Collapse analysis of externally prestressed structures 9783836622981, 9783836672986

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Jens Tandler

Copyright © 2009. Diplomica Verlag. All rights reserved.

Collapse analysis of externally prestressed structures

Diplomica Verlag

Jens Tandler Collapse analysis of externally prestressed structures ISBN: 978-3-8366-2298-1 Herstellung: Diplomica® Verlag GmbH, Hamburg, 2009

Copyright © 2009. Diplomica Verlag. All rights reserved.

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Abstract The use of external prestressing is becoming more popular throughout Europe due to their expected higher durability and the possibility of active maintenance of the prestressing cables. Questions have been raised about the behaviour of these structures beyond service loads. A comprehensive numerical analysis has been carried out comparing the behaviour of three different types of externally prestressed bridges to a conventionally internally prestressed bridge. The external types are a monolithically built bridge with external tendons, a monolithically built bridge with external tendons and blocked deviators, and a precast segmental bridge with external tendons. The internally prestressed bridge is monolithic. The primary objectives are to determine whether or not ductile failure occurs, i.e. the load-deflection response, and the tendon stress increase at ultimate stage. The results show that the monolithically built bridges have a considerable higher ultimate moment capacity as well as deflection. This shows the advantage

of

using

continuous

ordinary

reinforcement.

All

externally

prestressed types did not reach the capacities of the internally prestressed bridge. It was found that ductility depends mostly on the reinforcement within the cross-section. Externally prestressed girders have no prestressing cables in the cross-section and need sufficient ordinary reinforcement in order to deform ductile. Although the tendon stress increase up to failure in the actual analysis is remarkable, the discussion shows that the magnitude varies greatly

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depending on the layout of the whole structure.

KEYWORDS: EXTERNAL PRESTRESS, DUCTILITY, TENDON STRESS INCREASE, FINITE ELEMENT ANALYSIS

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TO ALL MY SUPPORTERS, ESPECIALLY TO KRISTIN

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Contents at a Glance

1

Introduction ........................................................................................ 1

2

Behaviour of externally prestressed structures ............................ 10

3

Collapse analysis ............................................................................. 23

4

Results .............................................................................................. 73

5

Discussion of the results ................................................................ 85

6

Conclusion and Recommendations ............................................... 98

VII

Contents

Acknowledgements .................................................................................. XI Notation ................................................................................................... XIII 1

Introduction ........................................................................................ 1 1.1

Definitions ..................................................................................... 1

1.2

Significance of this study .............................................................. 3

1.3

Scope of the project ...................................................................... 5

1.4

Historical overview and typical characteristics of external

prestressing .................................................................................................... 6 1.5 2

Further structural applications of external prestressing ................. 9

Behaviour of externally prestressed structures ............................ 10 2.1

Tendon layout considerations ..................................................... 10

2.2

Behaviour at serviceability stage................................................. 12

2.3

Fatigue problems ........................................................................ 14

2.4

Behaviour at ultimate limit stage ................................................. 14

2.4.1

Influence of tendon slip on the ultimate limit state ............................ 18

2.4.2

Influence of the arrangement of the deviators on the behaviour at

ultimate limit state .............................................................................................. 19 2.4.3

Influence of simply support and continuous support on the ultimate

limit state 20

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2.4.4

3

Precast segmental and monolithic bridges ....................................... 21

Collapse analysis ............................................................................. 23 3.1

Investigated bridge types and their differences ........................... 23

3.2

Original bridge data..................................................................... 28

3.3

Simplified bridge data as basis for the calculations..................... 30

IX

3.4

FE Calculation............................................................................. 32

3.4.1

Technical aspects ............................................................................. 33

3.4.2

General approach ............................................................................. 34

3.4.3

Geometric model............................................................................... 39

3.4.4

Element specifications ...................................................................... 40

3.4.5

Constitutive models........................................................................... 45

3.4.6

Ordinary reinforcement ..................................................................... 59

3.4.7

Prestress ........................................................................................... 60

3.4.8

Material and geometric non-linearity ................................................. 63

3.4.9

Kinematic constraints ........................................................................ 66

3.4.10

Discrete crack propagation analysis of the precast segmental type

with gap elements .............................................................................................. 68 3.4.11

Summary of the dividing features of the different structure types for

the FE analysis .................................................................................................. 72

4

5

Results .............................................................................................. 73 4.1

Load deflection behaviour ........................................................... 73

4.2

Tendon stress increase up to failure ........................................... 76

4.3

Other results ............................................................................... 78

Discussion of the results ................................................................ 85 5.1

Interpretation of the results ......................................................... 85

5.2

Discussion of the exactness of the FE calculations by comparing

to the full scale test ....................................................................................... 89 5.3

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6

Comparison to other FE calculations and test results ................. 93

Conclusion and Recommendations ............................................... 98 6.1

Concluding remarks .................................................................... 98

6.2

Recommendations ...................................................................... 99

References ............................................................................................. 101 Codes of practice .................................................................................. 107 Appendices

X

Acknowledgements I would like to thank the people, who helped me to do this Study. In particular, I would like to thank Tony Thorne, who set up the ABAQUS machine, assisted me with UNIX, and tried to solve patiently all the bugs related with the Preprocessing software, and also Jonathan Hulatt for his useful hints for ABAQUS. Jonathan had also a look at my English writing despite his own work-load. I am grateful to Nigel Hewson, who originally inspired me to the actual topic of this study and gave me some ideas to start with. Mike Ryall and Paul Mullord helped me through useful discussion about prestressing and Finite Element theory.

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Jens Tandler

XI

Notation Symbols Subscripts b

Biaxial

c

Concrete

m

Mean, hydrostatic

p

Prestressing steel

u

Ultimate

x,y,z

x,y,z directions

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Main symbols H

Strain

V

Normal stress

W

Shear stress

V1, V2, V3

Principal stresses

Vbp

Stress at bottom fibre of section caused by prestress

Wc

Part of a term describing pure shear strength

Oc

Hardening parameter from the concrete compression yield surface

Vcb

Negative principal stress

Vce’

Concrete stress at level of tendon from applied moment

Hpb

Bending initiated strain

Vpb

Bending initiated stress

Hpe

Direct strain from prestress in tendon

Vpe

Direct stress from prestress in tendon

Vs

Stress in ordinary reinforcement

Us

Percentage ordinary reinforcement

Vt

Hardening parameter of the crack detection surface

Vtbu

Biaxial tensile strength of the concrete

Vte

Applied tensile stress

XIII

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Vtp

Stress at top fibre of section caused by prestress

Ixx

Deflection angle of the tendon

Vy

Uniaxial yield strength

ao

Constant

bo

Constant

DL

Dead load

E

Young’s modulus

e

Tendon eccentricity before application of load

e’

Tendon eccentricity after load application

eb

Distance from tendon to the bottom of the section

F

Force

fc

Compression yield function of the concrete

fctm

Concrete tensile strength

fcu

Uniaxial concrete compression strength

fpu

Ultimate strength of the tendon

ft

Crack detection surface function of the concrete

Gf

Concrete fracture constant

hx

Coefficient EC2 DD ENV 1992-2:2001

I

Second moment of area

k

Coefficient EC2 DD ENV 1992-2:2001

k1

Coefficient EC2 DD ENV 1992-2:2001

kc

Coefficient EC2 DD ENV 1992-2:2001

M

Moment

Mapplied

Applied moment

Mcrbf, Mcrtf

Cracking moment bottom flange/ top flange

Me

Moment introduced form tendons into the end diaphragm

Nsd

Axial force on part of the section from the quasi permanent load and prestress

p

Hydrostatic pressure

q

Distorsional stress

rbcV

Ratio of the maximum biaxial compression strength to the maximum uniaxial compression strength of the concrete

XIV

r tV

Ratio of the negative uniaxial tensile strength to uniaxial compression strength of the concrete

Txx

Deflection force from the tendon

u

Crack width

z

Lever arm

Zb, Zt

Section modulus bottom fibre and top fibre

Sign convention All compressive actions are indicated with a minus sign and the tensile actions are shown with a positive sign or no sign respectively. There is one exception: p, the hydrostatic pressure, is negative in tension and positive in compression.

Units SI units are generally used. However, some values in graphs are given in

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imperial units. 1 in

=

25.4 mm

1 ft

=

0.305 m

1 kip force =

4448 N

1 psi

1/145 N/mm²

=

XV

1 Introduction This study is an investigation into the behaviour of externally prestressed structures, focusing on bridge box girders, at the ultimate limit state. The main objective is the ductility and the tendon stress increase up to failure of externally prestressed structures. Their behaviour will be compared to internally prestressed structures. This document may have valuable information for the first stages of the design process for medium span bridges as the study is concerned about the overall safety and efficiency of prestressed concrete bridges by the means of ductility. The aim is also to provide information about the tendon stress at failure, which is required for the detailed design.

1.1 Definitions External prestressing is a special technique of post-tensioning. Posttensioning is used to apply prestress forces to the concrete after hardening. (Hewson, 2000a). External tendons are placed outside of the section being stressed. The forces are only transferred at the anchorage blocks or deviators

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(Hewson, 2000b).

Figure 1-1: Typical view in box girder bridge with externally deflected tendons (modified from Krautwald, 1998)

1

Internal prestressing is defined in this dissertation, if tendons lie within the cross-section of the structure. Internal prestressing can be carried out using bond between the structure and the prestressing steel (grouted ducts). The other possibility is internal post-tensioning without bond between the duct and the tendon. The prestressing force is again only transferred through the anchorages and contact pressure against the surface of the duct. Throughout the dissertation only internal post-tensioning with bond and external prestressing is investigated. The figure below outlines the prestressing methods of interest for this dissertation.

Figure 1-2: Prestressing techniques

The figure shows the pure types. There are more techniques possible, which are the hybrid systems. Hybrid systems are combinations between different pure types. External prestressing in combination with internal post-tensioning is recommended in Germany for launched box girders, although it is not widely used. Pretensioning with internal post-tensioning has been used because of limited stressing capacity for the pretension. All these hybrid systems are only

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cost-effective in certain situations. The difference between monolithic constructions and a precast segmental constructions is that the precast segmental constructions have no ordinary reinforcement crossing the joints of the segments whereas monolithic bridge constructions have normal reinforcement along the whole bridge. Precast segmental bridges can be erected by lifting match cast segments into place by the means of different crane types. The segment is then stressed against the

2

rest of the structure or held in place before stressing all segments together. A monolithically cast concrete bridge can be lifted as a whole into place, launched from the abutments, or constructed by balanced cantilever construction with slip form.

1.2 Significance of this study Recent Problems on external prestressed structures show that there are still problems in the understanding of these structures. Accidents took place in South Africa in 1998 and in Guam in 1996. In the first case a box girder with external straight tendons collapsed during the launching process. The bridge dropped workers and a party of visitors 30m to the ground. 14 people were left dead, including the bridge designer, and 13 were seriously injured (NCE, 1998).

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Figure 1-3: Collapse Injaka Bridge in South Africa (NCE, 1998)

Another example was the catastrophic collapse of what was once the longest post-tensioned balanced cantilever bridge of the world with a span of 241m. The bridge in Guam suffered the destruction after an attempt to strengthen the bridge with external tendons. The project was supervised by an

3

American structural engineer carried out largely by a well-established posttensioning contractor (NCE, 1996). A considerable number of scientific papers have been published during the last two decades dealing with ductility and tendon stress increase in externally prestressed bridges. There are differences between the findings. Fundamental research and work in this field was done by B.G. Rabbat and K. Slowat (1987), J. Muller and Y. Gauthier (1989) and MacGregor R.J.G. et al. (1989). Many codes of practice are based on this American research, e.g. the BD 58/94 “Design of concrete highway bridges and structures with external unbonded tendons” for the UK. The connection to the above mentioned research work can be found in Development of BA and BD 58/94 by Jackson P.A. (1995). There have always been concerns about brittle failure of externally prestressed bridges (Hollingshurst, 1995), because there is only a small increase of the tension in the steel tendons until failure. Another concern was coming from the behaviour of external prestressed segmental structures with no passive reinforcement between the segments (Bruggeling, 1989).

Figure 1-4: Segmental box girder bridge with deflected external tendons and dry joints under extensive loading in first span (Muller and Gauthier, 1989)

It is possible that there will be a growing demand for externally prestressed structures in Europe because of their likely higher durability, which is obviously

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attractive to the authorities. An indication of this new demand might be the New Medway Bridge for widening of the M2 in Kent (WS Atkins, 2001). This bridge will be a balanced cantilever prestressed concrete construction with external tendons. For this reason it is thought to be necessary to make new considerations about the behaviour of these bridges at ultimate limit state with the background

4

of the concerns, the failures, and the latest research. Also the ultimate limit state might govern the check for such structures, because of the low increase of strain up to failure in the tendon. This is in contrast to bonded internally prestressed concrete structures, where the check at service governs the amount of prestressing steel. There might also be important implications regarding the cost efficiency of externally prestressed structures.

1.3 Scope of the project Three externally prestressed bridge types will be studied; these include an externally prestressed bridge monolithically built, an externally prestressed concrete bridge monolithically built with blocked deviators, and a precast segmental bridge with external tendons. A conventional internally prestressed bridge with bonded tendons monolithically built will also be analysed as a reference. All bridges are box girders. They are simply supported and have a span of 43.25m. The basic bridge data is taken from the Bangkok Second Stage Expressway. As part of this major project a full-scale destructive test was conducted by Takebayashi et al., (1994). The bridge was a precast segmental box girder with external tendons and dry joints. The data collected from this test will also be used to verify the results. The objectives of this investigation are to determine whether or not externally prestressed bridges fail ductile and the amount of increase in tendon stress up to failure. The analysis will be done by numerical methods using ABAQUS. Kong 1996 defines ductile failure as followed. ”The failure of an under-

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reinforced beam is characterised by large steel strains, and hence extensive cracking of the concrete and substantial deflection. The ductility of such a beam provides ample warning of impending failure….” On the other hand brittle failure occurs (Hurst, 1998), if the steel in the tension zone has not reached the yield strain. In this case the concrete crushes suddenly without showing big cracks in the tension zone. Such a section is also described as over-reinforced.

5

After the introduction, a outline of the recent research will be given explaining the key aspects of the structures concerned. The next section deals with the analysis. This includes the simplification of the original bridge data and statements of all the assumptions made. The explicit explanation of the differences of each of the analysed bridge types are also shown in this section. Thereafter, theoretical background regarding the Finite Element analysis is given together with a description of the actual analysis undertaken. Chapter 4 illustrates the results of the study, which are discussed in Chapter 5. The study will then conclude with the summary of the findings.

1.4 Historical overview and typical characteristics of external prestressing Looking back to the early days, it is surprising to recognise that the first prestressed concrete bridge was externally post-tensioned. This bridge was built from 1935 to 1937 in Aue, Germany, by Franz Dischinger. Steel with a tensile strength of about 500 N/mm² was used at the time. Considerable losses in the prestressing force have occurred due to the low tensile steel and the bridge was restressed twice in 1962 and 1980 (Virlogeux, 1989). The bridge

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was demolished in 1994 (Landschaftverband Westfalen-Lippe, 2001).

Figure 1-5: Elevation and cross-section of the Station Bridge Aue/ Germany with external tendons (Schönberg and Fichtner, 1939)

Although the prestressing bars were not performing so well, the drawback of the low tensile steel has been overcome by the advantage of external prestressing.

