Wedge splitting test for a steel–concrete interface

Engineering Fracture Mechanics 72 (2005) 2565–2583

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Wedge splitting test for a steel–concrete interface

Rasmus Walter *, Lennart Østergaard, John F. Olesen, Henrik Stang

Department of Civil Engineering, Technical University of Denmark, Brovej, DK-2800 Kgs. Lyngby, Denmark

Received 24 February 2005; received in revised form 10 June 2005; accepted 14 June 2005

Abstract

This paper presents a test method designated for the determination of the stress–crack opening relationship of asteel–concrete interface. The method is based on the well known wedge splitting test (WST), and it is illustrated howto obtain the stress–crack opening relationship through an inverse analysis. This inversion method utilizes the crackedhinge model, modified such that it describes the problem at hand. In this paper, pure concrete and steel–concrete com-posite specimens are tested and compared. It turns out that interfacial cracking of a bimaterial specimen usuallybehaves as one of the parent materials, in this case concrete. The stress–crack opening relationship of both the concreteand bimaterial specimens are obtained through the proposed inverse analysis. The results show, that interfacial crackingis dominated by the so-called wall-effect and its behavior can be described as quasi-brittle. However, due to the wall-effect, interfacial cracking is more brittle than for the pure concrete.� 2005 Elsevier Ltd. All rights reserved.

Keywords: Nonlinear fracture mechanics; Steel–concrete interface; Inverse analysis

1. Introduction

In composite structures and materials, the weakest part is often the interface between different materials.So, naturally, a lot of attention has been paid to the understanding of the characteristics of bimaterial inter-faces. In the present study a steel–concrete interface is the focus and is defined as the region of the concretemortar near the boundary between the two materials. Experimental experience shows that interfacial crack-ing of a steel–concrete interface usually occurs at a certain distance from the physical boundary. Physically,the interfacial transition zone between concrete and steel has a finite thickness on the microscale, whichis related to the penetration of the cement paste into the rough steel surface, cf. [1]. In the present study

0013-7944/$ - see front matter � 2005 Elsevier Ltd. All rights reserved.doi:10.1016/j.engfracmech.2005.06.001

* Corresponding author. Tel.: +45 452 51766; fax: +45 458 83282.E-mail address: [email protected] (R. Walter).

mailto:[email protected]

Nomenclature

a normalized depth of fictitious crackaw angle of wedgeb1, b2, f1, f2 normalized fracture parametersde, dg, dCOD contributions to CMODdn normal crack deformationds crack sliding deformation� strain�c, �s strain of concrete and steel partj curvaturej�c ; j�

s mean curvature of concrete and steel partl normalized cross-sectional momentq normalized cross-sectional normal forcer, rw normal stresss shear stressh normalized hinge rotationh0–I point of transition between phase 0 and IhI–II point of transition between phase I and IIhII–III point of transition between phase II and IIIu angular deformationuc, us angular deformation of concrete and steel parta0, am coordinates for the notcha1, a2, b1, b2 parameters in the bilinear stress–crack opening relationshipb distance to line of CMOD-measurementbm width of top specimenc constantd depth of crackd1, d2 coordinates for the load pointDij stiffness components for interface descriptionE* bimaterial elastic modulusEc elastic modulus of concreteEs elastic modulus of steelft tensile strengthg(w) normalized function describing shape of stress–crack opening relationshipGf fracture energyGI

f pure mode I energyGII

f pure mode II energyh height of hinge and ligamentK1c mode I stress intensity factorK2c mode II stress intensity factorL side length of WST-cubeM cross-sectional momentN cross-sectional normal forceNmax number of experimental observationsPv vertical load

2566 R. Walter et al. / Engineering Fracture Mechanics 72 (2005) 2565–2583

Psp split loads total hinge widthsc, ss hinge width—concrete and steel partt thickness of specimenu horizontal displacement of load pointw crack openingw1, w2 crack opening limits in the bilinear stress–crack opening relationshipy coordinate along crack directiony0 depth of neutral axis

R. Walter et al. / Engineering Fracture Mechanics 72 (2005) 2565–2583 2567

interfacial cracking is defined as taking place close to or inside the interfacial transition zone. This zone isdominated by the presence of cement paste and lack of aggregates caused by the wall-effect.

The first model describing an interfacial crack between two dissimilar materials was developed by Wil-liams [2]. Later, test set-ups to measure the fracture resistance of bimaterial interfaces using linear elasticfracture mechanics concepts have been developed, i.e. the notched four-point bending beam specimen pre-sented by Charalambides et al. [3]. Soares and Tang [4] proposed a set-up to characterize the linear elasticfracture mechanical properties of a rock/mortar interface using a test which they call: The Bimaterial Bra-zilian Specimen. Since the introduction of the nonlinear fracture mechanics based fictitious crack model byHillerborg et al. [5], papers have been published utilizing this theory. Test specimens have been proposed tomeasure the stress–crack opening relationship of a cement-based interface. Wang andMaji [6] investigated arock/mortar interface and proposed a hinged uniaxial tension set-up to determine the stress–crack openingrelationship. The stress–crack opening relationship was described by a single linear relationship. Verificationwas carried out comparing experiments on bimaterial interfaces using the so-called Compact Tensionspecimen with FE simulations using constitutive input obtained from the uniaxial tension tests. Recently,Tschegg et al. [7] and Chandra Kishen et al. [8] used the WST to evaluate the mode I fracture energy of lime-stone–concrete interfaces. Both studies included comparison between full concrete and limestone specimens,and as expected, in both cases, a more brittle behavior was observed in the bimaterial tests.

