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Mechanical properties of constitutive parameters in steel–concrete interface
Engineering Structures 33 (2011) 1277–1290
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Mechanical properties of constitutive parameters in steel–concrete interface Yong-Hak Lee a , Young Tae Joo b , Ta Lee c , Dong-Ho Ha a,∗ a
Department of Civil Engineering, Konkuk University, Gwangjin-Gu, Seoul 143-701, South Korea
b
MidasIT Co., Boondang-Gu, Seongnam 463-400, South Korea
c
Ssangyong Eng. & Const. Co., Songpa-Gu, Seoul 138-240, South Korea
article
info
Article history: Received 4 November 2008 Received in revised form 27 December 2010 Accepted 6 January 2011 Available online 5 February 2011 Keywords: Steel–concrete interface Bond and slip Fracture energy Elasto-plasticity
abstract Mechanical properties of steel–concrete interfaces are evaluated on the basis of three existing experimental evidences. The properties include bond strength, unbonded and bonded friction parameters, residual level of the friction parameter, normal fracture energy release rate, bonded and unbonded slip fracture energy release rates under different levels of normal stress, and shape parameters defining the geometrical shape of the failure envelope. For this purpose, a typical type of constitutive model for describing steel–concrete interface behavior is presented based on a hyperbolic three-parameter failure criterion. The constitutive model depicts the strong dependency of interface behavior on bonding condition of the interface, bonded or unbonded. Mechanical roles of the interface parameters are discussed with reference to those of the presented interface constitutive model. Values of the interface parameters are determined through interpretation of existing experimental results, geometry of the failure envelope and sensitivity analyses. These values are applied to push-out tests of concrete-infilled rectangular steel columns with three different cases of interface lengths. The failure process of concrete-infilled rectangular steel columns is discussed through comparison of experimental measurements with numerical results. © 2011 Elsevier Ltd. All rights reserved.
1. Introduction Numerical prediction of interface behavior using nonlinear finite element framework requires a constitutive model to describe the progressive failure localized in the interface region. Numerous research works have been made to model the localized failure process on the steel–concrete interface in the lumped manner ([1–12] among others). Most of the models except for the early work of [1] that used a Lagrange multiplier method have adopted the fracture energy release concept to depict the localized deformation process. However, even if those models are wellestablished based on a robust continuum theory of plasticity, the values of parameters defining the constitutive equation have to be properly determined from the experimental results for a practical purpose. The parameters may be bond strength, bonded and unbonded frictions, residual level of friction, normal (or mode I) fracture energy release rate, bonded and unbonded slip (or mode II) fracture energy release rates under different level of normal stress, and shape parameters defining the geometrical shape of the failure envelope. Little effort has been expended to extract the fundamental properties from post-peak measurements as only a limited
∗
Corresponding author. Tel.: +82 2 450 3514; fax: +82 2 2201 0783. E-mail addresses: [email protected] (Y.-H. Lee), [email protected] (Y.T. Joo), [email protected] (T. Lee), [email protected] (D.-H. Ha). 0141-0296/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.engstruct.2011.01.005
literature is currently available [13–15]. Consequently, the lack of quantitative definition of parameters has hampered the numerical implementation to predict the behavior of steel–concrete composite structures and the further development of constitutive formulation of steel–concrete interfaces. The aim of this paper is laid on defining the mechanical properties of constitutive parameters in steel–concrete interfaces. Steel–concrete interface behavior is defined in the normal and tangential directions to the interface plane. Once reaching their critical states, tensile and shear behaviors in the normal and tangential directions, respectively, follow mode I type of normal fracture (or debonding) and mode II type of slip fracture failure mechanisms. The slip on the interface strongly depends not only on the magnitude of the normal stress to the interface but also on the bonding condition of interface, bonded and unbonded, as [13,15] observed in their experiments performed on sandwichtyped steel–concrete interface specimen tests. To identify the constitutive parameters, a comprehensive constitutive model that accounts for the strong dependency of interface slip on the bonding condition of interface is presented by modifying the hyperbolic failure criterion of [5,7]. The modification is attained by expressing the bonded friction parameter in terms of the unbonded friction parameter and friction adjusting parameter. Values of parameters defining the model are determined through existing experimental results, geometry of failure envelope and sensitivity analyses. Only
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Notations b: bmax : bres : De : Dep : DN : DT : 1u: fI : GIf :
Bond parameter bond strength Residual bond parameter Elastic interface tangent matrix Elasto-plastic interface tangent matrix Elastic interface normal stiffness Elastic interface tangent stiffness Incremental relative displacement Nonassociativity intensity factor Normal fracture energy release rate
GIIfb :
Slip fracture energy release rate for bonded
GIIfu :
Slip fracture energy release rate for unbonded Curvature parameter Maximum curvature parameter Plastic flow vector Normal vector to failure surface Normal fracture work Slip fracture work Slip fracture work fraction parameter for friction Normal fracture work fraction parameter for bond Slip fracture work fraction parameter for bond Bonded friction parameter Maximum bonded friction parameter Unbonded friction parameter Maximum unbounded friction parameter Residual friction parameter Friction adjusting parameter Nonassociativity inflection parameter Normal stress Tangential stress Maximum tangential stress for bonded
k: kmax : m: n: WPN : WPT : a: βN : βT : φb : φbmax : φu : φumax : φ res : η: σIP : σN : σT : max σTb :
a few experimental results are available to evaluate mechanical properties of interface parameters because of highly pressuresensitive nature of the steel–concrete interface. Consequently, three existing experimental evidences of [13,16–18], and [15] are used to determine the values of parameters defining the model. Maximum value of unbonded friction parameter, residual level of friction parameter and unbonded slip-friction energy are calculated from the unbonded tests of [13]. Bond strength and normal fracture energy release rate are determined by geometrical relationships between parameters. Bonded slip fracture energy release rate and shape parameters of the failure envelope are determined through sensitivity analyses performed based on the experiments of [16,17,15]. As a result, those values are applied to pushout tests of concrete-infilled rectangular steel columns with three different cases of interface lengths of [16–18]. The failure process of a concrete-infilled rectangular steel column is discussed through the comparison of experimental measurements with numerical predictions. 2. Failure criterion for steel–concrete interface Interface behavior between steel and concrete can be decomposed into mode I type of normal fracture and mode II type of slip fracture. A number of failure criteria have been presented including a linear representation of [2], a parabolic representation of [3,4], and hyperbolic representations of [5,6]. The hyperbolic criterion that was initially developed by Zienkiewicz and Pande [19] take advantage of exhibiting C 1 -continuity throughout the entire response regime and is asymptotically equivalent to
Fig. 1. Three-parameter failure envelope of steel–concrete interface.
the Mohr–Coulomb criterion when compressive normal stress increases. In this paper, the hyperbolic criterion is adopted to define the onset of interface fracture and monitor the progressive fracture process. Slip on the interface strongly depends not only on the magnitude of the normal stress to the interface but also on the bonding condition of the interface, bonded and unbonded, as observed in the sandwich-typed steel–concrete interface specimen tests of [13,15]. To incorporate the behavior of unbonded as well as bonded interfaces, the hyperbolic representation of the failure envelope of [5,6] is modified as F (σN , σT , φ, b, k) = σT2 − φ 2 {(σN − b)2 − 2k(σN − b)} where σN is normal stress, σT =
(1)
σL2 + σM2 is tangential stress, the
subscripts N, L and M represent the local coordinates along normal and tangential directions to the interface plane, respectively. b and k are bond and curvature parameters, respectively, and φ is the friction parameter. The friction parameter φ is defined as φb and φu for bonded and unbonded interfaces, respectively. The bonded friction parameter φb is expressed in terms of the unbonded friction parameter φu and friction adjusting parameter η to account for the dependency of the slip behavior on interface bonding conditions as
φb = φu + ηb
(2)
where the friction adjusting parameter η bridges the gap between bonded and unbonded frictions. Fig. 1 shows the geometrical relationship among the three parameters. Evaluating the friction parameter value for the test specimens of [13,15] according to Coulomb’s friction law, it is smaller than 1.0 for the unbonded and greater than 1.0 for the bonded, which clearly shows a strong dependency of interface behavior on the bonding condition of the interface. Along the line of classical theory of plasticity, progressive interface failure can be depicted by the evolution of the failure envelope where the evolution process may be monitored by gradually decreasing the values of three parameters φ , b and k from the maximum values of φ max , bmax and kmax to the residual values of φ res , bres and kres , respectively. To resolve the missing link between the maxima and residues, the three parameters are commonly modeled in terms of energy fractions between fracture energy release
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rate and related fracture work [5,6]. In this paper, the bond parameter b is assumed to be dependent on slip fracture as well as normal fracture similar to the work of [5] to account for the effect of a slipinduced shear crack on normal cracks. The other two parameters of friction and curvature, φ and k, are assumed to be dependent only on slip fracture. Thereby, the three parameters are modeled in terms of energy fractions of normal and slip fracture energy release rates, and related fracture works as
b=b
res
+ (b
max
WPT
WPN
WPT
k = kres + (kmax − kres ) 1 − γ
+ ηb
(3)
WPT GIIfb
where βN and βT are bond-related normal and slip fracture work parameters, respectively, α and γ are friction- and curvaturerelated slip fracture work parameters, respectively. The values of βN , βT , α , and γ are determined by limiting the value of parentheses to range from 0 to 1. GIf and GIIfb are normal and bonded slip fracture energy release rates, respectively. It is noted in Eq. (3) that the evolutions of friction and curvature parameters are controlled by the energy fraction between slip fracture energy release rate and related fracture work rather than fracture work itself of [5] because the variation of slip fracture energy release rate under different level of normal stress can be obtained from existing experimental results. In cases of normal fracture and unbonded slip fracture, the three parameters of Eq. (3) are modified as; (1) Normal fracture: the bond parameter b governs the interface fracture process while the friction and curvature parameters of φ and k remain at their maxima.
b=b
φb = φ
+ (b
max u
max
a
− b ) 1 − βN I − βT II Gf Gfb res
φb = φ res + (φbmax − φ res ) 1 − α II Gfb
res
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− b ) 1 − βN res
+ ηb,
k=k
max
.
