Railway Concrete Arch Bridge over Kalix

River at LångforsenDynamic Properties and Load-Carrying Capacity

Natalia Sabourova, Niklas Grip, Yongming TuChao Wang, Ola Enoksson, Thomas BlanksvärdMartin Nilsson, Ulf Ohlsson and Lennart Elfgren

Railway Concrete Arch Bridge over

Kalix River at Långforsen

Dynamic Properties and Load-Carrying Capacity

Natalia Sabourova, Niklas Grip, Yongming TuChao Wang, Ola Enoksson, Thomas BlanksvärdMartin Nilsson, Ulf Ohlsson, and Lennart Elfgren

Luleå University of TechnologyDepartment of Civil, Environmental and Natural Resources

Division of Structural Engineering

Luleå University of Technology

ISSN 1402-1528 ISBN 978-91-7790-473-1 (pdf)

Luleå 2019

www.ltu.se

4

Abstract

The concrete arch bridge over Kalix River at Långforsen was built in 1960 and has a mid-span

of 89,5 m and a height of 13,7 m. The bridge owner, Trafikverket wanted to increase its

allowable axle load from 225 to 300 kN. Field tests were carried out under service condition

and with ambient vibrations. The test results were used to update and validate Finite Element

Models. At last, the refined models were used to check the possibility to increase the axle load.

According to earlier assessments, most parts of the bridge is capable of carrying an axle load

of 330 kN. The only critical sections are located in the beams carrying the rail on top of the

arch in the section where the beams are united with the arch. Here the stresses in the longitudinal

bottom reinforcement are slightly too high.

These sections have been studied in a FEM model for different loads and results show

maximum strains of about 50·10-6 corresponding to stresses of only about 10 MPa in the

reinforcement in the critical sections. Live load vertical deflections of the crown of the arch is

of the order of only ± 6 mm. Dynamic studies have also been made showing that fatigue is no

issue. Altogether the studies show that the bridge is able to carry an increased axle load of 300

kN without problems.

Photo on cover (2009-10-05): The bridge over Kalix River at Långforsen was built in 1960

and has a total length of 176,5 m and a mid-span length of 89,5 m. It is situated in the north

of Sweden some 10 km NW of Kalix and some 60 km W of the border to Finland.

5

Preface

The work presented in this report was initiated by Banverket (now Trafikverket) in order to

assess the possibility to increase the allowable loads on the railway concrete arch bridge over

Kalix River at Långforsen.

Measurements on the bridge were performed in 2009 and 2011 by Complab at Luleå

University of Technology. The measurements during 2009 were planned by Mr. Ola

Enoksson, Tekn. Lic., and analyzed by Associate Professors Martin Nilsson, Tekn. Dr., and

Thomas Blanksvärd, Tekn. Dr. The measurements during 2011 were planned and analyzed by

Ms. Natalia Sabourova, Fil. Lic., Associate Professor Niklas Grip, Fil. Dr., and Guest

Professor Yongming Tu, Ph. D. From Complab, Messrs. Georg Danielson, Håkan Johansson,

Mats Petersson and Lars Åström have taken part. Assistant Professor Ulf Ohlsson, Tekn. Dr.

and Senior Professor Lennart Elfgren, Tekn. Dr., have guided the work.

Different finite element models have been constructed during 2009-2012. Preliminary

models were constructed by guest student Huang Wei from Central South University of

Forestry and Technology in Changsha, China, and by Ph. D. student Arto Puurula, Tekn. Lic.,

from Savonia University of Applied Sciences, Kuopio, Finland. The final model has been

constructed by Guest Professor Yongming Tu from Southeast University in Nanjing, China.

During 2016-18 the model has been improved by Mr. Chao Wang, guest Ph D student at

LTU, also from Southeast University in Nanjing, China.

In this report Ola Enoksson and Lennart Elfgren have compiled most of chapters 1-3 and

Appendices A-D, Natalia Sabourova and Niklas Grip most of chapter 4 and Appendix E,

Yongming Tu most of chapter 5 and Appendix F, while all the authors have contributed to

chapter 7. Draft versions were prepared in 2012 and 2016. The final editing in 2019 has been

done by Lennart Elfgren

The work has in different phases been supported by Banverket (now Trafikverket), the

Swedish Research Council for Environment, Agricultural Sciences and Spatial Planning

(Formas, dnr 2012–1037), the Development Fund of the Swedish Construction Industry

(SBUF, id 12513, 13010), Luleå Centre for Risk Analysis and Risk Management (CRR),

Luleå Railway Research Centre (JVTC), the European Regional Funds and Luleå University

of Technology.

Luleå in September 2019

Natalia Sabourova, Niklas Grip, Yongming Tu, Chao Wang, Ola Enoksson

Thomas Blanksvärd, Martin Nilsson, Ulf Ohlsson and Lennart Elfgren

6

Table of Contents

Abstract ...................................................................................................................................... 4

Preface ........................................................................................................................................ 5

Table of Contents ....................................................................................................................... 6

1. Introduction ............................................................................................................................ 8

1.1 Background ....................................................................................................................... 8

1.2 Historical note on arch bridges ......................................................................................... 9

1.3 The arch line ................................................................................................................... 10

1.4 Management and assessment of bridges ......................................................................... 12

2. Design and construction ....................................................................................................... 13

2.1 General ............................................................................................................................ 13

2.2 Concrete arch .................................................................................................................. 13

2.3 Top beams, columns and abutments ............................................................................... 19

2.4 Construction .................................................................................................................... 19

3. Geometry and material properties ........................................................................................ 21

3.1 Geometry ........................................................................................................................ 21

3.2 Material parameters ........................................................................................................ 21

4. Assessments 2002 -2004 ...................................................................................................... 23

4.1 General ............................................................................................................................ 23

4.2 Results ............................................................................................................................ 23

5. Measurements ....................................................................................................................... 26

5.1 Measurements during 2009 ............................................................................................ 26

5.2 Measurements during 2011 ............................................................................................ 26

6. Finite element models (FEM) .............................................................................................. 29

6.1 General ............................................................................................................................ 29

6.2 Design of Models ............................................................................................................ 30

6.3 Calibration and updating of models 2009-2012 ............................................................. 32

6.3 Calibration and updating of models 2016-2018 ............................................................. 33

7. Conclusions .......................................................................................................................... 35

References ................................................................................................................................ 36

7

Appendix A. Drawings from 1958 -1960

Appendix B. Design calculations from 1958

Appendix C. Photos from construction 1959

Appendix D. Assessment 2002-2004

Appendix E. Measurements 2010- 2012

Appendix F Finite Element Models

Notation

The symbols are explained where they are introduced.

