Unified Triple-Shear Solution for Unsaturated Soil Pressure on Backfill Surface Inclination Under Rainfall Infiltration
XueyeCao1Emailcaoxueye@yau.edu.cn
EryuanZhang1✉Email18834871885@stu.yau.edu.cn
ZhonghuiLi1Emaillizhonghui@stu.yau.edu.cn
JielongSun1Emailsunjielong@126.com
MingmingQiu1Emailsxdfqiuming@163.com
1School of Civil EngineeringYan’an University716000Yan’anChina
Xueye Cao, Eryuan Zhang*, Zhonghui Li, Jielong Sun, Mingming Qiu
School of Civil Engineering, Yan’an University, Yan’an 716000, China; caoxueye@yau.edu.cn, 18834871885@stu.yau.edu.cn, lizhonghui@stu.yau.edu.cn, sunjielong@126.com, sxdfqiuming@163.com
A
Abstract
To refine the theoretical framework of earth pressure calculation and enrich its practical applications, a theoretical analysis of unsaturated Rankine’s earth pressure was conducted. Based on the triple-shear unified solution for unsaturated soil shear strength, the stress circle graphical method was employed to establish the triple-shear unified solution for unsaturated earth pressure on backfill surface inclination under rainfall infiltration. The derived formula was subsequently validated through degeneration analysis and example validation. Furthermore, the effects of rainfall infiltration, backfill surface angle 
on earth pressure were investigated. The results demonstrate that the formulas proposed in this study can be reduced to or approximate various existing formulae, making them applicable to earth pressure calculations under different working conditions. Both active earth pressure 
. With increasing duration of rainfall infiltration 
, , whereas . This study provides a theoretical basis for earth pressure analysis and optimal design of retaining walls, with practical implications for future research and engineering applications.
, soil unsaturation, and the intermediate principal stress parameter
and passive earth pressure
increase linearly with the depth of rainfall infiltration
initially decrease and then increase,
first increase and then decrease, ultimately all stabilizing over time. With an increase in the backfill surface angle
,
increases nonlinearly, while
decreases nonlinearly, and the trends become more pronounced as
grows larger. Moreover, a larger
leads to a greater influence of the unsaturated characteristic parameters
and n on both
and
. Additionally,
decreases with an increase in
increases with an increase in
Keywords:
Unsaturated soil
Rankine's earth pressure
Triple-shear strength criterion
Rainfall infiltration
Backfill surface inclination
1.Introduction
In geotechnical engineering, structures such as slope support, excavation support, and basement side walls are closely related to retaining wall design. The calculation of earth pressure serves as a critical basis for the design of these structures. Earth pressure refers to the lateral pressure exerted by soil on retaining structures. Its distribution and magnitude are significantly influenced by factors such as wall deformation, displacement patterns, properties of backfill soil, soil-structure interaction, and groundwater seepage [1–2]. Accurate calculation of earth pressure acting on retaining walls is paramount importance for ensuring its structural safety [3].
Classical earth pressure theories are widely applied in practical engineering; however, they were developed for saturated soils and cannot be applied to the ubiquitous unsaturated soils. Moreover, their fundamental assumptions—a vertical and smooth wall back with a horizontally infinite soil surface—deviate from realistic conditions, resulting in significant discrepancies with field measurements. Thus, retaining wall designs based on these theories may become uneconomical and potentially unsafe [4–5]. Consequently, researchers worldwide have conducted extensive studies on earth pressure computation, developing numerous theoretical models for the shear strength of unsaturated soils. Among these, Bishop's effective stress principle for shear strength and Fredlund's two stress state variables approach are the most widely adopted, providing a theoretical foundation for calculating unsaturated earth pressure.
