Seismic displacement of earth slopes incorporating co‐seismic accumulation of dynamic pore water pressure

The earthquake‐induced displacement of sloping soil mass is an important indicator of co‐seismic landslide initiation. In this paper, an improved Newmark displacement model that considers the accumulation of dynamic pore water pressure (DPWP) in the soil caused by both vertical and horizontal ground motions is proposed. The model can quantitatively describe the dynamic changes of the seismic yield acceleration of near‐saturated infinite soil slopes. Three different types of ground motion time histories are selected to compare the performance of the proposed Newmark displacement model, and the influence of DPWP accumulation on the slope yield acceleration and on the seismic displacement is obtained. The seismic slope displacement analyses indicate that the weakening effect of slope yield acceleration caused by bidirectional earthquake excitation‐induced DPWP is more obvious than when only considering the horizontal ground motion, when the slip surface soils are in near‐saturated state. The effect of initial saturation (Sr0) on the DPWP accumulation caused by vertical ground motion is also investigated. Furthermore, the accumulated seismic displacement can be reasonably explained by the frequency distribution characteristics of the ground motions. Finally, the numerical results of this paper show that the seismic displacement model seldomly considering the effect of DPWP, or only considering the DPWP induced by horizontal ground motion, can significantly underestimate the displacement value when the slip surface soils are in a near‐saturated state.


NOVELTY
1. Improved Newmark modeling of seismic displacement of shallow landslides; 2. Modeling the seismic effect of dynamic pore water pressure in the presence of both vertical and horizontal ground motions; 3. Quantitatively describing the dynamic changes of the seismic yield acceleration of near-saturated soil slopes; 4. Detailed explanation of the seismic slope displacement accumulation by investigating the frequency distribution characteristics of the ground motion.
general, the methods of seismic slope stability analysis can be roughly classified into three categories 1 : the pseudo-static approach, the stress-deformation approach, and the Newmark displacement approach. The pseudo-static approach is simple and easy to use, but the pseudo-static coefficient is difficult to determine and it ignores the time history characteristics of ground motion such as the frequency and the duration. 2 The stress-deformation approach is more accurate and it can reflect the real process of slope deformation and instability, whereas the computations are time-consuming and the modeling process is relatively complicated. 3 In contrast, the Newmark displacement approach overcomes these shortcomings by simplifying the landslide body as a sliding block on an inclined base. The seismic slope displacement can be obtained by the quadratic integration of the acceleration time history curve that exceeds the yield acceleration of the underlying sliding soil mass. 4 Since most earthquake-induced landslides are of the type of shallow and plane sliding, 5,6 the Newmark displacement model is an effective tool to deal with shallow landslide predictions. [7][8][9][10][11] For some specific shallow landslides formed by rock-soil mixtures of the slip surface, the cohesion of the shear strength used in the Newmark analysis is often disregarded. [12][13][14][15] It is worth pointing out that there are some potential assumptions associated with the Newmark displacement approach, which needs to be improved by taking various real-world conditions into account. For instance, the traditional Newmark displacement approach assumes that the direction of seismic movement is downward along the sliding surface, ignoring the fact that both vertical and horizontal seismic forces alternate in direction, and their magnitude also change with the excitation. Some previous studies show that the vertical ground motion can have an important influence on the seismic stability of soil slopes. [16][17][18][19][20] Zhang et al. 21 analyzed the effect of vertical ground motion on the Wenchuan earthquake landslides by FLAC modeling and concluded that calculated slope displacements are greater when both vertical and horizontal ground motions acted simultaneously than when only horizontal ground motion is considered. Korzec and Jankowski 22 conducted shaking table tests under bidirectional cyclic loads for thirty-five European seismic records, showing that the underestimation of permanent displacement can reach up to 19% if the vertical excitation is neglected.
In addition, the traditional Newmark displacement approach usually ignores the effect of dynamic pore water pressure (DPWP), which is an important parameter for significant damage to pore pressure-prone soil slopes. [23][24][25] For the seismic displacement analysis of soil slopes using Newmark model and the like, there are also some previous investigations that consider the DPWP, which showed that the yield acceleration will degrade and seismic displacement increases significantly in saturated sandy soil slopes as a result of the co-seismic DPWP buildup. [26][27][28][29][30] Huang et al. 31 investigated the influence of changing groundwater table and time-dependent DPWP on the dynamic stability of unsaturated sandy slope using finite element modeling and shaking table test, and the results demonstrated that the seismic failure of slope with high groundwater is mainly caused by the water flow around the slope toe where the DPWP is the largest. However, the above-mentioned studies only consider the DPWP accumulation caused by horizontal ground motion and ignore the contribution of the vertical one. Yang 32 used a fully coupled numerical procedure named SUMDES for analyzing the DPWP time history response subjected to various combinations of seismic loading and saturation conditions, and concluded that the effect of vertical excitation on sand liquefaction dependents on the saturation condition. The vertical excitation will only have a minor influence on the build-up of DPWP under full saturation conditions, apart from causing some high frequency oscillation. 33 Nevertheless, the vertical excitation induced DPWP increases by about 100% under semi-full saturation. The semi-full saturation conditions may occur in certain situations as a result of fluctuating groundwater tables associated with natural or man-made processes, such as the downward rainfall infiltration from ground surface or an upwards water flowing from the bedrock to the soil layer. 34 The cases of earthquake-induced shallow landslides in near-saturated soils have been reported elsewhere. [35][36][37] On the other hand, considering the Poisson's ratio of the soil decreases significantly when the saturation decreases from 100% to 80%, the deviatoric stress caused by vertical ground motion will lead to volumetric deformation of the soil, resulting in the accumulation of DPWP. 32 To the F I G U R E 1 Computational model of an infinite slope with DPWP accumulation on slip surface. best knowledge of the authors, it is still rare in literature to consider the accumulation of DPWP in near-saturated soils due to both vertical and horizontal ground motions.
In this paper, an improved Newmark displacement model for seismic slope stability analysis is proposed. Based on the proposed Newmark model, the effects of DPWP in the presence of both vertical and horizontal ground motions are considered. The dynamic changes of the seismic yield acceleration, which is caused by the DPWP accumulation and the corresponding seismic slope displacement, are quantitatively described for near-saturated soil slopes. It is worth noting that the effect of DPWP caused by vertical ground motion on seismic slope displacement is not negligible in soils of near-saturated state.

