Numerical investigation on the deformation of railway embankment under normal faulting

Abstract

Active faults in the earthquake region are consistently regarded as a potential geological hazard to the construction and operation of railway engineering. However, the effects of normal faulting on railway embankments have not been investigated thoroughly. For bridging this knowledge gap, three-dimensional finite element analysis considering the influence of faulting offset, the soil layer’s thickness, the fault dip angle and the embankment cross-fault angle are conducted to clarify the normal faulting effects on the railway embankment. Emphasis is given to the stress and strain characteristic in the fault rupture outcropping regions on the embankment, the deformation of the embankment centerline for design purposes, and the determination of the affected zones for railway embankment preservation. The analysis shows that the normal fault rupture outcropping regions on railway embankment are tensile yield in most cases. The existence of the soil layer and its thickening would widen the affected zones and the regions where the fault ruptures outcrops. The fault dip angle and the cross-fault angle of the embankment have a complex effect on the behaviors of the crossing embankment. The depth of the subsidence zone of the embankment would increase with the decrease of the fault dip angle and the large fault dip angle would change the primary fault rupture to be a compressive one directly above the fault line. If the embankment crosses the fault line obliquely, the curvature radius of the centerline would hardly meet the design code.

Introduction

Numerous faults have been created within the continental plates, especially at plate junctions, due to the interaction of the continental plates. Active fault controls the ground deformation and seismic hazards [1], which can induce disasters to the engineering structures such as buildings [2], embankment dams [3], bridges [4], tunnels [5], pipelines [6], roads [7], and railways [8], as shown in Fig. 1a. It can also induce geological hazards such as ground fissures, sand liquefaction, collapse, and landslides [9, 10]. Located at the junction of the Asia-Europe, Pacific, and Indian plates, China is an earthquake-prone country with numerous active faults [11,12,13], as shown in Fig. 1b. Since 1999, the Large-scale Development of the Western Region strategy of China has made the research of railway construction in the Tibet Plateau a hot topic.

Fig. 1

Research backgrounds. a Rail misalignment caused by February 6, 2023 Kahramanmaraş-Türkiye Earthquakes [14]. b Distribution map of active faults in China and its adjacent sea area [15]

In the realm of geotechnical earthquake engineering, scholars have been studying the effects of faulting on tunnels, pipelines and embankment structures by conducting 1g model tests, centrifuge model tests and numerical simulations. Most of the research are focused on tunneling in the mountain fault zones [16,17,18]. Much less effort has been devoted to understanding the effects of active faulting on the overlying earth structure, such as roadway embankments [19, 20], railway embankments [21, 22] and embankment dams [23, 24]. Scaled model tests and numerical simulations are commonly used in studying the deformation of embankment structures under earthquake faulting. Zhang et al. [25] and Yang et al. [26] evaluated the mitigation effects of the three measures (sand layer, geogrid, and CFG piles) with 1g scaled clay embankment models under reverse and strike-slip faulting by comparing the length of the ruptures and the displacement of the crest of the embankment. Petala and Klimis [19, 20] proposed two criteria (relative compaction and volume change) to map and quantify the damage of the 2D embankment caused by dip-slip faulting by using finite discrete modelling. The finite element method is the most commonly used technique. Qu et al. [27] investigated the best position of a raft foundation embedded in the embankment to mitigate the effects caused by dip-slip faulting. The longitudinal displacement and lateral displacement of the embankment centerline are the most commonly used indices [28,29,30,31]. However, the displacements of the embankment haven’t been discussed for the design purpose. The stress and strain characteristics of fault rupture outcropping regions on the embankment remain unclear. The affected zone of the embankment under faulting hasn’t been investigated quantitively considering the variation of the faulting offset, the thickness of the soil layer, the fault dip angle and the cross-fault angle of the embankment.

