Eindhoven University of Technology

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1 Eindhoven University of Technology MASTER Strengthening of reinforced concrete structures with externally bonded carbon fibre reinforcement experimental research on strengthening of structures in multispan or cantilever situations Bukman, L.M. Award date: 3 Link to publication Disclaimer This document contains a student thesis (bachelor's or master's), as authored by a student at Eindhoven University of Technology. Student theses are made available in the TU/e repository upon obtaining the required degree. The grade received is not published on the document as presented in the repository. The required complexity or quality of research of student theses may vary by program, and the required minimum study period may vary in duration. General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. Users may download and print one copy of any publication from the public portal for the purpose of private study or research. You may not further distribute the material or use it for any profit-making activity or commercial gain

2 Strengthening of reinforced concrete structures with externally bonded carbon fibre reinforcement Experimental research on strengthening of structures in multispan or cantilever situations Appendix Linda Bukman A-3.

3 Table of contents Appendix Serviceability limit state Appendix Peeling-off caused at shear cracks Appendix 3 M-κ relation 4 Appendix 4 Material properties single span situation Appendix 5 Critical cross-sections 4 Appendix 6 Mechanisms of failure 7 Appendix 7 Material properties multi span situation 5 Appendix 8 High-speed camera 6 Appendix 9 Data beam to 6 35 Appendix ESPI measurement beam 4 57 i Table of Contents

4 Appendix One of the requirements that have to be met when strengthening a structure with FRP EBR concerns the serviceability limit state. In many of the decisions on the final arrangement of the fibre composite reinforcement, the serviceability limit state appears to be a restricting factor. The following minor calculation makes this statement more conceivable. The stiffness of a concrete structure can be approached by: EI structure.5 E A d s s with: E s A s d modulus of elasticity of steel reinforcement cross-section of steel reinforcement depth of the member When strengthening a structure, material is added to the cross-section. This material has a certain cross-section and stiffness of its own. If FRP EBR is used, the added cross-section and stiffness of the added material can be expressed as A f =αa s and E f =βe s. The stiffness of the structure may now be approached by: EI EI structure structure.5 E A d.5 E A d s s s s +.5 βe αa d ( + αβ ) s s The added FRP EBR cross-section, necessary to reach a certain M s, is considerably less than the steel cross-section necessary to achieve the same strength increase. This implies α is less than. As the stiffness of FRP EBR is generally (slightly) less than that of steel, β is also less than This implies the factor αβ is usually smaller than. Which means that if a beam is strengthened to a certain capacity (M s ), the relative increase in stiffness of the structure using FRP EBR, is considerably less than when steel is used. The increase in stiffness, depending on the factor αβ, is graphically displayed in figure A... The red line represents the original unstrengthened structure (αβ=). If steel is added to strengthen a structure (αβ=), no increase in curvature is found. If, on the other hand, a certain type of FRP EBR is used to strengthen a structure (<αβ<) an increase in curvature is found. Increased curvature causes increased deflections. And increased deflections can cause the serviceability limit state to be the restricting factor in many decisions on the final arrangement of the FRP EBR. Ms Mu M (knm) κ ( ) αβ= αβ=.75 αβ=.5 αβ= αβ=.5 Figure A..: increase in stiffness of a structure, depending on αa and βe κ Appendix Serviceability limit state

5 Appendix Displacements of crack faces, relatively to each other, are considered to initiate debonding for mechanism A (peeling of at shear cracks). The cause and the results of the displacements are examined in this appendix. In figure A.., an uncracked beam is displayed. The moment over the beam, as well as the curvature over the beam are given. The curvature is linear over the beam; of constant value between the supports and a constant increase between the supports and the loading points. ds F F ds Μ ds In figure A.., a cracked beam is displayed. The moment over the beam is linear, like for the uncracked beam. Only the course of the curvature over the beam is different. As the cracks do not offer as much resistance to deformation as the rest of the beam, the curvature concentrates around cracks. For the area between the loading points, where the curvature is of constant value, this implies the crack faces move horizontally from each other. For the area between the loading point and the support, where the curvature increases, this implies the increase in curvature over the distance between the cracks is concentrated is the cracks. This concentrated curvature will cause the crack faces to move vertically as well as horizontally to each other. κ Figure A..: course of curvature over beam F F ds ds ds Μ κ Figure A..: course of curvature over cracked beam For the deriving of mechanism A (peeling-off at shear cracks), it is assumed the displacements of the crack faces cause tensile stress perpendicular to the FRP EBR, which initiates debonding. If a shear crack develops, and the crack faces move relatively from each other, the fibre will have to bridge the distance between the crack faces (figure A..3). This causes a very high strain in the fibre. As the fibre starts debonding, this strain will become less: l ( v + l) + w = ( l + l ) v + w vl + l + = ( l + l As v and w are very small: ) w v α l ( l + l + l) = v l and ( l + l) = v l + l l+ l ε = l l ( l + l) l = l P ε = vl + l l vl + l = l l T Figure A..3: displacement of the crack faces α K Appendix Peeling-off caused at shear cracks

6 As the modulus of elasticity and the cross-section of the fibre remain constant, the strain in the fibre is proportional to the angel α. The equilibrium of forces makes it possible to determine forces P and T from figure A..3. K = E A ε P = cos α K = cosα E A ε T = sin α K = sinα E A ε P T α 9 Figure A..4: force parallel to the FRP EBR α 9 Figure A..5: force perpendicular to the FRP EBR Appendix 3 Peeling-off caused at shear cracks