Calculation model of slope safety factor and yield acceleration
To facilitate the seismic slope displacement analysis accounting for more real-world conditions, an improved Newmark displacement model that considers the accumulation of DPWP in soil mass caused by both vertical and horizontal ground motions is proposed. It is considered that the direction and magnitude of the ground motion will cause the soil mass to have trends of moving both upslope and downslope, respectively, and the safety factor (FS) and the yield acceleration of the slope soil mass are derived based on the limit equilibrium analysis. The forces of the soil mass M in an infinite slope model, 38 as illustrated in Figure 1, are expressed as: where G and γ m are the weight of the sloping soil mass and the soil's unit weight, respectively; l and H are respectively the length and vertical height of the sloping soil mass; β is the slope angle; U 0 is the static pore water pressure; n is the ratio of groundwater table to the total height of the soil column, as shown in Figure 1; γ w is the unit weight of water; ΔU(t) is the DPWP caused by ground motion and Δu*(t) is the normalized DPWP to N; N is the reactive force perpendicular to the slip surface under static pore water pressure; k v and k h are the vertical and horizontal pseudo-static coefficients respectively; r u0 is the initial static pore water pressure ratio. The resultant force along the slip surface changes dynamically when an earthquake ground motion occurs, because the magnitude and direction of the vertical and horizontal ground motions change continuously. 16 When the resultant force of the sliding soil mass is in the downslope direction, the sliding soil mass tends downslope moving, and vice versa for upslope moving. The force balance of the sliding soil mass is illustrated in Figure 2. When the horizontal ground motion is in the upslope direction, the limit equilibrium condition of forces is illustrated in Figure 2 (A). The direction of resultant acceleration may occur both upward and downward along the slip surface. Considering the direction of the vertical seismic force, 39 there are two situations: The soil mass has a tendency to move upslope. The safety factor FS 1 and the yield acceleration k y1 of the sliding soil mass can be written as: The yield acceleration is the value of the horizontal seismic coefficient (k h ) when the safety factor is equal to 1.
where N' is the pressure perpendicular to the slip surface after considering the static and dynamic pore water pressures; c ' and φ ' are the effective cohesion and friction angle respectively; F Ri and F Di are the resistant force and the driving force, respectively; FS i is the safety factor; k yi is the yield acceleration (i = 1, 2, 3 which are used to distinguish between different situations).
Else ℎ − (1 ± ) tan < 0 The soil mass tends to move downslope. The safety factor FS 2 and the yield acceleration k y2 of the sliding soil mass can be written as: When the horizontal ground motion is in downslope direction, the limit equilibrium condition of forces is illustrated in Figure 2 (B). Regardless of the direction of the vertical ground motion, the direction of the resultant acceleration on the sliding surface is always in downslope direction, the safety factor FS 3 and the yield acceleration k y3 of the sliding soil mass can be written as: where N '' is the reactive forces perpendicular to the slip surface under the static and dynamic pore water pressures.

