Different Approaches for Remaining Fatigue Life Estimation of …C).pdf · 2018-01-08 · stress...

Steel Structures 7 (2007) 263-276 www.kssc.or.kr

Different Approaches for Remaining Fatigue Life Estimation

of Critical Members in Railway Bridges

Siriwardane Sudath Chaminda1, Mitao Ohga2,*, Ranjith Dissanayake3 and Kazuhiro Taniwaki4

1Graduate Student, Department of Civil and Environmental Engineering, Ehime University, Bunkyo-cho 3, Matsuyama 790-8577, Japan2Professor, Department of Civil and Environmental Engineering, Ehime University, Bunkyo-cho 3, Matsuyama 790-8577, Japan

3Senior Lecturer, Department of Civil Engineering, University of Peradeniya, Peradeniya 20400, Sri Lanka4Lecturer, Department of Civil and Environmental Engineering, Ehime University, Bunkyo-cho 3, Matsuyama 790-8577, Japan

Abstract

Rail authorities all over the world are paying attractive attentions to evaluate the remaining fatigue lives of riveted railwaybridges since most of these bridges are currently reaching the ends of their theoretical fatigue lives. This paper proposes threedifferent approaches to evaluate remaining fatigue life of an existing riveted railway bridge. First proposed approach is basedon combination of measured stress histories, Miner’s rule and partially known Wöhler curve. The second approach mainlyconsists of measured stress histories, recently developed sequential law and fully known Wöhler curve. The both mentionedapproaches are specially based on evaluation of primary stresses and code provided fatigue curve. Therefore, proposed thirdapproach is anchored in secondary (local) stresses, sequential law and experimental Wöhler curve. Finally the obtained fatiguelives are compared. Thus, it has been concluded that the second approach is more advisable for general use and third approachhas been recommended for detail studies.

Keywords: Fatigue life, Railway bridges, Structural appraisal, Stress concentration, Sequential law

1. Introduction

In the decades of 1960’s and 1970’s, civil engineering

was dominated by design and construction of new civil

infrastructures. Considering the management of structures

in terms of maintenance, member replacement had a wide

acceptance during this period (Junk et al., 2004). At

present, rail authorities all over the world are paying special

attention to evaluate the remaining fatigue life of riveted

railway bridges, since most of these bridges are nearing

the end of their theoretical fatigue lives. Furthermore, the

fatigue behaviour of wrought-iron and older steels, which

were mainly used for the construction of these bridges, is

not well known. These observations coupled with the lack

of information on loading history of these bridges raise

question about their fatigue performance (Imam et al.,

2005). As a result, the assessment of remaining fatigue

life of riveted railway bridges for continuing services has

become more important than ever, especially when

decision making regarding structure replacement, deck

replacement or other major retrofits.

Even though considerable amount of the past studies

have been done on this area, experiences from engineering

practices have indicated that fatigue analysis based on

specification loads and distribution factors usually

underestimates the remaining fatigue life of the existing

bridges by overestimating the live load stress ranges. In

this context fatigue evaluation based on field measured

stress range histograms under actual traffic load proves to

be a more accurate and efficient method for existing

bridges (Köröndi et al., 1998; Constantine et al., 2004).

Most of the present day used fatigue assessment approaches

of railway bridges are generally based on combination of

measured stress histories, Miner’s rule (Miner, 1945) and

railway code provided fatigue curve (also referred to as S-

N or Wöhler curve). However, the Miner’s rule does not

properly take account of loading sequence effect (Suresh,

1998; Dattoma et al., 2006; Mesmacque et al., 2005). As

a result, real fatigue life due to same loading pattern is

higher than the Miner’s expectation for increasing of

loads and it is lower than the Miner’s expectation for

decreasing of loads. Recently, a new damage indicator-

based sequential law (Mesmacque et al., 2005) was

originated to overcome this shortcoming of Miner’s rule

and it has been proved that sequential law gives more

realistic results than Miner’s rule when material is subjected

to variable amplitude loading. As for the authors view,

application of the sequential law to estimate the remaining

fatigue life of existing railway bridges is not yet published.

Main objective of this study is to apply above mentioned

sequential law to evaluate the remaining fatigue life of an

existing railway bridge based on both field measured

*Corresponding authorTel: +81-89-9279816, Fax: +81-89-9279816E-mail: [email protected]

264 Siriwardane Sudath Chaminda et al.

stresses and its current condition. To achieve this objective,

this paper presents methodologies to estimate remaining

fatigue life of a riveted railway bridge based on three

different approaches. The first approach is based on

combination of measured stress histories, Miner’s rule

and code provided Wöhler curve. This approach shows

proximity to present day available approach. The considered

fatigue curve only describes stress ranges, which are

corresponding to more than ten thousands of failure

cycles (usually called as partially known Wöhler curve).

