Path-dependent mechanical model for deformed reinforcing bars at RC interface under coupled cyclic shear and pullout tension

Transcription

1 Engineering Structures 30 (2008) Path-dependent mechanical model for deformed reinforcing bars at RC interface under coupled cyclic shear and pullout tension Masoud Soltani a,, Koichi Maekawa b a Department of Civil Engineering, Tarbiat Modares University, Jalaale-al Ahmad Ave., Tehran, Iran b Department of Civil Engineering, The University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo , Japan Received 10 January 2007; received in revised form 26 June 2007; accepted 27 June 2007 Available online 15 August 2007 Abstract Local discontinuities may arise in structural concrete due to abrupt changes in the sectional stiffness of connecting elements, construction joints or stress-induced interfaces in both cast-in-place and prefabricated structures. This paper aims at developing a unified path-dependent constitutive model for deformed reinforcing bars crossing such interfaces when subjected to generic three-dimensional displacement paths. The stress transfer mechanism is formulated from a microscopic point of view, and the engineering attention is chiefly directed at coupled cyclic pullout and transverse shear. Systematic verification of the model is carried out by comparing computational predictions with available experimental results. c 2007 Elsevier Ltd. All rights reserved. Keywords: RC interface; Dowel action; Cyclic shear; Bond; Joint interface 1. Introduction Structural interfaces may be present in pre-formed joints, which are common in precast concrete, construction joints in cast-in-place concrete and stress-induced interfaces or cracks, which commonly occur in beam column connections, brackets and corbels, etc. (Fig. 1). These interfaces may turn out to be critical planes in the operation of the load-resisting mechanism and govern the ultimate strength, ductility and energy absorption capability of an entire structure [1,2]. In RC interface planes under combined shear and open/closure dynamics, dowel action and pullout of reinforcement crossing the RC interface play a major role in the stress transfer mechanism, especially in the joints of prefabricated structures where aggregate interlock is not as active as dowel action. The mechanism of stress transfer in reinforcing bars crossing an RC interface is complex because mutually interacting mechanisms need to be treated in a unified manner: flaking of concrete that acts as bearing support for bars, development of plastic strain and hinges along reinforcing bars, deterioration of bond close to the interface and flexural localization along the bars. Corresponding author. Tel.: ; fax: address: msoltani@modares.ac.ir (M. Soltani). A number of investigations have considered ultimate capacity [3 6] and the ductility of dowel bars crossing RC interfaces [1,7 9]. However, three-dimensional (3D) pathdependent stress transfer across an RC interface is still under investigation. Extensive efforts at 2D and 3D constitutive modeling for RC have been undertaken [1,10] and the authors utilize these as the main basis of this study. The local bondslip strain model used here was originally developed by Shima et al. [11] and Shin et al. [12], which the relevant details can also be found in [1,10]. The model takes into account bond deterioration near crack planes and accurately predicts the behavior of a reinforcing bar that undergoes a high degree of plastic deformation. Qureshi and Maekawa [8] experimentally investigated the effect of dowel shear on bond deterioration by measuring the curvature-localized zone along reinforcing bars. By doing this, the complete monotonic response of reinforcing bars crossing the RC interface was simulated well, but the method can be applied only to two-dimensional analysis under monotonic loads. Here, the same concept is used, but by decomposing the overall deformation into elastic and plastic components, its applicability will be extended to a full 3D path-dependent RC interface model under multi-directional combined shear with open/closure dynamics. To make the model more consistent with nonlinear structural analysis, a /$ - see front matter c 2007 Elsevier Ltd. All rights reserved. doi: /j.engstruct

2 1080 M. Soltani, K. Maekawa / Engineering Structures 30 (2008) Fig. 1. Typical examples of interfaces in RC structures. numerically explicit type of formulation is aimed at. To clarify the versatility of the proposed method, verification is carried out by comparing analysis results with experiments. 2. Path-dependent slip strain model of reinforcing bar The axial tension force of a reinforcing bar is partly transferred to the concrete through bonding and consequently the local stress along a bar differs from that at the interface. Shima et al. proposed a bond stress slip strain model for reinforcement under uni-axial pullout states [1,10,11]. In their experiments, the bond between bar and concrete was intentionally removed for a distance of 10 times the bar diameter from the loaded end to eliminate the free-surface effect. Their model can, as a result, be used for bar sections somewhat apart from the interface plane in practice. A complete explanation of the basic concept and formulation can be found elsewhere [1,13,14]. For pullout specimens, the local slip (S) at any point (x ) along the bar is the summation of free end slip (S 