Triaxial Shear Behaviour and Strength Prediction Models of Recycled Tyre-Derived Grid-Reinforced Weathered Sand

Abstract

This paper presents a method of using recycled tyre-derived grids (RTDG) as reinforcement materials for mechanical stabilised weathered sand embankment. To demonstrate the effectiveness of the RTDG-reinforced weathered sand on shear behaviour, large-scale triaxial tests were conducted under different confining pressures and reinforcement layers. The test results indicate: (1) RTDG reinforcement significantly alters the shear failure mode of weathered sand, transitioning it from shear failure to bulging failure. (2) RTDG reinforcement enhances the ultimate deviatoric stress of specimens by 20–30%, transforming the stress–strain response from strain softening to strain hardening. (3) RTDG reinforcement causes apparent cohesion to increase at an approximate linear rate of 37.2% per reinforcement layer, whereas the internal friction angle exhibits only a gradual increase. (4) The coupled effect of RTDG reinforcement and confining pressure alter volumetric behaviour from shear contraction–dilation patterns to solely shear contraction. It also reduces maximum dilation strain by about 50% and the dilation angle by 23%. (5) Two shear strength prediction models based on confinement enhancement (CEB) and interface friction (IFB) were proposed. Comparative analysis shows that, within the present dataset, the IFB model exhibits lower prediction error and a more stable error distribution than the CEB model, with a WRAI value of 0.033. Accordingly, the IFB model provides an effective prediction approach for estimating the shear strength of RTDG-reinforced weathered sand within the investigated test range, and the corresponding prediction results may serve as a preliminary reference for the engineering evaluation of RTDG-reinforced weathered sand.

1. Introduction

With the rapid development of transport infrastructure construction, road engineering has progressively expanded into complex terrains, including hilly areas, karst regions, and sites with weak foundations. As the demand for embankment fill materials continues to increase, many regions are experiencing severe shortages of well-graded aggregates [1,2,3]. For example, in the northwest and north China, weathered sand is widely distributed over approximately 7.5 × 10⁵ km² [4], and is commonly characterised by poor gradation and a loose structure [5,6]. Considering economic efficiency and construction practicality, numerous projects therefore continue to use locally available weathered sand as embankment fill material [4,7]. However, weathered sand embankments are prone to insufficient bearing capacity during service [8], which may lead to engineering problems such as uneven embankment settlement [9,10] and collapse [11,12]. Accordingly, additional reinforcement measures are required in such projects to ensure embankment stability and mitigate potential safety risks.

Among existing reinforcement measures, incorporating reinforcing materials such as fibre [13,14], polymers [15,16], and construction waste [3,17] into embankment fill is widely recognised as an effective approach. These reinforcing materials exert tensile resistance, thereby redistributing stresses within the embankment fill and increasing the soil modulus, which contributes to improvements in embankment bearing capacity and service stability [18]. Among these materials, recent studies have indicated that waste tyres can serve as a promising reinforcement option, owing to their favourable mechanical properties, including high tensile strength, adequate frictional resistance, flexibility, and durability [19,20].

According to their geometric configuration, the utilisation of waste tyres in geotechnical engineering can be broadly categorised into three main types, as illustrated in Figure 1: (1) whole tyre reinforcement [21,22,23], (2) tyre chips and shreds reinforcement [24,25,26], and (3) tyre-derived strips and grids reinforcement [27,28,29]. Compared with whole tyre and tyre-derived particle, tyre-derived strip and grid are more suitable for embankment reinforcement. Their planar configuration allows flexible layout without significant spatial constraints, while enabling continuous interaction with the surrounding soil and providing continuous stress pathways throughout the entire loading process [30].

Figure 1. Schematic Diagram of a Reinforced Embankment.

Tyre-derived grids constitute a novel reinforcement structure formed by assembling tyre strips as longitudinal and transverse ribs [31]. This configuration enables efficient utilisation of the mechanical properties of tyres while achieving high material recycling efficiency [32,33]. To verify the advantage and reinforcement effect of tyre-derived grids, Tajabadipour et al. [34] investigated the pull-out resistance of geo scrap tyre (GST) strip with transverse. The pull-out resistance increased by 239–250% and 352–398%, respectively, compared with geosynthetic strips and steel strips. Zhang et al. [35] further investigated the interfacial characteristics, revealing that tyre grids exhibit maximum pull-out capacity ranging from 15.56 kPa to 22.82 kPa. These forces were attributed to friction and end resistance at the soil–ribs interface and remained relatively stable throughout the pull-out process. Esmaeili et al. [36] investigated the enhanced ballast shear behaviour using waste tyre strip grids in large-scale direct shear tests. Compared to the non-reinforced state, the shear strength and internal friction angle experienced respective increases of 20% and 7%, while the dilatancy angle decreased by 30%. Mehrjardi et al. [37] investigated the bidirectional sealing effect of tyre grids on slope soil, which can effectively enhance the slope bearing capacity under cyclic loading. This enhancement effect increases with the depth of the tyre grid embedded horizontally into the slope.

In summary, existing studies on tyre-derived grids reinforcement have primarily focused on local interface responses under pull-out and direct shear tests, with emphasis on frictional behaviour and localised shear resistance. However, for RTDG-reinforced weathered sand, the reinforcement effect may involve both soil–grid interface interaction and confinement-related stress redistribution under three-dimensional stress conditions. The relative contribution and mobilisation of these effects remain insufficiently clarified. To address these gaps, a series of large-scale triaxial tests were conducted in this study on recycled tyre-derived grid (RTDG)-reinforced weathered sand under different confining pressures (σ₃ = 200, 300, and 400 kPa) and reinforcement layer numbers (N = 0, 1, 2, and 3). The investigation focuses on the failure modes, mechanical response characteristics, volumetric behaviour evolution, and the development of key parameters, aiming to reveal the intrinsic RTDG-reinforced mechanisms under realistic three-dimensional stress conditions. Furthermore, two shear strength models based on confinement enhancement and interface friction are proposed to quantitatively describe the reinforcement effect of RTDG. These enable a theoretical basis for the design and prediction from structural effect characterisation to mechanism decomposition and engineering-oriented quantitative modelling.

