1. Model Studies of Bearing Capacity of Strip Footing on Sand Slope

KSCE Journal of Civil Engineering (2013) 17(4):699-711DOI 10.1007/s12205-013-0406-x

− 699 −

www.springer.com/12205

Geotechnical Engineering

Model Studies of Bearing Capacity of Strip Footing on Sand Slope

M. Salih Keskin* and Mustafa Laman**

Received December 16, 2011/Accepted August 27, 2012

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Abstract

An experimental investigation into the ultimate bearing capacity of strip footing on sand slope is reported. The parametersinvestigated are the effect of setback distance of the footing to the slope crest, slope angle, relative density of sand and footing widthon the ultimate bearing capacity of strip footings. A series of finite element analyses was additionally performed on a prototype slopeto ascertain the validity of the findings from the laboratory model tests and to supplement the results of the model tests. Theagreement between observed and computed results is found to be reasonably well in terms of load-settlement and general trend ofbehavior. The results show that the ultimate bearing capacity increases with increase in setback distance, relative density of sand,footing width and decrease in slope angle. At a setback distance of five times of the width of the footing, bearing capacity remainsconstant like that of a footing on level ground.Keywords: slope, bearing capacity, shallow foundation, laboratory test, finite element method

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1. Introduction

The bearing capacity of the foundations is a primary concernin the field of geotechnical engineering. Design of foundationson a horizontal ground surface depends on the mechanicalcharacteristics of the soil such as unit weight, shear strength etc.,and the physical properties of the foundation such as depth,width, and shape. There are two considerations to decide theallowable bearing pressures for shallow foundations; the safetyfactor against ultimate shear failure must be adequate and thesettlements under allowable bearing pressures should not exceedtolerable values. Several methods may correctly predict thebearing capacity of foundations resting on or in level grounds(Terzaghi, 1943; Meyerhof, 1963; Hansen, 1970; Vesic, 1975).They are either based on laboratory or in-situ test results.However, there are many circumstances where foundations mustbe built on or near a slope. Due to the land limitation, architecturaland economical purposes, structures are generally placed on theslope crest or at a setback distance from the slope crest. Examplesinclude bridge piers supported on approach embankments,foundations on electrical transmission towers and somebuildings. The stability of the slope and the bearing capacity of afoundation constructed near to the edge of a slope are importantfactors in the performance of the structure built near a slope.When a foundation is constructed on sloping ground, one side ofthe foundation is exposed to the sloping surface. Thus, as thefoundation soil approaches limit state the plastic region of failureis very limited and it significantly affect the mechanical stability

of slope and thereby the bearing capacity of the foundation.Consequently, the bearing capacity of a foundation near a slopedfill is less than that on a flat ground. Therefore, in manysituations, the conventional types of shallow foundations areinadequate even though they are very economical and easier toconstruct. Hence, researchers have paid attention to investigatethe bearing capacity and settlement behavior of footings onslopes.

The methods to predict the bearing capacity of footings on orin level grounds are well developed. However the bearingcapacity of footings near or on slopes, much still remains to beinvestigated. The theoretical methods on the bearing capacity offooting on a slope (Meyerhof, 1957; Hansen, 1970; Vesic, 1975;Saran et al., 1989) are generally developed based on the bearingcapacity equation proposed by Terzaghi (1943) and valid for alimited range of footing location and embedded depth. Meyerhof(1957) proposed a theoretical solution to determine the ultimatebearing capacity of a shallow foundation located on the face of aslope. According to Meyerhof, the ultimate bearing capacity canbe expressed as:

(1)

where,B = Width of footing c = Cohesion of soil

Ncq, Nγq = Bearing capacity factorsγ = Unit weight of soil

qu cNcq12---γBNγq+=

*Assistant Professor, Dept. of Civil Engineering, University of Dicle, Diyarbakir 21280, Turkey (Corresponding Author, E-mail: [email protected],[email protected])

**Professor, Dept. of Civil Engineering, University of Cukurova, Adana 01330, Turkey (E-mail: [email protected])

M. Salih Keskin and Mustafa Laman

− 700 − KSCE Journal of Civil Engineering

For the special case of a footing resting on the surface of asaturated, clean, granular material (apparent cohesion, c = 0), thefirst term of Eq. (1) become zero and Meyerhof’s equationreduces to

(2)

Graham et al. (1987) provided an analytical solution for thebearing capacity of a shallow footing on the top of a cohesionlessslope based on the method of stress characteristics. Gemperline(1988) and Shields et al. (1990) developed empirical equationsfor the ultimate bearing capacity factors for a footing on a slopebased on centrifuge tests. Narita and Yamaguchi (1990) used amethod of slices to determine the bearing capacity of a threedimensional footing located on top of a slope. Buhan andGarnier (1998) used yield design theory to evaluate the ultimatebearing capacity of a shallow rectangular footing located on topof a slope. Furthermore, experimental studies on the bearingcapacity and settlement behavior of footings on slopes arerelatively limited (Shields et al., 1977; Garnier et al., 1984;Gemperline, 1988).