The DPWP accumulation model considering horizontal ground motion
The co-seismic accumulation of DPWP has a great influence on the dynamic properties of the soil. 28,40,41 In the literature, some studies have been carried out to develop the DPWP accumulation model by considering the horizontal ground motion. [42][43][44][45] One of the DPWP accumulation models for sandy slopes considering the horizontal ground motion was proposed by Mele et al., 46 such that * where u* h is the normalized DPWP caused by horizontal ground motion; N is the number of equivalent cycles; N L is the number of cycles at liquefaction; θ is a parameter related to soil properties; S r0 denotes initial saturation (%). From a practical point of view, the model has been properly verified against independent centrifuge test results carried out on saturated and non-saturated soils. Polito et al. 45 proposed an equation to predict the parameter θ in the form of: where FC is the fine particle content (%); D r is the relative density (%) and CSR is the cyclic stress ratio; and c 1 , c 2 , c 3 and c 4 are regression coefficients which are summarized in Table 1.
The number of cycles at liquefaction N L is univocally related to the cyclic resistance ratio CRR 43 : where (N r , CSR r ) is a reference point on the cyclic resistance curve of CRR: N L , where N r is recommended to be 15 43 ; α is a curve fitting parameter; CSR t denotes the asymptotic value when the number of cycles at liquefaction N L tends to infinite. Following Chiaradonna et al., 47 regression equations for CSR r , CSR t , and α can be obtained from experiments, which are expressed as: where q c1Ncs is the corrected cone tip resistance; σ ' v0 is the initial vertical effective stress; p ' 0 represents the atmospheric pressure and the regression coefficients of Equation (25) are shown in Table 2. The corrected cone tip resistance is defined as 48 : where q c is the cone resistance measured during the cone penetration test; P a is the atmospheric pressure.
For an irregular shear loading history such as an earthquake excitation, Equation (22) requires the conversion into an equivalent cyclic load, which depends on the estimation of the number of equivalent cycles empirically. Chiaradonna et al. 43 applied the endochronic theory to express the relationship between the damage parameter and the number of equivalent cycles: where k is the damage parameter that reflects the effect of DPWP accumulation and the weakened resistance of soil in undrained cyclic conditions; k l is the damage parameter at liquefaction. Combining Equations (24) and (33) for CSR equal to CRR, the following expression holds: It is particularly convenient in engineering practice to substitute the number of equivalent cycles with the damage parameter for existing DPWP models. Thus, Equation (22) can be rewritten as: * For any irregular shear loading history, the expression of k with time becomes k(t) 43 : where k 0 is a stepwise function; dk is the increment of the damage parameter; t-dt is the time step; * ( ) anḋ * ( ) are the normalized shear stress and its rates of change respectively; τ(t) is the shear stress with time; σ ' 0 is the initial effective stress; * 0 ( ) = * max ( ) iḟ * ( ) < 0 and * 0 ( ) = otherwise; more details can be found in Chiaradonna et al. 43