The main goal of this paper is to present an in-depth numerical analysis of the deformation of cross-fault railway embankment under normal faulting, providing insights for engineers during railway route selection and preservation. A series of parametric studies are conducted to analyze the influences of the faulting offset, the thickness of the soil layer, the fault dip angle and the cross-fault angle of the embankment. In this study, the fault rupture outcropping regions and the stress–strain distribution on the embankment are clarified. The vertical and lateral displacement of the embankment centerline are studied for design purposes by converting them to the longitudinal slope and the radius of the curvature. Finally, the affected zones of the railway embankment under normal faulting are given for railway preservation.

Finite element modeling methodology

The Finite-Element code ABAQUS 2021 is utilized to simulate the deformation of the railway embankment subjected to normal faulting. Numerous previous studies concluded that finite element modelling can simulate the propagation of the fault rupture through soil successfully, on the condition that the refined mesh, boundary effects and the nonlinear behavior of the soil are considered properly [32,33,34,35]. The descriptions of the FE model are as follows.

Prototype of the railway embankment

The prototype is selected from the Ya’an-Linzhi railway connecting Sichuan and Tibet, China. The railway crosses the fault zone on a 640m long embankment and obliquely intersects with the Litang Fault at DK520 + 340 (the departure section of Litang Station in Litang Basin), as shown in Fig. 2. The design speed of the railway is 200km/h. The Litang fault is an active fault within the Sichuan-Yunnan rhombic block and once induced a moment magnitude 7.25 earthquake in 1948. At present, the faulting offset is still ongoing at a rate of 4 to 6 mm per year. In this region, the stratum of the soil layer is sedimentary rocks and shallow metamorphic rocks. The shallow metamorphic rocks are mostly sandstone, slate, shale, and microphyllite.

Fig. 2

The intersection of Litang Fault and railway embankment

Constitutive model validation

The constitutive model used in this study to simulate the behavior of the soil layer and the overlaying embankment is the elastic-perfectly plastic soil model with the Mohr–Coulomb failure criterion which is commonly used in previous studies [36,37,38]. To validate the effectiveness of the Mohr–Coulomb model, 3D numerical models are established and the results are used to compare with the centrifuge test results of Anastasopoulos et al. [39]. Figure 3a shows the geometry and definitions of the numerical validating model. The geometry parameters of the model are summarised in Table 1. Table 2 shows the soil layer’s properties. The element type is C3D8 and the size of the elements is 1m $\times$ 1m $\times$ 1m (long $\times$ width $\times$ height). The comparison between the numerical analysis and the centrifuge tests in terms of vertical displacement of the surface is shown in Fig. 3b. Evidently, the numerical results and the centrifuge results are successfully matched. The location of fault outcropping on the surface and the localization of the deformation within a narrow band exhibit the same trend in the numerical analysis and centrifuge test. Therefore, the results of numerical analysis are satisfactory and the Mohr–Coulomb model is effective in predicting the phenomenon of normal fault rupture propagation through the overlaying soil.

Fig. 3

Constitutive model validation modelling. a Definitions of the model. b Numerical simulation vs centrifuge test in terms of vertical displacement of the ground surface (centrifuge test data from Anastasopoulos et al. [39])

Table 1 The geometry parameters of the validating model

Table 2 Properties of the material of the soil layer adopted in the validating model

Geometry of the numerical modal

The geometry of the embankment section is based on the railway project in Tibet (Xinduqiao to Changdu section) and the “Code for design of railway earth structure” [40] at route DK520 + 341, as displayed in Fig. 4. Although the influence of bedrocks on the results is negligible proven by previous studies [41,42,43], bedrocks are established in the model to emphasize the existence of the fault. The dip-slip fault rupture propagation through the soil layer can be regarded as a plane strain problem [39, 44]. In this study, due to the existence of the railway embankment, the modelling can not be simplified as a 2D problem. The width ${W}_{S}$ and length ${L}_{T}$ of the domain are set to be sufficiently wide to eliminate the boundary effects, as shown in the results (Fig. 8). Recommended by Bray [45], ${L}_{T}/{H}_{S}$ are set to be larger than 4. The geometry parameters of the numerical model are summarised in Table 3. ${L}_{H}$ and ${L}_{F}$ are the upper length of the hanging wall and the footwall, respectively. ${H}_{R}$ and ${H}_{S}$ are the thickness of the bedrock and soil layers. ${D}_{f}$ is the faulting offset. ${\alpha }_{f}$ is the fault dip angle. ${\alpha }_{o}$ is the cross-fault angle of the embankment.

