*6.3.2 Generalized limit equilibrium method*

A generalized limit equilibrium (GLE) formulation was developed by Fredlundand and Krahn [52] and Fredlund et al. [53]. This method encompasses the key elements of all the methodslistedin **Table 2**. The interslice shear forces (Morgenstern and Price [49]) are

$$X = E\mathcal{X}f(\mathbf{x})\tag{4}$$

Where f(x) –a function, E and X –the interslice normal force and shear forces respectively and λ- function. GLE Method showing forces on slice and geometrical parameters is shown in **Figure 11**.

The factor of safety, Fm, with respect to moment equilibrium is

$$F\_{\rm \tiny \tiny \tiny \Delta} = \frac{\sum \Big[ C' \beta R + (N - \iota \beta \beta) R \tan \phi' \Big]}{\sum \mathcal{W} \mathcal{x} - \sum N \mathcal{f} \pm \sum Dd} \tag{5}$$

The factor of safety, Ff, with respect to horizontal force equilibrium is

$$F\_f = \frac{\sum \left[ C' \beta \cos \alpha + (N - u\beta) \tan \phi' \cos \alpha \right]}{\sum N \sin \alpha - D \cos \alpha} \tag{6}$$

where c' –effective cohesion, φ′-effective angle of friction, u –pore water pressure, N –normal force on slice base, W–slice weight, D–line load and β, R, x, f, and d – geometric parameters as detailed in **Figure 11**, and α - inclination of slice base. One of the key variables in both the equations is N –the normal force at the base of each slice. N is obtained by the summation of vertical forces as

$$N = \frac{W + (XR - XL) - \frac{C'\beta\sin\alpha + u\beta\sin\alpha\tan\phi}{F}}{\cos\alpha + \frac{\sin\alpha\tan\phi'}{F}}\tag{7}$$

The base normal force, N, is dependent on the interslice shear forces respectively XL and XR on the left and the right sides of the slice. The reinforcement force is accounted for in the analysis by GLE method.

#### *6.3.3 Finite element computed stress in limit equilibrium*

Krahn [45] suggested that normal stress determined from finite element stress analysis can be fed into General Limit Equilibrium Analysis. Thus, Limit Equilibrium and Finite Element method are integrated.

**25**

*Geoysynthetic Reinforced Embankment Slopes DOI: http://dx.doi.org/10.5772/intechopen.95106*

*6.3.4 Iterative GLE method for reinforced slope*

*Horizontal slice method (after Shahgholi and Fakher [39]).*

*6.3.5 Horizontal slice method*

**Figure 12.**

a horizontal slice are shown in **Figure 12**.

ering oblique pull by Reddy et al. [54].

**7. Reinforced slope with cohesive backfill**

Song et al. [28] proposed new approach based on LE principle to evaluate stability of reinforced slope. The effect of reinforcement is included as equivalent resisting force acting at slice base and added to GLE method. The corresponding equations are derived based on force equilibrium in the directions normal and parallel to slice base and moment equilibrium at the center of base of slice. The indeterminacy is resolved by assuming half sine function for inclination of interslice force in Eq. 4. The method satisfies both the force and the moment equilibrium

Shahgholi and Fakher [39] proposed horizontal slice method (HSM) in which

With failure wedge divided into N horizontal slices, 4Nunknowns can be determined by 4 N equations and a complete formulation is possible. The formulation is simplified if only vertical equilibrium is considered for individual slices together with overall horizontal equilibrium for the whole wedge, no account being taken of moment equilibrium. In this case the number of equations and the number of unknowns get reduced to 2 N + 1. HSM was extended to design of RE walls consid-

Reinforced Slopes are conventionally constructed with granular fill. However, this has limited the use of reinforced soil structures in locations where such free

horizontal slices are used in place of vertical ones to analyze the stability of reinforced and unreinforced slopes and walls. The limitation of the vertical slice method for the analysis of reinforced soil of unknown parameters being more than the number of equations available, is resolved by the horizontal slices method. The assumptions of HSM are (i) the vertical stress on an element in the soil mass is equal to the overburden pressure, (ii) the factor of safety (F.S.) is equal to the ratio of the available shear resistance to the mobilized shear stress along the failure surface, (iii) the factor of safety for all the slices is equal and (iv) the failure surface can have any arbitrary shape but does not pass below the toe of the slope or wall. Forces acting on

considerations applicable to arbitrary failure surfaces and is iterative.