Shields et al. (1977) have carried out series of experiments toobtain the bearing capacity factor Nγq for a footing oncohesionless slope. They showed that the theory of Meyerhof(1957) overestimates the magnitude of the bearing capacity.However, at shallow depths close to the edge of the slope thetheory is closer to the experimental values. Garnier et al. (1984)presented an experimental study on strip footing near a slope toevaluate the coefficient of reduction of bearing capacity due toslope effect. The tests were performed using three slope modelsand loads were applied on the model footing at differentdistances from the edge of slope. For different slope models, itwas found that the bearing capacity of the footing was notpractically different from the value of distance/width ratio (b/B)greater than 6 due to the effect of slope.

Research in the area of bearing capacity of footings on slopingground is very much in demand because of the significant effectof the slope on the bearing capacity. Physical modeling is one ofthe best approaches to overcome the limitations of analyticalmethods (Wood, 2004). It is common practice in geotechnicalengineering that the results gained from a physical model areused to validate analytical and numerical models. Although thelaboratory model tests have several drawbacks such as the scaleeffect, model tests do provide reasonable understanding of thebearing capacity of shallow foundations.

The object of this study is to investigate the ultimate bearingcapacity of strip footings located on top of a slope withlaboratory model tests. In the study, the relationship between thefooting response and the variable parameters including, edgedistance between the footing and the crest of slope, angle of theslope inclination, relative density of sand and width of thefooting were investigated. And also, numerical analyses on aprototype footing-slope system were conducted using a commercialfinite element program PLAXIS (professional version 8,

Brinkgreeve and Vermeer, 2002) to verify the model test results.

2. Experimental Study

2.1 Test Set-upA series of laboratory model tests were performed in a test box

made of a steel frame with inside dimensions of 1.140 m(length), 0.475 m (width) and 0.500 m (depth) as shown in Fig.1. The bottom and vertical edges of the box were stiffened usingangle sections to avoid lateral yielding during soil placement andloading of the model footing. The two sidewalls of the test boxwere made of 20 mm thick glass to see the sand sample duringpreparation and observe the sand particle deformations duringthe tests. The box was enough rigid to provide plane strainconditions for all model tests. Static vertical loads were appliedto the model footings by a motor-controlled hydraulic jacksystem. The system attached to the loading frame located abovethe test box has a loading rate of 0.5 mm/min. An electronic 15kN capacity load cell was used to measure applied loads.Settlements of the footing were measured using two LinearVariable Displacement Transducers (LVDTs) located at the twocorners of the model footing. The load cell and the displacementtransducers were connected to data acquisition system (ADU)for recording and data handling.

qu12---γBNγq=

Fig. 1. Schematic View of the Experimental Set-up: (a) Side view,(b) Plan view


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Loading tests were carried out on two model strip footings inorder to investigate the effect of footing width. The modelfootings were 70 mm and 50 mm in width, 465 mm in length and20 mm in thickness and fabricated from mild steel with a hole atits center to accommodate a ball bearing. The footings werelocated on the sand. The lengths of the footings were madealmost equal to the width of the test box of the tank to maintainplane strain conditions.

The load was applied to the model footing through a ballbearing which was placed between the model footing and theproving ring and allowed the footing to rotate freely as itapproached failure and eliminated any potential moment transferfrom the loading fixture.

2.2 Model GroundThe soil used for the model tests was uniform, clean and fine

sand obtained from Cakit River bed. The sand was washed, driedand sorted by particle size. The particle size distribution wasdetermined using the dry sieving method and the results areshown in Fig. 2. Using the Unified Soil Classification System,the material was determined to be poorly graded sand (SP). Thespecific gravity of the soil particles was determined bypicnometer test. The maximum and minimum dry densities ofthe sand were measured and corresponding values of theminimum and maximum void ratios were calculated. Table 1

summarizes the general physical characteristics of the sand.To obtain a reasonably homogeneous sand bed throughout the

experimental study, the same compaction procedure was used todeposit sand in 50 mm thick layers into the model box. In thismethod the quantity of sand for each layer, which was requiredto produce a specific relative density, was first weighed andplaced in the box and compacted by a hand-held vibratorycompactor until achieving the required layer height. Theexperimental tests were conducted on samples prepared withaverage unit weights of 16.5, 17.0 and 17.5 kN/m3. Correspondingrelative densities of the samples were 45, 65 and 85%, respectively.The estimated internal friction angles of the sand were 40.6, 41.8and 43.5o, respectively.