As a three-phase medium comprising solid, liquid, and gas, unsaturated soils exhibit mechanical behavior predominantly governed by matric suction, degree of saturation, and intermediate principal stress. Du [6] and Li et al. [7] demonstrated that rainfall infiltration alters matric suction in unsaturated soils, thereby reducing shear strength. Wang [8] derived an earth pressure formula for unsaturated soils under rainfall infiltration conditions based on the Mohr-Coulomb strength criterion, although this formulation does not account for the influence of the intermediate principal stress. Zhao et al. [9–12] analyzed existing experimental data and found that the variation in saturation significantly influences the shear strength of unsaturated soils. For sandy soils and certain clayey soils, the shear strength initially increases then decreases with decreasing saturation. In contrast, most cohesive soils exhibit a continuous increase in shear strength as saturation decreases. Zhao [13] and Zhang [14] demonstrated that the intermediate principal stress significantly influences the mechanical properties of unsaturated soils. Zhao [15] and Wang [16] established a unified solution for unsaturated Rankine earth pressure based on the unified strength theory. Their study demonstrated that the intermediate principal stress parameter significantly influences earth pressure distribution. However, this theory is constrained by issues including the assumption of double-slip angles and an abrupt change in the direction of the slip surface. Cui [17] and Hu et al. [18] utilized the triple-shear strength criterion to analyze earth pressure characteristics. Their results demonstrated that this criterion effectively addresses the double-slip angle issue and more accurately reflects soil strength behavior under true triaxial stress conditions.
Although extensive studies have focused on horizontal backfill conditions, practical engineering applications often involve inclined surfaces due to topographic constraints or construction requirements. As a prevalent engineering boundary condition, backfill surface inclination demonstrate markedly different earth pressure distribution patterns compared to horizontal fill surfaces. Zhang et al. [19–20] derived the Rankine earth pressure formula for cohesive backfill surface inclination behind retaining walls using the graphical stress circle method, accounting for the influence of the intermediate principal stress. Zhou et al. [21] derived an analytical formula for calculating active earth pressure on inclined rigid retaining walls with non-cohesive backfill, incorporating the soil-wall friction angle within the framework of Rankine's earth pressure theory. Liu et al. [22] derived an expression for Rankine earth pressure under backfill surface inclination conditions behind retaining walls based on the unified strength theory, incorporating the influence of the backfill surface angle. However, these studies are predominantly limited to saturated soils, with insufficient attention given to unsaturated soils under backfill surface inclination.
Therefore, this study comprehensively considers the effects of the intermediate principal stress, rainfall infiltration, unsaturated soil properties, and backfill surface angle. Based on the unified triple-shear strength theory for unsaturated soils, a formula for calculating the unsaturated Rankine earth pressure on an backfill surface inclination is derived. The proposed formula is verified through degeneration analysis and comparative case studies. Furthermore, the variation of earth pressure under rainfall infiltration on backfill surface inclination is investigated. The findings provide a theoretical basis for the optimal design of retaining walls.
2. Derivation of the Unified Triple-Shear Solution
2.1 Unified Triple-Shear Formulation for Shear Strength of Unsaturated Soils
The triple-shear strength criterion, which considers the combined action of the three principal shear stresses on a dodecahedral unit element, has been widely adopted in engineering applications [23–27]. Cao et al. [28] established a unified triple-shear solution for the shear strength of unsaturated soils, incorporating the effects of rainfall infiltration. This formulation is applicable to unsaturated soils under diverse complex stress states and overcomes the limitations inherent in the double-shear unified strength theory. The expression is given by:
1
where
2
3
4
5
6
where
are unsaturated characteristic parameters of soil; 
is intermediate principal stress parameter; 
and time 
(for slope angle ≠ 0°);
is rainfall duration; 
is steady-state water table in -direction; -direction; -direction;
is shear strength of unsaturated soil;
is triple-shear effective cohesion;
is triple-shear effective friction angle;
is triple-shear total cohesion;
is effective stress parameter, functionally related to soil saturation;
is net normal stress;
is matric suction;
,
is effective cohesion;
is effective friction angle;
is Lode stress parameter;
is pressure head of groundwater at depth
is complementary error function;
is vertical infiltration rate in
is hydraulic conductivity in
,
is saturated hydraulic diffusivity,
is backfill surface angle;
is gas diffusion coefficient, taken as 1.
2.2 Theoretical Model
Assuming a vertical and smooth wall back, the stress state at a point within the semi-infinite soil mass inclined at angle
to the horizontal plane is illustrated in Fig. 1.