The proposed second approach mainly consists of

measured stress histories, recently developed sequential

law and fully known Wöhler curve. In this approach it is

essential to use the fully known Wöhler curve as the

related fatigue curve. Therefore, the technique, which

utilizes to transfer the partially known Wöhler curve to

fully known curve, is also discussed. This paper also

describes the reasonably accurate procedure to obtain the

past stress histories from present day measured stress

histograms. This is of extreme importance because most

of the bridges do not have the past strain measurements.

The both mentioned approaches are specially based on

evaluation of primary stresses and code provided fatigue

curve. In reality secondary stress (local stress concentration)

effect in riveted connection between the primary members

of bridges was found to be one of main reasons for

fatigue damage (Fisher et al., 1984). Further it has been

identified that the rotational fixity of riveted connection

and the variation in the clamping force of rivets (Akesson,

1994) are the major causes leading to fatigue cracking in

riveted connection (Imam et al., 2005). However, in the

cases of both discussed approaches, some safety factors,

which are included in code provided fatigue curve, help

in some extent to capture the effect of fatigue damage due

to secondary stresses. Since these safety factors are

corresponding to the assessment code, they are mostly

applicable within the design life of the bridge. But the

applicability of these factors for old structures, which are

already exceeded the design age, has not yet been confirmed.

Therefore, third approach of the study is proposed to

evaluate the remaining lives of railway bridges based on

secondary stresses evaluation, multiaxial formulation of

sequential law and experimental Wöhler curve.

Initially paper describes the details of the considered

railway bridge and the appraisals related to condition

evaluation, finite element analysis, material testing,

experimental static and dynamic load testing. Secondly,

the remaining fatigue life of each critical component of

the bridge is discussed separately based on the above

mentioned approaches. Finally comparisons of the results

are made and the validity of obtained fatigue lives is

confirmed. Hence the applicability of the proposed

approaches is discussed.

2. Bridge Description and Present Condition

The selected bridge is one of the longest railway bridge

in Sri Lanka spanning 160 m (Fig. 1). It is a six span

riveted bridge with double lane rail tracks having warren

type semi through trusses, supported on cylindrical piers.

The bridge deck is made of wrought iron and the piers are

made of cast iron casings with infilled concrete. The

bridge was constructed in 1885. Details of trains carried

by the bridge at present and their frequencies illustrate

that the bridge is subjected to variable amplitude loading.

A condition survey was carried out to assess the present

condition of the bridge with the contribution of expert

practicing engineers. Basically it consisted of detailed visual

Figure 1. General views of the riveted bridge.

Figure 2. Some identified corroded locations of the bridge.

Different Approaches for Remaining Fatigue Life Estimation of Critical Members in Railway Bridges 265

examination and in situ measurements of each component

of the bridge. Further, the identified critical connections

of the bridge were subjected to X-Ray examination.

Although the identified critical members with respective

connections should be subjected to various non-destructive

field tests such as ultrasonic examination, magnetic particle

examination, radiographic examination and etc, the lack of

facilities with experiences hindered such detail investigation

of the bridge.

The visual examination revealed that some places of

the bridge have been subjected to mild corrosion due to

the absence of anti corrosive coating (see Fig. 2). Some

rivets connecting the diagonal bracings of the bridge deck

were found to be loose. No visual cracks were observed

in any component of the super structure. However, the

results of X- ray examination showed very small internal

gaps between rivet-plate contact surfaces at a few

connections. In situ measurements of member sizes,

connections and support bearings verified the fact that the

existing drawings were applicable and only few significant

variations were observed. Further the bridge components

have been categorized to several groups entitled “member

set” by considering similar cross sectional properties as

shown in Fig. 3. Finally it was revealed that comparative

maintenance work carried out on the bridge thus far is

satisfactory.

3. Structural Appraisal

Since fatigue evaluation based on field-measured stress

range histograms under actual traffic loads of the bridge is

a more accurate and efficient method for the existing

bridges (Köröndi et al., 1998; Constantine et al., 2004),

this section describes the evaluation methodology of real

performance in the bridge according to the current state.

Extensive laboratory and field-testing, as well as analytical

work were performed to assess the condition of the

superstructure and hence determine the remaining fatigue

life of each critical component of the bridge.

3.1. Testing of bridge materials

Laboratory tests were carried out to determine the

mechanical properties and chemical composition of the

materials that were used for the construction of the

bridge. As-built drawings of the bridge show that only

one type of material has been used for the superstructure

(main truss girder, secondary cross girders, stringers,

bracings and bearings). The sampling of materials, specimen

preparation and testing were carried out according to the

ASTM standards. The chemical analyses as well as

microscopic examinations lead to the conclusion that the

bridge super structure material is wrought iron. The

fatigue strength of the bridge material was obtained by

the rotating bend test. The rotating bending smooth

specimens were prepared from extracted members. From

the results of primary analysis, it was confirmed that one

of the extracted members is subjected to compressive

stress and the other one is operating well bellow the

fatigue strength of bridge constructed material (wrought

iron). Therefore, the test specimens are free of fatigue

damage. As a result, the obtained fatigue curve (from

rotating bend test) exhibits the full fatigue life of the

bridge-constructed material (wrought iron). Further, it

describes only the fatigue behavior of wrought iron

material while closely representing the detail class B

structure (BS5400, 1980). The obtained values for elastic

modulus, yield strength and fatigue strength are 195 GPa,

240 MPa and 155 MPa, respectively.

3.2. Field load testing

Static and dynamic load testing was performed to study

the real behavior of the bridge under various load

combinations. The obtained results were used to develop

a validated analytical model. The test results further

assisted in evaluating actual dynamic factors of each

structural component. The in situ measurements were

performed using two M8 engines, each weighting 1120

kN, which is the heaviest rail traffic in current operation.

