CRACK PROPAGATION CONTROL WITH PRE-EXISTING SLITS AND ..... fact that the loading condition of a crack may bring other cracks to loading, unloading ...
P.Amorn (W-270) 1/10 CRACK PROPAGATION CONTROL WITH PRE-EXISTING SLITS AND CRACKS IN CONCRETE Pimanmas Amorn
1)
2)
Maekawa Koichi
3)
Fukuura Naoyuki
1)
Asst. Prof., School of Building facilities and Civil Engineering, Sirindhorn International Institute of Technology, Thammasart University, Thailand 2) Prof., Department of Civil Engineering, University of Tokyo, JAPAN 3) Engineer, Taisei Corporation, JAPAN Keywords: shear failure, crack propagation control, pre-cracking, shear anisotropy, crack arrest
1 INTRODUCTION In general practice, growth of crack in reinforced concrete is mainly controlled by reinforcing bars. For a linear member like beam and column, flexural crack growth is controlled by longitudinal bars whereas the diagonal shear crack growth is controlled by transverse ones (stirrups). When sufficient amount of reinforcement is provided in directions where cracks are expected to form, for example, 2way reinforcement for arbitrary in-plane cracks and 3-way reinforcement for arbitrarily space crack, RC exhibits ductile failure. However, full use of reinforcing bars for controlling crack growth may bring about difficulty for some large 2- and 3-D structures such as RC slab, foundation, etc. For these structures, large amount of shear reinforcement cannot be placed physically and economically. In fact, there are some provisions that specify no placement of transverse bars in slab and foundation structures. These structures are therefore classified as lightly reinforced and are prone to localized failure characterized by the propagation of few dominant cracks. In this paper, the authors aim to propose a new method of controlling crack propagation without using reinforcing bars. We raise the idea of specifying paths of crack propagation by installing pre-slit made from pre-cracks and metallic/non-metallic plates embedded in RC members. These pre-slits will alter the stress distribution inside the member, and lead to a change in failure characters and the overall behavior of the member. Here the authors present the structural behavior of RC beams containing pre-slits and conduct the finite element analysis for these beams. The target is to study how pre-slits alter the path of shear crack propagation in RC members.
2 EXPERIMENTAL PROGRAM Two experimental programs are presented to investigate the structural behavior of beams containing pre-slits. The first one is the shear test on beam containing vertical pre-cracking and the second is the shear test on beam containing artificial pre-slits made from metal and acrylic plate. 2.1 Shear experiment of beam containing vertical pre-cracking The authors have conducted the shear experiment of RC beam containing vertical pre-cracking [1]. The experimental program is shown in Fig. 1. Dimension and cross sections of the beam are shown in Fig. 1(a). Compressive strength of concrete is 26.5 MPa. Yield strength of main reinforcement is 338.4 MPa. Pre-cracks penetrating the entire sections were introduced into the beam by means of reversed flexural loading (Fig. 1(b)). The reversed flexural loading was accomplished in two steps (Fig. 1(b)). After the beam was loaded in flexure in the first step, it was rotated by 180°, and then loaded in the second flexure. To apply shear loading, supports were moved towards beam mid-span such that shear span to effective depth ratio was 2.41 (Fig. 1(b)). Four beams have been tested in the experiment. The level of reversed flexure is different in these beams to create pre-cracks with different widths. As the experimental results and discussions were given in detail in [1], only the beam with the largest pre-crack width is discussed here in comparison with the control non pre-cracked beam. Loaddisplacement relationship of the pre-cracked beam under shear is shown in Fig. 2. It is noted that precracked beams can attain significantly higher loading capacity than the non pre-cracked one. Up to 50 % increase in capacity till yielding of main reinforcement was experimentally observed.
