shallow foundation response variability due to soil and model parameter uncertainty

RESEARCH ARTICLE

Shallow foundation response variability due to soil andmodel parameter uncertainty

Prishati RAYCHOWDHURY*, Sumit JINDAL

Department of Civil Engineering, Indian Institute of Technology Kanpur, Kanpur UP-208016, India*Corresponding author. E-mail: [email protected]

© Higher Education Press and Springer-Verlag Berlin Heidelberg 2014

ABSTRACT Geotechnical uncertainties may play crucial role in response prediction of a structure with substantialsoil-foundation-structure-interaction (SFSI) effects. Since the behavior of a soil-foundation system may significantly alterthe response of the structure supported by it, and consequently several design decisions, it is extremely important toidentify and characterize the relevant parameters. Moreover, the modeling approach and the parameters required for themodeling are also critically important for the response prediction. The present work intends to investigate the effect of soiland model parameter uncertainty on the response of shallow foundation-structure systems resting on dry dense sand. TheSFSI is modeled using a beam-on-nonlinear-winkler-foundation (BNWF) concept, where soil beneath the foundation isassumed to be an assembly of discrete, nonlinear elements composed of springs, dashpots and gap elements. Thesensitivity of both soil and model input parameters on shallow foundation responses are investigated using first-ordersecond-moment (FOSM) analysis and Monte Carlo simulation through Latin hypercube sampling technique. It has beenobserved that the degree of accuracy in predicting the responses of the shallow foundation is highly sensitive soilparameters, such as friction angle, Poisson’s ratio and shear modulus, rather than model parameters, such as stiffnessintensity ratio and spring spacing; indicating the importance of proper characterization of soil parameters for reliable soil-foundation response analysis.

KEYWORDS shallow foun dation, sensitivity analysis, centrifuge data, first-order-second-moment (FOSM) method,parameter uncertainty

1 Introduction

Uncertainty refers to situations where the outcome of anevent or the value of a parameter may differ from its truevalue. Uncertainty plays an important role in responsevariability of a soil-foundation systems. These uncertain-ties may arise from geotechnical and structural materialproperties, input loadings and modeling methods. For ananalysis of a structure with significant soil-foundation-structure-interaction (SFSI) effects, geotechnical uncer-tainties may play crucial role in overall system responsevariability. Uncertainties in geotechnical engineeringproperties are largely induced from estimation of basicsoil strength and stiffness parameters, knowledge ofgeology, judgments, and statistical reasoning. Most soilsare naturally formed in many different depositional

environments; thus showing variation in their physicalproperties from point to point. However, soil propertiesexhibit variations even within an apparently homogeneoussoil profile. Basic soil parameters that control the strengthand stiffness of the soil-foundation system are cohesion,friction angle, unit weight, shear modulus and Poisson’sratio of soil. These soil parameters can be delineated usingdeterministic or probabilistic models. Deterministic mod-els use a single discrete descriptor for the parameter ofinterest, whereas probabilistic models define parameters byusing discrete statistical descriptor or probability densityfunctions.Uncertainty in soil properties can be formally grouped

into aleatory and epistemic uncertaintyas described byLacasse and Nadim [1]. Aleatory uncertainty represents thenatural randomness of a property and is also a function ofspatial variability of the soil property. This type ofuncertainty is inherent to the variable and cannot beArticle history: Received Sep. 30, 2013; Accepted Apri. 6, 2014

