estimation on bearing capacity of shallow foundations in...

Scientia Iranica A (2014) 21(3), 505{515

Sharif University of TechnologyScientia Iranica

Transactions A: Civil Engineeringwww.scientiairanica.com

Estimation on bearing capacity of shallow foundationsin heterogeneous deposits using analytical andnumerical methods

R. Jamshidi Chenari�, N. Zhalehjoo and A. Karimian

Department of Civil Engineering, Faculty of Engineering, University of Guilan, Rasht, P.O. Box 3756, Iran.

Received 13 November 2012; received in revised form 28 June 2013; accepted 13 August 2013

KEYWORDSShear strength;Shallow foundations;Bearing capacity;Finite di�erencemethod;Heterogeneity;Equivalenthomogeneity.

Abstract. In situ soil properties are spatially variable parameters causing soil depositsto be heterogeneous. Heterogeneity consists of two components: (i) A deterministic trendand (ii) The residual component. This paper presents the e�ect of di�erent componentsof soil heterogeneity on the ultimate bearing capacity of a vertically loaded shallowfoundation resting on clay deposits. The numerical model used in this study is basedon �nite di�erence simulations, employing FLAC 5.0. Results of numerical analysis arecompared with other simple and analytical solutions. For heterogeneous soil deposits,considering both linear and bi-linear deterministic trends, �nite di�erence tools were foundto be able to re ect salient features of heterogeneity in bearing capacity estimation. Anequivalent homogeneous analysis solution is introduced, in order to allow for heterogeneity,by adopting a representative depth for shear strength measurements. Stochastic variationof shear strength is shown to induce under conservatism by solely relying upon deterministicestimations.c 2014 Sharif University of Technology. All rights reserved.

1. Introduction

Soil is a naturally occurring material exhibiting no-ticeable changes in its engineering properties, dueto spatial variability in physical and chemical en-vironments, which brings about various formations.This will introduce uncertainty and variability in theestimation of engineering parameters from geotech-nical engineering perspectives, de�ning the strengthand sti�ness characteristics of in situ soil, and alsoinduces uncertainty in the safety limits required forassessing the safety and performance characteristics ofthe structures [1].

The inherent variability of soil properties is usu-

*. Corresponding author. Tel.: +98 131 6690485;Fax: +98 131 6690271E-mail address: jamshidi [email protected] (R. JamshidiChenari)

ally separated into a depth-dependent spatial trendand uctuations around this mean value trend. Thespatial trend is identi�ed by regression analysis em-ploying a su�cient number of in-situ test data, andis usually removed from the subsequent stochasticanalyses.

Noticeable attempts have been made to study thee�ect of heterogeneity on the bearing capacity of shal-low foundations on clays under undrained conditions,' = 0 [2-5]. The majority of these studies found thatheterogeneity has a paramount e�ect on the bearingcapacity of clays.

Raymond [2] also studied the bearing capacity offootings and embankments on heterogeneous clay usingthe slip circle method. He presented a dimensionlessplot of failure criteria for footings. For some simplecases, where the strength bears linear and bi-linearvariations with depth, Davis and Booker [3] haveprovided approximate analytical solutions. However,

506 R. Jamshidi Chenari et al./Scientia Iranica, Transactions A: Civil Engineering 21 (2014) 505{515

for more general cases, such solutions do not exist.A general variation of undrained shear strength withdepth is relatively easy to simulate in numerical analy-sis. The undrained shear strength can be realized usingrandom �eld theory adopted in �nite element or �nitedi�erence formulations.

The current study investigates the capability ofdi�erent analytical and numerical methods in bearingcapacity estimation for shallow foundations. It focuseson �nite di�erence methods using the commercialsoftware, FLAC 5.0 [6], taking into account di�erentforms of heterogeneity. It will be �nally compared withother conventional methods and a new translation ofconservatism will be introduced.

2. Shear strength variability

Soil properties are well-known for variability frompoint to point in, so-called, heterogeneous naturalalluvial deposits. Variability in measured propertiesin these layers emanates from di�erent sources. Themost important component of variability is the in-herent spatial variability originating from the naturalgeological process that produces and gradually alterssoil mass. The results of various in-situ tests onthe variation of undrained shear strength of naturaldeposits with depth reveals that the ratio ( cu�0 ), where�0 is the e�ective overburden stress, is substantially aconstant for normally consolidated clays [7]. Therefore,a linear increasing of overburden stress with depthintroduces a linear increase in shear strength withdepth.