A decimal comma (,) is used to separate integers from fractional parts of a number as

customary in Sweden but contrary to the English use of a decimal point (.)

8

Figure 1.1. To the left a map of the Swedish

Railways. Above a detail of the north-eastern

part of the lines with Kalix located some 50 km

west of the border to Finland at Haparanda. The

bridge over Kalix River is situated at Långforsen

some 10 km north-east of Kalix. The location is

marked with a red dot.

1. Introduction

1.1 Background

The bridge over Kalix river is situated at Långforsen on the railway line between Kalix and

Morjärv in northern Sweden, see Figures 1.1 and 1.2. The line was built in 1960 to facilitate

transport to and from the timber, pulp and paper industry in Karlsborg outside Kalix. The line

was connected to the older Haparanda line, Boden - Morjärv - Haparanda, built 1900 – 1919,

some 30 km north of the coast, see Järnväg.net (2019). A new line along the coast between

Haparanda and Kalix was built during 2008-2012. At the same time the line between Kalix

and Morjärv was upgraded and electrified in order to be able to replace the old line between

Haparanda and Morjärv.

9

Figure 1.2. Elevation and plan of the bridge over Kalix River at Långforsen, km 24+701, built in

1959-1960 with a main span of 89,5 m. From Drawing No 1861-39 in Appendix A.

In order to upgrade the line between Kalix and Morjärv from the original axle load of 22,5 ton

(STAX 22,5 from Swedish “STörsta Tillåtna AXeltryck”, maximum allowable axle load) to

higher loads, the bridge over Kalix river at Långforsen is studied in this report to assess its

capacity.

1.2 Historical note on arch bridges

Arches as man-made structures have a long history. They first appeared as early as the 2nd

millennium BC as brick structures. This happened in Mesopotamia at the rivers Euphrates and

Tigris in present day Iraq. Their systematic use started with the ancient Romans who applied

the technique to a wide range of structures - a few of them still in use today.

The first concrete arch bridges were made of unreinforced mass concrete about 1850 and they

continued to be built for about a century. A well-known example is the “Harry Potter Bridge”

at Glenfinnan west of Fort William in Scotland built by Sir Robert McAlpine (“Concrete

Bob”) in 1897-1901. The railway viaduct consists of 21 arches each spanning 15 m with a

maximum height of 30m.

The first reinforced concrete arch bridge was built by Joseph Monier in 1875 in Chazelet in

France. It was a low-rise pedestrian footbridge with a span of 16,5 m. Longer spans soon

emerged and the Sandö Bridge over the Ångerman River in Sweden had the world record with

a span of 264 m when it was built in 1943. The world’s longest concrete arch in 1997 has a

span of 420 m belonging to the Wanxian Bridge over the Yangtze River in China. In 2017, a

10

span length of 445 m was reached at the Beipan River Bridge on the Shanghai-Kunming high

speed railway. The history of arch bridges is treated in many beautifully illustrated books as

e.g. Brown (1998), Graf (2002) and Fernández Troyano (2003). The theoretical development

is presented by e.g. Timoshenko (1953) and Kurrer (2008).

In Sweden reinforced concrete arch bridges started to be built around 1910, see Ahlberg-

Spade (2001). Early examples are the 50 m arch at Motala ström built in 1911 and the 90 m

arch built at Tällberg over Öre älv for the railway line to northern Sweden. The building of

this and other bridges is described in a Ph.D. thesis by Larsson (1997) about the bridge

builder Otto Linton.

Other railway arch bridges in northern Sweden are the 39,2 m bridge at Myrheden over Byske

älv from 1941 and the two 112,1 m arch bridges at Vindeln over the Vindel River from 1952

and at Ragunda over the Indal River from 1955. The bridge treated in this report, the 89,5 m

arch at Långforsen over the Kalix River was built in 1960. Assessments of the Byske bridge is

presented in Enochsson et al (2011) and of the Vindel Bridge in Enochsson et al (2005),

Bennitz (2006), He et al (2006, 2008) and Häggström and Martinsson (2009).

Other notable early Swedish road concrete arch bridges are the 181m span of the bridge at

Traneberg in Stockholm from 1934 and the earlier mentioned Sandö bridge from 1943 with

the world record span of 264 m

1.3 The arch line

Design of concrete arch bridges is treated by e.g. Strassner (1936), Gustafsson and

Müllersdorf (1961), Asplund (1966) and Lorentsen (1966).

The basic assumption for an arch bridge is to form it in such a way that there are no tensile

forces or bending moments in the arch. The principle is illustrated in Figure 1.3.

Figure 1.3. An arch acted on by a distributed load g and a horizontal thrust H can be seen as

the sum of two statically determined curved beams loaded by g and H respectively. In order to

avoid moments in the arch, M = 0, the form of it must be such that the moment 𝑀𝑔of the dead

load counteracts the moment 𝑀𝐻 = −𝐻𝑦 of the force H.

The form of the curve can be derived if we study a small element of it, see Figure 1.4.

11

Figure 1.4. A small element of an arch with height y and length x is acted on by a

distributed load 𝑔(𝑥) and a constant horizontal force H. At the ends of the element, section

forces appear as the horizontal force H, a vertical force V and a moment M. The section

forces V and M may change with a small amount on the right side.