Front. Struct. Civ. Eng. 2014, 8(3): 237–251DOI 10.1007/s11709-014-0242-1

reduced or eliminated by additional information. Epistemicuncertainty results from the lack of information andshortcomings in measurements and calculations [1].Epistemic uncertainty can usually be reduced by acquisi-tion of more information or improvements in measuringmethods.In last few decades, significant research has been carried

out in order to understand the behavior of a structureconsidering uncertainty in soil parameters [2–11]. In anearly work, Lumb [5] showed that the soil parameters canbe modeled as random variables confirming to theGaussian distribution within the framework of probabilitytheory. Ronold and Bjerager [11] observed that modeluncertainties are important in reliability analysis forprediction of stresses, capacities, deformation etc. instructure and foundation systems. Chakraborty and Dey[2] studied the stochastic structural responses consideringuncertainty in structural properties, soil properties andloadings using Monte Carlo simulation technique. Lutes etal. [6] evaluated the response of a seismically excitedstructural system with uncertain soil and structural proper-ties. Ray Chaudhuri and Gupta [8] investigated thevariability in seismic response of secondary systems dueto uncertain soil properties through a mode accelerationmethod. Foye et al. [3] performed a thorough study forassessment of variable uncertainties by defining theprobability density functions for uncertain design variablesin load resistance factor design (LRFD). Na et al. [7]investigated the effect of uncertainties of geotechnicalparameters on gravity type quay-wall in liquefiablecondition using tornado diagram and first-order second-moment (FOSM) analysis. Raychowdhury [9] studied theeffect of soil parameter uncertainty on seismic demand of

low-rise steel buildings supported by shallow foundationson dense silty sand with considering a set of 20 groundmotions. Raychowdhury and Hutchinson [10] carried outthe sensitivity analysis of shallow foundation response touncertain input parameters using simplified FOSM andtornado diagram methods.The present article focuses on studying the effect of

uncertainty in soil and model input parameters on theresponse of a shearwall-foundation system. To incorporatethe nonlinearity at the soil-foundation interface, a beam-on-nonlinear-winkler-foundation (BNWF) approach isadopted. To evaluate sensitivity of parameters on thepredictive capability of the numerical model, a series ofcentrifuge experiments conducted on shallow strip foot-ings at the University of California, Davis are utilized. Theuncertainty analysis has been carried out using FOSMmethod and Monte Carlo simulation through Latinhypercube sampling technique.

2 Modeling of SFSI

In this paper, a BNWF approach is used to model thenonlinear soil-structure-interaction of shallow foundationssubjected to lateral loads. The BNWF model includes asystem of closely spaced independent, mechanisticelements consisting of inelastic springs, dashpots andgap elements (Fig. 1). The vertical springs (q-z elements)are intended to capture the axial and rotational behavior ofthe footing, whereas the lateral springs, t-x element and p-xelement, are intended to capture the sliding and passiveresistance, respectively. The material models were origin-ally developed by Boulanger et al. [12] and modified by

Table 1 Details of selected uncertain parameters

parameters symbol range mean (μ) coefficient of variation (Cv)

friction angle/(°) f# 38 - 42 40 3%

Poisson’s ratio ν 0.3 - 0.5 0.4 16%

shear modulus/MPa Gs 12 - 20 16 15%

end length ratio/% Re 1 - 17 9 54%

stiffness intensity ratio Rk 1 - 9 5 48%

spring spacing/% Ss 1.0 - 3.0 2 30%

Table 2 Assumed correlation coefficient between the parameters

f# ν Gs Re Rk SS

f# 1 0.1 0.6 0 0 0

ν 1 0.2 0 0 0

Gs 1 0 0 0

Re 1 0.3 0.1

Rk 1 0.1

SS 1

238 Front. Struct. Civ. Eng. 2014, 8(3): 237–251

Raychowdhury and Hutchinson [13]. The BNWF model iscapable of reasonably capturing the experimentallyobserved behavior for various shallow foundation condi-tions. For more details regarding the BNWF modeling, onecan look into Refs. [13] and [14]. The material modelshave a nonlinear backbone curve with initial elasticportions followed by smooth nonlinear behavior (Fig. 2).In the elastic portion, the instantaneous load q is assumedto be linearly proportional with the instantaneous dis-placement z via the initial elastic (tangent) stiffness kin, i.e.,

q ¼ kinz, (1)

where kin is the initial elastic (tangent) stiffness. The rangeof the elastic region is defined by the following relation:

qo ¼ Crqult, (2)