0 ) and the integration of strain (ε s ) from the free end (x 0 ), as shown in Fig. 2: S = x x 0 ε s dx + S 0. (1) With this deformational compatibility, bar slip is clearly defined as displacement along the bar, measured relative to a far fixed point in the concrete. It should be noted that the definition of local slip is usually given as the relative displacement between the bar and the surrounding concrete. However the bond slip strain model developed by Shima et al. originally defines the relation between bond stress and relative displacement from fixed point in the concrete not relative displacement between bar and adjacent concrete. To specify the surrounding concrete is rather difficult, because the strain of concrete near the bar is very complicated due to the micro cracking induced by the bond effect of deformed bar [1,11]. The position of concrete, from which the relative displacement is measured, is usually specified as the position where the average strain is obtained in the section. In this case, the average strain of concrete at the section is necessary to specify the local slip. As the average strain of concrete in the section is difficult to be obtained in experiments, the surface of concrete is often used as the position. Then, the meaning of relative displacement Fig. 2. Definition of local slip and bond stress. becomes unclear. Therefore Shima et al. decided to define the slip as the relative displacement from a fixed point in the concrete [1,10]. In a pull out test the local slip is obtained by taking the summation of the integration of strain from the free end to the point concerned and the free-end slip (S 0 ). For the pull out case with adequate anchorage length the free end slip is zero (i.e. S 0 = 0). The local bond stress along the bar is proportional to the gradient of the stress profile as: τ = D 4 dσ dx (2) where τ is the local bond stress, D is bar diameter and dσ/dx is the gradient of the stress profile, which is also proportional to the slope of strain when it is elastic. The profile of steel stress and strain as well as the minimum required anchorage length (l min ) that also specifies upper bound of integral in Eq. (1) can be exactly computed from Shima s model through an iterative method as explained in [13,14]. Shima et al. logically showed that the bond stress, slip and strain of the steel are all in a unique correlation if an infinite anchorage length can be assumed [1,10]. Here, not only can a unique bond-slip relation be derived but there also exists a unique slip strain relation independent of location along the bar. So, the monotonic slip strain relation of elasticity is expressed as: s = ε s ( ε s ) (3)

3 M. Soltani, K. Maekawa / Engineering Structures 30 (2008) s = S D K f c (4) K f c = ( f c /20)2/3 (5) where s is defined as non-dimensional slip and f c is concrete compressive strength in MPa. When steel strain is greater than the strain at the onset of hardening, we have: s = s y = ε y ( ε y ) for ε y < ε s < ε sh. (6) In the hardening phase: Fig. 3. Bond deterioration close to the interface [8]. s = s y ( f u f y )(ε s ε sh ) for ε s > ε sh (7) where ε y is steel yield strain, ε sh is the strain at the onset of hardening; and f y and f u are the yield and tensile strengths of steel, respectively. Shin et al. [12] and Mishima and Maekawa [15] experimentally investigated the reliability of the above modeling by investigating the slip strain relation of steel crossing a RC interface. The original model by Shima et al. was confirmed to be highly accurate under the experimental condition where bond deterioration was avoided. However, the original model brings about an underestimation of steel pullout from real RC interface planes because of unavoidable bond deterioration just close to the interface. Qureshi and Maekawa [8] experimentally investigated the size of the bond deterioration zone and a detailed profile of bond stresses close to the free-end surface of the interface. Based on this work, the bond deterioration zone was found to extend about 5 times the diameter of the reinforcing bars under pure pullout and to increase in size when the RC interface is subjected to shear slide, which activates internal local curvature in the reinforcing bars (Fig. 3). The bond deterioration profile can be obtained by an exact iterative analysis along the reinforcing bars [16], but this implicit formulation may lead to some inefficiency in structural analysis where there is a large degree of freedom. For this reason, the authors sought an explicit formulation by simply modifying the original model of Shima et al. [11] to cover bond deterioration in a consistent manner as: a = 2(1 + (2x/L d ) 1.2 ) 2, x L d /2 (8) s = ε s (a ε s ) (9a) s = s y = ε y (a ε y ) for ε y < ε s < ε sh (9b) ( s = s y (2x L ) d) ( f u f y )(ε s ε sh ) L d for ε sh < ε s < ε 0 (9c) s = 0.007( f u f y )(ε s ε sh ) + s y / for ε s > ε s y /2 ε 0 = ε sh + ( (2x/L d 1))( f u f y ) (9d) (10) where x is the distance from the RC interface and L d is the size of the bond deterioration zone. Fig. 4 shows the envelope of this slip strain model (Eqs. (8) (10)). Fig. 4. Steel slip strain relation before yielding. The bond stress drops a great deal within some finite distance of the deterioration zone [1] due to radial bond micro cracks that reach the free interface plane, as shown in Fig. 3. Within this area (x < L d /2), the stress and strain of the