7 Appendix 3 A prediction of the M-κ relation of the different beams tested in the first set of tests is given in section 4.3 of chapter 4. The prediction for the strengthened cross-section is based on full composite action (figure A.3.). The concrete grade is assumed to be C35 and the yielding strength of the internal reinforcement is assumed to be 5 N/mm. The assumed σ-ε relations of concrete, steel and fibre reinforcement are given in figure A.3.. After the experiments were performed, the actual material properties could be used to adjust the predictions of section 4.3. These adjusted predictions can be found in section 4.4. To be able to quickly draw a M-κ relation for a certain beam, a model was made in MathCAD. An example of a M-κ calculation is given below. This is the first prediction of the M-κ relation. For the calculation of any other cross-section, the properties can be changed and the model will automatically calculate the new M-κ relation. ε'c N's N'b x ds df = h εs Ns fibre ε εf Nf Figure A.3.: full composite action σc (N/mm) f 'c σs (N/mm) f s σf (N/mm) f f ε'c (e-3) εy εs εsu εf εfu Figure A.3.: σ-ε relations of concrete, steel and fibre M-κ relations unstrengthened cross-section Concrete properties When a M-κ relation is calculated during the preparations of the experiments, the concrete strength is estimated, the modulus of elasticity is determined through VBC 995, and the yielding strain is taken equal to the theoretical yielding strain of.75. When a M-κ relation is calculated after the experiments are performed, the actual measured values for the concrete compressive stain, the concrete tensile strain and the modulus of elasticity are used. The yielding strain after the experiments is taken equal to the measured concrete compressive strain divided by the measured modulus of elasticity. Height cross-section h := 45 mm Width cross-section b := mm Concrete compressive cube strength B:= 5 N mm Appendix 3 4 M-κ relation

8 Modulus of elasticity of concrete E' b := 3475 N mm Yielding strain of concrete ε' by :=.75 3 Ultimate strain in concrete ε' bu := Steel reinforcement properties Yield stress of steel reinforcement FeB := 5 N mm Cross-sectional area of long. tensile steel reinforcement A st := 45 mm Concrete cover tensile steel reinforcement c st := 33 mm Diameter tensile steel reinforcement φ st := mm Cross-sectional area of long. compr. steel reinforcement A sc := mm Concrete cover compressive steel reinforcement c sc := 33 mm 4 x Diameter compressive steel reinforcement φ sc := mm x Modulus of elasticity of steel reinforcement E s := Ultimate strain in steel reinforcement ε su :=.35 N mm Derived parameters Cross-sectional area concrete A c := hb A c = 9 4 mm φ st Effective depth tensile reinforcement d st := h c st d st = 4 mm φ sc Effective depth compressive reinf. d sc := c sc + d sc = 39 mm Concrete compr. strength f' b :=.85 B f' b = 4.5 Concrete tensile strength f b := B f b = 3.55 N mm N mm f' b Modulus of elasticity ULS E' bu := ε' by E' bu =.49 4 N mm FeB Yield strain of steel reinforcement ε sy := ε E sy =.5 3 s Appendix 3 5 M-κ relation

9 Crack moment unstrengthened section Force definition: N' bcrack ( x, ε b ) b x E' x := b ε b ( h x) N bcrack ( x, ε b ) N sccrack ( x, ε b ) N stcrack ( x, ε b ) Guess value: := b E' b ε b ( h x) x c sc φ sc := ( h x) ( d st x) := ( h x) E s A st ε b E s A sc ε b f b x := ε b := ε E' b =. 4 b Given N' bcrack x, ε b ( ) N sccrack ( x, ε b ) ( ) + N bcrack x, ε b ( ) N stcrack x, ε b x := Find( x) x = 9.3 mm ε b =. 4 N' bcrack ( x, ε b ) N bcrack ( x, ε b ) N sccrack ( x, ε b ) N stcrack ( x, ε b ) M := N bcrack x, ε b = kn = kn =.775 kn = 7.65 kn ( ) h ( ) N sccrack x, ε b M =.6 7 Nmm x ( h x) x 3 c sc φ sc N stcrack x, ε b x ( ) d st 3 ε b κ := ( h x) κ = Appendix 3 6 M-κ relation

10 Yielding moment unstrengthened section Force definition: ( ) := bif ε' b < ε' bu N' b x, ε' b x ε' 3, b E' b, 4 x f' b N st := A st FeB ( ) := E s A sc N sc x, ε' b ε' b x c sc x φ sc Guess value: x := ε' b :=. Given x ε' b ε d st x sy ( ) N sc ( x, ε' b ) N' b x, ε' b + N st x ε' b N' b x, ε' b ( ) := Find x, ε' b x = mm ( ) =.8 5 ( ) = N sc x, ε' b N st =.6 5 N' b ( x, ε' b ) ( ) N sc x, ε' b N st = 6 kn = 8.6 kn = kn ε' b = x M := N st d st N' 3 b x, ε' b M = Nmm ( ) x 3 c sc φ sc κ := if ε' b <.75 3, κ = ε sy ( d st x) ε' b, x Ultimate moment unstrengthened section Guess value: x:= 5 Given ε' b ε' bu ( ) N sc ( x, ε' b ) N' b x, ε' b + N st Appendix 3 7 M-κ relation

11 x ε' b N' b x, ε' b ( ) := Find x, ε' b x =.739 mm ε' b = ( ) = ( ) = 5.56 N sc x, ε' b N st =.6 5 M 3 := d st N st 4 ( ) N' b x, ε' b M 3 = Nmm 7 8 x N' b ( x, ε' b ) ( ) N sc x, ε' b N st = 6 kn N sc x, ε' b = kn = 5.56 kn ( ) c sc + φ sc ε' b κ 3 := x κ 3 = M-κ relation strengthened cross-section Fibre properties Width of fibre d f := 6 mm Thickness of fibre t f :=. mm Cross-section of fibre A f := d f t f A f = 9 mm Modulus of elasticity of fibre E f := 65 Fibre tensile strength f f := 8 Crack moment strengthened section (before experiments) Force definition: N' bcrack ( x, ε b ) b x E' x := b ε b ( h x) N bcrack ( x, ε b ) N sccrack ( x, ε b ) N stcrack ( x, ε b ) := b E' b ε b ( h x) ( ) ε b E f N fcrack ε b := x c sc φ sc := ( h x) ( d st x) := ( h x) E s A st ε b A f E s A sc ε b N mm N mm Appendix 3 8 M-κ relation