where qo is the load at the yield point, Cr is a parametercontrolling the range of the elastic portion, and qult is theultimate load. In the nonlinear (post-yield) portion, thebackbone curve is described by

q ¼ qult – ðqult – qoÞcz50

cz50 þ zp – zpoj j� �n

, (3)

where z50 is the displacement at which 50% of the ultimateload is mobilized, zpo is the displacement at the yield point,zp is the displacement at any point in the post-yield region,and c and n are the constitutive parameters controlling theshape of the post-yield portion of the backbone curve. Theexpressions governing both p-x and t-x elements aresimilar to Eqs. (1) – (3), with variations in the constants n,c, and Cr, controlling the general shape of the curve.Furthermore, it may be noted that q-z material has a

Fig. 1 Shearwall resting on strip footing

Fig. 2 Material models. (a) q-z element; (b) p-x element; (c) t-x element

Prishati RAYCHOWDHURY et al. Shallow foundation response variability due to soil and model 239

reduced capacity in the tension side (Fig. 2(a)), whichallows the footing to uplift without losing contact with thesoil beneath it during a rocking movement. On the otherhand, the p-x material is characterized by a pinchedhysteretic behavior (Fig. 2(b)), whereas the t-x material ischaracterized by a large initial stiffness and a broadhysteresis (Fig. 2(c)).It is evident from Eqs. (1) – (3) that shape of spring

backbone curves which is basically controlling factor forthe soil-structure interaction behavior is mainly dependanton two physical parameters related to soil characteristics,namely, capacity (qult), and initial elastic stiffness (kin). Thecapacity and elastic stiffness of each spring is obtained bydistributing the global footing capacity and stiffnessutilizing proper tributary area of each spring. The footingcapacity is derived using general bearing capacity equationfrom Terzaghi [15] with shape, depth and inclinationfactors after Meyerhof [16] as shown in Eqs. (4) – (7).

qult ¼ cNcFcDcIc þ gDfNqFqDqIq

þ 0:5gBNγFγDγIγ, (4)

where qult is ultimate vertical bearing capacity of thefooting, c is cohesion, γ is unit weight of soil,Df is depth ofembedment, B is width of footing, Nc, Nq, Nγ are bearingcapacity factors, Fc, Fq, Fγ are shape factors,Dc, Dq, Dγ aredepth factors and Ic, Iq, Iγ are inclination factors. Bearingcapacity, shape, depth and inclination factors are calculatedbased on expressions given by Meyerhof [16]:

Nq ¼ eπ tanftan2 45þ f

2

� �, (5)

Nc ¼ ðNq – 1Þcotf, (6)

Ng ¼ ðNq – 1Þtanð1:4gÞ: (7)

For the p-x material, the ultimate lateral load capacity isdetermined as the total passive resisting force acting on thefront side of the embedded footing. For homogeneousbackfill against the footing, the passive resisting force canbe calculated using a linearly varying pressure distributionresulting in the following equation:

pult ¼ 0:5gD2fKp, (8)

where pult = passive earth pressure per unit length of

footing, Kp = passive earth pressure coefficient. For the t-xmaterial, the lateral load capacity is the total sliding(frictional) resistance which can be defined as the shearstrength between the soil and the footing as:

tult ¼ Wgtan δþ Abc, (9)

where, tult = frictional resistance per unit area offoundation, Wg = vertical force acting at the base of thefoundation, δ = angle of friction between the foundationand soil (typically varying from 1/3φ to 2/3φ) and Ab = thearea of the base of footing in contact with the soil ( = L x B).The initial elastic stiffnesses (vertical and lateral) of the

footing are derived from Gazetas [17] as follows:

kv ¼GL

1 – υ0:73þ 1:54

B

L

� �0:75� �, (10)

kh ¼GL

2 – υ2þ 2:5

B

L

� �0:85� �, (11)

where kv and kh are the vertical and lateral initial elasticstiffness of the footing, respectively; G is the shearmodulus of soil; vis the Poisson’s ratio of soil; B and L arethe footing width and length, respectively. The instanta-neous tangent stiffness kp, which describes the load-displacement relation within the post-yield or nonlinearregion of the backbone curves, may be expressed as:

kp ¼ nðqult – qoÞðcz50Þn

ðcz50 – zo þ zÞnþ1

� �: (12)