6

This construction type allows restressing and replacing of the prestressing strands. The replacement of the strands is even possible without full closure for the traffic crossing over the bridge. Such a replacement under traffic was done at the Braidley Road Bridge in the UK in 1980 (Clark, 1998).The replacement of the tendons was necessary because of corrosion problems. Most of the early externally prestressed bridges suffered from this problem. Corrosion was the main reason for caution to this technique and lead to preference of internal prestressing. In the meantime, reliable corrosion systems have been developed. The strands are commonly encased in high-density polyethylene ducts (HDPE) and the ducts are filled with grease or cement grout. The strand can also be separately encased again in the pipes. These days external prestressing is mostly used in France and in the USA. The reasons are significantly different. In the USA, external prestressing is used because of its cost-effectiveness, especially if it is used in combination with segmental construction. Major bridges were built with this technique, e.g. the Long key bridge with 101 spans with spans of 36m and an overall length of 3701m (Gallaway, 1980). The believed higher durability of certain types of externally prestressed bridges lead to a massive recovery of this construction technique in France. The French authorities believe, if the corroded tendons can be changed, the bridge will have a longer lifespan. And the possibility of inspection of the tendons should make such bridges more predictable and therefore safer. Virlogeux (1989) states, “…we can master the technique, it is no longer experimental for us, but the normal way of building large concrete bridges”. Although this is quite enthusiastic, it shows that external prestressing might have an important place in bridge construction of the future. The

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characteristics of this type of bridge construction seems to make them very cost-competitive for very long viaducts, e.g. the Second Severn Approach spans in the UK with about 4km length (NCE, 1994) and the Bangkok Second Stage Expressway with over 60km deck length (Hewson, 1993).

7

Figure 1-6: Sunshine Skyway (Florida) – span by span precast segmental, externally

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post– tensioned approach spans (courtesy of Figg and Muller Engineers Inc.)

8

1.5 Further structural applications of external prestressing External prestressing is not only used for bridge construction. It is also used for building constructions. There are reports about the strengthening of silos (Schallwig, 1998 and Hegger, 1998). In both cases cylindrical silos had unacceptable wide vertical cracks due to overloading in their outer vertical concrete walls. This was overcome by the use of external peripheral tendons.

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Figure 1-7: Silo strengthened by external tendons (Schallwig, 1998)

9

2 Behaviour of externally prestressed structures 2.1 Tendon layout considerations Although tendon layout considerations are not precisely the topic of the dissertation, it is believed that a short introduction to this field is necessary for a deeper understanding of the actual topic. Anchorage points have a much higher importance in external prestressed structures than in conventionally prestressed structures. The anchorage is the only point where the tension from the strands is contained. Grouted tendons in bonded prestressed structures transfer their force also along their length and therefore the anchorage is only during construction of such a high importance. Another difference of external prestressed structures is that the prestressing force is not directed towards the concrete mass but produce an eccentric force to the concrete cross-section. Thus, the anchorage points are typically massive concrete blocks.

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Figure 2-1: Deformation of an anchorage point and its surrounding in a box girder bridge (Standfuß et al., 1998)

First attempts leading the external tendons straight through the cross-section lead to a less satisfactory result. There are many anchorage points necessary

10

(marked with triangles in the next figures), which make this construction type uneconomic.

Figure 2-2: Straight tendons with high number of anchorages (Krautwald, 1998)

Research in behaviour of deflected external tendons allowed the next step. It followed a layout with deflected tendons aiming less anchorage points and a better conformity to the moments from the loads.

Figure 2-3: Deflected tendons with only few anchorage points

Further optimisation was carried out by improving the positions and numbers of deviators in the span.

Figure 2-4: Optimised tendon layout

Figure 2-5: Equivalent design with cross-

(Virlogeux, 1989)

beams, pseudo parabolic external tendon (Virlogeux, 1989)

Figure 2-4 shows a tendon layout, which is state of the art. Such a position Copyright © 2009. Diplomica Verlag. All rights reserved.

of the tendons produces a moment, which is very close to a parabola and therefore well suited to take effect against moments from loads. Also, it produces a higher shear reduction near the support than a tendon with only one deviator per span. Figure 2-5 shows an arrangement, which has the same effect than the design in Figure 2-4. It has the disadvantage of large cross-beams. It is preferable to deflect the tendons near the web to avoid transverse bending

11

moments. Hence, it is necessary to change their direction also in plan. Always the deflected tendons will be lead to the intersection web bottom slab (see also Figure 2-6).

Figure 2-6: Deviation in plan and elevation (Virlogeux, 1989)

2.2 Behaviour at serviceability stage The serviceability check is not of such a high importance for externally prestressed structures than for internally bonded prestressed structures. Traditionally the prestressed concrete is designed by controlling the tensile strength in order to avoid cracking. This is done in order to protect the prestressing cables within the concrete. If the cables are outside the concrete section, there will be no justification for this any longer and partially prestressing (UK: class 3) could be approached. Partially prestressing allows limited cracking under live load. Then, it is very likely that the check at ultimate limit state governs the design prestress force. However, partially prestressing is not Copyright © 2009. Diplomica Verlag. All rights reserved.

permitted in several countries, e.g. the UK, for highway and railway bridges (Jackson, 1995). The basic calculations at service of externally prestressed structure is very similar to the calculations of bonded post-tensioned structures. The prestressing force is introduced via a normal force along the bridge and via nearly vertical forces due to the sharp changes in tendon geometry at the deviators.

12

Beam with external tendon

Cross-section

Tendon

Tendon

Forces from tendons

Moments from tendons

Equivalent statical system with prestressing load only

Figure 2-7: Idealisation of prestressing load

If the tendon arrangement is favourable, the web thickness can be substantially smaller than the webs of conventional prestressed bridges, because the strands are outside the cross-section. It is also notable that losses due to friction are not as high as in internally stressed systems. Friction losses occur at the deviators only. Thus, a higher effective prestress can be reached. It was also found that the tendons do not slip at the deviator under service load (MacGregor et al., 1989). Cracking tends to be a more controversial aspect. A German research project is often mentioned in this context (Vielhaber, 1988). The following statement is made, “This project has shown that in the case of (pure) prestressing without bond no control of crack width is possible. In such structures, it should be tried to prevent cracking of concrete by increasing the prestressing force“. Although the last part of the statement is maybe a reason for some disagreement, the first part makes a definite comment. But the only reason for no passive reinforcement at certain points in a bridge might be in the case of a precast segmental construction, where the cracks will open at the

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joints. The reason behind crack control is durability. If there were a crack at the joint, i.e. the joint opens under loading, concrete cover at the longitudinal end of the segment would still protect the passive reinforcement within the precast segment. The situation becomes more difficult with glued joints, where the crack occurs next to the epoxy glue in the concrete.

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2.3 Fatigue problems As far as known by the author, there are only few identified fatigue problems with the external tendons. The tendons tend to swing under load. The vibration produces nearly no measurable prestress force changes (Standfuß et al., 1998). The BD58/94 requires dynamic checks when exceeding 12m between the lateral restraints of the tendon. The German code of practice limits the free length of the tendons to 35m (ARS Nr.28/1998) without further checks.

2.4 Behaviour at ultimate limit stage The ultimate limit state is for externally prestressed structures much more important than for internally bonded structures, because this check is very likely governed for the amount of prestress force needed. The reasons will be explained later. This is in contrast to internally bonded prestressed concrete structures, where the serviceability check governs the design. As the load on the section increases beyond the values of the serviceability state, the strain and the stress respectively in the section will increase following the appropriate stress-strain relationship. The constitutive laws from steel and concrete show different characteristics.

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Figure 2-8: Strains, stresses and forces acting on a bonded prestressed concrete section ignoring concrete tension

The strain due to bending in the strand and in the adjacent concrete has the same amount in bonded prestressed concrete structures, because they are

14

directly connected together via the grouted duct. The ultimate moment of resistance is calculated with the equation Mu = Fc * z . Furthermore, it is noticeable that Hpb (bonded)=

Mapplied * e' I * Ec

V ce' , Ec

where Vce’ is the stress in the concrete at the level of the tendon from an applied moment and e’ the eccentricity of the tendon. The situation is different with externally and therefore unbonded prestressed structures. The bending initiated strain in the tendon does not reach the same strain as in the adjacent concrete. The increase in length of the strands can be distributed between two adjacent anchorage points. Hence, the rotation of the beam producing the increase in strain at the bottom side leads not to such a high local prestressing force and ultimate moment of resistances as by bonded structures (Ramos and Aparicio, 1995). Hpb (bonded) > Hpb (unbonded) The statement above will be investigated later in the analysis. The extension of

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the external strands can be described as followed.

15

Figure 2-9: Extension of tendons under load

H pb (unbonded )

(2 * L1'L2' )  (2 * L1  L2) (2 * L1  L2)

(Ramos and Aparicio, 1995)

LX and LX’ respectively describe the length of the tendons before and after the application of the load. It can be seen that the strain in the strands is not directly connected to the strain in adjacent concrete. Another interesting characteristic is the second order effect of the tendon eccentricity. With increasing load the eccentricity of the tendon at midspan becomes gradually smaller. The next figure illustrates the fact.

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e

e’

Figure 2-10: Reduction of eccentricity

16

Hence, the lever arm of the tendon to the neutral axis of the section decreases, and therefore the ultimate moment of resistance as well (Tan and Ng, 1997). The two factors described above are the difficult parameters within the calculation of the ultimate moment of resistance. It can already be seen that these influences alter the resistance of section negatively in comparison to a bonded internally prestressed beam. The other factors influencing the eccentricity and the bending strain are the slip of the tendons at the deviators, the arrangement of the deviators, whether or not the structure is continuous, and whether the structure is a monolithic structure or a precast segmental bridge. A considerable effort has been made in the last two decades finding ways in order to evaluate all these factors satisfactory. It has been tried to find general solutions trying to include all factors in sufficient way. It is believed that a general solution cannot be found because of the numerous influences. It will be tried to describe a procedure rather than a general solution leading to the ultimate moment of resistance in this dissertation. The importance of the different factors will also be assessed. The general task is to calculate the bending strain in the tendon. Conservatively, it could be assumed that this strain is 0. The calculation is then relatively straightforward. Some codes of practice suggest this approach and sometimes with allowance for a small extra strain subjected to special conditions. The AASHTO 1996 (clause 9.17.4.1), the BD 58/94 (6.3.3.1(f)), and the EC2 part 1.5 (4.3.1.4 P (103)) comply with these specifications. Further increase will be allowed, if a non-linear calculation is conducted. The German code (ARS Nr. 7/1998, clause 8(2)) allows no increase

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in ultimate limit capacity due to rotation of the structure nor does it mentions a non-linear analysis. The non-linear analysis of externally prestressed structures at the ultimate limit state in bending is the subject of this dissertation.

17

Ultimate limit state shear is not a detailed part of this dissertation. However, some facts follow. Shear design tends to be empirical, based on tests carried out on beams with bonded tendons. Bonded tendons contribute to the shear resistance, since they cross the cracks. Concerns have to be expressed as to whether these rules are applicable or not. Alternatively, it could be approached as a reinforced concrete beam or column respectively subjected to external loads (Jackson, 1995).

2.4.1 Influence of tendon slip on the ultimate limit state It was found that it is reasonable to assume no tendon slip at the deviators at service as already mentioned above. But at ultimate conditions, the tendons move over the deviators (Takebayashi, 1994) or saddles releasing stress of the tendons. The release of stress leads to sudden fall in the strain in the tendon and hence to further deflection of the beam if not to failure. Failure will happen if the neutral axis shifts to high up in the section due to further deflection. The concrete will then fail in compression.

Figure 2-11: Forces at deviator (MacGregor et al., 1989)

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The slippage itself is very difficult to assess. On the one hand, there is the use of cement grouted external tendons and on the other hand, there are grease filled tendon ducts. These different encasements lead to different friction properties between the concrete section and the strand. Cement filled ducts produce a very good connection. The use of them makes sense from the statical point of view, but they are difficult to change in the case of corrosion.

18

Grease filled ducts have a much lower coefficient of friction. It is believed that a full fixity will normally not be reached, as even cement-grouted tendons slipped in a full-scale test (Takebayashi, 1994) at ultimate load. This might be due to the commonly used HDPE duct. An extensive analytical research project was conducted by Rao and Mathew (1996). They produced a finite element model capable of dealing with friction problems at multiple deviators. This is without doubt state of the art in finite element modelling. A model of such complexity is probably far beyond the possibilities of the normal practicing engineer. However, the possibility of the practical application of such solutions seems to be questionable. There will be the need for determination of the statical and dynamic friction coefficient. Rabinowicz (1995) writes on the prediction of the friction coefficient that this is only possible with a deviation of about r 20%. For reliable calculation, this might be hardly acceptable. In order to produce practical useful solutions, it is decided to use a calculation model, which allows free movement of the external tendons for the ultimate limit state at the deviation points. Free movement at the tendon deflection points produces a lower bound solution. Although this is slightly conservative, it is concluded that this is the only safe assumption possible for friction. Fixed connections will be only suitable in the calculation if there are blocking devices used to avoid tendon slip, as done at Second Severn approach spans (Jackson, 1995). Jackson states also that modelling free movement at the deviators is probably demanding. Finding ways of doing this in a satisfactory manner and with reasonable effort will be tried in the project.

2.4.2 Influence of the arrangement of the deviators on the behaviour

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at ultimate limit state The arrangement of the deviators mainly influences the change of eccentricity during application of load. Higher losses of eccentricity can be avoided by proper deviator positions, which can be seen below by comparing the eccentricity of the tendon at a deflected beam with different deviator arrangement.

19

e’1

e’2

Figure 2-12: Influence of a middle deviator on the eccentricity

The eccentricity in the first case (e’1) is smaller as in the second case (e’2). The introduction of an additional deviator at mid span helps to avoid higher eccentricity losses. Matupayont et al. (1994) found that the influence of the second order effect is not unimportant especially for longer unsupported tendon length.

2.4.3 Influence of simply support and continuous support on the ultimate limit state Generally, continuous structures have a higher redundancy and a failure at one section of the structure does not lead necessarily to failure of the whole structure. The released loads will be redistributed to any other point capable carrying this load. Only if enough pins are produced to create a mechanism, the

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structure will fail. But redistribution of the load is only possible by providing enough rotational capacity. Normally, this is done with an under-reinforced section, which means that the steel has yielded at ultimate limit state before the concrete reaches its load carrying limit. Over reinforced sections fail in simply supported and continuous structures suddenly, because the steel is not able to provide further rotation by yielding. The statements made above are not fully

20

true for externally prestressed structures. Their prestressing steel is less likely to reach yield. The ductility is provided by the rotation characteristics of the whole structure rather than yielding at a certain point in the structure (Takebayashi, 1994 and Muller and Gauthier, 1989). Also, it was found that the bending strain increase in the tendon at ultimate limit state is much less for continuous bridges than for simply supported bridges (Ramos and Aparicio, 1995). Although, the question about ductility and deformation will be an important part of this work, continuous structures are beyond the scope of the actual project.

2.4.4 Precast segmental and monolithic bridges Monolithic bridge types have small cracks almost arbitrary distributed along the length of the structure whereas precast segmental types are characterised by larger cracks propagating through the segment joints. The joints can be dry or glued together with epoxy resin. The use of glue only means that the cracks occur next to the epoxy glued joint because of the higher strength of the connector than the adjacent concrete (MacGregor et al., 1989). The cracks still have the same characteristics.

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Figure 2-13: Joint opening of precast segmental construction with external tendons

Bruggeling (1989) states that the carrying capacity at ultimate state may be reduced due to the behaviour of a precast segmental structure in shear. It was already mentioned above that it is nearly impossible to control the crack width in externally prestressed precast segmental bridges. Shear can then only be

21

transferred at a limited compression zone. He believes that there will be serious reduction in the ultimate carrying capacity. Bruggeling doubts also that there will be enough rotation capacity at ultimate limit state, because there will be no inclined shear cracks. On the other hand, Muller and Gauthier (1989) write, “It was found that structures prestressed with either internal or external tendons behave essentially in the same way at all loading stages up to ultimate”. This statement includes segmental bridges. They developed a computer program simulating a loading of such structures and made comparisons to tests. They found that both the load carrying capacity and the deformation behaviour are similar. Looking at the test results from a full-scale destructive test of a precast segmental box girder with dry joints and external tendons (Takebayashi et al., 1994), the failure was occurring at mid span by crushing of the concrete in the top slab. Hence, there was no obvious sign of shear problems. But Rabbat and Sowlat (1989) made different observations in a similar test. In the case of unbonded tendons the joints opened wider in a shear dominated area, and ultimately shear compression failure occurred in the top flange at the joint. However, both found considerable deflection before failure and reasonable ultimate carrying capacity. But the concerns of Bruggeling relating to shear

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problems might be therefore not unfounded.

