p75preEconomic Beam Designs Using FRP Rebar

8/2/2019 p75preEconomic Beam Designs Using FRP Rebar

1/48

Editorial Manager(tm) for Magazine of Concrete Research

Manuscript Draft

Manuscript Number:

Title: Economic Design Of Beams With FRP Rebar Or Prestress

Article Type: Paper

Corresponding Author: Ioannis Balafas, Ph.D.

Corresponding Author's Institution: University of Cyprus

First Author: Ioannis Balafas, Ph.D.

Order of Authors: Ioannis Balafas, Ph.D.;Chris J Burgoyne, MA, MSc, Ph.D.

Abstract: Fibre-Reinforced Polymers have not captured the market for reinforcing bars, or found useseven as external prestressing tendons, despite their obvious advantages in terms of lack of corrosion.

This paper considers the constraints that underlie the optimal design for beams reinforced or

prestressed with FRP, and compares the result with designs for beams with steel reinforcement. This

is done by means of a diagram that compares the depth of the beam with the area of reinforcement, on

which most of the governing constraints can be plotted. The method allows generic conclusions to be

drawn about the governing principles.


2/48

Ioannis BalafasDepartment of Civil andEnvironmental Engineering

University of Cypruse-mail: [email protected]: 00357 9671654915 June, 2011

ICE PublishingMagazine of Concrete Research40 Marsh Wall

London

E14 9TP

UK

Dear Sir/Madam,

This letter follows the submission of the article: Optimal design of beams with FRP bars orprestress to Magazine of Concrete Research. The paper studies the initial cost of beamsreinforced and prestressed with Fiber-Reinforced Polymers.

Fiber-Reinforced Polymers have not captured the market for reinforcing bars, or even

prestressing tendons, despite their obvious advantages in terms of lack of corrosion. Theconstraints that underlie the optimal design for beams reinforced or prestressed with FRPare determined, and the results are examined with designs for beams with steel. This isdone by means of a diagram that compares the depth of the beam with the area ofreinforcement, on which most of the governing constraints can be plotted. The methodallows generic conclusions to be drawn about the governing principles.

The paper is about 5000 words and includes five tables and fourteen figures.

Looking forward to hearing from you, concerning the reviews.

Ioannis Balafas Chris Burgoyne

ver Letter


3/48

Economic Design Of Beams With FRP Rebar Or Prestress12

I. Balafas*

and C. J. Burgoyne3

4

5

Fibre-Reinforced Polymers have not captured the market for reinforcing bars, or6

found uses even as external prestressing tendons, despite their obvious advantages in7

terms of lack of corrosion. This paper considers the constraints that underlie the8

optimal design for beams reinforced or prestressed with FRP, and compares the result9

with designs for beams with steel reinforcement. This is done by means of a diagram10

that compares the depth of the beam with the area of reinforcement, on which most of11

the governing constraints can be plotted. The method allows generic conclusions to12

be drawn about the governing principles.13

14

INTRODUCTION15

When high-strength fibres (glass, aramid and carbon) were first introduced to the16

structural engineering world it was assumed that they would be most suitable for use17

as prestressing tendons, which are the most highly-stressed of all structural elements:18

this would make maximum use of the high strength that was being provided so19

expensivly. It was considered that use as reinforcement would seldom be justified,20

since deflections would govern and the fibres would rarely be stressed to a high load.21

Experience has shown, however, that there have been few applications of prestressing,22

despite the added benefit that FRP tendons can be placed externally, and the limited23

use that has been made as reinforcement has been using GFRP, which has the lowest24

stiffness and is thus more lightly stressed than AFRP or CFRP bars would be. This25

*Department of Civil & Environmental Engineering, University of Cyprus, PO Box 20537,

1678 Nicosia, Cyprus, [email protected] of Engineering, University of Cambridge, Trumpington Street, Cambridge, CB2

1PZ, UK, [email protected].

n Text

k here to download Main Text: IC_14_6_2011.doc

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61
http://www.editorialmanager.com/macr/download.aspx?id=36111&guid=41489e65-c9ec-442c-ad66-1090b3782733&scheme=1http://www.editorialmanager.com/macr/download.aspx?id=36111&guid=41489e65-c9ec-442c-ad66-1090b3782733&scheme=1


4/48

paper seeks to produce some general principles that explain why this might be the1

case.2

The close fit between the strain capacities of steel and concrete means that they are3

ideally suited to produce a composite reinforced concrete where both materials are4

used economically, while strains and curvatures remain small. Steel is relatively cheap5

(aramid and carbon cost 5-6 times as much per unit of force), but corrosion of steel in6

concrete bridges is a major problem;1,2

by now FRPs should be in widespread use.37

However, applications of FRP are still scarce; the market is hesitant to adopt them,8

partly because of their high cost and partly because the design principles differ from9

those using conventional rebar.10

This paper addresses the question of how, or even whether, their properties should be11

included in the design process. The target is to identify ways in which FRP can be12

used at minimum cost, and to consider why FRPs are rarely used in practice. These13

questions will be addressed through the use of a diagram of depth (d) versus flexural14

bar area (Ap).15

The first sections below will show how the basic diagrams are built up. Additional16

constraints will then be added before specific questions are answered about the17

different options available to designers. Three case studies will be considered before18

final conclusions are drawn.19

20

DEPTH VERSUS AREA DIAGRAM21

The parameters affecting a design can be shown on a diagram of flexural bar area (Ap)22

versus section depth (d), which was first introduced by Al-Salloum et al4 for the23

optimal design of rectangular reinforced concrete beams. They considered only a24

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61


5/48

limited number of design constraints. The method will be expanded here to include1

many different factors.2

Detailed equations are not given below since the analysis follows standard procedures3

for the design of a cross-section on the assumption that plane-sections remain plane.4

Safety factors are included for ultimate loads, but not for serviceability conditions.5

Sections are designed from first principles, rather than a particular code of practice,6

although some use is made of factors that are also used in codes. Very similar plots7

would result from using codes, and a detailed discussion of the differences would8

distract from the main argument. For the same reason, although all the plots below9

relate to the same loading case, and have been produced as a result of specific10

calculations, details are not given here.11

The following sections show how the basic diagram is built up.12

Ultimate strength condition13

The ultimate strength condition divides the d-Ap diagram into two zones (Fig. 1(a));14

sections that are sufficiently strong lie above a boundary line while those that are15

inadequate lie below it. The boundary line is curved and there is a distinct change in16

slope when the limiting condition changes from failure by tendon snapping to failure17

by concrete crushing.18

Bar snapping condition19

Unlike sections reinforced with steel that can yield, sections with FRP should not be20

allowed to fail by reinforcement snapping, so a minimum limit on the amount of21

reinforcement has to be applied that reduces the size of the feasible zone (Fig. 1(b)).22