Note that Eq. (12) is obtained by rearranging Eq. (3).Also, note that the shape and instantaneous tangentstiffness of the nonlinear portion of the backbone curveis a function of the parameters c and n, which are derivedby calibrating against a set of shallow footing tests ondifferent types of soil. Details of the calibration study isdescribed in the PhD dissertation of the first author of thisarticle. It is evident from the preceding equations that thespring responses are primarily dependent on basic strengthand stiffness parameters of soil such as cohesion (c),friction angle (f), unit weight (γ), shear modulus (G) andPoisson’s ratio (ν) of soil. Therefore, proper characteriza-tion of these parameters are important for prediction ofresponses of the soil-foundation-structure system.In addition to the soil properties, parameters related to

the numerical modeling may also have significant effect on

Table 3 Details of the experiments considered in the study

test mass/mg length/m width/m height/m embedment/m FSv M/VL reference

SSG02_03 28 2.8 0.65 0.66 0 5.2 1.75 Gajan et al. [18]

SSG02_05 58 2.8 0.65 0.66 0 2.6 1.72 Gajan et al. [18]

SSG03_03 28 2.8 0.65 0.66 0.65 14 1.77 Gajan et al. [19]

SSG04_06 68 2.8 0.65 0.66 0 2.3 1.11 Gajan et al. [20]


the response prediction. The BNWF model considered inthis study has few such modeling parameters that might beconsidered as important, namely, stiffness intensity ratio,end length ratio and spring spacing. A distributed stiffnessintensity along the length of the foundation is provided inthe BNWF model in order to capture the rocking stiffnessof the footing when subjected to lateral or rotationalmovements. To achieve this, stiffness at the edges of thefooting is increased compared to the middle portion(Fig. 3). The ratio of the stiffness of the end zone to themiddle zone is defined as stiffness intensity ratio, while thelength of the increased stiffness zone as a fraction of thetotal footing length is defined as end length ratio. Springspacing is the spacing between two consecutive springs asa fraction of total footing length.

3 Selection of uncertain parameters

Based on the discussion provided in the previous sectionthe following parameters shown in Table 1 are consideredfor the sensitivity analysis. Since the simulation results arecompared with experiments conducted on shallow founda-tions resting on dry dense sand of relative density 80%, thechosen range of parameters corresponds to the propertiesof the same soil. The ranges of soil properties are chosenbased on discussions in Refs. [21] and [22], while thevalues of model properties are chosen based on Refs. [23–25]. It is assumed that all uncertain input parameters arerandom variables with a Gaussian distribution, having nonegative values. The upper and lower limits of the randomvariables are assumed to be in 95th and 5th percentile of itsprobability distribution. The corresponding mean (μ) andstandard deviation (σ) can be calculated as:

� ¼ ðLL þ LUÞ2

, (13)

� ¼ ðLL – LUÞ2k

, (14)

where LL and LU are the lower and upper limits,

respectively, and k depends on the probability level,(e.g., k = 1.645 for a probability of exceedance = 5%). Theparameters are assumed to be correlated following thecorrelation matrix shown in Table 2. The correlations areconsidered based on discussions in Refs. [4, 21], a numberof BNWF simulations and engineering judgment.

