Bearing capacity of foundation on slope

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1 Balachandra, A., Yang, Z. & Orense, R.P. (2013) Proc. 19 th NZGS Geotechnical Symposium. Ed. CY Chin, Queenstown A Balachandra Sinclair Knights Mertz Pty Ltd., Auckland, NZ (formerly University of Auckland) abalachandra@globalskm.com Z Yang Structex Ltd, Christchurch, NZ (formerly University of Auckland) JYang@structex.co.nz R P Orense Department of Civil Engineering, University of Auckland, NZ r.orense@auckland.ac.nz (Corresponding author) Keywords: bearing capacity, slope, discrete layout optimization ABSTRACT The problem of a footing on slope is encountered regularly in engineering practice, with some noteworthy examples being bridge abutments and basement excavations. This paper discusses the use of Discontinuity Layout Optimisation (DLO) technique in analysing the behaviour of shallow, strip footings near sloping ground problem through the LimitState:GEO software. Modification factors for N γ, N q and N c were proposed for a number of different combinations of variables, such as slope angle, edge distance, embedment depth, friction angle, slope length and cohesion. The results were then summarised in the form of design charts. Furthermore, the bearing capacity of embedded footings was also investigated and modification factors in terms of total bearing capacity were also recommended. It was found that sloping ground had significant effect on the bearing capacity, with the greatest reduction observed when footings were located at the edge of the slope. The validity of the results was confirmed by performing a thorough comparison with results using limit equilibrium method, upper bound solutions and FEM-based techniques. The design charts developed may be of significant interest to the profession, considering Auckland s topography, with many houses built along sloping grounds. 1 INTRODUCTION The natural topography of much of New Zealand can be described as either hilly or rolling terrain. Therefore, often in engineering projects, such those involving bridge abutments, building foundations and road pavements, foundations requiring to be constructed near sloping ground are encountered. The bearing capacity on sloping grounds has often been assessed using limit equilibrium method (Castelli and Motta, 2010; Hansen, 1970; Saran et al., 1989), upper bound method (Kumar and Rao, 2003) and finite element analysis (Loukidis and Salgado, 2009; Georgiadis, 2010). Of these methods, the Limit Equilibrium method (LEM) has been commonly used due to its simplicity; however, it is limited by the need to assume a failure mechanism. In this paper, the computer program LimitState:GEO, which is based on discontinuity layout optimisation, is used to model and analyse the problem of a footing located near sloping ground. Based on the results of analysis, the effects of different factors that contribute to the bearing capacity are analysed. The results of the analyses are compared to those available in the literature. Finally, the results obtained are presented in the form of design charts, which could be used in engineering practice for foundation design.

2 2 DISCONTINUITY LAYOUT OPTIMISATION LimitState:GEO, a computer program developed at Sheffield University, was used in this study. It uses the discontinuity layout optimisation (DLO) method for analysis and can be used to analyse a number of other geotechnical problems including slope failure, retaining walls, pipelines, tunnels and anchors (LimitState, 2010). The DLO procedure refers to an optimisation technique which is able to determine the critical layout of discontinuities, thereby producing the critical failure mechanism and the lowest collapse load. This is done by considering all connections between a node and all other nodes in the soil model. Therefore, far more connections are considered using the DLO method as opposed to say the finite element analysis. The connections themselves refer to the boundaries between the rigid blocks. Therefore, by performing a systematic search through all the connections available, a critical slip line could be determined (LimitState, 2009). The LimitState:GEO software uses the above DLO theory but modified slightly to reduce both memory usage and time required to perform the computations. The modification considers only connection between a node and neighbouring nodes, and performing a search through the available connections to find an upper bound solution. Using this upper bound solution an input, a search through all the discontinuities connecting each node with every other node is performed. In the process, the previously obtained solution is optimised, eliminating non-critical discontinuities until the correct solution is found (LimitState, 2009). An arrangement of nodal connections considered by LimitState:GEO is shown in Figure 1. 3 METHODOLOGY 3.1 Soil model Figure 1. Example of nodal connections (LimitState, 2009). The bearing capacity of footings is a function of the slope angle (β), distance between the footing and slope (d), embedment depth (D f ) and slope length (L). These are illustrated in Figure 2. The soil parameters include the friction angle (φ) and cohesion (c), as well as the unit weight of the soil (γ). For the parametric analysis, all the length parameters were normalised by the width of the footing, B. It was verified that that such normalisation did not have an effect on the bearing capacity as long as the nodal density was kept constant. The range of parameters considered in the study is presented in Table 1. The vertical and horizontal boundaries were specified a sufficient distance away from the footing so that they had no impact on the results. In this study, footings were analysed with a scale factor of 3 (i.e. 3 nodes for every 0.5m distance). This was determined considering both computational time and accuracy of the solution (less than 1% change in value).

