Arching effect and displacement on theoretical estimation for lateral force acting on retaining wall

A new approach is proposed to evaluate the non-limit active earth pressure in c–φ soil based on the horizontal slices method and limit equilibrium method, which takes into account arching effect, displacement, mobilized friction angle, tension cracks, and shear stress of the horizontal layer. The accuracy of the proposed method was demonstrated by comparing the experimental results and other theoretical methods. The comparison results showed that the proposed approach was appropriate for calculating the non-limit active earth pressure in c–φ soil and cohesionless soil. A parametric study was undertaken to access the effects of cohesion, mobilized friction angle, and shear stress of the horizontal layer for the lateral earth pressure, as well as the effects of displacement for the rupture angle and tension cracks. Moreover, by comparing calculated results of the different theoretical methods and numerical model, it indicated that the empirical formulations of the mobilized internal friction angle and soil–wall interface friction angle used to cohesionless soil were still suitable for c–φ soil.


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advanced procedures for predicting the nonlinear distribution of active earth pressure in sand, which considered arching effect (Handy 1985;Clough 1991;Paik and Salgado 2003;Li et al. 2017;Chen et al. 2019;Liu et al. 2020). Furthermore, various research methods have been applied to calculate the displacement-dependent lateral earth pressure. These calculated methods of the non-limit active earth pressure considering the displacement mainly utilize a fitting method of measured data (Mei et al. 2009;Ni et al. 2018;Ostad-Ali-Askari et al. 2020), a geotechnical parameter substitution method (Xu et al. 2013;Varol et al.2020), and a linear model (Bang 1985;Golam et al. 2015;Cetin et al. 2015)d. Nevertheless, the research results show that arching effect leads to the nonlinear distribution of active earth pressure (Fang 1986), which always appears with displacement, but the influence of arching effect and displacement on non-limit active earth pressure is rarely discussed at the same time in the existing literature.
Besides, arching effect is a common natural phenomenon, whether it is in cohesionless soil or c-φ soil (Rao et al. 2016). Due to the influence of arching effect and displacement on tension cracks and mobilized friction angle, it causes the distribution characteristics of the lateral earth pressure in c-φ soil to be significantly different from that of cohesionless soil. Although there have been many research achievements such as seismic, unsaturated, or reinforced active earth pressure in c-φ soil (Ahmadabadi and Ghanbari 2009;Kumar 2010;Vahedifard et al. 2015), the non-limit active earth pressure considering arching effect is still rarely studied in c-φ soil. In particular, the influence of arching effect on tension cracks can be not ignored in c-φ soil (Zhu et al. 2015). Moreover, many advanced methods were directly based on the assumption of Rankine rupture angle (45° + φ/2) or Coulomb rupture angle (Venanzio 2010;Tu and Jia 2012;Ostad-Ali-Askari et al. 2017), which ignored the influence of cohesion, soil-wall interface friction angle, and wall height on the rupture angle.
Hence, findings on this issue have been incomplete to date. In this paper, the relationship between the friction angle and displacement in c-φ soil is further studied with the numerical model. An analytical procedure based on static limit equilibrium methods and horizontal slices method is applied to evaluate the non-limit active earth pressure considering arching effect, displacement, horizontal average shear stress of the soil slice, and tension cracks.

Geometry, mesh, and boundary conditions
The finite element program Midas GTS was used to analyze the influence of the displacement on the non-limit active earth pressure. The model dimensions, adopted meshes, and boundary conditions are shown in Fig. 1. The height of the cantilever retaining wall was 10 m, which referred to the height of retaining wall in Matsuo's (1978) situ test. The total height of the mesh and the width at the left side of retaining wall are two times the height of retaining wall. As for the boundary conditions, the top surface of the mesh is free, the bottom surface is fixed, and other sides can only rotate. The thickness of the mesh is 1 m to simulate the plane strain state. The finite element mesh adopts the quadrilateral and triangular hybrid element. The element size is 0.5-0.7 m. The mesh boundary had been set sufficiently far to avoid impacts to the analysis results (Liu et al. 2020). The finite element mesh consisted of 1732 elements and 1813 nodes.