This paper proposes a test set-up to determine the fracture properties of a steel–concrete interface using aspecimen where one half is steel while the other half is concrete. The applied set-up is similar to the well-known WST originally proposed by Linsbauer and Tschegg [9]. This new test method, is designated forthe characterization of the mode I crack propagation taking place near the steel–concrete interface. It ishypothesized that this property may be described using the stress–crack opening relationship originally de-fined by Hillerborg et al. [5]. The most straight forward test to determine the stress–crack opening relation-ship would be the uniaxial test since no inverse analysis is necessary. However this test is rather demandingwith respect to laboratory equipment and staff experience. This gives reasons for using theWST which storeslittle elastic energy during testing, thus resulting in a stable experiment, and is well suited for inverse analysis,see e.g. Østergaard [10]. The need for this interfacial material description arises from an ongoing researchproject with focus on the application of strengthening steel bridge decks using a cement-based overlay.The goal is to cast a cement-based overlay on top of the steel bridge deck in order to form a top compositeslab. Hence the fracture properties of a steel–concrete interface is of great interest, see e.g. Walter et al. [11].

2. The bimaterial WST specimen

The proposed test method uses the shape of the well-known wedge splitting specimen as described by e.g.Bruhwiler and Wittmann [12]. This specimen is characterized by a groove and a starter notch, and the idea

(a) (b)

Fig. 1. (a) Geometry of the tested bimaterial WST specimen. The hatched part represents a steel block, specimens dimensions are inmm. (b) Load configuration.


is to replace half of the specimen with a steel block, cf. Fig. 1(a). The specimen is placed on a linear support.Two steel loading devices equipped with roller bearings are placed on top of the specimen. A wedge shapedsteel profile is placed between the bearings. Moving the actuator of the testing machine results in a splittingforce Psp acting between the two parts of the specimen, cf. Fig. 1(b).

During the test, the load in the vertical direction Pv and the crack mouth opening displacement (CMOD)are recorded. For stable experiments closed loop control with CMOD as the control parameter is applied.Frictional forces of the bearing are not taken into consideration since the contribution will be less than 2%[13]. The relation between the horizontal force and the splitting force is calculated using:

P sp ¼P v

2 tanðawÞð1Þ

In the current paper, an angle of aw = 15� is applied.It is furthermore assumed, that all cracking is pure mode I. However, since the bimaterial WST specimen

consist of two materials with different elastic moduli the interface is in fact not a plane of symmetry. Chan-dra Kishen and Saouma [8] analyzed the assumption of pure mode I cracking in a bimaterial WST speci-men. Their approach was to measure the stress intensity factors used in linear elastic fracture mechanics(LEFM) theory. In their investigation the fracture toughness of a limestone concrete interface was mea-sured to K1c ¼ 0.28 MPa

ffiffiffiffim

pand K2c ¼ 0.01 MPa

ffiffiffiffim

p. They conclude that the mode II fracture toughness

is relatively low.In the present study, the assumption of pure mode I cracking is analyzed using the commercial finite ele-

ment package DIANA [14]. The connection between steel and concrete is modelled using standard interfaceelements with a thickness of zero. Steel and concrete are both modelled assuming pure elastic behavior withan elastic modulus of 210 GPa and 30 GPa, respectively. The applied mesh is shown in Fig. 2(a) and isfound to be precise through a convergence analysis.

All nonlinear behavior is assumed to take place at the interface. The constitutive law of the interfaceemploys a mixed mode model implemented in the commercial finite element package using user-supplied

Fig. 2. (a) Applied mesh in the FE analysis to investigate the amount of pure mode I energy consumed using a bimaterial WSTspecimen. (b) Stress distribution at peak load for normal stress, r, and shear stress, s.


subroutines. The model is based on an coupling of the uniaxial r–dn relationship and s–ds relationship.Consider a two dimensional configuration, where the interface element relates the stresses acting on theinterface to the relative displacements, i.e. crack opening and crack sliding. The crack deformation in open-ing mode is denoted dn, while sliding mode is denoted by ds. Their relation to the normal, r, and shear, s,stress is given by:

r

s

� �¼

D11 D12

D21 D22

� �dn

ds

� �ð2Þ

In the cracked state a permanent coupling is present between the normal and shear deformation describedthrough the stiffness components Dij. Two bilinear curves in pure mode I and mode II deformation aregiven as input. The model utilizes the following criterion to describe the state of deformation for every pointon the curves:

dn

dmaxn

� �2

þ ds

dmaxs

� �2

¼ 1.0 ð3Þ

where dmaxn and dmax

s is the maximum crack deformation in normal and sliding mode given in the two bilin-ear curves, respectively. For further details on the model see Walter et al. [15] or Wernersson [16].