The bridge was instrumented with strain gauges placed

at selected locations to measure normal strains. In addition,

the triaxial vibrations were recorded at several locations

using accelerometers. In order to measure free vibration,

accelerations were recorded after the M8 engines had

crossed the bridge. Displacement transducers were used

to measures the vertical deflection at three places around

the mid span area of the bridge. The measured locations

Figure 3. Member set categorization (a) Main truss girder (b) Horizontal bridge deck.


are as shown in Fig. 4. The static and dynamic responses

of strains, displacements and accelerations were acquired

using the sophisticated static and dynamic data loggers.

The different type of load combinations, which are

critical to the bridge were obtained by placing as well as

moving the two test-engines under different speeds. The

considered three static load combinations are defined as

static load case (SLC) 1, 2 and 3 by considering criteria

of maximum shear effect, maximum bending effect

(maximum deflection) and maximum torsion effect to the

bridge deck respectively. The loading positions corresponding

to the mentioned three load cases are shown in Fig. 5.

The criteria, which were considered for dynamic load

combinations, basically illustrate that impact effect to the

bridge with different levels of speed and traction force

effect. Apart from the above mentioned formal field load

testing, the bridge was subjected to a two days continuous

field measurement program under present day actual

traffic. Even though under this investigation many types

of load combinations were considered, only the combinations,

which were used to evaluate the fatigue life of the bridge,

are only specify in the paper. When the bridge is affected

by maximum load due to the present day heaviest train

passage, the obtained sample measurements are shown in

Fig. 6. Finally the dynamic factors were obtained as 1.3,

1.4 and 1.4 for main truss girders, secondary cross girders

and stringers respectively by using the curves illustrated

in Fig. 7.

Figure 4. Locations of the strain gauges and displacement gauges (a) Main truss girder (b) Horizontal bridge deck.

Figure 5. Loading positions corresponding to three static load cases (a) SLC 1 (b) SLC 2 (c) SLC 3.


3.3. Development of the validated analytical model

The Bridge girder was analyzed using the finite element

(FE) method employing the general-purpose package

SAP 2000. A three-dimensional (3D) model (Fig. 8) of

one complete middle span of the bridge was analyzed

under test loadings and actual loadings to determine

stresses in members and deflections, as well as variations

of stresses under moving loads. The bridge deck was

modeled with 3D frame elements and the riveted

connections are assumed to be fully-fixed (Imam et al.,

2005). Even though the cross girders are ideally supported

on bottom chord of the main truss girder, the assumption

of rotational stiffness behavior with magnitude 18200

kNm/rad about second local axis for representative

connection of cross girder to truss were found to be in

better agreement with field measurements than the pinned

connection assumption. Every riveted connections of

cross girders with both stringers and bracings were

assumed to be fixed.

The validation of FE model was done by comparing the

results from analysis with those from field-tests as shown

in Table 1. From the results of static load cases it was

seen that there is good agreement among analytical

results of FE model and the measurements of the actual

bridge. Therefore, the considered 3D frame element

model was termed as “validated analytical model”. Then

the validated analytical model was used to analyze the

bridge for loading combinations involving the train types

Figure 6. Field measurements of the bridge due to heaviest load (a) Stresses at bottom chord of the main girder (b)Stresses at top chord of the main girder (c) Stresses at diagonal members which are usually subjecetd to tensile stress (d)Stresses at diagonal members which are usually subjecetd to compression stress (e) Stresses at stringers (f) Stresses atsecondry cross girders (g) Vertical displacemnt at midspan (h) Vertical acceleration at midspan.

Figure 7. Dynamic factor determination curves (Maximum responds variation with speed) (a) Main truss girder (b)Secondary cross girders (c) Stringers.


specified by the owner, in order to assess the strength,

stability and remaining fatigue life of the bridge.

4. Remaining Fatigue Life Estimation -Proposed Approach 1

Remaining fatigue life evaluation of the critical members

in each member set is discussed in this section. Evaluations

are specially based on primary stresses, which are

determined by the global analysis of whole structure.

Previously mentioned validated analytical model was used

to evaluate primary stresses and Miner’s rule (Miner,

1945) was employed to obtain remaining fatigue life for

the bridge.