P.Amorn (W-270) 2/10 D10@75
7D13
A-A
B-B
450
450
350 2300
2400 250
A-A
First stage - reversed flexure
Material properties
310 B-B
Unit:mm
Diagonal shear crack
Concrete compressive strength = 26.5 MPa Yield strength of main reinforcement = 338.4 MPa Yield strength of shear reinforcement = 338.4 MPa
Vertical pre-cracks
750
Unit: mm
Second stage - shear flexure
Fig. 1(a) Dimension and cross section of the beam
Fig. 1(b) Loading method
300 Yielding capacity = 233.5 KN
250 Non pre-cracked beam
200
100 50 0
Pre-cracked beam Reversed flexure
500
150 Load(kN)
Load (KN)
Failure crack pattern of non pre-cracked and pre-cracked beam is shown in Fig. 3. Substantial difference between these two beams is clearly identified. In non pre-cracked beam, the failure is governed by the propagation of a single diagonal crack. However, in pre-cracked beam, several discontinuous diagonal cracks are created. The propagation of diagonal crack is arrested at the pre-crack plane. This is because a low magnitude of stress can be transferred along the pre-crack interface. Consequently, the imposed total strain is localized into the shear slip along pre-crack without developing sufficient diagonal stress for further cracking.
Unit: mm
300 100 -100 -500
0
5
-20
-300
0
20
Center span deflection(mm)
10
Center deflection (mm)
15
20
Fig.2 Experimental load-displacement relationship for beam containing vertical pre-cracks
This explains the termination of diagonal crack at the pre-crack interface (Fig. 4) which is known as crack arrest and diversion mechanism. It is supposed that crack interaction between pre-crack and diagonal crack exists. Thus, cracks in the pre-cracked beam have Z-shaped pattern which reflects the mutual contribution of diagonal crack and pre-crack. Due to the presence of pre-cracks, the failure is not due to the propagation of a single diagonal crack as in non pre-cracked beam. Instead, the failure of the pre-cracked beam is caused by the combination of several diagonal cracks, which formed independently along the failure path. 2.2 Shear experiment of beams containing metallic and acrylic pre-slit It is seen that the crack arrest and diversion phenomenon shows another possible method for controlling the crack localization [1]. Based on this experimental result, the concept of crack arrest and diversion is applied as a means to control the crack localization proposed in this paper. Here, the authors intend to install the pre-slit made from metallic and acrylic plate inside the RC member to function as pre-cracking. The basic shear failure path in slender beam and corresponding loaddisplacement relation are shown in Fig. 5. The first step is the formation of the first diagonal crack and crack 2 around the web portion at the center of shear span. This diagonal crack is usually observed just above the nearby flexural crack (crack 1). The second step is the propagation of this crack towards loading point and backwards to support, and results in crack 3 and 3′ respectively. Once these cracks have been formed, the beam reaches failure and the loading capacity drops suddenly. Our target is to (1) obstruct the formation of this failure path and (2) divert the propagation of this crack path through a special device that performs as pre-cracking to alter the stress flow inside RC member. This is illustrated in Fig. 6.
P.Amorn (W-270) 3/10 Direction of shear crack propagation
Crack stops !
