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Buildings 2024, 14, 1533. https://doi.org/10.3390/buildings14061533 www.mdpi.com/journal/buildings
Article
Bearing Capacity and Reinforced Mechanisms of
Horizontal–Vertical Geogrid in Foundations: PFC
3D
Study
Jinjun Wu
1
, Fabin Zhang
1
, Liang Gao
1
and Juan Hou
2,
*
1
Qinghai Transportation Planning and Design Institute Co., Ltd., Xining 810016, China
2
School of Mechanics and Engineering Science, Shanghai University, Shanghai 200444, China
* Correspondence: juanhou@staff.shu.edu.cn
Abstract: The study presents a novel meshed horizontal–vertical (H–V) geogrid, offering promising
advancements in geotechnical structure performance. The study pioneers a modeling approach for
H–V geogrid foundation bearing capacity with discrete element method, expanding understanding
and optimizing design strategy. By analyzing the granular displacement, contact force distribution,
and vertical stress distribution within the foundation system, the study examines the impact of bur-
ial depth, vertical element height, and the number of vertical elements on H–V reinforced founda-
tions. The findings suggest that employing a burial depth equivalent to the width of the footing
enhances bearing capacity compared to conventional geogrid applications, with depths set at 0.4
times the width of the footing. This enhancement is aributed to forming a deeper slip surface in
H–V systems. Moreover, raising vertical elements to 0.6 times the width of the footing enhances
bearing capacity with minimal increase in geogrid usage, indicating a strategic approach to rein-
forcement. Increasing the number of vertical elements, particularly with three pairs, significantly
enhances bearing capacity by reinforcing lateral restraint on the soil and promoting stress homoge-
nization, thereby augmenting the “deep-footing” effect. The technical analysis underscores the effi-
cacy of H–V geogrids in bolstering the bearing capacity of reinforced foundations, which is at-
tributed to the robust grip and interlocking mechanism facilitated by these geogrids’ vertical ribs
and mesh structure, which augment lateral confinement and diminish horizontal soil displacement.
Keywords: horizontal–vertical geogrid; reinforced foundation; PFC
3D
; bearing capacity; mechanism
1. Introduction
The field of geotechnical engineering has long been concerned with the stability and
load-bearing capacity of soil structures reinforced with geogrids [1–5]. The interface
strength between the geogrid and soil is a critical factor in determining the effectiveness
of such reinforcement systems [6–8]. Insufficient interface strength can lead to premature
failure, posing risks to the stability of structures [9–11]. Researchers have explored various
strategies to enhance reinforcement effectiveness to address challenges associated with
inadequate geogrid–soil interface strength [12–15]. Chen et al. [16] have indicated that
geogrids can significantly enhance peak shear strength by restricting lateral expansion
within the reinforced area, effectively reducing the sensitivity of gravel to volumetric ex-
pansion. The research findings conducted by Poorahong et al. [17] indicate that the num-
ber of geogrid layers and their placement significantly affect the ultimate bearing capacity
and failure mechanism of reinforced pavements. Furthermore, the results by Du et al. [18]
show that as the number of geogrid layers and their width increase, there is a correspond-
ing increase in the ultimate bearing capacity of the embankment.
One promising avenue is the development of three-dimensional reinforcement tech-
niques. These techniques involve extending reinforcement both horizontally and verti-
cally within the soil mass, offering the potential for improved interaction with the
Citation: Wu, J.; Zhang, F.; Gao, L.;
Hou, J. Bearing Capacity and
Reinforced Mechanisms of
Horizontal–Vertical Geogrid in
Foundations: PFC3D Study.
Buildings 2024, 14, 1533. hps://
doi.org/10.3390/buildings14061533
Academic Editor: Eugeniusz Koda
Received: 25 April 2024
Revised: 15 May 2024
Accepted: 23 May 2024
Published: 25 May 2024
Copyright: © 2024 by the authors.
Submied for possible open access
publication under the terms and
conditions of the Creative Commons
Aribution (CC BY) license
(hps://creativecommons.org/license
s/by/4.0/).
