The damage constitutive model can reflect the fracturing and crushing of piles during horizontal loading, which is unavailable for the traditional elastic model. The reinforcement ratios of piles are represented by different tensile strengths. A reinforced concrete pile undergoes irreversible cracking when it reaches a critical state. The damage constitutive model simulates the destruction of piles in the reduction of tensile strength and shear strength. To simulate the cracking performance, both the tensile strength and the elastic modulus perpendicular to the crack are set to zero when the tensile stress of a zone exceeds the tensile strength of the concrete (Cedolin et al., 1982; Comodromos et al., 2009).

Furthermore, the shear strength of piles will be reduced due to the shear stress redistribution on cracks. The concrete interface crossed by a reinforcing bar in the pile will be subjected to a shear displacement when the landslide happens. A local increase in the crack width will occur due to the deformation of protruding asperities in the shear displacement (Maekawa and Qureshi, 1997). The lateral dilatancy leads to a pullout force that responded by reinforcing bars, which is balanced with the compressive force acting on the concrete near the bar (Murcia-Delso and Benson Shing, 2016). Based on the aforementioned phenomenon, Tassios and Vintzeleou (1987) have proposed a phenomenological model to describe the relationship between normal compressive stresses with steel stress: σ c = ρ σ s , (1) where σ_c is the normal compressive stress acting on the interface, σ_s is the steel stress, and ρ is the reinforcement percentage.

Significant dilatancies are expected when cracks through reinforced piles occur due to the interlocking of the protruding aggregates (Millard and Johnson, 1984). The considerable dilatancy causes high tensile stresses in the bars. It is precisely due to this dilatancy performance that a frictional resistance from the reinforcing bar is an evident part of the total shear resistance on the cracking interface (Harries et al., 2012). To obtain the shear strength in the reinforced pile, a model established by Tassios and Vintzeleou (1987) is applied. τ f r , max = μ σ c , max , (2) where τ_fr,max is the maximum shear stress due to friction, μ is the friction coefficient, and σ_c,max is the maximum normal compressive stress acting on the interface. The τ_fr,max is regarded as the shear strength in the numerical analysis. The above damage constitutive model was coded into FLAC3D by the FISH programming language.

Figure 3 indicates a three-dimensional finite difference computational model of pile-reinforced slopes. An idealized slope with a height (H) of 10 m and a length (L) of 15 m is simulated with the software FLAC3D. The bedrock is 15 m below the ground surface as a rigid base. A damage constitutive simulation models a pile with a diameter of 1 m. L _x is the horizontal distance from the slope toe to the pile. Hence, the ratio between L _x and L (L _x /L) can be represented by the installation location of the pile. Interface elements in FLAC3D are applied to assume the interface between the pile and the soil, and the stiffness (k _n ) can be determined as follows (Itasca, 2006): k n = 10 × max [ ( K + 4 3 G ) Δ Z min ] , (3) where K is the bulk moduli, G is the shear moduli, and the ΔZ _min is the minimum width of an adjacent zone in the normal direction. In the current study, the mechanical parameters were obtained from Cai and Ugai (2000). The mechanical parameters of the soil and the pile are summarized in Table 1.

Figure 4 indicates that the FOS of the pile-reinforced slope varied along the location of the pile. Three pile models are applied in the simulations, namely, the elastic constitutive model (ECM), damage constitutive model considering a balanced-reinforced (DCM-B) pile, and under-reinforced (DCM-U) pile, respectively. Moreover, the results calculated by a finite element method under the same parameters are plotted in Figure 4, where the piles are treated as linear elastic solid material (Cai and Ugai, 2000). A comparison of the findings with those of Cai and Ugai (2000) shows that they are similar, which confirms that the damage constitutive model is valid. It can be observed that the FOSs are in a parabolic shape along the L _x /L, and the most forceful pile location is in the middle of the slope. Compared to the FOS of the ECM, the FOS calculated by the DCM-U makes an approximately 12% reduction at L _x /L = 0.5. Nevertheless, the maximum FOS of DCM-B is just 2% smaller than that of the ECM, which makes a slight reduction. Interestingly, the FOS considered by DCM-B is approximately equal to the FOS of the ECM at the whole range of L _x /L. The FOSs obtained by DCM-B and DCM-U differ from those calculated by the ECM. The reason for this phenomenon is that the damage constitutive model can reflect the cracking development of the pile.

The observations are further expounded in Figures 5–7, which present the positional variation of a slip surface for elastic and damage constitutive models. The depth development of cracks on the pile is marked on the right side of the simulated pile. It can be seen in Figures 5–7 that the unstable slip surface will not pass through the pile when the pile installation location is close to the slope toe or crest. When the pile is positioned close to the middle of the slope (Figure 6), slope failure passes through the pile. Meanwhile, the different types of constitutive models reflect various responses of the piles. Cracks appear in the piles of DCM-B and DCM-U when the pile position is close to the middle of the slope, causing the reduction of the FOS. It can be seen that the level extension of cracks in the under-reinforced pile is much deeper than that in the balanced-reinforced pile. Cracks develop across the cross-section of the under-reinforced pile, while cracks are just presented on the concrete cover of the balance-reinforced pile. Hence, the FOS obtained from DCM-U is smaller than that calculated by the ECM and DCM-B. The damage constitutive model can reflect the influence of cracks on the tensile strength of the pile, which would make a more precise and realistic analysis of the slope stability. The existence of cracks on the pile is not negligible in slope engineering, which would decrease the FOS of the slope. The traditional elastic constitutive model regarded piles as a material without damage, and hence, the stability of the slope will be overestimated. The overestimated factor of slope stability is potentially dangerous in geotechnical engineering. Therefore, it is necessary to simulate piles by the damage constitutive model in pile–slope stability analysis.