However, numerous studies show that the deter-ministic trend of undrained shear strength decreaseswith depth in near-surface soils, and then, increases asthe depth increases [7-10].

This is due mainly to surface desiccation. Thedepth at which the trend of undrained shear strengthtransforms is called the transformation depth (Zt).Further discussion on the mentioned issue is providedelsewhere [11].

Other studies support the fact that the undrainedshear strength of natural deposits can be detrendedto extract the deterministic trend, which is a bi-linear trend as discussed earlier, and the stochasticor residual component, which has its own e�ect. Thecurrent study considers the e�ect of both deterministicand stochastic heterogeneity on the bearing capac-ity of shallow foundations. The conducted analyseslend support to the contention that for nonlinearproblems, like bearing capacity calculations and slopestability, the mutual contributions of di�erent compo-nents of heterogeneity should be taken into accountsimultaneously. Figure 1 schematically demonstratesdi�erent components of shear strength heterogene-ity.

Figure 1. Deterministic and stochastic trend ofundrained shear strength: (a) General variation; and (b)stochastic and deterministic components.

3. Bearing capacity

Limit equilibrium and slip line solution are two majoranalytical methods widely employed over the past fewdecades for considering the e�ect of \strength density"on bearing capacity problems. Strength density is theso-called increasing rate of the shear strength withdepth. Numerical analysis includes methods whichsatisfy all theoretical requirements, including equilib-rium, compatibility, constitutive behavior and bound-ary conditions. The Finite Element Method (FEM)and the Finite Di�erence Method (FDM) are twomore widely-used numerical analysis schemes adoptedin geotechnical engineering.

Raymond [2] studied the e�ect of the nonhomo-geneity of strength on bearing capacity with the limitequilibrium method. Figure 2 shows a typical slip circledrawn through a uniform surface strip load of intensity,q; the distance from the edge of the slip circle to theedge of load is B.

The undrained shearing strength at any depth isgiven by Eq. (1):

cu = cu0 + �Z; (1)

where � is the constant strength density obtainedexperimentally, Z is the depth, and cu0 is the shearingstrength at the surface. The results for the least loadproducing collapse are shown in Figure 3.

If a footing is analyzed according to the aforesaidassumption and the result plotted on the left sideof the critical line, the footing will be theoreticallyunsafe, and if the result falls on the right side, itwill be theoretically safe. As a numerical methodsolution, the commercially available �nite di�erencecode, FLAC 5.0, was used for the numerical modelingof the foundation model. For simplicity, the model wasassumed weightless and the soil behavior was soughtin an undrained condition. A plane-strain analysis wasperformed to model strip-footing on a heterogeneousstratum. Figure 4 provides the discretized model,

R. Jamshidi Chenari et al./Scientia Iranica, Transactions A: Civil Engineering 21 (2014) 505{515 507

Figure 2. Geometrical layout of a typical slip circle [2].

Figure 3. Dimensionless plot of failure criterion [2].

Figure 4. Model geometry and boundary conditions in�nite di�erence model, B = 10 m.

along with the boundary conditions at the sides andbottom. A downward velocity �eld was applied to thearea representing the footing. The value of the velocityapplied to the footing area was 2.5�10�2 mm/step foranalyses. This value is su�ciently small to minimizeany inertial e�ects. A rough strip footing was simulated

by �xing the x-velocity to zero for the grid pointsrepresenting the footing.

4. Deterministic heterogeneity

Due to the inherent variability of soil properties, thefailure surface under the footing will follow the weakestpath through the soil, which is not necessarily thelogarithmic spiral shape assumed by Terzaghi [12] andresearchers afterwards. Deterministic trends, as shownin Figure 5, are considered for undrained shear strengthto investigate the e�ects of deterministic heterogeneityon the bearing capacity of shallow foundations.

Undrained shear strength in Figure 5(a) increaseslinearly with depth and � is the strength density. Fig-ure 5(b) represents variations of shear strength whenthe undrained shear strength inherits a bi-linear trend.The undrained shear strength starts decreasing fromthe surface cohesion value of 30 kPa with a negativestrength density, �1, but turns to increasing with apositive strength density, �2, as the transformationdepth, Zt, is passed. Undrained shear strength inFigure 5(c) is constant, spatially, over the entire layer.

