FRP Repairs and Technique

Product Information

CE628

SLAB ANALYSIS

CE628

Given Loads on the Structure

≔LL 50 Live Load ≔Es 29000000 Modulus of Steel

≔SDL 10 Superimposed Dead Load ≔β 0.85 for fc=4000psi

≔fc 4000 Concrete Strength ≔fy 60000 Steel Strength

≔Ec =⋅57000 ‾‾‾‾‾⋅fc 3604.997 Modulus of Concrete

≔LLadd 50 Additional Live Load

≔ns =―Es

Ec

8.044 Modular Ratio

Structural Components:

One way Slab (1-ft Strip)

≔bs ⋅1 effective width ≔hs 4 Thickness of Slab

≔Ass =⋅―――⋅.11

2

1.330.083

2 Reinforcing in Slab in 1ft Section at the Bottom of SLAB

≔Ass.n =⋅―――⋅.11

2

1.50.073

2 Reinforcing in Slab in 1ft Section at the TOP of SLAB

≔ds 3 Ten. Steel to Extreme Comp. face at (Top)

≔ds.n 3 Ten. Steel to Extreme Comp. face at (Bottom)

≔ls 5 Design span Length Btw Joints

≔ss 14 Spacing Btw Shear Steel (center to center)

≔Asv ⋅.112 Area of Shear Reinforcement

≔B =―――bs

⋅ns Ass

216.435 ―1 Transformed Section Properties

≔kd =――――――−‾‾‾‾‾‾‾‾+⋅2 ds B 1 1

B0.524 Neutral Axia ≔k =―

kd

ds

0.175

≔Icr =+―――⋅bs ((kd))

3

3⋅⋅ns Ass ⎛⎝ −ds kd⎞⎠

24.654

4 Cracked moment of Inertia

Applied Loading:

≔DLslab =⋅⋅bs hs 150 ――3

0.05 ―― Dead Load of Slab

≔LLslab =⋅LL bs 0.05 ―― Live Load of Slab

≔LLs.new =⋅LLadd bs 0.05 ―― Additional Live Load of Slab

≔wDL =+DLslab ⋅SDL bs 0.06 ―― Total Dead Load

≔wLL =+LLslab LLs.new 0.1 ―― Total new Live Load

≔wser =+++DLslab ⋅SDL bs LLslab LLs.new 0.16 ―― Total Service Load with new loads

≔wslab =+1.2 ⎛⎝ +DLslab ⋅SDL bs⎞⎠ 1.6 ⎛⎝LLslab⎞⎠ 0.152 ―― Total Ultimate Load

≔ws.new =+1.2 ⎛⎝ +DLslab ⋅SDL bs⎞⎠ 1.6 ⎛⎝ +LLslab LLs.new⎞⎠ 0.232 ―― Total Ultimate Loadwith new Load

Moments:≔MDL.pos =⋅⋅.08 wDL ls

21.44 ⋅ Ultimate Positive Moment

≔MDL.neg =⋅⋅−.10 wDL ls

2−1.8 ⋅ Ultimate Negative Moment

≔Mser.pos =⋅⋅0.080 wser ls


≔Mser.neg =⋅⋅−0.10 wser ls


Max Shear and Moment (From figure above): ≔Vslab =⋅⋅0.600 wslab ls 0.456 Ultimate Shear

≔Mslab.pos =⋅⋅0.080 wslab ls


≔Mslab.neg =⋅⋅−0.10 wslab ls


Max Shear and Moment with New Load (From figure above): ≔Vs.new =⋅⋅0.600 ws.new ls 0.696 Ultimate Shear

≔Ms.pos.new =⋅⋅0.080 ws.new ls


≔Ms.nag.new =⋅⋅−0.10 ws.new ls

2−6.96 ⋅ Ultimate Nagetive Moment

Capacity Calculation Bases on Transformed Section (1-ft Strip)

Positve Moment Capacity of SLAB

＝⋅⎛⎝ ⋅ns Ass⎞⎠ ⎛⎝ −ds c⎞⎠ ⋅⎛⎝ ⋅c bs⎞⎠⎛⎜⎝―c

2

⎞⎟⎠

[Tension = Compression]

＝−⋅⋅ns Ass ds ⋅⋅ns Ass c ⋅c2

―bs

2

＝−+⋅c2 ⎛

⎜⎝―bs

2

⎞⎟⎠

⋅c ⎛⎝ ⋅ns Ass⎞⎠ ⎛⎝ ⋅⋅ns Ass ds⎞⎠ 0

Using Quadratic Equation:

≔a1 =―bs

26 ≔b1 =⋅ns Ass 0.005

2≔c1 =⎛⎝ ⋅⋅−ns Ass ds⎞⎠ −1.996

3

≔x1 =――――――――+−b1

‾‾‾‾‾‾‾‾‾‾‾‾‾−b12 (( ⋅⋅4 a1 c1))