The amount of fracture energy consumed in mode I, GIf , and mode II, GII

f , along the steel concrete inter-face for every node can be calculated using the following formulas

GIf ¼

ZC

rðdnÞddn GIIf ¼

ZC

sðdsÞdds ð4Þ

To investigate the amount of mode I and II energy which is consumed, three numerical simulations areinvestigated. In each case the mode I fracture energy of the bilinear curve is set to 0.1 N/mm, whereasthe bilinear curve in mode II is varied according to: GII

f 2 ½0.05; 0.1; 0.2 N/mm. The maximum normaland shear stress is set to 2 MPa in all three cases. Fig. 2(b) shows the normal and shear stress distributionalong the interface of the WST specimen at peak load. For each numerical simulation the amount of mode Iand II energy consumed are calculated using Eq. (4). In every case the mode I energy consumed accountsfor 99.5% of the total amount of energy consumed. Thus, when Es/Ec = 7 the pure mode I crackingassumption will produce a maximum error of 0.5%.

100

101

102

103

104

105

106

0

0.2

0.4

0.6

0.8

Es/E

c

Gf II /G

f I [%]

Fig. 3. The ratio of consumed energy GIIf =G

If in percent, according to formula (4), versus the elastic moduli ratio Es/Ec.


Further, an additional investigation, varying the ratio of the two elastic moduli are carried out to deter-mine the amount of mode II energy consumed for different Es/Ec ratios. The elastic modulus Ec is kept con-stant at 30 GPa, whereas Es is varied according to the following ratios: Es/Ec 2 [1; 7;70;700;700,000]. Theresults are shown in Fig. 3 as the ratio of consumed energy GII

f =GIf in percent versus the elastic moduli ratio

Es/Ec on a logarithmic scale. As observed the amount of mode II energy consumed grows for increasingelastic moduli ratio. However, at some point a threshold is observed.

3. The bimaterial cracked hinge model

The cracked hinge model may be employed to obtain the stress–crack opening relationship using exper-imental WST results. However, since the outcome of the WST is a global load–deflection curve rather thanthe stress–crack opening relationship directly, the latter has to be determined by an inverse analysis. Here,the advantage of the cracked hinge model is that it yields closed-form analytical solutions, which can beimplemented in a simple program written for the purpose of inverse analysis. The idea of a cracked hingewas originally developed by Ulfkjær et al. [17] and further developed by Pedersen [18], Stang and Olesen[19–21]. Furthermore, the proposed method has proven to be robust and accurate for normal concrete,see Østergaard [10].

The basic idea of the cracked hinge is to model the nonlinear behavior, in this case interfacial cracking,between two rigid boundaries. Inside the boundaries, independent horizontal spring elements are given afull elastic and nonlinear description, while the modelling outside the hinge is conducted using classical elas-tic theory. The specific hinge width of the steel and concrete parts are given as ss/2 and sc/2, respectively,while the sum is equal the total band width s, cf. Fig. 4. The constitutive behavior of the steel and concreteparts are modelled using elastic moduli of Es and Ec, respectively. In the cracked state the nonlinear behav-ior of the interface is described through a stress–crack opening relationship:

r ¼Es� steel part

Ec� concrete part

rwðwÞ ¼ gðwÞft interface in cracked state

8><>: ð5Þ

The stress–crack opening relationship is approximated by a bilinear softening curve, see Eq. (6) andFig. 5.

Fig. 4. Geometry, loading and deformation of the bimaterial hinge element.

Fig. 5. Definition of parameters of the bilinear stress–crack opening relationship.


gðwÞ ¼ bi aiw ¼b1 a1w 0 6 w < w1

b2 a2w w1 6 w 6 w2

�ð6Þ

where b1 = 1; while the limit w1 is given by the intersection of the two line segments. The intersection of thesecond line segment and the abscissa is denoted w2:

w1 ¼1 b2a1 a2

; w2 ¼b2a2

ð7Þ

The deformation of the hinge is described through the angular deformation u = us + uc, cf. Fig. 4. Themean value of the curvature for each part of the hinge can be expressed as

j�s ¼ 2

us

ss; j�

c ¼ 2uc

scð8Þ

The distribution of the longitudinal strains at the depth y can be expressed by the mean curvature

�s ¼ ðy y0Þj�s ; �c ¼ ðy y0Þj�

c ð9Þ
In the case where the strip has cracked, the deformation u(y), can be obtained as the sum of the elasticdeformation and the crack opening w.

Fig. 6specim


uðyÞ ¼ ss2

rw

Es

þ sc2

rw

Ec

þ w ð10Þ

By combining Eqs. (9) and (10), we may write

rw ¼ ðy y0Þðus þ ucÞ wð Þ2 ssEs

þ scEc

� �1

ð11Þ

This solution can easily be implemented in the original mono-material solution by Olesen [21], (brieflysketched in Appendix A). The ratio between the bimaterial elastic modulus, E*, and the total band widthis identified as

E�

s¼ 2

ssEs

þ scEc

� �1

ð12Þ

The original solution is presented in terms of normalized properties, where the only modification lies in thethird term.

l ¼ 6

fth2tM ; q ¼ 1

fthtN ; h ¼ hE�

sf t

u; a ¼ dh

ð13Þ

For any given value of the total angular deformation u, the bending moment M and the external force N

can be calculated. Since the crack propagation takes place in different stages, the solution is divided intofour phases. Phase 0 represents the elastic state, followed by three cracked phases. These three phases rep-resent softening: linear, bilinear, and bilinear with stress-free tail. The solution is given in [21].

Fig. 6 shows the geometry and implementation of the cracked hinge in the WST specimen. As seen, thehinge is placed up-side down. It is noted that the load line (the center point of the roller bearings) and thelevel where CMOD is measured do not coincide. The distance from the bottom of the specimen to the loadline is denoted d2, where as the vertical distance to the CMOD line is denoted b, cf. Fig. 6.

