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FRP Lamella
user manualfor design softwareversion 3
flexural and shear strengthening using FRP materials
Peter OnkenWiebke vom BergDirk Matzdorff
© bow ingenieure gmbhbraunschweig · hamburg
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FRP Lamella
design program
for flexural and shear strengthening with FRP materials
according to Eurocode 2
User Manual
version 3.4
Peter Onken, Wiebke vom Berg, Dirk Matzdorff
© 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
• DIN 1045 (7/88) (German DIN-Norm)
• British Standard 8110
• BAEL 91 (Normes Françaises)
• ACI (American Concrete Institute)
• KCI (Korean Concrete Institute)
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
effective flange width of upper flange b1
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 Iysection 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
anchorage length of rebars from the support front ls,A
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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FRP material modulus of elasticity of FRP material Ef
failure strain of FRP material εfu
strain limit of FRP material εf,limit
characteristic tensile strength of FRP material ffkpartial 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 tfwidth 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 determinated prestressing moment Mp0
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 lbdesign 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,AShear 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 ffkreduction 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
average lever arm of internal forces zm
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 twwidth 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
[1] Allgemeine bauaufsichtliche Zulassung für die Verstärkung von Stahlbetonbauteilen durch schubfest aufgeklebte S&P Kohlenfaserlamellen (Z-36.12-62); Deutschland.
[2] Richtlinie für das Verstärken von Betonbauteilen durch Ankleben von unidirektionalen kohlen-stoffaserverstärkten Kunststofflamellen (CFK-Lamellen), Anlage 2 der Zulassung [1], Deutsches Institut für Bautechnik, Berlin.
[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
Eurocode 2 1.35 1.5 1.5 1.15
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:
2MM
0Rd
fSdEC,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:
1MM
Skf
0RkEC >=θ (3)
The subscript k in the equation indicates characteristic values.
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7. Material behaviour
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
ckcd
ff
γα=α (α = 0.85; γC according to table 1) (4)
example: C 20/25 → ²mm/N33.115.1
2085.0fcd =⋅=α
The design shear strength can be determined from the characteristic concrete compressive strength using the following equation:
c
3/2ck
c
05.0,ctkRd
f0525.0
f25.0
γ=
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:
( ) 3/1ckcm 8f9500E +⋅= (6)
( ) 3/2ckctm 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
ε fEfk
Efd
εfuε f,l im
ffu
ffd
Es
εpy εsuεsy
σs
εs
αp • fpk
fykfykγs
αp • fpkγs
Ecm
σc
εcεcuεc1
fck
α • fckγc
Fig. 2 Stress-strain diagrams for concrete, reinforcing and prestressing steel as well as FRP Material
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8. Design aspects for FRP systems
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
sykitlim,f E
f5=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
MSd0,g+q
MSk0,g
MSdf,g+q
Fig. 3 Load cases before, during and after strengthening with FRP system
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10. Design procedure
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
In the next step the program determines the strain distribution of the initial strain state. At this point the FRP material is still unstressed. The required cross-sectional area of FRP Af,req is calculated for the additional demand at strengthened state by superposition of the strain profiles. A strain state is established, which leads to an equilibrium of the internal and external forces of the cross-section. Figure 4 shows the superposition of the strains and the internal forces acting on a reinforced and a prestressed or post-tensioned concrete cross-section respectively. Normally prestressed or post-tensioned concrete cross-sections are uncracked in unstrengthened state, unless the prestressing forces are very low. Commonly under additional loads at strengthened state the prestressed or post-tensioned cross-section turns over into a cracked stage.
=+
=+Stahl-beton
Spann-beton
x
x
ε εs s0 s = + ∆ε
ε εc c0 c = + ∆ε
ε εc c0 c = + ∆ε∆εc
∆εsεs0
∆εcεc0
εc,o
εc,u εf
εf εf
εf
ε εp p0 p = + ∆ε∆εpεp0 Fp
FsFf
FsFf
Fc
ac
ds df
dp ds df
ac
Fc
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
steel Fs = Es ⋅ As ⋅ εs ≤ ss
yk Af
⋅γ
(12)
FRP Ff = Ef ⋅ Af ⋅ εf εf ≤ εf,limit (13)
concrete c
ckRc
fxbF
γ⋅α
⋅⋅⋅α= (αR: parabolic form parameter) (14)
ΣH = 0
Fc – Fs – Fp – Ff = 0 (15)
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 < εs ≤ εsu ⇒ MRd0 is determined iteratively
0 < εc ≤ εcu
initial strain state:
MSk0 is known ⇒ εS0 and εC0 are determined iteratively
design:
MSdf is known
0 < εs ≤ εsu
0 < εc ≤ εcu ⇒ Af,req is determined iteratively
0 < εf ≤ εf,limit
strengthened cross-section:
Af,prov is known
0 < εs ≤ εsu
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
εf,lim 0 εf,l im
0 εcu
Fig. 5 Strain di
Strains in servic
The strain distriloads. In additiosteel as well as
10.5 Contr
If the design ansteel stresses istress limits for
concret
reinforc
prestres
Lamellendehnung ausgenutzt
design controlled by laminate strain limit
stribution in ultimate limit state
e state
bution in service state allows to contn 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 thethe rare combination of loads have to
e σc,limit = 0.6 fck
ing steel σs,limit = 0.8 fyk
sing steel σp,limit = 0.75 fpk
21
Betondehnung ausgenutzt
design controlled by concrete strain limit
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
lbd,maxlb
Fb
bond force
bond length
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)
[N/mm²] ff withc
csmcsd 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
tnE58.0l
csd
fffmax,bd
⋅⋅= (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/b1b/b2
06.1kf
fb +
−= [-]
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.
The bond force Fbd related to a bond length lb ≤ lbd,max can be calculated by the following equation:
⎟⎟⎠
⎞⎜⎜⎝
⎛−⋅⋅=
max,bd
b
max,bd
bmax,bdbd l
l2l
lFF (23)
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 ≥
solid slabs: dfbd F2.1
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
As1Af
aLlbd aL
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]).
EaL
ai
xE
lbf
M(x)
As1Af
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
AflbaLxE
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 bbb4
rd,Kffbd ≤⋅−⋅⋅⋅τ⋅⋅= (32)
( ) ]mm[115lfor115l70a
tanh065.02.26abmF bbr4
rd,Kffbd >⎟⎟⎠
⎞⎜⎜⎝
⎛−⋅⎟⎟
⎠
⎞⎜⎜⎝
⎛⋅+⋅⋅τ⋅⋅= (33)
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,Kc2
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[115lformmab0015.0
F000009.0
16.0003.04.0l bd4
rk,Kf
E,fdbd ≤⎟
⎟
⎠
⎞
⎜⎜
⎝
⎛
⋅τ⋅⋅−⎟
⎠⎞⎜
⎝⎛−⎟
⎠⎞⎜
⎝⎛= (36)
( ) ( ) [ ] ]mm[115lformm115tanh065.02.26
abtanh065.0
Fl bd
70a4
rk,Kf70a
E,fdbd
rr>+
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
⋅−
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
⋅τ⋅⋅⋅= (37)
28
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
ixxm
Σ= (38)
standard deviation )xx(1n
1s mi −Σ⋅−
= (39)
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
Test Nr. xi [N/mm²] (xi – xm)²
1 2.0 1.1025 2 2.2 0.7225
05.363.18xm ==
3 3.5 0.2025 4 4.0 0.9025 5 3.1 0.0025
792.0135.316
1s =⋅−
=
6 3.5 0.2025
Total 18.3 3.135 ²]mm/N[40.2792.0823.005.3fcsm =⋅−=
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,Sdfm
LA,SdfsR N
daVF +⎟
⎠
⎞⎜⎝
⎛⋅= (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:
bds
sA,sA,s f
dA4
lF ⋅⎟⎟⎠
⎞⎜⎜⎝
⎛ ⋅⋅= (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.