2.3 Preparation of Sand SlopeModel sand slopes with slope angles (β) of 20°, 25° and 30°

were prepared by using the same compaction procedure in layersof 50 mm thick sand. The inner surfaces of the test box were

Fig. 2. Grain Size Distribution of the Model Sand

Table 1. Properties of Sand BedProperty Value

Coarse sand fraction (%)Medium sand fraction (%)Fine sand fraction (%)D10 (mm)D30 (mm)D60 (mm)Uniformity coefficient, CuCoefficient of curvature, CcSpecific gravity (kN/m3)Maximum dry unit weight (kN/m3)Minimum dry unit weight (kN/m3)Maximum void ratioMinimum void ratioClassification (USCS)

00.046.453.600.1800.3000.5002.7801.0002.6817.915.500.72900.497SP

Fig. 3. Procedure for Construction of Sand Slope: (a) Soil Com-pacted Until Level Ground, (b) Apparatus Placed on LevelGround, (c) Compacting the First Sloped Layer, (d) Com-pacting the Second Sloped Layer, (e) Compacting the ThirdSloped Layer, (f) Compacting the Last Sloped Layer, (g)Removing the Apparatus, (h) Footing Placed at the Surface



marked at 50 mm intervals to make easy the preparation of thesand bed in layers and the geometry of the slope was marked onthe glass walls for reference. The sand was compacted in layersup to slope toe and then a special adjustable apparatus to get thepredetermined slope angles developed in this study was placed toobtain the sloping surface. The process continued layer by layeruntil the height of the slope was reached. Great care was given tolevel the slope face using special apparatus so that the relativedensity of the top surface was not affected (Fig. 3). In thismethod there is no need for excavation and seems to be useful toform a sloping surface with a desired angle and allows compactthe sand uniformly. The model strip footing was then placed onthe surface of the compacted sand and finally the load wasapplied until reaching failure.

2.4 Test ProgramAn experimental program was carried out to investigate the

effects of the variable parameters including, edge distance of thefooting to the slope crest (b), angle of the slope inclination (β),relative density of sand (Dr), and width of the footing (B) on thebearing capacity of strip footing on a sand slope. Model loadingtests were performed in seven test programs. Table 2 summarizesall the tests programs with constant and variable parametersused. Some tests were repeated at least twice to verify theconsistency of the test data.

3. Numerical Modeling

A series of two-dimensional Finite Element Analyses (FEA)on a prototype footing-slope system was carried out in order tovalidating the results of the laboratory model tests and providinginsights into the deformation behavior within the soil mass. Thefinite element analysis was performed using the commercialprogram PLAXIS (Brinkgreeve and Vermeer, 2002). The geometryof the prototype footing-slope system was assumed to be thesame as the laboratory model. The same angle of slopeinclination (β=20o, 25o, 30o) and the material of steel plate forfooting and sand were used in the numerical study.

An elasto-plastic hyperbolic model described as the HardeningSoil Model (HSM) was used from those available in PLAXIS todescribe the non-linear sand behavior in this study. Whensubjected to primary deviatoric loading, cohesionless soil shows

a decreasing stiffness and simultaneously irreversible plasticstrains develop. The observed relationship between the pressureand axial strain can be well approximated by a hyperbola as usedin the variable elastic, hyperbolic model (Duncan and Chang,1970). However the HSM is far superior to the hyperbolicmodel, being capable of simulating non-linear, inelastic, stressdependent material behaviour. Limiting states of stress describedby means of the friction angle (φ), the cohesion (c), and thedilatancy angle (ψ). In addition, the increase in soil stiffness withpressure is accounted for in all three stiffness used, i.e., thetriaxial loading stiffness E50, the triaxial unloading/reloadingstiffness Eur and the oedometer loading stiffness Eoed (Dickin andLaman, 2007). The model strip footing was modeled as elasticbeam elements based on Mindlin’s theory with flexural rigidityof EI=163 kNm²/m and normal stiffness of EA=3.4 × 105 kN/m.

The analyses were carried out using a plane strain model insand with three different densities as in the tests. During thegeneration of the mesh, 15-node triangular elements wereselected in preference to the alternative 6-noded versions in orderto provide greater accuracy in the determination of stresses.PLAXIS incorporates a fully automatic mesh generation procedure,in which the geometry is divided into elements of the basicelement type, and compatible structural elements. In the finiteelement modeling, as the slope surface is not horizontal, theinitial stress condition of the slope was established first byapplying the gravity force due to soil. A prescribed footing loadwas then applied in increments accompanied by iterativeanalysis up to failure. PLAXIS generates full fixity at the base ofthe geometry and smooth conditions at the vertical sides. Valuesof soil parameters used in the numerical investigation are shown

Table 2. Model Test ProgramSeries Constant parameters Variable parameters

1 Tests on level ground, β=0°, B=70 mm Dr = 45-65-85%

2 Test on level ground, β=0°, B=50 mm,Dr = 65% -

3 β=30°, B=70 mm, Dr = 65% b/B=0-1-2-3-4-54 β=25°, B=70 mm, Dr = 65% b/B=0-1-2-3-4-55 β=20°, B=70 mm, Dr = 65% b/B=0-1-2-3-4-56 β=30°, B=70 mm, Dr = 45% b/B=0-1-2-3-4-57 β=30°, B=70 mm, Dr = 85% b/B=0-1-2-3-4-58 β=30°, B=50 mm, Dr = 65% b/B=0-1-2-3-4-5