Derivation of the DPWP accumulation model considering vertical ground motion
In saturated soils, the accumulation of DPWP caused by vertical ground motion is regarded negligible, 33 but the situation could be completely different in near-saturated soils. The Poisson's ratio of the soil decreases significantly when the saturation decreases from 100% to 80%, 32 thus the deviatoric stress caused by vertical ground motion will lead to volumetric deformation of the soil, resulting in the accumulation of DPWP. The accumulation of normalized DPWP caused by vertical ground motion can be expressed as: where Δu v is the accumulation of DPWP caused by vertical ground motion; σ ' v0 is the initial effective stress; Δε vf is the volumetric strain increment of the fluid; K f and K w are the bulk modulus of the fluid and the pore water respectively; K w is the bulk modulus of pore water, which takes a value of 2×10 6 kPa 49 ; p a is the absolute pressure of the fluid; S r0 is the initial saturation.
The propagation of vertical ground motion has little influence on the nonlinear dynamic behavior of the soil, thus the elastic assumption can be used. 33 The volumetric strain increment of the fluid Δε vf can be computed by 50 : where Δε v is the volumetric strain increment of the soil; n is the porosity of the soil; υ is the Poisson's ratio of the soil; E is the elastic modulus; E/(1-2υ) is the bulk modulus of the soil; Δσ dv and Δσ dh are the vertical and horizontal stress increments.
According to the theory of elasticity, horizontal deformation is not allowed and the horizontal stress increment can be expressed as: For porous-elastic media, the Poisson's ratio of compressible unsaturated soil can be expressed as 49 : where n is the porosity; G is the shear modulus; K b and υ ' are bulk modulus and Poisson's ratio of the soil skeleton, υ ' takes a value of 0.3. 49 The equivalent tensile and compressive stress σ dv caused by vertical ground motion can be expressed as: where F denotes the force of the vertical ground motion on the soil; m and γ are mass and weight of the soil, respectively; g and a v are the gravity and vertical acceleration, respectively; A is the area of force F and σ z0 is the initial vertical stress at depth Z. Combining Equations (35) and (40), the accumulation of total normalized DPWP is derived as: The analysis of seismic slope stability with the DPWP accumulation subjected to bidirectional earthquake excitations incorporates four steps: (1) input geometry and geotechnical properties of slope, seismic input motion (i.e., the input); (2) calculate the normalized DPWP caused by horizontal ground motion; (3) calculate the normalized DPWP caused by vertical ground motion; and (4) calculate the safety factor, the yield acceleration and the seismic displacement (i.e., the output).
The entire methodology is illustrated in a flowchart shown in Figure 3, and the application of this method will be subsequently described in detail.