2. Materials

2.1. Recycled Tyre

A 175/70 R14 tyre was selected, as this size is widely available, exhibits stable mechanical properties, and is representative for engineering applications. The tyre sidewalls were first removed, after which tyre strips with a width of 2 cm and a thickness of 1.5 cm were produced using a rotary cutting method. Subsequently, the strips were mechanically assembled and fixed into a grid unit structure consisting of longitudinal and transverse ribs, with a centre-to-centre spacing of 8 cm and an effective aperture size of 6 cm, as shown in Figure 2. This geometric configuration facilitates specimen preparation while providing a reinforcement density sufficient to mobilise the reinforcement effect during triaxial testing.

Figure 2. Waste tyre processing method.

To characterise the fundamental mechanical properties of the waste tyre material, uniaxial tensile tests were conducted on the tyre strips, as shown in Figure 3. The tensile specimen used for strength calculation had an effective width of 60 mm, a thickness of 15 mm, and a load-bearing cross-sectional area of 900 mm². The gauge length between the grips was 250 mm. The test results are shown in Figure 4. The tyre strips exhibited an ultimate tensile strain of 18.9%, corresponding to a maximum tensile force of 21.2 kN, which is equivalent to an ultimate tensile strength of 23.5 MPa. At the initial stage of loading, the strip behaves in a linear elastic manner; once the ultimate load is reached, failure occurs, resulting in a rapid reduction in load-carrying capacity.

Figure 3. Diagram of the tensile test instrument.

Figure 4. Tyre strip tensile stress–strain curve.

These tensile properties indicate that recycled tyre-derived strips exhibit high tensile strength and adequate ductility, satisfying the requirements for strength and deformation compatibility of reinforcement materials in triaxial testing. As the RTDG primarily relies on its longitudinal ribs to mobilise tensile resistance during loading, the tensile properties of individual strips can serve as fundamental mechanical parameters for characterising the mechanical response of RTDG.

2.2. Weathered Sand

The weathered sand shown in Figure 5a was used as the fill material. A sieve analysis was conducted according to Chinese Standard JTG 3430-2020 [38], and the resulting grain size accumulation curve is shown in Figure 5b.

Figure 5. Weathered Sand and Its Particle Characteristic.

The physical and mechanical properties were obtained from specific gravity tests and relative density tests, with the corresponding parameters summarised in Table 1.

Table 1. Physical and Mechanical Parameters of Weathered Sand.

Effective Particle Size d10/mm	Median Particle Size d50/mm	Uniformity Coefficient Cu	Curvature Coefficient Cc	Specific Gravity of Soil Particles Gs	Maximum Dry Density ${\mathsf{\rho }}_{\mathbf{d}\mathbf{m}\mathbf{a}\mathbf{x}}$/$\mathbf{g}\cdot {\mathbf{cm}}^{-3}$	Minimum Dry Density ${\mathsf{\rho }}_{\mathbf{d}\mathbf{m}\mathbf{i}\mathbf{n}}$/$\mathbf{g}\cdot {\mathbf{cm}}^{-3}$
0.361	1.671	6.6	0.73	2.65	1.805	1.522

This weathered sand is characterised by poor gradation, a common unfavourable feature of it used as embankment fill materials, and is therefore appropriate for studying its strength behaviour.

2.3. Recycled Tyre-Weathered Sand Interface Pull-Out Characteristics

To quantitatively characterise the interaction at the interface between recycled tyre and weathered sand, interface pull-out tests were conducted on recycled tyre-derived strip-reinforced weathered sand using the apparatus shown in Figure 6, yielding the interface pull-out force $F_s$. The measured $F_s$ was used to characterise the basic interface friction behaviour of the longitudinal rib component without the influence of transverse-rib end-bearing resistance.

Figure 6. Pull-out test apparatus.

For application to the assembled RTDG, the strip-level interface response was further combined with the geometric parameters of the RTDG. According to Chinese Standard JTG 3430-2020 [38] and the methodologies of Moraci N et al. [39] and Bergado et al. [40], the interface shear strength ($\tau$), interface friction coefficient (f), and friction resistance ($P_{RS}$) were calculated by using Equations (1)–(3) based on pull-out tests. Here, $P_{RS}$ represents the surface-friction component mobilised along the longitudinal ribs of the RTDG.

Interface shear strength:

$$\tau ={\displaystyle \frac{F_s}{2W_r{L}_r}}$$

(1)

where $W_r$ and $L_r$ are the width and length of the reinforcement elements buried in the soil, respectively.

Interface friction coefficient:

$$f={\displaystyle \frac{\tau }{{\sigma }_n}}$$

(2)

where ${\sigma }_n$ is the normal stress.

Friction resistance:

$$P_{RS}=2{\alpha }_S{W}_r{L}_r{\sigma }_{n}tan \delta$$

(3)

where ${\alpha }_S$ is the effective interface coefficient of the tyre, which was taken as 0.6 in this study, and $\delta$ is the friction angle of the tyre–soil interface, which was taken as 29.8° based on direct shear test.

The measured pull-out force and the corresponding calculated interface friction parameters are summarised in Table 2.

Table 2. Recycled Tyre-Weathered Sand Interface Friction Parameters.

${\mathit{\sigma }}_{\mathit{n}}$ (kPa)	${\mathit{F}}_{\mathit{s}}$ (kN)	τ (kPa)	$\mathit{f}$	PRS (kN)
40	2.85	38.15	0.95	2.20
60	4.10	54.20	0.90	3.30
80	5.29	70.16	0.88	4.39
100	6.51	85.37	0.85	5.48

These parameters provide essential input for subsequent analysis of the mechanical behaviour and strength evolution of RTDG-reinforced weathered sand in triaxial testing.

