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FRP Lamella user manual for design software version 3 flexural and shear strengthening using FRP materials Peter Onken Wiebke vom Berg Dirk Matzdorff © bow ingenieure gmbh braunschweig · hamburg

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#$K+Übersicht© bow ingenieure gmbh braunschweig · hamburg

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according to Eurocode 2

© bow ingenieure gmbh · braunschweig / hamburg · germany

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Contents

Part I – Design concept

1. Introduction 5 2. Notation 6 3. Design program features 9 4. Basic assumptions 10 5. Safety concept 12 6. Degree of strengthening 13

6.1 Definition according to Eurocode 2 13 6.2 Failure of the FRP system 13

7. Material behaviour 14 7.1 Concrete 14 7.2 Reinforcing steel 14 7.3 Prestressing steel 14 7.4 FRP material 15

8. Design aspects for FRP systems 16 8.1 Externally bonded FRP laminates 16 8.2 Externally bonded carbon sheets 17 8.3 Near surface mounted CFRP laminates 17

9. Imposed actions 18 10. Design procedure 19

10.1 Capacity of the unstrengthened cross-section 19 10.2 Required cross-sectional area of FRP 19 10.3 Conditions of equilibrium 20 10.4 Control of strain profiles 21 10.5 Control of stresses 21

11. Bond check of the FRP system 22 11.1 Anchorage of externally bonded CFRP laminates 22 11.2 Anchorage of externally bonded carbon sheets 23 11.3 Calculation of the envelope line / verification of the anchorage 24 11.4 Anchorage of near surface mounted CFRP laminates 27 11.5 Surface tensile strength of concrete 29

12. Anchorage of bottom reinforcement at end support 30 13. Detailing provisions 31 14. Shear design 32

14.1 Shear capacity according to Eurocode 2 32 14.2 Design of the additional shear reinforcement 35 14.3 Anchorage of external stirrups 37

15. Further checkings 38 16. Fire protection 38

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Part II – Use of the program

17. Program user interface 39 17.1 Start of the program 39 17.2 Settings 39 17.3 Basic information about the FRP Lamella user interface 39 17.4 Data input 40 17.5 Output of results 41

18. Input and output windows 42 18.1 Input window project 42 18.2 Input window code 43 18.3 Input window geometry 44 18.4 Output window cross-section 45 18.5 Input window concrete 46 18.6 Input window steel 47 18.7 Input window main flexural reinforcement 48 18.8 rebar tables for the selection of reinforcement cross-sectional area 49 18.9 Input window flexural reinforcement at support 50 18.10 Input window loads in unstrengthened state 51 18.11 Input window loads in strengthened state 53 18.12 Input window FRP system 55 18.13 Input window FRP cross-section 56 18.14 Output window design 58 18.15 Output window strains in ultimate limit state 60 18.16 Output window strains / stresses in service state 61 18.17 Input window FRP end anchorage 62 18.18 Output window FRP end anchorage 64 18.19 Input window anchorage of flexural reinforcement at support 65 18.20 Output window anchorage of flexural reinforcement at support 66 18.21 Input window shear – reinforcement and loads 67 18.22 Input window shear strengthening 68 18.23 Output window shear strengthening 69 18.24 Output window shear strengthening – anchorage of additional external stirrups 70

19. Program menu and tool bar 72 19.1 Menu bar items 72 19.2 Tool bar symbols 74

20. Installation instructions 75 Appendix 1 example – T-beam according to Eurocode 2 Appendix 2 example – two-span slab according to Eurocode 2 Appendix 3 example – prestressed concrete beam according to Eurocode 2 Appendix 4 bow engineers – experts for strengthening design

1. Introduction

FRP Lamella is a design program for the strengthening of reinforced and prestressed or post- tensioned concrete structural members subjected to uniaxial flexure and axial forces using FRP materials (FRP – Fibre Reinforced Polymer). This program can be used for the predesign of strengthening measures as well as for complete calculations within the scope of structural analysis. The program provides the user with the required FRP cross-sectional area for the strengthened state and is performing the necessary verifications of the bond strength and the shear capacity of the concrete member based on the assumptions of the German Guidelines for the strengthening of concrete members using CFRP laminates [2], [3] and carbon sheets [4], (cf. [11]). The design concept according to Eurocode 2 is explained in [10].

The program FRP Lamella is used in almost 15 other countries, adapted to the relevant regulations, guidelines and national standards. Meanwhile different versions corresponding to the following international codes are available:

• Eurocode 2

• British Standard 8110

Fig. 1 Opening window of the FRP Lamella software

Note The software FRP Lamella is based on the material parameters of S&P FRP systems. If other types of reinforcing fibres or adhesive systems will be used, the results provided by the software will no longer be valid. Under these circumstances the system supplier S&P will refuse any liability for the application of S&P products.

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2. Notation

As a rule, in this manual the standard notations derived from Eurocode 2 are used. They may differ from notations of other national design codes and guidelines. The following list gives an overview of the notations used in this manual and in the software.

Notation Program Geometry width of the component b web width b0

overall height h, h0

thickness of upper flange h1

effective flange width of lower flange b2

thickness of lower flange h2

span l total cross-sectional area Ac

distance of the centroid from top edge of the member zcg

moment of inertia of concrete cross-section Iy section modulus above the gravity axis Wtop

section modulus below the gravity axis Wbottom

Reinforcement cross-sectional area of longitudinal rebars As

pre-strain of the longitudinal reinforcement due to prestressing εp0

cross-sectional area of internal stirrup rebars asw

distance from centroid of rebars to top edge of the member zs

diameter of rebars ds

concrete cover of the stirrups cw

Steel characteristic yield strength of reinforcing steel fyk

modulus of elasticity of reinforcing steel Es

strain limit of reinforcing steel εsu

characteristic tensile strength of prestressing steel fpk

modulus of elasticity of prestressing steel Ep

strain limit of prestressing steel εpu

reduction coefficient for the tensile strength of prestressing steel αp

partial safety factor for steel γs

Concrete characteristic compressive strength of concrete (EC 2) fck

strain limit of concrete εcu

strain at the axis of the parabolic curve of the stress-strain line of concrete εc1

reduction factor for the compressive strength of concrete (long term effects) α design shear strength of concrete τRd

average modulus of elasticity of concrete Ecm

average axial tensile strength of concrete fctm

partial safety factor for concrete γc

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failure strain of FRP material εfu

strain limit of FRP material εf,limit

characteristic tensile strength of FRP material ffk partial safety factor for FRP material γf

reduction factor for the strain limit of FRP material kε

number of FRP plies one on top of each other nf

number of FRP strips one next to each other mf

spacing of FRP strips sf

cross-sectional area of FRP strengthening Af

distance from centroid of FRP strip to top edge of the member zf

distance of FRP strips to the lateral edge of the member ar

thickness of FRP reinforcement tf width of FRP reinforcement bf

Design characteristic bending moment at time of strengthening MSk0

characteristic axial force at time of strengthening NSk0

characteristic prestressing force Np

statically indeterminated part of prestressing moment Mp’ design bending moment of strengthened state MSdf

design axial force of strengthened state NSdf

characteristic bending moment of strengthened state MSkf

characteristic axial force of strengthened state NSkf

average partial safety factor for bending moments caused by loads γM,m

average partial safety factor for axial forces caused by loads γN,m

design moment of resistance of unstrengthened cross-section MRd0

characteristic moment of resistance of unstrengthened cross-section MRk0

design moment of resistance of strengthened cross-section MRdf

strengthening ratio η remaining global safety in case of loss of FRP strengthening θ strain of extreme compression fibre of concrete εc

distance from neutral axis to extreme compression fibre x maximum strain of reinforcing steel εs

maximum strain of prestressing steel εp

maximum strain of FRP reinforcement εf

stress of extreme compression fibre of concrete σc

maximum stress of reinforcing steel σs

maximum stress of prestressing steel σp

maximum stress of FRP reinforcement σf

Anchorage substrate strength of concrete (median of the population) fcsm

design value of the substrate strength of concrete fcsd

characteristic compressive strength of adhesive fKc,k

characteristic tensile strength of adhesive fKt,k

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characteristic shear strength of adhesive τK,k

distance from calculated axis of support to edge of support ai

distance from end of the FRP strip to edge of support f horizontal displacement of the envelope line of tensile force aL

design moment of strengthened state in point E MSdf,E

design axial force of strengthened state in point E NSdf,E

distance from point E to theoretical axis of support xE

tensile force of FRP reinforcement in point E Ffd,E

design value of the maximum bond force Fbd,max

required bond length of maximum bond force lbd,max

recommended bond length lb design value of shear force at support VSdf,A

design value of axial force at support NSdf,A

total required anchorage force at support FA,req

anchorage force covered by internal reinforcement Fs,A

design bond strength of internal rebars fbd

force covered by FRP anchorage Ff,A

anchorage length of FRP reinforcement from the support front lf,A Shear cross-sectional area of internal stirrups asw

design shear force of strengthened state in relevant section X VSdf,X

design axial force of strengthened state in relevant section X NSdf,X

design bending moment of strengthened state in relevant section X MSdf,X

strain limit of additional shear reinforcement εlimit

characteristic modulus of elasticity of FRP Sheet Efk

characteristic tensile strength of FRP Sheet ffk reduction factor for modulus of elasticity due to manual lamination γE

modulus of elasticity of steel plates for shear strengthening Es

characteristic yield strength of steel plates for shear strengthening fyk

partial safety factor for shear strengthening steel plates γs

distance from resultant of concrete stress to extreme compression fibre ac

effective depth of internal steel rebars ds

effective depth of FRP reinforcement df

average effective depth dm

shear force limit of the strengthened cross-section Vmax

design shear resistance provided by concrete VRd1

design shear resistance of the concrete cross-section without web crushing VRd2

design shear resistance of concrete cross-section with internal stirrups VRd3

cross-sectional area of additional shear reinforcement aw

thickness of additional external stirrups tw width of additional external stirrups bw

cross-sectional area of one additional external stirrup Aw

spacing of additional external stirrups sw

stress of the internal shear reinforcement σsw

stress of the additional shear strengthening σfw

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3. Design program features

FRP Lamella is developed as a pure design program for the strengthening of reinforced and prestressed or post-tensioned concrete elements, i.e. the program does not perform any structural analysis. As a consequence the user has to determine the internal forces in advance with a calculation by hand or by using structural analysis software for instance. The updated version 3.x now also considers prestressed or post-tensioned elements or concrete structures subjected to axial forces.

The program supports 4 types of cross-section: slabs as well as rectangular beams, T-beams and double-T-beams. These options cover almost all reinforced or pre-stressed concrete components subjected to bending which will appear in practice.

There are 3 different FRP-systems for flexural strengthening: externally bonded CFRP laminates, near surface mounted (slot-in) CFRP laminates and externally bonded carbon sheets (unidirectional fabric). The program includes the complete range of S&P products for flexural and shear strengthening.

The required cross-sectional area of FRP strengthening is determined by variation of the strain profile within the limits defined in the regulations. The implementation of non-linear stress-strain relations for concrete as well as for reinforcing and prestressing steel and the iterative solution procedure lead to precise results. Compared with hand calculations the program provides particularly economic amounts of FRP strengthening. Additionally the strain and stress distributions can be controlled. The verification of the bond strength is based on the German Guideline for the strengthening of concrete components using CFRP laminates [2], [3] and carbon sheets [4]. The verification of anchorage lengths of the internal rebars as well as the design of the shear strengthening follows the concept of Eurocode 2.

For structures to be strengthened the geometry, internal reinforcement, steel grades, concrete compressive strength and bending moments can be derived from existing as-built documents. If not available this information has to be established by on-site testing.

In addition the program offers useful tools for the definition of the relevant national concrete strength, the reinforcing and prestressing steel grades and the selection of the existing rebar cross-sections.

The serviceability of the strengthened state cannot be proved by the program. If necessary, the user is responsible to check the deflections and crack widths of the strengthened structure.

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4. Basic assumptions

According to the regulations it can be assumed for the design at ULS (ultimate limit state) that externally bonded FRP reinforcement can be calculated like an additional layer of reinforcement within the provided strain limits. The calculations are based on the well-known assumptions of concrete design:

• For bending a linear strain distribution is assumed (Bernoulli hypothesis).

• For reinforced concrete elements it is assumed that concrete has no tensile strength. All tensile forces necessary for the equilibrium of the internal forces are covered by internal reinforcement and FRP strengthening.

• For prestressed or post-tensioned concrete elements the tensile strength of the concrete may be considered in the uncracked state.

• There is no slip between FRP strengthening and concrete. All cross-section fibres with the same distance to the neutral axis are subjected to the same strain.

The determination of the required FRP cross-sectional area and the resisting moments before and after strengthening result from calculation of the equilibrium of internal forces.

References

[3] Richtlinie für das Verstärken von Betonbauteilen durch Einkleben von unidirektionalen kohlen- stofffaserverstärkten Kunststofflamellen in Schlitze im Beton, Deutsches Institut für Bautechnik, Berlin.

[4] Richtlinie für das Verstärken von Betonbauteilen durch Auflaminieren von unidirektionalen Kohlenstofffaserlaminaten (CFK-Laminate), Deutsches Institut für Bautechnik, Berlin.

[5] CEB-FIP Model Code 1990; EPF Lausanne 1991.

[6] Eurocode 2: Planung von Stahlbeton- und Spannbetontragwerken; Teil 1: Grundlagen und Anwendungsregeln für den Hochbau; Juni 1992.

[7] Rostásy, F.S.; Holzenkämpfer, P. und Hankers, C.: Geklebte Bewehrung für die Verstärkung von Betonbauteilen. Beton-Kalender 1996, T.II, Berlin: Ernst & Sohn 1996.

[8] Holzenkämpfer, P.: Ingenieurmodelle des Verbunds geklebter Bewehrung für Betonbauteile. Dissertation TU Braunschweig, 1994.

[9] Onken, P.; vom Berg, W.; Matzdorff, D.; Nolte, T.: Bemessungsprogramm für CFK-Lamellen. Beton- und Stahlbetonbau 95, 9/2000, S. 551 – 552.

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[10] Onken, P.; vom Berg, W.: Biegezugverstärkung mit CFK-Lamellen – Neues Bemessungs- modell nach EC 2 und DIN 1045-1. Beton- und Stahlbetonbau 96, 2/2001, S. 61 – 70.

[11] Rostásy, F. S.: Expert Opinion No. 98/0322; S&P Reinforcement, Eisenstadt, Österreich.

[12] Blaschko, M. A.: Zum Tragverhalten von Betonbauteilen mit in Schlitze eingeklebten CFK- Lamellen, Dissertation an der TU München, 2001.

[13] Design guidance for strengthening concrete structures using fibre composite materials, Technical Report No. 55, The Concrete Society, Berkshire, UK, 2000.

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5. Safety concept

The ultimate limit state design is based on the following condition (cf. EC 2, section 2.3.2):

Sdf ≤ Rdf (1)

Sdf corresponds to the design value of internal forces or moments due to loads and Rdf to the design resistance, for instance the moment capacity of the cross-section at strengthened state. The index f defines the state after strengthening (with FRP). Both, Sdf and Rdf, are design values and associated with the partial safety factors for actions and materials properties as shown in table 1.

Loads Resistance dead loads live loads concrete reinforcing steelCode

γG γQ γC γS

DIN 1045-1 (Germany-new) 1.35 1.5 1.5 1.15

DIN 1045 (7/88) (Germany-old) 1.75 – 2.1 1.0

BS 8110 (UK) 1.4 1.6 1.5 1.15

BAEL 91 (France) 1.35 1.5 1.5 1.15

SIA 160 / 262 (Switzerland) 1.3 1.5 1.2

ACI 318 (USA) 1.4 1.7 1 / 0.9

KCI (Korea) 1.4 1.7 1 / 0.85

Tab. 1 Partial safety factors according to different design codes

For actions additionally the combination values for the probability of occurrence of several variable loads have to be considered. Additional partial safety factors for the FRP systems are missing in table 1 since different safety concepts are used according to the FRP system and the national guideline. In many cases, e.g. for externally bonded FRP laminates, the design strain is limited instead of introducing a partial safety factor for the tensile strength of the material. For further information see chapter 8.

6. Degree of strengthening

6.1 Definition according to Eurocode 2

In the German Guidelines for the strengthening of concrete members with external bonded FRP laminates [2], [3] and unidirectional carbon sheets [4] it is recommended, that the flexural capacity of the strengthened element should not exceed twice the flexural capacity of the unstrengthened element. This is expressed by the flexural strengthening ratio.

A limitation of the strengthening ratio is only mentioned in the German guidelines [2] – [4]. There exists no such limitation in other regulations or guidelines. One reason for the limitation of the strengthening ratio is the scant knowledge about the behaviour of highly strengthened structures. Other reasons were the insufficient design methods for strengthening with externally bonded FRP reinforcement in the past. Hand calculations do not allow the verification of strains and stresses at service state. On the other hand the design software FRP Lamella provides the strain distributions and stresses at strengthened state in all parts of the section. By this means yielding of internal reinforcement can be avoided, strain limits can be controlled. Therefore the limitation of the strengthening ratio based on the German guidelines [2] – [4] may not be applied very strictly but it is highly recommended not to increase the strengthening ratio far beyond the point which is twice the capacity of the unstrengthened element. The bond behaviour of externally bonded FRP strips will be influenced unfavourably by the increased formation of cracks in highly stressed concrete elements.