Fig. 1
Stress State Analysis at a Point within Soil Mass under Backfill Surface Inclination
As illustrated in Fig. 1, an infinitesimal soil element, taken at an arbitrary depth in the soil immediately behind the retaining wall, is analyzed. This element is bounded by top and bottom planes parallel to the ground surface and two vertical side planes, with an angle of
between these two sets of bounding planes. Under gravitational loading, equal vertical stresses
act on the top/bottom planes (inclined surfaces), while inclined stresses of equal magnitude
act on the vertical planes. When the wall displaces away from the backfill,
progressively decreases until the element reaches active limit equilibrium, corresponding to active earth pressure
. Conversely, when the wall displaces toward the backfill,
progressively increases until passive limit equilibrium is achieved, yielding passive earth pressure
.
As shown in Fig. 1, neither
nor
acts perpendicular to its respective plane, thus neither constitutes a principal stress. Stress
can represent either active or passive earth pressure, determinable through geometric analysis. This analysis employs the graphical method of Mohr's circle, following the approach in Refs. [19–20], and is illustrated in Fig. 2.
Fig. 2
Mohr's Circle Representation of Stress State at a Point in Backfill Surface Inclination
To determine the stress state in backfill surface inclination, lines
and
are drawn symmetrically about the horizontal
-axis, both forming angle
with the axis. Taking segment
on line
, which represents the stress on the inclined plane of the differential element, Point
must lie on Mohr's circle I corresponding to the stress state of the element. The center
is then located on the
-axis, and Mohr's circle I is constructed to pass through
while remaining tangent to the failure envelope. This circle thus represents the stress state at limit equilibrium. The intersections of Mohr’s circle I with the horizontal
-axis (from left to right : point
and
) represent the stress state of a soil element under horizontal backfill conditions. Circle I intersects the line
at point
, where the vector
corresponds to the stress
on the vertical plane, equivalent to active earth pressure
. Similarly, Mohr's circle II (representing the passive state) intersects
at point
, with the vector
denoting the passive earth pressure
.
2.3 Active Earth Pressure
From the geometric relationship between Mohr's circle I and the failure envelope illustrated in Fig. 2, the following expression can be obtained:
7
The governing equation is established based on the geometric configuration between Mohr's circle I and line segment
, yielding:
8
Equation (8) constitutes a quadratic equation in terms of
, yielding two roots:
9
10
These two roots correspond to the abscissas of points
and
, which are the intersection points between Mohr's circle I and line segment OH. As geometrically demonstrated in Fig. 2:
11
By combining Equations (7) through (11), the following expression is derived:
12
where
, which shares the same functional form as the classical Rankine solution but differs in the values assigned to cohesion and the active earth pressure coefficient;
represents the Rankine active earth pressure coefficient for unsaturated soils, given by
.
2.4 Passive Earth Pressure
From the geometric relationship between Mohr's circle II and the failure envelope illustrated in Fig. 2, the following expression for passive earth pressure can be derived:
13
From the geometric relationship between Mohr's circle II and the failure envelope illustrated in Fig. 2, the following expression for passive earth pressure can be derived:
14
where
, identical in functional form to the classical Rankine solution but differing in the values assigned to cohesion and friction angle. This formulation incorporates the effects of intermediate principal stress and matric suction;
represents the Rankine passive earth pressure coefficient for unsaturated soils, expressed as
.
2.5 Degeneration Analysis
Equations (12) and (14) represent the Rankine active and passive earth pressure formulations, respectively, incorporating the combined effects of intermediate principal stress, rainfall infiltration, soil unsaturation, and backfill surface angle. In these equations, parameters
and
account for intermediate principal stress effects; Terms
and
characterize rainfall infiltration and unsaturated soil behavior; The angle
quantifies backfill surface inclination influence. The proposed formulations degenerate or converge to multiple established solutions under specific limiting conditions.
When
(horizontal backfill surface), Equations (12) and (14) degenerate to:
15
While sharing identical functional form with Reference [16], this formulation is fundamentally distinguished by its theoretical framework: Reference [16] employs the unified strength theory to account for intermediate principal stress effects through parameters
,
and
, while modeling rainfall and evaporation via generalized matric suction
; conversely, the present study adopts the triple-shear strength criterion to characterize intermediate principal stress influence through parameters
,
and
, while explicitly incorporating rainfall infiltration mechanisms in unsaturated soils via the
.
