(a)
Length AC=2.466×10=24.66m
Length AB=√(〖(24.66)〗^2+〖(10)〗^2 )=26.61m
sinA/10=sin90/26.61
sin〖A=(10 sin90)/26.61〗
A=sin^(-1)((10 sin90)/26.61)
A=22.073°
β=22.073°
Assume φ=0 for undrained clay and su=cu (Xiao, 2015, p. 170)
F=c_u/NγH
1.2=c_u/(N×18×10)
c_u=216×N
Firm stratum depth not known ∴D=∞
From Fig 13.25 (Smith, 2014, p. 409) N=0.181
c_u=216×0.181
c_u=39.096kPa
c_u=39.1kPa
(b)
Group 1
φ^'=26°
c^'=2.5kPa
γ_d=18kN/m^3
γ_sat=19kN/m^3
γ_avg=18.5kN/m^3
r_u=0.28
H=10m
c^'/γH=2.5/(18.5×10)=0.0135
Using charts produced by Bishop and Morgenstern (1960) Figure 1 and Figure 2.
First use c^'/γH=0 from Figure 1.
Slope 2.466:1
φ^'=26°
n=1.4
m=1.2
F=m-nr_u
F=1.2-1.4(0.28)
F=0.808
Second use c^'/γH=0.025 D=1.00 from Figure 2.
Slope 2.466:1
φ^'=26°
r_u=0.28
From Figure 2 r_ue>0.8
r_ue>r_u∴ stop (Smith, 2014, p. 412)
n=1.5
m=1.65
F=m-nr_u
F=1.65-1.5(0.28)
F=1.23
For c^'/γH=0 F=0.808
For c^'/γH=0.025 F=1.23
Interpolate for c^'/γH=0.0135
F=0.808+((0.0135-0)/(0.025-0))(1.23-0.808)=1.036
(c)
Slice Area of Slice Saturated (m²) Area of Slice dry (m²) Total Area of Slice (m²) % of Slice Saturated % of Slice Dry γ_sat× % of Slice Saturated γ_dry× % of Slice Saturated γ_avg (kN/m3)
1 2.0029 7.6071 9.6100 20.84 79.16 3.9596 14.2488 18.2084
2 11.6344 8.7107 20.3451 57.19 42.81 10.8661 7.7058 18.5719
3 18.3438 4.9010 23.2448 78.92 21.08 14.9948 3.7944 18.7892
4 20.0854 4.0502 24.1357 83.22 16.78 15.8118 3.0204 18.8322
5 9.8914 2.9277 12.8191 77.16 22.84 14.6604 4.1112 18.7716
6 2.2649 2.9509 5.2157 43.42 56.58 8.2498 10.1844 18.4332
Areas taken from AutoCAD using Figure 3.
γ_sat=19kN/m3
γ_dry=18kN/m3
Slice γ_avg (kN/m3) Area of Slice (m²) Depth of Slice (m) Weight (kN) α° cosα N=Wcosα (kN) γ_water (kN/m3) Mean Height of Water Table Above Base of Slice (m)
1 18.2084 9.61 1 174.98 45 0.707 123.73 9.81 0.000
2 18.5719 20.47 1 380.17 33 0.839 318.83 9.81 2.820
3 18.7892 23.37 1 439.10 22 0.927 407.13 9.81 4.410
4 18.8322 24.25 1 456.68 9 0.988 451.06 9.81 4.050
5 18.7716 12.85 1 241.22 -2 0.999 241.07 9.81 2.625
6 18.4332 5.20 1 95.86 -12 0.978 93.76 9.81 0.375
Totals 1635.58
Slice U, Pore Water Pressure (kN/m2) L, Arc Length (m) UL (kN/m) sinα T=Wsinα (kN)
1 0.00 6.20 0.00 0.707 123.73
2 27.66 4.90 135.55 0.545 207.05
3 43.26 4.60 199.01 0.375 164.49
4 39.73 5.06 201.04 0.156 71.44
5 25.75 3.80 97.85 -0.035 -8.42
6 3.68 4.20 15.45 -0.208 -19.93
Totals 648.90 538.37
Group 1
φ^'=26°
tan〖φ^'=0.488〗
F=(Σ(N-UL) tan〖φ_c 〗)/ΣT=((1635.58-648.90)×0.488)/538.37=0.894
F=0.89
Swedish Method of Slices analysis (Smith, 2014, pp. 394-397)
(d)
WORST CASE
Centre at (105.00m,24.500m) Radius 25.000m
Iterations: 64 Horiz acceleration [%g]: 0.0
Net vertical force [kN/m]: 1.2279 Slip weight [kN/m] 1456.0
Net horiz force [kN/m]: 3.0792 Disturbing moment [kN/m]: 12266.
