Calculation Example. Strengthening for flexure

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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.

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

2 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 = 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 = 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.

3 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 mm

4 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 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 c MPa.1710 cu M M h y h y I I I c MPa.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

5 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 x mm A A A 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 mm 5

6 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 M x MPa I M01 d x MPa I With M 0 =170 knm during trengthening, the ollowing tree in concrete and teel are calculated: cö M x MPa I M0 d x MPa I Correponding train can then alo be calculated: cö cö c0 E e E d 1810 hx u, M dx

7 Step : Calculate trengthening need FRP Characteritic value Deign value k E k 10 GPa E 1. GPa Check I.C debonding (oten governing): cd d, ic ne dt Etimate the area o FRP A M A d d 0.9 yd Eh mm Thi correpond to two 100 x 1.4 mm CFRP laminate. Calculated the height o the compreive zone: A yd d, iceda x mm b cd w and then the moment capacity: MAydd x, d iceda h x kNm Thi exceed the moment capacity aked or 40 knm 7

8 Step 4: Check i the cro ection i normally reinorced (under reinorced) The ollowing hould be ulilled: bal bal d, ic u cu Maximum reinorced ection: A yd A d, iced b h 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 mm

9 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 mm 10 4 Calculate the bending tine: W c 10 Ic mm y Calculate the cracking moment: 8 Mx W 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 kn / m and conequently x = 94. mm R A ql kN 9

10 Step 5: Calculate the anchor length, cont. Calculate the diplacement, a l, and the bending moment M xa in ection x a : a 0.45d mm l and: Mx a 7.kNm 10

11 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 kN x a Ed A d Ed A h The orce or yielding in the tenile reinorcement i calculated a: F A kN yd Calculate the orce in the FRP when the teel yield: Mx a d 7. 0 F F kN 0.9h h 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 Nmm/ mm b ck ctm G , x Edt 11

12 Step 5: Calculate o the anchor length, cont. F, e, xaed kN Which i le then F. Calculate a new bending moment, M,e : M Ed A d 0.9hF 1 Ed A h, e, e and: d 0.9 M, e hf, e F h Chooe M,e =0.7 knm, and: knm knm qx Mx R,,, e Ax e x e mm Anchor length: l e E dt mm.5 ctm Chooe 50 mm. a x, ele mm a la FRP R a x,e 1

13 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 MPa Where l = L/ and z 0 = h - x and Gb 1 1 z Ed A Ecd Ac EcdWc a

14 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 kn/ m ater beore Support orce: R A ql kn Bending moment at a=100 mm a M ( ) x x RAaq 1.7kNm Normal tre calculation Mx x h y MPa I Failure criteria: 1 ctm x y x y 1 xy MPa Which i le than ctm =.5MPa 14

15 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

16 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

Shear Stress. Horizontal Shear in Beams. Average Shear Stress Across the Width. Maximum Transverse Shear Stress. = b h

Shear Stress. Horizontal Shear in Beams. Average Shear Stress Across the Width. Maximum Transverse Shear Stress. = b h Shear Stre Due to the preence of the hear force in beam and the fact that t xy = t yx a horizontal hear force exit in the beam that tend to force the beam fiber to lide. Horizontal Shear in Beam The horizontal