Since there is no experience with highly strengthened structures, the limitation of the strengthening ratio is also recommended for other national guideline or standards:

2 M M

fSd EC,B ≤=η . (2)

MSdf describes the imposed bending moment at strengthened state, MRd0 corresponds to the design moment of resistance of the unstrengthened cross-section. For MSdf, the combination principles of actions according to EC 2 have to be considered.

When the strengthening ratio exceeds the limit of 2, the design and detailing should be carried out with special care. For near surface mounted laminates there exist no requirements for the limitation of the strengthening ratio.

6.2 Failure of the FRP system

In other national regulations (e.g. ACI) there is a demand of minimum safety (γ > 1,0) after loss of the FRP strengthening under service (unfactored) loads. In this case the following equation is valid:

1 M M

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7.1 Concrete

For the determination of the concrete compressive stress a parabolic-rectangular stress-strain relationship can be assumed, as provided by Eurocode 2, shown in figure 2a. The parabolic curve ends at a strain value εc1 = 0.2 % and the maximum compressive strain is limited to εcu = 0.35 % (cf. [5]). However, the program also offers the possibility to modify the shape of the parabolic- rectangular stress-strain by adjusting the strain parameters (εc1, εcu). The design value of the concrete compressive strength fcd is determined by dividing the characteristic strength fck by the appropriate partial safety factor in table 1. The reduction factor takes into account the reduced compressive strength under long-term loading.

c

ck cd

f f

γ α=α (α = 0.85; γC according to table 1) (4)

example: C 20/25 → ²mm/N33.11 5.1

2085.0fcd =⋅=α

The design shear strength can be determined from the characteristic concrete compressive strength using the following equation:

c

γ =τ (5)

The average modulus of elasticity and the axial tensile strength of concrete are calculated from the concrete compressive strength. According to Eurocode 2 the following equations are used:

( ) 3/1 ckcm 8f9500E +⋅= (6)

( ) 3/2 ckctm f3.0f ⋅= (7)

7.2 Reinforcing steel

For the steel reinforcement, an idealised bilinear stress-strain relationship is considered with a design yield strength fyd as shown in fig. 2b. The appropriate parameters of strength and strains depend on the selected steel grade. For the design at strengthened state the strain of reinforcing steel is limited to εsu = 1 % according to Eurocode 2, section 4.2.2. The design value of the yield strength fyd is determined by dividing the characteristic strength fyk by the appropriate partial safety factor in table 1.

7.3 Prestressing steel

For prestressing steel the same bilinear line with a horizontal branch is applied (fig. 2b). The design value of the tensile strength fpd is determined by dividing the characteristic strength fpk by the appropriate partial safety factor in table 1. For the design of the member cross-section, the tensile strength is reduced to 90 % according to Eurocode 2. Therefore the reduction factor αp is introduced in fig. 2b.

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7.4 FRP material

The tensile stress-strain behaviour of FRP can be idealised by means of a linear response, defined in fig. 2c. The modulus of elasticity depends on the selected FRP system. It is quoted from the relevant national approval or guideline. At present, the characteristic values of the German General Approval [1] are applied, as far as national approvals or guidelines are not available:

modulus of elasticity:

CFRP laminate 150/2000 Ef = 164'000 [N/mm²] CFRP laminate 200/2000 Ef = 205'000 [N/mm²] C-Sheet 240 Ef = 240'000 [N/mm²] C-Sheet 640 Ef = 640'000 [N/mm²]

tensile strength:

CFRP laminate 150/2000 ffk = 2'500 [N/mm²] CFRP laminate 200/2000 ffk = 2'500 [N/mm²] C-Sheet 240 ffk = 3'800 [N/mm²] C-Sheet 640 ffk = 2'650 [N/mm²]

ultimate strain:

CFRP laminate 150/2000 εfu = 1.40 [%] CFRP laminate 200/2000 εfu = 1.30 [%] C-Sheet 240 εfu = 1.55 [%] C-Sheet 640 εfu = 0.40 [%]

For externally bonded FRP the ultimate tensile strength or the strain at failure are not significant for the design of strengthened structures, because other mechanisms like bond failure are prematurely responsible for the failure. Therefore, to determine the design moment of resistance for the strengthened state, the design strain of the external bonded FRP system will be limited to approximately 50 % of the ultimate elongation at failure (εfu).

concrete reinforcing / prestressing steel FRP material

σf

Fig. 2 Stress-strain diagrams for concrete, reinforcing and prestressing steel as well as FRP Material

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8.1 Externally bonded FRP laminates

Externally bonded FRP laminates cannot be stressed up to their tensile strength. Before externally bonded FRP reinforcement will reach the tensile strength, the strengthened system is failing, e.g. due to rupture of the anchorage or bond failure at shear or flexural cracks. For this reason, based on the German Guideline for the strengthening of concrete components using externally bonded CFRP laminates [2], the strain of externally bonded systems is limited. This design principle is meanwhile adopted by many other national guidelines.

The strain limits for CFRP laminates are defined in the national approvals and guidelines. Normally, the design strain is limited to about 50 % of the average ultimate strain in direction of the fibres. Below, the design strain limits for flexural strengthening are given according to the German General Approval [1]:

The lowest strain value εf,limit of the two following conditions is decisive:

Depending on the type of laminate and modulus of elasticity in fibre direction:

S&P CFRP laminate 150/2000 εf,limit = 0.75 [% ] S&P CFRP laminate 200/2000 εf,limit = 0.65 [% ]

Furthermore, the following condition is valid for reinforced concrete components:

s

f 5=ε (8)

fsyk/Es yield strain of the reinforcing steel (always refers to the outer layer of the internal reinforcement)

Given that the strains can hardly be controlled by hand calculation, the last condition indirectly helps to prevent yielding of the internal reinforcement at service state. However, the program offers the possibility to check the strains at service state. Anyway, this condition is only relevant for internal reinforcement with low steel strength.

Considering the low limits of the design strain there is no need for any additional partial safety factor (γ > 1.0) for externally bonded CFRP systems.

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8.2 Externally bonded carbon sheets

For externally bonded Carbon sheets the same principles as for externally bonded CFRP laminates are valid. The design strain at ultimate limit state is limited:

C-Sheet 240 εf,limit = 0.75 [%]

Following the German Guideline for the strengthening of concrete components using unidirectional Carbon sheets [4], the number of layers should not exceed 5 layers.

8.3 Near surface mounted CFRP laminates

CFRP laminates can also be glued into slots which will be cut into the concrete surface. Compared to externally bonded FRP strips, near surface mounted laminates have a higher anchoring capacity, therefore they can almost be stressed up to their tensile strength. The bond behaviour of near surface mounted CFRP laminates is comparable to embedded steel rebars. A sufficient bond length prevents bond failure and debonding problems will not occur. The design of near surface mounted laminates is based on recent investigations in Germany [12].

The design value of the tensile strength ffd and the ultimate strain εfd are determined by dividing the characteristic values by the following partial safety factors γf for FRP laminates:

ffd = ffk / γf (9)

εfd = εfu / γf (10)

The characteristic values of the tensile strength and the ultimate strain are quoted from chapter 7.4. According to the German Guideline for the strengthening of concrete elements using unidirectional CFRP laminates glued into slots [3] the following partial safety factors are valid:

γf = 1.2 for fundamental combinations

γf = 1.0 for accidental combinations

Additionally it has to be proved, that the maximum strain in the laminate does not exceed εf,max at ultimate limit state:

εf,max ≤ kε · εfd (11)

As a contribution to the reduced ductility of CFRP strengthened elements the reduction factor kε is assumed to 0.8.

9. Imposed actions

Similar to the design of new elements, the imposed actions of the reinforced concrete element to be strengthened must be known. The easiest way is to analyse the available as-built documents of the structure. If these documents are not available, geometry, idealised model of the structure and loads must be established by investigations on site. The bending moment, normal and shear force diagrams have to be determined considering the different type of loads and their combinations.

It is necessary to determine the imposed bending moment of the structure during application of the FRP strengthening system for the evaluation of the initial state of strain. Normally this will be the moment due to dead load of the structure and eventually to the prestressing force. In any case the bending moment of the initial state results from service loads (load safety factor = 1.0).

Furthermore, the characteristic and design bending moment due to expected future loads are required. The procedure of the determination of the moment curves is shown in figure 3.

The design value of the bending moment MSdf due to expected future loads must include the partial safety factors (table 1) and the additional combination values which consider the probability of occurrence of several variable loads. For prestressed or post-tensioned elements the statically indeterminate part of the prestressing moment Mp´ has to be considered for the determination of MSdf. The statically determinate part of the prestressing moment Mp0, which is defined by the prestressing force and the distance to centre of gravity of the concrete section, is considered by the design program.

Q

Q

G

G

G

MSdf,g+q

Fig. 3 Load cases before, during and after strengthening with FRP system

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10.1 Capacity of the unstrengthened cross-section

To check the strengthening ratio η the bending capacity of the unstrengthened cross-section has to be calculated first. The design resistance of the unstrengthened cross-section MRd0 is determined considering the existing geometry, reinforcement, prestressing steel, concrete quality as well as the partial safety factors for material properties listed in table 1. If as-built documents are not available this information has to be established by investigations on site. Samples may be taken to check the concrete compressive strength.

10.2 Required cross-sectional area of FRP

=+

ε εc c0 c = + εεc

εsεs0

εcεc0

εc,o

Fs Ff

Fs Ff

Fig. 4 Superposition of the strain profiles and internal forces

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10.3 Conditions of equilibrium

The unknown values like the capacity of the unstrengthened cross-section MRd0, the imposed initial strain state ε0, the required cross-sectional area of the FRP material Af,req and the resisting moment of the strengthened state MRdf are derived from the conditions of equilibrium ΣH = 0 and ΣM = 0 considering the mechanical behaviour of each material.

Internal forces

yk A f

concrete c

ck Rc

f xbF

γ ⋅α

ΣH = 0

ΣM = 0

Fc ⋅ ac – Fs ⋅ ds – Fp ⋅ dp – Ff ⋅ df = 0 (16)

The solution for the equilibrium conditions in equations 15 and 16 is found by variation of the strain profile. The strains are assumed to have linear distribution (Hypothesis of Bernoulli).The conditions for equilibrium are checked while running through the possible strain profiles within the defined limits:

unstrengthened cross-section:

0 < εc ≤ εcu

initial strain state:

design:

0 < εf ≤ εf,limit

0 < εc ≤ εcu ⇒ MRdf, ε and σ are determined iteratively

0 < εf ≤ εf,limit

The system of equations always leads to an unique solution.

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10.4 Control of strain profiles

Strains in ultimate limit state

The program provides the user with the calculated strain profiles. The determined strain values can be compared with the strain limits of concrete and FRP (Fig. 5). Normally the design will be controlled by highly stressed FRP material, i.e. the strain limit of CFRP. In cases where the design is limited by failure of the compression zone the user should check if strengthening with FRP is still reasonable

The design can be checked by hand calculation using the provided strain profiles and the equilibrium conditions in equations 15 and 16.

εcu

The strain distri loads. In additio steel as well as

10.5 Contr

If the design an steel stresses i stress limits for

concret

reinforc

prestres

stribution in ultimate limit state

e state

bution in service state allows to cont n stresses at service state are dete

the selected FRP cross-section.

ol of stresses

d detailing is not in compliance with n service state, a verification of the the rare combination of loads have to

e σc,limit = 0.6 fck

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rol yielding of internal reinforcement under service rmined for concrete, reinforcing and prestressing

the rules given in Eurocode 2 to limit concrete and stresses is necessary. In that case the following be respected.

(17)

(18)

(19)

11. Bond check of the FRP system

11.1 Anchorage of externally bonded CFRP laminates

The bonding characteristics of externally bonded CFRP laminates is totally different compared to embedded steel rebars. While steel rebars can be stressed up to the yielding point by increasing the bond length, the bond force of FRP laminates is limited. An increase in bond length above the length lbd does not result in an increase in resisting tensile stresses (see fig. 6). Based on tests a design model for the maximum bond failure force has been established in [8] for externally bonded steel plates on concrete structures.

Fbd,max

Fig. 6 Relationship between bond length and bond force

This model can also be applied to CFRP laminates in a modified form. It has become a substantial part of the German guideline [2] and is generally accepted as being the most up-to-date and straightforward to apply.

The maximum bond failure force Fbd,max (corresponds to Tk,max in the German guideline [2]) can be determined using the design value of the surface tensile strength fctd of the concrete:

[N] ftnEkkbm5.0F csdfffTbffmax,db ⋅⋅⋅⋅⋅⋅⋅= (20)

csm csd γ

= (21)

The subscript d in equation 20 indicates a design value considering the partial safety factor for concrete. Subscript f describes the properties of the FRP material while b corresponds to bond.

22

The associated bond length lbd,max can be derived from the following equation (cf. [10]):

[mm] f

tnE 58.0l

⋅⋅ = (22)

where:

mf number of laminates next to each other [ - ] bf laminate width [mm] nf number of laminates on top of each other [ - ] tf laminate thickness [mm] Ef modulus of elasticity of FRP laminates [N/mm²] fcsm surface tensile strength of the concrete [N/mm²], valid for: 1.5 N/mm² ≤ fctm ≤ 3.0 N/mm². γc partial safety factor for concrete [-] kT temperature reduction factor [-], taking into account temperature variations between -20°C and +30°C, 0.9 for exterior components, 1.0 for interior components. kb width factor according to the German guideline [2]

400/b1 b/b2

06.1k f

f b +

b beam width or laminate spacing for slabs [mm]

The factor 0.5 in equation 20 refers to the material characteristics of the adhesive bond. This explanation is also valid for equation 22. Additional information is given in the German guideline [2] or in the publications of Rostásy [7], [11] and Onken [10] respectively.

−⋅⋅=

11.2 Anchorage of externally bonded carbon sheets

The bond failure behaviour of FRP sheets is based on the same mechanical principles as CFRP laminates. In the equation stated above the thickness of the laminate tf is replaced by the theoretical fibre thickness of the selected sheet. The factor lf gives the number of layer glued one on top of each other. The width factor kb is set to 1 for the calculation of the bond failure force of carbon sheets.

Compared to CFRP laminates the surface of sheets is relatively large, so the bond behaviour is much better than for CFRP laminates.

23

11.3 Calculation of the envelope line / verification of the anchorage

The verification of the anchorage is carried out for ultimate limit state, considering the partial safety factors listed in table 1. For the application of externally bonded FRP systems the envelope line of the tensile forces has to be covered over the total length of the reinforced concrete element. Therefore the CFRP laminate or sheet should be extended to the support line as close as possible. It has to be proved that the design tensile force of the FRP material Ffd,E does not exceed the bond failure force Fbd,max that can be transmitted by the associated length lbd,max (see fig. 6). The tensile force of the FRP system is determined from the moment line in the same way as the force of the internal tension rebars.

A distinction has to be made between solid slabs and beams. Since CFRP laminates applied on slabs cannot be clasped by strap binders, the bond failure force has to be reduced. According to the German guidelines [2], [4] a reduction factor of 1.2 is introduced for slabs.

beams: (24) dfbd FF ≥

F ≥ (25)

Verification at an end support

To avoid the determination of the whole envelope line for the tensile forces the bond check can be carried out for a certain point E according to the German guidelines [2], [4]. For the end support of beams or slabs it is assumed that the first crack due to the imposed moment will appear at the point E, which corresponds to the associated bond length lbd,max of the maximum bond failure force Fbd,max. The maximum distance from the end of the CFRP laminates or sheets to the edge of support should not exceed 50 mm for sagging moments (cf. [11]).

xE

ME

E

fai

M(x)

Fig. 7 Verification of the FRP end anchorage at the end support

24

25

For reinforced concrete elements the definite point E for the calculation of the existing tensile force in the FRP strengthening can be simply determined by adding up the following lengths (see fig. 7):

xE = ai + f + lbd,max + aL (26)

xE distance from point E to the theoretical support axis ai distance from the support axis to the support front f distance from the end of FRP strengthening to the support front (≤ 50 mm) lbd,max bond length related to Fbd,max according to equations 20 and 22 aL horizontal displacement of the envelope line

According to Eurocode 2 the following values are valid for the horizontal displacement of the envelope line:

beams: aL = 0.5 · zm · (cot θ – cot α) (27) simplified for vertical stirrups and compression struts at 45°: (cot θ – cot α) = 1

T-beams: aL = 0.5 · zm · (cot θ – cot α) + x (28) with x = distance of reinforcement placed in the flange outside the web

slabs: aL = dm (29)

The distance xE is calculated by the program. The user must determine the corresponding bending moment MSdf,E from the moment line of the structure. The tensile force Ffd,E of the FRP material at point E is calculated from the entered design value of the bending moment by iteration of the equilibrium.

From the entered design value of the bending moment at point E the program calculates the tensile force of the FRP material Ffd,E by iteration of the equilibrium.

As prestressed or post-tensioned concrete members are usually uncracked near the support line the bond check has to be modified. Externally bonded FRP systems always have to be anchored beyond the last flexural crack. The design program determines the cracking moment of the prestressed or post-tensioned cross-section considering the tensile strength of the concrete. The user then has to enter the value xE which means the distance of the cracking moment from the support line, measured from the moment curve of the structure. Considering the selected FRP cross-section the program is calculating the tensile force Ffd,E in the section at the point where the first crack will occur. This force is compared to the maximum bond force Fbd,max of the selected FRP system.