⋅2 a10.524

≔x2 =――――――――−−b1

‾‾‾‾‾‾‾‾‾‾‾−b12

⋅⋅4 a1 c1

⋅2 a1−0.635

≔cs =max (( ,x1 x2)) 0.524 Neutral axis ≔as =⋅cs β 0.445 Comp. Block

≔εs.p =―――――.003 ⎛⎝ −ds cs⎞⎠

cs

0.014 Strain in Steel

≔ϕp =||||||||||

|

if

else if

else

≤cs ⋅―3

8ds

‖‖ .9

≥cs ⋅―3

5ds

‖‖ .65

‖‖‖

+.65 ⋅⎛⎝ −εs.p .002⎞⎠ ――250

3

0.9 Reduction Factor based on steel strain

≔fy =||||

|

if

else

≥ϕp .9‖‖ 60

‖‖ ⋅εs.p Es

60000 Steel Strength based on Steel Strain

≔phiMn.b.p =⋅⋅ϕp ⎛⎝ ⋅Ass fy⎞⎠⎛⎜⎝

−ds ―as

2

⎞⎟⎠

12.404 ⋅ Flexure Capacity of Beam

≔phiVn.b.p =⋅.75 ⎛⎝ ⋅⋅2 ‾‾‾‾‾⋅fc bs ds

⎞⎠ 3.415 Shear Capacity of Beam

Nagetive Moment Capacity of SLAB

＝⋅⎛⎝ ⋅ns Ass.n⎞⎠ ⎛⎝ −ds.n c⎞⎠ ⋅⎛⎝ ⋅c bs⎞⎠⎛⎜⎝―c

2

⎞⎟⎠

[Tension = Compression]

＝−⋅⋅ns Ass.n ds.n ⋅⋅ns Ass.n c ⋅c2

―bs

2

＝−+⋅c2 ⎛

⎜⎝―bs

2

⎞⎟⎠

⋅c ⎛⎝ ⋅ns Ass.n⎞⎠ ⎛⎝ ⋅⋅ns Ass.n ds.n⎞⎠ 0

Using Quadratic Equation:

≔a2 =―bs

26 ≔b2 =⋅ns Ass.n 0.004

2≔c2 =⎛⎝ ⋅⋅−ns Ass.n ds.n⎞⎠ −1.77

3

≔x1 =――――――――+−b2

‾‾‾‾‾‾‾‾‾‾‾‾‾−b22 (( ⋅⋅4 a2 c2))

⋅2 a20.496

≔x2 =――――――――−−b2

‾‾‾‾‾‾‾‾‾‾‾‾‾−b22 (( ⋅⋅4 a2 c2))

⋅2 a2−0.594

Neutral axis ≔as.n =⋅cs.n β 0.422 Comp. Block

≔εs.n =―――――.003 ⎛⎝ −ds cs⎞⎠

cs


≔ϕn =||||||||||

|

if

else if

else

≤cs.n ⋅―3

8ds.n

‖‖ .9

≥cs.n ⋅―3

5ds.n

‖‖ .65

‖‖‖

+.65 ⋅⎛⎝ −εs.n .002⎞⎠ ――250

3

0.9 Reduction Factor based on steel strain

≔fy =||||

|

if

else

≥ϕp .9‖‖ 60

‖‖ ⋅εs.n Es

60000 Steel Strength based on Steel Strain

≔cs.n =max (( ,x1 x2)) 0.496

≔phiMn.b.n =⋅⋅ϕn ⎛⎝ ⋅Ass.n fy⎞⎠⎛⎜⎝

−ds.n ――as.n

2

⎞⎟⎠


≔phiVn.b.n =⋅.75 ⎛⎝ ⋅⋅2 ‾‾‾‾‾⋅fc bs ds.n

⎞⎠ 3.415 Shear Capacity of Beam

BEAM ANALYSIS

CE628







≔ns =―Es

Ec

8.044 Modular Ratio

Structural Components:

One Way Slab known information:


≔Ass =―――⋅.11

2

1.330.083 ――

2

Reinforcing in Slab

≔ds 3 Ten. Steel to Extreme Comp. face ≔ls 5 Design span Length Btw Joints

BEAM:

≔bb 8 Beam width ≔hb 18 Beam Thickness

≔Asb =⋅⋅.312

3 0.932 Flewxure Reinforcing in Beam

≔db 15.4 Ten. Steel to Extreme Comp. face ≔widthbeam 5 Tributary width

≔lb 20 Design span Length



≔B =―――bb

⋅ns Asb

12.832 ―1 Transformed Section Properties

≔kd =――――――−‾‾‾‾‾‾‾‾‾+⋅2 db B 1 1


kd

db

0.293

≔Icr =+―――⋅bb ((kd))

3

3⋅⋅ns Asb ⎛⎝ −db kd⎞⎠

21131.844


≔DLbeam =⋅⋅bb hb 150 ――3

0.15 ―― Dead Load - Beam

≔DLs =⋅⋅⋅hs widthbeam 150 ――3

0.25 ―― Dead Load - Slab

≔LLbeam =⋅LL widthbeam 0.25 ―― Live Load - Beam

≔LLb.new =⋅LLadd widthbeam 0.25 ―― Additional - Live Load

≔wb.DL =++DLbeam DLs ⋅SDL widthbeam 0.45 ―― Total Dead Load

≔wb.LL =⎛⎝ +LLbeam LLb.new⎞⎠ 0.5 ―― Total Live Load

≔wb.ser =+⎛⎝ ++DLbeam DLs ⋅SDL widthbeam⎞⎠ ⎛⎝ +LLbeam LLb.new⎞⎠ 0.95 ―― Total New Service Load

