Probabilistic Analysis of Multi-layered Soil Effects on Shallow Foundation Settlement

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1 Probabilistic Analysis o Multi-layered Soil ects on Shallow Foundation Settlement 54 Y L Kuo B Postgraduate Student, School o Civil and nvironmental ngineering, University o Adelaide, Australia M B Jaksa B(Hons), PhD, M.I..Aust, CPng Senior Lecturer, School o Civil and nvironmental ngineering, University o Adelaide, Australia W S Kaggwa BSc(ng), MngSc, PhD, M.I..Aust Senior Lecturer, School o Civil and nvironmental ngineering, University o Adelaide, Australia G A Fenton Bng, Mng, MA, PhD, P ng, M. ASC Proessor, Department o ngineering Mathematics, Dalhousie University, Canada D V Griiths BSc, MS, PhD, DSc, C ng, P ng, Proessor, Department o ngineering, Colorado School o Mines, USA J S Goldsworthy B(Hons) Postgraduate Student, School o Civil and nvironmental ngineering, University o Adelaide, Australia Summary: The results o a preliminary investigation into the eects o multi-layered soil on oundation settlement are reported in this paper. In this study, the settlement o a square pad ooting placed on two-layered soil proile is examined. Using a combination o Finite lement Method (FM) analyses and random ields simulation, probabilistic analyses o the settlement o ooting ounded on two-layered soil proile is established through Monte Carlo simulation. The Young s modulus ield has been simulated via Local Average Subdivision method (LAS) with a ixed mean, various coeicients o variation and spatial correlation structures. The coeicients o variation and spatial correlation structures in each layer are set to be dierent. It is assumed that the boundary between two soil layers is abrupt as may be represented by aults and geological unconormities in the soil mass. The results o the analyses indicate a modest decrease in the coeicient o variation o settlement o the -layered soil proile compared to the settlement o the uniorm single layer soil mass. INTRODUCTION Foundations o engineering structures transer and distribute their loading to the underlying soil and rock. Foundations are mostly supported by multi-layered soil proiles. It is widely known that the spatial variability in soil properties has signiicant eects on oundation perormance (i.e. bearing capacity and settlement). Due to the variety o soil types and multi-layered soil proiles that exist in nature, establishing a probabilistic analysis experimentally without the aid o modern high-speed computers would be tedious, i not impossible. Thereore, stochastic numerical modelling combined with Monte Carlo simulation has been adopted in this study. Random ield simulation is incorporated into the inite element modelling to simulate the soil medium as spatially random ields, taking into consideration the correlation structures. In the present study, the soil medium is divided into two layers, each with dierent spatial statistics. A probabilistic study is carried out on the settlement o a uniormly loaded rigid ooting placed at the surace o the two-layered soil medium. PROBABILISTIC ANALYSIS OF MULTILAYR-SOIL PROFILS Previous research on investigating and establishing probabilistic perormance o oundations ounded on a single layer spatially random ield has been reported by Baecher and Ingra (98); Righetti and Harrop-Williams (988); Zeitoun and Baker (99); Paice et al. (994, 996) and Fenton and Griiths (00). These research works represent the pioneering works o developing probabilistic analysis on oundation settlements ound on a randomly distributed soil using a combination o stochastic inite element analysis and Monte Carlo simulation. The random soil medium is modelled based on random iled theory (Vanmarcke 983), which assumes stationarity (i.e. mean and variance are constant throughout the entire soil mass). Two-dimensional inite element analysis has been employed and the ooting is assumed to extend large distances perpendicular to the plane o the random ield. It is also assumed that the D elastic modulus ield has an ininite correlation length

