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Numerical approach for evaluating shear failure behavior of strain hardening cementitious composite member
Accepted Manuscript Numerical Approach for Evaluating Shear Failure Behavior of Strain Hardening Cementitious Composite Member Yongxing Zhang, Naoshi Ueda, Hikaru Nakamura, Minoru Kunieda PII: DOI: Reference:
S0013-7944(16)30010-8 http://dx.doi.org/10.1016/j.engfracmech.2016.02.008 EFM 5049
To appear in:
Engineering Fracture Mechanics
Received Date: Revised Date: Accepted Date:
24 September 2015 15 January 2016 1 February 2016
Please cite this article as: Zhang, Y., Ueda, N., Nakamura, H., Kunieda, M., Numerical Approach for Evaluating Shear Failure Behavior of Strain Hardening Cementitious Composite Member, Engineering Fracture Mechanics (2016), doi: http://dx.doi.org/10.1016/j.engfracmech.2016.02.008
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Numerical Approach for Evaluating Shear Failure Behavior of Strain Hardening Cementitious Composite Member
Yongxing Zhanga,b*, Naoshi Uedab, Hikaru Nakamurab, Minoru Kuniedab
a
School of Civil Engineering, Nanjing Forestry University, No. 159 Longpan Road, Nanjing 210037, China b
Department of Civil Engineering, Nagoya University, Chikusa-ku, Nagoya 464-8603, Japan
*Corresponding Author. Tel.: +86-25-83792021 E-mail addresses: [email protected]
Abstract: This paper presents a numerical approach for evaluating the shear failure behavior of strain hardening cementitious composite (SHCC) member. The developed approach can clarify the respective contributions of both contact stress and fiber bridging stress for the shear stress transfer behavior on crack surface of SHCC member, which thus can disclose the contribution of fibers to local shear of SHCC member. In the numerical approach, a shear stress transfer model and shear lattice system are proposed, in which the shear lattice system can appropriately express the shape of crack surface, and the shear stress transfer model can accurately quantify the transferred shear stress on the crack surface of SHCC member, focusing on the contribution from both matrix’s contact effect and fiber bridging effect respectively. The effectiveness of the numerical approach is confirmed through the comparison between the experimental and numerical results. Keywords: strain hardening cementitious composite member; shear failure behavior; shear stress transfer model; contact effect of matrix; fiber bridging effect.
1
Nomenclature ε = strain εA = strain at point A σA = stress at point A εB = strain at point B σB = stress at point B εC = strain at point C σC = stress at point C Lelm = element size Leq = equivalent element length Ls = crack spacing lm = measured length in uniaxial tensile test Gf = Fracture energy in uniaxial tensile behavior, equaling to the area of stress-displacement relationship after peak θ = angle H = asperity height S = contact rate w = crack width E* = elastic modulus of fiber contribution Vf = fiber content Ef = elastic module of fiber ffiber = elastic module of fiber σs1 = contact stress on tension part of crack surface σs2 = fiber bridging stress on tension part of crack surface Δ{εlcr} = strain field in local coordinate system (ξ ,η) [Tε,s1,s2] = strain rotation matrix used to rotate the incremental local strain into direction of contact stress Esi = gradient of equivalent uniaxial stress-strain relationship [Ω] = controlling matrix to neglect the influence of tensile stress in the direction of ξ
2
1. Introduction Strain hardening cementitious composite (SHCC) is an attractive construction material due to the wellknown advantages in terms of excellent mechanical and physical properties [1-3]. Several experimental investigations concerning the shear behavior of SHCC members [4,5] and SHCC for RC shear strengthening [6] have been carried out, which demonstrate more excellent shear resistance than ordinary concrete and fiber reinforced concrete. Moreover, several numerical models have been developed for evaluating the shear behavior of SHCC member, such as the proposed computational continuum damage formulation based model [7] and fixed smeared crack model [8]. In view of previous published literatures, the shear strength of SHCC is closely correlated with the flexural strength of SHCC [9,10], and the bridging action of flexible fibers in SHCC also has effect on the shear behavior of SHCC [11]. Therefore, the shear stress transfer behavior on crack surface of SHCC member or strengthening layer is contributed by not only contact stress but also fiber bridging stress on the crack surface, whereas the aforementioned numerical models haven’t sufficiently consider the aforementioned respective contributions. However, the contribution of fibers to local shear is today an open question as most of the current approaches consider only the shear component of the local tensile fiber behavior on the diagonal cracks. There is thus a need of developing numerical model for clarifying the respective contributions of both contact stress and fiber bridging stress for the shear stress transfer behavior on crack surface of SHCC member, which is also expected to disclose the contribution of fibers to local shear of SHCC member. In this paper, this type approach is proposed for evaluating the shear failure behavior of SHCC member, in which the shape of SHCC member’s crack surface is expressed by shear lattice system, and a shear stress transfer model is proposed for accurately quantifying the transferred shear stress on SHCC member’s crack surface, taking account of the respective contributions from both matrix’s contact effect and fiber bridging effect on crack surface. 2. Experimental investigation of shear failed SHCC member 2.1. Specimen characteristics and materials The experimental investigation of shear failed SHCC member is carried out. Fig. 1 shows the geometries of the experimental specimens, including effective depth d, width t and shear span length a, the details of which are also listed in Table 1. The SHCC members are 900mm and 1200mm long respectively, with a
3
cross-section of 50 × 200mm2. One deformed bar of 25mm diameter is arranged as longitudinal reinforcement for both SHCC members, and no deformed bars are arranged as stirrups. Moreover, the specimens are loaded under a three-point bending setup, in which the load and displacements (labeled as A and B in Fig. 1) are measured using displacement transducers and load cell respectively. 2.2. Material Property Fig. 2 demonstrates the uniaxial tensile stress-strain curves and ultimate crack pattern of SHCC, which are obtained from the dumbbell-shaped specimens in uniaxial tensile test, with the measured length of 100mm. All the specimens exhibit significant strain hardening behavior until ultimate tensile strength (labeled as B1 or B2 in Fig. 2), since multiple fine cracks occur and propagate after the initial tensile strength (labeled as A in Fig. 2) with an initial crack. The specimens are finally failed due to localization of some multiple fine cracks, illustrated by the crack pattern shown in Fig. 2. The mean compressive strength and Young’s modulus of SHCC are 91MPa and 29GPa respectively. The mean Young’s modulus and yield strength of the longitudinal reinforcement are 200GPa and 1050MPa respectively. 