A Modified Computational Method of Active State Earth Pressure behind Rigid Retaining Wall

Modes and magnitude of wall movement have a great influence on the distribution of Earth pressure behind retaining wall. For gravity retaining wall exerted by cohesionless soil rotating outward about the base, based on Dubrova’s method of redistribution of pressure and forming mechanism of Earth pressure, the distribution pattern of internal friction angle along the wall back was established for different displacements, and then, an improved method of calculating active Earth pressure was proposed, which considers the influence of displacement. It considers the influence of displacement and also reflects the gradual formation process of Earth pressure. When the retaining wall is in an at-rest state, the distribution of Earth pressure calculated in this paper is defined as initial active state Earth pressure, and it is not equal to static Earth pressure, because the retaining wall has a tendency of active displacement. As the wall movement develops, the mobilized internal friction angle of the quasi-sliding surface increases gradually from the initial internal friction angle ϕ 0 to the internal friction angle ϕ , and accordingly, the distribution of Earth pressure at the intersection of the sliding surface and the wall back decreases gradually from the distribution of initial active state Earth pressure in the static state to that of Coulomb active Earth pressure.


Introduction
Accurate calculation of the distribution of Earth pressure is the prerequisite for retaining wall design. e main methods of Earth pressure calculation in engineering are Rankine Earth pressure theory and Coulomb Earth pressure theory. Rankine's theory is based on the stress level in the elastic half-space body and the limit equilibrium condition of the soil, which requires that the back of the retaining wall is vertical and smooth, and the lling surface is horizontal. Coulomb's theory is based on the static equilibrium condition of sliding soil wedge when the soil behind the wall reaches the limit equilibrium state. Coulomb's method is more widely used.
Terzaghi [1], Sherif et al. [2], Fang and Ishibashi [3], and Zhou and Ren [4] studied the e ect of modes and magnitude of wall movement on the Earth pressure distribution behind the retaining wall through laboratory tests, which shows that Earth pressure distribution behind the wall will be signi cantly a ected by the modes and magnitude of wall movement whether the back ll is sand or clay. Chang put forward the concept of mobilized internal friction angle of back ll and mobilized external friction angle between back ll and the wall and established their displacement-related distribution patterns along the wall back. Based on Coulomb's Earth pressure theory, a simpli ed Earth pressure calculation method was proposed which shows the in uence of the modes and magnitude of wall movement [5]. Zhang and Chen established the basic equation of wall active Earth pressure under the condition of nonlimit state by taking the horizontal thin layer of the sliding wedge as the di erential element. Based on the moment equilibrium condition of the whole sliding wedge, the theoretical formula of Earth pressure distribution and resultant force point was set up [6]. From the stress of Mohr's circle of cohesive soil, Xu et al. deduced the relationship between the internal friction angle of cohesive soil and the wall displacement when the soil is in an intermediate state between the initial state and ultimate active equilibrium state [7]. Applying the horizontal-slice analysis method, the formula of Earth pressure in nonlimit active equilibrium state is obtained. Hu et al. studied the nonlimit active Earth pressure of retaining wall by thin-layer element method and obtained the theoretical formula of lateral coefficient, resultant force, and action point of Earth pressure [8]. Based on the calculation method of Bang's active Earth pressure, Wang et al. proposed an improved calculation method of nonlimit active Earth pressure considering the influence of wall displacement [9]. On the basis of the stress-strain concept, Chen et al. studied the nonlinear relationship between wall displacement and internal friction angle of soil and soil-wall external friction angle, and a theoretical calculation method of nonlimit Earth pressure was proposed considering the effect of retaining wall displacement [10]. Wang et al. deduced the analytical solution of active Earth pressure of cohesionless soil using the thin-layer element method [11]. Liu and Chen put forward an expression of active Earth pressure under translational mode by the modified horizontal layer method, which considers the influence of average shear stress between horizontal layers in a sliding wedge of backfill [12].
Only when the displacement of the retaining wall is large enough, can the backfill behind the wall be able to reach the limit equilibrium state represented in the classical theory of Earth pressure. However, in most cases, the wall displacement is not enough, and the backfill behind the retaining wall is often in a nonlimit equilibrium state. According to Chang's relationship between the wall displacement and the backfill friction angle [5] and Fang's experimental results [3], an improved distribution model of friction angle along the wall back is established for different wall movements. Based on Dubrova's method of redistribution of pressure [13], an improved calculation method of active Earth pressure for gravity retaining wall with cohesionless backfill soil rotating outward about the base is put forward, which considers the influence of wall displacement. [13]. Coulomb Earth pressure theory shows that the active Earth thrust acting on a vertical wall with a horizontal backfill of cohesionless soils is P a � cH 2 2 cos δ

