The Design of Fiber Reinforced Polymers for Structural Strengthening An Overview of ACI 440 Guidelines. Sarah Witt Fyfe Company November 7, 2008

Transcription

1 The Design o Fiber Reinored Polymers or Strutural Strengthening An Overview o ACI 440 Guidelines Sarah Witt Fye Company November 7,

2 GUIDE FOR THE DESIGN AND CONSTRUCTION OF EXTERNALLY BONDED FRP SYSTEMS FOR STRENGTHENING CONCRETE STRUCTURES ACI Doument 440.2R-08 Printed July

3 Outline 3 Strengthening Conrete Strutures Reasons or strengthening Types o FRP strengthening systems Materials and properties o FRP strengthening systems Substrate Preparation/FRP Appliation Repair Proper detailing and installation methods Quality ontrol Design Priniples Strengthening limits Flexural strengthening Shear strengthening Axial strengthening Reinorement Details Bond and delamination Detailing o laps and splies Design Examples and Case Studies

4 Reasons or Strengthening Change in use Constrution or design deets Code hanges Seismi retroit Deterioration 4

5 5 Exessive Loading

6 6 Flexural Craking

7 7 Overloading

8 8 Seismi Loads

9 9 Improper Steel Plaement

10 10 Impat Damage

11 Typial FRP Systems or Strengthening Strutures Setion 3.2, Guide: Wet lay-up systems Unidiretional iber sheets Multidiretional iber sheets Mehanially applied iber tows Prepreg systems Unidiretional iber sheets Multidiretional iber sheets Mehanially applied iber tows Preured systems Unidiretional laminates Multidiretional grids Shell elements Other orms not overed 11

12 Typial FRP Systems or Strengthening Strutures 12

13 Typial FRP Systems or Strengthening Strutures 13

14 Typial Fiber Properties Carbon Aramid E-Glass 14

15 Substrate Preparation / Repair Bond vs. Contat Critial Contat Critial Requires intimate ontat between the FRP System and the onrete Coninement o olumns Bond Critial Requires an adhesive bond between the FRP system and the onrete Beam, slab and wall strengthening 15

16 Substrate Preparation / Repair Setion 6.4, Guide: Substrate issues: ACI 503 ICRI psi (1.4 MPa) minimum tensile strength 2500 psi minimum ompressive strength o onrete Removal / replaement o unsound onrete 16

17 Setion 6.4, Guide: Substrate Preparation Minimum ICRI CSP 3 Preparation o onrete surae 17

18 Setion 6.4, Guide: Craks wider than in (0.3 mm) should be injeted prior to appliation o the FRP system. ACI Epoxy Injetion Smaller raks in aggressive environments may require sealing 18

19 Quality Control & Assurane During-onstrution: Bond testing ACI 503R ASTM D4541 Tension adhesion strengths should exeed 200 psi (1.4 MPa), exhibit ailure o the onrete substrate. Cured thikness Extrat small ore samples less than 0.5 in (13 mm) diameter Avoid sampling in high stress areas i possible Repair using overlapping sheets on illed ore. 19

20 Quality Control & Assurane Post-onstrution: General Aeptane Criteria or Delaminations Wet Layup Delaminations less than 2 in 2 (1300 mm 2 ) eah are permissible: No more than 10 delaminations per 10 t 2 o laminate area Total delamination area less than 5% o total laminate area Delaminations less than 25 in 2 (16,000 mm 2 ) may be repaired by resin injetion or ply replaement, depending upon the size, number and loation o delaminations. Delaminations greater than 25 in 2 (16,000 mm 2 ) should be repaired by seletively utting away the aeted sheet and applying an overlapping sheet path o equivalent plies. Preured systems Eah delamination must be inspeted and repaired in aordane with the engineer s diretion 20

21 21 Design Guidelines

22 FRP Strengthening Appliations Flexural Strengthening Beams, Slabs, Walls, et. Shear Strengthening Beams, Columns, Walls, et. Axial Enhanement Column Wrapping, Pressure Vessels 22

23 Strengthening Limits Setion 9.2, Guide: Limited by strength o other strutural omponents Columns, ootings, et. Limited by other ailure mehanisms Punhing shear Loss o FRP should not result in immediate ollapse ( φr ) ( 1.1S S ) n DL LL new existing (9-1) 23

