Cohesive Zone Model Based Numerical Analysis of Steel-Concrete ...

Research ArticleCohesive Zone Model Based Numerical Analysis ofSteel-Concrete Composite Structure Push-Out Tests

J P Lin J F Wang and R Q Xu

Department of Civil Engineering Zhejiang University Hangzhou 310058 China

Correspondence should be addressed to J F Wang wangjinfengzjueducn

Received 3 April 2014 Accepted 24 May 2014 Published 3 July 2014

Academic Editor Gianluca Ranzi

Copyright copy 2014 J P Lin et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Push-out tests were widely used to determine the shear bearing capacity and shear stiffness of shear connectors in steel-concretecomposite structuresThefinite elementmethodwas one efficient alternative to push-out testingThis paper focused on a simulationanalysis of the interface between concrete slabs and steel girder flanges as well as the interface of the shear connectors and thesurrounding concrete A cohesive zone model was used to simulate the tangential sliding and normal separation of the interfacesThen a zero-thickness cohesive element was implemented via the user-defined element subroutine UEL in the software ABAQUSand a multiple broken line mode was used to define the constitutive relations of the cohesive zone A three-dimensional numericalanalysis model was established for push-out testing to analyze the load-displacement curves of the push-out test process interfacerelative displacement and interface stress distribution This method was found to accurately calculate the shear capacity and shearstiffness of shear connectorsThe numerical results showed that themultiple broken linesmode cohesive zonemodel could describethe nonlinear mechanical behavior of the interface between steel and concrete and that a discontinuous deformation numericalsimulation could be implemented

1 Introduction

The shear stiffness and shear bearing capacity of shearconnectors in a steel-composite structure is usually assessedusing push-out test [1ndash7] But push-out test is time con-suming and expensive and its results can be affected byinterface bonding boundary conditions and other factorsFinite element methods can provide an efficient alternativeto full-scale push-out tests It can also be used to carryout parametrical analysis During push-out tests mechanicalbehavior of the interfaces between the concrete slab and thesteel girder flange and between the shear connectors andthe surrounding concrete is relatively complex This complexinterface mechanical behavior is one of the difficulties ofnonlinear numerical analysis involving push-out tests

One of the methods used in numerical analysis of push-out tests involves considering the elastic-plastic behaviorof concrete and steel and neglecting the interface slip andseparation Oguejiofor and Hosain have developed a three-dimensional numerical model using ANSYS software to

analyze push-out specimens with perfobond rib connectors[8] The push-out test specimen was modeled therein using3D reinforced concrete solid elements and shell elements forstructural steel of beam flanges and perfobond rib connec-tors by ignoring the interface mechanical behavior Al-Darziet al also used this method to analyze similar perfobondrib connectors [9] Mirza and Uy developed a 3D nonlinearfinite element model using ABAQUS to study the effectsof combination of axial and shear loading on the behaviorof headed stud steel connectors [10] Solid elements wereused for the concrete slab steel beam and shear connectorsConcrete and steel element nodes at the interface werecoupled and no interface elements were used

Based on an experimental study Johnson and Oehlersfound that separation between the stud and the concrete onthe surface of the stud shank opposite to the load can occureven at low load levels [4] To simulate this phenomenon theyassigned zero stiffness to the coincident concrete elementswith stud shank surface where separation will occur Asimilar method was used by Kalfas and Pavlidis [11] and by

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 175483 12 pageshttpdxdoiorg1011552014175483

2 Mathematical Problems in Engineering

Kim et al [12] Lam and El-Lobody developed a push-outtest finite element model using ABAQUS [13] To simulatethe separation between the stud root and its surroundingconcrete coincident stud nodes in the opposite directionof loading were detached from the surrounding concreteelements while nodes on the surface of the stud shank inthe direction of loading were connected to the surroundingconcrete nodes Ellobody and Young have used a similarmethod to analyze push-out test of composite beams withprofiled steel sheeting [14]

Guezouli and Lachal proposed a 2D nonlinear finiteelement model to study the influence of friction coefficientson push-out tests [15] This simplified 2D model showedstrong convergence but did not consider local damage to thesurrounding concrete of shear studs or spatial mechanicalcharacteristics of the structure Xu et al developed a 3Dfinite element model of push-out testing with group studs[16] Contact interactions available in ABAQUS were usedto simulate the interfaces between steel flanges and concreteslabs and between stud shafts and surrounding concrete[16] Okada et al performed push-out tests on compositestructures with grouped stud connectors and developed a3D numerical analysis model which considered nonlinearproperties of the material and interface bonding friction[17] The interface bonding model consisted of a linearlyincreasing curve rising curve and a peak platform line Apeak bonding stress of 09MPa corresponding to a slip valueof 006mm was used based on experimental results Theinterface bondingmodel did not consider the softening stage

Nguyen and Kim have used the bilinear cohesive zonemodel available in ABAQUS to simulate the mechanicalbehavior of the interface between the concrete slab and thesteel place of the push-out test specimen [18] Then 01119866cmand 01119864cm were used for tangential stiffness and normaltensile stiffness respectively where 119866cm is the shear modulusand119864cm is the elasticmodulus of concreteThe critical relativedisplacement corresponding to peak cohesive stress and themaximum displacement at which the cohesive layer failedwere determined by the authors to facilitate better agree-ment with experimental results Then tangential and normalcritical relative displacements were assigned as 05mm and01mm respectively and the displacement at failure wasassigned as 08mm There were obvious differences betweenpeak cohesive stresses adopted by the author and the actualcohesive stresses For example when C50 concrete was usedthe values of its tangential and normal peak stress were147MPa and 368MPa respectively In the finite elementmodel the shear stud nodes and its surrounding concretenodes were tied together and slip and separation of theinterface between the shear stud and its surrounding concretewere not considered

In summary numerical simulation studies of push-out tests have been conducted by various researchers anddocumented in the literature as discussed above Howeverdetailed numerical analysis taking the complex mechanicalbehavior of the interfaces between the concrete slab and thesteel girder flange and between the shear connectors and itssurrounding concrete into account are not well documentedIn this paper a multiple broken lines mode cohesive zone

nminus

n+tminus

t+

Ω

F

ΓF

Γ+c

Γminusc

Γu

u

Figure 1 Modeling of a cohesive crack

model was used to describe the tangent slip and normalcracking at the interface of steel and concrete Then a zero-thickness cohesive element was implemented using a user-defined element subroutine UEL in ABAQUS [19] Finally athree-dimensional numerical analysis model was presentedsimulating push-out testing The load-displacement curveof the push-out test process interface relative displacementand interface stress distribution were analyzed Numericalsimulation of discontinuous deformation at the interface wasachieved

2 Mechanical Description ofDiscontinuous Deformation

Consider a discontinuous physical domain Ω as shown inFigure 1The domain contains a cohesive crack and the cohe-sive interfaces can be denoted by Γ+

119888and Γminus119888 The prescribed

tractions F are imposed on boundary Γ119865and the prescribed

displacement u on Γ119906 The stress field inside the domain 120590

is related to the external loading F and the tractions t+ andtminus along the discontinuity through the equilibrium equations[20]

div 120590 + f = 0 (inΩ)

120590 sdot n = F (on Γ119865)

u = u (on Γ119906)

t+ = 120590 sdot n+ = t (on Γ+119888)

tminus = 120590 sdot nminus = minust (on Γminus119888)

(1)

Here the traction t is a function of the relative displacementw between Γ+

119888and Γminus119888 that is t = t(w)

The domain surrounding the discontinuity is assumed tobe elastic We further assume small strains and displacement

Mathematical Problems in Engineering 3

condition Thus the constitutive law and geometric equationfor the domain can be written as

120590 = C 120576 (inΩ Γ119888)

120576 = 120576 (u) =[nablau + (nablau)119879]

2(inΩ Γ

119888)

(2)

in which C denotes the material stiffness tensorThedisplacementumust be one of the set of kinematically

admissible displacement U

u isin U = k isin V k = 0 on Γ119906 (3)