WPN GIf
− βT
WPT
b
(4)
GIIfb
(2) Unbonded slip fracture: the friction and curvature parameters of φ and k decrease from their maxima to residues while the bond parameter b remains its residual
b=b ,
φu = φ
res
res
+ (φ
max u
k = kres + (kmax − kres ) 1 − γ
WPT
− φ ) 1 − α II Gfu res
WPT GIIfu
Fig. 2. Incremental plastic work. (a) Normal fracture. (b) Shear fracture.
(5)
8
.
7 4
Normal and slip fracture works of WPN and WPT of Eq. (3) (see Fig. 2) are defined in incremental formats of
σN ≥ 0 σN < 0 = (σT − σTres ) · duPT
dWPN = dWPT
M 5
σN duPN 0
zero-thickness
N 3
L 6
(6)
1 2
where uPN and uPT are relative normal and tangential displacements, duPT = (duPL )2 + (duPM )2 , and residual tangential stress
σTres = ±φ res (σN − bres )2 − 2kres (σN − bres ). Fig. 3. Description of 8-node zero-thickness interface element.
3. Elasto-plastic interface formulation In the case of infinitesimal deformation, the relative total dis˙ may be decomposed into independent placement rate vector u ˙ e and u˙ p , respectively, and leadelastic and plastic components of u ˙ = u˙ e + u˙ p . Adopting a zero-thickness finite interface eleing to u ment in Fig. 3, stress vs. relative displacement relationship in local coordinate is expressed as
where σ = {σN σL σM }T and ue = {uN uL uM }T . De = diag{DN DL DM } is a diagonal matrix to define the elastic stiffnesses of the interface along normal and tangential directions, and defined as normal and slip stiffnare the plasticesses in Fig. 4(a) and (b), respectively. Adopting a nonassociated flow rule, plastic displacement vector ˙ p is defined in terms of u
σ˙ = De : (u˙ − u˙ p )
˙ p = λ˙ m u
(7)
(8)
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a
a
b Unbonded behavior
b
Bonded behavior
Fig. 5. Nonassociated plastic flow. (a) Direction of plastic flow on failure surface. (b) Nonassociativity intensity factor fI .
Fig. 4. Stress vs. relative displacement relationship. (a) Normal direction. (b) Tangential direction.
˙ and m are the plastic multiplier and the plastic flow where λ vector, respectively. When the current stress predicted with elastic increments of Eq. (7) violates the current state of the failure criterion of Eq. (1), i.e. F > 0, plastic flow occurs. A truncated Taylor series expansion of Eq. (1) leads to the linearized format of ˙ consistency condition to calculate the plastic multiplier λ F˙ =
∂F ∂F : σ˙ + u˙ p = 0 ∂σ ∂ up
(9)
where equivalent plastic displacement u˙ p is obtained from the ˙ m‖. The plastic multiplier λ˙ is obtained Euclidean norm of u˙ p = λ‖ by substituting Eqs. (7) and (8) into (9) as
˙ n : De : u (10) −H ‖m‖ + n : De : m where n = ∂ F /∂σ , m = ∂ Q /∂σ , and Q is plastic potential. The
λ˙ =
hardening modulus H in Eq. (10) is calculated through the chain rule of differentiation as ∂ F ∂φb ∂φu ∂ WPT ∂ F ∂φb ∂F H = + + ∂φb ∂φu ∂ WPT ∂ uPT ∂φb ∂ b ∂b
× +
∂ b ∂ WPN ∂ b ∂ WPT + ∂ WPN ∂ uPN ∂ WPT ∂ uPT
∂ F ∂ k ∂ WPT . ∂ k ∂ WPT ∂ uPT
(11)
Shear dilatancy on interface is primarily due to the saw-tooth effect of an irregularly cracked interface plane and decreases with abrasion of the residue and confining effect of normal pressure. It is a common way to adopt nonassociated flow rule to control the excessive dilatancy in a computational plasticity. Fig. 5(a) schematically shows the change of plastic flow direction with increasing confining pressure. In this paper, the direction of nonassociated plastic flow m is determined by adjusting the normal vector n similarly to the formulation of [20]. Adopting the concept, the nonassociativity intensity factor fI is introduced as a function of the nonassociativity inflection parameter σIP . Thereby, the direction of plastic flow component along normal direction is modified with nonassociativity intensity factor fI while the other components along the tangential direction remain unchanged
∂F ∂Q ∂F ∂Q = , = ∂σL ∂σL ∂σ ∂σM M ∂Q ∂F φ − φ res = fI ∂σN ∂σN φ max − φ res
(12)
where nonassociativity intensity factor fI shown in Fig. 5(b) is defined as fI = {(σN − σIP )/σIP }2 . Elastoplastic tangent operator Dep can be obtained by substituting the plastic multiplier of Eq. (10) into (7) as Dep = De −
De : m ⊗ n : De . −H ‖m‖ + n : De : m
(13)
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Test point
a
Confining pressure 610
Slip force
340
150
360
25 710 Concrete Block Steel Plate
Fig. 8. Geometrical description of failure envelope in peak.
Slip force
b
Confining pressure
150
Confining pressure
20 150 150 LVDT
Conc. Steel
Fig. 6. Descriptions of test specimens. (a) Rabbat and Russell [13] and (b) Chiew et al. [15].
Fig. 9. Sensitivity analysis results with respect to normal fracture energy release rate of GIf .
4. Parameter values of steel–concrete interface Values of nine constitutive parameters defining the presented model were determined based on the existing experimental results, geometry of the failure envelope and sensitivity analyses. Nine parameters are the maximum and residual values of unbonded friction parameter φumax and φ res , unbonded and bonded slip fracture energy release rates GIIfu and GIIfb , normal fracture energy release rate GIf , bond strength bmax , maximum value of curvature parameter kmax , friction adjusting parameter η, and the nonassociativity inflection parameter σIP . Due to the limited number of available experiments, three cases of experimental results of [13,16,17,15] were used for determining the parameter values. 4.1. Experiments of Rabbat and Russell [13], Chiew et al. [15] and Shakir-Khalil [16,17]
Fig. 7. Unbonded slip test results of [13].
Fig. 6(a) shows a test specimen of a steel–concrete interface tested by Rabbat and Russell [13]. In the experiments, a concrete block with 150 mm thickness, 340 mm width and 610 mm length was cast on a 25 mm thick, 360 mm wide and 710 mm long steel plate of Grade 70 according to the recommendation of A516 of ASTM. The surface of the steel plate was treated according to the
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a
b
c
Fig. 10. Sensitivity analysis results with respect to φumax ; (a) σN = −0.5 MPa (b) σN = −1.0 MPa (c) σN = −1.5 MPa.
recommendation of SSPC-SP2-63 of the Steel Structures Council Specifications. At 7 days of age, the formwork of the concrete block was removed and cured in the laboratory with 23 °C temperature and 50% humidity. A confining pressure of σN = 0.06 MPa was applied to the specimen with fc′ = 30 MPa compressive strength of concrete and then, a bonded slip test was performed by applying a slip force until the interface crack was fully developed. After the bonded slip test, an unbonded slip test was carried out on the tested specimen in the same manner except for a confining pressure of σN = 0.41 MPa. Fig. 6(b) shows a test specimen of [15] where a 20 mm thick steel plate was placed between two concrete blocks that are 150 mm wide, 150 mm high and 65 mm thick. To investigate the effects of confining pressure on interface behaviors, three different levels of confining pressures including σN = 0.5, 1.0, and 1.5 MPa were applied to the specimens. In the case of σN = 1.5 MPa confining pressure, the bonded specimen was tested until an interface crack was fully developed and then, the tested specimen was retested for unbonded slip behavior with the same confining pressure to investigate the difference between bonded and unbonded. Shakir-Khalil [16,17] evaluated the failure strength of steel– concrete interfaces by testing forty concrete-infilled rectangular
steel columns. The effects of the number of shear studs on the failure strength were also addressed by considering three cases of shear stud arrangements including 2, 4 and 6, and no shear stud case. All the test specimens with 150 mm × 150 mm cross -sectional dimensions were made of 5 mm thick steel plate. Three different cases of interface lengths of 200, 400 and 600 mm were taken into account to investigate the effects of the interface area on the failure strength. 4.2. Determination of interface parameter values 4.2.1. Maximum and residual values of unbonded friction parameter φumax and φ res , unbonded slip fracture release rate GIIfu
Values of φumax and φ res were determined according to the definition of the friction parameter expressed by the ratio between shear and normal stresses. For this purpose, the maximum unmax bonded average shear stress of σTu = 0.28 MPa was obtained from the slip test results of [13]. The maximum value of the unbonded friction parameter φumax = 0.7 was calculated by dividing max the maximum shear stress σTu = 0.28 MPa by the confining pressure σN = 0.41 MPa. In a similar way to φumax , the residual value of the friction parameter φ res = 0.57 was calculated by dividing
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b
a
c
Fig. 11. Sensitivity analysis results with respect to φ res ; (a) σN = −0.5 MPa (b) σN = −1.0 MPa (c) σN = −1.5 MPa. Table 1 Experimental results of [13]. No.
