CFRP中等約束鋼筋混凝土方柱反復(fù)受壓本構(gòu)模型

王代玉+王震宇+喬鑫

文章編號(hào)：16742974(2014)04003908

收稿日期：20130713

基金項(xiàng)目：國(guó)家自然科學(xué)基金資助項(xiàng)目(51078109，51278150)；深圳市科技研發(fā)資金基礎(chǔ)研究重點(diǎn)項(xiàng)目(JC201005250051A)



作者簡(jiǎn)介：王代玉（1984-），男，山東鄒城人，哈爾濱工業(yè)大學(xué)講師，博士

通訊聯(lián)系人，E-mail:daiyuwang@hit.edu.cn

摘要：為填補(bǔ)FRP約束混凝土滯回本構(gòu)模型的空白，對(duì)邊長(zhǎng)305mm,高915mm及邊長(zhǎng)204mm,高612mm兩種尺寸的CFRP約束鋼筋混凝土方柱，采用500t四棱柱壓力試驗(yàn)機(jī)進(jìn)行了單調(diào)及反復(fù)軸壓試驗(yàn).試驗(yàn)結(jié)果表明,大尺寸CFRP約束鋼筋混凝土方柱的應(yīng)力應(yīng)變關(guān)系曲線存在軟化段，CFRP的約束作用為中等約束，且尺寸效應(yīng)對(duì)CFRP的有效約束作用有明顯影響；箍筋對(duì)單調(diào)受壓應(yīng)力應(yīng)變曲線形狀、極限壓應(yīng)變、反復(fù)受壓卸載曲線和殘余應(yīng)變影響均較大.基于試驗(yàn)結(jié)果提出了考慮鋼筋、尺寸效應(yīng)及CFRP包裹層數(shù)等影響參數(shù)的CFRP中等約束混凝土方柱的反復(fù)受壓本構(gòu)模型.模型由描述包絡(luò)線的單調(diào)受壓本構(gòu)模型、曲線形式的卸載曲線及直線形式的再加載曲線3部分組成，模型預(yù)測(cè)結(jié)果與試驗(yàn)結(jié)果吻合較好.

關(guān)鍵詞：碳纖維；約束混凝土；方柱；單軸受壓；應(yīng)力應(yīng)變模型

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中圖分類號(hào)：TU352；TU375.3 文獻(xiàn)標(biāo)識(shí)碼：A

CyclicStressstrainModelforCFRPModeratelyconfined

ReinforcedConcreteSquareColumns

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WANGDaiyu1,2,WANGZhenyu1,2,QIAOXin3



(1.KeyLabofStructuresDynamicBehaviorandControl(HarbinInstituteofTechnology),MinistryofEducation,Harbin,

Heilongjiang150090,China;2.SchoolofCivilEngineering,HarbinInstituteofTechnology,Harbin,Heilongjiang150090,China;

3.CCTEGShenyangEngineeringCoLtd,Shenyang,Liaoning110015,China)

Abstract:ToaddresstheknowledgegapsofthecyclicstressstrainmodelofFRPconfinedconcrete,thispaperpresentedtheexperimentalresultsofCFRPconfinedplainandRCsquarecolumnsundermonotonicandcyclicaxialcompression.Thespecimensweredividedintotwogroupsonthebasisofsize.Thewidthofthecrosssectionandtheheightofthetotalcolumnwere305mm,915mmand204mm,612mmforthetwogroups,respectively.TestresultsshowedthatthestressstraincurvesoftheCFRPconfinedlargercolumnsexhibitedalocalizedstrainsofteningbehavior,inwhichtheconfinementofCFRPwrapwasmoderatelyconfinedandsignificantlyinfluencedbysizeeffect.Moreover,theultimatecompressionstrain,unloadingcurvesandplasticstrainwereremarkablyinfluencedbyhoops.Basedonthetestdata,acyclicaxialstressstrainmodelforFRPconfinedsquareconcretecolumnswasproposed.Theproposedcyclicaxialstressstrainmodelconsistsofthreemaincomponents,namely(i)amonotonicstressstrainmodeltodescribetheenvelopecurve,(ii)apolynomialexpressionfortheunloadingcurve,and(iii)astraightlineforthereloadingpath.Theinfluencesofinternalsteelreinforcements,sizeeffectandaswellasthenumberoflayersofCFRPwrapwereconsidered.Thegoodagreementbetweenthepredictionsoftheproposedmodelandthetestresultsdemonstratesthecapabilityandaccuracyoftheproposedmodel.

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Keywords:carbonfiber;confinedconcrete;squaresectionRCcolumns;cycliccompression;stressstrainmodel

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FRP約束混凝土反復(fù)受壓滯回本構(gòu)模型是對(duì)FRP加固混凝土構(gòu)件及結(jié)構(gòu)抗震性能進(jìn)行研究和分析的基礎(chǔ).目前，國(guó)內(nèi)外學(xué)者對(duì)FRP約束混凝土單調(diào)受壓性能進(jìn)行了許多研究，并提出了多種單調(diào)受壓本構(gòu)模型[1-10]，但對(duì)反復(fù)受壓性能的研究卻較少；而已有的研究對(duì)象多為FRP約束小尺寸的素混凝土圓柱，考慮已有縱筋和箍筋的存在對(duì)反復(fù)受壓性能的研究則更少.