Weak plane of pre-crack Element near pre-crack Low traction transferred
Element far from pre-crack High traction transferred
Fig. 4 Mechanism of crack arrest and diversion
Fig. 3 Failure crack pattern (Top: non pre-cracked beam, Bottom: pre-cracked beam)
140
1
Crack Crack Crack Crack beam
Crack 2, crack 3,3’
120
2 3
4
1: Flexural crack 2: Advent of first diagonal crack 3,3’: Propagation of crack 2 4: Sudden energy release from the
Load(kN)
3
Path of shear crack propagation
100
Crack 1
80 60 40 20 0
0
1
2
3
Center span deflection(mm)
4
Fig. 5 Formation of shear failure path in slender beam and corresponding load-displacement relation Arrest of crack propagation
Diversion of crack propagation direction
Crack control device
Fig. 6 Control of crack propagation (Left: arrest of crack propagation, right: diversion of crack propagation) The authors have proposed Artificial Crack Device (ACD) to perform as pre-cracking [2]. ACD is fabricated from steel and acrylic plate, which has smooth surface and low stress transfer ability. It will be embedded in formwork before concrete is cast and will form a part of the concrete members. An experimental program [2] was conducted to study structural behavior of ACD-embedded beams. Here, the authors present a part of this experimental program. The detail of selected beams is shown in Fig. 7(a). Beam 1 is the control beam with no ACD. In beam 2, ACD is embedded in the inclined direction opposite to the expected crack propagation direction. In beam 3 and 4, ACD is placed horizontally along the centroidal axis. The ACD is made from steel plate in beam 3 and acrylic one in beam 4. The
P.Amorn (W-270) 4/10 shear span to effective depth ratio is 2.8. Reinforcement ratio of main bars is 1.548%. Tested compressive strength of concrete is 31.4 MPa. Tested yield strength of main reinforcement is 397.3 MPa. No shear reinforcements is arranged in any beam. The experimental load-displacement relationships for all beams are shown in Fig. 7(b). The failure crack pattern and sequence for all beams are shown in Fig. 8. For control beam, the failure follows the conventional path (Fig. 5). The first diagonal crack forms around the web portion, propagates towards loading point and support and immediately fails the beam. The loading capacity is 126.5 kN while Okamura-Higai equation [3] predicts 128.7 kN. 160
150
700
300
700
Control beam (Beam 1) 150
82.5 135 82.5
283 135 283
300
160
700
Beam with inclined ACD (Beam 2) 150
150
500 150 700
300 2100
700
Beam with horizontal ACD (Beam 3 and 4)
Fig. 7(a) Detail of tested beams
Unit : mm
Fig. 7(b) Experimental load-displacement relationships
Fig. 8 Failure crack pattern and sequence
P.Amorn (W-270) 5/10 In beam 2, steel ACD is placed in the direction normal to the crack propagation. Loading capacity significantly increases up to 56%. Firstly, flexural crack denoted by crack 1 forms and turns into diagonal crack. Then, diagonal crack (crack 2) forms independently but does not immediately propagate backwards support. This is the main difference from the behavior of diagonal crack in the control beam. Diagonal cracks in the control beam can propagate immediately but the diagonal crack in beam 5 is arrested by ACD. Thus, the failure path does not suddenly form and loading capacity is greatly increased. The failure of beam 5 is caused by the independent development of crack 3, which combines with crack 2 and results in the complete failure path. ACD can effectively arrest the crack propagation. In beam 3 and 4, ACD is placed horizontally at the centroidal axis of the beam. The crack diversion is shown in the experiment (Fig. 8). The failure process is described in Fig. 8. Firstly, flexural crack (crack 1) forms but cannot evolve into diagonal crack due to the existence of ACD. Instead, horizontal crack (crack 2) takes place when crack 1 reaches the ACD plane. Then, horizontal crack 2 propagates into diagonal crack 3 towards the loading point. The resulting crack path 1-2-3 is equivalent to the crack 1-2-3 in the control beam (Fig. 5). In beam 3 and 4, crack 2, instead of being diagonal, becomes horizontal by virtue of ACD. At the ACD interface, small stresses can be transferred. Therefore, any imposed shear deformation must be localized into horizontal shear slip. This explains why diagonal crack cannot form at the ACD interface. The failure in beam 3 and 4 is finalized by crack 4 (Fig. 8), which is the propagation of the interface crack backwards to support. It is noted that acrylic ACD is used in beam 4 but higher loading capacity is reached compared to beam 3 where steel ACD is embedded. It is supposed that the interface property of steel plate is not much different from that of acrylic plate; that is, the steel and acrylic surface would be equally smooth. The reason that beam 7 reaches higher shear capacity is not exactly clear from this experiment but the authors suppose that it may be due to the interaction between concrete and ACD. Steel ACD may react on concrete more strongly than acrylic one due to its higher stiffness.