12 Guess value: f b x := ε b := ε E' b =. 4 b Given N' bcrack x, ε b ( ) N sccrack ( x, ε b ) ( ) + N bcrack x, ε b N stcrack ( x, ε b ) N fcrack ( ε b ) x := Find( x) x = 3.73 ε b =. 4 N' bcrack ( x, ε b ) ( ) N bcrack x, ε b ( ) N fcrack ε b N sccrack x, ε b = kn = kn = 3.36 kn ( ) ( ) N stcrack x, ε b M s := N bcrack x, ε b =.8 kn = kn ( ) h ( ) h x ( h x) N stcrack x, ε b ( ) x + N fcrack ε b N 3 sccrack x, ε b M s = Nmm 3 x ( ) d st x 3 c sc φ sc ε b κ s := ( h x) κ s = Yielding moment strengthened section (before experiments) Force definition: ( ) := bif ε' b < ε' bu N' b x, ε' b N st := A st FeB N sc x, ε' b ( ) := E s A sc N f ( x, ε' b ) ( h x) := x ε' b ε' b x ε' 3, b E' b, 4 x f' b x c sc φ sc E f A f Guess value: x := 5 ε' b :=.8 x Appendix 3 9 M-κ relation

13 Given x ε' b ε d st x sy ( ) N sc ( x, ε' b ) N' b x, ε' b ( ) + N st N f x, ε' b x ε' b N' b x, ε' b ( ) := Find x, ε' b x = mm ε' b = ( ) = ( ) =.65 4 N sc x, ε' b N st =.6 5 ( ) = N f x, ε' b N' b ( x, ε' b ) ( ) N sc x, ε' b N st = 6 kn N f ( x, ε' b ) = kn =.65 kn = kn ( ) h x x M s := N st d st + N 3 f x, ε' b N' 3 b x, ε' b M s =.35 8 Nmm ( ) x 3 c sc φ sc ε sy κ s := ( d st x) κ s = Ultimate moment strengthened section (before experiments) Guess value: x:= 5 Given ε' b ε' bu ( ) N sc ( x, ε' b ) N' b x, ε' b ( ) + N f x, ε' b N st x ε' b N' b x, ε' b ( ) := Find x, ε' b x =. mm ( ) = ( ) = N sc x, ε' b N' b ( x, ε' b ) ( ) N sc x, ε' b = kn = kn ε' b = Appendix 3 M-κ relation

14 N st =.6 5 ( ) = N f x, ε' b N st = 6 kn ( ) N f x, ε' b = kn ( ) h 7 M 3s d st 8 x N 7 := st + N f x, ε' b 8 x N sc x, ε' b M 3s = Nmm ( ) 7 8 x c sc φ sc ε' b κ 3s := x κ 3s = Breaking fibre moment strengthened section (before experiments) Guess value: x:= 5 Given N f x, ε' b ( ) 688 ( ) N sc ( x, ε' b ) N' b x, ε' b x ε' b N' b x, ε' b ( ) + N f x, ε' b ( ) N st := Find x, ε' b x = mm ( ) = ( ) = N sc x, ε' b N st =.6 5 ( ) = N f x, ε' b N' b ( x, ε' b ) ( ) N sc x, ε' b N st = 6 kn ( ) N f x, ε' b = kn = kn = 68.8 kn ε' b = ( ) h 7 M fbs d st 8 x N 7 := st + N f x, ε' b 8 x N sc x, ε' b M fbs =.99 8 Nmm ( ) 7 8 x c sc φ sc ε' b κ fbs := x κ fbs =.79 5 Nmm Appendix 3 M-κ relation

15 Appendix 4 To be able to compare analytical and experimental data, material properties of the specimen are obtained. Concrete Three batches of concrete are used to pour the five beams tested in the single span situation. The properties of all three batches are tested. For this purpose, cylinders and cubes are poured from each batch. Only from batch two, no cylinders could be poured, since the amount of concrete was insufficient. An outside party delivered the first two batches of concrete to the laboratory, while the third batch is produced in the laboratory itself. Test results per batch are summarized in table A.4.. At age of tests Batch Age F c F split F t E (days) (kn) (kn) (kn) (N/mm) Table A.4.: test results per batch Batch is used to pour beam and. Batch was supposed to provide concrete for beam 3, 4 and 5. As mentioned, batch supplied an insufficient amount of concreted. Therefore, only beam 5,4 and about /3 of beam 3 could be poured with batch. Batch 3 is used to finish beam 3 and forms the compression zone of beam 3. Since the two concrete specifications for beam 3, make calculations very complicated, it is decided to simply use the specifications of batch for beam 3. Using the mean values of the obtained rsults, an overview of beam properties is given in table A.4.. The value for the mean concrete tensile stress (f bm ) of batch is derived from the splitting force values of batch. Since there aren t enough cubes to perform splitting tests for the second batch, f bm for batch is calculated according to VBC 995. As no cylinders are available, the modulus of elasticity of the beams produced with batch is also calculated according to VBC 995. Age at test f cm f b f bm E b (days) (N/mm ) (N/mm ) (N/mm ) (N/mm ) Beam Beam (preloading) Beam Beam Beam Beam Table A.4.: beam properties Appendix 4 Material properties single span situation

16 Steel For the internal steel reinforcement, steel bars S5 are used. No tests have been performed to verify the yielding stress. The M-κ relations, drawn of the beams before the test are performed, consider the yielding stress of the longitudinal steel reinforcement to be 5 N/mm. After beam was preloaded, it appeared the analytical and the actual yielding strength did not correspond. Both the modulus of elasticity and the yielding strength can be altered to reduce the difference. Correcting the yielding strength is the most transparent option. The corrected value for f y is found to be 55 N/mm. The corrected value is used to draw analytical the M-κ relations of the beams when tested. FRP EBR No tests have been performed to verify the tensile stress of the externally bonded carbon fibre reinforcement. Information provided by the supplier is used: f f = 8 N/mm E f = 65 N/mm Appendix 4 3 Material properties single span situation