Buildings 2024, 14, 1533 2 of 16
surrounding soil. Among the recent innovations in this area is introducing a new three-
dimensional horizontal–vertical (H–V) geogrid, as depicted in Figure 1. Experimental and
theoretical studies have demonstrated the superior interaction strength between H–V ge-
ogrids and soil compared to conventional geogrids [19]. However, despite these advance-
ments, there is still a limited understanding of the reinforcement mechanism of H–V ge-
ogrids. This study aimed to fill this knowledge gap by investigating the bearing capacity
and reinforcement mechanism of foundations reinforced with H–V geogrids. Leveraging
the advanced capabilities of Particle Flow Code in three dimensions (PFC
3D
), an analysis
of the complex behavior of H–V geogrids within the soil matrix was conducted.
Figure 1. Photographs of the H–V geogrid.
The study introduces a newly developed meshed H–V geogrid, which is promising
to enhance geotechnical structures’ performance and stability. The study also employs a
novel approach using Particle Flow Code in three dimensions (PFC
3D
) to model the bear-
ing capacity of H–V geogrid foundations, thereby expanding understanding of their be-
havior under load and offering insights into optimizing foundation design and perfor-
mance. PFC
3D
outperforms traditional methods like FEM and FDM for handling large de-
formations. Cheng et al. (2022) [20] highlighted CEL as an efficient approach. The discrete
element modeling accurately captures particle interactions, which are crucial for granular
materials, but it requires precise parameter calibration and entails computational costs.
The findings of this study are expected to contribute to the advancement of more robust
and reliable reinforcement solutions for civil infrastructure projects.
2. Methodology
2.1. DEM Model
The study utilizes PFC, a widely recognized method in geotechnical engineering re-
search grounded in Newton’s second law and force-displacement principles. This ap-
proach iteratively updates contact forces and particle positions through a temporal step-
ping process, applying these principles alternately [21].
The PFC model of the H–V geogrid-reinforced foundation is illustrated in Figure 2.
Establishing this PFC model involved four main steps. Initially, the boundaries were es-
tablished by five walls. Secondly, gravity deposition generated balls, representing the soil
in a randomized distribution. Thirdly, the H–V geogrid incorporated stiffness and parallel
bonds capable of transmiing tensile force and moment. Lastly, a steep nine-gradient
loading, referencing experimental conditions [22,23], was applied to the plate, represent-
ing the strip footing. Microparameters of the PFC models were determined through iter-
ative calculations in numerical tests, employing a trial-and-error approach [24,25]. Con-
currently, particle microparameters were continuously adjusted to align with the macro
phenomena observed in model tests, as determined through soil static load tests and ge-
ogrid tensile and bending tests [26–28]. Meanwhile, Chen et al. (2024) [29] and Cundall
and Strack (1979) [30] emphasized the importance of conducting convergent analyses on
parameters like mesh sizes in FEM or particle size in PFC to represent system behavior
while optimizing computational resources accurately. The radius of the actual soil particle
was increased by tenfold (4 mm) in the PFC model, compared to the model test, which
Buildings 2024, 14, 1533 3 of 16
was informed by insights from prior studies and convergence analyses by Zhao et al.
(2015) [31] and Feng et al. (2022) [32]. The stiffness of the loading plate was set at ten times
the stiffness of the soil granules [33,34]. The loading velocity was seled at a uniform
speed of 2 mm/min during the tests, according to Saha Roy and Deb (2017a, 2017b) [35,36].
The selection of geogrid was rooted in the model experimental findings, with a specific
focus on the steel–plastic composite geogrid [37]. All other essential mesoscopic parame-
ters for simulation in this study are summarized in Table 1.
Figure 2. PFC
3D
model of H–V reinforced foundation.
Table 1. Summary of mesoscopic parameters in PFC.