Using the principle of virtual work governing equationsin integral form can be written as follows [21]

intΩ

120590 120576 (k) dΩ + intΓ119888

t sdot w (k) dΓ = intΓ119865

F sdot k dΓ forallk isin U

(4)

3 Cohesive Zone Model

The interfaces between the concrete slabs and the steelgirder flanges and between the shear connectors and thesurrounding concrete of a typical push-out test are shownin Figure 2 Cohesive bonding stress exists at the interface ofconcrete and steel during the push-out test process No slipis expected at the interface when longitudinal shear stressesare lower than the bonding resistance As loads increase andlongitudinal shear stresses exceed the bonding resistanceinterface slippage occurs If normal stress exceeds the tensilestrength of the interface crack initiation and propagationtake place which causes uplift forces on the shear connectorsTo conduct continuous-discontinuous deformation analysisof a push-out test a cohesive zone model was used here todescribe the relationship between the interface shear stressand slip displacement and between the normal stress and thetensile displacement

Dugdale proposed the cohesive zone model to describethe relationship between cohesive stress and cracking dis-placement during the material fracture process [22] Yanget al developed a functional relationship between cohesivestress and the relative displacement to analyze mode-I andmode-II fracture using a criterion proposed byWang and Suo[23ndash26] Cohesive zone models have been used to analyzethe mechanical behavior of the bond interface between fiber-reinforced polymers (FRP) and concrete [27 28] Ling et alused a cohesive zone model based augmented finite elementand analyzed progressive failure at the soil-structure interface[20 29] If parameters are properly selected the cohesivezone model can indicate mechanical properties of the bondinterface such as modulus strength and toughness [30]

As shown in Figure 3 a multiple broken line modecohesive zonemodel was used in this paper In thismodelΔ119908and Δ119906 refer to normal displacement and slip displacementrespectively 120590 and 120591 represent the normal and shear stressesrespectively120590

1and 1205911are peak stresses formode-I andmode-

II fractures respectively

Figure 2 Interface between steel and concrete of typical push-outtest

Themultiple broken lines mode cohesive zone model canbe written as follows

120590 =

119870119899Δ119908 (Δ119908 le 0) 1205901 (0 lt Δ119908 le Δ119908

1)

1205901+

Δ119908 minus Δ1199081

Δ1199081minus Δ1199082

(1205901minus 1205902) (Δ119908

1lt Δ119908 le Δ119908

2)

1205902+

Δ119908 minus Δ1199082

Δ1199082minus Δ1199083

(1205902minus 1205903) (Δ119908

2lt Δ119908 le Δ119908

3)

0 Δ119908 gt Δ1199083

120591 =

sgn (Δ119906) 1205911 (0 le |Δ119906| le Δ119906

1)

sgn (Δ119906) [1205911+

|Δ119906| minus Δ1199061

Δ1199061minus Δ1199062

(1205911minus 1205912)] (Δ119906

1lt |Δ119906| le Δ119906

2)

sgn (Δ119906) 1205912 |Δ119906| gt Δ119906

2

(5)

4 Cohesive Interface Element

A zero-thickness cohesive interface element was imple-mented using a user-defined subroutineUEL inABAQUS [2127 31 32] In the user-defined element the element stiffnessmatrix (AMATRX) nodal residual force vector (RHS) andstate variables (SVARS) must be defined

The eight-node cohesive interface element used in thispaper is shown in Figure 4

The nodal displacements of cohesive interface elementin the global coordinate system are denoted by u then therelative displacement between the top and bottom nodes canbe given as follows

120575 (120585 120578) =

4

sum

119894=1

N119894(120585 120578) (u

119894+4minus u119894) (6)

Here N(120585 120578) is the standard shape function The matrixB(120585 120578) is defined as follows

B = (minusN1minusN2minusN3minusN4N1N2N3N4) (7)


120590

1205901

12059021205903

Δw1 Δw2 Δw3 Δw

Kn

(a) Mode-I cohesive law

Δu1 Δu2 Δu

120591

1205911

1205912

(b) Mode-II cohesive law

Figure 3 Cohesive zone model of interface

y

x

1

3

2

5

4

6

z

8 7

120585

120578

Figure 4 Cohesive interface element

Then the relative displacement of the interface can bewritten as follows

120575 (120585 120578) = B (120585 120578) u (8)

Calculation of the transform matrix that describes therelationship between the local and global coordinates isshown below During large deformations initial configura-tion is given by x and the reference surface state x119877 can becomputed using a linear interpolation between the top andbottom nodes in their deformed state as follows

x119877 (120585 120578) =sum4

119894=1N119894(120585 120578) (x + u)2

(9)

T1and T

2indicate unit tangent vectors of the local

coordinate element and T119899is used to denote unit normal

vector The unit normal vector T119899can be written as follows

T119899=

1

1003817100381710038171003817(120597x119877120597120585) times (120597x119877120597120578)1003817100381710038171003817

(120597x119877

120597120585times120597x119877

120597120578)

119879

(10)

Here sdot denotes the norm of a vector Then the unit tangentvector can be given as follows

T1=

1

1003817100381710038171003817120597x1198771205971205851003817100381710038171003817

120597x119877

120597120585

T2= T119899times T1

(11)

Then the transformmatrix that describes the relationshipbetween the local and global coordinates can be written asfollows

T = (T1T2T119899) (12)

Local displacements are then obtained as follows

120575loc = T119879120575 (13)

Cohesive stresses can be calculated using the specifiedcohesive laws (Figure 3) and the relative displacement of theinterface Then node force vector can be obtained as follows

F = int119860

B119879t 119889119860 = ∬1

minus1

B119879Ttloc |J| 119889120585 119889120578 (14)

Here |J| is Jacobi matrix value of the transform matrixThe tangent stiffness matrix of the cohesive interface

element can be written as follows

K119879=120597F120597d

= int119860

B119879T120597tloc120597u

119889119860

= int119860

B119879T 120597tloc120597120575loc

120597120575loc120597120575

120597120575

120597u119889119878

= ∬

1

minus1

B119879TD119862119879T119879B |J| 119889120585 119889120578

(15)

Here D119879= 120597tloc120597120575loc is tangent stiffness matrix of the

cohesive zone modelA solution algorithm of cohesive interface element is

shown in Figure 5In the finite element model cohesive interface elements

are utilized at the interface between concrete and steel tosimulate initiation and propagation of cracks Conventionalsolid elements can be used tomodel the concrete and the steelplate


Select element types

Calculate matrix B

Stiffness matrixAMATRX

Nodal residualforce vector RHS

Update state variablesSVARS

Global nodal stress t

Input CZM parameters

Global relativedisplacement δ

Transform matrix T

Coordinates within reference surface

Shape function N and nodal displacements u

Local nodal stress tloc andtangent stiffness matrix DT

Local relative displacement δloc

Figure 5 Solution algorithm of cohesive interface element

5 Finite Element Model of Push-Out Test

51 Geometry of Push-Out Testing Thegeometry of the push-out test specimen analyzed in this paper was the same as thatused in an experimental study performed by Guezouli andLachal [15] The geometry of the push-out test specimen isshown in Figure 6 The height and width of the steel beamwere 260mm the thicknesses of the flange plate and webplate was 175mm and 10mm respectivelyThe height widthand thickness of the concrete slab were 620mm 600mmand 150mm respectively The diameter of reinforcement inthe concrete slab was 10mm the lengths of the transverseand longitudinal reinforcement were 520mm and 550mmrespectively The height of the studs was 100mm The diam-eter of the stud shanks was 19mm and the diameter of thestud heads was 317mm

52 Material Parameters Constitutive relationship of theconcrete used in this paper is shown in Figure 7(a) Youngrsquosmodulus of the concrete slab 119864

119888= 36900MPa and Poissonrsquos

ratio was equal to 02 The cylinder strength in compression119891119888119896= 56MPa and the one in tension 119891