17 Appendix 5 To be able to apply the models that describe the mechanisms of failure, a critical section has to be agreed on. In this critical cross-section, the acting forces will be compared to the resisting forces. Since all models were derived from a single span situation, the critical cross-sections in single span structures are derived with the models. A translation to the critical sections in a multi span situation has to be made. Mechanism A; peeling-off caused at shear cracks This model describes a mechanism of failure caused by high shear forces. The shear force in the area in which the FRB EBR is present is therefore restricted to V odu. This implies the cross-section with maximum shear force is the critical cross-section. In single span situations, this section is situated at the end of the FRP EBR. The actual moment in this section of the beam can be determined from the shifted moment line. According to CUR 9, the critical shear force (V dmax ) is located at distance d s from the end of the FRP EBR (figure A.5.). Vd As Af ds Vdmax shifted moment M Figure A.5.: critical section in single span situation ds h In CUR 9, the translation to a multi span situation has been made. It is assumed all loads to a distance d s from the support are directly passed on to the support. In this case, the critical cross-section in a multi span situation is located at a distance d s from the support. According to CUR 9, this is where V dmax is located (figure A.5.). Vd Vdmax? ds Vdmax Af As ds h However, if the actual moment has to be determined from the shifted moment line, the actual shear force at a distance d s from the support is located in the section at the edge of the support (V dmax?). M shifted moment Figure A.5.: critical section at support in multi span situation Appendix 5 4 Critical cross-sections

18 Mechanism B; peeling-off caused by high shear stress This model describes a mechanism of failure caused by high shear stress. The shear stress, in the area in which the FRB EBR is present, is restricted to τ osu. In the model, the magnitude of the shear stress is derived from the shear force. The model only applies for cross-sections in which the internal reinforcement is yielding. This implies the cross-section with the highest shear force, in the area of the beam where the internal reinforcement is yielding, is the critical cross-section. The actual moment can be determined from the shifted moment line. According to CUR 9, for single span structures the critical shear force (V de ) is located at distance d s from the cross-section where the internal reinforcement starts yielding (figure A.5.3). In CUR 9, the translation to a multi span situation has been made. If assumed all loads to a distance d s from the support are directly passed on to the support, the critical crosssection is located at d s from the support. The actual shear force in this section is equal to the shear force at d s from that section, which is at the edge of the support. According to CUR 9, the highest shear force (V de ), in the area of the beam where the internal reinforcement is yielding, is located at the cross-section at the edge of the support (figure A.5.4). Vd shifted moment M ds Vde Figure A.5.3: critical section mechanism B in single span situation shifted moment Vd M ds Vde Figure A.5.4: critical section mechanism B at support in multi span situation Me Me As Af Af As ds ds h h Mechanism C; peeling-off at the end anchorage This model describes a mechanism of failure caused by insufficient anchorage length. This anchorage length (l f ) is the length of the fibre from cross-section x to the end of the fibre. Cross-section x is the point where the total force in the internal reinforcement and the FRP EBR (N r ) is equal to the yielding strength of the internal steel (A s f y ). According to CUR 9, the actual moment is determined from the shifted moment line (figure A.5.5). This single span situation can be translated to a multi span situation. This has not been done in CUR 9. Ny lf(x) Nf(x) derived from shifted moment ds F Ns Nf x Nr=Ns+Nf=Md/zr Figure A.5.5: critical section mechanism C in single span situation As Af Nr ds h Appendix 5 5 Critical cross-sections

19 The theoretical point at which the FRP EBR can be ended is the point where the total force in the internal reinforcement and the FRP EBR (N r ) is equal to the yielding strength of the internal steel (A s f y ). The anchorage length is the length of the fibre from this point to the end of the FRP EBR (figure A.5.6). The actual moment in this cross-section of the beam is assumed to be determined from the shifted moment line. derived from shifted moment Nr=Ns+Nf=Md/zr ds x Nf Ns Figure A.5.6: critical section mechanism C at support in multi span situation Nf lf Ny Af As ds h Mechanism D; end shear failure This model describes a mechanism of failure caused by a plate-end shear crack. The shear force is therefore restricted to V ouu. In CUR 9, it is assumed V ouu is equal to V dmax from mechanism A. In a single span situation, V dmax is located at the end of he FRP. This is the location a plate-end shear crack will occur (figure A.5.). The specific translation to a multi span situation has not been made in CUR 9. If however V dmax from mechanism A is used, as suggested, the critical cross-section will not be located near the end of the FRP in a multi span situation (see figure A.5.). The fact that for ω s the unplated region of the beam should be considered, can also lead to unclear situations for multi span structures, as the reinforcement in the unplated region of the beam could differ from section to section. Appendix 5 6 Critical cross-sections

20 Appendix 6 In order to determine the failure loads of second set of beams, using different fibre lengths, a model is made in MathCAD. This model is based on CUR 9. Even though all material properties and can be altered, the model is derived for the situation as tested in the second set of tests. It is only valid for this situation. Experimental set up Geometry of the beam Figure A.6.: beam and loading scheme Figure A.6.: loading scheme Forces applied to the beam (F) F := N From support to midspan b := 75 mm Cantilever a := 65 mm Moment over beam Figure A.6.3: moment over beam M( x) if x b 5 F a :=, + F x, guess value: x := 6 mm Appendix 6 7 Mechanisms of failure

21 Given M( x) x := Find( x) x = 5 mm b x a + b =.53 Location of M= in percentage of (a+b) Shear force over beam Shear force for x b V( x) := F V ( x ) =. 5 N Properties of the beam Concrete properties Height cross-section h con := 45 mm Width cross-section b con := mm Concrete compressive cube strength B:= 5 N mm Modulus of elasticity of concrete E' b := 5 N mm Yielding strain of concrete ε' by :=.75 3 Ultimate strain in concrete ε' bu := Steel reinforcement properties Yield stress of steel reinforcement FeB := 555 Modulus of elasticity of steel reinforcement E s := Ultimate strain in steel reinforcement ε su :=.35 N mm N mm Cross-sectional area of tensile steel reinforcement at C A st.c := 45 mm Diameter tensile steel reinforcement at C φ st.c := mm Cross-sectional area of tensile steel reinforcement at B A st.b := 63 mm Appendix 6 8 Mechanisms of failure