2 in this out-o-plane direction. In random ield modelling, since any abrupt change in the soil proile violates the basic assumption o stationarity, the soil proiles examined by previous researchers were limited to single layers. In the present study, the Local Average Subdivision (LAS) method (Fenton 990; Fenton and Vanmarcke 990, Fenton 994) has been employed due to its simplicity to generate realizations o possible elastic modulus random ields. A Gaussian random ield with zero mean, unit variance, and spatial correlation length, θ, is irst simulated and then transormed into a log-normal random ield using standard statistics techniques. A lognormal distribution is used to avoid negative values and because such distributions have been observed in practice. Similar to other random ield generators, LAS is also based on the stationarity assumption. Any abrupt change in the random process will contravene this basic assumption. In elastic settlement analyses, two constitutive parameters are required, or example, the Young s modulus,, and the Poisson s ratio, v. For the present study, the Poisson s ratio is ixed at 0.3 and the Young s modulus is modelled as a random ield with a constant mean, µ, o 0,000 kpa over the entire soil mass or all analyses. The coeicient o variation ( COV ) o the Young s modulus ield has been reported to vary between % and (Lee et al. 983). The statistical distribution o the Young s modulus ield is assumed to ollow a lognormal distribution. The correlation length, θ, describes the spatial continuity o the random ield; that is, values at adjacent locations are more correlated than those separated by larger distances. The spatial dependence is assumed to ollow an isotropic Markovian correlation unction: τ ρ ( τ ) exp () θln where ρ is the coeicient o correlation between two points separated by a distance τ. For simplicity, the spatial correlation length in the x, y and z directions are assumed to be the same, that is, isotropic. In the 3-dimensional inite element modelling, a two layered soil medium with a ixed total thickness,, o 30 m is assumed to overlay a rigid stratum. A rigid ooting carrying a uniorm load o, q 50 kpa, is placed 0 H T 54 at the centre o soil mass. Four sides o the inite element model are restrained against horizontal displacement but ree to deorm vertically. The base o the model is ully ixed. The soil mass is discretized into m m m eight-noded quadrilateral elements. The size o the inite element mesh in the horizontal direction is adjusted according to the width o the ooting. It is generally recognized that the mesh boundaries should be set at a distance at least ive times the loaded area to ensure that boundary eects do not inluence the results (Desai and Abel 97). The inite element program used in this study is 3-dimensional and is identical to that given by Smith and Griiths (998). Figure shows grey scale representations o a 3D random ield, each simulated using dierent correlation lengths by the LAS method. Darker areas in the simulated soil mass designate higher stiness, whilst lighter areas denote low values o elastic modulus. It can be seen that as the correlation length increases, the randomness decreases (there is transition rom darker areas to lighter areas). (i) (ii) (iii) Figure. Simulated Random Soil Proile Via LAS: (i) θ m ; (ii) θ 4 m ; (iii) θ 8 m. Figure shows a D x - z plane in grey-scale o a potential realization o the 3D Young s modulus ield. The values o Young s modulus,, taken vertically below the ooting are shown in the plot illustrating an abrupt change across the boundary. To simulate a two-layered random soil medium without deying the basic stationarity assumption, a simple approach is adopted. The soil medium is divided into two layers; each with