Restoring moment [kNm/m]: 12266.
Reinf.Rest.Moment [kNm/m]: 0.0
Factor of Safety: 1.0000
(e)
From parts (b)-(d) it can be seen that the slope is unstable or does in fact fail as the factor of safety is below 1. For highway embankments 1.25 as a factor of safety is used as a minimum (Samtani & Nowatzki, 2006, Chapter 6, p. 5) and all of the values within parts (b)-(d) fall well below this meaning the original design of the slope was not designed to standard.
From the Bishop and Morgenstern charts a factor of safety of 1.036 was calculated. This is an over-estimated factor of safety because the charts average the pore water pressure throughout the cross-section. This results in the ru value being reduced, thus pushing up the factor of safety overall (Barnes, 1999). Bromhead (1992, pp. 158-159) suggests the charts were created for soils with very little cohesion and more specifically “earth dams”, leading them to become “impaired” when dealing with a cut slope made of soft clay, which itself is cohesive.
The Swedish method of slices, also referred to as the Fellenius method, underestimates the factor of safety as it does not consider the forces between the slices (Knight, 2015, p. 39) and Craig (2004, p. 354) also mentions this method as having an error of “5-20%” and Lei, Chiu and Zheng (2011) describe how errors with this method “can reach 50%”. This is shown through the factor of safety being calculated at 0.89, which is well below the factor of safety given by any other method.
Bishop’s method through the use of Oasys slope as shown in part (d) gave a factor of safety of 1.00. This method gives the most accurate factor of safety as it considers the vertical forces, which changes for each slice (Yuen, n.d.). The software also analyses more slices in the slope and a range of possible failure surfaces can be located. A greater accuracy for the water table can also be inputted into the programme and this “is the most important factor in slope stability” (Abramson, Lee Sharma & Boyce, 2002, p. 48). The drained and saturated unit weights of the soil can also be used alongside the water table to give an accurate picture of the slope. Comparing Bishop’s method to the Bishop and Morgenstern charts that use an average pore water pressure, the charts are far less precise. Using a single ru value should not be used in detailed analysis (Chowdhury & Zhang, 1991, p. 27). The Swedish method of slices uses a ratio of dry and saturated unit weight based upon the areas calculated in a few slices and assuming the base of each slice is a straight line. In real life the base of each of the slices are curved and this is what Oasys uses in its analysis, providing a much higher level of accuracy in the factor of safety produced. Oasys slope also makes sure that all “the forces and/or moments acting on each slice are in equilibrium”, which is much more than any of the other methods provide in terms of accuracy (“Slope – Slope Stability Analysis Software”, 2018). However, it does underestimate by roughly “7%” (Craig, 2004, p. 355) and its error “does not exceed 10%” (Lei, Chiu and Zheng, 2011), which means there is still some error in the factor of safety calculated, but less so than other methods.
The long-term condition of the slope is when the pore water pressures are “in hydrostatic equilibrium with the ground water” (Skempton, 1964, p. 77) or in simpler terms some of the water will drain away over time matching the existing water table (Sivakugan, 2014). Long-term stability refers to the effective stress parameters of the soil (Craig, 2004, p. 426) and the soil being “fully drained” (Craig, 2004, p. 363). Bromhead (1992, p. 188) describes that for clay soils “pore-fluid pressures cannot escape” and in the long-term they can greatly affect the soil. Also, that drained refers to the “absence of stress-induced pore-pressure” responses and not that there is no water present.
The depth and location of cracking and swelling must also be considered when analysing the slope in the long-term and omission in calculations can significantly change the factor of safety generated (Chowdhury & Zhang, 1991, p. 27). Bromhead (1993, p. 139) indicates that for undrained clay soils tension cracks can be deeper than in other cases and can even reach the water table and cause failure.