reinforcing bar are constant and can be obtained from values computed at the boundary of the zone (x = L d /2). It is assumed that the bond characteristics are identical and the same strain slip model is applicable to both positive and negative loadings [15]. For the unloading paths, we have the relations of steel strain and axial slip in the elastic range (s < s y ) based on the works by Shin et al. [12] and the above modification on the loading states as: s = ε s (a 3500ε s ) in portion (2) (11a) s = s max 0.85(a ε )ε, ε = ε max ε s in portion (3) (11b) s = s min 0.85(a ε )ε, ε = ε min ε s in portion (4) (11c) s = 0.6(s p s max)(ε s 4/3 ε max ) 2 /εmax 2 + (16s max s p )/15 in portion (5) (11d) s = 0.6(s p s min )(ε s 4/3 ε min ) 2 /ε 2 min + (16s max s p )/15 in portion (6) (11e) where s max and ε max are the maximum slip and strain, and s min and ε min are the minimum slip and strain of steel before unloading, respectively. Notations s p and s p represent residual

4 1082 M. Soltani, K. Maekawa / Engineering Structures 30 (2008) reloading is almost uniquely related to the strain ε s at the crack interface [17,1] as: ε sp = s sh ξ(ε max ε s ) (14) where ξ is the ratio of maximum stress before unloading (σ max ) to the yield strength of steel bar (ξ = σ max /f y ). As the steel immediately develops a very large strain after yielding, the strain distribution is not continuous at the junction between the yielding and the elastic zone. However, steel stress must be continuous there, i.e. σ se = σ sp and the following expression may be written as: ε se = σ sp /E s (15) Fig. 5. Steel slip strain relation before for cyclic loading (before yielding). Fig. 6. Post-yielding distribution of strain along the bar and Kato s model. slips in the unloaded state defined as: s p = 0.15s max s p = 0.15s min. (12a) (12b) The different portions of the reversed cyclic path are shown in Fig. 5. The details of Eq. (11) and the experimental verification are available in [1,12], and the energy absorption of any closed loop paths was already verified to be positive definite. Thus, the second principle of thermodynamics is satisfied well. By considering the discontinuity in the strain distribution along the bar after yielding of the reinforcement, the unloading and reloading paths of slip strain after steel yield can also be obtained [1]. It is assumed that the normalized steel slip, denoted by s, can be expressed as the sum of slip s pl in the yield region and slip s e in the elastic region as: s = s pl + s e. (13) Here, steel strain at the interface, strain at the yield boundary point on the yield region, and strain at the yield boundary point on the elastic region are denoted by ε s, ε sp and ε se, respectively as shown in Fig. 6. Similarly, corresponding steel stresses are denoted by σ s, σ sp and σ se. Idequti et al. [17] pointed out that the strain ε sp at the yield boundary point during unloading and where E s is the elastic modulus of steel bar. Eq. (15) implies that ε se can be related to ε sp (and ε s ) through Eq. (14). As the elastic region is distant from the interface, it is assumed that bond deterioration does not reach the elastic boundary and the original model of Shin et al. [12] can be used to compute the elastic component of slip. So, with parameter a assumed equal to 2 and maximum slip as s y (slip at onset of yielding), Eqs. (8) (10) are used to obtain the elastic part of slip after yielding of the reinforcing bar. Through the above procedure, we can express the elastic steel slip s e as a function of steel strain ε s at the crack interface. The plastic part (s pl ) is computed based on the work of Shin et al. as [1,12]: s pl = (1 + ξ)ε s + ε sh ξε max l y (K f c /D) (16) 2 l y = 2 s max s y ε max + ε sh (D/k f c ) (17) where l y is the size of the steel yield zone as shown in Fig. 6. The steel strain can be explicitly computed from the aforementioned path-dependent slip strain model and the pull out slip. Kato s stress strain model [18] for the steel bar is used here to compute the steel stress from the given local strain. 3. Stress strain profile along the reinforcement With the steel slip at the precise location of the RC interface, the stress and strain profiles along the bar can be computed in an explicit non-iterative manner as stated in the previous section. The reinforcement is discretized into small finite segments and the compatibility requirement yields: s n 1 = s n K f c D ε sn x (18) where s n 1 and s n are the non-dimensional slip at the edge of the (n 1)th segment and the nth one, respectively, and ε sn is the strain already identical and assumed to be constant over the nth segment. Neglecting the elastic deformation of concrete, the interface opening ω is nearly twice the bar pullout slip S, defined from the single free end of the bar. Then, the nondimensional slip at the RC interface, denoted by s A, can be obtained as: s A = K f c 2D ω. (19)