Parameter Sand Geogrid Model Box Footing
Normal stiffness (N/m) 5×10
5
6 × 10
7
5 × 10
5
5 × 10
6
Shear stiffness (N/m) 5 × 10
5
1 × 10
7
5 × 10
5
5 × 10
6
Density (kg/m
3
) 2.63 × 10
3
3 × 10
3
– 2 × 10
3
Particle radius (m) 0.4 × 10
−2
2.5 × 10
−3
– 0.5 × 10
−2
Friction coefficient 0.7 0.5 0.7 0
Normal contact bond strength (N) 0 1 × 10
4
– –
Shear contact bond strength (N) 0 1 × 10
4
– –
Parallel bond gap 0 10 – –
Shear parallel bond strength (Pa) 0 1 × 10
8
– –
Normal parallel bond strength (Pa) 1 × 10
8
Shear parallel bond stiffness (Pa/m) 3 × 10
6
Normal parallel bond stiffness (Pa/m) 6 × 10
7
Table 2 summarizes all the test cases conducted in this study. Tests 1 and 2 were de-
signed to investigate the influence of the buried depth of the H–V geogrid on the bearing
capacity of foundations. Tests 1 and 4 were devised to scrutinize the effect of the number
of vertical elements in the H–V geogrid on the bearing capacity of the foundation. Tests 2
and 3 were structured to assess the impact of the height of the vertical element in the H–
V geogrid on the bearing capacity of foundations.
Table 2. Summary of study’s cases.
No. z/B L/B l/B v/B
1 0.4
2.7
1.5 0.3
2 1.0 1.5 0.3
3 1.0 1.5 0.6
4 0.4 0.3, 1.5, 2.7 0.3
Notes: The loading plate (B), which simulates the footing, was 0.15 m in width, 0.598 m in length,
and 0.025 m in height. The total depth of the H–V geogrid layers is z, the length of reinforcement is
L, the distance between the two vertical elements is l, and the vertical geogrid height is v, which is
based on our experimental findings [37].
Buildings 2024, 14, 1533 4 of 16
2.2. Boundary Condition
To expedite the PFC simulation process while staying within practical computational
limits, the numerical model was streamlined by halving the dimensions of the physical
specimen. This decision leveraged the inherent symmetry of the H–V geogrid-reinforced
foundation model, as advocated by Wang et al. (2020) [38]. Additionally, insights from
model tests indicate that soil particle displacement remained insignificant beyond 2.3
times the footing width (B) and a depth surpassing 2.3B [37]. Consequently, five walls
with equivalent normal and shear stiffness to the soil particles were integrated as bound-
ary conditions to address this issue. Figure 3 illustrates the details of the boundary condi-
tions. The loading plate measured 0.15 m (width) × 0.598 m (length) × 0.025 m (height),
while the box dimensions were 0.125 m (width) × 0.35 m (length) × 0.6 m (height). This
strategic substitution effectively minimized soil particle displacement and reduced com-
putation time.
Figure 3. Schematic diagram of sand bed reinforced with multilayer H–V geogrids.
2.3. Parameters Calibration
The soil particle parameters underwent calibration by establishing an unreinforced
foundation model in PFC. Through iterative adjustments of these parameters, the ob-
tained P-s curve from numerical simulations closely matched that of model experiments
(Figure 4a). Subsequently, the tensile parameters of the geogrid were calibrated using a
PPFC geogrid tensile test model. Tensile specimens, 200 mm long, were subjected to a
stretching rate of 2 mm/min. Fine-tuning the geogrid’s microscopic parameters enabled
the simulated sample’s tensile curve to closely align with the physical test results (see
Figure 4b). The close agreement between the numerical and experimental results suggests
that the calibrated PFC model parameters accurately reflect the tensile performance of the
geogrids (Figure 4b). Additionally, parameters governing the bending characteristics of
the geogrid were calibrated through multiple sets of bending tests (Figure 4c). After cali-
brating all these parameters, they were input into the PFC model of the H–V geogrid-
reinforced foundation.
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(a) (b) (c)
Figure 4. Parameter calibration results: (a) the P–s curve of unreinforced foundation; (b) experi-
mental and simulated tensile force–strain relationships of geogrid in a tensile test; (c) vertical dis-
placement–time step relationships of DEM geogrid in bending test.