3 SHEAR ANISOTROPY AND CRACK INTERACTION The common character of pre-crack and ACD is that they induce the local shear anisotropy in RC members. Under the imposed strain, only stresses of low magnitude can develop in concrete elements adjacent to ACD or pre-cracks because their surfaces are smooth, thus transmitting very small stresses. It results in crack arrest mechanism (Fig. 4). Instead, the imposed shear strain is mainly transformed into interface slip. The authors describe this behavior as ‘crack arrest and diversion’. Crack arrest means the arrest of diagonal crack whereas crack diversion means the localized slip at the interface (Fig. 4). All experimental results of shear tests of pre-cracked beams [1] and ACDembedded beams [2] can be consistently explained by the concept of shear anisotropy. In the pre-cracked beam problem, we have two systems of cracks; that is, vertical pre-cracks and diagonal cracks. The behavior of a multi-cracked element under arbitrary stress state is important to the simulation of pre-cracked RC members. Fig. 9 shows that the overall behavior of a multi-cracked element under a generic multi-directional stress condition can be computed by enforcing the strain compatibility and equilibrium to all cracks, concrete between cracks and the whole element simultaneously. The behavior of each crack is governed by the width and inclination of the crack itself, the global stress state and the interaction with neighboring cracks. The interaction arises due to the fact that the loading condition of a crack may bring other cracks to loading, unloading or reloading conditions depending on their geometrical and physical properties. As a result of equilibrium and compatibility of all cracks in the element, a certain set of cracks may become active whereas others are idle. Hence, the activation-dormancy of crack may vary from element to element. Then, we can view a concrete member as an assembly of several elements with different conditions of crack activation/dormancy. Hence, different behaviors may be expected depending on the spatial distribution of crack activation-dormancy for each specific loading condition and past history.
4 FOUR-WAY FIXED CRACK MODEL In this section, the authors apply the finite element method to simulate the structural behavior of pre-cracked beam and ACD-embedded beam. The four-way fixed crack model is adopted because it can deal with shear anisotropy and multi-cracking. Due to the shear anisotropy, principal stress vector will not coincide with principal strain vector. Consequently, the shear behavior is independent of normal action. In order to reproduce this behavior, the crack model has to explicitly consider the shear transfer model along ACD and pre-crack interface. The fixed crack approach explicitly considers Mode I normal crack stress release as well as Mode II shear traction transfer. Therefore, it enables the
P.Amorn (W-270) 6/10 independent treatment of shear and normal stress behaviors and fulfills the requirement of shear anisotropy. For the 2-D in-plane finite element, kinematic variables include tensile strain normal to a crack, compressive strain parallel to a crack and shear strain along crack interface. Corresponding static variables include tensile stress normal to a crack, compressive stress parallel to a crack and shear stress along crack interface. The relationships between kinematic and static variables are described by constitutive equations. Experiment showed that the formation of failure path in pre-cracked beam and ACD-embedded beam is caused by the combination of several independently formed diagonal cracks into a single failure crack. FEM should naturally capture the failure path through the inherent principle of minimum potential energy. MULTI-CRACKED ELEMENT SUBJECTED TO MULTI-STRESSES
Mechanics of multi-cracked element (GLOBAL ELEMENT RESPONSE) ACTIVATION/ DORMANCY OF EACH CRACK
THE WHOLE ELEMENT EQUI LI BRI UM OF EACH CRACK
PRE-CRACKED RC MEMBERS UNDER GENERIC LOADING CONDITION
THE WHOLE ELEMENT COMPATI BI LI TY OF EACH CRACK
of constituent crack ∑ Mechanics (LOCAL CRACK RESPONSE) Mechanics of constituent crack
FACTORS
t
s
KINEMATICS
ω
d
Shear transfer model Stress-opening relationship
Crack slip
LOADING STATE
• Width • Inclination • External loading direction • Interaction ωith neighboring crack
Crack opening
• Loading • Unloading • Reloading
CONSTITUTIVE STATICS LAWS Normal crack stress release
Crack opening Crack slip
Shear transfer