Verification of the proposed method using a simplified worked example
To benchmark the model performance, a simple sinusoidal ground motion with 0.15 g amplitude and 1 Hz frequency was employed, as shown in Figure 4. The "None-DPWP" means there is no effect of DPWP, "H-DPWP" means that DPWP is induced by horizontal sinusoidal ground motion only for fully saturated sandy slope. The effect of initial saturation and of vertical ground motion is neglected. Illustrated by performing seismic displacement analysis of infinite slopes, the F I G U R E 3 The flowchart of seismic stability analysis with DPWP accumulation subjected to bidirectional earthquake excitations.
parameters of the slope are shown in Table 3. The relationship between e max , e min , and FC is determined by the empirical model suggested by Chang et al. 51 Note that the DPWP generated on the slip surface has a more significant effect on the sliding displacement than the nonlinear change in soil material, 28 this study will focus on the seismic displacement analysis by modeling the DPWP accumulation effects only. The sinusoidal ground motion with 10 s duration and 0.02 s interval is considered first. The sinusoidal ground motion induced shear stress is calculated by the equivalent linear method. 52 The cyclic strength parameters are adopted from Chiaradonna et al. 43 for convenience, α = 1.17, CSR t = 0.09, CSR r = 0.122, k l = 1.07. The damage parameter k is calculated by Equations (36) to (39). Thus, the normalized DPWP caused by horizontal sinusoidal ground motion u* h can be derived from Equation (35). Finally, the FS and the yield acceleration are derived from Equations (6) to (21). The shear strength of the infinite sloping soil is assumed to be Mohr-Coulomb criterion. The corresponding seismic displacement can be obtained by the Newmark's double integration concept.  where ρ is the soil's natural density; e is the natural void ratio; e min is the minimum void ratio; e max is the maximum void ratio; V s is the shear wave velocity; θ is a parameter in the DPWP accumulation model considering horizontal ground motion.
As shown in Figure 4 (A), the slope yield acceleration remains horizontal in the case of "None-DPWP" while it is weakened significantly from 0.144 to 0.0014 in the "H-DPWP" case. Figure 4

Effects of vertical ground motion and DPWP on the improved model
To illustrate the application of the improved Newmark displacement model considering the DPWP accumulation caused by vertical ground motion, the typical infinite slope was investigated. The slope parameters are given in Table 4. Note that c* = c ' /γ m H is a dimensionless parameters representing soil properties. The movement tendency of the soil mass in the downslope direction and upslope direction are discussed respectively, as shown in Figure 5.  In the case of β = 25 • , k h = 0.15, the soil mass tends to move downslope as shown in Figure 5 (A)-(D). The FS of the slope decreases with the increase of vertical downward pseudo-static coefficient, whereas increases with the vertical upward pseudo-static coefficient. It was found that the FS is less affected by the vertical pseudo-static coefficient for the case c* = 0.
In the case of β = 15 • , k h = 0.40, as shown in Figure 5 (E,F), the soil mass tends to move upslope with the horizontal pseudo-static coefficient k h in the upslope direction. Because the soil mass always moves downslope when the horizontal ground motion is in the downslope direction regardless of the direction of the vertical ground motion, the FS of the slope increases in a systematic way with the vertical downward pseudo-static coefficient, and decreases with the increase of the vertical upward pseudo-static coefficient. In a word, the vertical ground motion may bring both positive and negative impacts to the stability of the slope, the case of which is determined by the magnitude and direction of the vertical and horizontal ground motions. Therefore, vertical and horizontal ground motions should be considered simultaneously in the seismic slope stability analysis.
The FS of the slope is reduced by a maximum of 1.3 with the increase of normalized DPWP Δu*(t) from 0 to 0.7 under constant soil properties, as depicted in Figure 5 (A)-(F). This result indicates that the co-seismic accumulation of DPWP causes a significant reduction in the FS, which cannot be neglected in seismic slope stability analysis.

Seismic displacement analysis of near-saturated sandy slope under bidirectional earthquake excitations
It has been well illustrated that the DPWP changes significantly with the initial state of soil saturation S r0 during an earthquake excitation. 32 In this paper, an improved Newmark displacement model that considers the DPWP accumulation in soils caused by both vertical and horizontal ground motions is proposed. Illustrated by seismic displacement analysis of infinite slopes, the model parameters are the same with Table 3. The relationship between the cone resistance and soil shear strength parameters can be determined by Muromachi. 53 To quantitatively compare the proposed model with others, three different ground motion histories were selected from the NGA-West2 database. In order to compare the magnitude of seismic displacement produced by different ground motions, all these ground motions were scaled to a peak horizontal acceleration (PHA) of 0.540 g, and the peak vertical acceleration (PVA) was scaled accordingly. More details are shown in Table 5 and Figure 6.