Shao等[11]對(duì)24個(gè)采用不同F(xiàn)RP材料和包裹層數(shù)約束的素混凝土圓柱進(jìn)行了不同加卸載水平的反復(fù)受壓試驗(yàn)，發(fā)現(xiàn)殘余應(yīng)變與卸載應(yīng)變具有很好的線性關(guān)系，建立了FRP約束素混凝土的加卸載規(guī)則，提出了FRP素混凝土的強(qiáng)化型滯回本構(gòu)模型.Lam和Teng等[12-13]進(jìn)行了FRP約束素混凝土圓柱的反復(fù)受壓試驗(yàn)，結(jié)果表明：加卸載歷史對(duì)反復(fù)受壓時(shí)的應(yīng)力應(yīng)變包絡(luò)線影響不大，可用單調(diào)受壓應(yīng)力應(yīng)變曲線代替反復(fù)受壓包絡(luò)線，反復(fù)加卸載可產(chǎn)生累積損傷，對(duì)殘余應(yīng)變及應(yīng)力的退化有影響.并在隨后的分析中建立了FRP約束混凝土圓柱的加、卸載規(guī)則，細(xì)化了再加載曲線的表述，將再加載曲線分為共同點(diǎn)之前的直線段和之后的曲線過(guò)渡段，建立了適于圓柱強(qiáng)約束的應(yīng)力應(yīng)變滯回模型.王震宇等[14]對(duì)12個(gè)CFRP約束素混凝土方柱進(jìn)行了反復(fù)受壓試驗(yàn)，研究發(fā)現(xiàn)FRP約束方柱與約束圓柱的反復(fù)受壓性能有明顯差異：兩者具有不同的殘余應(yīng)變與卸載應(yīng)變關(guān)系，且再加載曲線形式也不同，圓柱近似為直線，而方柱的再加載曲線與卸載曲線相似，為下凸的曲線形式.隨后基于試驗(yàn)結(jié)果，建立了FRP約束素混凝土方柱的滯回本構(gòu)模型.Abbasnia等［15］對(duì)10個(gè)CFRP約束混凝土方柱開(kāi)展了反復(fù)受壓試驗(yàn)，建立了殘余應(yīng)變與卸載應(yīng)變的線性關(guān)系，但并未提出相應(yīng)的反復(fù)受壓滯回本構(gòu)模型.

針對(duì)目前FRP約束混凝土反復(fù)受壓性能研究較少，且研究對(duì)象也多為小尺寸的素混凝土柱，沒(méi)有考慮尺寸效應(yīng)及已有鋼筋對(duì)反復(fù)受壓應(yīng)力應(yīng)變滯回模型影響的情況，本文對(duì)較大尺寸CFRP約束鋼筋混凝土方柱開(kāi)展了單調(diào)及反復(fù)受壓的試驗(yàn)研究，考察了尺寸效應(yīng)、縱筋、箍筋和CFRP包裹層數(shù)對(duì)反復(fù)受壓應(yīng)力應(yīng)變關(guān)系的影響，在材料層面上建立了可用于CFRP約束鋼筋混凝土柱非線性分析的滯回本構(gòu)模型.

1試驗(yàn)概況

1.1試件設(shè)計(jì)

共設(shè)計(jì)了30個(gè)試件，根據(jù)截面尺寸分為2個(gè)系列，其中S1系列試件為12個(gè)邊長(zhǎng)305mm，高915mm的方柱，包裹0~3層CFRP；S2系列試件為18個(gè)邊長(zhǎng)204mm，高612mm的方柱，分別包裹0~2層CFRP.未約束混凝土實(shí)測(cè)抗壓強(qiáng)度為25.5MPa，縱筋配筋率均為1.46%，箍筋體積配箍率分別為0，0.4%和0.8%；縱筋和箍筋分別采用HRB335和HPB235級(jí)鋼筋.為保證破壞出現(xiàn)在柱中，上下柱端采取箍筋加密并多包裹一層CFRP.試件尺寸及配筋如圖1所示，具體試件試驗(yàn)工況見(jiàn)表1.其中直徑分別為6,10和12mm鋼筋實(shí)測(cè)屈服強(qiáng)度分別為397,312和340MPa；CFRP的極限抗拉強(qiáng)度、彈性模量和單層厚度分別為4340MPa,2.4×105MPa和0.167mm；MS系列碳纖維配套粘結(jié)樹(shù)脂的抗拉強(qiáng)度及受拉彈性模量分別為46.3和2745MPa.

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圖1試件尺寸及配筋

Fig.1Dimensionandreinforcementofspecimens

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表1試件工況及主要試驗(yàn)結(jié)果

Tab.1Specimencharacteristicsandmaintestresults

試件編號(hào)