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3 Collapse analysis The following chapter describes the measures undertaken in order to evaluate the collapse behaviour of the bridge types of interest.

3.1 Investigated bridge types and their differences It is the objective of this dissertation to investigate the ultimate behaviour of the following types of bridges, o

an externally prestressed monolithically built bridge with blocked deviators

o

an externally prestressed concrete bridge monolithically built

o

an externally prestressed concrete bridge consisting of precast segments with dry joints

o

and an internally prestressed monolithically built bridge, which is the conventional prestressed bridge with bonded tendons, as a reference.

The following table shows the differences graphically. The table shows a beam elevation including the tendon with their special characteristics. All of these bridge types are currently in use; none of them is an exotic variant. Only the

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external prestressing with blocking devices at the deviators is not so common.

23

o

o

Internally stressed bonded tendon

(continuous reinforcement)

Monolithic construction

(conventional)

Internal bonded tendons

Characteristics:

1 Internal, monolithic

Type:

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o

o

Monolithically built

24

External unbonded tendon

(continuous reinforcement)

Monolithic construction

Precast segmental type

continuous reinforcement)

segments with dry joints, no

construction (precast

Precast segmental

friction)

friction)

can slip horizontally at

External tendons (tendon

(continuous reinforcement)

o

o

deviators assuming no

can slip horizontally at

External tendons (tendon

4 External, precast segmental

deviators assuming no o

o

3 External, monolithic

Monolithic construction

at the deviators

External tendons, blocked

monolithic

2 External, blocked deviators,

Table 3-1: Investigated bridge types

The following figures show the practical application of those different types. Type 1, the conventional version with bonded tendons and monolithic construction, can be seen below. It demonstrates the position of the tendons, which is within the concrete section. The tendons are typically parabolic draped for post-tensioned systems rather than deflected at one certain point. The structural concrete around the tendon allows this favourable arrangement. However, the tendon arrangement in this project is the same for all bridges. The strands will be deflected only at certain points, as it would be typically found in pretensioned beams.

Figure 3-1: Casting bed for post-tensioned beam (courtesy of Hyder Consulting Ltd.)

The next system illustrated is the Type 3, the external version, which allows movement at the deviators. It appears to be favourable to show Type 3 first, because Type 2 can then be easier understood. The Figure 3-2 and Figure 1-1 show the inside of a externally prestressed monolithically built box girder. It can

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be seen the tendons run through deviators. If the tendon force on one side of the deviator is bigger than the friction force inside the deviator and the tendon force on the other side, the tendon will slip. It is decided to assume zero friction within the deviators. Reasons for this were already comprehensively explained in 2.4.1. Assuming no slippage is a conservative approach.

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Figure 3-2: Bois De Rosset Viaduct (Switzerland), 15-span, 517-m long externally prestressed bridge (courtesy VSL International)

Type 2 is a monolithic externally prestressed construction with blocked deviators. This type was used in the UK for Second Severn approach spans. The designer wanted to avoid tendon slippage at any stage of loading at the deviators, because this allows some simplifications in the design calculations (Jackson, 1995). The construction is monolithically, i.e. it has continuous reinforcement throughout the structure.

Figure 3-3: Construction of the approach spans of the Second Severn crossing (courtesy

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of Gifford and Partners)

The last type (Type 4) investigated is precast segmental construction with dry joints and external tendons running outside the cross-section. The match cast segments are held in place only by shear connectors coined at the joints of the segments and contact pressure produced by prestressing. There is no continuous

ordinary

reinforcement

throughout

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the

whole

bridge.

The

reinforcement stops before the joints and starts again on the other side within the new segment.

Figure 3-4: Erection of a precast segmental bridge with overhead truss (courtesy of Hyder Consulting Ltd.)

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Figure 3-5: Segments with shear keys after production in fabrication yard (Rombach, 1995)

27

3.2 Original bridge data The bridge data is based on the externally prestressed concrete box girders used for the Bangkok Second Stage Expressway (see also Figure 3-4 and Figure 3-5). The whole bridge structure has a deck length of about 60km. A fullscale destructive test was carried out on a test span (Takebayashi et al, 1994). The next figures illustrate the original bridge data:

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43 250

Figure 3-6: Original cross-section and tendon layout (Takebayashi et al, 1994)

28

Average figure [N/m m ²]

Material

Properties item

Concrete

Com pressive strength Modulus of Elasticity Tensile strength Breaking strength Modulus of elasticity

Re-bars Tendons

55-62 43000 390 1920 193000

Table 3-2: Material properties (Takebayashi et al, 1994)

The bridge shown above is designed for AASTHO specifications. The relevant .

codes are the Standard Specifications of 1983 and the Guide Specifications for Segmental Bridges of 1989. It is a precast segmental bridge with external tendons and a span of 43.25m. The segments are typically 3.4m long. There are for different segments, two end diaphragm segments with anchorages, two side deviator segments, one middle deviator segment, and 9 intermediate segments (see Figure 3-6). The two end segments have a length of 1.725m. The bearing’s midpoint has a distance of 0.5m from the edge of the end segment. The cross-section is a typical box girder section. There are 6 pairs of tendons running along the inside of the box girder. Five of them are continuous and running along the whole girder and are deflected three times, two times at the side deviators and one time in the middle. The specification of these five tendons is 19K15. The other pair is anchored between the two the side deviator and is therefore only one time deflected in the middle. The specification is 12K15. They are further described as strands protected with high-density polyethylene ducts (HDPE) and are cement grouted. The total prestressing force was 38443kN after losses. This was found from back calculations of Takebayashi et al. (1994). The whole arrangement had been already installed

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two years before testing. The long-term losses occurred were 12% over a time period of 2 years. The supports were elastomeric bearings, and the bridge had no surfacing and no parapets (Takebayashi et al, 1994). The test span was loaded with steel billets as shown in the next figure over a time period of five days. The deflection of the span; several strains, e.g. the

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tendon strains and concrete strains; joint opening; and slippage at the deviators were measured. The load was being increased until failure. 8 025

10 200

6 800

10 200

8 025

[mm]

Figure 3-7 Test loading arrangement

3.3 Simplified bridge data as basis for the calculations The bridge data described above has then been simplified for the calculations. These calculations for the simplification of the tendon layout can be exactly followed in Appendix A (Derivation of the simplified tendon layout). The cross-section was drawn with a CAD program and the section parameters, the area and the second moment of area, were exactly calculated with this program. A hand check followed. Afterwards a simplification to a suitable cross-section for the calculations was carried out. The simplified crosssection together with the area, the second moment of area, and the section

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modulus are shown in the next figure.

[m]

Area Second m om ent of area about weaker axis Top section m odulus Bottom section m odulus

5.04 m ² 4 3.49 m 3 4.79 m 3 2.10 m

Figure 3-8: Section properties of concrete box

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The final tendon layout consists of 2 pairs of tendons. The originally 5 continuous tendon pairs are grouped together to one pair and the short pair is kept separately. There is no further simplification possible. All the tendon force after losses is given with 38443kN. The exact calculation regarding this topic can be found in Fehler! Verweisquelle konnte nicht gefunden werden..

Neutral axis of the beam 459 1034

1 335

1231 1660

Short tendon Long tendon

10200

9725

13600

9725

[mm]

Figure 3-9: Simplified tendon arrangement in elevation

Tendon stresses and forces after losses Long tendons 2*5*19K15 Short tendons 2*1*12K15 Sum Without losses Long term losses (measured) ultimate strength allowable anchorage stress (AASHTO)

Area 2*5*19*150mm²= 28500 mm² 2*1*12*150mm²= 3600 mm² 32100 mm²

fpu =

0.12 1920 N/mm²

0.70*fpu=

1344 N/mm²

(from measured data)

Force [kN] Stress [N/mm²] 38785.9 1360.9 4899.3 1360.9 43685.2

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Long tendons Short tendons Sum Possible jacking force possible temporary jacking stress (AASHTO)

0.80*fpu= Long tendons Short tendons Sum

Force [kN] Stress [N/mm²] 34131.6 1197.6 4311.4 1197.6 38443.0

1536 N/mm² Force [kN] Stress [N/mm²] 43776.0 1536.0 5529.6 1536.0 49305.6

Table 3-3: Forces and stresses in tendons

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The cross-section of the tendons is chosen in compliance with EN 138-79 and BS 5896: 1980. The 15mm diameter wire is given a 150mm² cross-section. The whole prestressing force of 38443kN, given from the paper about the test, is then distributed to the tendons by weighting their cross-sections. It is assumed all cables have the same stress. The losses given allow a back calculation to the initial stress in the tendons. The losses mentioned by the author of the test paper are long-term losses. Adding these long-term-losses to the tendon stress leads already 1% beyond the maximum tendon stress at the anchorage point of 0.7*fpu (AASHTO; Segmental constructions, 9.1.2). This can only be explained, if the force meant is a force in the tendon further inside the structure. The maximum allowable tendon stress at internal location immediately after transfer is 0.74*fpu (AASHTO; Segmental constructions, 9.1.3). Even in external prestressed structures, there have to be some short-term losses from friction. Hence, this might be a reasonable explanation of this problem. The maximum jacking force can be determined by considering 0.8*fpu tendon stress (AASHTO; Segmental constructions, 9.1.3). The other parameters needed for the calculations, which are assumption for the material properties, have undergone special consideration due to its importance. A special chapter (3.4.5 Constitutive models) is devoted only to explanations about material assumption in relation with strength and elasticity theory.

3.4 FE Calculation Finite element analysis is now by far the most used method for calculations of the continuum. The structure could also be simplified to a line model. Copyright © 2009. Diplomica Verlag. All rights reserved.

However, this is deemed to be not satisfying in this case. At collapse stage the structure will leave the elastic structure response. Considering the special case of concrete there will be large cracks, which are impossible to simulate with a line model. Therefore a continuum analysis is the choice for this investigation. FE methods have proved to be successful in many structural applications, especially in linear elastic calculations. However modelling of concrete

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structures up to collapse seems to be not always successful. One of the main reasons is the complex inelastic behaviour of concrete. Unlike steel components the material response departs at a very early stage from the linear elastic range. The peak stress, especially in tension, is then followed by a very rapid unloading, which makes the use of plastic flow questionable (Kotsovos and Pavlovi, 1995). A detailed discussion follows in a later chapter to constitutive modelling for the analysis. The aim of this project is a full non-linear analysis of this kind of concrete structure up to collapse. This includes nonlinear material behaviour and geometric non-linearity. Reinforced concrete elements taking into account cracking are used in this analysis as well as the non-linear characteristics of the tendons. The successful modelling of the reinforced concrete in the structure is assumed to be the key for a successful collapse analysis.

3.4.1 Technical aspects The finite element calculations were carried on a Sun Workstation with 4 GB RAM. The operational system was UNIX in combination with SOLARIS WINOWS. The geometric model was created with a CAD system. The geometric data was read into PATRAN 2000R2. Meshing and first simple property assignments were then done within PATRAN. Afterwards, an input file for ABAQUS was produced, which was edited with a conventional text editor in order to introduce more difficult elements and other calculation routines. These were the steps used for pre-processing. ABAQUS 5.8 did the calculations and Post-processing

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was done with ABAQUS POST.

33

Figure 3-10: Analysis Process

3.4.2 General approach This section describes the basic ideas to the configurations of the models. The models shall be able to simulate correct responses to vertical load action. It is not intended to investigate torsional action about the longitudinal axis of the bridge, e.g. warping effects or distorsional effects. The main action causing the failure will be bending, although the structures are also subjected to a high compression because of the prestress. Due to the concrete, the section parts are not slender and therefore not prone to local buckling. It is assumed that the failure will occur either by crushing of the concrete in compression caused by bending or the tension zone of the beam fails. Shear overloading is the other possible failure mode. Capability of real shear failure is not the purpose of the

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analysis. These general consideration lead to a model that represents the behaviour in the vertical direction as close as possible. This includes mainly vertical bending in connection with the exact compression representation of the prestress. This is realised firstly by idealising the cross-section in a way that all section properties governing these factors are reproduced in the FE model with

34

very little deviation. These section properties are the second moment of area about the weaker axis, the area, and the position of the neutral axis. Also, the vertical shear area is represented exactly. However, perfect shear behaviour cannot be approached with his data, especially in the case of the precast segmental bridge. The horizontal parameters such as horizontal second moment of area are not represented so close as the vertical one. A line model can satisfy all these parameters.

Figure 3-11: Line model of externally prestressed beam for a linear analysis

However, as soon as it gets to collapse analysis, local overloading is of interest. This can hardly be done with line model. Local overloading in our case is mainly cracking of the concrete. This causes parts of sections within the models, which are not longer perpendicular to the longitudinal axis. All these requirements can be met by a two dimensional analysis, if vertical bending is off interest. A twodimensional model for an external prestressed beam is shown below. Beam element for top slab Shell elements for web

Bar elements for tendon

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Beam element for bottom slab

Figure 3-12: Two-dimensional FE model of externally prestressed box girder

The problem with this model is the inaccurate representation of the non-uniform stresses in the top and bottom slab due to shear lag. The shear lag phenomenon is created by the box characteristics of the structure analysed. An

35

externally prestressed solid beam would be perfectly represented by the twodimensional analysis.

Figure 3-13: Three-dimensional mesh of the box girder

The four structure types approached in this dissertation have different characteristics of the embedding of the tendons in the model. Besides, the structure consisting of the precast segments has special areas where the cracks propagate. Type 1, the internal bonded tendon version, has grouted tendons. The cement grout is supposed to produce full fixity between the concrete and the duct. The representation is done by connecting the elements of the tendon with every node from the concrete web it passes. The next figure shows the symbol given to this type of bridge and the method how it is realised in the calculation.

Symbol

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FE representation

Figure 3-14: Bonded tendon in FE mesh

It can be seen that the tendon determines the position of certain points within the mesh. Thus, the mesh is not always so even as desirable. The tendon is

36

virtually in the same plane as the web. This is not of importance to the FE program. It needs mainly the connectivity data. The model with external tendons and blocked deviators is a further development of the model before. The tendon is only fixed to the points in the mesh where the tendon is deflected or anchored. The next figure shows this.

Symbol

FE representation

Figure 3-15: External tendon with blocked deviators

The figure shows, that the tendon does not determine the mesh spacing in the same way as with type 1. The mesh can be much more even. A typical characteristic of external tendons is, that they have the same stress between two adjacent deviators. Internal tendons have everywhere different stresses, because the stress is directly related to the curvature in the structure. The next type with external tendons, which are free to move at the deviators, is modelled by the means of fixity at the anchorages and kinematic constraints at the deviators. The kinematic constraints have the task to transfer vertical forces coming from the tendon to the structure, but it has to allow slip of the tendons at the deviators. This means it does not transfer horizontal force to Copyright © 2009. Diplomica Verlag. All rights reserved.

the structure. The modelling in Finite elements is nearly the same as the symbolic representation (Figure 3-16). A separate point explains the application of the kinematic constraint (3.4.9). It is essentially a release of one degree of freedom to the adjacent node from the web mesh.

37

Symbol

FE representation

Figure 3-16: External tendon in FE mesh

The main characteristics of this tendon are that the force within the tendon is the same along the whole length considering the deflection angle. The precast segmental bridge has the same tendon properties as the type just explained above. The difference is that it has independent segments cast separately. They are only stressed together by the prestressing force. The main task is the representation of the joint opening. A comprehensive cover of this crack propagation analysis together with a description of the joint opening mechanism can be found in a later chapter (3.4.10 Discrete crack propagation analysis of the precast segmental type with gap elements). The actual section is only meant to give a short introduction in the model approach and the main differences of the meshes.

Segment 1 Segment 2

Segment 3

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Symbol

FE representation

Figure 3-17: Precast segmental bridge in FE model

38

Consideration about symmetry is always useful due to eventual massive computation time savings. It is not used in this case. The structure is not symmetrical in the longitudinal direction (see also Figure 3-6). It was aimed to produce a middle joint from the designer of the test span. Hence the middle deviator is not exactly at midspan. The whole bridge could have been simplified. This is not done, because a comparison of the test results with the results from the analysis is intended. This might give a good check about the quality of the analysis. On the other hand, there is symmetry at the cross-section, which means half the model was sufficient for the calculations. This is also not done. This symmetry is used to check the symmetry of the output subjected to the appropriate load. Summarising, the symmetry is not used in order to allow checks of the output. This means in turn that computational time is sacrificed for this purpose.