Taiwan earthquake excitation
For an infinite slope with initial yield acceleration and FS of 0.144 and 1.423, respectively, the yield acceleration time histories in four different S r0 are shown in Figure 7 (A)-(D). The "None-DPWP" means there is no effect of DPWP, "H-DPWP" means that DPWP is induced by horizontal ground motion only, "H-DPWP (S r0 = 1.00)" means that DPWP is induced by horizontal ground motion only in fully saturated soil which is the same when the S r0 of the "H-DPWP" case is 1.00 and "HV-DPWP" means that DPWP is induced by both horizontal and vertical ground motions. The S r0 ranging from 0.80 to 1.00 in 0.05 intervals were analyzed. In the case of "None-DPWP," the slope yield acceleration fluctuates but approximately remains horizontal. The slope yield acceleration is weakened significantly at about 10 s in the "H-DPWP" case, but the weakness is no longer apparent after 11 s. In the case of "H-DPWP (S r0 = 1.00)," the weakness of slope yield acceleration is greater than the "H-DPWP" case, but this difference decreases with the increase of S r0 . Due to the accumulation effect of DPWP caused by vertical ground motion, the weakness of slope yield acceleration in the case of "HV-DPWP" is more obvious, the degree of weakness aggravates with the increase of S r0 . The weakness of yield acceleration is greatest at the S r0 = 0.95, which decreases from 0.144 to 0.075. Figure 8 (A)-(D) demonstrate the effects of S r0 on the slope seismic displacement. The seismic displacement is 4.4 cm in the "None-DPWP" case, which shows almost no change with S r0 . In the "H-DPWP" case, the seismic displacement increases from 5.0 to 5.4 cm with S r0 increasing from 0.80 to 0.95. In the "H-DPWP (S r0 = 1.00)" case, the seismic displacement is 5.5 cm. Overall, the S r0 has little effect on the seismic displacement in this condition. In the case of "HV-DPWP," the seismic displacement of the slope has enlarged from 5.2 to 6.0 cm with S r0 enlarging from 0.80 to 0.95.

Loma earthquake excitation
The Loma earthquake excitation, which has much more high and medium frequency components, was applied on the same infinite slope. The yield acceleration time histories in four different cases of the S r0 are shown in Figure 9 (A)-(D). The yield acceleration shows more dramatic fluctuations. In the case of "None-DPWP," the slope yield acceleration fluctuates sharply from 5 to 10 s and maintains a horizontal trend in general. In the case of "H-DPWP," the slope yield acceleration starts to weaken significantly at about 4 s until 12 s and then remains constant. In the case of "H-DPWP (S r0 = 1.00)," the weakness of yield acceleration is greater than the "H-DPWP" case, and this difference decreases with the increase of S r0 . The weakness of yield acceleration is more obvious in the case of "HV-DPWP" and it aggravates with the increase of S r0 . At S r0 = 0.95, the weakness of yield acceleration is greatest, which decreases from 0.144 to 0.029. The seismic displacement shown in Figure 10 (A)-(D) is 2.7 cm in the "None-DPWP" case, showing almost no change with S r0 . In the case of "H-DPWP", the seismic displacement increases from 4.3 to 5.2 cm with S r0 increasing from 0.80 to 0.95. The seismic displacement is 5.7 cm in the "H-DPWP (S r0 = 1.00)" case. In the case of "HV-DPWP," the seismic displacement increases from 5.2 to 13.6 cm with S r0 increasing from 0.80 to 0.95.

