## 1 Introduction  
Cement-based materials such as mortar and concrete have long been used as one of the major construction materials. However, these materials are characterized by a relatively low tensile strength, accompanied by a brittle failure. Among the various methods developed to overcome such weaknesses, the application of fibers exhibiting high resistance to tension enhances the tensile performance of cement-based materials [1,2]. Fiber reinforcements are known for improving the toughness of fiber-reinforced concrete (FRC) since the crack bridging effect will better distribute cracks and reduce crack width. However, controlling the cracking behavior with conventional macro fibers is difficult since these fibers do not engage until the concrete or mortar matrix has cracked. Recently, several researchers have developed a high-performance fiber-reinforced cementitious composite (HPFRCC) [3, 4, 5, 6], and they have shown that the HPFRCC can provide a possible solution for serious localized wide cracks, since it can induce a number of finely distributed cracks. In addition to this advantage, the salient feature of the HPFRCC lies in its strain hardening behavior with large deformation capacity, which conventional fiber-reinforced concrete cannot easily obtain. These multi-cracking and strain-hardening characteristics are achieved by many distributed cracks formed sequentially before a HPFRCC member failed after one of the cracks is localized. Since the tensile stress-crack width relationship of HPFRCC is much different from that of conventional FRC, many studies have been conducted to investigate and predict the behavior of HPFRCC, especially for tensile and flexural responses. During recent years, the stress-crack width relationship of concrete and HPFRCC has been widely investigated. The pseudo-strain hardening properties of Engineered Cemertitious Composite (ECC) under tension are often described as a bilinear stress-strain relation, since this assumption accurately describes the important characteristics of this material behavior. [PERSON] and [PERSON] [7] and JSCE TC 334 [8] use an elastic-plastic relation, which is a function of strain, without considering cracks. However, in order to describe the failure of HPFRCC, crack localizations need to be considered. [PERSON] _et al._[9] describe the localizing process of ECC using a bilinear softening relation, using the function of crack width to incorporate the influence of the localization in the flexural analysis. However, it should be noted that, in their model, a crack starts to occur when the crack is localized. Therefore the model cannot describe the initial crack width properly, although this can be ignored because the ECC material shows only distributed fine micro-cracks before the localization. For the flexural behavior of the HPFRCC beam, several methods have been proposed based on the geometry of a beam. For a notched flexural beam, a fracture-based method has usually been used. [PERSON] and [PERSON] [7] have estimated the flexural strength of fiber cementitious composites, and several researchers have predicted the flexural behavior of FRC [10] and ECC [11] using a fracture-based model. For an un-notched beam, a hinge model or section analysis method has commonly been used. [PERSON] _et al._[9] adapted a hinge model that includes tensile strain hardening and tension softening to predict the flexural behavior of the ECC. A section analysis method was also used for the analysis of convectional FRC [12] and ECC [13]. Recently, a meso-mechanical approach [14] has been developed to explicitly simulate the contribution of matrix and fibers and their mutual interaction. This approach was adopted by [PERSON] _et al._[15] to simulate the flexural behavior of FRC beams with a notch through being implemented in a cracked-hinge model [16]. Among many models, the analytical methods that consider the average cracks over the beams are effective for the investigation of the average tensile and flexural behavior of HPFRCC.