b/mm

h/mm

縱筋

箍筋

L

R/mm

fcc

/MPa

εcc/%

fcu/MPa

εcu/%

εh,rup/%

εfe/%

S1H1L0M

305

915

1212

6@80

0

0

32.1

0.373

25.7

0.662

-

-

S1H2L0M

305

915

1212

6@40

0

0

34.7

0.412

27.8

0.684

-

S1H0L1M

305

915

-

-

1

30

29.4

0.313

17.2

0.786

-0.843

-0.880

S1H0L2M

305

915

-

-

2

30

32.3

0.393

24.4

1.771

-0.951

-0.700

S1H1L1M

305

915

1212

6@80

1

30

35.1

0.428

24.1

1.538

-0.735

-0.370

S1H1L2M

305

915

1212

6@80

2

30

34.9

0.434

30.7

2.269

-1.257

-0.280

S1H1L2C

305

915

1212

6@80

2

30

33.9

0.387

27.9

2.507

-1.103

-0.810

S1H1L3M

305

915

1212

6@80

3

30

36.9

0.428

33.3

2.521

-1.186

-0.600

S1H1L3C

305

915

1212

6@80

3

30

36.5

0.487

33.9

3.830

-1.108

-1.210

S1H2L1M

305

915

1212

6@40

1

30

30.7

0.530

29.6

1.832

-1.307

-

S1H2L2M

305

915

1212

6@40

2

30

35.5

0.416

30.9

2.104

-1.002

-0.330

S1H2L3M

305

915

1212

6@40

3

30

37.2

0.529

35.8

2.584

-1.093

-1.240

S2H1L0M

204

612

810

6@120

0

0

29.9

0.364

23.9

0.834

-

S2H2L0M

204

612

810

6@60

0

0

32.5

0.406

26.0

1.352

-

S2H0L1M

204

612

-

-

1

20

28.7

0.391

25.0

1.740

-0.986

-0.860

S2H0L1P

204

612

-

-

1

20

28.8

0.308

24.6

1.699

-0.919

-0.990

S2H0L1C

204

612

-

-

1

20

31.2

0.355

23.5

2.220

-1.172

-1.190

S2H0L2M

204

612

-

-

2

20

30.8

0.481

31.4

2.287

-1.212

-0.620

S2H0L2P

204

612

-

-

2

20

31.9

0.446

27.9

2.810

-1.115

-0.990

S2H0L2C

204

612

-

-

2

20

32.4

1.512

32.1

3.162

-1.024

-1.090

S2H1L1M

204

612

810

6@120

1

20

35.5

0.704

34.4

2.969

-1.387

-

S2H1L1P

204

612

810

6@120

1

20

33.1

0.485

31.0

1.990

-0.868

-

S2H1L1C

204

612

810

6@120

1

20

33.5

0.535

32.8

3.043

-1.394

-

S2H1L2M

204

612

810

6@120

2

20

34.6

0.633

40.0

3.588

-1.064

-

S2H1L2P

204

612

810

6@120

2

20

37.7

0.921

43.8

3.801

-1.143

-

S2H1L2C

204

612

810

6@120

2

20

34.6

0.797

38.6

4.574

-1.202

-

S2H2L1M

204

612

810

6@60

1

20

33.0

0.465

31.6

2.562

-1.339

-

S2H2L1C

204

612

810

6@60

1

20

34.1

0.654

34.8

2.661

-1.358

-

S2H2L2M

204

612

810

6@60

2

20

34.3

0.652

40.8

4.230

-1.410

-1.420

S2H2L2C

204

612

810

6@60

2

20

36.0

0.801

41.4

4.839

-1.198

-0.960

注：試件編號(hào)中S表示方柱，數(shù)字1和2分別代表截面邊長(zhǎng)305mm和204mm,H表示配箍率,數(shù)字0,1,2分別代表無(wú)箍筋及配箍率為0.4%和0.8%，L1，L2和L3分別表示CFRP的包裹層數(shù)為1，2和3層，M，C，P分別表示單調(diào)加載及完全和部分加卸載；b為柱截面寬度；h為柱高；R為倒角半徑；fcc,εcc分別為峰值點(diǎn)時(shí)混凝土的應(yīng)力和應(yīng)變；fcu，εcu分別為約束后極限應(yīng)力和應(yīng)變，未包裹CFRP鋼筋混凝土柱的極限強(qiáng)度取為其峰值強(qiáng)度的80%；εh,rup為所有應(yīng)變片量測(cè)的FRP斷裂應(yīng)變的平均值；εfe為倒角處應(yīng)變片量測(cè)的FRP斷裂應(yīng)變的平均值.

1.2加載及量測(cè)裝置

軸向荷載采用5000kN壓力機(jī)加載，縱向位移采用4個(gè)LVDT測(cè)量，量測(cè)范圍為柱中間1/3高度，箍筋和CFRP應(yīng)變采用應(yīng)變片量測(cè)，量測(cè)方案如圖2所示.

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圖2LVDT及應(yīng)變片位置

Fig.2LocationsofLVDTandstrain

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2試驗(yàn)結(jié)果與分析

2.1破壞模式

試件最終破壞均是由于標(biāo)距范圍倒角附近區(qū)域的CFRP由于應(yīng)力集中被拉斷導(dǎo)致，典型破壞模式如圖3所示.CFRP約束素混凝土試件(圖3(a))破壞時(shí)表現(xiàn)出明顯的脆性，柱中區(qū)域CFRP幾乎被同時(shí)拉斷，試件迅速喪失承載力．而包裹相同CFRP層數(shù)的鋼筋混凝土試件(圖3(b))則表現(xiàn)出一定的延性特征，CFRP隨荷載增加而逐步拉斷，直至擴(kuò)展到柱中間較大區(qū)域后試件才最終喪失承載力.試驗(yàn)結(jié)束后剝掉CFRP發(fā)現(xiàn)，約束素混凝土柱(圖3(c))表面出現(xiàn)交叉斜裂縫；而約束鋼筋混凝土柱(圖3(d))則表現(xiàn)出明顯膨脹變形，箍筋由于混凝土的膨脹，受彎向外彎曲，縱筋受壓屈曲成燈籠狀.