4 Experiments considered

To study the effect of parameter uncertainty on thepredictive capability of the soil-foundation numericalmodel, a set of four centrifuge experiments are chosen.These experiments include cyclic tests on shearwallstructures supported by shallow strip footings resting ondense dry sand of relative density 80%. Details of thechosen experiments are provided in Table 3. It can be notedthat the experiments include a range of mass, vertical factorof safety (FSv), depth of embedment (Df), and moment toshear aspect ratio (M/VL).The shearwall-footing system as described in the

experiments are modeled using finite element softwareOpenSees [26]. The soil-foundation interface is modeledusing BNWF model as described in the earlier sections. Atotal number of 120 simulations have been conducted (4tests � 6 parameters � 5 simulations). Each parameter isvaried independently while keeping all other parametersfixed at their mean values. The simulation results arecompared with the experiments and absolute errors inpredicting peak responses are obtained for each test withvarying parameters. The responses considered are: abso-lute maximum values of moment, shear, rotation, andsettlement. Absolute error in predicting any demand isobtained using the following relationship:

δj jð%Þ ¼ DemandðexpÞ –DemandðsimÞDemandðexpÞ

�� 100: (15)

For a particular test, a total number of 30 simulations (5simulation each for six input parameters) have been carriedout. Let for ith experiment, the absolute error in predictingmoment is δM (i; j; k), where j = 1, 2,...,6 and k = 1, 2,...,5

Fig. 3 Variable stiffness distribution of BNWF model and model input parameters


corresponds to each of the six parameters and five values ofeach parameters, respectively. The range of input para-meters is defined in Table 1. The mean of absolute error inthat response is obtained assuming equal weight to allexperiments as shown below. The absolute error in shear,settlement and rotation demands are calculated byfollowing the same procedure.

δMðj,kÞ ¼1

4

X4

i¼1δMði,j,kÞ: (16)

5 Uncertainty analysis

To understand the effect of variability of uncertain inputparameters on footing-structure response, uncertaintyanalysis has been performed using simple FOSM methodand Monte Carlo Simulation through Latin hypercubesampling technique. Followed is a brief description of theadopted methods for the uncertainty analysis.

5.1 FOSM method

The FOSM method is a simplified probabilistic responseanalysis to evaluate the effect of variability of inputparameters on a resulting response variable. This method isbased on few assumptions: 1) all uncertain parameters arerandom variables with a Gaussian distribution, 2) therelationship between the response variables and theuncertain input parameters are assumed to be linear orlow-to-moderately nonlinear. The FOSM method uses aTaylor series expansion of the function to be evaluated andexpansion is truncated after the linear first order term. Thismethod does not account the form of the probabilitydensity function describing random variables, uses onlytheir mean and standard deviation. The response of thefoundation is considered as a random variable Q, whichhas been expressed as a function of the input randomvariables, Pi (for i = 1,2,...,N)denoting uncertain para-meters and Q given by,

Q ¼ hðP1,P2,:::,PN Þ: (17)

The random variable Pi has been characterized by itsmean, μP and variance �

2p. Now,Q can be expanded using a

Taylor series as follows:

Q ¼ hð�P1,�P2

,:::,�PNÞ þ 1

1!

XNi¼1

ðPi –�PiÞ∂h∂Pi

þ 1

2!

XNj¼1

XNi¼1

ðPi –�PiÞðPj –�PjÞ∂2h

∂Pi∂Pjþ � � � : (18)

Considering only first order terms of Eq. (13), andignoring the higher order terms, Q can be approximated as:

Q � hð�P1,�P2

,:::,�PNÞ þ

XNi¼1

ðPi –�PiÞ∂h∂Pi

, (19)

Taking expectation of both sides of Eq. (17), the mean ofQ can be expressed as:

�Q ¼ hð�P1,�P2

,:::,�PNÞ: (20)

Utilizing the second order moment of Q as expressed inEq. (19), the variance of Q can be derived as:

�2Q �

XNi¼1

XNj¼1

covarianceðPi,PjÞ

∂hðP1,P2,:::,PN Þ∂Pi

∂hðP1,P2,:::,PN Þ∂Pj

�XNi¼1

�2Pi


� �2

þXNi¼1

XNj≠1

�Pi,Pj


∂hðP1,P2,:::,PN Þ∂Pj

, (21)

where �Pi,Pjdenotes correlation coefficient for random

variables Pi and Pj. The partial derivative of h (P1, P2,...,PN) with respect to Pi has been calculated numericallyusing the finite difference method (central) as follows:


¼ hðp1,p2,:::,�i þ Δpi,pN Þ – hðp1,p2,:::,�i –Δpi,pN Þ2Δpi

:

(22)

FOSM analysis is carried out to investigate thesensitivity of the uncertain soil and model parameters onthe response of the foundation. It provides an approximatesense of sensitivity of the parameters. The parametersdiscussed previously have been varied one by one whilekeeping all others constant to study the isolated effect ofeach one. Number of simulations has been performedvarying each input parameter individually to approximatethe partial derivatives as given in Eq. (21) considering thecorrelation between uncertain input parameters as definedin Table 2.

5.2 Latin hypercube method

For probabilistic analysis of engineering structures havinguncertain input variables, Monte Carlo simulation (MCS)technique is considered as a reliable and accurate method.However, this method requires a large number of equallylikely random realizations and consequent computational


effort. To decrease the number of realizations required toprovide reliable results in MCS, Latin hypercube sampling(LHS) approach [27] is widely used in uncertaintyanalysis. LHS is a type of stratified MCS which providesa very efficient way of sampling variables from theirmultivariate distributions for estimating mean and standarddeviations of response variables [28]. It follows a generalidea of a Latin square, in which, there is only one sample ineach row and each column. In Latin hypercube sampling togenerate a sample of size K from N variables, theprobability distribution of each variable is divided intosegments with equal probability. The samples are thenchosen randomly in such a way, that each interval containsone sample. During the iteration process, the value of eachparameter is combined with the other parameter in such away, that all possible combinations of the segments aresampled. Finally, there are M samples, where the samples.In this study, to evaluate the response variability due to

the uncertainty in the input parameters, samples aregenerated using Stein’s approach [29]. This method isbased on the rank correlations among the input variablesdefined by Ref. [28] which follows Cholesky decomposi-tion of the covariance matrix. The previously mentionedsimple shearwall structure (Fig. 1) is considered for theLHS analysis, where the responses of the system areconsidered to be dependent on the six independentnormally distributed variables described in Table 1. Tofind out the correct sample size, a static pushover analysisis carried out using a number of 10, 20,...,100, 200 and 300samples. Figure 4 shows the responses corresponding toeach sample size normalized by the value corresponding toa sample size of 300. It has been observed that theresponses, as expected, tend to converge as the sample sizeincreases. However, the responses converges at a samplesize of 100, indicating that a sample size of at least 100 isrequired for a reasonably accurate estimation. Therefore, asample size of 100 has been used in this study for ananalysis with reduced computational effort yet reasonableaccuracy.

6 Results and discussion

The effect of uncertainty in soil and modeling parametersare investigated using a number of centrifuge experimentsof shallow foundation-supported shearwalls as describedin Table 3. First, comparison between the simulation andthe experimental behavior is studied with varying para-meters within the chosen ranges as mentioned in Table 1.Figure 5 shows the comparison results for the testSSG02_05 for two extreme values of friction angle, f#.The results include moment-rotation, settlement-rotation

and shear-rotation behavior with the BNWF simulationshown in black and experimental results shown in grayscale. These comparisons indicate that the BNWF model isable to capture the hysteretic features such as shape of theloop, peaks, unloading and reloading stiffness reasonablywell. It can also be observed from Fig. 5 that withincreasing the friction angle from 38° to 42°, peak momentand peak shear demands increase, whereas peak settlementdemand decreases. It can also be noted that the variation offriction angle from 38° to 42° has most significant effect onsettlement prediction (more than 100%). However, themoment, shear and rotation demands are moderatelyaffected by this parameter. This indicates that theuncertainty in one parameter may have significantlydifferent influence on the prediction of different responses,pointing out toward the importance of proper characteriza-tion of each parameters and conducting the sensitivityanalysis.For more comparison results for each experiment and

each parameter, one can look into Ref. [30]. A summarytable is provided (Table 4) for mean peak demandsconsidering all experiments for varying all parameters.Note that this responses are obtained by varying eachparameter at a time, while keeping other parameters fixedat their respective mean values.To investigate the effect of parameter uncertainty on the

predictive capability of the model, the amount of error inpredicting different force and displacement responses of