This will be referred to hereafter as the balanced-section constraint.23

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61


6/48

Deflection condition1

Sections with FRP reinforcement are more flexible than sections with steel rebar, so2

the deflection condition needs to be checked, which leads to a further reduction in the3

size of the feasible zone, as shown in Fig. 1(c).4

Cost Function5

The cost of the beam can be directly related to the cross-sectional area and the amount6

of reinforcement. Allowances can be made for the cost of the formwork, as well as7

for the shear reinforcement. Beams of equal cost are defined by a line of fixed slope8

on the d-Ap diagram; as the cost of the beam increases, this line moves to the right but9

the slope remains the same (Fig. 1(d)). The optimum solution is therefore where the10

cost function line first meets the feasible region; in the example shown in Fig 1 that11

occurs when the bar snapping and deflection constraints meet.12

Other constraints13

Other constraints can also be shown. Crack-width limits can be introduced; for FRP-14

reinforced structures these are associated with appearance, while for steel-reinforced15

structures they are associated with durability. A limit on concrete stress, to eliminate16

the possibility of excessive creep, can also be considered. The latter rarely governs,17

but Fig. 1(e) shows that the crack width is close to the limit for the cheapest design in18

this particular case and it can limit in other cases.19

The method does not specifically include the breadth of the beam. Only rarely is the20

breadth a governing condition; it will be assumed here that it is as small as21

practicable. The breadth does affect the amount of shear steel needed, and is reflected22

in the costs functions, but the details do not affect the main argument and it will not23

be discussed further.24

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61


7/48

1

APPLICATIONS OF THEd-Ap DIAGRAM2

Having derived the basic diagram, it is now possible to compare different alternative3

designs. This will be done by showing pairs of diagrams for different cases, which4

allows the distinctions to be highlighted.5

Steel versus FRP6

The most fundamental consideration when designing a reinforced concrete section is7

the choice of reinforcement material. To date, steel has been the obvious solution,8

and the d-Ap diagrams show why.9

Fig. 2 compares two designs (for the same loading case), one with steel rebar and one10

with FRP. A number of differences can be noted in the way the constraints apply.11

1. In the steel diagram, the strength condition is more critical than the deflection12condition, except for very shallow beams.13

2. There are minimum areas of steel reinforcement (to avoid snapping or14excessive cracking) as well as maximum areas (to avoid over-reinforcement).15

3. Steel is much cheaper (per unit volume) than FRP, so the constant cost line is16much flatter.17

4. The optimal design for use with FRP is almost always at the vertex where the18deflection (or occasionally the crack width) conditions meet the deflection19

limit.20

5. The flexural optimum solution for steel reinforcement is not at a vertex but lies21on a curved edge. There will thus be a range of designs that have very similar22

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61


8/48

costs and the optimum design will be very sensitive to variations in the relative1

costs of labour and materials. This can mean that the optimum solution can2

alter depending on where the structure is built.3

Furthermore, some conclusions can be drawn about the differences between the4

design with FRP and that with steel. Beams with FRP need to be significantly deeper5

than beams with steel in order to comply with the deflection conditions, but the6

amount of reinforcement required is less because the rebar is stronger.7

Prestressed Concrete Beams8

Prestressed concrete beams can be studied using the same basic principles, but9

different parameters become important.10

The most significant difference is the addition of upper and lower bounds on the11

section stresses at the working load. This has the corollary that there is an upper limit12

on the amount of prestress that can be applied, so the feasible zone is now bounded.13

Stress rupture of FRP tendons also has to be considered, but it appears in these14

diagrams indirectly since it affects the allowable stress in the tendon, which in turn15

affects the Ap required.16

Typical results are shown in Fig. 3, (a) (for steel) and (b) for FRP. As with the plots17

for reinforced concrete design, the optimal design for sections with FRP is usually at a18

vertex of the feasible zone (which makes it relatively insensitive to changes in relative19

costs of reinforcement and concrete), while the optimum for a section prestressed with20

steel lies along a curved edge of the zone.21

Deflection criteria are unlikely to govern for either material; the prestress prevents the22

section cracking at the working load so the beam stiffness is now almost entirely a23

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61


9/48

property of the concrete and not of the reinforcement, even for sections prestressed1

with FRP.2

Comparison of the two plots for prestressed beams yields conclusions that differ from3

those for reinforced sections. The optimum depth for both sections is similar, which4

is logical given that prestressing allows the concrete itself to carry the load. Less5

material is needed for prestressing with FRP because those tendons are generally6

stronger, while the stress-rupture and ultimate strength of the tendon enter via the7

balanced section criterion.8

9

CASE STUDIES10

The plots developed above are interesting in their own right, but having set out the11

basic principles of how the diagrams are produced, their use can be extended into a12

study which can map the design space and identify factors that most affect the13

economics of design. Three cases will be considered in more detail; a reinforced14

concrete beam and two prestressed concrete beams, one prestressed, the other15

statically indeterminate.16

Materials considered in the case studies17

Various FRP materials have been considered in this study. Glass fibre and carbon18

fibre reinforced polymers (GFRP) and (CFRP) are typically used in the form of19

pultruded rods of various diameters.20

Aramid fibres are available in various forms. Kevlar and Twaron fibres have very21

similar properties and are available in standard modulus and high modulus versions.22

The high modulus version is used in Fibra, which is a braided product embedded in23

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61


10/48

resin. The standard modulus version of a similar, but not identical product, Technora,1

is used to make a pultruded rod that is stronger but less stiff than the Fibra. Because2

aramid fibres are tougher than carbon, they can also be used without resin and are3

used as parallel-lay ropes under the name Parafil. These ropes can be fitted with4

termination systems that make them suitable for use as prestressing tendons, but they5

are not suitable for use as rebar.6

Typical values for the strength and stiffness are given in Table 1. Under long term7

loading strength may reduce due to creep rupture; the strength at time t,ft, is given by:8

101 logt i f f B t (1)9

fi is the short term strength, B is a constant determined by fitting to existing creep10

rupture data5,6 and its values are shown in Table 1. Note that t is in hours. Creep11

rupture for steel is insignificant and thus ignored.12

The unit costs for the different materials, obtained from recent quotations from13

manufacturers, are shown in Table 2.14

With these general values as a basis, it is now possible to investigate the effect of15

changing various aspects of the design.16

Case study 1. Reinforced Concrete17

The example chosen is a simply-supported T-beam as shown in Fig. 4. Several input18

parameters assumed in the analysis are shown on Table 3. The width of the web is19

assumed to be 250 mm. The beam and the slab are rigidly connected so the beams20

cross section is T-shaped. The effective width of the slab that acts compositely with21

the beam is difficult to determine. The normal calculations used in codes of practice22

