高強度鋼筋混凝土梁之高強度鋼筋混凝土梁之裂縫發展與控制裂縫發展與控制

邱建國Associate Professor,Department of Civil and Construction Engineering,Taiwan-TECH.

Life-Cycle ofStructure Engineering

2015/12/11

ContentContent

Destination

D i C dDesignCodes

Experimentp

ShearCrackControl

FlexuralCrackControl

ConclusionsConclusions

1

DestinationDestinationDestinationDestination

l l k lPerformance

① FlexuralCrackControl(ACI318)

② ShearCrackControl(AIJ2010)

③ Damage Evaluation

Service-ability

Repar-abilitySafety

③ DamageEvaluationCrackWidthsandDeformation

Crack widthExperiment

Design Damage Evaluation

Ref：日本建築學會‧鋼筋混凝土建築物之耐震性能評價指針,2004,5.梁部材の性能評価法,pp.129~168 2

Shear Crack ControlShear Crack ControlAllowable shear force corresponding to

Shear Crack ControlShear Crack ControlAIJ(2010)

i fAllowableshearforcecorrespondingtoserviceabilityensuring

Reinforcement

Allowableshearforcecorrespondingtoreparabilityensuring

Concrete

3

Shear Crack ControlShear Crack ControlShear Crack ControlShear Crack ControlAIJ(2010)According the research conducted by Simazaki (2009), if the long-termg y ( ), gloading is less than VAL1, the maximum shear crack width can becontrolled within 0.3 mm. If the short-term loading of a medium-magnitude is less than VSL1, the residual maximum shear crack widthcan be controlled within 0.3 mm.

4

Flexural Crack ControlFlexural Crack ControlFlexural Crack ControlFlexural Crack ControlACI-318 (2005) 280380( ) 2.5 cs cf

Ref Robert J Frosch (1999)

280300( )

s

s

f

sf

(1) Model:

Ref.RobertJ.Frosch (1999)

fsc s c c

s

fw S SE

CrackwidthgrowslinearlyN.A

h

c

dd c

1max c c

h cw w wd c

2

Ref：Robert J. Frosch, 1999, Another Look at Cracking Control in Reinforced Concrete, ACI Structural Journal 5

Flexural Crack ControlFlexural Crack ControlFlexural Crack ControlFlexural Crack ControlACI-318 (2002&2005)

(2) Crackspacing

Ref.RobertJ.Frosch(1999) Fromthebarcentertothefarthest

concrete element

d *d2*dC

concreteelement

sh c fw S * 2 21 c cd d d

d1*2dC

sControl !

maxs

cw cS

d E* 2 22 ( )2c

sd d

Control !

f s2 22 ( )2

smax c

s

f sw dE

Ref：Robert J. Frosch, 1999, Another Look at Cracking Control in Reinforced Concrete, ACI Structural Journal 6

Flexural Crack ControlFlexural Crack ControlC 318 2002 & 200ACI-318 (2002&2005)

Cs s

250 300 252s 380 ‐ 2.5Cf f

0280 300 28380 2 5C

0.41 mm

0 41 0 53

(3) Crackcontrol

Ref.RobertJ.Frosch (1999) Cs ss 380 ‐ 2.5C

f f 0.41‐0.53 mm

2 22 ( )2

smax c

s

f sw dE

2 22 ( )2max s

cw Es d

f

2 sf

Crackwidth：0.41 mmTensilestressofmainbars：0.6fy

Ref：Robert J. Frosch, 1999, Another Look at Cracking Control in Reinforced Concrete, ACI Structural Journal

cC 代替 cd7

Experiment Experiment ppSpecimenDesign

Chiuetal.(2013)Yielding stress 685、785MPaReinforcement size D32(#10)、D25(#8)

Stirrup ratioStirrup ratio

2000 mm2000 mm

2000 mm

Hinge

Roller

Number：10Concretecover：4cm

Compressive strength 70、100MPaSection size

Chiuetal.(2015) Chiuetal.(2014)