Failure will most likely occur in the short-term (undrained) when the clay can be treated as homogenous (Sivakugan, 2014), but as Bromhead (1992, p. 203) suggests both the short-term and long-term should be considered and “caution” used. The long-term is of importance for cuttings in particular because over time the factor of safety will reduce as the clay expands, leading to an increased water content, thus reducing effective stress, which is why the analysis should be carried out “using effective stress parameters” (Samtani & Nowatzki, 2006, Chapter 6, p. 4). Terzaghi (1967, p. 232) states that if there are any “undetected discontinuities” they “may completely invalidate the results” of any calculations, showing the importance for a detailed site investigation and collecting as much data on the slope as possible.
(f)
There are a range of solutions that could be implemented in order to stabilise the slope. The way in which the slope is constrained and the context in which it is situated must first be considered. The slope is part of a road alignment meaning that there is a road at the footing of the slope. Some of the solutions that could be used are not feasible in this case and so careful consideration must be taken when choosing the most appropriate solution.
Drainage is one of the possible solutions that can be used to help stabilise the slope. According to Abramson, Lee, Sharma and Boyce (2002 p. 487), drains in clay soils “can very efficiently lower pore water pressures” thus increasing the stability of the slope. This is echoed by D’Acunto and Urciuoli (2006, p. 764) who modelled seasonal variations in pore pressures in a slope and assessed that drainage eliminates high pore pressures. Farrar and Brady (2000, p. 14) comment on the fact that clay fill in modern construction is well compacted resulting in low permeability, thus water is not able to drain away easily over time.
There are different types of drains that could be implemented into a slope of this nature. Counterfort drains or rock ribs as explained by Carder, Watts, Campton and Motley (2008, pp. 15-17) enable the slope to be stabilised and an example of the A12 in Romford is given showing how slope drainage of this kind significantly reduced pore water pressures and that stability of the slope had been achieved.
The slopes slip circle that has been calculated and shown in question (d) makes counterfort drains a viable option. Price and Fish (2017, p. 2) highlight that counterfort drainage as a solution is for “shallow slips typically up to 6m depth” and from Oasys slope using Bishops variably inclined interface method the slip is seen to be just over 4m, as can be seen in Figure 4. Macdonald, Vooght, and Parkin (2012, p. 115) also suggest that this type of drainage can be used from “2-7m”, however they go on to state that at the upper end of this depth difficulties may present themselves during construction. They also give a general range of values for drainage centres and widths and suggest that within the drains, coarse material with geotextile should be used to stop finer soil particles from entering the drain. A typical counterfort drain section is shown by Hearn and Hunt (2011, pp. 232-233), whereby the granular backfill used cuts below the slip surface to “break up the continuity of the failure surface” and leads to a collector drain at the base of the slope that is perpendicular to the slope to take the water away from the potential failure area. However, a collector pipe may not be feasible in certain situations for safety reasons (Farrar & Brady, 2000, p. 9).
Counterfort drains are usually installed with excavators that dig the trenches to depth and fill with the coarse material and geotextile as they travel up the slope as pictured on the M26 Sevenoaks cutting (Macdonald, Vooght, and Parkin, 2012, p. 117). Site access and construction methods are laid out for installation of counterfort drains by Goldfingle (2010). The site presented was challenging and had a railway line at the base of the slope and the ground was boggy so inventive ways of carrying out the works was necessary. The idea of using a long-reach excavator and spider cranes along with traditional 9T excavators are presented, the selection of which is dependent on the specific site. As the slope that has been calculated is 22°, excavators are able to track up and down easily as Bennink (2017) describes and can reach 30° within industry limits. Usually drainage is installed in an “uphill direction”, however this may interfere with the steel framed building being constructed at the top of the slope, so downhill is preferable in this case (Price & Fish, 2017, p. 4). If it is seen to be too unstable to put plant on the slope a long-reach excavator could sit at the base of the slope and excavate the trenches. Operatives would use hand tools to lay the geotextile, and the excavator would then lay the coarse material on top. Price and Fish (2017, p. 4) give a sequence of works for how the trench should be dug saying that the trench should be dug in stages to prevent collapse and supports should be used where necessary. The works should also be carried out in the summer months to ensure the water table is at its lowest and it is suggested that if collapse becomes problematic then dewatering of the slope by “less hazardous methods” should take pace first.