2.3. Experimental result 2.3.1. Shear load-displacement curve and cracking behavior Fig. 3 shows the shear stress-displacement curves and cracking behavior obtained from the experiment, in which the shear stress is defined by the measured load dividing the area of effective section (d × t). The peak shear stresses are 11.6 MPa and 8.3 MPa in cases 1 and 2 respectively, which means the shear load carrying capacity of SHCC member is significantly influenced by the ratio of shear span length to effective depth. The shear stress-displacement curves of SHCC members demonstrate linear curves up to peak shear stress, which experience the cracking behavior from multi-cracking to localization of some multiple fine cracks as demonstrated in Fig. 3(b). Moreover, the investigated crack spacing between multiple fine cracks in diagonal shear direction is about 3mm. 2.3.2. Cracking pattern Fig. 4 demonstrates the cracking pattern of cases 1 and 2. It can be obviously seen some multiple fine cracks occur in diagonal shear direction and play dominant role in the cracking behavior of SHCC member. This also means the shear failed SHCC member experiences the multi-cracking processes in diagonal shear
4
direction. As illustrated in Fig. 4, the SHCC member is finally failed due to the localization of some multiple fine cracks in diagonal shear direction. Fig. 5 demonstrates the asperity investigation of one sample’s fracture surface, which is cut from the tested SHCC member. The investigation is implemented using a laser displacement meter in terms of assuming the asperity of the start and end positions as 0 mm. It is worth mentioning that an assumed smooth crack is adopted for reflecting the asperity of fracture surface, which has the angle of 15º and asperity height of 1.2mm. It can be obviously seen the assumed crack has a good fitting with the measured asperity, which implies the asperity along the measured longitudinal line is very smooth, and also verifies the smoothness of SHCC member’s fracture surface. The above experimental investigation illustrates the shear failed SHCC member experiences the multicracking processes, whereas the fracture surface of tested member is very smooth, which implies the shear transfer behavior on crack surface of SHCC member is contributed by not only the contact effect but also the fiber bridging effect on the crack surface. 3. Numerical approach for evaluating the shear failure behavior of SHCC member In this section, a numerical approach is developed for evaluating the shear failure behavior of SHCC member. In the numerical approach, the shape of SHCC member’s crack surface is expressed by a shear lattice system, and a shear stress transfer model is proposed for accurately quantifying the transferred shear stress on SHCC member’s crack surface. 3.1. Crack width of shear failed SHCC member Based on the above-described experimental investigation, the crack width of shear failed SHCC member can be demonstrated as Fig. 6, in which ε is the strain, εA and εB correspond to the strain with initial cracking and ultimate tensile strength respectively. The multiple fine cracks firstly occur in strain hardening phase (ε smaller than εB and larger than εA), and the crack width w is defined by the width of one multiple fine crack wm at this moment. Thereafter, some of the multiple fine cracks localize in strain softening phase (ε larger than εB), and the crack width w is correspondingly defined by the width of the localized crack among some multiple fine cracks ws, expressed by Eq. (1), in which Leq is the equivalent element length, and crack spacing ls is assumed as 3 mm, according to the experimental measurement.
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w=
0
A
A B A ls B B A ls + B Leq
(1)
3.2. Shear lattice system Fig. 7 shows the shear lattice for expressing the shape of SHCC member’s crack surface, which is described by a simple geometry with series of triangle and modeled by the parameters of angle (θ) and asperity height (H) [14]. The contact stress in contact area is compressive stress perpendicular to the incline surface and denotes as S1 in Fig. 7(b); the fiber bridging stress in the area without contacting is tensile stress along the perpendicular direction of crack surface’s inclined plane and denotes as S2 in Fig. 7(b). As illustrated in Fig. 5, the fracture surface of the experimental specimens can be expressed by series of triangle with 15º θ and 1.2mm H. 3.3. Shear stress transfer model Fig. 8 demonstrates the concept of the proposed shear stress transfer model, which can accurately quantify the transferred shear stress on SHCC member’s crack surface, taking account of the contribution from both matrix’s contact effect and fiber bridging effect in crack surface. 3.3.1. Contribution from matrix’s contact effect in crack surface As shown in Fig. 8, the shear stress transfer behavior is influenced by the contact stress on the compression part of crack surface. This influence is varied and affected by the crack width, since the contact area in crack surface’s compression part is significantly decreased with the increasing crack width, which can be expressed by the assumed relationship of contact rate (S) versus crack width (w), as demonstrated in Fig. 9. Moreover, the crack width (w) is also governed by above-demonstrated Eq. (1). 3.3.2. Contribution from fiber bridging effect in cack surface The shear stress transfer behavior is also influenced by the varied fiber bridging stress on the tension part of crack surface. The relationship between fiber bridging stress and strain in shear lattice direction is assumed as shown in Fig. 10, in which the uniaxial bond stress-slip curve is identified through the results of single fiber pull-out tests [13], and E* is the elastic modulus of fiber contribution. Moreover, fibers are randomly distributed in SHCC member and also related to fiber content in SHCC, and the elastic modulus of fiber contribution E* is thus assumed to be calculated by Eq.(2) as a simplified hypothesis for a first 6
study, whereas this assumption should be refined in future study. In the equation, Vf is fiber content in SHCC, Ef is elastic module of fiber, and the maximum shear fiber bridging stress (shear fiber bridging strength) ffiber in shear lattice is decided by the bond strength of fiber and matrix, since it depends on the pullout of fibers between the crack surfaces.
E * V f 3 E f 2
(2)
3.3.3. Transferred shear stress on crack surface of SHCC member Based on the above-described contact stress (σs1 ) on compression part and shear fiber bridging stress (σs2) on tension part of crack surface, the transferred shear stresses on SHCC member’s crack surface can be calculated by Eqs. (3) and (4), which also can rotate to local stress field by stress rotation matrix [Tε,s1,s2]-1 thereafter, and the incremental local stress thus can be calculated by Eq.(5). In the equations, Δ{εlcr} is strain field in local coordinate system (ξ ,η), [Tε,s1,s2] is strain rotation matrix used to rotate the incremental local strain into direction of contact stress (σs1 and σs2), Esi is gradient of equivalent uniaxial stress-strain relationship, Es2 equals to E* in Eq.(2), and [Ω] is the controlling matrix to neglect the influence of tensile stress in the direction of ξ.