Table 3. Values of Soil Parameters used in PLAXIS AnalysesParameter Value

Unit weight, γ (kN/m3) 16.5 17.0 17.5Primary loading stiffness, (kN/m2) 20000 28000 40000Initial stiffness, Eoed (kN/m2) 20000 28000 40000Unloading/reloading stiffness, Eur (kN/m2) 60000 84000 120000

Cohesion, c (kN/m2) 0.10 0.10 0.10Friction angle, φ (°) 40.6 41.8 43.5Dilatancy angle, ψ (°) 10.6 11.8 13.5Poisson’s ratio, ν 0.25 0.25 0.25Earth pressure coefficient at rest, K0 0.35 0.33 0.31

E50ref

Fig. 4. Prototype Slope Geometry, Finite Element Mesh, and Bound-ary Conditions


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in Table 3. Shear strength and stiffness parameters representingsand conditions derived from series of drained triaxial compressiontests. The prototype slope geometry, finite element mesh, and theboundary conditions are shown in Fig. 4.

4. Results and Discussions

A total of 40 model tests were conducted on model stripfooting on sand slope. The effect of the edge distance of thefooting to the slope crest, the slope angle, the relative density ofsand, and footing width on the ultimate bearing capacity wasobtained and discussed. Numerical study was also carried outusing the finite element model using PLAXIS to verify themodel tests.

The bearing capacity behavior of the footing on sand slope, isrepresented using a non-dimensional factor, called bearingcapacity reduction factor, iβ. This factor is defined as the ratio ofthe ultimate bearing capacity of footing on slope (quslope) to theultimate bearing capacity of footing on level ground (qu).

(3)

The footing settlement (s) is also expressed in non-dimensionalform in terms of the footing width, B as the ratio s/B (%).

The ultimate bearing capacities for the model are determinedfrom the load-displacement curves as the pronounced peaks,after which the footing collapses and the load decreases. In load-displacement curves which did not exhibit a definite failurepoint, choosing a single value of may be extremely subjective. Inthis study, 0.1B method, which takes the ultimate bearingcapacity at a settlement of 10% of the footing width, was used.Although this method is completely arbitrary, it may actually beclose to the average soil strain at failure, forces a fixed value atqu, and treats the displacement of all footing sizes the same(Cerato, 2005).

4.1 Effect of Footing Distance to the Slope CrestA series of tests was performed on strip footing resting on a

sand slope, in order to investigate the effect of distance of thefooting to the slope crest (b/B). Tests were conducted for b/Bratios of 0.0, 1.0, 2.0, 3.0, 4.0 and 5.0. Test was also conductedwith the footing on a level surface (β=0o) for the purpose ofcomparison. In these test series, the slope angle was β=30o, therelative density of sand was Dr=65% and the footing width wasB=70 mm.

Load settlement curves for five different b/B ratios obtainedfrom model tests and FE analyses are presented in Figs. 5 and 6.The results are also summarized in Table 4.

The results indicate that, the ultimate bearing capacity increaseswith increasing setback distance of the footing from the slopecrest and the finite element results provide a reasonable fit withthe experimental results. When the footing is moved away fromthe slope crest (b/B=0) to the setback distance of b/B=1.0, thereis a serious increase in bearing capacity (an average value of

70%). However, the rate of increase in bearing capacity decreaseswith increasing distance of setback until b/B=5.0 where theultimate bearing capacity of the footing on slope approaches thatof a footing on level ground. The effect of slope is minimizedwhen the footing is placed at an edge distance beyond five timeswidth of the footing. This change in bearing capacity of thefooting with its location relative to slope crest can be attributed tosoil passive resistance from the slope side. When, the footing isplaced far away of the slope, the passive resistance from theslope side to the failure wedge under the footing increases. Also,

iβquslope

qu-------------=

Fig. 5. Variations of q with s/B for Model Tests with Different Ratiosof b/B

Fig. 6. Variations of q with s/B for FE Analyses with DifferentRatios of b/B

Table 4. Results of Footing Located at Different Locations fromthe Slope Crest (β=30o, Dr=65%, B=70 mm)

qu (kN/m2) iβ

b/B Test Analyses Test Analyses0 31.50 36.27 0.24 0.271 56.70 58.64 0.43 0.442 72.86 74.27 0.56 0.553 90.27 89.88 0.69 0.674 109.11 107.12 0.84 0.805 124.40 123.83 0.95 0.92

Level ground 130.40 134.74 1.00 1.00



as b/B increases a much greater force is required for the failuresurface to reach the slope. Hence, the bearing capacity of thefooting increases.

