Taylor's slope stability chart for combined effects of horizontal and vertical seismic coefficients

Design standards and codes of practice on earth slope stability often recommend the pseudo-static method of analysis for determining the factor of safety of a slope subjected to seismic forces. In ...


INTRODUCTION
The stability of soil slope is always a great topic of debate for geotechnical engineers. In several routine applications, Taylor's stability chart (Taylor, 1937(Taylor, , 1948 is used as the main tool for the determination of the factor of safety F of finite homogeneous slopes consisting of c-ϕ soils under undrained conditions; c being the cohesion intercept and ϕ the angle of internal friction of soil under static conditions (e.g. Terzaghi et al., 1996;Das, 2010;Shukla, 2015). Apart from Taylor's slope stability chart, several analytical methods have been developed in the past in order to have more realistic estimation of the factor of safety of slopes under static (Janbu, 1954;Bishop, 1955;Morgenstern & Price, 1965;Spencer, 1967) and dynamic/seismic (Majumdar, 1971;Sarma, 1973Sarma, , 1979 conditions, considering different field situations and assumptions (Chowdhury et al., 2010;Cheng & Lau, 2014;Duncan et al., 2014). Under static loading condition, when the slope inclination and angle of internal friction of soil are known, then a stability number can be computed easily from Taylor's stability chart under static loading conditions. However, this design chart does not consider the estimation of factor of safety under the application of seismic force. It would be more advantageous to field engineers if Taylor's chart could further be used when both static and dynamic loading conditions exist, because stability of the slope becomes more critical under such circumstances. Majumdar (1971) explained how Taylor's stability chart can be used under the application of horizontal earthquake force by defining a modified friction angle of the c-ϕ soil, taking into account the horizontal seismic coefficient. Although most of the studies discuss the effect of horizontal earthquake force on stability of the slope, it has been found that the vertical component of earthquake force cannot be disregarded, as it greatly alters the effect of dynamic stress distribution (Chopra, 1966). Ling et al. (1997Ling et al. ( , 1999 and Shukha & Baker (2008) found that the vertical seismic force has significant effects on the stability of the slope. From past earthquake records, such as Loma Prieta (Lew, 1991), the Northridge earthquake (Stewart et al., 1994) and the Hanshin earthquake (Bardet et al., 1995), it has been found that the maximum vertical seismic force can be equal to or even greater than the horizontal seismic force. Aoi et al. (2008) reported that the vertical seismic acceleration was twice the horizontal seismic acceleration during the Iwate-Miyagi earthquake in Japan. Ling & Leshchinsky (1998) also reported that the vertical seismic acceleration was 30% larger than the horizontal seismic acceleration in the Hanshin earthquake. Ling et al. (1997) examined the stability and displacement of a slope under seismic action by using the log-spiral method and observed that, when the vertical acceleration was accompanied by horizontal acceleration, a prominent effect could be observed. Therefore, in this paper, an attempt is made to extend Majumdar's work (Majumdar, 1971) for the use of Taylor's stability chart under the combined actions of horizontal and vertical seismic forces. The developed approach will help practising engineers to analyse the stability of homogeneous slopes with simple profiles more realistically in earthquake-prone areas, using Taylor's stability chart in a simple way, without depending on the need for any commercial software or the development of an original numerical model.

ANALYTICAL FORMULATION
Using the modified friction circle method, Taylor (1937Taylor ( , 1948 presented a slope stability chart, as shown in Fig. 1, which provides a relationship between the stability number c/FγH and slope angle i for different values of the angle of internal friction ϕ of the soil, with c, γ, H and F as the cohesion intercept, total unit weight of soil, height of slope and factor of safety of the slope, respectively. This chart is based on the following assumptions. (a) The entire soil mass forming the slope is homogeneous. (b) The potential failure surface passes through the toe of the slope and is cylindrical (Fig. 2). (c) When the soil mass is on the verge of failure, the failure surface follows the limiting condition of equilibrium, and hence, the shearing strength of the soil s can be expressed in the form of the Mohr-Coulomb criterion as where σ is the total normal stress on the failure surface. (d ) The analysis is based on total stresses and assumes that the cohesion c is constant with depth.