SdfV1Vη−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
As1Af
internal stirrups
compression chord
tension chords
concrete compression strut
sw
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 SdfV1Vη−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)
33
x
VSdf < VRd3 VSdf > VRd3
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:
fwfreq,w z
Vaσ⋅
∆= (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:
ydsitlimfw fE ≤⋅ε=σ (62)
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
εfuεf
E fk E fd
εlim
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)
36
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)
hlbw
swbwswsw
sw,max = 0,5 h lb = 0,5 lbw
h1
lb
FRP LamelleFRP laminate
Laschenbügelexternal stirrup Schubrißshear crack
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:
zV
5.0f wdwd
∆⋅= per meter for one side of the web (69)
zsV
5.0F wwdwd
⋅∆⋅= 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)
37
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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. Program user interface
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. Input and output windows
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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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
SdfSkf
MM
γ=
m,N
SdfSkf
NN
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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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,min = maximum size of aggregate 32 mm (near surface mounted laminates)
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For each layer the cross-sectional area Af is calculated. The total sum of FRP strips in one layer is calculated from the number of plies lf multiplied by the number of strips nf.
For each layer determine the depth zf of the FRP reinforcement from the top edge of the concrete member. To guarantee the position of the bonded FRP system in the tension zone, keep the limit zf,min and zf,max respectively. The depth of the tension zone corresponds approximately to a fifth of the member height.
The lateral distance ar of FRP strips to the edge of the member is only required for the calculation of the bond force of near surface mounted laminates (s. chap. 11.4). For all types of FRP keep the minimum distance ar,min (s. chap. 13).
tip The program checks if the arrangement of FRP strips fits to the tension face of the member, taking into consideration the lateral distance to the edge. A message will appear if the strengthening does not fit.
note After the determination of the required FRP cross-section the outstanding information can be given to ensure the sufficient moment capacity of the strengthened member (s. chap. 18.14). All 3 FRP layers are taken into account for the determination of the resisting moment in strengthened state.
18.14 Output window design
After the design calculation the result window design will appear in the lower part of the user interface. The chosen cross-sectional area of the FRP system can be compared to the required cross-sectional area and the flexural capacity of the strengthened member is checked.
The required cross-sectional area A f,req is calculated based on the previously entered data. It considers the imposed bending moment Mu of the strengthened state and the depth zf of FRP layer 1 from the top edge of the member.
The provided cross-sectional area A f,prov is displayed. The total cross-sectional area results from the sum of all three layers entered on the window FRP cross-section.
note If you have entered several FRP layers with different depths zf, the bending moment proof can fail in spite of sufficient cross-sectional area of FRP, because the required cross-sectional area only considers the depth of FRP layer 1.
The design capacity of the strengthened section MRdf is determined by an additional iteration taking into account the chosen cross-sectional area of FRP strengthening.
In the proof line the load capacity is compared to the imposed bending moment
tip If the proof condition is met, the result fields will be highlighted in blue and you can access the results of the service limit state and all additional proofs.
On the right side of the window MRd0 describes the design capacity of the unstrengthened member, taking into account the partial safety factors for concrete and reinforcing steel. The determination of the moment capacity considers the design value of the axial force NSdf.
MRk0 is the characteristic capacity of the unstrengthened member. The calculation is based on characteristic strengths of the materials without partial safety factors. The determination of the moment capacity considers the characteristic value of the axial force NSkf.
The degree of strengthening η indicates the ratio between the applied design moment MSdf of the strengthened state and the design resistance MRd0 of the unstrengthened section. According to the German guidelines [2], [4] the degree of strengthening should not exceed 2.0 for externally bonded FRP strips.
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The program indicates the remaining global safety factor θ in case of loss of FRP strengthening. The characteristic capacity of the unstrengthened section MRk0 will be determined without any partial safety factors (γc = γs = 1.0). It is then compared to the imposed characteristic (service) moment MSkf (s. chap. 6.2, 16)
18.15 Output window strains in ultimate limit state
The output window strains in ultimate limit state (ULS) shows the strain diagram as superposition of the initial strain and an additional strain (s. chap.10.2). The strain diagram is displayed true to scale.
The left part of the window shows the strain distribution of the initial state resulting from the initial bending moment MSk0 .
tip Click the graphic and the strain diagram will be scaled up.
The right part of the window shows the strain profile at the ultimate limit state (ULS) considering the required cross-sectional area of FRP strengthening Af,req .
The strain of the extreme compressive fibre εc and the height of the compression zone x are displayed.
εp is the total strain of the prestressing steel including the initial strain due to prestressing.
The strains of the reinforcing steel and the strains of the FRP material are displayed as εs and εf respectively.
note Positive values describe expansion of the material, negative values indicate compression.
Control the design with a hand calculation
The compressive force of the concrete as well as the forces of the steel reinforcement and the CFRP strengthening can be determined using the corresponding strain of each material given by the program. For the cross-sectional area of CFRP you should apply the required cross-sectional area provided by the program in the design window. Then establish the sum of horizontal forces and the sum of bending moments taking into account the imposed bending moment MSdf in strengthened state (s. chap. 10.3).
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18.16 Output window strains / stresses in service state
The output window strain / stresses in service state (SLS) shows the strain diagram and the maximum stresses of the different materials at service state. The strain diagram is displayed true to scale
The left part of the window presents the strain distribution at service limit state (SLS) as a result of the characteristic load considering the provided cross-section of FRP Af,prov. The output of the strain distribution at service state serves particularly to control steel strains, which should not exceed the yielding point. Otherwise the program will give a warning and the selected FRP cross-section must be increased until no yielding of the reinforcement occurs at service state. (s. chap. 10.4).
tip The strain profile is displayed in the same scale as the strains in ultimate limit state. Click the graphic and the strain diagram will be scaled up.
The strain of the extreme compressive fibre εc and the height of the compression zone x are displayed.
εp is the total strain of the prestressing steel including the initial strain due to prestressing.
The strains of the reinforcing steel and the strains of the FRP material are displayed as εs and εf respectively.
The right part of the window shows the corresponding maximum stresses σmax of the different materials at service state.
The maximum stresses are compared with the stress limits σlimit of Eurocode 2. They are according to rare combination of loads. Limit values of other load combinations have to be proofed separately.
note If it is not necessary to check the stress limits (s. chap. 10.5), you can delete them by deselecting the check box consider stress limits.
note Positive values describe expansion of the material, negative values indicate compression.
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18.17 Input window FRP end anchorage
The window FRP end anchorage enables you to carry out the required checking for bond anchorage of the tensile force at the end of the FRP strips.
When strengthening a moment of span you have to choose the appropriate situation between the options end support and interior support. The graphic schematically displays the situation at support and the corresponding moment line.
For the bond check at an end support, enter the distance f between the end of the FRP strips and the front edge of the support. To prevent delamination this distance shall not exceed 50 mm (s. chap. 11.3).
Enter the distance ai between the theoretical support line and the support front.
Additionally the horizontal displacement of the tensile force line aL is given, which normally corresponds to the average effective depth of the internal reinforcement and the external FRP layer. (s. chap.11.3)
note You can modify the proposed value by using the key button in the tool bar.
For externally bonded FRP systems, enter the concrete surface tensile strength fcsm. This value has to be determined from several pull-off tests. The minimum value for prefabricated CFRP laminates is 1.5 N/mm², for carbon sheets 1.0 N/mm². For the calculation of the bond force a maximum of 3.0 N/mm² should be considered (s. chap. 11.5).
Near surface mounted laminates consider the characteristic shear strength of the adhesive τK,k. This value is determined in respect of the specific material properties of the adhesive system. (s. chap. 11.4)
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The bond check will be carried out at the point E as shown in the graphic. For the bond check at an end support the program will calculate the distance xE from point E to the support axis. For the bond check at an interior support the point E corresponds to the moment zero point. Now you have to enter the distance xE, which results from the moment line you have determined prior to the design of the FRP strengthening during the structural analysis (s. chap. 11.3).