22 Diameter tensile steel reinforcement at B φ st.b := 6 mm Concrete cover tensile steel reinforcement c st := 33 mm Cross-sectional area of compr. steel reinforcement at C A sc.c := mm Diameter compressive steel reinforcement at C φ sc.c := 8 mm Cross sectional area of compr. steel reinforcement at B A sc.b := mm Diameter compressive steel reinforcement at B φ sc.b := 8 mm Concrete cover compressive steel reinforcement c sc := 33 mm Fibre properties Length of fibre l f := 46 mm Width of fibre b f := 8 mm Thickness of fibre t f :=. mm Cross-section of fibre A f := b f t f A f = 96 mm Modulus of elasticity of fibre E f := 65 Fibre tensile strength f fu := 8 N mm N mm Derived parameters Cross-sectional area concrete A con := h con b con A con = 9 4 mm Effective depth tensile steel reinf. at C d st.c := h con c st d st.c = 4 mm φ st.c Effective depth compressive steel reinf. at C d sc.c := h con c sc d sc.c = 43 mm φ sc.c Effective depth tensile steel reinf. at B d st.b := h con c st d st.b = 49 mm φ st.b Effective depth compressive steel reinf. at B d sc.b := h con c sc d sc.b = 43 mm φ sc.b Effective depth fibre reinforcement d f := h con d f = 45 mm Design value of concrete compr. strength f' b :=.85 B f' b = 4.5 N mm Appendix 6 9 Mechanisms of failure

23 Design value of concrete tensile strength f b := B N f b = 3.55 mm Ultimate tensile force in steel reinforcement N stu.c := FeB A st.c Ultimate tensile force in fibre reinforcement N fu := f fu A f N stu.c =.59 5 N N fu = N Total ultimate tensile force N ru := N stu.c + N fu N ru = N Total cross-sectional area of steel reinf at C A C := A st.c + A sc.c A C = 553 mm A st.c Steel reinforcement ratio at C ω s.c := ω s.c =.55 d st.c b con Total cross-sectional area of steel reinf at B A B := A st.b + A sc.b A B = 74 mm A st.b Steel reinforcement ratio at B ω s.b := ω s.b =.737 d st.b b con A f Fibre reinforcement ratio ω f := A con ω f =.7 E f Equivalent reinforcement ratio ω eq.c := ω s.c + ω f E s ω eq.c = Depth of compression zone x u.c := 3 FeB A st.c f' b b con x u.c = mm 7 Seizing point of compressive force x s.c := 8 x u.c x s.c = 5.33 mm d st.c N stu.c + Effective depth of tensile force d r.c := N ru d r.c = mm Lever arm total tensile and compr. Force z r.c := d r.c x s.c z r.c = mm d f N fu Appendix 6 Mechanisms of failure

24 Mechanism A Peeling-off caused at shear cracks Resisting shear stress τ odrep := ω eq.c Material factor γ m := τ odrep =.343 τ odrep Design value of resisting shear stress τ odu := γ m N mm N τ odu =.343 mm Resisting shear force V odu := τ odu b con d st.c V odu =.4 5 N Unity check: V( x) =.996 V odu Mechanism B Peeling-off caused by high shear forces Design value of bond shear strength f hrep := f b Resisting shear stress τ osrep :=.8 f hrep Material factor γ m := τ osrep Design value of resistng shear stress τ osu := γ m Resisting shear force V osu := τ osu z r.c b f V osu =.6 5 N Unity check: V( x) =.57 V osu Appendix 6 Mechanisms of failure

25 Mechanism C Peeling-off at the end anchorage Shifted moment line 5 F dst.b Mb=5/3 Mc F dst.b Md=Mb=5/3 Mc 5 F A B C D E P(xshifted) x lan Mc dst.c Yielding moment of tensile steel reinforcement at C: ( ) N stu.c z r.c Px shifted := P( ) =.43 8 Nmm Function for non-shifted moment: Q( p) if p b 5 F a, + F p, := Function shifted moment: Mx ( shifted ) if x d st.c x shifted b d st.c F x shifted + F d :=, st.c 5 F a, Guess value: x shifted := Given ( ) Px ( shifted ) Find( x shifted ) Mx shifted x shifted := x shifted =.57 3 mm Available anchorage length l an := l f b + x shifted l an = mm Maximum FRP tensile force that can be anchored k :=.783 k :=.4 b f b con k b := max,.6 k b b =.4 f + 4 f hm := f b N vfmax := k k b b f k E f t f f hm N vfmax = N Appendix 6 Mechanisms of failure

26 Maximum anchorage length k E f t f l vfmax := l f vfmax = mm hm Force in FRP Mx ( shifted ) N f ( x shifted ) := N A st.c E f x shifted s z r.c + A f E f ( ) = N Required anchorage length Ql vf := l vf N vfmax l vfmax + N f x shifted ( ) ( l vf ) N vfmax guess value l vf := mm ( ) ( l vfmax ) Given ( ) Find( l vf ) Ql vf l vf := l vf = 7. mm Unity check: l vf =.99 l an N f ( x shifted ) =.9 N vfmax Mechanism D End shear failure Distance between the end of the FRP and the support L:= b l f L = 5 mm Factor k 3 := 4 Resisting shear stress τ ourep := k 3 f b Material factor γ m := τ ourep =.5 ω s.c 4 L N mm τ ourep Design value of resisting shear force τ ouu := γ m τ ouu =.5 N mm Appendix 6 3 Mechanisms of failure

27 Resisting design shear capacity V ouu := τ ouu b con d st.c V ouu =.83 5 N x shifted := b d st.c x shifted = ( ) Mx shifted = Nmm ( ) Mx shifted Slenderness ratio λ v := d st.c V( x) λ v =.993 Unity check: V( x) =.67 V ouu + L d st.c =.757 λ v Additional application restriction: 4 ω s.c a L := d ω st.c L 3 a L =.73 3 mm s.c b =.75 3 mm Unity check: a L =.99 b Appendix 6 4 Mechanisms of failure

28 Appendix 7 To be able to compare analytical and experimental data, material properties of the specimen are obtained. Concrete Two batches of concrete are used to pour the six beams tested in the multi span situation. The properties of both batches are tested. For this purpose, cylinders and cubes are poured from each batch. Concrete properties per batch are summarized in table A.7.. Batch is used to pour beam, 3 and 5, batch is used to pour beam, 4 and 6. At age of property tests Batch Age F c F split F t E (days) (kn) (kn) (kn) (N/mm ) Table A.7.: concrete properties per batch Using the mean values of the above specifications, an overview of beam properties is given in table A.7.. The values for the mean concrete tensile stress (f bm ) are derived from the splitting force values. Age at test f cm f b f bm E b (days) (N/mm ) (N/mm ) (N/mm ) (N/mm ) Beam Beam (preloading) Beam Beam Beam Beam Beam (preloading) Beam Table A.7.: beam properties Steel For the internal steel reinforcement, steel bars S5 are used. The M-κ relations drawn of the beams before the test are performed, consider the yielding stress of the longitudinal steel reinforcement to be 55 N/mm, as obtained from the single span tests. A test has been performed to verify the actual yielding stress. This appeared to be 5 N/mm. FRP EBR No tests have been performed to verify the tensile stress of the externally bonded carbon fibre reinforcement. Information provided by the supplier is used: f f = 8 N/mm E f = 65 N/mm Appendix 7 5 Material properties multi span situation