E s1 s1 s 2 0
0 T lcr Es 2 , s1, s 2
Esi si si 0 1 l n T , s1, s 2 s1 s 2
(3)
(4)
(5)
4. Evaluation of the shear failure behavior of SHCC member 4.1. Numerical model Fig. 11 shows the numerical models of the above experimental specimens, the mesh sizes of which are 30mm and 25mm in longitudinal and vertical directions respectively. 4.2. Material properties 4.2.1. Modeling uniaxial tensile property of SHCC The tri-linear curve model taking into account fracture energy is adopted for modeling the uniaxial tensile behavior of SHCC, the concept of which is demonstrated in Fig. 12(a). Especially, the fracture
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energy obtained from uniaxial tensile stress-strain curve is illustrated in Fig. 12(b), in order to avoid the element size dependency problem [14]. In the model, points A and B are defined as the position where initial crack occurs and the position with ultimate uniaxial tensile strength. The strain of point C, εC, is represented by Eq.(6), by accounting fracture energy in localized element:
C = B +
2G f
(6)
B Lelm
where, εB is strain at point B, σB is stress at point B, Lelm is element size (mm), εC is strain at point C, Gf is fracture energy in uniaxial tensile behavior, equaling to the area of stress-displacement relationship after peak (N/mm). As mentioned in Fig. 12(b), the uniaxial tensile stress-strain curve of SHCC obtained from experiment is abstracted by tri-linear curve with minimum ultimate tensile strength, due to damage generally occurs at weak area. Therefore, the corresponding parameters of points A, B, C for tri-linear curve model taking account of fracture energy are summarized in Table 2. 4.2.2. Modeling uniaxial compressive property of SHCC The compressive stress-strain relationship of SHCC is modeled as shown in Fig. 13, in which σc and fc are compressive stress and strength (MPa), ε is compressive strain, and Leq is equivalent element size (mm). In the relationship, saenz equation:
i
Ec cu
1 Ec
, is used up to the compressive
2 fc 0 2 cu 0 cu 0
strength and a linear softening branch is assumed thereafter. The slope of linear softening branch is defined by considering the compressive fracture energy (Gfc), in order to avoid the above-mentioned element size dependency problem [14]. 4.2.3. Modeling property of reinforcement The longitudinal reinforcement is modeled by truss elements, and bond property between rebar and SHCC is modeled by link element. Moreover, bond stress-slip relationship in link element is represented by Eq.(7), in which the proposed relationship is assumed until peak stress and linear softening behavior is assumed after peak stress. Eq.(7) has been calibrated by Suga et al. [15], the coefficients of which are demonstrated in Fig. 14.
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0.4 0.9 f c2 / 3 1 exp 40s / D 0.5 max max 0.1 max s s1 / s 2 s1 0.1 max
0 s s1 s1 s s 2 s2 s
(7)
Where, τ and s are bond stress (MPa) and slip (mm) respectively, τmax is bond strength corresponding to bond stress at 0.2 mm slip, D is diameter of rebar (mm). 4.3. Numerical result Fig. 15 shows the numerical load-displacement curves of the experimental specimens using the proposed numerical approach, in terms of shear fiber bridging strength ffiber equaling to 0.0 MPa, 1.0 MPa, 3.0 MPa and 6.0 MPa respectively. It can be clearly seen the load carrying capacities are obviously increased and post peak behaviors become more ductile while ffiber varies from 0.0 MPa to 3.0 MPa, and the result from ffiber with 0.0MPa is quite different from those obtained from experiment. The load carrying capacities and post peak behavior are almost the same while the ffiber varies from 3.0 MPa to 6.0 MPa. Moreover, brittle failures occur while ffiber are 0.0 MPa and 1.0 MPa, whereas softening branches occur while ffiber are 3.0 MPa and 6.0 MPa. This behavior illustrates fiber bridging contribution has significant effect for shear stress transfer behavior on crack surface of SHCC member. Besides, The certain overestimation of the initial structural rigidity of the SHCC member occurs, which may be due to some assumed equations adopted in the model and should be refined for enhancing the accuracy of numerical result in the future study. Figs. 16 and 17 demonstrate crack pattern and deformation of case 2 at positions marked in Fig. 15, under 1.0 MPa and 3.0 MPa fiber bridging strength. The solid and dotted lines in the crack pattern represent multiple fine cracks and localized cracks respectively. The crack pattern illustrates localized crack occurs in pre peak load region (point 1), and propagates in the web of shear span thereafter (point 2 at peak load). Moreover, localized cracks open and propagate along the rebar in post peak load region under 1.0 MPa fiber bridging strength (point 3), whereas those just along diagonal shear direction in post peak load region under 3.0 MPa fiber bridging strength (point 3), which also can be illustrated by the deformations shown in Fig. 17. Therefore, the numerical result under 3.0 MPa fiber bridging strength shows identical result with the experimental one, which is better than that under 1.0 MPa fiber bridging strength. 5. Conclusions
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In this paper, a numerical approach is proposed for evaluating the shear failure behavior of SHCC member, and the effectiveness of the developed numerical approach is confirmed through the comparison between the experimental and numerical results. (1) Although the fracture surface is very smooth, the shear failed SHCC member still experiences the multi-cracking process, and the multiple fine cracks in diagonal shear direction significantly increase and gradually play dominant role with the increasing load carrying capacity, since fiber bridging effect on crack surface has significant influence for shear failure behavior of SHCC member. (2) A numerical approach is proposed for evaluating the shear failure behavior of SHCC member, which can clarify the respective contributions of both contact stress and fiber bridging stress for the shear stress transfer behavior on crack surface of SHCC member, and thus can disclose the contribution of fibers to local shear of SHCC member. (3) In the developed approach, the shape of SHCC member’s crack surface is appropriately expressed by shear lattice system, and a shear stress transfer model is proposed for accurately quantifying the transferred shear stress on SHCC member’s crack surface, taking account of the contribution from both matrix’s contact effect and fiber bridging effect on crack surface. (4) The comparison between the experimental and numerical results confirms the effectiveness of the developed numerical approach, and the numerical result of the experimental specimens illustrates the contribution of fibers to local shear of SHCC member, which is affected by shear fiber bridging strength. Moreover, some assumed equations adopted in the model should be refined for enhancing the accuracy of numerical result in the future study. Acknowledgements The author would like to acknowledge the support from National Natural Science Foundation of China (No. 51408124, 51578292), Natural Science Foundation of Jiangsu Province (No. BK20140629).