3. Method

3.1. Triaxial Test Apparatus

The triaxial tests were conducted using a DJSZ-1000 triaxial testing system (Donghua Zhuoyue Technology Co., Chengdu, China). A photograph of the apparatus is shown in Figure 7. The system comprises five subsystems: control, loading, hydraulic, cooling, and water storage. The maximum axial load capacity is 1500 kN, the maximum confining pressure is 2 MPa, and the minimum loading rate is 0.01 mm/min. This system is capable of accommodating the loading requirements of large-scale reinforced soil specimens under different confining pressure conditions.

Figure 7. Large-scale triaxial test apparatus.

3.2. Triaxial Test Conditions

Triaxial tests were performed to evaluate the effects of different reinforcement layers and confining pressures on the shear behaviour of reinforced weathered sand. Cylindrical specimens with dimensions of 200 mm in diameter and 400 mm in height were prepared. The reinforcement layouts are shown in Figure 8, and the corresponding test conditions are summarised in Table 3. The test matrix consisted of 12 triaxial conditions, and one specimen was tested for each condition. For reinforced specimens, the volume occupied by the embedded RTDG was considered when calculating the required soil volume and corresponding soil mass. The volumetric reinforcement ratio $v_r$ was defined as $v_r=v_{RTDG}/v_{Specimen}$.

Figure 8. Reinforcement layouts in triaxial specimens.

Table 3. Test conditions.

Condition Number	Confining Pressure ${\mathit{\sigma }}_{\mathit{c}}$ (kPa)	Reinforcement Layer N	Volumetric Reinforcement Ratio ${\mathit{v}}_{\mathit{r}}$/%
S-1	200	0	0
S-2	300	0	0
S-3	400	0	0
I-1	200	1	3.05
I-2	300	1	3.05
I-3	400	1	3.05
II-1	200	2	6.11
II-2	300	2	6.11
II-3	400	2	6.11
III-1	200	3	9.17
III-2	300	3	9.17
III-3	400	3	9.17

3.3. Triaxial Specimen Preparation

Specimens were prepared using a layered compaction method as shown in Figure 9. To simulate the medium-density fill material commonly encountered in engineering projects, the weathered sand was mixed at an initial water content of 8%, and relative density was controlled at 0.7 throughout this study. Accordingly, each specimen was constructed in 5 layers, with a thickness of 8 cm for each layer. The required soil mass for each layer was calculated according to the target relative density and the corrected soil volume. During specimen preparation, each layer was compacted using a rammer, and the height of each compacted layer was measured using a steel ruler to ensure that the designed layer thickness and target relative density were achieved. When placing the reinforcement layers, the tyre grids were laid flat on the compacted sand surface, followed by placement and compaction of the subsequent soil layer.

Figure 9. Preparation of triaxial test specimens.

3.4. Triaxial Test Procedures

All tests were conducted according to the Chinese standard JTG 3430-2020 [38]. Each specimen was wrapped with a latex membrane and properly sealed before being placed into the pressure chamber. Prior to the application of confining pressure, a small initial pressure was applied to ensure specimen stability during the early stage of loading. The confining pressure was then applied incrementally to the target value and maintained constant. During shearing, no drainage was allowed, and pore-water pressure was recorded by the testing system. Axial loading was applied at a rate of 2 mm/min until an axial strain of 20% was achieved. After the test concluded, the specimen was unloaded and removed, and the specimen failure pattern was observed and recorded.

4. Results and Discussion

4.1. Triaxial Failure Mode

Figure 10 illustrates the shear failure modes of the RTDG-reinforced weathered sand specimen under different reinforcement layers and confining pressures.

Figure 10. Shear failure modes of specimens under different conditions.

For the unreinforced specimens under all confining pressure conditions, as shown in Figure 10a–c, shear failure consistently manifested as oblique shear planes that penetrated the specimen. The inclination angle of the failure plane with respect to the major principal stress direction is approximately given by ${\theta }_f={45}^{\circ }+{\displaystyle \frac{\phi }{2}}$, which conforms to the classical Mohr–Coulomb failure criterion. A schematic illustration of this failure mode is illustrated in Figure 11a. These indicate that unreinforced weathered sand predominantly undergoes general shear failure along a distinct shear plane.

Figure 11. Failure mode diagrams of specimens.

With the inclusion of one reinforcement layer of RTDG, the failure mode of the specimen changed noticeably. As shown in Figure 10d–f, the specimen still exhibited a shear failure mode under a confining pressure of 200 kPa, but the resulting shear plane was less continuous and distinct. When the confining pressure increased to 300 kPa and 400 kPa, the failure mode gradually transitioned from shear failure to a bulging failure. Bulging did not occur directly at the reinforcement location but rather at an adjacent position, indicating that the reinforcement effect of a single grid layer is spatially limited. Consequently, localised shear failure remains prone to occur even under relatively low confining pressure. The corresponding failure mode is illustrated in Figure 11b.

With a further increase in the number of reinforcement layers, the failure mode changed substantially. As shown in Figure 10g–l, specimens with two or three reinforcement layers did not display distinct shear planes; instead, they exhibited bulging failure. Furthermore, as the numbers of reinforcement layers and confining pressure increased, bulging deformation became smaller and more uniformly distributed between layers. This indicates that multi-layer RTDG reinforcement can form a more continuous confinement system within the specimen, effectively inhibiting the development of a shear plane and promoting stress redistribution over a wider region. Consequently, the failure mode shifts from localised shear-dominated failure to overall bulging, as illustrated in Figure 11c.

An analysis of the shift in failure mode suggests that the presence of RTDG introduces a barrier effect, which inhibits the further propagation of shear stresses into a continuous shear plane. Instead, the applied shear stresses are redistributed into interfacial horizontal stress and vertical compressive stress.

4.2. Triaxial Mechanical Response

4.2.1. Deviatoric Stress–Axial Strain Relationship

Figure 12 illustrates the stress–strain curves under different confining pressures and reinforcement layers. According to Chinese Standard JTG 3430-2020 [38], for stress–strain curves with a distinct peak followed by strain softening, the ultimate deviatoric stress was identified at the onset of strain softening. For curves without a distinct peak before the termination of the test, the deviatoric stress at 20% axial strain was taken as the comparative ultimate deviatoric stress under the adopted test termination criterion.

