01-08-1 Strengthening or lexure 1 Lat 1 L Sektion 1-1 (Skala :1) be h hw A bw FRP The beam i a part o a lab in a parking garage and need to be trengthened or additional load. Simply upported with L=8.0 m. Ditributed load. Max moment due to ervice load 00 knm and additional moment 40 knm in ultimate limit tate. The load during trengthening can be decreaed to 170 knm. 1
01-08-1 Geometrical Propertie Notatio Value Unit Decription n b = b e = 10 mm Eective lange (EC 5...1) h = 180 mm Heigth on lange h w = 50 mm Heigth on web h= 700 mm Total heigth c= 0 mm Concrete cover b w = 50 mm Width web A c = 599800 mm Cro ectional area concrete A = 15. mm Area teel reinordement Ø t = 0 mm Diameter teel reinorcement d= 0 mm Level arm L= 8000 mm Ditance between upport B = 5000 mm Ditance between beam A w = 157.1 mm Area o tirrup Ø = 10 mm Area hear reinorcement = 50 mm Internal ditance hear reinorcement Partical coeecient actor Concrete Steel FRP c =1.5 =1.15 rp =1. cc =0.85 ct =0.85 φ e =.0 ce =1.
01-08-1 Concrete Step 1: Invetigate exiting tage Characteritic value Characteritic value ck 40 MPa yk 500 MPa ctm.5 MPa E 10 GPa E cm 5 GPa Steel Concrete Deign value Deign value cd. MPa yd 45 MPa ctm.5 MPa E d 18 GPa Steel Calculation in SLS Proportional contant: α y Calculation in (SLS): Invetigate i the beam i cracked or not with conideration to the original loading. Calculate the ditance to the neutral axi. o ce, 1 φe 101 E E d 18.0 E E 5 cm h hw be h bw hw h 1Ad bh bh 1 A w w 180 50 10180 5050 180 18.0 1 15. 0 10180 5050 18.0 115. 18.9mm
01-08-1 Moment o inertia in tage I, I 1, can then be calculated: bh h bh w w hw 1 c 1 0 w w 0 I I I b h y b h y h 1 1 1A d y 0 10180 180 5050 10180 18.9 1 1 50 505018.9 180 18.0 1 15. 0 18.9 10 4.1710 mm The maximum tre in the teel reinorcement and bottom ibre o the concrete beam can then be calculated: M M d y d y I I I 0 0 0 0 c 1 1 0010 10 0 18.94.40MPa.1710 cu M M h y h y I I I 0 0 0 0 c 1 1 0010 10 700 18.94.7MPa.1710 The concrete can aumed being cracked when cu exceed ctm =.50 MPa. Since 4.7 >.50, the concrete i aumed crack and the ection i in tage II. 4
01-08-1 With the aumption that the neutral axi i in the lange the level arm x can then be calculated: bx b A dx x A x Ad A B C and x can then be calculated: B B C 5581 5581 1701541 x 104.5 mm A A A 105 105 105 The aumption that the neutral axi wa in the lange wa correct. We can then calculate the moment o inertia in tage II, I : bx x I Ic I bx 1Ad x 1 10104.5 104.5 10104.5 1 10 4 0. 1 15. 0 104.5 0.9010 mm 5
01-08-1 Step : Calculate the initial train and tree With M 01 =00 knm (ervice load) the tree in concrete and teel reinorcement can be calculated: cö M01 0010 x 10 104.5.MPa I 0.9010 0010 M01 d x 0. 10 0104.554.80 MPa I 0.9010 With M 0 =170 knm during trengthening, the ollowing tree in concrete and teel are calculated: cö M0 17010 x 10 104.5 1.98MPa I 0.9010 17010 M0 d x 0. 10 0 104.51.58 MPa I 0.9010 Correponding train can then alo be calculated: 1.98 0.17 cö cö c0 E e 11.710 1.58 1.18 0 E d 1810 hx 1.08700 104.5 u, M 1.1 0 dx 0 104.5
01-08-1 Step : Calculate trengthening need FRP Characteritic value Deign value k 15 1.50 E k 10 GPa E 1. GPa Check I.C debonding (oten governing):.7 0.41 0.41 4.5 11.10 1.4 cd d, ic ne dt Etimate the area o FRP A M A d d 0.9 yd 4010 0.9 15.44.780 77 Eh 4.510 1.10 700 mm Thi correpond to two 100 x 1.4 mm CFRP laminate. Calculated the height o the compreive zone: A yd d, iceda 15.44.78 4.51.80 x 157.7 mm b 0.81.0.750 cd w and then the moment capacity: MAydd x, d iceda h x 0.8 0.8 15.44.780 157.7 4.5 1. 80 700 157.7 4.kNm Thi exceed the moment capacity aked or 40 knm 7