Following possibilities are recommended, if the anchorage verification according to equations 24 and 25 may fail:

• increase the cross-sectional are of FRP,

• reduce the distance f between the end of FRP strengthening and the front of support,

• verify substrate strength (pull-off-test) and increase if possible,

• extend the FRP reinforcement beyond the support (e.g. slot-in end)

• increase the contact pressure of FRP reinforcement using additional anchorage devices

The design program gives adequate advice.

Verification at an intermediate support for moment of span

Since the position of the moment zero point varies with different load combinations the FRP system should be anchored at least 1 m beyond the zero-crossing of the horizontal displaced envelope line of the tensile forces. However, at least the bond length lbd,max related to the maximum bond force should be applied. According to the German guidelines [2], [4] the maximum distance from the end of FRP strengthening to the front of support should not exceed 50 mm for sagging moments (cf. [11]).

E aL

As1 Af

Fig. 8 anchoring verification for CFRP laminates and sheets at intermediate support

At the intermediate support the point E refers to the zero point of the bending moment line. From the distance xE, the program determines the distance f between the end of the CFRP laminates or sheets and the front line of the support (see fig. 8).

fmax = xE – ai – aL – lbd,max (30)

Recommendation: f = xE – ai – aL – lb with lb = 1 m

xE distance from the theoretical support line to the moment zero point aL horizontal displacement of the envelope line lb bond length f distance from the end of FRP strengthening to the support front

If the existing length at the intermediate support is not sufficient to anchor the external bonded FRP system with the minimum bond length lbd,max, the program will calculate a negative value for f. In this case the bond forces have to be proved equivalent to the bond check at the end support or the FRP system must be extended beyond the support line.

For prestressed or post-tensioned elements the verification of the anchorage at intermediate support can be performed by the same approach as for the end support.

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Verification at an intermediate support for the moment at support

The verification of the anchorage for the FRP top strengthening can be carried out in a similar way as for the FRP bottom strengthening at the intermediate support. In the German guidelines [2], [4] it is recommended to anchor the FRP system at least 1 m beyond the zero-crossing of the displaced envelope line. The program determines the distance from the intermediate support front line to the end of the FRP strengthening material (see fig. 9).

aL

As1

fai

E

M(x)

Fig. 9 Verification of the anchorage of FRP top strengthening at intermediate support

fmin = xE – ai + aL + lbd,max (31)

Recommendation: f = xE – ai + aL + lb with lb = 1 m

xE distance from point E to the theoretical support line aL horizontal displacement of the envelope line lb bond length f distance from the end of FRP strengthening to the support front line

11.4 Anchorage of near surface mounted CFRP laminates

As already mentioned, compared to externally bonded strips, near surface mounted laminates have a higher anchoring capacity. Therefore they can almost be stressed up to their tensile strength with increasing bond length. Based on the investigations in [12] a design model was established for the anchoring of near surface mounted FRP laminates in the surrounding concrete cover. According to [12] the bond force Fbd of the laminate depends on the bond length lb. It can be described by the following equations:

( ) ]mm[115lforl0015.04.0labmF bbb 4

27

where:

mf number of slots bf width of the FRP laminate [mm] ar distance of the laminate axis to the edge of the member [mm], (valid for ar ≤ 150 mm) τK,d design bond strength of the epoxy adhesive [N/mm²]

The characteristic bond strength τK,k of the epoxy adhesive is determined with the following equation:

( ) k,Ktk,Kck,Ktk,Kc 2

k,Ktk,Ktk,K fffff2f2 ⋅

+⋅+⋅−⋅=τ (34)

For the determination of the design bond strength τK,d according to the German Guideline for the strengthening of concrete components using unidirectional CFRP laminates glued into slots in the concrete [3] the following partial safety factors are valid:

τK,d = τK,k / γb (35)

γb = 1.3 for fundamental combinations

γb = 1.05 for accidental combinations

The envelope line of the tensile forces is carried out at ultimate limit state under consideration of the horizontal displacement in the same way as for externally bonded FRP systems. The envelope line has to be covered by the envelope line of the resisting tensile force considering the internal steel reinforcement and the near surface mounted FRP laminates. Unlike externally bonded FRP systems near surface mounted laminates can be anchored from the point where they are theoretically no longer required to cover the entire tensile force.

The required bond length for near surface mounted FRP laminates results from transformation of equations 32 and 33:

[ ] ]mm[115lformm ab0015.0

11.5 Surface tensile strength of concrete

A proper execution of strengthening with externally bonded FRP systems on site will always lead to a failure in the concrete covering layer (bond failure) and not to a failure in the adhesive substance. An essential parameter for the adhesive bond of FRP systems is the surface tensile strength fcsm of the existing concrete cover. The testing of the surface tensile strength has to be carried out according to the relevant national regulation. Due to the German guidelines [2], [4], at least five tests should be performed on each concrete element.

For the calculation of the bond failure force the surface tensile strength can be applied according to the German guidelines [2], [4] as the median value of the population. This value can be derived from test results under consideration of the student distribution with a statistical safety of 95%.

arithmetic median n

median of the population (40) skxf mcsm ⋅−=

xi strength of test i n number of tests k reduction factor

Depending on the number of tests, the reduction factor k can be taken from the following table (see German DIN 1048)

n 5 6 7 8 9 10 15 20 25 30 35

k 0.953 0.823 0.734 0.67 0.62 0.58 0.455 0.387 0.342 0.31 0.286

Tab. 2 Reduction factor k for the calculation of the median of the population (German DIN 1048)

Example

1 2.0 1.1025 2 2.2 0.7225

05.36 3.18xm ==

792.0135.3 16

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12. Anchorage of bottom reinforcement at end support

On account of the truss model used for the shear design the bottom reinforcement has to be properly anchored at end and intermediate supports. The required conditions are stated in Eurocode 2. It is necessary to

a) retain not less than 25% (or 50% for slabs) of the required steel section present in the span

b) cover the tension force, which is derived from the truss analogy considering the shear and normal force at the support

A,Sdf m

⋅= (41)

The maximum value of both conditions is valid for the anchorage of the bottom reinforcement.

Normally the first condition is only valid for the new design of structural concrete elements, not for strengthening. Applied to existing concrete elements to be strengthened 25% (or 50%) of the tensional force due to the maximum moment of span has to be anchored at the supports. For strengthened systems the combined maximum tensile force of the internal steel reinforcement and the external FRP system has to be considered.

As long as the existing bottom reinforcement is not curtailed, the internal rebars extended beyond the support line are normally sufficiently anchored an strengthened state. The program calculates the required anchorage force at the support from the two conditions mentioned above and determines the part of the tensile force covered by the internal rebars. It is calculated from the circumference of the bar and the bond strength fbd:

bd s

⋅ ⋅= (42)

If the internal reinforcement is not sufficient for the anchorage an strengthened state, a part of the FRP strengthening has to be extended beyond the support front line. The software determines the tensile force to be anchored and the required bond length.

In practice the anchorage of FRP systems beyond the support line is very difficult and questionable. Under slabs supported by masonry walls, externally bonded CFRP laminates can be extended to the adjacent span by removing one brick of the wall.

At concrete walls and beams the only adequate solution is to slot-in the end of the laminate and inject the slot with epoxy paste. If the slot is sufficiently thin (d ≤ 10 mm), a verification of the anchorage according to chapter 11.4 is possible.

In contrast to externally bonded FRP systems, an additional anchorage of near surface mounted laminates beyond the support front line can hardly be realised in practice.

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31

13. Detailing provisions

For externally bonded FRP strengthening the spacing of the strips is limited. According to the German guidelines [2] – [4] different limits are valid for the FRP systems.

Externally bonded laminates and sheets

edge distance: ar,min = cw (43)

axial spacing: sf,max = 0.2 · l (bearing distance) (44) sf,max = 5 · h (slabs) (45) sf,max = 0.4 · l (cantilevering length)

Near surface mounted laminates

edge distance: ar,min = 2 · bf (46) ar,min = dk (47)

axial spacing: sf,min = dk (48) sf,min = bf (bei as > 2 · ds) (49)

slot: ts,max = cw – h (50) bs,min = tf + 1 [mm] (51) bs,max = tf + 3 [mm] (52)

where:

cw concrete cover of internal stirrups dk maximum diameter of aggregates in concrete as axial spacing of internal longitudinal rebars ds diameter of internal longitudinal rebars ts, depth of the slot bs width of the slot h allowance for tolerances of the concrete cover

14. Shear design

14.1 Shear capacity according to Eurocode 2

In most cases of flexural strengthening with FRP systems, it is necessary to check the shear capacity of the concrete structure as well. Especially beams they also require shear strengthening. On the other hand for concrete solid slabs it may be proven that shear reinforcement is dispensable for expected future loads. If not, other strengthening methods have to be considered.

According to Eurocode 2 the imposed shear force VSd can either be transferred by the concrete alone or in combination with shear reinforcement. The shear resistance is described by the design values VRd1 to VRd3. For the calculation of the shear reinforcement the standard method is applied considering vertical stirrups and an inclination of the compression struts of 45°. The relevant equations according to Eurocode 2 can be taken from the following flowchart (fig. 11).

VRd1 shear resistance without shear reinforcement – shear force is transferred by concrete alone,

VRd2 maximum shear resistance – the capacity of the inclined compression struts is decisive for the shear resistance,

VRd3 shear resistance with shear reinforcement – the shear force transmission results from concrete and shear reinforcement.

The lower design value VRd1 is the relevant value for slabs which are usually constructed without any shear reinforcement. Presenting the uppermost limit, VRd2 must not be exceeded by the imposed shear force.

Following the conditions given in the German guidelines [2] – [4] and according to Eurocode 2, a distinction has to be made between four different cases with regard to the shear force capacity of a strengthened concrete structure (see [10]):

1. In case that the existing shear force VSdf of the structure at strengthened state is lower than the shear resistance VRd1 of the concrete alone, no shear strengthening is necessary. This case generally applies to slabs.

1RdSdf VV ≤ (53)

VSdf and VRd1 are determined considering the partial safety factors in Table 1.

2. If the shear force at strengthened state can be completely covered by the existing internal stirrups, minimum shear strengthening is necessary to complete the mechanical truss model

VV 3RdSdf ≤ (54)

The additional shear reinforcement has to clasp the flexural strengthening and is designed for the shear force difference V depending on the strengthening ratio.

SdfV1V η −η= (55)

In this case, anchorage of the shear strengthening in the compression zone can be omitted (see Fig. 11).

32

The fact that additional shear reinforcement in the form of external bonded stirrups is necessary despite sufficient internal shear reinforcement is justified by the concrete beam design truss analogy. The additional tension chord of the external flexural strengthening must be connected to the tension struts of the internal stirrups for completion of the truss model (fig. 10, see [10]).

Fig. 10 Connection of the FRP flexural strengthening to the internal truss structure

As1 Af

internal stirrups

compression chord

tension chords

shear strengthening

3. If the shear force demand at strengthened state exceeds the shear capacity of the existing cross- section, the shear strengthening has to be designed for the remaining amount of shear force.

VV 3RdSdf > (56)

VVV 3RdSdf −= or SdfV1V η −η= (57)

The higher value V of both conditions in equation 57 is valid for the design. Since the additional shear reinforcement is necessary to cover the total shear force of the cross-section, the external stirrups have to be anchored in the compression zone (see fig. 11).

4. The maximum shear resistance VRd2 provides the upper limit of the shear force also for strengthened state. However, the German guidelines [2] – [4] do not permit shear strengthening of high stressed beams. Therefore in the program it is recommended to limit the maximum shear capacity in the design concept of Eurocode 2 as well. Reducing the maximum shear capacity to 50 % (Vmax = 0.5 VRd2) corresponds approximately to the limitation given in the German guidelines [2] – [4]

VV maxSdf ≤ (58)

V5.0V 2RdSdf ⋅≤ (59)

x

Fig. 11 Anchorage of the additional shear reinforcement depending on the imposed shear force

Fig. 12 Flowchart for the shear check of strengthened beams according to Eurocode 2

34

Even if additional external stirrups are not required to cover the imposed shear force (case 1), it is recommended for beams to clasp the FRP flexural strengthening system at least with 2 external stirrups at the end of the beam. For strengthening the moment of span these external stirrups should be anchored in the compression zone. For moments of support an additional anchorage of the flexural strengthening with external stirrups can be omitted.

14.2 Design of the additional shear reinforcement

As additional shear reinforcement, steel plates as well as high modulus carbon sheets (unidirectional fabrics) can be used. The sheets have a modulus of elasticity of about 640'000 N/mm² (C-Sheet 640), they are easier to handle than steel plates and therefore the application is more economical despite the price of the material.

The required cross-section of the additional shear reinforcement in form of steel plates or sheets has to be determined for the remaining shear force difference V (eq. 55 and 57).

The internal steel stirrups and the external stirrups made of steel plates or carbon sheets are considered as parallel connected elastic or elastic-plastic elements. The strain conformity of these elements must be ensured also at strengthened state. Therefore the same strain limit of εlimit = 0.2 % according to [7] will be taken as a basis for the design.

The stress of the internal stirrups to determine the shear capacity VRd3 must be limited according to equation 60:

ydsitlimsw fE ≤⋅ε=σ (60)

The required cross-sectional area of the additional external shear reinforcement results from the following equation:

fwf req,w z

= (61)

where:

V shear force difference covered by the additional external shear reinforcement zf internal lever arm between the concrete compressive force and the flexural strengthening is iteratively determined by the program σfw stress of the additional external shear reinforcement (at εlimit = 0.2 %)

Steel plates:

Carbon sheets:

fditlimfw E⋅ε=σ (63)

Since the carbon sheets are rather weak during handling and are applied by hand lamination at building site conditions, it is doubtful that the high modulus of elasticity of about 640'000 N/mm² will be

35

achieved in practice. Therefore the modulus of elasticity in equation 63 should be reduced for carbon sheets as shown in figure 13. For this a reduction factor of γE = 1.2 is recommended in [10]:

σf

Efd = Efk / γE (64)

Fig. 13 Reduction of the modulus of elasticity of carbon sheets (design value)

Spacing of external stirrups

Steel plates:

The maximum spacing of the plates sw,max results from the truss analogy and is approximately equal to the effective depth of the FRP flexural strengthening (see fig. 10) which corresponds to the overall height of the beam.

sw,max = h (65)

Carbon sheets:

When strengthening with carbon sheets C-Sheets 640, a maximum strip spacing of 80 % of the overall height of the beam is recommended.

sw,max = 0.8 · h (66)

14.3 Anchorage of external stirrups

If only minimum external stirrups are required without anchorage in the compression zone, (case 2: Vsdf ≤ VRd3), it has to be proved, that the adhesive bond of the external stirrups provides a sufficient anchorage. For this check the bond behaviour as stated in chapter 11 can be applied, where the material properties of steel plates or carbon sheets have to be considered.

As the form an the location of shear cracks can not be predicted, according to the German guideline [2] it has to be proved, that the tensile force of the additional external stirrups does not exceed 50 % of the bond force Fbd as given in equation 23:

Fwd ≤ 0.5 Fbd (67)

The external stirrups always have to be bonded over the whole height of the web. But only half of the existing bond length lbw at the side of the web can be considered for lb in equation 23 (see fig. 14):

lb ≤ 0.5 lbw (68)

h1

lb

Fig. 14 Adhesive bond anchorage of minimum external stirrups

The tensile force of the external stirrups is determined according to Eurocode 2 using the following equations:

z V

z sV

⋅ ⋅= each leg of a stirrup (70)

For external stirrups that are only anchored by adhesive bond, differing from equations 65 and 66 the following is valid:

sw,max = 0.5 · h (71)

15. Further checkings

In addition to the calculations which can be carried out with the program FRP Lamella, the structural engineer should also check cracks and deformations of the strengthened structure if necessary. According to the most national regulations or guidelines, the control of crack widths at strengthened state is not required. Nevertheless, in a special case where it may be necessary, you should make sure that durability and serviceability of the strengthened concrete structure are ensured. Please note that strengthening with FRP products has no significant influence on the deformations of a strengthened concrete structure. In case of deflection problems, preference should be given to other strengthening methods like for instance sprayed concrete.

16. Fire protection

If fire protection is required, the program enables the user to check the remaining safety θ for service loads under the condition that the external bonded FRP system and the external stirrups will completely fail. Please note that epoxy resins may loose their load bearing capacity when the temperature approaches 80° C. If necessary it has to be proved in special cases by an approval or an expert opinion that the FRP system and the external stirrups are sufficiently protected against fire, using additional protective measures as for instance fire protection plates.

39

17.1 Start of the program

The program is developed for the following Windows operating systems: Win 9x, 2000, NT.

After successful installation of FRP Lamella you start the program by either clicking the FRP Lamella icon on the Desktop or choosing the option FRP Lamella in the Windows start menu of your computer. First the disclaimer will appear. You have to accept to start the FRP Lamella program window.

To exit the program FRP Lamella and close the window, click the cross in the title bar at the upper right corner of the program window. Instead you can either choose the option exit of the file menu.

17.2 Settings

To ensure optimum display performance of the FRP Lamella program on your screen a minimum screen resolution of 800 x 600 pixel is assumed.

The display font size is also essential for a proper display of the program window. In the Windows menu Start point to Settings, click Control Panel, and then double click Display. On the Settings tab, click Advanced, then you will find the Font Size list on the General tab. Small fonts (standard) should be selected, otherwise several items might not be completely displayed.

17.3 Basic information about the FRP Lamella user interface

You will find general information about Windows user interfaces in your Windows manual or in the online help function of your Windows operating system.

Title bar The uppermost line of the FRP Lamella program window shows information about program, data file and path.