Material properties
Retaining wall is the perfect linear elastic material in the numerical model; the soil mass destruction obeys Mohr-Coulomb failure criterion. The soil mass and retaining wall are homogenous. Because the soil stiffness only affected the convergence rate of the computed lateral earth pressure, its role was not investigated (Pirone et al. 2018). The material parameters properties are listed in Table 1. The parameters of c-φ soil were the same as silty sand in Matsuo's situ test.

Analysis of numerical results
The nonlinear static analysis method was used to solve the numerical model. The max horizontal displacement, resultant forces of the lateral earth pressure, and shear forces were recorded from the at-rest to active state. The number of load iteration steps in the computational process was set to 21. The shear force (T) and lateral resultant force (N) acting on retaining wall can be obtained by the finite element method with the different displacement in Table 2. Meanwhile, Fig. 2 shows that the resultant forces of the lateral earth pressure in FEM gradually decrease and remain at the same pace with the measured values. It indicates that the model parameters, boundary, and convergence accuracy in the numerical model are reasonable.
The lateral earth pressure with the wall movement is gradually variation from an at-rest state to an active state (Bang 1985). The internal friction angle (φ m ), soil-wall interface friction angle (δ m ), and cohesion (c m ) with displacement are gradually mobilized from an initial value to a peak value; the empirical formulas applied to cohesionless soil had also been given (Chang 1997). However, the relationship between the friction angle and displacement is still ambiguous in c-φ soil. For further illustration, the mobilizations of the internal friction angle and soil-wall interface friction angle in c-φ soil are presented based on the numerical results. According to the shear force T = N tan m +(H − z) c w and the cohesion of the soil-wall interface c w = c m tan m tan m (Zhu et al 2015), the mobilized soil-wall interface friction angle (δ m ) based on Rankine's theory can be obtained by Eq. (1). where φ m and δ m are the mobilized friction angle, φ, is the limit friction angle, T and N are the total vertical shear force and lateral resultant force in the finite element method, which acts on the back of retaining wall, respectively. Meanwhile, the resultant force of the lateral earth pressure (N) based on Rankine's theory can be expressed by Eq. 2.
where H is the height of the retraining wall, and γ is the unit weight of the soil.
For the relationship between the mobilized cohesion (c m ) and displacement, Xu et al. (2013) obtained Eq. (3) by the geometric relationship in the stress of the Mohr's circle.
By substituting Eq. (3) into Eq. (2), the mobilized internal friction angle (φ m ) can be obtained by Eq. (4). Li et al. (2017) had proposed an empirical model about the mobilized friction angle (φ m , δ m ), the limit friction angle (φ 0 , δ 0 ), and the limit friction angle (φ, δ) obtained by Eqs. (5) and (6) in cohesionless soil. where s is the horizontal displacement of the wall at the arbitrary depth; s a is the critical displacement condition corresponding to the maximum internal friction angle; and s c is the critical displacement corresponding to the maximum soil-wall interface friction angle. For normally consolidated soil, the internal friction angle (φ 0 ) under the initial state can be obtained by Eq. (7) when the influence of the initial soil-wall interface friction angle is not considered.
where k 0 is the coefficient of the at-rest earth pressure, which can be estimated by the empirical formulation (for example, cohesive soil k 0 = 0.95 − sin � , cohesionless soil k 0 = 1 − sin � , and overconsolidated soil k 0 = 1 − sin � (OCR) sin ) (Mayne and Kulhawy 1983;Chen et al. 2019). Figure 3 shows that the mobilized soil-wall interface friction angle always keeps increasing with the displacement until the soil is at the limit state. The initial value of the soil-wall interface friction angle (δ 0 ) obtained by the finite element method is 3.28°, which is not consistent with half of the limit internal friction angle in Fang's (1986) experiment. Various errors in the model experiment may have caused certain deformation of the wall and soil, which make the experimental initial value of the soil-wall interface friction angle larger than the ideal initial value. However, the measured values (δ m ) of Matsuo's (1978) situ test were too bigger than the theoretical value. By checking the Mohr's circle of stress for an arbitrary point near the retaining wall, this phenomenon is impossible (Paik and Salgado 2003). Moreover, according to the numerical calculation results in Table 2, the mobilized soil-wall interface friction angle (δ m ) is always smaller than the limit internal friction angle (φ), which is consistent with the change of Fang's (1986) and Li's (2017) experimental values in cohesionless soil. By comparing the calculated values between the theoretical method and numerical model under different conditions, it indicates that Eq. (5) is also applicable for c-φ soil. The initial soil-wall interface friction angle is advised to be 0°, and the critical displacement (s c ) is approximately 0.01 H.
The variation of the mobilized internal friction angle with the wall movement is presented in Fig. 4. Comparing the calculating values of the three theoretical methods, it shows that Eq. (5) is also applied to cohesive-frictional soil. The initial internal friction angle can be appropriately calculated by Eq. (3). The critical displacement (s a ) is approximately 0.01 H, which is in agreement with s c . It shows that the internal friction angle and soil-wall interface friction angle will reach the limit value at the same displacement.