A range for the Young's modulus for undrainedloading can be considered between 300cu and1500cu [13]. However, in this study, E = 300cu wasconsidered. Therefore Young's modulus would alsoimitate the same deterministic trend as adopted forundrained shear strength. A value of 0.49, appropriatefor undrained conditions, was considered for the Pois-son ratio.

By adopting the trend shown in Figure 5(a)for undrained shear strength, the ultimate bearingcapacity of heterogeneous soil, qult(N.H.), increases asthe strength density, �, increases (Figure 6).

From Figure 6, it is evident that surface cohesioncontributes most to the undrained bearing capacity ofshallow foundations, explaining why full heterogeneitywith zero surface cohesion renders a trivial bearingcapacity and it is thought unrealistic to make a zerosurface cohesion assumption.

Bi-linear heterogeneity, which is usually observed


Figure 5. Assumed deterministic trends for undrained shear strength: (a) Linear heterogeneity (Gibson soil); (b) bi-linearheterogeneity; and (c) homogeneous �eld.

Figure 6. Ultimate bearing capacity of shallowfoundation on heterogeneous Gibson-soil, qultN:H , vs.strength density, �.

for soft soils due to surface desiccation, adopts twoinversely varying linear trends which intersect in theso-called transformation depth, Zt. According to thegeneral trend introduced in Figure 5(b), �nite di�er-ence analyses were conducted for di�erent strengthdensity values, �1 and �2, and di�erent footingwidths.

The results illustrated in Figure 7(a) show thatwith increasing the rate at which the undrained cohe-sion, cu, decreases (�1), the ultimate bearing capacitydecreases. This e�ect is justi�ed by the fact that theslip failure zones extending downward depend on thefooting width. The ultimate bearing capacity of thefooting is indeed the accumulation of shear strengthalong the failure surface, which is masked by thenegative strength density in shallow depths.

Another observation from Figure 7 is that thebearing capacity increases with the increase of positivestrength density (�2), which happens in the zone fol-

lowing the surface crust zone. However, it is seen thatthe bearing capacity of the footing with small width(B = 2 m) is, in e�ect, unin uenced by the strengthdensity of the second zone (�2). This is explained bythe fact that failure zone dimensions are controlled bythe footing width, and when the transformation depthis deeper than the footing width, the increase of shearstrength in lower depths has actually no in uence onthe bearing capacity.

The e�ect of transformation depth, Zt, is studiedby performing a bearing capacity analysis for di�erenttransformation depths, while other strength parame-ters are assumed constant between di�erent analyses.Figure 8 demonstrates the variation in the load bearingcapacity of footings with two di�erent widths and withtransformation depth, Zt. It is derived that for bothsmall and large footings, the bearing capacity decreasesas the transformation depth moves downward. Asexpected, the increase of transformation depth pushesthe failure zone to be laid more widely within thesurface crust zone where the undrained cohesion de-creases. However, this e�ect fades after a speci�ctransformation depth, which is beyond the failure zonefor each speci�c footing width, and the bearing capacityconverges to a constant value (Figure 8).

Superimposed on Figure 8 is the result of bearingcapacity calculations with Limit Equilibrium (LE)analysis, which shows how the bearing capacity offooting underlain by a bi-linear heterogeneous stratumvaries with the transformation depth. Calculationswere made according to Eq. (2) by assuming a circularfailure surface:

qu = 2�cu0 � 4�1B Zt � B;

qu=2�cu0�4�1B � 4(�1+�2)��2��t�

qB2�Z2

t

�Zt < B; (2)


Figure 7. Ultimate bearing capacity of bi-linear heterogeneous soil, Zt = 2 m: (a) �1 variation; and (b) �2 variation.

Figure 8. E�ect of the transformation depth on bearingcapacity (Zt); �2 = �1 = 1 kPa/m and cu0 = 30 kPa.

where:

�t = sin�1�ZtB

�:

A critical line based on the results of numericalFDM analyses is drawn in Figure 9 in a dimensionlessfashion, along with the Raymond's line.