4.1. Primary stress evaluation

To apply the Miner’s rule, it is essential to determine

the primary stress ranges generated by the passage of

trains over the bridge. Therefore, it is required to know

the stress cycles (stress histories) distributions of all the

critical members of each member sets for trains that are

included in the timetables obtained from bridge owner for

present and past rail traffic over the bridge. Since the used

types of trains are changed with age of the bridge, the age

had to be divided in to eight periods. According to the

past and present time tables of the bridge, it could be

decided that the traffic sequence is almost constant during

a single week of each period of age. Then the validated

analytical model was used to obtain the static stress

histories of each critical member during a single week of

each period. Due to the dynamic effect of moving trains,

the actual working stresses should be higher than the

analytical static stress. Therefore, the dynamic factor of

each member, which was found experimentally in sub

section 3.2, was used to multiply the static stress to get

the service stresses. Then the stress histories were

converted into stress ranges by using the reservoir

counting method (BS 5400 part 10,1980) and hence the

stress range histogram can be obtained for all the

members. The stress range histograms for one critical

member (critical member of member set DT 3) are drawn

in Fig. 9 for all the periods of its age.

4.2. Determination of fatigue curve

In order to capture the fatigue damage due to the

secondary stresses near the riveted connection or

discontinuities, detail class (BS 5400 part 10, 1980) of

riveted connection based S-N curves are considered for

life estimation. The detail class is determined by considering

the quality of the workmanship and current condition of

the riveted connection. Since field investigations reveal

that the connections of the bridge represent the lapped or

spliced connection behavior with normal clamping force,

riveted connections were classified as class Wrought-iron

(WI) which is proposed by the UK railway assessment

code (Network Rail, 2001). Hence the S-N curve, which

Figure 8. 3D frame element model for single span (a) Deflected shape for SLC 2 (b) Axial force diagram at SLC 2 (c)Bending moment diagram at SLC 2.

Table 1. Comparison of FE analytical results with load test results

Static load caseDisplacement (mm) Stress (MPa)

Location of measurement Load test FEM Location of measurement Load test FEM

SLC 1 Main girder mid span 19.4 21.0 Critical member of DC3 -40.2 -40.6

Critical member of DT3 51.4 57.3

Critical member of MT3 47.3 48.2

SLC 2 Main girder mid span 21.3 22.5 Critical member of DC3 -37.8 -37.7



SLC 3 Main girder mid span - 19.1 Critical member of DC3 -39.5 -39.9




is mentioned under the UK railway assessment code for

WI detail class connection (Network Rail, 2001), was

considered as the suitable fatigue curve for this evaluation.

4.3. Fatigue life estimation

With help of stress cycle distributions and the daily rail

frequency, the specified number of yearly repetition of

various stress ranges in the stress histories (n) for each

time period can be found. For the particular ith period of

age, Miner’s summation of fatigue damage, αi = (Σn/N)i x

(Number of years for period where sequence of operations

was constant throughout). Miner’s summation of the total

cumulative fatigue damage for all periods α = Σαi where

i = 1 to k which is equal to number of divided periods of

age. For this bridge, value of k was 8. Therefore remaining

Miner’s summation of fatigue damage at present = (1 −

α). Hence the remaining life of each member, assuming

that the future sequence of passage is the same as that for

Figure 9. Sample stress range histograms for critical member of set DT3 for (a) Period from 1995 to date (b) Period from1985 to 1994 (c) Period from 1975 to 1984 (d) Period from 1970 to 1974 (e) Period from 1946 to 1969 (f) Period from1928 to 1945 (g) Period from 1910 to 1927 (h) Period from 1885 to 1909.


the present period = [(1 − α)/αpresent )] years, where αpresent

= (Σn/N)i.

The remaining fatigue lives for each critical member of

corresponding member set are calculated by following the

above methodology and summarized as shown in Table 2.

Since the Miner’s rule does not properly take account of

loading sequence effect (Suresh, 1998; Dattoma et al.,

2006; Mesmacque et al., 2005), the validity of the obtained

remaining lives cannot be assured. Therefore, the recently

developed new damage indicator based on sequential law

(Mesmacque et al., 2005), is utilized as second approach

for remaining fatigue life estimation of this bridge.

5. Remaining Fatigue Life Estimation - Proposed Approach 2

This section also discuses remaining fatigue life evaluation

of the critical members in each member set. Evaluations

are specially based on primary stresses, which are determined

by the global analysis of whole structure. Previously

mentioned validated analytical model was used to

evaluate primary stresses as above section and uniaxail

sequential law (Mesmacque et al., 2005) was employed

to obtain a more realistic fatigue life for the bridge.

5.1. Primary stress evaluation

In order to apply the uniaxil sequential law, it is

essential to determine the primary stress ranges generated

by the passage of trains over the bridge. Primary stress

evaluation is as same as previous approach and the age of

the bridge also divided in to same number of periods as

before. The stress ranges, which correspond to each

period of age, were obtained by using reservoir counting

method as similar to proposed approach 1 (see subsection

4.1).