24 Strutural Fire Endurane Glass Transition Temperatures o most FRP systems is typially in the range o o F (60-80 o C) Use o an insulation system an improve the overall ire rating o the strengthened reinored onrete member Insulation system an delay strength degradation o onrete and steel, inreasing the ire rating o the member The ontribution o the FRP system an be onsidered i it is demonstrated that the FRP temperature remains below a ritial temperature 24

25 Rational Fire Endurane Chek ACI 216R: Given over and ire endurane requirement Find the temperature o the steel & onrete Find a redued steel & onrete material strength Find the assoiated redued setion strength Redued strength > Unatored demand No phi ators or load ators 25

26 Rational Fire Endurane Chek Setion 9.2.1, Guide: From ACI 216R - Redue material strengths at elevated temperature: Steel: y y θ Conrete: ' ' θ FRP: 0* u ( R ) ( S + S ) n existing DL LL (9-2) 26

27 Maximum Servie Temperature Setion 1.3.3, Guide: Typial glass transition temperature (T g ) or epoxy o F (60-80 o C) Above T g mehanial properties start to degrade Servie temperature should not exeed T g - 27 F (T g 15 C) 27

28 Flexural Strengthening Chapter 10, Guide Typial lexural strength inreases up to 40% This limit is based on the Guide s requirements Positive and negative moment strengthening Add strength to RC and PC members Redue rak widths Seismi loadings not overed φ M > M (10-1) n u 28

29 Setion , Guide: Assumptions Design alulations are based on atual dimensions and material properties. Plane setions remain plane (inluding FRP). Maximum ompressive strain ε u Tensile strength o onrete is ignored. FRP has linear-elasti relation to ailure. Peret bond between FRP and onrete (no slip). The shear deormation within the adhesive layer is negleted. 29

30 Veriiation o Shear Capaity Setion , Guide: Setion shear apaity must be suiient to handle shear ores assoiated with inreased lexural apaity. 30

31 Setion , Guide: Failure Modes 1. rushing o onrete prior to steel yield 2. yield o steel ollowed by onrete rushing 3. yield o steel ollowed by FRP ailure 4. shear / tension delamination in onrete over 5. FRP debonding rom substrate The desired mode o ailure is usually mode 2 or 3. 31

32 Eetive Strain in FRP Rupture Strain Stress (ksi) Eetive Strain Strain (in/in)

33 Limitation on Strain in FRP To prevent debonding in regions away rom FRP Termination (10-2) US ' ε d ε u ne t ' ε d ε u ne t (10-2) SI ε e ε u h ε bi ε d (10-3) 33

34 Calulation Proedure Determine initial strain in substrate Estimate neutral axis, Determine ailure mode Compute Moment Capaity Calulate material strain Chek servie onditions Calulate stresses and ores Chek Equilibrium (Calulate ) No Estimated or Equilibrium? Yes 34

35 Estimate the Neutral Axis Depth No losed orm solution exists Must ind depth to the neutral axis by trial and error As a starting point, a good rule o thumb is 20% o the eetive setion depth 0. 20d ε s ε e ε bi ε b ε 35

36 Determine Mode o Failure ε e ε u h ε bi ε d (10-3) ε e ε u h ε bi ε d Conrete Crushing Controls ε e ε u h ε bi ε d FRP Rupture Controls 36

37 Conrete Stress Blok Whitney stress blok is valid only when onrete rushing governs ailure (i.e., ε 0.003) β 1 γ' I FRP rupture ontrols, a stress blok appropriate or the onrete strain level should be used Atual Stress Distribution Equivalent Stress Distribution 37

38 Conrete Stress Blok ε < β [( ) ( )] 1 ε ε tan ε ε ( ) ( 2 2 ε ε ln 1+ ε ε ) γ' γ 0. 90ln β ε 1 ( ε ε ) ε β 1 ε E 38

39 Calulation o Flexural Strain Assume strain ompatibility Based on ailure mode Calulate the strain in eah material by similar triangles ε ε s s d ( ε ) e + ε bi d ε (10-10) ε e ε bi ε b 39

40 Calulation O Stress Steel Elasti / Plasti: s E ε s s y (10-11) Stress FRP FRP Elasti: e E s ε e (10-9) Steel Strain 40

41 Chek Fore Equilibrium Sum ores in the horizontal diretion I ores do not equilibrate, revise Repeat previous steps β 1 α 1 ' est A s s + β1α 1 A b e A s s A 41