119905= 396MPa were used

in themodel Based on the information provided in literaturethe proportional limit stress was set at 08 119891

119888119896= 448MPa

and the corresponding strain was set at 00012 [14 17 18]The compressive strain associated with ultimate strength wasequal to 00022 The ultimate strain of concrete at failurein compression and in tension was equal to 001 and 0005

respectively A damage plasticitymodel available in ABAQUSwas utilized for the concrete element

Youngrsquos moduli of the steel beam shear stud and rein-forcement were all equal to 210000MPa Poissonrsquos ratio wasequal to 03 for the steel An ideal elastic-plastic modelwas used for the steel beam The yield strength of the steelbeam was equal to 355MPa The constitutive relationship ofshear stud and reinforcement is shown in Figure 7(b) Theyield stress and ultimate stress were 500MPa and 550MParespectively Based on information available in the literature[10 17 33] strain before strain hardening and strain whenultimate stress is reached are set at 002 and 010 respectively

Coefficients for the cohesive law were derived fromexperimental results published in the literature [2 17 34ndash38]Parameters for mode-II fracture were set as 119888

1= 041MPa

1198882= 0MPa Δ119906

1= 01mm and Δ119906

2= 06mm The tensile

strength of the interface between concrete and steel plate waslow so a small value can be used for peak stress 120590

1 In this

paper parameters formode-I fracture were 1205901= 01MPa 120590

2=

005MPa 1205903= 0001MPa Δ119908

1= 0003mm Δ119908

2= 003mm

and Δ1199083= 015mm The compressive stiffness 119870

119899was set as

20 times 107MPa

53 Finite Element Model The whole geometric model ofthe push-out specimen is shown in Figure 8(a) Because ofthe symmetry it was only necessary to model half of theactual structure using the ABAQUS program as shown inFigure 8(b) Then 3D solid elements were used for concreteslabs steel beams and shear studs Reinforcement was mod-eled using truss elementsThe user-defined cohesive interfaceelements were implemented at the interfaces between theconcrete slab and steel girder flange and between the shearconnectors and the surrounding concrete The finite elementmesh is shown in Figure 8(c) in which the highlighted regionis the position where the cohesive interface elements wereimplemented

In this paper two models with different boundary con-ditions were considered In one of the simulation modelsthe concrete slab at the bottom was allowed to slide freely inlateral direction Degree of freedom U2 was not constrainedThis is hereafter referred to as the lateral free model In theother simulation model the concrete slab at the bottom wasconstrained in lateral direction This is hereafter referredto as the lateral fixed model For the actual push-out testexperiment the real boundary conditions of the concrete slabat the bottom involve contact with the base support Theload-bearing capacity of the shear stud in the experimentwas found to be in between the values observed in the twosimulation models

6 Numerical Analysis

61 Shear Capacity and Shear Stiffness of the Shear ConnectorThe load-slip curves of the push-out test process are shownin Figure 9(a)The ordinate value is the average force per studdefined as the total action load divided by the total number ofstudsThe abscissa is the average value of the slip at the top ofthe interface (point U in Figure 6) and the slip at the bottom


150 260 150

780 100

100

160

160

(a)

600

80

620

A

C

B

U

D

(b)

Figure 6 Push-out test model (unit mm)

minus10

0

10

20

30

40

50

60

minus0006 minus0004 minus0002 0000 0002 0004 0006 0008 0010Strain

Stre

ss (N

mm2)

(a) Concrete

0

100

200

300

400

500

600

0 004 008 012 016Strain

Stre

ss (N

mm2)

(b) Studs and reinforcements

Figure 7 Constitutive laws for concrete studs and reinforcements

XY

Z

(a) Full geometric model

X

Y

Z

(b) Half geometric model

X

Y

Z

(c) Finite element mesh

Figure 8 Finite element model


of the interface (point D in Figure 6) The experimentalresults shown in Figure 9(a) were reported by Guezouli andLachal [15] Results of the shear strength of shear connectorscalculated by Eurocode-4 and AASHTO LRFD are alsoshown in Figure 9(a) [39 40] As shown slip values calculatedusing the two different boundarymodels are similar when theapplied load was relatively small (lt60 kN)The shear capacityof lateral fixed model and lateral free model was 156 kN and138 kN respectively The applied load when the slip valuereached 5mm was adopted as the shear capacity because theapplied load did not increase evidently when the slip valuesexceed 5mm The shear capacity of lateral fixed model wasthe same as the experimental result while the shear capacityof lateral free model was 135 smaller than experimentalvalue Results indicated that the boundary conditions of theconcrete slab at the bottom could influence the shear strengthof the shear connectors and friction at the concrete slab baseincreased the bearing capacity This was consistent with testresults reported by Johnson and Oehlers [4] Results of thesecant shear stiffness are shown in Figure 9(b) These resultsindicate that the secant shear stiffness values calculated usingthe lateral fixed model and the lateral free model were bothsimilar to the experimental results At relatively small slipvalues shear stiffness decreased rapidly as slip increased

62 Separation between the Stud Root and Its SurroundingConcrete For the middle-row studs (as shown in Figure 6stud A (hereafter referred to as the top stud) stud B (hereafterreferred to as the middle stud) stud C (hereafter referred toas the bottom stud)) and the separation between the studroot and the concrete on the surface opposite to the load isshown in Figure 10 As shown separation took place evenat low load levels This is consistent with the experimentalresults reported in the literature [3 4] Separation valuescalculated using the lateral fixedmodel and lateral free modelwere similar when the applied loads were relatively smallThe differences among separation values calculated by thetwo simulation models became larger as the applied loadincreased When the applied load reached 138 kN separa-tion values at the top stud middle stud and bottom studcalculated using the lateral free model were 21 23 and 25times the values calculated using the lateral fixed modelrespectively In the direction of loading separation valuesat the three studs were not equal Separation values werelargest at the top stud and were lowest at the middle studThetop stud was the closest to the applied load and carried themaximum shear force thus separation values at the top studwere largest Separation values at the middle stud were closeto its values at the bottom stud Although the bottom studwas far away from the applied load it was closer than themiddle stud to the bottom of concrete slab and shear forceof the bottom stud was slightly larger than the middle studbecause of the influence of boundary

63 Normal Separation between the Concrete Slab and SteelBeam Normal separation at the bottom centre of the inter-face between the concrete slab and the steel plate (point D inFigure 6) is shown in Figure 11 Results indicated that normal

separation values calculated using the lateral free model werelarger than those calculated using the lateral fixed modelWhen the applied load reached 138 kN the normal separationvalue calculated using the lateral free model (234mm) wasabout 35 times the value calculated using the lateral fixedmodel (0066mm) When the concrete slab at the bottomwas able to slide freely in the lateral direction the upliftphenomenon was more obvious and the shear studs weresubjected to tensile forcesWith this phenomenon it reducedthe shear bearing capacity of the shear connectors The axialstress of the stud when the applied load reached 138 kN isshown in Figure 12

Results of the normal separation distribution at themiddle position (connection line between point U and pointD in Figure 6) and at the side position of the interface inthe push out direction are shown in Figure 13 It must benoted that the separation distribution is not uniform alongthe transverse direction and the simplified 2D model isdifficult to simulate this mechanical behavior accurately [15]It can be seen from Figure 13 that the separation value at themiddle position is larger than at the side position aroundthe top row shear studs Around the bottom of the interfacewhen the concrete slab base can slide freely in the lateraldirection the normal separation value at the middle positionis larger than at the side position The normal separationvalues were relatively smaller when the concrete slab basewas constrained in the lateral direction Displacements in thelateral direction (U2) are shown in Figure 14when the appliedload reached 80 kN

64 Slip Distribution of the Interface between the ConcreteSlab and Steel Beam Results of slip distribution of theinterface between the concrete slab and steel beam in pushout direction are shown in Figure 15 The slip value of theinterface calculated using the lateral free model was largerthan that of the lateral fixed model When the applied loadreached 138 kN the slip value at the top of the interface (pointU in Figure 6) calculated using the lateral free model was 192times the value calculated using the lateral fixed model