3. Results and Discussion
3.1. P–s Curves
Figure 5 depicts the pressure–selement (P–s) curve relationship using closed sym-
bols for all four simulated cases. The experimental test results, represented by open sym-
bols and consistent with case 1, are provided for comparison. The observed alignment
between experimental and simulation data indicates a firm agreement. The findings une-
quivocally demonstrate that increasing the H–V geogrid’s buried depth enhanced the
bearing capacity of foundations, as evidenced by comparing the red and blue lines. Foun-
dations buried at a depth of 1B demonstrated beer bearing capacity than those buried at
a depth of 0.4B, which has been identified as the optimal value in conventional geogrid–
reinforced foundations [13,39]. Moreover, increasing the height of the vertical elements
also played a significant role in improving bearing capacity. For example, foundations
with vertical elements at 0.6B height demonstrated a 1.2 times greater bearing capacity
than those with elements at 0.3B height, as indicated by the comparison between the black
and red lines, with minimal increase in geogrid consumption. Furthermore, increasing the
number of vertical elements led to a significant enhancement in bearing capacity. For in-
stance, foundations equipped with three pairs of vertical elements exhibited a substantial
1.3 times increase compared to those featuring only one pair, as demonstrated by compar-
ing the green and blue lines. It should be noted that this enhancement required only a
modest 12% increase in geogrid consumption.
Figure 5. The P–s curve of reinforced foundations with different H–V geogrid arrangement.
0 20 40 60 80 100 120 140 160 180 200
-16
-14
-12
-10
-8
-6
-4
-2
0
s/mm
P/kPa
No. z/B l /B ν/B
1 0.4 1.5 0.3
2 1.0 1.5 0.3
3 1.0 1.5 0.6
4 0.4 0.3, 1.5, 2.7 0.3
Experimental data
−
−
−
−
−
−
−
−
Buildings 2024, 14, 1533 6 of 16
3.2. Displacement in the H–V Geogrid-Reinforced Foundation
Figure 6 illustrates the displacement distribution of the reinforced foundation with
the H–V geogrid at embedment depths of 0.4B and 1.0B, respectively. Magnified views of
areas A and B were further enlarged to facilitate the analysis of soil granule displacement.
Comparing the orange solid arrow in Figure 6a to the dashed arrow in Figure 6b shows
that increasing the embedment depth of the H–V geogrid resulted in a deeper slip surface
for the footing. This is aributed to the upper soil behaving as a softer layer compared to
the lower soil and significant deformation induced by shallow embedment depths poten-
tially warping the soil surface and affecting the effectiveness of the vertical geogrid, espe-
cially near the basement. Furthermore, comparing the magnified views of Areas A and B
in Figure 6c,d reveals significant differences in soil displacement paerns. When the em-
bedment depth was 0.4B, soil deflection was observed below the H–V geogrid, gradually
forming a prominent shear zone as illustrated by the orange solid arrow. However, when
the embedment depth was 1B, soil predominantly moved vertically downward, with mi-
nor lateral movement towards the sides of the vertical geogrid, as illustrated by the orange
dashed arrows. This phenomenon can be aributed to the confinement effect exerted by
the vertical elements, suggesting that augmenting the embedment depths of the vertical
H–V geogrid can bolster the shear strength of the soil, thereby potentially mitigating shear
failure surfaces. This highlights that the H–V geogrid not only influences soil movement
in its immediate vicinity but also alters the trajectory of soil over a wider area.
Figure 6. Displacement distribution of reinforced footings with different H–V geogrid arrange-
ments: (a) z = 0.4B; (b) z = 1.0B; (c) area A; (d) area B.