Raymond considered both positive and negativestrength densities. However, the current study con-sists of two major sorts of heterogeneity, which arelinear heterogeneity with positive strength density andbi-linear heterogeneity with mutual negative-positivestrength densities. Therefore, the pure negativestrength density as considered by Raymond is notrealistic and it was shown in the previous section that

Figure 9. Finite di�erence results in comparison toRaymond's results [2].

the bearing capacity of footing on soft soils bears themixed e�ects of crust and the layer underneath. Theaforesaid discrepancy was resolved by solely choosingthe positive side of Figure 3 and taking only thelinear heterogeneity into consideration. From the linesprovided in Figure 9, it is clear that the Raymondcriterion is quite unconservative and overestimates thebearing capacity of footings on linear heterogeneousstrata.

5. Stochastic bearing capacity

Random �elds have been employed by many re-searchers to study a wide range of geotechnical issues.Among them, many adopt the �nite di�erence method


joined with random �eld theory. For instance, Srivas-tava and Babu [1] studied the e�ect of soil variability ontwo cases of bearing capacity and slope stability withthe aid of the �nite di�erence numerical code, FLAC5.0. More recently, Zhalehjoo et al. [14] conducteda study, in which the so-called stochastic bearingcapacity was compared to classic methods.

In this study, the stochastic variation of undrainedshear strength is modeled by adopting a log-normaldistribution with the aid of three recognized repre-sentative statistical parameters: mean value, standarddeviation and correlation length. Also, there areother distributions, like Normal, Beta, etc., in practice.However, the use of log-normal distribution lies in thefact that shear strength is strictly non-negative and, inlog-normal distribution, there is no possibility of theexistence of negative values. For a detailed descriptionof proper distributions in geomaterials, Harr [15] andLee et al. [16] can be referred to. In practice, it ismore common to use the dimensionless Coe�cient OfVariation (COV) instead of standard deviation, whichcan be de�ned as the standard deviation divided by themean. Typical values for the COV of the undrainedshear strength have been suggested by several investi-gators [16,17]. The suggested values are based on insitu or laboratory tests and the recommended range is0.1-0.5 for the COV of the undrained shear strength.The third important feature of a random �eld is itscorrelation structure. It is obvious that if two samplesare close together, they will be usually more correlatedcompared to the case where they are widely separated.It is common in literature to use a correlation functionin the following single exponential form, which is knownas the Markovain spatial correlation function [18]:

� = exp��2j� j

�

�; (3)

where � is called the correlation length or the scale of uctuation and � is the lag distance. The correlationlength is the parameter which describes the degree ofcorrelation of a soil property, and is de�ned as thedistance beyond which the random values will be nolonger correlated. We should note that in the case of alarge correlation length, the random �eld tends to besmooth, and oppositely, when it is small, the random�eld tends to be rough [19]. The �eld is assumed to becharacterized by a correlated log-normal distributionand is generated by the matrix decomposition algo-rithm, which is based on the Choleski method [20].The technique is based on decomposing a symmetric,positive de�nite matrix into a lower triangular matrix.

Since the undrained shear strength �eld is log-normally distributed, taking its logarithm yields anormally distributed random �eld, meaning that lncuis normally distributed. The values of the undrained

shear strength are estimated from:

ln cu = L:"+ �ln cu ; (4)

where �ln cu is the mean of ln cu, " is a Gaussian vector�eld (having zero mean and unit variance), and L is alower triangular matrix de�ned by:

A = LLT ; (5)

where A is the covariance matrix, which will be formedby using a speci�ed form of the covariance function.The approach can allow consideration of anisotropy;however, in the present study, the isotropic �eld isadopted. The isotropic covariance matrix is givenby [20]:

A = �2e� 2j�j� ; (6)

where �2 is the variance of ln cu, � is the autocorrelationlength and matrix � is the lag distance, which tends tobe the distance matrix. Figure 10 illustrates a samplerealization of undrained shear strength with assumedstochastic properties.

As stated before, the bearing capacity will becalculated by a FISH program developed by the au-thors, which adopts the �nite di�erence method mergedwith random �eld theory (RFDM). In this code, thePoisson ratio (v) is assumed to be constant, whilethe undrained shear strength (cu) and the undrainedYoung's modulus (Eu) are randomized throughout thedomain. The undrained Young's modulus is assumedto be fully correlated to the undrained shear strengthby adopting a Eu=cu ratio of 300.