For the bond check at an end support enter the corresponding design bending moment MSdf,E of the strengthened state at point E. This value results from the moment line in strengthened state, which has to be determined by structural analysis prior to the FRP design. For the bond check at an interior support the point E is equivalent to the moment zero point and the program automatically sets MSdf,E = 0.
The internal rebars at point E are only relevant for the anchorage at an end support. Specify whether or not the bending reinforcement is curtailed. The program will calculate the forces at point E considering the reduced reinforcement area entered in the window flexural reinforcement at support. Otherwise the program will apply the area of reinforcement at midspan.
tip If you click the button curtailed bending reinforcement, the window flexural reinforcement at support will be opened. You can check the position and the cross-sectional area of reinforcement at support. Use the button back below the graphic to return to the anchorage check.
After you have entered all required values, you can start the bond check either by clicking the proof button on the bottom right of the window or by clicking the tick symbol in the tool bar.
18.18 Output window FRP end anchorage
On the left part of the result window FRP end anchorage the bond forces are compared. The right side of the window gives further information about the bond length and the recommendations of the German guidelines [2] [1]– [4].
At point E the remaining tensile force of the FRP strips is called Ffd,E. It is determined iteratively from the bending moment MSdf,E at point E. For the bond check at an interior support Ffd,E = 0 is applies, since point E is the moment zero point.
Fbd,max indicates the design value of the maximum bond failure force at point E. This value results from the material properties of the provided FRP product and the substrate strength of the concrete and the shear strength of the adhesive respectively. For slabs additionally a reduction factor of 1.2 is taken into account (s. chap. 11.1, 11.2, 11.5).
For the anchorage check of externally bonded FRP systems the bond length lbd,max related to the maximum bond force Fbd,max is applied (s. chap. 11.1, 11.2). This value is calculated for each layer of FRP strengthening depending on the chosen cross-section. The value lbd,max should always be considered as the minimum bond length.
According to the German guidelines [2] – Fehler! Verweisquelle konnte nicht gefunden werden. in some cases is a higher bond length lb should be applied. For the anchorage at a moment zero point a bond length of at least 1 m is recommended (s. chap.11.3).
The values fmax, fmin and f define the distance from the end of the FRP strips to the edge of the support. They are calculated taking into account the appropriate bond length and enable you to determine the position of the FRP strengthening and the required length. For underside strengthening of beams and slabs, the distance f from the support front to the starting point of the FRP strengthening should not exceed 50 mm. (s. chap.11.3)
note The distance f may also be a negative value. In this case the FRP strengthening must be extended beyond the support line. Additionally the field is highlighted in red colour.
Short information terms about the checking are displayed in the two text boxes at the bottom of the window. If the end anchorage check is ok, the background of the bottom lines will be highlighted in blue. A bond failure will be indicated by a red background and further information how to improve the anchorage will be given.
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18.19 Input window anchorage of flexural reinforcement at support
The window anchorage at support enables you to enter the required data, to prove a sufficient anchorage of the total bending reinforcement at support. This proof is only necessary for strengthening of a positive moment (moment of span).
Choose direct or indirect support by clicking the option field. The illustrated graphic schematically shows the selected support condition.
For indirect support the calculation considers the distance ai between the theoretical support line and the support front. This value is copied from the input window FRP end anchorage (s. chap. 0).
For prefabricated laminates at a direct support, additionally choose the type of bonding at the end of the laminates. Slot-in laminate ends must be completely surrounded by adhesive in a narrow slot (s. chap. 12).
The concrete surface tensile strength fcsm is required for the calculation of an externally bonded FRP strengthening. This value is taken from the window FRP end anchorage (s. chap. 0).
Enter the design shear force at support VSdf,A.
Enter the design axial force at support NSdf,A.
To check the given reinforcement at support click the corresponding button at the bottom of the window. The window reinforcement at support will be opened. Use the button back below the graphic to return to the anchorage check.
After you have entered all required values, you can start the anchorage check either by clicking the proof button on the bottom right of the window or by clicking the tick symbol in the tool bar.
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18.20 Output window anchorage of flexural reinforcement at support
The left part of the window anchorage of flexural reinforcement at support shows the different bond forces. On the right part of the window you can enter the bond length for each strengthening layer at support. The anchorage at support is schematically represented in the graphic of the input window.
According to Eurocode 2 a specific part of the bending tensile force has to be anchored at support. This is the total required tension force FA,erf (s. chap. 12).
The part of the tensile force Fs,A already covered by the internal reinforcement is calculated in dependence of the rebar diameter ds and bond length ls,A entered in the window rebars at support (s. chap. 12)
Ff,A,req is the additionally required anchorage force at support. This value is the difference of FA,req and Fs,A and must be covered by FRP.
Below, the provided anchorage force Ff,A,prov is compared to the required FRP reinforcement at support. This value depends on the provided bond length lf,A entered on the right part of the window.
In the check line at the bottom left of the window, the provided and the required anchorage force are compared.
note If the anchorage check is ok, the background of the bottom lines will be highlighted in blue. An anchorage failure will be indicated by a red background.
On the right side of the output window enter the bond length lf,A of each strengthening layer measured from the support front.
For each strengthening layer, the bond force Ffd,A at support is calculated depending on the given bond length. The sum of these forces is displayed on the left side of the window in the field Ff,A,prov.
For information the maximum design bond force Fbd,max as well as the maximum bond length lbd,max are displayed (s. chap. 11.1, 11.2).
Short information terms about the anchorage of FRP are displayed in the right text box at the bottom of the window. If the flexural strengthening can end in front of the support the background of the text box is highlighted in blue.
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18.21 Input window shear – reinforcement and loads
In the window shear – reinforcement and loads you enter the existing internal shear reinforcement as well as the imposed loads at strengthened state.
For beams and T-beams enter the cross-sectional area of the internal stirrups asw per meter.
tip Click the button table and an additional window will appear. Here you can select from a table of rebar diameters the cross-sectional area of double-shear stirrups per meter and you can take over the selected value.
Choose the steel grade of the existing shear reinforcement from the list. The list shows the two grades of reinforcing steel defined in the window steel. For unclassified steel grades the defined yield strength will be displayed. The field next to the list shows the appropriate yield strength fyk.
Specify whether or not the bending reinforcement is curtailed. The program will calculate the reinforcement ratio of the longitudinal internal rebars considering the reduced reinforcement entered in the window flexural reinforcement at support. Otherwise the program will calculate the reinforcement ratio at midspan. The longitudinal reinforcement ratio is needed to evaluate VRd1.
tip If you click the button curtailed bending reinforcement, the window flexural reinforcement at support will be opened. You can check the position and the cross-sectional area of reinforcement at support. Use the button back below the graphic to return to the shear check.
Enter the design shear force VSdf in strengthened state at the point X. According to Eurocode 2 you can evaluate VSdf at the distance d (effective depth of reinforcement) from the front of a direct support on beams with continuously distributed loading.
Determine the axial force NSdf,X at the same position X. An axial compressive force increases the shear capacity of the concrete.
The bending moment MSdf,X is needed to determine the lever z of the internal forces.
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18.22 Input window shear strengthening
In the window shear strengthening the material properties of the strengthening material are defined. You can choose between S&P C-Sheets and steel plates.
Select the shear strengthening material from the list. According to your choice, input fields for the material parameters will appear.
Below the FRP material list the appropriate adhesive is mentioned.
Enter the appropriate modulus of elasticity Es of the structural steel. For C-Sheets 640 the characteristic modulus of elasticity Efk is a fixed value.
Enter the characteristic strength fyk according to the structural steel you want to apply. For C-Sheets 640 the characteristic strength ffk is a fixed value.
