## 10 Discussion
### How Design Codes Treat Dynamic Earth Pressures on Retaining Structures  
As mentioned previously, the methods preferred by the various design codes for the calculation of active and passive earth pressures under seismic conditions are the M-O and M-O-K. The various design codes recognize the limitations of these methods and they either attempt an interpretation, dictate a modification of these equations, or propose alternatives.  
In this respect, Eurocode 8 [36], for \(\beta>\varphi^{\prime}-\psi\), dictates (without any explanation) the whole square root in the M-O formula (Equation (56)) to be arbitrarily replaced by unity! This produces the paradox, as seen in Figure 8, the earth pressure coefficient for given \(k_{h}\) value (see dashed lines) to appear smaller for smaller \(\varphi^{\prime}\) values. The passive case Eurocode 8 [36] does not suggest or dictate something. However, doing the same for the M-O-K solution, a similar paradox appears. For example, in Figure 10 (dashed lines), for \(\varphi^{\prime}=25^{\circ}\), the \(K_{pe,M-O-K}\) coefficient is smaller for \(k_{h}=0.4\) as compared to \(k_{h}=0.5\).  
On the other hand, for AASHTO [37], the condition of \(\varphi^{\prime}\geq\)\(\beta+\psi=\)\(\beta+\arctan(k_{h}/(1-k_{v}))\) is thought of as specifying a limit to the horizontal acceleration coefficient that could be sustained by any structure in a given soil. Solving the above as for \(k_{h}/(1-k_{v})\), the limiting condition is  
\[k_{h}/(1-k_{v})\leq\tan(\varphi^{\prime}-\beta). \tag{59}\]  
Apparently, the above is not true, since the amount of horizontal acceleration sustained by a structure must depend on the structure itself. More recently, [PERSON] et al. [69] in a Federal Highway Administration course (FHWA-NHI-11-032), discussed the number of instances where the M-O and the M-O-K methods should not be applied. Instead, they suggest that the Trial Wedge Method or the General Limit Equilibrium Method be used. However, these methods have their own serious disadvantages e.g., the one related with the position of surcharge on the surface, i.e., if it is on the trial wedge or not, and more importantly the fact that the peak shear strength along the failure plane is again used.  
Finally, [PERSON] and [PERSON] [70] in the \"GEO Report No. 45\" of Geotechnical Engineering Office of Hong Kong connected the limitation of the M-O method with the allowable angle of backfill slope (\(\beta\leq\varphi^{\prime}-\psi\)), mentioning that the last will be unstable unless the soil has sufficient cohesive strength. This point of view, indeed, appears in most geotechnical engineering and soil mechanics books. However, how does [PERSON]'s purely geometrical solution understand that \(\varphi^{\prime}\) is the angle of repose of (dry) frictional soils? Should we not be able to calculate the earth pressure on a wall even when, for example, the earthquake causes the backfill slope above the higher point of the wall to fail? Apparently, the restrictions caused by the square root are only coincidental and the answer to these questions can be found in the setup of [PERSON]'s problem, where, the angle of the \"rapture surface\" (according to [PERSON] and [PERSON]) must austerely be greater than the angle of the backslope fill (both measured from horizontal). Otherwise, [PERSON]'s wedge cannot be formed. It is reminded that [PERSON] and [PERSON] treated the (rigid) soil wedge as being in the state of failure. However, this is not true (see Section 8.2) and this is the main fault of the method in question. The problematic \(\varphi^{\prime}-\psi\mp\beta\) term just derives from a series of mathematical operations with trigonometric functions. For the history, should this case appear, the Geotechnical Engineering Office of Hong Kong dictates the use of the trial wedge method.