Figure 7a,b showcase the displacement distribution with vertical element heights of
0.3B and 0.6B in the H–V geogrid. The soil displacement with 0.6B height vertical elements
(Figure 7b) displayed minimal variation and improved uniformity compared to that with
0.3B height vertical elements in Figure 7a. Further comparative analysis between the two
Buildings 2024, 14, 1533 7 of 16
figures reveals that the soil in Figure 7a exhibits a more pronounced slip surface (see the
orange solid arrow) than that in Figure 7b (see the orange dashed arrow). This suggests
that increasing the vertical element height mitigated soil displacement, aiding in the ad-
justment of uneven selement and delaying the formation of slip surfaces. The magnified
views of areas A and B in Figure 7c,d further highlight that soil within 0.3B height vertical
elements tended to move upward to the right in Figure 7c, whereas, in Figure 7d, displace-
ment was bifurcated, with some granules moving upward to the right and the majority
moving downward to the right. This distinction underscores how the vertical elements of
H–V geogrid alter the direction of soil movement, leading to a broader dispersion of soil
granules. Consequently, this slows down the formation of slip surfaces and enhances the
“deep-footing” effect of the H–V geogrid, as suggested by Huang et al. (1997) [40].
Figure 7. Displacement distribution of reinforced foundations with different numbers and heights
of vertical geogrids: (a) No. 2 ν = 0.3B; (b) No. 3 ν = 0.6B; (c) area A; (d) area B.
Figure 8a,b illustrate the displacement distribution with one pair and three pairs of
vertical elements in the H–V geogrid, respectively. A comparison between the two figures
shows that with three pairs of vertical elements (Figure 8b), there were notably smaller
soil displacement values and surface uplift than the case with only one pair of vertical
elements (Figure 8a). This indicates that increasing the number of vertical elements slowed
down the formation of slip surfaces, thereby enhancing the bearing capacity of the foun-
dations. Additionally, the deformation and inclination angle of the vertical elements are
less pronounced in Figure 8b than in Figure 8a (compare solid and dashed orange arrows).
This difference arose because, despite the substantial pressure on the upper part of the
vertical element, the minimal difference in soil stress on both sides prevented lateral slip-
ping. Conversely, the vertical element at the edge experienced minimal influence from soil
movement due to its distance from the footing. These observations highlight the
Buildings 2024, 14, 1533 8 of 16
significant impact of the number and positioning of vertical elements in the H–V geogrid.
The magnified views of areas A and B in Figure 8c,d further provide insights into the
observation that the included angle between the movement trajectory of soil granules near
the vertical elements and the vertical direction is smaller in Figure 8d compared to Figure
8c (compare solid and dashed orange ellipses). This difference is aributed to the in-
creased number of vertical elements in Figure 8d, which exerted a more notable lateral
restraint on the soil, directing its movement vertically.
Figure 8. Displacement distribution of reinforced foundations with different numbers and heights
of vertical geogrid: (a) No. 1, 1 pair; (b) No. 4, 3 pairs; (c) area A; (d) area B.
3.3. Stress Distribution in the H–V Geogrid-Reinforced Foundation
Figure 9a,b illustrate the vertical stress distribution diagrams of the reinforced foun-
dation with H–V geogrid within embedment depths of 0.4B and 1.0B, respectively. The
black doed line indicates the location of the loading plate, as seen in Figures 10 and 11.
It is indicated that the H–V geogrid-reinforced foundation with an embedment depth of
1B exhibited beer stress redistribution across a wider area than the foundation with an
embedment depth of 0.4B. This phenomenon may be aributed to the tighter soil com-
pression between the footing and the H–V geogrid at smaller embedded depths. As the
embedded depth of the H–V geogrid increased, the vertical stress distribution in the soil
became more uniform. This uniformity can be aributed to the redistribution of overbur-
den loading. Conversely, when the embedment depth of the H–V geogrid was shallow, it
could only bear the loading within the local range of the boom of the footing. This limi-
tation highlights the importance of considering the appropriate embedded depth of the
Buildings 2024, 14, 1533 9 of 16
H–V geogrid to ensure effective load redistribution and uniform stress distribution within
the reinforced foundation.
Figure 9. Vertical stress distribution of reinforced foundations with different H–V geogrid arrange-
ments (the black doed line is where the loading plate was initially located): (a) No.1 z = 0.4B; (b)
No. 2 z = 1.0B.