For carbon sheets the ultimate strain εfu of the fibres is shown. The ultimate limit strain of the steel is not relevant for the shear design.
For steel plates as additional shear reinforcement the partial safety factor γs is preselected to γs = 1.15. It is the same value as for reinforcing steel which is also used for internal stirrup reinforcement.
For the design of hand lay-up carbon sheets a reduction factor γE for the modulus of elasticity is recommended (s. chap. 14.2). The value is preselected to 1.2. The factor takes into account the softening effects of the hand lay-up on construction site.
According to [1] the design will be carried out with a strain limit εlimit of 0.2 %. (s. chap. 14.2)
After you have entered all required data, you can start the shear check either by clicking the proof button on the bottom right of the window or by clicking the tick symbol in the tool bar.
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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18.23 Output window shear strengthening
The calculation of the shear capacity is derived from the standard method described in Eurocode 2 assuming vertical stirrups and a compression strut inclination of 45°. On the left part of the result window shear, the imposed shear force is compared to the design values of the shear capacity. The right part of the result window shear shows the required additional shear reinforcement and gives the recommendation for the anchorage of the external shear reinforcement.
The recommended maximum shear resistance Vmax shall not exceed 50 % of the shear capacity VRd2 corresponding to the failure of the concrete compression struts (s. chap. 14.1).
The shear resistance provided just by concrete VRd1 is calculated taking into account the design shear strength τRd and the longitudinal reinforcement ratio. In case the imposed shear force is inferior to this value, shear strengthening is not necessary (s. chap. 14.1).
The shear capacity of the section with internal shear reinforcement VRd3 indicates whether external shear strengthening is required or not to cover the whole imposed shear force. If the imposed shear force VSdf does not exceed the shear capacity VRd3 of the unstrengthened cross-section, additional shear reinforcement is only needed to link the flexural strengthening with the internal stirrups. In this case the external shear reinforcement has not to be anchored in the compression zone (s. chap. 14.1).
The proof conditions of the shear check are displayed in the left bottom line
note If the shear check is ok, the background of the bottom line will be highlighted blue. If the shear force exceeds the maximum permissible value the line will change to red.
Choose the thickness tw of the additional external stirrups. If you apply carbon sheets you can choose up to 5 layers.
Decide the width bw of the additional external stirrups. Select a value from the list or enter a value into the field.
The cross-sectional area Aw of one stirrup leg is calculated and shown in the field next to the list.
Enter the spacing sw of the additional external stirrups. The maximum spacing sw,max is displayed below (s. chap. 14.2).
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The provided cross-sectional area aw,prov of the additional shear reinforcement is calculated per meter. As the shear strengthening is usually attached to both sides of the web, the dimensioning of the external strap binder considers two-leg stirrups.
The required cross-sectional area aw,req is compared to the selected shear strengthening (s. chap. 14.2).
The right bottom line of the window gives additional recommendations for the anchorage of the external shear strengthening. If the external stirrups have to be anchored in the compression zone, you can use the height of the compression zone x shown in the output window strains in ultimate limit state as a reference.
note If shear strengthening is necessary, the bottom line will be highlighted in red. If no external shear strengthening is required, it will be highlighted in blue.
18.24 Output window shear strengthening – anchorage of additional external stirrups
If an external shear strengthening is required, the anchorage of the additional stirrups in the compression zone or the adhesive bond on the side of the web has to be proved. The left part of the result window shows the tensile force of the external stirrups, the right part displays the bond force of the adhesive bond. The right part of the output window is only enabled if an adhesive bond anchorage is admissible.
The shear resistance of shear strengthening ∆Vwd is the difference of VSdf and VRd3 of the previous window or a minimum value according to the German guidelines [2] [1]– [4] (s. chap. 14.1).
The calculation considers two-leg stirrups (both sides of the web). The tensile force of one stirrup leg Fwd is shown (s. chap. 14.3).
Additionally the tension force per meter fwd is displayed. This value is related to one side of the web.
The left bottom line of the window gives additional recommendations for the anchorage of the external shear strengthening.
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On the right part of the window the bond length lbw of the additional external stirrups has to be entered as shown in the graphic of the corresponding input window shear strengthening. According to the German guidelines [2] – [4] the bond length should tally with the height of the web. (s. chap. 14.3). The appropriate value is preselected.
note This value can be modified using the key button in the toolbar.
The program calculates the design bond force Fbd of the adhesive bond, applying the properties of the external stirrups and the concrete surface. According to the German guideline [2] the calculation only considers half of the adhesive length lbw (s. chap. 14.3).
The maximum bond force Fbd,max as well as the related bond length lbd,max are displayed (s. chap. 11.1, 11.2).
In the right bottom line the tensile force to be anchored is compared to the bond force. According to the German guideline [2][1] only half of the bond force is allowed to be considered (s. chap. 14.3).
note If the check is ok, the background of the results and the bottom line will be highlighted in blue, otherwise it will change to red.
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19. Program menu and tool bar
19.1 Menu bar items
Menu File
open Open an existing data file or search for a file. The windows dialogue open is displayed. Select the folder where the required data file is located.
save Save the opened data file using the existing file name. If no file is opened the menu save as will be called.
save as Save entered data by indicating a file name, the folder and the drive.
print Print the currently displayed input data and results. If no calculation has been carried out yet, only the pages containing the input data will be printed. The windows dialogue print is called. You have the possibility to select and print each page individually.
end Terminate the program FRP Lamella and close the program window. If you made changes in the opened data file, a corresponding inquiry appears which enables you to save those changes.
last files A list of the last 4 files you have been working on is displayed in the lower part of the menu window. It is possible to call one of these files by a simple mouse click.
Menu Calculation
dimensioning Starts the FRP design, after you have entered all relevant values required for the calculation or after opening an input data file.
strains After you have selected the required FRP cross-section, you can control the strain distributions at ultimate limit state and at service state.
anchorage Once you carried out a flexural strengthening design, you can move to the check of the anchorage with this menu level. When all required data is entered, you can start the calculation using the submenu proof.
shear Once you carried out a flexural strengthening design, you can move to the shear check with this menu level. When all required data is entered, you can start the calculation using the submenu proof.
Menu Extras
company letterhead Displays a window where you can enter and save the name and address of your company. This information will appear in the head line of the program printout. It will still be available at the next program start. The default name and address is S&P Clever Reinforcement Company.
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always on top If you are working on several windows at the same time, you can predefine that FRP Lamella should always appear in the foreground. A tick symbol next to this menu level indicates the current state of this option.
Menu Info
info Indicates the number of the program version as well as the name of the program authors and information about your computer and your operating system.
contact Shows the address of S&P Clever Reinforcement Company, Switzerland.
? Opens the FRP Lamella online help function (not available yet).
19.2 Tool bar symbols
new: Delete the results and clear the input data fields. The program automatically switches to the first window.
open : Open an existing data file or search for a file. The windows dialogue open is displayed. Select the folder where the required data file is located.
save : Save the open data file using the existing file name. If no file is opened the menu save as will be called.
print: Print the currently displayed input data and results. If no calculation has been carried out yet, only the first page with the input data will be printed. The windows dialogue print is called. You have the possibility to select and print each page individually.
unlock: Modify the preselected values. Attention: when you change the preselected values, the calculation is no longer based on the chosen design code and guideline.
calculate: Start the FRP design, after you have entered all relevant values required for the calculation or after opening an input data file.
stop iteration: Stop the running iteration. A dialog box opens to ask again if you really want to stop the calculation.
checkings: Carry out the checkings for anchorage and shear. This button is only enabled after the design of the flexural strengthening.
info: Indicates the number of the program version as well as the name of the program authors and information about your computer and your operating system.
contact: Shows the address of S&P Clever Reinforcement Company, Switzerland
help: Opens the FRP Lamella online help function (not available yet).