42 Ultimate Strength Model ε ε s s A n t w ε e ε bi e E ε e M n A s s d β 2 + ψ h β1 2 1 A e (10-13) 42

43 Loss in Dutility ACI 318 : A setion with lower dutility should ompensate with a higher reserve o strength φ ρ b 0.75ρ b 43 φ ( ε ε ) ε 0.65 t sy sy or or or ε sy ε sy ε t < ε < t t ε ε sy Steel Strain at Ultimate (10-5)

44 Design Flexural Strength φm n φ A s s d β 2 + ψa h β1 2 1 e Redution ator or FRP ontribution: ψ φm > M n u (10-13) (10-1) 44

45 Servieability At servie, stress in steel should be limited to 80% o yield strength: M u FRP Strengthened Moment M y M s. 80 Unstrengthened s, s y (10-6) 45 Curvature

46 Prestressed Conrete Members Assumptions Assumptions or onrete members apply strain ompatibility or strain or hange in strain in the prestressing steel prestressing steel rupture mode should be investigated where prestressing steel is draped several setions should be evaluated 46

47 Prestressed Conrete Members Failure Modes 1. Strain level in FRP governed by strain limitations due: 1. onrete rushing 2. FRP rupture 3. FRP debonding 4. Prestressing steel ailure 47

48 Prestressed Conrete Members Strength Redution Fator To maintain a suiient dutility the nominal strain in the prestressing steel should be higher than I this strain is not ahieved a lower strength ator should be used φ ( ε 0.010) ( ) 0.65 ps or or or ε ps < ε ε ps ps < (10-19) 48

49 Prestressed Conrete Members Servieability In servie stress in the prestressing steel should be prevented rom yielding: 82 ps, s 0. py (10-20a) ps, s pu (10-20b) 49

50 Prestressed Conrete Members Nominal Strength The alulation proedure or nominal strength: satisy should strain ompatibility satisy ore equilibrium onsider mode o ailure similar to method or reinored members 50

51 Prestressed Conrete Members Nominal Strength For a given value o the neutral axis, : Stress level in the FRP e E ε e (10-21) Strain in the tendon ε ps P e ε pe ε pnet A E r e (10-22) 51

52 Prestressed Conrete Members Nominal Strength The value o e net depends on the mode o ailure onrete rushing ε pnet d p (10-23a) FRP rupture or ε pnet + ε ( ) debonding ε e bi d d P (10-23b) 52

53 Prestressed Conrete Members Nominal Strength Fore equilibrium an be heked by satisying: A p ps + ' A α β b 1 1 e (10-25) 53

54 Case Study Slab Upgrade P/T lat slab live load inrease: ps 54

55 Case Study Slab Upgrade Positive moment upgrade to olumn strip 55

56 Shear Strengthening Chapter 11, Guide Inrease shear apaity o beams or olumns Amount o inrease depends on setion geometry, existing reinorement, and a variety o additional ators. Change ailure mode to lexural Typially results in a more dutile ailure φv > V n u (11-1) 56

57 Wrapping Shemes Overlap Fully Wrapped U-wrap Two sides bonded 57

58 Eetive Strain in FRP Maximum strain that an be ahieved in the FRP system at the ultimate load stage Governed by the ailure mode o the FRP system and the strengthened member. ε e ε u or ompletely wrapped members (11-6a) ε e κ ε v u or bonded U wraps or ae plies (11-6b) 58

59 Eetive Strain Limitations or FRP Determination o bond-redution oeiient κ v : κ κ v v k 1 k 2 468ε L e u k k2le 11,900ε 1 u (11-7) US (11-7) SI k k 1 1 ' / 3 (11-9) US ' 27 2 / 3 (11-9) SI k 2 d d d 2L d L e e oru wraps or two sides bonded (11-10) 59

60 Eetive Strain Limitations or FRP Determination o ative bond length L e : L e 2500 ( n t E ) (11-8) US L e 23,300 ( n t E ) (11-8) SI L e 60

61 Eetive Strain Limitations or FRP Determination o bond-redution oeiient κ v : κ κ k v v 2 k 1 k 2 468ε L e u k k2le 11,900ε d d 1 d 2L d L e e u (11-7) US (11-7) SI oru wraps (11-10) or two sides bonded k k 1 1 ε ' / 3 (11-9) US e ' 27 2 / 3 κvε u (11-9) SI

62 Pertinent Shear Dimensions d s s βα w w V A v A v 2nt e w ( sin α + os α) s d (11-3) (11-4) e ε e E (11-5) 62

63 Design Shear Capaity n ( ) V V ψ V φ V φ + + s (11-2) φv n φ ( V + V + ψv ) s φ 0.85 ( ACI 318) ψ 0.95 ( ully wrapped) ψ 0.85 ( bonded U wraps or ae plies) 63