Theoretical considerations
The detailed derivation procedures for the non-limit active earth pressure in c-φ soil are described in the following section. To facilitate the earth pressure for c-φ soil considering arching effect and displacement, the present study makes the following assumptions: (1) The top of retaining wall is horizontal, and there is no load on it.
(2) The rigid plate, whose initial condition is upright, rotates around the heel of the wall.  (3) It is assumed that there is a polyline potential slip surface in cohesive-frictional soil, when retaining wall does not reach the limit equilibrium condition. (4) The potential slip surface passes through the heel of the wall. (5) The trajectory of the minor principal stress is an arc (Paik and Salgado 2003). Quinlan (1987) and Kingsley (1989) proved that the shape of the soil arching is closer to the arc. According to Paik and Salgado's (2003) method about soil arching, minor principal stress is always tangent to the arc, and major principal stress is always vertical to the tangent. As shown in Fig. 5, the radius of the arc for the minor principal stress is r, and the center of the arc is point O. The depth of tension cracks is z. The length of the trapezoidal layer element is L. The distance between the trapezoidal layer element and the top of the wall is y. The deflection angle between the major principal stress and the horizontal surface is θ A at point A. For point B, the deflection angle between the major principal stress and the horizontal surface is θ B . For the arbitrary point D, the deflection angle between the major principal stress and the horizontal surface is θ.

