## 2 Design Method Development
### Ultimate Limit State Analyses  
Recommendations for the design of GRPE systems are given above; the design method not only gives the maximum strain but also can be used to perform the ultimate limit state analyses of reinforcement in GRPEs. The following two methods exemplify how the design charts can be used.  
Figure 7: Design charts for a GRPE with over-consolidated soil (\(\Delta p\leq(p_{c}-p_{1})\)); (a) Full-scale chart; (b) Enlarged chart.  
(a)BS8006-recommended method  
The maximum strain of the reinforcement was obtained from the charts. With the axial stiffness of the reinforcement, the maximum tensile force (\(T_{\text{max}}\)) in the reinforcement can then be obtained.  
\[T_{\text{max}}=J\varepsilon \tag{26}\]  
The maximum tensile force of the reinforcement should remain less than its allowable resistances:  
\[T_{\text{max}}\cdot\gamma_{T}\leq R_{G,d}, \tag{27}\]  
Figure 8: Design charts for a GRPE with over-consolidated soil (\(\Delta p>(p_{c}-p_{1})\)); (a) Full-scale chart; (b) Enlarged chart.  
where \(\gamma_{T}\) is the load partial factor, and \(R_{G,\text{d}}\) represents the design value of the resistance of the reinforcement.  
BS8006 [15] gives the formula to calculate the tensile load, \(T_{\text{rp}}\), per meter run in the reinforcement induced by the distributed load, \(W_{T}\), as shown in Equation (28).  
\[T_{\text{rp}}=\frac{W_{T}(S-b)}{2b}\sqrt{1+\frac{1}{6\epsilon}} \tag{28}\]  
When substituting \(T_{\text{max}}\) into Equation (28), replacing \(T_{\text{rp}}\) by \(T_{\text{max}}\), the distributed load, \(W_{T}\), acting across the reinforcement is then obtained. The minimum distributed load may be assumed to depend on the area coverage ratio of the pile caps relative to the pile spacing (\(b^{2}/s^{2}\)) and the relative height of the embankment, i.e., \(H/(s-b)\), which is the embankment height (\(H\)) normalized by the clear span between the piles (\(s-b\)). Irrespective of the arching theory, pile layout, and embankment geometry, the geosynthetic reinforcement needs to carry, at least, a practical minimum proportion (15%) of the embankment loading.  
\[W_{T}\geq 0.15S\cdot(f_{s}\gamma H+f_{q}q), \tag{29}\]  
where the \(f_{s}\) is the partial factor for soil unit weight, and \(f_{q}\) is the partial load factor for an external surcharge.  
EBGEO-recommended method  
Linearizing the short-term load-extension curve for the reinforcement at \(\epsilon=2.5\%\), and simplifying the short-term stiffness of reinforcement to long-term,  
\[J_{L}=\frac{U}{2.5}\cdot J_{S}, \tag{30}\]  
where \(J_{L}\) and \(J_{S}\) are the long-term and short-term stiffness of the reinforcement, respectively. \(U\) is the utilization factor of the reinforcement corresponding to \(\epsilon=2.5\%\) in the short-term load-extension curve.  
The tensile force in the reinforcement as a result of membrane action is given by Equation (31).  
\[E_{M}=\epsilon\cdot J_{L} \tag{31}\]  
The reinforcement in the ultimate limit state was then analyzed. The tensile force in the reinforcement should remain less than its design resistance:  
\[E_{M}\cdot\gamma_{E}+\Delta E_{\text{d}}\leq R_{G,\text{d}}, \tag{32}\]  
where \(\gamma_{E}\) is load partial factor, \(\Delta E_{\text{d}}\) is the design value of additional effects in the reinforcement as a result of spreading force, and \(R_{G,\text{d}}\) represents the design value of the resistance of reinforcement.  
The adequate tensile strength of geosynthetic reinforcement is, thus, demonstrated. If adequate tensile strength cannot meet the requirement, and then the analysis must be repeated by altering embankment parameters or reinforcement type until the adequate tensile strength meets the requirements of serviceability.