3 Table 1: Range of parameters considered Parameters Range of values φ 25, 30, 35 and 40 0 c 10, 50, 100 and 150 kpa β * d/b 0 8 * D f /B 0, 0.5 and 1.0 L/B 2, 4, 5 and 7 Note:* - or until the threshold value is reached. Figure 2. Soil Parameters investigated 3.2 Determination of modification factor for N γ The output from the software LimitState:GEO was in the form of an adequacy factor. This can be multiplied by the applied load to determine the failure load. Therefore, knowing the area on which the ultimate load was applied, the bearing capacity of the soil at failure can be determined. However, as the adequacy factor does not differentiate the ultimate bearing capacity into three components as proposed by Terzaghi (1943), a purely cohesionless soil with no surcharge was considered so that the entire ultimate bearing capacity was solely due to the selfweight of the soil. Based on the parametric analysis, the modification factors were determined by computing the ratio between the reduced bearing capacity due to the presence of the sloping ground and the bearing capacity for level ground. Thus, the modification factor applies to N γ, where it modifies the N γ factor to take into account the effect of the sloping ground. 3.3 Determination of modification factor for N q As the software was unable to analyse bearing capacity problems involving weightless soils, a cohesionless soil with a known surcharge was used to determine the modification factors for N q. Following the same process as above, the total bearing capacity and the bearing capacity due to the self-weight of the soil were determined. Though simple subtraction it was possible to find the bearing capacity due to the surcharge component. As before, the modification factors for N q were calculated as the ratio of the bearing capacity for sloping ground and that of level ground. Note that in the software, only normal or shear loads are permitted to be applied and therefore, the surcharge load was applied normal to the slope surface. Likewise, a constant surcharge was applied on the surface boundary for the entire soil profile.

4 3.4 Determination of modification factor for N c Similar procedure as outline above was followed with the only difference being a purely cohesive soil under short-term conditions was considered. Again no surcharge was applied and as only short term undrained conditions were investigated, the friction angle of the soil was equal to zero. The modification factors for N c was determined by calculating the ratio of bearing capacity of soil when sloping ground is present over the level ground bearing capacity. 3.5 Effect of slope length For any slope angle there is a direct relationship between the slope height and slope length. Therefore, as opposed to slope height, the slope length was the parameter investigated. The effect on bearing capacity of different slope lengths was due to significant differences in the failure mechanism. Figure 3 shows the failure mechanism observed for slope lengths, L/B of 2 and 4, where for the former, the failure mechanism does not end at the toe of the slope but instead ending some distance to the right of the toe. (a) (b) Figure 3. Failure mechanism for (a) L/B = 2; and (b) L/B = Effect due to embedment The footings were modelled as simple footing with a uniform load applied to the top face. This was done so that it was valid for comparison with footings located above ground. However, in practice, embedded footings will be generally loaded with a column and therefore checks were performed to determine if the solutions obtained from the simple model were similar to the solutions obtained from the column model. It was found that the difference in the results obtained was very small; therefore, the simple model was deemed to be suitable. The effect of embedment depth on N γ and N q was difficult to determine as the final output was in terms of total bearing capacity. In this study, the modification factors for embedded footings were given in terms of factors that modify the total bearing capacity as opposed to modifying the individual bearing capacity factors. The factors were calculated as a ratio of the total bearing capacity for an embedded footing near a sloping ground over the bearing capacity of an embedded footing on level ground. 4 RESULTS AND DISCUSSION In the preliminary analysis, the N γ values obtained in the present study were compared with well-established solutions in the literature to gauge the validity of the model. Figure 4 shows the results where it is seen that the present solutions are in general agreement with those in the literature.