Kobe earthquake excitation
The results of yield acceleration time histories obtained by applying Kobe earthquake excitation on the same infinite slope in the cases of different S r0 are shown in Figure 11  acceleration is greater than the "H-DPWP" case. Due to the DPWP accumulation caused by vertical ground motion, the weakness of yield acceleration in the case of "HV-DPWP" is more significant with the increase of S r0 . The weakness of yield acceleration is greatest at the S r0 = 0.95, which decreases from 0.144 to 0.017. Figure 12 (A)-(D) show that the seismic displacement of the slope in the "None-DPWP" case is 5.6 cm, which does not change with S r0 . When the S r0 increases from 0.80 to 0.95 in the case of "H-DPWP," the seismic displacement of the slope increases from 8.7 to 10.3 cm, which is only a small change with the increase of S r0 . The seismic displacement is 11.0 cm in the "H-DPWP (S r0 = 1.00)" case. In the case of "HV-DPWP," the seismic displacement of the slope increases from 10.4 to 28.1 cm with S r0 increasing from 0.80 to 0.95.
The above three-time history analysis results indicate that in the near-saturation state, the weakening effect of slope yield acceleration caused by DPWP is more obvious than when only considering the horizontal ground motion. The seismic slope displacement is larger and increases gradually with S r0 . The traditional Newmark displacement model seldomly considers the effect of DPWP, or only considers the DPWP induced by horizontal ground motion, which can underestimate the seismic displacement of the slope when the slip surface soils are in a near-saturated state. In this study, we have shown that the DPWP induced by vertical ground motion under different values of S r0 has a great effect on the calculated seismic slope displacement.
Note that the resistant forces under static and dynamic conditions are the same and constant in the traditional Newmark displacement model. 4,54,55 In this paper, both resistant and driving forces of the soil are assumed changing with ground motions and the DPWP accumulation. Based on the expression of FS in Section 2.1, the DPWP accumulation will reduce the FS until the end of ground motion. Figure 13 shows the FS of the slope under three different earthquake excitations. It can be found that the FS decreases more significantly with the increases of S r0 from 0.80 to 0.95. For example, the FS decreases from 1.423 to 1.079 under Kobe earthquake excitation when the S r0 is 0.95. The Loma earthquake excitation has the greatest decrease in the FS from 1.423 to 1.010, which can be attributed to the fact that the accumulation of DPWP  in the Loma case is most significant, as shown in Figure 14. The decrease in the FS will be less significant in the case of "H-DPWP (S r0 = 1.00)." In a word, the traditional Newmark displacement model may overestimate the stability of seismic slopes when the soil at the slip surface is in a near-saturated state.

Frequency content analysis
Further, the Fourier transform is performed for the three excitation histories. The Fourier spectrum obtained by converting the time data to frequency data is shown in Figure 15. The average period proposed by Rathje and Bray 56 was used to evaluate the frequency characteristics of the ground motions, the results are shown in Table 6. = ∑ 2 (1∕ ) where C i is defined as the square root of the sum of the squares of the real and imaginary parts of the Fourier coefficient; f i is the discrete Fourier transform frequencies between 0.25 and 20 Hz. The Kobe and Taiwan earthquakes are mainly concentrated in low frequencies for horizontal ground motion. The Fourier amplitude of the Kobe earthquake is the largest, and the frequency distribution of the Loma earthquake is broader with a smaller amplitude. As for vertical ground motion, both Kobe and Loma earthquakes have a wide range of frequency distribution, with the Kobe earthquake having the largest Fourier amplitude, while the Taiwan earthquake is concentrated at low frequencies. Table 6 shows that the larger the frequency, the smaller the average period of the ground motions.
Comparing Figures 8, 10, 12, and 15, the seismic slope displacement increases sustainably considering the effect of DPWP accumulation caused by both horizontal and vertical ground motions. The seismic slope displacement ignoring the effect of DPWP accumulation increases to a specified magnitude and then remains constant, which means the DPWP accumulation caused by horizontal and vertical ground motions has different effects on the seismic slope displacement. The seismic displacement is accumulated almost instantaneously for the low-frequency concentrated Taiwan earthquake. However, the accumulation of seismic displacement duration is longer (about 10s) for the Kobe earthquake with a wider distribution frequency of vertical ground motion, both the PVA/PHA ratio and the seismic displacement are the largest. As for the Loma earthquake, horizontal and vertical ground motion frequencies are concentrated in the range of medium and high. The accumulation of seismic displacement only takes about 5 s, which means that the wider frequency range for horizontal ground motion, the faster accumulation rate of seismic slope displacement.
The results demonstrate that the accumulation rate of the seismic slope displacement is affected by the frequency distribution of vertical ground motion. The wider the frequency distribution, the slower the accumulation rate of seismic slope displacement, and the longer the duration of excitation damage to the soil slope.