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61


11/48

(which typically give effective breadths = span/5) are designed to give a beam with an1

effective value of second moment of area, but the present analysis is more concerned2

with the peak compressive stress, which gives lower effective breadth.13

The value3

used in this analysis is span/7 = 0.85 m.4

Proposals have been made to improve the ductility of beams with FRP by adding5

confinement reinforcement to the compression zone,14,15

but there is little benefit for6

reinforced concrete since the curvatures at failure are very large and the compression7

area after cracking is small, so the addition of confinement reinforcement would have8

little effect.9

Deflections are assumed to be limited to span/250 unless otherwise stated while the10

concrete cover is taken as 50 mm for FRPs and 75 mm for steel due to durability11

concerns. The crack widths are limited to 0.3 mm for all FRPs and 0.2 mm for steel.12

The concrete compression stress under service loads is limited to 40% of the13

characteristic concrete strength.14

Stirrups are provided according to standard practice and are assumed to be made from15

the same material as the flexural reinforcement.16

Deflection limit and material17

Varying the deflection limit from the normal span/250 shows how the governing18

criteria can change. Deflection limits were imposed from span/180 to span/480. All19

the other values were taken as in Table 1. Fig. 5 shows the effect on a beam reinforced20

with relatively flexible Technora bars. In Fig 5(a) the most relaxed deflection limit is21

imposed. The optimal solution is found to lie at the intersection of the crack width and22

deflection constraints. On the other hand, in Fig 5(b) the most strict deflection limit23

(span/480) is applied; the deflection constraint moves higher, reducing beam24

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61


12/48

deflections, while all the other constraints remain unchanged. Consequently, the1

optimal solution is also at the intersection between crack width and deflections2

constraints.3

For steel-reinforced beams the ultimate load condition governs except when the most4

severe deflection constraint is applied. However, for the FRP-reinforced beams, due5

to their much lower modulus, the optimum solution is almost invariably governed by6

deflection constraints at the working load. For relaxed deflection limits and low7

elastic modulus FRPs like glass and Technora, the optimal solution lies at the8

intersection with the crack width limit, whereas for stiffer FRPs or tighter deflection9

conditions, the optimum condition is usually at the intersection of the deflection and10

balanced-section constraints.11

In Fig. 6, cost ratios relative to the cost of the steel-reinforced beams are shown. Total12

costs include formwork, concrete, flexural and shear reinforcements costs. The figure13

shows that GFRP reinforcement is only marginally more expensive than steel-14

reinforced sections, although of course with much deeper sections. The use of the15

higher-modulus FRPs seems never to be cost-effective for reinforced sections, and in16

general applying more severe deflection constraints increases the cost ratio.17

Sensitivity to price reduction18

It has long been assumed that FRP reinforcement would become more economic as19

the price dropped by the economy of scale theory.16

Fig. 7 shows that, because the20

optimal design lies at a sharp vertex of the feasible zone, reducing the cost of the21

reinforcement (in this case, Fibra) by 40% does not alter the optimal design, although22

it does, of course, reduce the cost of the section. The cost of the reinforcement needs23

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61


13/48

to be almost as cheap as steel, on the basis of a cost per unit volume, before there is1

any effect on the position of optimal solution in the diagram.2

Bond3

Bond strength does not normally affect the design of a cross section, provided there is4

enough to grip the rebar at its ends. However, it can have an effect on the crack5

widths, and because FRPs have low stiffness, those limits can be critical.6

Manufacturers have produced bars with various surface characteristics with a view to7

improving bond; representative bond-slip curves, which are assumed to apply to all8

types of FRP, are shown in Fig. 8.17

9

Two d-Ap diagrams are shown. Fig. 9(a) shows the effect of having a smooth surface,10

where the crack-width criterion dominates, while Fig. 9(b) shows the result for a beam11

with braided and indented GFRP bars, which bond very strongly to concrete. As a12

result, the crack width constraint moves to the left on the figure and no longer13

governs. The best solution is then found at the deflection and balanced-section 14

crossing point.15

The effect of the different types of bond are shown in Fig. 10. For beams with low16

elastic modulus fibres, like glass and aramid-Technora, low bond strength is a17

disadvantage, whereas stiffer bars, such as Fibra and CFRP are not affected by the18

crack width criterion.19

Partial material safety factor20

Lack of knowledge on durability and long term behaviour of FRPs has led to various21

proposals on partial material safety factors.18 To assess the effect of the partial22

material safety factor on the optimal solution, values between 1.3 and 2.0 were23

studied and the optimal solutions were found in each case. The partial material safety24

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61


14/48

factors will only affect the ultimate condition constraints; unity safety factors are1

assumed in service conditions. However, since the balanced section condition is there2

to prevent the bar snapping, increasing the safety factor makes this constraint much3

more severe.4

The balanced-section constraint rotates clockwise on the depth versus bar area5

diagram and the ultimate constraint moves upwards as shown in Fig. 11, which results6

in a substantial increase in total cost when the safety factor is raised. Since the7

deflection constraint still governs, more bars have to be provided to prevent bar8

snapping at failure but the beam can be less deep.9

Concrete strength10

Due to FRP's low elastic modulus, the results up to this point have shown that the11

deflection constraint usually governs the optimal design solution. The use of stiffer12

high strength concrete has been proposed to limit the problem of excessive deflections13

of structures reinforced with these materials.19

14

Analyses have been performed to assess the effect of high strength concrete in FRP-15

RC design. The deflection constraint does indeed yield lower depths but this is16

balanced by a clockwise rotation of the balanced-section line on the d-Ap figure; more17

bars are needed to crush the stronger concrete. In general, the latter affect more than18

offsets the cost reduction due to the stiffer concrete and there is no benefit in using19

stronger concrete.20

Flexural versus total costs21

The d-Ap diagrams give flexural optimum results by inspection, which for expensive22

FRPs are normally located at the intersection between the deflection and the23

balanced-section conditions. The total cost has to include the shear reinforcement as24

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61


15/48

well and these can be determined from the flexural design. The optimal beam depths1

for low elastic modulus FRPs are normally high to avoid extensive deflections;2

increasing beam depth enhances shear capacity and fewer stirrups may be needed. In3

certain circumstances, however, a default minimum amount of shear links have to be4

provided; consequently stirrup volume may increase.5

When stirrup volume increases with beam depth, the total cost function reduces its6

slope on the d-Ap diagram and the optimal solution may lie at a reduced beam depth.7