3.4.3 Geometric model This point deals with the geometric modelling and shows the geometric model used for the analysis. The geometric data was created with a CAD program. It consists of point and line data, which are positioned on the axes of each element in the simplified cross-section. The geometric mesh has some more lines as needed for the description of the boundaries. The other lines impose restriction to the mesh in order place mesh nodes later at certain points.

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The next figures show the geometric input and some important measures.

39

Figure 3-18: Geometric model

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Figure 3-19: Distance of the tendons to the neutral axis in the geometric model

3.4.4 Element specifications The next section explains the elements used in the analysis. There are two main element types in this model. The first one is the plane element for the concrete regions and second one are one-dimensional elements for the

40

tendons. These elements are obviously positioned in three-dimensions. ABAQUS uses the terms scalar element, which is only a node; a onedimensional element is an element with only one dimension between two nodes, e.g. a bar; and a two dimensional element is an element with two dimensions between the nodes. Although the concrete surfaces from the bridge have three-dimensions, they are modelled as two-dimensional element. The third dimension, the thickness, is assigned as a property. The other option was taking solid elements. This is not considered to be necessary, because the shell elements get integration points over their thickness. This means they can have different stresses along their thickness. Hence, solid elements are not necessary. The meshing of the surfaces was done with PATRAN R2000. This is a program, which can be used for pre-processing of ABAQUS. The other elements are meshed manually by the use of a text editor. There are two element families, which could be used for the concrete surfaces. The first option is the use of membrane elements and the second option is the use of a shell elements. The membrane element has 3 translational degrees of freedom at each node. The simplest shell element in ABAQUS available has 5 degrees of freedom at each node. Hence, the quality of the calculations should be better with the shell element, especially because there are regions where out of plane bending takes place. The use of shell elements tends to be tricky in ABAQUS. They have a top and bottom surface. If the elements are randomly orientated in one single surface, e.g. in the web, the output is wrong. Therefore, extreme care has to be taken by the use of these elements. ABAQUS offers 11 different shell elements for non-symmetrical use Copyright © 2009. Diplomica Verlag. All rights reserved.

in three dimensions and for structural use. There are basically three sub groups. Some shell elements are general-purpose elements. The other groups are thin shell elements and thick shell elements. Thin shell elements are capable of dealing mainly with classic Kirchhoff bending. Thick shell elements are made mainly for shear deformation (Mindlin theory). General-purpose elements give sensible results for both actions (Hibbitt et al. 1998c). The main disadvantage of

41

the general-purpose elements is the much higher computational time. The two other types need to be used in very specific situations, otherwise the serve with wrong results. The calculation undertaken is a non-linear analysis. Hence, computational time is a major factor of consideration, and the use of special purpose elements seems to be desirable. Four node shell elements were chosen. The possible four node shell elements are the S4 element, which is a four node doubly curved general-purpose element, the S4R element, which has similar characteristics except a reduced integration, and S4R5, which has only 5 degrees of freedom at the nodes. The missing degree of freedom is the in plane bending. The last element is not recommended for “thick shells” by ABAQUS. A thick shell has a thickness bigger 1/15 of its characteristic length. S4R and S4 are also finite-strain elements, whereas S4R5 is a small strain element. The latter gives for arbitrary large rotations only small stains. Reduced integration tends to give more accurate results (S4R) than full integration (S4) (Hibbitt et al. 1998c). Fully integrated elements are sometimes to stiff. Based on these explanations, the S4R element seems to be the most suitable element. However, this element had convergence problems during the non-linear algorithm with this particular task. Convergence is used here in the sense, that it does not allow a unique solution with certain parameters. The S4 element is therefore the only element which can be used. 4

4 3

3

4 3

1 1

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1

2

2

1

2

S4R: 4-node reduced

S4: 4-node full

integration element

integration element

Figure 3-20: Nodes and integration points of shell elements

The calculation time is four times longer with this element than with the reduced integration. The assigned thickness to the shells is 0.348m for the web shells,

42

0.298m for the top slab, and 0.18m for the bottom slab. The shell elements have 5 integration point layers throughout the thickness. The outer integration points show always the element output. It might be worth to mention that the four-node element is classified by ABAQUS as a full integration element (Figure 3-20). In fact, this is mathematically speaking not correct, because it is still a reduced integration with discrete integration points and involves still approximations. There is no mathematical proof that some elements work. Some elements have only shown in the past that they give good results. The shell elements are superposed with elements imitating the ordinary reinforcement. This reinforcement is placed at the midsurface of the concrete. ABAQUS creates a “smeared” orthotropic layer parallel to midsurface, in this case at the midsurface, which is based on the modulus and the dimensions of the rebars. The rebar element has only resistance against normal force. This is done by giving the elements only areas in the two reinforcement directions r1 and r2. The element has no response to in-plane shear or to any out of plane action. The symbol for these elements is shown below. The rebar definition was done by hand meshing. Additional information about the amount of ordinary reinforcement can be found in point 3.4.6 (Ordinary reinforcement). The ordinary reinforcement has high importance as it supports the convergence of the concrete shells and it is responsible for major differences in the results.

4 3 4

r2

3 r1

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1

2

1

2

Figure 3-21: Rebar shell element

43

The other basic elements in this mesh are the tendons. These elements have mainly the task to transfer normal forces. Hence, a bar element is sufficient for the purpose. But it was not possible to prestress bar elements. Further explanations to prestress follow in section 3.4.7 (Prestress). Thus, a beam element is used. The name of this element is B31. This name stays for a beam element in three dimensions and with one integration point.

1

1

2

Figure 3-22: Beam element

The actual chapter includes the description of the main element types. The gap element, further properties of the tendons and the rebars follow in later sections. The mesh of the structures can be seen in the next figures. There are two different mesh families looking on the concrete and the connection to the tendons only. The first figure shows the mesh of the internal prestressed structure. The second one illustrates the fundamental mesh of all the external types. The differences of the external types are explained latter. The pure external type has extra kinematic constraints and the precast segmental type has also “gap” elements modelling the joints. The following figures show the

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differently meshed concrete parts of the structure

Figure 3-23: Mesh with internal tendons

44

Figure 3-24: Basic mesh fitted with external tendons

3.4.5 Constitutive models The simulation of a process until failure makes it necessary to introduce enhanced constitutive models. Basic features of non-linear constitutive models are behaviour at elastic range if applicable, information about yielding, plastic flow, strain hardening, and failure characteristics.

Figure 3-25: Elastic-plastic stress-strain relationship for the uniaxial load case

The elastic range can be described by the E-modulus. The yield point is defined with a yield function in the n-dimensional stress space. Hence this function is Copyright © 2009. Diplomica Verlag. All rights reserved.

defined as an equation, which tells us whether or not the material has yielded. A common yield function is the von Mises yield function. This surface is shown in Figure 3-26.

45

V3 Hydrostatic axis

V2 plastic

(V1=V2=V3)

range elastic range

V3

V2

V1

V1

Figure 3-26: Different views on the von Mises yield surface

Materials described with this yield function are assumed to have no yielding dependency from the hydrostatic pressure or mean stress. This stress describes the distance of a stress state at the hydrostatic axis from the origin. This stress is described with the three-dimensional stress tensor as, ªV m «0 « «¬ 0

0 Vm 0

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where -p=Vm=

0º 0 »» , Vm »¼

1 *(V1+V2+V3) . 3

The equivalent of the hydrostatic pressure is often also used with the letter p. The letter p gets here a negative sign because ABAQUS uses p in this way. This negative sign does not concur with the common use. If the material behaviour shows dependency from the hydrostatic pressure, this tensor will determine the starting point of the stress tensor on the hydrostatic axis. This is

46

of importance for the yield surface of the concrete because the radius of yield function reduces with decreasing hydrostatic pressure. The deviatoric stress tensor with the distorsional stresses is then given by ªV x  V m « « W yx « W zx ¬

º ª V1  V m » « » or « 0 «¬ 0 V z  Vm »¼

W xy V y  Vm

W xz W yz

W zy

º 0 »» V 3  V m »¼

0

0

V 2  Vm 0

(Ugural and Fenster, 1995). The deviatoric stress tensor can then be evaluated for its position within the stress space and compared with boundaries of the yield surface. The equation relating the principal stresses to the uniaxial strength is given by ( V 1  V 2 ) 2  V 2  V 3  ( V 3  V 1 ) 2 2

2 V 2y .

This relationship can be rewritten as

q

Vy



1 2 (V1  V 2 )2  V 2  V3  (V3  V1 )2 2



1 2

,

where q is the von Mises equivalent stress. The von Mises equivalent stress can be directly compared with uniaxial material strength. Also, the plane stress tensor will be explained here. This tends to be of particular useful for visualising of sections of surfaces in the continuum, which will be needed for the description of the concrete. Additionally, the idea of the pq stress space will be introduced in this description. The plane stress case or

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two-dimensional case is defined as follows. The equivalent to the hydrostatic pressure is again ªV m «0 ¬

0º Vm »¼

47

-p = Vm =

1 *(V1+V2) . 2

The deviatoric stress tensor is then given by, ªV x  Vm « W yx ¬

W xy

º ªV  V m or « 1 » V y  Vm ¼ ¬ 0

º . V 2  Vm »¼ 0

The definition for q can be expressed from components of this deviatoric tensor, 1 1 q = V1  Vm = V1 - *(V1+V2) = *(V1-V2) 2 2

or

q = V 2  Vm =

1 *(V2-V1) 2

(Fenner, 1987). The second expression changes the sign of q because of the change of V2 and V1. This is correct by looking on the graph below, which represents finally a Mohr circle. The two – dimensional stress state is shown in the next figure.

V2 A 1 2

ª3 0º «0 7» ¬ ¼

*q

1 2

*

1 * * ( V1  V 2 ) 2 2

1

1 * (V1  V 2 ) 2

A

1 2

*-p

5 2

V1

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45°

3

Figure 3-27: Example of a 2D tensor in different coordinate systems

The distorsional stress describes the distance normal to the hydrostatic axis. A similar relationship exists for the p-q surface in three dimensions. The p axis runs then along the three-dimensional hydrostatic axis. Thus, it can be used for

48

assessing, whether or not the material has yielded by comparing the value of q with the diameter of the tube (Figure 3-26) at a particular hydrostatic pressure. The p-q stress space for the von Mises function can be visualised as shown in Figure 3-28. q Von Mises yield line

p

Figure 3-28: Von Mises yield line in p-q stress space

It exists a further parameter “t“ for three dimensional yield functions. This value alters the radius of the yield function around the hydrostatic axis. It is needed for yield functions such as the Durcker-Prager model. However, all functions used in this dissertation have a constant radius around the hydrostatic axis at one particular hydrostatic pressure and this parameter is therefore not needed and will be not further explained. Thereafter the flow rule determines the direction of the plastic strain after yielding in the stress space and the hardening law relates the changing stress to the amount of plastic strain (Woods, 2001). The following two capitals explain the use of the provided constitutive models within ABAQUS and the adjustments made. 3.4.5.1 Constitutive model for steel

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The stress-strain relationships for steel have been defined as follows.

49

Figure 3-29: Tendon stress-strain

Figure 3-30: Ordinary reinforcement stress-

relationship for the distorsional stress

strain relationship for the distorsional

component

stress component

The tendon properties ultimate strength and Young’s modulus are given in Takebayashi et al. (1994). The rest of the curve follows the advice of BS 5400 Part 4: 1990. The behaviour of the reinforcement is governed by the yield strength given in the paper described above. A Young’s modulus of 200000 N/mm² is been assumed as well as full plastic flow after yielding. The properties are the same for tension and compression. The yield function is chosen as the von Mises yield surface. This theory assumes

that

yielding

is

independent

from

the

hydrostatic

pressure

1/3(V1+V2+V3). The function in the p-q space is then simply given by

q

Vy .

This assumption is confirmed experimentally for most metals (except voided metals) (Hibbit et al., 1998a). Yielding is followed by perfect plasticity. Associated plastic flow is used, which means the inelastic deformation occurs in Copyright © 2009. Diplomica Verlag. All rights reserved.

the direction of the normal to the yield surface. 3.4.5.2 Inelastic constitutive model for concrete

The concrete is reinforced concrete in this case. The reinforced concrete is modelled by standard elements associated with the plain concrete model, which are superposed with rebar elements. Concrete/rebar interaction such as bond

50

slip and dowel action is taken into account by modifying some aspects of the plain concrete model as described later. Cracking is believed to be the most important feature of the concrete. Hence, it dominates the considerations. The yield function consists of an isotropically hardening yield surface, which is active in the case of dominantly compressive stresses and an independent “crack detection surface”. The latter determines if the sample fails by cracking (Hibbitt et al., 1998). The concrete model is a smeared crack model, which means that it does not track each individual crack. The stress strain relationship of the concrete, which is used for the analysis, in tension and compression is shown below for the uniaxial response. The compression stress – strain response is based widely on the BS 5400 Part 4 Figure 1 ignoring any factor of safety. The values taken are shown in Figure 3-31 and Table 3-4.

60 50

IC

Input Compression (IC)

ES

Stress [N/mm²]*-1

40

IT

TS

30 20

TS IC

ES

10

-0.0010

IT

0 0.0000

0.0010

Input Tension (IT)

Total strain BS 5400 (TS) Elastic strain component BS 5400 (ES)

0.0020

0.0030

-10

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Strain [-]*-1

Figure 3-31: Stress – strain relationship concrete

51

0.0040

Strain BS 5400 [-] *-1

0.00000 0.00025 0.00050 0.00075 0.00100 0.00125 0.00150 0.00175 0.00181 .. 0.00350

Stress BS 5400 Input strain [- Input stress [N/m m ²] *-1 ] *-1 [N/m m ²] *-1 -0.00076 0 -0.00014 -4.95 0.00 0.00000 0 14.17 26.23 36.18 .. .. 44.03 49.78 53.41 54.95 0.00157 55 55.00 .. .. .. 55.00 0.00350 55

Young's m odulus [N/m m ²] BS 5400 (tangent) Input (secant) 60879 35000

BS 5400 values: Perfect plastic behaviour at strain: 2.44 * 10 4 * fcu Tangent Young’s modulus: 5.5 * fcu * 0.67 1 § § 5500 2 Stress up to perfect plasticity: ¨ 5500 * fcu * H  ¨ ¨ 2.68 ¨ © ©

· 2· ¸ * H ¸ * 0.67 1 ¸ ¸ ¹ ¹

Table 3-4: Numerical stress and strain values for concrete

After several trials an ideal elastic, perfectly plastic relationship has been chosen for compression. The trails included also the exact stress-strain relationships from BS 5400. The drawback from the BS version was a worse convergence of the solution with more computational time by qualitative similar solutions. The ascending part of the relationship is approximated with a secant Young’s modulus providing an ideal elastic behaviour rather than a tangent Young’s modulus with a plastic strain component as suggested in BS 5400 Part 4 Figure 1. This is also done following experience with non-linear collapse analysis in concrete structures (Hibbitt et al., 1998b). Hence, there is a definite yield point in the diagram at 55N/mm². The ascending part in the diagram is Copyright © 2009. Diplomica Verlag. All rights reserved.

followed by a perfectly plastic flow. The plastic behaviour has been chosen because of a lack of other reliable data for the concrete. The tension part of the diagram follows recommendations from Hibbitt et al. (1998a). The fracture ratio to the compressive strength has been chosen to rtV =0.09, which yields in a tension strength of 4.95 N/mm². Normally, a very rapid unloading follows this peak in tension. Actually, it is very difficult to examine the behaviour of concrete

52

in tension because even the stiffest test machine can not exactly follow the path in the stress – strain diagram after the peak stress. As explained above, the reinforced concrete parts consist of the underlying shell elements associated with plain concrete properties superposed by a reinforcement mesh, which consists of rebar rods modelled with the rebar element. An interaction of both elements providing simulation of bond slip has to be done with the descending part in the tension region of the stress – strain relationship of the concrete. The peak value is typical at a tension strain of 10-4. It was found that zero stress at about 10-3 simulates the interaction in a reasonable way (Hibbitt et al., 1998c). The yield and failure surface consists of a “crack detection surface” for the tension region and isotropically hardening yield surface for the compression region. Both functions are independent. The following figure shows a section in the V1-V2 plane through those surfaces. uniaxial tension 4.95 N/mm² V2 uniaxial compression 55 N/mm² biaxial tension crack detection surface

compression surface

biaxial compression 63.8 N/mm²

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V1

Figure 3-32: Yield and failure surface for plane stress for concrete

Some other constants shall be mentioned here. The ratio of biaxial ultimate compressive stress to uniaxial ultimate compressive is 1.16. The magnitude of a principal component of plastic strain at ultimate stress in biaxial compression to

53

the plastic strain at ultimate stress in uniaxial compression is set to 1.28. These constants follow the recommendations from ABAQUS in absence of any other reliable data. The biaxial tension strength can be calculated from the function for the crack detection surface. The value is 2.82 N/mm². Derivations for the functions follow. Construction of the crack surface in three dimensions shows a surprisingly simple shape. It is a funnel covering the principal stress axis in the negative directions. The function is shown in Figure 3-33.

