.7 ×10–2 2.0 ×10–4
Variation [-] -×2.37 ×0.97 >>
Tensile Diagonal (TD)
nTOT,i
ki[-] 1.08 1.91 di=nTOT,i/N*i
∆σ∆σnet ∆σ*SED [13,14] [22,39] [21]
[-] [N/mm2][-]
2.6 ×1055.5 5.9 11.3 5.7 ×10–7 7.5 ×10–5 4.1 ×10–6 1.4 ×10–7
3.3 ×1063.2 3.5 6.6 1.7 ×10–6 1.9 ×10–4 3.5 ×10–6 1.3 ×10–7
2.2 ×1054.9 5.3 10.1 2.5 ×10–7 4.5 ×10–5 1.9 ×10–6 7.0 ×10–8
2.6 ×1063.0 3.2 6.2 8.6 ×10–7 1.3 ×10–4 2.0 ×10–6 7.2 ×10–8
3.7 ×1049.8 10.6 20.2 1.1 ×10–6 6.1 ×10–5 1.0 ×10–5 3.7 ×10–7
8.0 ×1051.1 1.2 2.3 2.2 ×10–8 1.9 ×10–6 4.1 ×10–9 1.5 ×10–10
3.7 ×1044.4 4.8 9.1 3.3 ×10–8 5.5 ×10–6 1.9 ×10–7 6.8 ×10–9
2.2 ×1052.2 2.4 4.5 2.3 ×10–8 4.1 ×10–6 3.6 ×10–8 1.3 ×10–9
DTOT [-] 4.5 ×10–6 5.1 ×10–4 2.2 ×10–5 7.9 ×10–7
Variation [-] - << ×0.20 ×5.70
6. Conclusions
In the present work, the advanced fatigue assessment of a riveted railway bridge
located in Italy is illustrated. For this purpose, refined multi-scale modelling of the case
study was developed, and fatigue analyses were carried out via the advanced strain energy
density (SED) method. Derived results were thus compared against the literature and
normative fatigue provisions for hot-driven riveted connections. Based on the derived
outcomes, the following conclusive remarks can be pointed out:
•
Global-scale FE analyses of the 3D truss structure highlighted that most critical details
from a fatigue perspective were represented by the lower chord (LC) midspan connec-
tion and by the most stressed tensile diagonal (TD) connection. This condition resulted
from the adopted structural scheme and the presence of significant mean tensile stress;
•
Local-scale FEMs were developed to accurately represent the geometric details. Subse-
quently, SED-based fatigue demand was derived with reference to equivalent stress
cycles to be endured during the reference service life Lref = 100 years;
•
According to the well-known Miner criterion, both connections were deemed safe
with respect to fatigue issues, although with significantly different margins of safety
(DTOT = 1.6 ×10−2and 4.5 ×10−6, respectively);
•
SED-based outcomes were compared against nominal stress-life approaches drawn
from the literature and normative prescriptions. Accordingly, the SED method pre-
dicted higher fatigue damage in the LC compared with other methods, due to its
ability to capture local stress amplification induced by secondary bending;

Buildings 2024,14, 2271 16 of 17
•
Conversely, lower damage was observed for TD with respect to EN1993-1-9 and
prEN1993-1-9:2023, due to the peculiar profiles and connection configuration in which
out-of-plane bending was inhibited by longitudinal symmetry;
•
Further studies will be carried out to assess the validity of SED-based fatigue assess-
ment with reference to full-scale structures.
Author Contributions: Conceptualization, A.M. and M.D.; methodology, A.M.; software, A.M.; valida-
tion, A.M. and M.D.; formal analysis, A.M.; resources, R.L.; data curation, A.M.; writing—original draft
preparation, A.M.; writing—review and editing, M.D. and R.L.; visualization, A.M.; supervision, M.D.
and R.L.; project administration, R.L.; funding acquisition, R.L. All authors have read and agreed to
the published version of the manuscript.
Funding: This research received no external funding.
Data Availability Statement: All relevant data are available within the body of the manuscript.
Acknowledgments: The authors would like to express their gratitude to Giuliano Cipolletta and Erika
Sasso for their precious work during their Master’s theses. Their help is gratefully acknowledged.
Conflicts of Interest: The authors declare no conflicts of interest.
