Jóhannes Helgi Jóhannesson

Design of a 170 m span bridge over the fjord Thorskafjordur in Iceland

Jóhannes Helgi Jóhannesson

Avdelningen för Konstruktionsteknik Lunds Tekniska Högskola Lunds Universitet, 2010

Rapport TVBK - 5185

i

Avdelningen för Konstruktionsteknik Lunds Tekniska Högskola Box 118 221 00 LUND Department of Structural Engineering Lund Institute of Technology Box 118 S-221 00 LUND Sweden Design of a 170 m long bridge over the fjord Þorskafjörður in Iceland Dimensionering av 170 m lång bro över fjorden Þorskafjörður på Island Jóhannes Helgi Jóhannesson 2010 Abstract Determining the structural type of a bridge is often a difficult task. The purpose of this thesis is to preliminary design three bridge alternatives. The bridge shall cross the fjord Þorskafjörður in Iceland. The goal is to determine the most favorable option. That decision will be based on economy, construction and aesthetics. Following that a more detailed design of the superstructure is performed for the chosen alternative. All calculations are performed according to Eurocode.

Keywords: concrete girder bridge; arch bridge; cable-stayed bridge; concrete; reinforcement; prestress

ii

Rapport TVBK-5185 ISSN 0349-4969 ISRN: LUTVDG/TVBK-10/5185+92p Examensarbete Supervisor: Dr. Fredrik Carlsson Examinator: Prof. Sven Thelandersson October 2010

iii

Foreword This thesis was written under the administration of the Division of Structural Engineering at the University of Lund. It was written during the period September 2009 - September 2010 under the supervision of Dr. Fredrik Carlsson.

I especially want to thank my supervisor, Dr. Fredrik Carlsson, for all his help with making this thesis become real. I also want to thank Einar Hafliðason, the head of the bridge division of the Icelandic Road Administration, for the help with finding a subject for this thesis and for giving me necessary information regarding this subject. For the cost of various structural materials I would like to thank Oskar Bruneby, a site manager at Peab, for his contribution. In addition I would like to thank a good friend from Iceland, Ástþór Ingvason, for making 3D animations of the three bridge alternatives presented in this thesis. Finally, I want to thank my friends at LTH: Bzav Abdulkarim, Daniel Honfi, Ívar Björnsson and Valdimar Örn Helgason for all their help and last but not least other friends and family for their moral support.

Lund, October 2010 Jóhannes Helgi Jóhannesson

v

Table of Contents 1 Introduction ..................................................................................................................................... 1

1.1 Background ............................................................................................................................. 1

1.2 Objectives ............................................................................................................................... 1

1.3 Outline of the thesis ................................................................................................................ 1

2 Bridge types .................................................................................................................................... 2

2.1 Concrete bridge ....................................................................................................................... 2

2.2 Arch bridge ............................................................................................................................. 3

2.3 Cable-stayed bridge................................................................................................................. 4

3 The actual project – geometry and boundary conditions ................................................................ 5

4 Preliminary design .......................................................................................................................... 7

4.1 Introduction ............................................................................................................................. 7

4.2 Loads ....................................................................................................................................... 7

4.2.1 Permanent loads .............................................................................................................. 7

4.2.2 Variable loads ................................................................................................................. 7

4.2.3 Load combinations .......................................................................................................... 8

4.3 Material cost ............................................................................................................................ 9

4.4 The concrete beam bridge ..................................................................................................... 10

4.4.1 Geometry for type 1 ...................................................................................................... 10

4.4.2 Size estimation .............................................................................................................. 10

4.4.3 Supports ........................................................................................................................ 11

4.4.4 Construction .................................................................................................................. 13

4.4.5 Cost estimation/conclusions .......................................................................................... 13

4.5 The arch bridge ..................................................................................................................... 16


4.5.2 Arch ............................................................................................................................... 17

4.5.3 Bridge Deck .................................................................................................................. 22

4.5.4 Hangers ......................................................................................................................... 23

4.5.5 Transversal Bracing ...................................................................................................... 23

4.5.6 Foundations ................................................................................................................... 24

4.5.7 Construction .................................................................................................................. 24


4.6 The cable-stayed bridge ........................................................................................................ 26

4.6.1 Aesthetics of cable-stayed bridges ................................................................................ 26


4.6.3 Design ........................................................................................................................... 27

vi

4.6.4 Deck .............................................................................................................................. 32

4.6.5 Pylons ............................................................................................................................ 33

4.6.6 Foundation .................................................................................................................... 34

4.6.7 Construction .................................................................................................................. 34


4.7 Summary and choice of bridge type...................................................................................... 37

5 Final design ................................................................................................................................... 38

5.1 Introduction ........................................................................................................................... 38

5.2 Design ................................................................................................................................... 39

5.2.1 Building codes .............................................................................................................. 39

5.2.2 Loading ......................................................................................................................... 39

5.2.3 Materials ....................................................................................................................... 39

5.2.4 Exposure classes and service life .................................................................................. 40

5.2.5 Tendon alignment and prestress force ........................................................................... 40

5.2.6 Prestress losses .............................................................................................................. 46

5.2.7 Secondary effects of prestress ....................................................................................... 51

5.3 Ultimate moment capacity .................................................................................................... 55

6 References ..................................................................................................................................... 57

6.1 Literature ............................................................................................................................... 57

6.2 Computer programs............................................................................................................... 58

6.3 Other references .................................................................................................................... 58

Appendix A ........................................................................................................................................... 59

Appendix B ........................................................................................................................................... 74

Design of a 170 m long bridge over the fjord Þorskafjörður in Iceland

1

1 Introduction

1.1 Background The motivation for writing this thesis is an interest in bridges that the author has acquired during his studies in structural engineering. Many people consider bridges to be state of the art of all civil structures. That can be for many reasons; f. ex. bridges sometimes cross a difficult passing or because of their aesthetic aspects.

During the time the subject for this thesis was under consideration the author decided to contact the bridge division of the Icelandic Road Administration (ICERA). Einar Hafliðason, the head of the bridge division of ICERA, was contacted and he was more than willing to help. He came up with a few options to look into which were all considered. Following that, a decision was made and a bridge that is to be constructed to cross the fjord Þorskafjörður in Iceland was chosen as a subject for this thesis.

1.2 Objectives The main purpose for a bridge over the fjord Þorskafjörður is to shorten the distance of the route on the way to the northwestern part of Iceland. With this bridge the route will shorten of about 10 km. Another purpose is to increase traffic security by eliminating all one-lane bridges on this 10 km sector.

The main objective of this thesis is divided into two parts. First, a preliminary design of three bridge alternatives is made. A rough cost estimation and an estimation of quantity of materials is made based on the preliminary design for these three alternatives. Secondly, a more detailed design is made of the most appropriate bridge type. The choice of a bridge type is based on the conclusions from the first part. These conclusions will primarily be based on economy, aesthetics and construction method.

1.3 Outline of the thesis Chapter 2 consists of a general discussion about aesthetics, advantages and disadvantages and other aspects for the three bridge types that are chosen to be analyzed.

Chapter 3 displays the bridge location and describes the boundary conditions and geometry at the construction site. It also includes information about why this bridge is to be built.

Chapter 4 includes preliminary design and cost estimations of the three chosen bridge alternatives with respect to the quantity of materials needed for each type. That chapter also includes conclusions of the preliminary design, that is, which type of bridge is chosen for a more detailed design with respect to the limits that are set.

Chapter 5 includes more detailed structural analysis for the superstructure of the chosen bridge alternative.

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7

4 Preliminary design

4.1 Introduction This chapter contains preliminary design of the three bridge types, a concrete girder bridge, an arch bridge and a cable-stayed bridge. The aim of the preliminary design is to determine the most suitable bridge type for the purpose of crossing the fjord Þorskafjörður. The chapter is divided in to two different sections. The first sections (sections 4.2 and 4.3) treat factors that are common for all three bridge types, i.e. loads, load combinations and materials. Sections 4.4 to 4.6 treat the three different bridge types respectively. In these sections are sizes of important bridge elements for each bridge type estimated. These sections also contain rough cost estimations and construction methods for each bridge type. Finally, in the last section of this chapter, the most suitable bridge type is determined based on the preliminary design.

4.2 Loads For the preliminary design of this project only three loads are considered. Two permanent loads, self-weight and pavement, and one variable load, traffic load. The loads are determined according to Eurocode 1 (EC1).

4.2.1 Permanent loads Self-weight for reinforced concrete is set to 25 kN/m3. The self-weight of pavement and structural steel are set to 2.1 kN/m2and 78.5 kN/m3 respectively.

4.2.2 Variable loads The variable actions, which are taken into account in this thesis, are traffic loads in vertical direction. After some discussion with the head of the bridge division of ICERA it seemed reasonable to do this simplification in the preliminary analysis.

Traffic Loads In EC1-2, chapter 4, there are defined four different load models for traffic loads. In this case Load Model 1 (LM1) is used with two partial systems, one including axle loads (Tandem system TS) and the other including uniformly distributed loads (UDL system), see figure 4-1. LM1 is considered to cover most of the effects from traffic of lorries and cars and should be used for general and local verifications while the other load models are considered for dynamic effects, special vehicles and other situations. LM1 should be applied on each notional lane and on the remaining areas. On notional lane number i, the load magnitudes are referred to as αQiQik and αqiqik, axle load and distributed load respectively. On the remaining areas, the load magnitude is referred to as αqrqrk. According to chapter 4.3.2(3) in EC1-2 the recommended minimum values for the adjustment factors are: ≥ 0,8 ≥ 1,0

The national annex for Sweden recommends the following minimum values for the adjustment factors: = 0,9 = 0,9 = 0

Hence, Icelandicharacte

These lomost unf

4.2.3 Several when a state is mlimit sta

Des

these valuesc national an

eristic vertica

oad arrangemfavorable res

Load combload combinbridge is anamade for the ate is accordin

ign of a 170

s are used fnnex for this al distributed

Table

ments are dispsult.

Figure 4

binations nations need talyzed and dpreliminary

ng to EC0, s

,

m long bridg

for this situapart of Euro

d loads, qik, ar

e 4-1: Axle load

played on fig

4-1: Load arra

to be taken idesigned. Bu design of alection 6.4.3.

, " + "

ge over the f

8

= 1,0= 1,0ation. The S

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ds and uniform

gure 4-1 and

angements for l

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fjord Þorskaf

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mly distributed

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mpletely corree been inves

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mate the price(SEK/kg). T

ry to find thehuan (1998) aones that wance or othegn of that is

ign of a 170

r for perman

ic value of p

r for prestres

presentative v

r for variable

ic value of th

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ombination v

ic value of th

γP=1.0 and γpectively. Th

ost zes unit priceect they will stigated. Theresources liis included

Table

e of the steelTo come upe unit weightabout stay ca

were chosen r factors, winot done in

m long bridg

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permanent ac

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value of a pr

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4-2: Unit pric

l hangers in tp with a prit (kg/m) of thables where here to use

ll be estimatthe prelimin

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able action i

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partial safetyin this equa

aterials used.d perspective cost estimat

ons with conlues and high

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the arch bridice for the he cables. Unthe unit wei

e and that vted and cost oary phase. H

fjord Þorskaf

ction

n 1

le action i

y factors foration is not

. Even thoug on prices fotion and pricntractors andher values a

structural mat

dge, see sectistay cables nit weight foght of cablesvalue is 24of foundatio

Here estimatio

fjörður in Ice

r permanent required sin

gh values of or comparisoces is from cd engineersre chosen w

terials.

ion 4.5.4, uniof the cable

or cables wass that has a s.1 kg/m. Nons is a factoron is used to

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action, prestnce there is o

various expeon of the bridourses the auboth in Ice

where a price

it price for soe-stayed brids found in a similar breako lifetime cr of uncertain

o calculate th

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enses are dge types uthor has land and range is

olid steel dge it is literature king load cost, like nty since

he cost of

the subselements

4.4 TBridge tsupportithe bridg

4.4.1 This bricontinuodown to

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structures wis. Included in

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Size estimmentioned bwo internal -2. This choith of the exte

st step is to s system is 0 mm. 3 cabadditional th

ght of the gL/h, of the ghould be choic reasons (some up probl

nter distancedersson (2009

B is the free w

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th a method n the substru

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for type 1 ill have a cridge with 4

ation efore, the totspans with tice of span leernal spans i

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irder is detegirder. A recosen in the rasection 7.2.1lems later in

e between t9):

width of the b

slab is chose

-section of th

m long bridg

developed bucture are the

m bridgepost-tension

ll be two mae figure 4-3.

oncrete slabspans, see fi

tal span of ththe length 4engths is mas about 80 %

Figure 4-2: Sp

prestressing The dimens

sen in each r300 mm is de

ermined by tcommended ange of 12-3

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ℎ = 20the two ma

= 1,8 ∙bridge.

en to be 250

he bridge, ba

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10

by Menn (19e piers and fu

ned girder bain girders wSupports wil

b supported igure 4-2. Th

he bridge is 18 m and twde to get an

% of the lengt

pan lengths for

system to esions for anrow which retermined ov

the slenderneslenderness

35 and some1986)) and So the estim

⟹ ℎ = 4820ain girders,

⟹ = 11,mm to be ab

ased on the m

fjord Þorskaf

986) as a 23.undaments.

bridge with fwith post-tens

ll be founded

on two conthe girders ar

170 m. The bwo external sp

even momenth of the inte

r bridge type 1

estimate the chorage blo

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ess, a ratio bratio for a clower rangea low value

mated height b

= 2,4

a, is chose

0,8 = 5,6

ble to resist s

methods abov

fjörður in Ice

.5% of the to

four spans ansioning cabled on concrete

tinuous girdre supported

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.

size of the cks for thatminimum wiorts.

between the conventional

e should be ce should alsobecomes:

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otal cost of s

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ided into foulength of 3

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structural

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chieved if

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ts.

4-3.

4.4.3 MaximuTo estimhas to bloads haTo find figure 4-

The GD

And the

Des

Supports um load on amate the mosbe determineave to be loc

GDF, the m-4.