Figure 3-33: Different views on the crack detection surface

The crack detection surface and the compression surface can also be shown in

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the p-q plane (Figure 3-34).

54

q

“compression“ surface q=0.363*p+48.33

“crack detection“ surface with hardening characteristics q

Vt=4.95

Vt · § § 0.38207 V t · ¸ * V t with * ¨¨ 3  0.38207 * 4.95 ¸¸ * p  ¨¨ 2  3 4.95 ¸¹ © © ¹ p

0 d V t d 4.95

Vt=0

Figure 3-34: Concrete failure surfaces in p-q plane

The function of the compression yield is fc

q  3a0p  3Wc

0.

Equation 3-1, (Hibbit et al., 1998c)

The definition of q and p is the same as in 3.4.5. ; a0 is a constant, which is derived from the ratio ultimate strength in uniaxial compression to ultimate strength in biaxial compression; and Wc=f(Oc) is a hardening parameter. There is no stress hardening in this case. For this reason, Wc is a constant and describes with the last part of the equation the pure shear strength of the material. That is, if all stresses are zero except Wxy=Wxz =Wyz z0. The surface is a straight line in the p-q stress space and provides a good match to experimental data (Hibbitt et al.,

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1998c). The value of a0 can be derived by considering a uniaxial compression of p=1/3*Vc and q=Vc. Vc is any compression stress. Putting these relationships into Equation 3-1 leads to

Wc Vc

§ 1 a0 ·  ¸. ¨ © 3 3¹

55

The other part needed for solving the equation is given by considering two negative equal principal stresses and one zero principal stress with q=2/3*Vcb and q=Vcb This ends with the following expression by putting it into Equation 3-1.

Wc Vcb

§ 1 2a0 ·  ¨ ¸ 3 ¹ © 3

Assuming Vcb is the maximum biaxial principal stress and Vc is maximum uniaxial stress, the ratio rbcV of Vcb to Vc is 1.16. 1.16 was already determined above. Then, the constant a0 can be calculated from

a0

3*

1  rbcV = 0.21 1  2rbcV

(Hibbit et al., 1998c).

Wc can now determined by considering uniaxial compression with again p= 1/3*Vc and q=-Vc but with Vc=-55 N/mm², the uniaxial compression strength. After solving Equation 3-1 the value becomes 27.9 N/mm². The final expression for compression yield is therefore q

3 * 0.21 * p  3 * 27.9

0.363 * p  48.32 .

As already shown in the stress-strain relationship of the concrete (Figure 3-31) no hardening is assumed. The compression yield surface is then used with associated flow.

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The crack assessment follows the guidelines of Hilleborg et al. (1976). Hilleborg suggests that a certain fracture energy

Gf

³V

te

du

is needed to produce a crack. The applied tensile stress is Vte and the width of the open crack is u. Gf is a material constant in this case, which is at about 180

56

N/m for a concrete grade 55. By assuming a constant G for the specimen the crack width is fixed for a certain displacement u and tensile stress respectively. This does not mean there is single crack across the concrete. It is not possible to determine discrete cracks with this model. It is a smeared crack model, but it is possible determine the crack width of all cracks in the material and their directions. A displacement consists then of a displacement u=ucr+uel, where ucr is the crack width and uel is the elastic displacement. The crack detection is formulated with the equation

ft

§ § V · b V · q  ¨¨ 3  b0 ut ¸¸ * p  ¨¨ 2  0 ut ¸¸V t Vt ¹ 3 Vt ¹ © ©

0,

Equation 3-2, (Hibbit et al., 1998c)

where bo is a constant. b0 is defined from the tensile failure stress, if the second principal stress has the value Vuc and the third principal stress is zero. This condition can be seen in Figure 3-32. V t is a function of O t , which describes the hardening characteristics of the concrete in tension. This time, the hardening parameter is active. The hardening process is shown in the next Figure. Tension stress, V

Failure point

“Tension stiffening”

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curve (hardening part)

Reloading curve, if failure

Tension strain, H

point was overcome

Figure 3-35: Tension softening for the concrete model

57

Actually, it is a softening process. Hardening means in this context that there is a change in ultimate strength after the yield point. Hardening will end, if Vt=0 and the Equation 3-2 becomes q-3p=0. The value of bo can derived by considering maximum stress at one principal axis in compression ( Vuc ) and the maximum possible tensile stress at the same time (f* Vut =f* rtV *- Vuc ). f is determined by the user. It relates the absolute maximum tensile strength ( Vut ) to the former condition. The chosen value is 1/3, which is recommended from Hibbitt (1998a). The other value needed is rtV , which was already mentioned above. This value relates the maximum uniaxial tensile strength to the maximum uniaxial compression strength Vut = rtV *- Vuc . It is chosen to 0.09. Cracking failure occurs now at the principal stresses V uc ,f* rtV *- Vuc and 0 (see also Figure 3-32). The hydrostatic pressure is then 

p





1 1  f * rtV Vuc 3

and the von Mises stress is given by q

2



V uc  V uc * f * rtV



2

2

 V uc * f * rtV .

bo can then be derived by assuming no hardening has occurred so far ( Vut

V t ).

Putting all these relations above into Equation 3-2 leads to b0=0.381. Finally the expression of the crack detection surface is

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q

V · V · § § ¨ 3  0.381 t ¸p  ¨ 2  0.127 t ¸V t . 4.95 ¹ 4.95 ¹ © ©

The so far only stated biaxial tensile strength can now calculated from p=  2 / 3 * Vbu and q= Vbu t t . Equating the above expression for the crack detection surface with these relationships and allowing no hardening ( Vut

V t ) gives

Vbu t =2.82 N/mm². The model is now able to detect the crack. If the first time the crack detection surface is overcome, the calculation will assume cracking has

58

occurred. Cracking is irrecoverable. The normal of the crack runs along the maximum principal strain. There are effectively three possible crack directions in the three-dimensional stress state, one for each principal strain. The crack influences the model as it damages the elasticity. After cracking a principal strain component is set to be a crack strain, which means, following Hilleborg’s (1976) relationship, it can be converted to a crack width and all stress components following this principal strain direction are removed. This is done by modifying the D-matrix, which relates the strain to the stresses in an element. It was found with trails using different quantities for the concrete model that such changes significantly change the quantities of the results. These findings underline again the importance of the concrete model and also the need for reliable data of the concrete strength, which is not available in this case. The calculation can therefore only aim to reach qualitatively exact solutions. Hence, the different load - deflection curves for the different prestressed concrete types can have the right ratio to each other, but the absolute values derived with the model will not necessarily consistent with those of the full-scale destructive test of the precast segmental bridge.

3.4.6 Ordinary reinforcement The ordinary reinforcement is provided by the means of rebar elements. These elements can be defined singly or can be embedded in oriented surfaces within an at least two-dimensional element. The latter method is used here. Section 3.4.4 explains the configuration of the element type. This section deals with the amount of reinforcement needed and how it is applied. The ordinary reinforcement is of high importance for the model. There are two reasons. The

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first one is the nature of the structure to simulate. It is a reinforced concrete structure. This might sound trivial, but it is often tried to model the concrete as a homogeneous isotropic material. As long as there is no “serious” crack in it, there is every justification to do so. However, reinforced concrete and prestressed concrete overcoming the tensile strength of the concrete must be modelled with any reinforcement. “Serious” cracks have therefore two

59

characteristics. They have to be open, and they do not transfer any stress component from the principal strain normal to the crack propagation. Both of these conditions are true in certain areas of this model. The other reason, why reinforcement is needed, is the constitutive model for concrete used in this analysis. This model does not or hardly converge without any reinforcement. The manuals of ABAQUS (Hibbitt et al. 1998a) mention this phenomenon. Smeared reinforced concrete models own this characteristic. This was also found during the analysis. The amount of ordinary reinforcement is taken as 1.5%. The magnitude is calculated from EC2 (DD ENV 1992-2:2001). The calculation can be followed in the Appendix. The same amount is taken for the flanges as well as for the webs. The reinforcement mesh is orthogonal placed in the midsurface of the concrete shell. For reasons of simplicity, no attempt is made to place the reinforcement at its right position closer to the surfaces of the shells. Although the reinforcement input has the information of the diameter and the spacing, the actual calculation is performed only with a smeared orthotropic layer. Hence, the value 1.5% reinforcement and the orthogonal directions are sufficient to describe the actual input. But this amount is important because it describes a major difference between the bridge types. The monolithic types have this reinforcement throughout the whole structure. The precast segmental type has the continuous reinforcement only in the segments. It stops at the joints. This is an important fact, because this investigation aims to find out, if this has any significance.

3.4.7 Prestress

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The section deals with the application of the prestress in the FE models. There are two methods of doing it. The prestress can be applied as external forces on the model and the constitutive model of the prestressing steel has to be adjusted by lowering the yield point of the prestressing steel. This is the recommended method from several authors, for instances Kotsovos and Pavlovic (1995). The other method is to apply a real prestress to the tendons

60

with initial conditions. The second method is chosen in this analysis. The first method represents the classical method of doing it and involves deviations from the original model, which are not necessary with FE technique available nowadays. The main shortfall is the unsatisfactory modelling of the behaviour of the model at the anchorages. The anchorage load will be typically underestimated. This might be satisfactory in a high number of cases, because it can be considered as a local effect only. The second approach, applying an initial prestress to the tendon, is the more rigorous analysis. One of the aims of this analysis is to find out the tendon stress increase at ultimate state. This can be easily done with this method. The tendon stress can be read at failure and compared to the initial stress. Besides, this is exactly, what is found in reality. The method with the substitute forces allows not such a direct comparison. However, the direct method has also several difficulties. The first is applying the right prestress. The problem is the elastic shortening of the structure, which means if the prestress is applied to the tendon and via the anchorages to the structure, the concrete bridges shorten elastically. The effect can be fond in reality in pretensioned structures only. This does not happen with post-tensioned structures. Typically the jack is used to apply the prestress wanted. In fact, the elastic shortening is even not noticed during the jacking process, because the jacking force is applied directly against the structure. Elastic shortening is also depended from the dead load and the position of the anchorage itself. ABAQUS offers here an easy way of keeping the prestress constant during the initial equilibrium stage. A function is available, which is called “*Prestress hold”. This method works excellent with the internal prestressed version and the externally prestressed version with blocked

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deviators. The prestress can be set to the wished value and is kept constant during the initial equilibrium state. The initial equilibrium state consists mainly of applying the prestress, the dead load and occurrence of any kind of shortening. However, this does not work for the pure external type and the precast segmental version. The allowance of movement at the deviators makes the difference. If an initial stress is applied to the tendon and the first equilibrium step is started, the tendon moves at the deviator as long as it reaches

61

equilibrium for any reason. This lead in trails to tendon movements of several meters by applying the prestress and the dead load only (Figure 3-36). Of course, this does not happen in reality. The solution for this problem is a more empirical search for the initial prestress. The tendon is stressed considering elastic shortening, which means higher than original wished. The function *Prestress hold” is omitted, elastic shortening occurs and the prestress reduces. The Starting prestress can be found with some trials or by simple hand calculations considering elastic shortening. It might be worth mentioning that the distribution of tendon stress in reality even at initial state is varying along the length mainly due to friction and deflecting forces, and is still more complicated than simulated in this model. But it is believed that the calculation undertaken are a reasonable approximation. 1N

Deviatior symbol 1N D 1N

1N

Tendon Equilibrium step:

Start in first equilibrium step: Unbalanced prestress forces at deviator

*Prestress hold function holds the prestress constant and system tries to establish equilibrium (D=180°), tendon moves

Figure 3-36: Equilibrium problems with *Prestress hold function at deviation point

The second problem initiated by this calculation approach is the concentration of stresses at the anchorages. This might sound logical and happens in reality. Hence, it is also an advantage, because it is close to the real

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condition. But the high stress at the anchorages caused a series of anchorage failures during the analysis. The anchorage zone had to be heavily reinforced. The amount of reinforcement is not further specified. The amount at the anchorage zones was simply increased until the anchorage were save, because it is not the aim of this calculation to investigate the anchorage zones.

62

The elements for the tendons are beam elements as specified in 3.4.4 with a very low second moment of area. The tendon element itself has a Young’s modulus of 0.1 N/mm². It is effectively a rubber tendon. But this rubber tendon is reinforced with a rebar element, which has the properties of the real tendons. The combination is caused from the restrictions imposed of ABAQUS. It was not possible to prestress any element. But it was possible to prestress a rebar element, but a rebar element needs a parent element. The area for each long tendon

is

28500mm²/2=14250mm²

and

for

each

short

tendon

is

3600mm²/2=1800mm². The Young’s modulus is 193000 N/mm². These properties are further explained in section 3.3. The applied prestress for the internal type and the external type with blocked deviators is 1197.6 N/mm². The applied prestress for the other two versions is 1275 N/mm². The effect is the same and reasons explained above. Other information about the time dependent application of dead load and prestress is available in the next point.

3.4.8 Material and geometric non-linearity All calculations have been carried out firstly linear. This tends to be of significant importance. Firstly, the mesh can be verified for sensible stress magnitude output and sensible boundaries of the stress isobars. Critical points for the later non-linear analysis can also be seen in this pre-analysis. The source of non-linearity in this case is mainly material non-linearity. However, considerable deformation can also be expected. This makes the use of a combined non-linear calculation algorithm necessary. A non-linear

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calculation is characterised by serious of linear steps divided into increments and iterative steps. The prerequisites for the calculation of material non-linear analysis are prepared by introducing non-linear constitutive models (3.4.5). This analysis is divided into two steps. The dead load and prestress are applied in step 1. The second step is the application of two point loads over the webs and the middle deviators until the structure fails.

63

Figure 3-37: Loading arrangement

The first step uses the method of Newton and the second step uses the method of Riks. The reason for changing the methods is the higher capability of the Riks algorithm. This method is even able to cope with sign change of the stiffness. As already mentioned, the first analysis step involves applying of the prestress and the self-weight. This has to be done careful because an unbalanced apply of those components could already destroy the bridge or damage the elasticity of the model with unreasonable high cracking at an early stage. Cracks are stored for the whole calculation, which means in practice they do not vanish during the life of structure but can only growth. Further explanation on damaged elasticity can be found in section 3.4.5.2. The introduction of the prestress does not typically damage the elasticity of structure in reality. The economical Newton’s method can be used here. This algorithm converges quite rapidly but is not capable of dealing with qualitative stiffness change. Such a process is not expected by application of the prestress, because the concrete tensile strength will not overcome anywhere in the structure and no cracks influencing the calculations should growth. The original Copyright © 2009. Diplomica Verlag. All rights reserved.

method of Newton is slightly modified in this calculation. The calculation is divided into increments. The next figure demonstrates the convergence scheme used. Only a short introduction to the method will be given. Further formulations are omitted here, because they are readily available in Finite Element textbooks, for instance Zienkiewics and Taylor (1991).