With notation as in Figure 1.4, we may write two equations of equilibrium, one vertical and

one as the moment around the right end of the element:

𝑔∆𝑥 + ∆𝑉 = 0 → ∆𝑉

∆𝑥 = - 𝑔 →

𝑑𝑉

𝑑𝑥= −𝑔

(1.1)

𝐻(−∆𝑦) + 𝑉∆𝑥 − ∆𝑀 − 𝑔∆𝑥∆𝑥

2= 0 (1.2)

For small x the last term 𝑔∆𝑥∆𝑥

2 will approach zero. For an arch we want M = M = 0 so we

obtain:

∆𝑦

∆𝑥=

𝑉

𝐻 →

𝑑𝑦

𝑑𝑥=

𝑉

𝐻 →

𝑑2𝑦

𝑑𝑥2 = −𝑔𝐻 (1.3)

For the special case when the dead load 𝒈(𝒙) is constant, the moment 𝑀𝑔caused by it will

have the maximum value gl2/8 and the form of a second-degree parabola

𝑀𝑔 = 𝑔𝑙2

8⌊1 − (

2𝑥

𝑙)

2

⌋ (1.4)

In order to obtain zero moment M everywhere in the arch, the arch line must also be formed

as a parabola

𝑦 = 𝑓 ⌊1 − (2𝑥

𝑙)

2

⌋ (1.5)

12

and the moment of the horizontal force 𝑀𝐻 = −𝐻𝑦 must be equal to 𝑀𝑔. This gives with

equations (1.4) and (1.5) that the height of the curve f shall be proportional to the square of its

length l in the form 𝑓 = 𝑔𝑙2 8⁄

We further obtain from Eq (1.5) that

𝑑𝑦

𝑑𝑥 = −2𝑓

2𝑥

𝑙∙

2

𝑙= −8𝑓

𝑥

𝑙2 →

𝑑2𝑦

𝑑𝑥2= −8

𝑓

𝑙2

With d2y/dx2 = - g/H from Eq.(1.3) we again obtain 𝑓 = 𝑔𝑙2 8.⁄

In most arches the dead load will vary and be higher at the supports than at the center.

Consequently, the arch line will deviate from the simple second-degree parabola and be

somewhat steeper close to the supports, see e.g. Strassner (1936) and Lorentsen (1966).

1.4 Management and assessment of bridges

Management and assessment of bridges are treated by e.g. Yanev (2007), Cremona (2011)

and in reports from the European Research Projects Sustainable Bridges, SB (2008), and

MAINLINE (2011).

In northern Sweden, the Iron Ore Railway Line was built around 1900 and has more than 100

bridges. It has a length of ca 500 km and runs from Kiruna and Malmberget to the ice-free

harbour in Narvik in Norway on the Atlantic and to Luleå in Sweden on the Baltic. The

original axle load was 14 ton. The axle load has gradually been increased to 25 ton in 1955, to

30 ton in 1998 and to 32,5 ton in 2017. The increases in axle loads have been preceded by

monitoring and assessment studies of the bridges. The capacity and need for strengthening or

replacement of the bridges have been evaluated. Many of the bridges could carry a higher

load than what it was designed for, Coric et al. (2018), Elfgren et al. (2009, 2019), Enochsson

et al. (2002, 2005, 2008, 2011), Paulsson (1998), Paulsson – Töyrä (1996), Paulsson et al.

(1996, 1997, 2016), Puurula (2004, 2012), Thun et al. (1999, 2000, 2006, 2011).

Testing to assess bridges has also been used by e.g. Bagge et al. (2017, 2018), Elfgren et al.

(2009, 2015, 2018, 2019), Enochsson et al. (2002, 2004, 2005, 2008), Grip et al. (2013,

2016), Huang et al. (2016), Häggström et al. (2016, 2017), Nilimaa (2015), Nilsson et al.

(1999), Stenlund (2008) and Wang et al.(2016, 2019).

13

2. Design and construction

2.1 General

The railway bridge over Kalix River at Långforsen, km 24+701, has a total length of 177.3 m

with a central arch of 89,5 m and two side spans of 42 m, see Figures 1.2 and 2.1. The free

spans have lengths of 13,0 + 12,8 +12,6 + 87,92 +12,6 + 12,8 + 13,0 m = 164,7 m.

The bridge was designed by Uno Nordstrand at Kungliga Järnvägsstyrelsen during 1956-58

and built in 1960 by Nya Asfalt AB. The bridge consists of an arch which carries a reinforced

concrete slab via underlying longitudinal and transversal concrete beams, connected through

fixed columns. The arch consists of a reinforced concrete hollow box girder with two hollow

spaces. The cross section is lowest at the crown of the arc and highest at the connection to the

arch abutment. The original drawings of the bridge are given in Appendix A and the original

design calculations in Appendix B.

The design train load corresponds to an axle load of 250 kN for the locomotive and a

distributed load of 85 kN/m for the wagons.

The design was based on Strassner (1936) and Uno Nordstrand made the following

assumptions (see page B6, i.e. page 6 in Appendix B):

Length of the span l = 89,5 m

Height of arch f = 13,45 m which gives f/l = 0,15

The arch is formed after the compression line for the dead load. The moment of inertia is

varied according to Strassner:

𝐼𝑠

𝐼𝑧 ∙ cos 𝜑 = 1 − (1 − 𝜂)𝜉 or 𝐼𝑧 =

𝐼𝑠

[1(1−𝜂)𝜉]𝑐𝑜𝑠𝜑 with

Iz = the moment of inertia at the horizontal coordinate z measured from the center of the arch

Is = the moment of inertia at the center of the arch (index s possibly from superior)

Ik = the moment of inertia at the support (index k possibly from German kant = end)

φ = is the angle of the arch in relation to the horizontal axis

𝜉 = 2s/s0 is the relative distance along the arch from the center. The total arch length is s0

𝜂 = 𝐼𝑠

𝐼𝑘 ∙ cos 𝜑𝑘

2.2 Concrete arch

Nordstrand (1958) assumes that the ratio m of the dead loads at the end and the center of the

arch is m = gk/gs = 2 which according to Strassner gives him cos φk = 0,8254 and φk = 34,4 o.