Considering all these assumptions and following the approach explained by Majumdar (1971), a modified friction angle (ϕ m ) of the c-ϕ soil can be derived as explained below, for the generalised seismic conditions, so that Fig. 1 can also be used as the design chart for determining the factor of safety of the soil slope subjected to both horizontal and vertical seismic loads. Figure 2 shows a soil slope of height H with inclination i to the horizontal with a soil mass tending to slide over a cylindrical failure surface with its cross-section FE as the circular arc having its radius R and centre at O (h, k). The position of the centre O (h, k) is defined by two angles, namely the central angle of the failure arc (2β 0 ) and the angle made by the chord EF with the horizontal of the slope (α 0 ). The forces acting on the sliding mass FGE are as follows (Fig. 2) (a) weight, W, of the sliding mass, acting vertically downward at the centre of gravity (CG) (b) horizontal seismic load, F h acting outward, and vertical seismic load, F v acting either in upward (") or downward (#) directions on the sliding mass at the centre of gravity (c) the total resultant cohesion C of the slope along the failure arc (d ) the resultant (P resultant ) of normal force and frictional force intersecting with the line of action of W, which makes an angle ϕ with the normal to the failure circle. As a result, the line of action of the resultant force will remain tangent to the circle of radius Rsinϕ, also known as the friction circle shown in the figure.
It is assumed that if the weight W E of the sliding mass, including both horizontal and vertical seismic loads, F h and F v , is expressed in terms of an equivalent total unit weight γ E ; then the total overturning moment due to the weight Wof the potential sliding mass and combined seismic loads, F h and F v , will give the same overturning moment as produced by the equivalent weight W E . As γ E is the equivalent total unit weight, this gives where V is the volume of the potential sliding soil mass. It may be noted that the weight W is given by where γ is the total unit weight of soil of the slope. Horizontal seismic/earthquake force where k h is the pseudo-static horizontal seismic coefficient. Vertical seismic/earthquake force where k v is the pseudo-static vertical seismic coefficient. It may be noted that, in equation (5), the positive (+) sign indicates that F v acts in the vertically downward direction (#), whereas the negative (À) sign indicates that F v acts in the vertically upward direction ("). Taking the moment of forces about the centre of the circle O where l is the moment arm of the force F h and d is the moment arm of the force F v and W.
Using equations (2)-(5), equation (6) becomes ig. 1. Stability chart for c-ϕ soil slope (after Taylor, 1937Taylor, , 1948 Δx Fig. 2. Stability analysis due to the horizontal and vertical earthquake forces on the slope (adapted from Majumdar, 1971) Considering a vertical jth slice of the sliding soil mass of unit thickness with weight W j , width Δx and slice side lengths y 2 and y 1 , as shown in Fig. 2, and subjected to both horizontal seismic force F h and vertical seismic force F v , the total resisting force S besides cohesion (assumed to be independent of seismic/dynamic conditions) can be expressed as follows where θ j is the angle made by the base of the jth slice with the horizontal and b denotes the limit of the area bounded by the sliding mass, as shown in the figure, and is expressed as follows The total effect due to combined seismic load can be represented in the form of total equivalent unit weight γ E and modified friction angle ϕ m . In this case, the resisting force is From equations (8) and (10), one has where m is the ratio of tangent of modified angle of internal friction of soil to the tangent of angle of internal friction of soil, and it may be termed the friction reduction factor. It may be noted that m ranges from 0 to 1. Substituting the value of γ E from equation (7) into where and For k v ¼ 0, equation (12) or (13) reduces to the case of consideration of only horizontal seismic force as Majumdar (1971) has considered. In his work, no attempt was made to present the simplified form of summation terms. So, in the following paragraph, an attempt is made in this direction, for which the integration concept has been used in place of summation for simplicity.