Fs for the tw

following tr

ign of a 170

Figu

a column wost unfavorab

ed. GDF tellscated in the mmoment is ca

Figure

wo traffic loa

raffic loads th

m long bridg

ure 4-3: The cr

ould be whenble load actins how the trmost unfavoalculated aro

e 4-4: Location

ds, axle and = 1,37= 1hat act on on

ge over the f

11

ross-section at

n the given tng on a singraffic load israble positio

ound B with

n of traffic load

distributed l7 For the ta1,16 For U

ne beam are:

fjord Þorskaf

preliminary st

traffic load igle beam the s distributed on on the bri

the lever ar

ds to determine

oads, are det

andem system

DL system

fjörður in Ice

tage.

s located as girder distribetween the

idge deck in rm for each

e GDF.

termined to b

m

eland

shown in figibution factoe girders. Ththe lateral dload as disp

be:

gure 4-4. or (GDF) he traffic direction. played in

where thanalyzedsection. the colum

The maxstate. At

The widcalculate

with theas fck=45the mini

11So for a

For the p

Precaste

Des

he axle loadd in the lengFigure 4-5 smns and the

Figure 4-5: P

ximum normt this stage th

dth of a sined from a for

yield streng5 MPa, the pimum thickn

1180 10single suppo

piles, given t

ed 270x270 p

ign of a 170

ds are changth directionshows the acbridge sectio

Position of traf

mal force in the concrete q

ngle girder ormula for pre

=gth of the reinpercentage oess of the su

= 1380ort the total s

that each pile

piles with 12

m long bridg

nged into onn for one girctions and thon in the ulti

ffic loads when

the middle suquality is assu

over supporeliminary de

0.44nforcement a

of reinforcemupport should

0.44 45size is determ

e resists 400 11400 mm spaci

ge over the f

12

= 1012= 37 ne concentrarder. Calcula

he position oimate limit st

n the largest no

upport is calumed to be C

rts is 1.380 sign given in

+ 100 0.6as fy=500 MP

ment p=2% ad be:

+ 2100 (0.67mined to be b

kN, gives: 18000 28 ing and a fou

fjord Þorskaf

2 /

ated force (aations are mf actions to tate.

ormal force at

lculated to beC45/55.

mm. The thn the ISE ma

67 − 0.44Pa, characterand Ac as the

7 500 − 0b x t = 7500 x

undation of t

fjörður in Ice

assumption).made for half

decide the la

the middle sup

e 11.180 kN

hickness of anual (1985):

istic cylindere gross cross

.44 45) ⇒x 450 mm.

the size 5x10

eland

Then the bf of the bridargest shear

pport occurs.

N in the ultim

the support:

r strength of -sectional ar

⇒ ≥ 311

0 m.

bridge is dge cross force for

mate limit

t wall is

f concrete rea. Thus


13

4.4.4 Construction The construction method of a concrete girder bridge is relatively easy to perform. Concrete girder bridges are one of the most common bridges in Iceland. This bridge alternative is often chosen for similar conditions as are in this case because of economic and constructional reasons, that is, when a shallow fjord is to be crossed.

Supports and columns below the superstructure will be constructed first. They will be founded on piles. Since the level of sea depth at is shallow at the construction site the superstructure of the bridge will be casted in forms that are supported on a temporary filling under the bridge. The filling will finally be removed when the concrete has hardened and can then be used as road material.

4.4.5 Cost estimation/conclusions To estimate the cost of this bridge type a method from Menn (1986) is used. The following is an explanation of this method.

The superstructure’s costs can be reliably estimated with the help of the geometrical average span length, lm, defined as:

= ∑∑

where li is the length of span i.

The empirical equations given below give the quantities of concrete, reinforcing steel, and prestressing steel as functions of lm and have been derived from samples of recently constructed bridges.

By this method the volume of concrete in the whole superstructure is obtained by multiplying the total deck surface by the effective girder depth, hm, defined by the following expression: ℎ = 0.35 + 0.0045 ∙

where hm and lm are in meters. This equation is valid provided the actual girder depth, h, satisfies the following inequality: 120 ≤ ℎ ≤ 116

which fulfills the criteria used earlier, l/h=20. The quantity of reinforcing steel is obtained by multiplying the total volume of concrete by the mass of steel per unit volume of concrete, ms. The parameter ms is estimated using the equation: = 90 + 0.35 ∙

where lm is in meters and ms is in kilograms per cubic meter of concrete (kg/m3). This expression is valid provided the deck slab is not transversely prestressed. Between 65 and 70 kg/m3 of reinforcement is required for stability during construction and crack control; this quantity is independent of span length, see Menn (1986). The transverse reinforcement required to resist loads is primarily a function of cross-section dimensions. An additional 20 to 25 kg/m3 is required for commonly used cross-sections, regardless of span length. Most of the steel required above the

minimumattention

The maconstrucequation

where lm

girders multiply

The estiobtained(scaffold(temporainto accomaterialfalseworsea leveincreaseestimate

Here is ois a tabcalculati

Des

m 65 to 70 kn in the desig

ss of prestrection methodn:

m is in meterthat are no

ying mp by th

imated cost d by multiplding systemary structureount the propl costs, anotrk and formwl the cost pe

ed. These vaed using table

Table 4

only listed thble that sumions as well

ign of a 170

kg/m3 is locagn and arrang

essing steel d. For girders

s and mp is iot transversehe total volum

of concretelying the es

ms that are ue used to retposed constrther construwork costs yiercentage of alues are choe 4-3, from M

4-3: Table from

he quantity omarizes thoas the empiri

m long bridg

ated in the dgement of th

per unit vos that are cas

in kilogramsely prestressme of concre

e, reinforcinstimated quaused to temtain unhardenruction sequeuction methoields the totathe total cos

osen to be 2Menn (1986)

m Menn (1986)

of super- andse results. Tical equation

ge over the f

14

deck slab. Thhe superstruct

lume of consted on conv

= 0.4 ∙s per cubic med. The qu

ete.

ng steel andantities withmporarily suned concrete

ence; if it is god should bal superstructst of those st5% and 23%

).

) to estimate co

d substructureThese quantins above. The

fjord Þorskaf

he deck slab ture reinforc

ncrete, mp, iventional fals

meter of concuantity of pr

d prestressinunit materi

upport permae until hardegreater than 6be considereture cost. Sintructural part% respective

osts for variou

e materials thities are acqe calculated

fjörður in Ice

should thereement.

s a functionsework, mp i

crete. This exrestressing s

ng steel in tial costs. Thanent structuened) should65 percent o

ed. Adding nce abutmentts as well as ely. The rem

us structural el

hat will be dquired basedquantities ar

eland

efore be the

n of span lens estimated u

xpression is steel is obta

the superstruhe cost of fures) and fo

d be estimatef the superstthe bridge ts and piers afalsework/fo

maining cost

lements.

determined and on the prere given in ta

focus of

ngth and using the

valid for ained by

ucture is falsework formwork ed taking tructure´s material, are under

formwork ts can be

nd below eliminary able 4-4.

For the fsub- andcost for

This resutype of bISK/SEK

Des

T

figures givend superstructthis bridge ty

ult is consistbridge was 6K. Note that

ign of a 170

Table 4-4: Amo

n in table 4-4ture and the type the total

tent with a d638.500.000 this draft ass

F

m long bridg

ounts of structu

4, abutments total cost basvalues for th

Table 4-5: T

draft for this pISK which isumes the tot

Figure 4-6: An

ge over the f

15

ural materials

and columnsed on the prhe structural

Total cost of br

project madeis around 38tal length of

n overview of th

fjord Þorskaf

for the concre

s are includerice values felements are

ridge type 1.

e by ICERA.700.000 SE

f 182 m inste

he beam bridg

fjörður in Ice

ete beam bridg

ed. Below is from section e doubled.

where the eEK with the e

ad of 170 m.

e.

eland

ge.

a table with 4.3 and to g

estimated cosexchange rat.

prices of get a total

st for this e of 16.5

4.5 TBridge tEach arcwill be odeck wilmain girwhich arthe main

4.5.1 The chochosen bwhich wconditio

To conna drawin

To deterdetermin

Des

The arch btype no. 2 is ch will be ofof zero hingell be of comrders with shre connected

n girders, see

Geometry oice of the rbetween 4 an

would result ons on site (n

nect the deckng of the stru

rmine the sened accordin

ign of a 170

bridge a conventio

f a steel box ed type with

mposite steel/chear studs. Td to the arche figure 4-7.

F

for type 2 rise of the and 8 see, Loin less horiz

no rock – only

k to the arch uctural mode

Figur

ction forces ng to figure 4

m long bridg

nal steel arccross-sectioX-bracing bconcrete. In

To connect thhes with hang

igure 4-7: A cr

rch is basedoretsen and Szontal reactiy sediment s

= 4 ⇒vertical steel of the bridg

e 4-8: A struct

in the arche4-9.

ge over the f

16

h bridge witon with steelbetween the a

the longitudhe two maingers. A reinf

ross-section of

d on the ratiSundquist (19ion forces. Tsoil layers). T

⇒ = 174l wire hangege.

tural model of

es the GDF n

fjord Þorskaf

th two separastiffeners in

arches to incdinal direction girders therforced concr

the bridge dec

io between s995). This ra

This ratio is That results i704 = 42,5 ers with c/c 2

the tied arch b

needs to be

fjörður in Ice

ate arches abnside, see figrease lateral

on of the bridre will be trarete slab wil

ck.

span and risatio is in thissuitable becn

25 m are cho

bridge.

determined a

eland

bove the bridgure 4-11. Th

stiffness. Thdge there wiansversal stel be casted o

e which is gs case chosencause of geot

osen. On figu

again. The G

dge deck. he arches he bridge ll be two el beams on top of

generally n to be 4 technical

ure 4-8 is

GDFs are

The follo

And the

4.5.2 To desiginvestigamoving moment

Des

owing GDFs

following tr

Arch gn the arch tated: abutmea point load

t, shear force

ign of a 170

Figure 4

s are acquire

raffic loads th

the influenceent, ¼ of thed of 100 kN

e and normal

m long bridg

4-9: Position of

d: = 1,13= 1hat act on on

e lines for the arch and tN in 10 m i

force. Thes

ge over the f

17

f traffic loads w

3 For the ta1,00 For U

ne girder are:= 694= 27 he arch needhe middle. Iintervals ovese influence d

fjord Þorskaf

when calculati

andem system

DL system

: 4 /

d to be determInfluence liner the deck diagrams can

fjörður in Ice

ing GDF.

m

mined. 3 secnes for each in the longi

n be seen in f

eland

ctions in the section are

itudinal direfigure 4-10.

arch are made by ction for


18

Figure 4-10: Influence lines for various section forces at most critical placements.

To calculate the important section forces for design of the cross section of the arch the traffic loads are placed on the most unfavorable position corresponding to these influence diagrams in a program called PCFrame. PCFrame is a commercial program for structural analysis of frames.

Cross Section To design the arch in the ultimate limit state the section forces are required. The highest moment in the arch is reached when the traffic load is located in the middle of the span. The position of the point load at the first quarter of the span gave the highest normal force. So these corresponding section forces are used to design the cross section and are shown in table 4-6.

-1.00

-0.50

0.00

0.50

1.00

0.00 0.20 0.40 0.60 0.80 1.00

M

x/L

Influence Lines for Moment in the Arch

Abutment 1/4 of span

Middle of the span

-1.00

-0.50

0.00

0.50

1.00

0.00 0.20 0.40 0.60 0.80 1.00

V

x/L

Influence Lines for Shear Force in the Arch


Middle of the span

-1.00

-0.50

0.00

0.50

1.00

0.00 0.20 0.40 0.60 0.80 1.00

N

x/L

Influence Lines for Normal Force in the Arch


Middle of the span

The mat

The follodirection

The crodimensioA.

StabilityTo checwell as wload and

Des

terial qualitie

owing cross-n and resistan

oss-section ions of the cr

y ck for stabiliwith the helpd the maximu

ign of a 170

T

es of the stee

-section was nce, see figu

F

s a welded ross section (

ity in longitup of PCFramum normal f

m long bridg

Table 4-6: Des

el are given in

Table 4-7:

determinedure 4-11:

Figure 4-11: Ch

box section(moment of

udinal directme. To fulfillforce needs to

ge over the f

19

sign section for

n table 4-7.

Material quali

after few tria

hosen cross-sec

n with trapeinertia, secti

tion two invel the requiremo be in the ra

fjord Þorskaf

rces in the arch

ities of steel.

als with resp

ction of the arc

ezoidal stiffeion modulus

estigations aments for staange from 4

fjörður in Ice

h.

pect to stabili

ch.

feners. Furthetc.) can be

are made; caability the rato 5, see Lo

eland

ity in the lon

her details aseen in the a

alculation bytio between

oretsen and S

ngitudinal

about the appendix

y hand as buckling

Sundquist

(1995). critical b

where S case S isdetermin

From anwell widetermin

EC3 givis given

with Nd

Here χ is

with e

and α, astronger

, the no

Des

To calculatebuckling load

S is the one-hs 98525 mm ned to be 458

nalysis in PCith the calcnation of the

ves another min section 5.

as the design

s the reductio

equal to

an imperfectir axis and a w

on-dimension

ign of a 170

e the bucklind is given as

half length ofand k is 0.70

845 kN and t

CFrame for thculations by

cross section

Fi

method to de.5.1 in EC3.

n normal forc

on factor for

ion factor, owelded box s

nal slenderne

m long bridg

ng load by ha:

f the arch and0 for a fixedthe ratio betw

his case this y hand. Then. The buckl

igure 4-12: Bu

etermine the EC3 defines

ce and the ca

.r the relevant

= += 0,5

obtained fromsection with b

ess, is define

ge over the f

20

and formulas

= ( )d k is the effed arch with a ween the buc

= 4.8

factor is deteese calculatiling mode sh

uckling mode sh

buckling ress the followin≤ .apacity of the=t buckling m1− ,

1 + − 0m an appropb/tf<30, α be= 0.49ed as

fjord Þorskaf

s presented b

)

ective lengthrise-span rat

ckling load an

ermined to bions were t

hape for this

hape of the arc

sistance criteng criteria:

e cross-sectio/

ode and is de

10,2 +

riate bucklincomes

fjörður in Ice

by Austin (1

h factor, see Atio as 0.25. Fnd the maxim

be 5.0. That mthe most crarch is show

ch.

eria for comp

on, Nb.Rd , as

efined as

ng curve. Fo

eland

971) were u

Austin (1971For these valmum normal

matches consritical ones

wn in figure 4

pressed mem

or buckling a

used. The

1). In this lues Pe is l force is

siderably for the

4-12.

mbers and

about the

where βA

In this c

Table 4-

It can bearch.

CompreTo detedetermin

In this c

The cros

Here, Np

and Wpl factor isseen in aforce are

The resu

Des

A is dependin

ase βA is equ

-8 summarize

e seen in tabl

ession and ermine the cned. To decid

case the cros

ss-section ca

Npl.Rd is the plthe plastic s defined as appendix A. e made accor

ults of this an

ign of a 170

ng on the cro

ual to 1. For r

es the results

Table

le 4-8 that th

bending cacompression de the cross

s section cla

apacities are d

lastic designsection modu= 1,1 inNext a chec

rding to secti

nalysis are su

m long bridg

=oss-section a= 1 for Cl= /resistance of

s of this anal

e 4-8: Criteria

he buckling re

pacities and bendin

section class

ass is determi

defined in EC

.n resistance fulus. For res

n section 5.1.ck for bendinion 5.4 in EC

.ummarized in

ge over the f

21

= /s below:

lass 1, 2 or 3

for Class 4

f member to = 1,1ysis.

for buckling r

esistance is w

ng capacities plastic stres

ined to be 1.