Fig. 9 Behavior of multi-cracked element One of the recent advances in line with the fixed crack scheme is the four-way crack approach [4,5]. This crack model was installed as the basic platform in WCOMD [6] general nonlinear pathdependent finite element program. Four-way fixed crack model can be regarded as the generalization of two-way cracking approach [6], implemented such that up to four cracks in arbitrary orientations at any Gauss point can be covered. In this model, the stress computation is carried out along the active crack in the active co-ordinate. The crack with largest width is considered to be the active one [4,5]. For the 2-D RC in-plane element, the local constitutive laws are required for both concrete and reinforcement. Local constitutive laws of cracked concrete are formulated along the crack axis. Local constitutive law of reinforcing bar is formulated along the bar axis. The local distribution of stresses and strains in the cracked RC element is not uniform due to bond action at the concrete-steel interface and local traction transfer along the cracks. In the smeared crack approach, the constitutive laws are expressed in terms of average stress-average strain relations. The constitutive laws for cracked concrete include the tensile stress perpendicular to the crack axis, the compressive stress parallel to the crack and the shear transfer model along crack. The concrete tensile stress perpendicular to the crack axis is described by the coupled tension stiffening and bridging softening relationship in line with zoning approach [7]. In the model, concrete near reinforcement is assumed to have gradual stress release after cracking due to tension stiffening effect. In contrast, concrete far from reinforcement follows the plain concrete behavior with sharp stress release due to bridging softening at the crack interface. In compression, the elasto-plastic fracture model [6] is used to compute compressive stress parallel to the crack axis. The model combines the non-linearity of plasticity and fracturing damages to account for the permanent deformation and loss of elastic strain energy capacity, respectively. The effect of orthogonal tensile strain on the reduction of the compressive stress transfer is also taken into account [8]. To model shear transfer along the crack interface, the shear transfer model based on the contact density theory [9] is adopted. Under the multi-cracking situation, the total imposed shear strain is
P.Amorn (W-270) 7/10 decomposed into the shear strain due to cracks and shear strain due to un-cracked concrete. Hence, the computation of the overall shear modulus of the cracked element must consider the contribution of the continuum concrete part and all cracks in the active crack co-ordinate [4,5]. The model for reinforcing bars considers the effect of localized plasticity [10] at the vicinity of the cracks. The reduced yield strength due to bond transfer between concrete and steel bar is taken into account. Since cracks in reinforced concrete element need not be orthogonal to the reinforcement direction, the bond effect will not be fully functional in such a case. Reinforcing bar orthogonal to crack is supposed to have bond effect. On the contrary, reinforcing bar parallel to crack is supposed to follow bare bar behavior. Therefore, the computation of mean yield strength has to take into account the angular deviation of normal to crack from reinforcing bar direction [10]. These local constitutive laws were reformulated and detailed in the work of Fukuura and Maekawa [4,5].
Fig. 10 Finite element mesh and loaddisplacement relation
Fig. 11 Crack pattern
5 NUMERICAL ANALYSIS 5.1 Numerical simulation of RC Beam containing perpendicular pre-cracks The numerical analysis of reinforced concrete beam containing pre-cracks perpendicular to the beam axis has been reported in detail by the authors [11]. Here, only the finite element analysis of non pre-cracked beam and beam with largest pre-crack width is presented. The size and dimension of beam are shown in Fig. 1(a). The finite element mesh of the pre-cracked beam is shown in Fig. 10(a). Discrete joint elements [12] are placed in the mesh to represent pre-crack. Due to the symmetry of the problem, the analysis of half-beam is sufficient. The load is numerically applied by incrementing the forced displacement to the specified node. Load-displacement relationships under the shear loading (Fig. 10(b)) show that the behavior of pre-cracked beam significantly differs from the non pre-cracked one. Pre-cracked beam reaches considerably higher loading capacity, displacement ductility and energy consumption, but with much lower initial stiffness as compared to the non pre-cracked beam. It is noted that numerical loaddisplacement curve is not smooth but serrated instead. In fact, experimental observation also demonstrated this characteristic, which is caused by the crack arrest and diversion mechanism [1]. Once a diagonal crack forms, load drops. However, since diagonal crack cannot propagate continuously across pre-crack planes, higher load can be resisted. The crack pattern during the initial stage is shown in Fig. 11(a). As seen, the FEM can reproduce the formation of Z-cracks which form as
P.Amorn (W-270) 8/10 a result of relative deformational contribution between pre-crack and diagonal crack [1]. The failure crack also matches the experimental one (Fig. 11(b)).