Equations (14a) and (14b) can be presented considering Δx ! 0 as In Fig. 2, it is noticed that the potential sliding mass is bounded by line FG, line GE and circular arc FE (y 1 ) having centre at O (h, k). The equation of the line FG (y 2 ) can be defined as a function of x as As line GE is parallel to the x-axis The equation of the circle with centre O can be defined as where h and k are the x and y coordinates of the centre O with their values as Differentiating equation (19), the slope of the line can be obtained as and cos θ ¼ À Using equations (9) and (16)-(21b), equations (15a) and (15b) are expressed as and To calculate m from equation (12), it is essential to compute the moment arm ratio (l/d ) for the given sliding soil mass. This can be done by considering the slope geometry as shown in Fig. 2. IfȲ is the distance from the centre O to the centre of gravity CG of the circular segment, then the expression for Y is derived as where D is the perpendicular distance from centre O to the chord EF of the slope, and θ 1 is the subtended angle of the circular segment. The value θ 1 depends upon the inclination angle of the slope, and can be determined from the geometry of the slope in Fig. 2. The moment arm d of F v and W can be estimated by considering the circular segment (area A 1 ) and the triangular segment (area A 2 ) as given by the following expression (Arredi, 1966) d where a 1 and a 2 represent the moment arms from the centre of the circle to the centre of gravity of the areas A 1 and A 2 , respectively, which are expressed as follows Substituting values from equations (25a)-(25d) into equation (24), d is estimated, and using equations (23) and (24), the moment arm l is calculated for a given angle of inclination (i) and height of the slope (H ). All the calculations are made in the spreadsheet for a set of slope angles as Taylor (1937) considered when developing his chart. By substituting the values of P and Q from equations (22a) and (22b) into equation (12), m is calculated for any failure surface within the given soil slope. With known value of m for the specific case, ϕ m is determined using the angle of internal friction of soil ϕ. For use of Taylor's stability chart (Fig. 1) to determine the factor of safety of the slope under effect of horizontal and vertical seismic loads, ϕ m is used in place of ϕ.
The design value of the seismic coefficients can be determined by referring to the earthquake design manuals or standards as applicable at a particular location. The U.S. Army Corps of Engineers (USACE, 1989) recommends In the present work, k v ¼ 0·5k h has been considered.

RESULTS AND DISCUSSION
In the present study, the use of Taylor's chart can be made under the effect of combined seismic action by evaluating the modified friction angle of the soil from equation (11) and thus the factor of safety for a given soil slope can be determined. For this purpose, the height of the slope has been considered as 50 m only in order to observe the effect of combined seismic pseudo-static coefficients; although Majumdar (1971) considered about 40 different cases to analyse the effect of horizontal earthquake force for two different heights (50 ft (15·24 m) and 75 ft (22·86 m)). However, for different heights, the plot of dimensionless quantity l/d ratio in equation (12) against i with respect to different angles of internal friction of soil is found to be the same. Figure 3 shows the variation of the moment arm ratio l/d with slope inclination i for different angles of internal friction of the soil. It is noticed that l/d does not depend on the seismic coefficients, as Majumdar (1971) also reported. Figure 4 shows the variation of the friction reduction factor m with the horizontal seismic coefficient k h for vertical seismic coefficient, k v ¼ 0, the slope angle i ¼ 30°, and internal friction angle of the soil, ϕ = 5-25°. It is observed that for any value of ϕ, m decreases non-linearly with an increase in k h , the rate of decrease in m being significantly higher for lower values of k h . For example, for ϕ ¼ 5°, as k h increases from 0 to 0·1, m decreases by 0·24, whereas the decrease in m is 0·06 for an increase in k h from 0·4 to 0·5. It should be noted that the m does not vary significantly with an increase in ϕ from 10°to 25°, and this variation has also been observed by Majumdar (1971). Figures 5, 6, 7 and 8 show the variation of m with k h as the design charts for the slope angles i ¼ 30°, 45°, 60°and 75°, respectively, with ϕ ¼ 5°, 10°, 15°, 20°, 25°and k v ¼ 0·5k h . In these design charts, both vertically upward and downward directions for k v have been considered. In all these design plots, it may be noted that for any slope angle, as k h increases, the value of m decreases, resulting in a lower value of ϕ m compared to soil friction angle ϕ as evident from the relationship in equation (12). Also, it is observed that for any value of k h , the value of m is smaller when k v acts vertically upward compared with the values when k v ¼ 0 or k v is downward. For example, in Fig. 8(a), for k h ¼ 0·4, the value of m is found to be 0·38 when k v is upward but 0·52 with k v downward. The value of m is further noted to be 0·46 in the absence of the vertical seismic coefficient. Thus, the variation of m depends upon the values of k h and k v as well as the direction of k v . It is also noted from Figs 7(c)-7(e) and 8 that, with increase of k h , m tends to zero for k v being upward or ig. 4. Variation of friction reduction factor m with horizontal seismic coefficient k h for different angles of internal friction of soil for slope angle i = 30°and k v = 0 k v ¼ 0, which means that the friction angle as well as the modified friction angle of the soil under seismic conditions is zero. This suggests that soil may behave as an undrained case under earthquake conditions. This aspect can be used in Taylor's chart for the undrained case of c-ϕ soil slope under combined earthquake conditions.