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20. Installation instructions
Make sure that you are provided with administrative rights under Windows NT or Windows 2000 and close all current applications. Uninstall older versions of FRP Lamella (Start >> Settings >> Operating system >> Software >> FRP Lamella >> Delete)
Insert the CD in your CD-ROM drive. Click Start, then select Execute.
Enter the following under Open : D:\SETUP\SETUP.EXE (when the letter D corresponds to your CD-ROM drive)
Welcome: Click Next.
Select Installation Folder: Click Next to install FRP Lamella in the indicated folder or Browse to choose or create another folder.
Confirm Installation: Click Next to start the installation.
Installation complete: Click Close.
Installation of the required components
FRP Lamella requires special Microsoft components to be perfectly installed on your Windows operating system. The FRP Lamella installation routine searches for those components in your system. In case these components do not exist you will get an error message and the FRP Lamella installation will abort.
Components:
Microsoft Data Access Component 2.5: Mdac_typ.exe
Database Communication: DCOM95 für Windows 95 DCOM98 für Windows 98
The required data can be found in the components folder of the CD-Rom. It is possible for you to download the Data Access Components SDK Version 2.5 (Mdac_typ.exe) file in any language from http://www.microsoft.com/downloads.
To install these files:
1. Insert the FRP Lamella CD in your CD-ROM drive. Double click Windows-Explorer.
Double click your CD-ROM drive.
Double click the FRP Lamella directory, then the components folder. You will see three files: Dcom95, Dcom98 and Mdac_typ
Double click the required file to execute it:
Once the required components have been installed on your operating system, restart the FRP Lamella installation.
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DESIGN EXAMPLE T-beam according to Eurocode 2
1. Idealised structure and loads
For the T-beam shown in fig. 2 an increase of live loads from 17.5 kN/m to 50 kN/m is planed.
pk,new = 50 kN/m
10 m
g = 35 kN/mk
pk,old = 17.5 kN/m
Fig. 1: Idealised structure and loads
2. Cross-section and materials
50
900
2000
180
720
600
2 16ø
5 28ø
ø 8 / 200
452 28ø C 20/25
S 500
Fig. 2: Cross-section of unstrengthened state
As1 = 3080 [mm²] (bottom) As2 = 1230 [mm²] (bottom) As3 = 402 [mm²] (top) Asw = 502 [mm²] (stirrups) concrete: C 20/25 reinforcing steel: S 500 (swiss) FRP laminate: S&P CFK 150/2000 adhesive: S&P Resin 220 external stirrups: S&P C-Sheet 640 adhesive: S&P Resin 55
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3. Imposed moments
initial strain state (at time of application of the FRP strengthening)
g = 35 kN/mk
10 m
MSk0 = 437.5 kNm Fig. 3: Load and internal moments of initial strain state
During the application of the FRP strengthening only the dead load of the T-beam and the slab have an effect on the cross-section. The resulting bending moment in service state leads to an initial state of strain that has to be considered in the design.
strengthened state
10 m
+
+
1.5 p = 75 kN/m
1.35 g = 47.25 kN/m
+ += 1.35 g + 1.5 p = 122.25 kN/m qd
MSdf = 1528 kNm
MSdf,E = 483 kNm
xE = 0.87 mxE
611 kN 611 kN
Fig. 4: Loads and internal moments of strengthened state
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For the design of the FRP strengthening the bending moments of ultimate limit state and service state can be determined using a structural analysis program or by a hand calculation. Considering the partial safety factors for actions, the maximum design moment amounts to MSdf = 1528 [kNm]. The maximum bending moment in service state results to MSkf = 1062.5 [kNm]. 4. Design
flexural resistance of the unstrengthened cross-section: MRd0 = 1376.5 [kNm] imposed design moment in strengthened state (see above): MSdf = 1528 [kNm] strengthening ratio: η = MSdf / MRd0 = 1528 / 1376.5 = 1.11 [-] remaining global safety in case of loss of the FRP strengthening: θ = MRk0 / MSdf = 1596.9 / 1062.5 = 1.5 [-] required cross-sectional area of FRP strengthening: Af,req = 166 [mm²] 5. Selection of the FRP cross-section
Two prefabricated laminates of type CFK 150/2000 are chosen. number = 2 cross-section: bf / tf = 80 / 1.2 [mm/mm] Af,prov = 192 [mm²] > Af,req = 166 [mm²] MRdf = 1553.9 [kNm] > 1528 [kNm] = MSdf → design ok !
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6. Strain profiles
Ultimate limit state In ultimate limit state the FRP material reaches its strain limit. Due to the high strain of the FRP strengthening and the internal steel reinforcement the ductility of the member is ensured. εf = 7.5 [‰] = εf,limit εs = 7.674 [‰] >> εsy = 2.3 [‰] However, the compression zone is not stressed up to the limit εc = 1.543 [‰] < εcu = 3.5 [‰] service limit state To avoid uncontrollable high deformations in service state, it must be ensured that the internal steel reinforcement does not yield under characteristic loads. εs = 1.584 [‰] > 2.3 [‰] = εsy → strain check ok ! 7. Stresses
If the design and detailing rules given in EC 2 to restrict concrete and steel stresses in service state are not satisfied, the following stress limits for the rare combination of loads are valid: σc,limit = 0.6 · fck = 0.6 · 20 = 12 [N/mm²] σs,limit = 0.8 · fyk = 0.8 · 460 = 368 [N/mm²] σc,max = 6.83 [N/mm²] < 12 [N/mm²] = σc,limit → stress check ok ! σs,max = 316.89 [N/mm²] < 368 [N/mm²] = σs,limit → stress check ok !
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8. FRP end anchorage
geometry of support: f = 50 [mm] distance from end of FRP strengthening to edge of support ai = 150 [mm] distance from support axis to edge of support aL = 408 [mm] horizontal displacement of tensile force line The mimimum value of the substrate strength is assumed. This value must be proved by sampling on site.
fcsm = 1.5 [N/mm²] ²]mm/N[0,15,15,1f
fc
csmcsd ==γ=
The maximum bond force and the required bond length can be derived from the formulas given in the German guideline. The modified coefficients result from the adaption to the partial safety factor concept of EC 2.
[ ]mm3.2570.1
2.116400058.0f
tE58.0l
csd
ffmax,bd =⋅⋅=
⋅⋅=
[ ] [ ]kN86.41N418590.12.11640001179.18025.0ftEkkb5.0F ctdffTbfmax,bd ==⋅⋅⋅⋅⋅⋅⋅=⋅⋅⋅⋅⋅⋅=
179.1400/8021600/802206.1
400/b1b/b2
06.1kf
wfb =
⋅+⋅−⋅=
+−
⋅=
The anchorage check is carried out in point E, which is determined from the geometry of support and the bond length of Fbd. xE = ai + f + lbd,max + aL = 150 + 50 + 257 + 408 = 865 [mm] Considering a distance xE = 0.865 m and a reaction at support of A = 611 kN the bending moment in point E in ultimate limit state amounts to: MSdf,E = 483 [kNm] (from structural analysis) The strain of the FRP strips in point E is determined by iteration of the static equilibrium and the resulting force of the FRP strips is calculated. εf,E = 0.7729 ‰ (determined by iteration) Ffd,E = Ef · Af · εf,E = 164000 · 2 · 80 · 1.2 · 0.0007729 = 24337 [N] = 24.34 [kN] Fbd,max = 41.86 [kN] > 24.34 [kN] = Ffd,E → anchorage check ok !