1 2 3
Comp. strength (MPa)
25.0 25.6 25.6
Averaging shear stress (MPa) for unbonded
Friction coefficient
Maximum
Residual
Maximum
0.28 0.32 0.25
0.23 0.24 0.23
0.68 0.77 0.62
the average residual shear stress σTres = 0.23 MPa by the confining pressure σN = 0.41 MPa. Table 1 summarizes the test results and the calculated friction parameter values. Residual shear stresses in Table 1 were defined with the shear stress levels to which the shear stresses were converged. The value of the unbonded slipfracture release rate GIIfu = 1.21σN N mm/mm2 was obtained by measuring the dotted triangular area (12 mm × 0.08 MPa × 0.5 = 0.48 N mm/mm2 ) of Test-3 in Fig. 7. 4.2.2. Bond strength bmax and curvature parameter kmax Considering the asymptotic convergence of failure envelope with the increase of compression as shown in Fig. 1, the maximum failure envelope may be linearly approximated as φbmax (|σN | + max bmax + kmax ) = σTb . Fig. 8 shows geometrical relationship among
Average
0.69
Minimum 0.56 0.58 0.57
Average
0.57
max σN , φbmax , kmax , bmax , and σTb . Substituting φbmax = φumax + ηbmax max max of Eq. (2) and b = 2k obtained from the sensitivity anal-
yses for the test results of [13,15] into the linearly approximated equation, a second order polynomial equation with respect to bond strength bmax is derived as η(bmax )2 +{φumax +η(|σN |+ kmax )}bmax + max φumax (|σN | + kmax ) − σTb = 0. Bond strength bmax can be readily calculated with the equation when a relatively high value of max normal stress σN and corresponding σTb defining a test point on the maximum failure envelope are known from a bonded interface experiment. Bond strength bmax was calculated and compared to the test results of [13,15] to examine the degree of sensitivity with respect to the level of shear stress. In case of [13], the maximum bonded shear max stress of σTb = 0.41 MPa was obtained by averaging the three
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a
b
c
Fig. 12. Sensitivity analysis results with respect to bmax ; (a) σN = −0.5 MPa (b) σN = −1.0 MPa (c) σN = −1.5 MPa. max test results of σTb = 0.46, 0.37, 0.4 MPa. Substituting the average max shear stress σTb = 0.41 MPa and the corresponding normal stress σN = 0.06 MPa into the second order polynomial equation and solving with respect to bmax , a bond strength of bmax = 0.13 MPa and a curvature parameter of kmax = 0.65 MPa were obtained. The friction adjusting parameter η = 6.5 was used in the calculation which was obtained through sensitivity analyses for the test results of [15]. In case of [15], the maximum bonded shear max stresses of σTb = 0.79, 1.51, and 2.25 MPa were obtained from the three tests considering three confining pressures of σN = 0.5, 1.0, and 1.5 MPa, respectively. Bond strengths of bmax = 0.086, 0.096, max and 0.108 MPa corresponding to each of σTb = 0.79, 1.51, and 2.25 MPa, respectively, were calculated by solving the second order polynomial equation. As a result of the calculation, bond strength bmax = 0.1 MPa and curvature parameter kmax = 0.05 MPa were obtained by averaging the three bond strengths. Considering the different material properties, testing methods, curing environments, and measuring instruments, 30% of difference between the two calculated bond strengths may ensure that the presented approach could be a proper tool to determine the value of bond strength without performing a direct tension test.
4.2.3. Normal fracture energy release rate GIf , bonded slip fracture en-
ergy release rate GIIfb , friction adjusting parameter η, and nonassociativity inflection parameter σIP The values of GIf , GIIfb , η, and σIP were determined through numerical failure tests for the test results of [13,15–17]. Parameter values of φumax = 0.7, φ res = 0.57, GIIfu = 1.21σN N mm/mm2 , and bmax = 0.1 MPa obtained from the slip test results of [13] were used for the analyses. Since the residual shear stress σTres = 1.2 MPa measured from the test results of [15] showed relatively large difference with the residual shear stress σTres = 0.85 MPa calculated from σTres =
±φ res (σN − bres )2 − 2kres (σN − bres ) of Eq. (6),the bonded slip
fracture energy release rate GIIfb was determined through sensitivity analyses. The initial value of GIfb = 9 N mm/mm2 required for the analysis was obtained by measuring the area of shear stress vs. relative slip displacement curve and residual shear stress level, and used for the analyses. The bonded slip fracture energy release rate of GIIfb = (6.5σN + 0.12) N mm/mm2 was obtained by linearly interpolating the three values of GIIfb measured from shear stress vs. relative slip displacement curves obtained for the three cases of σN = 0.5, 1.0, and 1.5 MPa. A normal fracture energy release rate
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b
a
c
Fig. 13. Sensitivity analysis results with respect to kmax ; (a) σN = −0.5 MPa (b) σN = −1.0 MPa (c) σN = −1.5 MPa.
of GIf = 0.2N mm/mm2 was obtained through sensitivity analyses that considered three cases of normal fracture energy release rates of GIf = 0.05, 0.2, 0.35 N mm/mm2 . The initial value of friction adjusting parameter η was calculated from the maximum failure envelope expression of Eq. (2), φbmax = φumax + ηini bmax . The value of φbmax = 1.47 used in this calculation was obtained by dividing the maximum bonded shear max stress σTb = 2.2 MPa by the confining pressure σN = 1.5 MPa. In addition, the value of φumax = 0.77 was obtained by dividing the max maximum unbonded shear stress σTu = 1.15 MPa by the confining pressure σN = 1.5 MPa. As a result, the friction adjusting parameter η = 6.5 was determined through sensitivity analysis with reference to the initial value of η = 7. Nonassociativity inflection parameter σIP = 0.25 MPa was obtained through the sensitivity analysis performed for the test results of [16,17]. All the sensitivity analysis results are discussed in the subsequent section. 4.2.4. Sensitivity analyses with respect to constitutive parameter values Sensitivity analyses were carried out to investigate the applicability of the obtained parameter values and the influence on the overall interface behavior. The analyses were referenced on the
test results of [15] and performed by giving a perturbation for the reference value of a constitutive parameter while the other constitutive parameters were kept as reference values. The reference values were GIf = 0.2 N mm/mm2 , φumax = 0.7, φ res = 0.57, bmax = 0.1 MPa, kmax = 0.05 MPa, η = 6.5, GIIfb = (6.5σN +
0.12)N mm/mm2 , σIP = 0.25 MPa, and GIIfu = 1.21σN N mm/mm2 . The effects of confining pressure were also investigated by taking account of three cases of normal stress, σN = 0.5, 1.0, 2.5 MPa, as in the case of [15]. Figs. 9–16 show sensitivity analysis results and compare these with experimental results. Fig. 9 shows three normal stress vs. relative normal fracture displacement curves corresponding to three normal fracture energy release rates of GIf = 0.05, 0.2, and 0.35 N mm/mm2 . It is interesting to note that the descending branch becomes brittle when the fracture energy release rate decreases. This is primarily due to the role of βN /GIf in Eq. (3) where the value of bond parameter b decreases according the decrease of fracture energy release rate GIf . In this case, the numerical results were not compared with the experimental results because of no available experimental data regarding the direct tension test for the steel–concrete interface. The effects of unbonded friction parameter φumax were investigated by considering three cases of φumax including 0.6, 0.7, and
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a
b
c
Fig. 14. Sensitivity analysis results with respect to η; (a) σN = −0.5 MPa (b) σN = −1.0 MPa (c) σN = −1.5 MPa.
0.9. Fig. 10 shows that the increase of φumax increases the maximum value of shear stress. This is because the slope of the dotted line to which failure envelope is asymptotically converged in Fig. 1 becomes stiff when φumax increases. The slope change due to the increase of φumax is relatively high when compared to the slope change due to the increase of σN . This is because the increase of φumax induces a shape change of the failure envelope, however the increase of σN moves the failure point to left side without allowing a shape change. Fig. 11 shows the effects of residual friction parameter φ res on the interface behavior. The reference value of residual friction parameter φ res = 0.57 was determined from the test results of [13] and there is no difference between bonded and unbonded because the surface conditions for both cases become similar to each other when the interface slip is fully developed. The residual shear stress level increases or decreases depending on the increase or decrease of residual friction parameter φ res because it geometrically defines the minimum slope of the failure envelope in softening. Since the increase of φb expands the failure envelope and results in the increase of peak shear stress σTmax , bond strength bmax to define the magnitude of φb through the definition of Eq. (2) has an important role on the onset of interface cracking. In Fig. 12, when
bmax increases, a sudden drop of shear stress σT at the initial stage of slip fracture is observed due to the inherent nature of bmax . It is shown in Fig. 13 that the curvature parameter kmax has little influence on the overall interface behavior because geometrically has it nothing to do with the slope change of the failure envelope. Friction adjusting parameter η has a role to bridge the gap between the two parameter values of unbonded and bonded which is the same role as bond strength bmax as explained in Eq. (2). Thereby, sensitivity analysis results with respect to η shown in Fig. 14 are well-aligned with those with respect to bmax in Fig. 12. It is shown in Fig. 15 that the increase of slip fracture energy release rate GIIfb lifts up the softening curve to preserve the energy equivalence between the increased slip fracture energy release rate GIIfb and the area surrounded by shear stress vs. relative slip displacement curve and residual shear stress level. Sensitivity analysis results with respect to nonassociativity inflection parameter σIP are shown in Fig. 16. It is noted that the inflection parameter σIP has a strong influence on the interface behavior when normal stress is relatively low as σN = −0.5 MPa. On the other hand, when the normal stress level is relatively high as at 1.5 MPa, the interface behavior becomes independent of the change of σIP . It is noted through the sensitivity analyses that the major parameters to govern the interface behavior are bond strength bmax
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b
a
c
Fig. 15. Sensitivity analysis results with respect to GIIfb ; (a) σN = −0.5 MPa (b) σN = −1.0 MPa (c) σN = −1.5 MPa.
and unbonded friction parameter φumax . In case of an unbonded interface where bmax = 0, the dominant parameter is the unbonded friction parameter φumax .