ILLUSTRATIVE EXAMPLE
Consider a 10 m high soil slope with an inclination of 60°t o the horizontal (Fig. 9). The soil has the following properties total unit weight, γ = 16 kN/m 3 cohesion, c ¼ 20 kPa angle of internal friction, ϕ = 25°D etermine the factor of safety under seismic conditions considering both horizontal and vertical seismic coefficients, for their following values for both the vertically downward and upward directions of k v .
or ϕ m % 7Á5°F rom Taylor's chart (Fig. 1), for slope angle, i ¼ 60°and or ϕ m ¼ 0°F rom Taylor's chart (Fig. 1), for slope angle, i ¼ 60°and For the comparison point of view, the stability of slopes considered in the illustrative example has been analysed by finite-element modelling using Plaxis 2D, which is a wellaccepted commercially available software. The details are presented below.

Finite-element model and analysis
A two-dimensional (2D) plane-strain analysis with elasticperfectly plastic Mohr-Coulomb soil criterion was used to model the slope considered in the illustrative example. Fifteen-noded triangular elements with 12 Gaussian points were used for the gravity load generation, the stiffness matrix generation and stress redistribution in order to simulate the accurate behaviour of soil slope under dynamic loading conditions (Al-Defae et al., 2013). Boundary conditions for the model were selected in such a manner so as to minimise the influence of stress distribution. In order to simulate a semi-infinite soil condition, full fixity was allowed to the base of the slope, while vertical boundaries were restrained in the horizontal direction. The soil model in Plaxis consists of six soil parameters, namely, effective shear strength parameters c′ and ϕ′, dilation angle ψ, total unit weight γ and elastic parameters as Young's modulus E′ and Poisson ratio ν′. The elastic parameters of soil have very little influence in the computation of factor of safety of the slope when compared to the deformation characteristic of soil. Therefore, in the absence of meaningful data, nominal values E′ ¼ 10 5 kN/m 2 and ν′ ¼ 0·3 were considered in the simulation process (Griffiths & Lane, 1999). As the stability analysis of slopes is relatively unconfined and the main objective of the current research is to predict the factor of safety, the dilation angle ψ was taken as zero considering no volume change during yielding of soil. It has been observed that the parameters of the finite-element soil model are the same as the parameters used in the traditional approach of the limit equilibrium method, namely, total unit weight γ and total shear strength parameters c and ϕ for a given geometry of the problem definition (Griffiths & Lane, 1999;Duncan et al., 2014). However, in the soil model, a foundation of depth 3 m was included to check whether the slip surface passes beyond the toe of the slope, and slope soil properties have been assigned to the foundation soil. Once the geometry was ready, the finite-element mesh was generated using the default mediumsize mesh. The mesh should be fine enough to obtain accurate numerical results. Nevertheless, very fine meshing needs to be avoided considering its excessive calculation time (Plaxis, 2016). Fig. 10 shows the generated mesh of the soil slope. In order to evaluate the factor of safety of the soil slope considering pseudo-static conditions, the following two cases were considered in the numerical modelling (a) gravity loading (b) safety analysis. ig. 9. Illustrative example for a homogeneous c-ϕ soil slope The initial state of stress of the slope was evaluated by considering the gravity-loading option in the preliminary phase of the modelling, where the forces of each element of mesh due to gravity were assembled into a global gravity force vector. The factor of safety of the slope was then calculated by using the principle of strength reduction method (Matsui & San, 1992), where the original strength parameters are divided by a factor to bring the sliding mass of the slope to the point of failure. It should be noted that, for computing the factor of safety using the pseudo-static option, it is essential to include the modal acceleration in the gravity-loading option. This is because Plaxis does not allow the user to include pseudo-static acceleration in the safety phase analysis. Figure 11 shows the deformed shape of the slope under effect of combined seismic action with the seismic coefficient k h ¼ 0·1 and k v ¼ 0·05(#). In the analysis, the iteration numbers were set to 60 for the convergence of the factor of safety. It has been observed that the factor of safety obtained for k h ¼ 0·1 is 1·084, which is in good agreement with the analytical result having a factor of safety of 1·09, as mentioned in case (a) of the illustrative example with vertically downward direction of application of vertical seismic coefficient k v . Although a factor of safety of 1·034 was evaluated for the vertically upward direction of seismic coefficient k v ¼ 0·05("), for ϕ ¼ 20°and k v ¼ 0·25("#) of case (b) of the illustrative example, the slope collapses and Plaxis is unable to deliver the result if the factor of safety is less than unity.