The variations of bearing capacity reduction factor, iβ, atdifferent edge distance of the footing width (b/B) for slope angleof β = 30o obtained from model tests and FE analyses are shownin Fig. 7. The figure clearly shows that the general trend of FEanalyses is agree fairly well with this of the model tests. Theresults show that the values of iβ increases almost linearly withan increase in setback distance. iβ values obtained from modeltests and FE analysis for the strip footing located on the setbackdistance of 5.0B are 0.95 and 0.92, respectively. Hence, it isinferred that the ultimate bearing capacity for a footing at asetback distance larger than 5.0B may be close enough to that inthe level ground case and there would be no benefit from movingthe footing any further from the slope.

4.2 Effect of Slope AngleIn addition to the series of model tests for footing located on

slope with slope angle of β = 30o, another series of model testswere carried out for footing on slope with slope angles of β = 25o

and 20o to examine the influence of the slope angle on thebearing capacity of the strip footing. For each slope angle, thefooting was placed at five different setback distances such thatthe setback distance to footing width ratio b/B = 0.0, 1.0, 2.0, 3.0,4.0 and 5.0. Dr = 65% and B = 70 mm were constant for allmodel tests. Table 5 summarizes the results of model tests andFE analysis. The variations of qu and iβ with slope angle β fordifferent setback distances obtained from load-displacementcurves both experimentally and numerically are shown in Figs. 8and 9, respectively.

As seen from Figs. 8 and 9, the agreement between experimentaland numerical results is reasonably well. Fig. 8 clearly showsthat, the ultimate bearing capacity decreases with an increase inslope angle. The bearing capacity also increases with increasingsetback distance and at setback distance of 5.0B; is notinfluenced by the slope gradient. A similar trend was observedfor variations of iβ with slope angle β for different setback

distances as seen in Fig. 9. The value of iβ increases withdecrease of slope angle and increase of setback distance.

Fig. 7. Variations of iβ with b/B

Fig. 8. Variations of qu with Slope Angle β for Different SetbackDistances

Fig. 9. Variations of iβ with Slope Angle β for Different Setback Dis-tances

Table 5. Results of Footing on Slope with Different Slope Angles(Dr=65%, B=70 mm)

qu (kN/m2) iβ

β b/B Test Analyses Test Analyses

25°

0 44.45 48.81 0.34 0.361 67.81 74.08 0.52 0.552 84.43 90.76 0.65 0.673 100.41 106.42 0.77 0.794 114.80 118.10 0.88 0.885 125.27 127.30 0.96 0.94

20°

0 60.80 63.53 0.47 0.471 81.19 81.94 0.62 0.612 97.82 100.84 0.75 0.753 107.25 112.84 0.82 0.844 120.02 122.49 0.92 0.915 127.78 133.59 0.98 0.99


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It is known that when a slope angle is increased, e.g. byexcavation, it becomes less stable. This is evident in that thebearing capacity is significantly reduced as the slope angleincreases. As the slope angle is increased the area of the slope isdecreased therefore the failure zone is decreased resulting in amuch smaller bearing capacity. To overcome this reduction Fig.8 shows that while increasing the slope angle but moving thefooting further from the slope a greater bearing capacity offooting is achieved. For example a 20 degree slope with b/B=0.0has approximately the same bearing capacity as a 30 degreeslope with b/B=1.0. Also, it can be concluded that the effect ofsetback distance on the bearing capacity of footing is regardlessthe value of slope angles ranged between 20o and 30o and atsetback distance of b/B=5 the slope has no influence on thebearing capacity and the problem is considered as a level groundproblem.

4.3 Effect of Relative Density of SandIn order to study the effect of relative density, in addition to the

series of model tests for footing on sand slope with relativedensity of Dr = 65%, two series of tests were carried out on modelfooting located on different setback distances of sand slope withthe densities of Dr = 45% and 85%. b = 30o and B = 70 mm wereconstant for all model tests of this series. The results aresummarized in Table 6. The variations of qu and iβ with relativedensity of Dr for different setback distances are shown in Figs.10, 11 and 12.

Table 6 and Fig. 10 clearly show that the bearing capacity ofthe footing on a slope is significantly increases with an increaseof relative density of sand and at setback distance of 5.0B theeffect of slope is minimized. A good agreement was observedbetween the results of model tests and FE analysis. The datatrends in Fig. 10 show that as relative density of sand increasesthe ultimate bearing capacity of the footing increases linearly.This increase in bearing capacity of footing with relative densityof sand can be attributed with soil-footing interaction. As the

relative density increases, the angle of friction of the sandincreases and hence the adhesion and friction between soil andfooting increases. This causes a larger failure surface and leadingto greater bearing capacity.