5 M. Soltani, K. Maekawa / Engineering Structures 30 (2008) Also, using this non-iterative computation, the distribution of bond stress along the bar can be inversely computed by using Eq. (2), as shown in Fig. 9(b). It can be seen that the above formulation gives a good representation of local bond deterioration close to the crack interface. 4. Response of embedded bar under coupled axial and dowel shear displacements 4.1. Simulation under monotonic loading Fig. 7. Linear relation between local slip at the interface plane and middle of bond deterioration zone. Regarding dowel action, a beam resting on an elastic foundation (the BEF model) has been the most expedient way to simulate the response of an embedded reinforcing bar crossing the interface under monotonic loading [6,9]. However, the BEF analogy does not encompass the whole mechanism, because both the embedded bar and the supporting concrete actually behave nonlinearly [8,9]. Qureshi and Maekawa [8] experimentally investigated the induced curvature of embedded reinforcing bars under large applied deformations. The shape of the curvature distribution φ within the flexural zone is modeled as (see Fig. 10(a)): φ(x) = 3φ max(l c x) 2 L 2 for L c c 2 x L c [ φ(x) = 3φ ( ) 2 ( ) ] max Lc 3 L 2 3 c 2 x L c 4 L c x Fig. 8. The comparison between steel slip strain model at different location along the bar with available experimental results. Fig. 7 shows the slip at the center of the bond deterioration zone s B (point B in Fig. 3) versus the pull out slip at the interface plane s A (point A in Fig. 3), which is computed from the local stress analysis proposed by the authors [14]. Based on this result, we have the non-dimensional slip at point B simply expressed by: s B = s A [ (L d /D 5)] (20) which is treated as the boundary condition for the first finite segment. By using Eq. (20) and Eqs. (1) (4), the slip strain relation at three different locations, i.e. the interface (point A ), the center of the bond deterioration zone (point B ) and the onset of the bond deterioration zone (point C ) are shown in Fig. 8 in comparison with experimental data. This confirms the reliability of the assumptions made above. The steel strain and the stress of the reinforcing bar are sequentially computed (by explicit backward analysis) segment by segment from the slip based on the path-dependent constitutive models of slip strain and the steel stress strain relationships previously explained. Typical computed profiles of stress and strain and the local response of the bar at different locations are shown in Fig. 9(a). for 0 x L c 2. (21) When both reinforcing bars and surrounding concrete are in elasticity of small strains, the length of the curvature-induced zone (L c ) can be obtained from the BEF model as: L c0 = 3π 4 4Es I b (22) 4 k D where I b is the moment of inertia of the bar section and k is the supporting stiffness assumed as: k = 150 f 0.85 c. (23) D However, outside the elastic range, the flexural zone of the steel increases with increased shear displacement as a result of inelasticity and local crushing of the supporting concrete. This can be mathematically expressed by a non-dimensional damage parameter, DI, as: ( DI = S ) δ (24) D D L c = L c0 for DI 0.02 L c = L c0 (1 + 3(DI 0.02) 0.8 ) for DI > 0.02 (25) where δ is the local deflection at the RC interface. This is equal to one-half the shear slip of the interface plane denoted by δ t. The maximum curvature, φ max, along the flexural zone can be

6 1084 M. Soltani, K. Maekawa / Engineering Structures 30 (2008) Fig. 9. (a) Computed path-dependent profile of stress and strain of steel; (b) distribution of bond stress along the bar. obtained by satisfying the following boundary condition: δ = φ(x)dx (26) L c which yields: φ max = 64δ 11L 2. (27) c Comparison of computed induced curvature with experimental results [8] for three different transverse displacements and constant axial pullout slip is shown in Fig. 10 which confirms the reliability of the above simulation. As curvature and axial deformation along the reinforcing bar proceed together, some parts of the bar reach the local yield point. Considering y z as the coordinate axes in the plane of the RC interface and x as the axis of the reinforcing bar (Fig. 11), the local strain at each cell over the cross section is expressed as: ε (x,y,z) = ε s (x) + φ y (x) z + φ z (x) y (28) where ε s (x) is the average steel strain computed from the axial slip of the reinforcing bar as explained before and φ y (x) and φ z (x) are projections of the local curvature, φ(x), on y and z axes as: φ y (x) = φ(x) cos(θ) φ z (x) = φ(x) sin(θ) (29a) (29b) where θ is the angle between displacement direction and the y axis, as shown in Fig. 11. The effect of shear on yielding of the steel bar (kinking of the bar) is taken into consideration by applying Von Mises yield criterion and the isotropic hardening rule [14]. The bending moments M y (x), M z (x) are computed from the local stresses. Consequently, the shear forces along the bar axes, Fig. 10. Curvature profile along the bar; (a) model of induced curvature in the vicinity of RC interface; (b) comparison of model with test results [8]. V y (x), V z (x), are computed as: V y (x) = dm y(x) dx (30a) V z (x) = dm z(x). (30b) dx Thus, the shear force carried by the reinforcing bar, the socalled dowel action, is directly calculated at the plane of the RC interface where x = 0. The computed profiles of steel