5 Figure 4. Comparison of N γ for level ground with existing solutions. Modification factors for N γ for friction angles of 30 and 35 degrees are presented in Figures 5. As explained earlier, the modification factors are the multipliers applied to level ground values due to the presence of the slope. It can be observed that both figures show very similar trend. For cohesionless soils, the threshold slope angle was found to be the same as the friction angle, with even a slight increase above the limit resulting in an unstable slope. This is indicative of a change in mode of failure, from bearing capacity to slope failure. Moreover, for high slope angles and when the footing was located at the very edge of the slope, a dramatic reduction in bearing capacity occurred. At high slope angles, the modification factor followed more or less a linear behaviour. Furthermore, the threshold distance at which sloping ground had no effect on the bearing capacity also increased with increasing sloping angle. In addition, the effect due to different slope lengths was pronounced for large slope angles and small d/b values. However, it could be seen that the effect due to slope length decreases as d/b value is increased. Likewise, the effect of slope length or the difference in bearing capacity observed for L/B = 2 and L/B 5 was greater when higher friction angles were considered. (a) (b) Figure 5. Modification factor for N γ for D f /B=0 and: (a) φ = 30 o ; and (b) φ = 35 o. The calculated modification factors for N q are shown in Figure 6 for φ = 30 o and 35 o. At higher friction angles and slope lengths, there was approximately a linear relationship between N q * /N q and d/b, with a very slight decrease in gradient at lower d/b values. However, the decrease in gradient, when moving from higher d/b values to zero, was much more pronounced for shorter slope length and lower friction angles. Like N γ modification factors, there was a significant loss

6 in bearing strength for footings located near the slope. However, unlike N γ, recovery of much of the bearing strength does not occur until about three d/b units away from the slope. (a) (b) Figure 6. Modification factor for N q for D f /B=0 and: (a) φ = 30 o ; and (b) φ = 35 o. In determining the modification factors for N c, only the short term bearing capacity behaviour of cohesive soil and undrained conditions were assumed (friction angle was assumed as zero). Modification factors for representative undrained shear strength are presented in Figure 7 for L/B=4. The sloping ground did not have the same drastic effect in reducing the N c component as it had in reducing the N γ and N q terms, even for footings located very near the sloping ground. Much of the cohesive component of the bearing capacity was recovered when the slope was a few d/b units away from the footing. In addition, threshold distances for cohesive soils were far smaller than that observed for cohesionless soils. Therefore, it could be concluded that for soils that are predominantly clayey, a sloping ground will have very little effect on the bearing capacity if it was built even a fairly small distance away from the sloping ground. Figure 7. Modification factor for N c with D f /B=0. As mentioned earlier, the modification factors for embedded footings were given in the form of factors that modified the total bearing capacity. In Figure 8(a), modification factors for embedded footings near sloping ground relative to footings with the same embedment ratio on level ground are provided. Figure 8(b) on the other hand shows modification factors for embedded footing near sloping ground relative to footings located above surface on level ground. Therefore, Figure 8(b) is useful in determining the amount of bearing strength that was gained through embedment and the percentage of which that was lost as a result of sloping ground. From Figure 8(a), it could be seen that there was a similar trend to N γ for lower slope