Figure 12. Stress–strain curves under different conditions.

For the unreinforced specimens, strain softening occurred as axial strain increased. With increasing axial strain, the deviatoric stress rapidly reaches a peak value, followed by a gradual reduction and eventual stabilisation at a residual strength level. The measured ultimate deviatoric stresses were 789.3 kPa, 1143.7 kPa, and 1357.6 kPa at confining pressures of 200, 300, and 400 kPa, respectively. The corresponding residual strengths were 652.2 kPa, 1023.7 kPa, and 1295.5 kPa, representing decreases of 17.4%, 10.5%, and 4.6%. It can be observed that as confining pressure increases, the degree of strain softening in the specimen markedly diminishes. This indicates that higher confining pressure enhances lateral constraint, effectively inhibiting the rapid initiation and propagation of shear planes [41]. The bearing capacity of the specimen is therefore enhanced.

In contrast, the reinforced specimens exhibited strain-hardening behaviour. Their ultimate deviatoric stress was significantly higher than those of the unreinforced specimens under the same confining pressures. For example, with two reinforcement layers, ultimate deviatoric stresses reached 963.7 kPa, 1371.9 kPa, and 1785.7 kPa at confining pressures of 200, 300, and 400 kPa, respectively, representing an increase of approximately 20–30% compared with the corresponding unreinforced specimens. This demonstrates that RTDG reinforcement significantly enhances the bearing capacity of specimens, fundamentally altering their mechanical behaviour. Analysis suggests that the incorporation of RTDG leads to a more cohesive and integrated internal structure, which strengthens interparticle interactions and confinement. The reinforcement facilitates more efficient stress transfer, thereby effectively enhancing the resistance of the specimens to shear failure.

Further analysis indicates that under identical confining pressure conditions, the ultimate deviatoric stress exhibits a positive correlation with the number of reinforcement layers, though the rate of increase gradually diminishes. Taking a confining pressure of 300 kPa as an example, when the number of reinforcement layers increases from 0 to 3, the ultimate deviatoric stress rises from 1143.7 kPa to 1257.3 kPa, 1372.0 kPa, and 1477.0 kPa respectively, corresponding to increases of 9.9%, 9.1%, and 7.7%. Although the ultimate deviatoric stress increased with the number of reinforcement layers, the marginal gain became less pronounced as the reinforcement layers increased.

4.2.2. Mechanical Parameters

(1)

The predicted ultimate deviatoric stress

Figure 13 depicts a contour heatmap of ultimate deviatoric stress under different test conditions, aiming to quantitatively characterise the synergistic effect of the two primary factors—confining pressure and reinforcement layer—on ultimate deviatoric stress.

Figure 13. Contour heatmap of ultimate deviatoric stress.

A multiple regression analysis was conducted to quantify their combined influence, resulting in the empirical relationship shown in Equation (4). For the present test dataset, the regression yielded a correlation coefficient of R² = 0.982. A comparison of calculated and measured values is shown in Figure 14. To avoid dimensional inconsistency, the ultimate deviatoric stress and confining pressure were normalised by the reference atmospheric pressure $P_a=100$ kPa before regression.

$$P^{*}=\mathrm{A}{{\mathsf{\sigma }}_3}^{*}+\mathrm{B}$$

(4)

where $P^{*}=P/P_a$ is the normalised ultimate deviatoric stress, ${{{\mathsf{\sigma }}_3}^{*}=\mathsf{\sigma }}_3/P_a$ is the normalised confining pressure, and N is reinforcement layers. The fitting coefficients are expressed as A(N) = $0.3N+3.3$, B(N) = $0.381N+0.932$, where A(N) reflects the sensitivity of the normalised ultimate deviatoric stress to the normalised confining pressure, while B(N) represents an empirical intercept term within the tested confining-pressure range.

Figure 14. Comparison between calculated and measured values of ultimate deviatoric stress.

(2)

The Shear Strength Parameters

Based on ultimate deviatoric stress data, Mohr circles and corresponding strength envelopes were constructed to obtain the shear strength parameters—apparent cohesion c_r and apparent friction angle φ_r. These quantify the contributions of interface friction and interlocking effects within the specimens, serving as key indicators for assessing the overall bearing capacity and stability of reinforced structures. The results are illustrated in Figure 15.

Figure 15. The shear strength parameter values.

It can be observed that unreinforced specimens exhibited relatively low values of apparent cohesion and apparent friction angles. In contrast, reinforced specimens demonstrated a pronounced increase in apparent cohesion, exhibiting an approximately linear trend with increasing reinforcement layers, and the average growth rate can reach 37.2%. The apparent friction angle exhibited only a modest increase with increasing reinforcement layers.

These results indicate that the shear strength enhancement of RTDG-reinforced weathered sand is primarily reflected in an increase in apparent cohesion. In contrast, the apparent friction angle exhibits weaker sensitivity to RTDG reinforcement and remains largely governed by the inherent properties of the soil particles.

4.3. Triaxial Volumetric Strain Characteristics

4.3.1. Volumetric Strain–Axial Strain Relationship

The apparent volumetric strain was derived from the measured axial and radial deformations and was used to describe the contraction–dilation tendency of the specimens during shearing. Figure 16 illustrates the volumetric stress–strain curves under different confining pressures and reinforcement layers.

Figure 16. Volumetric strain responses under different conditions.

For unreinforced specimens, the volumetric behaviour under all confining pressure conditions is characterised by an initial shear contraction followed by shear dilation with increasing axial strain. During the initial loading stage, the lateral confinement is relatively weak, and the loose weathered sand predominantly undergoes particle slippage and pore filling, resulting in shear contraction. With continued loading, the development of lateral confinement promotes closer particle contact. In this stage, particles’ movement and rearrangement require overcoming increased interlocking resistance and interface friction leading to shear dilation.