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References [1] V.C. Li, H.C. Wu. Conditions for pseudo strain hardening in fiber reinforced brittle matrix composites, Applied Mechanics Review 1992; 45(8): 390-398. [2] M. Kunieda, K. Rokugo. Recent progress of HPFRCC in Japan-required performance and applications, Journal of Advanced Concrete Technology 2006; 4(3): 19-33. [3] Q.H. Li, S.L. Xu. Performance and application of ultra high toughness cementitious composite: a review, Journal of Engineering Mechanics 2009; 26(Sup.2): 23-67(in Chinese). [4] S. L. Xu, L. J. Hou and X.F. Zhang. Shear behavior of reinforced ultrahigh toughness cementitious composite beams without transverse reinforcement, Journal of Materials in Civil Engineering 2012; 24: 1283-1294. [5] T. Kanda, S. Watanabe, V.C. Li. Application of Pseudo strain hardening cementitious composites to shear resistant structural elements, Proceedings of the 3th Fracture Mechanics for Concrete and Concrete Structures (H. Mihashi and K. Rokugo (eds)), 1998, Freiburg, Germany. [6] Y.X. Zhang, B Shu, Q.B. Zhang, H.B. Xie, X.M. Zhang. Failure behavior of strain hardening cementitious composites for shear strengthening RC member, Construction and Building Materials 2015; 78: 470-473. [7] W. P. Boshoff, G. P. A. G. van Zijl. A computational model for strain-hardening fibre-reinforced cement based composites, Journal of the South African Institution of Civil Engineering 2007; 49(2): 24-31. [8] E. Suryanto, K. Nagai, K. Maekawa. Modeling and Analysis of Shear-critical ECC Members with Anisotropic Stress and Strain Fields, Journal of Advanced Concrete Technology 2010; 8(2): 239-258. [9] H. Higashiyama, N. Banthia. Correlating Flexural and Shear Toughness of Lightweight FiberReinforced Concrete, ACI Materials Journal 2008; 105(3): 251-257. [10] V. C. Li, D. K. Mishra, A. E. Naaman, J. K. Wight, J. M. LaFave, H.C. Wu and Y. Inada. On the shear behavior of engineered cementitious composites, Advanced Cement Based Materials 1994; 1(3): 142-149. [11] T. Kanakubo, K. Shimizu, S. Nagai, T. Kanda. Shear transmission on crack surface of ECC, Proceedings of the 7th Fracture Mechanics for Concrete and Concrete Structures (B.H. Oh, O.C. Choi and L. Chung (eds)), 2010, Jeju, Korea.
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[12] A. Itoh, P. Kongkeo, H. Nakamura, T. Tanabe. FEM analysis of RC members based on lattice equivalent continuum model, Journal of Materials, Concrete Structure and Pavements 2004; 767(64): 115129. (In Japanese) [13] M. Kunieda, H. Ogura, N. Ueda, H. Nakamura. Tensile fracture process of strain hardening cementitious composites by means of three-dimensional meso-scale analysis, Journal of Cement & Concrete Composites 2011; 33(9): 956-965 [14] Z.P. Bazant. Concrete fracture models: testing and practice, Journal of Engineering Fracture Mechanics 2002; 69(1):165-205. [15] M. Suga, H. Nakamura, T. Higai, S. Saito. Effect of bond properties on the mechanical behavior of RC beam, Proceedings of the Japan Concrete Institute 2001; 23(3): 295-300. (In Japanese)
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TABLES AND FIGURES List of Figures: Fig. 1 –Geometry of experimental specimens. Fig. 2 –Result obtained from uniaxial tensile test: (a) Stress–strain relationship; (b) Crack pattern. Fig. 3 –Experimental shear stress-displacement curves. Fig. 4 –cracking pattern of experimental specimens: (a) Case 1; (b) Case 2. Fig. 5 –Investigation of fracture surface’s asperity for SHCC member. Fig. 6 –Crack width of shear failed SHCC member. Fig. 7 –Modeling SHCC member’s cracked surface: (a) Crack surface; (b) Shear lattice. Fig. 8 –Concept of the proposed shear stress transfer model. Fig. 9 –Contact rate versus crack width. Fig. 10 –Fiber bridging stress-strain relationship. Fig. 11 –Numerical models of cases 1 and 2: (a) Case 1; (b) Case 2. Fig. 12 –Tri-linear curve model taking into account fracture energy of SHCC: (a) Proposed tri-linear curve model; (b) Fracture energy. Fig. 13 –Uniaxial compressive behavior of SHCC. Fig. 14 –Bond stress-slip relationship of rebar. Fig. 15 –Numerical load-displacement curves of cases 1 and 2: (a) Case 1; (b) Case 2. Fig. 16 –Crack pattern of case 2: (a) ffiber=1.0 MPa; (b) ffiber=3.0 MPa. Fig. 17 –Deformation of case 2: (a) ffiber=1.0 MPa; (b) ffiber=3.0 MPa.
List of Tables: Table 1 –Details of tested members. Table 2 –Stress and strain values in tri-linear curve model.
13
(a) Case 1
(b) Case 2 Fig. 1 Geometry of experimental specimens
14
(a) Stress–strain relationship
(b) Crack pattern
Fig. 2 Result obtained from uniaxial tensile test
15
Fig. 3 Experimental shear stress-displacement curves
16
(a) Case 1
(b) Case 2
Fig. 4 cracking pattern of experimental specimens
17
Fig. 5 Investigation of fracture surface’s asperity for SHCC member
18
Fig. 6 Crack width of shear failed SHCC member
19
(a) Crack surface
(b) Shear lattice
Fig. 7 Modeling SHCC member’s cracked surface
20
Fig. 8 Concept of the proposed shear stress transfer model
21
Fig. 9 Contact rate versus crack width
22
Fig. 10 Fiber bridging stress-strain relationship
23
(a) Case 1
(b) Case 2 Fig. 11 Numerical models of cases 1 and 2
24
(a) Proposed tri-linear curve model
(b) Fracture energy
Fig. 12 Tri-linear curve model taking into account fracture energy of SHCC
25
Fig. 13 Uniaxial compressive behavior of SHCC
26
Bond stress (MPa)
1.2 1.0 0.8 0.6 0.4 0.2 0
0.1
0.2
0.3 0.4 Slip (mm)
0.5
0.6
Fig. 14 Bond stress-slip relationship of rebar
27
(a) Case 1
(b) Case 2
Fig. 15 Numerical load-displacement curves of cases 1 and 2
28
(a) ffiber=1.0 MPa
(b) ffiber=3.0 MPa Fig. 16 Crack pattern of case 2
29
(a) ffiber=1.0 MPa
(b) ffiber=3.0 MPa Fig. 17 Deformation of case 2
30
Table 1 Details of tested members case
a/d
a (mm)
d (mm)
t (mm)
1
2
300
150
50
2
3
450
150
50
* a, d and t are the shear span length, effective depth and width of specimens respectively
31
Table 2 Stress and strain values in tri-linear curve model Point A
Point B
Point C
σ
ε
σ
ε
σ
ε
4.15
0.00016
5.33
0.0085
0
0.039
* The units of σ is MPa.