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圖3試件破壞模式

Fig.3Failuremodesofspecimens

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2.2應(yīng)力應(yīng)變?cè)囼?yàn)曲線

反復(fù)受壓試件及對(duì)應(yīng)工況下單調(diào)受壓試件的應(yīng)力應(yīng)變?cè)囼?yàn)曲線如圖4所示.圖中應(yīng)力為實(shí)測(cè)軸力

除以試件截面面積，應(yīng)變?yōu)長(zhǎng)VDT量測(cè)位移平均值除以標(biāo)距，且應(yīng)力、應(yīng)變均以受壓為正受拉為負(fù).由圖4可知:1)與未約束鋼筋混凝土試件相比，F(xiàn)RP約束柱的極限壓應(yīng)變得到顯著提高，但承載力的提高幅度不大；2)應(yīng)力應(yīng)變關(guān)系試驗(yàn)曲線多為峰值點(diǎn)后存在軟化段的中等約束情況；3)單調(diào)受壓應(yīng)力應(yīng)變曲線與反復(fù)受壓時(shí)的包絡(luò)線整體趨勢(shì)基本一致；4)對(duì)S1和S2系列柱，全部CFRP應(yīng)變片量測(cè)的橫向斷裂應(yīng)變平均值分別為CFRP極限拉應(yīng)變的57%和68%，兩者相差不大；而柱倒角部位CFRP應(yīng)變片量測(cè)的橫向斷裂應(yīng)變平均值分別為CFRP極限拉應(yīng)變的40%和60%，由于方柱僅角部混凝土受到了FRP的有效約束，故計(jì)算FRP的有效約束作用時(shí)應(yīng)取倒角部位應(yīng)變片的量測(cè)結(jié)果平均值.



圖4應(yīng)力應(yīng)變?cè)囼?yàn)曲線

Fig.4Stressstraintestcurves



2.3鋼筋對(duì)應(yīng)力應(yīng)變曲線的影響?yīng)?/p>

單調(diào)受壓試驗(yàn)結(jié)果表明，鋼筋的存在對(duì)單調(diào)受壓應(yīng)力應(yīng)變關(guān)系曲線形狀、峰值應(yīng)力及極限應(yīng)變等有明顯影響，詳細(xì)結(jié)果討論見(jiàn)文獻(xiàn)[16-17].

本文反復(fù)受壓試驗(yàn)結(jié)果表明：鋼筋的存在對(duì)加卸載曲線同樣有較大影響.部分包裹層數(shù)相同的鋼筋混凝土柱和素混凝土柱反復(fù)受壓應(yīng)力應(yīng)變關(guān)系試驗(yàn)曲線的比較如圖5所示.由圖5可知，卸載曲線前期近似為直線，后期呈明顯的曲線形式，素混凝土方柱在卸載后期的模量變化很大，而鋼筋混凝土方柱的卸載模量變化較??；在卸載應(yīng)變相同時(shí)，鋼筋混凝土柱的殘余應(yīng)變明顯大于素混凝土柱.素混凝土方柱與鋼筋混凝土方柱的再加載曲線也不同，鋼筋混凝土方柱為直線，而素混凝土方柱為曲線.因此，在建立反復(fù)受壓應(yīng)力應(yīng)變關(guān)系模型時(shí)不應(yīng)忽略鋼筋的影響.

軸向應(yīng)變

圖5鋼筋對(duì)加卸載曲線的影響?yīng)?/p>

Fig.5Influenceofsteelbarsonunloading/reloadingcurves



3反復(fù)受壓應(yīng)力應(yīng)變模型

本文試驗(yàn)結(jié)果表明，由于鋼筋的存在導(dǎo)致柱的倒角半徑不能做到很大，此時(shí)對(duì)截面尺寸較大的柱其應(yīng)力應(yīng)變關(guān)系曲線存在軟化段，CFRP的約束作用降低，為中等約束.已有FRP約束混凝土本構(gòu)模型多針對(duì)應(yīng)力應(yīng)變關(guān)系曲線為單調(diào)上升的強(qiáng)約束情況提出，中等約束本構(gòu)模型很少，考慮鋼筋影響的反復(fù)受壓本構(gòu)模型則更少．而在實(shí)際工程中，由于構(gòu)件截面尺寸較大且均為鋼筋混凝土，此時(shí)約束混凝土應(yīng)多為中等約束情況.故本文針對(duì)FRP中等約束鋼筋混凝土方柱建立反復(fù)受壓應(yīng)力應(yīng)變關(guān)系模型.

3.1有效側(cè)向約束應(yīng)力

前文試驗(yàn)結(jié)果表明，F(xiàn)RP約束鋼筋混凝土柱應(yīng)考慮箍筋約束對(duì)有效側(cè)向約束應(yīng)力的影響，故本文所建立的修正有效約束應(yīng)力模型為：

flm=flf+fls，(1)

flf=κa2EfntfεfeB=0.5κaρfEfεfe，(2)

fls=0.5keskvρstfyt．(3)

式中：flm為修正后有效側(cè)向約束應(yīng)力；flf為FRP有效側(cè)向約束應(yīng)力；fls為箍筋約束應(yīng)力；Ef為FRP彈性模量；n為FRP包裹層數(shù)；tf為FRP單層厚度；ρf為FRP體積含纖率；εfe為FRP有效斷裂應(yīng)變，由前文可知應(yīng)取倒角部位應(yīng)變片量測(cè)結(jié)果平均值且應(yīng)考慮截面尺寸的影響，基于試驗(yàn)結(jié)果本文建議：當(dāng)約束方柱截面尺寸大于300mm時(shí)，εfe=0.4εfu，當(dāng)截面尺寸小于300mm時(shí)取εfe=0.6εfu；ρst為體積配箍率；fyt為箍筋屈服強(qiáng)度；截面形狀系數(shù)κa及箍筋有效約束系數(shù)kes和kv分別為：

κa=1－2B－2rc23Ag－ρg1－ρg，(4)

Ag=B－4－πr2c，(5)

kes=1－∑w2xi+w2yi/6B－2rc21－ρcc，(6)

kv=1－s′/2B－2rc2．(7)

式中：ρg和ρcc分別為全截面和核心區(qū)截面縱筋配筋率；rc為截面倒角半徑；wxi和wyi分別為沿截面兩垂直方向第i個(gè)縱筋間凈距.