Furthermore, increasing confining pressure effectively suppresses the development of shear dilation. At confining pressures of 200, 300, and 400 kPa, the corresponding maximum shear dilation strains were 3.0%, 1.8%, and 1.4%, respectively. This indicates that increasing confining pressure effectively suppresses shear dilation by restricting volumetric expansion. Under high confining pressure conditions, particle rearrangement is increasingly constrained, which limits dilation.

Reinforced specimens similarly exhibited a transition from shear contraction to shear dilation in their volumetric response. However, the magnitude of volumetric change was substantially lower than that of unreinforced specimens. Taking the two-layer reinforcement conditions as an example, under confining pressures of 200, 300, and 400 kPa, the maximum shear dilation strains were 1.6%, 1.1%, and 0.5%, respectively, amounting to approximately 50% of the corresponding values for unreinforced specimens.

Further comparison of volumetric responses under identical confining pressure conditions reveals that, with increasing numbers of reinforcement layers, the maximum shear dilation strain decreases markedly, while the onset strain for shear dilation increases. At a confining pressure of 300 kPa, as the number of reinforcement layers increased from 0 to 3, the maximum shear dilation strain decreased from 1.8% to 1.0%, whereas the onset axial strain for shear dilation increased from 4.4% to 5.6%.

It is noteworthy that under a confining pressure of 400 kPa combined with three reinforcement layers, shear dilation was no longer observed, and the volumetric response was characterised solely by shear contraction. Under the combined effects of high confining pressure and multiple reinforcement layers, the displacement and rearrangement of weathered sand particles were severely restricted. Consequently, volumetric response was dominated by weathered sand particle compression and pore filling, which is consistent with the studies of Chen et al. [42].

In summary, both confining pressure and reinforcement layers exert significant influences on the volumetric response of the specimen. As these two factors increase, the amplitude of shear dilation progressively diminishes, and the volumetric response gradually transitions from shear contraction–dilation to shear contraction alone.

4.3.2. Peak Dilation Angle

The dilation angle characterises the volumetric expansion of granular materials during shearing and reflects particle arrangement and deformation mechanisms. It is an important indicator of the mechanical behaviour of RTDG-reinforced weathered sand. The values for each condition were calculated using Equation (5) [43] and are presented in Table 4.

$$\sin {\mathsf{\psi }}_{\mathrm{P}}={\displaystyle \frac{(\mathrm{d}{\mathsf{\epsilon }}_{\mathsf{\nu }}-\mathrm{d}{\mathsf{\epsilon }}_1)}{2-(\mathrm{d}{\mathsf{\epsilon }}_{\mathsf{\nu }}-\mathrm{d}{\mathsf{\epsilon }}_1)}}$$

(5)

where $\mathrm{d}{\mathsf{\epsilon }}_1$ is axial strain increment, $\mathrm{d}{\mathsf{\epsilon }}_{\mathsf{\nu }}$ is volumetric strain increment, and peak dilation angle is the value corresponding to the maximum slope of the ${\mathsf{\epsilon }}_{\mathsf{\nu }}-{\mathsf{\epsilon }}_1$ curve.

Table 4. Peak dilation angle of specimen under different reinforcement conditions.

	${\mathsf{\sigma }}_3$ (kPa)	$(\mathbf{d}{\mathsf{\epsilon }}_{\mathsf{\nu }}-\mathbf{d}{\mathsf{\epsilon }}_1)$	${\mathsf{\psi }}_{\mathbf{P}}$ (°)
Unreinforcement	200	0.4394	16.3
	300	0.4012	14.5
	400	0.3621	12.8
One-Layer Reinforcement	200	0.4236	15.6
	300	0.3785	13.5
	400	0.3418	11.9
Two-Layer Reinforcement	200	0.3872	13.9
	300	0.3267	11.3
	400	0.3056	10.4
Three-Layer Reinforcement	200	0.3528	12.3
	300	0.2984	10.1
	400	- -	- -

As shown in Table 4, increases in both confining pressure and reinforcement layers correspond to decreases in peak dilation angle. At a confining pressure of 300 kPa, the peak dilation angle decreases progressively as the number of reinforcement layers increases from 0 to 3, falling from 14.5° to 13.5°, 11.3°, and 10.1°, representing reductions of 6.9%, 22.1%, and 30.3%, respectively. For specimens with two reinforcement layers, the peak dilation angles under confining pressures of 200 kPa, 300 kPa, and 400 kPa were 13.9°, 11.3°, and 10.4° respectively, corresponding to reductions of 18.7% and 8.0% across successive confining pressure intervals.

These observations suggest that the internal restraint provided by RTDG, combined with external confining pressure, limits particle displacement during shearing. This interaction reduces volumetric expansion and leads to the reduction in peak dilation angle. The results are consistent with the volumetric strain trends identified in Section 4.3.1, further confirming the effectiveness of tyre grids in controlling deformation.

5. Analysis and Modelling of RTDG-Reinforced Weathered Sand

5.1. RTDG-Reinforced Mechanisms

To elucidate the evolution of strength and deformation characteristics in RTDG-reinforced weathered sand in triaxial testing, the reinforcement mechanisms are discussed from the following two perspectives.

(1)

Confinement enhancement induced by mesh interlocking

As illustrated in Figure 17, during shear loading, the mesh structure of the RTDG restricts the free movement and rearrangement of weathered sand particles, thereby generating a mesh interlocking effect that enhances the bearing capacity of the specimen. Simultaneously, under shear stress, the transverse and longitudinal ribs of the RTDG exert compressive interactions on the surrounding weathered sand, promoting the diffusion of localised stresses over a broader region [44]. This confinement enhancement mechanism induced by mesh interlocking increases the resistance to deformation, which manifests macroscopically as an increase in the equivalent strength parameters of the specimen. When considering the additional normal stress increment induced by the non-dilative force zone, the resulting confinement resistance can be expressed as Equation (6):

$$F_c=2W_r{L}_r{\sigma }_n\left(1+2\Delta {\sigma }_n\right)tan \delta$$

(6)

Figure 17. Diagram of mesh interlocking.