64

Load

Increment 2

Load

Pi Ka Ko

Increment 1

Ia

Displacement

Displacement uo

ua

Figure 3-38: Newton method

The black graph represents the true load deflection response and the blue set of linear curves is the path, which the calculation takes. The first displacement is gained with the initial stiffness matrix Ko (uo=Ko-1*Pi), where Pi is the load change in the current increment. The structure nodes are then updated and a new stiffness matrix (Ka) is established. A new force is then calculated with this stiffness matrix (Ia=Ka*u0). The difference between Ia and Pi is the out of equilibrium force or residual force. This residual force is applied to the structure, and the same process is repeated until the residual has a reasonable small value. The second step is done with the Riks algorithm, which is a special calculation algorithm capable of working with very high non-linearity even with qualitative stiffness change. This algorithm is not as common as the former, but is typically used for every kind of instable problems such as collapse and buckling analysis. For instance, the phenomenon of snap through in dome

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structures can be also investigated with this approach. The basic algorithm stays the Newton method. The Newton method has typically problems with the limit points (Figure 3-39). The solution tends to diverge, as the stiffness matrix gets very small.

65

Load

Limit points

Displacement

Figure 3-39: Typical structure response suitable for the Riks algorithm

The Riks method uses the tangent of current solution and moves forward a definite length on this tangent, also called a certain arc length. The arc length replaces the load increment. A normal to this tangent is calculated and the Newton method iterates between this normal and the true solution. The next increment starts again with the tangent of the former equilibrium point (Hibbitt et al. 1998c). Load Normal Tangent

Newton iterations Equilibrium solution Displacement

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Figure 3-40: Riks iteration

3.4.9 Kinematic constraints All the features so far were applied to all the models. This and the next section explain characteristics, which are only applied to certain models. The kinematic coupling constraint is used for the pure external type and the precast

66

segmental version. The task of these elements is to release one degree of freedom of the tendons at the deviators in order to allow movement. The other models have tendons, which are simply fixed to the nodes of the webs (compare also 3.4.2). The tendon movement at the deviators is expected to be in the region of some centimetres. This expectation is based on measures of the full-scale test from Takebayashi et al. (1994).

Slip at deviator

Slip at deviator

Figure 3-41: Free movement of the cables at the deviators (symbolically)

As the beam deflects, the cables are tensioned. But the slip at the deviators allows he redistribution of the cable force along the whole length. The cables force is constant along the whole length considering the horizontal part only. A cable, which is fixed has the highest horizontal force in the middle and the outer parts of the cable are less stretched. Friction is neglected for reasons of safety. This is explained in point 2.4.1 Influence of tendon slip on the ultimate limit state. Because the tendon movement is considered to be very small, simple remove off one degree of freedom (the horizontal translation) at the deviation point seems to be sufficient. This can be done by the means of kinematic constraint, which allows constraints to be set in relation to other nodes at the Copyright © 2009. Diplomica Verlag. All rights reserved.

model. The kinematic constrain chosen acts effectively as a link, without to bother about cross-section and deformation of this link. The kinematic constrain has no weight and is perfectly stiff. It relates the movement of a node to other with a rotational constraint.

67

Nodes at web Link Node at tendon

Figure 3-42: Links connecting web nodes and tendon nodes at model

3.4.10

Discrete crack propagation analysis of the precast

segmental type with gap elements The model of the precast segmental type with external tendon is probably the most advanced model of this investigation. The separate segments of this type are only stressed together with the tendons. Also there are coined shear connectors, which match each other on the adjacent segments. The task is to model the behaviour of a joint, which does not transfer any tensile stress, but does transfer compression and shear. If the pressure on the joint gets smaller, the segments extend and the joints will open up after overcoming the zero stress. The parts of the joint, which are open, do not transfer any more stress.

Segment 1 Segment 2

Segment

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3

FE representation

Figure 3-43: Joint opening

The crack does not open in the sense that the segments rotate about the top node. But, the crack is still closed at parts where zero stress is not overcome.

68

Figure 3-44: Joint propagation

This particular form of behaviour can be modelled with contact analysis. It can be either performed with surface interaction or contact elements. The contact element option is chosen, because ABAQUS has an element, which is suited for this problem. It is a gap element, which allows specifying the direction, in which the gap should open. If the gap has opened, it does not transfer any force. The problem is to specify the direction of the gap. This has to be done in global coordinates. As the bridge deflects, the normal to the longitudinal bridge axis rotates as well. Hence, the gap direction changes with every calculation increment and it is difficult to determine the crack opening direction. By providing no gap direction, the different segments slide along each other due to their weight, because the gap element opens then directly in the direction,

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where the load is applied.

Figure 3-45: Problems with uniaxial gap element

69

This problem was overcome by using a continuous connection at the top joint. The connection acts as a conductor that introduces the direction of the gap.

Figure 3-46: Final use of the gap elements

The use of a considerable number of gap elements along the joint allows the satisfactory simulation of the crack propagation process (Figure 3-44). The

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finished model is shown below.

70

Figure 3-47: Mesh showing segmental joints

Detail A

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Figure 3-48: Joint more detailed

Figure 3-49: Detail A, Gap elements between segments

71

3.4.11

Summary of the dividing features of the different

structure types for the FE analysis The following section summarises briefly the dividing features used for the FE models. The symbols used are the same as in the Table 3-1: Investigated

bridge types, which describes the differences of the types in reality. The internally prestressed concrete bridge: o Tendon connected to all nodes it passes

The externally prestressed concrete bridge with blocked deviators: o Tendon is only connected to the nodes at the

deviation points The pure externally prestressed bridge: o Tendon is free to translate horizontally at the

deviators The externally prestressed precast segmental bridge: o Tendon is free to translate horizontally at the

deviators o Large cracks propagate through the joints

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between the segments

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4 Results This chapter summarises the main results from the analysis conducted according to the objects. The purpose was to investigate the ductility or loaddeflection behaviour respectively of four different prestressed concrete bridge types. The four bridges have the same properties except their dividing features described in 3.4.2 General approach and 3.4.11 Summary of the dividing features of the different structure types for the FE analysis.

4.1 Load deflection behaviour The behaviour of the bridges was calculated up to failure. The recorded load deflection curves are compared in Table 4-1 and Figure 4-1. Table 4-1 shows the deflection of the different types as a function of the applied midspan live load moment. The load is applied as increasing point loads above the webs and the middle deviators of the box girders besides the dead and prestress load. This loading arrangement yields to a flexural failure. The table shows the range in which the structures respond in an elastic fashion and followed by a plastic part until failure. The visualisation of those responses is done in Figure 4-1. The elastic range has similar characteristics in all cases. The only exception is the externally prestressed precast segmental bridge. The plastic behaviour starts already at 50MNm applied midspan moment (live load). The others stay linear up to 60MNm. In general, all types show the same load deflection modulus. A common feature of the load-deflection lines is, that all start of with approximately the same negative deflection. This is because of the

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prestress which bows the bridge up, as only dead load and prestress is applied. The region up to 60MNm (50MNm) could be called, the working or service state. Thereafter, substantial differences occur.

73

Applied m idspan m om ent [MNm ] 0.00 5.00 10.00 15.00 20.00 25.00 30.00 35.00 40.00 45.00 50.00 55.00 60.00 65.00 67.75 70.00 75.00 80.00 85.00 90.00 95.00 100.00 100.38 102.68 105.00 110.00 113.19

Internal

External, blocked deviators

0.0339 0.0247 0.0193 0.0139 0.0085 0.0031 -0.0023 -0.0077 -0.0131 -0.0189 -0.0249 -0.0308 -0.0381 -0.0517

Deflection [m ] 0.0338 0.0346 0.0246 0.0291 0.0192 0.0237 0.0138 0.0182 0.0084 0.0128 0.0031 0.0073 -0.0023 0.0019 -0.0077 -0.0036 -0.0131 -0.0091 -0.0185 -0.0147 -0.0256 -0.0205 -0.0319 -0.0262 -0.0470 -0.0332 -0.0659 -0.0519

-0.0709 -0.0923 -0.1121 -0.1377 -0.1744 -0.2318 -0.3011

-0.0952 -0.1187 -0.1501 -0.1841 -0.2216 -0.2694 -0.3368 -0.3573

External

External, precast segm ental 0.0341 0.0285 0.0229 0.0172 0.0116 0.0060 0.0003 -0.0053 -0.0110 -0.0168 -0.0284 -0.0709 -0.1357 -0.2202 -0.2811

Deform ation characteristics

Elastic range

Plastic range Failure

-0.0790 -0.1124 -0.1493 -0.1952 -0.2499 -0.3119 -0.3832 -0.4299

-0.3795 -0.4680 -0.5304

Table 4-1: Load deflection behaviour up to flexural failure

The precast segmental bridge leaves first the linear response path dropping down fairly suddenly in comparison to the others. The computer stopped the calculation for this bridge type first. This happens, if the computer cannot establish equilibrium any more. The related failure load to the maximum

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deflection of 0.28m is shown in bold numbers with 67.75 MNm. The highest ultimate load was found with the internally prestressed monolithic construction. The structure collapsed at 113.19 MNm and a deflection of 0.53m. The plastic response of the internally prestressed concrete bridge is much softer than the non-linear behaviour of the externally prestressed precast segmental bridge. The overall behaviour of the others is in between these two extreme types.

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-0 .6 0

-0 .5 0

-0 .4 0

-0 .3 0

-0 .2 0

-0 .1 0

0 .0 0

0 .1 0

0

I

EB

E

EP

In te rn a l (I)

E xte rn a l , b lo c k e d d e via to rs (E B )

E xte rn a l (E )

E xte rn a l, p re c a s t s e g m e n ta l (E P )

20

40

75

EP

60

80

A p p lie d m id s p a n m o m e n t [M N m ] (L ive lo a d o n ly)

Wobble

EB

E

100

I

Figure 4-1: Comparison of load deflection behaviour of different types of prestressed concrete bridges for flexural failure

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Deflection [m]

120

The magnitude of the ultimate moment of the two external monolithic types is approximately the same. The externally prestressed concrete bridge with blocked deviators has an ultimate moment of resistance of 100.38 MNm at a deflection of 0.357m. The externally prestressed concrete type reaches its capacity at 102.68 MNm but at a much higher deflection of 0.43m. Also, it was found a wobble in the curve in both monolithic external types by leaving the linear response. The wobble will be also a topic of the discussion.

4.2 Tendon stress increase up to failure The other important result aimed with this investigation is the increase in tendon stress up to failure. The first figure on the next pages illustrates the initial tendon stress and the tendon stress increase until immediately before failure. The stress distribution can also be seen there. The internally prestressed bridge shows the highest stress increase as it might be expected. The increase is proportional to the moment in the beam at every point in the bridge beam. Hence, the stress increase flattens out in the direction to the supports. The maximum increase is 52.8% for the long tendon. The second version with the external tendons, which are blocked at the deviators, is characterised by a lower stress increase. The value at the longer tendon is 30.1%. The tendon stress is evenly distributed along the middle span of the strand. The pure external type shows slightly less tendon stress increase. However, the stress is uniformly distributed along the whole length of the tendon because it is able to redistribute the stress over the deviators. The maximum increase is 29.3% for the long tendon. The external version, which consists of precast segments, has the same general characteristics. The magnitude of the stress increase is only Copyright © 2009. Diplomica Verlag. All rights reserved.

20.0% for the long tendon. Figure 4-3 compares the stress increase in a column chart, in order to show the key differences in an appropriate scale.

76

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Figure 4-2: Tendon stress increase for all bridge types

77

60 52.7

Increase in %

50

40 30

29.3

30

20 20

10

0 Internal, bonded

External, blocked deviators

External

External, precast segmental

Figure 4-3: Comparison increases of tendon stress at ultimate state at midspan in % for the long tendon

4.3 Other results There are of course other results following this investigation besides the main objectives, such as the tendon slip at the deviators, the joint opening of the precast segmental bridge, and the anchorage forces of the external types. These three interesting measures will be shown in this section. Also some stress output will be shown in order to give some feeling for the capabilities of

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the models, although it is not necessary for the objectives of the project. The first measure is the tendon slip at the side deviator for the externally prestressed monolithic construction. The slip for the precast segmental construction is essentially the same. But this represents not the true tendon slip behaviour, as it is no friction assumed and it is an idealised tendon. The validity will be later discussed. The tendon starts of with a negative slip, because it had

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been stressed and slipped towards the anchorages. Then, the tendon moved towards the midspan as the bridge was increasingly loaded. The maximum magnitude is 18mm. The wobble in the graph is the same as in the moment deflection curves of the external constructions. This will also be discussed later.

Side deviator of long tendon Negative slip

Positive slip

Figure 4-4: Position of the measure and direction signs

20 18 16

Slip [mm]

14 12 10

Tendon Slip at side deviator of externally prestressed bridge

8 6 4 2 0 -2 0

20

40

60

80

100

120

Applied midspan moment [MNm] (Live load only)

Figure 4-5: Moment - tendon slip curve

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The next measure is the joint opening at midspan of the precast segmental construction. The opening is recorded at the lower edge of the web. The abscissa is marked again with the applied mid-span moment. The exact position of the distance concerned can be also seen on the next figure.

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45 40

Joint opening [mm]

35 30 25 20 15 10 5 0 -5

0

20

40

60

80

Applied midspan moment [MNm] (Live load only)

Location of measurement

Figure 4-6: Joint opening of middle joint

The anchorage stress is the other point worth to mention, because the results are somewhat surprising. The highest anchorage stress can be found in the pure external version. This stress overcomes even the stress of 0.8*fpu, which is the maximum stress during the jacking process. The magnitude of 0.8*fpu is 1536 N/mm² (see Table 3-3: Forces and stresses in tendons). The maximum stress in the tendon at the anchorage is 1540.4 N/mm² for the long one. The short tendon has an even higher anchorage stress of 1592.7 N/mm².

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This can be seen in Figure 4-2. The magnitude is the same along the whole length of the structure, not considering deflection effect. Although, it is not the main objective, some stress output graphics will be shown. The intention is to give the possibility to verify to a certain degree the performance of the finite element models.

80

[N/mm²]

81

Figure 4-7: Internally prestressed bridge immediately before failure, average membrane stress in the longitudinal direction

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82

direction [N/mm²]

Figure 4-8: Externally prestressed bridge with blocked deviators immediately before failure, average membrane stress in the longitudinal

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83

Figure 4-9: Externally prestressed bridge immediately before failure, average membrane stress in the longitudinal direction [N/mm²]

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84

direction [N/mm²]

Figure 4-10: Externally prestressed precast segmental bridge immediately before failure, average membrane stress in the longitudinal

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5 Discussion of the results This chapter is devoted to the discussion and interpretation of results. The first part explains possible reasons for the different characteristics and the other points deal with the validation of the finite element model. Furthermore, comparisons will be made to other investigations and the full-scale test on which the bridge data is based.

5.1 Interpretation of the results The results are surprisingly clear and comply with instinctive expectations, which one might have. The results show significant differences in the behaviour of the four investigated bridges. The three monolithic bridge types behave essentially the same. They have a similar linear response and nearly the same starting point for the plastic deformation. The plastic branch of the graph is descending in a ductile manner. The deformations at failure are relatively high, which is desirable. The precast segmental bridge is a different matter. Although the linear part in the load deflection diagram is similar, the non-linear response starts earlier. The slope in the diagram (Figure 4-1) towards failure decreases rapidly in the diagram. The deflection and the ultimate capacity cannot compete with the characteristics of the monolithic types.