Results indicated that the interface slip was not evenlydistributed For the region between the top row studs andthe bottom row studs the slip value at the side position waslarger than at the middle position Slip values were higher inthe regions above the top row studs and below the bottomrow studs because of the compression deformation of the steelflange and concrete slab For regions around the three rowstuds in the push-out direction slip values of the interfacearound the top row studs were larger than in the other tworowsThis is because the top row studs carry a higher share ofthe total shear force When the applied load reached 80 kNthe slip at the top of the interface (point U in Figure 6) andthe slip at the bottom of the interface (point D in Figure 6)calculated using the lateral free model were 058mm and038mm respectively The former value is about 53 largerthan the latter The slip values calculated using the lateralfixed model were 041mm (point U in Figure 6) and 031mm(point U in Figure 6) respectively The former value is about52 larger than the latter When the applied load reached


0

20

40

60

80

100

120

140

160

180

0 1 2 3 4 5 6Slip (mm)

Load

per

stud

(kN

)

Eurocode-4Lateral freeLateral fixedExperiment

AASHTO LRFD

(a) Load-slip curve

0

100

200

300

400

500

600

0 1 2 3 4 5 6Slip (mm)

Lateral freeLateral fixedExperiment

Ks

(kN

mm

)

(b) Secant shear stiffness-slip curve

Figure 9 Load-slip and scant shear stiffness-slip curves

0

20

40

60

80

100

120

140

160

180

0 1 2 3 4 5Relative displacement (mm)

Load

per

stud

(kN

)

Top stud (lateral free)Middle stud (lateral free)Bottom stud (lateral free)

Top stud (lateral fixed)Middle stud (lateral fixed)Bottom stud (lateral fixed)

Figure 10 Separation between the stud root and the surroundingconcrete

138 kN the slip values at the top of the interface calculatedusing the lateral freemodel and lateral fixedmodelwere about9 and 8 larger than the slip values at the bottom of theinterface respectively Results of displacement in the push-out direction (U3) are shown in Figure 16 when the appliedload reached 80 kN

65 Plastic State The magnitude of the plastic strain whenthe applied load reached 138 kN is shown in Figure 17Equivalent plastic strains in tension of the concrete slabare shown in Figure 18 Results indicate that the concretearound the stud root in the load direction has plastic strain

0

20

40

60

80

100

120

140

160

180

0 1 2 251505Relative displacement (mm)

Load

per

stud

(kN

)

Lateral freeLateral fixed

Figure 11 Normal separation at the bottom central portion of theinterface

due to compressive stress As calculated using the lateralfixed model the load values corresponding to the concretedeformation values of 201000 and 351000 were 23 kNand 32 kN respectively While its values calculated usingthe lateral free model were 24 kN and 34 kN respectivelyConcrete around the stud head can undergo plastic strainin tension because of excessive principal tensile stresses Ascalculated using the lateral freemodel themost serious crackstate appears on the concrete around the top stud head Ascalculated using the lateral fixed model the most seriouscrack state appears on the concrete around the bottom studhead Failure modes calculated using the proposed models


+6323e + 08+5241e + 08+4159e + 08+3077e + 08+1995e + 08+9126e + 07minus1694e + 07minus1251e + 08minus2333e + 08minus3415e + 08minus4497e + 08minus5579e + 08minus6661e + 08

(avg 75)S S22

(a) Lateral free model


(avg 75)S S22

(b) Lateral fixed model

Figure 12 Axial stress of the stud when the applied load reached 138 kN

00

01

01

02

02

03

03

04

00 01 02 03 04Distance from top (m)

Relat

ive d

ispla

cem

ent (

mm

)

Middle (lateral fixed)Side (lateral fixed)

Middle (lateral free)Side (lateral free)

(a) When the applied load reached 80 kN

00 01 02 03 04



00

05

10

15

20

25

30

Distance from top (m)

Relat

ive d

ispla

cem

ent (

mm

)

(b) When the applied load reached 138 kN

Figure 13 Normal separation of the interface in push out direction

can agree with the experimental results reported by Ollgaardet al [5]

7 Conclusions

Amultiple broken lines mode cohesive zone model was usedin this study to describe the tangent slip and normal crack atthe interfaces between concrete slab and steel girder flangeand between shear connectors and surrounding concreteA zero-thickness cohesive element was incorporated into afinite element model using the user-defined element sub-routine UEL in ABAQUS A three-dimensional numericalanalysis model was established for push-out testing and aload-displacement curve of the push-out test process inter-face relative displacement and interface stress distributionwere analyzedThe following conclusions are drawn from thisstudy

(1) The method proposed in this paper can accuratelycalculate (a) the shear strength and shear stiffness ofthe shear connectors (b) the normal separation andtangential slip of the interfaces between concrete slaband steel girder flange and (c) the normal separationand tangential slop of the interfaces between shearconnectors and its surrounding concrete Separation

between the stud root and the concrete on the surfaceopposite to the load took place even at low load levelsFor the push-out test model analyzed in this paperseparation values were largest at the top stud lowestat the middle stud Results indicated that normalseparation and tangential slip of the interface betweenconcrete slab and steel girder flange are not evenlydistributed

(2) The boundary conditions of the concrete slab at thebottom can influence the shear strength of the shearconnectors For the push-out test model analyzedin this paper shear capacity when the concrete slabbase was constrained (156 kN) was about 14 largerthan and when the concrete slab base was free inthe lateral direction (138 kN) The separation valuesbetween the stud root and the concrete on the surfaceopposite to the load calculated using the lateral fixedmodel and lateral free model were similar when theapplied loads were relatively small and the differencein the separation values calculated using the twosimulation models became lagrer as the applied loadincreased Constraints of the concrete slab base werefound tomarkedly influence the normal separation ofthe interface between concrete slab and steel girder


+3950e minus 04+3501e minus 04+3053e minus 04+2604e minus 04+2156e minus 04+1707e minus 04+1259e minus 04+8104e minus 05+3619e minus 05minus8657e minus 06minus5351e minus 05minus9835e minus 05minus1432e minus 04

U U2


+8318e minus 05+6375e minus 05+4432e minus 05+2489e minus 05+5466e minus 06minus1396e minus 05minus3339e minus 05minus5282e minus 05minus7224e minus 05minus9167e minus 05minus1111e minus 04minus1305e minus 04minus1500e minus 04

U U2


Figure 14 Displacement in lateral direction when the applied load reached 80 kN

00

01

02

03

04

05

06

07


Relat

ive s

lip (m

m)




0

1

2

3

4

5

6

7

8

Relat

ive s

lip (m

m)





Figure 15 Slip distribution of the interface in push out direction

+3242e minus 05minus3176e minus 05minus9594e minus 05minus1601e minus 04minus2243e minus 04minus2885e minus 04minus3527e minus 04minus4168e minus 04minus4810e minus 04minus5452e minus 04minus6094e minus 04minus6736e minus 04minus7377e minus 04

UU3



UU3


Figure 16 Displacement in push out direction when the applied load reached 80 kN

+3559e minus 01+1000e minus 02+9167e minus 03+8333e minus 03+7500e minus 03+6667e minus 03+5833e minus 03+5000e minus 03+4167e minus 03+3333e minus 03+2500e minus 03+1667e minus 03+8333e minus 04+0000e + 00

PEMAG(avg 75)



PEMAG(avg 75)


Figure 17 Plastic strain magnitude when the applied load reached 138 kN



PEEQT(avg 75)



PEEQT(avg 75)


Figure 18 Equivalent plastic strain in tension when the applied load reached 138 kN

flange When the concrete slab at the bottom wasable to slide freely in the lateral direction normalseparation values were much larger and the upliftphenomenon was more obvious than when it wasnot Because of this phenomenon a free concrete slabreduces the shear bearing capacity of shear connec-tors The tangential slip of the interface between theconcrete slab and the steel girder flange calculatedusing the lateral free model was larger than that of thelateral fixed model