Figure 8 represents the relationship between the pile’s reinforcement ratio and the FOS on the slope. The reinforcement ratio could be equivalent to the tensile strength of the pile (f _t ) in this study. The five damage constitutive models with pile positions (L _x /L = 0.1, 0.3, 0.5, 0.7, and 0.9) were compared. When the pile was installed near the top or toe of the slope (L _x /L = 0.1 and 0.9), the f _t affected the FOS slightly. It can be indicated that the FOS, which is the pile position close to the middle of the slope, increases to a constant value in the whole range of f _t . The peak value of the FOS appeared when the f _t was approximately 30 MPa, which means the pile is a balance-reinforced pile. When the pile is under-reinforced, the f _t is less than that of the balance-reinforced pile, and hence, the FOS would be reduced. In addition, the over-reinforced pile made a small contribution to the FOS, which is uneconomical in engineering. This is because the tensile failure would occur on the concrete of the pile instead of the bar. The optimum reinforcement ratio of the pile could be determined by the damage constitutive model at specific slope engineering, which can avoid the appearance of the under-reinforced and over-reinforced piles.

Figure 9 summarizes the distribution of pile behavior characteristics along the depth under the action of landslide, including 1) horizontal displacement, 2) bending moment, 3) shear force, and 4) soil pressure. Pile responses at the specific position were obtained from elastic (ECM) and damage constitutive simulation (DCM-B). The results observed from the ECM and DCM-B are consistent with each other only at L _x /L = 0.1 and 0.9, respectively.

It can be seen from Figure 9A that the lateral deformation is reduced to zero at a depth of approximately 13 m. The horizontal deformation of the pile top of the DCM-B at L _x /L = 0.7 is the biggest, with a value of 210 mm. The slightest lateral deformation of the pile toe is 7.2 mm at L _x /L = 0.1. The displacement of the ECM is smaller by 57% than that of the DCM-B at L _x /L = 0.7, which is an enormous difference. The pile deformation calculated by the DCM-B is more significant than the ECM. Nevertheless, the bending moments of the pile obtained from DCM-B are smaller than that from the ECM at L _x /L = 0.3, 0.5, and 0.7.

Figure 9B shows that the bending moment increases at first and then decreases along the depth. The maximum bending moments at all pile positions occur at the boundary between the natural soil and bedrock. The reason for this phenomenon is that Young’s modulus (E) of natural soil (200 MPa) is relatively smaller than that of bedrock (3000 MPa), which has been shown in Table 1. The apparent distinction of E at the boundary would lead to the stress concentration of the pile, and hence, the maximum bending moment was caused. Additionally, the difference in internal forces between the DCM-B (2.0 MN⋅m) and the EMC (2.6 MN⋅m) is most significant at L _x /L = 0.5.

A similar observation can be found in Figure 9C, which plots the distribution of shear force along the depth. At L _x /L = 0.5, the maximum negative and positive shear force of DCM-B is 17% and 18%, respectively, smaller than those of the ECM. It can be observed that the distribution law of shear force with different pile positions and constitutive models is similar. With the increase of depth, the maximum negative shear force occurs first, and then the maximum positive shear force appears. The depths, where the shear force of piles located at Lx/L = 0.1, 0.3, 0.5, 0.7, and 0.9 is zero, are approximately 5.5, 7.5, 9.5, 10, and 8 m, respectively. Concurrently, the bending moment reaches the maximum value at the corresponding depth where the shear force is zero.

The variations of soil pressure along the depth are presented in Figure 9D. The maximum soil pressure of the pile at L _x /L = 0.1 (0.37 MPa) and 0.9 (0.12 MPa) is smaller than that at other positions, which indicates that the reinforced effect of the pile on the soil is slightly at the top or toe of the slope. At L _x /L = 0.5, the DCM-B results for soil pressures disagree slightly with the data observed by the ECM at 9.0-m and 11.7-m depth. It is expected that the pile is to be installed at the middle of the slope (L _x /L = 0.5), there is an apparent agreement of the simulated data for the soil pressure in the DCM-B and the ECM.

Figure 9 further explains the dissimilarity of FOS in DCM-B and ECM piles. The neglect of pile plastic failure rigidity tends to overestimate the pile capacity. The damage constitutive model can reflect the irreversible plastic deformation of the balance-reinforced pile. Therefore, compared to the elastic constitutive model, the damage model predicted larger lateral displacements and a smaller internal force in the pile at a specific condition.