7 M. Soltani, K. Maekawa / Engineering Structures 30 (2008) Fig. 13. Effect of cover on capacity of dowel bars: comparison of model and experimental results. Fig. 11. Discrimination of reinforcing bar and computing the strain of each cell. Fig. 12. Typical profile of stress strain and sectional forces along the bar. strain, stress and sectional forces under applied shear are shown in Fig. 12 with local bond stress along the bar. For the interface under a complex in-plane shear displacement path, the damage parameter formulated in Eq. (24) can be expressed independently for the two coordinate axes of the RC interface (y and z) as: ( DI y = S ) δy (31a) D D ( DI z = S ) δz (31b) D D where δ y and δ z are the components of shear slip at the interface in its local coordinate axes. We then have: DI = DIy 2 + DI z 2. (32) 4.2. Splitting of concrete cover When an embedded bar applies pressure on the concrete cover, the transverse force in the bar is in equilibrium with the radial tensile stress over the concrete cover. If the concrete strength is low, the bar diameter large and the concrete cover insufficient, splitting of the cover concrete may take place. According to an experiment by Vintzeleou and Tassios [3], a splitting crack can be initiated when the dowel force reaches: V cr = 5D f ct c c + D (33) where f ct is the tensile strength of concrete in MPa and c is the concrete cover in the loading direction in mm. Given the inability of the cover concrete to hold the bar, the force displacement relationship shows lower stiffness. To take this influence into consideration, a simple formula is presented with the original damage parameter as: DI = DF (damage parameter in Eq. (31)) (34) DF = 1 1 (1 c/12d) 5 for V V cr (35a) DF = 1 for V < V cr (35b) which is computed for both local axes in the interface plane (y and z) with the cover size. A comparison of computed results with the experiment [3,19] is shown in Fig. 13. It should be noted that the location of stirrups and their spacing significantly influence the response of embedded bars just after cover splitting. This influence is not considered in this study and remains a future development Response under load reversal To extend the formula above to unloading and reloading cases, we divide the total deformation into two components, the reversible elasticity and the residual plasticity of dowel action. Experiments by Vintzeleou and Tassios [3] as well as by Soroushian et al. [19] under cyclic deformation show that the remaining plastic deformation of a dowel bar is within the range times the maximum (or minimum, depending on the direction of loading) deformation of dowel bars just before unloading. In this paper, direct proportionality between the residual and maximum (or minimum) shear sliding is assumed as: β y = δ py /δ my = 20DI ym 0.4 β z = δ pz /δ mz = 20DI zm 0.4 (36a) (36b)

8 1086 M. Soltani, K. Maekawa / Engineering Structures 30 (2008) Fig. 14. Progressive inelasticity and fracturing in supporting concrete. where δ py and δ pz are the plastic components of deformation and δ my and δ mz are the maximum (or minimum) deformation before unloading in the y and z directions, respectively. DI ym and DI zm are the components of damage parameters in the y and z directions just before unloading (corresponding to δ my and δ mz ). The source of nonlinearity is rooted in the plasticity and continuum fracturing of the concrete that bears the force imposed by the bar (Fig. 14), as well as the plasticity across the steel bar section. From this analogy, the shear force versus transverse deformation relation on the reloading path is: V (δ) = K u (δ δ p ) = (1 β)k u δ (37) where V and K u are, respectively, the shear force carried by a reinforcing bar and the elastic stiffness before unloading in the direction of the displacement component (y or z) and β is the plastic deformation ratio as defined in Eq. (36). Instead of subtracting the plastic term and computing the changes in elastic stiffness, the complete load deformation of an embedded bar throughout the unloading and reloading paths can be obtained by a simply defined equivalent shear displacement, δ eq as: δ eq = δ λδ m (38) where λ is expressed as (based on Eq. (37)): K u (δ βδ m ) = K u (δ λδ m ) (1 β) (39) where δ βδ m represents the elastic component of deformation before unloading and (δ λδ m )(1 β) is the elastic component of the proposed equivalent shear displacement. From Eq. (39), λ is expressed by: λ = β (δ δ m ). (40) β 1 δ m Considering different unloading and reloading stiffness, with reference to the available experimental results [3,5], in the case of unloading, we have, (δ βδ m ) = γ (δ) (δ λδ m ) (1 β) (41) λ = [1 γ (δ) (1 β)]δ βδ m γ (δ) (β 1)δ m (42) Fig. 15. Definition of path dependent parameter λ to consider plasticity. where γ (δ) is the relation of unloading and reloading stiffness and [γ (δ) = 1] yields Eq. (40). The value of λ for loading, unloading and reloading conditions with a simple bilinear form for γ (δ) is shown in Fig. 15. The value of γ (δ) is assumed to linearly increase from 1 to 2.3 and kept constant after the change of loading direction as shown in Fig. 15. It should be noted that the proposed bilinear form for γ (δ) is an engineering approximation based on the available experimental results [3,5, 8]. The authors understand that more experimental works will lead to enhancement of the model concerning applicability. The shear force derives from the proposed equivalent shear deformation as: V (δ) = (1 β)k u δ eq. (43) With regard to the original model of Qureshi and Maekawa [6] for computing V (δ), represented by Eq. (37), the reloading unloading response with Eq. (43) can be computed in the same manner as the monotonic loads. Also, the damage parameter with respect to total deformation can be defined to reflect the change in elastic stiffness of the system. Before elastic deformation is completely removed in the unloading reloading phase, it is assumed that the damage parameter is the same as that computed just before unloading, i.e. it is equal to DI ym or DI zm depending on the direction of the displacement concerned. This makes the stiffness constant within the reloading path. In other paths, Eq. (31) based on the equivalent shear displacement is used to compute the damage parameter in the direction of the displacement component (z or y direction). This is expressed as: ( DI y = S ) δeqy (44a) D D ( DI z = S ) δeqz (44b) D D