References
1.
de Jesus, A.M.P.; da Silva, A.L.L.; Figueiredo, M.V.; Correia, J.A.F.O.; Ribeiro, A.S.; Fernandes, A.A. Strain-life and crack
propagation fatigue data from several Portuguese old metallic riveted bridges. Eng. Fail. Anal. 2011,18, 148–163. [CrossRef]
2.
Pedrosa, B.; Correia, J.A.F.O.; Rebelo, C.; Lesiuk, G.; de Jesus, A.M.P.; Fernandes, A.A.; Duda, M.; Calcada, R.; Veljkovic, M.
Fatigue resistance curves for single and double shear riveted joints from old Portuguese metallic bridges. Eng. Fail. Anal. 2019,96,
255–273. [CrossRef]
3.
Sieber, L.; Urbanek, R.; Bar, J. Crack-Detection in old riveted steel bridge structures. Procedia Struct. Integr. 2019,17, 339–346.
[CrossRef]
4.
Marmo, R. Numerical and Experimental Investigation on Shear Behaviour of Riveted Connections. Ph.D. Thesis, University of
Naples “Federico II”, Naples, Italy, 2015.
5.
Milone, A. On the Performance of Lap-Shear Hot-Driven Riveted Connections: Experimental and Numerical Study. Ph.D. Thesis,
University of Naples “Federico II”, Naples, Italy, 2023.
6.
Collette, Q.; Sire, S.; Wouters, I. Lap shear tests on repaired wrought-iron riveted connections. Eng. Struct. 2015,85, 170–181.
[CrossRef]
7.
Bertolesi, E.; Buitrago, M.; Adam, J.M.; Calderon, P.A. Fatigue assessment of steel riveted railway bridges: Full-scale tests and
analytical approach. J. Constr. Steel Res. 2021,182, 106664. [CrossRef]
8.
Liu, Z.; Hebdon, M.H.; Correia, J.A.F.O.; Carvalho, H.; Vilela, P.M.L.; de Jesus, A.M.P.; Calcada, R.A.B. Fatigue Assessment of
Critical Connections in a Historic Eyebar Suspension Bridge. J. Perform. Constr. Facil. 2019,33, 04018091. [CrossRef]
9.
Milone, A.; D’Aniello, M.; Landolfo, R. Influence of camming imperfections on the resistance of lap shear riveted connections. J.
Constr. Steel Res. 2023,203, 107833. [CrossRef]
10.
D’Aniello, M.; Fiorino, L.; Portioli, F.; Landolfo, R. Experimental investigation on shear behaviour of riveted connections in steel
structures. Eng. Struct. 2011,33, 516–531. [CrossRef]
11.
Kuzawa, M.; Kaminski, T.; Bien, J. Fatigue assessment procedure for old riveted road bridges. Arch. Civ. Mech. Eng. 2018,18,
1259–1274. [CrossRef]
12.
Marques, F.; Correia, J.A.; de Jesus, A.M.; Cunha, A.; Caetano, E.; Fernandes, A.A. Fatigue analysis of a railway bridge based on
fracture mechanics and local modelling of riveted connections. Eng. Fail. Anal. 2018,94, 121–144. [CrossRef]
13. EN1993-1-9; Eurocode 3: Design of Steel Structures—Part 1–9: Fatigue. CEN: Brussels, Belgium, 2005.
14. EN1993-1-9; Background Document for EN1993-1-9. CEN: Brussels, Belgium, 2005.
15.
Leonetti, D.; Maljaars, J.; Pasquarelli, G.; Brando, G. Rivet clamping force of as-built hot-riveted connections in steel bridges. J.
Constr. Steel Res. 2020,167, 105955. [CrossRef]
16.
Landolfo, R.; Cascini, L.; D’Aniello, M.; Portioli, F. Gli effetti del degrado da fatica e corrosione sui ponti ferroviari in carpenteria
metallica: Un approccio integrato per la valutazione della vita residua. Riv. Ital. Della Saldatura 2011,3, 367–377. (In Italian)
17.
De Jesus, A.M.P.; Figueiredo, M.A.V.; Ribeiro, A.S.; de Castro, P.M.S.T.; Fernandes, A.A. Residual Lifetime Assessment of an
Ancient Riveted Steel Road Bridge. Strain 2011,47, 402–415. [CrossRef]
18.
Lazzarin, P.; Zambardi, R. A finite-volume-energy based approach to predict the static and fatigue behavior of components with
sharp V-shaped notches. Int. J. Fract. 2001,112, 275–298. [CrossRef]
19.