Dubrova's Redistribution of Pressure
where c is the soil unit weight, H is the height of the wall, ϕ is the internal friction angle, and δ is the external friction angle between wall and soil. e active Earth pressure can be obtained by equation (1) only when the sliding wedge reaches the limit equilibrium state, and the friction angle on the sliding surface is ϕ.
Dubrova considers that a series of quasi-Coulombsliding surfaces are formed in the backfill which is in the nonlimit equilibrium state, and the mobilized internal friction angle on the sliding surface is ψ.
Replacing ϕ in equation (1) with ψ, the active Earth thrust at any depth z after the wall is P � cz 2 2 cos δ (2) Dubrova points out that the external wall friction angle δ is only related to ϕ and independent of ψ.
Differentiating P in equation (2) with respect to z, the lateral active Earth pressure p a (z) at any depth z will be expressed as where m � (1 + tan δ/tan ψ) 1/2 . Dubrova thinks that m is related to ϕ, and take m � (1 + tan δ/tan ϕ) 1/2 . At the same time, taking (1 + m 2 )/2m � m, which will lead to little loss, (3) is further simplified as For any kind of active movement mode, once the distribution of ψ along the back of the retaining wall is specified, the corresponding distribution of active state Earth pressure can be determined by equation (4).

Distribution of Friction Angle along the Back of the Wall.
For the case of a wall rotating outward about the base, according to Dubrova's redistribution of pressure, the mobilized internal friction angle of backfill at the top of the wall is ϕ, and the limiting active state exists only at the very top. e mobilized internal friction angle of backfill at the bottom of the wall is 0, and at any depth zψ � ϕ(1 − z/H).
at is to say, the distribution of ψ along the back of the wall is linear as shown in Figure 1(a), which takes into account the effect of the mode of wall movement but does not consider the effect of the magnitude.
Chang also assumed that a series of quasi-Coulombsliding surfaces are formed in the backfill behind the wall. e mobilized internal friction angle ψ of the sliding surface is related to the displacement, s, of the intersection point of the sliding surface and retaining wall, as shown in Figure 1(b), where H is the wall height, the critical displacement, s c , is needed for shearing resistance to mobilize fully, uniform along the wall and about 0.0003H, and s a is the maximum wall movement at the top of the wall. For the upper portion AC of the wall in Figure 1(b), the horizontal movement s ≥ s c at any depth z, where 0 ≤ z ≤ z 0 and z 0 � (s a − s c )/s a H, and the mobilized internal friction angle ψ of the sliding surface intersecting the wall at this depth are ϕ.
e horizontal displacement at the bottom of the wall is 0, and the mobilized internal friction angle of the sliding surface intersecting the wall at point B is defined as the initial internal friction angle ϕ 0 . For the lower part CB of the wall in Figure 1(b), the horizontal movement 0 ≤ s ≤ s c at any depth z, where z 0 ≤ z ≤ H, and the variation of the mobilized internal friction angle ψ of the sliding surface with z are linear, which is obtained by linear interpolation between ϕ at point C and ϕ 0 at point B as illustrated in Figure 1 e distribution of internal friction angle along the back of the wall introduced by Chang considers the influence of the mode and magnitude of wall movement [5]. Fang's indoor model test shows that the distribution of Earth pressure at the bottom of the wall gradually converges to Coulomb's active Earth pressure as the retaining wall rotates outward around the wall base [3]. However, Chang considered that when the wall rotates outward around the base, the mobilized internal friction angle on the sliding surface intersecting with the wall at point B always takes the initial internal friction angle value ϕ 0 . erefore, the corresponding calculated Earth pressure distribution acting at point B is always constant, and it cannot embody the process of gradual decrease of Earth pressure at this point which Fang's experiment shows.
Based on Chang's relationship between the mobilized internal friction angle of the backfill and the wall displacement, for the retaining wall rotating outward around the wall base, a modified distribution pattern of the mobilized internal friction angle ψ along the back of the wall considering the influence of displacement is established. Two cases are considered.
e mobilized internal friction angle ϕ A of backfill behind point A at the top of the wall could be calculated by the following equation: e mobilized internal friction angle ϕ B of the sliding surface behind point B at the bottom of the wall could be calculated by the following equation: (6) can also be used to calculate ϕ B when S a > S c . (6) shows that when S a � 0, ϕ B � ϕ 0 , and when S a � ∞, ϕ B � ϕ. It means that as the displacement of the retaining wall rotating outward around the base develops, the mobilized internal friction angle ϕ B of the sliding surface intersecting the bottom of the wall gradually increased from ϕ 0 to ϕ. e mobilized friction angle ψ of the sliding surface at any depth z behind the wall could be obtained by linear interpolation between ϕ A at the top of the wall and ϕ B at the bottom of the wall as follows: e distribution of ψ along the back of the wall is illustrated in Figure 2(a). Equations (5) and (6) show that when S a is known, ϕ A and ϕ B are constants, and the firstorder derivative of ψ in (7) on z at any depth along the wall back is dψ dz Figure 2(b), the displacement, s, of point C is equal to critical displacement, S c , and the mobilized internal friction angle of the sliding surface intersecting the wall at this point is ϕ. e displacement, s, of the upper part AC is not less than S c , and as mentioned above, the mobilized internal friction angle of the sliding surface intersecting with this portion is ϕ. e length, z 0 , of AC is determined as follows:

Mathematical Problems in Engineering
e mobilized internal friction angle ϕ B at point B is still calculated by equation (6). e mobilized friction angle ψ of the sliding surface intersecting with the lower portion BC could be obtained by linear interpolation between ϕ at point C and ϕ B at the bottom of the wall, which is shown as follows: Similarly, when the wall top movement, S a , is known, the first-order derivative of ψ on any depth, z, along the wall back is (8), (11), or (12) into (4) yields the distribution of active state Earth pressure for any wall displacement.

