## 2 Compaction-Induced Stress  
During the construction of GRS walls, fill compaction is typically carried out by operating a vibratory or roller compactor, which in turns imposed a compaction load to the newly placed soil lift in the form of a moving point or a moving strip load. References [6; 12] used the stress path proposed by References [9; 10] to show that the CIS at any given location, as a result of multiple passes of a compactor moving towards or moving away from the wall, could be assessed by applying a compaction load directly over the location considered.  
When a soil mass is subjected to a surcharge, there would generally be an increase in the vertical and horizontal stresses in the soil mass. When a compaction load is added to I-I (Figure 1a), its horizontal stress at depth \(z\) would increase corresponding to the stress path AB (Figure 1b). As the compaction load moves away from I-I, the horizontal stress would begin to decrease following the stress path BC. AC is thus the increment of the horizontal stress after the compaction load is removed. This increase in horizontal stress and, hence, the confining pressure would lead to an increase in the stiffness and strength of the soil mass ([6; 12]).  
References [6; 12] proposed a procedure to calculate the increase in lateral stress in a reinforced soil mass caused by backfill compaction:  
\[\Delta\sigma^{\prime}_{\text{h,c}}=(\Delta\sigma^{\prime}_{\text{v,c max}})(K_{ \text{i,c}})(F)\sqrt{1+\frac{0.7E_{\text{r}}}{E_{\text{s}}S_{\text{v}}-0.7E_{ \text{r}}}}, \tag{1}\]where \(\Delta\sigma^{\prime}_{\text{v,c max}}\) is the maximum increase of the \"vertical\" stress (point B in Figure 1b) due to fill compaction, which can be calculated using the Westergaard solution ([27]), for simplicity it is taken as the compaction load \(q\) in this study; \(K_{\text{i,c}}\) is the coefficient of lateral earth pressure of a GRS mass for initial loading, which is generally taken as:  
\[K_{\text{a}}\leq K_{\text{i,c}}\leq K_{0}\qquad\text{or}\qquad\tan^{2}\left(45 ^{\text{o}}-\frac{\phi^{\prime}}{2}\right)\leq K_{\text{i,c}}(=\beta K_{\text {a}})\leq 1-\sin\phi^{\prime}, \tag{2}\]  
where \(1.0\leq\beta\leq 1.5\) ([12]); \(E_{\text{s}}\) is the stiffness of the soil (kPa), \(E_{\text{r}}\) and \(S_{\text{v}}\) is, respectively, the stiffness (kN/m) and spacing (m) of the reinforcement. The value of \(F\) has been given by References [9; 28] using:  
\[F=1-\frac{\text{OCR}-\text{OCR}^{\sin\phi}}{\text{OCR}-1}, \tag{3}\]  
where OCR and \(\phi^{\prime}\) is the over-consolidation ratio, and the internal friction angle, respectively.  
The lower and upper limits of the compaction-induced stress for \"un-reinforced\" soil has been proposed by Reference [29] as:  
\[0.45\sqrt{\gamma Q}\leq\text{CIS}\leq 0.90\sqrt{\gamma Q}, \tag{4}\]  
where \(\gamma\) is the as-compacted unit weight of the fill, and \(Q\) is the maximum force per unit length of the compactor due to the combined action of gravity and cyclic accelerations. The compaction stresses should not then exceed the larger of \(0.45\sqrt{\gamma Q}\) for the free field, and \(0.90\sqrt{\gamma Q}\) for the zone near the wall ([29]). Reference [26] has recommended the use of Bolton's upper and lower limits of CIS value could be adopted for the preliminary analysis and design of GRS mass structures.  
Figure 1: (**a**) Schematic diagram of a moving compaction load. (**b**) Stress paths of a soil element at depth \(z\) as the compaction load moves away from I-I.  
compaction equipment) and 30.3 kPa (for heavy compaction equipment) corresponded, respectively, to an equivalent total force per unit width \(Q\) of 53.2 kN/m and 61 kN/m. Note that, these CIS values corresponding to equipment with plate or roller width \(w\) ranges from 0.25 to 0.75 m, and from 0.391 to 2.080 m, for vibratory plate soil compactor and vibratory roller soil compactor, respectively. Clearly, for geosynthetic reinforced soil (GRS), the values of the compaction-induced stresses are likely to be higher.