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Internally prestressed bridge (monlithic) Externally prestressed bridge with blocked deviators (monolithic) Externally prestressed bridge (monolithic) Externally prestressed precast segmental bridge

Maximum moment from live load[kNm] 113.19

Maximum deflection [m] -0.5304

100.38 102.68 67.75

-0.3573 -0.4299 -0.2811

Table 5-1: Ultimate moment of resistance and maximum deflection

The main reason for this seems to be the ordinary reinforcement. The monolithic types have a continuous reinforcement throughout the structure. The ordinary reinforcement of the precast segmental type stops before the joint. This underlines the importance of the continuous ordinary reinforcement as well as

85

its substantial contribution to the load capacity and the overall behaviour of a structure. It was found that the different tendon arrangements have not such a high influence as the normal reinforcement. However, it makes a difference if the tendon is an internal one or an external one. The internally prestressed bridge reaches the highest ultimate moment and the highest deflection overall within the monolithic types. The two external versions, one with blocked deviators and the other with tendon slip at the deviators, show less ultimate capacity and deflection as the internally prestressed bridge and behave similarly. But the pure external type has a higher deflection. The higher deflection might be explained by considering the increasing stress in the middle part of the tendon in a real externally prestressed structure. As the friction force is overcome at the deviator the tendon slips through it. The structure needs then further deflection to increase the tendon stress again. The question of ductile failure can now be approached. Ductile failure was defined in section 1.3 as “providing ample warning of impending failure”. Ample warning might be related to visibility. If it is assumed that a sag of L/250 is noticeable or worrying (EC2 and Hurst, 1998), a deflection of 17.3cm will be the ample warning for a girder of a span of 43.25m. The precast segmental bridge has reached a midspan live load moment of approximately 62.34MNm at this moment. It was found in the analysis that the girder fails under 67.75MNm. Hence, the load increase until failure needs to increase by only 8.6%. The weakest monolithic girder, which is the externally prestressed concrete bridge with blocked deviators, needs an increase of 20.4% from the warning until failure. This is more than double the

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value of the segmental one.

Internally prestressed bridge (monlithic) Externally prestressed bridge with blocked deviators (monolithic) Externally prestressed bridge (monolithic) Externally prestressed precast segmental bridge

Applied live load midspan moment at a deflection of 17.3cm in [MNm] 89.83 83.37 82.70 62.34

Moment at failure Difference [MNm] in [%] 113.19 26.00 100.38 102.68 67.75

Table 5-2: Difference between applied moment at L/250 and failure

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20.40 24.15 8.69

Although the assessment of ductility is subjective and this is certainly a difficult point to decide, the segmental bridge is categorised in this study as a structure with brittle failure. This is done specifically with respect to the performance of the other bridges. Another description would not illustrate the notably qualitative difference. All the other types show the warning much earlier before failure and they are categorised to fail ductile. The tendon stress increase produced from bending in the bridge, which is a concern in the codes of practice, is in all cases higher than the 100 N/mm². 100 N/mm² is suggested as a calculation value for hand calculations of the ultimate moment of resistance in the BD 58/94 and the AASHTO for externally prestressed concrete structures. All bridge types, even the precast segmental version, overtake this value easily.

Internally prestressed bridge (monlithic)

Intial tendon stress [N/mm²] 1197.9

Tendon Tendon stress at stress midspan before increase failure [N/mm²] [N/mm²] 1830.1 632.2

Tendon stress increase [%] 52.8

Externally prestressed bridge with blocked deviators (monolithic) Externally prestressed bridge (monolithic)

1197.9 1190.9

1558.0 1540.4

360.1 349.5

30.1 29.3

Externally prestressed precast segmental bridge

1192.9

1431.6

238.7

20.0

Table 5-3: Overview tendon stress increase of the longer tendon

The findings of this investigation allow no statements about tendon stress increase in continuous structures, as the tendons have not to be anchored over the support in continuous structures. The worst bending action on one span can be typically reached with load at one span and no load on the adjacent. The Copyright © 2009. Diplomica Verlag. All rights reserved.

higher loaded span extends the tendon more than the other, and might pull the tendon literally in this higher loaded span in an externally prestressed structure. Such interaction needs further research. The load-deflection graphs have some interesting characteristics. These characteristics will now be interpreted. The wobble of the load-deflection graph

87

of the externally prestressed monolithically built bridges follows directly the linear part of the structure response.

0.10 Applied midspan moment [MNm]

0.00

-0.10

20

Deflection [m]

0

40

60

80

100

12

wobble EP

-0.20 EP

-0.30

-0.40

E

EB

External, precast segmental (EP)

EB

External (E)

E

External , blocked deviators (EB) I

-0.50

I

Internal (I)

-0.60

Figure 5-1: Special characteristics in the load-deflection curve

The internally prestressed as well as the precast segmental structure do not show such behaviour. This wobble might be caused by propagation of serious cracks in the tension zone of the structure. The tensile strength of the concrete is overtaken at this stage and the load is suddenly transferred to the tendon and the ordinary reinforcement respectively, which might cause this phenomenon. However, this is probably not the case in reality. The discrete mesh has a linear strain between the nodes. If the load is increased a whole concrete element fails in tension. A real concrete structure has no linear strain along a bigger distance and the concrete cracks more gradual. A much finer mesh could Copyright © 2009. Diplomica Verlag. All rights reserved.

probably improve this behaviour, but this would cause a much longer computational time. The precast segmental bridge does not show this behaviour. The joints open gradually, as there is no tensile strength between the segments. The internal prestressed bridge does also not show such behaviour. The tendon is nearly continuously connected to the mesh. Therefore,

88

the mesh has to be finer than the mesh of the external types. The result is the smoother curve.

5.2 Discussion of the exactness of the FE calculations by comparing to the full scale test A comparison to the results of the full-scale test of an externally prestressed precast segmental bridge will be made in this section. The basic bridge data is based on the properties from this test (Takebayashi et al., 1994). This allows some statements about the value of the finite element analysis undertaken. 0.1 Applied midspan moment [MNm] (Live load only)

0.05

0 0

10

20

30

40

50

60

70

Deflection [m]

-0.05

-0.1 TDF

-0.15

-0.2

TDF MRS

MRS

-0.25 MRF

-0.3

Test data: externally prestressed, precast segmental, flexural failure (TDF) Model results: externally prestressed, precast segmental, shear failure (MRF)

MRF

Model results: externally prestressed, precast segmental, flexural failure (MRS)

-0.35

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Figure 5-2: Comparison to test results

Figure 5-2 shows the direct comparison. It can be seen that the ultimate moment carrying capacity from the finite element is higher than the measured data from the test. Hence, the model overestimates the ultimate moment of resistance. However, the basic response mode of the model complies with the structure. There is firstly a nearly linear branch followed by a sharp drop of the

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structure response introducing the non-linear behaviour up to failure. The deflection is modelled nearly accurately. Modelling of a collapse tends to be difficult. The models do typically not exactly comply with reality. One way of overcoming this is to adjust the model until it matches the curves from tests. This was not done in this case. It is preferred to discuss reasons why the model departs slightly from the test results. Possible reasons for the higher ultimate moment capacity of the model will now be explained. One reason can be the material assumption of the concrete. The paper, on which the test is based, does not give enough information about the concrete. The strength class of the concrete is given with 55-62 N/mm². It is not possible to produce an exact collapse analysis with this concrete strength data. The stress-strain relationship for the concrete was simplified to a perfectly elastic, perfectly plastic behaviour in compression and some additions for the tensile behaviour (compare Figure 3-31). Although, Hibbitt et al. (1998c) have successfully used this method to simulate reinforced concrete destruction, the use of nearly realistic stress-strain relationship as suggested in EC2 could maybe lead to a slight decrease in the strength of model. Another reason might be the nature of the finite element model. Finite element models represent the continuum as a series of discrete elements. It is assumed that the behaviour of the discrete model converges to the behaviour of the continuum with an acceptable tolerance by a choice of sufficient small elements. The mesh can never be never fine enough to approach the structural continuum. Normally this problem is resolved by the introduction of other softening factor into the mesh such as reduced integration. It was not possible to use reduced integration elements in this case because the computer could not find a unique solution for

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reduced integration elements. Hence, a possible solution to this problem is further refinement of the mesh. However, the calculation seems to pick the right mechanism of the collapse by comparison to the real behaviour. Hence, the model seems to be suited for comparison of different prestressed structures. The total magnitude of the results might have tolerances to the reality. The ultimate moments might be

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slightly too high but the deflection prediction should give exact values within a certain tolerance. The next comparison is dealing with the tendon stress increase up to failure. Takebayashi et al. (1994) have measured several strain differences. The full original tendon layout can be seen in Figure 3-6.

Tendon T1R:

Tendon T5L::

Tendon T6L:

193000.0

Strain increase [-]

Full-scale test data

Stress increase [N/m m²]

M odel data

Tendon T1R Maxim um change at point 31 approx. Maxim um change at points 20,36,37 approx. Maxim um change at points 42,43,26,27 approx.

0.00192 0.00152 0.00090

370.6 293.4 173.7

Tendon T5L

Maxim um change at all points

approx.

0.00140

270.2

Tendon T6L

Maxim um change at point 40 Maxim um change at points 40,34 Maxim um change at points 35

approx. approx. approx.

0.00255 0.00225 0.00195

492.2 434.3 376.4

Stress increase [N/mm²]

long tendon

238.7

short tendon

372.2

Figure 5-3: Comparison tendon stress increase (full-scale test data from Takebayashi et al., 1994)

The figure above illustrates the limitations of the FE model. The FE model consists of only two pairs of tendons, the real bridge has 6 pairs. The position Copyright © 2009. Diplomica Verlag. All rights reserved.

has to be different within the FE model as in the real bridge. The real tendons are also influenced by the friction from the deviation points, which can be seen at Tendon T1R. Hence, the tendon stress increase is different and shows only some

common

features.

Nevertheless,

the

model

shows

the

same

characteristics regarding the short and the long tendon. The short tendon has a higher stress increase. Also, the increase is notable. Thus, the suggestions of

91

some codes of practices to allow no tendon stress increase for externally prestressed bridges seems to be too conservative.

45 40 MO

Joint opening [mm]

35

Model, joint opening (MO) TO Test data, joint opening (TO)

30 25

TO MO

20 15 10 5 0 -5

0

10

20

30

40

50

60

70

80

Applied midspan moment [MNm] (Live load only)

Location of measurement

Figure 5-4: Comparison joint opening

There is also other data, which can be compared. Figure 5-4 shows the opening of the middle joint under increasing load. The same response can be seen as before. The opening characteristics are the same in both cases. But the model has a higher strength and the joint opens later. The paper of Takebayashi et al. (1994) also delivers data of the tendon slip. This data cannot

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be directly compared because the test data shows the relative movement between the deviator and the duct, whereas the computer model shows the relative movement between the deviator and the tendon itself. Also, the computer model has only 4 simplified tendons. The test model had 12 tendons. These tendons are also differently placed within the section compared to the computer model.

92

5.3 Comparison to other FE calculations and test results As already described in the literature review in chapter 2, a significant number of tests and calculations concerning this topic have already been undertaken by a number of researchers. The results did not always correspond. This section will be used to show some of these results and compare them to the results of this investigation. There are a number of tests and numerical analyses available. The tests are naturally restricted to one or two models and cannot always produce enough data for comparison in a scale, which is needed for this investigation. Lots of the recent researchers claim their calculation method is general and can be used for almost every prestressing technique, but not all of them deliver graphs and numerical results for all the methods. The preference is given to those papers which show clearly the different structural response of different prestressing techniques in direct comparison together with numerical results. Kreger et al. (1989) show a load displacement curve for a monolithic construction and a segmental construction, both externally post-tensioned. They

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conducted a finite element analysis for those two types.

Figure 5-5: Comparison of load-deflection curves, monolithic and precast segmental, both externally post-tensioned from Kreger et al. (1989)

93

The investigated bridge was a continuous beam consisting of three spans of 7.62m (25inch). The graph is directly taken from the source in order to avoid any biased interpretation by redrawing. Although the bridge was continuous, there are some characteristics, which are comparable to this dissertation. The monolithic bridge shows a far better performance, but the precast segmental version has not such a strong descending path as in the full-scale test of Takebayashi et al. (1994) and in the actual calculation. However, it was a continuous girder and a different behaviour can be expected. The next comparison will be made to a research conducted by Rabbat and Sowlat (1987). They did destructive testing on three “match cast” segmental girders with different tendon layouts. The primary objective was to evaluate the behaviour of prestressed concrete girders with different locations of the ducts. The first girder was fitted with conventional internal tendons. The second girder had partially embedded ducts and the last one was a girder with external tendons. Their second option is not of further interest for the actual investigation as such arrangements have not gained practical importance nor was it simulated in the actual investigation. Hence, it is focused on option 1, the bonded girder, and option 3, the unbonded girder. All were simply supported

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girders with a span of 9.45m.

Figure 5-6: Comparison to tests of Rabbat and Sowlat (1987)

94

The girders were subjected to two load circles. The first load circle was stopped, if serious non-linear behaviour was observed. This point was determined to be 76mm deflection. The test was done by applying small load increments. The girders were then completely unloaded and some anchorage wedges removed in order to simulate anchorage loss through an earthquake. The next circle (circle 2) included loading until destruction. The general essence valuable for the actual investigation is that the internally prestressed girder had a higher ultimate moment of resistance and a higher maximum deflection. The next research chosen for the comparison is a computer analysis of Muller and Gauthier (1989). They developed a computer program, which has the name “Deflect” in order to prove that the behaviour of the segmental bridges is essentially the same as that of monolithic constructions and to prove that continuous ordinary reinforcement is not necessary. The chosen graph from this paper includes calculation of simply supported beams with different distances of

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the deviation points of the tendon and an internal bonded girder.

Figure 5-7: Comparison numerical analysis of Muller and Gauthier (1989)

Muller and Gauthier conclude in their study that all girders behave similarly based on their analysis shown with this graphic. This is in contradiction with all

95

the other results introduced in Chapter 4 and in Chapter 5 of this study. Neither is it entirely clear how they conducted the computer analysis of the internally prestressed beam, especially whether this beam is monolithic. In summation, there are two studies, which found similar results as in the actual dissertation. The type of prestressing and the continuous ordinary reinforcement makes a difference to the load response of the girder. The monolithic girder tends to have more rotational capacity and a higher ultimate moment of resistance. The same was found to be true for the internally prestressed girder with the higher capacity versus the externally prestressed girder. And there is one study, which says it makes nearly no difference. The next verification will deal with the increase in tendon stress up to failure. The paper used for the comparison is from Ramos and Aparicio (1995). Their primary objective was the stress prediction at any load stage in the tendons for all prestressing techniques for bridge girders including the precast segmental version. The full range of the results is not included in the paper. The numerical results in the paper are about externally prestressed monolithic girders only. A computer program was developed to simulate the behaviour.

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The graphs shown here are for a simply supported girder with a span of 30m.

Figure 5-8: Increase tendon stress from Ramos and Aparicio (1995)

96

The left hand graph shows the stress increase of the tendon, which is deflected at the middle deviator. The graph on the right hand side shows the stress increase of the tendon deflected at the quarter points. The maximum tendon stress for the two times deflected cable is approximately 150 N/mm² higher than the initial. The stress in the tendon of the precast segmental brigde of the full-scale test was already compared in Figure 5-3: Comparison tendon stress increase (full-scale test data from Takebayashi et al., 1994). It was found that the actual model underestimates the stress increase. Hewson (2000b) gives a 250 N/mm² increase for the Bangkok Second Stage Expressway, which is the same structure as used in this analysis. A 238 N/mm² increase was found in this analysis. The match is relatively good to the actual findings. Other precast segmental bridges described in his notes are the Western Bypass in Melbourne Australia with a 300 N/mm² increase and the Hung Hom Bypass, Hong Kong with a 250 N/mm² increase . Other bridges with much lower increases are the New Tagus Bridge in Portugal (50N/mm²) and the Rambler Channel “MTRC” Bridge in Hong Kong with a 0N/mm² increase. It can be seen that a general statement seems to be difficult, as the variation in values is notable. The stress might depend strongly from the actual tendon arrangement which cannot be proved with the actual study, as no parametric investigations were conducted. The calculated tendon stress increase in this analysis compares satisfactory with other precast segmental

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bridges mentioned by Hewson (2000b).