C3 in section

. = /= /for compresssistance of C1 in EC3. Th

ng moment, cC3. The follo≤ .≤ .+ .n table 4-9.

fjord Þorskaf

,

cross sectio

4 cross sectio

buckling the

resistance from

well above th

es the class ss distributio

ns 5.4.4 and

/

sion, Mpl.Rd thClass 1, 2 or he calculatiocompression owing design

Ben

Com

≤ 1 Com

fjörður in Ice

ons

ons

e safety facto

m EC3.

he calculated

of the croson is assumed

5.4.5 respect

he plastic re3 cross-sect

ons for Npl.Rd,and combin

n criteria are

nding

mpression

mbined bend

eland

or is

d normal forc

ss-section had.

tively as

esistance for tion the parti Mpl.Rd and W

ned bending achecked:

ing and axia

ces in the

as to be

bending ial safety

Wpl can be and axial

l force

All the cso small

4.5.3 The bridwork aswill only

TransveThe tranloading them. Thmomentcross-se

Figure 4

The highthese secgiven in

Des

criteria are ful and are ther

Bridge Decdge deck wi composite dy make the st

ersal beamsnsversal beamfor these behese positiont and shear ction in the u

4-13: Positions

hest momentction forces

n sections 5.4

ign of a 170

Tabl

ulfilled. Alsorefore neglec

ck ill consist ofdeck with sttructure mor

s ms will be m

eams will bens are show iforce in the ultimate limi

s of traffic load

t and shear fothe cross-sec

4.5 and 5.4.6 ≤

m long bridg

le 4-9: Design v

o, the cross-scted. The arc

f main steel teel shear sture rigid. Estim

made of stee when the ain figure 4-1transversal

it state.

ds and shear- a

force are detection can be in EC3 as

. =

ge over the f

22

values and resi

section is assch is mostly i

girders, tranuds. The commated thickn

el and will axle load fro13. This locat

beams. The

and moment didetermined.

ermined to bdetermined.

/

fjord Þorskaf

istance for the

sumed to resin compressi

nsversal beammposite effeness of the co

be placed wom the traffiction of the ax

ese section f

iagrams when

e 3025 kNm The design

For class 1 o

fjörður in Ice

arch.

ist the shear on.

ms and concct is not cal

oncrete slab i

with 5 m spac is placed exle load willforces are us

the cross-secti

m and 1359 kNcriteria for s

or 2 cross sec

eland

forces since

crete slab. Tculated hereis 200 mm.

acing. Worstexactly abovl generate thsed to determ

ions of cross be

N respectiveshear- and m

ctions

e they are

They will e but that

t case of ve one of e highest mine the

eams are

ely. From moment is

where thbeams is

Main giThe maihangers.7241 kNthe trancalculati

4.5.4 The hanhangers ultimateexactly carbon s

4.5.5 The chothe X-tyanalyzed

Des

he parameters shown in fi

rders in girders w. The largest

Nm respectivnsversal beaions can be s

Hangers ngers are the

are designee limit state. at hanger nu

steel with the

Transversaoice of X-typype, see Bund as a truss sy

ign of a 170

rs are explainigure 4-14. F

Fi

will be connet shear force

vely. Here thems. The de

seen in appen

Fig

elements thed to resist t

This force umber 1 cloe design yield

al Bracing e bracing ratnner and Wrystem. Later

m long bridg

ned in the lasFor detailed c

igure 4-14: Est

ected to the te and momene same criteretermined crndix A.

gure 4-15: Dete

hat connect thhe largest teis acquired

osest to the d strength as

ther than Vieright (2006)ral braces wi

ge over the f

23

≤ . =st section. Thcalculations,

timated size of

transversal bnt in the mairia are checkross-section

ermined size of

he bridge deension force,when the axsupport. The

s 3605 kN.

erendeel brac, which resull not be calc

fjord Þorskaf

(√ )/he determinesee appendix

f the cross beam

beams whichin girders ar

ked, shear- ancan be see

f the main gird

eck to the arc, which is dxle force froe chosen ma

cing is that thults in less lculated at thi

fjörður in Ice

ed cross-sectx A.

ms.

h are hanginre determinednd moment ren in figure

ders.

ch. They aredetermined toom the traffaterial of the

he system wilateral deflecis time.

eland

tion of the tra

ng from the ad to be 2087resistance, ase 4-15 and

e vertical cabo be 3393 kfic load is poese hangers

ill be more rctions and w

ansversal

arches in 7 kN and s was for

detailed

bles. The kN in the ositioned is M100

igid with would be

4.5.6 The soilconcrete(2010). length oestimate

for eachbe 35 pifundamesupport direction

4.5.7 My propFirst, foube placesegmentsegmentplaced in

Next thebeams thwelded casted.

4.5.8 In table quantitieon the m

Des

Foundatiol under the e piles that wThe piles wi

of the piles wed amount of

h arch. The spiles under eacent itself. Ththe fundam

n and the app

Constructiposal of a coundations foed under thets, sizes that t at a time, sun right positi

e deck is cohat are hangtogether and

Cost estim4-10 the to

es are based method in sec

ign of a 170

ns foundations

will be driveill have an in

will be 14 m.f piles is

pacing betwech abutmenthe filling beh

ments. A conproximate siz

Fig

ion onstruction mr the arches

e bridge. Theare possible

upported by ions the temp

onstructed. Tging in the had finally, wh

mation/conctal quantity on the preli

ction 4.4.2 fo

m long bridg

s is sedimenen down to nclination do. The largest

124een the piles. The piles whind the brid

ntinuous founze of it will b

gure 4-16: Plac

method for thwill be consten the archee to transporfalsework stporary worki

The main girangers. The

hen all the w

clusions of materialsminary calcu

or bridge type

ge over the f

24

nt layers. Ththe ground.

own in the dit reaction for

60100 32 will be 1.2

will also be adge will alsondation is chbe 7.2 x 4 x 1

cing of piles at

his bridge typtructed and t

es will be rart. These segtanding on thing plane is r

rders come hangers con

work with the

s for the arculations. Thee one. These

fjord Þorskaf

he foundatioEach pile r

irection to thrce in the arc

m. The totalable to resist o be able to hosen under16.8 m, see f

the arch suppo

pe is similarthen a tempoaised. Each agments will bhe working premoved and

in segmentsnnect the arche structural s

hes and bride cost estimae cost figures

fjörður in Ice

n will be foresists about he arch’s direch’s direction

number of pthe risk of turesist some

r the whole figure 4-16.

orts.

r to the methorary workingarch will be be welded toplane. When d can be used

s and are cohes to the desteel is done

dge deck areation for the s are given in

eland

ounded on p400 kN, Ha

ection. The en is 12601 k

piles is deterurning alongexternal actbridge in th

hod for bridgg plane of grdivided into

ogether in stethe arches h

d as a road fil

onnected to teck. Each se, the concret

e summarizefoundations

n table 4-11.

precasted afliðason estimated kN so the

rmined to g with the tions and he lateral

ge type 1. ravel will o several eps, each

have been lling.

the cross egment is te slab is

ed. These is based

To get a

This bridsteel cos

Des

a total cost th

dge type is lst and compl

ign of a 170

Table 4-10:

he total mater

little less thaexity of the s

F

m long bridg

Quantities of

rial cost is do

Table 4-11:

an twice as exstructure. An

Figure 4-17: An

ge over the f

25

structural mat

oubled, whic

Total cost of b

xpensive as n overview o

n overview of t

fjord Þorskaf

terials for the

ch is a rough

bridge type 2.

the first bridof the arch br

the arch bridg

fjörður in Ice

arch bridge.

estimation.

dge type. Thiridge can be

ge.

eland

is mainly depseen in figur

pends on re 4-17.

4.6 TBridge tpylons steel/conabout thcable-sta

4.6.1 Bridge tcable stconfigur4-20. Thstabilize

Nowadaorthotrop

Many cofor concconstrucconstrucyard. Thweight obecause

A propeHoweveexcept fconcrete

AlthougcomposicomposiMost poHoweveend span

In this conditiothere is transferrand a cocable sy

Des

The cable-type no. 3 ison each sidncrete deck.

he choice of tayed bridges

Aestheticstype no. 3 istayed bridgerations, see fhe pylons caed by the cab

ays the pylonpic or as a co

oncrete cablecrete cable-stction is a fction it is pohe segments,of the segmeit is stiffer a

erly designeder, with incrfor very longe. Thus the re

gh the steel orite deck witite with the ortions of ther, tensile strns. Post-tensi

a self-anchoons on site, th

a so called red to the suombination

ystem can be

ign of a 170

-stayed bs a back and de, similar The cross s

the superstrus are discusse

s of cable-ss a cable stay is to be defigure 4-18. Tan either be

bles that are a

Figure 4-18

ns are most omposite ste

e-stayed bridtayed bridge

further deveossible to use, however, shent is limitedand easier to

d and fabricreasing laborg spans. Theeduced self-w

rthotropic deth a concretesteel girder he girder arresses may oioning is usu

oring systemhe foundationearth anchor

upports at thof self-anchseen on figu

m long bridg

bridge front cable-to the Öres

section of thucture is in thed.

stayed bridyed bridge. Tesigned. TheThe number rigid, work

anchored into

: Configuratio

often madeel/concrete d

dges have bees: cast-in-plelopment of e a more comhould all be d by the tranerect.

cated orthotrr costs, the use of steelweight of the

eck is too expe slab on a by shear stu

re under higoccur in the mually used in

m is preferans will be bered system we ends of thoring and ea

ure 4-19.

ge over the f

26

-stayed bridgsund bridge

he deck is sihe next chap

dges There are see cables canof spans can

k as a cantileo the ground

ons of the cable

e of concretedeck.

een built. In lace construc

the free camplicated crosimilar to av

nsportation c

ropic deck isorthotropic

l in the decke deck slab m

pensive for csteel frame

uds reduces gh compressmiddle portiothese areas t

able, see figelow sea levewhere the ho

he bridge wharth anchorin

fjord Þorskaf

ge, see figuree. The bridgimilar to bridpter where ae

everal types n be arrangen either be twever, or the

d.

es for cable-sta

e. The deck

general, thection or precantilever cooss-section bvoid adjustmcapability. B

s a good soldeck becom

k is, today, twmust result in

construction can be very

the steel quasion, which on of the cento keep the c

gure 4-19. Tel and sedimorizontal comhich requiresng system. T

fjörður in Ice

e 4-21. Thisge will condge type 2. esthetics and

that can comed in a harpwo or three, deck is stif

ayed bridges.

k can be mad

ere are two ccast construconstruction mbecause prec

ment in the pBox is the pr

lution for a mes less comwo to four ti

n appreciable

in most couny competitivantity of theis good for

nter span anconcrete unde

That dependment layers ar

mponents of favorable foThe principl

eland

bridge typesist of a coA further di

d structural s

me into mindp-, fan- or c

see figures ff and the py

de of concre

construction ction. A castmethod. Forcasting is donprecasting forreferred cros

cable-stayedmmercially aimes as expe

e savings.

ntries at this ve. Making e girder signr concrete m

nd at both ener compressi

s on the fore the main sf the cable fofoundation cole of a self-a

e has two omposite iscussion

system of

d when a combined 4-19 and ylons are

ete, steel

methods t-in-place r precast ne in the rms. The s section

d bridge. attractive ensive as

time, the the deck

nificantly. members. nds of the ion.

oundation soil. Also orces are onditions anchored

Let us cobe a gooseen on

But, an a self-an

4.6.2 Based oHere, a hand delithe viewin the pyThe outestays womodelleanalysis two pylbridge dhorizont

4.6.3 In this se

Des

onsider an asod choice witfigure 4-20.

asymmetricanchoring syst

Geometry on the discusharp shape ccate appeara

wing angle. Itylon begin aer spans lengon’t exceed d in SAP200 of structuresons are conndeck to eachtal.

Design ection the ele

ign of a 170

symmetricalth respect to

al system hatem.

for type 3 ssion above configurationance becauset also allows

at a lower elegths should bits limits. Fi

00 for 3D anas. The deck, nected togeth pylon. Eac

ements in the

m long bridg

Figure 4-1

system withfoundation c

Figure 4-2

s earth-anch

self-anchorin of the cablee an array of an earlier stevation so thbe around 30igure 4-21 dalysis. SAP2see figure 4-

ther with onch cable wil

Figure 4-21

e following t

ge over the f

27

19: Self anchor

h only one pyconstruction

20: Asymmetri

ored cables

ing cable syes is chosen.parallel cabl

tart of the dehat fastening 0-40% of theisplays the g

2000 is a com-26, is 10 m

ne cross-beamll be connec

1: Model for br

table will be

fjord Þorskaf

red system.

ylon as can bn. An exampl

ical system.

and requires

ystem is pref Harp shape les will alwa

eck constructof cables ca

e main spangeometry ofmmercial fini

wide with fom for stabilcted to trans

ridge type 3.

checked in t

fjörður in Ice

be seen in figle of that stru

s better found

ferable in thconfiguratio

ays appear pation because an start beforlength so thethe chosen m

ite element pour pylons, twity. Ten cab

sversal beam

the ultimate

eland

gure 4-19. Thuctural system

dation condi

his specific son offers a vearallel irrespthe cable an

re the pylon e stresses in model. The

program for swo on each s

bles will conms with 30°

limit state.

hat could m can be

tion than

situation. ery clean

pective of nchorages

is ready. the back bridge is structural side. The nnect the angle to

To begielementstraffic lois movedisplayeindicate

Des

in with the s of the bridoads are placd in 5 m inc

ed for those the position

ign of a 170

Tab

necessary sge that are u

ced by using crements alon

parts of theof the pylon

m long bridg

ble 4-12: Eleme

section forceunder investithese influenng the bridge bridge thatns.

ge over the f

28

ents that will b

es and reactgation. SAP2nce lines. To

ge deck. On ft will be an

fjord Þorskaf

be checked in U

tions are de2000 is used

o create influfigures 4-22

nalyzed at th

fjörður in Ice

ULS.

etermined fod to create inuence lines a

to 4-24 are his stage. Th

eland

or the corresnfluence linepoint load othese influe

he dark verti

sponding s and the f 100 kN nce lines ical lines


29

Figure 4-22: Influence lines for moments in the deck and at pylon supports.

-1.00-0.90-0.80-0.70-0.60-0.50-0.40-0.30-0.20-0.100.000.100.200.300.400.500.600.700.800.901.00

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

M

x/L

Influence Lines for Moment in the main girders

@Pylon Center

-1.00-0.90-0.80-0.70-0.60-0.50-0.40-0.30-0.20-0.100.000.100.200.300.400.500.600.700.800.901.00

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

M

x/L

Influence Lines for Moment in Pylon Supports

Left Pylon Right Pylon


30

Figure 4-23: Influence lines for normal forces in pylon support and moment at 55% of the height of the pylons.