After the drains have been installed maintenance is key because clogging is a potential issue that in the long term can reduce water flows through the drain resulting in a decrease in effectiveness (Hutchinson, 1977, p. 148).
Another viable option is using lime to stabilise the slope. Rogers and Bruce (1991, pp. 395) describe how the addition of lime to a clay slope has almost an immediate effect on stability, as within “24-72 hours” there is a reduction in the presence of water and the clay becomes more granular. This is of significant importance to the slope in question as the water table is high, which is leading to instability. Lime columns are the most appropriate form of lime stabilisation measure for a slip depth as outlined in (d), as they can penetrate down to “10m deep”, which is below the failure slip circle meaning that it can increase the shear resistance of the slope, preventing possible failure (Rogers & Bruce, 1991, pp. 396). The process reduces the bulk density of the clay and increases the permeability, resulting in quicker flows of water through the slope (Rogers and Bruce, 1991, pp. 397). This is linked to the solution of counterfort drainage previously presented as the water flows can feed into the drainage and be moved out of the slope, reducing the water table more effectively. The lime columns effectively act as vertical drainage through the slope, reducing the pore water pressures (Abramson, Lee Sharma & Boyce, 2002, pp. 550-553). This solution uses “small equipment” that can be used on slopes that are difficult to access and are unstable (Rogers & Bruce, 1991, pp. 402). This also means that the cost of this slope stabilisation measure is less than that of other possible piling techniques. It also has the added benefit of not making any of the conditions of the slope worse during installation compared to that of driven piles which can have a detrimental effect on the properties of the slope (Abramson, Lee Sharma & Boyce, 2002, p. 552). Rogers and Glendinning (1997, pp. 246-247) highlight the further benefits of this method of slope stabilisation and how the pile itself can increase the bearing capacity of the soil. Lime can also stabilise the soil to prevent swelling and reduce the size of shrinkage cracks (Bell, 1993, p. 259), which is one of the long-term failure modes of a cut slope.
Using equations from Abramson, Lee Sharma & Boyce (2002, p. 552) to calculate what the increase in soil strength is from installing lime columns.
c_avg=c_u (1-a)+S_col/a
a=(πD^2)/(4S^2 )=(π〖×0.25〗^2)/(4〖×1.5〗^2 )=0.0218
c_avg=39.1(1-0.0218)+300/0.0218
c_avg=13789
Taking D as 250mm and S as 1.5m as outlined by (Bell, 1993, p. 262) and undrained shear strength from (a) and average shear strength of stabilised clay within columns from…
References
Abramson, L. W., Lee, T. S., Sharma, S., & Boyce, G. M. (2002). Slope stability and stabilization methods. (2nd ed.). New York: John Wiley & Sons.
Barnes, G (1999). Synopsis. Retrieved from https://www.newcivilengineer.com/synopsis/676801.article
Bell, F. G. (1993). Engineering Treatment of Soils. London: E & FN Spon
Bennink, C. (2017). Ensure Safe, Effective Excavator Operation on Slopes. Retrieved from https://www.forconstructionpros.com/equipment/earthmoving/excavators/article/12285554/ensure-safe-effective-excavator-operation-on-slopes
Bishop, A. W., & Morgenstern, N. R. (1960). Stability coefficients for earth slopes. Geotechnique, 10(4), 129-150. https://doi.org/10.1680/geot.1960.10.4.129
Bromhead, E. (1992). The stability of slopes. (2nd ed.). Retrieved from https://www.scribd.com/doc/93754887/Bromhead-1992-the-Stability-of-Slopes-2nd-Ed
Carder, D. R., Watts, G., Campton, L., & Motley, S. (2008). Drainage of earthworks slopes. Retrieved from https://trl.co.uk/sites/default/files/PPR341.pdf
Chowdhury, R. N., & Zhang, S. (1991). Tension cracks and slope failure. In R. J. Chandler (Ed.), Slope stability engineering developments and applications: Proceedings of the international conference on slope stability organized by the Institution of Civil Engineers and held on the Isle of Wight on 15–18 April 1991 (pp. 27-32). https://doi.org/10.1680/ssedaa.16606.0005
Craig, R. F. (2004). Craig’s Soil Mechanics. (7th ed.). Retrieved from http://www.icivil-hu.com/Civil-team/3rd/Geo-technical%20Engineering%20(Soil)/SOIL%20CRAIG.PDF
D’Acunto, B., & Urciuoli, G. (2006). Groundwater regime in a slope stabilized by drain trenches. Mathematical and computer modelling, 43(7-8), 754-765.