(3) The multiple broken lines mode cohesive zone modelused in this paper was found to effectively describethe nonlinear mechanical properties of the inter-face between the concrete and the steel Thus thediscontinuous deformation numerical simulation ofthe interface was achieved Shear strength and shearstiffness of the shear connectors were calculatedaccurately

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors gratefully acknowledge the support from theZhejiang Provincial Natural Science Foundation (Grant noY1110181) National Natural Science Foundation Projects ofChina (Grant nos 51108411 and 11172266) Project of ZhejiangEducation Department (no N20110091) and the Key Sci-ence and Technology Innovation Team Program of ZhejiangProvince (no 2010R50034)

References

[1] L An and K Cederwall ldquoPush-out tests on studs in highstrength and normal strength concreterdquo Journal of Construc-tional Steel Research vol 36 no 1 pp 15ndash29 1996

[2] J C Chapman and S Balakrishnan ldquoExperiments on compositebeamsrdquo Structural Engineer vol 42 no 11 pp 369ndash383 1964

[3] B S Jayas and M U Hosain ldquoBehaviour of headed studs incomposite beams push-out testsrdquo Canadian Journal of CivilEngineering vol 15 no 2 pp 240ndash253 1988

[4] R P Johnson and D J Oehlers ldquoAnalysis and design forlongitudinal shear in composite T-beamsrdquo in Proceedings of theInstitution of Civil Engineers pp 989ndash1021 1981

[5] J G Ollgaard R G Slutter and J W Fisher ldquoShear strengthof stud connectors in lightweight and normalweight concreterdquoEngineering Journal vol 8 no 2 pp 55ndash64 1971

[6] L Pallares and J F Hajjar ldquoHeaded steel stud anchors incomposite structures I Shearrdquo Journal of Constructional SteelResearch vol 66 no 2 pp 198ndash212 2010

[7] D Xue Y Liu Z Yu and J He ldquoStatic behavior of multi-studshear connectors for steel-concrete composite bridgerdquo Journalof Constructional Steel Research vol 74 pp 1ndash7 2012

[8] ECOguejiofor andMUHosain ldquoNumerical analysis of push-out specimens with perfobond rib connectorsrdquo Computers andStructures vol 62 no 4 pp 617ndash624 1997

[9] S Al-Darzi A R Chen and Y Q Liu ldquoParametric studies ofpush-out test with perfobond rib connectorrdquo in Proceedings ofChina-Japan Joint Seminar on Steel and Composite Bridges pp103ndash111 2007

[10] O Mirza and B Uy ldquoEffects of the combination of axial andshear loading on the behaviour of headed stud steel anchorsrdquoEngineering Structures vol 32 no 1 pp 93ndash105 2010

[11] C Kalfas and P Pavlidis ldquoLoad-slip curve of shear connectorsevaluated by FEM analysisrdquo in Proceedings of the InternationalConference Composite Construction-Conventional and Innova-tive pp 151ndash156 Innsbruck Austria 1997

[12] BKimHDWright andRCairns ldquoThebehaviour of through-deck welded shear connectors an experimental and numericalstudyrdquo Journal of Constructional Steel Research vol 57 no 12pp 1359ndash1380 2001

[13] D Lam and E El-Lobody ldquoBehavior of headed stud shear con-nectors in composite beamrdquo Journal of Structural Engineeringvol 131 no 1 pp 96ndash107 2005

[14] E Ellobody and B Young ldquoPerformance of shear connectionin composite beams with profiled steel sheetingrdquo Journal ofConstructional Steel Research vol 62 no 7 pp 682ndash694 2006

[15] S Guezouli and A Lachal ldquoNumerical analysis of frictionalcontact effects in push-out testsrdquo Engineering Structures vol 40pp 39ndash50 2012

[16] C Xu K Sugiura C Wu and Q Su ldquoParametrical staticanalysis on group studs with typical push-out testsrdquo Journal ofConstructional Steel Research vol 72 pp 84ndash96 2012

[17] J Okada T Yoda and J Lebet ldquoA study of the groupedarrangements of stud connectors on shear strength behaviorrdquoStructural EngineeringEarthquake Engineering vol 23 no 1pp 75sndash89s 2006

[18] H T Nguyen and S E Kim ldquoFinite element modeling ofpush-out tests for large stud shear connectorsrdquo Journal of


Constructional Steel Research vol 65 no 10-11 pp 1909ndash19202009

[19] ABAQUSUserrsquos Manual 610 Dassault Systemes Simulia Prov-idence RI USA 2010

[20] D Ling Q Yang and B Cox ldquoAn augmented finite elementmethod for modeling arbitrary discontinuities in compositematerialsrdquo International Journal of Fracture vol 156 no 1 pp53ndash73 2009

[21] NMoes andT Belytschko ldquoExtendedfinite elementmethod forcohesive crack growthrdquo Engineering FractureMechanics vol 69no 7 pp 813ndash833 2002

[22] D S Dugdale ldquoYielding of steel sheets containing slitsrdquo Journalof the Mechanics and Physics of Solids vol 8 no 2 pp 100ndash1041960

[23] Q D Yang andM DThouless ldquoMixed-mode fracture analysesof plastically-deforming adhesive jointsrdquo International Journalof Fracture vol 110 no 2 pp 175ndash187 2001

[24] Q D Yang M D Thouless and S M Ward ldquoNumericalsimulations of adhesively-bonded beams failing with extensiveplastic deformationrdquo Journal of the Mechanics and Physics ofSolids vol 47 no 6 pp 1337ndash1353 1999

[25] Q Yang and B Cox ldquoCohesive models for damage evolutionin laminated compositesrdquo International Journal of Fracture vol133 no 2 pp 107ndash137 2005

[26] J-S Wang and Z Suo ldquoExperimental determination of inter-facial toughness curves using Brazil-nut-sandwichesrdquo ActaMetallurgica Et Materialia vol 38 no 7 pp 1279ndash1290 1990

[27] Y Wu and W Chen ldquoCohesive zone model based on analysisof bond strength between FRP and concreterdquo EngineeringMechanics vol 27 no 7 pp 113ndash119 2010 (Chinese)

[28] JWang ldquoCohesive-bridging zonemodel of FRP-concrete inter-face debondingrdquo Engineering Fracture Mechanics vol 74 no 17pp 2643ndash2658 2007

[29] D S Ling C Han Y M Chen and C X Lin ldquoInterfacialcohesive zone model and progressive failure of soil-structureinterfacerdquo Chinese Journal of Geotechnical Engineering vol 33no 9 pp 1405ndash1411 2011 (Chinese)

[30] C Y Zhou W Yang and D N Fang ldquoCohesive interfaceelement and interfacial damage analysis of compositesrdquo ActaMechanica Sinica vol 31 no 3 pp 372ndash377 1999 (Chinese)

[31] A De-Andres J L Perez and M Ortiz ldquoElastoplastic finiteelement analysis of three-dimensional fatigue crack growthin aluminum shafts subjected to axial loadingrdquo InternationalJournal of Solids and Structures vol 36 no 15 pp 2231ndash22581999

[32] S FeihDevelopment of aUser Element inABAQUS forModellingof Cohesive Laws Pitney BowesManagement ServicesDenmarkAS Broslashndby Denmark 2005

[33] N H Burns and C P Siess ldquoLoad-deformation character-istics of beam-column connections in reinforced concreterdquoCivil Engineering Studies SRS No 234 University of IllinoisUrbana Ill USA 1962

[34] M Yamada S Pengphon C Miki A Ichikawa and T IrubeldquoShear strength of slab-anchor and adhesion fixing a non-composite girder bridgersquos slabrdquo Journal of Structural Engineer-ing vol 47 no 3 pp 1161ndash1168 2001 (Japanese)