N b

Shear span to depth ratioShear span to depth ratioConcrete coverConcrete cover

Chengetal.(2014)400~2000Hinge

Roller Number：10Concretecover：4cm

g2000 mm

Number：3

Number：2Concretecover：2、3cm

Cyclic loadingCyclic loading

u be ：3Concretecover：3cm

8

Experiment Experiment ppMeasurementsystem

1. Crackmeasurement 2. NDI

等剪力段 等彎矩段Moment-equivalentShear-

l等剪力段 等彎矩段

剪力裂縫量測處regionequivalent

region

撓曲裂縫量測處3. Strainofreinforcement

9

Shear CrackExperimental DataExperimental DataShearCrack

TensileStrengthofConcrete

ShearCrackStrengthofC tConcrete

10

Shear CrackExperimental DataExperimental DataShearCrack

UltimateShearStrengthofConcrete

11

Shear CrackExperimental DataExperimental DataShearCrack

12

Shear Crack ControlShear Crack ControlDesign FormulaDesign Formula

10

Averageshearstress andMaximum Shear crack width

v/v c

r

MaximumShearcrackwidth(Reinforcement785MPa)

1v

1

1E-008 1E-007 1E-006 1E-005 0.0001(wps,max/d)*(pw)

1313

Shear CrackExperimental DataExperimental Data

AllowableShearStress(Maximum shear crack width0 3 mm)

ShearCrack

1.2

(Maximumshearcrackwidth0.3 mm)

1

0.6

0.8

P wf s

0.4P

0.2

140 0.004 0.008 0.012pw

0(MPa)

Shear CrackExperimental DataExperimental DataShearCrack

Allowable Shear Stress Allowable Shear StressAllowableShearStress(Maximumshearcrackwidth 1.0 mm)

AllowableShearStress(Maximumresidual

shear crack width 0 415

shearcrackwidth 0.4mm)

Shear Crack ControlShear Crack ControlDesign FormulaDesign Formula

Performance Allowable shear stress (MPa)

Long- Serviceability

Shear crackstrength

-term

ServiceabilityMaximumMaximum shearshear

crackwidthcrackwidth0 40 40.4mm0.4mm

ShortR bilit

Maximum sheark idth-

termReparability crackwidth

1.0mm

16

Shear Crack ControlShear Crack ControlDesign Formula

1.8

2.0

pwu,b/D=0.50pwu,b/D=0.55

Design Formula

1 4

1.6

pwu,b/D=0.60pwu,b/D=0.65pwa,b/D=0.50pwa,b/D=0.55

Serviceability

1.2

1.4

Rat

io(%

) pwa,b/D=0.60pwa,b/D=0.65

0.8

1.0

Stir

rups

R

SeismicDesign

:2.5%685

Assumption

f MP

0.4

0.6

S 6850.125 ( 785 )550

y

sa yt yt

f MPaf f f MPaf MPa

0.22

2

5501300 /1000 1.2 300 /

su

a

u

f MPaw kgf mw kgf m

0

10 11 12 13 14 15 16 17 18

Ln/D 17

Shear Crack ControlShear Crack ControlDesign FormulaDesign Formula

0 4 mmb/D:0.45-0.55B/L: 0.5-1.0

Randomvariables

0.4 mm B/L:0.5 1.0DeadLoad:800-1100kgf/mLiveLoad:200-400kgf/m2

fy:685Mpafyt:785Mpaf 550 MP

0.3 mm

fsu:550MPafsa:0.125or0.15fyt

L/DMCSmax (L/D) = 12max (L/D) = 12 (0.9997, 0.40mm)(0.9997, 0.40mm)

Reliability

18

Flexural Crack ControlFlexural Crack Control

'0 35 0 33 f A f手冊建議, 4.3.2

'0.351.5

c v sserw

f A fvb s

0.35 0.331.5

c v sserw

f A fvb s

f為剪力鋼筋之容許應力，最大值為0 15f[解說]設計剪應力之計算可參考ACI318之建議，v=V/(bwd)。計算梁構件於靜載重與活載重下之斷面剪應力時，不須考慮兩載

fs為剪力鋼筋之容許應力，最大值為0.15fyt

重之載重放大因子。式(4.3.1)之前項為混凝土之容許應力，依實驗結果可知，其與構件之剪力跨距與深度比有密切相關，就一

般中長梁而言則建議採用0.35[3]。由式(4.3.1)可知，梁構件可藉由剪力鋼筋之用量以提升其容許應力。若梁構件之跨深比增加，

為有效控制靜載重與活載重下之剪力裂縫，則具所需之剪力鋼

筋用量亦會增加，並會大於耐震設計下剪力鋼筋量；反之，若

19

能將梁構件之跨深比降至12以下，則耐震設計下剪力鋼筋需求會主控設計，而不須檢核本節規定[3]。

Experimental DataExperimental DataExperimental DataExperimental DataFlexuralCrack