-1.00

-0.90

-0.80

-0.70

-0.60

-0.50

-0.40

-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

N

x/L

Influence Lines for Normal Force in Pylon Supports


-1.00

-0.90

-0.80

-0.70

-0.60

-0.50

-0.40

-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

M

x/L

Moment in 55% of the Height of the Pylons



31

Figure 4-24: Influence lines for various parts of the structural system of the cable-stayed bridge.

After the influence lines have been created the bridge is modeled as a 3D model in SAP2000 with the forces positioned at the corresponding positions. The slab is modeled as area section elements with a meshing of 0.5 m so that the axle traffic loads can be positioned right. Main girders and cross beams in the bridge deck are modeled as frame elements as well as the pylons. The cables are modeled as cable elements with high tensional stiffness. The only supports of the model are the fixed supports under the pylons because the pylons and cables should be able to carry its self-weight under construction.

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

N

x/L

Axial Force in cables - Back Stays

Cable 1 Cable 2 Cable 3

Cable 4 Cable 5

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

N

x/L

Axial Force in cables - Front Stays

Cable 1 Cable 2 Cable 3

Cable 4 Cable 5

4.6.4

ConcreThe conconcreteshear stu

Cross BThe crodesignedloads for

When ththe crossof the flsee appe

Main GLargest span andreached is chosenthe widtcalculati

Des

Deck

te Slab ncrete slab we casted on suds will be w

Beams ss beams wd to resist thr the largest

Figu

he largest mos beams. Theanges is 300

endix A.

irder moment in td is determinwhen the tran. From thesth of the flanion, see appe

ign of a 170

will be 250 mite and a me

welded on cro

will be made he self-weighmoment can

ure 4-25: Posit

oment and she cross sectio

0 mm and the

the main girned to be 73affic loads arse design valnges is 400 mendix A. Figu

m long bridg

mm thick andetal deck benoss beams an

of steel andht of the co

n be seen in f

ion of the traff

hear force areon of the beae thickness o

rders is when11 kNm. Thre applied wlues the size mm and the ure 4-26 disp

ge over the f

32

d the concretneath of trapend main girde

d are placedncrete slab

figure 4-25.

fic loads to esti

e determinedam is I shapeof the web an

n the traffic he largest she

where the pyloof the girderthickness of

plays the cho

fjord Þorskaf

te quality is ezoidal profilers.

d in 5 m intand the traff

imate the size

d in ULS, it’sed with the tond flanges is

loads are apear force is dons are positr is determinf the web anosen bridge d

fjörður in Ice

C35/45. Theles. To achie

tervals alongffic loads. Lo

of the cross be

s possible to otal height of30 mm. For

pplied at the determined ttioned. An I-ned. The totand flanges is deck.

eland

e slab will ceve composit

g the deck. Tocation of th

eams.

determine thf 860 mm. Tr detailed cal

middle of thto be 1247 k-shaped crosal height is 130 mm. For

consist of te effects

They are he traffic

he size of The width lculation,

he bridge kN and is ss section 200 mm, r detailed

4.6.5 The pylC40/50. beams. normal determinpylons iinteractidesign pappendix

The cho

Des

Figure 4-26

Pylons lons will be

The towersEach pylon force were dned 1.5 x 2.0is determineion diagram tpoint for thex A.

sen cross-sec

N(k

N)

ign of a 170

6: Configuratio

a concrete s have two v

is designed determined t0 m with a wd to be 1.2 to estimate the above men

Figu

ction of the p

-30,000

-20,000

-10,000

0

10,000

20,000

30,000

40,000

50,000

60,000

70,000

0

N (k

N)

m long bridg

on of the deck

hollow sectivertical cable

for combinto be 27292 wall thicknes

x 2.0 m withe capacity ontioned sect

re 4-27: N-M i

pylon is show

5,000 10,

N

ge over the f

33

and girders w

ion. The cone planes andned moment

kNm and 5ss as 0.3 m. Tth a wall thiof the pylon.tion forces.

interaction dia

wn in figure

,000 15,000 2

M (kN

N-M Diagram

fjord Þorskaf

with shear studs

ncrete qualitd are connec

and normal5121 kN respThe size of tickness of 0 The star on Calculations

agram for the p

4-28.

20,000 25,000

Nm)

m

fjörður in Ice

s and trapezoi

ty in the pyted togetherl force. The pectively. Ththe cross-bea0.25 m. Figu

the inside ofs of the pylo

pylon.

0 30,000 35,0

eland

dal profiles.

lons is chosr with two tr

design momhe size of a ams that conure 4-27 disf the curve sons are disp

000

sen to be ransverse ment and

pylon is nnects the splays an hows the

played in

4.6.5.1 Where tmain girtension f2000 wi

Finally a

4.6.6 The founbridge tykN and pylon. T

4.6.7 The founpylon wthe deckconstrucmain girthe beammain girof the sp

Des

Cables the cables arrders. The cforce in the th tendon un

a model of th

Foundationdation undeypes. The lar27292 kNm

The fundamen

Constructindations are

will be construk can start wcted further urders and croms. Finally thrders weldedpan. This wa

ign of a 170

Figure 4

re connectedcables are mcables was d

nits as 6-61 w

he bridge tha

Figure 4-2

n er each pylonrgest vertical

m respectivelnt along with

ion to be constr

ucted up to awhere the decup. During toss-beams arhe slab is casd togeather uay, the struct

m long bridg

4-28: Cross sec

d to the bridgmodeled as cdetermined towith a design

at was constru

29: 3-D model

n will consisl reaction forly. That imph the piles is

ructed first. Aa height wheck is connecthis the crosre in place thsted. This pr

until the top rture works a

ge over the f

34

ction of the pyl

ge deck croscable elemeno be 15888 k

n capacity of

ucted in SAP

of the cable-st

st of concretrce and momplies that app

assumed to

After that, thre the lowestcted to the cs-beams arehe trapezoidaocess is donerow is reacheas a self-anch

fjord Þorskaf

lon with reinfo

ss beams arents with highkN. A propo17019 kN.

P2000 is disp

tayed bridge in

te footings anment under on

proximatelyresist the ris

he work witht cable is con

cables in seg also set in al profiles are for each caed and the dehored system

fjörður in Ice

orcement.

e placed to rh tensional sosal of a cabl

played in figu

n SAP2000.

nd cohesive ne pylon is d

13 piles arek for overtur

h the pylons nnected. The

gments meanplace. After

re fastened oable row witheck structure

m with the de

eland

reduce torsiostiffness. Thle system is V

ure 4-29.

piles as for determined toe needed unrning.

can start when the construnwhile the pyr the segmenon the upper h the segmene meets in theck hanging

on in the he largest VSL SSI

the other o be 5121 nder each

here each ruction of ylons are nts of the

edges of nts of the he middle

from the

cables ostayed b

Figure

4.6.8 The totaprice vasubstruccalculatibridge ty

Finally t

Des

on each side bridge in the w

4-30: The Ston

Cost estimal quantities alues given icture are lisions but the ype one.

T

the total cost

ign of a 170

of the pylonworld that di

necutters Brid

mation/concof materials in section 4.sted for the

cost estimat

Table 4-13: Am

t of this bridg

m long bridg

ns. Below, fisplays how

dge (currently t

clusions are summar3. In table 4 cost estimtion for the

mounts of struc

ge type is sum

Table 4-14:

ge over the f

35

figure 4-30, ithis principl

the largest cab

rized in table4-14 the mai

mation. Thesfoundations

ctural materia

mmarized in

Total cost of b

fjord Þorskaf

is an exampe works.

ble-stayed brid

e 4-13 and coin materials e quantitiesis based on

als for the cable

n the table 4-

bridge type 3.

fjörður in Ice

le from one

ge in the world

ost estimatioand quantiti are based

n the method

e-stayed bridg

14.

eland

of the large

d) under const

on made baseies of the su

on the pred in section

e.

est cable-

truction.

ed on the uper- and eliminary 4.4.2 for

This bridependsthere is qalso a la

Des

idge type is s on the samequite much q

arge factor. A

ign of a 170

little less the factors as fquantity of co

An overview

Figure 4-31

m long bridg

han three timfor bridge tyoncrete and rof the arch b

: An overview

ge over the f

36

mes as expeype 2, steel creinforcemenbridge can be

of my proposa

fjord Þorskaf

ensive as thcost and comnt that is usee seen in figu

al of a cable-st

fjörður in Ice

he first bridgmplexity of thed in the pyloure 4-31.

tayed bridge.

eland

ge type. Thihe structure. ons and the c

s mainly But also

cables are


37

4.7 Summary and choice of bridge type From the total cost estimations for the bridge types it is clear that the cable-stayed bridge is the most expensive one. Also the arch bridge is quite expensive compared to the prestressed concrete bridge that is the least expensive one. From a construction point of view the prestressed bridge is also the most favorable. From these perspectives a concrete girder bridge is the obvious choice.

But, there are also other aspects that need to be taken into consideration when choosing a bridge type; aesthetics, method of construction and construction time are obvious factors that can affect which choice is made. The author will leave those decisions for others to make at later stages but chooses to design the concrete beam bridge in a more detailed manner. In the following chapter more detailed calculations will be performed for the superstructure of bridge type 1. Calculations of the post-tensioned cables are performed where the prestress force and eccentricity of the cable profile are determined. Following that all cable losses are determined and then the secondary effects of prestress. Finally the ultimate moment capacity is determined for relevant members of the superstructure.


38

5 Final design

5.1 Introduction Prestressed concrete structures, using high-strength materials to improve serviceability and durability, are an attractive alternative for long-span bridges, and have been used worldwide since the 1950s. The presence of cracks that can develop in tensile members can lead to corrosion of the reinforcement due to its exposure to water and chemical contaminants. Corrosion is generally only a problem for structures in aggressive exterior environments (bridges, marine structures, etc.) and is not critical in the majority of buildings. The effect of cracking of members can lead to substantial loss in stiffness which occurs after cracking and the second moment of area of the cracked section is far less than the second moment of area before cracking. Thus, allowing cracks to develop can cause a large increase in the deformation of the member. For prestressed concrete, compressive stresses are introduced into a member to reduce or nullify the tensile stresses which result from bending due to the applied loads. The compressive stresses are generated in a member by tensioned steel anchored at the ends of the members and/or bonded to the concrete.

There are two types of prestressing systems: pre-tensioning and post-tensioning systems. Pre-tensioning systems are methods in which the strands are tensioned before the concrete is placed. This method is generally used for mass production of prefabricated members. Post-tensioning systems are methods in which the tendons are tensioned after concrete has reached a specified strength. This technique is often used in projects with very large elements. The main advantage of post-tensioning is its ability to post-tension cast-in-place members. Mechanical prestressing jacking is the most common method used in bridge structures.

The post-tensioning process involves three fundamental stages. In the first stage of the process, the concrete is cast around a hollow duct. After the concrete has set or hardened, a tendon, consisting of a number of strands, is pushed through the duct (alternatively, the tendon can be placed in the duct before casting). Thus, the tendon can be fixed in any desired linear or curved profile along the member. By varying the eccentricity of the tendon from the centroid, the maximum effectiveness of a constant prestressing force can be utilized by applying the prestress only where it is required. Once the concrete has achieved sufficient strength in compression, the tendon is jacked from one or both ends using hydraulic jacks, thus putting the concrete into compression. When the required level of prestress is achieved, the tendon is anchored at the ends of the member. After anchorage, the ducts are usually filled with grout under pressure. The grout is provided mainly to prevent corrosion of the tendon but it also forms a bond between the tendon and the concrete which reduces the dependence of the beam on the integrity of the anchor and hence improves its robustness.

When prestressed concrete elements are designed the following factors need to be considered:

• The prestressing reinforcement is determined by concrete stress limits under service load. • Bending and shear capacities are determined for the ultimate limit state • Deformations are determined in the serviceability limit state.

5.2 DThroughloads arthe girdemm.

5.2.1 The brid

• •

5.2.2 Self-weithe load

5.2.3 In this se

5.2.3.1 High strconcreteconcrete

The effedefinitioconstantEcm, is re

where φ

In this cconcrete

Des

Design hout the desie calculated ers is change

Building cdge shall be d

FS ENV 199FS ENV 199

Loading ight and trafgenerated by

Materials ection the m

Concreterength concre quality is e compressio

fects of creeon of creep t stress. To taeduced to an

is the creep

case, there ise is loaded is

ign of a 170

ign process, with respect

ed from what

odes designed acc

91 92

ffic loads arey prestressin

ost common

e rete is alwayC45/55. For

on strength is

ep need to bis that undeake account

n effective ela

coefficient.

Table 5-

s a humid at 28 days. Th

m long bridg

one girder it to that. Aftt was chosen

cording to the

Eurocode 1 Eurocode 2

e the same asng is taken in

n physical pro

ys used in por this qualitys fck=45 MPa

be taken inter compressifor creep in tastic modulu

φ is taken fr

-1: Creep coeff

tmospheric che circumfere

ge over the f

39

is designed wfter iteration n in chapter 4

e following s

(Loading) (Concrete D

s in the pre-dnto account.

operties of al

ost-tensionedy the modul. The self-we

to account fon the concthe design of

us, Ec,eff. Ec,eff

, = 1 +rom EC2 and

ficient, φ, for n

condition at tence of the se= 27700

fjord Þorskaf

with propertiof the calcu

4.4 with a he

standards:

Design)

design phase

ll materials u

d structural mlus of elastieight of reinf

for calculatiorete membef concrete mf is determine

d shown in ta

normal weight

the bridge siection, u, is:

fjörður in Ice

ies of half thlations in th

eight of 1800

e. But in this

used are sum

members. In city is Ecm=forced concr

on of long-tr will contra

members the med with the f

able 5-1.

concrete.

ite of 80% a

eland

he cross-sectis chapter th

0 mm instead

s detailed de

mmarized.

this case th=36000 MParete is 25 kN/

term deflectact with timmodulus of efollowing for

and the age w

tion. The he size of d of 2400

sign also

he chosen a and the /m3.

tion. The me due to elasticity, rmula:

when the


40

The cross sectional area of the concrete is: = 6460000

The notional size is calculated to be: 2 = 466

which gives after interpolation the creep coefficient as φ =1.56, see table 5-1, and the effective modulus of elasticity is determined to Ec,eff=14066 MPa.

5.2.3.2 Reinforcement The quality of the reinforcement is chosen to be B500B with characteristic yield strength fyk=500 MPa.