Several runs are performed to investigate thee�ects of COV and � (strength density) separately. Thevalues of COV vary from 0.1 to 0.75. Adopted valuesof � are 1, 3 and 5 (kPa/m), where the surface shearstrength value is adopted as 30 (kPa). It should benoted that the correlation length is assumed constant,being equal to the footing width. For each set ofadopted COV and � values, Monte Carlo simulationshave been conducted involving 500 realizations ofthe shear strength random �eld and the subsequentnumerical analysis of the bearing capacity.

The results of the random �nite di�erence methodfor each set of input parameters have been preparedby the explained method. Figure 11 shows the e�ect

Figure 10. Realization of the undrained shear strengthwith COV = 0:75, cu0 = 30 kPa and � = 10 m.


Figure 11. Bearing capacity variation with COV and �.

of COV on the bearing capacity for di�erent levels ofstrength densities, �. It is evident from Figure 11that the COV has a decremental e�ect on the bearingcapacity of shallow foundations. It means that byan increase in the variation of strength parameters,the possibility of weak zone formation in underlyingstratum increases, and the bearing capacity of theoverlying foundation decreases. This �nding is inconformity with other research [1,18].

6. Comparison with \equivalent"homogeneous soil

According to most modern building codes [21], thebearing capacity calculation for shallow foundations isbased on uniform shear strength parameters acquiredfrom laboratory tests. Regulations essentially refer to ahomogenous or inhomogeneous pro�le, without stronggradients in shear strength parameters with depth.

On the other hand, in the case of inhomogeneouslayers, such as those described in Eq. (1), the choiceof pertinent \representative" shear strength has notyet been established. A simple approach to considerheterogeneity in the bearing capacity calculation is topick up the shear strength parameters from a speci�cdepth below the ground surface and then to follow theconventional homogeneous formulations. Towards thisaim, the proposed numerical solution for a heteroge-neous system is compared to the bearing capacity of anequivalent homogenous soil possessing shear strengthparameters taken from a representative depth. Threehomogenous layers are examined to this end, using thefollowing alternative representative depths:

� cuhom1 , equal to the undrained shear strength at thedepth of B/3 beneath the foundation base.

cuhom1 = cu�B3

�: (7)

Raymond [2] was the �rst to make such an assump-tion by evaluating the stability of surface footings,assuming a uniform strength equal to the strengthat a depth of one third of the footing (B/3).

� cuhom2 , calculated if a linear variation for shearstrength is assumed and a circular failure mechanismis adopted. By choosing a circular failure mecha-nism, as shown in Figure 2, such that the radius ofthe failure is B, the representative depth for shearstrength will be equal to 2B

� .

cuhom2 = cu(2B�

): (8)

This means that the bearing capacity calculationfor a linearly varying soil stratum with surfacecohesion, cu0 , and strength density, �, holds inanalogy with the homogeneous condition of thebearing capacity calculation when the shear strengthat the representative depth introduced in Eq. (8) ispicked.

� cuhom3 , equal to the undrained shear strength at thedepth of B beneath the foundation base.

cuhom3 = cu(B): (9)

This is indeed believed to be the farther depth in u-enced by the shallow foundation stress bulbs. This isfurther con�rmed if someone assumes a slip circle withradius B.

If someone takes the equivalent uniform shearstrength value from a representative depth, Zrep, then,the ultimate bearing capacity of the heterogeneous soilstratum calculated by the �nite di�erence method canbe compared to the equivalent homogenous solution byassuming the above mentioned representative depths,Zrep = B=3, 2B=� and B, where B is the width of thefooting.

The �rst adopted representative depth is to eval-uate the Raymond assumption. The second one isre ecting the depth calculated by the limit equilibriummethod, as discussed earlier, and the last is only tocover the maximum depth at which the slip surface forshallow foundation extends.

In Figure 12, the inhomogeneous system is com-pared to the above equivalent homogeneous soil interms of its bearing capacity. NH analysis in this �gurerefers to non-homogeneous analysis.