Table 2 Parameter values for concrete-infilled steel column. Properties
Values Ec (MPa)
4.3. Prediction of test results of Shakir-Khalil [16,17] The determined parameter values were used to predict the interface behavior of concrete-infilled rectangular steel column tests of [16,17]. Fig. 17 shows the undeformed and deformed configurations of a finite element quarter model of test specimens with three different interface lengths of 200, 400, and 600 mm, i.e. length ratio of 1:2:3. A three dimensional 8-node brick element, 4-node Mindlin-type shell element and 8-node zero-thickness element were used for the finite element modeling of concrete, steel plate and interface, respectively. A conventional three dimensional elasto-plastic constitutive model based on a four-parameter failure criterion [21] was used to depict the material nonlinearity of concrete, and the von Mises failure criterion was used to depict the material nonlinearity of the steel plate. Material properties and interface parameters used in the analysis are summarized in Table 2. The values of βN , βT , α , and γ in Table 2 that were defined
Conc.
DT (N/mm3 ) DN (N/mm3 )
φumax φ res bmax bres kmax kres
νc
fck ft
27,900 0.18 41 4.1 30 10,000 0.7 0.57 0.1 0 0.05 0
Properties
Values Es
Steel
νs fy
η α βN βT γ σIP
GIf (N mm/mm2 ) GIIfb (N mm/mm2 )
196,000 0.3 295 6.5 1.0 50.0 1.0 1.0 −0.25 1.21σN 6.5σN + 0.12
in Eq. (3) were obtained through numerical analyses for the three tests of [13,15–17]. Fig. 18 compares the measured and predicted results of the applied load vs. slip displacement curves for the three different interface length cases. The predicted results are in agreement with the measured results except for the shortest interface length of
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a
b
c
Fig. 16. Sensitivity analysis results with respect to σIP ; (a) σN = −0.5 MPa (b) σN = −1.0 MPa (c) σN = −1.5 MPa.
a
b
Fig. 17. Deformed and undeformed configurations of test specimens of ShakirKhalil (200, 400 and 600 mm interface lengths) ; (a) Undeformed geometry (b) Deformed geometry.
200 mm where the measured peak load is much higher than the predicted one. It is interesting to note that the predicted peak loads are approximately in a 1:2:3 ratio that is same as the interface
Fig. 18. Comparison of slip displacements for the test specimens of [16,17].
length ratio of 1:2:3 for 200, 400, 600 mm lengths. However, the ratio is not preserved for the measured results because the peak load for the 200 mm length case is almost same as that of the 400 mm length case. Another interesting point is that post-peak
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5. Conclusion
Fig. 19. Location of strain gages attached on steel plate.
Fig. 20. Comparison of axial strains for the test specimens of [16,17].
behaviors of the predicted results are similar to one another and residual levels are in a 1:2:3 ratio. Based on the observations obtained from the predicted results, the resistance on the interface of a concrete-infilled rectangular hollow section to a pushout force is proportional to the interface area, which means that slip fracture energy release rate on interface is constant regardless of the interface area in this particular case. However, it may be controversial because the measured results show that changing the interface length does not have a direct and proportional effect on the loadcarrying capacity. Shakir-Khalil [16,17] explained the reason for no relation between interface resistance and area in terms of interface condition, local imperfections and shrinkage of infilled concrete. Restricting the scope of interest into the performance of the current constitutive model and parameters, it may be worthwhile to discuss a constant value of slip fracture energy release rate, which could be acceptable in a mechanical sense. Fig. 20 compares the axial strains measured at the four locations of a 600 mm interface length specimen with the predicted axial strains. Locations of strain gages attached on the specimen are shown in Fig. 19. Both the measured and predicted strains gradually increase from the top to the base. This is due to the load transfer from the core concrete to the steel hollow section by slip resistance acting on the interface. Slopes of load vs. strain curves for gages 1 and 2 gradually increase upward while those of gages 3 and 4 gradually decrease downward as the applied load level approaches its peak. These slope changes indicate that the interface slip occurred in gages 1 and 2 before reaching peak, the interface slip occurred at the vicinity of the peak in gage 3, and the interface slip in gage 4 led to the sudden global failure, which resulted in strain recovery in the steel. The measured and predicted results are in good agreement in depicting the failure process as well as in comparing absolute values.
A method to determine the values of constitutive parameters for steel–concrete interfaces was presented which is applicable to the interface model based on the Mohr–Coulomb type failure criterion combined with normal and slip fracture energy concepts. For this purpose, a comprehensive constitutive model to depict the strong dependency of interface behavior on the bonding condition of the interface, bonded or unbonded, was introduced based on the conventional flow theory of plasticity. To bridge the missing link from unbonded to bonded interfaces, a friction adjusting parameter was introduced and combined with bond strength to account for the slip resistance by bonding on the interface. Values of nine constitutive parameters defining the presented model were determined based on existing experimental results, geometry of failure envelope and sensitivity analyses. Three values of friction parameter for unbonded φumax , residual friction parameter φ res , and slip friction energy release rate for unbonded GIIfu were determined from the test results of [13]. Values of bond strength bmax and curvature parameter kmax were calculated through geometrical interpretation of the failure envelope. Five values of normal fracture energy release rate GIf , slip fracture energy for bonded GIIfb , friction adjusting parameter η, and nonassociativity inflection parameter σIP were determined through sensitivity analyses for the existing test results. It was found through sensitivity analyses that the most influencial parameters on interface behavior were bond strength bmax and unbonded friction parameter φumax for the particular case. The parameter values were applied to predict the slip resistance of a concrete-infilled rectangular steel column subjected to a push-out force. It was observed that the interface resistance of the concrete-infilled rectangular hollow section was proportional to the interface area, which led to a constant slip fracture energy release rate independent of the interface area. In addition, enough load transfer between the steel plate and infilled-concrete was observed until the column reached global failure, and upon reaching peak load, the failure was sudden and resulted in strain recovery in the steel plate part by releasing the energies stored in the steel plate as well as in the infilled-concrete. Acknowledgement This research was funded by the Konkuk University Research grant Committee that made it possible for the first author to stay at the University of Nevada at Reno, USA, to complete this research work during his sabbatical year. References [1] Katona MG. A simple contact-friction interface element with applications to buried culverts. Int J Numer Anal Methods Geomech 1983;7:371–84. [2] Plesha ME. Constitutive models for rock discontinuities with dilatancy and surface degradation. Int J Numer and Anal Meth in Geomech 1987;11:345–62. [3] Stankowski T, Runesson K, Sture S. Fracture and slip of interfaces in cementitious composites. I: characteristics. J Engrg Mech, ASCE 1993;119(2): 315–27. [4] Stankowski T, Runesson K, Sture S. Fracture and slip of interfaces in cementitious composites. II : implementation. J Engrg Mech, ASCE 1993; 119(2):315–27. [5] Lotfi HR, Shing PB. Interface model applied to fracture of masonry structures. J Struct Engrg, ASCE 1994;120(1):63–79. [6] Carol I, Prat PC, Lopez CM. Normal/shear cracking model: application to discrete crack analysis. J Engrg Mech, ASCE 1997;123(8):765–73. [7] Carol I, Lopez CM, Roa O. Micromechanical analysis of quasi-brittle materials using fracture-based interface elements. Int J Numer Methods Engrg 2001;52: 193–215. [8] Hajjar JF, Schiller PH, Molodan A. A distributed plasticity model for concretefilled steel tube beam–coulmns with interlayer slip. Eng Struct 1998;20(8): 663–76. [9] Soh CK, Chiew SP, Dong YX. Damage model for concrete-steel interface. J Engrg Mech, ASCE 1999;125(8):979–83. [10] Lei XY. Contact friction analysis with a simple interface element. Comput Methods Appl Mech Engrg 2001;190:1955–65.
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[11] Cai CS, Nie J, Shi XM. Interface slip effect on bonded plate repairs of concrete beams. Eng Struct 2007;29(6):1084–95. [12] Han L-H, Wang W-D, Zhao X-L. Behavior of steel beam to concrete-filled SHS column frames: finite element model and verifications. Eng Struct 2008;30(6): 1647–58. [13] Rabbat BG, Russell HG. Friction coeffcient of steel on concrete or grout. J Struct Engrg, ASCE 1985;111(3):505–15. [14] Chajes MJ, Finch Jr WW, Januszka TF, Thomson Jr TA. Bond and force transfer of composite material plates bonded to concrete. ACI Struct J 1996;93(2):208–17. [15] Chiew SP, Dong YX, Sho CK. Concrete–steel plate interface characteristics for composite construction. Computing Developments in Civil and Structural Engineering 1999;35–40. [16] Shakir-Khalil H. Pushout strength of concrete-filled steel hollow sections. The Struct Eng 1993;71(13):230–3.