目前多以FRP側(cè)向約束應(yīng)力與未約束混凝土峰值應(yīng)力之比即約束比進(jìn)行FRP強(qiáng)弱約束的界定，但已有界定標(biāo)準(zhǔn)大都未考慮鋼筋及截面尺寸對(duì)側(cè)向約束作用的影響；故本文基于建立的修正后有效側(cè)向約束應(yīng)力模型(式（1）)與未約束混凝土峰值應(yīng)力之比對(duì)強(qiáng)弱約束重新進(jìn)行了界定：即當(dāng)修正約束比大于0.17時(shí)為強(qiáng)約束，小于0.09時(shí)為弱約束，介于兩者之間時(shí)為中等約束.對(duì)于FRP約束鋼筋混凝土強(qiáng)弱約束的劃分已有另文介紹，限于篇幅，本文不再重復(fù)介紹，詳見(jiàn)文獻(xiàn)[16-17].

3.2包絡(luò)線

本文試驗(yàn)及已有研究[11-12]均表明，F(xiàn)RP約束混凝土反復(fù)受壓時(shí)的包絡(luò)線可用其單調(diào)受壓時(shí)的應(yīng)力應(yīng)變曲線代替.故本文采用文獻(xiàn)[16]已提出的FRP中等約束鋼筋混凝土柱單調(diào)受壓應(yīng)力應(yīng)變模型作為反復(fù)受壓時(shí)的包絡(luò)線，其方程形式為：

y=Ax+3－2Ax2+A－2x3，x≤1.0；y=x/αx－12+x，1.0＜x≤xct；y=ycu，xct＜x≤xcu．(8)

式中：y=fc/fcc；ycu=fcu/fcc；x=εc/εcc；xct=εct/εcc；xcu=εcu/εcc；fc，εc分別為混凝土軸向應(yīng)力和應(yīng)變；fcc，εcc分別為峰值點(diǎn)應(yīng)力和應(yīng)變；fcu，εcu分別為極限點(diǎn)應(yīng)力和應(yīng)變；εct為轉(zhuǎn)折點(diǎn)應(yīng)變；A=Ec/Ep為初始上升段曲線控制參數(shù)，Ec=4730fc0為混凝土初始彈性模量，fc0為素混凝土峰值應(yīng)力，Ep=fcc/εcc為約束后峰值點(diǎn)割線模量；α為下降段控制參數(shù).

上式中各參數(shù)的詳細(xì)計(jì)算公式在文獻(xiàn)[16-17]中根據(jù)單調(diào)受壓試驗(yàn)結(jié)果已經(jīng)回歸得到.本文基于反復(fù)受壓試驗(yàn)結(jié)果，又對(duì)各參數(shù)進(jìn)行了重新修正，修正后的極限應(yīng)力和應(yīng)變及下降段控制參數(shù)表達(dá)式分別為：

fcu=fc00.2+3.47flffc00.64+0.59flsfc00.20，(9)

εcu=εc02+73.31flffc01.07+5.06flsfc00.03，(10)

α=1－fcu/fcc1.66．(11)

式中：εc0=0.002為未約束混凝土峰值點(diǎn)應(yīng)變.

3.3卸載曲線

根據(jù)試驗(yàn)卸載曲線的形狀特征，CFRP約束鋼筋混凝土方柱的卸載曲線描述如下：

fcσun=B0εc－εpεun－εpB1+1－B1εc－εpεun－εp．(12)

式中：σun和εun分別為卸載點(diǎn)應(yīng)力和應(yīng)變；εp為卸載殘余應(yīng)變；B0和B1為卸載曲線形狀系數(shù)，由試驗(yàn)數(shù)據(jù)回歸分析可得：

B0=0.5+0.48flffc00.32－0.16flsfc00.04．(13)

對(duì)約束鋼筋混凝土柱：

B1=－0.1εunεc01.28+0.33εunεc0+2.15．(14)

對(duì)約束素混凝土柱：

B1=－0.26εunεc01.31+0.89εunεc0+1.51．(15)

已有研究表明[11-15]，殘余應(yīng)變與卸載應(yīng)變成線性關(guān)系，本文試驗(yàn)得到同樣結(jié)論，但鋼筋的存在對(duì)殘余應(yīng)變有較大影響，如圖6所示.

由試驗(yàn)結(jié)果可知，當(dāng)卸載應(yīng)變小于0.001時(shí)，試件處于彈性段無(wú)殘余應(yīng)變產(chǎn)生，當(dāng)卸載應(yīng)變大于0.001時(shí)，回歸分析得到的殘余應(yīng)變表達(dá)式如下.