32
S0013-7944(16)30010-8 http://dx.doi.org/10.1016/j.engfracmech.2016.02.008 EFM 5049
To appear in:
Engineering Fracture Mechanics
Received Date: Revised Date: Accepted Date:
24 September 2015 15 January 2016 1 February 2016
Please cite this article as: Zhang, Y., Ueda, N., Nakamura, H., Kunieda, M., Numerical Approach for Evaluating Shear Failure Behavior of Strain Hardening Cementitious Composite Member, Engineering Fracture Mechanics (2016), doi: http://dx.doi.org/10.1016/j.engfracmech.2016.02.008
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Numerical Approach for Evaluating Shear Failure Behavior of Strain Hardening Cementitious Composite Member
Yongxing Zhanga,b*, Naoshi Uedab, Hikaru Nakamurab, Minoru Kuniedab
a
School of Civil Engineering, Nanjing Forestry University, No. 159 Longpan Road, Nanjing 210037, China b
Department of Civil Engineering, Nagoya University, Chikusa-ku, Nagoya 464-8603, Japan
*Corresponding Author. Tel.: +86-25-83792021 E-mail addresses: [email protected]
Abstract: This paper presents a numerical approach for evaluating the shear failure behavior of strain hardening cementitious composite (SHCC) member. The developed approach can clarify the respective contributions of both contact stress and fiber bridging stress for the shear stress transfer behavior on crack surface of SHCC member, which thus can disclose the contribution of fibers to local shear of SHCC member. In the numerical approach, a shear stress transfer model and shear lattice system are proposed, in which the shear lattice system can appropriately express the shape of crack surface, and the shear stress transfer model can accurately quantify the transferred shear stress on the crack surface of SHCC member, focusing on the contribution from both matrix’s contact effect and fiber bridging effect respectively. The effectiveness of the numerical approach is confirmed through the comparison between the experimental and numerical results. Keywords: strain hardening cementitious composite member; shear failure behavior; shear stress transfer model; contact effect of matrix; fiber bridging effect.
1
Nomenclature ε = strain εA = strain at point A σA = stress at point A εB = strain at point B σB = stress at point B εC = strain at point C σC = stress at point C Lelm = element size Leq = equivalent element length Ls = crack spacing lm = measured length in uniaxial tensile test Gf = Fracture energy in uniaxial tensile behavior, equaling to the area of stress-displacement relationship after peak θ = angle H = asperity height S = contact rate w = crack width E* = elastic modulus of fiber contribution Vf = fiber content Ef = elastic module of fiber ffiber = elastic module of fiber σs1 = contact stress on tension part of crack surface σs2 = fiber bridging stress on tension part of crack surface Δ{εlcr} = strain field in local coordinate system (ξ ,η) [Tε,s1,s2] = strain rotation matrix used to rotate the incremental local strain into direction of contact stress Esi = gradient of equivalent uniaxial stress-strain relationship [Ω] = controlling matrix to neglect the influence of tensile stress in the direction of ξ
2
1. Introduction Strain hardening cementitious composite (SHCC) is an attractive construction material due to the wellknown advantages in terms of excellent mechanical and physical properties [1-3]. Several experimental investigations concerning the shear behavior of SHCC members [4,5] and SHCC for RC shear strengthening [6] have been carried out, which demonstrate more excellent shear resistance than ordinary concrete and fiber reinforced concrete. Moreover, several numerical models have been developed for evaluating the shear behavior of SHCC member, such as the proposed computational continuum damage formulation based model [7] and fixed smeared crack model [8]. In view of previous published literatures, the shear strength of SHCC is closely correlated with the flexural strength of SHCC [9,10], and the bridging action of flexible fibers in SHCC also has effect on the shear behavior of SHCC [11]. Therefore, the shear stress transfer behavior on crack surface of SHCC member or strengthening layer is contributed by not only contact stress but also fiber bridging stress on the crack surface, whereas the aforementioned numerical models haven’t sufficiently consider the aforementioned respective contributions. However, the contribution of fibers to local shear is today an open question as most of the current approaches consider only the shear component of the local tensile fiber behavior on the diagonal cracks. There is thus a need of developing numerical model for clarifying the respective contributions of both contact stress and fiber bridging stress for the shear stress transfer behavior on crack surface of SHCC member, which is also expected to disclose the contribution of fibers to local shear of SHCC member. In this paper, this type approach is proposed for evaluating the shear failure behavior of SHCC member, in which the shape of SHCC member’s crack surface is expressed by shear lattice system, and a shear stress transfer model is proposed for accurately quantifying the transferred shear stress on SHCC member’s crack surface, taking account of the respective contributions from both matrix’s contact effect and fiber bridging effect on crack surface. 2. Experimental investigation of shear failed SHCC member 2.1. Specimen characteristics and materials The experimental investigation of shear failed SHCC member is carried out. Fig. 1 shows the geometries of the experimental specimens, including effective depth d, width t and shear span length a, the details of which are also listed in Table 1. The SHCC members are 900mm and 1200mm long respectively, with a
3
cross-section of 50 × 200mm2. One deformed bar of 25mm diameter is arranged as longitudinal reinforcement for both SHCC members, and no deformed bars are arranged as stirrups. Moreover, the specimens are loaded under a three-point bending setup, in which the load and displacements (labeled as A and B in Fig. 1) are measured using displacement transducers and load cell respectively. 2.2. Material Property Fig. 2 demonstrates the uniaxial tensile stress-strain curves and ultimate crack pattern of SHCC, which are obtained from the dumbbell-shaped specimens in uniaxial tensile test, with the measured length of 100mm. All the specimens exhibit significant strain hardening behavior until ultimate tensile strength (labeled as B1 or B2 in Fig. 2), since multiple fine cracks occur and propagate after the initial tensile strength (labeled as A in Fig. 2) with an initial crack. The specimens are finally failed due to localization of some multiple fine cracks, illustrated by the crack pattern shown in Fig. 2. The mean compressive strength and Young’s modulus of SHCC are 91MPa and 29GPa respectively. The mean Young’s modulus and yield strength of the longitudinal reinforcement are 200GPa and 1050MPa respectively. 2.3. Experimental result 2.3.1. Shear load-displacement curve and cracking behavior Fig. 3 shows the shear stress-displacement curves and cracking behavior obtained from the experiment, in which the shear stress is defined by the measured load dividing the area of effective section (d × t). The peak shear stresses are 11.6 MPa and 8.3 MPa in cases 1 and 2 respectively, which means the shear load carrying capacity of SHCC member is significantly influenced by the ratio of shear span length to effective depth. The shear stress-displacement curves of SHCC members demonstrate linear curves up to peak shear stress, which experience the cracking behavior from multi-cracking to localization of some multiple fine cracks as demonstrated in Fig. 3(b). Moreover, the investigated crack spacing between multiple fine cracks in diagonal shear direction is about 3mm. 2.3.2. Cracking pattern Fig. 4 demonstrates the cracking pattern of cases 1 and 2. It can be obviously seen some multiple fine cracks occur in diagonal shear direction and play dominant role in the cracking behavior of SHCC member. This also means the shear failed SHCC member experiences the multi-cracking processes in diagonal shear