Under the dual conditions of
(horizontal backfill) and
(neglecting rainfall infiltration), Equations (12) and (14) degenerate to:
16
While functionally analogous to Reference [34], this formulation distinguishes itself through fundamental theoretical advancements: Reference [34] derives cohesion and friction angle parameters from the unified strength theory, whereas the present study determines these parameters via the triple-shear strength criterion—thereby eliminating the double-slip angle limitation inherent in unified theory while further incorporating backfill surface angle effects that are neglected in Reference [34]'s horizontal backfill assumption.
When
,
,
,
,
, the formulation degenerates to
17
Equation (17) constitutes the classical Rankine earth pressure formulation.
3 Example Validation and Discussion of Influencing Factors
3.1 Example Validation
The case study adopts the benchmark configuration from Reference [8]: a retaining wall of height
with vertical and smooth back face; soil properties
,
,
,
,
; hydrogeological conditions featuring a steady-state water table at
depth and rainfall duration
with sufficient intensity (
). The earth pressure analysis is treated as a plane strain problem, yielding
and
per standard geotechnical assumptions.
When determining the rainfall infiltration depth
, the results can be obtained from Eq. (12) and Eq. (14), and the comparison with literature results is presented in Table 1.
Earth Pressure | Varying Conditions | b | |||||
0 | 0.25 | 0.5 | 0.75 | 1 | |||
(Rainfall + Triple-Shear Strength Criterion + Unsaturated Soils) | 23.93 | 19.79 | 16.60 | 14.07 | 12.02 | ||
24.12 | 19.92 | 16.70 | 14.15 | 12.09 | |||
24.70 | 20.34 | 17.02 | 14.40 | 12.29 | |||
Ref. [19] (M-C Criterion + Saturated Soils) | 21.17 | ||||||
21.87 | |||||||
24.00 | |||||||
Ref. [20] (Unified Strength Theory + Saturated Soils) | 23.98 | 20.97 | 18.75 | 17.04 | 15.69 | ||
24.33 | 21.25 | 18.98 | 17.24 | 15.86 | |||
25.47 | 22.14 | 19.72 | 17.87 | 16.42 | |||
Ref. [29](Unified Strength Theory + Unsaturated Soils) | 23.68 | 20.65 | 18.42 | 16.70 | 15.34 | ||
Classical Theory (M-C Criterion + Saturated Soils) | 23.98 | ||||||
(Rainfall + Triple-Shear Strength Criterion + Unsaturated Soils) | 153.64 | 166.37 | 177.32 | 186.84 | 195.19 | ||
152.50 | 165.18 | 176.09 | 185.57 | 193.89 | |||
149.06 | 161.60 | 172.39 | 181.77 | 189.99 | |||
Ref. [19] (M-C Criterion + Saturated Soils) | 153.56 | ||||||
150.26 | |||||||
140.46 | |||||||
Ref. [20] (Unified Strength Theory + Saturated Soils) | 152.53 | 161.30 | 168.30 | 174.01 | 178.76 | ||
149.26 | 158.07 | 165.08 | 170.79 | 175.54 | |||
139.58 | 148.51 | 155.57 | 161.30 | 166.05 | |||
Ref. [29](Unified Strength Theory + Unsaturated Soils) | 154.07 | 163.19 | 170.49 | 176.45 | 181.43 | ||
Classical Theory (M-C Criterion + Saturated Soils) | 153.56 | ||||||
As can be observed from Table 1, with an increase in 
, increases from 0 to 1, increases from 0 to 1, the increase in
gradually decreases while
gradually increases. As the backfill surface angle
increases,
gradually increases while
gradually decreases. When
and
, the earth pressures obtained in this study (
,
) are in close agreement with the classical Rankine solution (
,
). When
and
in this study decreases by 49.77%, compared to decreases of 34.57% in Reference [20] and 35.22% in Reference [29]. Moreover, our results yield relatively smaller values, indicating that the triple-shear strength criterion more fully mobilizes the intermediate principal stress effect in soil. This leads to a reduction in the active earth pressure design load, optimizing the cross-section design of retaining walls and consequently lowering construction costs. The passive earth pressure exhibits the opposite trend: the
in this study increases by 27.04%, compared to increases of 17.20% in Reference [20] and 17.76% in Reference [29]. This is attributed to the triple-shear strength criterion accounting for the combined action of three pairs of principal shear stresses on the dodecahedral unit cell, thereby more fully mobilizing the strength potential of unsaturated soils in the passive zone. When
ranges from 26.23kPa to 41.55kPa. When
increases from 0° to 10°, the decrease in
ranges from 4.58kPa to 13.10kPa. When the soil transitions from unsaturated to saturated conditions, the increase in
ranges from 1.54kPa to 2.67kPa. When rainfall infiltration is considered, the fluctuation in
ranges from 0 to 1 kPa. Consequently, the Rankine earth pressure parameter sensitivity, ranked in descending order, is: intermediate principal stress parameter > backfill slope angle > soil unsaturation > rainfall infiltration.