Farrar, D. M., & Brady, K. C. (2000). Drainage of Earthwork Slopes: A Review. Retrieved from https://trl.co.uk/sites/default/files/TRL454.pdf
Goldfingle, G. (2010). Drainage Experience. Retrieved from https://www.newcivilengineer.com/latest/draining-experience/8624216.article
Hearn, G. J., & Hunt, T. (2011). C6 Slope and Road Drainage. Geological Society, London, Engineering Geology Special Publications, 24(1), 231-242. https://doi.org/10.1144/EGSP24.15
Hutchinson, N. N. (1977). Assessment of the effectiveness of corrective measures in relation to geological conditions and types of slope movement. Bulletin of the International Association of Engineering Geology-Bulletin de l'Association Internationale de Géologie de l'Ingénieur, 16(1), 131-155. https://doi.org/10.1007/BF02591469
Knight, Z. (2015). Slope stability analysis of tailings dam embankments. (Unpublished undergraduate dissertation). Charles Darwin University, Australia. Retrieved from https://core.ac.uk/download/pdf/47204314.pdf
Lei, G. H., Chiu, A. C. F. & Zheng, Q. (2011). Understanding the limitations of the Swedish method of slices from the stress perspective. Retrieved from http://geoserver.ing.puc.cl/info/conferences/PanAm2011/panam2011/pdfs/GEO11Paper780.pdf
Macdonald, G. J., Vooght, A. R., & Parkin, S. (2012). The use of deep counterfort drains as an effective method of stabilizing cuttings constructed in overconsolidated clays. Geological Society, London, Engineering Geology Special Publications, 26(1), 115-124. https://doi.org/10.1144/EGSP26.14
Price, S. L., & Fitch, N. R. (2017). Counterfort drains–design, installation and long-term performance in soils of Greater Auckland. Retrieved from https://fl-nzgs-media.s3.amazonaws.com/uploads/2017/11/NZGS_Symposium_20_Price1.pdf
Rogers, C. D. F., & Bruce, C. J. (1991). Slope stabilisation using lime. In R. J. Chandler (Ed.), Slope stability engineering developments and applications: Proceedings of the international conference on slope stability organized by the Institution of Civil Engineers and held on the Isle of Wight on 15–18 April 1991 (pp. 395-402). https://doi.org/10.1680/ssedaa.16606.0061
Rogers, C. D. F., & Glendinning, S. (1997). Improvement of clay soils in situ using lime piles in the UK. Engineering geology, 47(3), 243-257. https://doi.org/10.1016/S0013-7952(97)00022-7
Samtani, N. C. & Nowatzki, E. A. (2006). Soils and Foundations Reference manual – Volume I. Retrieved from https://www.fhwa.dot.gov/engineering/geotech/pubs/nhi06088.pdf
Sivakugan, S. (2014). Drained vs Undrained Loadings in Geotechnical Engineering. Retrieved from https://www.linkedin.com/pulse/20140618014822-6961529-drained-vs-undrained-loadings-in-geotechnical-engineering/
Skempton, A. W. (1964). Long-term stability of clay slopes. Geotechnique, 14(2), 77-102. https://doi.org/10.1680/geot.1964.14.2.77
Slope – Slope Stability Analysis Software. (2018). Retrieved from the Oasys website: https://www.oasys-software.com/products/geotechnical/slope/
Smith, I. (2014). Smith's elements of soil mechanics. Retrieved from https://ebookcentral.proquest.com
Terzaghi, K., & Peck, R. B. (1967). Soil mechanics in engineering practice (2nd ed.). New York: John Wiley & Sons.
Yuen, S. T. S. (n.d.). THE STABILITY OF SLOPES. Retrieved from https://people.eng.unimelb.edu.au/stsy/geomechanics_text/Ch11_Slope.pdf
Xiao, M. (2015). Geotechnical engineering design. Retrieved from https://ebookcentral.proquest.com