[17] Shakir-Khalil H. Resistance of concrete-filled steel tubes to pushout forces. The Struct Eng 1993;71(13):234–43. [18] Shakir-Khalil H, Hassan NKA. Push-out resistance of concrete-filled tubes. In: Grundy P, Holgate A, Wong. W, editors. Tubular structures VI. Australia (A.A. Balkema, Rotterdam, The Netherlands): Melbourne; 1994. p. 285–91. [19] Zienkiewicz O, Pande G. Some useful forms of isotropic yield surfaces for soil and rock mechanics. In: Gudehus G, editor. Finite elements in geomechanics. New York (NY): John Wiley & Sons; 1977. p. 179–90. [20] Lee Y-H, Willam K. Anisotropic vertex plasticity formulation of concrete inplane stress. J Engrg Mech, ASCE 1995;123(7):714–26. [21] Lee Y-H, Joo Y-T, Seong W-J. Anisotropic loading criterion for depicting loading induced anisotropy in concrete. In: 7th int. conf. on fracture mech. of concrete and concrete structures, FRAMCOS-7, held in Jeju, Korea; 2010. p. 621–8.
Contents lists available at ScienceDirect
Engineering Structures journal homepage: www.elsevier.com/locate/engstruct
Mechanical properties of constitutive parameters in steel–concrete interface Yong-Hak Lee a , Young Tae Joo b , Ta Lee c , Dong-Ho Ha a,∗ a
Department of Civil Engineering, Konkuk University, Gwangjin-Gu, Seoul 143-701, South Korea
b
MidasIT Co., Boondang-Gu, Seongnam 463-400, South Korea
c
Ssangyong Eng. & Const. Co., Songpa-Gu, Seoul 138-240, South Korea
article
info
Article history: Received 4 November 2008 Received in revised form 27 December 2010 Accepted 6 January 2011 Available online 5 February 2011 Keywords: Steel–concrete interface Bond and slip Fracture energy Elasto-plasticity
abstract Mechanical properties of steel–concrete interfaces are evaluated on the basis of three existing experimental evidences. The properties include bond strength, unbonded and bonded friction parameters, residual level of the friction parameter, normal fracture energy release rate, bonded and unbonded slip fracture energy release rates under different levels of normal stress, and shape parameters defining the geometrical shape of the failure envelope. For this purpose, a typical type of constitutive model for describing steel–concrete interface behavior is presented based on a hyperbolic three-parameter failure criterion. The constitutive model depicts the strong dependency of interface behavior on bonding condition of the interface, bonded or unbonded. Mechanical roles of the interface parameters are discussed with reference to those of the presented interface constitutive model. Values of the interface parameters are determined through interpretation of existing experimental results, geometry of the failure envelope and sensitivity analyses. These values are applied to push-out tests of concrete-infilled rectangular steel columns with three different cases of interface lengths. The failure process of concrete-infilled rectangular steel columns is discussed through comparison of experimental measurements with numerical results. © 2011 Elsevier Ltd. All rights reserved.
1. Introduction Numerical prediction of interface behavior using nonlinear finite element framework requires a constitutive model to describe the progressive failure localized in the interface region. Numerous research works have been made to model the localized failure process on the steel–concrete interface in the lumped manner ([1–12] among others). Most of the models except for the early work of [1] that used a Lagrange multiplier method have adopted the fracture energy release concept to depict the localized deformation process. However, even if those models are wellestablished based on a robust continuum theory of plasticity, the values of parameters defining the constitutive equation have to be properly determined from the experimental results for a practical purpose. The parameters may be bond strength, bonded and unbonded frictions, residual level of friction, normal (or mode I) fracture energy release rate, bonded and unbonded slip (or mode II) fracture energy release rates under different level of normal stress, and shape parameters defining the geometrical shape of the failure envelope. Little effort has been expended to extract the fundamental properties from post-peak measurements as only a limited
∗
Corresponding author. Tel.: +82 2 450 3514; fax: +82 2 2201 0783. E-mail addresses: [email protected] (Y.-H. Lee), [email protected] (Y.T. Joo), [email protected] (T. Lee), [email protected] (D.-H. Ha). 0141-0296/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.engstruct.2011.01.005
literature is currently available [13–15]. Consequently, the lack of quantitative definition of parameters has hampered the numerical implementation to predict the behavior of steel–concrete composite structures and the further development of constitutive formulation of steel–concrete interfaces. The aim of this paper is laid on defining the mechanical properties of constitutive parameters in steel–concrete interfaces. Steel–concrete interface behavior is defined in the normal and tangential directions to the interface plane. Once reaching their critical states, tensile and shear behaviors in the normal and tangential directions, respectively, follow mode I type of normal fracture (or debonding) and mode II type of slip fracture failure mechanisms. The slip on the interface strongly depends not only on the magnitude of the normal stress to the interface but also on the bonding condition of interface, bonded and unbonded, as [13,15] observed in their experiments performed on sandwichtyped steel–concrete interface specimen tests. To identify the constitutive parameters, a comprehensive constitutive model that accounts for the strong dependency of interface slip on the bonding condition of interface is presented by modifying the hyperbolic failure criterion of [5,7]. The modification is attained by expressing the bonded friction parameter in terms of the unbonded friction parameter and friction adjusting parameter. Values of parameters defining the model are determined through existing experimental results, geometry of failure envelope and sensitivity analyses. Only
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Notations b: bmax : bres : De : Dep : DN : DT : 1u: fI : GIf :
Bond parameter bond strength Residual bond parameter Elastic interface tangent matrix Elasto-plastic interface tangent matrix Elastic interface normal stiffness Elastic interface tangent stiffness Incremental relative displacement Nonassociativity intensity factor Normal fracture energy release rate
GIIfb :
Slip fracture energy release rate for bonded
GIIfu :
Slip fracture energy release rate for unbonded Curvature parameter Maximum curvature parameter Plastic flow vector Normal vector to failure surface Normal fracture work Slip fracture work Slip fracture work fraction parameter for friction Normal fracture work fraction parameter for bond Slip fracture work fraction parameter for bond Bonded friction parameter Maximum bonded friction parameter Unbonded friction parameter Maximum unbounded friction parameter Residual friction parameter Friction adjusting parameter Nonassociativity inflection parameter Normal stress Tangential stress Maximum tangential stress for bonded
k: kmax : m: n: WPN : WPT : a: βN : βT : φb : φbmax : φu : φumax : φ res : η: σIP : σN : σT : max σTb :
a few experimental results are available to evaluate mechanical properties of interface parameters because of highly pressuresensitive nature of the steel–concrete interface. Consequently, three existing experimental evidences of [13,16–18], and [15] are used to determine the values of parameters defining the model. Maximum value of unbonded friction parameter, residual level of friction parameter and unbonded slip-friction energy are calculated from the unbonded tests of [13]. Bond strength and normal fracture energy release rate are determined by geometrical relationships between parameters. Bonded slip fracture energy release rate and shape parameters of the failure envelope are determined through sensitivity analyses performed based on the experiments of [16,17,15]. As a result, those values are applied to pushout tests of concrete-infilled rectangular steel columns with three different cases of interface lengths of [16–18]. The failure process of a concrete-infilled rectangular steel column is discussed through the comparison of experimental measurements with numerical predictions. 2. Failure criterion for steel–concrete interface Interface behavior between steel and concrete can be decomposed into mode I type of normal fracture and mode II type of slip fracture. A number of failure criteria have been presented including a linear representation of [2], a parabolic representation of [3,4], and hyperbolic representations of [5,6]. The hyperbolic criterion that was initially developed by Zienkiewicz and Pande [19] take advantage of exhibiting C 1 -continuity throughout the entire response regime and is asymptotically equivalent to
Fig. 1. Three-parameter failure envelope of steel–concrete interface.
the Mohr–Coulomb criterion when compressive normal stress increases. In this paper, the hyperbolic criterion is adopted to define the onset of interface fracture and monitor the progressive fracture process. Slip on the interface strongly depends not only on the magnitude of the normal stress to the interface but also on the bonding condition of the interface, bonded and unbonded, as observed in the sandwich-typed steel–concrete interface specimen tests of [13,15]. To incorporate the behavior of unbonded as well as bonded interfaces, the hyperbolic representation of the failure envelope of [5,6] is modified as F (σN , σT , φ, b, k) = σT2 − φ 2 {(σN − b)2 − 2k(σN − b)} where σN is normal stress, σT =
(1)
σL2 + σM2 is tangential stress, the
subscripts N, L and M represent the local coordinates along normal and tangential directions to the interface plane, respectively. b and k are bond and curvature parameters, respectively, and φ is the friction parameter. The friction parameter φ is defined as φb and φu for bonded and unbonded interfaces, respectively. The bonded friction parameter φb is expressed in terms of the unbonded friction parameter φu and friction adjusting parameter η to account for the dependency of the slip behavior on interface bonding conditions as
φb = φu + ηb
(2)
where the friction adjusting parameter η bridges the gap between bonded and unbonded frictions. Fig. 1 shows the geometrical relationship among the three parameters. Evaluating the friction parameter value for the test specimens of [13,15] according to Coulomb’s friction law, it is smaller than 1.0 for the unbonded and greater than 1.0 for the bonded, which clearly shows a strong dependency of interface behavior on the bonding condition of the interface. Along the line of classical theory of plasticity, progressive interface failure can be depicted by the evolution of the failure envelope where the evolution process may be monitored by gradually decreasing the values of three parameters φ , b and k from the maximum values of φ max , bmax and kmax to the residual values of φ res , bres and kres , respectively. To resolve the missing link between the maxima and residues, the three parameters are commonly modeled in terms of energy fractions between fracture energy release
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rate and related fracture work [5,6]. In this paper, the bond parameter b is assumed to be dependent on slip fracture as well as normal fracture similar to the work of [5] to account for the effect of a slipinduced shear crack on normal cracks. The other two parameters of friction and curvature, φ and k, are assumed to be dependent only on slip fracture. Thereby, the three parameters are modeled in terms of energy fractions of normal and slip fracture energy release rates, and related fracture works as
b=b
res
+ (b
max
WPT
WPN
WPT
k = kres + (kmax − kres ) 1 − γ
+ ηb
(3)
WPT GIIfb
where βN and βT are bond-related normal and slip fracture work parameters, respectively, α and γ are friction- and curvaturerelated slip fracture work parameters, respectively. The values of βN , βT , α , and γ are determined by limiting the value of parentheses to range from 0 to 1. GIf and GIIfb are normal and bonded slip fracture energy release rates, respectively. It is noted in Eq. (3) that the evolutions of friction and curvature parameters are controlled by the energy fraction between slip fracture energy release rate and related fracture work rather than fracture work itself of [5] because the variation of slip fracture energy release rate under different level of normal stress can be obtained from existing experimental results. In cases of normal fracture and unbonded slip fracture, the three parameters of Eq. (3) are modified as; (1) Normal fracture: the bond parameter b governs the interface fracture process while the friction and curvature parameters of φ and k remain at their maxima.