Menu bar The items File, Calculation, Extras and Info on the menu bar lead to different submenus. You will find a detailed list of all menu items in chapter 19.1.

Tool bar The most frequently required functions can easily be called from the toolbar by clicking one of the symbols. You will find a detailed list of all tool bar symbols in chapter 19.2.

Tree view The tree views in the left part of the program window enable you to call the different input and output windows directly. Click the + symbol in front of a heading to display the subordinated items. A click on the – symbol hides the subordinated items again.

Quick info Positioning the mouse pointer on one of the input or output fields an explanation (tooltiptext) will appear after a few seconds. Proceed in the same way to get explanations for the functions of the tool bar.

17.4 Data input

The required data is entered on several input windows shown in the upper part of the program user interface. The titles of the different windows are listed in the opposite tree view. They are classified according to the topics general information, cross-section, loads, strengthening and proofs. Every input window shows a graphic to illustrate the essential data.

To show the different input windows click the related title in the tree view. If one heading includes several subordinated windows, the first window will be displayed automatically. Input windows that are not accessible yet are displayed in light grey.

Use the button on the right below the picture box of each input window to display the next window. It is recommended to follow the sequence of the windows to make sure that no window is left out. The button on the left below the picture box leads to the previous input window.

Enter the required data in the provided text boxes of each window. If necessary overwrite the entry 0. Text boxes with a grey font are locked and cannot be modified. Disabled text boxes having a dark background are not considered in the calculations.

The key button in the toolbar enables you to unlock input fields having a grey font and modify the preselected values.

For some items you can choose from a list of different values.

After you have entered all required data you can start the calculation either by clicking the calculation button below the graphic of the input window FRP cross-section or by clicking the calculator symbol in the toolbar.

Start the proofs of anchorage and shear capacity by clicking the proof button below the graphic of the corresponding input window. A click on the tick symbol in the toolbar will carry out all proofs successively. This function is useful after reopening an existing input data file.

40

17.5 Output of results

The results are displayed in additional windows located in the lower part of the program user interface. The titles of the different windows are listed in the opposite tree view. They are classified according to the topics general information, strengthening and proofs.

To show the different output windows click the related title in the tree view. If one heading includes several subordinated windows, the first window will be displayed automatically. Output windows that are not accessible yet are displayed in light grey

The result values are displayed in text fields having a light grey or coloured background. These values cannot be modified.

Pay attention to the output fields highlighted in blue or red colour. They will show you if the proof conditions are met and if special details of construction have to be followed.

The output windows of strains in ultimate limit state and strains in service state additionally show a graphical representation of the strain distribution (s. chap. 18.15 and 18.16). You can change the scale by clicking the picture.

After the calculation of certain proofs you have to design a sufficient strengthening. The required input data is entered in the white text boxes of the output window (s. chap. 18.20 and 18.23).

The performed calculations can be printed on any printer installed under Windows operating systems. You can modify the content of the heading line in the menu Extras >> company letterhead. It is possible to print each page individually.

pages 1 – 4 design of flexural strengthening page 5 proof of FRP end anchorage page 6 proof of anchorage of flexural reinforcement at support page 7 design of shear strengthening

41

18.1 Input window project

When you start the program the first window project opens. It offers the possibility to enter some general project and structural element data. This information appears on each page of the printout and will help you administrating your projects.

Enter the project number and the project name.

For each structural element you can enter a number and an appropriate description.

tip Use the button on the right below the picture box to display the next window.

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18.2 Input window code

In this window you can choose the underlying code and guideline for the design. Additionally you determine the material properties as well as the unit measurement for the input and output.

Choose the code according to which standard you want to perform the design of the strengthening measure simply by clicking the related option button (not every version offers the possibility to choose the code).

Afterwards choose the guideline on which the design of FRP strengthening shall base.

Select a country for the available steel and concrete grades based on the national standard that will appear on the steel and concrete window, respectively. The appropriate national flag appears next to the selection box.

You have the choice from different units of measurement: the unit of lengths and forces as well as the unit of strains.

43

18.3 Input window geometry

FRP Lamella offers the possibility to strengthen the most frequent types of cross-section: slabs, rectangular beams, T-beams and I-beams. According to your choice FRP Lamella will present an appropriate illustrating graphic.

Click the geometry list and choose the type of cross-section.

Enter the dimensions of the cross-section in the corresponding data fields.

note After entering all data the graphic turns into a true to scale graphic to allow a visual control of the values.

Please indicate whether you want to strengthen an exterior or interior structural member. According to the German guidelines [2], [3] for external concrete members the program will consider a temperature reduction factor kT for the bond strength of externally bonded FRP strips due to temperature variations from –20° C to 30° C (s. chap. 11.1, 11.2).

For slabs please indicate the span or the cantilever length. This input data field will be displayed when you select a slab. The length is required to calculate the maximum spacing of FRP strips. (s. chap. 18.13).

44

18.4 Output window cross-section

After entering the size of the structural member the cross-section values are calculated and shown in the lower part of the user interface.

The upper field shows the gross cross-sectional area Ag of the structural member.

The gravity axis zcg of the cross-section is related to the top of the member.

Furthermore the moment of inertia Iy of the cross-section is given.

The section modulus Wtop and Wbottom apply to the top and the button of the cross-section.

note The cross-section values are related to the gross cross-section of the member.

45

18.5 Input window concrete

The window concrete indicates the material properties of the concrete. You can select the concrete classes according to the chosen code.

Select the existing concrete class of the member from the list. The appropriate characteristic compressive strength fck will be displayed in the next field. If the required concrete class is not available in the list, select the option other, this enables you to define the characteristic strength (s. chap. 7.1).

The concrete maximum strain εcu is limited to 0.35 [%] according to Eurocode 2.

The strain at the axis of the parabolic curve εc1 is assumed to be 0.2 [%] according to Eurocode 2 (Model Code).

The reduction factor α is a coefficient taking into account long term effects on the compressive strength. It is generally assumed to be α = 0.85.

The basic value of the design shear strength τRd is calculated from the characteristic cylinder compressive strength (s. chap. 7.1).

The average modulus of elasticity of concrete Ecm is necessary for the calculation of the uncracked state of the structural member.

The characteristic tensile strength of the concrete fctm defines the transition between the uncracked and the cracked state of the cross-section.

The partial safety factor γc for concrete is preselected as γc = 1.5 according to Eurocode 2. (s. chap. [1])

note You can modify the proposed values by using the key button in the tool bar.

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18.6 Input window steel

The material properties of the reinforcing steel are entered in the window steel. The program offers the possibility to define 2 types of reinforcing steel as well as 2 types of prestressing steel. The graphic shows both stress-strain diagrams.

Select the steel grade of the reinforcing steel and of the prestressing steel from the steel list. The appropriate characteristic yield stress fyk and the appropriate characteristic tensile strength fpk of the prestressing steel will be displayed in the field next to the steel list. If the required steel grade is not available in the list, select the entry other, which enables you to define the characteristic strength.

note The available steel grades in the list depend on the country selected in the window code.

For rebars select between plain and ribbed sections, for prestressing steel it is to distinguish between strands and wires. The choice has an effect on the bond of the rebars or the modulus of elasticity of the prestressing steel, respectively.

The modulus of elasticity Es for reinforcing steel is preselected as Es = 200'000 [N/mm²] in accordance to Eurocode 2. For prestressing steel the modulus of elasticity Ep is preselected depending on the chosen section in accordance to Eurocode 2.

The maximum steel strain εsu and εpu is assumed as 2 [%] according to Eurocode 2. It is only needed for the termination of the iteration. Before this strain limit will be reached, the design will be controlled by the strain limit of the FRP material.

For the calculation of the prestressing force the horizontal line of the design stress-strain diagram is used. According to Eurocode 2 the characteristic tensile strength of the prestressing steel is reduced by αp = 0.9.

The partial safety factor γs for reinforcing steel is preselected to 1.15 according to Eurocode 2 (s chap. [1]).

note You can modify the proposed values by using the key button in the tool bar.

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18.7 Input window main flexural reinforcement

Details about the existing rebars of the concrete member at the position of the maximum bending moment can be entered in the window main flexural reinforcement. FRP Lamella allows the input of six reinforcement layers in the tension or compression zone. You must at least enter one layer of internal reinforcement since the program does not design unreinforced members.

Enter the cross-sectional area As of the existing tension rebars.

tip Click the number button at the beginning of each line and an additional window shows a table of rebars diameters. You can select the number and the cross-section for groups of rebars and copy the total sum of the cross-sectional area of reinforcement to the input window (s. chap. 18.8).

The position of the reinforcement is given by the depth zs measured from the top edge of the structural member to the axis of the rebars.

Afterwards you choose the steel grade. The list shows the four grades defined in the previous window steel. For unclassified steel grades the defined yield strength or tensile strength will be displayed.

For prestressed steel you have to enter the prestress after creep and shrinkage σp0. The resulting prestressing force is shown in the input window loads in unstrengthened state (s. chap. 18.10).

The option bonded is only enabled for prestressing steel. You can define if the tendon is fixed to the surrounding concrete or if it slides without bond through a sheath. Rebars are always bonded to the concrete.

The input of a concrete cover cw is optionally and just serves to if the arrangement of FRP strips fits to the width of the member. The minimum distance between the lateral edge of the FRP strips and the edge of the concrete member is equivalent to the concrete cover.

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18.8 rebar tables for the selection of reinforcement cross-sectional area

The cross-sectional area of reinforcement can be copied from a table which offers a wide selection of rebar diameters. The cross-sectional area is depending on the number of rebars in a beam or the spacing of rebars in a slab. To open the rebar table just click the number button in the input windows main reinforcement and reinforcement at support, respectively.

Choose a cross-sectional area by clicking a white field in the table. The background of a selected field turns into blue. For beams, a multiple choice is possible. The sum of the cross-sectional area As of the selected rebars is displayed below the table.

You cancel the selection by clicking the blue field again.

You can copy the selection to the reinforcement window by clicking the copy button. The rebar table will be closed.

Close the table without copying the value by clicking the cancel button.

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18.9 Input window flexural reinforcement at support

Details about the existing rebars at support can be entered in the window flexural reinforcement at support. Additional information is needed for the proof of anchorage at support.

Enter the cross-sectional area As for every layer of reinforcement. The value is automatically copied from the previous input window main reinforcement.

tip Click the number button at the beginning of each line and an additional window shows a table of rebars diameters. You can select the number and the cross-section for groups of rebars and copy the total sum of the cross-sectional area of reinforcement to the input window (s. chap. 18.8).

The position of the reinforcement is given by the depth zs measured from the top edge of the structural member to the axis of the rebars. The default values are copied from the previous input window main flexural reinforcement.

For prestressed steel layers the prestress σp0 at the region of support is needed. The default values are copied from the previous input window main flexural reinforcement.

Enter the diameter ds of the rebars.

For determination of the anchorage force the anchorage length ls,A from the support front is needed.

Choose the coefficient for effectiveness of anchorage αa. According to Eurocode 2 the value is 1.0 for straight bars. For hooks, bends, loops and welded transverse bars in the anchorage zone the value can be reduced to 0.7.

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18.10 Input window loads in unstrengthened state

The window loads in unstrengthened state defines the imposed actions just before strengthening with FRP. The resulting strains in the concrete member are taken into consideration for the strengthening design. The graphic schematically presents the initial state as a bridge girder, which is closed for traffic during application of the FRP strengthening and only loaded by the dead load of the structure.

Choose the design for a positive (moment of span) or negative moment (moment at support).

note This choice is linked with the type of moment in the window loads in strengthened state. It is not possible to choose different types of moments in both windows.

Enter the characteristic bending moment MSk0 which is imposed to the structure during the application of the FRP strengthening. Commonly this will be the dead load moment. This value defines the initial state of strain in the cross-section. For statically indeterminated systems you may have to add the secondary moment from prestressing Mp'.

If the member is subjected to an external axial force, e.g. the dead load of an inclined beam, you have to choose between compressive and tensile force.

Enter the characteristic axial force NSk0 resulting from the imposed load. Compressive forces have a positive effect and can be ignored.

For prestressed members the program displays the prestressing force NP as well as the prestressing moment Mp0' that are considered in the design. The values result from the cross- sectional area, prestress and position of the prestressing steel entered in the window main flexural reinforcement (s. chap. 18.7).

note The program considers only the statically determinated prestressing moment. If the support conditions of the prestressed member are statically indeterminated, add the secondary moment Mp' to MSk0, MSdf and MSkf.

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52

To define the transition between the uncracked and the cracked state of the cross-section, decide if the bending tension zone of the cross-section is already cracked under service loads. Reinforced concrete members are usually always cracked, for prestressing members it depends on the degree of prestress and the history of loading. As a rule the cross-section is uncracked for maximum prestress (no tensile forces in service state).

18.11 Input window loads in strengthened state

The window loads in strengthened state defines the future actions. You have to enter the bending moments which are imposed to the concrete member after strengthening with FRP (s. chap. 9). The graphic schematically presents the strengthened state as a bridge girder, which is loaded by dead load of the structure and an additional high live load.

Choose the design for a positive (moment of span) or negative moment (moment at support).

note This choice is linked with the type of moment in the window loads in strengthened state. It is not possible to choose different types of moments in both windows.

If the member is subjected to an external axial force, e.g. the dead load of an inclined beam, you have to choose between compressive and tensile force.

Enter the design bending moment MSdf for the expected loads considering the partial safety factors γ for permanent and variable loads as well as the combination value ψ. For statically indeterminated systems you may have to add the secondary moment from prestressing Mp'. (s. chap. 18.10).

Proceed in the same way for the design axial force NSdf. Take also into consideration the different partial safety factors γ and the combination value ψ .

You can either enter the exact values of imposed actions in service state or choose the option approximate. In this case the characteristic values of the bending moment and axial force are calculated from the given design values as follows:

m,M

γ =

note The use of the option approximate is especially recommended if the design actions are determined by a complicated structural analysis (e.g. finite elements). Using the average safety factors you can avoid another analysis applying characteristic loads without partial safety factors.

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54

If you have chosen the option exact, enter the characteristic (service) moment of strengthened state MSkf. Use the load combination applicable for the proof of the service limit state. For statically indeterminated systems you may have to add the secondary moment from prestressing Mp' (s. chap. 18.10).

If the member is subjected to an external axial force, enter the characteristic axial force NSkf.

If you have chosen the option approximate, enter an average partial safety factor γM,m for bending moments. The characteristic moment MSkf is then calculated with above-mentioned equation from the design moment MSdf. The valid range for the average partial safety factors is 1.0 to 2.0 (1.35 to 1.5 according to Eurocode 2).

Is the member loaded with an axial force, also enter the average partial safety factor γN,m.

18.12 Input window FRP system

The window FRP system shows the material properties of the CFRP laminates and sheets. The displayed safety and reduction factors as well as the limited design strain are preselected according to the guideline you have chosen in the window code.

Select the FRP product from the list. There are two types of prefabricated S&P FRP laminates and one hand lay-up S&P C-Sheet in different grammages available.

For prefabricated laminates choose the type of bonding. While externally bonded laminates provide a larger cross-sectional area, near surface mounted (slot-in) laminates show a higher bond resistance (s. chap. 11).

Below the FRP material list the appropriate adhesive is mentioned. For some systems a selection of different adhesives is available. S&P Resin 50 is used for pouring slots in negative moment regions of slabs.

Depending on the chosen type of FRP the modulus of elasticity Efk, the tensile strength ffk and the ultimate strain εfu of the material will be displayed (s. chap. 7.4). For FRP linear-elastic material behaviour is assumed.

The partial safety factor γf depends on the guideline. The reduction factor for strain limit kε enables an additional reduction of the design strain limit εf,limit (s. chap. 8.3).

For carbon sheets a reduction factor γE for the characteristic modulus of elasticity is recommended (s. chap. 14.2). The value is preselected to γE = 1.2.

Some guidelines prescribe a strain limit εf,limit for the design of FRP (s. chap. 8.1, 8.2).

note You cannot modify the material properties of FRP. But you can change the preselected partial safety and reduction factors by using the key button in the tool bar.

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18.13 Input window FRP cross-section

On the window FRP cross-section you are asked to choose the number and arrangement of the FRP products. Observe the minimum and maximum spacing respectively. You can choose up to three layers with different cross-sectional areas as well as different effective depths.

Before selecting the FRP cross-section, start the iteration either by clicking the calculation button on the bottom right of the window or by clicking the calculator symbol in the toolbar. The required FRP cross-sectional area will be calculated and displayed in the output window design below.

note To carry out the design calculation, in this window only the depth zf1 of the FRP layer 1 from the top edge of the concrete member has to be given. As the design iteration can only determine the cross-sectional area of one FRP layer (one unknown), FRP layers 2 and 3 are initially locked, but they are enabled after the design calculation.

For each layer choose an FRP cross-section. The available selection depends on the chosen FRP product in the window FRP System. For sheets the theoretical fibre thickness tf is given and you can choose the width bf of the sheet. The delivery width of the sheet is preselected.

Enter the number of FRP plies nf lying on top of each other. One single ply is preselected. The maximum is two plies of laminates or five plies of sheets.

For beams enter the number mf of FRP strips lying next to each other. The spacing sf of the strips is calculated.