He then assumes a geometry of the arch section, see Figure 2.2, and starts at the end with a

height of Hk = 2,8 m, a width Bk = 6,3 m, an area Ak = 8,193 m2 and a moment of inertia Ik

= 8,896 m4 and obtains at the centre (head) of the arch that Hs = 1,8 m, Bs = 5,4 m, As =

4,830 m2 and Is = 2,169 m4 = 0,24 Ik. For the deck he assumes an area Af = 5,474 m2 and a

moment of inertia If = 1,606 m4. These values are later refined,

14

10000 5000 5000 6040

6e 7e

6900

3600

18

70

Figure 2.1. The Arch Bridge over Kalix River at Långforsen

To the distributed dead load of the arch is added the load of the deck which enters the arch

through the columns. This is illustrated in Figure 2.3. It results in a horizontal force H = 24,15

MN and a vertical force at each support V = 15,55 MN.

15

Figure 2.2 Typical arch section according to Appendix B, page B9.

Figure 2.3. Moments from the deck influencing the arch, p B51.

The traffic load is called 0,85F + Load Group 1 and has an axle load of 250 kN, see TRV

Tåglaster (2010) and Figure 2.4.

16

Figure 2.4. Traffic load with a maximum axle load of 250 kN. 1 ton = 10 kN. Figure text in

Swedish.

The maximum moment M and the corresponding horizontal force from traffic loads are

illustrated in Figure 2.5. The traffic live load gives a maximum negative moment at the

support of -25 MNm (tension in the top of the arch) and a maximum positive moment of 21,5

MNm (tension in the bottom of the arch). At the center the corresponding values are -2,8

MNm and 7,9 MNm. The corresponding horizontal forces are -5,8 MN and - 6,9 MN at the

end of the arch and -4,1 MN and - 5,4 MN at the center.

When dead load and exceptional loads from wind, temperature, creep and settlements are

added, the following moments are obtained: at the support -75,4 MNm and +28,75 MNm for

an exceptional load case and – 36,0 MNm and +10 MNm for a normal load case. At the center

he obtained -3 MNm and +21,5 MNm for the exceptional load and 0,5 MNm and 11 MNm

for the normal load case (see pages B108-151), see Figure 2.6.

17

Figure 2.5. Maximum moments M and horizontal force N from traffic loads, p B92.

1 ton = 10 kN; 1 tonm = 10 kNm

18

Figure 2.6. Total moments from normal and exceptional loads, p 150. At the arch top the max

and min exceptional moments are 21,5 MNm and -3 MNm and at the supports they are 28,8

MNm and -75,4 MNm respectively. A positive moment gives tension in the bottom of the arch.

19

The weight of the reinforcement in the arch is summarized on page B213 and in Figure 2.7 to

53,9 ton.

Figure 2.7. Summary of reinforcement in the arch, page B213-B214

Longitudinal:

- SAS 70: a) 965 cm2 length 5 m; b) 965 cm2 length 5 m

- Ks 40: c) 600 cm2 length 10 m; d, e, f, g) 250 cm2 length 10 m

- Total weight 7,9 (9,65·2·50·2 + 6,00·100·2 + 2,50·4·100·2) kg = 15 300 + 9 480 + 15 800 kg

= 40,8 ton

Shear reinforcement: about 3 ton. Transverse reinforcement: about 10 ton

Transverse walls in arch: about 300 kg

Total 15,3 (SAS70) + 38,6 (Ks 40) = 53,9 ton

2.3 Top beams, columns and abutments

The design of the rest of the bridge is presented on pages B218 –B477. Drawings are

presented in Appendix A.

2.4 Construction

The construction of the bridge is illustrated by photos in Appendix C. Dredging for the

abutments were carried out in March 1959. The spring flood in May got rather high and the

abutments were overflown.

During casting of the arch, a fire broke out in the morning of November 26th 1959 at 05:30,

see Figure 2.8. The fire was caused by over-heating of burners which were used to prevent the

newly cast concrete from freezing and to preheat the formwork for the next casting. The fire

was extinguished already at 8 o’clock in the morning by fire brigades from Kalix and Töre,

which fought the fire from each side of the river. Much of the formwork for the central part

of the arch was destroyed and had to be replaced together with damaged concrete. In the

beginning of December, the casting of the arch walls continued. The top beam was casted in

the spring of 1960.

20

Figure 2.8. Top: Formwork September 29, 1956. Bottom: Due to overheating of burners a fire

broke out on November 26th 1959 during the casting of the arch. Part of the formwork and the

concrete had to be removed and replaced, see Appendix C.

21

3. Geometry and material properties

3.1 Geometry

The main geometry can be seen in Figures 1.2 and 2.1 and in Appendix A. Some essential

data are collected in Table 3.1 below.

Table 3.1. Geometrical properties of arch and deck

Arch [m] Deck [m]

Length, L0 87,92 Height 1,87

Rise, f0 13,706 Width of deck 6,90

Height of crown, h0 1,80 Width of beam 1,00

Height of abutment, hb 2,80

Width of crown 5,40

Width of abutment 6,30

3.2 Material parameters

Concrete of class Btg K400, with a nominal compression strength of 400 kp/cm2 (40 MPa),

was used for everything but the foundations, where Btg K300 was used with nominal

compression strength of 300 kp/cm2 (30 MPa). As reinforcement, steel of quality Ks 40 and

Ss 70 was used with a nominal yield stress of 40 kp/mm2 (400 MPa) and 70 kp/mm2 (700

MPa). respectively

The characteristic concrete material properties were obtained from chapter 13 in BV Bärighet

(2000) according to Table 3.2.