3.2 Discussion of Influencing Factors
3.2.1 Effect of Rainfall Infiltration
To compare with classical Rankine earth pressure solutions, when 
under different backfill surface angles
and
, the variation of unsaturated active earth pressure
and passive earth pressure
with rainfall infiltration depth
are shown in Fig. 3.
Fig. 3
Variation of earth pressures with rainfall infiltration depth z
As observed from Fig. 3, both . Compared with classical earth pressures, increases, the influence of increases from 5m to 7m, exhibiting more pronounced influence than
and
increase linearly with increasing
is lower while
is higher. Due to the existence of the critical depth
,
exhibits a triangular distribution along the wall height with its resultant force acting at
, whereas
shows a trapezoidal distribution with the resultant force at the centroid of the trapezoid, consistent with classical earth pressure distribution trends. When
, the influence of
on
is negligible. Moreover, when
(i.e., at the surface layer of the backfill), its impact on
remains minimal. As
on both
and
progressively intensifies, indicating that the backfill surface angle exerts greater effects on deeper soil layers. At
, when
increases from 0° to 20°,
increases by 5.22kPa while
decreases by 35.11kPa. At
, when
increases by 18.10kPa and
increases by 71.78kPa. This demonstrates that both backfill surface angle
and depth z significantly affect
, with
.
When 
under different backfill surface angles
and
, the variation of unsaturated earth pressures with rainfall infiltration time
is shown in Fig. 4.
Fig. 4
Variation of earth pressures with rainfall infiltration time t
As observed from Fig. 4, when rainfall duration is constant, earth pressures vary nonlinearly with infiltration time 
. Fluctuations are observed during the initial stage of rainfall and infiltration. This phenomenon is attributed to the variation in pore-air pressure induced by rainwater infiltration, which subsequently alters the matric suction and effective stress parameters of the unsaturated soil, resulting in corresponding changes in soil pressure. Under different backfill surface angles
, the variation trends of earth pressures are generally consistent:
initially decreases then increases, while
first increases then decreases, both eventually stabilizing. As rainfall infiltration progresses, the increased water content in the unsaturated soil reduces interparticle frictional resistance and cohesion, consequently diminishing soil shear strength. This results in an increase in
and a decrease in
. After sustained infiltration, as the water content stabilizes, both A and B also tend to stabilize. When rainfall ceases at
, earth pressures stabilize at
. The cessation of rainfall exhibits a hysteresis effect due to the delayed response of soil suction during hydraulic redistribution, leading to dynamic earth pressure adjustments. As these adjustments persist post-rainfall, continuous safety monitoring of retaining walls remains necessary in engineering practice.
3.2.2 Effect of Backfill Surface Angle
When
,
, and
, the variation of unsaturated active earth pressure
and passive earth pressure
with backfill surface angle
is shown in Fig. 5.