b=b
φb = φ
+ (b
max u
max
a
− b ) 1 − βN I − βT II Gf Gfb res
φb = φ res + (φbmax − φ res ) 1 − α II Gfb
res
1279
− b ) 1 − βN res
+ ηb,
k=k
max
.
WPN GIf
− βT
WPT
b
(4)
GIIfb
(2) Unbonded slip fracture: the friction and curvature parameters of φ and k decrease from their maxima to residues while the bond parameter b remains its residual
b=b ,
φu = φ
res
res
+ (φ
max u
k = kres + (kmax − kres ) 1 − γ
WPT
− φ ) 1 − α II Gfu res
WPT GIIfu
Fig. 2. Incremental plastic work. (a) Normal fracture. (b) Shear fracture.
(5)
8
.
7 4
Normal and slip fracture works of WPN and WPT of Eq. (3) (see Fig. 2) are defined in incremental formats of
σN ≥ 0 σN < 0 = (σT − σTres ) · duPT
dWPN = dWPT
M 5
σN duPN 0
zero-thickness
N 3
L 6
(6)
1 2
where uPN and uPT are relative normal and tangential displacements, duPT = (duPL )2 + (duPM )2 , and residual tangential stress
σTres = ±φ res (σN − bres )2 − 2kres (σN − bres ). Fig. 3. Description of 8-node zero-thickness interface element.
3. Elasto-plastic interface formulation In the case of infinitesimal deformation, the relative total dis˙ may be decomposed into independent placement rate vector u ˙ e and u˙ p , respectively, and leadelastic and plastic components of u ˙ = u˙ e + u˙ p . Adopting a zero-thickness finite interface eleing to u ment in Fig. 3, stress vs. relative displacement relationship in local coordinate is expressed as
where σ = {σN σL σM }T and ue = {uN uL uM }T . De = diag{DN DL DM } is a diagonal matrix to define the elastic stiffnesses of the interface along normal and tangential directions, and defined as normal and slip stiffnare the plasticesses in Fig. 4(a) and (b), respectively. Adopting a nonassociated flow rule, plastic displacement vector ˙ p is defined in terms of u
σ˙ = De : (u˙ − u˙ p )
˙ p = λ˙ m u
(7)
(8)
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a
a
b Unbonded behavior
b
Bonded behavior
Fig. 5. Nonassociated plastic flow. (a) Direction of plastic flow on failure surface. (b) Nonassociativity intensity factor fI .
Fig. 4. Stress vs. relative displacement relationship. (a) Normal direction. (b) Tangential direction.
˙ and m are the plastic multiplier and the plastic flow where λ vector, respectively. When the current stress predicted with elastic increments of Eq. (7) violates the current state of the failure criterion of Eq. (1), i.e. F > 0, plastic flow occurs. A truncated Taylor series expansion of Eq. (1) leads to the linearized format of ˙ consistency condition to calculate the plastic multiplier λ F˙ =
∂F ∂F : σ˙ + u˙ p = 0 ∂σ ∂ up
(9)
where equivalent plastic displacement u˙ p is obtained from the ˙ m‖. The plastic multiplier λ˙ is obtained Euclidean norm of u˙ p = λ‖ by substituting Eqs. (7) and (8) into (9) as
˙ n : De : u (10) −H ‖m‖ + n : De : m where n = ∂ F /∂σ , m = ∂ Q /∂σ , and Q is plastic potential. The
λ˙ =
hardening modulus H in Eq. (10) is calculated through the chain rule of differentiation as ∂ F ∂φb ∂φu ∂ WPT ∂ F ∂φb ∂F H = + + ∂φb ∂φu ∂ WPT ∂ uPT ∂φb ∂ b ∂b
× +
∂ b ∂ WPN ∂ b ∂ WPT + ∂ WPN ∂ uPN ∂ WPT ∂ uPT
∂ F ∂ k ∂ WPT . ∂ k ∂ WPT ∂ uPT
(11)
Shear dilatancy on interface is primarily due to the saw-tooth effect of an irregularly cracked interface plane and decreases with abrasion of the residue and confining effect of normal pressure. It is a common way to adopt nonassociated flow rule to control the excessive dilatancy in a computational plasticity. Fig. 5(a) schematically shows the change of plastic flow direction with increasing confining pressure. In this paper, the direction of nonassociated plastic flow m is determined by adjusting the normal vector n similarly to the formulation of [20]. Adopting the concept, the nonassociativity intensity factor fI is introduced as a function of the nonassociativity inflection parameter σIP . Thereby, the direction of plastic flow component along normal direction is modified with nonassociativity intensity factor fI while the other components along the tangential direction remain unchanged
∂F ∂Q ∂F ∂Q = , = ∂σL ∂σL ∂σ ∂σM M ∂Q ∂F φ − φ res = fI ∂σN ∂σN φ max − φ res
(12)
where nonassociativity intensity factor fI shown in Fig. 5(b) is defined as fI = {(σN − σIP )/σIP }2 . Elastoplastic tangent operator Dep can be obtained by substituting the plastic multiplier of Eq. (10) into (7) as Dep = De −
De : m ⊗ n : De . −H ‖m‖ + n : De : m
(13)
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Test point
a
Confining pressure 610
Slip force
340
150
360
25 710 Concrete Block Steel Plate
Fig. 8. Geometrical description of failure envelope in peak.
Slip force
b
Confining pressure
150
Confining pressure
20 150 150 LVDT
Conc. Steel
Fig. 6. Descriptions of test specimens. (a) Rabbat and Russell [13] and (b) Chiew et al. [15].
Fig. 9. Sensitivity analysis results with respect to normal fracture energy release rate of GIf .
4. Parameter values of steel–concrete interface Values of nine constitutive parameters defining the presented model were determined based on the existing experimental results, geometry of the failure envelope and sensitivity analyses. Nine parameters are the maximum and residual values of unbonded friction parameter φumax and φ res , unbonded and bonded slip fracture energy release rates GIIfu and GIIfb , normal fracture energy release rate GIf , bond strength bmax , maximum value of curvature parameter kmax , friction adjusting parameter η, and the nonassociativity inflection parameter σIP . Due to the limited number of available experiments, three cases of experimental results of [13,16,17,15] were used for determining the parameter values. 4.1. Experiments of Rabbat and Russell [13], Chiew et al. [15] and Shakir-Khalil [16,17]
Fig. 7. Unbonded slip test results of [13].
Fig. 6(a) shows a test specimen of a steel–concrete interface tested by Rabbat and Russell [13]. In the experiments, a concrete block with 150 mm thickness, 340 mm width and 610 mm length was cast on a 25 mm thick, 360 mm wide and 710 mm long steel plate of Grade 70 according to the recommendation of A516 of ASTM. The surface of the steel plate was treated according to the
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a
b
c
Fig. 10. Sensitivity analysis results with respect to φumax ; (a) σN = −0.5 MPa (b) σN = −1.0 MPa (c) σN = −1.5 MPa.
recommendation of SSPC-SP2-63 of the Steel Structures Council Specifications. At 7 days of age, the formwork of the concrete block was removed and cured in the laboratory with 23 °C temperature and 50% humidity. A confining pressure of σN = 0.06 MPa was applied to the specimen with fc′ = 30 MPa compressive strength of concrete and then, a bonded slip test was performed by applying a slip force until the interface crack was fully developed. After the bonded slip test, an unbonded slip test was carried out on the tested specimen in the same manner except for a confining pressure of σN = 0.41 MPa. Fig. 6(b) shows a test specimen of [15] where a 20 mm thick steel plate was placed between two concrete blocks that are 150 mm wide, 150 mm high and 65 mm thick. To investigate the effects of confining pressure on interface behaviors, three different levels of confining pressures including σN = 0.5, 1.0, and 1.5 MPa were applied to the specimens. In the case of σN = 1.5 MPa confining pressure, the bonded specimen was tested until an interface crack was fully developed and then, the tested specimen was retested for unbonded slip behavior with the same confining pressure to investigate the difference between bonded and unbonded. Shakir-Khalil [16,17] evaluated the failure strength of steel– concrete interfaces by testing forty concrete-infilled rectangular
steel columns. The effects of the number of shear studs on the failure strength were also addressed by considering three cases of shear stud arrangements including 2, 4 and 6, and no shear stud case. All the test specimens with 150 mm × 150 mm cross -sectional dimensions were made of 5 mm thick steel plate. Three different cases of interface lengths of 200, 400 and 600 mm were taken into account to investigate the effects of the interface area on the failure strength. 4.2. Determination of interface parameter values 4.2.1. Maximum and residual values of unbonded friction parameter φumax and φ res , unbonded slip fracture release rate GIIfu
Values of φumax and φ res were determined according to the definition of the friction parameter expressed by the ratio between shear and normal stresses. For this purpose, the maximum unmax bonded average shear stress of σTu = 0.28 MPa was obtained from the slip test results of [13]. The maximum value of the unbonded friction parameter φumax = 0.7 was calculated by dividing max the maximum shear stress σTu = 0.28 MPa by the confining pressure σN = 0.41 MPa. In a similar way to φumax , the residual value of the friction parameter φ res = 0.57 was calculated by dividing
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1283
b
a
c
Fig. 11. Sensitivity analysis results with respect to φ res ; (a) σN = −0.5 MPa (b) σN = −1.0 MPa (c) σN = −1.5 MPa. Table 1 Experimental results of [13]. No.