Very good agreement between the FDM het-erogeneous solution and the equivalent homogeneoussolution at the representative depth of B/3 is found,con�rming Raymond's assumption. The same level ofconformity is observed when the soil layer beneath thefoundation is soft and probably bears a bi-linear shearstrength variation trend, as assumed in Figure 5(b).Figure 12(b) clearly shows this agreement. The nu-merical analyses results, illustrated in Figure 12, also


Figure 12. Load-displacement curves, B = 10: (a) Partial heterogeneous Gibson soil, � = 2 kPa/m; and (b) bi-lineardeterministic trend with �1 = 3 kPa/m, �2 = 3 kPa/m and Zt = 2 m.

show that assuming a deeper representative depth forshear strength will lead to under-conservatism in theshallow foundation bearing capacity estimation. Thisbecomes more important when paying attention to thefact that the adoption of a safety factor of 3 or evenmore does not necessarily guarantee safety when therepresentative depth for shear strength is not realistic.

The implication of the aforesaid consideration ofrepresentative depth for shear strength, in practice, isto properly select the sampling depth for conductingshear strength tests, and apply a consistent value ofcon�ning stress in laboratory test schemes that bestrepresents the actual condition at that depth.

7. Implication to design

The current study examines di�erent approaches lead-ing to bearing capacity calculation for shallow footings.It was shown that �nite di�erence analysis, among all,is an e�cient tool for performing such calculations. Itwas used to investigate the e�ect of di�erent compo-nents of heterogeneity on the ultimate bearing capacityof footings. The normal practice in a construction�eld is to conduct some limited numbers of laboratorytests and calculate the bearing capacity of foundationsfrom conventional equations, which are mainly derivedby adopting limit equilibrium analysis. Terzaghi [12],Meyerhof [22,23] and Hansen [24] are good examples ofsuch equations. These equations have found general usein geotechnical practice. However, none of them con-sider heterogeneity as explained and analyzed here. Weshould now investigate the implication of heterogeneityon engineering design and the e�ect of its negligenceon design conservatism. Short-term bearing capacity

(undrained condition) was considered in the course ofthis research; however, it can be extended to a long-term (drained) condition in a similar manner. Theequivalent homogeneous ultimate bearing capacity cal-culated from homogeneous FDM analyses, qult(EH) , wascompared with heterogeneous FDM analysis results.A factor of conservatism, FOC, de�ned as the ratioof non-homogeneous to homogeneous bearing capacity(Eq. (10)), is employed herewith to re ect the level ofconservatism.

FOC =qult(NH)

qult(EH)

: (10)

The equivalent homogeneous bearing capacity calcu-lation (qult(EH)) is based on shear strength at therepresentative depth, considered as B=3, where B isthe width of footing.

Figure 13 shows the variation of FOC values withthe strength density, �, for footings of two di�erentwidths embedded on a Gibson-soil stratum with partialheterogeneity.

It is concluded that for medium to sti� clay pos-sessing linear heterogeneity, performing homogeneousbearing capacity analysis, based on the strength datafrom the representative depth (B/3), will not lead tosigni�cant over conservatism, as the FOC value remainsalmost invariable and independent of strength density,�, and is close to 1.

For soft soils, the variation of undrained cohesionwas stated to be of a bi-linear nature. The resultsof FOC variation with di�erent strength parameters,namely, densities and transformation depth, are plot-ted in Figure 14.


Figure 13. Variation of FOC values with strengthdensity, �, for partially heterogeneous Gibson-soil.

It is evident that for small footing widths, theFOC values do not vary signi�cantly. However, it isslightly less than 1, which implies under conservatism.As the footing width becomes large enough (10 m inthis case), FOC values grow and homogeneous analysiswill lead to over conservatism. However, the largefooting width is not usually the case in engineeringpractice, as the strip footing assumed in this study isusually of small size and bears values even less thanthose adopted in this study (2 m).

Another observation is that the introduced rep-resentative depth for undrained cohesion proposed byRaymond [2] is a competent approximation, which doesnot lead to signi�cant over or, more importantly, underconservatism.

Stochastic variation of soil properties will induce

Figure 15. Variation of FOC values with COV forstochastic variation of undrained cohesion.

further uncertainty in bearing capacity estimation.Uncertainty is translated into unconservatism in prac-tice when relying upon the homogeneous formulationfor bearing capacity estimation, as found in classicfoundation engineering literature. Figure 15 illustrateshow the spatial variability of undrained shear strengthrepresented by COV a�ects design conservatism.