Table 3.2 Nominal Characteristic Material Properties

Concrete Characteristic

Properties

[MPa] Steel

Characteristic Properties

[MPa]

Compression strength, fcck 30,775 Yield stress Ks40 390

Tensile strength, fctk 1,95 Yield stress Ss70 720

Modulus of Elasticity, Ec 32 000 Modulus of Elasticity, Es 200 000

Testing of six drilled out cores from the lower part of the arch was carried out at Luleå

University of Technology in 2009. The cores had a diameter of 95 mm and results according

to Table 3.3 were obtained.

Table 3.3 Tested concrete properties in 2009, compression, fcc, and tensile splitting, fct.

fcc [MPa] fct [MPa]

Sample 1 76,7 Sample 4 1,85

Sample 2 79,8 Sample 5 4,17

Sample 3 65,2 Sample 6 3,77

Mean value, m3 73,9 Mean value 3,26

22

According to BBK 94 (1994) this corresponds to a concrete with fKK ≤ m3 – 4 = 73,9 – 4 =

69,9 MPa, fKK ≤ xmin + 5MPa = 65,2 + 5 =70,2 and fKK ≤ xmin/0,8 = 65,2/0,8 = 81,4 which all

are > 62 MPa of class K80 with characteristic properties according to Table 3.4

Table 3.4. Characteristic Concrete Properties for K80 based on testing

Concrete Characteristic

Properties

[MPa]

Compression strength, fcck 56,5

Tensile strength, fctk 2,65

Modulus of Elasticity, Ec 38,5

This characteristic compression strength value fcck = 56,5 MPa is much higher than the value

30,78 MPa used in an earlier assessment by Leander and Fredriksson (2003, 2004), see

chapter 4.

In the Finite Element Calculations properties according to Table 3.5 were used.

Table 3.5 Properties used in FEM calculations

Property

Modulus of Elasticity, Ec 40 000 MPa

Mass density of concrete, ρc 2,5 t/m3

Mass density of gravel, ρg 2,0 t/m3

The soil deformation modulus was assumed by Nordstrand (1958) to 10 kg/cm3 = 100 GN/m3

(p B390)

23

4. Assessments 2002 -2004

4.1 General

In order to determine if the axle load 330 kN (BV-2000) could be used for the bridge, a

preliminary assessment was made during 2002. This assessment was reviewed by the

consulting company Reinertsen, see Kolster (2002). The programs Strip Step (1971) [see

further Bengtsson & Wolf (1969, 1970) and Lundin (1971)] and Solvia (2000) were used and

it was recommended that a more complete finite element model (FEM) should be applied.

Banverket (The Swedish Rail Administration) then asked the consulting firm Tyréns to carry

out such an improved analysis and an early version of the FEM program Brigade (2012) was

used, Leander-Fredriksson (2003, 2004). A summary of some of the results are presented

below. More details are given in Appendix D.

4.2 Results

The calculations were based on nominal design values, concrete Btg K400 (nominal

compressive strength 400 kp/cm2 = 40 MPa) for the arch, columns, top beams and slab and

Btg K300 (30 MPa) for the foundations. The concrete compressive strength was adjusted

according to BV Bärighet (2000) to a characteristic compressive strength fcck = 30,78 MPa

(corresponding to ≈ C40), a characteristic tensile strength fctk = 1,95 MPa and a modulus of

elasticity Ec = 32 GPa. Characteristic yield stresses used for the steel reinforcement were, fyk =

390 MPa for Ks 40 and fyk = 720 MPa for Ss70.

Several train loads were tested. The different train loads are given in TRV Tåglaster (2010).

The most severe one was tested first. If the section could not stand this load a lesser load was

tried.

BV2000 has an axle load of 330 kN, a line load of 110 kN/m, and an axle distance for bogies

and buffers of 1,6m

UIC 71 has an axle load of 250 kN, a line load of 80 kN/m; and an axle distance for bogies

and buffers of 1,6m

BV-3 has an axle load of 250 kN, a line load of 80 kN/m, an axle distance for bogies of 1,8 m

and for buffers of 3,0m

C3 has an axle load of 200 kN, a line load of 72 kN/m, an axle distance for bogies of 1,8 m

and for buffers of 3,0m.

The results from Leander and Fredriksson (2003, 2004) are summarized in Table 4.1.

24

Table 4.1 Maximum utility ratios (load/capacity) for different parts of the bridge and different

load classes, Leander-Fredriksson (2003, 2004).

Structural Part Section Line Class according to BV Tåglaster

(2009)

Train load class BV 2000 UIC71 BV-3 C3

Axle load (stax), kN 330 250 250 200

Line load, kN/m 110 80 80 72

Distance between

bogies/buffers, m/m

1,6/1,6 1,6/1,6 1,8/3,0 1,8/3,0

Arch Mmin in mid-section 0,92 (1,64*) - - -

Top Beams Mmin where the slab is

connected to the arch

1,087 1,035 1,000 -

Slab, built in Mmax in mid-slab 1,109 0,877 - -

(Slab, simply

supported)

Mmax in mid-slab (2,008) (1,608) (1,135) (0,980)

Cross beam M and V 0,817

*The number in parenthesis is for the case when a full temperature load is acting,

Table 4.1 shows that the arch can carry Load Class BV2000 with an axle load of 330 kN if

the influence of temperature changes is reduced. The most stressed section is the bottom of

the mid-section (head) of the arch. Other sections have lesser strains.

The top beams carrying the concrete slab, the ballast and the rail can in the longitudinal

direction carry Line Load BV 2000 in all sections but the section where the slab is connected

to the arch, see Section M in figure 4.1. Here the amount of bottom reinforcement allows the

load BV 3, with an axle load of 250 kN. However, the linear modelling of the joint M is not

made in much detail and it is probable that a higher axle load (330 instead of 250 kN), which

formally increases the utility rate with 8,7% would only give less, or at most, about the same

increase in the steel stress in the reinforcement. The design steel stress is 283 MPa for Ks 40

and an increase with ca 10% would still be far from yielding (at 400 MPa).

Figure 4.1. Sections of top beams controlled by Leander-Fredriksson (2003), The most critical

section is Section M, where the top beams are joining the arch.