1 2 3
Comp. strength (MPa)
25.0 25.6 25.6
Averaging shear stress (MPa) for unbonded
Friction coefficient
Maximum
Residual
Maximum
0.28 0.32 0.25
0.23 0.24 0.23
0.68 0.77 0.62
the average residual shear stress σTres = 0.23 MPa by the confining pressure σN = 0.41 MPa. Table 1 summarizes the test results and the calculated friction parameter values. Residual shear stresses in Table 1 were defined with the shear stress levels to which the shear stresses were converged. The value of the unbonded slipfracture release rate GIIfu = 1.21σN N mm/mm2 was obtained by measuring the dotted triangular area (12 mm × 0.08 MPa × 0.5 = 0.48 N mm/mm2 ) of Test-3 in Fig. 7. 4.2.2. Bond strength bmax and curvature parameter kmax Considering the asymptotic convergence of failure envelope with the increase of compression as shown in Fig. 1, the maximum failure envelope may be linearly approximated as φbmax (|σN | + max bmax + kmax ) = σTb . Fig. 8 shows geometrical relationship among
Average
0.69
Minimum 0.56 0.58 0.57
Average
0.57
max σN , φbmax , kmax , bmax , and σTb . Substituting φbmax = φumax + ηbmax max max of Eq. (2) and b = 2k obtained from the sensitivity anal-
yses for the test results of [13,15] into the linearly approximated equation, a second order polynomial equation with respect to bond strength bmax is derived as η(bmax )2 +{φumax +η(|σN |+ kmax )}bmax + max φumax (|σN | + kmax ) − σTb = 0. Bond strength bmax can be readily calculated with the equation when a relatively high value of max normal stress σN and corresponding σTb defining a test point on the maximum failure envelope are known from a bonded interface experiment. Bond strength bmax was calculated and compared to the test results of [13,15] to examine the degree of sensitivity with respect to the level of shear stress. In case of [13], the maximum bonded shear max stress of σTb = 0.41 MPa was obtained by averaging the three
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a
b
c
Fig. 12. Sensitivity analysis results with respect to bmax ; (a) σN = −0.5 MPa (b) σN = −1.0 MPa (c) σN = −1.5 MPa. max test results of σTb = 0.46, 0.37, 0.4 MPa. Substituting the average max shear stress σTb = 0.41 MPa and the corresponding normal stress σN = 0.06 MPa into the second order polynomial equation and solving with respect to bmax , a bond strength of bmax = 0.13 MPa and a curvature parameter of kmax = 0.65 MPa were obtained. The friction adjusting parameter η = 6.5 was used in the calculation which was obtained through sensitivity analyses for the test results of [15]. In case of [15], the maximum bonded shear max stresses of σTb = 0.79, 1.51, and 2.25 MPa were obtained from the three tests considering three confining pressures of σN = 0.5, 1.0, and 1.5 MPa, respectively. Bond strengths of bmax = 0.086, 0.096, max and 0.108 MPa corresponding to each of σTb = 0.79, 1.51, and 2.25 MPa, respectively, were calculated by solving the second order polynomial equation. As a result of the calculation, bond strength bmax = 0.1 MPa and curvature parameter kmax = 0.05 MPa were obtained by averaging the three bond strengths. Considering the different material properties, testing methods, curing environments, and measuring instruments, 30% of difference between the two calculated bond strengths may ensure that the presented approach could be a proper tool to determine the value of bond strength without performing a direct tension test.
4.2.3. Normal fracture energy release rate GIf , bonded slip fracture en-
ergy release rate GIIfb , friction adjusting parameter η, and nonassociativity inflection parameter σIP The values of GIf , GIIfb , η, and σIP were determined through numerical failure tests for the test results of [13,15–17]. Parameter values of φumax = 0.7, φ res = 0.57, GIIfu = 1.21σN N mm/mm2 , and bmax = 0.1 MPa obtained from the slip test results of [13] were used for the analyses. Since the residual shear stress σTres = 1.2 MPa measured from the test results of [15] showed relatively large difference with the residual shear stress σTres = 0.85 MPa calculated from σTres =
±φ res (σN − bres )2 − 2kres (σN − bres ) of Eq. (6),the bonded slip
fracture energy release rate GIIfb was determined through sensitivity analyses. The initial value of GIfb = 9 N mm/mm2 required for the analysis was obtained by measuring the area of shear stress vs. relative slip displacement curve and residual shear stress level, and used for the analyses. The bonded slip fracture energy release rate of GIIfb = (6.5σN + 0.12) N mm/mm2 was obtained by linearly interpolating the three values of GIIfb measured from shear stress vs. relative slip displacement curves obtained for the three cases of σN = 0.5, 1.0, and 1.5 MPa. A normal fracture energy release rate
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b
a
c
Fig. 13. Sensitivity analysis results with respect to kmax ; (a) σN = −0.5 MPa (b) σN = −1.0 MPa (c) σN = −1.5 MPa.
of GIf = 0.2N mm/mm2 was obtained through sensitivity analyses that considered three cases of normal fracture energy release rates of GIf = 0.05, 0.2, 0.35 N mm/mm2 . The initial value of friction adjusting parameter η was calculated from the maximum failure envelope expression of Eq. (2), φbmax = φumax + ηini bmax . The value of φbmax = 1.47 used in this calculation was obtained by dividing the maximum bonded shear max stress σTb = 2.2 MPa by the confining pressure σN = 1.5 MPa. In addition, the value of φumax = 0.77 was obtained by dividing the max maximum unbonded shear stress σTu = 1.15 MPa by the confining pressure σN = 1.5 MPa. As a result, the friction adjusting parameter η = 6.5 was determined through sensitivity analysis with reference to the initial value of η = 7. Nonassociativity inflection parameter σIP = 0.25 MPa was obtained through the sensitivity analysis performed for the test results of [16,17]. All the sensitivity analysis results are discussed in the subsequent section. 4.2.4. Sensitivity analyses with respect to constitutive parameter values Sensitivity analyses were carried out to investigate the applicability of the obtained parameter values and the influence on the overall interface behavior. The analyses were referenced on the
test results of [15] and performed by giving a perturbation for the reference value of a constitutive parameter while the other constitutive parameters were kept as reference values. The reference values were GIf = 0.2 N mm/mm2 , φumax = 0.7, φ res = 0.57, bmax = 0.1 MPa, kmax = 0.05 MPa, η = 6.5, GIIfb = (6.5σN +
0.12)N mm/mm2 , σIP = 0.25 MPa, and GIIfu = 1.21σN N mm/mm2 . The effects of confining pressure were also investigated by taking account of three cases of normal stress, σN = 0.5, 1.0, 2.5 MPa, as in the case of [15]. Figs. 9–16 show sensitivity analysis results and compare these with experimental results. Fig. 9 shows three normal stress vs. relative normal fracture displacement curves corresponding to three normal fracture energy release rates of GIf = 0.05, 0.2, and 0.35 N mm/mm2 . It is interesting to note that the descending branch becomes brittle when the fracture energy release rate decreases. This is primarily due to the role of βN /GIf in Eq. (3) where the value of bond parameter b decreases according the decrease of fracture energy release rate GIf . In this case, the numerical results were not compared with the experimental results because of no available experimental data regarding the direct tension test for the steel–concrete interface. The effects of unbonded friction parameter φumax were investigated by considering three cases of φumax including 0.6, 0.7, and
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a
b
c
Fig. 14. Sensitivity analysis results with respect to η; (a) σN = −0.5 MPa (b) σN = −1.0 MPa (c) σN = −1.5 MPa.