5.2.3.3 Prestress system As mentioned in chapter 4 for bridge type 1, the prestressing system is VSL 6-31. All technical information concerning VSL systems are acquired from technical brochures published by VSL International Ltd. (2010). Each tendon consists of 28 strands each consisting of 7 wires with a diameter of 15.7 mm with a total nominal cross-section Ap=4200 mm2, where each strand has a nominal cross section of 150 mm2. The steel quality of the wires is fp0,1k/fpk=1640/1860 MPa. The breaking load of each tendon is 7812 kN. The cables will be placed before casting in a group around a centerline that counteracts the moment from the self-weight of the bridge. The cables will be anchored individually with conical devices at each end of the bridge. When the concrete has achieved sufficient strength large multi-cable hydraulic jacks are used at both ends of the bridge to prestress the structure. When the required level of prestress is achieved the tendons are anchored at each end of the member. After anchorage, the ducts around the tendons are filled with cement grout under pressure, called bonded tendons. The cement grout is provided mainly to prevent corrosion of the tendons, but also it forms bond in the integrity of the anchor and hence improves its robustness. The prestressing process will be done in that manner that first the two internal spans will be constructed and prestressed and then finally the two external spans will be constructed and prestressed. To simplify the calculations it is assumed that the cables are calculated as one element, stressed from both ends of the bridge.

5.2.4 Exposure classes and service life According to Eurocode 2 (EN 1992-1-1) a structure should be classified after environmental conditions, chemical and physical. This structure will be classified in the following classes:

• Corrosion induced by carbonation: XC4 • Corrosion induced by chlorides: XD3 • Corrosion induced by chlorides from sea water: XS3 • Freeze/Thaw Attack: XF4

These classifications give a structural class of S4 according to table 4.3N in EC2. For that class the minimum concrete cover for reinforcement steel is cmin,dur=45 mm and for prestressing steel cmin,dur=55 mm. Also, for post-tensioned members, the concrete cover should not be less than the diameter of the duct. In this case the external diameter of the duct is 117 mm. According to EC0, table 2.1, the service life (indicative design working life) for bridges is 100 years.

5.2.5 Tendon alignment and prestress force The position of the centroid of the tendons should be chosen to give the highest effective depth. The alignment is based on concrete cover, the number and size of the tendons, the size of the cable ducts

and stresection afiber at s

These vgiven inmm. Macenter odecided displayssuperstru

At each section)different

Immediaas the sm

At the sacceptabsuggests

with fck(fck(t) is d

Des

sses. Roughlare chosen asupports. Th

values will bn section 8.10aximum momof the interna

with respecs the momenucture will b

end of the . Eurocode 2t times. The

ately after thmaller of:

same time, eble value deps that an acce

(t) as the chardefined in sec

ign of a 170

ly, a good mas 0.17h frome total heigh

be used in th0.1.3 in EC2ments from al bays. Thest to minimum

nt curve frombe taken into

Figure 5-1

bridge the t2, part 1.1, smaximum al

≤he cables are

≤excessive copends on theeptable comp

racteristic coction 3.1.2 in( )

m long bridg

moment distrim the bottomht of the cross0.170.1he beginning, and shouldself-weight ise values wim concrete c

m self-weightaccount.

1: Moment dist

tendons are section 5.10.2llowed pre-te

≤ 0.8 0.9 .released fro

0.75 0.85 .ompressive ae length of tipressive stres

ompression sn Eurocde 2 = ( ) −( ) =

ge over the f

41

ibution is obm extreme fibs-section, h, 7ℎ = 348.512ℎ = 246

g of the calcnot be less t

in external bill be checkecover and mt. In the pre

tribution in the

placed at th2, is used toensioning str= 0.8 ∙ 1860= 0.9 ∙ 164

om the jacks

= 0.75 ∙ 186= 0.85 ∙ 16and tensile sime during wss is: ≤ 0.60 strength as a and is found8 ( )

for t

fjord Þorskaf

btained if theber at bay anis 2050 mm.

ulations. Mithan the diambays occur aed later when

minimum diststressing sta

e girder from s

he centroids o determine tress during te0 = 1488 40 = 1476 the maximum

60 = 1395 640 = 1394stresses muswhich the co

( )

function of td with the fol

for 3 < t <

≥ 28 days

fjörður in Ice

e positions ond as 0.12h f. Hence,

inimum spacmeter of the t 0.375L or n the amountances betwe

age only the

self-weight.

of the T-secthe maximumensioning is

m stresses in

st not arise oncrete has h

time of the pllowing meth

28 days

eland

f cables in thfrom the top

cing of the cduct, in this 13.875 m an

nt of cables een ducts. Fiself-weight

ction (half thm allowed stthe smaller o

n the cables a

in the concrhardened. Eu

prestressing ohod:

he cross-p extreme

cables, is case 117 nd in the has been igure 5-1 from the

he cross-tresses at of:

are given

rete. The urocode 2

operation.


42

fcm(t) is estimated from the following expressions: ( ) = ( )

With βcc(t) as:

( ) = exp 1 − 28 = exp 0.25 1 − 2810 = 0.85

where s is chosen to 0.25 which is valid for cement of strength classes CEM Class N at 10 days and fcm is 53 MPa. From this fcm(t) is determined to 44.79 MPa and fck(t) to 36.79 MPa. Hence, the compressive strength at transfer becomes: ≤ 0.6 ( ) = 0.6 ∙ 36.79 = 22.07

And the concrete strength at service time is: ≤ 0.6 = 0.6 ∙ 45 = 27

EC2 does not lay down any compulsory permissible tension stresses so the choice of concrete tension stress limits is left to the discretion of the designer. Hence the design is restricted by not allowing high tensile stresses to develop at service and only the concrete tensile strength at transfer:

. , ≤ 2.7

And at service the maximum tensile stress is chosen to: ≤ 2.0

The stresses in the cross-section are calculated according to Navier’s formula:

= − + −

where P is the normal force, A is the area of the cross-section and I is the moment of inertia. Mg is the moment generated by self-weight, es is the eccentricity of the normal force and y is the location in the section where the stresses are calculated. In this equation P and es are unknown and have to be determined. The prestress force and eccentricity will be determined by developing a Magnel diagram, a method to determine a Magnel diagram is descriped in O’Brien (1999). Magnel diagrams are determined for the critical sections where the maximum transfer- and service moments occur. An estimation of the ratio between the prestress force at service and the prestress force at transfer, ρ, is made (generally from 0.75-0.90) and is chosen here to be 0.75. The critical sections that will be checked are external and internal spans and supports B and C. Figure 5-2 displays the load arrangements to establish the largest moments at each section at service stage. These load arrangements are determined based on influence lines that are created by the same method as in section 4 and can be seen in appendix B.

Figu

These mdevelope

Table 5

Below isfollow.

σtmin: min

σtmax: ma

σbmin: mi

σbmax: ma

These st

Des

ure 5-2: Mome

moments are ed for each s

5-2: Maximum

s a list that d

nimum stres

aximum stres

inimum stres

aximum stre

tresses are de

ign of a 170

ent distribution

calculated isection and th

m moments at e

defines neces

s at extreme

ss at extreme

ss at extreme

ss at extreme

etermined ac

=

m long bridg

ns and load arr

in the compuhe moments

each section wh

ssary notatio

top fibre.

e top fibre.

e bottom fibr

e bottom fibr

ccording to:

ge over the f

43

rangement for

uter programcan be seen

here M0 is the m

n, sign conv

e.

re.

−

fjord Þorskaf

r moments at se

m PCFRAMEin table 5-2.

moment at tra

ventions and

1

fjörður in Ice

ervice for the c

E after influe

ansfer and Ms t

formulas for

−

eland

calculated sect

ence lines h

the moment at

r the calculat

tions.

ave been

service.

tions that

where Wpositive centroidMS at seare show

ρomin: mi

ρomax: ma

ρSmin: mi

ρSmax: ma

In table

And fina

Note thnegative

The follthe Mag

Des

W is the sectivalue, A is

d is positive aervice), see twn in append

inimum perm

aximum perm

inimum perm

aximum perm

5-3 the num

ally the resul

at for these e.

lowing four gnel diagrams

ign of a 170

===

on modulus area of the

and below neable 5-2. Sag

dix B. The no

missible stres

missible stre

missible stres

missible stre

erical values

lts for these t

Table

calculations

inequalities s: 1

m long bridg

(I/y). Wb is a

e cross-sectioegative) and g moments aotation for pe

ss at transfer.

ss at transfer

ss at service.

ss at service

s for these pe

Table 5-

top and botto

5-4: Maximum

s compressiv

are used to

1 ≤ 1

ge over the f

44

−−

−at bottom anon, e is the M are applieare positive ermissible str

.

r.

.

ermissible str

-3: Permissible

om stresses a

m stresses at to

ve stresses

determine th

+

fjord Þorskaf

1 1

1nd is a negati

eccentricity ed moments and hog momresses in thes

resses are giv

e stresses.

are listed in t

op and bottom

are consider

he feasible z

fjörður in Ice

−−

−ive value and

of prestress(M0 is the mments are nese equations

ven.

table 5-4.

fibers.

red positive

zone for the p

1

eland

d Wt is at tops tendons (aoment at tranegative. Thesis the follow

and tensile

prestressing

p and is a above the nsfer and se values

wing:

e stresses

force on


45

1 + ≤ 1 2 1 ≤ 1 + 3 1 + ≤ 1 4

These inequalities represent half-planes bounded by the line on which the stress limits are just satisfied. To determine which half-plane represents the inequality, the origin of the Magnel diagram is substituted into inequality 1, 1/P=0 and e=0, and gives the following:

0 ≤ 1 1

If this is true (A is positive), the correct half-plane is the one containing the origin and the same procedure is done for the three remaining inequalities. 1 ≤ 0 2

0 ≤ 1 3 1 ≤ 0 4

Thus, for inequality 1 the half-plane contains the origin, for inequality 2 it doesn’t, for inequality 3 the half-plane contains the origin and for inequality 4 it doesn’t. For further information see appendix B and O’Brien (1999).

After some calculation the Magnel diagrams are established (Appendix B) and the appropriate prestress force and eccentricity are chosen. The chosen prestress force is 46.512 kN (1/P=2.15 x 10-8 N-1) which is based on the maximum allowable eccentricity that is at supports B and C. That requires approximately 8 tendons each with a breaking load of 7812 kN. The eccentricity limits at each section are given below, based on these Magnel diagrams.

Section at support A: -180 mm ≤ e ≤ 283.6 mm

Section at span 1: -210 mm ≤ e ≤ -410 mm

Section at support B: 200 mm ≤ e ≤ 283.6 mm

Section at span 2: -300 mm ≤ e ≤ -420 mm

Section at support C: 250 mm ≤ e ≤ 283.6 mm

On figure 5-3 is the longitudinal feasible zone and the cable layout displayed. That zone is based on the calculations above.

Fig

On figur

5.2.6 The effethe lengtransfer losses coccur prand lossThe pre-

• •

The pos

•

•

Des

gure 5-3: Longi

re 5-4 the ch

Prestress ective prestregth of the m

and servicean be dividerior to the poses which occ-transfer loss

Elastic shorFriction betw

st-transfer lo

Time dependthe concreteLoss due to tension).

ign of a 170

itudinal feasib

osen eccentr

Figu

losses ess force is n

member. There, the losses ed into two oint in time cur after the ses consist of

rtening of theween the ten

sses consist

dent losses, . slippage of

m long bridg

ble zone of the c

ricities are di

ure 5-4: Chose

not generally refore, in orin prestress

groups in acwhen stress prestress is

f:

e cross-sectiondons and the

of:

which are fr

the tendons

ge over the f

46

cable (the beam

isplayed.

n eccentricitie

equal to therder to deters must first ccordance wis first felt btransferred t

on. e surroundin

rom relaxatio

at the ancho

fjord Þorskaf

m size has the

s for the cable

e applied jackrmine the effbe calculate

with the time by the concrto the concre

g ducts (in c

on of the ste

orage known

fjörður in Ice

scale of 10 in v

line.

king force noffective stresed at each d

when they rete are calleete are called

ase of post-t

el and by cr

n as draw-in

eland

vertical directi

or is it constas due to pre

design sectiooccur. Losse

ed pre-transfd post-transfe

tension).

reep and shri

loss (in case

ion).

ant along estress at n. These es which fer losses er losses.

inkage of

e of post-

5.2.6.1 The loss

• •

The frict

where μ section friction used:

where yhorizont

The frict

k is a wosupportssystem tto sectio

The resu

Des

Friction ls of prestress

Friction dueFriction due

tion loss at s

is a friction i and n is tcoefficient a

yi is the vertal distance b

Figure 5-5:

tion loss at s

obble coeffics, the degree the produceron number i.

ults of the ca

ign of a 170

losses s force from f

to curvature to unintentio

section i due

coefficient athe total numas μ=0.2. To

rtical changebetween the c

Sections of the

section i due

cient which of vibration

r recommendHence, the t lculated frict

m long bridg

friction is ca

e of the tendoonal variatio

to curvature and θi is the amber of sect

calculate th

e in height correspondin

e beam for calc

to wobble is depends on t

n used in placds k=0.0008 otal friction tion losses ar

ge over the f

47

aused by two

on on of the duct

is: = (aggregate chtions. The sye angle chan

= 2( )between the

ng sections. T

culations of fri

s: = (1the quality ocing the concm-1. The termloss become= (1re displayed

fjord Þorskaf

sources:

t from its pre

(1 − ∑hange in slopystem produ

nges at each

e sections thThese section

iction losses an

− ∑of workmanscrete and them xi is the d

es: 1 − ∑in table 5-5.

fjörður in Ice

escribed prof

)

pe in radians ucer recommsection the f

hat are calcns are display

nd elastic short

)

ship, the diste type of the distance (in m

( ))

eland

file or ‘wobb

between the mends a valufollowing eq

culated and yed on figur

tening losses.

ance betweeduct. For th

meters) from

ble’.

jack and ue of the quation is

xi is the e 5-5.

en tendon he chosen

m the jack

5.2.6.2 As the pcause a shortenintendon ttendon t

where ineccentricshortenin

5.2.6.3 After jaanchoragthe anch

Des

Elastic shprestress is tslackening ong in each that is releasthat is release

ndex 1 and city respecting loss for th

Draw-in lcking the teges to the co

hors. The ´dr

ign of a 170

hortening ltransferred tof the strandtendon of p

sed first and ed first is:

ℎ2 indicate t

ively. Ag is he whole cab

losses endons are reoncrete. Thisraw-in´ of the

m long bridg

Table

losses to the concrd which resupost-tensione

then less in

endon 1 andthe area of

ble group in o

Table 5-6:

eleased from process resue tendons res

ge over the f

48

e 5-5: Friction

ete, the conults in a lossed members n the next on

d 2 respectivf the cross-seone girder ar

Elastic shorte

m the jacks aults in a loss sults in a los

fjord Þorskaf

losses.

crete undergs of prestress

are differenne and etc. T

1 =vely. P and ection. The re displayed

ening losses.

and the forcdue to slipp

ss of strain o

fjörður in Ice

goes elastic s force. The nt. The maxThe elastic s

1 +e are the prresults of tin table 5-6.

e is applied page or ‘drawf εjack at the a

eland

shortening. losses due t

ximum loss shortening lo

restressing fthe calculate

directly thrw-in’ of the sanchorage. H

This can to elastic is in the

oss in the

force and ed elastic

ough the strands at However,


49

because of friction, this loss decreases along the length of the member to zero at a distance Ld from the jack. The extent of draw-in losses is determined with the following equation:

= ∆( − )/

where (Pjack-PL)/L is the slope of the distribution of prestress force if the friction loss is assumed to vary linearly. Δs is the total shortening of the tendon. The magnitude of the draw-in loss at the anchorage is then given by the following equation:

∆ = 2 ( − )

where ΔP is the draw-in loss. The distribution of the prestress force after the draw-in loss is displayed in figure 5-6 where the decrease of this loss along the length of the member becomes zero at distance Ld approximately 15 m from the jack.