The chosen fatigue curve in previous approach only

describes stress ranges, which are corresponding to more

than ten thousands of failure cycles (usually called as

partially known Wöhler curve). In the case of sequential

law it is essential to know the Wöhler curve for full range

of number of cycles. Therefore, the chosen partially

known Wöhler curve in proposed approach 1, which is

mentioned under the UK railway assessment code

(Network Rail, 2001), was transferred to fully known

Wöhler curve by using Kohout and Vechet Wöhler curve

modeling technique (Kohout et al., 2001). The obtained

function and the geometrical shape of new fatigue curve,

which corresponds to class WI riveted connection, are

illustrated in Fig. 10 (a).


A new damage indicator based sequential law in

multiaxial fatigue, (Mesmacque et al., 2005), was used to

obtain a more realistic fatigue life for the bridge. A

detailed description of the damage stress model and the

definition of damage indicator, Di is available in the

corresponding paper (Mesmacque et al., 2005). In

appendix A, only the concept is summarized with an

algorithm for understanding.

The new damage indicator (present Di value) was

calculated from the date of bridge construction to the

present by considering the sequence of stress ranges of

each critical member. Assuming that future sequence of

passage is similar to that of the present day, the time

taken to reach the present day’s Di values to one (when

Di = 1 is considered as fatigue failure) was considered as

the remaining fatigue life for each critical member. The

calculated remaining fatigue lives for critical members of

each member set are shown in Table 2. The critical

members of which the stress range is entirely in

compression zone, the effect of fatigue damage were

ignored (BS 5400 part 10,1980).

6. Remaining Fatigue Life Estimation - Proposed Approach 3

Remaining fatigue life evaluation of a critical member

of one set is discussed in this section and evaluations are

Figure 10. W?hler curve for wrought iron material (a) Predicted from UK railway assessment code (b) Predicted fromfatigue test data.


especially based on secondary stresses, which are

generated around the riveted connection due to stress

concentration effect of primary stress. The considered

secondary stresses are usually subjected to multiaxial

state of stress. To evaluate the so-called secondary

stresses, the riveted area of the member was subjected to

further fine mesh FEM analysis. The Wöhler curve,

which was determined from the rotating bend tests of the

material, was utilized for the remaining fatigue life

calculation. Since this evaluation considers more realistic

critical stress histories (secondary stress) and real fatigue

damage behavior of the construction material (experimental

Wöhler curve), it can be said that the fatigue life which

are calculated using this method gives more realistic

predictions to fatigue damage. Since this secondary stress

evaluation approach is also a macroscopic scale and the

highly stressed area has been subjected to various types

of micro structural changes with age, actual fatigue life

may somewhat deviate from this estimation. Therefore,

main objective of this section is not to give an exact result

for fatigue life but to apply the sequential law to estimate

remaining life due to multiaxial fatigue.

6.1. Secondary stress evaluation

When a hot rivet is inserted into the hole of plates in

order to connect them and when the second head is

formed from the protruding shank, the rivet gets shortened

in length due to cooling. However most of shrinkage of

the free rivets is restricted by the connected plates, which

consequently are compressed through the thickness. The

residual tensile force in the rivet and the compressive

force in the plates get balanced each other; i.e. called as

clamping force. Therefore the clamping force from the

rivet generates a complex tri-axial stress state in the

connected plate in the vicinity of the rivet hole (Akesson.,

1994). Finally the clamping force seems to be affected by

the mechanism of distribution of stresses along the

connection. The experience from the field practice reveals

that resulting clamping force could vary substantially due

to normal conditions. Therefore, it could consequently

not be given a reliable value. Furthermore, one can

assume a certain relaxation of the rivet clamping force

due to creep, fretting of the interfacing plate surfaces,

overloading (due to residual plastic deformation) and etc

with the time (when bridge is getting older). However, the

secondary stress analysis, which corresponds to normal

clamping force, becomes more difficult. Because the

geometry of the problem consists of all the rivets with all

members and the connected ply. But considered criterion

of this section corresponds to the low or high clamping

force at the rivets. Therefore, to obtain reasonable

accurate results only critical member without rivets can

be considered as relevant geometry for secondary stress

analysis in this section.

Figure 11. (a) Critical riveted connection of the main truss girder (b) Close view of the critical connection and the criticalmember (c) Schematic representation of the critical member and related areas for primary and local stresses.

Table 2. Summary of remaining fatigue lives for critical members in member sets

Bridge component Member setRemaining Fatigue life from today (years)

Approach 1 Approach 2

Main girder bottom chord MT1 305 323



Cross girders CG 20 12

Stringers ST 24 13

Truss diagonal (tension member) DT1 179 191





The most critical member of the truss girder (see Fig.

11), which belongs to DT 3, was subjected to further

analysis of secondary stress evaluation. Usually, applied

load transfers to selected member through six rivets. The

nine-node isoperimetric shell elements were used for the

FE mesh as shown in Fig. 12. To represent the effect of

no clamping forces in the rivets, the actual air gap restraint

conditions were applied to represent the unilateral contact

between rivet and plate. Similarly, to simulate the effect

of friction forces due to clamping of the rivet (rotational

fixity of riveted connection), the fully bonding unilateral

contact behavior between rivet and plate was implemented.