The top slab can carry Load Class BV2000 2000 in all sections but the one where the slab

rests on the longitudinal beams, see Section R in Figure 4.2. Here the amount of top

25

reinforcement allows the load UIC 71 with an axle load of 250 kN and slightly smaller

distances between bogies and buffers, 1,6 m, than for the load BV3, where the distances

between bogies are 1,8 m and between buffers 3,0 m.

Figure 4.2. Sections in the top slab controlled by Leander-Fredriksson (2004), The most

critical section is the top of Section R, where slab rests on the top beams.

26

5. Measurements

5.1 Measurements during 2009

During October, 8-9, 2009, measurements were carried out by Luleå University of

Technology. Material properties were determined. Six cylinders with a diameter of 95 mm

were drilled out from the lower part of the arch as presented in Chapter 3.

Measurements were done when a locomotive moved over the bridge. Strains were studied in

one of the cross beams, see Figure 5.1, and the curvature κ was determined in one of the main

top beams. Very small strains and curvatures were observed. The maximum strain was 8·10-6

corresponding to a concrete stress of about c = E ≈ 30·109 · 8·10-6 Pa = 0,24 MPa. The

maximum curvature was of the order of κ = 0,2·10-4 1/m corresponding to a live moment of 5

MNm as compared to a design moment of 8 MNm.

Figure 5.1 Left: View of arch, columns, top beams and top slab, Right: Strain gauges in a

cross-beam ca 20 m from the top (head) of the arch between sections 6e and 7e.

Concrete (top, blue) and steel (bottom, red).

5.2 Measurements during 2011

In 2011 more systematic measurements were carried out. In Figure 5.2 a reinforcement strain

gauge is marked, in Figure 5.3 deformation measurement points and in Figure 5.4 acceleration

measurement points are given. Two locomotives used are shown in Figure 5.5. More

information about the measurements is given in Appendices E and F and in the following

chapters.

27

Figure 5.2 Reinforcement strain gauge in the longitudinal direction in the bottom of one of

the concrete top beams close to section where the beam joins the arch.

Figure 5.3. Measurements of deformations. Blue dots = optical measurements; green dots

(between 3e and 4e = vertical LVDT; orange dots (4w & 4e) = horizontal LVDT.

Figure 5.4. Measurements of accelerations. Position of accelerometers.

28

Locomotive type T43

Locomotive type T66

Figure 5.5. Two locomotives used during the measurements. The T43 has an axle load of 180

kN and a total load of 720 kN and the T66 has an axle load of 210 kN and a total load of 1260

kN.

29

6. Finite element models (FEM)

6.1 General

A first beam model was made by Huang Wei in 2009. Its intention was to estimate global

dynamic characteristics before the first measurements were carried out. The model was made

with the program Lusas (2005).

The second model was made by Arto Puurula in 2010 after the first measurements. It was

made in Brigade (2012) and consisted of shell elements in the arch and deck, and beam

elements in the deck beams and columns.

In order to investigate the dynamic behavior of the Långforsen Bridge more accurately and

comprehensively, two new bridge models were developed during 2011 by Yongming Tu with

Abaqus/Brigade: A comprehensive model with foundations (Type I) and a simplified model

where the foundations have been exchanged by springs (Type II), see Figure 6.1. The models

have in 2016-2018 been refined by Chao Wang.

(a) without displaying of ballast mass points

(b) with displaying of ballast mass points

(c) with displaying of thickness of shell element and profile of beam

Type I: Bridge models with solid element

foundations

Type II: Bridge models without solid element

foundations

Figure.6.1 Global view of bridge models

30

6.2 Design of Models

The advantage of the Type I model is that it is closer to the real bridge structure, and the

predicted results from it should be more reliable and closer to the 'real results', but the

disadvantage is that the number of elements is rather high. The number of elements is 93 910

in type I and decreases to 47 438 in type II and the number of variables decreases from 438

800 to 282 808. Both models give very close predictions of eigenfrequencies and deflections.

Both types of models are mainly developed with shell elements except that 16 circular

columns were constructed with beam elements and the two bridge foundations were

constructed with solid elements if the foundations were included in the model. The track was

modelled with point mass/inertia element and flexible boundary conditions with spring

elements (only for the type II model). In the following the type I model will also be named as

the Modal Benchmark model, and type II as the Dynamic Benchmark model)

Beam elements: 16 circular columns, of which 8 are located in the main span, and the other 8

are located in the side spans.

Solid elements: the two arch abutments (end foundations) if they are included in the model

Point mass/Inertia elements: the track including ballast was modeled with equivalent mass

points attached to the top surface of the bridge decks.

Spring elements: for the models without solid element foundations, the translational and

rotational boundary conditions as well as rotational springs as extra flexible rotational

boundary conditions on the bottoms of arch ends and pier bottoms.

Shell elements: all the other parts of the bridge structure except the 16 circular columns and

two arch end foundations

According to the drilled core test, the concrete Young's modulus of the bridge model was set

as 40 GPa. The density of concrete was assumed as 2500 kg/m3, and the density of ballast was

set as 2000 kg/m3. The longitudinal steel rebars embedded in the concrete was considered in

the updated Abaqus/Brigade model using the embedded Abaqus function 'rebar layer' of shell

element according to the reinforcement drawings of the bridge, although the reinforcement

rebars have very small effect on the results for the modal analysis and dynamic responses in

the linear elastic stage of the bridge structure. In the Abaqus/Brigade models, everything was

constructed according to drawings but of course, some simplifications had to be made.

Furthermore, the joints between shell and shell, or beam and shell, or shell and solid, were

overlapped by connecting parts, which were partial sources of modeling error. Fortunately,

parts of such errors were counterbalanced by the over-reinforcement at these joints.

One principle is required when constructing these models: simplifications of the real bridge

should be as few as possible, and the structure simplified should be equivalent to the original

structure (at least mass equivalent, sectional area equivalent and moment inertia equivalent,

etc.). The simplifications and assumptions brought into the FE bridge model in

Brigade/Abaqus could be summarized as follows:

(i) To simplify cantilever parts, which are of trapezoidal shape, rectangular forms are used.