9 M. Soltani, K. Maekawa / Engineering Structures 30 (2008) Fig. 16. Cycling effect on response degradation; (a) change in secant stiffness; (b) different portion in hysteretic response; (c) comparison with experiment [3]. where δ eqy and δ eqz are the equivalent components of shear deformation of the interface in its local coordinate axes as computed by Eq. (38). Based on the definition of equivalent shear displacement, Eq. (44) yields the same damage parameter before unloading as the loading condition Degradation due to cyclic action Some experiments on dowel action have shown extensive degradation of stiffness and strength with load repetition [3,19]. This cyclic degradation is highly nonlinear in nature [3]. The effect of cyclic loading is thought to be a degradation of surrounding concrete stiffness represented by non-damage elastic springs in Fig. 14, which in turn influences the size of the curvature influencing zone, as mathematically expressed in Eq. (22). Vintzeleou and Tassios [3] experimentally investigated the effect of cyclic loading on stiffness degradation. They showed that the stiffness decay is greater in the case of full reversal and that it is also particularly independent of compressive strength, concrete cover and bar diameter. Based on this experimental result the following relation for the reduction in subgrade stiffness is adopted (see Fig. 16(a) and Fig. 16(b)): ( ) D k = η 1 (45) CP + CN η = 1.0, for the first cycle (46a) η = 0.4, for the second and subsequent cycles, in path 1, 3 η = δ/δ m, in path 2, 4 (46b) for the second and subsequent cycles, (46c) where CP and CN are the number of unloadings from the maximum positive and negative deformations, respectively, (CP + CN is equal to 2 for one full cycle) and δ m is the maximum or minimum deformation before unloading. To check the reliability of these empirical formulae, we consider the effect of degradation on dowel action. Based on the work of Dei Poli et al. [9], which is also compatible with the BEF analogy, the following proportionality is known to exist between the secant stiffness (K s ), (in turn the effective elastic stiffness in the response of the dowel bar) and the stiffness of the concrete (k) reacting against it, as: K s k (47) Thus, Eq. (45) can be rearranged for the secant stiffness just before unloading (where η = 1) as: ( ) 1 1/3 D K s =. (48) CP + CN Fig. 16(c) compares the adopted formulation with the experimental results obtained by Vintzeleou and Tassios [3] for dowel bars subjected to reversal of transverse shear. Fair agreement between experiment and the model is obtained.

10 1088 M. Soltani, K. Maekawa / Engineering Structures 30 (2008) Fig. 17. Comparison of test results and model; (a) comparison with test results reported in Ref. [9]; (b) comparison with test results in Ref. [4]. Fig. 19. Secant stiffness degradation of embedded bar; (a) comparison with model of Dei Poli et al. [9] in case of no axial pullout; (b) stiffness reduction due to axial pull out. Fig. 18. Comparison of experimental [8] and numerical results for coupled shear and axial displacement. 5. Computational procedure The computation procedure can be summarized as below. 1. The axial steel slip and shear displacements in the local axes of the interface (x, y and z) are computed from the opening and shear sliding of the interface. 2. By using the path-dependent slip strain and stress strain relationships given in Eqs. (8) (17), the local steel strain and stress along the bar are computed. The bond deterioration zone is initially five times the diameter of the reinforcing bars and increases as curvature takes place close to the interface [1,8]. Here it is assumed that, for the case of coupled axial and shear deformation, the size of the bond deterioration zone is equal to Fig. 20. Comparison between experimental [15] and numerical results for embedded bar under cyclic pull out. the maximum size of the flexure-induced zone of the bar during deformation reversal, but not less than 5D. 3. From the computed shear sliding in the y and z directions, the path dependent damage parameters and degradation factors due to cycling are derived in each direction by using Eqs. (44) and (45). 4. The damage parameter in the direction of the resultant deformation is computed from Eq. (32) and for the effective degradation factor the simple interpolation below is applied: D k = D ky cos 2 (θ) + D kz sin 2 (θ). (49)