For slabs of a standard width (1 [m] or 12 [in]), enter the spacing sf of the strips. The number mf of FRP strips lying next to each other is calculated. The limit spacing sf,max or sf,min respectively, is calculated according to the German guidelines [2] – [4] with following conditions (s. chap. 13):

sf,max = 0,2- times span sf,max = 5- times slab thickness sf,max = 0,4- times cantilevering length sf,mi

2

according to Eurocode 2

© bow ingenieure gmbh · braunschweig / hamburg · germany

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Contents

Part I – Design concept

1. Introduction 5 2. Notation 6 3. Design program features 9 4. Basic assumptions 10 5. Safety concept 12 6. Degree of strengthening 13

6.1 Definition according to Eurocode 2 13 6.2 Failure of the FRP system 13

7. Material behaviour 14 7.1 Concrete 14 7.2 Reinforcing steel 14 7.3 Prestressing steel 14 7.4 FRP material 15

8. Design aspects for FRP systems 16 8.1 Externally bonded FRP laminates 16 8.2 Externally bonded carbon sheets 17 8.3 Near surface mounted CFRP laminates 17

9. Imposed actions 18 10. Design procedure 19

10.1 Capacity of the unstrengthened cross-section 19 10.2 Required cross-sectional area of FRP 19 10.3 Conditions of equilibrium 20 10.4 Control of strain profiles 21 10.5 Control of stresses 21

11. Bond check of the FRP system 22 11.1 Anchorage of externally bonded CFRP laminates 22 11.2 Anchorage of externally bonded carbon sheets 23 11.3 Calculation of the envelope line / verification of the anchorage 24 11.4 Anchorage of near surface mounted CFRP laminates 27 11.5 Surface tensile strength of concrete 29

12. Anchorage of bottom reinforcement at end support 30 13. Detailing provisions 31 14. Shear design 32

14.1 Shear capacity according to Eurocode 2 32 14.2 Design of the additional shear reinforcement 35 14.3 Anchorage of external stirrups 37

15. Further checkings 38 16. Fire protection 38

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Part II – Use of the program

17. Program user interface 39 17.1 Start of the program 39 17.2 Settings 39 17.3 Basic information about the FRP Lamella user interface 39 17.4 Data input 40 17.5 Output of results 41

18. Input and output windows 42 18.1 Input window project 42 18.2 Input window code 43 18.3 Input window geometry 44 18.4 Output window cross-section 45 18.5 Input window concrete 46 18.6 Input window steel 47 18.7 Input window main flexural reinforcement 48 18.8 rebar tables for the selection of reinforcement cross-sectional area 49 18.9 Input window flexural reinforcement at support 50 18.10 Input window loads in unstrengthened state 51 18.11 Input window loads in strengthened state 53 18.12 Input window FRP system 55 18.13 Input window FRP cross-section 56 18.14 Output window design 58 18.15 Output window strains in ultimate limit state 60 18.16 Output window strains / stresses in service state 61 18.17 Input window FRP end anchorage 62 18.18 Output window FRP end anchorage 64 18.19 Input window anchorage of flexural reinforcement at support 65 18.20 Output window anchorage of flexural reinforcement at support 66 18.21 Input window shear – reinforcement and loads 67 18.22 Input window shear strengthening 68 18.23 Output window shear strengthening 69 18.24 Output window shear strengthening – anchorage of additional external stirrups 70

19. Program menu and tool bar 72 19.1 Menu bar items 72 19.2 Tool bar symbols 74

20. Installation instructions 75 Appendix 1 example – T-beam according to Eurocode 2 Appendix 2 example – two-span slab according to Eurocode 2 Appendix 3 example – prestressed concrete beam according to Eurocode 2 Appendix 4 bow engineers – experts for strengthening design

1. Introduction

FRP Lamella is a design program for the strengthening of reinforced and prestressed or post- tensioned concrete structural members subjected to uniaxial flexure and axial forces using FRP materials (FRP – Fibre Reinforced Polymer). This program can be used for the predesign of strengthening measures as well as for complete calculations within the scope of structural analysis. The program provides the user with the required FRP cross-sectional area for the strengthened state and is performing the necessary verifications of the bond strength and the shear capacity of the concrete member based on the assumptions of the German Guidelines for the strengthening of concrete members using CFRP laminates [2], [3] and carbon sheets [4], (cf. [11]). The design concept according to Eurocode 2 is explained in [10].

The program FRP Lamella is used in almost 15 other countries, adapted to the relevant regulations, guidelines and national standards. Meanwhile different versions corresponding to the following international codes are available:

• Eurocode 2

• British Standard 8110

Fig. 1 Opening window of the FRP Lamella software

Note The software FRP Lamella is based on the material parameters of S&P FRP systems. If other types of reinforcing fibres or adhesive systems will be used, the results provided by the software will no longer be valid. Under these circumstances the system supplier S&P will refuse any liability for the application of S&P products.

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6

2. Notation

As a rule, in this manual the standard notations derived from Eurocode 2 are used. They may differ from notations of other national design codes and guidelines. The following list gives an overview of the notations used in this manual and in the software.

Notation Program Geometry width of the component b web width b0

overall height h, h0

thickness of upper flange h1

effective flange width of lower flange b2

thickness of lower flange h2

span l total cross-sectional area Ac

distance of the centroid from top edge of the member zcg

moment of inertia of concrete cross-section Iy section modulus above the gravity axis Wtop

section modulus below the gravity axis Wbottom

Reinforcement cross-sectional area of longitudinal rebars As

pre-strain of the longitudinal reinforcement due to prestressing εp0

cross-sectional area of internal stirrup rebars asw

distance from centroid of rebars to top edge of the member zs

diameter of rebars ds

concrete cover of the stirrups cw

Steel characteristic yield strength of reinforcing steel fyk

modulus of elasticity of reinforcing steel Es

strain limit of reinforcing steel εsu

characteristic tensile strength of prestressing steel fpk

modulus of elasticity of prestressing steel Ep

strain limit of prestressing steel εpu

reduction coefficient for the tensile strength of prestressing steel αp

partial safety factor for steel γs

Concrete characteristic compressive strength of concrete (EC 2) fck

strain limit of concrete εcu

strain at the axis of the parabolic curve of the stress-strain line of concrete εc1

reduction factor for the compressive strength of concrete (long term effects) α design shear strength of concrete τRd

average modulus of elasticity of concrete Ecm

average axial tensile strength of concrete fctm

partial safety factor for concrete γc

7

failure strain of FRP material εfu

strain limit of FRP material εf,limit

characteristic tensile strength of FRP material ffk partial safety factor for FRP material γf

reduction factor for the strain limit of FRP material kε

number of FRP plies one on top of each other nf

number of FRP strips one next to each other mf

spacing of FRP strips sf

cross-sectional area of FRP strengthening Af

distance from centroid of FRP strip to top edge of the member zf

distance of FRP strips to the lateral edge of the member ar

thickness of FRP reinforcement tf width of FRP reinforcement bf

Design characteristic bending moment at time of strengthening MSk0

characteristic axial force at time of strengthening NSk0

characteristic prestressing force Np

statically indeterminated part of prestressing moment Mp’ design bending moment of strengthened state MSdf

design axial force of strengthened state NSdf

characteristic bending moment of strengthened state MSkf

characteristic axial force of strengthened state NSkf

average partial safety factor for bending moments caused by loads γM,m

average partial safety factor for axial forces caused by loads γN,m

design moment of resistance of unstrengthened cross-section MRd0

characteristic moment of resistance of unstrengthened cross-section MRk0

design moment of resistance of strengthened cross-section MRdf

strengthening ratio η remaining global safety in case of loss of FRP strengthening θ strain of extreme compression fibre of concrete εc

distance from neutral axis to extreme compression fibre x maximum strain of reinforcing steel εs

maximum strain of prestressing steel εp

maximum strain of FRP reinforcement εf

stress of extreme compression fibre of concrete σc

maximum stress of reinforcing steel σs

maximum stress of prestressing steel σp

maximum stress of FRP reinforcement σf

Anchorage substrate strength of concrete (median of the population) fcsm

design value of the substrate strength of concrete fcsd

characteristic compressive strength of adhesive fKc,k

characteristic tensile strength of adhesive fKt,k

8

characteristic shear strength of adhesive τK,k

distance from calculated axis of support to edge of support ai

distance from end of the FRP strip to edge of support f horizontal displacement of the envelope line of tensile force aL

design moment of strengthened state in point E MSdf,E

design axial force of strengthened state in point E NSdf,E

distance from point E to theoretical axis of support xE

tensile force of FRP reinforcement in point E Ffd,E

design value of the maximum bond force Fbd,max

required bond length of maximum bond force lbd,max

recommended bond length lb design value of shear force at support VSdf,A

design value of axial force at support NSdf,A

total required anchorage force at support FA,req

anchorage force covered by internal reinforcement Fs,A

design bond strength of internal rebars fbd

force covered by FRP anchorage Ff,A

anchorage length of FRP reinforcement from the support front lf,A Shear cross-sectional area of internal stirrups asw

design shear force of strengthened state in relevant section X VSdf,X

design axial force of strengthened state in relevant section X NSdf,X

design bending moment of strengthened state in relevant section X MSdf,X

strain limit of additional shear reinforcement εlimit

characteristic modulus of elasticity of FRP Sheet Efk

characteristic tensile strength of FRP Sheet ffk reduction factor for modulus of elasticity due to manual lamination γE

modulus of elasticity of steel plates for shear strengthening Es

characteristic yield strength of steel plates for shear strengthening fyk

partial safety factor for shear strengthening steel plates γs

distance from resultant of concrete stress to extreme compression fibre ac

effective depth of internal steel rebars ds

effective depth of FRP reinforcement df

average effective depth dm

shear force limit of the strengthened cross-section Vmax

design shear resistance provided by concrete VRd1

design shear resistance of the concrete cross-section without web crushing VRd2

design shear resistance of concrete cross-section with internal stirrups VRd3

cross-sectional area of additional shear reinforcement aw

thickness of additional external stirrups tw width of additional external stirrups bw

cross-sectional area of one additional external stirrup Aw

spacing of additional external stirrups sw

stress of the internal shear reinforcement σsw

stress of the additional shear strengthening σfw

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3. Design program features

FRP Lamella is developed as a pure design program for the strengthening of reinforced and prestressed or post-tensioned concrete elements, i.e. the program does not perform any structural analysis. As a consequence the user has to determine the internal forces in advance with a calculation by hand or by using structural analysis software for instance. The updated version 3.x now also considers prestressed or post-tensioned elements or concrete structures subjected to axial forces.

The program supports 4 types of cross-section: slabs as well as rectangular beams, T-beams and double-T-beams. These options cover almost all reinforced or pre-stressed concrete components subjected to bending which will appear in practice.

There are 3 different FRP-systems for flexural strengthening: externally bonded CFRP laminates, near surface mounted (slot-in) CFRP laminates and externally bonded carbon sheets (unidirectional fabric). The program includes the complete range of S&P products for flexural and shear strengthening.

The required cross-sectional area of FRP strengthening is determined by variation of the strain profile within the limits defined in the regulations. The implementation of non-linear stress-strain relations for concrete as well as for reinforcing and prestressing steel and the iterative solution procedure lead to precise results. Compared with hand calculations the program provides particularly economic amounts of FRP strengthening. Additionally the strain and stress distributions can be controlled. The verification of the bond strength is based on the German Guideline for the strengthening of concrete components using CFRP laminates [2], [3] and carbon sheets [4]. The verification of anchorage lengths of the internal rebars as well as the design of the shear strengthening follows the concept of Eurocode 2.

For structures to be strengthened the geometry, internal reinforcement, steel grades, concrete compressive strength and bending moments can be derived from existing as-built documents. If not available this information has to be established by on-site testing.

In addition the program offers useful tools for the definition of the relevant national concrete strength, the reinforcing and prestressing steel grades and the selection of the existing rebar cross-sections.

The serviceability of the strengthened state cannot be proved by the program. If necessary, the user is responsible to check the deflections and crack widths of the strengthened structure.

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4. Basic assumptions

According to the regulations it can be assumed for the design at ULS (ultimate limit state) that externally bonded FRP reinforcement can be calculated like an additional layer of reinforcement within the provided strain limits. The calculations are based on the well-known assumptions of concrete design:

• For bending a linear strain distribution is assumed (Bernoulli hypothesis).

• For reinforced concrete elements it is assumed that concrete has no tensile strength. All tensile forces necessary for the equilibrium of the internal forces are covered by internal reinforcement and FRP strengthening.

• For prestressed or post-tensioned concrete elements the tensile strength of the concrete may be considered in the uncracked state.

• There is no slip between FRP strengthening and concrete. All cross-section fibres with the same distance to the neutral axis are subjected to the same strain.

The determination of the required FRP cross-sectional area and the resisting moments before and after strengthening result from calculation of the equilibrium of internal forces.

References

[3] Richtlinie für das Verstärken von Betonbauteilen durch Einkleben von unidirektionalen kohlen- stofffaserverstärkten Kunststofflamellen in Schlitze im Beton, Deutsches Institut für Bautechnik, Berlin.

[4] Richtlinie für das Verstärken von Betonbauteilen durch Auflaminieren von unidirektionalen Kohlenstofffaserlaminaten (CFK-Laminate), Deutsches Institut für Bautechnik, Berlin.

[5] CEB-FIP Model Code 1990; EPF Lausanne 1991.

[6] Eurocode 2: Planung von Stahlbeton- und Spannbetontragwerken; Teil 1: Grundlagen und Anwendungsregeln für den Hochbau; Juni 1992.

[7] Rostásy, F.S.; Holzenkämpfer, P. und Hankers, C.: Geklebte Bewehrung für die Verstärkung von Betonbauteilen. Beton-Kalender 1996, T.II, Berlin: Ernst & Sohn 1996.

[8] Holzenkämpfer, P.: Ingenieurmodelle des Verbunds geklebter Bewehrung für Betonbauteile. Dissertation TU Braunschweig, 1994.

[9] Onken, P.; vom Berg, W.; Matzdorff, D.; Nolte, T.: Bemessungsprogramm für CFK-Lamellen. Beton- und Stahlbetonbau 95, 9/2000, S. 551 – 552.

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[10] Onken, P.; vom Berg, W.: Biegezugverstärkung mit CFK-Lamellen – Neues Bemessungs- modell nach EC 2 und DIN 1045-1. Beton- und Stahlbetonbau 96, 2/2001, S. 61 – 70.

[11] Rostásy, F. S.: Expert Opinion No. 98/0322; S&P Reinforcement, Eisenstadt, Österreich.

[12] Blaschko, M. A.: Zum Tragverhalten von Betonbauteilen mit in Schlitze eingeklebten CFK- Lamellen, Dissertation an der TU München, 2001.

[13] Design guidance for strengthening concrete structures using fibre composite materials, Technical Report No. 55, The Concrete Society, Berkshire, UK, 2000.

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5. Safety concept

The ultimate limit state design is based on the following condition (cf. EC 2, section 2.3.2):

Sdf ≤ Rdf (1)

Sdf corresponds to the design value of internal forces or moments due to loads and Rdf to the design resistance, for instance the moment capacity of the cross-section at strengthened state. The index f defines the state after strengthening (with FRP). Both, Sdf and Rdf, are design values and associated with the partial safety factors for actions and materials properties as shown in table 1.

Loads Resistance dead loads live loads concrete reinforcing steelCode

γG γQ γC γS

DIN 1045-1 (Germany-new) 1.35 1.5 1.5 1.15

DIN 1045 (7/88) (Germany-old) 1.75 – 2.1 1.0

BS 8110 (UK) 1.4 1.6 1.5 1.15

BAEL 91 (France) 1.35 1.5 1.5 1.15

SIA 160 / 262 (Switzerland) 1.3 1.5 1.2

ACI 318 (USA) 1.4 1.7 1 / 0.9

KCI (Korea) 1.4 1.7 1 / 0.85

Tab. 1 Partial safety factors according to different design codes

For actions additionally the combination values for the probability of occurrence of several variable loads have to be considered. Additional partial safety factors for the FRP systems are missing in table 1 since different safety concepts are used according to the FRP system and the national guideline. In many cases, e.g. for externally bonded FRP laminates, the design strain is limited instead of introducing a partial safety factor for the tensile strength of the material. For further information see chapter 8.

6. Degree of strengthening

6.1 Definition according to Eurocode 2

In the German Guidelines for the strengthening of concrete members with external bonded FRP laminates [2], [3] and unidirectional carbon sheets [4] it is recommended, that the flexural capacity of the strengthened element should not exceed twice the flexural capacity of the unstrengthened element. This is expressed by the flexural strengthening ratio.

A limitation of the strengthening ratio is only mentioned in the German guidelines [2] – [4]. There exists no such limitation in other regulations or guidelines. One reason for the limitation of the strengthening ratio is the scant knowledge about the behaviour of highly strengthened structures. Other reasons were the insufficient design methods for strengthening with externally bonded FRP reinforcement in the past. Hand calculations do not allow the verification of strains and stresses at service state. On the other hand the design software FRP Lamella provides the strain distributions and stresses at strengthened state in all parts of the section. By this means yielding of internal reinforcement can be avoided, strain limits can be controlled. Therefore the limitation of the strengthening ratio based on the German guidelines [2] – [4] may not be applied very strictly but it is highly recommended not to increase the strengthening ratio far beyond the point which is twice the capacity of the unstrengthened element. The bond behaviour of externally bonded FRP strips will be influenced unfavourably by the increased formation of cracks in highly stressed concrete elements.

Since there is no experience with highly strengthened structures, the limitation of the strengthening ratio is also recommended for other national guideline or standards:

2 M M

fSd EC,B ≤=η . (2)

MSdf describes the imposed bending moment at strengthened state, MRd0 corresponds to the design moment of resistance of the unstrengthened cross-section. For MSdf, the combination principles of actions according to EC 2 have to be considered.