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direction. As illustrated in Fig. 4, the SHCC member is finally failed due to the localization of some multiple fine cracks in diagonal shear direction. Fig. 5 demonstrates the asperity investigation of one sample’s fracture surface, which is cut from the tested SHCC member. The investigation is implemented using a laser displacement meter in terms of assuming the asperity of the start and end positions as 0 mm. It is worth mentioning that an assumed smooth crack is adopted for reflecting the asperity of fracture surface, which has the angle of 15º and asperity height of 1.2mm. It can be obviously seen the assumed crack has a good fitting with the measured asperity, which implies the asperity along the measured longitudinal line is very smooth, and also verifies the smoothness of SHCC member’s fracture surface. The above experimental investigation illustrates the shear failed SHCC member experiences the multicracking processes, whereas the fracture surface of tested member is very smooth, which implies the shear transfer behavior on crack surface of SHCC member is contributed by not only the contact effect but also the fiber bridging effect on the crack surface. 3. Numerical approach for evaluating the shear failure behavior of SHCC member In this section, a numerical approach is developed for evaluating the shear failure behavior of SHCC member. In the numerical approach, the shape of SHCC member’s crack surface is expressed by a shear lattice system, and a shear stress transfer model is proposed for accurately quantifying the transferred shear stress on SHCC member’s crack surface. 3.1. Crack width of shear failed SHCC member Based on the above-described experimental investigation, the crack width of shear failed SHCC member can be demonstrated as Fig. 6, in which ε is the strain, εA and εB correspond to the strain with initial cracking and ultimate tensile strength respectively. The multiple fine cracks firstly occur in strain hardening phase (ε smaller than εB and larger than εA), and the crack width w is defined by the width of one multiple fine crack wm at this moment. Thereafter, some of the multiple fine cracks localize in strain softening phase (ε larger than εB), and the crack width w is correspondingly defined by the width of the localized crack among some multiple fine cracks ws, expressed by Eq. (1), in which Leq is the equivalent element length, and crack spacing ls is assumed as 3 mm, according to the experimental measurement.
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w=
0
A
A B A ls B B A ls + B Leq
(1)
3.2. Shear lattice system Fig. 7 shows the shear lattice for expressing the shape of SHCC member’s crack surface, which is described by a simple geometry with series of triangle and modeled by the parameters of angle (θ) and asperity height (H) [14]. The contact stress in contact area is compressive stress perpendicular to the incline surface and denotes as S1 in Fig. 7(b); the fiber bridging stress in the area without contacting is tensile stress along the perpendicular direction of crack surface’s inclined plane and denotes as S2 in Fig. 7(b). As illustrated in Fig. 5, the fracture surface of the experimental specimens can be expressed by series of triangle with 15º θ and 1.2mm H. 3.3. Shear stress transfer model Fig. 8 demonstrates the concept of the proposed shear stress transfer model, which can accurately quantify the transferred shear stress on SHCC member’s crack surface, taking account of the contribution from both matrix’s contact effect and fiber bridging effect in crack surface. 3.3.1. Contribution from matrix’s contact effect in crack surface As shown in Fig. 8, the shear stress transfer behavior is influenced by the contact stress on the compression part of crack surface. This influence is varied and affected by the crack width, since the contact area in crack surface’s compression part is significantly decreased with the increasing crack width, which can be expressed by the assumed relationship of contact rate (S) versus crack width (w), as demonstrated in Fig. 9. Moreover, the crack width (w) is also governed by above-demonstrated Eq. (1). 3.3.2. Contribution from fiber bridging effect in cack surface The shear stress transfer behavior is also influenced by the varied fiber bridging stress on the tension part of crack surface. The relationship between fiber bridging stress and strain in shear lattice direction is assumed as shown in Fig. 10, in which the uniaxial bond stress-slip curve is identified through the results of single fiber pull-out tests [13], and E* is the elastic modulus of fiber contribution. Moreover, fibers are randomly distributed in SHCC member and also related to fiber content in SHCC, and the elastic modulus of fiber contribution E* is thus assumed to be calculated by Eq.(2) as a simplified hypothesis for a first 6
study, whereas this assumption should be refined in future study. In the equation, Vf is fiber content in SHCC, Ef is elastic module of fiber, and the maximum shear fiber bridging stress (shear fiber bridging strength) ffiber in shear lattice is decided by the bond strength of fiber and matrix, since it depends on the pullout of fibers between the crack surfaces.
E * V f 3 E f 2
(2)
3.3.3. Transferred shear stress on crack surface of SHCC member Based on the above-described contact stress (σs1 ) on compression part and shear fiber bridging stress (σs2) on tension part of crack surface, the transferred shear stresses on SHCC member’s crack surface can be calculated by Eqs. (3) and (4), which also can rotate to local stress field by stress rotation matrix [Tε,s1,s2]-1 thereafter, and the incremental local stress thus can be calculated by Eq.(5). In the equations, Δ{εlcr} is strain field in local coordinate system (ξ ,η), [Tε,s1,s2] is strain rotation matrix used to rotate the incremental local strain into direction of contact stress (σs1 and σs2), Esi is gradient of equivalent uniaxial stress-strain relationship, Es2 equals to E* in Eq.(2), and [Ω] is the controlling matrix to neglect the influence of tensile stress in the direction of ξ.