97

6 Conclusion and Recommendations 6.1 Concluding remarks It was found that the position of the tendons is less important to the ductility of prestressed concrete beam as the existence of continuous reinforcement throughout the beam. All the monolithic bridges behave essentially the same. The precast segmental beam is a totally different matter. The monolithic bridges provide a much higher ultimate moment of resistance and substantial bigger deflections than the precast segmental version of the bridge. Also, the first signs of plastic deformation are followed by a wide swing in the load deflection curve (Figure 4-2: Tendon stress increase for all bridge types), which means they are able to carry much more extra load after the first signs of overloading. The precast segmental beam fails fairly suddenly after the first signs of overloading. The extra load needed to destroy it is only a fragment of what is needed for the monolithic bridges. The evaluation of whether or not this is ductile is subjective. The term ductile behaviour was defined in section 1.3 as providing ample warning of impending failure. This term does not appear to be applicable to the precast segmental bridge as it fails shortly after warning (5.1 Interpretation of the results). However, the position of the tendon itself also makes a difference, none of the externally prestressed bridges reached the load carrying capacity and the rotational capacity of the internally prestressed beam with bonded tendons. The difference between the monolithic types seems to be acceptable, as the monolithic externally prestressed girders reach sufficient deflection and ultimate moment capacity.

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The other objective of this study was the tendon stress increase up to failure. The analysis shows notable stress increases in all models using external tendons. All tendons exceed the 100 N/mm² as suggested in the BD 58/94 and the AASHTO. It implies that almost double the value is normally reached with simply supported precast segmental bridges, and 300 N/mm² seems to be typical for externally prestressed bridges monolithically built with similar tendon

98

arrangement. However, the discussion (5.3 Comparison to other FE calculations and test results) has shown that the tendon stress increase might strongly depend on the tendon layout; this implies that a exact calculation is always necessary and general statements about the magnitude of the stress increase are difficult.

6.2 Recommendations As externally prestressed girders have no tendons within the cross-section, sufficient continuous ordinary reinforcement will be necessary in order to gain similar ductility as with internally prestressed concrete bridges. 1.5% ordinary reinforcement throughout the girder was used in this analysis for the monolithic externally prestressed bridges. An exact non-linear calculation will be always necessary for the tendon stress increase up to failure in externally prestressed bridges, unless no

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increase is used.

99

References Bruggeling, A.S.G. (1989)

External prestressing – a state of the art, External

prestressing

in

bridges,

American

Concrete

Institute, pp. 61-77 Clark, G (1998)

Past and Present Experience in the United Kingdom with Prestressing of Bridges in Eibl, J.

(ed)

Externe

Vorspannung

und

Segmentbauweise, Berlin, Ernst und Sohn, pp. 121-132 Fenner, D. N. (1987)

Engineering stress analysis – a finite element approach, Chichester, England, Ellis Horwood

Limited Gallaway, T.M. (1980)

Design Features and Prestressing Aspects of the Long Key Bridge, PCI Journal, Nov-Dec 1980, pp.

84-111 Hegger, J. et al. (1998)

Ertüchtigung von zwei Rohmehl-Silos mit externer Vorspannung

in

Eibl,

Vorspannung

und

J.

(ed)

Externe

Segmentbauweise,

Berlin,

Ernst und Sohn, pp. 357-366 Hewson, N. R. (1993)

The use of external tendons for the Bangkok Second

Stage

Expressway,

The

Structural

Copyright © 2009. Diplomica Verlag. All rights reserved.

Engineer, Vol. 71, Dec 1993, pp. 412-415 Hewson, N.R. (2000a)

Post-tensioned concrete bridges - notes with the lecture at the University of Surrey, University of

Surrey, Guildford, England Hewson, N.R. (2000b)

External

prestressing

101

tendons

in

Bridge

Engineering- notes with the lecture at the University

of

Surrey,

University

of

Surrey,

Guildford, England Hibbitt et al. (ed) (1998a)

Abaqus/ Standard user’s manual Volume I,

Hibbitt, Kalson & Sorensen Inc, USA Hibbitt et al. (ed) (1998b)

Collapse analysis of a reinforced concrete slab,

Abaqus/ Standard example problems manual Volume II, Hibbitt, Kalson & Sorensen Inc, USA, pp. 4.2.7-1 – 4.2.7-8 Hibbitt et al. (ed) (1998c)

Abaqus/

Theory

manual,

Hibbitt,Kalson

&

Sorensen Inc, USA Hilleborg, A. et al. (1976)

Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements, Cement and concrete research

(USA), 1976, Vol. 6, pp. 773-782 Hollingshurst, E. (1995)

River Camel Viaduct Wadebridge, The structural

Engineer, April 1995, pp.99-104 Hurst, M.K. (1998)

Prestressed concrete design, Second Edition,

London and New York, E & FN Spoon Jackson P.A. (1995)

Development of BA and BD 58/94, Materials of

the British Cement Association training course,

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July 1995, pp. 1-10 Kong, F: K. and Evans R. H. Reinforced (1996)

and

prestressed

concrete,

Third

Edition, London, Chapman & Hall

Kotsovos, M.D. and Pavlovi, Structural concrete - Finite Element analysis for M.N. (1995)

limit-state design, London, Thomas Telford

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Krautwald, W. (1998)

Extern vorgespannte Brücken- Erfahrungsbericht eines Bauausführenden in Eibl, J. (ed) Externe

Vorspannung

und

Segmentbauweise,

Berlin,

Ernst und Sohn, pp. 141-149 Kreger et al. (1989)

Finite

element

analysis

of

externally

post-

tensioned segmental box girder construction,

External

prestressing

in

bridges,

American

Concrete Institute, pp.389-407 Landschaftsverband

Erste Spannbeton-Brücken in Westfalen-Lippe,

Westfalen Lippe (2001)

Available

from:

http://www.lwl.org/HTML/STRASSE/bruecken/1415.HTM [Accessed 17 May 2001] MacGregor R.J.G et al.

Strength and Ductility of a Three-Span Externally

(1989)

Post-Tensioned Segmental Box Girder Bridge Model, External prestressing in bridges, American

Concrete Institute, pp. 315-338 Matupayont, S. et al. (1994)

Loss

of

tendon’s

eccentricity

in

externally

prestressed concrete beam, Transactions of the

Japan concrete institute, Vol. 16, pp. 403-410 Muller, J. and Gauthier Y. Ultimate Behaviour of precast segmental box (1989)

girder

with

prestressing

external

in

bridges,

tendons,

External

American

Concrete

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Institute, pp.355-374 New Civil Engineer (1994)

Severn in the Swing, New Civil Engineer, March

1994, pp. 20-23 New Civil Engineer (1996)

Tropical overload, NCE, 26 December 1996, London, Thomas Telford, pp.18-21

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New Civil Engineer (1998)

Injaka Bridge collapse, NCE, 16 July 1998, London, Thomas Telford, pp.1-5

Rabbat, B.G. and Slowat K. Testing of segmental concrete girders with (1987)

external tendons, PCI Journal, March/April 1987,

pp.86-107 Rabinowicz, E (1995)

Friction and Wear of Materials, Second edition,

USA, John Wiley & Sons Ramos, G. and Aparicio, A.C. Ultimate Behaviour of Externally Prestressed (1995)

Concrete

Structural

Bridges,

Engineering

International, 3/95, pp. 172-177 Rao, P. S. and Mathew, G. Behavior of Externally Prestressed Concrete (1996)

Beams with Multiple Deviators, ACI Structural

Journal, July-August 1996, pp. 387-396 Rombach, G (1995)

Bangkok Expressway – Segmentbrücken contra Verkehrschaos, in Hilsdorf and Kobler (ed.) Aus

dem Massivbau und dem Umfeld, Proceedings of the

Institut

für

Massivbau

und

Baustofftechnologie, Universität Karlsruhe, pp. 645-656 Ryall, M.J. et al. (ed) (2000)

Manual of bridge engineering, London, Thomas

Telford

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Schallwig, K. (1998)

Sanierung eines 600m3 Flugaschesilos in Eibl, J.

(ed)

Externe

Vorspannung

und

Segmentbauweise, Berlin, Ernst und Sohn, pp. 347-356 Schönberg, M. and Fichtner, Die Adolf-Hitler-Brücke in Aue/Saale, Bautechnik, F. (1939)

Vol 17, p. 97

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Standfuß, F. (1998)

Erläuterungen zur Richtlinie für Betonbrücken mit externen

Beton-

Spanngliedern,

und

Stahlbetonbau 93, Heft 9, pp. 264-272 Takebayashi, T. et al. (1994)

A full scale destructive test of a precast segmental

box

girder

bridge

with

external

tendons, Proceedings of the Institution of Civil

Engineers Structures and Buildings, August 1994, pp.297-315 Tan, K. -H. and Ng, C.-K.

Effects of Deviators and Tendon Configuration in

(1997)

the Behavior of Externally Prestressed Beams

ACI Structural Journal, January-February 1997, pp. 13-22 Ugural, A. C. and Fenster, S. Advanced strength and applied elasticity, 3rd K. (1995)

edition, New Jersey, USA, Prentice Hall PTR

Vielhaber, J. (1988)

Teilweise Vorspannung ohne Verbindung im segmentierten

Brückenbau,

Proceedings

21

DAfStb Forschungskolloquium Hannover, pp. 7782 Virlogeux, M.P. (1989)

External Prestressing: from Construction History to Modern Technique and Technology, External

prestressing

in

bridges,

American

Concrete

Institute, pp.1-60

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Woods, R. (2001)

Soil-structure interaction - notes with the lecture at the University of Surrey, University of Surrey,

Guildford, England WS Attkins (2001)

Company

Information,

England

105

WS

Atkins,

Epsom,

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Zienkiewicz, O. C. and

The finite element method, Volume 2, 4th edition,

Taylor, R. L. (1991)

London, McGraw-Hill Book Company

106

Codes of practice AASHTO 1996

Standard Specifications for Highway Bridges

(Design standard), American Association of State Highway and Transportation Offices, USA, 1996 AASHTO: Guide

Guide Specifications for Segmental Bridges,

Specifications for Segmental

(Design Standard), American Association of State

Bridges 1989

Highway and Transportation Officials, USA, 1989

ARS Nr.28/1998

Spannbetonbrückenbau-Richtlinie

für

Beton-

brücken mit externen Spanngliedern (Design

standard), The Ministry of Traffic, Germany, 1998 BD 58/94

The design of concrete highway bridges and structures

with

external

and

unbonded

prestressing (Design standard), The Highway

Agency, United Kingdom, 1994 BS 5400 Part 4: 1990

Code of practice for design of concrete bridges

(Design

standard),

British

Standard,

United

Kingdom, 1990 EC 2

Eurocode 2: Design of concrete structures.

(DD ENV 1992-2:2001)

Concrete

bridges

(Prestandard),

European

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Committee for Standardization, EU, 2001 EC 2

Eurocode 2: Design of concrete structures.

(DD ENV 1992-1-5:1996)

General rules. Structures with unbonded and external

prestressed

tendons

(Prestandard),

European Committee for Standardization, EU, 1996

107

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Appendix A: Derivation of the simplified tendon layout

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Appendix B: Calculation of the minimum reinforcement

The reinforcement will be derived following the suggestion of EC2 (DD ENV 1992-2:2001). Minimum reinforcement from internal crack forces (4.4.2.2.3): Determine internal forces from prestress

o

4458 kNm

4458 kNm

2216 kN

15666 kNm 9 725

852 kN

10 200

2337 kN

13 600

15666 kNm

9 725

51371.9 46403.7 46407.7

41945.7

41949.7 15666

15666

Moments in kNm

+ Prestress force 38 443 kN Minimum reinforcement web: Stresses from prestress: σ tp

σ bp

38.443

51.3719. 0.729

5.0414

3.4912

38.443

. 51.37191.662

5.0414

3.4912

σ tp = 3.102

N/mm²

σ bp = 32.081

N/mm²

DL = 0.126

MN/m

M DL = 29.461

MNm

σ tDL = 6.152

N/mm²

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Stresses from dead load: Dead load:

DL

0.025. 5.04 DL. 43.25

2

Dead load moment:

σ tDL

σ bDL

M DL. 0.729 3.4912 M DL. 1.662 3.4912

M DL

8

σ bDL = 14.025 N/mm²

Final stresses: σt

σ tDL

σb

σ bDL

σ tp σ bp

σ t = 3.05

N/mm²

σ b = 18.056

N/mm²

10.2

-3.05

0.298

-4.92 0.346

0.346

1.913

-16.93

0.18 -18.06 3.767

Calculate Nsd: axial force on web under quasi-permanent loading and prestress: -> compression force in the web

4.92 16.93. 1.913. 0.692 2

Nsd w

Nsd w = 14.462 MN (both webs)

hx f ctm

1.0

h

1.913

k1

1.5

k

1.0 (Hurst,1998)

4.1 N/mm²

reinforcement steel diameter 12mm -> table 4.120: σ s kc

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ρs

Nsd w

0.4. 1

k1.0.692. h . 0.8. k c . k .f ctm σs

k c = 0.029

h . f ctm hx

ρ s = 3.903 10

240

4

~0.04% minimum reinforcement

minimum reinforcement bottom slab: calculate tensile force in flange immediately prior to cracking bottom flange: f ctm .3.4912 M crbf 1.662

4.1. 0.729 1.662

M crbf = 8.612

MNm

-1.798 N/mm²

= 1.798 0.729 m 1.662 m 3.66 N/mm²

tension force bottom slab: 4.1 3.66. 0.18. 3.767 = 2.631 2

0.9.

kc

2.631 0.18. 3.767.4.1

fctm MN

k c = 0.852

minimum reinforcement bottom slab: 0.8. k c .k . f ctm

ρs

σs

ρ s = 0.012

~1.2% minimum reinforcement

minimum reinforcement top slab: calculate tensile force in flange immediately prior to cracking top flange: M crtf

f ctm .3.4912

4.1. 1.662 0.729

M crtf = 19.635 MNm

0.729

fctm = 9.347

0.729 m

2.42 N/mm²

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1.662 m

tension force top slab: -9.43 N/mm²

4.1 2.42. 0.298. 10.2 = 9.909 2

kc

0.9.

9.909 0.298. 10.2. 4.1

MN

k c = 0.716

minimum reinforcement top slab:

ρs

0.8. k c .k . f ctm

σs

ρ s = 9.78 10

3

~1.0% minimum reinforcement

Minimum reinforcement considering maximum spacing and reinforcement diameter Diameter 12

max steel stress: 240 N/mm²

Table 4.120

240 N/mm²

max spacing: 100mm

Table 4.121

Diameter 12 has 113.09mm² area -> 113.09/0.01*2=1130.9mm²*2 (both sides) =2261.8mm²/m Minimum reinforcement bottom slab: 2261.8*10-6/(0.18*1.0)=0.013

~1.3% minimum reinforcement

Minimum reinforcement top slab: 2261.8*10-6/(0.298*1.0)=0.008

~0.8% minimum reinforcement

Minimum reinforcement web: 2261.8*10-6/(0.346*1.0)=0.006

~0.6% minimum reinforcement

Decision

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Add some reinforcement for curtailment and overlapping. Take 1.5% reinforcement for whole structure.

Appendix C: ABAQUS Input file for the precast

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segmental externally prestressed box girder

The following pages show the print of the input file for the precast segmental bridge, which is externally prestressed in this case. There are of course four input files for each of the analysed types. However, the printout of only this input file seems to be sufficient, because this is the most complex analysis. The other types can be created by removing of certain parts of this input file or reconnecting of the some nodes, which can be done based on the description in the main part. The next lines summarize the content. •

The first pages contain mainly geometrical data including the position of the GAP elements.

•

This part is followed by geometric properties ( *.. SECTION). The crack propagation direction is determined in this part followed by Multi-point constraints (*MPC), which are used for link connections.

•

Then the materials are defined under the MATERIAL commands.

•

The BOUNDARY commands assign afterwards the boundary conditions.

•

Thereafter the reinforcement is defined (*REBAR).

•

The prestress is applied with *INTIAL CONDITIONS

•

Then follows the description for the first calculations step, which includes self-weight and prestress (*STEP). There are also the information about the maximum of increments and that this is a combined material and

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geometric non-linear step besides other information. •

Finally step 2 follows, which applies the test load with the RIKS algorithm.

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