Fig. 4

Geometry of the numerical model

Table 3 Geometry parameters of the numerical model

Interactions and boundary conditions

There are four pairs of interactions in the model, as shown in Fig. 5(a). Between the footwall and the hanging wall, the tangential behavior is assumed to be frictionless (the stick–slip phenomenon on the fault planes is not considered in this study) and the normal behavior is set to be hard contact (the contacting surfaces are not allowed to penetrate each other). The embankment’s contact with the soil layer and the soil layer’s contact with the bedrock are set to be bonded together which can prevent the detachment between the contact surfaces when the faulting happens.

Fig. 5

Basic assumptions of the interactions and boundary conditions

Two analysis steps are set in the simulation procedure: geostatic and faulting. In the step of geostatic, the initial stress and displacement field are obtained by applying global gravity to the model. In the step of faulting, the simulations are assumed to be quasi-static and the inertial effects due to rupture propagation in the soil layer are ignored as the slip rate diminishes gradually when the rupture approaches the ground surface [44]. Figure 5(b) shows the boundary conditions of the model in the faulting step. The normal faulting is simulated by applying displacement excitation to the bottom of the hanging wall and the sectional boundary on the hanging wall side. The bottom surface and the sectional boundary on the footwall side are fixed, and the sides of the model are set to be roller-supported.

Material properties

The embankment, soil layer, and bedrock are considered as isotropic homogeneous materials. The former validated Mohr–Coulomb constitutive model is used for the modelling of the embankment and the soil layer. Because of the distinguished difference in the stiffness between the soil and bedrock, the bedrock hardly deforms during faulting. Therefore, the bedrocks are regarded as linear elastic with relatively large elastic modulus. Figure 6 shows the variation of the vertical displacement of the embankment centerline with the properties of the soil layer. It is true that soil properties can influence the formation of the subsidence region over the fault line. However, their influences are relatively limited compared with the influences of the faulting offset ${D}_{f}$, the thickness of the soil layer ${H}_{S}$, the fault dip angle ${\alpha }_{f}$ and the cross-fault angle of the embankment ${\alpha }_{o}$. Therefore the soil properties are not the main focus of this study. For conservative concerns, the dilation of the soil is not considered in the following modelling. A relatively small value of cohesion and an intermediate value of friction angle is used in the modelling of the soil layer and embankment. Table 4 summaries the material properties utilized in the following modelling, including the cohesion $c$, internal friction angle $\varphi$, dilation angle $\psi$ the density $\rho$, modulus of elasticity $E$, and Poisson's ratio $\nu$.

Fig. 6

Influence of the soil layer’s properties

Table 4 Properties of the embankment, soil layer, and bedrock

Mesh dependency

To minimize the mesh dependency, a parametric study is conducted to examine the influence of the element size on the normal faulting-induced vertical displacement of the embankment centerline as shown in Fig. 7(a). The influence of the mesh size on the simulation in this study is nearly negligible. Considering the calculation efficiency, 5 m mesh size is used for the discretization of the soil layer in the following analysis. The mesh size is about 1 m on the embankment crest and about 2 m along the embankment slope. In the longitudinal direction (X-axis), the mesh size of the embankment above the fault line within a 200 m wide region is finer to be 2 m and of the rest is 5 m. The mesh size of the bedrock is 5 m. The hexahedral structural element (C3D8) is used in the model. This kind of element is appropriate for simulating problems with severe mesh distortion and can effectively simulate the large deformation of the embankment under the faulting effect [38, 42, 46, 47].

Fig. 7

Mesh sensitivity analysis

Modelling scheme

As in previous studies, the thickness of the soil deposit [48, 49], the angle of the fault dip [43, 50] and the orientation of the fault relative to the earth structure [45, 51] will exert considerable influence on the fault rupture propagation. To discover the influence of these factors on the deformation of the railway embankment under normal faulting, a parametric study is conducted. The modelling scheme is summarized in Table 5.