(2)

Interface friction effect

As illustrated in Figure 18, interfacial friction is mobilised through two primary mechanisms [45]: (i) frictional resistance generated by direct contact between the irregular surface texture of the RTDG and weathered sand particles; (ii) additional frictional resistance arising from the rib and edge features of the RTDG that impede particle movement. The total frictional resistance generated by the interfacial interactions can be calculated using Equation (7). The mobilisation of frictions results in stress redistribution along the RTDG–weathered sand interface, effectively suppressing the initiation and development of shear planes, thereby enhancing the specimen’s shear strength.

$$F_f=W_r{L}_r{\sigma }_n\left(f+2{\alpha }_{S}tan \delta \right)$$

(7)

Figure 18. Diagram of interface friction effect.

In summary, RTDG alters the stress transmission paths and deformation characteristics of weathered sand through the effects of confinement enhancement induced by mesh interlocking and interfacial friction mobilisation. Consequently, RTDG-reinforced weathered sand exhibits improved bearing capacity and more stable deformation behaviour in triaxial loading testing.

5.2. RTDG-Reinforced Modelling of Shear Strength

5.2.1. Overview of RTDG-Reinforced Model

Numerous theoretical models have been proposed in the literature for calculating the shear strength of grid-reinforced soils, based on different underlying mechanical assumptions [46,47,48,49]. These models can be broadly classified into two representative modelling approaches. One approach focuses on the improvement of the overall stress state and deformation behaviour of the soil induced by reinforcement, whereas the other emphasises the contribution of soil–reinforcement interface interaction to shear strength enhancement. Based on the reinforcement mechanism analysis presented in Section 5.1, this paper establishes two shear strength models under the aforementioned mechanical assumptions, namely the Confinement–Enhancement-Based (CEB) model and the Interface–Friction-Based (IFB) model. Model parameters are determined and validated using triaxial test data, enabling a further investigation into the appropriate representation of the reinforcement effect and the engineering applicability of the aforementioned two models under different mechanistic assumptions.

5.2.2. Confinement–Enhancement-Based (CEB) Model

The CEB model assumes that the primary role of RTDG is to modify the stress state of the weathered sand through mesh interlocking, resulting in a confinement effect that enhances the specimen’s shear strength. In this model, the stress diffusion and deformation restraint induced by the RTDG are equivalently idealised as an overall confinement enhancement within the reinforced specimen. Based on this assumption, the restraint generated by RTDG–soil interlocking was represented as an equivalent confining-pressure increment $\Delta {\sigma }_3^{\prime }$ (Equation (8)). This parameter provides a macroscopic measure of the confinement enhancement induced by RTDG reinforcement and was incorporated into the Mohr–Coulomb stress framework to quantify the shear strength improvement of RTDG-reinforced weathered sand.

$$\Delta {\sigma }_3^{\prime }={\sigma }_{3fU}\left({\displaystyle \frac{{\Delta \sigma }_{1fR-U}}{{\sigma }_{1fU}}}\right)$$

(8)

where $\Delta {\sigma }_3^{\prime }$ is the equivalent confining pressure increment, ${\sigma }_{3fU}$ is the initial confining pressure, and ${\Delta \sigma }_{1fR-U}={\sigma }_{1fR}-{\sigma }_{1fU}$ is the increase in the major principal stress caused by RTDG reinforcement under the same confining pressure. Here, ${\sigma }_{1fR}$ and ${\sigma }_{1fU}$ denote the major principal stresses of the reinforced and unreinforced specimens, respectively.

Figure 19 presents the results of additional confining pressure calculated using Equation (8). It increases rapidly within the low to medium confining pressure range, while the rate of increase gradually diminishes at higher confining pressures. This trend is consistent with the observations in Section 4.

Figure 19. Determination of confinement–enhancement-based model parameters.

The characterisation of the combined contribution from the equivalent confining pressure increment $\Delta {\sigma }_3^{\prime }$ and the initial confining pressure ${\sigma }_{3fU}$ is based on the principle of stress compensation, incorporating a correction function k as Equation (9):

$${\sigma }_{3fR}={\sigma }_{3fU}+k{\displaystyle \frac{\Delta {\sigma }_3^{\prime }}{{\sigma }_{3fU}}}$$

(9)

Here, $\Delta {\sigma }_3^{\prime }/{\sigma }_{3fU}$ represents a relative confinement-enhancement ratio, in which the equivalent confining pressure increment is normalised by the reference confining pressure to describe the reinforcement-induced stress compensation effect under different confining pressure levels. The coefficient k is a stress correction coefficient with the unit of kPa. Therefore, $k(\Delta {\sigma }_3^{\prime }/{\sigma }_{3fU})$ represents the corrected equivalent confinement contribution introduced by RTDG reinforcement.

The functional form of the correction function k was determined through MATLAB R2022b. Calibration inversion of triaxial test data revealed that the correction function k exhibits strong sensitivity to confining pressure. Accordingly, $k\left({\sigma }_{3fU}\right)$ was selected as the dominant control variable for model formulation, with the influence of reinforcement layers implicitly incorporated. To ensure that the overall evolution trend of $k\left({\sigma }_{3fU}\right)$ remains consistent with the triaxial tests response while avoiding reduced generalisation capability caused by over-parameterisation, a linear functional form $k\left({\sigma }_{3fU}\right)=m{\sigma }_{3fU}+n$ was adopted.

By solving the Ordinary Least Squares (OLS) regression for all test condition samples, the regression coefficients were obtained as m = −0.012 and n = 14.222. Consequently, the optimal correction function k corresponding to confining pressures of 200, 300, and 400 kPa are 11.816, 10.612, and 9.409, respectively, as illustrated in Figure 20. The corrected additional stress values $\Delta {\mathsf{\sigma }}_3^{\prime }$ are summarised in Table 5.

Figure 20. The optimal correction factors k.

Table 5. Additional confining pressure $\Delta {\mathsf{\sigma }}_3^{\prime }$ for CEB model.