01-08-1 Step 4: Check i the cro ection i normally reinorced (under reinorced) The ollowing hould be ulilled: bal bal 0.8 0.8 0.05 d, ic u0 4.5 1.1 1 1.5 cu Maximum reinorced ection: A yd A d, iced 15. 44.78 804.51. 0.017 b h 10700.7 e cd and bal Step 5: Calculate the anchor length Calculate the ditance to the lat crack, x cr, where the ection tenile capacity correpond to the cracking moment. On ae ide the bending tine or the concrete only, neglect the teel reinorcement, can be calculated: y 0 h hw be h bw hw h b h b h e w w 180 50 10180 5050 180 15.9mm 101805050 8
01-08-1 Step 5: Calculate the anchor length, cont. be h h bh w w hw c e 0 w w 0 I b h y b h y h 1 1 10180 180 5050 10180 15.9 1 1 50 505015.9 180 1.710 mm 10 4 Calculate the bending tine: W c 10 Ic 1.710 1.0110 mm y 15.9 0 8 Calculate the cracking moment: 8 Mx W 1.01 10.5 51.80 cr cctm knm Calculate the ditance to the lat crack. The beam i placed on ree upport. We can then calculate: Wit h x Mx( x) RAx q Vx( x) RAqx q 5.8.9 kn / m and conequently x = 94. mm R A ql 5.8810 15kN 9
01-08-1 Step 5: Calculate the anchor length, cont. Calculate the diplacement, a l, and the bending moment M xa in ection x a : a 0.45d0.450 97.0mm l and: Mx a 7.kNm 10
01-08-1 Step 5: Calculate the anchor length, cont. Calculate the tenile orce in the FRP that together with the tenile reinorcement can carry the bending moment M xa : F 7. M 0.9h 0.9700 85.0kN x a Ed A d 0010 15. 0 1 1 1. 10 Ed A h 80 700 The orce or yielding in the tenile reinorcement i calculated a: F A 15.44.78 54.4kN yd Calculate the orce in the FRP when the teel yield: Mx a d 7. 0 F F 54.10 8.7kN 0.9h h 0.9700 700 Choe the larget load, i.e. F = 85, kn. Check that the load in the FRP in the tudied ection doe not exceed: F A E, e, x d 100 b b c k 50 b 0.8 1b 100 b c 1 50 G 0.0k 0.01.0 40.5 0.5 Nmm/ mm b ck ctm G 0.5 1.95 1. 10 1.4, x Edt 11
01-08-1 Step 5: Calculate o the anchor length, cont. F, e, xaed1.95801. 7.81kN Which i le then F. Calculate a new bending moment, M,e : M Ed A d 0.9hF 1 Ed A h, e, e 1.10 80 700 0010 15. 0 0.9 700 7.81 10 1 0.7 and: d 0.9 M, e hf, e F h 0 700 0.9 700 7.81 10 54. 10 70.41 Chooe M,e =0.7 knm, and: knm knm qx Mx R,,, 1980. e Ax e x e mm Anchor length: l e E dt 1.10 1.4 1.mm.5 ctm Chooe 50 mm. a x, ele1980.50 170.mm a la FRP R a x,e 1
01-08-1 Step : Calculate hear- and peeling tree Chooe a=100 mm (anchor the laminate a cloe a poible to the upport). The maximum hear tre or a imply upported beam or a ditributed load can then be calculated: q G a al l a max EcdWc 100 1004000 0.1094000 8 4.710.9 0.1MPa 9.1710 1.0010 0.109 Where l = L/ and z 0 = h - x and Gb 1 1 z Ed A Ecd Ac EcdWc a 0 1 1 595.5 8 9.1710 1.0010 4.7 10 100 1. 10 80 9.17 10 599800 0.109 1
01-08-1 Step : Calculate hear- and peeling tree Calculate the normal tre. Shear and normal tree i only aected by additional load, i.e. the load that are added ater trengthening. qq q 5.751.5.5 kn/ m ater beore Support orce: R A ql.58000 10 kn Bending moment at a=100 mm a M ( ) 10 10 100.5 10 100 x x RAaq 1.7kNm Normal tre calculation Mx 1.710 x h y 0 10 700 18.9 0.0 MPa I.1710 1 Failure criteria: 1 ctm x y x y 1 xy 0.0 0.1 0.0 0.1 0.1 0.40 MPa Which i le than ctm =.5MPa 14
01-08-1 Deign or trengthening, calculation tep Step 1: Invetigate exiting tage Step : Calculate the initial train and tree Step : Calculate trengthening need Step 4: Check i the cro ection i normally reinorced (under reinorced) Step 5: Calculate the anchor length Step : Calculate hear and peeling tree 15
01-08-1 Calculation tep Calculation in (SLS) I the ection i cracked Stree and train Exiting train ield Deign or trengthening (ULS) Etimate the area o the FRP Calculated the moment capacity (iteration) Calculate the anchorage length Calculate the peeling tree Check ailure criteria 1