約束鋼筋混凝土柱時(shí)：

εp=0.580εun－0.0006，0.001≤εun≤0.004;

0.930εun－0.002，εun>0.004．(16)

約束素混凝土柱時(shí)：

εp=0.443εun－0.0004，0.001≤εun≤0.004;

0.836εun－0.002，εun>0.004．(17)

卸載應(yīng)變εun

圖6殘余應(yīng)變和卸載應(yīng)變關(guān)系



Fig.6Lasticstrainsversusunloadingstrains



3.4再加載曲線

根據(jù)試驗(yàn)得到的再加載曲線特征，采用直線模型描述再加載曲線，其表達(dá)式為:

fc=σnewεnew－εpεc－εp．(18)

式中：σnew和εnew分別為卸載曲線與再加載曲線交點(diǎn)處的應(yīng)力和應(yīng)變，由試驗(yàn)結(jié)果分析可知其分別與σun和εun成線性關(guān)系，如圖7和圖8所示.

卸載點(diǎn)應(yīng)力σun/MPa

圖7共同點(diǎn)應(yīng)力和卸載應(yīng)力關(guān)系



Fig.7Stressofcommonpointsversusunloadstress



卸載點(diǎn)應(yīng)變εun

圖8共同點(diǎn)應(yīng)變和卸載應(yīng)變關(guān)系



Fig.8Strainsofcommonpointsversusunloadstrains



基于試驗(yàn)數(shù)據(jù)的回歸，σnew和εnew確定如下：

σnew=0.921σun，(19)

εnew=εun．(20)

3.5本文模型與試驗(yàn)結(jié)果的比較驗(yàn)證

以前述單調(diào)受壓應(yīng)力應(yīng)變曲線作為骨架曲線，結(jié)合加卸載曲線模型，即可建立CFRP中等約束鋼筋混凝土方柱的滯回本構(gòu)模型.部分卸載時(shí)先按完全卸載曲線卸載至殘余應(yīng)力點(diǎn)，然后從殘余應(yīng)力點(diǎn)以直線加載至共同點(diǎn)，并與骨架曲線延伸相交.部分計(jì)算結(jié)果與試驗(yàn)結(jié)果的比較，如圖9所示.由圖9可以看出，本文所提滯回本構(gòu)模型對(duì)CFRP約束鋼筋混凝土及素混凝土柱在完全卸載和部分卸載時(shí)均與試驗(yàn)結(jié)果吻合較好，模型精度較高.



圖9計(jì)算結(jié)果與試驗(yàn)結(jié)果的比較

Fig.9Comparisonofcalculationresultsversustestdata

4結(jié)論

本文對(duì)CFRP約束鋼筋混凝土方柱單調(diào)及反復(fù)受壓性能進(jìn)行了試驗(yàn)研究，在此基礎(chǔ)上建立了反復(fù)受壓應(yīng)力應(yīng)變滯回本構(gòu)模型，得到以下主要結(jié)論：

1)對(duì)大尺寸鋼筋混凝土方柱CFRP約束后明顯改善了柱的延性，但對(duì)應(yīng)力提高幅度不大，其應(yīng)力應(yīng)變關(guān)系曲線多為峰值點(diǎn)后存在軟化段的中等約束情況．

2)單調(diào)受壓試件的應(yīng)力應(yīng)變關(guān)系曲線與相同工況反復(fù)受壓試件的包絡(luò)線基本一致，鋼筋對(duì)約束混凝土反復(fù)受壓時(shí)的加卸載曲線形狀及殘余應(yīng)變大小有明顯影響，殘余應(yīng)變與卸載應(yīng)變成很好的線性關(guān)系．

3)提出了CFRP中等約束鋼筋混凝土方柱單調(diào)受壓應(yīng)力應(yīng)變曲線、卸載曲線和再加載曲線的數(shù)學(xué)描述，在此基礎(chǔ)上建立了反復(fù)受壓應(yīng)力應(yīng)變滯回本構(gòu)模型，模型預(yù)測(cè)結(jié)果與試驗(yàn)結(jié)果吻合較好，可用于CFRP約束鋼筋混凝土結(jié)構(gòu)及構(gòu)件的非線性分析.

參考文獻(xiàn)［1］ XIAO Y, WU H. Compressive behavior of concrete confined by fiber composite jackets[J]. Journal of Materials in Civil Engineering, 2000, 12(2):139-146.