Figure 10a,b illustrate the vertical stress distribution with vertical element heights of
0.3B and 0.6B. In Figure 10a, the vertical stress isolines enclosed by the horizontal and
vertical elements are almost vertical. However, increasing the height of the vertical ele-
ment to 0.6B in Figure 10b resulted in a significantly smaller stress concentration area than
0.3B in Figure 10a. This is aributed to the restriction of soil movement to the right after
increasing the height of the vertical element, directing more sand downward (compare
solid and dashed orange arrows).
Figure 10. Vertical stress distribution of reinforced foundations with different H–V geogrid arrange-
ments: (a) No. 2 v = 0.3B; (b) No. 3 v = 0.6B.
Figure 11a,b present the vertical stress distribution with one pair and three vertical
elements in the H–V geogrid, respectively. Comparison of the two figures reveals that
increased vertical elements promoted stress homogenization within the foundation. The
vertical stress on the edge geogrid was the smallest, while the vertical stress on the vertical
elements directly below the footing was the largest. The middle vertical elements experi-
enced the densest stress isolines in three pairs of vertical elements, indicating a substantial
deformation gradient in the soil within this region.
Buildings 2024, 14, 1533 10 of 16
Figure 11. Vertical stress distribution of reinforced foundations with different H–V geogrid arrange-
ments: (a) No. 1, 1 pair; (b) No. 4, 3 pairs.
3.4. Contact Force distribution in the H–V Geogrid-Reinforced Foundation
Figure 12a,b depict the contact force diagrams for reinforced foundations with H–V
geogrids embedded at depths of 0.4B and 1.0B, respectively. Comparing area A to area B,
it exhibited significant differences in stress homogenization between the depths. The em-
bedded depth of 1.0B demonstrated superior stress distribution, particularly in regions
distanced from the footing. This indicates that the deeper embedding enhanced the influ-
ence of H–V geogrid on the base, creating a more cohesive and effective reinforced cush-
ion [41]. Force chains exhibited almost-vertical alignment at 1.0B depth, while deflection
and divergence occurred at 0.4B depth (compare solid and dashed orange arrows). This
deviation from previous findings suggests that the mesh structure and vertical element
components significantly minimized the force chain deflection. Increasing the H–V ge-
ogrid embedded depth enhanced the lateral restraint of reinforced area, similar to the
“deep-footing” effects observed in conventional geogrids [39,40].
Figure 12. The contact force of reinforced footings with different H–V geogrid arrangements: (a) No.
1 z = 0.4B; (b) No. 2 z = 1.0B; (c) area A; (d) area B.
Buildings 2024, 14, 1533 11 of 16
Figure 13a,b depict the contact force diagram for vertical geogrid heights of 0.3B and
0.6B in the H–V geogrid. The magnified views of areas A and B in both figures provide
detailed insights into the observations. Comparing Figure 13a,b, it is apparent that the
contact force of soil in the area below the H–V geogrid showed lile difference with the
increase in the height of the vertical elements. However, the contact force of soil above the
geogrid appeared more uniform, aligning with the stress analysis. Further examination
between Figure 13c,d reveals notable differences in the contact force distribution. The con-
tact force between the geogrid and soil primarily concentrated in the lower-left part of the
vertical element at lower vertical element heights. Conversely, as the height of the vertical
element increased, the contact force between the soil on the left side of the vertical element
became more homogenized. This indicates that increasing the height of the vertical ge-
ogrid led to a more uniform distribution of contact force, reflecting a more even transmis-
sion of the force chain.
Figure 13. Contact force of reinforced footing with different heights of vertical geogrid: (a) No. 2 v =
0.3B; (b) No. 3 v = 0.6B; (c) area A; (d) area B.
It should be noted that increasing the height of vertical geogrid elements plays a piv-
otal role in achieving a more uniform contact force distribution within a reinforced foun-
dation system. As vertical elements are extended, they interact with a larger soil volume,
enabling beer stress distribution over a broader vertical section. Taller elements offer an
increased surface area for load transfer and stress absorption, effectively spreading forces
Buildings 2024, 14, 1533 12 of 16
from the overlying structure more evenly and reducing stress concentrations that may
lead to localized failures. Moreover, taller elements more effectively restrict lateral soil
movements, maintaining soil structure under loading conditions and ensuring a uniform
contact force distribution. This restriction is vital when soil selement under load may
occur unevenly.