[35] N Gattesco ldquoAnalytical modeling of nonlinear behavior ofcomposite beams with deformable connectionrdquo Journal ofConstructional Steel Research vol 52 no 2 pp 195ndash218 1999

[36] Y Lee Y T Joo T Lee and D Ha ldquoMechanical properties ofconstitutive parameters in steel-concrete interfacerdquo EngineeringStructures vol 33 no 4 pp 1277ndash1290 2011

[37] B G Rabbat and H G Russell ldquoFriction coefficient of steel onconcrete or groutrdquo Journal of Structural Engineering vol 111 no3 pp 505ndash515 1985

[38] K Dorr Ein Beitrag zur Berechnung von Stahlbeton-scheibenunter besonderer Berucksichtigung des Verbund-verhaltens[PhD thesis] University of Darmstadt Darmstadt Germany1980 (German)

[39] ENV 1994-2Eurocode-4Design of Composite Steel andConcreteStructuresmdashpart 2 General Rules and Rules for Bridges CEN2005

[40] AASHTO LRFD Bridge Design Specifications American Asso-ciation of State Highway and Transportation Officials 4thedition 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Kim et al [12] Lam and El-Lobody developed a push-outtest finite element model using ABAQUS [13] To simulatethe separation between the stud root and its surroundingconcrete coincident stud nodes in the opposite directionof loading were detached from the surrounding concreteelements while nodes on the surface of the stud shank inthe direction of loading were connected to the surroundingconcrete nodes Ellobody and Young have used a similarmethod to analyze push-out test of composite beams withprofiled steel sheeting [14]

Guezouli and Lachal proposed a 2D nonlinear finiteelement model to study the influence of friction coefficientson push-out tests [15] This simplified 2D model showedstrong convergence but did not consider local damage to thesurrounding concrete of shear studs or spatial mechanicalcharacteristics of the structure Xu et al developed a 3Dfinite element model of push-out testing with group studs[16] Contact interactions available in ABAQUS were usedto simulate the interfaces between steel flanges and concreteslabs and between stud shafts and surrounding concrete[16] Okada et al performed push-out tests on compositestructures with grouped stud connectors and developed a3D numerical analysis model which considered nonlinearproperties of the material and interface bonding friction[17] The interface bonding model consisted of a linearlyincreasing curve rising curve and a peak platform line Apeak bonding stress of 09MPa corresponding to a slip valueof 006mm was used based on experimental results Theinterface bondingmodel did not consider the softening stage

Nguyen and Kim have used the bilinear cohesive zonemodel available in ABAQUS to simulate the mechanicalbehavior of the interface between the concrete slab and thesteel place of the push-out test specimen [18] Then 01119866cmand 01119864cm were used for tangential stiffness and normaltensile stiffness respectively where 119866cm is the shear modulusand119864cm is the elasticmodulus of concreteThe critical relativedisplacement corresponding to peak cohesive stress and themaximum displacement at which the cohesive layer failedwere determined by the authors to facilitate better agree-ment with experimental results Then tangential and normalcritical relative displacements were assigned as 05mm and01mm respectively and the displacement at failure wasassigned as 08mm There were obvious differences betweenpeak cohesive stresses adopted by the author and the actualcohesive stresses For example when C50 concrete was usedthe values of its tangential and normal peak stress were147MPa and 368MPa respectively In the finite elementmodel the shear stud nodes and its surrounding concretenodes were tied together and slip and separation of theinterface between the shear stud and its surrounding concretewere not considered

In summary numerical simulation studies of push-out tests have been conducted by various researchers anddocumented in the literature as discussed above Howeverdetailed numerical analysis taking the complex mechanicalbehavior of the interfaces between the concrete slab and thesteel girder flange and between the shear connectors and itssurrounding concrete into account are not well documentedIn this paper a multiple broken lines mode cohesive zone

nminus

n+tminus

t+

Ω

F

ΓF

Γ+c

Γminusc

Γu

u

Figure 1 Modeling of a cohesive crack

model was used to describe the tangent slip and normalcracking at the interface of steel and concrete Then a zero-thickness cohesive element was implemented using a user-defined element subroutine UEL in ABAQUS [19] Finally athree-dimensional numerical analysis model was presentedsimulating push-out testing The load-displacement curveof the push-out test process interface relative displacementand interface stress distribution were analyzed Numericalsimulation of discontinuous deformation at the interface wasachieved

2 Mechanical Description ofDiscontinuous Deformation

Consider a discontinuous physical domain Ω as shown inFigure 1The domain contains a cohesive crack and the cohe-sive interfaces can be denoted by Γ+

119888and Γminus119888 The prescribed

tractions F are imposed on boundary Γ119865and the prescribed

displacement u on Γ119906 The stress field inside the domain 120590

is related to the external loading F and the tractions t+ andtminus along the discontinuity through the equilibrium equations[20]

div 120590 + f = 0 (inΩ)

120590 sdot n = F (on Γ119865)

u = u (on Γ119906)

t+ = 120590 sdot n+ = t (on Γ+119888)

tminus = 120590 sdot nminus = minust (on Γminus119888)

(1)

Here the traction t is a function of the relative displacementw between Γ+

119888and Γminus119888 that is t = t(w)

The domain surrounding the discontinuity is assumed tobe elastic We further assume small strains and displacement



120590 = C 120576 (inΩ Γ119888)

120576 = 120576 (u) =[nablau + (nablau)119879]

2(inΩ Γ

119888)

(2)





intΩ

120590 120576 (k) dΩ + intΓ119888



(4)









120590 =

119870119899Δ119908 (Δ119908 le 0) 1205901 (0 lt Δ119908 le Δ119908

1)

1205901+

Δ119908 minus Δ1199081

Δ1199081minus Δ1199082

(1205901minus 1205902) (Δ119908

1lt Δ119908 le Δ119908

2)

1205902+

Δ119908 minus Δ1199082

Δ1199082minus Δ1199083

(1205902minus 1205903) (Δ119908

2lt Δ119908 le Δ119908

3)

0 Δ119908 gt Δ1199083

120591 =

sgn (Δ119906) 1205911 (0 le |Δ119906| le Δ119906

1)

sgn (Δ119906) [1205911+

|Δ119906| minus Δ1199061

Δ1199061minus Δ1199062

(1205911minus 1205912)] (Δ119906

1lt |Δ119906| le Δ119906

2)

sgn (Δ119906) 1205912 |Δ119906| gt Δ119906

2

(5)





120575 (120585 120578) =

4

sum

119894=1

N119894(120585 120578) (u

119894+4minus u119894) (6)




120590

1205901

12059021205903

Δw1 Δw2 Δw3 Δw

Kn


Δu1 Δu2 Δu

120591

1205911

1205912



y

x

1

3

2

5

4

6

z

8 7

120585

120578



120575 (120585 120578) = B (120585 120578) u (8)


x119877 (120585 120578) =sum4

119894=1N119894(120585 120578) (x + u)2

(9)

T1and T




T119899=

1

1003817100381710038171003817(120597x119877120597120585) times (120597x119877120597120578)1003817100381710038171003817

(120597x119877

120597120585times120597x119877

120597120578)

119879

(10)


T1=

1

1003817100381710038171003817120597x1198771205971205851003817100381710038171003817

120597x119877

120597120585

T2= T119899times T1

(11)


T = (T1T2T119899) (12)


120575loc = T119879120575 (13)


F = int119860

B119879t 119889119860 = ∬1

minus1

B119879Ttloc |J| 119889120585 119889120578 (14)



K119879=120597F120597d

= int119860

B119879T120597tloc120597u

119889119860

= int119860

B119879T 120597tloc120597120575loc

120597120575loc120597120575

120597120575

120597u119889119878

= ∬

1

minus1

B119879TD119862119879T119879B |J| 119889120585 119889120578

(15)