≔wbeam =+1.2 ⎛⎝ ++DLbeam DLs ⋅SDL widthbeam⎞⎠ 1.6 ⎛⎝LLbeam⎞⎠ 0.94 ―― Total Ultimate Load

≔wb.new =+1.2 ⎛⎝ ++DLbeam DLs ⋅SDL widthbeam⎞⎠ 1.6 ⎛⎝ +LLbeam LLb.new⎞⎠ 1.34 ―― Total New Ultimate Load

Moments:

≔Mb.DL =――――⋅wb.DL lb

2

8270 ⋅ Dead Load Moment

≔Mb.LL =―――⋅wb.LL lb

2

8300 ⋅ Live Load Moment

≔Mb.ser =―――⋅wb.ser lb

2

8570 ⋅ Service Load Moment with New Load

Ultimate Shear and Moment:

≔Vbeam =―――⋅wbeam lb

29.4 Ultimate Shear

≔Mbeam =――――⋅wbeam lb

2

8564 ⋅ Ultimate Moment

Max Shear and Moment with New Load:

≔Vb.new =―――⋅wb.new lb

213.4 New Ultimate Shear

≔Mb.new =――――⋅wb.new lb

2

8804 ⋅ New Ultimate Moment

Capacity Calculation with Steel Reinforcement only: (Assumed Steel has Yielded)

≔Tb =⋅Asb fy 55.8 ＝ ⋅⋅⋅0.85 fc ab bb Also ＝T C

≔ab =――――⋅Asb fy

⋅0.85 fc bb

2.051 Compression Block

≔cb =―ab

β2.413 Neutral axis

＝――cb

0.003―――

−db cb

εs.b

≔εs.b =―――――.003 ⎛⎝ −db cb⎞⎠

cb

0.016

≔Check1 if

else

≥―――――.003 ⎛⎝ −db cb⎞⎠

cb

.005

‖‖ “Assumption that steel has yielded is correct”

‖‖ “Assumption that steel has yielded is NOT correct”

=Check1 “Assumption that steel has yielded is correct”

≔ϕ .9 Tension Controlled

≔phiMn.b =⋅⋅ϕ Tb

⎛⎜⎝

−db ―ab

2

⎞⎟⎠


≔phiVn.b =⋅.75⎛⎜⎝

+⎛⎝ ⋅⋅2 ‾‾‾‾‾⋅fc bb db

⎞⎠ ⋅

⎛⎜⎝――――

⋅⋅Asv fy db

ss

⎞⎟⎠

2⎞⎟⎠

22.578 Shear Capacity of Beam

ϕ

FLEXURE STRENGTHENING of BEAM with CFRP Laminates:

The existing reinforced concrete beam is Strengthened with the Unidirectional Carbon Fabric CSS-CUCF44 saturated with Epoxy Resin CSS-ES to produce CFRP composite. Roll Size used for this job will be 12inX150ft. Field cut 7in wide X 19ft Long piles and apply to the soffit of the beam using wet layup technique.

CFRP system we are using consists of Dry, unidirectrional sheets that will be installed by hand with an adhesive resin (Epoxy) using the Wet layup technique.

CFRP Material Properties:≔wf 7 width of the Laminate ≔n 2 # of Piles

≔tf ⋅0.08 Thickness per ply

≔ffu1 128000 Ultimate Tensile Strength ≔df =hb 18 Eff. Depth of FRP

≔εfu1 .09 Rupture Strain ≔Ef 14200000 Modulus of Elasticity

Strength of the Structure without FRP - (Equation 9-1) [Integrity Check for Beam]

≔wb.check =+1.1 ⎛⎝wb.DL⎞⎠ .75 ⎛⎝wb.LL⎞⎠ 0.87 ――

≔Mb.check =――――⋅wb.check lb

2

8522 ⋅ Max Moment (Check)

≔Check2 if

else

≤Mb.check phiMn.b

‖‖ “Beam Strength is OK to Strengthen with CFRP”

‖‖ “Beam Strength is NOT OK to Strengthen with CFRP”

=Check2 “Beam Strength is OK to Strengthen with CFRP”

STEPS TO PERFORM FLEXURE STRENGTHENING OF BEAM AS PER ACI 440.2R-08

Step-1

≔CE .85 for Carbon Fiber; Exterior Exposure [Table 9.1]≔ffu =⋅CE ffu1 108800 Ultimate Strength of CFRP≔εfu =⋅CE εfu1 0.077 Ultimate Strain of CFRP