The shear demand for the beams reinforced with FRP stirrups was determined by8

assuming truss analogy and stress limitation on the FRP stirrups.20,21 Studies were9

carried out to see the effect of this on the amount of FRP provided, but it was found10

that this would only be significant if FRP stirrups were many times more expensive11

than the cost of straight FRP rebar. This was beyond the scope of this study since it12

depends on the methods adopted to produce the links.13

14

Case Study 2. Determinate Prestressed Concrete15

Prestressed concrete ought to be a logical application for FRPs, and the initial cost16

model can also be used identify ways in which they can best be used in determinate17

prestressed concrete design.18

It has also been identified that elements prestressed with FRP can benefit from19

confinement22

because this provides ductility to the compression zone of the concrete.20

It is assumed here that the FRP prestressed beam's top flange can be confined with21

100 mm diameter AFRP spirals.23

Spirals are placed in regions where the moment is22

highest, and thus where crushing is more likely. The spiral pitch is assumed to be 10023

mm, which is the minimum needed to ensure that the spirals are effective in ensuring24

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61


16/48

that concrete does not fail explosively. As moment redistribution is not required,1

denser confinement spacing is not needed.2

The structure used for comparison purposes is similar to the reinforced structure3

considered above, but with a 12 m span and an I-shaped cross section. Various inputs4

assumed in the analysis, such as structural and material properties, as well as design5

limits, are summarized in Table 4.6

The prestressing tendons are initially prestressed to 60% for steel and CFRP tendon.7

A lower value is chosen for FiBRA, Technora and Parafil tendons due to creep-8

rupture concerns (Table 4). Concrete creep, shrinkage and tendon relaxation are also9

taken into account.10

Bonding to concrete11

It is not clear whether FRP tendons should be connected to concrete .24,25

Bond12

conditions have a considerable influence on tendon effectiveness and cost.26

The13

method described there was applied to determine the effect of bond on the final14

optimal solution. Three different scenarios were assumed: one with the tendons fully15

bonded to the concrete, one fully unbonded, while the last has partially bonded16

tendons, similar to those tested by Lees and Burgoyne.26 This reduces the local strain17

that is attracted to the tendon, thus reducing the likelihood that the tendon will snap.18

When fully-bonded conditions do not exist, the plane-section-remains-plane19

assumption no longer applies since the tendon sees lower strains than the concrete.20

The effect depends on tendon prestress, tendon profile, tendon length/static depth and21

loading type.27

To allow for these effects, strain reduction factors () are introduced22

for the uncracked as well as for the cracked plane section analysis.3,28

23

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61


17/48

Strain reduction factors do not exist in the literature for partially bonded prestressed1

beams. A value for halfway between unity and that determined for the unbonded2

tendons is assumed to represent strain sensitivity in partially bonded beam.3

The plot shown in Fig. 3(a) shows that for steel tendons the optimal solution was4

governed by concrete tensile stresses at the working load and, due to the cost function5

slope, it was located towards higher tendon areas and away from the balanced-6

sections constraint.7

In contrast, FRP tendons are expensive; thus the cost function is steeper on the d-Ap8

diagram. Flexural optimal solutions for the FRP prestressed beams are almost always9

located where the concrete tensile stress limits and the balanced-section conditions10

intersect. Typical plots for a beam with bonded, unbonded, and partially bonded Fibra11

tendons are shown in Fig. 12.12

The differences between the three cases are clear. Most of the constraints do not13

change significantly, but the balanced section constraint is much more severe for the14

fully-bonded case. The ultimate strength, creep rupture and deflection constraints15

shift to lower section depths and tendon areas as the bond increases, while the transfer16

constraints are the same for both cases. For all of these reasons, bonded beams are17

always more expensive than those with unbonded tendons.18

Because the cheapest way to use FRPs in prestressed concrete design is by debonding19

them from concrete, all the FRP beams examined in the sections below are unbonded,20

unless stated otherwise.21

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61


18/48

Price reduction1

When designing beams with FRP rebar, the optimal design was at a sharp vertex of2

the feasible region, and very large changes in cost of FRP would be needed to alter the3

optimum. However, the shape of the feasible zone with prestressed beams means4

that, although it still occurs at a vertex, one of the lines at that intersection has a5

similar slope to the cost function. Thus, changes to the cost of the material would6

lead to alterations in the optimal design. These effects are illustrated in Fig. 13(a) and7

Fig. 13(b). A 40% reduction in the cost of CFRP would not alter the optimal solution8

it would remain at the vertex, but for Technora a 40% reduction in price would9

move the optimal solution to a lower depth and higher tendon area (and thus higher10

prestress). It is still clear, however, that significant price changes would be needed11

before the cost of structures with FRP matched the initial cost of structures with steel12

tendons.13

Concrete strength - deflection limits14

Concrete strength has little effect on the total cost. When the concrete strength15

increases, more tendon area is needed to eliminate the tendon snapping mode of16

failure. Thus the balanced-section constraint rotates to require a greater tendon area,17

but this is partly offset by the better shear capacity of stronger concrete.18

The imposition of stricter deflection limits does not affect the optimal solution for19

prestressed concrete since pre-straining flexible FRPs eliminates the problem of20

deflections, as the concrete remains un-cracked. Even for extreme deflection limits as21

much as span/480, the same criteria (tensile stresses in the concrete at the working22

load and balanced-section constraints) still governs.23

24

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61


19/48

Case Study 3. Indeterminate Prestressed Concrete1

The design of indeterminate prestressed concrete structures differs from the2

determinate beams as more constraints come into play. The amount of prestress has3

to be satisfactory for both hogging and sagging bending, which have an effect both on4

the amount of prestress and the depth of the beam. It is also necessary to ensure that5

the beam has adequate ductility to take benefit from the moment redistribution that6

can occur in such beams. It is often assumed that beams with FRP are less ductile7

than beams with steel, but the large strain capacity of FRPs usually means that large8

rotations can occur as the concrete cracks.9

A continuous 3-span prestressed concrete beam supporting a loaded slab is assumed.10

Structural and section dimensions, as well as loading properties, are shown in Table 5.11