The individual deformations of rivets due to loading were

not captured in this model. To make the continuity of

stress field between global model (shown in Fig. 8) and

the sub-model (shown in Fig. 12), it is required to use any

interface between the two models at every iterative step.

In this model, the tensile stress history of the critical

member of member set DT3, which has been obtained

from global model (shown in Fig. 8) in section 3, is

applied on the bottom face (ab of Fig. 12 (a)) as a uniform

pressure P (Imam et al., Article in press, Kiss et al., 2000,

Liu et al., 2006). The position of the ab boundary of the

sub-model (shown in Fig. 12) was determined based on

far field primary stress of the member. In house FEM

code was employed to perform a two surface plasticity

theory based nonlinear kinematic hardening elasto-plastic

analysis. Corresponding increment of far field stress histories

(P) of the member was imposed for the elasto-plastic

incremental analysis. The obtained maximum stress

contours are shown in Fig. 12 for two considered features

of riveted connection. Since related fatigue theory (new

damage indicator based sequential law) describes the

stress field at critical locations in terms of equivalent von

Mises stress (Mesmacque et al., 2005), from this analysis

the von Mises stress histories at critical location due to

daily passage of trains was obtained for fatigue life

evaluation. The von Mises stress histories are converted

in to stress ranges as similar to previous case by using the

reservoir counting method (BS 5400 part 10,1980).


The Wöhler curve was obtained from the rotating bend

test results of bridge material (subsection 3.1). The

rotating bending smooth specimens were prepared from

extracted members. From the results of structural analysis

and field load tests (section 3.2 and 3.3), it was realized

that one of the extracted member is subjected to

compressive stress and the other one is operating well

bellow the fatigue strength of bridge constructed material

(wrought iron). Therefore it could be concluded that the

test specimens are still free of fatigue damage. As a

result, the obtained fatigue curve (from rotating bend test)

exhibits the full fatigue life of the bridge-constructed

material (wrought iron). Further, it describes only the

material behavior of wrought iron and does not represent

any riveted feature of Wrought iron connection. The

mathematical expression for the test curve was obtained

using Kohout and Vechet fatigue curve modeling technique

(Kohout et al., 2001) and the obtained expression with

geometrical shape of the curve is illustrated in Fig. 10 (b).

This new function is proposed to describe the fatigue

curves in both low and high cycle fatigue regions i.e for

the whole region of cycles from tensile strength to

permanent fatigue limit.


A new damage indicator based on sequential law in

multiaxial fatigue (Mesmacque et al., 2005) was utilized

in this section to obtain a more realistic service life for the

Figure 12. (a) Fine FE mesh (b) Maximum von Mises stress contour when all six rivets are active (c) Maximum vonMises stress contour when five rivets are active (d) Maximum von Mises stress contour when four rivets are active (e)Maximum von Mises stress contour when three rivets are active (f) Maximum von Mises stress contour when two rivetsare active.


bridge. Although the hypothesis behind the fatigue model

is the same as the previous case, only difference is that

the active stress field is considered as the equivalent von

Misses stress for multiaxial state of stress. Therefore, it is

possible to use the previous fatigue concept with algorithms,

which has been described under Appendix A, for fatigue

life estimation by replacing active stress by equivalent

von Misses stress. The critical member of member set

DT3, which is the most critical member of main girder,

was subjected to fatigue evaluation by considering three

major steps. These steps describe the variation of

remaining fatigue life with condition of riveted connection,

such as level of clamping force effect, degree of surface

contact-ness of rivets and etc.

Step 1: In the first step the significance of remaining

life with effect of clamping force on all the rivets was

considered at once. When a rivet is subjected to plastic

loading, the effect of clamping force begins to deviate

from initial value. But considered connection is operating

in a elastic state of stress when it has a high clamping

force effect. (The maximum von Mises stress value is

132.2 MPa). When only the low clamping forces are

presents at all six rivets, the maximum stresses are

subjected to plastic state (The maximum von Mises stress

value is 256.7 MPa). In this case, there is no clamping

force as described before. Therefore it is possible to

assume that the clamping force of the considered connection

is not significantly changing with the time. As a result,

fatigue life was obtained by considering that the deemed

feature of clamping force remains the same from bridge

construction to the date of failure. The calculated results

are shown in Table 3.

Step 2: In this step, the fatigue life is obtained

considering that the effect of clamping force disappears

from one rivet to the next and the calculated results are

shown in Table 4. Maximum stresses for first five features

of the Table 4 (until five rivets have low clamping force,

remaining one has high clamping force) are operating

bellow the yield limits. (The maximum von Mises stress

values are 194.3, 196.1, 198.2, 208.4, 217 MPa respectively

from top feature of the Table 4). Only the last case (all six

rivets have low clamping force) is subjected to plastic

state of stress (The maximum von Mises stress value is

256.7 MPa). Therefore, here also it is possible to assume

that there is no significant change of clamping force with

the time and it is remained constant (at high clamping

force rivets) or disappeared (at low clamping force rivets)

in the elasto-plastic analysis of this step. The defined

fatigue life in this section describes the time duration

from the date when considered feature of riveted connection

appeared, to the date of fatigue failure. Further it was

considered that the sequence and density of rail passage

were similar to present period of operation.