${\mathsf{\sigma }}_3$ (kPa)	One-Layer Reinforcement	Two-Layer Reinforcement	Three-Layer Reinforcement
200	10.36	43.76	82.13
300	35.98	65.89	93.45
400	95.66	126.32	170.78

Furthermore, based on the Mohr–Coulomb failure criterion, the shear strength calculation equations derived for the CEB model is presented in Equations (10) and (11):

$${\tau }_{CEB}=\sigma tan {\varphi }_{mob}$$

(10)

where

$${\varphi }_{mob}={sin }^{-1}({\displaystyle \frac{{\sigma }_{1fR}-{\sigma }_{3fR}}{{\sigma }_{1fR}+{\sigma }_{3fR}}})$$

(11)

5.2.3. Interface–Friction-Based (IFB) Model

The IFB model treats the shear strength of RTDG-reinforced weathered sand as the combined contribution of the soil’s inherent strength and the additional resistance mobilised at the soil–reinforcement interface. At the macroscale, it is assumed that the shear strength parameters of the weathered sand remain constant, with the reinforcement effect primarily manifested through interfacial friction. Under this assumption, the Mohr–Coulomb strength of the unreinforced weathered sand was taken as the baseline, and the RTDG contribution was incorporated through an interface-related term. The reinforcement contribution is quantified by introducing a dimensionless interfacial efficiency factor Rint (Equation (12)), which describes the relative contribution of the soil–RTDG interface friction compared with the frictional strength component of the unreinforced weathered sand.

$$R_{int}={\displaystyle \frac{tan \delta }{tan {\varphi }_{fU}}}$$

(12)

where δ = 29.8° from Section 2.3, ${\varphi }_{fU}$ = 38.2° from Figure 15, and $R_{int}=0.728$ from the calculation.

According to the Mohr–Coulomb theory, the shear strength provided by weathered sand and that generated at the soil–RTDG interface may be expressed as Equations (13) and (14):

$${\tau }_{fU}={\sigma }_{1}tan {\varphi }_{fU},$$

(13)

$${\tau }_{int}={\sigma }_{1}tan \delta ={\sigma }_{1}tan {\varphi }_{fU}·R_{int}$$

(14)

However, triaxial test results indicate that the reinforcement effect is not governed by simple linear superposition. To address the influence of multi-layers reinforcement, a layer efficiency function $\eta \left(N\right)$ is introduced, and a dynamic coupling function $\beta \left({\sigma }_3\right)$ is used to account for the coupling influence of confining pressure on the mobilisation of reinforcement effect. Here, $\eta \left(N\right)$ and $\beta \left({\sigma }_3\right)$ are dimensionless empirical functions. The shear strength of RTDG-reinforced weathered sand can be expressed as Equation (15):

$${\tau }_{IFB}={\sigma }_{1}tan {\varphi }_{fU}·(1+{\eta \left(N\right)·R}_{int})·\beta \left({\sigma }_3\right)$$

(15)

For parameter determination of the $\eta \left(N\right)$ and $\beta \left({\sigma }_3\right)$ functions, a loop iteration algorithm was employed using MATLAB for parameter inversion. By invoking predefined anonymous functions, shear strength predictions were computed based on state variables. The Levenberg–Marquardt algorithm was applied to minimise the squared error objective function between triaxial test values and predicted values, as defined in Equation (16):

$$min J={\displaystyle {\sum }_{i=1}^m}{\left({\tau }_{pred,i}-{\tau }_{test,i}\right)}^2$$

(16)

As shown in Figure 21, the parameter inversion results under the present test conditions yield the following calibrated functions: $\eta \left(N\right)=0.158+0.074N$, $\beta \left({\sigma }_3\right)=1.062-0.015·ln\left({\sigma }_3\right)$.

Figure 21. Expression of Nonlinear Effect Function.

To validate the robustness and physical consistency of the selected functional form, a sensitivity analysis was further conducted. The predicted shear strength values under arbitrary operating conditions are shown in Figure 22. It can be observed that shear strength increases approximately linearly with increasing reinforcement layers, whereas the effect of confining pressure gradually approaches saturation. This trend aligns with the assumed functional structure, demonstrating the validity of the IFB model within a certain stress path range.

Figure 22. Sensitivity Analysis of Nonlinear Effect Function.

5.2.4. Shear Strength Calculation and Applicability Analysis

To comprehensively evaluate the applicability of the aforementioned predictive models under RTDG-reinforced weathered sand, this study systematically compares the prediction error characteristics of the CEB model and IFB model based on triaxial test values and Equations (10) and (15) prediction values.

Figure 23 illustrates the overall correspondence between predicted and test values. It can be observed that the IFB model exhibits greater accuracy across the entire data range, with predictions clustering more consistently near the ideal reference line (y = x). In contrast, the CEB model demonstrates noticeable dispersion under certain operating conditions.

Figure 23. Comparison of predicted and experimental values.

The relative error distribution in Figure 24 reveals that the IFB model exhibits significantly more convergent error intervals, with an extreme error of merely 3.02, substantially lower than the CEB model’s 5.95. Moreover, the IFB model’s median of 2.01 closely approximates the mean average relative error of 2.40, indicating minimal error fluctuation and robust consistency. This contrasts with the CEB model, which displays a broad error distribution with a single pronounced outlier, rendering its mean average relative error statistically unrepresentative.

Figure 24. Relative Error Plot of Prediction Models.

Further quantitative error metrics were calculated to evaluate the prediction accuracy and error dispersion of the CEB and IFB models within the present dataset. These metrics are summarised in Table 6 and include RMSE, MAE, MAPE, maximum relative error (Max RE), and standard deviation of relative error (STD).

Table 6. Quantitative error indicators of the CEB and IFB models.