Step-2

≔Ef =――ffu1

εfu1

1422222.222 Modulus of Elasticity of FRP

≔Af =⋅⋅n tf wf 1.122 Area of FRP piles

Step-3

≔εbi =―――――⋅Mb.DL ⎛⎝ −df kd⎞⎠

⋅Icr Ec

0.001

Step-4

≔εfd =|||||||

|

if

else

≤⋅⋅0.083‾‾‾‾‾‾‾‾‾‾‾‾‾――――――

⋅fc

⋅⋅⋅⋅n Ef tf

⋅0.7 εfu

‖‖‖‖

⋅⋅0.083‾‾‾‾‾‾‾‾‾‾‾‾‾――――――

⋅fc

⋅⋅⋅⋅n Ef tf

‖‖ ⋅0.9 εfu

0.011 Limit on the Design Strain recomended by committee, based on studies performed[Rizkall 2003,De lorenzis et al 2004, Kotynia 2005]

≔Check3

|||||||

if

else

≤⋅⋅0.083‾‾‾‾‾‾‾‾‾‾‾‾‾――――――

⋅fc

⋅⋅⋅⋅n Ef tf

⋅0.7 εfu

‖‖ “FRP Debonding CONTROL the design”

‖‖ “FRP Debonding DO NOT CONTROL the design”

=Check3 “FRP Debonding CONTROL the design”

Step-5 [to find Neutral Axis]

≔nf =―Ef

Ec

0.395

≔ns =―Es

Ec

8.044

＝++⋅x2 ⎛

⎜⎝―bb

2

⎞⎟⎠

x ⎛⎝ +⎛⎝ ⋅ns Asb⎞⎠ ⎛⎝ ⋅nf Af⎞⎠⎞⎠ ⎛⎝ −⋅⋅−ns Asb db ⋅⋅nf Af df⎞⎠ 0 Transformed Section

≔a1 =―bb

24 ≔b1 =+⎛⎝ ⋅ns Asb⎞⎠ ⎛⎝ ⋅nf Af⎞⎠ 7.923

2 Quadratic Equation

≔c1 =⎛⎝ −⋅⋅−ns Asb db ⋅⋅nf Af df⎞⎠ −123.1653

≔x1 =――――――――+−b1

‾‾‾‾‾‾‾‾‾‾‾‾‾−b12 (( ⋅⋅4 a1 c1))

⋅2 a14.646

≔x2 =――――――――−−b1

‾‾‾‾‾‾‾‾‾‾‾−b12

⋅⋅4 a1 c1

⋅2 a1−6.627

≔c =max (( ,x1 x2)) 4.646

Step-6

≔εfe =||||||||

if

else

≥−0.003⎛⎜⎝――

−df c

c

⎞⎟⎠

εbi εfd

‖‖ εfd

‖‖‖

0.003⎛⎜⎝――

−df c

c

⎞⎟⎠

0.009

≔εc =||||||||

if

else

>+εfe ⋅εbi

⎛⎜⎝――

c

−c df

⎞⎟⎠

.003

‖‖ .003

‖‖‖

+εfe ⋅εbi

⎛⎜⎝――

c

−c df

⎞⎟⎠

0.003Strain in Concrete

Step-7

≔εs =+εfe ⋅εbi

⎛⎜⎝――

−db c

−df c

⎞⎟⎠


Step-8

≔fs =||||

|

if

else

≥⋅Es εs fy

‖‖ fy

‖‖ ⋅Es εs

60Stress in Steel

≔ffe =⋅Ef εfe 12262.656

Step-9

≔cf =―――――+⋅Asb fs ⋅Af ffe

⋅⋅⋅.85 fc β bb

3.008 Assumed value of "c" is correct

Step-11 and 12

≔Ψf 0.85

≔Mns =⋅⋅Asb fy

⎛⎜⎝

−db ――⋅β c

2

⎞⎟⎠

749.133 ⋅ Capcity with just Steel

≔Mnf =⋅⋅⋅Ψf Af ffe

⎛⎜⎝

−df ――⋅β c

2

⎞⎟⎠

187.08 ⋅ Capcity with CFRP

≔phiMntotal =0.9 ⎛⎝ +Mns Mnf⎞⎠ 842.592 ⋅ New Strengthened section that is cable of sustaining the new required Moments.

≔Check4 ||||

|

if

else

≥phiMntotal Mb.new

‖‖ “Section is OK, Strengthened Section is capable of sustaining New Loads ”

‖‖ “Section is NOT OK, Try to Add more FRP Layers”

=Check4 “Section is OK, Strengthened Section is capable of sustaining New Loads ”

Step-13

≔fss =――――――――――――――――――

⋅⋅⎛⎜⎝

+Mb.ser ⋅⋅⋅εbi Af Ef

⎛⎜⎝

−df ―kd

3

⎞⎟⎠

⎞⎟⎠

⎛⎝ −db kd⎞⎠ Es

+⋅⋅⋅Asb Es

⎛⎜⎝

−db ―kd

3

⎞⎟⎠

⎛⎝ −db kd⎞⎠ ⋅⋅⋅Af Ef

⎛⎜⎝

−df ―kd

3

⎞⎟⎠

⎛⎝ −df kd⎞⎠

42251.869

≔Check5 ||||

|

if

else

≤fss ⋅.8 fy

‖‖ “Stress in the Reinforcing steel is in the limits”