DISCUSSION ON SEISMIC DISPLACEMENT OF SILTY SAND SLOPE
The above analyses are based on sandy slopes without considering the effective cohesion. It is necessary to analyze the near-saturated sandy slope with low effective cohesion when S r0 changes. The parameters of four conditions are shown in Table 7, in which β = 20 • , H = 5.3 m, r u0 = 0.15, V s = 174 m/s, FC = 45%. All the soil slope parameters used in this study are adapted from the China's Engineering Geology Handbook. 57 They are typical and reliable empirical data for sand or silty sand slopes obtained through many geotechnical tests. In the next, the Loma earthquake was selected for time history analysis.
Comparing the condition I with the condition II in Figures 16 and 17, respectively, it can be found that the difference in the seismic slope displacement is 2.4 cm in the case of "H-DPWP," and 7.0 cm in the case of "HV-DPWP." For conditions III and IV, the difference of the seismic displacement is 0.6 cm in the "H-DPWP" case and 4.1 cm in the "HV-DPWP" case. In a word, the sandy slope with lower effective cohesion has a more obvious tendency of weakening effect for the yield acceleration caused by the co-seismic DPWP accumulation. The seismic slope displacement ignoring the DPWP accumulation varies negligibly with the changing S r0 . Thus, the corresponding seismic displacement can be significantly underestimated by ignoring the effect of DPWP caused by earthquake excitations.

CONCLUSION
In this paper, an improved Newmark seismic displacement model that considers the accumulation of dynamic pore water pressure (DPWP) in the soil caused by both vertical and horizontal ground motions is proposed. Illustrated by performing seismic displacement analysis of infinite slopes, three different types of ground motion time histories are selected to compare the performances of the proposed seismic displacement model by analyzing the influence of DPWP accumulation on the slope yield acceleration and the seismic displacement. Based on this study, the following conclusions are drawn: 1. Depending on the magnitude and direction, the vertical ground motion may give rise to both negative and positive impacts on the seismic stability of the soil slope. In the case of co-seismic DPWP accumulation conditions, the vertical and horizontal ground motions should be considered simultaneously in the seismic slope stability analysis for a more accurate prediction of the seismic displacement. 2. In the near-saturation state of soils, the weakening effect of the yield acceleration for soil slopes caused by DPWP accumulation is more obvious than when only considering the horizontal ground motion. The slope seismic displacement tends to increase when the initial saturation is at a higher value. Compared with the proposed improved model, the traditional Newmark displacement model that ignores the DPWP effect or only considers the DPWP induced by horizontal ground motion may underestimate the seismic displacement of the slope when the soil at the slip surface is in a near-saturated state. Also, the effect of DPWP induced by vertical ground motion on the seismic displacement varies significantly under different initial saturations. 3. The accumulation rate of the slope seismic displacement is affected by the frequency distribution of vertical ground motion. The wider the frequency distribution, the slower the accumulation rate of seismic displacement, and the longer the duration of excitation damage to the soil slope.
4. The sandy slope with lower effective cohesion has a more obvious tendency of weakening effect for the yield acceleration caused by the co-seismic DPWP accumulation. Thus, the corresponding seismic displacement can be significantly underestimated by ignoring the effect of DPWP caused by earthquake excitations.
Note that this paper is mainly based on the improved rigid Newmark displacement analysis, which means the flexible deformation inside the soil mass and the nonlinear dynamic response of the soil are not considered. The coupling of nonlinear dynamics and flexible soil mass under the influence of DPWP by earthquake excitations will require further investigation.

D ATA AVA I L A B I L I T Y S TAT E M E N T
The data that support the findings of this study are available from the corresponding author upon reasonable request.