E s1 s1 s 2 0
0 T lcr Es 2 , s1, s 2
Esi si si 0 1 l n T , s1, s 2 s1 s 2
(3)
(4)
(5)
4. Evaluation of the shear failure behavior of SHCC member 4.1. Numerical model Fig. 11 shows the numerical models of the above experimental specimens, the mesh sizes of which are 30mm and 25mm in longitudinal and vertical directions respectively. 4.2. Material properties 4.2.1. Modeling uniaxial tensile property of SHCC The tri-linear curve model taking into account fracture energy is adopted for modeling the uniaxial tensile behavior of SHCC, the concept of which is demonstrated in Fig. 12(a). Especially, the fracture
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energy obtained from uniaxial tensile stress-strain curve is illustrated in Fig. 12(b), in order to avoid the element size dependency problem [14]. In the model, points A and B are defined as the position where initial crack occurs and the position with ultimate uniaxial tensile strength. The strain of point C, εC, is represented by Eq.(6), by accounting fracture energy in localized element:
C = B +
2G f
(6)
B Lelm
where, εB is strain at point B, σB is stress at point B, Lelm is element size (mm), εC is strain at point C, Gf is fracture energy in uniaxial tensile behavior, equaling to the area of stress-displacement relationship after peak (N/mm). As mentioned in Fig. 12(b), the uniaxial tensile stress-strain curve of SHCC obtained from experiment is abstracted by tri-linear curve with minimum ultimate tensile strength, due to damage generally occurs at weak area. Therefore, the corresponding parameters of points A, B, C for tri-linear curve model taking account of fracture energy are summarized in Table 2. 4.2.2. Modeling uniaxial compressive property of SHCC The compressive stress-strain relationship of SHCC is modeled as shown in Fig. 13, in which σc and fc are compressive stress and strength (MPa), ε is compressive strain, and Leq is equivalent element size (mm). In the relationship, saenz equation:
i
Ec cu
1 Ec
, is used up to the compressive
2 fc 0 2 cu 0 cu 0
strength and a linear softening branch is assumed thereafter. The slope of linear softening branch is defined by considering the compressive fracture energy (Gfc), in order to avoid the above-mentioned element size dependency problem [14]. 4.2.3. Modeling property of reinforcement The longitudinal reinforcement is modeled by truss elements, and bond property between rebar and SHCC is modeled by link element. Moreover, bond stress-slip relationship in link element is represented by Eq.(7), in which the proposed relationship is assumed until peak stress and linear softening behavior is assumed after peak stress. Eq.(7) has been calibrated by Suga et al. [15], the coefficients of which are demonstrated in Fig. 14.
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0.4 0.9 f c2 / 3 1 exp 40s / D 0.5 max max 0.1 max s s1 / s 2 s1 0.1 max
0 s s1 s1 s s 2 s2 s
(7)
Where, τ and s are bond stress (MPa) and slip (mm) respectively, τmax is bond strength corresponding to bond stress at 0.2 mm slip, D is diameter of rebar (mm). 4.3. Numerical result Fig. 15 shows the numerical load-displacement curves of the experimental specimens using the proposed numerical approach, in terms of shear fiber bridging strength ffiber equaling to 0.0 MPa, 1.0 MPa, 3.0 MPa and 6.0 MPa respectively. It can be clearly seen the load carrying capacities are obviously increased and post peak behaviors become more ductile while ffiber varies from 0.0 MPa to 3.0 MPa, and the result from ffiber with 0.0MPa is quite different from those obtained from experiment. The load carrying capacities and post peak behavior are almost the same while the ffiber varies from 3.0 MPa to 6.0 MPa. Moreover, brittle failures occur while ffiber are 0.0 MPa and 1.0 MPa, whereas softening branches occur while ffiber are 3.0 MPa and 6.0 MPa. This behavior illustrates fiber bridging contribution has significant effect for shear stress transfer behavior on crack surface of SHCC member. Besides, The certain overestimation of the initial structural rigidity of the SHCC member occurs, which may be due to some assumed equations adopted in the model and should be refined for enhancing the accuracy of numerical result in the future study. Figs. 16 and 17 demonstrate crack pattern and deformation of case 2 at positions marked in Fig. 15, under 1.0 MPa and 3.0 MPa fiber bridging strength. The solid and dotted lines in the crack pattern represent multiple fine cracks and localized cracks respectively. The crack pattern illustrates localized crack occurs in pre peak load region (point 1), and propagates in the web of shear span thereafter (point 2 at peak load). Moreover, localized cracks open and propagate along the rebar in post peak load region under 1.0 MPa fiber bridging strength (point 3), whereas those just along diagonal shear direction in post peak load region under 3.0 MPa fiber bridging strength (point 3), which also can be illustrated by the deformations shown in Fig. 17. Therefore, the numerical result under 3.0 MPa fiber bridging strength shows identical result with the experimental one, which is better than that under 1.0 MPa fiber bridging strength. 5. Conclusions
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In this paper, a numerical approach is proposed for evaluating the shear failure behavior of SHCC member, and the effectiveness of the developed numerical approach is confirmed through the comparison between the experimental and numerical results. (1) Although the fracture surface is very smooth, the shear failed SHCC member still experiences the multi-cracking process, and the multiple fine cracks in diagonal shear direction significantly increase and gradually play dominant role with the increasing load carrying capacity, since fiber bridging effect on crack surface has significant influence for shear failure behavior of SHCC member. (2) A numerical approach is proposed for evaluating the shear failure behavior of SHCC member, which can clarify the respective contributions of both contact stress and fiber bridging stress for the shear stress transfer behavior on crack surface of SHCC member, and thus can disclose the contribution of fibers to local shear of SHCC member. (3) In the developed approach, the shape of SHCC member’s crack surface is appropriately expressed by shear lattice system, and a shear stress transfer model is proposed for accurately quantifying the transferred shear stress on SHCC member’s crack surface, taking account of the contribution from both matrix’s contact effect and fiber bridging effect on crack surface. (4) The comparison between the experimental and numerical results confirms the effectiveness of the developed numerical approach, and the numerical result of the experimental specimens illustrates the contribution of fibers to local shear of SHCC member, which is affected by shear fiber bridging strength. Moreover, some assumed equations adopted in the model should be refined for enhancing the accuracy of numerical result in the future study. Acknowledgements The author would like to acknowledge the support from National Natural Science Foundation of China (No. 51408124, 51578292), Natural Science Foundation of Jiangsu Province (No. BK20140629).