Secondary moments are not considered since these would require the complete cable12

profile to be calculated which is beyond the scope of this paper.13

New design constraints were introduced using the theory of Low29: P2 limits which14

ensure that tendon eccentricities fit in the beam depth and P 3 limits which ensure that15

concordant tendon profiles exist in allowable tendon eccentricities; detailed16

description of those limits can be found elsewhere.317

In particular, for continuous beams prestressed with brittle FRPs, provision of18

confinement in the compression zone is essential.30

The structure has to behave non-19

linearly so that moment redistribution takes place and failure does not occur when one20

of the points on the structure fails. Non-linear behaviour is also needed for design21

calculations to be justified through the safe theorem of plasticity.22

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61


20/48

The regions where concrete is likely to fail in compression are confined with AFRP1

spirals similar to those tested by Burgoyne and Leung.30The spirals are placed in the2

top flange in sagging moment regions (mid-span) and in the bottom flange in hogging3

moment regions. The length over which they are extended is chosen to be 70% of the4

span's length for top flange confinement and 30% of the span for the bottom flange.5

Four different spiral pitches were chosen: 100, 50, 35 and 10 mm. A moment6

redistribution of 0, 5, 10 and 20% is also assumed to occur respectively before the7

structure fails. There is not yet sufficient data to allow an accurate calculation of the8

moment redistribution extent according to the spiral spacing provision. No9

redistribution is assumed to take place at 100mm spiral spacing, as this is equal to10

spiral diameter and tests have shown that at this spacing concrete fails in a gentler11

manner, but with limited non-linearity.12

When the confinement is increased, the concrete failure strain increases and thus the13

danger of snapping of tendon before concrete crushing is increased. This can be seen14

on Fig. 14 where two d-Ap diagrams are plotted for beams prestressed with unbonded15

Parafil tendons with 100mm and 10mm spiral pitch. The balanced-section constraint16

moves to greater tendon areas when confinement is increased. In addition, the17

ultimate constraint relaxes to lower section depths and tendon areas as redistribution18

takes place. The net effect is that the tendon area has to increase with a corresponding19

increase in the optimal cost.20

It can be concluded that there is not much benefit in providing confinement in the21

compression zone since it can change the mode of failure, threaten the safety of the22

structure and increase cost considerably. Further research is needed in this area.23

24

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61


21/48

CONCLUSIONS1

A method has been introduced that allows the optimal design, and hence initial costs,2

of concrete beams reinforced and prestressed with FRPs to be studied. Beams with3

steel and FRP reinforcements were optimised and comparisons were made. Some4

conclusions can be drawn from the analytical results.5

For the reinforced structures:6

the serviceability limit state condition, in most cases deflections or crack7widths, govern the design due to the relatively high flexibility of FRP as8

concrete section reinforcement.9

because the FRP-snapping mode of failure has to be eliminated, the optimum10solution will normally be at the intersection of the above condition and the11

balanced-section constraints12

for the most flexible FRP bars (GFRP and Technora-AFRP) with poor bond13properties (sandblasted or twisted), crack widths may be wide under service14

loads and these will be the govern for aesthetic reasons. For stiffer bars15

(AFRP-Fibra and CFRP) the deflection criterion governs.16

using stronger concrete to diminish the excessive deflections does not seem to17solve the problem. The stronger concrete is harder to crush and more bars are18

needed for that purpose. As stronger concrete is more expensive a cost19

increase was observed.20

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61


22/48

beams with FRP reinforcement are usually deeper than their steel-reinforced1counterparts and have higher deflections, but use less reinforcing material2

because it is stronger.3

because the optimal design is at a vertex of the feasible region, small changes4to the design do not reduce the cost. Beams with GFRP are slightly more5

expensive than beams with steel but it is unlikely that beams designed with6

stiffer FRP reinforcement will ever be economic on a first-cost basis.7

For simply supported and continuous prestressed concrete structures:8

optimum solutions are governed by the tension stress limits in the cross9section under working load conditions and the balanced-section constraint, as10

the tendon snapping mode has to be prevented11

in contrast to steel-prestressed bridges, less tendon bond to concrete produces12cheaper solutions. When tendons are bonded they snap more easily and the13

designer will be forced to use more FRPs,14

spiral provision to enhance concrete plasticity triggers tendon snapping and15more flexural bars are needed to avoid this mode of failure,16

for bonded FRP tendons higher prestressing does not necessarily mean17reduced cost,18

designs with unbonded prestressing tendons are the most likely application for19economic use of FRP in concrete20

For reinforced and prestressed structures, the ultimate moment capacity constraint is21

very unlikely to be one of the governing constraints and the optimum structural22

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61


23/48

dimensions remain the same even for the most optimistic future FRP price1

predictions. On the other hand, the balanced-section constraint proves to be a2

governing constraint in most cases. Hence, the design calculations should:3

1. determine the material volumes for a balanced section and42. check:5

deflections for reinforced concrete designs or,6

stress limits in prestressed concrete design.7

The outcome will then be very close to optimal in terms of cost.8

From the above study it is clear that the initial cost of FRP reinforced or prestressed9

concrete structures is higher than those with steel, which explains why there are so10

few commercial applications of FRP to date. These additional costs cannot be justified11

unless life cycle costs are taken into account. Whole-life costs for bridges can be12

enormous, especially if the users costs are taken into account as well as the owners,13

and if they are ignored, the cost estimation is far from being complete.3,3114

AKNOWLEDGEMENTS15

The EU-TMR network ConFibrecrete is gratefully acknowledged for funding this16

work.17

18

REFERENCES19

1. FICKELHORN, M. Editorial.Materials and Structures, RILEM, 1990, 23, No. 137,20317.21

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61


24/48

2. MAKHTOUF, H. M., ABMADI, B. H. and AL-JABAL, J. Preventing Reinforced1Concrete Deterioration in the Arabian Gulf. Concrete International, 1991, 13,2

No. 5, 65-67.3

3. BALAFAS, I. Fibre-reinforced-polymers vs Steel in Concrete Bridges: Structural4Design and Economic Viability. PhD thesis, University of Cambridge, 2003.5

4. AL-SALLOUM, Y. A. and SIDDIQI G. H. Cost Optimum Design of Reinforced6Concrete Beams. ACI: Structural Journal, 1994, 91, No. 6, 647-655.7

5. GIANNOPOULOS, I. P. and BURGOYNE, C. J. Stress limits for aramid fibres.8Proceedings of the Institution of Civil Engineers: Structures and Buildings ,9

2009, SB4, 221-232.10

6. YAMAGUCHI,T.,NISHIMURA,T.andUOMOTO,T. Creep model of FRP rods based11on fiber damaging rate. Proceedings of the First International Conference12

(CDCC 1998), Durability of Fibre Reinfoced Polymer Composites for13

Construction, eds. B. Benmokrane, H. Rahman, 1998, 427-438.14

7. KALLITSIS CONSULTANTS. GlyfadaAthens. Personal Communication, 2008.158. Republic of Cyprus - Ministry of Public Works. Personal Communication, 2008.169. HUGHES BROTHERS,INC. Personal Communication, 2008.17

10.