‖‖ “Stress in the Reinforcing steel is NOT in the limits”

=Check5 “Stress in the Reinforcing steel is in the limits”

Step-14

≔ffs =−⋅⋅fss

⎛⎜⎝―Ef

Es

⎞⎟⎠

⎛⎜⎝―――

−df kd

−db kd

⎞⎟⎠

⋅εbi Ef 1297.643

≔Check6 ||||

|

if

else

≤ffs ⋅.55 ffu

‖‖ “Stress in the CFRP is within the limits”

‖‖ “Stress in the CFRP is NOT within the limits”

=Check6 “Stress in the CFRP is within the limits”

STEPS TO PERFORM SHEAR STRENGTHENING OF BEAM AS PER ACI 440.2R-08

We will use 2-side scheme for Shear Strengthening [11.4.1.2]. Same FRP Product as used in Flexure Strengthening.

≔n 1 Number of Piles

=tf 0.08 Thickness of pile

≔wf 12 Width of pile

≔dfv =−−db hs 1 10.4 Effective Depth of FRP for shear (for Two side or U-wrap)

≔Ψf 0.95 For U-wrap or 2-sides wrap [Table 11.1]

≔sf 8 Provided Spacing Btw CFRP Laminate (center to center)

≤＝εfe ⋅kv εfu .004 [11-6b]

≤＝kv ―――⋅⋅k1 k2 Le

468 εfu

.75 [11-7]

=εfu 0.077 Rupture Strain of FRP

≔Le =⋅⎛⎜⎜⎜⎝

―――――2500

⎛⎜⎝

⋅⋅n ―tf

――Ef ⎞

⎟⎠

0.58

⎞⎟⎟⎟⎠

2.92 Active Bond Length

≔k1 =

⎛⎜⎜⎝―――

⋅fc ――1

4000

⎞⎟⎟⎠

―2

3

1 [11-9]

≔k2 =―――−dfv Le

dfv

0.719 [11-10]

≔kv =|||||||

|

if

else

≤――――⋅⋅k1 k2 Le

⋅468 εfu

.75

‖‖‖‖

――――⋅⋅k1 k2 Le

⋅468 εfu

‖‖ .75

0.059 [11-7](Bond reduction coefficient applicable to Shear)

≔εfe =||||

|

if

else

≤⋅kv εfu .004‖‖ ⋅kv εfu

‖‖ .004

0.004 [11-6b]

≔ffe =⋅εfe Ef 5688.889 Tensile Stress in FRP

≔Afv =⋅⋅n tf wf 0.962 Area of FRP Shear reinforcement

≔Vf =――――⋅⋅Afv ffe dfv

sf

7.1 Shear contribution of FRP

≔Vs =――――⋅⋅Asv fy db

ss

7.26 Shear Strength of Steel

≔Vc =⋅⋅⋅2 ‾‾‾‾‾⋅fc bb db 15.584 Shear Strength of Concrete

≔phiVntotal =.75 ⎛⎝ ++⋅Ψf Vf Vc ⋅Vs 2⎞⎠ 27.636 Shear Strength of Section with FRP wraps

We will only strengthen the 3ft section from end. we donot need to strengthen the beam for Shear ( Vn > Vu)ϕ

GIRDER ANALYSIS

CE628







≔ns =―Es

Ec

8.044 Modular Ratio

Structural Components:One way Slab Known information


≔Ass =―――⋅.11

2

1.330.083 ――

2

Reinforcing in Slab

≔ds 3 Ten. Steel to Extreme Comp. face ≔ls 5 Design span Length Btw Joints

BEAM Known information:≔bb 8 Beam width ≔hb 18 Beam Thickness

≔Asb =⋅⋅.312

3 0.932 Flewxure Reinforcing in Beam

≔db 15.4 Ten. Steel to Extreme Comp. face ≔widthbeam 5 Tributary width

≔lb 20 Design span Length

≔DLbeam =⋅⋅bb hb 150 ――3

0.15 ―― Dead Load - Beam

≔DLs =⋅⋅⋅hs widthbeam 150 ――3

0.25 ―― Dead Load - Slab

≔LLbeam =⋅LL widthbeam 0.25 ―― Live Load - Beam

≔LLb.new =⋅LLadd widthbeam 0.125 ―― Additional - Live Load

≔wb.DL =++DLbeam DLs ⋅SDL widthbeam 0.45 ―― Total Dead Load

≔wb.ser =+⎛⎝ ++DLbeam DLs ⋅SDL widthbeam⎞⎠ ⎛⎝ +LLbeam LLb.new⎞⎠ 0.825 ―― Total New Service Load

≔wbeam =+1.2 ⎛⎝ ++DLbeam DLs ⋅SDL widthbeam⎞⎠ 1.6 ⎛⎝LLbeam⎞⎠ 0.94 ―― Total Ultimate Load