Figures 11 and 12 show the variations of iβ with relativedensity of sand at different setback distances of footing obtainedfrom model tests and FE analysis, respectively. The results ofmodel tests and FE analyses show that the values of bearingcapacity reduction factor, iβ increases almost linearly withincrease of setback distance ratio of b/B. However, the relativedensity of sand has not considerable effect on the behavior of iβ.As mentioned before the bearing capacity factor, iβ is the ratio ofthe ultimate bearing capacity of footing on slope to the ultimatebearing capacity of footing on level ground. When the relativedensity of sand increases the ultimate bearing capacity of footingon slope and level ground increase almost equally. Hence, thereare slight differences between iβ values and it can be concludedthat the relative density of sand has negligible effect on the

Table 6. Results of Footing on Slope with Different Relative Den-sities of Sand (β=30o, B=70 mm)

qu (kN/m2) iβ

Dr b/B Test Analyses Test Analyses

45%

0 27.61 26.01 0.32 0.271 41.22 43.70 0.48 0.462 50.31 54.32 0.59 0.573 62.77 67.14 0.74 0.704 73.05 76.68 0.86 0.805 82.41 88.32 0.96 0.92

85%

0 44.40 45.83 0.26 0.281 76.62 74.85 0.45 0.462 94.51 95.89 0.56 0.593 118.14 115.87 0.70 0.714 143.67 138.19 0.85 0.855 163.34 153.53 0.97 0.94

Fig. 10. Variations of qu with Relative Density for Different SetbackDistances

Fig. 11. Variations of iβ with Relative Density for Different SetbackDistances (Model Test)



bearing capacity reduction factor, iβ.

4.4 Effect of Footing WidthA series of test were carried out on a model footing (B = 50

mm) located on different setback distances of sand slope in orderto investigate the effect of footing width on the bearing capacity.The other parameters were β = 30o and Dr = 65% in model tests.Table 7 summarizes the results of model tests and FE analyses.The variations of qu and iβ with footing width of B at differentsetback distances are shown in Figs. 13, 14 and 15.

As seen from Table 7 and Fig. 13, although the ultimate bearingcapacity values obtained from numerical analyses do not fitcompletely with the experimental results and give greaterbearing capacity, especially for footing of B = 50 mm, but theagreement is reasonably well. This discrepancy may be related tothe model and foundation parameters chosen in numerical andexperimental model. Based on the numerical and experimentalresults, it is clear that, as the footing width increases the ultimatebearing capacity of footing on a slope increases and effect of

slope is minimized at setback distance of b/B = 5.0. This increasein bearing capacity can be attributed to the larger contact areabetween soil and footing. Larger displacements and verticalstresses occur below the larger footing. Therefore, the failure

Fig. 12. Variations of iβ with Relative Density for Different SetbackDistances (FEA)

Table 7. Results of Footing on Slope with Different Footing width(β=30o, Dr=65%)

qu (kN/m2) iβ

B b/B Test Analyses Test Analyses

50 mm

0 15.54 29.20 0.17 0.261 40.02 47.33 0.43 0.422 51.27 58.71 0.55 0.523 69.00 72.02 0.73 0.644 80.50 84.12 0.86 0.755 89.88 102.42 0.96 0.91

70 mm

0 31.50 36.27 0.24 0.271 56.70 58.64 0.43 0.442 72.86 74.27 0.56 0.553 90.27 89.88 0.69 0.674 109.11 107.12 0.84 0.805 124.40 123.83 0.95 0.92

Fig. 13. Variations of qu with Footing width for Different SetbackDistances

Fig. 14.Variations of iβ with Footing Width for Different SetbackDistances (Model Test)

Fig. 15. Variations of iβ with Footing Width for Different SetbackDistances (FEA)


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wedge becomes larger and the frictional resistance on failureplanes becomes greater and bearing capacity of the footingincreases.

Figures 14 and 15 show the variations of iβ with footing widthat different setback distances obtained from model tests and FEanalysis, respectively. The results of model tests and FE analysesshow that the values of bearing capacity reduction factor, iβ

increases with increase of setback distance ratio of b/B. Thebehavior of iβ is similar to that behavior observed in test series onthe effect of relative density of sand and seems to be footingwidth has not considerable effect on the bearing capacity reductionfactor, iβ.

4.5 Parametric Study

4.5.1 Effect of Footing WidthIn order to evaluate the effect of footing width on the bearing

capacity factor Nγq, finite element analyses were performed onstrip footings with widths of 0.05, 0.06, 0.07, 0.08, 0.09, 0.10,0.11, 0.12 and 0.13 m. The strip footings were located on twodifferent distances from the edge of the slope (b/B=0 and 1)having slope angle of β=30° and relative density of Dr=65%. Theultimate bearing capacity, qu and bearing capacity factor, Nγq

values obtained from the analyses are summarized in Figs. 16and 17, respectively. A significant effect of footing width may beseen in the results. The qu values increase and Nγq values decreasewith an increase in B. Also, it is clear that qu and Nγq valuesincrease with an increase in setback distance ratio of b/B. Fig. 16shows that the value of qu can be expressed as a linear function ofB, as follow:

(4)

in which, x1=14.507, x2=310.55 for b/B=0 and x1=22.09, x2=

506.33 for b/B=1. Fig. 17 shows that curve-fitting using Eq. (5)generates good values of correlation coefficient.