Moving the wedge vertically downwards creates a force acting on the specimen. This force can be splitinto two forces acting at the center point of the roller bearing on each side of the specimen. The total loadon the specimen consists of a vertical and a horizontal force, Pv and Psp respectively. The self weight of thespecimen is ignored since the influence can be neglected, see e.g. Østergaard [10].

The moment M and the normal force N acting at the cracking plane can be calculated using equilibrium.From Fig. 6(c) it is seen that the moment and normal force is given by

(a) (b) (c)

. (a) Geometry and loading of the bimaterial WST specimen, (b) application of the hinge element to the bimaterial WSTen, (c) loading.


N ¼ P sp ð14Þ

M ¼ P sp d2 h2

� �þ P spd1 tanðawÞ ð15Þ

The CMOD, depends on three different contributions. The elastic deformation of the specimen causes anopening of CMOD, denoted de. Second, a contribution dCOD denotes the crack opening of the specimen.Finally, a contribution dg has to be taken into account since the line where CMOD is measured in the exper-iment is located differently than the actual crack mouth. Thus, CMOD is given by:

CMOD ¼ de þ dCOD þ dg ð16Þ
Having a wedge splitting specimen consisting of one material, de can be determined using handbooks onstress analysis, e.g. Tada et al. [22]. However, since the specimen differs in geometry and consist of twomaterials having different elastic moduli, the approximation formulaes by Tada et al. [22] are not appli-cable. Using these formulas will produce an error on the determination of Ec of approximately 20%.For higher precision de may simply be determined using a FE model of the adopted WST-geometry. Thisapproach has been used here. Thus, the elastic contribution to the CMOD, de, is calculated according to thefollowing formula
de ¼1

2

P sp

tm2

1

Es

þ 1

Ec

� �ð17Þ

where t is the thickness of the specimen, and the term v2 is determined from a linear elastic FE analysis.Opening of the crack mouth caused by the presence of the crack has been derived by Olesen [21]:

dCOD ¼ sf t

E�1 bi þ 2ah

1 bi; bi ¼

ftaisE� i 2 ½1; 2 ð18Þ

where the terms bi and bi must be chosen according to one of the three cracking phases.The third and final term dg, which is due to a geometrical amplification since there is a certain distance,

b h, from the crack mouth, located at h, to the line where CMOD is measured, located at b. The contri-bution can be found from

dg ¼ 2ðb hÞ dCOD

2ah sf t

hE�h0–I

1 b1

� �ð19Þ

where h0–I represents the normalized angular deformation of the hinge at initiation of the crack. For furtherexplanation see Østergaard [10].

4. Inverse analysis

Methods for the extraction of the constitutive properties of the cracked region of quasi-brittle materialshas been investigated since Hillerborg et al. [5] proposed the fictitious crack model in 1976. The methodshave been analytically as well as numerically based and usually applied to pure concrete, employing thethree point bending specimen. A successful approach was reported by Wittmann et al. [23] in which a sim-ple finite element based method proved applicable for the three point bending specimen. The output was abilinear stress–crack opening curve. Another strategy presented by Kitsutaka [24] yields a multi-linear curvebased on an incremental approach linking specific parts of the load–displacement curve to specific parts ofthe stress–crack opening displacement curve. Lately, the proposed methods of inverse analysis tend to bemore and more complex. Strategies based on evolutionary algorithms have been proposed [25], while alsothe extended Kalman filter procedure (EKF) has been applied by Bolzon et al. [26]. While complex methods


may be justified for complex problems, inverse analysis of pure mode I cracking of concrete, assumingvalidity of the fictitious crack model, is in fact a simple problem and should be treated as such.

A simple approach to the inverse analysis of the concrete–steel interface is application of the semi-ana-lytical hinge model proposed by Olesen [21], utilizing e.g. the familiar simplex optimization algorithm. Amethod based on the above techniques applied to pure concrete employing the WST was proven to be fastand reliable in [10]. As will be shown in the present paper, this is also the case for the concrete–steel inter-facial problem. The method proposed generally yields the solution to the inverse problem in a matter ofminutes rather than hours or days for the more complex methods or FE-methods.

Eqs. (1)–(18) form the basis for interpretation and inverse analysis of the WST. The interpretation pro-ceeds by balancing the internal moment and normal force with the external ones. The idea in the inverse ana-lysis is to use a stepwise algorithm, where the optimization problem is solved in steps corresponding to thedifferent phases of crack propagation. First, the optimization is conducted in the elastic phase with themodulus of elasticity of the concrete part as the only free parameter, and only considering the observationsbelonging to the elastic phase. This first part, named Step I, see Eq. (20), will result in a fast and reliable deter-mination of the modulus of elasticity. It is important to realize that the initial guess on the tensile strength willdetermine how many observations to include in the optimization. But with reasonable initial guesses and byglobally re-running the optimization process (including all steps) this is aminor problem, since the global iter-ations will converge at the true phase change point. Having determined an estimate for the modulus of elas-ticity, the next step is to formulate an optimization strategy for the cracked phases. It turns out that the beststrategy is to separate the problem into two, such that ft and a1 are determined first (Step II), while a2 and b2are determined subsequently (Step III). This is due to results showing that local minimamay be avoided usingthis approach. Note in this context that many methods for inverse analysis are prone to finding local minima,see e.g. Ulfkjær and Brincker [27], Villmann et al. [25] and Bolzon et al. [26]. With the proposed method, localminima are entirely avoided. In contrast to Step I, all observations must be included in the optimization forStep II and III. If only observations belonging to the actual phase (e.g. phase I) were considered, a spurioussolution may be found. This solution represents the minimum where the constitutive parameters have beenselected such that no observation belongs to the considered phase.