29 Appendix 8 8. High-speed images beam In the following figure, some images of the high-speed camera recording of beam are displayed. As not all 3 images can be displayed, only the images around the actual moment of failure are selected. The images are taken every.4 milliseconds. Figure A.8.: image ; beam Figure A.8.: image ; beam Figure A.8.3: image 3; beam Figure A.8.4: image 4; beam Figure A.8.5: image 5; beam Appendix 8 6 High-speed camera

30 Figure A.8.6: image 6; beam Figure A.8.7: image 7; beam Figure A.8.8: image 8; beam Figure A.8.9: image 9; beam Figure A.8.: image ; beam Figure A.8.: image ; beam Appendix 8 7 High-speed camera

31 Figure A.8.: image ; beam Figure A.8.3: image 3; beam Figure A.8.4: image 4; beam Figure A.8.5: image 5; beam Figure A.8.6: image 6; beam Figure A.8.7: image 7; beam Appendix 8 8 High-speed camera

32 8. High speed images beam In the following figure, some images of the high-speed camera recording of beam are displayed. As not all 3 images can be displayed, only the images around the actual moment of failure are selected. The images are taken every.8 milliseconds. Figure A.8.8: image ; beam Figure A.8.9: image ; beam Figure A.8.: image 3; beam Figure A.8.: image 4; beam Figure A.8.: image 5; beam Appendix 8 9 High-speed camera

33 Figure A.8.3: image 6; beam Figure A.8.4: image 7; beam Figure A.8.5: image 8; beam Figure A.8.6: image 9; beam Figure A.8.7: image ; beam Figure A.8.8: image ; beam Appendix 8 3 High-speed camera

34 Figure A.8.9: image ; beam Figure A.8.3: image 3; beam Figure A.8.3: image 4; beam Figure A.8.3: image 5; beam Figure A.8.33: image 6; beam Figure A.8.34: image 7; beam Appendix 8 3 High-speed camera

35 8.3 High-speed images beam 5 In the following figure, some images of the high-speed camera recording of beam are displayed. As not all 3 images can be displayed, only the images around the actual moment of failure are selected. The images are taken every.8 milliseconds. Figure A.8.35: image ; beam 5 Figure A.8.36: image ; beam 5 Figure A.8.37: image 3; beam 5 Figure A.8.38: image 4; beam 5 Figure A.8.39: image 5; beam 5 Appendix 8 3 High-speed camera

36 Figure A.8.4: image 6; beam 5 Figure A.8.4: image 7; beam 5 Figure A.8.4: image 8; beam 5 Figure A.8.43: image 9; beam 5 Figure A.8.44: image ; beam 5 Figure A.8.45: image ; beam 5 Appendix 8 33 High-speed camera

37 Figure A.8.46: image ; beam 5 Figure A.8.47: image 3; beam 5 Figure A.8.48: image 4; beam 5 Figure A.8.49: image 5; beam 5 Figure A.8.49: image 6; beam 5 Appendix 8 34 High-speed camera

38 Appendix 9 9. Data beam In the following figures, all data from beam are displayed. If data are given in relation to a moment, the concerning moment is the moment at midspan x 5 mm Figure A.9.: place of strain gauges on beam 8 6 M-strain beam moment [knm] strain [mm/m] Figure A.9.3: strain against moment at midspan of beam strain [mm/m] strain over fibre; beam place on beam [mm] 6 knm 6 knm 5 knm 4 knm 3 knm knm knm knm 9 knm 8 knm 7 knm 6 knm 5 knm 4 knm 3 knm knm knm knm Figure A.9.: strain over fibre beam Appendix 9 35 Data beam to 6

39 A B C D E Figure A.9.4: place of displacement measurements on beam 8 6 displacements beam A E B D C 4 moment [knm] displacements [mm] Figure A.9.5: displacements of beam displacement [mm] displacements beam A E B C D place on beam [mm] knm knm 3 knm 4 knm 5 knm 6 knm 7 knm 8 knm 9 knm knm knm knm 3 knm 4 knm 5 knm 6 knm 6 knm Figure A.9.6: displacements over beam Appendix 9 36 Data beam to 6

40 Figure A.9.7: place of LVDT s on beam deformation LVDT's beam moment [knm] deformation [mm] Figure A.9.8: deformation of LVDT s on beam (length of LVDT=3 mm) 5 Beam moment [knm] kappa [*e-6 mm-] Figure A.9.9: M-κ relation beam Appendix 9 37 Data beam to 6

41 9. Data beam In the following figures, all data from beam are displayed. If data are given in relation to a moment, the concerning moment is the moment at midspan x 5 mm Figure A.9.: place of strain gauges on beam 8 M-strain beam moment [knm] strain [mm/m] Figure A.9.: strain against moment at midspan of beam strain [mm/m] strain over fibre; beam place on beam [mm] 55 knm 5 knm 4 knm 3 knm knm knm knm 9 knm 8 knm 7 knm 6 knm 5 knm 4 knm 3 knm knm knm knm Figure A.9.: strain over fibre beam Appendix 9 38 Data beam to 6

42 A B C D E Figure A.9.3: place of displacement measurements on beam 8 displacements beam 6 A E B D C 4 moment [knm] displacement [mm] Figure A.9.4: displacements of beam displacements 5 A B C D E knm knm 3 knm 4 knm displacement [mm] knm 6 knm 7 knm 8 knm 9 knm knm knm knm 3 knm - 4 knm 5 knm place on beam [mm] Figure A.9.5: displacements over beam 55 knm Appendix 9 39 Data beam to 6