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curtailed bending reinforcement If the bending reinforcement is curtailed, so that only an area of 30.8 cm² is retained near the support, the resulting tensile forces of the FRP strips are higher. εf = 1.0159 [‰] (determined by iteration) Ffd,E = 164000 · 2 · 80 · 1.2 · 0.0010159 = 31989 [N] = 31.99 [kN] Fbd,max = 41.86 [kN] > 31.99 [kN] = Ffd,E → bond check ok ! 9. Anchorage of flexural reinforcement at the support
According to EC 2 at least 25 % of the bottom reinforcement has to be retained over the support. Additionally, the anchorage of the reinforcement should be capable of resisting the tensile force FsR. The total tensile force at midspan amounts to: Fmax = 1928.4 [kN] (determined by iteration) The proportion that has to be anchored at the supports results to: 25 % Fmax = 25 % · 1928.4 = 482.1 [kN] → decisive The tensile force of the reinforcement is calculated from the design shear and axial force as follows: FsR = VSdf,A · (aL / dm) + NSdf,A = 611 · (408 / 848) + 0 = 293.98 [kN] Ferf,A = 482.1 [kN] The proportion of the tensile force covered by the internal reinforcement is calculated from the surface shell of the rebar and the bond stress fbd. fbd = 2.32 [N/mm²] (bond stress and bond conditions according to EC 2) Due to the high lateral pressure in a direct support, the bond strength is increased by the factor 3/2 according to EC 2. Fs,A = ls,A · (4 · As / ds) · fbd · 3/2 = 40 · (4 · 30.8 / 2.8) · 0.232 · 3/2 = 612.73 [kN] Fs,A = 612.7 [kN] > 482.1 [kN] = Ferf,A → anchorage check ok ! The internal reinforcement ensures a sufficient anchorage at the support. An extension of the FRP strengthening beyond the support line is consequently not required and the FRP strips can be cut-off at most 50 mm in front of the support front.
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10. Shear strengthening design
The design shear force is assumed at a distace dm from the edge of support according to EC 2: xX = ai + dm = 150 + (900 + 850) / 2 = 1025 [mm] VSdf,X = 486.0 [kN] (from structural analysis) MSdf,X = 562.0 [kNm] (from structural analysis) The program uses the exact lever arms of the internal forces, which are iteratively determined from the given moment in point X. zs = ds – ac = 837.8 – 72.7 = 765.1 [mm] zf = df – ac = 900 – 72.7 = 827.3 [mm] zm = 768.2 [mm] dm = 840.2 [mm] The maximum shear resistance amounts to 0.5 · VRd2. This value corresponds to the limit given in the German guideline.
]kN[7.18432.7686005.1
206.05.0zbf5.0V mwcd2Rd =⋅⋅⋅⋅=⋅⋅⋅ν⋅= with 6.0200f
7.0 ck =−=ν
Vmax = 0.5 · VRd2 = 0.5 · 1843.7 = 921.8 [kN] VSdf = 486.0 [kN] < 921.8 [kN] = Vmax → shear check ok ! The part of the shear force transmitted by the concrete alone can be derived from the following equations:
mw1Rd1Rd db)402.1(kV ⋅⋅ρ⋅+⋅⋅τ=
]kN[25.202840600)00855.0402.1(126.0V 1Rd =⋅⋅⋅+⋅⋅=
00858.083760012303080
dbA
sw
sl1 =
⋅+=⋅=ρ
VSdf = 486 [kN] > 202 [kN] = VRd1 → external shear reinforcement required Considering the existing internal stirrups the cross-section can transfer the shear force VRd3, which is the sum of the concrete shear capacity and the resistance of the stirrups.
]kN[92.15376515.1
460503.0zfaV sydswwd =⋅⋅=⋅⋅=
]kN[2.3569.1532.202VVV wd1Rd3Rd =+=+=
VSdf = 486 [kN] > 356 [kN] = VRd3 → anchorage of external stirrups in compression zone required
The shear strengthening is designed for the remaining shear force proportion ∆V, however the value given in the German guideline is assumed as minimum.
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]kN[16.4848611.1
111.1V1V Sdfmin =⋅−=⋅η−η=
∆V = VSdf - VRd3 = 486.0 – 356.2 = 129.8 [kN] > Vmin The strain in the cross-section is limited to εlimit = 0.2 % to ensure an even deformation of the cross-section and to avoid shear offset. The lower stiffness of the sheets due to manual lamination on-site is considered by the reduction factor γE = 1.2 for the modulus of elasticity.
²]mm/N[5333332.1
640000EE
E
fkfd ==
γ=
The stress of the carbon sheets at a strain of 0.2 % amounts to:
²]mm/N[67.1066533333002.0Efditlimfw =⋅=⋅ε=σ
The theoretical fibre section of the required shear strengthening is determined as follows:
[ ] [ ] m/²mm147mm/²mm147.082767.1066
129800z
Vaffw
req,w ==⋅
=⋅σ
∆=
Carbon sheets of 300 mm width are selected (delivery width). To achieve the required cross-section, single-ply FRP stirrups are chosen. Each leg of a stirrup provides the following cross-sectional area: Aw = tf · bf = 1 · 0.19 · 300 = 57 [mm²] For two-leg stirrups with a spacing of sw = 700 mm the resulting cross-sectional area per running meter results to: aw,prov = 2 · Aw · 1 m / sw = 2 · 57 · 1000 / 700 = 163 [mm²/m] The maximum spacing comes to: sw,max = 0.8 · h0 = 0.8 · 900 = 720 [mm] aw,prov = 163 [mm²/m] > aw,req = 147 [mm²/m] → shear design ok ! As the internal shear reinforcement is not sufficient to transfer the imposed shear force, the external FRP stirrups have to be anchored in the concrete compression zone. The tensile force of the stirrups to be anchored on one side of the web per running meter amounts to:
[ ]m/kN44.788273.0
8.1295.0zV5.0f
f
wdwd =⋅=
∆⋅=
The force of each stirrup leg results to:
[ ]kN91.547.044.78sfF wwdwd =⋅=⋅= curtailed bending reinforcement
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If the bending reinforcement is curtailed, so that only an area of 3080 mm² is retained near the support, this must be considered in the calculation of VRd1
00604.0850600
3080db
Asw
sl1 =
⋅=⋅=ρ
Additionally a slight alteration of the internal lever arms will result.
]kN[7.1916.852600)00604.0402.1(126.0V 1Rd =⋅⋅⋅+⋅⋅=
Vwd = asw · fyd · zs = 0.503 · 460/1.15 · 784 = 157.8 [kN]
]kN[5.3498.1577.191VVV wd1Rd3Rd =+=+=
∆V = VSdf - VRd3 = 486 - 349.5 = 136.5 [kN] > Vmin
The theoretical fibre section of the required shear strengthening then amounts to: aw,req = 153 [mm²/m] The same shear strengthening as before is sufficient. aw,prov = 163 [mm²/m] > aw,req = 153 [mm²/m] → shear design ok ! As the internal shear reinforcement is not sufficient to transfer the imposed shear force, the external FRP stirrups have to be anchored in the concrete compression zone. The tensile force of the stirrups to be anchored on one side of the web per running meter amounts to:
[ ]m/kN83.818341.0
51.1365.0zV5.0f
f
wdwd =⋅=
∆⋅=
The force of each stirrup leg results to:
[ ]kN28.577.083.81sfF wwdwd =⋅=⋅=
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DESIGN EXAMPLE two-span slab according to Eurocode 2
1. Idealised structure and loads
For the floor slab shown in fig. 2 an increase of live loads from 3.5 kN/m to 7.5 kN/m is planed.
pk,new = 7.5 kN/m²pk,old = 3.5 kN/m²
g = 5 kN/m²k
5 m 5 m
q pk k k= g + = 12.5 kN/m²
Fig. 1: Idealised structure and loads
2. Cross-section and materials
1000
200
R 513
C 20/25 S 500
R 589
Fig. 2: Cross-section of unstrengthened state
bottom reinforcement: As1 = 513 [mm²] top reinforcement: As2 = 589 [mm²]
concrete: C 20/25 reinforcing steel: S 500 FRP laminate: S&P CFK 150/2000 adhesive: S&P Resin 220
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3. Imposed moments
initial strain state (at time of application of the FRP strengthening)
g = 5 kN/m²k
MSk0,span = 8.8 kNm/m
5 m 5 m
MSk0,support = 13.79 kNm/m
Fig. 3: Load and internal moments of initial strain state
During the application of the FRP strengthening only the dead load of the slab acts on the cross-section. The resulting bending moment in service state leads to an initial state of strain that has to be considered in the design. strengthened state
g = 1.35 5 = 6.75 kN/m²d
MSdf,span = 38.6 kNm/m
5 m 5 m
MSdf,support = 41.4 kNm/m
pd = 1.5 7.5 = 11.25 kN/m²q pd d d= g + = 18 kN/m²
M = 17.52 kNm/mSdf,E
x = 0.54 mExE
+
+
Fig. 4: Loads and internal moments of strengthened state
For the design of the FRP strengthening the bending moments of ultimate limit state and service state can be determined using a structural analysis program or by a hand calculation.