Kn= 58800 GPa
Shear stress
Normal stress
5.2 Numerical simulation of ACD-embedded Beams In this section, the analysis of ACD-embedded beams subjected to shear is presented. Interfacial joint elements are used for representing the smooth interface at the concrete-ACD contact. Since the exact constitutive properties of ACD-concrete interface are not available, the authors assume the constitutive properties of the joint element in both tangential and normal directions as shown in Fig. 12. The normal stiffness is assigned very low value (no bond) in tension but very high value in compression (to avoid overlapping with full contact). The shear stiffness is assigned very low value in both tension and compression.
Tension
Tension
Kn= 2.94 GPa
Kt = 2.94 GPa
Normal relative displacement
Shear relative displacement Kt = 4.90 GPa
Compression
Compression (a) Normal direction
(b) Shear direction
Fig. 12 Constitutive properties of interfacial joint element representing ACD-concrete interface
Fig. 13(a) Finite element mesh of beam 2
Fig. 13(b) Finite element mesh of beam 3 and 4
The finite element meshes for beam 2 and beam 3 and 4 are shown in Fig. 13 (see Fig. 7(a) for experimental program). In this paper, no difference is made between steel and acrylic plate. The comparison of numerical and experimental load-displacement relationships for beam 2 is shown in Fig. 14. The prediction of crack arrest is demonstrated in Fig. 15. Initial cracks (Fig. 15(a)) form at the load around 121 kN. The crack is developed at the web portion at the right side of ACD. This crack tries to propagate downwards to support but the propagation is obstructed by the inclined ACD. The crack arrest is simulated in the analysis. It is seen than the computed crack becomes widening locally without further propagation (Fig. 15(b)). Because of this crack arrest mechanism, the beam can reach much higher capacity compared with control beam. The failure of this beam is caused by the formation of a new diagonal crack in the left side of ACD. This crack propagates upwards and combines with the arrested crack, forming the complete failure path. For beam 3 and 4, comparison of numerical and experimental load-displacement relationships is shown in Fig. 16. Fair agreement is observed. The numerical crack pattern is shown in Fig. 17. Fair agreement with the experiment can be noticed. The analysis predicts no diagonal cracks above nor across the ACD. Instead, due to the local shear anisotropy at the ACD interface, the analysis reproduces the diversion of diagonal cracks.
P.Amorn (W-270) 9/10 Loading capacity can be fairly predicted but the point where stiffness noticeably decreases in the load-displacement relation is overestimated. In the experiment, this stiffness change is due to the interface crack along ACD. However, in the analysis, the interfacial joint elements are used to represent the ACD interface. Since the constitutive laws of these joint elements are assumed to be linearly elastic, FEM cannot capture the immediate occurrence of the interface crack as in the experiment. In the experiment, the formation of the interface cracks does not cause the failure yet. Load can continue increasing while more diagonal cracks are generated. This results in the gradual reduction in stiffness as loading progresses. The analysis can reproduce this tendency too. As load increases, the analysis predicts many cracks below ACD (Fig. 17) similar to the experimental observation. These cracks lead to the gradual decrease in stiffness. Regarding the failure process, the experimental observation indicates that beams 3 and 4 reached failure when diagonal crack d (Fig. 17) formed and connected the interface crack to the support. It is noted that the failure path is defined by the crack c-b-d, not by the initially formed crack a-b-c. The analysis can also reproduce this experimental observation.