Coefficient estimation of non-limit active earth pressure
To simplify the following theoretical derivation, the stress state will be analyzed by the translational coordinate system in Fig. 6. The transition formulation of the old stress and the new stress is as follows in Eq. (8).
Trajectory of the minor principal stress in the differential element of the sliding wedge where σ and τ are the normal stress and shear stress in the old coordinate system (τOσ), σ´ and τ´ is the normal stress and shear stress in the new coordinate system (τ´Oσ´), respectively.
According to Mohr's circle representation of stresses and stress analysis in Fig. 6, the normal stress (σ w ′) and shear stress (τ w ′) acting on retaining wall at point A can be expressed by Eq. (9) (Tu et al. 2012;Lou 2015). where σ 1 ′ and σ 3 ′ are the major principal stress and minor principal stress in the new coordinate system, respectively; α is the angle between the direction of the major principal stress and the vertical direction of the retaining wall. According to equation � w = � w tan m , α can be obtained from the following relationship with the soil-wall interface friction angle.
Then, the horizontal stress (σ Ah ′) at point A can be obtained by Eq. (11). where ε is the rotation angle of the wall.
Similarly, according to the stress analysis in Fig. 6c, the normal stress (σ s ′) and shear stress (τ s ′) acting on retaining wall at point B can be expressed by Eq. (12).
According to Mohr's circle representation of the stresses in Fig. 6a, the horizontal normal stress (σ v ′) is related to the vertical normal stress (σ h ′) by Eq. (13). Then, the relationship can also be applied to an arbitrary point D in the AB trapezoidal layer element.
Although the direction of the principal stress deflects after arching effect appears, the horizontal stress of the arbitrary position is nearly equal in the AB trapezoidal layer element of the sliding wedge. This is different from the vertical stress (Chen et al. 2019).
Similarly, the horizontal normal stress, the vertical normal stress, and the shear stress at an arbitrary point D of the trapezoidal layer element can be obtained from the following relationship: In Fig. 5, the average vertical stress (σ av ′) in the AB trapezoidal layer element can be derived from the total vertical stress divided by the width (L) (Rao et al. 2016).
where the radius of the arc can be solved by Eq. (16), and the deflection angle of the stress (θ A , θ B ) can be obtained by Eq. (17).
where β is the angle between the potential slip surface and horizontal surface. The coefficient of the non-limit active earth pressure (K aw ) is defined as the ratio of the horizontal stress to the average vertical stress at a wall with soil-wall friction and the effects of the soil arching (Handy, 1985;Zhu 2015). Thus, K aw can be derived from Eq. (18).
When a point in cohesionless soil reaches the limit state, the relationship between the major principal stress and the minor principal stress is given as follows (Rao et al. 2016;Chen et al. 2019).
As shown in the above formulation, K aw is related to the change with the depth, and it will cause difficulties when solving the differential equation later. However, when the wall is in the initial position, the coefficient of the non-limit active earth pressure (K aw ) should keep pace with the coefficient of the at-rest earth pressure (K 0 ). In reality, K aw is inconsistent with the actual initial situation in Eq. (18), and it continues to increase with the increase in depth in c-φ soil. Besides, there are generally tension cracks in c-φ soil on the top of retaining wall. When y ≥ z and ε = 0, the coefficient of the non-limit active earth pressure (K aw ) is equal at the arbitrary depths after the displacement is determined. If y = H, it can be substituted into Eq. (18), and it can be regarded as the value as the initial coefficient (K o ) of every point along the depth. Then, the coefficient of the average non-limit active earth pressure (K awn ) of every point along the depth shows the same change as the wall movement.

Coefficient estimation of horizontal shear stress
The horizontal slices method has been widely used for determining active earth pressure at the intermediate state (Ahmadabadi et al. 2009;Chen et al. 2019), but the horizontal shear stress of the trapezoidal layer element is hardly considered. Actually, if the horizontal displacement of each point on the retaining wall is inconsistent, there must be a mutual dislocation among the soil slices. On the other hand, the horizontal shear stress of each point in the trapezoidal layer element is not equal because of arching effect. Therefore, the horizontal average shear stress (τ am ′) of the trapezoidal layer element can similarly be obtained from Eq. (22).
The coefficient of the average shear stress (K) can be defined as the ratio of the average shear stress (τ am ′) of the trapezoidal layer element to the average vertical stress. Similarly, K is also related to the depth (y), but it can be replaced with the coefficient (K ′ ).