Model	RMSE	MAE	MAEP	Max RE	STD
CEB	41.6	8.42	2.7	8.8	44.1
IFB	22.8	8.10	2.4	3.0	7.8

In addition, the Weighted Radar Area Index (WRAI) was introduced to provide an integrated comparison based on the normalised error indicators. The WRAI was calculated as:

$$\mathrm{WRAI}={\displaystyle \frac{1}{2}}\sin \left({\displaystyle \frac{2\pi }{n}}\right){\displaystyle {\sum }_{i=1}^n}{r}_i{r}_{i+1}$$

(17)

where n is the number of error indicators, $r_i$ is the normalised value of the i-th error indicator, and $r_{n+1}=r_1$.

A smaller WRAI indicates a smaller enclosed area of the normalised error indicators, corresponding to lower overall prediction error and better error-distribution stability. As shown in Figure 25, the WRAI of the IFB model is 0.033, which is significantly smaller than the 0.076 of the CEB model. This indicates that, within the present dataset, the IFB model not only achieves a lower overall error level but also exhibits a more balanced error distribution across different performance metrics. In contrast, the CEB model demonstrates pronounced weaknesses in several indicators, particularly RMSE, Max RE, and STD.

Figure 25. Multi-error Indicator Radar Chart.

Figure 26 presents a three-dimensional error response surface analysis for both models. Compared to the uneven fluctuating characteristics exhibited by the CEB model’s error surface, the IFB model’s error surface demonstrates a more continuous error distribution with a smaller overall amplitude. This reflects its more stable predictive performance under the coupled effects of multiple factors.

Figure 26. Three-dimensional error response surface of the predictive model.

From a mechanistic perspective, the aforementioned discrepancies can be attributed to the introduction of the η–β parameter system within the IFB model. By explicitly characterising the coupling relationship between confining pressure levels and the RTDG reinforcement interface effect, the reinforcement contribution can be progressively mobilised according to the stress state, rather than being simplified as an empirical gain term. This enables the IFB model to better reproduce the observed variation in reinforcement effects under different confining pressures and reinforcement layers within the present dataset. Consequently, the IFB model provides an effective prediction approach for estimating the shear strength of RTDG-reinforced weathered sand within the investigated test range. The corresponding prediction results may serve as a preliminary reference for the engineering evaluation of RTDG-reinforced weathered sand.

6. Conclusions

This study investigates the shear behaviour of RTDG-reinforced weathered sand under different confining pressures and reinforcement layers through large-scale triaxial tests. The analysis focuses on failure mode, mechanical response, volumetric behaviour, and the evolution of key parameters. Accordingly, two shear strength models based on different reinforcement mechanism assumptions are further proposed. The main conclusions are as follows:

(1)

RTDG reinforcement significantly alters the shear failure mode of weathered sand by inhibiting the development of shear planes and promoting a transition towards bulging failure. Unreinforced specimens consistently exhibited general shear failure under all confining pressure conditions. By contrast, reinforced specimens no longer developed fully penetrating shear planes. Under one reinforcement layer at a confining pressure of 200 kPa, the specimen exhibited localised shear failure. With further increases in confining pressure and reinforcement layers, the failure mode transitioned to bulging failure.

(2)

RTDG reinforcement modifies the mechanical response of weathered sand and effectively enhances the ultimate deviatoric stress of the specimens. Compared with unreinforced specimens, the stress–strain behaviour of reinforced specimens shifts from strain softening to strain hardening, with peak deviatoric stress increases of approximately 20–30%. However, the incremental gain in ultimate deviatoric stress diminishes with increasing reinforcement layers. It is speculated that the reinforcement efficiency of RTDG may gradually decrease at higher reinforcement densities.

(3)

The RTDG reinforcement exerts a certain influence on shear strength, manifested in the enhancement of apparent cohesion and internal friction angle. Specifically, the apparent cohesion exhibits near-linear growth with an increase rate of 37.2% as the number of reinforcement layers increases. Conversely, the internal friction angle exhibits weaker sensitivity to reinforcement effects and remains largely governed by the inherent properties of the soil particles.

(4)

Volumetric behaviour is manifested in the coupled effects of confining pressure and reinforcement layers. Unreinforced specimens exhibited a transition from shear contraction to shear dilation. Increasing confining pressure reduced the maximum dilation strain from 3.0% to 1.4%, and the peak dilation angle from 16.3° to 12.8°. In reinforced specimens, shear dilation was markedly suppressed. The maximum dilation strain and peak dilation angle were reduced to approximately 50% and 77%, respectively, of unreinforced values. Increasing reinforcement layers further decreased the maximum dilation strain from 1.8% to 1.0% and delayed its onset axial strain from 4.4% to 5.6%. Under high confining pressure and multiple reinforcement layers, shear dilation was completely suppressed, and volumetric behaviour was dominated by shear contraction.

(5)

Two shear strength prediction models were proposed to quantify the reinforcement effect of RTDG. Comparative analysis indicates that, within the present dataset, the IFB model exhibits lower prediction error and dispersion than the CEB model, with a WRAI value of 0.0331. Accordingly, the IFB model provides an effective prediction approach for estimating the shear strength of RTDG-reinforced weathered sand within the investigated test range. The corresponding prediction results may serve as a preliminary reference for the engineering evaluation of RTDG-reinforced weathered sand.

(6)

Although the present study confirms the reinforcement effect of RTDG and develops two strength prediction models, the findings should be interpreted within the current experimental scope. The models were calibrated using compacted weathered sand with an initial water content of 8% and a relative density of Dr = 0.7, reinforced with RTDG having an effective aperture of 60 mm and arranged in 0–3 layers, under triaxial loading conditions with confining pressures of 200, 300, and 400 kPa. Therefore, the calibrated model parameters and error comparisons are mainly applicable to RTDG-reinforced weathered sand with similar soil states, reinforcement configurations, and laboratory loading conditions. Future studies should further consider the effects of different soil types and densities, RTDG apertures and layer spacings, wider stress ranges, loading rates, and cyclic loading conditions. In addition, the environmental compatibility and long-term durability of RTDG should be further investigated, including leaching behaviour, rubber ageing, creep, wet–dry and freeze–thaw effects, construction-induced damage, and field-scale embankment performance.