600

700

(MPa

) Flexural crack

400

500

stre

ss f

s(

200

300

rcem

ent s

本研究試體

102年試體[17]

103年試體[18]

0

100

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Rei

nfor 103年試體[18]

文獻[21]

Maximum crack width (mm)

Crack width is proportional to reinforcement stress

20

Experimental DataExperimental DataExperimental DataExperimental DataStable →Crack number not increased →Crack spacing turns to be constantFlexuralCrackStable →Crack number not increased →Crack spacing turns to be constant

400

500

acin

g

2C100

400500

acin

g

3C100

0

100

200

300

Avg.

Cra

ck S

pa

0100200300

0 2 4 6 8 10 12 14 16 18 20Avg.

Cra

ck S

pa

300ng

175R100500g

275R100500

325R70

0 2 4 6 8 10 12 14 16 18 20Av

Cuvature : rad/m ()0 2 4 6 8 10 12 14 16 18 20Av

Cuvature : rad/m ()

0100150200250300

Cra

ck S

paci

n

100

200

300

400

500

Cra

ck S

paci

ng

200

300

400

500

rack

Spa

cing

A k i C k b ／S f h i l i

050

0 2 4 6 8 10 12 14

Avg.

C

Cuvature : rad/m (‰)

0

100

0 2 4 6 8 10 12 14 16 18 20Avg

. C

Cuvature : rad/m ()0

100

0 3 6 9 12 15 18 21 24 27 30Avg

. Cr

Cuvature : rad/m ()

Average crack spacing＝Crack numbers／Span of the moment-equivalent region

21

AverageAverage CCrack Spacingrack SpacingAverage Average CCrack Spacingrack SpacingCrack width＝(strain)(crack spacing)

ACI 318(2002&2005)

AIJ-2010(附錄)

JSCE-2007

CEB-FIP

Broms

趙唯堅與丸山

22

Fl l C k A C k S i

Experimental DataExperimental Data

ACI-318： 2 2( )sd * *1 2[ ]

Flexural Crack-Average Crack Spacing

ACI 318：

本研究試體 文獻[17] 文獻[18] 文獻[21] 本研究試體 文獻[17] 文獻[18] 文獻[21]

( )2c

d 1 2[ , ]

140

160

180

paci

ng(m

m)

140

160

180

g. S

paci

ng ModifiedUnconservative

80

100

120

rim

ent A

vg. S

p

80

100

120

xper

imen

t Avg

6060 80 100 120 140 160 180E

xper

ACI -318 Avg. Spacing

6060 80 100 120 140 160 180

Ex

ACI -318 Avg. Spacing (Modified)

d1*d2*dC

s 23

Fl l C k A C k S i

Experimental DataExperimental Data

AIJ 2010 (附錄) JSCE 2007

Flexural Crack-Average Crack Spacing

AIJ-2010 (附錄) JSCE-2007

本研究試體 文獻[17] 文獻[18] 文獻[21] 本研究試體 文獻[17] 文獻[18] 文獻[21]

140

160

180

Spac

ing

[ ] [ ] [ ]

140

160

180

Spac

ing

[ ] [ ] [ ]

80

100

120

erim

ent A

vg.

60

80

100

120

rim

ent A

vg.

60

80

60 80 100 120 140 160 180

Exp

e

AIJ - 2010 Avg. Spacing

40

60

40 60 80 100 120 140 160 180Exp

erJSCE -2007 Avg. Spacing

24

Fl l C k A C k S i

Experimental DataExperimental Data

180.

本研究試體 文獻[17] 文獻[18] 文獻[21]

Flexural Crack-Average Crack Spacing

CEB-FIP100120140160

peri

men

t Avg

.Sp

acin

g

6080

60 80 100 120 140 160 180

Exp

Model Code 2010 Avg. Spacing

Broms 趙唯堅與丸山

160180

vg.