64 Spaing, Reinoring Limits s w +,max d 4 Setion 11.1, Guide: Based on ACI , Setion : V s + V 8 ' b w d (11-11) US Vs + V 0.66 bd (11-11) SI 64

65 Case Study Preast Garage Installed FRP U Wraps 65

66 Coninement Chapter 12, Guide Inrease in member axial ompressive strength Enhane the dutility o members subjeted to ombined axial and bending ores Inrease the strength o members subjeted to ombined axial and bending ores 66

67 Axial Compression Fibers oriented transverse to the longitudinal axis o the member Contribution o any longitudinal ibers to axial strength is negligible Results in an inrease in the apparent strength o the onrete and in the maximum usable ompressive strain in the onrete Passive oninement Intimate ontat between FRP system and member is ritial 67

68 Coninement Conining Pressure

69 FRP Conined Conrete Behavior Stress FRP Conined Conrete ε L ε T 0.85 Unonined Conrete Transverse Strain ε u ε e ε ε u ε u Longitudinal Strain Transverse Strain (Dilation) 69 Longitudinal Strain

70 FRP Conined Conrete Strain Limitation For pure axial loading: ε κ (12-5) e εε u ε κε 0.55 For ombined axial + bending: κ e εε u Reommended value (aounts or premature ailure strain o FRP) (12-12) Transverse Strain (Dilation) 70 Limit to maintain shear integrity o onrete Longitudinal Strain

71 FRP Coninement Model Stress E E 2 Unonined Conrete FRP Conined Conrete Eε + E2ε ( E E ) 4 Where, E ε or 0 ε u or ε ε ε ε ε t t u (12-2a) (12-2b) ε ε ε t u Strain ε t 2 E E 2 (12-2) Transverse Strain (Dilation) 71 Longitudinal Strain

72 FRP Coninement Model Stress E 2 FRP Conined Conrete + ψ 3. 3κ a l (12-3) Unonined Conrete ε u ε κ b ε l ε e 0.45 (12-6) E ε ε ε t u Strain Where, l is the onining pressure exerted by the FRP jaket κ a and κ b are shape ators Transverse Strain (Dilation) 72 Longitudinal Strain

73 Cirular Setions FRP Jaket E ε e l l Conrete E ε e Conining pressure: l 2E Shape ators: κ a κ b nt D ε 1.0 e (12-4) 73 l

74 Retangular Setions Equivalent irular olumn D 2 2 D b + h Conining pressure: b l 2E nt D ε e (12-4) h 74

75 Retangular Setions Shape ators: Ae b κ a A h Eetive oninement b 0. 5 area, A e Ae h κ b A b 2 (12-9) (12-10) h Conining stress onentrated at orners 75

76 76 Retangular Setions Ratio o eetive oninement area to total area o onrete ( ) ( ) g g g e A r b b h r h h b A A ρ ρ (12-11)

77 Using the Coninement Model Compressive Strength: with existing steel spiral reinoring n [ ( A A ) A ] ' 0.85 φp 0.85φ + with existing steel-tie reinoring: n g [ ( A A ) A ] ' 0.85 φp 0.80φ + g st st y y st st (12-1a) (12-1b) Use the onined onrete ompressive strength in ACI 318 equations 77

78 Servieability Considerations- Setion , Guide: Axial Compression To ensure radial raking will not our under servie loads, 0.65 ' To avoid plasti deormation under sustained or yli loads, s y 78

79 Reinorement Details Chapter 13, Guide General Guidelines: Do not turn inside orners; Provide a minimum 1/2 in. (13 mm) radius when the sheet is wrapped around outside orners Provide adequate development length Provide suiient overlap when spliing FRP plies. 79

80 Allowable Termination Points Setion , Guide Simply Supported Beams Plies should extend a distane equal at least to l d past the point along the span orresponding to the raking moment, M r, I V u > 0.67V at the termination point the FRP laminate should be anhored with transverse ( lamping ) reinorement 80

81 Bond and Delamination Transverse ( lamping ) reinoement Area o transverse ( lamping ) FRP U-Wrap reinorement to prevent onrete over layer rom splitting: A anhor ( A ) u longitudinal ( E ) κ ε v u anhor (13-1) 81

82 Development Length The bond apaity o FRP is developed over a ritial length: l d ne t ' in in.-lb units (13-2) l d ne t ' in SI units 82