Note that it is not possible in the cracked stages to determine both the bimaterial modulus of elasticity,E* and the bimaterial hinge width s from Eq. (12). However, this is no problem since only the ratio E*/senters the calculations, see Olesen [21]. Thus, assuming that the hinge width parameters sc and ss are knownfrom an initial FEM-calibration, Ec is known from the initial optimization of the first slope of the load-CMOD curve, and that the steel modulus of elasticity is known a priori, the E*/s-ratio may be calculated.This result is used for the determination of the constitutive parameters of the crack.

Utilizing all observations in the cracked phases, and using the mean square of differences between obser-vations and predictions as an error norm, the optimization problem reads:

Step I. Determination of Ec

minE

1

N 0max

XN0max

0

P sp P sp

�2 ð20Þ

subject to E > 0

Step II. Determination of ft and a1

minðft;a1Þ

1

Nmax

XNmax

0

P sp P sp

�2 ð21Þ

subject to f t > 0


Step III. Determination of a2 and b2

TableResult

Materi

1234

56

789

10

1112

The nunumbegiven t

minða2;b2Þ

1

Nmax

XNmax

0

P sp P sp

�2 ð22Þ

where N 0max and Nmax represent the last observation belonging to phase 0 and the total number of obser-

vations, respectively. The optimization is restricted such that only physical meaningful solutions are found(Ec > 0, ft > 0 etc.). More details concerning the inverse analysis and the validation with regard to finite ele-ment analysis may be found in [10]. The existence of local minima has been investigated by a large numberof start guesses of the constitutive parameters and the results are clear: All starting guesses will eventuallyresult in a convergence at the global minimum.

The cracked hinge model is dependent on the width of the hinge in both the steel and the concrete parts.Thus, initially, a calibration is conducted in order to identify the most suitable values of these width para-meters. This is done using a finite element model of the WST from which load–CMOD curves are generatedfor a number of different stress–crack opening curves. The selected values of the stress–crack opening pro-file and the error on the determination of these parameters through an inverse analysis using the optimalwidth parameters are given in Table 1. The optimal s-parameters have been determined by an optimizationin which the squared sum of differences between the predicted parameters and the input parameters hasbeen minimized:

minðsc;ssÞ

X f t ftft

!2

þ a1 a1a1

� �2

þ � � � þ Ec Ec

Ec

� �20@

1A ð23Þ

where ðf t; a1; a2; b2; EcÞ are the output values from the usual inverse analysis, see Eqs. (20)–(22), for theactual values of (sc, ss), while (ft,a1,a2,b2,Ec) are the true values which have been entered into the finiteelement code to generate the load–CMOD curve.

The performance of the method of inverse analysis is striking. Only a few results produce errors above5% while the majority of the results are within a error margin of 4% or better. It is not surprising since the

1s from inverse analysis with 12 different materials together with the optimal choices of sc and ss

al ft [MPa] a1 [mm1] a2 [mm1] b2 [–] Ec [GPa] Es [GPa]sch

ssh

2 (8%) 10 (3%) 0.2 (5%) 0.1 (7%) 30 (2%) 210 1.04 0.082 (2) 20 (3) 0.2 (7) 0.1 (4) 30 (2) 210 0.74 0.22 (4) 30 (2) 0.2 (0) 0.1 (1) 30 (2) 210 0.86 0.252 (5) 40 (0) 0.2 (5) 0.1 (1) 30 (3) 210 0.98 0.15

2 (3) 20 (2) 0.6 (0) 0.1 (4) 30 (3) 210 0.8 0.22 (3) 20 (2) 1.2 (2) 0.1 (6) 30 (3) 210 0.74 0.21

2 (4) 20 (3) 0.2 (1) 0.25 (1) 30 (2) 210 0.84 0.22 (4) 20 (4) 0.2 (0) 0.4 (1) 30 (2) 210 0.9 0.182 (2) 20 (22) 0.2 (1) 0.7 (1) 30 (3) 210 0.83 0.18

2 (2) 20 (3) 0.2 (9) 0.1 (6) 40 (3) 210 0.77 0.19

2 (2) 20 (3) 0.2 (2) 0.1 (2) 30 (5) 60 1.22 0.112 (3) 20 (3) 0.2 (5) 0.1 (4) 30 (3) 120 0.94 0.17

mber given on each place in the middle part of the table refers to the input value of the relevant constitutive parameter, while ther in parenthesis is the error on the determination of this value using the method of inverse analysis utilizing the s-parameterso the right.


hinge and the FE-models are quite similar. Disregarded effects in the hinge model are the stress concentra-tion at the tip of the pre-sawn/cast notch and the shear stiffness of the material. The latter is not very impor-tant due the vanishing shear stresses (the problem is virtually symmetric), while the disregarded stressconcentration results in an insignificant change of the curve at the load level of first crack (the crack initi-ates earlier in the FE-model, but this is almost invisible on the load–CMOD curve).