PULTRALL,INC. Personal Communication, 2008.18

11. SIREG S.p.A. Personal Communication, 2008.1912. LINEAR COMPOSITES LIMITED. Personal Communication, 2008.2013. HAMBLY,E.C. Bridge Deck Behaviour. 2nd edition, Taylor & Francis, 1991.2114. LEUNG H. Y. Study of Concrete Confined with Aramid Spirals. PhD Thesis,22

University of Cambridge, 2000.23

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61


25/48

15. MIKAMI H., KATOH M., TAKEUCHI H. and TAMURA T. Flexural and Shear1Behaviour of RC Beams Reinforced with Braided FRP rods in Spiral Shape.2

Transaction of the Japan Concrete Institute, 1991, 11, No. 1, 119-206.3

16. VARIAN, H. R. Intermediate Microeconomics. 5th edition, W. W. Norton &4Company Ltd, 1999.5

17. COSENZA E.,MANFREDI G.andREALFONZO R. Behavior and modelling of bond6of FRP rebars to concrete.ASCE Journal of Composites for Construction, 1997,7

1, No. 2, 40-51.8

18. Fib-TG-9.3. Chapter 3: Reinforced Concrete. fib Model-Code on Fibre9Reinforced Polymer Reinforcement (Riga Draft). CEB Committee TG9.3, 2003.10

19. NANNI, A. Flexural Behaviour and Design of RC Memebers Using FRP11Reinforcement. ASCE, Journal of Structural Engineering, 1993, 119, No. 11,12

3344-3359.13

20. GUADAGNINI M., PILAKOUTAS K. and WALDRON P. Shear Performance of FRP14Reinforced Concrete Beams. Journal of Reinforced Plastics and Composites,15

2003, 22, No. 15, 1389-1407.16

21. GUADAGNINI M.,PILAKOUTAS K.andWALDRON P. Shear Resistance of FRP RC17Beams: Experimental Study. Journal of Composites for Construction, ASCE,18

2006, 10, No 6, 464-473.19

22. BURGOYNE C. J. Rational Use of Advanced Composites in Concrete.20Proceedings of the Institution of Civil Engineers: Structures and Buildings, 2001,21

146, 253-262.22

23. LEUNG H. Y. and BURGOYNE C. J. The uniaxial stress-strain relationship of23spirally-confined concrete.ACI Materials Journal, 2005, 102, No. 6, 445-453.24

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61
http://online.sagepub.comhttp/jrp.sagepub.com/content/22/15/1389.abstracthttp://online.sagepub.comhttp/jrp.sagepub.com/content/22/15/1389.abstracthttp://link.aip.org/link/?QCC/10/464/1http://link.aip.org/link/?QCC/10/464/1http://link.aip.org/link/?QCC/10/464/1http://link.aip.org/link/?QCC/10/464/1http://link.aip.org/link/?QCC/10/464/1http://online.sagepub.comhttp/jrp.sagepub.com/content/22/15/1389.abstracthttp://online.sagepub.comhttp/jrp.sagepub.com/content/22/15/1389.abstract


26/48

24. BURGOYNE,C.J. Should FRP be Bonded to Concrete?. In A. Nanni and C. W.1Dolan, editors, Non-Metallic (FRP) Reinforcement for Concrete Structures,2

FRPRCS-1, 1993, SP-138, 367-380.3

25. ABDELREHMAN, A. A. and S. H. RIZKALLA. Serviceability of Concrete Beams4Prestressed by Carbon-Fibre-Reinforced-Plastics Bars. ACI Structural Journal,5

1997, 94, No. 4, 447-457.6

26. LEES J.M.andBURGOYNE C.J. Experimental study of the Influence of Bond on7the Flexural Behaviour of Concrete Beams Pre-tensioned with Aramid Fibre8

Reinforced Plastics.ACI Structural Journal, 1999, 96, No. 3, 377-385.9

27. JANNEY,J.R.,HOGNESTAD,E.andMCHENRY,D. Ultimate Flexural Strength of10Prestressed and Conventionally Reinforced Concrete Beams. American Concrete11

Institute, 1956, 27, No. 6, 601-620.12

28. NAAMAN, A. E. and ALKHAIRI, F. M. Stress at Ultimate Post-Temsioned13Tendons: Part 2 Proposed Methodology. American Concrete Institute:14

Structural Journal, 1992, 88, No. 6, 683-692.15

29. LOW, A. M. The preliminary design of prestressed concrete viaducts.16Proceedings of the Institution of Civil Engineers, 1982, 73, No. 2, 351-364.17

30. BURGOYNE C.J.andLEUNG H.Y.Test on Prestressed Concrete Beam with AFRP18Spiral Confinement and External Aramid Tendons, Antoine E. Naaman19

Symposium - Four Decades of Progress in Prestressed Concrete, Fiber20

Reinforced Concrete, and Thin Laminate Composites, SP272-04, American21

Concrete Institute, 2010, 272, 69-86.22

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61
http://www-civ.eng.cam.ac.uk/cjb/papers/p36.pdfhttp://www-civ.eng.cam.ac.uk/cjb/papers/p36.pdfhttp://www-civ.eng.cam.ac.uk/cjb/papers/p36.pdfhttp://www-civ.eng.cam.ac.uk/cjb/papers/p36.pdfhttp://www-civ.eng.cam.ac.uk/cjb/papers/p36.pdfhttp://www-civ.eng.cam.ac.uk/cjb/papers/p36.pdf


27/48

31. BALAFAS I. and BURGOYNE C.J. Economic viability of structures with FRP1reinforcement or prestress,4

thConference on Advanced Composite Materials in2

Bridges and Structures, Calgary, Canada, 2004, Paper 186.3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61
http://www-civ.eng.cam.ac.uk/cjb/papers/cp67.pdfhttp://www-civ.eng.cam.ac.uk/cjb/papers/cp67.pdfhttp://www-civ.eng.cam.ac.uk/cjb/papers/cp67.pdfhttp://www-civ.eng.cam.ac.uk/cjb/papers/cp67.pdf


28/48

LIST OF TABLES1

Table 1. Material Properties .....................................................................................272

Table 2. Material costs.............................................................................................283

Table 3. Parameters for reinforced beam comparisons .............................................294

Table 4. Parameters for prestressed beam comparisons ............................................305

Table 5. Parameters for continuous prestressed beam comparisons ..........................316

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61


29/48

Material FormStrength(N/mm2)

ElasticModulus

[kN/mm2]

SnappingStrain

PartialSafetyFactor

B1

GFRPPultruded

rod1200 40 0.030 1.3 0.101

AFRP-

Fibra

Braided

rod 1480 68 0.022 1.3 0.069AFRP-

TechnoraPultruded

rod1900 56 0.034 1.3 0.053

CFRPPultruded

rod2200 140 0.012 1.3 0.016

ParafilParallel-lay

rope1900 120 0.016 1.3 0.069

Steel

rebarRod 460 210 0.1 1.15 -

Steeltendon

7-wirestrand

1700 210 0.1 1.15 -

1determined by fitting on data shown in Giannopoulos and Burgoyne 20095and Yamaguchi1

et al6.