Fig. 4 Response variation with different sample sizes


the footing are calculated. The deviations from theexperimental responses to that obtained from simulationsare presented in the Table 5. The absolute errors inpredicting peak moment, peak shear, peak rotation andpeak settlement (|δM|, |δV|,|δθ|, and |δS|, respectively) areconsidered. These are calculated as per Eq. (4) for eachexperiment and the mean of these are shown in Table 5.The variation of each response parameter with respect toeach input parameter are presented in Fig. 6. The error inpredicting each response is normalized by the samecorresponding to the lowest value of each input parameter.It can be observed from Table 5 and Fig. 6 that most of the

parameters have significant effects on the responseestimation with respect to the experimental observation.For example, increasing friction angle from 38° to 42°reduces error in moment prediction from 22% to 5%,whereas the same variation increases the error in settlementprediction from 27% to 50%. Similar observations aremade from varying other parameters, indicating that eachparameter has different effect on each response variable.Moreover, it can be observed from Fig. 6 that most of the

variations are linear and moderately nonlinear, with fewexceptions such as shear modulus versus shear demand,stiffness intensity ratio versus moment and shear demand.

Table 4 Absolute mean demands with varying input parameters

parameters rangemean absolute demands from simulation

moment/(KN-m) shear/KN rotation/rad settlement/mm

f#/(°) 38.0 368.56 87.73 0.055 111.50

39.0 391.04 93.47 0.055 93.05

40.0 414.75 99.49 0.054 72.55

41.0 437.22 105.21 0.054 60.67

42.0 459.08 110.14 0.054 52.70

ν 0.30 406.49 97.59 0.054 73.64

0.35 410.48 98.48 0.055 72.58

0.40 414.75 99.49 0.054 72.55

0.45 418.20 100.24 0.055 72.36

0.50 423.27 101.51 0.054 72.22

Gs/MPa 12.0 399.31 96.03 0.055 74.32

14.0 407.71 97.92 0.054 73.43

16.0 414.75 99.49 0.054 72.55

18.0 420.88 101.00 0.054 72.33

20.0 428.59 103.18 0.055 60.80

Re/% 1.0 419.61 101.06 0.055 69.27

5.0 412.60 98.96 0.055 70.22

9.0 414.75 99.49 0.054 72.55

13.0 416.61 99.89 0.054 75.40

17.0 418.30 100.31 0.054 78.51

Rk 1.00 418.23 100.52 0.055 69.88

3.00 418.76 100.50 0.055 70.15

5.00 414.75 99.49 0.054 72.55

7.00 408.53 97.91 0.054 74.72

9.00 402.79 96.47 0.054 76.64

SS/% 1.00 412.65 98.96 0.055 73.04

1.50 413.56 99.18 0.055 72.42

2.00 414.75 99.49 0.054 72.55

2.50 416.26 99.87 0.055 71.99

3.00 410.41 98.00 0.055 65.85

experimental value 477.09 103.89 0.052 140.83


This indicates that the assumption of linear relationshipbetween input parameters and response variables aregenerally satisfied for this system, and therefore, theFOSM method can be applicable herein. Figure 7 showsthe results of the FOSM analysis carried out to evaluate therelative importance of each input parameter on thepredictive capability of the model in obtaining fourimportant response parameters: peak moment, peakshear, peak rotation and peak settlement. It can be observedfrom Fig. 7(a) that the error in predicting moment demandis highly sensitive to friction angle (67% relative variance);moderately sensitive to Poisson’s ratio and shear modulus(18% and 12%, respectively); and least sensitive to the

model parameters (less than 3% relative variance).Figure 7(b) provides sensitivity scenario for predictingpeak shear demand. The findings of the shear demand arequite similar with that of moment demand. Figure 7(c) and(d) show the sensitivity for errors in predicting rotation andsettlement demands, respectively. It can observed fromthese figures, that friction angle has a sensitivity of 51%and 84% on the rotation and settlement prediction,respectively. End length ratio, Re is found to have moderateeffect on the rotational demand (33%). This makes sense asthe rotational stiffness is controlled by the change instiffness distribution along the length of the footing. Thesefindings indicate that friction angle is the most important