≔wb.new =+1.2 ⎛⎝ ++DLbeam DLs ⋅SDL widthbeam⎞⎠ 1.6 ⎛⎝ +LLbeam LLb.new⎞⎠ 1.14 ―― Total New Ultimate Load

M t

Moments:

≔Mb.DL =――――⋅wb.DL lb

2

8270 ⋅ Dead Load Moment

≔Mb.ser =―――⋅wb.ser lb

2

8495 ⋅ Service Load Moment with New Load

Ultimate Shear and Moment:

≔Vbeam =―――⋅wbeam lb

29.4 Ultimate Shear

≔Mbeam =――――⋅wbeam lb

2

8564 ⋅ Ultimate Moment

Max Shear and Moment with New Load:

≔Vb.new =―――⋅wb.new lb

211.4 New Ultimate Shear

≔Mb.new =――――⋅wb.new lb

2

8684 ⋅ New Ultimate Moment

GIRDER:

≔bg 10 Beam width ≔hg 18 Beam Thickness

≔Asg =⋅⋅.442

3 1.322 Flexure Reinforcing in Beam

≔dg 15.3 Distan from Ten to Extreme Comp. Face

≔lg 15 Design Span Length



≔B =―――bg

⋅ns Asg

11.301 ―1 Transfromed Section Properties

≔kd =――――――−‾‾‾‾‾‾‾‾‾+⋅2 dg B 1 1


kd

dg

0.31

≔Icr =+―――⋅bg ((kd))

3

3⋅⋅ns Asg ⎛⎝ −dg kd⎞⎠

21539.105


≔wg.DL =⎛⎜⎝

⋅⋅bg hg 150 ――3

⎞⎟⎠

0.188 ―― Uniform Dead Load

≔PDL =⋅wb.DL lb 9 DL Reaction from Beam

≔PLL =⋅LLbeam lb 5 LL Reaction from Beam

≔PLL.new =⋅LLb.new lb 2.5 LL Reaction from Beam

≔Pser =++PDL PLL PLL.new 16.5 Service Reaction from Beam

≔Pbeam =+1.2 ⎛⎝PDL⎞⎠ ⋅1.6 ⎛⎝PLL⎞⎠ 18.8 Ultimate Reaction from Beam

≔Pb.new =+1.2 ⎛⎝PDL⎞⎠ ⋅1.6 ⎛⎝ +PLL PLL.new⎞⎠ 22.8 Ultimate Reaction from Beam with New Loads

Moments:

≔MLL =+⋅PLL ls ⋅PLL.new ls 450 ⋅ Service Load Moment on Girder

≔MDL =+――――⋅wg.DL lg

2

8⋅PDL ls 603.281 ⋅ Total DL Moment on Girder

≔Mser =+MDL MLL 1053.281 ⋅ Total Service Moment on Girder

Max Shear and Moment (From figure above):

≔Vgirder =+Pbeam 1.2⎛⎜⎝―――

⋅wg.DL lg

2

⎞⎟⎠

20.488

≔Mgirder =+⋅Pbeam ls 1.2⎛⎜⎝――――

⋅wg.DL lg

2

8

⎞⎟⎠

1203.938 ⋅ where =ls 5 Dist Btw Supports

Max Shear and Moment with New Load (From figure above):

≔Vg.new =+Pb.new 1.2⎛⎜⎝―――

⋅wg.DL lg

2

⎞⎟⎠

24.488

≔Mg.new =+⋅Pb.new ls 1.2⎛⎜⎝――――

⋅wg.DL lg

2

8

⎞⎟⎠

1443.938 ⋅ where =ls 5 Dist Btw Supports

Capacity Calculation with Steel Reinforcement only: (Assumed Steel has Yielded)

≔Tg =⋅Asg fy 79.2 ＝C ⋅⋅⋅0.85 fc ag bg Also ＝T C

≔ag =――――⋅Asg fy

⋅0.85 fc bg

2.329 Compression Block

≔cg =―ag

β2.74 Neutral axis

＝――cg

0.003―――

−dg cg

εs.g

≔εs.g =―――――.003 ⎛⎝ −dg cg⎞⎠

cg

0.014

≔Check1 if

else

≥―――――.003 ⎛⎝ −dg cg⎞⎠

cg

.005

‖‖ “Assumption that steel has yielded is correct”

‖‖ “Assumption that steel has yielded is NOT correct”

=Check1 “Assumption that steel has yielded is correct”

≔ϕ .9 Tension Controlled

≔phiMn.g =⋅⋅ϕ Tg

⎛⎜⎝

−dg ―ag

2

⎞⎟⎠

1007.564 ⋅ Flexure Capacity of Girder

≔phiVn.g =⋅.75⎛⎜⎝

+⎛⎝ ⋅⋅2 ‾‾‾‾‾⋅fc bg dg

⎞⎠ ⋅

⎛⎜⎝――――

⋅⋅Asv fy dg

ss

⎞⎟⎠

2⎞⎟⎠

25.334 Shear Capacity of Girder

FLEXURE STRENGTHENING of GIRDER with CFRP Laminates:

The existing reinforced concrete GIRDER is Strengthened with the Unidirectional Carbon Fabric CSS-CUCF44 saturated with Epoxy Resin CSS-ES to produce CFRP composite. Roll Size used for this job will be 12inX150ft. Field cut 9in wide X 14ft Long piles and apply to the soffit of the beam using wet layup technique.