Figure 14a,b depict the contact force diagram for one pair and three pairs of vertical
elements within the H–V geogrid. The magnified views of areas A and B provide detailed
insights into the distribution of force chains. A comparison between Figure 14a,b illus-
trates that the force chain within three pairs of vertical elements in Figure 14b appeared
thinner and more uniform than that within one pair of vertical elements in Figure 14a,
both beneath the strip footing and in the lower right area distanced from the footing. This
observation suggests that increasing the vertical elements of the H–V geogrid contributed
to a more homogenized stress distribution within the foundation. Further comparison be-
tween Figure 14c,d reveals significant differences in the stress distribution of soil sur-
rounded by three pairs of vertical elements in Figure 14d compared to one pair of vertical
elements in Figure 14c. The force chains on the left and right sides of the three pairs of
vertical elements in Figure 14d exhibited minimal penetration, with most force chains by-
passing the upper part of the vertical elements. Additionally, some force chains deflected
when passing through the mesh of the vertical elements, as illustrated in Figure 14d, re-
sembling a membrane reinforcement mechanism. This behavior is aributed to the occlu-
sion and locking between the soil and the geogrid meshes, effectively integrating the ver-
tical elements and adjacent soil into a cohesive whole (compare solid and dashed orange
ellipses).
Figure 14. Contact force of reinforced footing with different number of vertical bars: (a) No. 1, 1 pair;
(b) No. 4, 3 pairs; (c) area A; (d) area B.
Buildings 2024, 14, 1533 13 of 16
3.5. Failure Theories
The failure of foundations under load, often aributed to insufficient subsoil bearing
capacity, typically presents a general and local shear failure [42]. When the vertical rib
was small, the number was few, and the burial depth was shallow (see Figure 6a), the H–
V geogrid tended to experience general shear failure while interacting with the soil. How-
ever, it is essential to note that more extended vertical ribs may induce significant uplift
forces. When the vertical rib was small, the number was few, but the burial depth was
deep (see Figure 6b), the H–V geogrid still leaned towards general shear failure while
working with the soil. Nonetheless, there is a higher likelihood of potential local shear
failure in the soil above the geogrid. When the vertical rib was more prominent, the num-
ber was few, but the burial depth was deep (see Figure 7b), the H–V geogrid primarily
experienced general shear failure at deeper distances while interacting with the soil. When
the vertical rib was small, the number was more, and the burial depth was shallow (see
Figure 8b), the H–V geogrid behaved similarly to a “deep-footing” within the soil. Hence,
in practical applications, careful consideration should be given to factors such as buried
depth, vertical element height of the H–V geogrid, and the number of vertical elements.
Meanwhile, it is essential to note that soil properties, such as cohesion, internal friction
angle, and water content, type, and stiffness of the geogrid, were not the main focus of
this study. However, these factors still need to be considered when using H–V geogrids
in real-world practical engineering applications.
3.6. The Deformation of the H–V Geogrid
Figure 15a provides insights into the deformation characteristics of the H–V geogrid,
focusing on both transverse and longitudinal deformation within the orange box area (the
dashed red box outlines the location of the strip footing above). It is suggested that the
horizontal deformation of the transverse rib was more pronounced than that of the longi-
tudinal rib. Furthermore, comparing the deformation among the transverse ribs, it is in-
dicated that the geogrids closer to the center of the footing experienced more significant
deformation. The confinement effect of the vertical element diminished as the horizontal
space from the center exceeded 1.5B. Meanwhile, the displacement paern of the vertical
element, as observed in Figure 15b, showed a gradual increase from the boom to the top,
with the displacement predominantly to the right. This movement was influenced by the
connection of the boom of the vertical element to the longitudinal geogrid, causing a
clockwise movement around its boom. Additionally, two inflection points can be ob-
served in Figure 15b, approximately 1B (solid circle and arrow) and 1.5B (dashed circle
and arrow) away from the center of the footing. These points are aributed to the rigid
connections between the vertical and horizontal elements of the H–V geogrid. The bend-
ing of the vertical element caused a specific rotation angle at the rigid connection points,
altering the distribution of the inverse bending points of the geogrid. This reduced the
buckling degree of the horizontal element, equalizing the deformation difference of the
horizontal element in the vertical direction and enhancing the uniformity of the reinforce-
ment effect provided by the H–V geogrid.