7 angles while an increasingly linear trend was observed as slope angles were increased. In comparing the modification factors obtained for embedded footings, with factors found for N γ and N q, it could be seen that for the same friction angle, the modification factors for embedded footing were generally greater than the respective N γ modification factor while being significantly lower than the respective N q modification factors. From Figures 8(a) and other calculations made, it was also noted that slope length had (a) (b) Figure 8. Modification factor for ultimate bearing capacity considering embedment depth: (a) φ = 35 o and D f /B=0.5; and (b) cohesionless soil with φ = 35 o. significant effect on the modification factors, with the effect increasing with increasing embedment depth. For very small slope lengths, the toe of the slope was almost level or only a small distance beneath the bottom of the footing. Therefore, if a line joining the bottom corner of the footing with the toe of the slope could be drawn, the relative slope angle of that to the footing was significantly less than the actual slope angle. Thus, a considerably greater bearing capacity was calculated for smaller slope lengths as opposed to larger slope lengths. On the other hand, Figure 8(b) shows the extent of increased bearing capacity that was obtained through embedment. This could be seen by examining the case of a footing with the relevant parameters being D f /B of 1, β = 35 and d/b of 0. For this footing the modification factor is almost equal to one. This indicates that it has the same bearing strength as a footing placed above ground that is not affected by sloping ground. Finally, as modification factors provided in Figure 8(a) were relative to footing embedded on level ground and therefore the effect due to depth was already accounted for, the modification factors only account for the presence of the sloping ground and should be used together with depth correction factor for the general bearing capacity equation. 5 CONCLUSIONS LimitState:GEO software was used to analyse the bearing capacity of footing near a slope for a number of different combination of parameters. It was found that the presence of the slope had significant effect on the bearing capacity and played an important role in foundation design. The major conclusions observed are as follows. The greatest reduction in bearing capacity was seen when footings were located at the edge of the slope. The reduction, however, decreased with increasing edge distance until the sloping ground had no effect. The threshold distance also increased with increasing slope angles and friction angles. For cohesionless soils the threshold slope angle was equal to the soil friction angle. Shorter slope lengths had significant effect on N γ and N q. This effect, however, diminished at larger slope lengths.

8 Slope length, L/B = 2 generally produced a less conservative solution than when higher slope lengths was used. Due to difficulty in separating the bearing capacity components for embedded footings, modification factors for total bearing capacity were proposed. ACKNOWLEDGEMENTS The authors would like to thank Dr Colin Smith of Sheffield University for allowing them to use the software with academic license. This study was the Part 4 research project of the first two authors in 2010, and they would like to thank all people involved in the research. REFERENCES Castelli, F. & Motta, E. (2010) Bearing capacity of strip footings near slopes. Geotechnical and Geological Engineering, Vol. 28, Georgiadis, K. (2010) Undrained bearing capacity of strip footings on slopes. Journal of Geotechnical and Geoenvironmental Engineering, 136(5), Hansen, J. (1970) A revised and extended formula for bearing capacity. Bulletin No. 28, Danish Geotechnical Institute, Copenhagen, Kumar, J. & Rao, V. (2003) Seismic bearing capacity of foundations on slopes. Geotechnique, 53, Limitstate (2009) LimitState:GEO Manual, Version 2.0. Limitstate (2010) Analysis & Design Software for Engineers, Loukidis, D. & Salgado, R. (2009) Bearing capacity of strip and circular footings in sand using finite elements. Computers and Geotechnics, 36, Michalowski, R. (1997). An estimate of the influence of soil weight on bearing capacity using limit analysis. Soils and Foundations, 37, Saran, S., Sud, V. & Handa, S. (1989) Bearing capacity of footings adjacent to slopes. Journal of Geotechnical Engineering, 115, 553. Terzaghi, K. (1943) Theoretical Soil Mechanics, Wiley New York.