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References [1] V.C. Li, H.C. Wu. Conditions for pseudo strain hardening in fiber reinforced brittle matrix composites, Applied Mechanics Review 1992; 45(8): 390-398. [2] M. Kunieda, K. Rokugo. Recent progress of HPFRCC in Japan-required performance and applications, Journal of Advanced Concrete Technology 2006; 4(3): 19-33. [3] Q.H. Li, S.L. Xu. Performance and application of ultra high toughness cementitious composite: a review, Journal of Engineering Mechanics 2009; 26(Sup.2): 23-67(in Chinese). [4] S. L. Xu, L. J. Hou and X.F. Zhang. Shear behavior of reinforced ultrahigh toughness cementitious composite beams without transverse reinforcement, Journal of Materials in Civil Engineering 2012; 24: 1283-1294. [5] T. Kanda, S. Watanabe, V.C. Li. Application of Pseudo strain hardening cementitious composites to shear resistant structural elements, Proceedings of the 3th Fracture Mechanics for Concrete and Concrete Structures (H. Mihashi and K. Rokugo (eds)), 1998, Freiburg, Germany. [6] Y.X. Zhang, B Shu, Q.B. Zhang, H.B. Xie, X.M. Zhang. Failure behavior of strain hardening cementitious composites for shear strengthening RC member, Construction and Building Materials 2015; 78: 470-473. [7] W. P. Boshoff, G. P. A. G. van Zijl. A computational model for strain-hardening fibre-reinforced cement based composites, Journal of the South African Institution of Civil Engineering 2007; 49(2): 24-31. [8] E. Suryanto, K. Nagai, K. Maekawa. Modeling and Analysis of Shear-critical ECC Members with Anisotropic Stress and Strain Fields, Journal of Advanced Concrete Technology 2010; 8(2): 239-258. [9] H. Higashiyama, N. Banthia. Correlating Flexural and Shear Toughness of Lightweight FiberReinforced Concrete, ACI Materials Journal 2008; 105(3): 251-257. [10] V. C. Li, D. K. Mishra, A. E. Naaman, J. K. Wight, J. M. LaFave, H.C. Wu and Y. Inada. On the shear behavior of engineered cementitious composites, Advanced Cement Based Materials 1994; 1(3): 142-149. [11] T. Kanakubo, K. Shimizu, S. Nagai, T. Kanda. Shear transmission on crack surface of ECC, Proceedings of the 7th Fracture Mechanics for Concrete and Concrete Structures (B.H. Oh, O.C. Choi and L. Chung (eds)), 2010, Jeju, Korea.
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[12] A. Itoh, P. Kongkeo, H. Nakamura, T. Tanabe. FEM analysis of RC members based on lattice equivalent continuum model, Journal of Materials, Concrete Structure and Pavements 2004; 767(64): 115129. (In Japanese) [13] M. Kunieda, H. Ogura, N. Ueda, H. Nakamura. Tensile fracture process of strain hardening cementitious composites by means of three-dimensional meso-scale analysis, Journal of Cement & Concrete Composites 2011; 33(9): 956-965 [14] Z.P. Bazant. Concrete fracture models: testing and practice, Journal of Engineering Fracture Mechanics 2002; 69(1):165-205. [15] M. Suga, H. Nakamura, T. Higai, S. Saito. Effect of bond properties on the mechanical behavior of RC beam, Proceedings of the Japan Concrete Institute 2001; 23(3): 295-300. (In Japanese)
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TABLES AND FIGURES List of Figures: Fig. 1 –Geometry of experimental specimens. Fig. 2 –Result obtained from uniaxial tensile test: (a) Stress–strain relationship; (b) Crack pattern. Fig. 3 –Experimental shear stress-displacement curves. Fig. 4 –cracking pattern of experimental specimens: (a) Case 1; (b) Case 2. Fig. 5 –Investigation of fracture surface’s asperity for SHCC member. Fig. 6 –Crack width of shear failed SHCC member. Fig. 7 –Modeling SHCC member’s cracked surface: (a) Crack surface; (b) Shear lattice. Fig. 8 –Concept of the proposed shear stress transfer model. Fig. 9 –Contact rate versus crack width. Fig. 10 –Fiber bridging stress-strain relationship. Fig. 11 –Numerical models of cases 1 and 2: (a) Case 1; (b) Case 2. Fig. 12 –Tri-linear curve model taking into account fracture energy of SHCC: (a) Proposed tri-linear curve model; (b) Fracture energy. Fig. 13 –Uniaxial compressive behavior of SHCC. Fig. 14 –Bond stress-slip relationship of rebar. Fig. 15 –Numerical load-displacement curves of cases 1 and 2: (a) Case 1; (b) Case 2. Fig. 16 –Crack pattern of case 2: (a) ffiber=1.0 MPa; (b) ffiber=3.0 MPa. Fig. 17 –Deformation of case 2: (a) ffiber=1.0 MPa; (b) ffiber=3.0 MPa.
List of Tables: Table 1 –Details of tested members. Table 2 –Stress and strain values in tri-linear curve model.
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(a) Case 1
(b) Case 2 Fig. 1 Geometry of experimental specimens
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(a) Stress–strain relationship
(b) Crack pattern
Fig. 2 Result obtained from uniaxial tensile test
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Fig. 3 Experimental shear stress-displacement curves
16
(a) Case 1
(b) Case 2
Fig. 4 cracking pattern of experimental specimens
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Fig. 5 Investigation of fracture surface’s asperity for SHCC member
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Fig. 6 Crack width of shear failed SHCC member
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(a) Crack surface
(b) Shear lattice
Fig. 7 Modeling SHCC member’s cracked surface
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Fig. 8 Concept of the proposed shear stress transfer model
21
Fig. 9 Contact rate versus crack width
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Fig. 10 Fiber bridging stress-strain relationship
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(a) Case 1
(b) Case 2 Fig. 11 Numerical models of cases 1 and 2
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(a) Proposed tri-linear curve model
(b) Fracture energy
Fig. 12 Tri-linear curve model taking into account fracture energy of SHCC
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Fig. 13 Uniaxial compressive behavior of SHCC
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Bond stress (MPa)
1.2 1.0 0.8 0.6 0.4 0.2 0
0.1
0.2
0.3 0.4 Slip (mm)
0.5
0.6
Fig. 14 Bond stress-slip relationship of rebar
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(a) Case 1
(b) Case 2
Fig. 15 Numerical load-displacement curves of cases 1 and 2
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(a) ffiber=1.0 MPa
(b) ffiber=3.0 MPa Fig. 16 Crack pattern of case 2
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(a) ffiber=1.0 MPa
(b) ffiber=3.0 MPa Fig. 17 Deformation of case 2
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Table 1 Details of tested members case
a/d
a (mm)
d (mm)
t (mm)
1
2
300
150
50
2
3
450
150
50
* a, d and t are the shear span length, effective depth and width of specimens respectively
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Table 2 Stress and strain values in tri-linear curve model Point A
Point B
Point C
σ
ε
σ
ε
σ
ε
4.15
0.00016
5.33
0.0085
0
0.039
* The units of σ is MPa.
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