We choose them so that the equivalent moment of inertia of the cross section of the bridge

deck remains the same after the simplification, see Figure 6.2.

31

10000 5000 5000 6040

6e 7e

6900

3600

1870

(a) Standard cross section of bridge

deck before simplification

(b) Standard cross section of bridge

deck after simplification in

Abaqus/Brigade model

Figure 6.2 Standard cross section of the bridge deck

(ii) To assume the discrete reinforcement rebars as a continuous steel layer and to embed

these steel layers into bridge deck, arch and piers.

(iii) Boundary conditions and connections between main span and side spans of the bridge.

Here the models were built in steps:

- Side spans, see Figure 6.3

- Main span, see Figure 6.4

- Arch ends, foundations and piers, see Figure 6.5

Figure 6.3. Design of Side Span Models

32

Figure 6.4. Detail of Main Span.

Figure 6.5. The combination of arch end foundations, piers and (half) main span

Details of the models are given in Appendix F.

6.3 Calibration and updating of models 2009-2012

Two models were built as already has been mentioned:

I A Modal Benchmark Model with 93 910 elements, which was calibrated towards

measurements

II A Dynamic Benchmark Model, with 47 438 elements, which was studied for accelerations

and displacements. In this model the arch end foundation in Figure 6.5 was modelled by

spring elements.

A summary of the first eigenfrequencies are given in Table 6.1 and further results are

presented in Appendix F. Predicted and measured strains were compared for the

33

reinforcement situated in the bottom of the top beams close to the section where they join the

arch. This section is close to the most critical section according to the Assessment in 2003-

2004. The maximum measured strain was = 50·10-6 which only gives a low live steel stress

of s= Ess ≈ 200·109 · 50·10-6 Pa = 10 MPa.

Table 6.1. Comparison between measured and predicted eigenfrequencies (Hz)

Mode no Measured

values Sept

2011

Type I

(with

foundation)

Type II

(without

foundation)

1. Swaying, Axial-1 0,34185 0,34465

2. Swaying, Axial-2 0,34767 0,35042

3. Symmetric, Transverse-1 1,790±0,002 1,7819 1,7771

4. Asymmetric, Vertical-1 3,062±0,016 2,5528 2,5563

5. Asymmetric, Transverse-2 3,184±0,005 3,1697 3,2274

6. Symmetric, Transverse-3 3,436±0,248 3,5023 3,5131

7. Symmetric, Vertical-2 4,158±0,024 4,2924 4,3991

8. Asymmetric, Transverse-4 5,015±0,038 5,0425 5,0228

9. Symmetric, Vertical-3 5,964±0,022 5,8469 5,9792

The maximum predicted deflections vary from 1,5 to 6 mm, see Appendix F. This is less than

half the value 14 mm assessed in 2003 and much less than the allowable deflection = L/800=

89,5/800 m = 112 mm, where L is the span length

6.3 Calibration and updating of models 2016-2018

During 2016-2017 Mr Chao Wang was a visiting Ph D student at LTU working with FEM

modelling of the Långforsen bridge. He presented preliminary results in Wang et al (2016)

and refined results in Wang et al (2019).

The 2019 paper proposes a method for assessing fatigue damage in reinforced concrete

bridges based on a train-bridge coupled dynamic analysis system. The analysis system is used

to perform coupled vibration analyses of the bridge’s behavior during train passages. The

Palmgren-Miner rule can be used to estimate the linear accumulation of fatigue damage based

on recommended S-N relationships for reinforced concrete from various codes. The method

can be used to evaluate the dynamic performance of a bridge in conjunction with moving load

(ML) and moving spring-mass-damper (MSMD) vehicle models. Calculated dynamic

stresses are used to evaluate the fatigue damage at critical locations.

Fatigue assessments are conducted for the Långforsen Bridge by combining the presented

method with empirical measurements. The effects of train speed and axle load on fatigue

damage are investigated. Train-bridge coupled dynamic analysis using the proposed method

provides a feasibly accurate solution for fatigue damage prediction in four studied load cases.

For low speeds and light loads, the two vehicle models yielded identical predictions of

cumulative fatigue damage. However, their predictions diverged significantly for higher

speeds.

34

A main result of the analysis is that there is no fatigue risk for the bridge for axle loads up to

350 kN.

35

7. Conclusions

The bridge in very good condition.

The arch can carry a Line Load BV 2000 (330 kN axle load). Most sections were cleared in an

assessment by Leanader-Fredriksson (2003, 2004). The remaining sections have been studied

by measurements of deformations and accelerations of the bridge under loads from running

trains and by calibrated finite element models. According to these measurements and models

only small stresses occur in the critical sections of the bridge and the studied sections have no

problems to carry the live load BV 2000. Also, the maximum live load deflections of up to 6

mm are much lower than the allowable deflection = L/800= 89,5/800 m = 112 mm, where L is

the span length

It would be interesting to monitor critical sections for cracking and strain development to

further calibrate the constructed models and to be able to use them as twin digital models in

future management of the bridge

36

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Thesis 2009:152 ISSN 1402-1617, Division of Structural Engineering, Luleå University of

Technology, 100 pp. http://www.diva-portal.org/smash/get/diva2:1031858/FULLTEXT01.pdf

(Read 2019-07-22)

Häggström, Jens (2016). Evaluation of the Load Carrying Capacity of a Steel Truss Railway

Bridge: Testing, Theory and Evaluation. Lic. Thesis, Luleå University of Technology, 142 pp.

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Häggström, Jens; Blanksvärd, Thomas; Täljsten, Björn (2017). Bridge over Åby River –

Evaluation of full-scale testing. Research Report, Div. of Structural Engineering, Luleå

University of Technology, 180 pp. http://ltu.diva-portal.org/

Järnväg.net (2019): Historia Haparandabanan (In Swedish)

http://www.jarnvag.net/banguide/boden-haparanda (Read 2019-07-22) See also:

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Kolster (2002). Bro över Kalix älv vid Långforsen, Morjärv – Kalixborgs bruk, km 24+703.