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Considering the partial safety factors for actions, the maximum design moments at midspan and at the middle support amount to: MSdf,span = 38.6 [kNm] MSdf,support = 41.4 [kNm]. The maximum bending moments in service state result to MSkf,span = 24.9 [kNm] MSkf,support = 28.75 [kNm].
4. Design for the moment of span
flexural resistance of the unstrengthened cross-section: MRd0 = 30.9 [kNm] imposed design moment in strengthened state (see above): MSdf = 38.6 [kNm] strengthening ratio: η = MSdf / MRd0 = 38.6 / 30.9 = 1.25 [-] remaining global safety in case of loss of the FRP strengthening: θ = MRk0 / MSkf = 36.0 / 24.9 = 1.45 [-] required cross-sectional area of FRP strengthening: Af,req = 37 [mm²/m] 5. Selection of the bottom FRP strengthening
Prefabricated laminates of type CFK 150/2000 with a spacing of 800 mm are chosen cross-section: bf / tf = 80 / 1.2 [mm/mm] spacing: sf = 800 [mm] smax = 0.2 time span = 0.2 · 5000 = 1000 [mm] smax = 5 time slab thickness = 5 · 200 = 1000 [mm] Af,prov = 75 [mm²/m] > 37 [mm²/m] = Af,req MRdf = 46.9 [kNm ] > 38.6 [kNm] = MSdf → design ok !
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6. Design for the moment of support
flexural resistance of the unstrengthened cross-section: MRd0 = 35.2 [kNm] imposed design moment in strengthened state (see above): MSdf = 41.4 [kNm] strengthening ratio: η = MSdf / MRd0 = 41.4 / 35.2 = 1.18 [-] remaining global safety in case of loss of the FRP strengthening: θ = MRk0 / MSkf = 41.1 / 28.8 = 1.43 [-] required cross-sectional area of FRP strengthening: Af,req = 30 [mm²/m] 7. Selection of the top FRP strengthening
The same type of prefabricated laminates CFK 150/2000 with a spacing of 1 m is chosen. cross-section: bf / tf = 50 / 1.2 [mm/mm] spacing: sf = 1000 [mm] smax = 0.2 time span = 0.2 · 5000 = 1000 [mm] smax = 5 time slab thickness = 5 · 200 = 1000 [mm] Af,prov = 60 [mm²/m] > Af,req = 30 [mm²/m] MRdf = 47.8 [kNm ] > 41.4 [kNm] = MSdf → design ok !
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8. FRP end anchorage
M = 16.8 kNm/mSdf,E
2830780
x = 505E
37.3 kN/m
Fig. 5: Position of the moment zero point and point E
Bottom reinforcement at end support geometry of support: f = 30 [mm] distance from end of FRP strengthening to edge of support ai = 80 [mm] distance from support axis to edge of support aL = 172.4 [mm] horizontal displacement of tensile force line (aL = dm EC 2) In this case the mimimum value of the substrate strength is not sufficient. A greater value is assumed. This value must be proved by sampling on site.
fcsm = 2.0 [N/mm²] ²]mm/N[33.15.10.2ff
ccsm
csd ==γ=
The maximum bond force and the required bond length can be derived from the formulas given in the German guideline. The modified coefficients result from the adaption to the partial safety concept of EC 2.
[ ]mm9.22233.1
2.116400058.0f
tE58.0l
csd
ffmax,bd =⋅⋅=
⋅⋅=
[ ] [ ]kN56.18N1855633.12.11640001391.150800
10002.15.0
ftEkkbn2.15.0F csdffTbffmax,bd
==⋅⋅⋅⋅⋅⋅⋅=
⋅⋅⋅⋅⋅⋅⋅=
391.1400/501800/50206.1
400/b1s/b2
06.1kf
ffb =
+−⋅=
+−
⋅=
The anchorage check is carried out in point E, which is determined from the geometry of support and the bond length of Fbd. xE = ai + f + lbd,max + aL = 80 + 30 + 222.9 + 172.4 = 505.3 [mm]
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Considering a distance xE = 0.505 m the bending moment in point E in ultimate limit state amounts to: MSdf,E = 16.8 [kNm] (from structural analysis) The strain of the FRP strips in point E is determined by iteration of the static equilibrium and the resulting force of the FRP strips is calculated. εf,E = 0.1656 [%] (determined by iteration) Ffd,E = Ef · Af · εf = 164000 · 50 · 1.2 · 0.001656 = 16298 [N] = 16.30 [kN] Fbd,max = 18.56 [kN] > 16.30 [kN] = Ffd,E → anchorage check ok ! Bottom reinforcement at intermediate support At the intermediate support the anchorage length of the bottom laminates is measured from the zero point of the horizontally displaced moment line. According to the German guidelines the distance between the laminates' end and the support front should not exceed 50 mm. geometry of support: xE = 780 [mm] distance of the moment zero point (from structural analysis) ai = 120 [mm] distance from support axis to edge of support aL = 172.4 [mm] horizontal displacement of tensile force line (aL = dm EC 2) lbd,max = 22,29 [cm] design anchorage length (see above) maximum distance between the end of laminate and the support front: fmax = xE - ai - lbd,max- aL = 780 – 12 – 222.9 – 172.4 = 264.7 [mm] according to the guideline: f = 50 [mm] Top reinforcement at intermediate support At the intermediate support the anchorage length of the top laminates is measured from the zero point of the horizontally displaced moment line. According to the German guidelines the minimum anchorage length is 1 m. geometry of support: xE = 2380 [mm] distance of the moment zero point (from structural analysis) ai = 120 [mm] distance from support axis to edge of support aL = 169.5 [mm] horizontal displacement of tensile force line (aL = dm EC 2) minimum distance between the end of laminate and the support front:
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fmin = xE + aL + lbd,max - ai = 2830 + 169.5 + 222.9 - 120 = 3102.4 [mm] according to the guideline: f = xE + aL + lb - ai = 2830 + 169.5 + 1000 - 120 = 3879.5 [mm] 9. Anchorage of flexural reinforcement at the support
Bottom reinforcement at the end support According to EC 2 at least 25 % of the bottom reinforcement has to be retained over the support. Additionally, the anchorage of the reinforcement should be capable of resisting the tensile force FsR. The total tensile force at midspan amounts to: Fmax = 500.68 [kN] (determined by iteration) The proportion that has to be anchored at the supports results to: 25 % Fmax = 25 % · 500.68 = 125.17 [kN] → decisive The tensile force of the reinforcement is calculated from the design shear and axial force as follows: FsR = VSdf,A · (aL / dm) + NSdf,A = 37.8 · (172.4 / 172.4) + 0 = 37.8 [kN] Freq,A = 125.17 [kN] The proportion of the tensile force covered by the internal reinforcement is calculated from the surface shell of the rebar and the bond stress fbd. fbd = 2.32 [N/mm²] (bond stress and bond conditions according to EC 2) Due to the high lateral pressure in a direct support, the bond strength is increased by the factor 3/2 according to EC 2. Fs,A = ls,A · (4 · As / ds) · fbd · 3/2 = 100 · (4 · 513 / 6.5) · 2.32 · 3/2 = 109.91 [kN] The required additional anchorage force, that has to be covered by the FRP reinforcement results to: ∆Freq,A = Freq,A – Fs,A = 123.14 – 109.91 = 15.27 [kN] The FRP strengthening must be anchored beyond the support front. At each position of a laminate one brick is of the supporting wall is taken out. The end of the laminates is assumed as externally bonded. The required anchorage length is determined by trial and error. lf,A ≥ 130 [mm]
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[ ]kN34.152231302
22313056.18
ll
2l
lFF
max,bd
b
max,bd
bmax,bdA,f =⎟
⎠⎞⎜
⎝⎛ −⋅⋅=⎟⎟
⎠
⎞⎜⎜⎝
⎛−⋅⋅=
Ff,A = 15.34 [kN] > 15.27 [kN] = ∆Freq,A → anchorage check ok ! Bottom reinforcement at the intermediate support According to EC 2 at intermediate supports the same conditions as for end supports are valid. The calculation of required anchorage forces is presented above. As the internal bottom reinforcement is continuous, the bars are anchored up to the yield strength with the basic anchorage length lb.