Tension cracks and rupture angle in cohesive-frictional soil
The depth of tension cracks in c-φ soil has a great influence on the distribution of lateral earth pressure (Zhu et al. 2015). However, when the influence of arching effect on the lateral earth pressure was considered in some existing literature, tension cracks were still ignored in c-φ soil. This would lead to a larger calculation error. The depth of tension cracks at the top of retaining wall is generally related to the mechanical parameters of the soil (Rankine 1857), but it is also related to the mechanical parameters of the soil-wall interface and arching effect in Fig. 7. Moreover, tension cracks at the top of retaining wall are more likely to occur at the contact position of the wall-soil interface in engineering practice. Hence, the cohesion (c m ) of the soil should be replaced by cohesion of soil-wall interface (c w ) in practical engineering application. According to the boundary conditions: y = z, P am | |y=z = 0 , and K aw = 0 ; substituting the conditions into Eq. (18), the depth (z) of tension cracks considering arching effect can be calculated by Eq. (25).
If the mechanical parameters of the soil were already known, the calculated values for tension cracks could be compared with different theoretical methods in Fig. 8. Because tan m cos (cos 2 + K a sin 2 ) Rankine' s theory considered the influence of the soil cohesion (c m ), Lou (2015) ignored the influence of the rotation angle (ε), while the proposed method considered the smaller cohesion of the soil-wall interface (c w ), the calculated values of the proposed method are smaller than other two approaches. Moreover, the depth of tension cracks for Rankine and Lou's method was about 1.9 m in the initial state, but it should be closer to 0 m in the practical initial situation. Therefore, the proposed method is more applicable for calculating the depth of tension cracks under the intermediate state.
It is well known that the rupture angle can be simply assumed to remain at 45° + φ/2 in Rankine's theory. Based on this assumption, some scholars researched the nonlinear distribution of active earth pressure (Paik and Salgado 2003;Tu and Jia 2012). However, Kumar (2010) and Zhu et al. (2015) believed that the rupture angle was related to the wall height and only derived a complex formulation. However, a special distribution relationship between the rupture angle and wall height was still indistinct. Hence, we still need to find a simple formulation for the rupture angle considering arching effect.
According to Appendix Eq. (49), cot β is monotonic functions about β, if only one value exists for cot β which can make p a reach the extreme value, it is obvious that only an existed value for β can make p a reach an extreme value. Since β satisfies the conditions: 0 • < < 90 • , cot β cannot be negative (Kumar 2010). When In Fig. 9, several theoretical methods for determining the rupture angle were compared. The results show that the potential rupture angle (β) continues to increase with the increase in displacement from a potential slip surface to the critical failure surface except for Rao's (2016) method; the calculated values of the proposed method are consistent with Rankine solution and Zhu's method. When the shear strength of the soil will reach the limit state, the rupture angle for the proposed method will gradually approach 45° + φ/2. Although Rao et al. (2016) derived a formulation of rupture angle based on a geometrical relationship of arching effect, other methods obtained rupture angle with the limit equilibrium method, but the calculated values of Rao's method were significantly larger than other methods. Moreover, the proposed method and Zhu's method considered the influence of the wall height, cohesion, and unit weight, but other methods only considered the internal friction angle and the soil-wall friction angle.

Lateral earth pressure at the intermediate active state
To obtain active earth pressure at the intermediate state, a trapezoidal layer element is subjected to mechanical analysis, as shown in Fig. 10.
By establishing the static equilibrium in the horizontal direction, Eq. (28) can be obtained. Equations (29), (30), and (31) are substituted into Eq. (28). Considering that the second derivative and rotation angle (ε) are generally sufficiently small, their influence can be ignored to simplify the formulation. Therefore, Eq. (32) can be derived as follows: For the trapezoidal layer element in Fig. 10, the static equilibrium in the vertical direction requires the use of Eq. (33): The following Eq. (34) can be obtained by simplifying the above formulation.
Equations (30) and (32)  where A, B, and C are the constants, and they yield the following expressions: Because the boundary condition is the average vertical stress σ av = γz, when y = z. Substituting them into Eq. (35), the average vertical stress can be expressed by Eq. (36).
When 0 ≤ y < z , the horizontal stress σ h = 0 because of tension cracks; when z ≤ y < H , the horizontal stress h = K aw av .
In Fig. 7, the angle between active earth pressure and the horizontal surface is m + (Zhao and Bai 2015).Therefore, the non-limit active earth pressure (p am ) is derived from Eq. (38).
According to Eq. (38), the non-limit active earth pressure (p am ) is possibly negative on the bottom of the wall as the increasing displacement because of arching effect. Therefore, this part of the negative earth pressure should be ignored. Assuming that y 0 is a critical point of the lateral earth pressure from the bottom of the wall, the non-limit lateral resultant force (Q m ) is given as follows:

Comparison with experimental data and other methods
To check the applicability of the proposed method in c-φ soil or cohesionless soil, the predictions were compared with experimental values and other theoretical methods.
First, in c-φ soil, Yue (1992) conducted 11 sets of centrifuge model tests by a homemade hydraulic control device, which could control the movement of the wall. The parameters of the soil were based on the unit weight of the soil γ = 18.6 kN/m 3 , cohesion c = 38.2 kPa, and internal friction angle φ = 22.7°. The wall height (H) was 250 mm, and the model rate was 80.18. In the test, the horizontal displacement of retaining wall arriving at the limit state was s = 2.7 mm. Figure 11 shows the comparison between the predicted and measured values about the lateral earth pressure in c-φ soil. Because the lateral earth pressure at tension cracks zone is 0 kPa, the calculated values of the lateral earth pressure for the proposed method are not uniformly marked throughout this paper. When the displacement is 0 or 2.7 mm, the calculated values of the proposed method were consistent with the measured values. According to the distribution diagram of the active earth pressure in Fig. 11 and the lateral resultant force in Table 3, it showed that the calculated results of the proposed method were closer to the measured values than Rankine solution; the relative error of the proposed method is 2.9%. The results showed that the calculated values of the proposed method were in good agreement with the measured values in c-φ soil.
Secondly, in cohesionless soil, Tsagareli (1965) measured the distribution of active earth pressure under the translation mode. The parameters of the soil were based on the unit weight of the soil γ = 18.0 kN/m 3 , cohesion c = 0 kPa, internal friction angle φ = 37°, wall height H = 4 m, and critical soil-wall interface angle δ = 2φ/3. Figure 12 shows the distributions of active earth pressure between the different theoretical methods and measured values in cohesionless soil. According to the distribution diagram of active earth pressure in Fig. 11 and the lateral resultant force in Table 3, it showed that the proposed method and Rankine solution were closer to measured values than other theoretical methods; the relative error of the proposed method is 9.9%. The results showed that the proposed method was still well applicable to cohesionless soil.

Fig. 11
Comparison between predicted and experimental values obtained about lateral earth pressure in cohesive-frictional soil According to the above analysis, the proposed method is applied to predict the non-limit active earth pressure with the displacement and arching effect in c-φ soil or cohesionless soil. Figure 13 shows the distribution of active earth pressure along the depth with different values of cohesion. The calculated parameters of the soil were based on the unit weight of the soil γ = 19.0kN/m 3 , internal friction angle φ = 30°, wall height H = 10 m, and limit soil-wall interface angle δ = 2φ/3. According to Fig. 13, active earth pressure gradually decreases with the increase in cohesion. The distribution of active earth pressure still keeps linearly increasing at 0 ≤ y ≤ 0.6 H it does not significantly decrease until arching effect obviously appears near the bottom of the wall. In addition, because tension cracks are considered in the proposed method, which is different from the Rankine theory, the changes of active earth pressure near the top are slight with the increase in cohesion. With the increase in depth, the effect of the cohesion on active earth pressure gradually increases. From Fig. 13, active earth pressure at the bottom appears a negative value with the increase in cohesion, which shows that tension cracks appear when cohesion c ≥ 20 kPa . The peak point of active earth pressure keeps along the depth continually moves up with increasing cohesion, which shows that the position of the soil arching also keeps moving up. The results show that the cohesion is an important factor for arching effect.  Figure 14 shows the distribution of active earth pressure along the depth with different friction angles. The calculated parameters of the soil are based on the unit weight of the soil γ = 19.0 kN/m 3 , cohesion c = 10 kPa, wall height H = 10 m, and limit soil-wall interface angle δ = 2φ/3.