3 uniorm thickness, lying horizontally and separated by an abrupt boundary. ach layer is assigned dierent spatial statistics. It is assumed that each soil layer is unrelated to the other and any spatial continuity across the boundary is ignored. This boundary may represent aults or geological unconormities present in the soil mass. ach layer is then simulated independently and the two merged based on the location o the boundary and thickness o the layers. The simulated Young s modulus ields are assigned to the inite element mesh ater which an analysis o settlement is undertaken. q o 50 kpa 543 Rigid Footing 0 Layer Layer Boundary Depth Below Surace (m) Young's Modulus (kpa) Figure. Two-dimensional representation o a single ( m m) pad ooting ounded on a two-layered soil medium; The variation o elastic modulus with depth under the pad ooting (shown with white dots in ). For the parametric study undertaken in the present study, the ranges o coeicient o variation, COV, the scale o luctuation, θ, the width o ooting, W, and the ratio o the soil layer thickness, H / H, are summarised in Table. It has been reported that equilibrium solutions can be obtained by analysing the results o Monte Carlo simulations consisting o,000 realisations (Paice et al. 996; Fenton and Griiths 00). Table. Parametric Study: Varying Input Parameters in the Study. Parameter Values Considered COV, θ (m).0,.0, 4.0, 8.0 W (m).0, 4.0 H / H 0., 0.5,.0,.0, 5.0 RSULTS AND DISCUSSIONS In order to assess the eect o the abrupt boundary in the soil mass on the probabilistic settlement o a pad ooting, irst, a single soil layer was analysed and the results are presented in Figures 3 and 4. Two ooting widths, W {.0 m, 4.0 m}, soil correlation lengths, θ {.0 m,.0 m, 4.0 m, 8.0 m}, and COV {, }, were considered in this case. Figure 3 shows the variation o COV o ooting settlement, µ δ / σ δ or COV δ, with the variation o θ / W whilst Figure 4 shows the variation o the COV δ with the variation o θ / H T. These indings are validated by the results o previous analytical evaluations by other researchers (i.e. Paice et al. 996; Fenton and Griiths 00). Figures 3 and 4 show that, as the correlation length increases, the COV δ also increases or all ooting sizes. Also, as W increases, COV δ decreases.

4 544 COV W θ / W ln Figure 3. Coeicient o Variation o Settlement with Varying θ / W or a Single Soil Layer Proile. W COV θ / H T Figure 4. Coeicient o Variation o Settlement with Varying θ / H or a Single Soil Layer Proile. Figure 5 shows the variation o COV o ooting settlement in a soil proile with a discontinuity located 5 m below surace compared with the variation o COV o ooting settlement in a uniorm layer. The results indicate that COV δ decreases in a -layered soil proile compared to a single uniorm soil proile. It also shows the decrement is signiicant i the location o the discontinuity, H /W, is near to the ooting. T W W.0 % 5 m Discontinuity 5 m 8% Uniorm layer, ooting width 4 m Discontinuous layer located 5 m below surace, ooting width 4 m Uniorm layer, ooting width m Discontinuous layer located 5 m below surace, ooting width m θ / W Figure 5. Variation o COV o Settlement in Soil Layer with Discontinuity Located 5 m Below Surace. Figures 6 to 9 show some o the results o the parametric analyses perormed. The results are grouped by the ratio o the correlation length o the top layer over the correlation length o the bottom layer, θ / Ln θ Ln, and the width o the ooting, W. It can be seen that the top layer ( COV ) has the biggest inluence on COV δ or all the cases considered. This is to be expected as the ooting is ounded on this layer resulting in larger stress and strain increments within the upper layer.

5 545 θ / θ 0.5; W.0 m ln θ / θ 0.5; W 4.0 m ln COV ; COV COV ; COV COV ; COV % % % COV ; COV COV ; COV COV ; COV 0. 0 % 0. 0 H /H H /H Figure 6. Variation o the COV o Settlement or Various Layer Thickness ( θ. 0, θ 8. 0 and θ / θ 0.5 ). These results show the variation o COV δ o ooting settlement with the variation o the ratio o the thickness o the two soil layers, H / H, which indicates the location o discontinuity below the surace. Whilst the results show that the COV δ o a m wide ooting is higher than that or a 4 m wide ooting, the results also suggest that the variation o COV δ o a 4 m wide ooting, which has a greater inluence zone with respect to H / H, is greatly inluenced by the location o discontinuity in the soil mass compare to a m wide ooting. The indings also show that as H / H increases, COV δ approaches an asymptotic value, suggesting that there is little inluence on COV δ as the discontinuity is located urther rom the surace o the soil mass and beyond the inluence zone o the ooting. θ / θ 0.5; W.0 m ln θ / θ 0.5; W 4.0 m ln 8% 8% % COV ; COV COV ; COV COV ; COV 0. 0 % COV ; COV COV ; COV COV ; COV 0. 0 H /H H /H Figure 7. Variation o the COV o Settlement or Various Layer Thickness ( θ 4. 0, θ 8. 0 and θ θ 0.5 ). /