Figure 5-6: Distribution of the prestress force after the draw-in loss for half of the total span.

5.2.6.4 Time-dependent losses Losses in prestress which occur gradually over time are caused by shortening of the concrete due to creep and shrinkage and due to relaxation of the prestressing steel. Shrinkage is a time-dependent strain which occurs as the concrete sets and for a period after setting. The shrinkage strain approaches final value at infinite time. When maintained at a constant tensile strain, steel gradually loses its stress with time due to relaxation. The extent of the loss of stress in prestressing strands due to relaxation is determined by the stress to which the steel is tensioned, the ambient temperature and the class of steel. Maximum allowable value for relaxation losses of the prestressing steel are specified by the manufacturer as 2.5% for 15 mm strands. To determine the loss EC2 suggests that it should be based on the 1000 hour values given in figure 5-7. These values should be trebled to determine the final relaxation losses.

0, 46512

85, 40830

0, 4327024, 44891

40000

45000

50000

0 20 40 60 80

P [k

N]

x [m]

Draw-in loss

Prestress dueto friction loss

Draw-in loss

The pherelaxatiooccurs athe conc

EC2 recfollowin

where

The finaautogenoauthor r

Des

enomenon oon refers to tat constant stcrete.

commends thng formula:

Δσp,tot = loss

εcs = final shof this strain

Ep = modulu

Δσp,r = loss o

m = modular

φ = final cre

σc = stress in

Ap = area of

Ag, Ig = gros

e = eccentric

al shrinkage ous shrinkagefers to sect

ign of a 170

F

of creep is the loss of sttress. For pre

hat these two

∆of stress in s

hrinkage stran is εcs=0.000

us of elasticit

of stress in th

r ratio, Ep/Ec

eep coefficien

n the concret

prestressing

s area and se

city of tendo

strain is comge strain, εca. tion 3.1.4 (6)

m long bridg

Figure 5-7: Re

essentially ttress under cestressed con

o prestress lo

, = 1 +steel due to c

ain calculated0256.

ty of the pres

he tendons at

c.

nt taken from

te at the level

g steel.

econd mome

ns from cent

mposed of tw For detailed) in EC2. Th

ge over the f

50

elaxation of pre

the same asconstant straincrete, relaxa

osses, creep

+ ∆1 +creep, shrink

d according

stressing stee

t the design s

m table 5-1.

l of the tendo

nt of area of

troid section.

wo componed explanationhe loss of str

fjord Þorskaf

estressing steel

s that of relin while creeation occurs

and shrinkag

, (1 + 0.kage and rela

to section 3.

el.

section due to

ons due to pe

f section.

.

ents, the dryns of calcularess in the te

fjörður in Ice

l.

laxation. Thep is the incin the steel w

ge, should b

.8 )

xation.

.1.4 (6) in E

o relaxation.

ermanent loa

ying shrinkagations of the endons due t

eland

he distinctionrease of strawhile creep o

e calculated

C2. Calculat

ads plus prest

ge strain, εcd

final shrinkato relaxation

n is that ain which occurs in

with the

ted value

tress.

, and the age strain , Δσp.r, is

derived the secti

and the l

The strecalculate

FollowinFrom ththen be c

5.2.7 For statmomenteccentricduring pdisplaceThe equparaboliM1=Pe. maximu

Since thparaboli

Applicat

Des

from figure ion that is ca

loss of stress

ess, σc, in thed using the

ng these calcese results acalculated. T

Secondaryically indetet and secondcity, the secoprestressing. ement methoduivalent load ic tendon proIf an equiva

um moment a

he equation ic, the mome

tion of this e

ign of a 170

5-7 with thelculated as:

s in the tendo

he concrete following eq

culations theand from the The results of

y effects oferminate strudary momenondary mom

To determid and equivais a uniform

ofile. The bealent uniformat mid-span i

of the bendents must be

equation at th

m long bridg

e ratio of stee

ons due to re

∆ , = 3at the leve

quation:

=e total time-dcalculations

f the calculat

Table 5-7

f prestressuctures the pnt. The priment is momene the secon

alent load memly distributeending momem upward loais equal to:

ding momenequal in the

he mid-span g

ge over the f

51

el stress/char

= +laxation as:

el of tendon

+ ( +dependent los of the drawted time-dep

: Time-depend

prestress mo

mary momennt caused byndary momeethod. In thised load estabent at any poad, w, is ass

= 8nt for unifortwo beams a=gives

fjord Þorskaf

racteristic ten

ns due to pe

+ )

oss of stress w-in losses th

endent losse

dent losses.

oment consisnt is the pry reactions deent there ares thesis the eblished by coint due to thsumed to act

rmly loaded at any point.

fjörður in Ice

nsile strength

ermanent loa

in the tendonhe applied pres are display

sts of two coduct of preveloped at ie two commquivalent loaonsidering ahis prestresson the beam

simply supThus

eland

h as (σp/fpk) w

ads plus pre

ns can be caestress at ser

yed in table 5

components, restressing fointermediate on methods;ad method isa simple beaming force is

m, see figure

pported beam

with σp at

estress is

alculated. rvice can 5-7.

primary force and

supports ; support s applied. m with a given by

e 5-8, the

m is also

and

where e

In the cae is the cable, se

Thus, th

To deteris used.

• • • •

Des

is the distan

ase when theeccentricity

ee figure 5-9

he eccentricit

rmine the moThe calculat

Draw a free Label the spUse the threMove one sp

ign of a 170

nce from the c

Figure 5-8:

e vertical posat mid span

.

Figure 5-9: D

ty becomes:

oment at the tion procedur

body diagrampans L1 and Le-moment eqpan further a

m long bridg

cable to the e

Calculation of

sition of the cn from the c

Determination

intermediatere of the thre

m of the firsL2 and the supquation to so

and repeat the

ge over the f

52

= 8= 8

eccentricity o

f the equivalen

cables at supcable to the

n of e for cable

= + +2e supports theee-moment e

t two spans.pports A, B a

olve the unkne procedure a

fjord Þorskaf

of the beam

nt load for curv

pports is not tline that con

with eccentric

e three-momquation is in

and C. nown momenabove.

fjörður in Ice

at mid-span.

ved tendons.

the same at tnnects these

city e1 and e2.

ment equationn the followin

nts.

eland

.

two adjacent two position

n (Clapeyronng order:

supports ns of the

n´s theory)

•

•

In this cMA, MB a

The nota

The termwith uni

In table momentMore de

Des

Repeat as nequations. Solve 3 simu

case these moand MC. The

ation for the

ms on the rigiformly distri

5-8 the calct diagrams foetailed calcul

ign of a 170

needed, alwa

ultaneous eq

oment equatiey are the fol+++formulas abo

Figure 5-10: N

ght hand side ibuted load o

culated valueor the corresplations to obt

m long bridg

ays moving o

quations for 4

ions are threlowing: 2 ( +2 ( +2 ( +ove is displa

Notation for ca

of these equon span no. n

es for w andponding momtain the secon

Table 5

ge over the f

53

one span to

4 spans to get

e since there

) +) +) +ayed on figur

lculations with

uations are acn:

= 24d the prestresments. The sndary mome

5-8: Prestress m

fjord Þorskaf

the right an

t the internal

e are four spa

= −6(= −6(= −6(re 5-10.

h the three-mo

cquired with

ss momentsecondary mo

ent are in app

moment.

fjörður in Ice

nd writing a

l moments.

ans and three

+ ) + ) + )

ment equation

h the followin

are listed anoment is linependix B.

eland

new set of

e unknown m

n.

ng formula fo

nd on figure ear between s

f moment

moments,

for beams

5-11 the supports.

When thcalculate

Des

he total moed. That is do

ign of a 170

Figure 5

oment has bone in the ne

m long bridg

5-11: Diagrams

been obtaineext chapter.

ge over the f

54

s of primary an

ed in the be

fjord Þorskaf

nd secondary m

eam, the ult

fjörður in Ice

moments.

timate mom

eland

ment capacityy can be


55

5.3 Ultimate moment capacity When the member has been designed to satisfy the transfer and service stress limits it is necessary to check for moment capacity in the ultimate limit state. As for ordinary reinforced concrete, the ultimate moment capacity of a prestressed section is calculated by equilibrium of forces at the corresponding section. The compressive strain in the concrete due to prestress, εce, at the level of the prestressing tendon is given with the following formula:

= 1 +

where Mp is the moment due to prestress and P is the prestress force, both after all losses, at the section that is calculated. The tendon strain is defined as (contraction positive):

= − + −

where Ap is the area of the prestress reinforcement and Ep is its modulus of elasticity. εct is the total ultimate strain in the concrete and is found with the following equation:

= − ( − )

EC2 recommends that εult should be taken as 0.0035. Here d is the effective depth, distance from extreme fibre in compression to the center of tension reinforcement, and x is the distance from extreme fibre in compression to neutral axis. To calculate the equilibrium of forces the force in the steel, Fp, and the compressive force acting on the concrete in compression, Fc, are required: = ( )

= 0.8

Here b is the width of the section which Fc is acting on. α is a coefficient which takes into account the long-term effects on the compressive strength and is 1.0. γc is the partial safety factor for concrete strength and is equal to 1.5 according to EC2. Now the following equation for equilibrium of forces is established and solved to find x: + = 0

When x is found a check is made to see if the steel has yielded by substituting x into the equation for the tendon strain defined above. The initial yield strain for prestressing steel is ⁄ where γp is the partial safety factor for reinforcement strength and is equal to 1.15 according to EC2. εpu cannot exceed the initial strain, otherwise the steel has yielded. Finally the ultimate moment capacity can be determined with the following formula: = =

with z as: = ( − 0.4 )

In appenare the b

Tab

Finally,

Des

ndix B are debending mom

ble 5-9: Total d

the cross-sec

ign of a 170

etailed calcuments in ULS

design moment

ction with th

Fi

m long bridg

ulations of theS with prestre

t in the ULS w

he cable eleva

igure 5-12: Cro

ge over the f

56

e moment caess and the m

ith prestress (M

ation is displ

oss-section and

fjord Þorskaf

apacity of eamoment capa

Mtotal) and the

layed for eac

d cable elevatio

fjörður in Ice

ach section dacity summar

ultimate mom

ch section in

on.

eland

isplayed. In rized for eac

ent capacity (M

figure 5-12.

table 5-9 h section.

Mult).


57

6 References

6.1 Literature Austin, W.J., (1971), “IN-PLANE BENDING AND BUCKLING OF ARCHES“, Journal of the Structural Division, May 1971, pp. 1575-1591.

Bunner, M. and Wright, K., (2006), “Selecting the Shape of a Steel Arch”, Bridgeline, Volume 15, NO. 1, April 2006.

Chen, W. and Duan, L., (2000), “Bridge Engineering Handbook”, CRC Press LLC, Florida.

EN 1990:2002. Eurocode - Basis of Structural Design. CEN, 2002.

EN 1991:2002. Eurocode - Actions on Structures. CEN, 2002.

EN 1992:2004. Eurocode – Design of Concrete Structures. CEN, 2004.

EN 1993:1992. Eurocode – Design of Steel Structures. CEN, 1992.

Ghoneim, M. and El-Mihilmy, M., (2008), “Design of Reinforced Concrete Structures”, Vol. 3, Cairo University, Cairo.

Gilbert, R. I., and Mickleborough, N. C., (2004), “Design of Prestressed Concrete”, Spon Press, London.

ISE manual, (1985), I. Struct. E./ICE Joint committee, “Manual for the Design of Reinforced Concrete Building Structures”, Institution of Structural Engineers, London.

Loretsen, M. and Sundquist, H., (1995), ”Bågkonstruktioner”, KTH.

Loretsen, M. and Sundquist, H., (1995), ”Hängkonstruktioner”, KTH.

Menn, C., (1986), “Prestressed Concrete Bridges”, Springer-Verlag, Wien.

Nawy, E. G., (2008), “Concrete Construction Engineering Handbook”, 2nd edition, CRC Press LLC, Florida.

O’Brien, E. J. and Dixon, A. S., (1999). “Reinforced and Prestressed Concrete Design”, Pearson Education Limited, Edinburgh.

“Prestress Manual”, (2005), State of California, Department of Transportation, Engineering Services.

Thelandersson, S., (2009), “Design of bridges – Structural design project”, Lund University, Structural engineering.

Zhuan M. Q. and Guo. X. J., (1998), “PE Shelled Large Pitch Twisted Stay Cable, For Construction Use”, Shanghai PuJiang Cable Co. Ltd. Shanghai.


58

6.2 Computer programs AutoCAD 2009, Autodesk.

CSI SAP2000, version 14, Computers & Structures.

Microsoft Office Excel 2007, Microsoft.

Microsoft Office Word 2007, Microsoft.

PCFrame, version 2.15, AEC AB.

6.3 Other references Hafliðason, E. (2010), personal communication with Hafliðason at ICERA.

Earthquake Engineering Research Centre, University of Iceland.