Friction angle
Similarly, active earth pressure gradually decreases with the increase in internal friction angle. The upper active earth pressure basically increases linearly, and active earth pressure does not decrease rapidly until the stress deflection obviously occurs near the bottom. Moreover, as the internal friction angle increases, the depth of tension cracks also increases gradually in Fig. 14. Figure 15 shows the variation of the average vertical stress with different depth and soil-wall interface friction angles. The average vertical stress can be obtained from Eq. (48) as a function of the depth and soil-wall interface friction angle. The calculated parameters of the soil were based on the unit weight of the soil γ = 19.0 kN/m 3 , cohesion c = 20 kPa, internal friction angle φ = 27°, initial soil-wall interface friction angle δ 0 = 0°, limit soil-wall interface friction angle δ = 18°, and wall height H = 10 m.
According to Fig. 15, when δ m = 0°, the average vertical stress linearly increases along the depth until the depth y = 0.9 H. When δ m ≠ 0°, the average vertical stress appears nonlinear distribution with the depth and significantly decreases with the increase in soil-wall interface friction angle. At the same time, it shows that δ m is also a crucial factor, due to causing a significant change for the upward movement of the soil arching. Figure 16 shows the coefficient variation of the average non-limit active earth pressure with the different soil-wall interface friction angles and displacements. K awn can be obtained from Eq. (22) as a function of the soil-wall interface friction angle and displacement. The calculated parameters of the soil were based on the unit weight of the  Fig. 16, K awn is nonlinearly decreasing with the displacement; as the critical soil-wall interface friction angle (δ) increases, the unit decrease in K awn also decreases.   Figure 17 shows the coefficient variation of the average shear stress with different depths and displacements. The coefficient of the average shear stress (K) can be obtained from Eqs. (17), (23), and (24) as a function of the depth and displacement. The calculated parameters of the soil were based on the unit weight of the soil γ = 19.0 kN/m 3 , cohesion c = 20 kPa, internal friction angle φ = 27°, initial soil-wall interface friction angle δ 0 = 0°, limit soil-wall interface friction angle δ = 18°, and the wall height H = 10 m. The coefficient of the average shear stress at a certain depth continuously increases with increasing displacement, and it gradually tends to a constant value until the soil-wall mass arrives at the limit state. In addition, the coefficient of the average shear stress increases significantly at the depth y/H = 0.1 than other depths with the increase in height of tension cracks. From Fig. 17, the coefficient (K) is always smaller than tan φ m , which shows that horizontal shear failure will not occur under normal conditions. Figure 18 shows the distribution of active earth pressure along the depth considering soil arching and horizontal shear stress or only soil arching. The parameters used for the soil are the same as those mentioned above.

Horizontal shear stress between the slices
According to Fig. 18, active earth pressure considering arching effect and horizontal shear stress is larger than that only considering arching effect, when the displacement is the same. This indicates that the horizontal shear stress between the slices hinders the destruction of the soil; the influence of the average shear stress gradually weakens with the increase in displacement. Therefore, the horizontal shear stress of the slice can be used as a safe reserve for retaining wall's stability under normal conditions.