Calculate matrix B







Transform matrix T













119888119896= 448MPa





1= 041MPa

1198882= 0MPa Δ119906

1= 01mm and Δ119906

2= 06mm The tensile


1 In this


2=

005MPa 1205903= 0001MPa Δ119908

1= 0003mm Δ119908

2= 003mm


119899was set as

20 times 107MPa






150 260 150

780 100

100

160

160

(a)

600

80

620

A

C

B

U

D

(b)


minus10

0

10

20

30

40

50

60


Stre

ss (N

mm2)

(a) Concrete

0

100

200

300

400

500

600

0 004 008 012 016Strain

Stre

ss (N

mm2)



XY

Z


X

Y

Z


X

Y

Z












0

20

40

60

80

100

120

140

160

180

0 1 2 3 4 5 6Slip (mm)

Load

per

stud

(kN

)


AASHTO LRFD

(a) Load-slip curve

0

100

200

300

400

500

600

0 1 2 3 4 5 6Slip (mm)


Ks

(kN

mm

)



0

20

40

60

80

100

120

140

160

180


Load

per

stud

(kN

)






0

20

40

60

80

100

120

140

160

180


Load

per

stud

(kN

)






(avg 75)S S22



(avg 75)S S22



00

01

01

02

02

03

03

04


Relat

ive d

ispla

cem

ent (

mm

)




00 01 02 03 04



00

05

10

15

20

25

30


Relat

ive d

ispla

cem

ent (

mm

)




7 Conclusions







U U2



U U2



00

01

02

03

04

05

06

07


Relat

ive s

lip (m

m)




0

1

2

3

4

5

6

7

8

Relat

ive s

lip (m

m)







UU3



UU3




PEMAG(avg 75)



PEMAG(avg 75)





PEEQT(avg 75)



PEEQT(avg 75)







Acknowledgments


References


















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120590 = C 120576 (inΩ Γ119888)

120576 = 120576 (u) =[nablau + (nablau)119879]

2(inΩ Γ

119888)

(2)





intΩ

120590 120576 (k) dΩ + intΓ119888



(4)









120590 =

119870119899Δ119908 (Δ119908 le 0) 1205901 (0 lt Δ119908 le Δ119908

1)

1205901+

Δ119908 minus Δ1199081

Δ1199081minus Δ1199082

(1205901minus 1205902) (Δ119908

1lt Δ119908 le Δ119908

2)

1205902+

Δ119908 minus Δ1199082

Δ1199082minus Δ1199083

(1205902minus 1205903) (Δ119908

2lt Δ119908 le Δ119908

3)

0 Δ119908 gt Δ1199083

120591 =

sgn (Δ119906) 1205911 (0 le |Δ119906| le Δ119906

1)

sgn (Δ119906) [1205911+

|Δ119906| minus Δ1199061

Δ1199061minus Δ1199062

(1205911minus 1205912)] (Δ119906

1lt |Δ119906| le Δ119906

2)

sgn (Δ119906) 1205912 |Δ119906| gt Δ119906

2

(5)





120575 (120585 120578) =

4

sum

119894=1

N119894(120585 120578) (u

119894+4minus u119894) (6)




120590

1205901

12059021205903

Δw1 Δw2 Δw3 Δw

Kn


Δu1 Δu2 Δu

120591

1205911

1205912



y

x

1

3

2

5

4

6

z

8 7

120585

120578



120575 (120585 120578) = B (120585 120578) u (8)


x119877 (120585 120578) =sum4

119894=1N119894(120585 120578) (x + u)2

(9)

T1and T




T119899=

1

1003817100381710038171003817(120597x119877120597120585) times (120597x119877120597120578)1003817100381710038171003817

(120597x119877

120597120585times120597x119877

120597120578)

119879

(10)


T1=

1

1003817100381710038171003817120597x1198771205971205851003817100381710038171003817

120597x119877

120597120585

T2= T119899times T1

(11)


T = (T1T2T119899) (12)


120575loc = T119879120575 (13)


F = int119860

B119879t 119889119860 = ∬1

minus1

B119879Ttloc |J| 119889120585 119889120578 (14)



K119879=120597F120597d

= int119860

B119879T120597tloc120597u

119889119860

= int119860

B119879T 120597tloc120597120575loc

120597120575loc120597120575

120597120575

120597u119889119878

= ∬

1

minus1

B119879TD119862119879T119879B |J| 119889120585 119889120578

(15)







Calculate matrix B







Transform matrix T













119888119896= 448MPa





1= 041MPa

1198882= 0MPa Δ119906

1= 01mm and Δ119906

2= 06mm The tensile


1 In this


2=

005MPa 1205903= 0001MPa Δ119908

1= 0003mm Δ119908

2= 003mm


119899was set as

20 times 107MPa






150 260 150

780 100

100

160

160

(a)

600

80

620

A

C

B

U

D

(b)


minus10

0

10

20

30

40

50

60


Stre

ss (N

mm2)

(a) Concrete

0

100

200

300

400

500

600

0 004 008 012 016Strain

Stre

ss (N

mm2)



XY

Z


X

Y

Z


X

Y

Z












0

20

40

60

80

100

120

140

160

180

0 1 2 3 4 5 6Slip (mm)

Load

per

stud

(kN

)


AASHTO LRFD

(a) Load-slip curve

0

100

200

300

400

500

600

0 1 2 3 4 5 6Slip (mm)


Ks

(kN

mm

)



0

20

40

60

80

100

120

140

160

180


Load

per

stud

(kN

)






0

20

40

60

80

100

120

140

160

180


Load

per

stud

(kN

)






(avg 75)S S22



(avg 75)S S22



00

01

01

02

02

03

03

04


Relat

ive d

ispla

cem

ent (

mm

)




00 01 02 03 04



00

05

10

15

20

25

30


Relat

ive d

ispla

cem

ent (

mm

)




7 Conclusions







U U2



U U2



00

01

02

03

04

05

06

07


Relat

ive s

lip (m

m)




0

1

2

3

4

5

6

7

8

Relat

ive s

lip (m

m)







UU3



UU3




PEMAG(avg 75)



PEMAG(avg 75)





PEEQT(avg 75)



PEEQT(avg 75)







Acknowledgments


References


















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










120590

1205901

12059021205903

Δw1 Δw2 Δw3 Δw

Kn


Δu1 Δu2 Δu

120591

1205911

1205912



y

x

1

3

2

5

4

6

z

8 7

120585

120578



120575 (120585 120578) = B (120585 120578) u (8)


x119877 (120585 120578) =sum4

119894=1N119894(120585 120578) (x + u)2

(9)

T1and T




T119899=

1

1003817100381710038171003817(120597x119877120597120585) times (120597x119877120597120578)1003817100381710038171003817

(120597x119877

120597120585times120597x119877

120597120578)

119879

(10)


T1=

1

1003817100381710038171003817120597x1198771205971205851003817100381710038171003817

120597x119877

120597120585

T2= T119899times T1

(11)


T = (T1T2T119899) (12)


120575loc = T119879120575 (13)


F = int119860

B119879t 119889119860 = ∬1

minus1

B119879Ttloc |J| 119889120585 119889120578 (14)



K119879=120597F120597d

= int119860

B119879T120597tloc120597u

119889119860

= int119860

B119879T 120597tloc120597120575loc

120597120575loc120597120575

120597120575

120597u119889119878

= ∬

1

minus1

B119879TD119862119879T119879B |J| 119889120585 119889120578

(15)







Calculate matrix B







Transform matrix T













119888119896= 448MPa





1= 041MPa

1198882= 0MPa Δ119906

1= 01mm and Δ119906

2= 06mm The tensile


1 In this


2=

005MPa 1205903= 0001MPa Δ119908

1= 0003mm Δ119908

2= 003mm


119899was set as

20 times 107MPa






150 260 150

780 100

100

160

160

(a)

600

80

620

A

C

B

U

D

(b)


minus10

0

10

20

30

40

50

60


Stre

ss (N

mm2)