2

Table 1. Material Properties3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61


30/48

Material [/m 3 ] [/kN/m]

Concrete1,2 (cf in N/mm

2 ) 48.145e0.0178f

c

steel reinforcement [1],[2] 6700 0.015

steel tendon [1],[2] 20000 0.012GFRP [3],[4],[5] 11500 0.010

FiBRA [6] 82000 0.055TechnoraParafil [6] 82000 0.043

CFRP bars [3],[4],[5] 90000 0.041Formwork [1],[2] [/m

2] 11

[1]7, [2]

8, [3]

9, [4]

10, [5]

11, [6]

12.

Table 2. Material costs12

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61


31/48

Structure propertiesdesired lifetime [y] 120

span [m] 6beam web width [m] 0.25

slab depth [m] 0.15

slab width [m] 5

beam top flange effective width [m] 0.85cover [mm] 50 (FRP) 75 (steel)

Loading propertiesslab's short-term live load [kN/m2] 1.0

slab's long-term live load [kN/m2] 1.0

partial dead load safety factor (ultimate) 1.4partial live load safety factor (ultimate) 1.6

partial dead, live load safety factor (working) 1.0

Concrete propertiesstrength [N/mm

2] 30

ultimate strain 0.0035

partial safety factor 1.5

Table 3. Parameters for reinforced beam comparisons12

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61


32/48

Structure propertiesdesired lifetime [y] 120

span [m] 15beam web width [m] 0.30

beam top flange width [m] 1.5beam bottom flange width [m] 0.55

beam bottom flange depth [m] 0.2slab depth [m] 0.15slab width [m] 5

cover [mm] 100

Loading propertiesslab uniform live load [kN/m

2] 2.0

slab uniform superimposed dead load [kN/m2] 1.0

partial dead load safety factor (ultimate) 1.4partial superimposed dead load safety factor (ultimate) 1.2

partial live load safety factor (ultimate) 1.6partial dead, live load safety factor (working) 1.0

Concrete properties

strength [N/mm2] 60confinement spacing [mm] 100

partial safety factor 1.5

Initial prestressing (t=0)Steel [%] 60

CFRP [%] 60AFRP (Fibra and Technora), Parafil rope [%] 55

Design limitsdeflections [m] span/250

transfer stress limits [N/mm2] -1 (tension), 18 (compression)

working stress limits [N/mm2] 0 , 25 (compression)

Table 4. Parameters for prestressed beam comparisons12

3

4

5

6

7

8

9

10

11

12

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61


33/48

Structure properties

desired lifetime [y] 120

spans [m] 15 - 20 - 15

beam web width [m] 0.3

beam top flange width [m] 1.5

beam bottom flange width [m] 0.9beam bottom flange depth [m] 0.2

slab depth [m] 0.2slab width [m] 5

cover [mm] 100

Loading properties

slab uniform live load [kN/m 2 ] 2.0

slab uniform superimposed dead load [kN/m 2 ] 1.0

Table 5. Parameters for continuous prestressed beam comparisons1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61


34/48

LIST OF FIGURES1

Fig. 1. Depth versus Ap diagram for reinforced concrete sections2

Fig. 2. Beam with: (a) steel rebar; (b) FRP rebar3

Fig. 3. Beam with: (a) steel tendon; (b) FRP tendon4

Fig. 4. Structural configuration5

Fig. 5. Technora reinforced beam with deflection limits: (a) span/180; (b) span/4806

Fig. 6. Cost of RC beams with different materials and different deflection limits7

Fig. 7. Effect of reducing cost of FRP (Fibra) by 40%8

Fig. 8. Bond-slip curves (Cosenza et al 1997)9

Fig. 9. Design curves for GFRP bars: (a) smooth; (b) braided and indented10

Fig. 10. Effect of bond on amount of FRP needed11

Fig. 11. Effect of varying Partial Material Safety Factor for CFRP-reinforced beams12

(a) = 1.3; (b) = 2.013

Fig. 12. Prestressed Beams with Fibra tendons: (a) fully bonded; (b) partially bonded;14

(c) unbonded15

Fig. 13. Prestressed Beam. Effect of price on: (a) CFRP; (b) Technora16

Fig. 14. Indeterminate Prestressed Beam. Technora spirals: (a) 100 mm pitch; (b) 1017

mm pitch18

19

20

21

22

23

24

1

2

3

4

5

67

8

9

10

11

12

13

14

15

16

17

18

1920

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61


35/48

0 1000 2000 3000 4000 5000 6000 70000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Ap

mm[2

]

p

[

]

Feasible zoneBeams stronger to

design specifications

constraintStrength

Concretecrushing

Barsn

app

ing

Balanced section

Non Feasible zoneBeams weaker to

design specifications

0 500 1000 1500 2000 25000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Ap

mm[2

]

depth

m[

] Feasible zone

sconstraint

trength

Balanced section

minimum reinforcementratio constraint

concrete crushingzone

bar snappingzone

bala

nced

secti

ons

0 500 1000 1500 2000 25000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Ap

mm[2

]

p

[

]

Feasible zone

sconstraint

trength

minimum reinforcementratio constraint

deflectionsconstraint

bala

nced

secti

ons

0 500 1000 1500 2000 25000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Ap

mm[2

]

depth

m[

]

Feasible zone

sconstraint

trength

deflectionsconstraint

Flexural Optimolution

alS

ba

lanced

sections

Costfu

nctio

n

0 500 1000 1500 2000 25000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Ap

mm[2

]

depth

m[

]

Feasible zone

strength

deflections

concrete stresses

crack width

Flexural Optimolution alS

bala

nced

secti

ons

Costfu

nctio

n

Fig. 1. Assembly of the section depth-Ap diagram for FRP reinforced sections

a. Ultimate strength condition b. Bar snapping condition

c. Deflection constraint d. Cost function

e. Complete diagram for FRP-reinforced section


36/48

0 500 1000 1500 2000 25000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Ap