Table 5 Modelling scheme

Results and discussion

The potential fault rupture outcropping region

By using the elastic half-space theory of dislocation, Han et al. [52] derived the threshold of differential displacement of generating surface fault rupture by sudden dislocation of the buried fault: when the original distance between two points on the earth surface is 5 m, the differential displacement threshold is 0.1 m which is based on the "Code for seismic design of buildings" [53]. However, the displacement difference in a large-scale field is not easy to get. In this paper, for practical consideration, the equivalent plastic strain ($PEEQ$) 2% is used as the threshold of fault rupture generation to locate the potential regions where the fault rupture outcropping might occur. Figure 8(a1) shows the PEEQ of the model. The grey region is the potential region where the primary fault rupture outcrops with PEEQexceeds 2%. In the figures, there is also a sign of the secondary fault rupture which is the light blue zone in Fig. 8(a2).

Fig. 8

The PEEQ in the model (D_f = 0.6m,H_S = 40m, α_f = 45°, α_o = 90°)

Figure 8(b)(c) shows the distribution of the maximum principal stress ${S}_{max}$ and the minimum principal stress ${S}_{min}$ in the whole model and the surface of the embankment. The regions where the value of ${S}_{max}$ is high represent the tensile zones and where the value of ${S}_{min}$ is high represent the compressive zones. The areas of the tensile and compressive zones on the embankment surface are marked in the figures. A compressive zone is over the fault line with the shape of a beetle, surrounded by the tensile zones, as shown in Fig. 8(b2). The maximum value of the ${S}_{min}$ appears on the footwall side, as shown in Fig. 8(c2). The tensile zones on both sides of the fault line cover the majority of the cross-fault region above the fault line. The tensile regions are also the places that generate plastic strains, which means that the potential fault rupture outcroppings in these regions are tensile cracks.

To find out the variation of the potential location of the fault rupture outcropping with the factors, the $PEEQ$ and ${S}_{mises}$ on the embankment surface and centerline are plotted in Fig. 9. The axis $\text{X}$ is the longitudinal direction of the railway embankment and the axis $Y$ is the lateral direction. The ${S}_{mises}$ is an index that represents the distortional energy of the material which can be used to analyze the yield state of embankment in the cross-fault region. In the compressive and tensile zones, the regions where the ${S}_{mises}$ does not vary with the variation of the factors are the places in the yield state. Soil is a material with low tensile strength and high compressive strength, so the distortional energy is easier to accumulate in the compressive zones than in the tensile zones. The feet of the embankment slopes on the footwall side are the places having the highest level of $PEEQ$ and the second high level of $PEEQ$ appears on the center of the embankment crest, which means that the primary fault rupture outcrops firstly on the feet of the embankment slopes and then the embankment crest.

Fig. 9

Strain and stress distribution on the embankment with the factors

The increase of the faulting offset ${D}_{f}$ doesn’t change the location of the fault rupture outcropping, as shown in Fig. 9(a). The reason is that once the fault rupture outcrops with the faulting offset reaching a certain amount, the soil layer block above the fault line will only move along the fault rupture with the later increase of the faulting offset. With the increase of the faulting offset ${D}_{f}$, the tensile zone where the potential primary fault rupture outcrops yields firstly when ${D}_{f}$ is still small. After ${D}_{f}$ exceeds a certain level, the tensile zone where the potential secondary fault rupture outcrops yields. The compressive zone is only on the crest of the embankment and it needs a relatively large amount of faulting offset to get into the yield state.