Fig. 5
Variation of earth pressures with backfill surface angle
As observed from Fig. 5,
increases nonlinearly with increasing backfill surface angle
, while
decreases nonlinearly. When
rises from 0° to 25°, the growth rates of
are 0.02, 0.08, 0.13, 0.21, and 0.31 kPa/° respectively, with progressively increasing rates. The reduction rates of
are 0.32, 0.95, 1.61, 2.31, and 3.10 kPa/° respectively, also accelerating. Notably,
exerts greater influence on
than on
. This indicates that higher backfill surface angles intensify earth pressure variations, with abrupt rate changes occurring when
. This explains why retaining walls in mountainous areas frequently fail at moderate-to-steep slopes. Consequently, backfill surface angle should be strictly controlled in design, and terraced retaining walls are recommended when angle limits are exceeded.
3.2.3 Effect of Soil Unsaturation
When
are shown in Figs. 6 and 7, respectively.
,
, and
, the variations of active earth pressure
and passive earth pressure
with
and
Fig. 6
Variation of earth pressures with unsaturated characteristic parameter
As observed from Fig. 6, with increasing
,
increases nonlinearly while
decreases nonlinearly. When
and
rises from 0.00 to 0.05,
increases by 7.83kPa, 6.94kPa, 2.74kPa, and 1.32kPa respectively, with progressively decreasing rates. Meanwhile,
decreases by 26.91kPa, 38.30kPa, 13.69kPa, 5.38kPa, 2.58kPa, and 1.42kPa respectively. Lower
values amplify the matric suction effect in unsaturated soils. When
>0.05, changes in both
and
remain within 1kPa, curves gradually flattening as suction contribution approaches saturation. Under different backfill surface angles
, the variation trends of
and
with increasing
are fundamentally consistent, with higher
intensifying these changes. Significant pressure variations occur across different backfill surface angles when
differs. Therefore, in practical engineering,
should be maintained within an appropriate range based on backfill conditions.
Fig. 7
Variation of earth pressures with unsaturated characteristic parameter 
As observed from Fig. 7, with increasing n,
initially increases then stabilizes, while
first decreases then stabilizes. Under
, when n increases from 2.0 to 3.0,
increases by 3.49kPa, 1.18kPa respectively, and
decreases by 6.83kPa, 2.31kPa. When n further rises from 3.0 to 4.0,
increases by 0.40kPa, 0.13kPa, while
decreases by 0.78kPa, 0.26kPa. This indicates that lower n values exert more significant influence on both
and
. At n = 3.0, when
increases from 0° to 20°,
increases by 0.50kPa, 1.71kPa respectively, while
decreases by 6.37kPa, 19.68kPa. Consequently, higher backfill surface angles amplify the effects on earth pressures. Therefore, in practical engineering, n should be strictly controlled based on soil unsaturation characteristics.
3.2.4 Effect of Intermediate Principal Stress Parameter b
When 
are shown in Figs. 8 and 9, respectively.
, the variations of active earth pressure
and passive earth pressure
with intermediate principal stress parameter
Fig. 8
Variation of earth pressures with b under different backfill surface angle
values: (a) active earth pressure; (b) passive earth pressure.
As observed from Fig. 8, for different while increases from 0 to 1, = 0, 0.5, and 1, when diminishes the influence of
values,
consistently decreases with increasing
increases. At
,
, and
, when
decreases by 40.61%, 41.23%, and 44.01% respectively. Concurrently,
increases by 26.13%, 26.54%, and 28.19% respectively. As
increases, both
and
exhibit progressively increasing trends. This indicates that the backfill surface angle enhances the intermediate principal stress effect in soils, and considering this effect allows better mobilization of the strength potential in unsaturated soils. At
increases from 0° to 20°,
increases by 36.59%, 22.66%, and 18.56% respectively, while
decreases by 20.27%, 18.67%, and 17.75% respectively. Thus, increasing
on both
and
, as heightened intermediate principal stress effects counteract the impact of backfill surface angle.