It is also encouraging that the s-values of the hinge are practically constant. The average values and thestandard deviation, based on the results given in Table 1 are (sc, ss) = (0.89 ± 0.18,0.14 ± 0.05). This makesit possible to determine the width parameters once for all thereby making the method practically applicable.The actual results using constant s-values for the present materials are given in Table 2. Here it is noted thata few results are significantly less precisely determined, but also that the majority of results are still deter-mined with a high precision (i.e. an error in the range of 5% or better).

It is important to realize that the results obtained by the method of inverse analysis are based on theentire load–CMOD curves, i.e. from onset of loading until the specimen has been fully separated. If the tailsof the load–CMOD curves are omitted, the errors on the determination of the constitutive parameters willincrease significantly. This is demonstrated in Fig. 7, where the FEM generated P–CMOD curve has beentruncated at different values of CMOD––and the inverse analysis subsequently has been conducted on theremaining curve. The parameter b2 has been selected since it only enters the equations after the crack profilehas turned into a bilinear function—thus a determination of this parameter (likewise a2) requires more P–CMOD curve to be known than e.g. the tensile strength, ft, or in particular modulus of elasticity, Ec, does.As Fig. 7 shows, the determination of b2 by the method of inverse analysis is spurious as long as the P–CMOD curve is truncated before the second point of transition. However, only a few data points after thistransition point are necessary to make the algorithm come up with a reasonable result, but the quality ofthe result continues to improve each time a new portion of the P–CMOD curve is included. It is thereforerecommended to obtain as much of the P–CMOD curve as possible in order to increase the accuracy of theresults. Note also that where a total CMOD of 4 mm seems sufficient for material no. 8, this will not be thecase for e.g. material no. 9 where the high value of b2 results in a far more ductile response of the material.It is more relevant to require that for example the experiment is continued until the load is less than 0.5% ofthe peak load.

Table 2Results from inverse analysis with 12 different materials calculated using constant values of sc and ss

Material ft [MPa] a1 [mm1] a2 [mm1] b2 [–] Ec [GPa] Es [GPa]sch

ssh

1 2 (6%) 10 (1%) 0.2 (11%) 0.1 (9%) 30 (3%) 210 0.89 0.142 2 (5) 20 (0) 0.2 (8) 0.1 (6) 30 (3) 210 0.89 0.143 2 (4) 30 (1) 0.2 (0) 0.1 (2) 30 (3) 210 0.89 0.144 2 (3) 40 (2) 0.2 (6) 0.1 (2) 30 (2) 210 0.89 0.14

5 2 (5) 20 (0) 0.6 (5) 0.1 (2) 30 (3) 210 0.89 0.146 2 (5) 20 (0) 1.2 (61) 0.1 (39) 30 (3) 210 0.89 0.14

7 2 (5) 20 (1) 0.2 (1) 0.25 (2) 30 (3) 210 0.89 0.148 2 (4) 20 (4) 0.2 (0) 0.4 (1) 30 (3) 210 0.89 0.149 2 (3) 20 (20) 0.2 (0) 0.7 (0) 30 (2) 210 0.89 0.14

10 2 (4) 20 (1) 0.2 (9) 0.1 (7) 40 (4) 210 0.89 0.14

11 2 (7) 20 (15) 0.2 (27) 0.1 (12) 30 (7) 60 0.89 0.1412 2 (1) 20 (6) 0.2 (15) 0.1 (8) 30 (5) 120 0.89 0.14

The number given on each place in the middle part of the table refers to the input value of the relevant constitutive parameter, while thenumber in parenthesis is the error on the determination of this value using the method of inverse analysis utilizing the s-parametersgiven to the right.

1E-4 1E-3 0.01 0.1 1 10 1000

500

1000

1500

2000

2500

Split

ting

load

[N

]

CMOD [mm]

0.1

1

10

100

1000

10000

Err

or o

n de

term

inat

ion

of b

2 [%

]

P-CMOD curveChange of stress distrib. Errors on b2

Fig. 7. Load–CMOD curve for material no. 8 as generated by FEM. The hollow circles mark the transition points where, respectively,the crack initiates, changes from linear to bilinear stress distribution, and changes from bilinear to bilinear stress distribution with astress free tail. The solid squares mark the error on the determination of b2, given that the FEM-curve is truncated corresponding to thex-coordinate of the square.


5. Experimental results

The use of the WST to determine the fracture properties of a steel–concrete interface using the inverseanalysis described in the previous section is illustrated. Full concrete and bimaterial WST specimens aretested and their fracture properties are compared.

The concrete used in this study was composed of cement, coarse aggregates 16 mm maximum grain size,fine aggregates, water and various admixtures, such as, fly ash, silica fume and plasticizer to improve thefresh properties of the mixture. The mix design is given in Table 3.

The mix is optimized for self-compacting properties and does therefore not need mechanical vibrationafter casting. This is believed to optimize the steel–concrete bond, see e.g. Schiessl and Zilch [28]. In thecase of the bimaterial WST specimens, the steel surface was sandblasted prior to casting to improve thebond and minimize the risk of defects. The test program consists of 12 test specimens in total, six concreteand six concrete–steel specimens. Specimens where cast in two batches with 3 of each type per batch.

Sample results of the WST experiments are given in Fig. 8, where the splitting force Psp versus CMOD isplotted. The figure also compares results on both the full concrete and bimaterial WST specimens.