Buildings 2024, 14, 1533 14 of 16
Figure 15. The deformation of the H–V geogrid: (a) top view; (b) elevation.
4. Conclusions
This paper employed PFC to investigate the bearing capacity and reinforced mecha-
nism of H–V reinforced geogrid foundation. The results are as follows:
(1) An embedment depth equivalent to the footing width improves the bearing capacity
of the H–V geogrid-reinforced foundation compared to depths of 0.4 times the foot-
ing width, which has been identified as the optimal value in conventional geogrid-
reinforced foundations. This increased embedment depth creates a deeper slip sur-
face, impacting soil movement trajectory beyond immediate effects.
(2) Boosting the height of vertical elements and introducing more of them significantly
improves the bearing capacity of the H–V reinforced foundation. Foundations
equipped with three pairs of vertical elements experience a significant boost, reach-
ing 1.3 times the capacity compared to just one pair, and raising the vertical elements
to 0.6 times the width of the footing results in 1.2 times the increase in capacity com-
pared to those set at 0.3 times the footing width, with only a modest 12% uptick in
geogrid usage. This underscores the strategic and economically beneficial reinforce-
ment of vertical elements.
(3) The vertical elements increase lateral restraint on the soil, reducing its movement and
enhancing the “deep-footing” effect of the H–V geogrid. Heightened vertical ele-
ments also decrease soil displacement, aid in seling unevenness, and delay slip sur-
face formation. They ensure uniform horizontal stress distribution during footing
loading and minimize force chain deflection. Increasing the embedded depth of the
H–V geogrid further strengthens lateral restraint, akin to conventional geogrids.
(4) The H–V geogrid analysis reveals that transverse ribs deform more prominently near
the footing’s center than longitudinal ribs. The confinement effect of vertical elements
weakens with increasing horizontal distance from the center. Displacement of verti-
cal elements rises from boom to top, mainly towards the right, influenced by their
connection to the longitudinal geogrid. Inflection points at around 1B and 1.5B from
the center result from rigid connections between vertical and horizontal elements,
reducing buckling and ensuring uniform reinforcement across the geogrid.
(5) The approach provides a comprehensive understanding of how H–V geogrids differ
from conventional geogrids and influence the overall performance of the reinforced
foundation system. However, it is essential to note that soil properties such as cohe-
sion, internal friction angle, water content, and the type and properties of the geogrid
can significantly influence the bearing capacity of the H–V geogrid-reinforced foun-
dation. Further investigation is warranted before its application in real-world practi-
cal engineering projects.
Buildings 2024, 14, 1533 15 of 16
Author Contributions: Conceptualization, J.H.; methodology, J.W., F.Z., and J.H.; data curation
writing—original draft preparation, J.W., F.Z., L.G., and J.H.; writing—review and editing, J.H.; vis-
ualization, J.W. and F.Z.; project administration, J.W. All authors have read and agreed to the pub-
lished version of the manuscript. All authors have read and agreed to the published version of the
manuscript.
Funding: Qinghai Provincial Department of Transportation Science and Technology Project (2022–
02) provided financial support for Wu’s contributions to this study.
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement: The data presented in this study are available on request from the
corresponding author.
Conflicts of Interest: Authors Jinjun Wu, Fabin Zhang and Liang Gao were employed by the com-
pany Qinghai Transportation Planning and Design Institute Co., Ltd. The remaining author declares
that the research was conducted in the absence of any commercial or financial relationships that
could be construed as a potential conflict of interest.
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