Step 3: In the third step the fatigue damage is evaluated

based on a criterion called critical state of stress due to

release of contact-ness of rivet while all the rivets have

low clamping force. When the rivet is not properly in

contact with the plate, particular rivets tend to transfer

less or zero amount of total load, and this leads to

unexpected stress redistribution around the riveted area.

In this stage other rivets, which are carrying the load, are

called as active rivets. The fatigue lives were evaluated

stepwise by reducing the contribution of active number of

rivets in the connection. The calculated lives are shown in

Table 5 and the defined fatigue life in this section

describes the time duration from date when considered

feature of riveted connections appeared, to the date of

fatigue failure. Further it was considered that the sequence

and density of rail passage was similar to present period

of operation as similar to previous step.

7. Comparisons and Discussions of Results

Obtained remaining fatigue lives of approach 1 and 2

for critical members of each member sets were compared

each other as shown in Table 2. Even though the

Table 3. Comparison of fatigue lives obtained from three approaches

Member set

Remaining fatigue life (years)

Approach 1 Approach 2Approach 3

High clamping force Low clamping force

DT3 131 138Infinite

(Upper bound)Failed 26 years before

(Lower bound)

Table 4. Fatigue life variation when the effect of clamping force is gradually decreasing

Considered features of the riveted connection Fatigue life (years)

One rivet has low clamping force, remaining five have high clamping force Infinite fatigue life

Two rivets have low clamping force, remaining four have high clamping force Infinite fatigue life

Three rivets have low clamping force, remaining three have high clamping force Infinite fatigue life

Four rivets have low clamping force, remaining two have high clamping force Infinite fatigue life

Five rivets have low clamping force, remaining one has high clamping force Infinite fatigue life

All six rivets have low clamping force 18


predicted lives of two approaches illustrate some amount

of deviation from each other, it can be said that both

approaches have highlighted cross girders become the

most critical members to fatigue failure. Further it reveals

that in case of main girder consisted truss member, the

sequential law-based approach 2 gives higher values than

approach 1 values. But it is the opposite for bridge deck

members. Since the Miner’s rule estimation produces

pessimistic results with increasing of loads and optimistic

results with decreasing loads (Mesmacque et al., 2005), it

can be said that in case of truss members, the global

increment of live load of trains with each period of age

has greater effect on fatigue damage than local variation

(increase and decrease of loading during a week) of

loading in each week. Similarly it can be seen that in the

case of bridge deck members (cross girders, stringers and

bracings), the local variation of loading has a greater

effect on fatigue damage than global increment of loading.

Although these types of conclusions are particular to this

bridge and fatigue criticality of structure varies from

bridge to bridge.

Considered critical member DT3 has been subjected to

three types of approaches to find out remaining fatigue

life. (section 4, section 5 and section 6).The obtained

results are summarized as shown in Table 3 and it shows

that lives, which are obtained through approach 1 and 2,

lie in between upper and lower bound of approach 3.

Therefore, it is possible to confirm that the UK railway

code provided S-N curve, (Network rail, 2001), represents

the normal or intermediate effect of clamping force for

wrought iron riveted connections of existing bridges.

Further, the particular results which are related to the

second step of approach 3 (see Table 4), illustrated that

the effect of clamping force in riveted connection tends to

deviate the fatigue life considerably. Likewise, fatigue life

results, which are shown in Table 5, reveal that the active

number of rivets, which are able to transfer the load, also

changes the fatigue life significantly.

8. Conclusions

Remaining fatigue life estimation of a riveted railway

bridge has been presented based on structural appraisal.

Handling of sequential law in both uniaxail and maltiaxial

fatigue of a railway bridge were described. Hence the

fatigue damages due to both the primary stresses and

secondary stresses of a riveted railway bridge have been

considered and the corresponding approaches were clearly

indicated. Reasonably accurate procedure, which is based

on structural appraisal, was presented to obtain the past

and present stress histories. Finally the study flows to

highlight the major conclusions as follows.

Condition evaluation of the bridge exhibits that the

overall maintenance of the considered bridge is satisfactory,

but there are localized mild corrosion at few places, and

these need immediate attention. Due to fatigue, under

current loadings, speeds and frequencies of operation, the

lowest remaining life found for a member is 12 years.

Thus it may be concluded that the bridge deck can be

used for another 12 years provided that the speed,

frequency, and weight of the trains are not increased. If

proper maintenance work is carried out and the critical

members are replaced with new members with longer

life, the bridge will be able to provide further service.

There is a 10 to 15 years variation among the estimated

remaining lives when comparing of approach 1 and

approach 2. Obviously the effect of members where the

lives are very low (eg. cross girders, stringers) becomes

more significant in percentage terms. These observation

and the phenomenological validity of the new damage

indicator-based sequential law tend to conclude that the

application of sequential law is much advisable for the

evaluation of remaining fatigue life of riveted railway

bridges where the detailed stress histories are known.