In the case of bimaterial WST specimens, cracking took place close to the physical boundary between thesteel and concrete block. This behavior indicates that the actual boundary is stronger than the cement pasteitself. In each bimaterial test, patches of cement paste was left on the steel block. Fig. 9 shows an actual

Table 3Mix design

Mix kg/m3

Cement (Portland, CEM I 52.5) 245Fly ash 94.5Silica fume 10.5Water 142.9Air entraining agent 0.4Plasticizer 4.2Sand, 00–04 mm 752.6Aggregates, 04–08 mm 450.6Aggregates, 08–16 mm 594.0

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

Psp

[kN

]

CMOD [mm]

Two examples on full concrete WST

Two examples on bimaterial WST

Fig. 8. Example on experimental data shown as the splitting load Psp versus CMOD.

Fig. 9. Actual picture of a bimaterial WST specimen after testing. The left part is the steel block.


picture of a bimaterial specimen after testing. The left part of the WST specimen is the steel block. It isclearly seen that some cement paste is still bonded to the steel block.

5.1. Results from inverse analysis

The results of the inverse analysis on the bimaterial WST specimens are shown in Table 4 while the re-sults from the tests on the full concrete specimens are shown in Table 5. The inverse analysis of the fullconcrete specimens are performed according to the method described in [10]. Note that for two tests inbatch no. 1 bearing failure occurred and no results are available.

In general, experimental results obtained in the present study are satisfactory. As a general result thebimaterial WST specimens produced a lower fracture energy in comparison with the full concrete specimen.Same observations have been made by Chandra Kishen and Saouma [8] using the WST. They observed thatlimestone–concrete specimens exhibited a lower fracture energy than comparable full concrete specimens.Regarding the results in Tables 4 and 5, it should be noted that there exist some scatter between the twobatches. No direct evidence exist in order to fully understand the difference but some indications regardingthe behavior of the mix design might explain the difference. By comparing the results of the tests on theconcrete specimens it is noted that the tensile strength ft is higher in batch no. 2 than in batch no. 1.The opposite is the case with the bimaterial tests. A poor bond, due to possible shrinkage, at the edge

Table 5Results from inverse analysis on full concrete WST specimens

Batch ft [MPa] a1 [mm1] a2 [mm1] b2 [–] Ec [GPa] Gf [J/mm2]

1 3.0 15 1.5 0.35 31 0.171 3.1 13 1.7 0.38 36 0.181a – – – – – –Mean 1 3.1 14 1.6 0.36 34 0.17

2 3.6 16 4.2 0.62 21 0.192 3.4 15 3.0 0.46 23 0.162 3.6 11 0.8 0.20 24 0.21Mean 2 3.6 14 2.7 0.42 23 0.19

a Note failure of bearing.

Table 4Results from inverse analysis on the steel–concrete WST specimens

Batch ft [MPa] a1 [mm1] a2 [mm1] b2 [–] Ec [GPa] Gf [J/mm2]

1 3.1 21 1.5 0.26 33 0.121 2.9 20 1.7 0.31 33 0.121a – – – – – –Mean 1 3.0 20.5 1.6 0.29 33 0.12

2 2.3 21 1.4 0.26 29 0.0882 1.8 15 2.7 0.38 26 0.0762 2.1 22 3.3 0.40 35 0.071Mean 2 2.1 19 2.5 0.35 30 0.078

a Note failure of bearing.


of the bimaterial specimens might be the explanation. This is further supported by the visual inspection ofthe bimaterial specimens after testing, which revealed clear signs of shrinkage on the bimaterial specimensin batch no. 2.

5.2. Discussion

The results from the bimaterial and concrete specimens can be viewed in a normalized stress versus crackopening diagram. Fig. 10 shows, for each specimen type, the average stress–crack opening relationship forbatch no. 1 and batch no. 2, respectively.

0 0.05 0.1 0.15 0.2 0.250

0.2

0.4

0.6

0.8

1

w [mm]

(w)/

f t [M

Pa/

MP

a]

0 0.05 0.1 0.15 0.2 0.250

0.2

0.4

0.6

0.8

1

w [mm]

σ(w

)/f t

[MP

a/M

Pa]

InterfaceConcrete

InterfaceConcreteBatch 1 Batch 2

Fig. 10. Stress–crack opening relationship for (a) batch no. 1 and (b) batch no. 2 determined using the inverse analysis.

Fig. 11. Schematic illustration of the wall-effect. A crack path (Path 1) close to a steel wall might be dominated by matrix crackpropagation where as in the case of crack propagation in the concrete material (path 2) crack propagation takes place in combinationof aggregates and matrix.