Fig. 14 Comparison of load-displacement relation of beam 2
Fig. 16 Comparison of load-displacement relation of beam 3 and 4
Fig. 15 Crack arrest
Fig. 17 Crack pattern of beam 3 and 4
P.Amorn (W-270) 10/10 6 CONCLUSION This paper presents a new method for controlling the propagation of crack by pre-slits such as precracks and metallic/nonmetallic plates. These pre-slits will alter the stress distribution within the RC member. As a result, the failure path can be changed, leading to the increase in loading capacity. The paper demonstrates two basic crack controls; that is, arrest of crack and diversion of crack propagation direction. Through the crack arrest mechanism, a considerable increase in shear capacity up to more than 50% has been realized in the experiment without any stiffness degradation. Moreover, the acrylic ACD also shows good performance compared to the steel one. The use of non-metallic ACD may be more attractive from the viewpoint of long-term durability and the easiness in manipulation and fabrication. The finite element method is applied to analyze the pre-cracked beam and ACD-embedded beams. Four-way fixed crack model is adopted because it can deal with shear anisotropy and multi-cracking. The combination of interfacial discrete elements and smeared crack modeling can work as the simulator for failure of RC beams with pre-cracks, metallic and non-metallic sheets that cause artificial shear anisotropy in the member.
REFERENCES
[1] Pimanmas, A., and Maekawa, K.: Influence of pre-cracking on reinforced concrete behavior in shear, Concrete Library of JSCE, No.38, pp.207-223, Dec 2001 [2] Pimanmas, A., and Maekawa, K.: Control of crack localization and formation of failure path in RC members containing artificial crack device, J.Material, Conc. Struct. Pavements, JSCE, No.683/V52, pp.173-186, Aug 2001 [3] Okamura, H. and Higai, T.: Proposed design equation for shear strength of reinforced concrete beams without web reinforcement, Proc. of JSCE, Vol.300, pp.131-141,1980 [4] Fukuura, N., and Maekawa, K.: Re-formulation of spatially averaged RC constitutive model with quasi-orthogonal bi-directional cracking, Proc. of JSCE, Vol. 45, pp. 157-176, 1999 (in Japanese) [5] Fukuura, N., and Maekawa, K.: Spatially averaged constitutive law for RC in-plane elements with non-orthogonal cracking as far as 4-way directions, Proceeding of JSCE, Vol.45, pp. 177-195, 1999 (in Japanese) [6] Okamura, H., and Maekawa, K.: Nonlinear analysis and constitutive models of reinforced concrete, Gihodo-Shuppan Co. Tokyo, 1991 [7] An, X., Maekawa, K. and Okamura, H.: Numerical simulation of size effect in shear strength of RC beams. J. Materials Conc. Struct., Pavements, JSCE. 1997, Vol.35, No. 564, 297-316 [8] Miyahara, T., Kawakami, T., and Maekawa, K.: Nonlinear behavior of cracked concrete plate element under uniaxial compression, Concrete Library International, JSCE, Vol. 11, pp.306-319, 1988 [9] Li, B., Maekawa, K. and Okamura, H.: Contact density model for stress transfer across cracks in concrete, J. Faculty of Eng. Univ. of Tokyo (B), 1989, Vol.40, 9-52 [10] Salem, H. and Maekawa, K.: Spatially averaged tensile mechanics for cracked concrete and reinforcement under highly inelastic range, J. Mater. Conc. Struct. Pavement, JSCE, 1999, 277293 [11] Pimanmas, A. and Maekawa, K.: Multi-directional fixed crack approach for highly anisotropic shear behavior in pre-cracked RC members, J. Material. Conc. Struct., Pavements, JSCE, 2001, Vol.50, No.669, pp.293-307 [12] Mishima, T., and Maekawa, K.: Development of RC discrete crack model under reversed cyclic loads and verification of its applicable range, Concrete library of JSCE, No.20, 1992.