CFRP system we are using consists of Dry, unidirectrional sheets that will be installed by hand with an adhesive resin (Epoxy) using the Wet layup technique.

CFRP Material Properties:≔wf 9 width of the Laminate ≔n 6 # of Piles

≔tf ⋅0.08 Thickness per ply

≔ffu1 128000 Ultimate Tensile Strength ≔df =hg 18 Eff. Depth of FRP

≔εfu1 .09 Rupture Strain ≔Ef 14200000 Modulus of Elasticity

Strength of the Structure without FRP - (Equation 9-1) [Integrity of Check]

Max Moment (Check):

≔Mb.check =+1.1 ⎛⎝MDL⎞⎠ .75 ⎛⎝MLL⎞⎠ 1001.109 ⋅

≔Check2 if

else

≤Mb.check phiMn.g

‖‖ “Beam Strength is OK to Strengthen with CFRP”

‖‖ “Beam Strength is NOT OK to Strengthen with CFRP”

=Check2 “Beam Strength is OK to Strengthen with CFRP”

STEPS TO PERFORM FLEXURE STRENGTHENING OF GIRDER AS PER ACI 440.2R-08

Step-1

Assumed Open Building.≔CE .85for Carbon Fiber; Exterior Exposure [Table 9.1]≔ffu =⋅CE ffu1 108800Ultimate Strength of CFRP≔εfu =⋅CE εfu1 0.077Ultimate Strain of CFRP

Step-2

≔Ef =――ffu1

εfu1

1422222.222 Modulus of Elasticity of CFRP

≔Af =⋅⋅n tf wf 4.322 Area of CFRP piles

Step-3

≔εbi =―――――⋅MDL ⎛⎝ −df kd⎞⎠

⋅Icr Ec

0.001

St 4

Step-4

≔εfd =|||||||

|

if

else

≤⋅⋅0.083‾‾‾‾‾‾‾‾‾‾‾‾‾――――――

⋅fc

⋅⋅⋅⋅n Ef tf

⋅0.7 εfu

‖‖‖‖

⋅⋅0.083‾‾‾‾‾‾‾‾‾‾‾‾‾――――――

⋅fc

⋅⋅⋅⋅n Ef tf

‖‖ ⋅0.9 εfu

0.006 Limit on the Design Strain recomended by committee, based on studies performed[Rizkall 2003,De lorenzis et al 2004, Kotynia 2005]

≔Check3

|||||||

if

else

≤⋅⋅0.083‾‾‾‾‾‾‾‾‾‾‾‾‾――――――

⋅fc

⋅⋅⋅⋅n Ef tf

⋅0.7 εfu

‖‖ “FRP Debonding CONTROL the design”

‖‖ “FRP Debonding DO NOT CONTROL the design”

=Check3 “FRP Debonding CONTROL the design”

Step-5 to find Neutral Axis

≔nf =―Ef

Ec

0.395

≔ns =―Es

Ec

8.044

＝++⋅x2 ⎛

⎜⎝―bg

2

⎞⎟⎠

x ⎛⎝ +⎛⎝ ⋅ns Asg⎞⎠ ⎛⎝ ⋅nf Af⎞⎠⎞⎠ ⎛⎝ −⋅⋅−ns Asg dg ⋅⋅nf Af df⎞⎠ 0 Transformed Section

≔a1 =―bg

25 ≔b1 =+⎛⎝ ⋅ns Asg⎞⎠ ⎛⎝ ⋅nf Af⎞⎠ 12.323

2 Quadratic Equation

≔c1 =⎛⎝ −⋅⋅−ns Asg dg ⋅⋅nf Af df⎞⎠ −193.1423

≔x1 =――――――――+−b1

‾‾‾‾‾‾‾‾‾‾‾‾‾−b12 (( ⋅⋅4 a1 c1))