參考文獻(xiàn)[2] ACI 440.2R-08 Guide for the design and construction of externally bonded FRP systems for strengthening concrete structures[S]. MI, USA：American Concrete Institute(ACI), Farmington Hills,2008.[3] 丁洪濤, 易偉建, 冼巧玲. 碳纖維布(CFRP)加固壓彎構(gòu)件全過(guò)程分析[J]. 湖南大學(xué)學(xué)報(bào)：自然科學(xué)版, 2003, 30(3):139-141.DING Hongtao, YI Weijian, XIAN Qiaoling. Nonlinear analysis of carbon fiber sheets (CFRP) strengthened members subjected to axial load and lateral shear[J]. Journal of Hunan University:Natural Sciences, 2003, 30(3):139-141.(In Chinese)[4] JIANG T,TENG J G.Analysisoriented stressstrain models for FRPconfined concrete[J]. Engineering Structures, 2007, 29(11): 2968-2986.[5] TENG J G,JIANG T,LAM L,et al.Refinement of a designoriented stressstrain model for FRPconfined concrete[J]. Journal of Composites for Construction, 2009, 13(4):269-278.[6] HARAJLI M H. Axial stressstrain relationship for FRP confined circular and rectangular concrete columns[J]. Cement & Concrete Composites, 2006, 28(10):938-948.[7] YOUSSEF M N,FENG M Q,MOSALLAM A S.Stressstrain model for concrete confined by FRP composites[J]. Composites: Part B, 2007, 38(5/6):614-628.[8] EID R,PAULTRE P.Analytical model for FRPconfined circular reinforced concrete columns[J]. Journal of Composites for Construction, 2008, 12(5):541-552.[9]TURGAY T,POLAT Z,KOKSAL H O,et al.Compressive behavior of largescale square reinforced concrete columns confined with carbon fiber reinforced polymer wraps[J]. Materials and Design, 2010, 31(1):357-364.[10]吳剛, 呂志濤. FRP約束混凝土圓柱無(wú)軟化段時(shí)的應(yīng)力應(yīng)變關(guān)系研究[J].建筑結(jié)構(gòu)學(xué)報(bào), 2003, 24(5):1-9.WU Gang, LV Zhitao. Study on the stressstrain relationship of FRPconfined concrete circular column without a strainsoftening response[J].Journal of Building Structures,2003,24(5):1-9.(In Chinease)[11]SHAO Y,ZHU Z,MIRMIRAN A.Cyclic modeling of FRPconfined concrete with improved ductility[J].Cement & Concrete Composites, 2006, 28(10):959-968.[12]LAM L,TENG J G,CHEUNG C H,et al.FRPconfined concrete under axial cyclic compression[J].Cement & Concrete Composites, 2006, 28(10):949-958.[13]LAM L,TENG J G.Stressstrain model for FRPconfined concrete under cyclic axial compression[J]. Engineering Structures, 2009, 31(2):308-321.[14]王震宇, 李洪鵬. 重復(fù)荷載作用下碳纖維預(yù)設(shè)混凝土加卸載準(zhǔn)則 [J]. 建筑結(jié)構(gòu), 2009, 39(7):100-103. WANG Zhenyu, LI Hongpeng. Loading and unloading criteria of FRPconfined concrete under cyclic compression[J]. Building Structure, 2009, 39(7): 100-103. (In Chinese)[15]ABBASNIA R,ZIAADIAN H.Behavior of concrete prisms confined with FRP composites under axial cyclic compression[J].Engineering Structures, 2010, 32(3):648-655.[16]王震宇, 王代玉, 呂大剛,等.CFRP中等約束鋼筋混凝土方柱單軸受壓應(yīng)力應(yīng)變模型[J]. 建筑結(jié)構(gòu)學(xué)報(bào), 2011, 32(4):101-109. WANG Zhenyu,WANG Daiyu,LV Dagang,et al.Stressstrain model for CFRP moderatelyconfined reinforced concrete square columns[J]. Journal of Building Structures, 2011, 32(4):101-109. (In Chinese)[17]WANG Z Y,WANG D Y,SMITH S T,et al.CFRPconfined square RC columns. I: experimental investigation [J]. Journal of Composites for Construction, ASCE, 2012, 16(2):150-160.endprint

參考文獻(xiàn)[2] ACI 440.2R-08 Guide for the design and construction of externally bonded FRP systems for strengthening concrete structures[S]. MI, USA：American Concrete Institute(ACI), Farmington Hills,2008.[3] 丁洪濤, 易偉建, 冼巧玲. 碳纖維布(CFRP)加固壓彎構(gòu)件全過(guò)程分析[J]. 湖南大學(xué)學(xué)報(bào)：自然科學(xué)版, 2003, 30(3):139-141.DING Hongtao, YI Weijian, XIAN Qiaoling. Nonlinear analysis of carbon fiber sheets (CFRP) strengthened members subjected to axial load and lateral shear[J]. Journal of Hunan University:Natural Sciences, 2003, 30(3):139-141.(In Chinese)[4] JIANG T,TENG J G.Analysisoriented stressstrain models for FRPconfined concrete[J]. Engineering Structures, 2007, 29(11): 2968-2986.[5] TENG J G,JIANG T,LAM L,et al.Refinement of a designoriented stressstrain model for FRPconfined concrete[J]. Journal of Composites for Construction, 2009, 13(4):269-278.[6] HARAJLI M H. Axial stressstrain relationship for FRP confined circular and rectangular concrete columns[J]. Cement & Concrete Composites, 2006, 28(10):938-948.[7] YOUSSEF M N,FENG M Q,MOSALLAM A S.Stressstrain model for concrete confined by FRP composites[J]. Composites: Part B, 2007, 38(5/6):614-628.[8] EID R,PAULTRE P.Analytical model for FRPconfined circular reinforced concrete columns[J]. Journal of Composites for Construction, 2008, 12(5):541-552.[9]TURGAY T,POLAT Z,KOKSAL H O,et al.Compressive behavior of largescale square reinforced concrete columns confined with carbon fiber reinforced polymer wraps[J]. Materials and Design, 2010, 31(1):357-364.[10]吳剛, 呂志濤. FRP約束混凝土圓柱無(wú)軟化段時(shí)的應(yīng)力應(yīng)變關(guān)系研究[J].建筑結(jié)構(gòu)學(xué)報(bào), 2003, 24(5):1-9.WU Gang, LV Zhitao. Study on the stressstrain relationship of FRPconfined concrete circular column without a strainsoftening response[J].Journal of Building Structures,2003,24(5):1-9.(In Chinease)[11]SHAO Y,ZHU Z,MIRMIRAN A.Cyclic modeling of FRPconfined concrete with improved ductility[J].Cement & Concrete Composites, 2006, 28(10):959-968.[12]LAM L,TENG J G,CHEUNG C H,et al.FRPconfined concrete under axial cyclic compression[J].Cement & Concrete Composites, 2006, 28(10):949-958.[13]LAM L,TENG J G.Stressstrain model for FRPconfined concrete under cyclic axial compression[J]. Engineering Structures, 2009, 31(2):308-321.[14]王震宇, 李洪鵬. 重復(fù)荷載作用下碳纖維預(yù)設(shè)混凝土加卸載準(zhǔn)則 [J]. 建筑結(jié)構(gòu), 2009, 39(7):100-103. WANG Zhenyu, LI Hongpeng. Loading and unloading criteria of FRPconfined concrete under cyclic compression[J]. Building Structure, 2009, 39(7): 100-103. (In Chinese)[15]ABBASNIA R,ZIAADIAN H.Behavior of concrete prisms confined with FRP composites under axial cyclic compression[J].Engineering Structures, 2010, 32(3):648-655.[16]王震宇, 王代玉, 呂大剛,等.CFRP中等約束鋼筋混凝土方柱單軸受壓應(yīng)力應(yīng)變模型[J]. 建筑結(jié)構(gòu)學(xué)報(bào), 2011, 32(4):101-109. WANG Zhenyu,WANG Daiyu,LV Dagang,et al.Stressstrain model for CFRP moderatelyconfined reinforced concrete square columns[J]. Journal of Building Structures, 2011, 32(4):101-109. (In Chinese)[17]WANG Z Y,WANG D Y,SMITH S T,et al.CFRPconfined square RC columns. I: experimental investigation [J]. Journal of Composites for Construction, ASCE, 2012, 16(2):150-160.endprint