11 M. Soltani, K. Maekawa / Engineering Structures 30 (2008) Fig. 21. Response of dowel bar under cyclic deformation path, comparison with experimental results [3]. (a) large cover in both sides; (b) different cover size in positive and negative directions. 5. The length of the curvature zone and the profile of curvature along the bar are computed with the effective damage parameter and degradation factor. 6. Using the curvature distribution along the bar and the average axial strain, the local strain and stress distribution over the bar cross section and the local bending moment can be computed. The shear force in the plane of interface is computed using Eq. (30). 6. Experimental verification 6.1. Monotonic displacement The experimental and computed dowel action provided by different diameters of bar under pure shear [9] are shown in Fig. 17(a). Predictions for the case described in reference [4] are also shown for different bar diameters in Fig. 17(b) and fair agreement with the experimental results is obtained. The nonlinearity under pure shear is much higher than in the case of a bar subjected to a coupled shear displacement and axial pullout. The failure mode is highly ductile, with significant plasticity along the bar axis and considerable depth of plasticization across the bar section [1,20]. Qureshi and Maekawa [8] experimentally investigated the effect of pullout deformation on the shear capacity and ductility Fig. 22. Comparison of numerical and experimental results [19] for response of dowel bar under cyclic deformation path. of embedded bars. A comparison of their experimental data extracted from [8] with the results predicted for the case of a coupled displacement path is shown in Fig. 18. In contrast with the case of pure transverse shear, there is greatly reduced stiffness and capacity of the embedded bar due to axial pullout. Dei Poli et al. extensively studied the subgrade stiffness and deformation of dowel bars under pure shear. A comparison between the secant stiffness computed by Dei Poli s model and the current procedure is shown in Fig. 19(a) for different bar diameters. There is good agreement between the two sets of results. However, the model described here is also able to show the change in stiffness influenced by axial pullout. The predicted stiffness reduction for different axial deformations is shown in Fig. 19(b) Cyclic displacement path Mishima and Maekawa [15] experimentally investigated the hysteretic response of RC interfaces with inclined reinforcement when subject to a repeated opening closing path. The experimental data [15] and analytical results are shown in Fig. 20. The slip at the center of the bond deterioration zone is calculated using Eq. (20), from which the steel stress at the interface was computed to check the reliability of the proposed bond deterioration formula. Good agreement between the analytical and test results is confirmed.

12 1090 M. Soltani, K. Maekawa / Engineering Structures 30 (2008) D-Hysteretic response of RC interface Fig. 23. Numerical example: effect of axial pull out on cyclic response of embedded bars. The experiment carried out by Vintzeleou and Tassios [3] is used to examine the versatility of the proposed modeling method under cyclic displacement paths. The computed hysteresis loops for two specimens made of different material and with different bar locations are summarized in Fig. 21. The cyclic degradation of stiffness is well simulated by the proposed model. Also, different responses to action against the cover concrete and core concrete of the specimen are shown to be in almost perfect agreement with the experimental data. Similar predictions are also applied to reference [19], with different bar diameters and displacements and different concrete properties from [3]. In the analysis, the reported interface opening was taken roughly into account based on the paths of complexity in reference [19]. Predictions are in good agreement with the experiment, as shown in Fig. 22. Degradation of stiffness and capacity under cyclic loading are shown in Fig. 23. Different axial deformation (opening) was considered in the analysis and it can be seen that the cyclic decay of stiffness as well as axial force can be numerically simulated. The main source of stiffness degradation is the inelasticity and fracturing of the support concrete, which would be more severe in the presence of axial force. The computational model can also be used for the interface under arbitrary shear displacement paths in the plane of the RC interface. In such a case, the path-dependent damage parameter and stiffness degradation factor in the local coordinate axes of the interface plane are computed as previously explained. To check the versatility of the proposed methodology, the interface as shown in Fig. 23 was reanalyzed under a displacement path applied at an angle of 30 with the y axis, but with the same resultant deformation path as in the previous case. Here, the path dependent parameters are separately computed in the local coordinate system (y z). As can be seen in Fig. 24, the computed resultant response is the same as that computed in Fig. 23, which verifies the method used to compute the accumulated damage in the coordinate axes. The interface in the other example was subjected to cyclic deformation at 45 prior to the cyclic deformation in the y direction. It is obvious from Fig. 25 that the response in the y direction is highly affected by the damage accumulated in the first displacement path. 8. Conclusions A micro-mechanism based approach was applied to the behavioral simulation of steel deformed bars embedded in concrete under combined shear and open/closure dynamics at RC joint interfaces. The hysteretic path dependency was Fig. 24. Numerical example: Embedded bar under displacement cyclic paths in x and y directions.