It is seen that for each batch a consistent pattern is observed. The bimaterial tests exhibits a higher slopein the first part of the stress–crack opening relationship, denoted a1, cf. Fig. 5. In general, the shape of ther–w curve is influenced by the size of aggregates and the matrix/aggregate bond. When a crack propagatesthrough the concrete the fracture strength of the individual parts, matrix/aggregates and their bond, deter-mines if the crack propagates through the aggregate, the matrix or through a combination of both. In thecase of a poor cement paste or a poor matrix/aggregate bond, aggregate pull out is often observed. Thisbehavior leads to a long tail of the r–w curve. The present mix design, Table 3, consist of a mix with opti-mized bonding properties and as a result the cracking has lead to aggregate breakage. As observed inFig. 10 the bimaterial specimens show higher stress drop in the beginning of the r–w curve. This is dueto the wall-effect, which is present in the region close to the steel–concrete boundary. Due to the wall-effect,crack propagation in the bimaterial specimens takes place in the matrix whereas in the full concrete spec-imens cracking takes place in a combination of aggregates/matrix. The higher stress drop in the beginningof the stress–crack opening relationship might be due to the fact that the matrix might be more brittle com-pared to that of aggregates. The wall-effect and the two different crack paths for concrete and bimaterialspecimens are illustrated in Fig. 11. As illustrated, close to the steel block a crack will mainly propagatein the matrix. On the contrary, path 2, consist of a combination of matrix and aggregates as observed inthe full concrete WST specimens.

6. Conclusions

A test set-up combined with a fast and simple inverse analysis to determine the fracture properties of abimaterial interface has been presented. Using the well known WST specimen as the basis set-up, a stabletest has been established. The bimaterial cracked hinge model has proven applicable in this connection.

It is concluded that the optimization strategy is always able to find the global minimum for the FEMcurves analyzed in this study. Two important issues are the width of the hinge and the amount of data re-corded in the experiment. Using FEM, an optimum hinge width ratio was estimated for each of the 12 dif-ferent theoretical materials analyzed. Applying the average hinge width ratio, the majority of the resultsshow an acceptable error. Finally, the study shows the effect of varying the amount of recorded data, whichconcludes the significance of recording data throughout the entire experiment, from the very start to almostzero load.


The proposed set-up and inverse analysis has been employed to a test program aimed at determining thesteel–concrete interface properties. The experimental set-up and the method of inverse analysis could wellbe applied for the investigation of other interface problems, e.g. mortar/rock, etc.

The experimental program consisted of full concrete and steel–concrete specimens. It can be concludedfrom the tests that interfacial cracking is influenced by the wall-effect. The bimaterial specimens show asteeper slope in the beginning of the r–w relationship due to the more brittle behavior of the matrix com-pared to that of the aggregates used in this study.

Appendix A

Considering the solution of the cracked hinge model, this appendix gives a short description of the ori-ginal solution by Olesen [21]. Analysis of the hinge element makes it possible to determine the external nor-mal force N and bending moment M for any given value of the angular hinge deformation u, as shown forthe bimaterial specimen in Fig. 4. The problem is solved in four phases, one for each state of crack prop-agation. Phase 0 represents the elastic state, where no crack has formed, while phases I, II and III representdifferent stages of propagation (linear, bilinear and bilinear with stress-free tail). The solution is presentedin terms of normalized properties, where the following normalizations are used:

b1 ¼fta1sE

; b2 ¼fta2sE

; c ¼ ð1 b2Þð1 b1Þb2 b1

ð24Þ

l ¼ 6M

fth2t; q ¼ N

ftht; h ¼ hEu

sf t

; a ¼ dh

ð25Þ

The hinge solution is expressed in terms of the normalized crack depth a and the normalized moment l asfunctions of the normalized hinge deformation h and the normalized normal force q:Elastic phase: 0 6 h 6 1 q

a ¼ 0 ð26Þl ¼ h ð27Þ

Phase I: 1 q < h 6 hI–II:

a ¼ 1 b1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1 b1Þ

1 qh

b1

� �sð28Þ

l ¼ 4 1 3a þ 3a2 a3

1 b1

� �h þ ð6a 3Þð1 qÞ ð29Þ

Phase II: hI–II < h 6 hII–III:

a ¼ 1 b2 1 b22h

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1 b2Þ

ð1 b2Þ2

4h2ðb1 b2Þ b2 þ

b2 qh

!vuut ð30Þ

l ¼ 4 1 3a þ 3a2 a3

1 b2

� �h þ ð6a 3Þð1 qÞ

ð1 b2Þ 3a2 c2h

� �2� �1 b2

ð31Þ


Phase III: hII–III < h:

a ¼ 1 1

2h1þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1 b2Þ2

b1 b2

þ b22b2

4qh

s0@

1A ð32Þ

l ¼ 4 1 3a þ 3a2 a3 �

h þ ð6a 3Þð1 qÞ 3a2

þ 1

4h21 b2

b2

� �1 b2

b2

þ c� �

1þ b1c1 b1

� �þ c

2h

� �2ð33Þ

In terms of h the points of transitions between phases are given as follows:

h0–I ¼ 1 q ð34Þ

hI–II ¼1

21 q cþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1 q cÞ2 þ c2

b1 1

s !ð35Þ

hII–III ¼1

2qðb2 1Þ þ b2

b2

þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiq2ðb2 1Þ2 þ 2qðb2 1Þ b2

b2

þ ð1 b2Þ2

b1 b2

þ b22b2

s0@

1A ð36Þ

The crack mouth opening displacement CMOD is given by:

CMOD ¼ sf t

E1 bi þ 2ah

1 bið37Þ

where:

ðbi; biÞ ¼ð1; b1Þ in Phase I

ðb2; b2Þ in Phase II

ð0; 0Þ in Phase III

8><>: ð38Þ

References

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Wedge splitting test for a steel–concrete interface

Documents

stresscrack

stresscrack

wedge splitting

ctitious crack

bimaterial

relevant constitutive

bilinear stress

cracked hinge