83 Detailing o NSM bars groove dimensions shall be at least 1.5 times the diameter o the bar For a retangular bar the minimum groove size shall be 3a b x 1.5b b 83

84 Development Length o NSM bars Development length o NSM bar: l db db 4 τ d ( 0.5 ) max or irular bars (13-3) l db abdb 2 τ d ( a + b )( 0.5 ) b b max or retangular bars (13-4) 84

85 QUESTIONS? Thank You 85

86 Design Example Flexural Strengthening o Interior Beam 86

87 Design Example: Flexural Strengthening o an Interior Beam x 23-0 FRP plies ELEVATION DL,wLL w 5000 psi 3-#9 bars y60 ksi φ Mn266 k-t (w/o FRP) FRP SECTION Manuaturer s reported FRP-system properties Thikness per ply, in mm Ultimate tensile strength 90 ksi 0.62 kn/mm 2 Rupture strain, Modulus o elastiity o FRP laminates, 5360 ksi 37 kn/mm 2 87

88 Design Example: Flexural Strengthening o an Interior Beam 88 Loadings and orresponding moments Loading/Moment Existing loads Antiipated loads Dead loads, wdl 1.00 k/t 14 N/mm 1.00 k/t 14 N/mm Live load, wll 1.20 k/t 17 N/mm 1.80 k/t 26 N/mm Unatored loads, 2.20 k/t 32.1 N/mm 2.80 k/t 40.9 N/mm (wdl + wll) Unstrengthened load limit (1.1wDL +0.75wLL) Fatored loads, (1.2wDL +1.6wLL) n/a n/a 2.45 k/t 34.9 N/mm 3.12 k/t 50.2 N/mm 4.46 k/t 65.1 N/mm Dead-load moment, MDL 72 k-t 96.2 kn-m 72 k-t 96.2 kn-m Live-load moment, MLL 86 k-t kn-m 130 k-t kn-m Servie-load moment, Ms 158 k-t kn-m 202 k-t kn-m Unstrengthened moment limit (1.1MDL +0.75MLL) n/a n/a 177 k-t 240 kn-m Fatored moment, Mu 224 k-t kn-m k-t kn-m Two, 12 in. wide by 23 t. long plies are to be bonded to the soit o the beam using the wet-lay-up tehnique.

89 Design Example: Flexural Strengthening o an Interior Beam Step 1 - Compute the FRP-system design material properties For an interior beam, an environmental-redution ator (C E ) o 0.95 is suggested. C u E * u u (0.95)(90 ksi) 85ksi ε u C ε E * u ε u (0.95)(0.015in./in.) in./in. 89

90 Design Example: Flexural Strengthening o an Interior Beam Step 2 - Preliminary alulations Properties o the onrete: β 1 rom ACI , Setion ' β E 57, psi 4,030,000psi 90

91 Design Example: Flexural Strengthening o an Interior Beam Step 2 - Preliminary alulations Properties o existing reinoring steel: A s 2 3(1.00 in. ) 3.00 in. 2 ρ s A s bd ρ s in. 12in. ( )( 21.5in. )

92 Design Example: Flexural Strengthening o an Interior Beam Step 2 - Preliminary alulations Properties o the externally bonded FRP reinorement: A nt w A 2 ( 2 plies)( in. )( 12 in. ) 0.96in. ply ρ A bd ρ in. 12 in. ( )( 21.5 in. )

93 Design Example: Flexural Strengthening o an Interior Beam Step 3 - Determine the existing state o the strain on the soit The existing state o strain is alulated assuming the beam is raked and the only loads ating on the beam at the time o the FRP installation are dead loads. A raked setion analysis o the existing beam gives k0.334 and I r 5937 in. 4 ε bi M ( h kd) DL I r E ε ε bi bi ( 864 k in. )[ 24in. ( 0.334)( 21.5in. )] 4 ( 5,937 in. )( 4,030 ksi)

94 Design Example: Flexural Strengthening o an Interior Beam Step 4 Determine the design strain o the FRP System 5000 psi ε d ε 2 ( )( ) u psi 0.04in ε d (0.0142)

95 Design Example: Flexural Strengthening o an Interior Beam Step 5 - Estimate, the depth to the neutral axis A reasonable initial estimate o is 0.20d. The value o is adjusted ater heking equilibrium d ( 0.20 )( 21.5 in. ) 4.30in. 95