(5)

in which, x1=96.631, x2=-664.34, x3=2334.6 for b/B=0 and x1=179.16, x2=-1662.1, x3=7048 for b/B=1.

For the evaluation of bearing capacity factor (Nγq) limitedfooting widths were used in the analysis. This part of study aimsto only show the effect of the footing width on the bearingcapacity factor Nγq. Therefore, the derived equations (Eqs. 4 and5) are valid within the content of current study. These equationsshould be considered as designating the relation between footingwidth and Nγq. Hence, when these equations compared with theother studies larger values up to 2 times are obtained.

4.5.2 Effect of Footing RigidityIn this study, the strip footing was used and the footing was

considered as a flexible body. A series of finite element analysiswere performed considering the footing as a rigid body. One wayof rigid body modeling is to assign prescribed uniformdisplacements to the footing which is the case for the rigid bodybehavior. In the analysis, the same geometry was considered andthe settlement of the footing is simulated by means of a uniform

qu x1 x2B+=

Nγq x1 x2B x3B2+ +=

Fig. 16. Variation of qu with B

Fig. 17. Variation of Nγq with B

Fig. 18. Rigid Model



indentation at the top of the sand layer instead of modeling thefooting itself as seen in Fig. 18. The prescribed displacement wastaken as s = 0.1B = 0.0070 m. In the analysis the slope angle wasβ = 30°, the relative density was Dr = 45%, the footing widthwas B = 0.070 m, and the distance of the footing to the slopecrest were varied from b = 0 to 5B.

Figures 19 and 20 show the variations between qu-b/B and iβ-b/B values, respectively. When comparing the qu results obtainedby flexible and rigid assumptions, it can be noticed that for thesame displacement the rigid footing, exhibited more bearingcapacity (about 8-10%) than that from flexible footing.

4.5.3 Effect of 3D ModelingA series of 3D finite element analysis were performed using

PLAXIS 3D to take care of 3D effects on the test results. In theanalysis the same geometry of the experimental set-up was used.b/B ratios were 0, 1, 2, 3, 4 and 5, footing width was B=70 mm,slope angle was β=30° and the relative density of sand wasDr = 65%. The used mesh and boundary conditions for 3Dmodel is seen in Fig. 21. Figs. 22 and 23 show the variationsbetween the qu and iβ values obtained from experimental, 2D and3D analysis with b/B ratios, respectively.

As seen from Fig. 22, the qu values obtained from 3D analysis

are somewhat greater (about 2%) than the values obtained from2D plane strain analysis. However, the iβ values obtained from3D and 2D analysis which shown in Fig. 23 are very close.

5. Failure Mechanism

For soil slopes, the results of previous studies showed that thefailure model could be categorized as perfect plastic failure witha circular slip surface. In common, the failure region was dividedinto active zone, passive zone, and radial shear zone; however,the radial shear zone was in global shear failure and bounded bya logarithmic spiral line. This section is devoted to discuss the

Fig. 19. Variations of qu with b/B


Fig. 21. Mesh and Boundary Conditions for 3D Model

Fig. 22. Variations of qu with b/B



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failure mechanism observed in experimental and numericalstudies of bearing capacity of strip footings adjacent to the crestof a sand slope.

The form of failure plane below the footing can illustrate theinfluence of the slope angle and the setback distance from theslope crest on the bearing capacity. In order to clearlydemonstrate the differences in the failure plane due to effect ofslope angle β, and b/B ratio, the case of soil with β = 20°, 25°and 30° with different b/B ratios of 0, 1, 2, 3, 4 and 5 are chosenfor comparison. The failure mechanisms obtained fromnumerical analyses for different slope angles are shown in Figs.24 through 26. Fig. 27(a) and 27(b) show the picture of themodel slope at the peak footing load for β = 30° with setbackdistance ratio of b/B = 0 and 1, respectively.

At the ultimate stage, the bearing pressure reaches the peakload, and begins to decrease suddenly with increasing settlement.For all slope angles, the failure initialing from the left edge of thefooting propagated into the slope surface and resulted in the totalloss of passive resistance. As a result, the footing collapses andthe soil deform laterally toward the sloping side. Once a fullyconnected failure surface was created, the soil begins to pushtoward the sloping side. As mentioned before, from the comparisonof the results of model tests and numerical analyses, the ultimatebearing capacity consistently decreases as the slope angle increases.Unlike the level ground case, the failure mode was notsymmetrical and only developed in the sloping side. Accordingto figures, it appears that the failure process for the 20°, 25° and30° slopes are very similar. A starting at the left edge of thefooting propagated with increasing load.