6 546 θ / θ.0; W.0 m ln θ / θ.0; W 4.0 m ln % % COV ; COV COV ; COV COV ; COV 0. 0 COV ; COV COV ; COV COV ; COV 0. 0 H /H H /H Figure 8. Variation o the COV o Settlement or Various Layer Thickness ( θ 8. 0, θ 4. 0 and θ / θ.0 ). θ / θ 4.0; W ln.0 m θ / θ 4.0; W 4.0 m ln % % COV ; COV COV ; COV COV ; COV 0. 0 COV ; COV COV ; COV COV ; COV 0. 0 H /H H /H Figure 9. Variation o the COV o Settlement or Various Layer Thickness ( θ 8. 0, θ. 0 and θ θ 4.0 ). / CONCLUSIONS In the present study, the response o a rigid ooting ounded on a two-layered spatially random soil proile has been investigated using combined stochastic inite element analysis with Monte Carlo simulation. Parametric studies have been carried out or a number o COV o elastic modulus, correlation length, width o the ooting and thickness o top and bottom layers, while the mean o elastic modulus was kept constant. The results indicate that the COV o ooting settlement is lower in soil layer with a discontinuity compared to a ooting ounded on a single uniorm soil layer. The decrement is more signiicant i the discontinuity is located near the surace o the soil proile and within the inluence zone o the ooting. It is suggested that the variation in the

7 547 soil layer that is nearest the oundation has the most dominating eect on oundation settlement. The results also suggest that COV δ o a wider ooting, which has a deeper zone o inluence, is ound to be more sensitive to the location o the discontinuity in the soil mass. ACKNOWLDGMNTS The research presented in this paper is part o ongoing work at the University o Adelaide unded by the Australia Research Council. Their support is grateully acknowledged. RFRNCS Baecher, G.B. and Ingra, T.S. (98). Stochastic FM in settlement predictions. Journal o the Geotechnical ngineering Division, ASC, Vol. 07, No GT4, pp Desai, C.S. and Abel, J.F. (97). Introduction to the Finite lement Method, Van Nostrand Reinhold, New York. Fenton, G.A. (990). Simulation and analysis o random ields, Ph.D. Thesis, Princeton University, Dept. o Civil ngineering and Op. Res., Princeton, New Jersey. Fenton, G.A. (994). rror evaluation o three random ield generators, Journal o ngineering Mechanics, ASC, Vol. 0(), pp Fenton, G.A. and Vanmarcke,.H. (990). Simulation o random ields via local average subdivision, Journal o ngineering Mechanics, ASC, Vol. 6(8), pp Fenton, G.A. and Griiths, D.V. (00). Probabilistic oundation settlement on spatially random soil, Journal o Geotechnical and Geoenvironmental ngineering, ASC, Vol. 8(5), pp Lee, I.K., White, W. and Ingles, O.G. (983). Geotechnical ngineering. Pitman Publishing Ltd, London. Paice, G.M. Griiths, D.V. and Fenton, G.A. (994). Inluence o spatially random soil stiness on oundation settlements. ASC Spec. Con.-Settlement 94, A.T. Yeung and G.Y. Felio, eds., ASC, New York, pp Paice, G.M. Griiths, D.V. and Fenton, G.A. (996). Finite element modelling o settlements on spatially random soil, Journal o Geotechnical ngineering, ASC, Vol. (9), pp Righetti, G., and Harrop-Williams, K. (988). Finite element analysis o random soil media, Journal o Geotechnical ngineering, ASC, Vol. 4(), pp Smith, I.M. and Griiths, D.V. (998). Programming the inite element method (3 rd d.). John Wiley & Sons, New York. Vanmarcke.H. (984). Random ields: analysis and synthesis. MIT Press, Cambridge, Mass. Zeitoun, D.G. and Baker, R. (99). A stochastic approach or settlement predictions o shallow oundations, Géotechnique, Thomas Telord Ltd, London, Vol. 4(4), pp