Based on both use of the developed design chart and finite-element modelling, the illustrative example presented here clearly shows that, as k h increases, the factor of safety of the slope decreases. Also, the factor of safety is found to be lower when k v acts vertically upward compared to the downward case. Therefore, for the design of a soil slope during earthquakes, it is important to consider the effect of the vertical seismic coefficient with its proper direction along with the horizontal seismic coefficient.

CONCLUSIONS
In this paper, an attempt is made to explain how Taylor's chart for the homogeneous c-ϕ soil slope can be used during earthquakes considering the effect of both horizontal and vertical seismic loads. On the basis of the results and discussion presented in the previous section, the following general conclusions can be drawn.
(a) An analytical expression for the modified friction angle of the slope soil has been developed by incorporating the combined pseudo-static seismic coefficients k h and k v for use of Taylor's slope stability chart under generalised earthquake conditions. (b) The value of friction reduction factor m is found to vary between 0 and 1 for a given slope angle and internal friction angle of the soil, irrespective of the value of k h and k v , as well as the direction of k v . (c) The greater the value of horizontal seismic coefficient with vertical seismic coefficient, the lower is the friction reduction factor m for a given angle of internal friction, which in turn gives rise to a higher stability number from Taylor's chart, resulting in a lower factor of safety. This implies that, under seismic loads, a soil slope is more unstable under the effect of vertical seismic load. (d ) The factor of safety of a slope with upward direction of k v is more critical as compared to the k v being in the downward direction or when k v is not considered. (e) For a higher slope inclination (e.g. i ¼ 75°), an increase in horizontal seismic coefficient with vertical seismic coefficient being vertically upward, the friction reduction factor m for the slope may decrease significantly for modified friction angle, ϕ m , to become zero. In this condition, soil may undergo undrained saturated loading condition, and hence it is recommended to consider the vertical seismic coefficient with its proper direction in order to have a safe design of the slope. ( f ) The illustrative example solved based on both use of the developed design chart and the finite-element modelling may help practising engineers to design any specific homogeneous c-ϕ soil slope with simple profiles under generalised conditions of horizontal and vertical seismic loads using the analytical concepts and design charts presented in this paper with full confidence. It is important to note that the developed design charts as presented here are not applicable to seismic stability of slopes having non-standard profiles and consisting of non-homogeneous soil properties. In such cases, the numerical analysis will be of great help.
NOTATION c cohesion of the soil (kPa) c/FγH stability number (dimensionless) F factor of safety (dimensionless) Factor of safety = 1·084  Fig. 11. Illustrative example: failure mechanism for homogeneous c-ϕ soil slope corresponding to k h = 0·1 and k v = 0·05(#). A full-colour version of this figure can be found on the ICE Virtual Library (www. icevirtuallibrary.com) 10 m 60°Y X Fig. 10. Illustrative example: 2D numerical mesh for homogeneous c-ϕ soil slope F h horizontal force due to earthquake (N/m) F v vertical force due to earthquake (N/m) H height of the slope (m) i angle of inclination of the slope with horizontal (degrees) k h horizontal seismic coefficient (dimensionless) k v vertical seismic coefficient (dimensionless) l/d moment arm ratio (dimensionless) m friction reduction factor (dimensionless) P area component (m 2 ) Q area component (m 2 ) R radius of circular arc (m) S total resisting force of the soil (N/m) s shear strength of the soil (kPa) W weight of the sliding mass (N/m) α 0 angle with the horizontal chord of the slope (degrees) β 0 central angle of the slope (degrees) γ unit weight of the soil (N/m 3 ) γ E effective total unit weight of the soil (N/m 3 ) σ total normal stress (kPa) ϕ angle of initial friction (degrees) ϕ m modified friction angle due to generalised earthquake (degrees)