As expected, the increase in COV values will leadto the probability of weak zone formations and, conse-quently, to reducing the mean bearing capacity. FOCvalues as de�ned earlier, will then show a decrease.

8. Conclusion

This study introduced and compared di�erent tech-niques to compute the ultimate bearing capacity of

Figure 14. FOC values vs. di�erent strength parameters: (a) �1 variation, �2=5 kPa/m and Zt=2 m; and (b) �2

variation, �1=6 kPa/m and Zt=2 m.


strip footings, inclusive of simple analyses and numeri-cal methods representing complex analysis techniques.For heterogeneous soil deposits, the following resultswere obtained:

� In linear heterogeneous medium to sti� soil deposits,the bearing capacity increases as the strength den-sity, �, increases.

� For partial heterogeneous soil, which inherits non-zero surface cohesion and positive strength density,surface cohesion was found to contribute most to theundrained bearing capacity.

� For soft soils where the shear strength variationis of a bi-linear nature, by increasing the rate atwhich the undrained cohesion decreases, �1, theultimate bearing capacity decreases. However, thepositive strength density, �2, belonging to the regionunderlying the surface desiccation zones, has an in-cremental e�ect, especially for larger footing width,where the stress bulbs extend deeper and beyondthe transformation depth.

� For small and large footings, the bearing capacitydecreases with the increase in transformation depth.However, this e�ect disappears when a thresholdtransformation depth is reached. The thresholdtransformation depth is footing width dependentand is indeed beyond the failure zone.

� In order to avoid under or, most importantly,over estimation of the ultimate bearing capacity ofshallow footings, it was found that performing anequivalent homogenous �nite di�erence analysis byadopting shear strength parameters from a repre-sentative depth would be a competent alternative.The formerly introduced representative depth forundrained cohesion proposed by Raymond, Zt =B=3, is still deemed to be valid. For both smalland large footing widths, it was seen that theresults of equivalent homogeneous bearing capacityanalyses are close enough to actual heterogeneouscalculations. This leads to the contention that FOCvalues remain near 1 for the practical ranges ofstrength density.

� Faced with common geotechnical issues, a practi-tioner usually refers to classic methods, such as limitequilibrium and slip line solution, available in litera-ture. Homogenous conditions or in its most complexform of study, deterministic heterogeneity, are recog-nized usually by practitioners or even academicians.However, the results of the current study implythat in real world situations, the spatial variabilityof strength reduces the estimated bearing capacity,especially by increasing the COV of parameters.This means that neglecting the spatial variability ofsoil properties leads to an overestimation in bearingcapacity prediction.

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Biographies

Reza Jamshidi Chenari received his BS (�rst class)and MS degrees from Sharif University of Technology,Tehran, Iran, and his PhD degree from Iran University

of Science and Technology, Tehran, Iran. He iscurrently Assistant Professor of Civil Engineering atthe University of Guilan, Iran. His research interestsinclude the dynamic behavior of retaining walls back-�lled with geotextile �ber reinforced sand.

Jamshidi's current research focuses on risk andreliability in geo-engineering and geohazards, appli-cation of Random Numerical Methods (RNM) indi�erent geotechnical problems, including foundationsettlement, shallow foundation bearing capacity, slopestability, consolidation of natural alluvial deposits,seismic ampli�cation of alluviums and the e�ect ofheterogeneity and anisotropy in the deformation andstrength properties of natural alluviums on these prob-lems. Dr. Jamshidi has published more than 20scienti�c papers.

Negin Zhalehjoo received her BS and MS degrees inCivil Engineering and Geotechnical Engineering fromthe University of Guilan, Iran, in 2010 and 2013,respectively. Her research interests include computa-tional geomechanics and the application of numericalmodeling in geotechnical engineering practice. She hasauthored two scienti�c journal papers focusing on therandom �eld theory.

Akram Karimian received her BS (�rst class) andMS (Geotechnical Engineering) degrees from the De-partment of Engineering at the University of Guilan,Iran. Her research interest includes application ofnumerical methods in classical geotechnical engineeringproblems, and she has co-authored 2 scienti�c publica-tions related to this �eld.

estimation on bearing capacity of shallow foundations in...

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