## 10 Discussion
### Comparison of the Proposed Coefficients with Existing Solutions and Experimental Results  
The proposed coefficients will be compared with those given by Mononobe-Okabe (M-O) [7; 8] and Kapila [9] (Mononobe-Okabe-Kapila or M-O-K) for the active and the passive state, respectively. These coefficients are not only the most widely used in academia and in practice, but also they are included in numerous seismic design codes worldwide (e.g., [36; 37]), since being suggested as standard methods by [PERSON] and [PERSON] [38]. A set of \(K_{ae}\) and \(K_{pe}\) versus \(\varphi^{\prime}\) curves were drawn for comparison purposes for various \(k_{h}\) values and \(k_{v}=k_{h}/2\); see Figures 7-10 (the curves drawn are for cohesionless soils). Regarding M-O and M-O-K solutions, it is very weird that both are not applicable to all \(k_{h}-\varphi^{\prime}\) combinations (see Figures 8 and 10). The problems with these two solutions are known (e.g., see [39]) with the most important one being the square root of negative number in Equations (56) and (57) for \(\varphi^{\prime}<\psi\mp\beta\) (the minus sign stands for the active case whilst the plus sign stands for the passive case); this issue is further discussed in Section 10.4. \[K_{ae,M-O}=\frac{\cos^{2}(\varphi^{\prime}-\psi-\beta)}{\cos\psi\cos^{2}\beta \cos(\delta+\beta+\psi)\bigg{(}1+\sqrt{\frac{\sin(\varphi^{\prime}+\delta) \sin(\varphi^{\prime}-\psi-\alpha_{s})}{\cos(\delta+\beta+\psi)\cos(\alpha_{s} -\beta)}}\bigg{)}^{2}}, \tag{56}\]  
\[K_{pe,M-O-K}=\frac{\cos^{2}(\varphi^{\prime}-\psi+\beta)}{\cos\psi\cos^{2} \beta\cos(\delta-\beta+\psi)\bigg{(}1-\sqrt{\frac{\sin(\varphi^{\prime}+ \delta)\sin(\varphi^{\prime}-\psi+\alpha_{s})}{\cos(\delta-\beta+\psi)\cos( \alpha_{s}-\beta)}}\bigg{)}^{2}}, \tag{57}\]  
where \(\psi\) is the angle of the resultant body force in the soil from vertical (Figure 11):  
\[\psi=\arctan\bigg{(}\frac{k_{h}}{1-k_{v}}\bigg{)}. \tag{58}\]  
On the other hand, the proposed solution gives smooth and continuous curves, being able to calculate the active and passive earth pressure on the wall in every case (Figures 7 and 9). It is very interesting that, following the M-O-K method, a catastrophic earthquake with \(k_{h}=0.5\) and \(k_{v}=0.25\), seems to have a relatively minor effect on the reduction of passive resistance. On the contrary, when the proposed procedure is followed the reduction of passive resistance appears to be more rational. Indeed, negative values may be obtained under extreme seismic accelerations. In this special case of passive failure, the horizontal pressure \(\sigma_{3}\) in [PERSON]'s diagram is not only to the left of \(\sigma_{1}\) (where the circles for the active state are traditionally drawn), but also to the left of the point of origin. Another interesting point is that, for \(\varphi^{\prime}=0^{o}\), the proposed coefficients of active and passive earth pressures equal unity. Actually, as shown in Figure 12, this value is the transition point between the active and the passive state. Moreover, despite the noncontinuity of the original M-O and M-O-K solutions, they are both defined for \(\varphi^{\prime}=0^{o}\), giving \(K_{ae}\) and \(K_{pe}\) values also equal to unity for any \(k_{h}\)-\(k_{v}\) combination. Using the (arbitrary) correction given in Eurocode 8-5 [36] for the M-O and M-O-K solutions, unity is also the transition point between the active and the passive state (see dashed lines in Figures 8 and 10). Figure 8: \(K_{ae}\) vs. \(\varphi^{\prime}\) curves for the active state obtained by Mononobe–Okabe (M-O) method for various \(k_{h}\) values and \(k_{v}=k_{h}/2\) (for 100% of PGA). The dashed lines were drawn based on the modification of the M-O method given in Eurocode 8-5. Values in brackets correspond to seismic coefficient equal to the 2/3 and 1/2 of PGA (first and second value, respectively). Figure 7: \(K_{ae}\) vs. \(\varphi^{\prime}\) curves for the active state obtained by the proposed solution for various \(k_{h}\) values and \(k_{v}=k_{h}/2\) (curves referring to cohesionless soils).