When the embankment is constructed directly on the fault without soil layer covering, the strain in the primary faulting rupture outcropping region is significantly high, and the region is the only place that reaches the yield state and it is extremely narrow, as shown in Fig. 9(b). The fault rupture outcrops easily with a relatively small amount of faulting offset as the fault rupture generates directly in the overlaying embankment when the faulting begins. The soil layer can lower the strain in the embankment effectively. With the increase of the thickness of the soil layer ${H}_{S}$, the fault rupture outcropping regions expand and move away from the fault line on both sides of the fault line and the value of the $PEEQ$ drops because the increase of ${H}_{S}$ lengthens the path of the fault rupture generation and the soil layer absorbs more energy that created by the faulting offset. In the meantime, the region of the primary fault rupture outcropping expands because the energy caused by the faulting diffuses along the path of the fault rupture. When ${H}_{S}$ exceeds 70m, the width of the tensile zone where the potential secondary fault rupture outcrops tends to narrow and the ${S}_{mises}$ in this region tends to decrease. The compressive zone appears not only on the embankment crest but also on the foot of the embankment slopes. With the increase of ${H}_{S}$, the ${S}_{mises}$ in the embankment crest tends to decrease and the ${S}_{mises}$ in the foot of the embankment slope tends to increase.

The secondary fault rupture is negligible when the fault dip angle is between 40˚ and 50˚, as shown in Fig. 9(c). When the fault dip angle ${\alpha }_{f}$ is smaller than 50˚, the primary fault rupture outcropping region and the strain in this place don’t change with the variation of ${\alpha }_{f}$. The sign of the secondary fault rupture outcropping becomes more significant when ${\alpha }_{f}$ decreases. When the fault dip angle exceeds 60˚, the secondary fault rupture outcrops near the fault line on the footwall side and both of the fault rupture outcroppings expand and move toward the hanging wall with the increase of ${\alpha }_{f}$. In the meantime, the strain in the fault rupture outcropping on the right decreases and conversely, the strain in the fault rupture outcropping on the left increases. The primary fault rupture outcropping region appears on the hanging wall side and becomes compressive yield when ${\alpha }_{f}$ is larger than 70˚. The secondary fault rupture outcropping region appears on the footwall side and remains tensile yield.

As shown in Fig. 9(d), when the cross-fault angle ${\alpha }_{o}$ of the embankment is larger than 40˚, the fault rupture outcropping region and ${S}_{mises}$ distribution in this region barely change with the variation of ${\alpha }_{o}$. The strain in the primary fault rupture outcropping region slightly decreases with the decrease of ${\alpha }_{o}$. When ${\alpha }_{o}$ is smaller than 40˚, the primary fault rupture outcropping region expands and moves towards the footwall side significantly and the compressive zone expands to the embankment slopes with the decrease of ${\alpha }_{o}$. In the meantime, the $PEEQ$ and ${S}_{mises}$ in the primary fault rupture outcropping region drop and the secondary fault rupture would outcrops less likely. The reason is that the smaller ${\alpha }_{o}$ is, the wider the region of the fault rupture outcrops underneath the embankment will be. The wider region of the fault rupture generated underneath the embankment can absorb more energy so that the strain and stress in the fault rupture outcropping regions are smaller when ${\alpha }_{o}$ decreases.

Deformation patterns of embankment centerline

Vertical displacement and the longitudinal slope

Figure 10 shows the longitudinal section and vertical view of the model. A subsidence region appears over the fault line generating longitudinal slopes on both sides of the fault line. There is also a necking phenomenon accrues on the longitudinal slops, which can explain the valley phenomenon in the longitudinal slope curve displayed in Fig. 11. The necking phenomenon of the embankment is a tensile behavior resulting from the positive longitudinal movement of the hanging wall. These multi-peak phenomena on the longitudinal slope indicate that normal faulting can induce unevenness to the crossing embankment. The peaks and valleys of the longitudinal curve of the embankment centerline are plotted in Fig. 10. The higher peaks are at the center of the necking regions. The valleys are in the middle of the longitudinal slopes which is the transitional region between the lower and higher peaks.

