Modeling of Strain Penetration Effects in Fiber-Based Analysis of Reinforced Concrete Structures

ACI Structural Journal, Mar/Apr 2007 by Zhao, Jian, Sritharan, Sri

According to Eq. (1), as the bar stress approaches the yield strength, (...) becomes zero, the slip approaches the yield slip (s^sub y^), and the slope of the curve approaches the initial slope (bK). Furthermore, as the bar stress approaches the ultimate strength, (...) becomes infinity, the slip approaches the ultimate slip (s^sub u^), and the slope of the curve approaches zero. To maintain a zero slope near the ultimate strength of the bar, the value of factor R^sub e^ should be slightly greater than one and was taken as 1.01 for the analyses reported in this paper. The remaining parameters that are required to construct the bar stress versus slip response envelope are s^sub y^, s^sub u^, and b.

The pull-out test data available in the literature for deformed steel reinforcing bars were used to establish a suitable value for s^sub y^. Ensuring that the bar had sufficient anchorage during testing, only the pull-out tests that used a bar embedment length equal to or greater than the minimum anchorage length (l^sub a,min^) specified by Eq. (2) were selected for this purpose (refer to Table 128-32). The minimum anchorage length was determined equating the bar stress to f^sub y^ at the loaded end and assuming an average bond stress of 1.75 [the square root of]f'^sub c^ (where f'^sub c^ is the concrete compressive strength in MPa) over l^sub a,min^.33 This average bond stress, which is comparable to that used by Lowes and Altoontash,24 was established assuming a linear slip distribution along l^sub a,min^ and the local bond stress reaching a maximum value of 2.5 [the square root of]f'^sub c^ (MPa) at the loaded end.7 Accordingly

... (2)

where d^sub b^ is the bar diameter (mm).

Given the different values for variables d^sub b^, f^sub y^, and f'^sub c^ in the tests summarized in Table 1 and the dependency of the yield slip on these variables, Eq. (3) was established from a linear regression analysis as represented in Fig. 5 to determine the suitable value for s^sub y^

... (3)

where α is the parameter used in the local bond-slip relation, as illustrated in Fig. 2, and was taken as 0.4 in this study in accordance with CEB-FIP Model Code 90.34

As observed for the yield slip, it is conceivable that the loaded-end slip at the bar ultimate strength s^sub u^ and the stiffness reduction factor b are also functions of steel and concrete properties as well as the bar diameter. Sufficient experimental data, however, were not available to establish these functions from regression analyses; most of the tests summarized in Table 1 were terminated soon after reaching the yield slip. The limited test information available in the literature indicated that s^sub u^ = 30 ~ 40s^sub y^ and b = 0.3 ~ 0.5 would be appropriate. Furthermore, in the absence of sufficient experimental data, it is suggested that Eq. (1) and (3) be used for sufficiently anchored bars with both straight and hooked ends under tension and compression loads. It is believed that this suggestion should not introduce any significant error in the simulation of flexural members subjected to low axial loads (for example, bridge columns and concrete walls in low- and mid-rise build-ings). As more data become available, appropriate empirical equations suitable for defining s^sub u^ and b can be developed.