0.9. Fig. 10 shows that the increase of φumax increases the maximum value of shear stress. This is because the slope of the dotted line to which failure envelope is asymptotically converged in Fig. 1 becomes stiff when φumax increases. The slope change due to the increase of φumax is relatively high when compared to the slope change due to the increase of σN . This is because the increase of φumax induces a shape change of the failure envelope, however the increase of σN moves the failure point to left side without allowing a shape change. Fig. 11 shows the effects of residual friction parameter φ res on the interface behavior. The reference value of residual friction parameter φ res = 0.57 was determined from the test results of [13] and there is no difference between bonded and unbonded because the surface conditions for both cases become similar to each other when the interface slip is fully developed. The residual shear stress level increases or decreases depending on the increase or decrease of residual friction parameter φ res because it geometrically defines the minimum slope of the failure envelope in softening. Since the increase of φb expands the failure envelope and results in the increase of peak shear stress σTmax , bond strength bmax to define the magnitude of φb through the definition of Eq. (2) has an important role on the onset of interface cracking. In Fig. 12, when
bmax increases, a sudden drop of shear stress σT at the initial stage of slip fracture is observed due to the inherent nature of bmax . It is shown in Fig. 13 that the curvature parameter kmax has little influence on the overall interface behavior because geometrically has it nothing to do with the slope change of the failure envelope. Friction adjusting parameter η has a role to bridge the gap between the two parameter values of unbonded and bonded which is the same role as bond strength bmax as explained in Eq. (2). Thereby, sensitivity analysis results with respect to η shown in Fig. 14 are well-aligned with those with respect to bmax in Fig. 12. It is shown in Fig. 15 that the increase of slip fracture energy release rate GIIfb lifts up the softening curve to preserve the energy equivalence between the increased slip fracture energy release rate GIIfb and the area surrounded by shear stress vs. relative slip displacement curve and residual shear stress level. Sensitivity analysis results with respect to nonassociativity inflection parameter σIP are shown in Fig. 16. It is noted that the inflection parameter σIP has a strong influence on the interface behavior when normal stress is relatively low as σN = −0.5 MPa. On the other hand, when the normal stress level is relatively high as at 1.5 MPa, the interface behavior becomes independent of the change of σIP . It is noted through the sensitivity analyses that the major parameters to govern the interface behavior are bond strength bmax
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b
a
c
Fig. 15. Sensitivity analysis results with respect to GIIfb ; (a) σN = −0.5 MPa (b) σN = −1.0 MPa (c) σN = −1.5 MPa.
and unbonded friction parameter φumax . In case of an unbonded interface where bmax = 0, the dominant parameter is the unbonded friction parameter φumax .
Table 2 Parameter values for concrete-infilled steel column. Properties
Values Ec (MPa)
4.3. Prediction of test results of Shakir-Khalil [16,17] The determined parameter values were used to predict the interface behavior of concrete-infilled rectangular steel column tests of [16,17]. Fig. 17 shows the undeformed and deformed configurations of a finite element quarter model of test specimens with three different interface lengths of 200, 400, and 600 mm, i.e. length ratio of 1:2:3. A three dimensional 8-node brick element, 4-node Mindlin-type shell element and 8-node zero-thickness element were used for the finite element modeling of concrete, steel plate and interface, respectively. A conventional three dimensional elasto-plastic constitutive model based on a four-parameter failure criterion [21] was used to depict the material nonlinearity of concrete, and the von Mises failure criterion was used to depict the material nonlinearity of the steel plate. Material properties and interface parameters used in the analysis are summarized in Table 2. The values of βN , βT , α , and γ in Table 2 that were defined
Conc.
DT (N/mm3 ) DN (N/mm3 )
φumax φ res bmax bres kmax kres
νc
fck ft
27,900 0.18 41 4.1 30 10,000 0.7 0.57 0.1 0 0.05 0
Properties
Values Es
Steel
νs fy
η α βN βT γ σIP
GIf (N mm/mm2 ) GIIfb (N mm/mm2 )
196,000 0.3 295 6.5 1.0 50.0 1.0 1.0 −0.25 1.21σN 6.5σN + 0.12
in Eq. (3) were obtained through numerical analyses for the three tests of [13,15–17]. Fig. 18 compares the measured and predicted results of the applied load vs. slip displacement curves for the three different interface length cases. The predicted results are in agreement with the measured results except for the shortest interface length of
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a
b
c
Fig. 16. Sensitivity analysis results with respect to σIP ; (a) σN = −0.5 MPa (b) σN = −1.0 MPa (c) σN = −1.5 MPa.
a
b
Fig. 17. Deformed and undeformed configurations of test specimens of ShakirKhalil (200, 400 and 600 mm interface lengths) ; (a) Undeformed geometry (b) Deformed geometry.
200 mm where the measured peak load is much higher than the predicted one. It is interesting to note that the predicted peak loads are approximately in a 1:2:3 ratio that is same as the interface
Fig. 18. Comparison of slip displacements for the test specimens of [16,17].
length ratio of 1:2:3 for 200, 400, 600 mm lengths. However, the ratio is not preserved for the measured results because the peak load for the 200 mm length case is almost same as that of the 400 mm length case. Another interesting point is that post-peak
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5. Conclusion
Fig. 19. Location of strain gages attached on steel plate.
Fig. 20. Comparison of axial strains for the test specimens of [16,17].
behaviors of the predicted results are similar to one another and residual levels are in a 1:2:3 ratio. Based on the observations obtained from the predicted results, the resistance on the interface of a concrete-infilled rectangular hollow section to a pushout force is proportional to the interface area, which means that slip fracture energy release rate on interface is constant regardless of the interface area in this particular case. However, it may be controversial because the measured results show that changing the interface length does not have a direct and proportional effect on the loadcarrying capacity. Shakir-Khalil [16,17] explained the reason for no relation between interface resistance and area in terms of interface condition, local imperfections and shrinkage of infilled concrete. Restricting the scope of interest into the performance of the current constitutive model and parameters, it may be worthwhile to discuss a constant value of slip fracture energy release rate, which could be acceptable in a mechanical sense. Fig. 20 compares the axial strains measured at the four locations of a 600 mm interface length specimen with the predicted axial strains. Locations of strain gages attached on the specimen are shown in Fig. 19. Both the measured and predicted strains gradually increase from the top to the base. This is due to the load transfer from the core concrete to the steel hollow section by slip resistance acting on the interface. Slopes of load vs. strain curves for gages 1 and 2 gradually increase upward while those of gages 3 and 4 gradually decrease downward as the applied load level approaches its peak. These slope changes indicate that the interface slip occurred in gages 1 and 2 before reaching peak, the interface slip occurred at the vicinity of the peak in gage 3, and the interface slip in gage 4 led to the sudden global failure, which resulted in strain recovery in the steel. The measured and predicted results are in good agreement in depicting the failure process as well as in comparing absolute values.
A method to determine the values of constitutive parameters for steel–concrete interfaces was presented which is applicable to the interface model based on the Mohr–Coulomb type failure criterion combined with normal and slip fracture energy concepts. For this purpose, a comprehensive constitutive model to depict the strong dependency of interface behavior on the bonding condition of the interface, bonded or unbonded, was introduced based on the conventional flow theory of plasticity. To bridge the missing link from unbonded to bonded interfaces, a friction adjusting parameter was introduced and combined with bond strength to account for the slip resistance by bonding on the interface. Values of nine constitutive parameters defining the presented model were determined based on existing experimental results, geometry of failure envelope and sensitivity analyses. Three values of friction parameter for unbonded φumax , residual friction parameter φ res , and slip friction energy release rate for unbonded GIIfu were determined from the test results of [13]. Values of bond strength bmax and curvature parameter kmax were calculated through geometrical interpretation of the failure envelope. Five values of normal fracture energy release rate GIf , slip fracture energy for bonded GIIfb , friction adjusting parameter η, and nonassociativity inflection parameter σIP were determined through sensitivity analyses for the existing test results. It was found through sensitivity analyses that the most influencial parameters on interface behavior were bond strength bmax and unbonded friction parameter φumax for the particular case. The parameter values were applied to predict the slip resistance of a concrete-infilled rectangular steel column subjected to a push-out force. It was observed that the interface resistance of the concrete-infilled rectangular hollow section was proportional to the interface area, which led to a constant slip fracture energy release rate independent of the interface area. In addition, enough load transfer between the steel plate and infilled-concrete was observed until the column reached global failure, and upon reaching peak load, the failure was sudden and resulted in strain recovery in the steel plate part by releasing the energies stored in the steel plate as well as in the infilled-concrete. Acknowledgement This research was funded by the Konkuk University Research grant Committee that made it possible for the first author to stay at the University of Nevada at Reno, USA, to complete this research work during his sabbatical year. References [1] Katona MG. A simple contact-friction interface element with applications to buried culverts. Int J Numer Anal Methods Geomech 1983;7:371–84. [2] Plesha ME. Constitutive models for rock discontinuities with dilatancy and surface degradation. Int J Numer and Anal Meth in Geomech 1987;11:345–62. [3] Stankowski T, Runesson K, Sture S. Fracture and slip of interfaces in cementitious composites. I: characteristics. J Engrg Mech, ASCE 1993;119(2): 315–27. [4] Stankowski T, Runesson K, Sture S. Fracture and slip of interfaces in cementitious composites. II : implementation. J Engrg Mech, ASCE 1993; 119(2):315–27. [5] Lotfi HR, Shing PB. Interface model applied to fracture of masonry structures. J Struct Engrg, ASCE 1994;120(1):63–79. [6] Carol I, Prat PC, Lopez CM. Normal/shear cracking model: application to discrete crack analysis. J Engrg Mech, ASCE 1997;123(8):765–73. [7] Carol I, Lopez CM, Roa O. Micromechanical analysis of quasi-brittle materials using fracture-based interface elements. Int J Numer Methods Engrg 2001;52: 193–215. [8] Hajjar JF, Schiller PH, Molodan A. A distributed plasticity model for concretefilled steel tube beam–coulmns with interlayer slip. Eng Struct 1998;20(8): 663–76. [9] Soh CK, Chiew SP, Dong YX. Damage model for concrete-steel interface. J Engrg Mech, ASCE 1999;125(8):979–83. [10] Lei XY. Contact friction analysis with a simple interface element. Comput Methods Appl Mech Engrg 2001;190:1955–65.
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