Fig. 10

Local subsidence and necking of embankment caused by the normal faulting (Case 10)

Fig. 11

Vertical displacement and the longitudinal slope of the embankment centerline with the factors

The operation of the railway requires the longitudinal slope rate of the railway line to meet the design requirement. The absolute value of the longitudinal slope of the embankment centerline is calculated from the vertical displacement of the route. The longitudinal slope ${i}_{l}$ is calculated by using Eq. (1). $\Delta L$ is the unit length of the embankment centerline. $\Delta {U}_{Z}$ is the variation of vertical displacement in this unit length. The ${U}_{Z}$ and ${i}_{l}$ of the embankment centerline is shown in Fig. 11. The maximum longitudinal slope ${i}_{lmax}$ in the route is plotted in Fig. 12, as it is usually of primary concern when designing the tractive ability of the train. In the prototype project, the ${i}_{lmax}$ should be smaller than $30{\permil }$, as the red dot line plotted in Fig. 12.

$$i_l=\vert\triangle U_Z/\triangle L\vert$$

(1)

Fig. 12

i_lmaxwith the factors

According to the ${U}_{Z}$ of the embankment centerline, a subsidence region appears around the fault location which is caused by the refraction and variation of the dip of the fault rupture as it approaches the ground surface [54]. The normal fault rupture tends to refract at the soil–bedrock contact and to increase in the dip as they approach the ground surface. All the results have a zero-point in the longitudinal slope ${i}_{l}$ at the fault location, resulting from the subsidence reaching the extremum.

The depth of the subsidence zone and ${i}_{l}$ tends to accumulate with the increase of ${D}_{f}$, as shown in Fig. 11(a). The main rupture and secondary rupture were formed when ${D}_{f}$ is small, which is the reason why the subsidence zone doesn’t expand with the increase of ${D}_{f}$.

The maximum longitudinal slope ${i}_{lmax}$ is significantly high when the embankment is constructed on the bedrocks without soil layer covering, as shown in Fig. 11(b). When the soil layer thickness ${H}_{S}$ increases, the subsidence zone tends to expand and the peak of ${i}_{l}$ in the embankment centerline drops dramatically. It can be concluded that the thickening of the soil layer can mitigate the faulting effects on the embankment efficiently, but the price is the expansion of the slope region.

As shown in Fig. 11(c), when the fault dip angle ${\alpha }_{f}$ is larger than 60°, there is no subsidence zone above the fault line and the location of the longitudinal slope varies with ${\alpha }_{f}$. When ${\alpha }_{f}$ is smaller than 50°, the subsidence zone is formed and the depth of the subsidence zone increases with the decrease of ${\alpha }_{f}$. The affected area expands significantly once the subsidence zone is formed. It is an interesting finding that the subsidence zone doesn’t expand with the decrease of ${\alpha }_{f}$ when ${\alpha }_{f}$ is smaller than 50°.

When the cross-fault angle ${\alpha }_{o}$ of the embankment is larger than 40°, the ${U}_{Z}$ of the embankment centerline hardly changes with ${\alpha }_{o}$ and maximum ${i}_{l}$ slightly decreases with ${\alpha }_{o}$ decreasing, as shown in Fig. 11(d). When ${\alpha }_{o}$ is smaller than 40°, with the decrease of ${\alpha }_{o}$, the subsidence zone is widened significantly and the maximum ${i}_{l}$ greatly drops. While it is true that the will be small when the embankment crosses the fault line with a small angle, the generated lateral displacement of the embankment will also affect the function of the train. So it is not recommended that the embankment crosses the fault line with an angle smaller than 30° unless the railway embankment has to cross the fault line obliquely.

In conclusion, there are some ways to lower the maximum longitudinal slope ${i}_{lmax}$ of the railway embankment normal faulting: (1) Cross a fault with limited offset potential; (2) Choose a place where the covering soil layer is relatively thick; (3) Avoid crossing a fault with nearly vertical fault dip; (4) Choose a smaller cross-fault angle.

Lateral displacement and the radius of the curvature

Faulting offset along the fault strike is the characteristic of strike-slip fault which will cause lateral displacement to the railway embankment. If the railway embankment crosses through the normal fault or reverse fault underneath with an angle other than 90°, the faulting can also induce lateral displacements. Figure 13(a) shows the lateral displacement ${U}_{Y}$ of the embankment centerline when the railway embankment crosses the fault line with an angle ${\alpha }_{o}$ varying from 10 to 90°. In the plan design of the railway project, it is necessary to limit the maximum radius of the curvature of the lateral curve according to the design speed, the locomotive type, and other conditions to avoid train overturning and derailment. The radius of the curvature $R$ is calculated by Eq. (2) and (3). $K$ is the curvature. Figure 13(b) is the calculated curvature of the lateral displacement curve of the embankment. It can be seen from the figure that the places that have a great amount of curvature are mainly located on the footwall side, and they are exactly the places where the longitudinal slope ${i}_{l}$ is high.