(a) Concrete

0

100

200

300

400

500

600

0 004 008 012 016Strain

Stre

ss (N

mm2)



XY

Z


X

Y

Z


X

Y

Z












0

20

40

60

80

100

120

140

160

180

0 1 2 3 4 5 6Slip (mm)

Load

per

stud

(kN

)


AASHTO LRFD

(a) Load-slip curve

0

100

200

300

400

500

600

0 1 2 3 4 5 6Slip (mm)


Ks

(kN

mm

)



0

20

40

60

80

100

120

140

160

180


Load

per

stud

(kN

)






0

20

40

60

80

100

120

140

160

180


Load

per

stud

(kN

)






(avg 75)S S22



(avg 75)S S22



00

01

01

02

02

03

03

04


Relat

ive d

ispla

cem

ent (

mm

)




00 01 02 03 04



00

05

10

15

20

25

30


Relat

ive d

ispla

cem

ent (

mm

)




7 Conclusions







U U2



U U2



00

01

02

03

04

05

06

07


Relat

ive s

lip (m

m)




0

1

2

3

4

5

6

7

8

Relat

ive s

lip (m

m)







UU3



UU3




PEMAG(avg 75)



PEMAG(avg 75)





PEEQT(avg 75)



PEEQT(avg 75)







Acknowledgments


References


















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Calculate matrix B







Transform matrix T













119888119896= 448MPa





1= 041MPa

1198882= 0MPa Δ119906

1= 01mm and Δ119906

2= 06mm The tensile


1 In this


2=

005MPa 1205903= 0001MPa Δ119908

1= 0003mm Δ119908

2= 003mm


119899was set as

20 times 107MPa






150 260 150

780 100

100

160

160

(a)

600

80

620

A

C

B

U

D

(b)


minus10

0

10

20

30

40

50

60


Stre

ss (N

mm2)

(a) Concrete

0

100

200

300

400

500

600

0 004 008 012 016Strain

Stre

ss (N

mm2)



XY

Z


X

Y

Z


X

Y

Z












0

20

40

60

80

100

120

140

160

180

0 1 2 3 4 5 6Slip (mm)

Load

per

stud

(kN

)


AASHTO LRFD

(a) Load-slip curve

0

100

200

300

400

500

600

0 1 2 3 4 5 6Slip (mm)


Ks

(kN

mm

)



0

20

40

60

80

100

120

140

160

180


Load

per

stud

(kN

)






0

20

40

60

80

100

120

140

160

180


Load

per

stud

(kN

)






(avg 75)S S22



(avg 75)S S22



00

01

01

02

02

03

03

04


Relat

ive d

ispla

cem

ent (

mm

)




00 01 02 03 04



00

05

10

15

20

25

30


Relat

ive d

ispla

cem

ent (

mm

)




7 Conclusions







U U2



U U2



00

01

02

03

04

05

06

07


Relat

ive s

lip (m

m)




0

1

2

3

4

5

6

7

8

Relat

ive s

lip (m

m)







UU3



UU3




PEMAG(avg 75)



PEMAG(avg 75)





PEEQT(avg 75)



PEEQT(avg 75)







Acknowledgments


References


















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










150 260 150

780 100

100

160

160

(a)

600

80

620

A

C

B

U

D

(b)


minus10

0

10

20

30

40

50

60


Stre

ss (N

mm2)

(a) Concrete

0

100

200

300

400

500

600

0 004 008 012 016Strain

Stre

ss (N

mm2)



XY

Z


X

Y

Z


X

Y

Z












0

20

40

60

80

100

120

140

160

180

0 1 2 3 4 5 6Slip (mm)

Load

per

stud

(kN

)


AASHTO LRFD

(a) Load-slip curve

0

100

200

300

400

500

600

0 1 2 3 4 5 6Slip (mm)


Ks

(kN

mm

)



0

20

40

60

80

100

120

140

160

180


Load

per

stud

(kN

)






0

20

40

60

80

100

120

140

160

180


Load

per

stud

(kN

)






(avg 75)S S22



(avg 75)S S22



00

01

01

02

02

03

03

04


Relat

ive d

ispla

cem

ent (

mm

)




00 01 02 03 04



00

05

10

15

20

25

30


Relat

ive d

ispla

cem

ent (

mm

)




7 Conclusions







U U2



U U2



00

01

02

03

04

05

06

07


Relat

ive s

lip (m

m)




0

1

2

3

4

5

6

7

8

Relat

ive s

lip (m

m)







UU3



UU3




PEMAG(avg 75)



PEMAG(avg 75)





PEEQT(avg 75)



PEEQT(avg 75)







Acknowledgments


References


















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra


















0

20

40

60

80

100

120

140

160

180

0 1 2 3 4 5 6Slip (mm)

Load

per

stud

(kN

)


AASHTO LRFD

(a) Load-slip curve

0

100

200

300

400

500

600

0 1 2 3 4 5 6Slip (mm)


Ks

(kN

mm

)



0

20

40

60

80

100

120

140

160

180


Load

per

stud

(kN

)






0

20

40

60

80

100

120

140

160

180


Load

per

stud

(kN

)






(avg 75)S S22



(avg 75)S S22



00

01

01

02

02

03

03

04


Relat

ive d

ispla

cem

ent (

mm

)




00 01 02 03 04



00

05

10

15

20

25

30


Relat

ive d

ispla

cem

ent (

mm

)




7 Conclusions







U U2



U U2



00

01

02

03

04

05

06

07


Relat

ive s

lip (m

m)




0

1

2

3

4

5

6

7

8

Relat

ive s

lip (m

m)







UU3



UU3




PEMAG(avg 75)



PEMAG(avg 75)





PEEQT(avg 75)



PEEQT(avg 75)







Acknowledgments


References


















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

20

40

60

80

100

120

140

160

180

0 1 2 3 4 5 6Slip (mm)

Load

per

stud

(kN

)


AASHTO LRFD

(a) Load-slip curve

0

100

200

300

400

500

600

0 1 2 3 4 5 6Slip (mm)


Ks

(kN

mm

)



0

20

40

60

80

100

120

140

160

180


Load

per

stud

(kN

)






0

20

40

60

80

100

120

140

160

180


Load

per

stud

(kN

)






(avg 75)S S22



(avg 75)S S22



00

01

01

02

02

03

03

04


Relat

ive d

ispla

cem

ent (

mm

)




00 01 02 03 04



00

05

10

15

20

25

30


Relat

ive d

ispla

cem

ent (

mm

)




7 Conclusions







U U2



U U2



00

01

02

03

04

05

06

07


Relat

ive s

lip (m

m)




0

1

2

3

4

5

6

7

8

Relat

ive s

lip (m

m)







UU3



UU3




PEMAG(avg 75)



PEMAG(avg 75)





PEEQT(avg 75)



PEEQT(avg 75)







Acknowledgments


References


















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











(avg 75)S S22



(avg 75)S S22



00

01

01

02

02

03

03

04


Relat

ive d

ispla

cem

ent (

mm

)




00 01 02 03 04



00

05

10

15

20

25

30


Relat

ive d

ispla

cem

ent (

mm

)




7 Conclusions







U U2



U U2



00

01

02

03

04

05

06

07


Relat

ive s

lip (m

m)




0

1

2

3

4

5

6

7

8

Relat

ive s

lip (m

m)







UU3



UU3




PEMAG(avg 75)



PEMAG(avg 75)





PEEQT(avg 75)



PEEQT(avg 75)







Acknowledgments


References


















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











U U2



U U2



00

01

02

03

04

05

06

07


Relat

ive s

lip (m

m)




0

1

2

3

4

5

6

7

8

Relat

ive s

lip (m

m)







UU3



UU3




PEMAG(avg 75)



PEMAG(avg 75)





PEEQT(avg 75)



PEEQT(avg 75)







Acknowledgments


References


















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











PEEQT(avg 75)



PEEQT(avg 75)







Acknowledgments


References


















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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