13 M. Soltani, K. Maekawa / Engineering Structures 30 (2008) Fig. 25. Numerical example: response of an embedded bar under cyclic displacement path in different directions. successfully taken into account. The following items were newly introduced into the two-dimensional formulation built up through previous research to enhance the dowel action model and to extend the simulation into three dimensions. (1) Explicit formulation of the bond-slip strain relation and backward analysis of local bond slip and strain; (2) Multi-directional flexure of steel bar and 3D yield criterion; (3) Cyclic deterioration. A systematic experimental verification was performed to examine the reliability of the new approach and reasonable accuracy was confirmed. Although further research is expected to improve the empirical relations and assumptions used in the paper, the algorithm and formulations presented herein demonstrated potential ability to accurately model complex hysteretic response of RC Interface. Currently, a computational approach is being developed for the assessment of safety and fatigue life of RC PC joints placed between pre-cast and castin-situ concrete which can be used for (1) The seismic and fatigue analysis of precast concrete structures, (2) Design of structural rehabilitation of existing structures which will be strengthened or retrofitted through the post-anchored steel reinforcing bars connecting to newly constructed RC members, (3) Underground slurry wall and RC/PC shells connected to improve the seismic performance. Acknowledgments The authors express their gratitude to Dr. N. Fukuura of Taisei Corporation for helping with intensive discussions. This study was financially supported by Grant-in-Aid for Scientific Research (S) No References [1] Maekawa K, Pimanmas A, Okamura H. Nonlinear mechanics of reinforced concrete. SPON Press; [2] Berg VB, Stratta JL. Anchorage and the Alaska earthquake of March 27, New York: American Iron and Steel Institute; [3] Vintzeleou EN, Tassios TP. Behavior of dowels under cyclic deformations. ACI Structural Journal 1987;84: [4] Paulay T, Park R, Phillips MH. Horizontal construction joints in castin-place reinforced concrete, shear in reinforced concrete. ACI Special Publication 1974;SP 42-27(2): [5] Soroushian P, Obaseki K, Rojas M, Najm HS. Behavior of bars in dowel action against concrete cover. ACI Structural Journal 1987;84: [6] Ince R, Yalcin E, Arslan A. Size-dependent response of dowel action in R.C. members. Engineering Structures 2007;29(6): [7] Martin-Perez B, Pantazopoulou SJ. Effect of bond, Aggregate interlock and dowel action on the shear strength degradation of reinforced concrete. Engineering Structures 2001;23(2): [8] Qureshi J, Maekawa K. Computational model for steel embedded in concrete under combined axial pullout and transverse shear displacement. Proceedings of the JCI 1993;15(2): [9] Dei Poli S, Di Prisco M, Gambarova PG. Shear response, deformations and subgrade stiffness of a dowel bar embedded in concrete. ACI Structural Journal 1992;89(6): [10] Okamura H, Maekawa K. Nonlinear analysis and constitutive models of reinforced concrete. Tokyo (Japan): Gihodo-Shuppan; [11] Shima H, Chou L, Okamura H. Micro and macro models for bond in reinforced concrete. Journal of Faculty of Engineering, The University of Tokyo (B) 1987;39(2): [12] Shin H, Maekawa K, Okamura H. Analytical approach of RC members subjected to reversed cyclic in-plane loading. Proceedings of JCI Colloquium on Ductility of Concrete Structures and its Evaluation 1988; 2:45 6. [13] Salem MMH, Maekawa K. Pre and post-yield FEM simulation of bond of ribbed reinforcing bars. Journal of Structural Engineering, ASCE 2004; 130(4): [14] Soltani M, An X, Maekawa K. Computational model for post cracking analysis of RC membrane elements based on local stress strain characteristics. Engineering Structures 2003;25(8): [15] Mishima T, Maekawa K. Development of RC discrete crack model under reversed cyclic loads and verification of its applicable range. Concrete Library of JSCE 1992;20: [16] Soltani M, An X, Maekawa K. Localized nonlinearity and size-dependent mechanics of in-plane RC element in shear. Engineering Structures 2005; 25: [17] Ideguti H, Matsumoto S, Maemura M. Study on distribution of plastic strain along a steel bar subjected to reversed cyclic loading and calculating of the steel slip. In: Proceedings of the 3rd annual conference of JSCE V-5, p [18] Kato B. Mechanical properties of steel under load cycles idealizing seismic action. In: AICAP-CEB Symposium: Structural concrete under seismic action. CEB Bulletin Information 1979;131:7 27. [19] Soroushian P, Obaseki K, Baiyasi MI, El-Swedian B, Choi K. Inelastic cyclic behavior of Dowel bars. ACI Structural Journal 1988;85:23 9. [20] Suzuki M, Nakamura T, Horiuchi M, Ozaka Y. Experimental study on the influence of tension force on dowel effect of axial bars. Proceeding of the JSCE 1991;426(14): [in Japanese].