本研究試體 文獻[17] 文獻[18] 文獻[21]

140160180

Avg.

g

本研究試體 文獻[17] 文獻[18] 文獻[21]

80100120140

Exp

erim

ent A

vSp

acin

g

6080

100120140

Exp

erim

ent

Spac

ing

6060 80 100 120 140 160 180

E

Broms. 研究建議 Avg. Spacing

6060 80 100 120 140 160 180趙唯堅與丸山久一研究建議 Avg. Spacing

25

Fl l C k Width

Experimental DataExperimental DataFlexural Crack Width

ACI 318 (2002 & 2005)

ACI 318

4

5

h R

atio

ode)

ACI - 318

4

5

h R

atio

ode)

ACI - 318 (Modified)

Modified. 1.15ExpCode

wAvg

W . 1.06Exp

Code

wAvg

W

1

2

3

Cra

ck W

idth

(Exp

. / C

o

1

2

3

Cra

ck W

idth

(Exp

. / C

o

Unconservative

0

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

f / fd2* using

0

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

fs / fyusing max{d1*,d2*} fs / fy fs / fy

26

Fl l C k A C k S i

Experimental DataExperimental Data

AIJ-2010(附

Flexural Crack-Average Crack Spacing

(錄)JSCE-2007

5

o

AIJ - 2010

h 0 0002

5

o

JSCE - 2007

h 0 0002

2

3

4

k W

idth

Rat

ioE

xp. /

Cod

e)

εsh=0.0002εsh=0.0004

2

3

4

k W

idth

Rat

ioE

xp. /

Cod

e)

εsh=0.0002εsh=0.0004

0

1

2

0 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 1

Cra

ck (E

0

1

2

0 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 1C

rack (E

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

fs / fy

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

fs / fy

Good prediction Good prediction in low stress0.002, . 0.87Expsh

Code

wAvg

w

Good prediction generally

Good prediction in low stress0.004, . 0.68Expsh

Code

wAvg

w

27

Fl l C k A C k S i

Experimental DataExperimental Data

CEB-FIP 5CEB-FIP 2010

Flexural Crack-Average Crack Spacing

CEB-FIP

2

3

4

Wid

th R

atio

p. /

Cod

e)

0

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Cra

ck

(Ex

fs / fy

Broms 趙唯堅與丸山fs fy

4

5

Rat

iode

)

Broms. 相關研究

4

5

atio

)

趙唯堅 & 丸山久一相關研究

εsh=0.0002εsh=0 0004

0

1

2

3

Cra

ck W

idth

R(E

xp. /

Cod

1

2

3ra

ck W

idth

Ra

(Exp

. / C

ode) εsh=0.0004

00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

C

fs / fy

00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Cr

fs / fy 28

Flexural Crack ControlFlexural Crack ControlDesign Formula

Problems： 5

o

ACI - 318

0 4 0 88Expswf f A

Design Formula

2

3

4

k W

idth

Rat

ioE

xp. /

Cod

e)(1) Low stress level

(2) Section parameter

0.4 , . 0.88Exps yy code

f f Avgf w

0

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Cra

ck (E

d * d

Improving：Crack spacing

Assumption：Bi linear fs / fyd2* used

275R100

Bi-linearAverage stress：263MPa

300

400

500

ack

Spac

ing

3

ϕ209sf MPa

0

100

200

0 2 4 6 8 10 12 14 16 18 20

Avg.

Cra

1

0 0 4

ϕ

Cuvature : rad/m () 0 0.4fs / fy

29

Flexural Crack ControlFlexural Crack ControlDesign Formula

4

5

o

Modified

0 82Expw

Avg

Design Formula

Problems：

2

3

4

ck W

idth

Rat

iE

xp. /

Cod

e)

. 0.82Code

Avgw

( ) sma cf sw d 2 22

(1) Low stress level

0

1

0 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 1

Cra

c (E

( ) max cs

w dE2 2

( / ), / . s y s yf f f f3 5 0 0 40 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

fs / fy(2) Section parameter0.8

d1*d2*dC ：預測偏低

* *2 1d d

0.4

0.6

0.8

縫寬

度(m

m)

using d1*

s * *

( )

d dsd d d

2 1

2 2 2 20

0.2

0 100 200 300 400 500 600 700

最大

裂縫

ACI-318

30

( )

c c

c

c

d

d d

s

d

22

0 100 200 300 400 500 600 700最大鋼筋應力 (MPa)