⋅2 a15.104

≔x2 =――――――――−−b1

‾‾‾‾‾‾‾‾‾‾‾−b12

⋅⋅4 a1 c1

⋅2 a1−7.568

≔c =max (( ,x1 x2)) 5.104

Step-6

≔εfe =||||||||

if

else

≥−0.003⎛⎜⎝――

−df c

c

⎞⎟⎠

εbi εfd

‖‖ εfd

‖‖‖

0.003⎛⎜⎝――

−df c

c

⎞⎟⎠

0.008

≔εc =||||||||

if

else

>+εfe ⋅εbi

⎛⎜⎝――

c

−c df

⎞⎟⎠

.003

‖‖ .003

‖‖‖

+εfe ⋅εbi

⎛⎜⎝――

c

−c df

⎞⎟⎠

0.003Strain in Concrete

Step-7

≔εs =+εfe ⋅εbi

⎛⎜⎝――

−dg c

−df c

⎞⎟⎠


Step-8

≔fs =||||

|

if

else

≥⋅Es εs fy

‖‖ fy

‖‖ ⋅Es εs

60Stress in Steel

≔ffe =⋅Ef εfe 10780.737

Step-9

≔cf =―――――+⋅Asg fs ⋅Af ffe

⋅⋅⋅.85 fc β bg

4.352 Assumed value of "c" is correct

Step-11 and 12

≔Ψf 0.85

≔Mns =⋅⋅Asg fy

⎛⎜⎝

−dg ――⋅β c

2

⎞⎟⎠

1039.964 ⋅ Capcity with just Steel

≔Mnf =⋅⋅⋅Ψf Af ffe

⎛⎜⎝

−df ――⋅β c

2

⎞⎟⎠

626.694 ⋅ Capcity with CFRP

≔phiMntotal =0.9 ⎛⎝ +Mns Mnf⎞⎠ 1499.992 ⋅ New Strengthened section that is cable of sustaining the new required Moments.

≔Check4 ||||

|

if

else

≥phiMntotal Mg.new




Step-13

≔fss =――――――――――――――――――

⋅⋅⎛⎜⎝

+Mser ⋅⋅⋅εbi Af Ef

⎛⎜⎝

−df ―kd

3

⎞⎟⎠

⎞⎟⎠

⎛⎝ −dg kd⎞⎠ Es

+⋅⋅⋅Asg Es

⎛⎜⎝

−dg ―kd

3

⎞⎟⎠

⎛⎝ −dg kd⎞⎠ ⋅⋅⋅Af Ef

⎛⎜⎝

−df ―kd

3

⎞⎟⎠

⎛⎝ −df kd⎞⎠

53326.053

≔Check5 if

else

≤fss ⋅.8 fy

‖‖ “Stress in the Reinforcing steel is in the limits”

‖‖ “Stress in the Reinforcing steel is NOT in the recomended limits”

=Check5 “Stress in the Reinforcing steel is NOT in the recomended limits”

Step-14

≔ffs =−⋅⋅fss

⎛⎜⎝―Ef

Es

⎞⎟⎠

⎛⎜⎝―――

−df kd

−dg kd

⎞⎟⎠

⋅εbi Ef 1232.627

≔Check6 if

else

≤ffs ⋅.55 ffu

‖‖ “Stress in the CFRP is within the limits”

‖‖ “Stress in the CFRP is NOT within the limits”

=Check6 “Stress in the CFRP is within the limits”

STEPS TO PERFORM SHEAR STRENGTHENING OF GRIDER AS PER ACI 440.2R-08

We will use 2-side scheme for Shear Strengthening [11.4.1.2]. Same Product as used for Flexure Strengthening.

≔n 1 Number of Piles

=tf 0.08 Thickness of pile

≔wf 12 Width of pile

≔dfv =−−dg hs 1 10.3 Effective Depth of FRP for shear (for Two side or U-wrap)

≔Ψf 0.95 For U-wrap or 2-sides wrap [Table 11.1]

≔sf 8 Provided Spacing Btw CFRP Laminate (center to center)

≤＝εfe ⋅kv εfu .004 [11-6b]

≤＝kv ―――⋅⋅k1 k2 Le

468 εfu

.75 [11-7]

=εfu 0.077 Rupture Strain of FRP

≔Le =⋅⎛⎜⎜⎜⎝

―――――2500

⎛⎜⎝

⋅⋅n ―tf

――Ef ⎞

⎟⎠

0.58

⎞⎟⎟⎟⎠

2.92 Active Bond Length

≔k1 =

⎛⎜⎜⎝―――

⋅fc ――1

4000

⎞⎟⎟⎠

―2

3

1 [11-9]

≔k2 =―――−dfv Le

dfv

0.716 [11-10]

≔kv =|||||||

|

if

else

≤――――⋅⋅k1 k2 Le

⋅468 εfu

.75

‖‖‖‖

――――⋅⋅k1 k2 Le

⋅468 εfu

‖‖ .75

0.058 [11-7](Bond reduction coefficient applicable to Shear)

≔εfe =||||

|

if

else

≤⋅kv εfu .004‖‖ ⋅kv εfu

‖‖ .004

0.004 [11-6b]

≔ffe =⋅εfe Ef 5688.889 Tensile Stress in FRP

≔Afv =⋅⋅n tf wf 0.962 Area of FRP Shear reinforcement

≔Vf =――――⋅⋅Afv ffe dfv

sf

7.031 Shear contribution of FRP

≔Vs =――――⋅⋅Asv fy dg

ss

7.213 Shear Strength of Steel

≔Vc =⋅⋅⋅2 ‾‾‾‾‾⋅fc bg dg 19.353 Shear Strength of Concrete

≔phiVntotal =.75 ⎛⎝ ++⋅Ψf Vf Vc ⋅Vs 2⎞⎠ 30.344 Shear Strength of Section with FRP wraps

≔Check7 if

else

≥phiVntotal Vg.new




We will Strengthen the Girder ends, 6'-0" from end at both ends.

FRP Repairs and Technique

Documents

Transcript of FRP Repairs and Technique