VSL Post-Tensioning Systems, Technical Data, (2010), VSL International Ltd.

http://www.ingvason.com/

Appendix A

Appendix A

59

Q1k= 300 αQ1= 0,9 αQ1∙Q1k= 270 kN

Q2k= 200 αQ2= 0,9 αQ2∙Q2k= 180 kN

q1k= 9 αq1= 1 αq1∙q1k= 9 kN/m2

q2k= 2,5 αq2= 1 αq2∙q2k= 2,5 kN/m2

q3k= 2,5 αq3= 1 αq3∙q3k= 2,5 kN/m2

F z M q l z MFor Q For q

CALCULATIONS OF GDF FOR BRIDGE TYPE 1

F z M q l z MRA 5,6 RA∙x 5,6 RA∙x

135 6,8 918 9 3 5,8 156,6135 4,8 648 2,5 3 2,8 2190 3,8 342 2,5 3 ‐0,2 ‐1,590 1,8 162 ∑M= 176,1

∑M= 2070 RA= 31,45RA= 369,64 GDFq= 1,16

GDFQ= 1,37 RB= 10,55

RB= 80,36

Total actions on half of the cross section for traffic loads in the length direction1012 kN

37 kN/mq=GDFq*RA=

RA

Q=2*GDFQ*RA=

Appendix A

60

Part b h A y0 S=Ay0 y‐y0 I0 A(y‐y0)2 Ix

nr. mm mm mm2 mm mm3 mm mm4 mm4 mm4

1 1080 2400 2592000 1200 3,11E+09 431,1 1,24E+12 4,82E+11 1,73E+122 1080 2400 2592000 1200 3,11E+09 431,1 1,24E+12 4,82E+11 1,73E+123 10000 250 2500000 2525 6,31E+09 ‐893,9 1,30E+10 2,00E+12 2,01E+12

∑ 7684000 1,25E+10 5,46E+12

y= 1631,1 mm WTop= mm3

h= 2650 mm WBottom= mm3

u= 30100 mm2A/u= 511 mmφ= 1,54Ec,eff= 14175 N/mm2



1 5000 250 1250000 2525 3,16E+09 ‐893,9 6,51E+09 9,99E+11 1,01E+122 1080 2400 2592000 1200 3,11E+09 431,1 1,24E+12 4,82E+11 1,73E+12

∑ 3842000 6,27E+09 2,73E+12

y= 1631,1 mm WTop= 2,68E+09 mm3

h= 2650 mm WBottom= 1,67E+09 mm3

In the middle of the span ‐ Half cross section

In the middle of the span

5,36E+09

3,35E+09

Perimeter:Notional size:Creep coefficient:

Effective elastic modulus:

Creep ‐ Effective elastic modulus

Note: All cross‐section calculations are done from the bottom fibreCalculations for Area, Moment of Inertia and Section Modulus for Bridge Type 1

Appendix A

61



1 1380 2400 3312000 1200 3,97E+09 363,1 1,59E+12 4,36548E+11 2,03E+122 1380 2400 3312000 1200 3,97E+09 363,1 1,59E+12 4,36548E+11 2,03E+123 10000 250 2500000 2525 6,31E+09 ‐961,9 1,30E+10 2,31335E+12 2,33E+12

∑ 9124000 1,43E+10 6,38E+12

y= 1563,1 mm WTop= 5,87E+09 mm3




1 5000 250 1250000 2525 3,16E+09 ‐961,9 6,51E+09 1,16E+12 1,16E+122 1380 2400 3312000 1200 3,97E+09 363,1 1,59E+12 4,37E+11 2,03E+12

∑ 4562000 7,13E+09 3,19E+12

y= 1563,1 mm WTop= 2,51E+09 mm3


u= 30100 mm2A/u= 606 mmφ= 1,50Ec,eff= 14416 N/mm2

A W,Top W,Bottom I Weight

mm2 mm3 mm3 mm4 kN/m7684000 5,36E+09 3,35E+09 5,46E+12 192,13842000 2,68E+09 1,67E+09 2,73E+12 96,059124000 5,87E+09 4,08E+09 6,38E+12 228,14562000 2,51E+09 2,04E+09 3,19E+12 114,05

25 kN/m3

2,1 kN/m2

Self weight of concrete for half of the bridge section96,1 kN/m

10,5 kN/m

106,6 kN/m

36000 N/mm2

In the Middle of the Span

Over a support

Over a support

Summary

Ec=

Perimeter:Notional size:Creep coefficient:

Effective elastic modulus:

Creep ‐ Effective elastic modulus

Over a support ‐ Half cross section

Over a support ‐ half

In the middle of the Span ‐ half

γconcrete=

γpavement=

gconcrete=

gpavement=

gtot=

Appendix A

62

Q1k= 300 αQ1= 0,9 αQ1∙Q1k= 270 kN

Q2k= 200 αQ2= 0,9 αQ2∙Q2k= 180 kN

q1k= 9 αq1= 1 αq1∙q1k= 9 kN/m2

q2k= 2,5 αq2= 1 αq2∙q2k= 2,5 kN/m2

q3k= 2,5 αq3= 1 αq3∙q3k= 2,5 kN/m2

F z M q l z MFor Q For q


F z M q l z MRA 10 RA∙x 10 RA∙x

135 9 1215 9 3 8 216135 7 945 2,5 3 5 37,590 6 540 2,5 3 2 1590 4 360 ∑M 268,5

∑M= 3060 RA 26,85RA= 306,00 GDFq= 0,99

GDFQ= 1,13 RB= 15,15

RB= 144,00


27 kN/mq=GDFq*RA=

RA

Q=2*GDFQ*RA=

Appendix A

63

Relevant distances fo

r cross‐section calculations of the

arch

Appendix A

64

Part

bh

tA

y 0S=Ay 0

y‐y 0

I 0A(y‐y

0)2

I x

nr.

mm

mm

mm

mm

2mm

mm

3mm

mm

4mm

4mm

4

Box Section

900

1700

5025

0.00

085

0,00

2,13

E+08

0,0

9,54

E+10

0,00

E+00

9,54

E+10

Stiffen

ers Nr. 1

‐‐

‐3.90

011

8,69

4,63

E+05

731,3

5,60

E+06

2,09

E+09

2,09

E+09

Stiffen

ers Nr. 2

‐‐

‐3.90

017

5,00

6,82

E+05

675,0

1,34

E+07

1,78

E+09

1,79

E+09

Stiffen

ers Nr. 3

‐‐

‐3.90

040

0,00

1,56

E+06

450,0

1,34

E+07

7,90

E+08

8,03

E+08

Stiffen

ers Nr. 4

‐‐

‐3.90

062

5,00

2,44

E+06

225,0

1,34

E+07

1,97

E+08

2,11

E+08

Stiffen

ers Nr. 5

‐‐

‐3.90

085

0,00

3,31

E+06

0,0

1,34

E+07

0,00

E+00

1,34

E+07

A19

49,8

mm

2Stiffen

ers Nr. 6

‐‐

‐3.90

010

75,00

4,19

E+06

‐225

,01,34

E+07

1,97

E+08

2,11

E+08

I x2,80

E+06

mm

4Stiffen

ers Nr. 7

‐‐

‐3.90

013

00,00

5,07

E+06

‐450

,01,34

E+07

7,90

E+08

8,03

E+08

I y6,70

E+06

mm

4Stiffen

ers Nr. 8

‐‐

‐3.90

015

25,00

5,95

E+06

‐675

,01,34

E+07

1,78

E+09

1,79

E+09

Stiffen

ers Nr. 9

‐‐

‐3.90

015

81,31

6,17

E+06

‐731

,35,60

E+06

2,09

E+09

2,09

E+09

∑28

5.09

72,42

E+08

1,05

E+11

y=85

0,0

mm

h=17

00mm

Wel=

1,24

E+08

mm

3

Stiffen

ers

Calculations fo

r Area, M

omen

t of Ine

rtia and

Sectio

n Mod

ulus For th

e Stronger Axis

Appendix A

65

Part

Ay

nW

xiType

nr.

mm

2mm

mm

3

119

49,8

731,31

457

0369

0Stiffen

er2

1949

,867

5,00

452

6451

8Stiffen

er3

1949

,845

0,00

435

0967

9Stiffen

er4

1949

,822

54

1754

839

Stiffen

er5

8000

053

,00

41,7E+07

Stiffen

er6

4500

082

5,00

27,4E+07

Flange

780

000

400

41,3E+08

Web

2,35

E+08

ENV 199

3‐1‐1

5.4.5 an

d 5.4.5.2 Be

nding Mom

ent (pa

ge 65)

5.4.4 Co

mpression

((1) and

(2)) (p

age 65

)

Msd=

1587

5kN

m‐109

01kN

a) Design plastic

resistance compression

f y=

355MPa

9200

9kN

a) Design plastic

resistance mom

ent

Calculations of p

lastic sectio

n mod

ulus of the

sectio

n

Wxpl=

NSd=

Npl.Rd=

.Sd

cRd

MM

≤ pl,Rd

pl

yM0

M=W

f/γ

⋅

.Sd

cRd

NN

≤

.0

/pl

Rdy

MN

Afγ

=

2

pl,Rd

.

1M

Sd

Sd

pl

Rd

MN

N⎡

⎤+

≤⎢

⎥⎢

⎥⎣

⎦

y35

5MPa

900

9kN

γ M0=

1,1

Mpl.Rd=

7598

4kN

m

5.4.8.1 Be

nding an

d axial force

Design

1587

5OK!

0,22

1090

1OK!

0,21

0,22

OK!

@ 1/4 of arch:

@ cen

ter of arch:

Compression

[kN]

Bend

ing [kNm]

Combine

d

Resistan

ce75

984

9200

91,00

Npl.Rd

.Sd

cRd

MM

≤ pl,Rd

pl

yM0

M=W

f/γ

⋅

.Sd

cRd

NN

≤

.0

/pl

Rdy

MN

Afγ

=

2

pl,Rd

.

1M

Sd

Sd

pl

Rd

MN

N⎡

⎤+

≤⎢

⎥⎢

⎥⎣

⎦

Appendix A

66

Part b h A y0 S=Ay0 y‐y0 I0 A(y‐y0)2 Iy

[mm2] [mm2] [mm2] [mm] [mm2] [mm] [mm4] [mm4] [mm4]Top Flange 300 40 12000 1160 1,39E+07 ‐570 1,60E+06 3,90E+09 3,90E+09

Web 40 1100 44000 590 2,60E+07 0 4,44E+09 0,00E+00 4,44E+09Bottom Flange 300 40 12000 20 2,40E+05 570 1,60E+06 3,90E+09 3,90E+09

∑ 68000 4,01E+07 1,22E+10

y= 590 [mm]h= 1180 [mm]Wel= 2,07E+07 [mm3]

Part y A Wpl

2 3

5.4 Restistance of cross‐sections in EC3

Main girders ‐ Bridge type 2

[mm] [mm2] [mm3]Top Flange 570 12000 6,84E+06 γM0= 1,1 fy= 355 MPa

Web 1 275 22000 6,05E+06Web 2 275 22000 6,05E+06

Bottom Flange 570 12000 6,84E+06

∑ 2,58E+07 Where

44000 mm2

MSd,max= ‐7241 kNm

ε 0,81 VSd,max= 2087 kN

d 1100tw 40

α 0,5 8320 kNm OK!d/tw 27,5 8198 kN OK!Class Class 1 →d/tw ≤ 33ε No reduction needed!c 150tf 40

c/tf 3,75Class Class 1 →c/tf ≤ 9ε

Design values:

Web

Flan

ge

Cross Section Class, table 5.3.1 in EC3

Wpl=

Combined

Provided that the design value of the shear force V sd does not exceed 50% of the design plastic shear resistance V pl.Rd no reduction need be

made

ShearMoment

Resistance:

Shear area: Av=Σ(dtw)=

.Sd c RdM M≤

.Sd pl RdV V≤

. 0( / 3) /Sd pl Rd v y MV V A f γ≤ =

Appendix A

67




∑ 36000 1,55E+07 3,35E+09

y= 430 [mm]h= 860 [mm]Wel= 7,79E+06 [mm3]

Part y A Wpl 5.4 Restistance of cross‐sections in EC3

Cross‐beams ‐ Bridge type 2

[mm] [mm2] [mm3]

Top Flange 415 6000 2,49E+06 γM0= 1,1 fy= 355 MPa

Web 1 200 12000 2,40E+06Web 2 200 12000 2,40E+06


∑ 9,78E+06 Where

24000 mm2

MSd,max= 3025 kNm

ε 0,81 VSd,max= 1359 kN

d 800tw 30

α 0,5 3156 kNm OK!d/tw 26,67 4472 kN OK!Class Class 1 No reduction needed!c 100tf 30

c/tf 3,333Class Class 1

Design values:

Wpl=

MomentShear

Combined

Resistance:

Provided that the design value of the shear force V sd does not exceed 50% of the design plastic shear resistance V pl.Rd no reduction need be

made

Flan

geWeb

Cross Section Class, table 5.3.1 in EC3


.Sd c RdM M≤

.Sd pl RdV V≤

. 0( / 3) /Sd pl Rd v y MV V A f γ≤ =

Appendix A

68

Q1k= 300 αQ1= 0,9 αQ1∙Q1k= 270 kN

Q2k= 200 αQ2= 0,9 αQ2∙Q2k= 180 kN

q1k= 9 αq1= 1 αq1∙q1k= 9 kN/m2

q2k= 2,5 αq2= 1 αq2∙q2k= 2,5 kN/m2

q3k= 2,5 αq3= 1 αq3∙q3k= 2,5 kN/m2

F z M q l z M


For Q For qF z M q l z MRA 10 RA∙x 10 RA∙x

135 9 1215 9 3 8 216135 7 945 2,5 3 5 37,590 6 540 2,5 3 2 1590 4 360 ∑M 268,5

∑M= 3060 RA 26,85RA= 306,00 GDFq= 0,99

GDFQ= 1,13 RB= 15,15

RB= 144,00


27 kN/m

Q=2*GDFQ*RA=

q=GDFq*RA=

RA

Appendix A

69

Mmax

Vmax




∑ 42000 1,81E+07 4,38E+09

y= 430 [mm]h= 860 [mm]Wel= 1,02E+07 [mm3]


Cross‐beams ‐ Bridge type 3

Part y A Wpl

[mm] [mm2] [mm3]


Web 1 200 12000 2,40E+06Web 2 200 12000 2,40E+06

Bottom Flange 415 9000 3,74E+06∑ 1,23E+07 Where

24000 mm2

Cross Section Class MSd,max= 3237 kNmε 0,81 VSd,max= 672 kN

d 800tw 30

α 0,5 3960 kNm OK!d/tw 26,67 4472 kN OK!Class Class 1 →d/tw ≤ 33ε No reduction needed!c 150tf 30

c/tf 5Class Class 1 →c/tf ≤ 9ε

5.4 Restistance of cross sections in EC3

Provided that the design value of the shear force V sd

does not exceed 50% of the design plastic shear resistance V pl.Rd no reduction need be made

MomentShear

Combined

Resistance:

Design values:

Web


Flan

ge

Wpl=

.Sd c RdM M≤

.Sd pl RdV V≤

. 0( / 3) /Sd pl Rd v y MV V A f γ≤ =

Appendix A

70




∑ 58200 3,49E+07 1,19E+10

y= 600 [mm]h= 1200 [mm]Wel= 1,99E+07 [mm3]


Main girder ‐ Bridge type 3

[mm] [mm2] [mm3]


Web 1 285 17100 4,87E+06Web 2 285 17100 4,87E+06


∑ 2,38E+07 Where

34200 mm2

MSd,max= 7311 kNm

ε 0,81 VSd,max= 1247 kN

d 1140tw 30

α 0,5 7677 kNm OK!d/tw 38 6372 kN OK!Class Class 1 →d/tw ≤ 33ε Combined No reduction needed!c 200tf 30

c/tf 6,67Class Class 1 →c/tf ≤ 9ε

Provided that the design value of the shear force V sd does not exceed 50% of the design plastic

shear resistance V pl.Rd no reduction need be made

Wpl=

Design values:


Resistance

MomentShear

Web

Flan

ge

Cross Section Class

.Sd c RdM M≤

.Sd pl RdV V≤

. 0( / 3) /Sd pl Rd v y MV V A f γ≤ =

Appendix A

71

Column

tx L l ty B bmm 300 1500 900 300 2300 1700

A 1920000 mm2

Reinforcement

Cover x (Pcs.) Sox y (Pcs.) Soy x (Pcs.) Sox y (Pcs.) Soymm 50 18 80,9 26 80,6 13 85,4 21 83,0

Outside Bars Inside Bars

3000

Cross‐SectionReshape Chart for Right Scaling

1500

2000

2500

3000


0

500

1000

1500

2000

2500

3000

0 500 1000 1500 2000


0

500

1000

1500

2000

2500

3000

0 500 1000 1500 2000


Appendix A

72

Materials

Safety Class 3 ⇒ γn 1,2Concrete C40/50

Outside Bars 1 ∅ 25 ⇒ Aslo 490,9 mm2/Bar2

Inside Bars 1 ∅ 25 ⇒ Aslo 490,9 mm2/Bar

Stirrups 2 ∅ 10 ⇒ Asv 78,5 mm2/Bar

Creep, RH % 95 φ 1

Reinforcement #Ks40 1 55Ks60 2 Normalt utomhus samt inomhus i icke

Creep, RH %Innomhus i uppvärmde lokaler

Ks60 2Ss260S 3B500B 4 ⩾ 95Ks600S 5Ns500 6Nps500 7

Mycket fuktig miljö

75Normalt utomhus samt inomhus i icke uppvärmde lokaler

fcck fcc fctk fct Eck Ec Ec,eff εcu

MPa MPa MPa MPa MPa MPa MPa -38 19,8 2,4 1,33 35.000 24.306 12.153 0,0035

fyk fst Esk Es εsy

MPa MPa MPa MPa -O id B 500 435 200 000 200 000 0 00217

Rebars

Concrete

Outside Bars 500 435 200.000 200.000 0,00217Inside Bars 500 435 200.000 200.000 0,00217Stirrups 500 435 200.000 200.000 0,00217

Load

k 60.000

70.000

X‐Axis

NSd= 5121 kNMSd= 27292 kNm

10.000

20.000

30.000

40.000

50.000

60.000

70.000

N (kN)

X‐Axis

‐30.000

‐20.000

‐10.000

0

10.000

20.000

30.000

40.000

50.000

60.000

70.000

0 5.000 10.000 15.000 20.000 25.000 30.000 35.000

N (kN)

M (kNm)

X‐Axis

‐30.000

‐20.000

‐10.000

0

10.000

20.000

30.000

40.000

50.000

60.000

70.000

0 5.000 10.000 15.000 20.000 25.000 30.000 35.000

N (kN)

M (kNm)

X‐Axis

Appendix A

73

Appendix B

Appendix B

74

‐1.00

‐0.80

‐0.60

‐0.40

‐0.20

0.00

0.20

0.40

0.60

0.80

1.00

0.00 0.20 0.40 0.60 0.80 1.00M/M

max

x/Ltot

Influence Lines for Moment in External Spans

0.2*37 0.4*37 0.6*37 0.8*37 1.0*37

‐1.00

‐0.80

‐0.60

‐0.40

‐0.20

0.00

0.20

0.40

0.60

0.80

1.00

0.00 0.20 0.40 0.60 0.80 1.00M/M

max

x/Ltot

Influence Lines for Moment in Internal spans

0.2*48 0.4*48 0.6*48 0.8*48 1.0*48

Appendix B

75

‐1.00

‐0.80

‐0.60

‐0.40

‐0.20

0.00

0.20

0.40

0.60

0.80

1.00

0.00 0.20 0.40 0.60 0.80 1.00R/Rm

ax

x/L

Influence Lines for Reaction Forces at Supports

RA RB RC

Appendix B

76

Appendix B

77

Appendix B

78

Appendix B

79

Appendix B

80

Appendix B

81

Calculations of secondary moment with the three-momentequation

w1 99.023−kNm

:= w2 79.685−kNm

:=

Span lengths:

L1 37m:= L2 48m:=

L4 L1:= L3 L2:=

MA 0kN m⋅:= ME 0kN m⋅:=

MB 1kN m⋅:=

MC 1kN m⋅:=

MD 1kN m⋅:=

EIθ1w1 L1

3⋅

24:= EIθ2

w2 L23

⋅

24:=

EIθ4 EIθ1:= EIθ3 EIθ2:=

Given

MA L1⋅ 2 MB⋅ L1 L2+( )⋅+ MC L2⋅+ 6− EIθ1 EIθ2+( )⋅=

MB L2⋅ 2 MC⋅ L2 L3+( )⋅+ MD L3⋅+ 6− EIθ2 EIθ3+( )⋅=

MC L3⋅ 2 MD⋅ L3 L4+( )⋅+ ME L4⋅+ 6− EIθ3 EIθ4+( )⋅=

Thus, solving these equations, the total moment, with primary and secondary effects, are:

MB

MC

MD

⎛⎜⎜⎜⎝

⎞⎟⎟⎟⎠

Find MB MC, MD, ( )16134

14882

16134

⎛⎜⎜⎝

⎞⎟⎟⎠

kN m⋅⋅=:=

Appendix B

82

Secondary moment from prestress

Prestress forces at service and eccentrities at corresponding positions after losses:

PsupportA 35890kN:= eA 0mm:= MprimaryA PsupportA eA⋅ 0 kN m⋅⋅=:=

Pspan1 36875kN:= e1 410− mm:= Mprimaryspan1 Pspan1 e1⋅ 15119− kN m⋅⋅=:=

PsupportB 37441kN:= eB 283.6mm:= MprimaryB PsupportB eB⋅ 10618 kN m⋅⋅=:=

Pspan2 35482kN:= e2 420− mm:= Mprimaryspan2 Pspan2 e2⋅ 14902− kN m⋅⋅=:=

PsupportC 34795kN:= eC 283.6mm:= MprimaryC PsupportC eC⋅ 9868 kN m⋅⋅=:=

Hence the secondary moments become:

MsecondaryA 0kN m⋅:=

MsecondaryB MB MprimaryB− 5515 kN m⋅⋅=:=

Msecondaryspan11387537000

MsecondaryB⋅ 2068 kN m⋅⋅=:=

MsecondaryC MC MprimaryC− 5015 kN m⋅⋅=:=

Msecondaryspan2 MsecondaryBMsecondaryC MsecondaryB−

2+ 5265 kN m⋅⋅=:=

Total bending from prestress:

MpA MsecondaryA MprimaryA+ 0 kN m⋅⋅=:=

Mp1 Msecondaryspan1 Mprimaryspan1+ 13050− kN m⋅⋅=:=

MpB MsecondaryB MprimaryB+ 16134 kN m⋅⋅=:=

Mp2 Msecondaryspan2 Mprimaryspan2+ 9637− kN m⋅⋅=:=

MpC MsecondaryC MprimaryC+ 14882 kN m⋅⋅=:=

Appendix B

83

Calculations of ultimate moment capacity

Ap1 4200mm2:= Ep 195000MPa:=

Ap 8 Ap1⋅:= Ec 36000MPa:=

Igsupport 1.5648 1012⋅ mm4

:= Agsupport 3770000mm2:=

Igspan 1.3462 1012⋅ mm4

:= Agspan 3230000mm2:=

Bending from self-weight and traffic loads in ULS:

MsupportA1 0kN m⋅:=

Mspan11 30314kN m⋅:=

MsupportB1 40838− kN m⋅:=

Mspan21 31954kN m⋅:=

MsupportC1 43737− kN m⋅:=

Bending in ULS with prestress:

MA.ULS MsupportA1 MA+ 0 kN m⋅⋅=:=

M1.ULS Mspan11 Msecondaryspan1+ Mprimaryspan1+ 17264 kN m⋅⋅=:=

MB.ULS MsupportB1 MsecondaryB+ MprimaryB+ 24704− kN m⋅⋅=:=

M2.ULS Mspan21 Msecondaryspan2+ Mprimaryspan2+ 22317 kN m⋅⋅=:=

MC.ULS MsupportC1 MsecondaryC+ MprimaryC+ 28855− kN m⋅⋅=:=

Appendix B

84

Effective depth, d: Width of the girder:

d1 1163.3mm:= bspan 1100mm:=

dB 1523.5mm:= bsupport 1400mm:=

Allowable yield strength of the steel (figure 3.10 in EC2):d2 1173.3mm:=

fpk 1860MPa:= fp0.1k 1640MPa:=

dC 1523.5mm:=γp 1.15:=

fck 45MPa:= fpdfp0.1k

γp1426 MPa⋅=:=

fpdEp

0.007313=

γc 1.5:=εud 0.02:= Strain limit recommended by EC2.

α 1:=

Appendix B

85

Ultimate moment capacity at span 1:Ultimate compressive strain in concrete:

εult 0.0035:=

Concrete strain due to prestress at the level of the tendon:

εce.11Ec

Pspan1Agspan

Mp1 e1⋅

Igspan+

⎛⎜⎝

⎞⎟⎠

⋅ 0.000428=:=

Total ultimate strain in concrete at the level of the tendons:

εct.1εult− d1 x1−( )⋅

x1=

Tendon strain:

εpu.1Pspan1−

Ap Ep⋅εct.1+ εce.1−=

Concrete compressive force:

Fc.1 0.8 x1⋅ bspan⋅α fck⋅

γc⋅=

Total force in tendon:

Fp.1 Ap Ep εpu.1⋅( )⋅=

Equilibrium of forces:

Fp.1 Fc.1+ 0=

Which leads to:

x1 1mm:=

Given

Ap EpPspan1−

Ap Ep⋅

εult− d1 x1−( )⋅

x1+ εce.1−

⎡⎢⎣

⎤⎥⎦

⋅⎡⎢⎣

⎤⎥⎦

⋅ 0.8 bspan⋅ x1⋅α fck⋅

γc⋅+ 0=

x1 Find x1( ) 1.371 m=:=

Check if steel stress is ok:


x10.000531=:=

εpu.1Pspan1−

Ap Ep⋅εct.1+ εce.1− 0.005525−=:= Less than fpd/Ep=0.007313

Appendix B

86

Ultimate moment capacity:

z1 d1 0.4 x1⋅−( ) 0.615 m⋅=:=

Fc1 0.8 bspan⋅ x1⋅α fck⋅

γc⋅ 36199 kN⋅=:=

Mult1 Fc1 z1⋅ 22256 kN m⋅⋅=:=

Appendix B

87

Ultimate moment capacity at support B:Ultimate compressive strain in concrete:

εult 0.0035=


εce.B1Ec

PsupportBAgsupport

MpB eB⋅

Igsupport+

⎛⎜⎝

⎞⎟⎠

⋅ 0.000357=:=


εct.Bεult− dB xB−( )⋅

xB=

Tendon strain:

εpu.BPsupportB−

Ap Ep⋅εct.B+ εce.B−=


Fc.B 0.8 xB⋅ bsupport⋅α fck⋅

γc⋅=


Fp.B Ap Ep εpu.B⋅( )⋅=


Fp.B Fc.B+ 0=

Which leads to:

xB 1mm:=

Given

Ap EpPsupportB−

Ap Ep⋅

εult− dB xB−( )⋅

xB+ εce.B−

⎡⎢⎣

⎤⎥⎦

⋅⎡⎢⎣

⎤⎥⎦

⋅ 0.8 bsupport⋅ xB⋅α fck⋅

γc⋅+ 0=

xB Find xB( ) 1.301 m=:=


εct.Bεult− dB xB−( )⋅

xB0.000599−=:=

εpu.BPsupportB−

Ap Ep⋅εct.B+ εce.B− 0.006671−=:= Less than fpd/Ep=0.007313

Appendix B

88


zB dB 0.4 xB⋅−( ) 1.003 m⋅=:=

FcB 0.8 bsupport⋅ xB⋅α fck⋅

γc⋅ 43707 kN⋅=:=

MultB FcB zB⋅ 43846 kN m⋅⋅=:=

Appendix B

89

Ultimate moment capacity at span 2:Ultimate compressive strain in concrete:

εult 0.0035=


εce.21Ec

Pspan2Agspan

Mp2 e2⋅

Igspan+

⎛⎜⎝

⎞⎟⎠

⋅ 0.000389=:=



x2=

Tendon strain:

εpu.2Pspan2−

Ap Ep⋅εct.2+ εce.2−=


Fc.2 0.8 x2⋅ bspan⋅α fck⋅

γc⋅=


Fp.2 Ap Ep εpu.2⋅( )⋅=


Fp.2 Fc.2+ 0=

Which leads to:

x2 1mm:=

Given

Ap EpPspan2−

Ap Ep⋅

εult− d2 x2−( )⋅

x2+ εce.2−

⎡⎢⎣

⎤⎥⎦

⋅⎡⎢⎣

⎤⎥⎦

⋅ 0.8 bspan⋅ x2⋅α fck⋅

γc⋅+ 0=

x2 Find x2( ) 1.335 m=:=



x20.000424=:=

εpu.2Pspan2−

Ap Ep⋅εct.2+ εce.2− 0.00538−=:= Less than fpd/Ep=0.007313

Appendix B

90


z2 d2 0.4 x2⋅−( ) 0.639 m⋅=:=

Fc2 0.8 bspan⋅ x2⋅α fck⋅

γc⋅ 35248 kN⋅=:=

Mult2 Fc2 z2⋅ 22532 kN m⋅⋅=:=

Appendix B

91

Ultimate moment capacity at support C:Ultimate compressive strain in concrete:

εult 0.0035=


εce.C1Ec

PsupportCAgsupport

MpC eC⋅

Igsupport+

⎛⎜⎝

⎞⎟⎠

⋅ =:=


εct.Cεult− dC xC−( )⋅

xC=

Tendon strain:

εpu.CPsupportC−

Ap Ep⋅εct.C+ εce.C−=


Fc.C 0.8 xC⋅ bsupport⋅α fck⋅

γc⋅=


Fp.C Ap Ep εpu.C⋅( )⋅=


Fp.C Fc.C+ 0=

Which leads to:

xC 1mm:=

Given

Ap EpPsupportC−

Ap Ep⋅

εult− dC xC−( )⋅

xC+ εce.C−

⎡⎢⎣

⎤⎥⎦

⋅⎡⎢⎣

⎤⎥⎦

⋅ 0.8 bsupport⋅ xC⋅α fck⋅

γc⋅+ 0=

xC Find xC( ) 1.25 m=:=


εct.Cεult− dC xC−( )⋅

xC0.000767−=:=

εpu.CPsupportC−

Ap Ep⋅εct.C+ εce.C− 0.006409−=:= Less than fpd/Ep=0.007313

Appendix B

92


zC dC 0.4 xC⋅−( ) 1.024 m⋅=:=

FcC 0.8 bsupport⋅ xC⋅α fck⋅

γc⋅ 41990 kN⋅=:=

MultC FcC zC⋅ 42982 kN m⋅⋅=:=

Appendix B

93

Jóhannes Helgi Jóhannesson

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