## 3 Analysis Model for Cracking and Flexural Behavior of HPFRCC Beams
### Constitutive Models for the Materials  
HPFRCC beams have been modeled with several segments, and the length of each segment is assumed to be equal to the final crack spacing, so that each segment consists of a single crack and elastic composite parts, as shown in Figure 7b. The crack is assumed to occur when the tensile stress exceeds the tensile strength of the matrix. The elastic part behaves like an elastic material while the crack is activated after the tensile crack occurs. Therefore, when the stress is below the tensile strength of the matrix, the crack width is zero and the segment behaves as an elastic one. After the tensile stress exceeds the tensile strength of the matrix, the total strain of the segment is assumed to follow the relationship shown in Figure 8 while the elastic part still follows the elastic behavior. The total tensile strain of a segment can be calculated from the sum of the strains for the elastic parts and crack width, as illustrated in Figure 8. In Figure 8, Region I represents the elastic state before a first cracking. Region II represents the status when the crack is activated but not localized. In Region II, both the elastic parts and the crack can resist an increased load, which reflects the fiber-bridging and strain-hardening behaviors of HPFRCC in tension. The localization process is described in Region III. For the segment with a localized crack, with an increase in total tensile strain, the tensile stress decreases and the tensile strain of the elastic parts decreases while the width of a localized crack increases. The crack strain in the segment can be calculated by subtracting the elastic strain from the total strain, so the crack width is calculated by multiplying the length of the segment:  
\[\delta_{\text{crack}}=s\cdot\left(\varepsilon_{\text{tot}}-\varepsilon_{\text {elastic}}\right) \tag{3}\]  
Thus, the present model can include the influence of the tensile stress-crack width relationship from the moment of micro crack initiation to the localization. For the apparent total tensile stress-strain relationship of HPFRCC, the multi-linear model shown in Figure 8 has been used in this paper. In the model, the following values are estimated from the test results shown in Figure 3: \(\varepsilon_{{}_{t1}}=0.0004\); \(\varepsilon_{{}_{t2}}=0.0193\); \(\varepsilon_{{}_{t3}}=0.0212\).  
When dealing with compressive behavior, a parabolic equation was used, as is commonly done with plain concrete [25] and with a conventional FRC [12; 26; 27]. On the other hand, although JSCE TC 334 [8] proposed a parabolic equation without a descending region, a linear relation has been also  
Figure 8: Total tensile stress–strain relationship in uniaxial tension.  
commonly used to describe the compressive behavior of HPFRCC. Several researchers [7; 28; 29] have proposed a bi-linear equation for tensile strain hardening in fiber-reinforced composites such as ECC.  
From the compressive test results shown in Figure 1, the constitutive model for the compressive behavior is expressed as:  
\[f_{c}=f_{c}^{{}^{\prime}}(\epsilon_{c}\ /\ \epsilon_{0})\ \ \text{for}\ \ \epsilon_{c}\leq\epsilon_{0} \tag{4a}\] \[f_{c}=f_{c}^{{}^{\prime}}(1-0.5\frac{\epsilon_{c}-\epsilon_{0}}{\epsilon_{f}- \epsilon_{0}})\ \ \text{for}\ \ \epsilon_{c}>\epsilon_{0} \tag{4b}\]  
where \(f_{c}\) is the concrete stress at strain \(\epsilon_{c}\), \(f_{c}^{{}^{\prime}}\) is the maximum compressive strength at strain \(\epsilon_{0}\), and \(\epsilon_{f}\) is the strain at the compression failure. All values are positive for the compression. The test results show the average compressive strength and the corresponding strain was measured at 76.28 MPa and 0.005, respectively.