Modified Active State Earth Pressure eory. Substitution of equations
2.3.1. 0 ≤ S a ≤ S c . Substituting (8) into (4), the lateral active Earth pressure will be where ϕ A and ϕ B are calculated by (5) and (6), respectively, and ψ is calculated by equation (7). By (13), the active state Earth pressure distribution behind the wall can be determined under the condition of 0 ≤ S a ≤ S c . Figure 2(b), substituting (11) into (4), the following (14) will be obtained:

S a > S c . For the upper part AC of the wall in
For the lower part CB of the wall, substituting (12) into (4), the following equation will also be obtained: In (15), ϕ B is calculated by (6), and ψ is calculated by equation (10). e first term describes the effect of the magnitude of wall movement on the distribution of Earth pressure, while the second term shows the effect of the mode of wall movement.
When S a > S c , the lateral active Earth pressure behind the upper portion AC of the wall can be calculated by (14), and the lateral active Earth pressure behind the lower part CB can be calculated by equation (15).

Analytical Results and Discussions
Fang investigated the distribution of active state Earth pressure behind retaining wall by indoor model test [3]. Based on Fang's model test data, a comparison of theoretical solutions presented in this paper with model test results was made. e soil used for the model test is air-dried sand. e height of the retaining wall with vertical back H � 1.016 m, the weight of backfill c � 15.34 kN/m 3 , the internal friction angle ϕ � 33.4°, the initial internal friction angle ϕ 0 � 7.5°, and the external friction angle between the retaining wall and soil δ � 16.7°. Figure 3 gives the distribution of active state Earth pressure calculated by the theoretical formula and also shows Fang's indoor model test results.
Comparisons of the curves are shown in Figure 3. When S a /S c � 0, the retaining wall is in a static state, but there is a trend of rotating outward, and the distribution of Earth pressure behind the wall is defined as the initial active state Earth pressure. When S a /S c � 1, only the soil behind the very top of the wall reaches the limit equilibrium state, the shearing resistance of which is mobilized fully. As the wall continues its rotation, S a will increase accordingly, the full mobilization of shearing resistance will propagate downwards, and the Earth pressure acting on the corresponding part of the wall will gradually converge to Coulomb's active Earth pressure. When S a /S c � ∞, the theoretical analysis shows that the entire soil mass behind the wall reaches the limit equilibrium state, and the Earth pressure acting on the wall is equal to Coulomb's active Earth pressure. Both theoretical and experimental results show that with the increase of displacement S a of the top of the wall, the value of Earth pressure at the bottom gradually decreases from the initial Earth pressure to Coulomb's active Earth pressure. However, Chang's method cannot show the process of the gradually decreasing of the active Earth pressure at the bottom of the wall as the displacement S a of the top of the wall increases from 0 to ∞. e theoretical and experimental values of the resultant forces and application points of lateral thrusts for different wall displacements are illustrated in Table 1, in which h 0 /H is the ratio of the height, h 0 , of the total lateral force from the base of the wall to the height, H, of the retaining wall. e data in Table 1 show that as the displacement of the retaining wall increases, the resultant force of Earth pressure from theoretical analysis or model tests decreases gradually and finally converges to Coulomb's active Earth pressure. e height of the application point of the active thrust from the wall base is first decreased and then increased. When the displacement of the retaining wall is large enough, the point of application of the active thrust is located 1/3H above the base of the wall. It can be seen that the displacement of the retaining wall significantly affects the active thrust behind the wall and the position of the point of application. Figure 4 shows the results calculated by the proposed method and the results obtained by Chang's method [5] when S a /S c � 1 and 5. e comparison of the data in the figure shows that the calculation result of Chang's method is relatively large, mainly because Chang's method does not  consider the progressive change process of the play internal friction angle on the potential sliding surface intersecting with the base of the wall during the outward rotation of the retaining wall around the base of the wall. And the value of play internal friction angle can significantly affect the Earth pressure distribution behind the wall.

Conclusions
(1) For a gravity retaining wall with cohesionless backfill soil rotating outward about the base of the wall, Dubrova's method of redistribution of pressure was modified, and an improved method of calculating active Earth pressure was proposed, which can consider the influence of modes and magnitude of wall movement and the progressive rupture mechanisms (2) e calculation method presented in this paper can reflect the developing process of the Earth pressure and shear strength of the soil behind the wall with the increase of the displacement of the retaining wall, and the Earth pressure at the bottom of the wall gradually converges to the Coulomb's active Earth pressure (3) e idea of derivation could also be expanded to develop a method of calculating passive Earth pressure taking into account the influence of deformation patterns

Data Availability
e experimental data used to support the results of this study are included within the article.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.