Summary and conclusions
In this paper, a new formulation considering arching effect, displacement, horizontal average shear stress of the slice, and tension cracks has been derived about the non-limit active earth pressure in c-φ soil, which is based on the horizontal slices method and limit equilibrium method. The relationship between the friction angle and displacement is investigated in c-φ soil with the numerical model. Based on the analysis, the conclusions are summarized below: (1) The numerical results and calculated values of the theoretical method indicate that the empirical formulas about the mobilized friction angle used to cohesionless soil are still applicable to c-φ soil; the internal friction angle and soil-wall interface friction angle of c-φ soil will reach the limit value at the same displacement, which is approximately 0.01 H. (2) By comparing the experimental results and other theoretical methods, it showed that the proposed analytical procedures could reliably evaluate the non-limit active earth pressure, rupture angle, depth of tension cracks in c-φ soil with arching effect and displacement; the potential rupture angle (β) keeps increasing with the increasing displacement from a potential rupture surface to the critical failure surface; the depth of tension cracks under the intermediate state is related to arching effect and wall-soil interface cohesion.
(3) The influence of the different cohesion (c) and friction angle (φ, δ) on active earth pressure and the average vertical stress was analyzed by the proposed method. The results show that active earth pressure decreases with the increase in cohesion (c) and internal friction angle (φ); the cohesion and wall-soil interface friction angle (δ) are Fig. 18 Change of the lateral earth pressure distribution considering arching effect and shear stress or only arching effect two critical factors to cause the change of the position of the soil arching; when cohesion c ≥ 20 kPa , the bottom of retaining wall in c-φ soil also appears tension cracks zone due to arching effect and displacement. (4) The influence of the displacement on K awn and K was also analyzed by the proposed method. The results show that K awn keeps decreasing with the increasing displacement, while K increases, and eventually both tend to stabilize. (5) The main limitations of the work are: the load at the top of retaining wall always exists, but the influence of the load on arching effect and tension cracks is ignored in the proposed method; the failure surface is a polyline potential slip surface and always passes through the heel of the wall, but the failure surface of c-φ soil is generally the arc failure surface in practice; moreover, the position of the toe of the failure surface is difficult to determine accurately.

Appendix: the derivation process about the rupture angle
To obtain a simple formulation for the potential rupture angle (β) in c-φ soil, the proposed method needs to be assumed that the total weight of the slip zone (ABC) always remains constant from an at-rest state to an active state. In Fig. 7, T 1 and T 2 are the shear forces in the BC and AB plane, N 1 and N 2 are both the normal forces across the contact surface BC and EB, respectively, and Q and R are both the resultant forces acting on the sliding wedge.
According to the static equilibrium condition of the sliding wedge in Fig. 7, then Based on Mohr-Coulomb criterion, the shear forces (T 1 , T 2 ) can be obtained by Eq. (41).
The normal force (N 1 , N 2 ) along the contact surface can be expressed by Eq. (42).
The total weight (G) of the soil wedge (EBCD) can be obtained by Eq. (43). Because there is no cohesion in cohesionless soil, the angle between the reaction force (R) and the normal BC plane is equal to the internal friction angle (φ m ). However, some scholars still ignored the influence of cohesion on the reaction force (Xu et al 2013;Zhu et al. 2015). Actually, the relationship between the angle (ψ) and other factors is intricate in c-φ soil. Based on the equivalence principle of the shear strength (Zhao and Bai, 2015), the angle (ψ) of the reaction force (R), internal soil friction (φ), and cohesion (c) can be established using the following relationship for Eq. (46). By substituting Eqs. (42) and (43) into (46), the following Eq. (47) can be obtained.
If the soil-wall interface is upright and smooth, the rupture angle (β) is 45° + φ/2, and this condition can be substituted into Eq. (47); therefore, the angle (ψ) of the reaction force R will become a known parameter taking into account the influence of the cohesion by Eq. (48).
According to Eq. (44), Q is a function about β. The depth (z) of tension cracks is a known quantity. It is difficult to obtain a formulation about β by directly differentiating β from Eq. (44). However, if by substituting H = y (depth) to Eq. (44), active earth pressure (p a ) can first be obtained through p a = Q y .
Because the failure surface always passes through point B, the rupture angle (β) is related to the wall height (H) instead of the depth (y). Substituting Eq. (45)