When the strengthening ratio exceeds the limit of 2, the design and detailing should be carried out with special care. For near surface mounted laminates there exist no requirements for the limitation of the strengthening ratio.

6.2 Failure of the FRP system

In other national regulations (e.g. ACI) there is a demand of minimum safety (γ > 1,0) after loss of the FRP strengthening under service (unfactored) loads. In this case the following equation is valid:

1 M M

13

7.1 Concrete

For the determination of the concrete compressive stress a parabolic-rectangular stress-strain relationship can be assumed, as provided by Eurocode 2, shown in figure 2a. The parabolic curve ends at a strain value εc1 = 0.2 % and the maximum compressive strain is limited to εcu = 0.35 % (cf. [5]). However, the program also offers the possibility to modify the shape of the parabolic- rectangular stress-strain by adjusting the strain parameters (εc1, εcu). The design value of the concrete compressive strength fcd is determined by dividing the characteristic strength fck by the appropriate partial safety factor in table 1. The reduction factor takes into account the reduced compressive strength under long-term loading.

c

ck cd

f f

γ α=α (α = 0.85; γC according to table 1) (4)

example: C 20/25 → ²mm/N33.11 5.1

2085.0fcd =⋅=α

The design shear strength can be determined from the characteristic concrete compressive strength using the following equation:

c

γ =τ (5)

The average modulus of elasticity and the axial tensile strength of concrete are calculated from the concrete compressive strength. According to Eurocode 2 the following equations are used:

( ) 3/1 ckcm 8f9500E +⋅= (6)

( ) 3/2 ckctm f3.0f ⋅= (7)

7.2 Reinforcing steel

For the steel reinforcement, an idealised bilinear stress-strain relationship is considered with a design yield strength fyd as shown in fig. 2b. The appropriate parameters of strength and strains depend on the selected steel grade. For the design at strengthened state the strain of reinforcing steel is limited to εsu = 1 % according to Eurocode 2, section 4.2.2. The design value of the yield strength fyd is determined by dividing the characteristic strength fyk by the appropriate partial safety factor in table 1.

7.3 Prestressing steel

For prestressing steel the same bilinear line with a horizontal branch is applied (fig. 2b). The design value of the tensile strength fpd is determined by dividing the characteristic strength fpk by the appropriate partial safety factor in table 1. For the design of the member cross-section, the tensile strength is reduced to 90 % according to Eurocode 2. Therefore the reduction factor αp is introduced in fig. 2b.

14

7.4 FRP material

The tensile stress-strain behaviour of FRP can be idealised by means of a linear response, defined in fig. 2c. The modulus of elasticity depends on the selected FRP system. It is quoted from the relevant national approval or guideline. At present, the characteristic values of the German General Approval [1] are applied, as far as national approvals or guidelines are not available:

modulus of elasticity:

CFRP laminate 150/2000 Ef = 164'000 [N/mm²] CFRP laminate 200/2000 Ef = 205'000 [N/mm²] C-Sheet 240 Ef = 240'000 [N/mm²] C-Sheet 640 Ef = 640'000 [N/mm²]

tensile strength:

CFRP laminate 150/2000 ffk = 2'500 [N/mm²] CFRP laminate 200/2000 ffk = 2'500 [N/mm²] C-Sheet 240 ffk = 3'800 [N/mm²] C-Sheet 640 ffk = 2'650 [N/mm²]

ultimate strain:

CFRP laminate 150/2000 εfu = 1.40 [%] CFRP laminate 200/2000 εfu = 1.30 [%] C-Sheet 240 εfu = 1.55 [%] C-Sheet 640 εfu = 0.40 [%]

For externally bonded FRP the ultimate tensile strength or the strain at failure are not significant for the design of strengthened structures, because other mechanisms like bond failure are prematurely responsible for the failure. Therefore, to determine the design moment of resistance for the strengthened state, the design strain of the external bonded FRP system will be limited to approximately 50 % of the ultimate elongation at failure (εfu).

concrete reinforcing / prestressing steel FRP material

σf

Fig. 2 Stress-strain diagrams for concrete, reinforcing and prestressing steel as well as FRP Material

15

8.1 Externally bonded FRP laminates

Externally bonded FRP laminates cannot be stressed up to their tensile strength. Before externally bonded FRP reinforcement will reach the tensile strength, the strengthened system is failing, e.g. due to rupture of the anchorage or bond failure at shear or flexural cracks. For this reason, based on the German Guideline for the strengthening of concrete components using externally bonded CFRP laminates [2], the strain of externally bonded systems is limited. This design principle is meanwhile adopted by many other national guidelines.

The strain limits for CFRP laminates are defined in the national approvals and guidelines. Normally, the design strain is limited to about 50 % of the average ultimate strain in direction of the fibres. Below, the design strain limits for flexural strengthening are given according to the German General Approval [1]:

The lowest strain value εf,limit of the two following conditions is decisive:

Depending on the type of laminate and modulus of elasticity in fibre direction:

S&P CFRP laminate 150/2000 εf,limit = 0.75 [% ] S&P CFRP laminate 200/2000 εf,limit = 0.65 [% ]

Furthermore, the following condition is valid for reinforced concrete components:

s

f 5=ε (8)

fsyk/Es yield strain of the reinforcing steel (always refers to the outer layer of the internal reinforcement)

Given that the strains can hardly be controlled by hand calculation, the last condition indirectly helps to prevent yielding of the internal reinforcement at service state. However, the program offers the possibility to check the strains at service state. Anyway, this condition is only relevant for internal reinforcement with low steel strength.

Considering the low limits of the design strain there is no need for any additional partial safety factor (γ > 1.0) for externally bonded CFRP systems.

16

17

8.2 Externally bonded carbon sheets

For externally bonded Carbon sheets the same principles as for externally bonded CFRP laminates are valid. The design strain at ultimate limit state is limited:

C-Sheet 240 εf,limit = 0.75 [%]

Following the German Guideline for the strengthening of concrete components using unidirectional Carbon sheets [4], the number of layers should not exceed 5 layers.

8.3 Near surface mounted CFRP laminates

CFRP laminates can also be glued into slots which will be cut into the concrete surface. Compared to externally bonded FRP strips, near surface mounted laminates have a higher anchoring capacity, therefore they can almost be stressed up to their tensile strength. The bond behaviour of near surface mounted CFRP laminates is comparable to embedded steel rebars. A sufficient bond length prevents bond failure and debonding problems will not occur. The design of near surface mounted laminates is based on recent investigations in Germany [12].

The design value of the tensile strength ffd and the ultimate strain εfd are determined by dividing the characteristic values by the following partial safety factors γf for FRP laminates:

ffd = ffk / γf (9)

εfd = εfu / γf (10)

The characteristic values of the tensile strength and the ultimate strain are quoted from chapter 7.4. According to the German Guideline for the strengthening of concrete elements using unidirectional CFRP laminates glued into slots [3] the following partial safety factors are valid:

γf = 1.2 for fundamental combinations

γf = 1.0 for accidental combinations

Additionally it has to be proved, that the maximum strain in the laminate does not exceed εf,max at ultimate limit state:

εf,max ≤ kε · εfd (11)

As a contribution to the reduced ductility of CFRP strengthened elements the reduction factor kε is assumed to 0.8.

9. Imposed actions

Similar to the design of new elements, the imposed actions of the reinforced concrete element to be strengthened must be known. The easiest way is to analyse the available as-built documents of the structure. If these documents are not available, geometry, idealised model of the structure and loads must be established by investigations on site. The bending moment, normal and shear force diagrams have to be determined considering the different type of loads and their combinations.

It is necessary to determine the imposed bending moment of the structure during application of the FRP strengthening system for the evaluation of the initial state of strain. Normally this will be the moment due to dead load of the structure and eventually to the prestressing force. In any case the bending moment of the initial state results from service loads (load safety factor = 1.0).

Furthermore, the characteristic and design bending moment due to expected future loads are required. The procedure of the determination of the moment curves is shown in figure 3.

The design value of the bending moment MSdf due to expected future loads must include the partial safety factors (table 1) and the additional combination values which consider the probability of occurrence of several variable loads. For prestressed or post-tensioned elements the statically indeterminate part of the prestressing moment Mp´ has to be considered for the determination of MSdf. The statically determinate part of the prestressing moment Mp0, which is defined by the prestressing force and the distance to centre of gravity of the concrete section, is considered by the design program.

Q

Q

G

G

G

MSdf,g+q

Fig. 3 Load cases before, during and after strengthening with FRP system

18

10.1 Capacity of the unstrengthened cross-section

To check the strengthening ratio η the bending capacity of the unstrengthened cross-section has to be calculated first. The design resistance of the unstrengthened cross-section MRd0 is determined considering the existing geometry, reinforcement, prestressing steel, concrete quality as well as the partial safety factors for material properties listed in table 1. If as-built documents are not available this information has to be established by investigations on site. Samples may be taken to check the concrete compressive strength.

10.2 Required cross-sectional area of FRP

=+

ε εc c0 c = + εεc

εsεs0

εcεc0

εc,o

Fs Ff

Fs Ff

Fig. 4 Superposition of the strain profiles and internal forces

19

10.3 Conditions of equilibrium

The unknown values like the capacity of the unstrengthened cross-section MRd0, the imposed initial strain state ε0, the required cross-sectional area of the FRP material Af,req and the resisting moment of the strengthened state MRdf are derived from the conditions of equilibrium ΣH = 0 and ΣM = 0 considering the mechanical behaviour of each material.

Internal forces

yk A f

concrete c

ck Rc

f xbF

γ ⋅α

ΣH = 0

ΣM = 0

Fc ⋅ ac – Fs ⋅ ds – Fp ⋅ dp – Ff ⋅ df = 0 (16)

The solution for the equilibrium conditions in equations 15 and 16 is found by variation of the strain profile. The strains are assumed to have linear distribution (Hypothesis of Bernoulli).The conditions for equilibrium are checked while running through the possible strain profiles within the defined limits:

unstrengthened cross-section:

0 < εc ≤ εcu

initial strain state:

design:

0 < εf ≤ εf,limit

0 < εc ≤ εcu ⇒ MRdf, ε and σ are determined iteratively

0 < εf ≤ εf,limit

The system of equations always leads to an unique solution.

20

10.4 Control of strain profiles

Strains in ultimate limit state

The program provides the user with the calculated strain profiles. The determined strain values can be compared with the strain limits of concrete and FRP (Fig. 5). Normally the design will be controlled by highly stressed FRP material, i.e. the strain limit of CFRP. In cases where the design is limited by failure of the compression zone the user should check if strengthening with FRP is still reasonable

The design can be checked by hand calculation using the provided strain profiles and the equilibrium conditions in equations 15 and 16.

εcu

The strain distri loads. In additio steel as well as

10.5 Contr

If the design an steel stresses i stress limits for

concret

reinforc

prestres

stribution in ultimate limit state

e state

bution in service state allows to cont n stresses at service state are dete

the selected FRP cross-section.

ol of stresses

d detailing is not in compliance with n service state, a verification of the the rare combination of loads have to

e σc,limit = 0.6 fck

21

rol yielding of internal reinforcement under service rmined for concrete, reinforcing and prestressing

the rules given in Eurocode 2 to limit concrete and stresses is necessary. In that case the following be respected.

(17)

(18)

(19)

11. Bond check of the FRP system

11.1 Anchorage of externally bonded CFRP laminates

The bonding characteristics of externally bonded CFRP laminates is totally different compared to embedded steel rebars. While steel rebars can be stressed up to the yielding point by increasing the bond length, the bond force of FRP laminates is limited. An increase in bond length above the length lbd does not result in an increase in resisting tensile stresses (see fig. 6). Based on tests a design model for the maximum bond failure force has been established in [8] for externally bonded steel plates on concrete structures.

Fbd,max

Fig. 6 Relationship between bond length and bond force

This model can also be applied to CFRP laminates in a modified form. It has become a substantial part of the German guideline [2] and is generally accepted as being the most up-to-date and straightforward to apply.

The maximum bond failure force Fbd,max (corresponds to Tk,max in the German guideline [2]) can be determined using the design value of the surface tensile strength fctd of the concrete:

[N] ftnEkkbm5.0F csdfffTbffmax,db ⋅⋅⋅⋅⋅⋅⋅= (20)

csm csd γ

= (21)

The subscript d in equation 20 indicates a design value considering the partial safety factor for concrete. Subscript f describes the properties of the FRP material while b corresponds to bond.

22

The associated bond length lbd,max can be derived from the following equation (cf. [10]):

[mm] f

tnE 58.0l

⋅⋅ = (22)

where:

mf number of laminates next to each other [ - ] bf laminate width [mm] nf number of laminates on top of each other [ - ] tf laminate thickness [mm] Ef modulus of elasticity of FRP laminates [N/mm²] fcsm surface tensile strength of the concrete [N/mm²], valid for: 1.5 N/mm² ≤ fctm ≤ 3.0 N/mm². γc partial safety factor for concrete [-] kT temperature reduction factor [-], taking into account temperature variations between -20°C and +30°C, 0.9 for exterior components, 1.0 for interior components. kb width factor according to the German guideline [2]

400/b1 b/b2

06.1k f

f b +

b beam width or laminate spacing for slabs [mm]

The factor 0.5 in equation 20 refers to the material characteristics of the adhesive bond. This explanation is also valid for equation 22. Additional information is given in the German guideline [2] or in the publications of Rostásy [7], [11] and Onken [10] respectively.

−⋅⋅=

11.2 Anchorage of externally bonded carbon sheets

The bond failure behaviour of FRP sheets is based on the same mechanical principles as CFRP laminates. In the equation stated above the thickness of the laminate tf is replaced by the theoretical fibre thickness of the selected sheet. The factor lf gives the number of layer glued one on top of each other. The width factor kb is set to 1 for the calculation of the bond failure force of carbon sheets.

Compared to CFRP laminates the surface of sheets is relatively large, so the bond behaviour is much better than for CFRP laminates.

23

11.3 Calculation of the envelope line / verification of the anchorage

The verification of the anchorage is carried out for ultimate limit state, considering the partial safety factors listed in table 1. For the application of externally bonded FRP systems the envelope line of the tensile forces has to be covered over the total length of the reinforced concrete element. Therefore the CFRP laminate or sheet should be extended to the support line as close as possible. It has to be proved that the design tensile force of the FRP material Ffd,E does not exceed the bond failure force Fbd,max that can be transmitted by the associated length lbd,max (see fig. 6). The tensile force of the FRP system is determined from the moment line in the same way as the force of the internal tension rebars.

A distinction has to be made between solid slabs and beams. Since CFRP laminates applied on slabs cannot be clasped by strap binders, the bond failure force has to be reduced. According to the German guidelines [2], [4] a reduction factor of 1.2 is introduced for slabs.

beams: (24) dfbd FF ≥

F ≥ (25)

Verification at an end support

To avoid the determination of the whole envelope line for the tensile forces the bond check can be carried out for a certain point E according to the German guidelines [2], [4]. For the end support of beams or slabs it is assumed that the first crack due to the imposed moment will appear at the point E, which corresponds to the associated bond length lbd,max of the maximum bond failure force Fbd,max. The maximum distance from the end of the CFRP laminates or sheets to the edge of support should not exceed 50 mm for sagging moments (cf. [11]).

xE

ME

E

fai

M(x)

Fig. 7 Verification of the FRP end anchorage at the end support

24

25

For reinforced concrete elements the definite point E for the calculation of the existing tensile force in the FRP strengthening can be simply determined by adding up the following lengths (see fig. 7):

xE = ai + f + lbd,max + aL (26)

xE distance from point E to the theoretical support axis ai distance from the support axis to the support front f distance from the end of FRP strengthening to the support front (≤ 50 mm) lbd,max bond length related to Fbd,max according to equations 20 and 22 aL horizontal displacement of the envelope line

According to Eurocode 2 the following values are valid for the horizontal displacement of the envelope line:

beams: aL = 0.5 · zm · (cot θ – cot α) (27) simplified for vertical stirrups and compression struts at 45°: (cot θ – cot α) = 1

T-beams: aL = 0.5 · zm · (cot θ – cot α) + x (28) with x = distance of reinforcement placed in the flange outside the web

slabs: aL = dm (29)

The distance xE is calculated by the program. The user must determine the corresponding bending moment MSdf,E from the moment line of the structure. The tensile force Ffd,E of the FRP material at point E is calculated from the entered design value of the bending moment by iteration of the equilibrium.

From the entered design value of the bending moment at point E the program calculates the tensile force of the FRP material Ffd,E by iteration of the equilibrium.

As prestressed or post-tensioned concrete members are usually uncracked near the support line the bond check has to be modified. Externally bonded FRP systems always have to be anchored beyond the last flexural crack. The design program determines the cracking moment of the prestressed or post-tensioned cross-section considering the tensile strength of the concrete. The user then has to enter the value xE which means the distance of the cracking moment from the support line, measured from the moment curve of the structure. Considering the selected FRP cross-section the program is calculating the tensile force Ffd,E in the section at the point where the first crack will occur. This force is compared to the maximum bond force Fbd,max of the selected FRP system.

Following possibilities are recommended, if the anchorage verification according to equations 24 and 25 may fail:

• increase the cross-sectional are of FRP,

• reduce the distance f between the end of FRP strengthening and the front of support,

• verify substrate strength (pull-off-test) and increase if possible,

• extend the FRP reinforcement beyond the support (e.g. slot-in end)

• increase the contact pressure of FRP reinforcement using additional anchorage devices

The design program gives adequate advice.