Berto, F.; Lazzarin, P. Recent developments in brittle and quasi-brittle failure assessment of engineering materials by means of
local approaches. Mater. Sci. Eng. R Rep. 2014,75, 1–48. [CrossRef]
20. Radaj, D.; Vormwald, M. Advanced Methods of Fatigue Assessment; Springer: New York, NY, USA, 2013.

Buildings 2024,14, 2271 17 of 17
21.
Taras, A.; Greiner, R. Development and Application of a Fatigue Class Catalogue for Riveted Bridge Components. Struct. Eng. Int.
2019,20, 91–103. [CrossRef]
22. prEN1993-1-9:2023; Eurocode 3: Design of Steel Structures—Part 1-9: Fatigue—2023 Draft. CEN: Brussels, Belgium, 2023.
23.
Mercuri, M.; Pathirage, M.; Gregori, A.; Cusatis, G. Influence of self-weight on size effect of quasi-brittle materials: Generalized
analytical formulation and application to the failure of irregular masonry arches. Int. J. Fract. 2023,246, 117–144. [CrossRef]
24. Computer and Structures, Inc. (CSI). SAP200 v23 User’s Manual; CSI: Walnut Creek, CA, USA, 2021.
25. D.M. 17/01/2018; Aggiornamento delle “Norme tecniche per le costruzioni”. MIT: Rome, Italy, 2018. (In Italian)
26.
Circolare 21/01/2019 n.7; Istruzioni per l’applicazione dell’«Aggiornamento delle “Norme Tecniche per le Costruzioni”. CSLLPP:
Rome, Italy, 2019. (In Italian)
27. RFI DTC INC PO SP IFS 003 A; Specifica per la Verifica a Fatica dei Ponti Ferroviari. RFI: Rome, Italy, 2011. (In Italian)
28. Matsuishi, M.; Endo, T. Fatigue of metals subjected to varying stress. Jpn. Soc. Mech. Eng. 1968,68, 37–40.
29. Anderson, T.L. Fracture Mechanics: Fundamentals and Applications, 4th ed.; CRC Press: Boca Raton, FL, USA, 2017.
30.
European Convention for Constructional Steelwork (ECCS). Fatigue Design of Steel and Composite Structures, 2nd ed.; ECCS:
Bruxelles, Belgium, 2018.
31. Reis, A.J.; Oliveira Pedro, J.J. Bridge Design: Concepts and Analysis; John Wiley & Sons: Hoboken, NJ, USA, 2019.
32.
Milone, A.; Foti, P.; Filippi, S.; Landolfo, R.; Berto, F. Evaluation of the influence of mean stress on the fatigue behavior of notched
and smooth medium carbon steel components through an energetic local approach. Fatigue Fract. Eng. Mater. Struct. 2024,46,
4315–4332. [CrossRef]
33.
Milone, A.; Foti, P.; Viespoli, L.M.; Wan, D.; Mutignani, F.; Landolfo, R.; Berto, F. Influence of hot-dip galvanization on the fatigue
performance of high-strength bolted connections. Eng. Struct. 2024,299, 117136. [CrossRef]
34. Simulia. ABAQUS v6.14 User’s Manual; Dassault Systemes Simulia Corp.: Providence, RI, USA, 2014.
35.
Lazzarin, P.; Berto, F.; Zappalorto, M. Rapid calculations of notch stress intensity factors based on averaged strain energy density
from coarse meshes: Theoretical bases and applications. Int. J. Fatigue 2010,32, 1559–1567. [CrossRef]
36.
Foti, P.; Berto, F. Strain Energy Density evaluation with free coarse mesh model. Mater. Des. Process. Commun. 2020,2, e116.
[CrossRef]
37.
Foti, P.; Razavi, N.; Ayatollahi, M.R.; Marsavina, L.; Berto, F. On the application of the volume free strain energy density method
to blunt V-notches under mixed mode condition. Eng. Struct. 2021,230, 111716. [CrossRef]
38. Miner, M.A. Cumulative Damage in Fatigue. J. Appl. Mech. 1945,3, 159–164. [CrossRef]
39.
Maljaars, J.; Euler, M. Fatigue S-N curves of bolts and bolted connections for application in civil engineering structures. Int. J.
Fatigue 2021,151, 106355. [CrossRef]
40. Pilkey, D.F.; Pilkey, W.D.; Bi, Z. Peterson’s Stress Concentration Factors, 4th ed.; John Wiley & Sons: Hoboken, NJ, USA, 2020.
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