Flexural Crack ControlFlexural Crack ControlDesign Formula

( ) smax cs

f sw dE2 22

2

Design Formula

Proposed formulap

. cs cf 280380 2 5

s

s

f

s f280300

31

s

Flexural Crack ControlFlexural Crack Control

280380 2 5 s C手冊建議, 4.3.2 300 280

s380 2.5 cs

s Cf

s

sf

fs為在使用載重下算得之鋼筋應力，若其小於0 4f時 須以0 4f取代之於0.4fy時，須以0.4fy取代之

[解說] 依現階段之實驗結果可知，梁撓曲裂縫控制可延用ACI318之規定。然[ ]而於使用載重下梁之鋼筋應力偏低，則依式計算所得之撓曲裂縫間距會有過

小之疑慮，而造成裂縫寬度之計算值亦偏小而不保守；因此，fs若其小於0.4fy時 須以0 4f代入式 以決定鋼筋中心距 之最大容許值[4]時，須以0.4fy代入式，以決定鋼筋中心距s之最大容許值[4]。

[解說] 依ACI318-02可知，式之成立條件為假設鋼筋中心距s之一半會大於撓曲受拉鋼筋中心至最近受拉面之距離 且依現階段之實驗結果可知 若鋼筋曲受拉鋼筋中心至最近受拉面之距離，且依現階段之實驗結果可知，若鋼筋

中心距s之一半小於撓曲受拉鋼筋中心至最近受拉面之距離時，則依式計算所得之撓曲裂縫間距會有過小之疑慮；因此，本手冊建議鋼筋中心距s之選取不

32宜小於2Cc[4]。

ConclusionsConclusions

( ) 280380 2 5

Spacingofreinforcement

( ) .

( )

cs

s cf

s f

380 2 5

280300

33

( )sf

Th kThank you

Life-Cycle ofStructure EngineeringStructure Engineering

34

Past ResearchesPast ResearchesFlexuralCrackPast ResearchesPast Researches

Bengt.B.Broms (1965)

Specimens

1.Stablestate：140~210MPa2 C k i 2d2.Crackspacing：2dc3.Crackwidthisproportionaltothesteelstress 35

文獻回顧撓曲裂縫發展

文獻回顧各國撓曲裂縫控制規範 剪力裂縫控制 損傷評估

美國ACI-318規範 歐洲EuroCode 2規範 日本AIJ-2010附錄 日本JSCE-2007

(1)滿足最小鋼筋量(2)應力限制(3)限制鋼筋直徑及間距

不計算裂縫寬度

鋼筋應力

(MPa) wk=0.4mm wk=0.3mm wk=0.2mm160 40 32 25

鋼筋最大直徑 (mm)鋼筋應力(MPa) wk=0.4mm wk=0.3mm wk=0.2mm

鋼筋最大間距 (mm)

160 40 32 25220 32 25 16240 20 16 12280 16 12 8320 12 10 6

160 300 300 200220 300 250 150240 250 200 100280 200 150 50320 150 100 -

360 10 8 5400 8 6 4450 6 5 -

360 100 50 -

36

文獻回顧撓曲裂縫發展

文獻回顧各國撓曲裂縫控制規範 剪力裂縫控制 損傷評估

美國ACI-318規範 歐洲EuroCode 2規範 日本AIJ-2010附錄 日本JSCE-2007

縫寬度

抗壓及抗拉 抗剪

SD295 195 195

長期容許應力鋼筋型號

裂縫寬度0.3~0.4mm 附錄鋼筋混凝土標準規範

SD295 195 195SD345 215(*195) 195SD390 215(*195) 195SD490 215(*195) 195SD785 580 580

以上之鋼筋為()內數值

可適用混凝土抗壓強度*D29以上之鋼筋為()內數值 60~100MPa

Ace=(2Cb+ϕ)b

xn

Ace (2Cb+ϕ)b

D sd

ϕ

cb

bcs

37