It can be noted that if the footing is located close to the edge ofthe slope, the soil below the footing tends to move toward theslope, since it has less shear resistance. The form of failuresurfaces are affected by the setback distance from the slope crest.With increasing setback distance, the failure zone is larger andwill provide a higher bearing capacity. In Fig. 28(a), the

Fig. 24. Output for β = 20°, b/B = 0: (a) Deformed Mesh, (b) TotalDisplacements, (c) Total Displacement Contours, (d) ShearStrains



Fig. 27. Pictures of the Model Sand at the Peak Footing Load: (a)β = 30o, b/B = 1, (b) β = 30o, b/B=0

Fig. 28. Contours of Shear Strain (β = 30°): (a) b/B = 0, (b) b/B = 1,(c) b/B = 2, (d) b/B = 3, (e) b/B = 4, (f) b/B = 5



influence area concentrates mainly on the side of the slope.When the footing is located away from the slope crest, thebearing capacity of the footing increases as expected. It can besaid that, the degree of confinement on the side of the slopeincreases and part of the stress due to the footing begins beinggoverned by soil on the side of the ground surface. According tothe Figs. 28(a-f), the displacement of the soil occurs on the sideof ground surface and the influence of the slope is vanishing bylocating the footing further away from the slope crest. At theratio of b/B = 5.0, the failure plane becomes symmetric, whichillustrates that the stress spread on both side of the footing.

In order to analyze the failure mechanism based on LimitEquilibrium (LE) method a series of analyses was performed.The limit equilibrium analysis was carried out using the commercialprogram GeoStudio (GeoStudio, 2012. Geo-Slope Int. Ltd.). Thegeometry of the model was taken to be the same as the laboratorymodel (Fig. 29). Using limit equilibrium, GeoStudio can modelheterogeneous soil types, complex slip surface geometry, andvariable pore-water pressure conditions.

The slip surface obtained from limit equilibrium analysis isshown in Fig. 30. It can be seen from Fig. 30 that a circularsurface propagates towards the slope until the sloping ground isreached. It should also be noted that when the factor of safetywas equal to 1 the ultimate load was about 60 kN/m2 which isreasonable close to the ultimate bearing capacity value of 56.70kN/m2 obtained from experimental study.

6. Conclusions

A series of model tests and numerical analyses has been carried

out to investigate the bearing capacity of a strip footing restingon sand slopes. The study primarily aimed at determining theeffect of setback distance of the footing to the slope crest and itsbehavior with various s slope angles, relative density of sand andwidth of the footing. Based on the experimental and numericalstudies, the following main conclusions are made:

1. The results show that the bearing capacity and bearingcapacity reduction factor iβ, increase almost linearly with anincrease in setback distance up to setback distance to foot-ing width ratio of b/B = 5.0. Beyond this value, the ultimatebearing capacity remains constant like that of a footinglocated on a level ground.

2. The bearing capacity of strip footing on sand slope is signifi-cantly dependent on the slope angle, relative density of sand,and width of the footing.

3. The results clearly show that as the slope angle increasesor slope becomes steeper the ultimate bearing capacity offooting decreases. Moving the footing further from theslope overcomes this reduction due to the increase inslope angle.

4. The bearing capacity of the footing on a slope is signifi-cantly increases with an increase of relative density of sand.However, the relative density of sand has not considerableeffect on the behavior of iβ.

5. Based on the numerical and experimental results, as thefooting width increases the ultimate bearing capacityincreases and bearing capacity factor decreases. Further-more, it seems to be footing width has a negligible effect onthe bearing capacity reduction factor, iβ.

6. The results show that the value of iβ characterizing the linearrelationship between the effect of setback distance of thefooting to footing width ratio (b/B) and the slope angle (β).It can be expressed as functions of b/B and β, independentfrom relative density of sand (Dr) and footing width (B).

7. From series of numerical studies (using finite element anal-yses) with the variation in the footing size, empirical equa-tions have been derived between the bearing capacitycharacteristics (qu and Nγq) and footing width for the stripfootings located on two different distances from the edge ofthe slope (b/B=0 and 1). A significant effect has been seenin the results. The qu values increase linearly and Nγq valuesdecrease parabolically with an increase in B. Also, the val-ues of qu and Nγq increase with an increase in setback dis-tance ratio of b/B.

8. The results of the parametric study show that for the samedisplacement the rigid footing, exhibited more bearingcapacity than that from flexible footing. Also, 3D analysisgives somewhat greater bearing capacity values than the val-ues obtained from 2D plane strain analysis.

9. A close agreement between the experimental and numericalresults on general trend of behavior is observed. However,the ultimate bearing capacity values obtained from FEAappears to be greater than that obtained from the modeltests.

Fig. 29. Geometry of Model used in LE Analysis

Fig. 30. Slip Surface Obtained from LE Analysis


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Acknowledgements

The authors thank the Scientific Research Project Directorateof Cukurova University for supporting this study (Project no:MMF2006D1).

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Documents

1. Model Studies of Bearing Capacity of Strip Footing on Sand Slope