Remaining life is 130-140 years for critical member in

set DT3 when all six rivets have normal clamping force

and it has failed before 26 years when it does not have

clamping force at all six rivets. Further, it increases to

infinity when it has high clamping force (see Table 3).

Therefore, it is possible to give an assurance to some

extent that the UK railway code provided design S-N

curve (Fig. 10-(a)) captures the stress concentration effect

of riveted connection when it has a normal or intermediate

effect of clamping force. Therefore obtained function and

the geometrical shape of this fully known design S-N

curve can be employed to assess the fatigue damages of

other wrought iron bridges, which have riveted connections

with normal clamping force.

The fatigue life evaluation based on secondary stress

analysis revealed that the clamping force and the activeness

of rivets play a big role in fatigue damage. Hence, it can

be concluded that it is great important to investigate

accurately the condition of places where the stress

concentration effect is severe such as notch, crack or

Table 5. Fatigue life variation when active numbers of rivets are gradually decreasing while low clamping force

Considered features of the riveted connection Maximum von Mises stress (MPa) Fatigue life (years)

All six rivets are active 256.7 18

Five rivets are active 257.2 17

Four rivets are active 258.4 15

Three rivets are active 260.5 11

Two rivets are active 263.1 9


connection area especially in old bridges for good

judgment of fatigue life. Finally it can be concluded that

the second approach is more advisable for general use

and approach three has been recommended for detail

studies.

Since this investigation has not been captured the effect

of various type of micro structural changes and the effect

of mesoscopic damage variables of particular material at

highly stressed locations, comparisons of above approach

with microsopic level fatigue theories are currently on the

way.

Acknowledgments

The authors wish to express their sincere gratitude to

Senior Professor M.P Ranaweera and other team of

experts who works in the Sri Lankan Railway Bridge

project, for their great advices to carry out this research.

The kind support given by the Sri Lanka Railways (SLR)

is also appreciated.

Appendix A

Here only the concept of the new damage indicator

based sequential law in multiaxial fatigue (Mesmacque et

al., 2005), is summarized with an algorithm for understanding

(see flow chart described in Fig. A-1).

The hypothesis behind the model is that if the physical

state of damage is the same, then fatigue life depends

only on loading condition. Therefore, the life can be

assessed using the Wöhler curve for new structures,

which are still free of damage. At load level i, a certain

stress amplitude σi is applied for a number of cycles ni.

Here the number of cycles to failure from the Wöhler

curve for σi is Ni. Thus, after ni applied cycles, the

residual life is considered as (Ni-ni) for load level i. From

the Wöhler curve, σ(i)ed is said to be ith level damage

stress (otherwise can be introduced as stress relevant to

the residual life) which corresponds to the failure life (Ni-

ni), (see Fig. A-2). Hence, the damage stress, Di is defined

as,

(A-1)

Where σu is the magnitude of ultimate stress. The stress

field can be considered in terms of equivalent von Mises

stress and in this way the model can be applied to the

multiaxial fatigue. In the case of uniaxial loading

condition, the stress field can be considered in terms of

corresponding stress values. The σ(i)ed is equal to σ1 at

first cycle when damage indicator Di = 0 and σ(i)ed is

equal to σu at the last cycle when Di = 1. Therefore, the

damage indicator is normalized to 1 at the failure of

material.

Same damage is then transformed to load level i + 1

and hence damage equivalent stress at level i + 1 is

calculated with the relation,

(A-2)

Further simplification of Eq. (A-2),

(A-3)

where is damage equivalent stress at the level

i + 1. Thus the corresponding equivalent number of

cycles to failure, can be obtained from the Wöhler

Di

σi( )ed σ

i–

σu

σi

–--------------------=

Di

σi( )ed σ

i–

σu

σi

–--------------------

σ'i 1+( )ed σ

i 1+–

σu

σi 1+

–-------------------------------= =

σ'i 1+( )ed D

iσu

σi 1+

–( ) σi 1+

+=

σ'i 1+( )ed

N' i 1+( )R

Figure A-1. Flow chart for damage stress based sequentiallaw.

Figure A-2. Schematic representation of parameters inWöhler curve.


curve as shown in Fig. A-2. The is the magnitude of

applied stress and it is subjected to number of

cycles at the level i + 1. Then the corresponding residual

life at the load level i + 1, is calculated as,

(A-4)

Hence the damage stress σ(i+1)ed, which corresponds to

N(i+1)R at loading level i + 1, can be obtained from the

Wöhler curve as shown in Fig. A-2. Then the cumulative

damage at loading level i + 1 is defined as,

(A-5)

The same procedure is followed until the failure of

material, that is, when damage indicator Di = 1.

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σi 1+

ni 1+( )

Ni 1+( )R

Ni 1+( )R N' i 1+( )R n

i 1+( )–=

Di 1+( )

σi 1+( )ed σ

i 1+–

σuσi 1+

–

------------------------------=

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