96 Design Example: Flexural Strengthening o an Interior Beam Step 6 - Determine the eetive level o strain in the FRP reinorement ε e d ε bi ε d ε e ε e ε e

97 Design Example: Flexural Strengthening o an Interior Beam Sine FRP ontrols the setion ailure, the onrete strain is less than 0.003: ε ( ε ) + ε e bi d ε ( )

98 Design Example: Flexural Strengthening o an Interior Beam Step 7 - Calulate the strain in the existing reinoring steel ε s ( ε + ) d e ε bi d ε s ( )

99 Design Example: Flexural Strengthening o an Interior Beam Step 8 - Calulate the stress level in the reinoring steel and FRP s E s ε s y s s (29,000 ksi)(0.0084) 348 ksi 60 ksi 60 ksi e E ε e e ( 5,360 ksi)( 0.009) 48.2 ksi 99

100 Design Example: Flexural Strengthening o an Interior Beam Step 9a - Calulate the internal ore resultants Approximate stress blok ators may be alulated using the paraboli stress-strain relationship o onrete as ollows: ε ' 1.7 E ' 1.7 (5,000) 6 4, β ' 4ε ε ' 6ε 2ε 4(0.0021) (0.0021) 2(0.0021) α ' 2 3ε ε ε '2 3γε 3(0.0021)(0.0021) 2 3(0.749)(0.0021)

101 Design Example: Flexural Strengthening o an Interior Beam Step 9b Chek equilibrium Fore equilibrium is veriied by heking the initial estimate o the neutral axis, A s s α 1 + ' A A β b 1 e (3)(60) + (0.96)(48.2) (0.886)(5)(0.749)(12) 5.87in 5.87in 4. 30in NG 101

102 Design Example: Flexural Strengthening o an Interior Beam Step 10 Iterate on until ore equilibrium is satisied ε s 5.17in s y ε d ε e ksi 49.8 ksi β α ( 3.00in. )( 60ksi) + ( 0.96in. )( 48.2ksi) ( 0.928)( 5ksi)( 0.786)( 12in. ) 5.17 in. OK. The value o seleted or the inal iteration is orret. 102

103 Design Example: Flexural Strengthening o an Interior Beam Step 11 Calulate reinorement and FRP ontribution to strength M ns β As s d (5.17) (3.00)(60) ,504k in 292k 2 2 t M s β (5.17) A e d (0.96)(48.2) 24 1,017k in 85k 2 2 t 103

104 Design Example: Flexural Strengthening o an Interior Beam Step 11 Calulate design lexural strength o the setion The lexural strength is alulated using the redution ator. Sine ε s >0.005, the value o Ф is 0.9 φ M n [ M + ψ M ] 0.9( (85) ) k t φ 327 ns n φ M n 327k t M 294 k u t The strengthened setion is apable o sustaining the new required moment strength 104

105 Design Example: Flexural Strengthening o an Interior Beam Step 12 Chek servie stresses in the reinoring steel and the FRP Calulate the elasti depth to the raked neutral axis by summing the irst moment o the areas o the transormed setion. 105

106 106 Step 13 Chek servie stresses in the reinoring steel and the FRP s s s s s s E E E E d h E E E E E E E E k ρ ρ ρ ρ ρ ρ 2 2 k ( )( ) in in kd Design Example: Flexural Strengthening o an Interior Beam

107 Design Example: Flexural Strengthening o an Interior Beam Step 13 Chek servie stresses in the reinoring steel and the FRP Calulate the stress level in the reinoring steel: kd M s + ε bi A E h ( d kd ) Es 3 s, s 0. 8 kd kd A ( ) ( ) ses d d kd + A E d d kd 3 3 y s ( 0.80)( 60 ksi) 48 ksi OK, 40.4 ksi s The stress level in the reinoring steel is within the reommended limit 107

108 Design Example: Flexural Strengthening o an Interior Beam Step 14 Chek reep rupture limit t servie or the FRP Calulate the stress level in the FRP: E d kd, s s, s bie E s d kd ε u s, 5.60 ksi (0.55)(85) 50ksi OK 108

109 Design Example: Flexural Strengthening o an Interior Beam Step 14 Detailing Requirements Detail the FRP reinorement as ollow: 1. Chek that shear ore at termination is less than shear ore that auses end-peeling (estimate as 2/3 o onrete shear strength). 2. Terminate FRP at ld (per Eq. 12.2) past raking moment. a) I shear ore is higher extend FRP beyond and/or use FRP U-wraps. 109