(mm2

)

depth(m)

Flexural Optimumsolution

defle

ction

s

Feasible zone

max

strength

Costfunction

b

ala

nced

secti

ons

0 500 1000 1500 2000 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Ap

(mm2

)

depth(m)

Flexural Optimumsolution

Feasible zone

deflections

strength

bala

nced

sect

ions

Costfu

nctio

n

Fig. 2. Beam with: (a) steel rebar; (b) FRP rebar


37/48

0 1000 2000 3000 4000 5000 6000 70000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Ap

(mm2

)

p

(

)

Feasiblezone

strength

Flexural optimumsolution

transfertension

workingtension

ransfercompression

t

Costfunction

orkingcompression

w

bala

nced

sections

! max

0 1000 2000 3000 4000 5000 6000 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Ap

(mm2

)

depth(m)

Feasiblezone

strength

transfercompression

workingtension

transfertension

Costfu

nctio

norking

compressionw

Flexuraloptimsolution

al

bala

nced

sections

Fig. 3. Beam with: (a) steel tendon; (b) FRP tendon


38/48

5.00

15

Ap = ?

d= ?

0.2

6.00

0.85

Fig. 4. Structural configuration


39/48

0 500 1000 1500 2000 25000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

A

p

(mm2

)

Feasible zone

Flexural Optimumsolution deflections

crackwidth

creep-rupture

concretestresses

b

ala

nced

secti

ons

costfunctio

n

0 500 1000 1500 2000 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

A

p

(mm2

)

depth(m)

Feasible zone

Flexural Optimumsolution deflectionscrack

width

creep-rupture

concretestresses

bal

anced

secti

ons

costf

unctio

n

Fig. 5. Technora reinforced beam with deflection limits: (a) span/180; (b) span


40/48

0

0,5

1

1,5

2

2,5

3

3,5

4

4,5

l/180

l/250

l/360

l/480

l/180

l/250

l/360

l/480

l/180

l/250

l/360

l/480

l/180

l/250

l/360

l/480

glass FiBRA Technora Carbon

cos

tr

atios

FRP

/stee

l

. 6. Cost of RC beams reinforced with different materials and different deflection


41/48

0 500 1000 1500 2000 25000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Ap

(mm2

)

depth(m)

Optimal solution

deflections

crack width

Feasible zone

strength

bala

nced

sectio

ns

curre

ntp

rice

40%barpri

cereduction

ig. 7. Effect of reducing cost of FRP (FiBRA) by 4


42/48

ribbed

twisted

braided and sanded

sand blasted

smooth

Fig. 8. Bond-slip curves 17


43/48

0 500 1000 1500 20000

0.2

0.4

0.6

0.8

1

1.2

Ap

(mm2

)

p

(

)

Feasible zone

Optimal solution

crackwidth

deflections

strength

bala

nced

sectio

ns

costfunction

creep rupture

concrete

stresses

0 500 1000 1500 0

0.2

0.4

0.6

0.8

1

1.2

Ap

(mm2

)

depth(m

)

Optimal solution

crackwidthdeflections

strength

bala

nced

sectio

ns

Feasible zone

costfunction

creeprupture

concrete stresses

Fig. 9. Design curves for GFRP bars: (a) smooth; (b) braided and indented


44/48

0

200

400

600

800

1000

1200

1400

smooth

sandblasted

twisted

ind-braided

ribbed

graincoated

braidedsanded

smooth

sandblasted

twisted

ind-braided

ribbed

graincoated

braidedsanded

smooth

sandblasted

twisted

ind-braided

ribbed

graincoated

braidedsanded

smooth

sandblasted

twisted

ind-braided

ribbed

graincoated

glass FiBRA Technora Carbon

flexuralbararea[m

m2]

Fig. 10. Effect of bond on amount of FRP needed


45/48

0 500 1000 1500 2000 25000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Ap

(mm2

)

p

(

)

crackwidth

deflections

strength

Optimal solution

Feasible zonebalan

ced

costfunctio

n

sections

cree

p

rup

ture

concrete stresses

0 500 1000 1500 2000 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Ap

(mm2

)

depth(m)

crackwidth

deflections

strength

Optimal solution

bala

nced

Feasible

costfunctio

n

sections

zone

creep rupture

concrete stresses

g. 11. Effect for varying Partial Material Safety Factor for CFRP-reinforcedams: (a) =1.3; (b) =2.0g g


46/48g. 12. Prestressed beams with FiBRA tendons: (a) fully bonded;unbonded

(b) partially bond

0 1000 2000 3000 4000 5000 6000 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

A (mm2

)

depth(m)

Feasiblezone

streng

th

transfercompression

workingtension

orkingcompression

w

transfertension

balan

cedse

ction

s

Optimal

solution

Costfu

nctio

n

d

eflections

creeprupture

0 1000 2000 3000 4000 5000 6000 70000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

A (mm2

)

dep

th(m) Feasible

zone

Optimalsolution

transfertension

ultimate

transfercompression

workingtensiondeflections

creeprupture

orkingcompression

w

balan

ceds

ectio

ns

Cos

tfunctio

n

(a) (b)

(c)

0 1000 2000 3000 4000 5000 6000 70000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Ap

(mm2

)

depth(m)

Feasiblezone

Optimal solution

transfer tension

ultimate

transfercompression

workingtension

deflectio

ns

creeprupture

orkingcompression

wbal

anced

section

s

Costfu

nctio

n


47/48

0 1000 2000 3000 4000 5000 60000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Ap

(mm2

)

p

()

Optimalsolution

transfer tension

strength

transfercompression

workingtension

deflections

Feasiblezone

orkingcompression

w

current

price

40%barpricereduction

creeprupture

balan

ceds

ectio

ns

0 1000 2000 3000 4000 5000 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Ap

(mm2

)

depth(m)

Feasible

zone

strength

transfertension

transfercompression

workingtension

creeprupture

orkingcompression

w

deflectio

ns

bala

nced

sectio

ns

currentp

rice

40%bar

pricereduction

Optimalsolution

Fig. 13. Prestressed beam - Effect of price on: (a) CFRP; (b) Technora


48/48

0 5

1

1.5

2

2.5

3

depth

(m)

Feasiblezone

Optimalsolution

P3min

Z limits

P1min

P2strength

balan

ced s

ectio

ns

Costfu

nctio

n

5

1

5

2

5

3

Feasiblezone

strength

P3min

Z limits

P1min

P2

creep

balan

ced

sectio

ns

Co

stfunctio

n

Optimalsolution

Documents

p75preEconomic Beam Designs Using FRP Rebar