參考文獻(xiàn)[2] ACI 440.2R-08 Guide for the design and construction of externally bonded FRP systems for strengthening concrete structures[S]. MI, USA：American Concrete Institute(ACI), Farmington Hills,2008.[3] 丁洪濤, 易偉建, 冼巧玲. 碳纖維布(CFRP)加固壓彎構(gòu)件全過(guò)程分析[J]. 湖南大學(xué)學(xué)報(bào)：自然科學(xué)版, 2003, 30(3):139-141.DING Hongtao, YI Weijian, XIAN Qiaoling. Nonlinear analysis of carbon fiber sheets (CFRP) strengthened members subjected to axial load and lateral shear[J]. Journal of Hunan University:Natural Sciences, 2003, 30(3):139-141.(In Chinese)[4] JIANG T,TENG J G.Analysisoriented stressstrain models for FRPconfined concrete[J]. Engineering Structures, 2007, 29(11): 2968-2986.[5] TENG J G,JIANG T,LAM L,et al.Refinement of a designoriented stressstrain model for FRPconfined concrete[J]. Journal of Composites for Construction, 2009, 13(4):269-278.[6] HARAJLI M H. Axial stressstrain relationship for FRP confined circular and rectangular concrete columns[J]. Cement & Concrete Composites, 2006, 28(10):938-948.[7] YOUSSEF M N,FENG M Q,MOSALLAM A S.Stressstrain model for concrete confined by FRP composites[J]. Composites: Part B, 2007, 38(5/6):614-628.[8] EID R,PAULTRE P.Analytical model for FRPconfined circular reinforced concrete columns[J]. Journal of Composites for Construction, 2008, 12(5):541-552.[9]TURGAY T,POLAT Z,KOKSAL H O,et al.Compressive behavior of largescale square reinforced concrete columns confined with carbon fiber reinforced polymer wraps[J]. Materials and Design, 2010, 31(1):357-364.[10]吳剛, 呂志濤. FRP約束混凝土圓柱無(wú)軟化段時(shí)的應(yīng)力應(yīng)變關(guān)系研究[J].建筑結(jié)構(gòu)學(xué)報(bào), 2003, 24(5):1-9.WU Gang, LV Zhitao. Study on the stressstrain relationship of FRPconfined concrete circular column without a strainsoftening response[J].Journal of Building Structures,2003,24(5):1-9.(In Chinease)[11]SHAO Y,ZHU Z,MIRMIRAN A.Cyclic modeling of FRPconfined concrete with improved ductility[J].Cement & Concrete Composites, 2006, 28(10):959-968.[12]LAM L,TENG J G,CHEUNG C H,et al.FRPconfined concrete under axial cyclic compression[J].Cement & Concrete Composites, 2006, 28(10):949-958.[13]LAM L,TENG J G.Stressstrain model for FRPconfined concrete under cyclic axial compression[J]. Engineering Structures, 2009, 31(2):308-321.[14]王震宇, 李洪鵬. 重復(fù)荷載作用下碳纖維預(yù)設(shè)混凝土加卸載準(zhǔn)則 [J]. 建筑結(jié)構(gòu), 2009, 39(7):100-103. WANG Zhenyu, LI Hongpeng. Loading and unloading criteria of FRPconfined concrete under cyclic compression[J]. Building Structure, 2009, 39(7): 100-103. (In Chinese)[15]ABBASNIA R,ZIAADIAN H.Behavior of concrete prisms confined with FRP composites under axial cyclic compression[J].Engineering Structures, 2010, 32(3):648-655.[16]王震宇, 王代玉, 呂大剛,等.CFRP中等約束鋼筋混凝土方柱單軸受壓應(yīng)力應(yīng)變模型[J]. 建筑結(jié)構(gòu)學(xué)報(bào), 2011, 32(4):101-109. WANG Zhenyu,WANG Daiyu,LV Dagang,et al.Stressstrain model for CFRP moderatelyconfined reinforced concrete square columns[J]. Journal of Building Structures, 2011, 32(4):101-109. (In Chinese)[17]WANG Z Y,WANG D Y,SMITH S T,et al.CFRPconfined square RC columns. I: experimental investigation [J]. Journal of Composites for Construction, ASCE, 2012, 16(2):150-160.endprint