Granskningsrapport av Bärighetsberäkning. (Comments on Assessment of the Bridge at

Långforsen. In Swedish). Reinertsen Projekt nr 20020760, 4 pp.

Kurrer, Karl-Eugen (2008). The History of the Theory of Structures. From Arch Analysis to

Computational Mechanics. Berlin: Ernst & Sohn, 848 pp. ISBN 978-3-433-01838-5

Larsson, Ulf (1997). Brobyggaren. Otto Linton, byggnadskonsten och dess profession i

Norden under första delen av 1900-talet. (Bridge Builder Otto Linton.Building in

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Summary).Stockholm: Carlsson Bokförlag, 398 pp, ISBN 91 7203 2022.

Leander, John and Fredriksson, Rune (2003). Morjärv-Kalix, Järnvägsbro över Kalix älv vid

Långforsen, km 24+701 (Assessment of the Bridge at Långforsen). Tyréns Projekt nummer

204609, 11 pp + Appendix with calculations, some 200 pp.

Leander, John and Fredriksson, Rune (2004). Morjärv-Kalix, Järnvägsbro över Kalix älv vid

Långforsen, km 24+701 Bärighetsberäkning; farbaneplatta (Assessment of the Bridge at

Långforsen, Concrete Deck). Tyréns Projektnummer 204644, 10 pp + Appendix with

calculations, some 100 pp.

Lorentsen, Mogens (1966). Bågbroar (Arch Bridges. In Swedish). Chapter 934 in Bygg, Vol

6, Part 9 Civil Engineering, 3rd Ed., Edited by Wilhelm Tell, Stockholm: Byggmästarens

Förlag, pp 386-422.

41

Lundin, Per, Editor (2006). Databehandling vid Väg- och vattenbyggnadsstyrelsen/Vägverket

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mars 2006 (Computational Methods used at the Swedish National Road Administration 1957–

1980. In Swedish). Working paper from the Div. of History of Science and Technology, KTH,

Stockholm, TRITA-HST 2007/2, 42 pp. https://www.diva-

portal.org/smash/get/diva2:12336/FULLTEXT01.pdf

(Read 2019-07-24)

LUSAS (2005). Powerful FE technology for specialist application, Modeller User Manual,

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MAINLINE (2011). A European Community 7th Framework Program research project with

the full title: MAINtenance, renewaL and Improvement of rail transport iNfrastructure to

reduce Economic and environmental impacts. Research Project 2011-2014 with 19 partners.

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Nilimaa, Jonny (2015). Concrete bridges: Improved load capacity. (Ph.D. Thesis),

Luleå University of Technology, Luleå, Sweden, ISBN 978-91-7583-345-3, 176 pp.

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i betongbroar längs Malmbanan. (Partial coefficients for concrete strength in bridges on

Malmbanan. In Swedish). Teknisk Rapport 1999:03, Avd för konstruktionsteknik, Luleå tekn.

universitet, 38 + 38 sid. Can be downloded from

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Nordstrand, Uno (1958). Dimensioneringsberäkningar (Design Calculations. In Swedish)

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Load on the Iron Ore Line. In Swedish). ). In the project several studies were carried out

regarding different parts of the infra-structure. The studies are presented in a summary report

and in seven detailed reports. A similar study was carried out by Jernbaneverket in Norway

for the line Riksgränsen – Narvik.

Summary Report of the following reports:

- 3.0 Infrastruktur - Broar och Geoteknik (Bridges and Geotechnology. Summary Report in

Swedish), 34 pp

- 3.1 Infrastruktur - Inventering broar (Inventory of Bridges), 22 pp + 8 appendices

- 3.2 Infrastruktur - Beräkningar och konsekvenser – broar (Bridges – Calculations), 48 pp +

10 appendices

- 3.3 Infrastruktur - Forsknings- och utvecklingsprojekt avseende betongbroars bärighet

(Research Project on Bridges) 51 pp + 5 appendices

- 3.4 Infrastruktur - Geoteknisk inventering (Geotechnical Inventroy), 53pp. + 14 app.

- 3.5 Infrastruktur – Stabilitetsutredning (Stability of Embankments), 19 pp + 5 app

- 3.6 Infrastruktur - FoU Beräkningsmodell för grundläggning på torv Fondations on Peat), 53

pp + 11 app.

- 3.7 Infrastruktur - Geotekniska åtgärder Geotechnical Actions) 50 pp + 5 app.

42

In Norway a corresponding study was made by Jernbaneverket on Ofotenbanan from

Riksgränsen to Narvik:

- 3.11 Infrastruktur -Kontrollberegning av stål- og steinhvelvbruer. (Steel and arch bridges. In

Norwegian) 8 pp + 22 App. Jernbaneverket, December 1996.

-3.12 Infrastruktur -Kontrollberegning av Nordalsbruene (Nordal bridges). 6 pp + 7 App.

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Paulsson, Björn; Töyrä, Björn; Elfgren, Lennart; Ohlsson, Ulf; Danielsson, Georg; Johansson,

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trough bridges and a laboratory fatigue test on one trough bridge. In Swedish). Banverket och

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001653, see https://ltu.diva-portal.org . Four guidelines have been prepared:

- Inspection and Condition Assessment (SB-ICA), 259 pp;

- Load and Resistance Assessment of Railway Bridges (SB-LRA), 428 pp;

- Guideline for Monitoring of Railway Bridges (SB-MON), 93 pp; and

- Guide for use of Repair and Strengthening methods for Railway Bridges (SB-STR), 139 pp.

Background documents with state-of-art reports, analytical and numerical analyses and test

results are also provided.

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portal.org

43

Solvia (2000) A FEM program: SOLVIA-PRE for stress analysis and SOLVIA-POST for

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44

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förstärkning av betongkonstruktioner med pålimmade fiberkompositer (Design Guideline for

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Available on line: https://doi.org/10.1080/15732479.2019.1671467