Fig. 13

Lateral displacement and the radius of the curvature with the cross-fault angle (Case 32 ~ 43)

Taking the maximum of the curvature, the minimum of the radius of the curvature can be calculated. Figure 13(c) shows the variation of ${R}_{min}$ with ${\alpha }_{o}$. The red dot line in the figure is the design requirement of the prototype railway embankment (${R}_{min}\ge 2$ 800m). The ${R}_{min}$ can not meet the design requirement when the cross fault angle ${\alpha }_{o}=20 \sim 80^\circ .$ Some measures should be taken to lower the curvature of the embankment, otherwise, the train should lower the speed to ensure that the train will not derail when passing by this region. It is strongly recommended that the railway embankment crosses the normal fault perpendicularly. Crossing the normal fault with an angle smaller than 15° can lower the ${K}_{max}$, but the price is that the affected area of the railway embankment would be very wide.

$$R=1/K$$

(2)

$$K=\left|\mathrm d\theta/\mathrm{ds}\right|=\left|U_Y^{"}\left(X\right)/\left\{1+\left[U_Y^{\prime}\left(X\right)\right]^2\right\}^{3/2}\right|$$

(3)

Determination of the affected zones

Since the existing research and design code do not have a precise definition of the range of the affected zone of faulting in railway engineering, a definition of the affected zone based on the deformation of the embankment centerline is proposed and conservative range of affected zones of the cross-fault embankment is suggested for the design consideration.

Figure 14 shows the displacement result ${U}_{magnitude}$ of the model. During normal faulting, the hanging wall moves downward along the fault dip and a localized displacement transitional zone (borrowing the concept from Choi et al. [55]) is formed through the soil layer and the embankment above the footwall.

Fig. 14

U_magnitudeof the model (Case 10)

The ${U}_{magnitude}$ of the embankment centerline are displayed in Fig. 15. With the faulting offset ${D}_{f}$ increasing from 0 to 0.6m, the displacement of the embankment on the hanging wall side tends to accumulate, but the position and span of ${Z}_{DT}$ remains. With the thickness of the soil layer ${H}_{S}$ increasing from 0 to 100 m, ${Z}_{DT}$ tends to gradually expand and move forward to the footwall side. When the fault dip angle ${\alpha }_{f}$ is smaller than 40˚, the sign of secondary fault rupture on the hanging wall side is apparent. With ${\alpha }_{f}$ increasing from 10 to 90˚, ${Z}_{DT}$ tends to become a narrow zone when ${\alpha }_{f}$ exceeds 40˚ and to move forward to the hanging wall side when ${\alpha }_{f}$ keeps increasing. When ${\alpha }_{f}$ is 90˚, ${Z}_{DT}$ is almost symmetric along the fault line. When the cross-fault angle ${\alpha }_{o}$ is larger than 40˚, ${Z}_{DT}$ barely changes. However, when ${\alpha }_{o}$ is smaller than 40˚, ${Z}_{DT}$ dramatically expands with ${\alpha }_{o}$ decreasing.

Fig. 15

U_magnitude of embankment centerline with the factors

As discussed by Loukidis et al. [44] and Shi et al. [56], because of the localization of the outcropping fault rupture, the differences in the value of the zone with significant ground distortion due to the choice of the ground inclination threshold value are relatively small. Therefore, to define the zone of embankment affected by faulting, the terminals of the route are taken as the reference point. The displacement thresholds of 1, 5, and 10 mm are taken to define the avoidance zone ${l}_{a}$, the deformation zone ${l}_{d}$ and the significantly affected zone ${l}_{s}$, as illustrated in Fig. 16 (reference to the definition in He et al. [57]). The reason why using ${U}_{magnitude}$ rather than the vertical displacement ${U}_{Z}$ to define the range of the affected zones is that ${U}_{magnitude}$ is more representative than ${U}_{Z}$ because not only ${U}_{Z}$ but also the horizontal displacement (${U}_{Y}$ and ${U}_{X}$) are considered in ${U}_{magnitude}$.