Verification at an intermediate support for moment of span

Since the position of the moment zero point varies with different load combinations the FRP system should be anchored at least 1 m beyond the zero-crossing of the horizontal displaced envelope line of the tensile forces. However, at least the bond length lbd,max related to the maximum bond force should be applied. According to the German guidelines [2], [4] the maximum distance from the end of FRP strengthening to the front of support should not exceed 50 mm for sagging moments (cf. [11]).

E aL

As1 Af

Fig. 8 anchoring verification for CFRP laminates and sheets at intermediate support

At the intermediate support the point E refers to the zero point of the bending moment line. From the distance xE, the program determines the distance f between the end of the CFRP laminates or sheets and the front line of the support (see fig. 8).

fmax = xE – ai – aL – lbd,max (30)

Recommendation: f = xE – ai – aL – lb with lb = 1 m

xE distance from the theoretical support line to the moment zero point aL horizontal displacement of the envelope line lb bond length f distance from the end of FRP strengthening to the support front

If the existing length at the intermediate support is not sufficient to anchor the external bonded FRP system with the minimum bond length lbd,max, the program will calculate a negative value for f. In this case the bond forces have to be proved equivalent to the bond check at the end support or the FRP system must be extended beyond the support line.

For prestressed or post-tensioned elements the verification of the anchorage at intermediate support can be performed by the same approach as for the end support.

26

Verification at an intermediate support for the moment at support

The verification of the anchorage for the FRP top strengthening can be carried out in a similar way as for the FRP bottom strengthening at the intermediate support. In the German guidelines [2], [4] it is recommended to anchor the FRP system at least 1 m beyond the zero-crossing of the displaced envelope line. The program determines the distance from the intermediate support front line to the end of the FRP strengthening material (see fig. 9).

aL

As1

fai

E

M(x)

Fig. 9 Verification of the anchorage of FRP top strengthening at intermediate support

fmin = xE – ai + aL + lbd,max (31)

Recommendation: f = xE – ai + aL + lb with lb = 1 m

xE distance from point E to the theoretical support line aL horizontal displacement of the envelope line lb bond length f distance from the end of FRP strengthening to the support front line

11.4 Anchorage of near surface mounted CFRP laminates

As already mentioned, compared to externally bonded strips, near surface mounted laminates have a higher anchoring capacity. Therefore they can almost be stressed up to their tensile strength with increasing bond length. Based on the investigations in [12] a design model was established for the anchoring of near surface mounted FRP laminates in the surrounding concrete cover. According to [12] the bond force Fbd of the laminate depends on the bond length lb. It can be described by the following equations:

( ) ]mm[115lforl0015.04.0labmF bbb 4

27

where:

mf number of slots bf width of the FRP laminate [mm] ar distance of the laminate axis to the edge of the member [mm], (valid for ar ≤ 150 mm) τK,d design bond strength of the epoxy adhesive [N/mm²]

The characteristic bond strength τK,k of the epoxy adhesive is determined with the following equation:

( ) k,Ktk,Kck,Ktk,Kc 2

k,Ktk,Ktk,K fffff2f2 ⋅

+⋅+⋅−⋅=τ (34)

For the determination of the design bond strength τK,d according to the German Guideline for the strengthening of concrete components using unidirectional CFRP laminates glued into slots in the concrete [3] the following partial safety factors are valid:

τK,d = τK,k / γb (35)

γb = 1.3 for fundamental combinations

γb = 1.05 for accidental combinations

The envelope line of the tensile forces is carried out at ultimate limit state under consideration of the horizontal displacement in the same way as for externally bonded FRP systems. The envelope line has to be covered by the envelope line of the resisting tensile force considering the internal steel reinforcement and the near surface mounted FRP laminates. Unlike externally bonded FRP systems near surface mounted laminates can be anchored from the point where they are theoretically no longer required to cover the entire tensile force.

The required bond length for near surface mounted FRP laminates results from transformation of equations 32 and 33:

[ ] ]mm[115lformm ab0015.0

11.5 Surface tensile strength of concrete

A proper execution of strengthening with externally bonded FRP systems on site will always lead to a failure in the concrete covering layer (bond failure) and not to a failure in the adhesive substance. An essential parameter for the adhesive bond of FRP systems is the surface tensile strength fcsm of the existing concrete cover. The testing of the surface tensile strength has to be carried out according to the relevant national regulation. Due to the German guidelines [2], [4], at least five tests should be performed on each concrete element.

For the calculation of the bond failure force the surface tensile strength can be applied according to the German guidelines [2], [4] as the median value of the population. This value can be derived from test results under consideration of the student distribution with a statistical safety of 95%.

arithmetic median n

median of the population (40) skxf mcsm ⋅−=

xi strength of test i n number of tests k reduction factor

Depending on the number of tests, the reduction factor k can be taken from the following table (see German DIN 1048)

n 5 6 7 8 9 10 15 20 25 30 35

k 0.953 0.823 0.734 0.67 0.62 0.58 0.455 0.387 0.342 0.31 0.286

Tab. 2 Reduction factor k for the calculation of the median of the population (German DIN 1048)

Example

1 2.0 1.1025 2 2.2 0.7225

05.36 3.18xm ==

792.0135.3 16

29

12. Anchorage of bottom reinforcement at end support

On account of the truss model used for the shear design the bottom reinforcement has to be properly anchored at end and intermediate supports. The required conditions are stated in Eurocode 2. It is necessary to

a) retain not less than 25% (or 50% for slabs) of the required steel section present in the span

b) cover the tension force, which is derived from the truss analogy considering the shear and normal force at the support

A,Sdf m

⋅= (41)

The maximum value of both conditions is valid for the anchorage of the bottom reinforcement.

Normally the first condition is only valid for the new design of structural concrete elements, not for strengthening. Applied to existing concrete elements to be strengthened 25% (or 50%) of the tensional force due to the maximum moment of span has to be anchored at the supports. For strengthened systems the combined maximum tensile force of the internal steel reinforcement and the external FRP system has to be considered.

As long as the existing bottom reinforcement is not curtailed, the internal rebars extended beyond the support line are normally sufficiently anchored an strengthened state. The program calculates the required anchorage force at the support from the two conditions mentioned above and determines the part of the tensile force covered by the internal rebars. It is calculated from the circumference of the bar and the bond strength fbd:

bd s

⋅ ⋅= (42)

If the internal reinforcement is not sufficient for the anchorage an strengthened state, a part of the FRP strengthening has to be extended beyond the support front line. The software determines the tensile force to be anchored and the required bond length.

In practice the anchorage of FRP systems beyond the support line is very difficult and questionable. Under slabs supported by masonry walls, externally bonded CFRP laminates can be extended to the adjacent span by removing one brick of the wall.

At concrete walls and beams the only adequate solution is to slot-in the end of the laminate and inject the slot with epoxy paste. If the slot is sufficiently thin (d ≤ 10 mm), a verification of the anchorage according to chapter 11.4 is possible.

In contrast to externally bonded FRP systems, an additional anchorage of near surface mounted laminates beyond the support front line can hardly be realised in practice.

30

31

13. Detailing provisions

For externally bonded FRP strengthening the spacing of the strips is limited. According to the German guidelines [2] – [4] different limits are valid for the FRP systems.

Externally bonded laminates and sheets

edge distance: ar,min = cw (43)

axial spacing: sf,max = 0.2 · l (bearing distance) (44) sf,max = 5 · h (slabs) (45) sf,max = 0.4 · l (cantilevering length)

Near surface mounted laminates

edge distance: ar,min = 2 · bf (46) ar,min = dk (47)

axial spacing: sf,min = dk (48) sf,min = bf (bei as > 2 · ds) (49)

slot: ts,max = cw – h (50) bs,min = tf + 1 [mm] (51) bs,max = tf + 3 [mm] (52)

where:

cw concrete cover of internal stirrups dk maximum diameter of aggregates in concrete as axial spacing of internal longitudinal rebars ds diameter of internal longitudinal rebars ts, depth of the slot bs width of the slot h allowance for tolerances of the concrete cover

14. Shear design

14.1 Shear capacity according to Eurocode 2

In most cases of flexural strengthening with FRP systems, it is necessary to check the shear capacity of the concrete structure as well. Especially beams they also require shear strengthening. On the other hand for concrete solid slabs it may be proven that shear reinforcement is dispensable for expected future loads. If not, other strengthening methods have to be considered.

According to Eurocode 2 the imposed shear force VSd can either be transferred by the concrete alone or in combination with shear reinforcement. The shear resistance is described by the design values VRd1 to VRd3. For the calculation of the shear reinforcement the standard method is applied considering vertical stirrups and an inclination of the compression struts of 45°. The relevant equations according to Eurocode 2 can be taken from the following flowchart (fig. 11).

VRd1 shear resistance without shear reinforcement – shear force is transferred by concrete alone,

VRd2 maximum shear resistance – the capacity of the inclined compression struts is decisive for the shear resistance,

VRd3 shear resistance with shear reinforcement – the shear force transmission results from concrete and shear reinforcement.

The lower design value VRd1 is the relevant value for slabs which are usually constructed without any shear reinforcement. Presenting the uppermost limit, VRd2 must not be exceeded by the imposed shear force.

Following the conditions given in the German guidelines [2] – [4] and according to Eurocode 2, a distinction has to be made between four different cases with regard to the shear force capacity of a strengthened concrete structure (see [10]):

1. In case that the existing shear force VSdf of the structure at strengthened state is lower than the shear resistance VRd1 of the concrete alone, no shear strengthening is necessary. This case generally applies to slabs.

1RdSdf VV ≤ (53)

VSdf and VRd1 are determined considering the partial safety factors in Table 1.

2. If the shear force at strengthened state can be completely covered by the existing internal stirrups, minimum shear strengthening is necessary to complete the mechanical truss model

VV 3RdSdf ≤ (54)

The additional shear reinforcement has to clasp the flexural strengthening and is designed for the shear force difference V depending on the strengthening ratio.

SdfV1V η −η= (55)

In this case, anchorage of the shear strengthening in the compression zone can be omitted (see Fig. 11).

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The fact that additional shear reinforcement in the form of external bonded stirrups is necessary despite sufficient internal shear reinforcement is justified by the concrete beam design truss analogy. The additional tension chord of the external flexural strengthening must be connected to the tension struts of the internal stirrups for completion of the truss model (fig. 10, see [10]).

Fig. 10 Connection of the FRP flexural strengthening to the internal truss structure

As1 Af

internal stirrups

compression chord

tension chords

shear strengthening

3. If the shear force demand at strengthened state exceeds the shear capacity of the existing cross- section, the shear strengthening has to be designed for the remaining amount of shear force.

VV 3RdSdf > (56)

VVV 3RdSdf −= or SdfV1V η −η= (57)

The higher value V of both conditions in equation 57 is valid for the design. Since the additional shear reinforcement is necessary to cover the total shear force of the cross-section, the external stirrups have to be anchored in the compression zone (see fig. 11).

4. The maximum shear resistance VRd2 provides the upper limit of the shear force also for strengthened state. However, the German guidelines [2] – [4] do not permit shear strengthening of high stressed beams. Therefore in the program it is recommended to limit the maximum shear capacity in the design concept of Eurocode 2 as well. Reducing the maximum shear capacity to 50 % (Vmax = 0.5 VRd2) corresponds approximately to the limitation given in the German guidelines [2] – [4]

VV maxSdf ≤ (58)

V5.0V 2RdSdf ⋅≤ (59)

x

Fig. 11 Anchorage of the additional shear reinforcement depending on the imposed shear force

Fig. 12 Flowchart for the shear check of strengthened beams according to Eurocode 2

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Even if additional external stirrups are not required to cover the imposed shear force (case 1), it is recommended for beams to clasp the FRP flexural strengthening system at least with 2 external stirrups at the end of the beam. For strengthening the moment of span these external stirrups should be anchored in the compression zone. For moments of support an additional anchorage of the flexural strengthening with external stirrups can be omitted.

14.2 Design of the additional shear reinforcement

As additional shear reinforcement, steel plates as well as high modulus carbon sheets (unidirectional fabrics) can be used. The sheets have a modulus of elasticity of about 640'000 N/mm² (C-Sheet 640), they are easier to handle than steel plates and therefore the application is more economical despite the price of the material.

The required cross-section of the additional shear reinforcement in form of steel plates or sheets has to be determined for the remaining shear force difference V (eq. 55 and 57).

The internal steel stirrups and the external stirrups made of steel plates or carbon sheets are considered as parallel connected elastic or elastic-plastic elements. The strain conformity of these elements must be ensured also at strengthened state. Therefore the same strain limit of εlimit = 0.2 % according to [7] will be taken as a basis for the design.

The stress of the internal stirrups to determine the shear capacity VRd3 must be limited according to equation 60:

ydsitlimsw fE ≤⋅ε=σ (60)

The required cross-sectional area of the additional external shear reinforcement results from the following equation:

fwf req,w z

= (61)

where:

V shear force difference covered by the additional external shear reinforcement zf internal lever arm between the concrete compressive force and the flexural strengthening is iteratively determined by the program σfw stress of the additional external shear reinforcement (at εlimit = 0.2 %)

Steel plates:

Carbon sheets:

fditlimfw E⋅ε=σ (63)

Since the carbon sheets are rather weak during handling and are applied by hand lamination at building site conditions, it is doubtful that the high modulus of elasticity of about 640'000 N/mm² will be

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achieved in practice. Therefore the modulus of elasticity in equation 63 should be reduced for carbon sheets as shown in figure 13. For this a reduction factor of γE = 1.2 is recommended in [10]:

σf

Efd = Efk / γE (64)

Fig. 13 Reduction of the modulus of elasticity of carbon sheets (design value)

Spacing of external stirrups

Steel plates:

The maximum spacing of the plates sw,max results from the truss analogy and is approximately equal to the effective depth of the FRP flexural strengthening (see fig. 10) which corresponds to the overall height of the beam.

sw,max = h (65)

Carbon sheets:

When strengthening with carbon sheets C-Sheets 640, a maximum strip spacing of 80 % of the overall height of the beam is recommended.

sw,max = 0.8 · h (66)

14.3 Anchorage of external stirrups

If only minimum external stirrups are required without anchorage in the compression zone, (case 2: Vsdf ≤ VRd3), it has to be proved, that the adhesive bond of the external stirrups provides a sufficient anchorage. For this check the bond behaviour as stated in chapter 11 can be applied, where the material properties of steel plates or carbon sheets have to be considered.

As the form an the location of shear cracks can not be predicted, according to the German guideline [2] it has to be proved, that the tensile force of the additional external stirrups does not exceed 50 % of the bond force Fbd as given in equation 23:

Fwd ≤ 0.5 Fbd (67)

The external stirrups always have to be bonded over the whole height of the web. But only half of the existing bond length lbw at the side of the web can be considered for lb in equation 23 (see fig. 14):

lb ≤ 0.5 lbw (68)

h1

lb

Fig. 14 Adhesive bond anchorage of minimum external stirrups

The tensile force of the external stirrups is determined according to Eurocode 2 using the following equations:

z V

z sV

⋅ ⋅= each leg of a stirrup (70)

For external stirrups that are only anchored by adhesive bond, differing from equations 65 and 66 the following is valid:

sw,max = 0.5 · h (71)

15. Further checkings

In addition to the calculations which can be carried out with the program FRP Lamella, the structural engineer should also check cracks and deformations of the strengthened structure if necessary. According to the most national regulations or guidelines, the control of crack widths at strengthened state is not required. Nevertheless, in a special case where it may be necessary, you should make sure that durability and serviceability of the strengthened concrete structure are ensured. Please note that strengthening with FRP products has no significant influence on the deformations of a strengthened concrete structure. In case of deflection problems, preference should be given to other strengthening methods like for instance sprayed concrete.

16. Fire protection

If fire protection is required, the program enables the user to check the remaining safety θ for service loads under the condition that the external bonded FRP system and the external stirrups will completely fail. Please note that epoxy resins may loose their load bearing capacity when the temperature approaches 80° C. If necessary it has to be proved in special cases by an approval or an expert opinion that the FRP system and the external stirrups are sufficiently protected against fire, using additional protective measures as for instance fire protection plates.

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17.1 Start of the program

The program is developed for the following Windows operating systems: Win 9x, 2000, NT.

After successful installation of FRP Lamella you start the program by either clicking the FRP Lamella icon on the Desktop or choosing the option FRP Lamella in the Windows start menu of your computer. First the disclaimer will appear. You have to accept to start the FRP Lamella program window.

To exit the program FRP Lamella and close the window, click the cross in the title bar at the upper right corner of the program window. Instead you can either choose the option exit of the file menu.

17.2 Settings

To ensure optimum display performance of the FRP Lamella program on your screen a minimum screen resolution of 800 x 600 pixel is assumed.

The display font size is also essential for a proper display of the program window. In the Windows menu Start point to Settings, click Control Panel, and then double click Display. On the Settings tab, click Advanced, then you will find the Font Size list on the General tab. Small fonts (standard) should be selected, otherwise several items might not be completely displayed.

17.3 Basic information about the FRP Lamella user interface

You will find general information about Windows user interfaces in your Windows manual or in the online help function of your Windows operating system.

Title bar The uppermost line of the FRP Lamella program window shows information about program, data file and path.

Menu bar The items File, Calculation, Extras and Info on the menu bar lead to different submenus. You will find a detailed list of all menu items in chapter 19.1.