43 Figure A.9.6: place of LVDT s on beam deformations LVDT's beam moment [knm] deformation [mm] Figure A.9.7: deformation of LVDT s on beam (length of LVDT=3 mm) 5 Beam moment [knm] kappa [*e-6 mm-] Figure A.9.8: M-κ relation beam Appendix 9 4 Data beam to 6

44 9.3 Combinations beam and In the following figures, a combination of the data from beam and are displayed. If data are given in relation to a moment, the concerning moment is the moment at midspan. 8 6 strain at midspan beam and 4 moment [knm] strain [mm/m] Figure A.9.9: M-strain in FRP EBR at midspan beam and 8 displacements at midspan beam and 6 4 moment [knm] displacements at midspan [mm] - -5 Figure A.9.: M-displacement at midspan relation beam and Appendix 9 4 Data beam to 6

45 8 6 4 M-kappa and moment [knm] kappa [*e-6 mm-] Figure A.9.: M-κ at midspan relation beam and displacements beam and 5 A E B C D displacement [mm] kNmbeam 5kNmbeam knm beam knm beam 5 knm beam 5 knm beam 6 knm beam 55 knm beam place on beam [mm] Figure A.9.: displacements over beam and Appendix 9 4 Data beam to 6

46 9.4 Data beam 3 In the following figures, all data from beam 3 are displayed. If data are given in relation to a moment, the concerning moment is the moment at midspan x 6 mm Figure A.9.3: place of strain gauges on beam 3 8 M-strain beam 3 6 moment [knm] strain [mm/m] Figure A.9.4: strain against moment at midspan of beam 3 strain [mm/m] strain over fibre; beam place on beam [mm] 8 knm knm knm knm 9 knm 8 knm 7 knm 6 knm 5 knm 4 knm 3 knm knm knm knm Figure A.9.5: strain over fibre beam 3 Appendix 9 43 Data beam to 6

47 A B C D E Figure A.9.6: place of displacement measurements on beam 8 displacements beam EA DB C moment [knm] displacements [mm] Figure A.9.7: displacements of beam 3 displacements beam 3 displacement [mm] A B C D E knm knm 3 knm 4 knm 5 knm 6 knm 7 knm 8 knm 9 knm knm knm - knm 8 knm place on beam [mm] Figure A.9.8: displacements over beam 3 Appendix 9 44 Data beam to 6

48 Figure A.9.9: place of LVDT s on beam 3 deformations LVDT's beam moment [knm] deformations [mm] Figure A.9.3: deformation of LVDT s on beam 3 (length of LVDT=3 mm) 5 Beam 3 moment [knm] kappa [*e-6 mm-] Figure A.9.3: M-κ relation beam 3 Appendix 9 45 Data beam to 6

49 9.5 Data beam 4 In the following figures, all data from beam 4 are displayed. If data are given in relation to a moment, the concerning moment is the moment at midspan x 6 mm Figure A.9.3: place of strain gauges on beam 4 8 M-strain beam moment [knm] strain [mm/m] Figure A.9.33: strain against moment at midspan of beam 4 4 strain over fibre; beam 4 strain [mm/m] place on beam [mm] 6 knm knm knm knm 9 knm 8 knm 7 knm 6 knm 5 knm 4 knm 3 knm knm knm knm Figure A.9.34: strain over fibre beam 4 Appendix 9 46 Data beam to 6

50 A B C D E Figure A.9.35: place of displacement measurements on beam 8 displacements beam A E B D C moment [knm] displacement [mm] Figure A.9.36: displacements of beam 4 displacements beam 4 5 A B C D E knm knm 3 knm 4 knm displacement [mm] -5-5 knm 6 knm 7 knm 8 knm 9 knm knm -5 knm knm 5 knm - 6 knm place on beam [mm] Figure A.9.37: displacements over beam 4 Appendix 9 47 Data beam to 6

51 Figure A.9.38: place of LVDT s on beam 4 deformations LVDT's beam moment [knm] deformations [mm] Figure A.9.39: deformation of LVDT s on beam 4 (length of LVDT=3 mm) 5 Beam 4 moment [knm] kappa [*e-6 mm-] Figure A.9.4: M-κ relation beam 4 Appendix 9 48 Data beam to 6

52 9.6 Data beam 5 In the following figures, all data from beam 5 are displayed. If data are given in relation to a moment, the concerning moment is the moment at midspan x 3 mm Figure A.9.4: place of strain gauges on beam 5 8 M-strain beam moment [knm] strain [mm/m] Figure A.9.4: strain against moment at midspan of beam 5 4 strain over fibre; beam 5 strain [mm/m] place on beam [mm] 9 knm knm knm knm 9 knm 8 knm 7 knm 6 knm 5 knm 4 knm 3 knm knm knm knm Figure A.9.43: strain over fibre beam 5 Appendix 9 49 Data beam to 6

53 A B C D E Figure A.9.44: place of displacement measurements on beam 8 displacements beam 5 6 moment [knm] 4 8 E A D B C displacement [mm] Figure A.9.45: displacements of beam 5 displacements over beam 5 5 A B C D E knm knm 3 knm 4 knm displacement [mm] knm 6 knm 7 knm 8 knm 9 knm knm knm knm 9 knm place on beam [mm] Figure A.9.46: displacements over beam 5 Appendix 9 5 Data beam to 6

54 Figure A.9.47: place of LVDT s on beam 5 deformations LVDT's beam moment [knm] deformation [mm] Figure A.9.48: deformation of LVDT s on beam 5 (length of LVDT=3 mm) 5 Beam 5 moment [knm] kappa [*e-6 mm-] Figure A.9.49: M-κ relation beam 5 Appendix 9 5 Data beam to 6

55 9.7 Data beam 6 In the following figures, all data from beam 6 are displayed. If data are given in relation to a moment, the concerning moment is the moment at midspan x 3 mm Figure A.9.5: place of strain gauges on beam 6 8 M-strain beam moment [knm] strain [m/m] Figure A.9.5: strain against moment at midspan of beam 6 strain [mm/m] strain over fibre; beam place on beam [mm] 37 knm 3 knm knm knm knm 9 knm 8 knm 7 knm 6 knm 5 knm 4 knm 3 knm knm knm knm Figure A.9.5: strain over fibre beam 6 Appendix 9 5 Data beam to 6