Fig. 9
Variation of earth pressures with infiltration depth z under different b values
As observed from Fig. 9, for different and 
. At and increases, the slope of increases to 1, resultant increases, the active earth pressure decreases and its point of application moves downward, thereby enhancing the stability of the retaining wall; conversely, the passive earth pressure increases and its point of application rises, which compromises the stability of the wall. Therefore, engineering practice requires comprehensive evaluation of the coupled effects between intermediate principal stress and backfill surface angle on the stress state of the backfill soil.
values, earth pressures increase linearly with depth
, when
and 1, the cracking depths
are 1.741m and 2.678m respectively, while at
,
measures 1.645m and 2.598m. This indicates that increasing b significantly enlarges cracking depth in the tensile zone, whereas
exerts relatively minor influence. Deeper zones exhibit more pronounced effects of
due to confining pressure intensifying intermediate principal stress and backfill surface angle impacts. The slope of the linear fit indicates that as
decreases while that of
increases;
variations cause negligible slope changes. At
and
,
, the resultant active pressures are 146.42kPa and 184.51kPa with application points 1.753m and 1.785m above the wall base. Resultant passive pressures are 968.61kPa and 850.95kPa at 2.597m and 2.616m. When
becomes 82.678kPa and 95.696kPa at 1.441m and 1.467m, while
reaches 1237.15kPa and 1104.27kPa at 2.649m and 2.650m. In other words, with increasing backfill surface angle
, active earth pressure increases, its application point rises, thereby compromising the overturning stability of the retaining wall. Conversely, passive earth pressure decreases, its application point rises, collectively enhance the overturning resistance of the retaining wall. As the value of
4. Conclusions
1) The unsaturated Rankine earth pressure triple-shear unified solution established in this paper comprehensively incorporates the influences of intermediate principal stress, rainfall infiltration, soil unsaturation, and backfill surface inclination. It can degenerate to or approximate various existing formulas, making it suitable for calculating earth pressure under different working conditions.
2) Both active earth pressure
and passive earth pressure
increase linearly with the depth of rainfall infiltration z. Under different backfill surface angles, the variation trends of earth pressures with rainfall infiltration time t are generally consistent:
first decreases and then increases, while
first increases and then decreases, both eventually stabilizing over time. There is a certain time lag after rainfall cessation. In practical engineering, safety monitoring of retaining walls should be continued even after rainfall has stopped.
3)
increases nonlinearly with the rise of backfill surface angles
, while
decreases nonlinearly. The larger the value of
, the more significant the change in earth pressure becomes. Therefore, the backfill surface angles must be strictly controlled in the design of retaining walls. When the backfill surface angle exceeds the specified limit, the use of a stepped retaining wall system is recommended.
4) With the increase of 
, the more pronounced the influence of 
becomes. Therefore, in practical engineering applications, the unsaturated characteristic parameters should be prudently selected based on the unsaturated properties of the soil.
,
increases nonlinearly while
decreases nonlinearly. As n increases,
initially rises and then plateaus, whereas
first declines and subsequently stabilizes. The larger backfill surface angle
and
on
and
and
5) increases, the resultant active earth pressure decreases with a lowered application point, whereas the resultant passive earth pressure increases with a raised one. With increasing
decreases with increasing b, while
increases. As
, the resultant active earth pressure increases with a raised application point, while the resultant passive earth pressure decreases, also with a raised application point. Therefore, in engineering practice, it is essential to comprehensively evaluate the coupled effects of the intermediate principal stress and the backfill slope angle on the stress state of the soil behind the wall.
A
Author Contribution
X.C.: Validation, writing—review and editing, Formal analysis, Supervi-sion; E.Z.: Writing-original draft, Data curation, Investigation, Formal analysis; Z.L.: Validation; J.S.: data curation; M.Q.: Supervision. All authors have read and agreed to the published version of the manuscript.
A
Funding:
This research was funded by the National Natural Science Foundation of China (grant nos. 12462032, 12102379).
A
Data Availability
Data is provided within the manuscript . Further inquiries can be directed to the corresponding author.
Conflicts of Interest:
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Author Identification
Xueye Cao is currently a Lecturer and Master's Supervisor at the School of Civil Engineering, Yan’an University. She received her Bachelor of Engineering in Civil Engi-neering in 2012 and her Ph.D. in Structural Engineering in 2017, both from Chang’an University, where she completed a combined Master’s and Doctoral program. Since December 2017, she has been affiliated with Yan’an University. Her current research focuses on strength theory and its applications, as well as geotechnical engineering. Dr. Cao can be contacted via email at cao-xueye@yau.edu.cn or by phone at + 86 15002964015.
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