Tool bar The most frequently required functions can easily be called from the toolbar by clicking one of the symbols. You will find a detailed list of all tool bar symbols in chapter 19.2.

Tree view The tree views in the left part of the program window enable you to call the different input and output windows directly. Click the + symbol in front of a heading to display the subordinated items. A click on the – symbol hides the subordinated items again.

Quick info Positioning the mouse pointer on one of the input or output fields an explanation (tooltiptext) will appear after a few seconds. Proceed in the same way to get explanations for the functions of the tool bar.

17.4 Data input

The required data is entered on several input windows shown in the upper part of the program user interface. The titles of the different windows are listed in the opposite tree view. They are classified according to the topics general information, cross-section, loads, strengthening and proofs. Every input window shows a graphic to illustrate the essential data.

To show the different input windows click the related title in the tree view. If one heading includes several subordinated windows, the first window will be displayed automatically. Input windows that are not accessible yet are displayed in light grey.

Use the button on the right below the picture box of each input window to display the next window. It is recommended to follow the sequence of the windows to make sure that no window is left out. The button on the left below the picture box leads to the previous input window.

Enter the required data in the provided text boxes of each window. If necessary overwrite the entry 0. Text boxes with a grey font are locked and cannot be modified. Disabled text boxes having a dark background are not considered in the calculations.

The key button in the toolbar enables you to unlock input fields having a grey font and modify the preselected values.

For some items you can choose from a list of different values.

After you have entered all required data you can start the calculation either by clicking the calculation button below the graphic of the input window FRP cross-section or by clicking the calculator symbol in the toolbar.

Start the proofs of anchorage and shear capacity by clicking the proof button below the graphic of the corresponding input window. A click on the tick symbol in the toolbar will carry out all proofs successively. This function is useful after reopening an existing input data file.

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17.5 Output of results

The results are displayed in additional windows located in the lower part of the program user interface. The titles of the different windows are listed in the opposite tree view. They are classified according to the topics general information, strengthening and proofs.

To show the different output windows click the related title in the tree view. If one heading includes several subordinated windows, the first window will be displayed automatically. Output windows that are not accessible yet are displayed in light grey

The result values are displayed in text fields having a light grey or coloured background. These values cannot be modified.

Pay attention to the output fields highlighted in blue or red colour. They will show you if the proof conditions are met and if special details of construction have to be followed.

The output windows of strains in ultimate limit state and strains in service state additionally show a graphical representation of the strain distribution (s. chap. 18.15 and 18.16). You can change the scale by clicking the picture.

After the calculation of certain proofs you have to design a sufficient strengthening. The required input data is entered in the white text boxes of the output window (s. chap. 18.20 and 18.23).

The performed calculations can be printed on any printer installed under Windows operating systems. You can modify the content of the heading line in the menu Extras >> company letterhead. It is possible to print each page individually.

pages 1 – 4 design of flexural strengthening page 5 proof of FRP end anchorage page 6 proof of anchorage of flexural reinforcement at support page 7 design of shear strengthening

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18.1 Input window project

When you start the program the first window project opens. It offers the possibility to enter some general project and structural element data. This information appears on each page of the printout and will help you administrating your projects.

Enter the project number and the project name.

For each structural element you can enter a number and an appropriate description.

tip Use the button on the right below the picture box to display the next window.

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18.2 Input window code

In this window you can choose the underlying code and guideline for the design. Additionally you determine the material properties as well as the unit measurement for the input and output.

Choose the code according to which standard you want to perform the design of the strengthening measure simply by clicking the related option button (not every version offers the possibility to choose the code).

Afterwards choose the guideline on which the design of FRP strengthening shall base.

Select a country for the available steel and concrete grades based on the national standard that will appear on the steel and concrete window, respectively. The appropriate national flag appears next to the selection box.

You have the choice from different units of measurement: the unit of lengths and forces as well as the unit of strains.

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18.3 Input window geometry

FRP Lamella offers the possibility to strengthen the most frequent types of cross-section: slabs, rectangular beams, T-beams and I-beams. According to your choice FRP Lamella will present an appropriate illustrating graphic.

Click the geometry list and choose the type of cross-section.

Enter the dimensions of the cross-section in the corresponding data fields.

note After entering all data the graphic turns into a true to scale graphic to allow a visual control of the values.

Please indicate whether you want to strengthen an exterior or interior structural member. According to the German guidelines [2], [3] for external concrete members the program will consider a temperature reduction factor kT for the bond strength of externally bonded FRP strips due to temperature variations from –20° C to 30° C (s. chap. 11.1, 11.2).

For slabs please indicate the span or the cantilever length. This input data field will be displayed when you select a slab. The length is required to calculate the maximum spacing of FRP strips. (s. chap. 18.13).

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18.4 Output window cross-section

After entering the size of the structural member the cross-section values are calculated and shown in the lower part of the user interface.

The upper field shows the gross cross-sectional area Ag of the structural member.

The gravity axis zcg of the cross-section is related to the top of the member.

Furthermore the moment of inertia Iy of the cross-section is given.

The section modulus Wtop and Wbottom apply to the top and the button of the cross-section.

note The cross-section values are related to the gross cross-section of the member.

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18.5 Input window concrete

The window concrete indicates the material properties of the concrete. You can select the concrete classes according to the chosen code.

Select the existing concrete class of the member from the list. The appropriate characteristic compressive strength fck will be displayed in the next field. If the required concrete class is not available in the list, select the option other, this enables you to define the characteristic strength (s. chap. 7.1).

The concrete maximum strain εcu is limited to 0.35 [%] according to Eurocode 2.

The strain at the axis of the parabolic curve εc1 is assumed to be 0.2 [%] according to Eurocode 2 (Model Code).

The reduction factor α is a coefficient taking into account long term effects on the compressive strength. It is generally assumed to be α = 0.85.

The basic value of the design shear strength τRd is calculated from the characteristic cylinder compressive strength (s. chap. 7.1).

The average modulus of elasticity of concrete Ecm is necessary for the calculation of the uncracked state of the structural member.

The characteristic tensile strength of the concrete fctm defines the transition between the uncracked and the cracked state of the cross-section.

The partial safety factor γc for concrete is preselected as γc = 1.5 according to Eurocode 2. (s. chap. [1])

note You can modify the proposed values by using the key button in the tool bar.

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18.6 Input window steel

The material properties of the reinforcing steel are entered in the window steel. The program offers the possibility to define 2 types of reinforcing steel as well as 2 types of prestressing steel. The graphic shows both stress-strain diagrams.

Select the steel grade of the reinforcing steel and of the prestressing steel from the steel list. The appropriate characteristic yield stress fyk and the appropriate characteristic tensile strength fpk of the prestressing steel will be displayed in the field next to the steel list. If the required steel grade is not available in the list, select the entry other, which enables you to define the characteristic strength.

note The available steel grades in the list depend on the country selected in the window code.

For rebars select between plain and ribbed sections, for prestressing steel it is to distinguish between strands and wires. The choice has an effect on the bond of the rebars or the modulus of elasticity of the prestressing steel, respectively.

The modulus of elasticity Es for reinforcing steel is preselected as Es = 200'000 [N/mm²] in accordance to Eurocode 2. For prestressing steel the modulus of elasticity Ep is preselected depending on the chosen section in accordance to Eurocode 2.

The maximum steel strain εsu and εpu is assumed as 2 [%] according to Eurocode 2. It is only needed for the termination of the iteration. Before this strain limit will be reached, the design will be controlled by the strain limit of the FRP material.

For the calculation of the prestressing force the horizontal line of the design stress-strain diagram is used. According to Eurocode 2 the characteristic tensile strength of the prestressing steel is reduced by αp = 0.9.

The partial safety factor γs for reinforcing steel is preselected to 1.15 according to Eurocode 2 (s chap. [1]).

note You can modify the proposed values by using the key button in the tool bar.

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18.7 Input window main flexural reinforcement

Details about the existing rebars of the concrete member at the position of the maximum bending moment can be entered in the window main flexural reinforcement. FRP Lamella allows the input of six reinforcement layers in the tension or compression zone. You must at least enter one layer of internal reinforcement since the program does not design unreinforced members.

Enter the cross-sectional area As of the existing tension rebars.

tip Click the number button at the beginning of each line and an additional window shows a table of rebars diameters. You can select the number and the cross-section for groups of rebars and copy the total sum of the cross-sectional area of reinforcement to the input window (s. chap. 18.8).

The position of the reinforcement is given by the depth zs measured from the top edge of the structural member to the axis of the rebars.

Afterwards you choose the steel grade. The list shows the four grades defined in the previous window steel. For unclassified steel grades the defined yield strength or tensile strength will be displayed.

For prestressed steel you have to enter the prestress after creep and shrinkage σp0. The resulting prestressing force is shown in the input window loads in unstrengthened state (s. chap. 18.10).

The option bonded is only enabled for prestressing steel. You can define if the tendon is fixed to the surrounding concrete or if it slides without bond through a sheath. Rebars are always bonded to the concrete.

The input of a concrete cover cw is optionally and just serves to if the arrangement of FRP strips fits to the width of the member. The minimum distance between the lateral edge of the FRP strips and the edge of the concrete member is equivalent to the concrete cover.

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18.8 rebar tables for the selection of reinforcement cross-sectional area

The cross-sectional area of reinforcement can be copied from a table which offers a wide selection of rebar diameters. The cross-sectional area is depending on the number of rebars in a beam or the spacing of rebars in a slab. To open the rebar table just click the number button in the input windows main reinforcement and reinforcement at support, respectively.

Choose a cross-sectional area by clicking a white field in the table. The background of a selected field turns into blue. For beams, a multiple choice is possible. The sum of the cross-sectional area As of the selected rebars is displayed below the table.

You cancel the selection by clicking the blue field again.

You can copy the selection to the reinforcement window by clicking the copy button. The rebar table will be closed.

Close the table without copying the value by clicking the cancel button.

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18.9 Input window flexural reinforcement at support

Details about the existing rebars at support can be entered in the window flexural reinforcement at support. Additional information is needed for the proof of anchorage at support.

Enter the cross-sectional area As for every layer of reinforcement. The value is automatically copied from the previous input window main reinforcement.

tip Click the number button at the beginning of each line and an additional window shows a table of rebars diameters. You can select the number and the cross-section for groups of rebars and copy the total sum of the cross-sectional area of reinforcement to the input window (s. chap. 18.8).

The position of the reinforcement is given by the depth zs measured from the top edge of the structural member to the axis of the rebars. The default values are copied from the previous input window main flexural reinforcement.

For prestressed steel layers the prestress σp0 at the region of support is needed. The default values are copied from the previous input window main flexural reinforcement.

Enter the diameter ds of the rebars.

For determination of the anchorage force the anchorage length ls,A from the support front is needed.

Choose the coefficient for effectiveness of anchorage αa. According to Eurocode 2 the value is 1.0 for straight bars. For hooks, bends, loops and welded transverse bars in the anchorage zone the value can be reduced to 0.7.

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18.10 Input window loads in unstrengthened state

The window loads in unstrengthened state defines the imposed actions just before strengthening with FRP. The resulting strains in the concrete member are taken into consideration for the strengthening design. The graphic schematically presents the initial state as a bridge girder, which is closed for traffic during application of the FRP strengthening and only loaded by the dead load of the structure.

Choose the design for a positive (moment of span) or negative moment (moment at support).

note This choice is linked with the type of moment in the window loads in strengthened state. It is not possible to choose different types of moments in both windows.

Enter the characteristic bending moment MSk0 which is imposed to the structure during the application of the FRP strengthening. Commonly this will be the dead load moment. This value defines the initial state of strain in the cross-section. For statically indeterminated systems you may have to add the secondary moment from prestressing Mp'.

If the member is subjected to an external axial force, e.g. the dead load of an inclined beam, you have to choose between compressive and tensile force.

Enter the characteristic axial force NSk0 resulting from the imposed load. Compressive forces have a positive effect and can be ignored.

For prestressed members the program displays the prestressing force NP as well as the prestressing moment Mp0' that are considered in the design. The values result from the cross- sectional area, prestress and position of the prestressing steel entered in the window main flexural reinforcement (s. chap. 18.7).

note The program considers only the statically determinated prestressing moment. If the support conditions of the prestressed member are statically indeterminated, add the secondary moment Mp' to MSk0, MSdf and MSkf.

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52

To define the transition between the uncracked and the cracked state of the cross-section, decide if the bending tension zone of the cross-section is already cracked under service loads. Reinforced concrete members are usually always cracked, for prestressing members it depends on the degree of prestress and the history of loading. As a rule the cross-section is uncracked for maximum prestress (no tensile forces in service state).

18.11 Input window loads in strengthened state

The window loads in strengthened state defines the future actions. You have to enter the bending moments which are imposed to the concrete member after strengthening with FRP (s. chap. 9). The graphic schematically presents the strengthened state as a bridge girder, which is loaded by dead load of the structure and an additional high live load.

Choose the design for a positive (moment of span) or negative moment (moment at support).

note This choice is linked with the type of moment in the window loads in strengthened state. It is not possible to choose different types of moments in both windows.

If the member is subjected to an external axial force, e.g. the dead load of an inclined beam, you have to choose between compressive and tensile force.

Enter the design bending moment MSdf for the expected loads considering the partial safety factors γ for permanent and variable loads as well as the combination value ψ. For statically indeterminated systems you may have to add the secondary moment from prestressing Mp'. (s. chap. 18.10).

Proceed in the same way for the design axial force NSdf. Take also into consideration the different partial safety factors γ and the combination value ψ .

You can either enter the exact values of imposed actions in service state or choose the option approximate. In this case the characteristic values of the bending moment and axial force are calculated from the given design values as follows:

m,M

γ =

note The use of the option approximate is especially recommended if the design actions are determined by a complicated structural analysis (e.g. finite elements). Using the average safety factors you can avoid another analysis applying characteristic loads without partial safety factors.

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54

If you have chosen the option exact, enter the characteristic (service) moment of strengthened state MSkf. Use the load combination applicable for the proof of the service limit state. For statically indeterminated systems you may have to add the secondary moment from prestressing Mp' (s. chap. 18.10).

If the member is subjected to an external axial force, enter the characteristic axial force NSkf.

If you have chosen the option approximate, enter an average partial safety factor γM,m for bending moments. The characteristic moment MSkf is then calculated with above-mentioned equation from the design moment MSdf. The valid range for the average partial safety factors is 1.0 to 2.0 (1.35 to 1.5 according to Eurocode 2).

Is the member loaded with an axial force, also enter the average partial safety factor γN,m.

18.12 Input window FRP system

The window FRP system shows the material properties of the CFRP laminates and sheets. The displayed safety and reduction factors as well as the limited design strain are preselected according to the guideline you have chosen in the window code.

Select the FRP product from the list. There are two types of prefabricated S&P FRP laminates and one hand lay-up S&P C-Sheet in different grammages available.

For prefabricated laminates choose the type of bonding. While externally bonded laminates provide a larger cross-sectional area, near surface mounted (slot-in) laminates show a higher bond resistance (s. chap. 11).

Below the FRP material list the appropriate adhesive is mentioned. For some systems a selection of different adhesives is available. S&P Resin 50 is used for pouring slots in negative moment regions of slabs.

Depending on the chosen type of FRP the modulus of elasticity Efk, the tensile strength ffk and the ultimate strain εfu of the material will be displayed (s. chap. 7.4). For FRP linear-elastic material behaviour is assumed.

The partial safety factor γf depends on the guideline. The reduction factor for strain limit kε enables an additional reduction of the design strain limit εf,limit (s. chap. 8.3).

For carbon sheets a reduction factor γE for the characteristic modulus of elasticity is recommended (s. chap. 14.2). The value is preselected to γE = 1.2.

Some guidelines prescribe a strain limit εf,limit for the design of FRP (s. chap. 8.1, 8.2).

note You cannot modify the material properties of FRP. But you can change the preselected partial safety and reduction factors by using the key button in the tool bar.

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18.13 Input window FRP cross-section

On the window FRP cross-section you are asked to choose the number and arrangement of the FRP products. Observe the minimum and maximum spacing respectively. You can choose up to three layers with different cross-sectional areas as well as different effective depths.

Before selecting the FRP cross-section, start the iteration either by clicking the calculation button on the bottom right of the window or by clicking the calculator symbol in the toolbar. The required FRP cross-sectional area will be calculated and displayed in the output window design below.

note To carry out the design calculation, in this window only the depth zf1 of the FRP layer 1 from the top edge of the concrete member has to be given. As the design iteration can only determine the cross-sectional area of one FRP layer (one unknown), FRP layers 2 and 3 are initially locked, but they are enabled after the design calculation.

For each layer choose an FRP cross-section. The available selection depends on the chosen FRP product in the window FRP System. For sheets the theoretical fibre thickness tf is given and you can choose the width bf of the sheet. The delivery width of the sheet is preselected.

Enter the number of FRP plies nf lying on top of each other. One single ply is preselected. The maximum is two plies of laminates or five plies of sheets.

For beams enter the number mf of FRP strips lying next to each other. The spacing sf of the strips is calculated.

For slabs of a standard width (1 [m] or 12 [in]), enter the spacing sf of the strips. The number mf of FRP strips lying next to each other is calculated. The limit spacing sf,max or sf,min respectively, is calculated according to the German guidelines [2] – [4] with following conditions (s. chap. 13):

sf,max = 0,2- times span sf,max = 5- times slab thickness sf,max = 0,4- times cantilevering length sf,mi