56 A B C D E Figure A.9.53: place of displacement measurements on beam 8 displacements beam EA DB C moment [knm] displacement [mm] Figure A.9.54: displacements of beam 6 displacements beam 6 displacement [mm] A B C D E knm knm 3 knm 4 knm 5 knm 6 knm 7 knm 8 knm 9 knm knm knm knm - 3 knm 36 knm place on beam [mm] Figure A.9.55: displacements over beam 6 Appendix 9 53 Data beam to 6

57 Figure A.9.56: place of LVDT s on beam 5 deformations LVDT's beam moment [knm] deformation [mm] Figure A.9.57: deformation of LVDT s on beam 6 (length of LVDT=3 mm) 5 Beam 6 moment [knm] kappa [*e-6 mm-] Figure A.9.58: M-κ relation beam 6 Appendix 9 54 Data beam to 6

58 9.8 Combinations beam 3 and 6 In the following figures, a combination of the data from beam 3 to 6 is displayed. If data are given in relation to a moment, the concerning moment is the moment at midspan. 8 M-strain beam 3 to End of fibre Midspan moment [knm] = beam 3 = beam 4 = beam 5 = beam strain [m/m] Figure A.9.59: M-strain in FRP EBR at midspan beam 3 to 6 4 strain over fibre; beam 3 to strain [mm/m] = beam 3 = beam 4 = beam 5 = beam place on beam [mm] Figure A.9.6: Strain over fibre beam 3 to 6 Appendix 9 55 Data beam to 6

59 8 displacements midspan beam 3 to moment [knm] = beam 3 = beam 4 = beam 5 = beam displacement [mm] Figure A.9.6: displacements at midspan beam 3 to 6 8 Beam 3 to moment [knm] kappa [*e-6 mm-] Figure A.9.6: M-κ relation at midspan beam 3 to 6 = beam 3 = beam 4 = beam 5 = beam 6 Appendix 9 56 Data beam to 6

60 Appendix One of the cracks around midspan of beam 4 is monitored with ESPI. The concerning crack is marked in figure A.., and displayed in figure A.. and A..3. In this appendix, all data obtained from the measurement can be found. The aim of the ESPI measurement and the verification of the results can be found in Chapter 5, section Figure A..: crack monitored by ESPI Figure A..: monitored surface at kn (just after cracking) Figure A..3: monitored surface at 3 kn (just before failure). Strain in x-direction around monitored crack In the following figures, the stain in the x-direction around the monitored crack is displayed. The figures on the left visualize the surface around the crack after a certain load increase. Reference is taken from a point somewhere in the dark blue area. The displacements over the lines B, G and R are separately displayed in the figures on the right; the letter representing the color of the line (Black, Green and Red). The location of the lines is given in figure A..4. Figure A..4: Strain in x-direction step Figure A..5: Plot strain in x-direction step Figure A..6: Strain in x-direction step Figure A..7: Plot strain in x-direction step Appendix 57 ESPI measurement beam 4

61 Figure A..8: Strain in x-direction step 3 Figure A..9: Plot strain in x-direction step 3 Figure A..: Strain in x-direction step 4 Figure A..: Plot strain in x-direction step 4 Figure A..: Strain in x-direction step 5 Figure A..3: Plot strain in x-direction step 5. Displacements in x-direction around monitored crack In the following figures, the displacements in the x-direction around the monitored crack are displayed. The figures on the left visualize the surface around the crack after a certain load increase. Reference is taken from a point somewhere in the green area. The displacements over the lines B, G and R are separately displayed in the figure on the right; the letter representing the color of the line (Black, Green and Red). The location of the lines is given in figure A..4. Appendix 58 ESPI measurement beam 4

62 Figure A..4: Displacements in x-direction step Figure A..5: Plot displacement sin x-direction step Figure A..6: Displacements in x-direction step Figure A..7: Plot displacements in x-direction step Figure A..8: Displacements in x-direction step 3 Figure A..9: Plot displacements in x-direction step 3 Appendix 59 ESPI measurement beam 4

63 Figure A..: Displacements in x-direction step 4 Figure A..: Plot displacements in x-direction step 4 Figure A..: Displacements in x-direction step 5 Figure A..3: Plot displacements in x-direction step 5.3 Strain in y-direction around monitored crack In the following figures, the strain in the y-direction around the monitored crack is displayed. The figures on the left visualize the surface around the crack after a certain load increase. Reference is taken from a point somewhere in the soft blue area. The displacements over the lines B, G and R are separately displayed in the figures on the; the letter representing the color of the line (Black, Green and Red). The location of the lines is given in figure A..4. Figure A..4: Strain in y-direction step Figure A..5: Plot strain in y-direction step Appendix 6 ESPI measurement beam 4

64 Figure A..6: Strain in y-direction step Figure A..7: Plot strain in y-direction step Figure A..8: Strain in y-direction step 3 Figure A..9: Plot strain in y-direction step 3 Figure A..3: Strain in y-direction step 4 Figure A..3: Plot strain in y-direction step 4 Appendix 6 ESPI measurement beam 4

65 Figure A..3: Strain in y-direction step 5 Figure A..33: Plot strain in y-direction step 5.4 Displacements in y-direction around monitored crack In the following figures, the displacements in the y-direction around the monitored crack are displayed. The figures on the left visualize the surface around the crack after a certain load increase. Reference is taken from a point somewhere in the yellow area. The displacements over the lines B, G and R are separately displayed in the figures on the right; the letter representing the color of the line (Black, Green and Red). The location of the lines is given in figure A..34. Figure A..34: Displacements in y-direction step Figure A..35: Plot displacements in y-direction step Figure A..36: Displacements in y-direction step Figure A..37: Plot displacements in y-direction step Appendix 6 ESPI measurement beam 4

66 Figure A..38: Displacements in y-direction step 3 Figure A..39: Plot displacements in y-direction step 3 Figure A..4: Displacements in y-direction step 4 Figure A..4: Plot displacements in y-direction step 4 Figure A..4: Displacements in y-direction step 5 Figure A..43: Plot displacements in y-direction step 5 Appendix 63 ESPI measurement beam 4