**Figure 11.** *Forces acting on sliding mass with circular slip surface (after [47]).*

*Geoysynthetic Reinforced Embankment Slopes DOI: http://dx.doi.org/10.5772/intechopen.95106*

*Slope Engineering*

respectively and

λ

parameters is shown in **Figure 11**.

Where f(x) –a function, E and X –the interslice normal force and shear forces

∑ +− ′

*Wx Nf Dd*

( ) <sup>=</sup> ∑ ∑

α

*N D*

pressure, N –normal force on slice base, W–slice weight, D–line load and β, R, x, f, and d – geometric parameters as detailed in **Figure 11**, and α - inclination of slice base. One of the key variables in both the equations is N –the normal force at the

α

XL and XR on the left and the right sides of the slice. The reinforcement force is

Krahn [45] suggested that normal stress determined from finite element stress analysis can be fed into General Limit Equilibrium Analysis. Thus, Limit

<sup>=</sup> ′ <sup>+</sup>

*C u W XR XL F N*

cos

∑′ + − ′ <sup>=</sup> ∑ − cos tan cos

*CR Nu R*

∑− ±

 βφ

 β

( )

The factor of safety, Fm, with respect to moment equilibrium is

β

The factor of safety, Ff, with respect to horizontal force equilibrium is

′

βα

φ

base of each slice. N is obtained by the summation of vertical forces as

( )

+−−

sin cos *<sup>f</sup> C Nu*

*m*

*F*

*F*

accounted for in the analysis by GLE method.

*6.3.3 Finite element computed stress in limit equilibrium*

Equilibrium and Finite Element method are integrated.

*Forces acting on sliding mass with circular slip surface (after [47]).*

where c' –effective cohesion,


tan

 φ

′-effective angle of friction, u –pore water

 α

βαβα

sin sin tan

α φ

The base normal force, N, is dependent on the interslice shear forces respectively

sin tan

*F*

′ +

 α

> φ

(5)

(6)

(7)

**24**

**Figure 11.**

**Figure 12.** *Horizontal slice method (after Shahgholi and Fakher [39]).*

#### *6.3.4 Iterative GLE method for reinforced slope*

Song et al. [28] proposed new approach based on LE principle to evaluate stability of reinforced slope. The effect of reinforcement is included as equivalent resisting force acting at slice base and added to GLE method. The corresponding equations are derived based on force equilibrium in the directions normal and parallel to slice base and moment equilibrium at the center of base of slice. The indeterminacy is resolved by assuming half sine function for inclination of interslice force in Eq. 4. The method satisfies both the force and the moment equilibrium considerations applicable to arbitrary failure surfaces and is iterative.

#### *6.3.5 Horizontal slice method*

Shahgholi and Fakher [39] proposed horizontal slice method (HSM) in which horizontal slices are used in place of vertical ones to analyze the stability of reinforced and unreinforced slopes and walls. The limitation of the vertical slice method for the analysis of reinforced soil of unknown parameters being more than the number of equations available, is resolved by the horizontal slices method. The assumptions of HSM are (i) the vertical stress on an element in the soil mass is equal to the overburden pressure, (ii) the factor of safety (F.S.) is equal to the ratio of the available shear resistance to the mobilized shear stress along the failure surface, (iii) the factor of safety for all the slices is equal and (iv) the failure surface can have any arbitrary shape but does not pass below the toe of the slope or wall. Forces acting on a horizontal slice are shown in **Figure 12**.

With failure wedge divided into N horizontal slices, 4Nunknowns can be determined by 4 N equations and a complete formulation is possible. The formulation is simplified if only vertical equilibrium is considered for individual slices together with overall horizontal equilibrium for the whole wedge, no account being taken of moment equilibrium. In this case the number of equations and the number of unknowns get reduced to 2 N + 1. HSM was extended to design of RE walls considering oblique pull by Reddy et al. [54].
