**7.3 Semi-analytical design method for cohesive backfill reinforced slope**

Abd and Utili [33] developed a semi-analytical method for uniform slope with c-ф soil using Limit Analysis (LA). The method provides the amount of reinforcement needed as a function of cohesion, tensile strength, angle of shearing resistance and slope inclination. Climate induced crack and cracks that form due to slope collapse are accounted for in this method. Both soil and reinforcement are assumed to be rigid-plastic and follow normality rule i.e., associated plastic flow.

#### **Figure 14.**

*Variation of dissipation time (t90) with coefficient of consolidation (cv) for reinforcement length equal to height of slope (after Naughton et al. [32]).*

**29**

**Figure 16.**

*Abd. and Utili [57]).*

*7.3.1 Design*

**Figure 15.**

Increasing Distribution - LID) (**Figure 16**).

yield and H - the slope height.

Average strength of reinforcement, Kt, for UD case is

Traction free slopes with slope angle of 40–90° reinforced with geosynthetic layers are considered. Reinforcement is equally spaced throughout (uniform distribution - UD) or at a spacing decreasing from top to bottom of slope (Linearly

*Variation of required transmissivity of the geogrids,* θ*, with coefficient of volume compressibility, mv, for a* 

*range of coefficients of consolidation, reinforcement length = 1.0H (after Naughton et al. [32]).*

where n- the number of reinforcement layers, T - the strength of a single layer at

*Geosynthetic-reinforcement layouts: (a) uniform and (b) linearly increasing distribution with depth (after* 

K nT / H <sup>t</sup> = (12)

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

**Figure 15.**

*Slope Engineering*

reduces to less than 12 hours (**Figure 14**). For soils with cv up to 50 m2

draining geogrid is less than 1.2x10−6 m2

vertical spacing of the draining geogrid.

and achievable transmissivity in the draining geogrid.

soil improves, i.e., cv > 75 m2

ranges of soil parameters.

30 m2

m2

pressure dissipation. Studies reveal that this is achievable unless cv is less than

/year and vertical spacing is more than 0.5 m. For cv values greater than 50

/year, the unfactored required transmissivity in the

/year, the required transmissivity increases rapidly by

/s. As the drainage characteristic of the

/year and thickness of fill not exceeding 0.5 m, the dissipation time required

orders of magnitude (**Figure 15**). The required transmissivity also depends on the

Smaller vertical spacings and longer reinforcement lengths require larger transmissivity in the draining geogrid. Reinforcement spacing of 0.5 m optimizes the time for dissipation of pore pressures while, at the same time, requiring realistic

At this spacing a draining geogrid will dissipate pore pressures over the full range of likely values encountered in low permeability fills within 24 hours, with the further advantage that the required transmissivity is independent of reinforcement length. With above spacing, reinforced slope with poorly draining backfill can be analyzed as a normal slope for both internal and external stability for normal

**7.3 Semi-analytical design method for cohesive backfill reinforced slope**

to be rigid-plastic and follow normality rule i.e., associated plastic flow.

Abd and Utili [33] developed a semi-analytical method for uniform slope with c-ф soil using Limit Analysis (LA). The method provides the amount of reinforcement needed as a function of cohesion, tensile strength, angle of shearing resistance and slope inclination. Climate induced crack and cracks that form due to slope collapse are accounted for in this method. Both soil and reinforcement are assumed

*Variation of dissipation time (t90) with coefficient of consolidation (cv) for reinforcement length equal to height* 

**28**

**Figure 14.**

*of slope (after Naughton et al. [32]).*

*Variation of required transmissivity of the geogrids,* θ*, with coefficient of volume compressibility, mv, for a range of coefficients of consolidation, reinforcement length = 1.0H (after Naughton et al. [32]).*

#### *7.3.1 Design*

Traction free slopes with slope angle of 40–90° reinforced with geosynthetic layers are considered. Reinforcement is equally spaced throughout (uniform distribution - UD) or at a spacing decreasing from top to bottom of slope (Linearly Increasing Distribution - LID) (**Figure 16**).

Average strength of reinforcement, Kt, for UD case is

$$\mathbf{K}\_t = \mathbf{n} \mathbf{T} / \mathbf{H} \tag{12}$$

where n- the number of reinforcement layers, T - the strength of a single layer at yield and H - the slope height.

#### **Figure 16.**

*Geosynthetic-reinforcement layouts: (a) uniform and (b) linearly increasing distribution with depth (after Abd. and Utili [57]).*

For LID case local reinforcement strength, K, is

$$\mathbf{K} = 2\mathbf{K}\_\mathbf{r} \left(\mathbf{H} - \mathbf{y}\right) / \mathbf{H} \tag{13}$$

where y - the vertical coordinate from the slope toe. Maximum depth of crack is limited from the requirement that remaining slope remains stable. Upper bound maximum crack depth to be adopted [59] is

#### **Figure 17.**

*Design charts for intact slopes not subject to crack formation (t = 1), intact slopes subject to crack formation (limited tensile strength of t =0.5, t = 0.2 & t = 0) and cracked slopes. (a) & (b) are for c/*γ*H = 0.05 while (c) & (d) are for c/*γ*H = 0.1. (after Abd and Utili [33]).*

**31**

**Figure 18.**

*(a) UD of reinforcement; and (b) LID.*

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

slope angles β and ϕ for specified value of c/γH.

φ

provides Kt/γH for slope inclinations of 400

for this case.

*Design charts for the required reinforcement for intact and cracked slopes (with* 

two (after Abd and Utili [33]).

φof 200

becomes important. Above charts are for fully drained slopes.

Design charts (**Figure 17**) provide the reinforcement strength and embedment length for uniform and linearly increasing reinforcement distributions for different

In **Figure 17**, 't' is dimensionless coefficient representing soil tensile strength and is defined as ratio of ground tensile strength to be measured experimentally over maximum unconfined tensile strength consistent with Mohr-Coulomb criteria. Considering the case of intact slopes, it can be observed that for relatively low values of cohesion, c/ϒH = 0.05, the tensile strength, t, has negligible effect on the required reinforcement force. But for higher values of cohesion (c/ϒH = 0.1), t

Using **Figure 17**, Kt/γH can be determined for given slope angle, β, and angle of

Eq. 10 for given number of layers. Influence of porewater pressure on required amount of reinforcement is analyzed using ru method [60]. A uniform value of ru is assumed throughout the slope and effective stress analysis carried out. **Figure 18**

Gray and black lines in **Figure 17** indicate respectively active and inactive constraint of maximum crack depth. The mark + signals the boundary between

Gray and black lines in **Figure 18** indicate respectively active and inactive constraint of maximum crack depth. The mark + signals the boundary between the

A combined failure mode consisting of pullout in some layers and rupture (tensile failure) in others, also needs to be considered to calculate the minimum length of the reinforcement layers. **Figure 19** provides Lr/H as a function of slope

, of soil. Tensile strength of reinforcement is calculated using

to 90°, ru = 0, 0.25 and 0.5 for UD and

φ

 *= 20° and c/*γ*H =0.1)* 

*7.3.2 Required reinforcement*

shearing resistance,

LID cases.

the two.

angle, β for