Fig. 16

Definition of the affected zone of the cross-fault embankment (Example case: 10)

The range of the affected zones of the embankment against ${D}_{f}$, ${H}_{S}$, ${\alpha }_{f}$ and ${\alpha }_{o}$ are shown in Fig. 17. ${l}_{a}$, ${l}_{d}$ and ${l}_{s}$ barely change with the increase of ${D}_{f}$ after ${D}_{f}$ exceeds 0.18m (Fig. 17(a)), so it can be concluded that the affected zone of the cross-fault embankment is less relevant to ${D}_{f}$. With the increase of ${H}_{S}$, ${l}_{a}$, ${l}_{d}$ and ${l}_{s}$ expand significantly until ${H}_{S}$ exceeds 80m (Fig. 17(b)). As depicted in Fig. 17(c), ${l}_{a}$, ${l}_{d}$ and ${l}_{s}$ suddenly narrow when ${\alpha }_{f}$ exceeds 60˚. ${l}_{a}$, ${l}_{d}$ and ${l}_{s}$ expand gradually with the decrease of ${\alpha }_{o}$ and the speed of expending increases significantly when ${\alpha }_{o}$ is less than 30˚, as shown in Fig. 17(d).

Fig. 17

Affected zones of the railway embankment under normal faulting

To give a suggestion for railway engineers when assessing the affected zone of the cross-fault embankment, a conservative range of the affected zone is given in Fig. 18 based on results in Fig. 17. The worst condition, deep buried fault with a small angle, is considered in the suggested affected zones. However, the suggested affected zone is not applicable when the embankment crosses the fault line with a small angle ($\alpha_o<30^\circ$).

Fig. 18

The suggested affected zones for cross-fault embankment in railway engineering (m)

Conclusion

In this study, the normal faulting effects on railway embankment are thoroughly studied considering the influence of the faulting offset, the soil layer’s thickness, the fault dip angle, and the embankment cross-fault angle. Emphasis is given to the stress and strain characteristic in the fault rupture outcropping regions on the embankment, the deformation of the embankment centerline for design purposes, and the affected zones for the preservation of the railway embankment. The main conclusions are as follows:

• (1) Both of the regions where the primary and secondary fault rupture outcrops are tensile yield in most cases. When the fault dip angle exceeds 60°, the primary and secondary fault rupture exchange their positions and the primary fault rupture outcropping region becomes compressive yield. The strain in the fault rupture outcropping regions accumulates with the increase of the faulting offset. The increase of the soil layer’s thickness and the decrease of the cross-fault angle can lower the strain in the primary fault rupture outcropping region, and expand and move this region away from the fault line towards the footwall direction.

• (2) A subsidence zone is formed on the embankment over the fault line under normal faulting generating longitudinal slopes on both sides of the fault line. The depth of the subsidence zone increases with the increase of the faulting offset and the decrease of the fault dip angle. The width of the subsidence zone grows with the increase of the soil layer’s thickness and the decrease of the cross-fault angle of the embankment. Two necking regions are formed on the embankment crest on both sides of the fault line and they are also the places where the longitudinal slope is relatively high. The increase of the soil layer’s thickness and the decrease of the fault dip angle can lower the maximum longitudinal slope. When the railway embankment crosses the normal fault with an angle other than 90°, the minimum value of the curvature cannot meet the design requirement unless ${\alpha }_{o}$ is smaller than 20°.

• (3) The variation of faulting offset has the smallest effect on the affected zones of the embankment. The affected zones expand significantly with the increase of the soil layer’s thickness. The large fault dip angle (${\alpha }_{f}$>50°) narrows half of the affected zones. When the embankment crosses the normal fault with a small angle (${\alpha }_{o}$<30°), the affected zones cover a wide range of area.