# Predicting Behavior of Grounded Dowel Connections Using Interfacial Cohesive Elements
## 3 Interfacial Modeling
### Bond-Slip Law  
The bond-slip law adopted in this paper was motivated by the experimental results shown in Figure 3. The law was derived based on the well-known Bertero, Popov, and Eligehausen (BPE) model presented by [PERSON] et al. [27], one of the most widely reported bond-slip models in the open literature. This model is mathematically expressed as follows:  
\[\tau_{2}(s_{2})=\left\{ \begin{array}{ll}\tau_{max}\left(\frac{s_{2}}{s_{a}} \right)^{\alpha}&s_{2}<s_{a}\\ \tau_{max}&s_{a}\leq s_{2}<s_{b}\\ \tau_{max}-\left(\frac{s_{b}-s_{2}}{s_{b}-s_{r}}\right)\left(\tau_{max}-\tau_{ f}\right)&s_{b}\leq s_{2}<s_{r}\\ \tau_{f}&s_{2}\geq s_{r}\end{array} \right., \tag{3}\]  
where \(\tau_{max}\) = maximum bond stress, \(\alpha\) = curve fitting parameter reflecting the degree of confinement, \(s_{a}\) and \(s_{b}\) = are bar slips corresponding, respectively, to the beginning and end of the bond strength plateau, \(s_{r}\) = is the rib spacing, and \(\tau_{f}\) = residual bond stresses. Equation (3) presents the local bond-slip law, whose variables depend on four main parameters that need calibration based on experimental results, namely \(\tau_{max}\); \(s_{a}\); \(s_{r}\), and \(\tau_{f}\). \(\tau_{f}\) was found to be in the range of 18-20 MPa. The curve fitting parameter \(\alpha\) was found to be in the range of 0.21-0.28. \(s_{a}\) fell in the 1.4-1.55-mm range. \(s_{b}\) was approximately equal to 1.2 \(s_{a}\) (1.86 mm). \(s_{r}\) was 9.75 mm. \(\tau_{f}\) (bond stress corresponding to a slippage equal to the spacing between the lugs) was found to be in the range of 12-13 MPa, approximately 50-60% of \(\tau_{max}\). The values of \(\tau_{max}\) and \(s_{a}\) (\(s_{max}\)) should be determined experimentally for each case, and no accurate theoretical model could be used to predict their values. In the absence of experimental data, [PERSON] and [PERSON] proposed Equations (4) and (5) (based on bond observation of bars embedded in confined concrete). \[\tau_{max}=1.163f_{c}^{\prime}\ 0.75; \tag{4}\]  
\[s_{max}=0.07\ d_{b}. \tag{5}\]  
To account for the damage of the anchorage due to influencing factors, the effective bond stress is calculated as shown in Equation (6). \[{\tau_{2}}^{*}=m_{n}(s_{1})\cdot\ m_{s}(\varepsilon_{s})\cdot\ \tau_{2}, \tag{6}\]  
where \(\tau_{f}^{*}\) = effective bond stress, \(m_{n}\) = modifier to account for concrete splitting (function of \(s_{1}\)), and \(m_{s}\) = modifier to account for post-yield behavior. \(m_{n}\) reflects the severity of splitting stresses in the concrete cover, which is a result of the radial tensile stresses induced unto the cover due to bar slip. This generates hoop tension, which forces the concrete to dilatate in the absence of transverse reinforcement. Consequently, regions where splitting occurs suffer from bond reduction. These failures were not observed when grouted connections were tested, irrespective of the intensity of the bond stresses developed. As such, \(m_{n}\) was taken equal to 1 since the corrugated duct provided a continuous compressive field restraining the grout. It is also worth noting that unconfined grout cones generally observed with the pull-out of bars from grout filled connections (e.g., observed in [PERSON] et al. [30], and [PERSON] et al. [2]) were not observed during the present experiments. It is not clear whether these grout cones were formed at the onset of the bonded length but were not observed due to de-bonding of the bars at the bar extremities, which shifted the unconfined length of the grout deeper inside the panels. \(m_{s}\) accounts for the reduction in bearing area resulting from inelastic strains, as shown in Figure 5a. [PERSON] et al. [31] observed bond reductions of as much as 25% at the onset of yielding, with further reductions as inelastic strains continuing to accumulate. This modifier is only active when the elastic strain limit is exceeded. The value of \(m_{s}\) depends on whether bars have yielded when the experimental bond-slip relationship is obtained.