]mm[8.1862/332.2
15.1/46045.6
ff
4d
llbd
ydsbA,s =
⋅⋅=⋅=≥ → ls,A = 200 [mm]
The proportion of the tensile force covered by the internal reinforcement is calculated from the cross-sectional area and the yield strength: Fs,A = As · fyd = 513 · 460/1.15 = 205.2 [kN] Fs,A = 205.2 [kN] > 125.17 [kN] = Freq,A → anchorage check ok ! The internal reinforcement ensures a sufficient anchorage at the support. An extension of the FRP strengthening beyond the support line is consequently not required and the FRP strips can be cut-off at most 50 mm in front of the support front. 10. Shear check
The design shear force is assumed at a distace dm from the edge of support according to EC 2. In this case the shear force near the intermediate support is decisive. xX = ai + dm = 120 + 164 = 284 [mm] VSdf,X = 51.94 [kN] (from structural analysis) MSdf,X = 15.48 [kN] (from structural analysis) The part of the shear force transmitted by the concrete alone results to:
mw1Rd1Rd db)40,.1(kV ⋅⋅ρ⋅+⋅⋅τ=
00368.01601000
589db
Asw
sl1 =
⋅=⋅=ρ [ ] 436.1164.06.1md6.1k m =−=−=
VRd1 = 0.26 · 1.436 · (1.2 + 40 · 0.00368) · 1000 · 164 = 82.54 [kN] VSdf = 51.94 [kN] < 82.54 [kN] = VRd1 → shear check ok ! The shear force does not exceed the limit of EC 2given for slabs without shear reinforcement.
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DESIGN EXAMPLE
Prestressed concrete beam according to Eurocode 2
1. Idealised structure and loads
The superstructure of a three-span footbridge required bottom strengthening at the end spans. The T-beam is prestressed with 5 parabolic tendons. The internal forces from the structural analysis are given.
2. Cross-section and materials
850
3000
770
8505 stranded tendons
C 35/45SSt 1420/1570
175
Fig. 1: Cross-section of unstrengthened state (at midspan)
Ap = 5 · 980 [mm²] = 4900 [mm²]
Ac = 1'098'750 [mm²] Woben = 234'762'516 [mm³] zcg = 309 [mm] Wunten = 134'380'281 [mm³]
concrete: C 35/45 prestressing steel: SSt 1420/1570 FRP laminate: S&P CFK 150/2000 adhesive: S&P Resin 220
3. Internal forces
The bending moments of ultimate limit state and service state are determined using a structural analysis program. The maximum characteristic values of the end span moment are given below:
dead load + additional load (surfacing): Mgk = 1'637.5 [kNm]
live load: Mqk = 1'253.7 [kNm]
prestress (after shrinkage and creep): Mpk = -1'266.8 [kNm]
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4. Prestress
To consider the prestress of the cross-section you have to give the pre-strain of each layer of prestressing steel. The admissible stress results from the admissible prestressing force of each tendon:
σp,max = Fp,zdm / Ap = 954 / 980 = 973.5 [N/mm²]
The loss of friction (ca. 10 %) as well as creep and shrinkage (ca. 15 %) are considered:
σp∞ = 0.9 · 0.85 · σpmax = 0.9 · 0.85 · 973.5 = 744.7 [N/mm²]
Subsequently the pre-strain of the tendons is determined and to be entered in the software:
εp0 = σp∞ / Ep = 744.7 / 190'000 = 0.00392 = 0.392 %
The program calculates the entire prestressing force:
PSk0 = 3'649.52 [kN]
The pre-strain of the tendons already considers the prestressing force and the statically determinated part of the prestressing moment, calculated from the prestressing force and the lever arm of each tendon.
Mp0 = PSk0 · (zp – zcg) = –3'649.52 · (0.77 – 0.3094) = –1'680.8 [kNm]
The secondary moment of prestress, resulting from the statically undeterminated supporting is not included. It always has to be added to the imposed moments.
Mp' = Mpk – Mp0 = –1'266.8 – (–1'680.8) = 414 [kNm]
5. Imposed moments
initial strain state (at time of application of the FRP strengthening)
During the application of the FRP strengthening only the dead loads of the cross-section and the surfacing act on the cross-section. The resulting bending moment in service state leads to an initial state of strain that has to be considered in the design. The secondary moment from prestressing has to be added.
MSk0 = Mgk + Mp' = 1'637.5 + 414 = 2'051.5 [kNm]
As the cross-section is prestressed, the tension zone is assumed to be uncracked.
ultimate limit state
For the design of the strengthening, the partial safety factors for actions according to EC 2 have to be considered. Also the secondary moment from prestressing has to be added.
MSdf = 1.35 · Mgk + 1.5 · Mqk + 1.0 · Mp' = 1.35 · 1'637.5 + 1.5 · 1'253.7 + 1.0 · 414 = 4'505.2 [kNm]
service state
In service state the following load combination is considered:
MSkf = 1.0 · Mgk + 1.0 · Mqk + 1.0 · Mp' = 1'637.5 + 1'253.7 + 414 = 3'305.2 [kNm]
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6. Design
flexural resistance of the unstrengthened cross-section:
MRd0 = 4'320.7 [kNm] MRk0 = 5'045.7 [kNm]
imposed design moment in strengthened state (see above):
MSdf = 4'505.2 [kNm]
strengthening ratio:
η = MSdf / MRd0 = 4'505.2 / 4'320.7 = 1.04 [–]
remaining global safety in case of loss of the FRP strengthening:
θ = MRk0 / MSkf = 5'045.7 / 3'305.2 = 1.53 [–]
required cross-sectional area of FRP strengthening:
Af,req = 244 [mm²]
7. Selection of the FRP strengthening
Two prefabricated laminates of type CFK 150/2000 are chosen.
Af,prov = nf · bf · tf = 2 · 100 · 1.4 = 280 [mm²]
flexural resistance of the strengthened cross-section:
MRdf = 4'533.4 [kNm]
MRdf = 4'533.4 [kNm] > MSdf = 4'505.2 [kNm] → design ok !
8. Service limit state
In service state the following stress limits are given according to EC 2:
σc,limit = 0.6 · fck = 0.6 · 35 = 21 [N/mm²] σp,limit = 0.75 · fpk = 0.75 · 1570 = 1'177.5 [N/mm²]
The software calculates the maximum service stresses of the materials as follows:
σc,max = 13.27 [N/mm²] < σc,limit = 21 [N/mm²] → stress check ok ! σp,max = 957.37 [N/mm²] < σp,limit 1'177.5 [N/mm²] → stress check ok !
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experts for strengthening design
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