Table 5 Absolute mean error in demands with varying input parameters

parameters rangeabsolute error in BNWF simulation/%

δM�� δV

�� δθ�� δS

�� f#/(°) 38.0 22.05 15.04 7.26 27.33

39.0 17.60 10.16 7.29 38.11

40.0 12.88 6.64 7.23 46.91

41.0 8.37 7.41 7.22 49.85

42.0 4.82 8.09 7.16 50.16

ν 0.30 14.62 7.61 7.25 46.83

0.35 13.78 7.11 7.26 47.10

0.40 12.88 6.64 7.23 46.91

0.45 12.12 6.47 7.26 46.95

0.50 11.08 6.52 7.22 46.82

Gs/MPa 12.0 16.13 8.53 7.33 47.00

14.0 14.36 7.46 7.25 46.88

16.0 12.88 6.64 7.23 46.91

18.0 11.59 6.54 7.23 46.92

20.0 10.06 7.22 7.25 50.57

Re/% 1.0 11.92 7.20 7.40 49.08

5.0 13.34 6.76 7.27 48.89

9.0 12.88 6.64 7.23 46.91

13.0 12.43 6.58 7.25 43.81

17.0 12.05 6.76 7.25 40.78

Rk 1.00 12.23 7.17 7.26 48.85

3.00 12.06 6.82 7.28 47.90

5.00 12.88 6.64 7.23 46.91

7.00 14.13 6.88 7.21 46.26

9.00 15.28 7.62 7.24 45.48

SS/% 1.00 13.30 6.75 7.29 46.86

1.50 13.12 6.71 7.30 47.04

2.00 12.88 6.64 7.23 46.91

2.50 12.58 6.66 7.27 47.15

3.00 13.64 5.96 7.28 49.59


Fig. 5 Footing response comparison for test SSG02 05 with varying friction angle. (a) Moment-rotation; (b) settlement-rotation;(c) shear-rotation


Fig. 6 Variability of response prediction due to parameter uncertainty. (a) Friction angle; (b) Poisson’s ratio; (c) shear modulus; (d) endlength ratio; (e) stiffness intensity ratio; (f) spring spacing


Fig. 7 Sensitivity analysis using FOSM method. (a) Moment; (b) shear; (c) rotation; (d) settlement


parameter for most of the responses, and characterizationof this parameter should be given utmost importance.Table 6 provides the results of the LHS analysis

described in the previous section. Simulations areperformed for four tests as mentioned in Table 3 byusing the Latin hypercube sample of size 100. Response ofthe each simulation is recorded for each test in the term offour absolute maximum demands: moment, shear, rotation,and settlement. The responses from simulation arecompared with the experimental response values for eachtest and the absolute errors are obtained. The mean andcoefficient of variation of the absolute errors in predictionare evaluated then. It is observed from Table 6 that theuncertainties in friction angle, Poisson’s ratio, shearmodulus, end length ratio, stiffness intensity ratio andspring spacing with coefficients of variation (Cv) of 3%,16%, 15%, 54%, 48% and 30%, respectively, result insignificant variation in response prediction with Cv as 54%,72%, 114% and 111% for the absolute error in maximummoment, shear, rotation and settlement demands, respec-tively. This indicates that 1) response variations (in termsof Cv) are greater than the input parameter variationsindicating significant effect of parameter uncertainty oneach response; 2) friction angle has largest effect on theresponses as it had lowest Cv among all input parameters;and 3) Other two soil parameters, Poisson’s ratio and shearmodulus have moderate effect, and the model parametershave least effect.Note that the findings of this analysis is in accordance

with the findings of FOSM analysis. It can also beobserved from Table 6 that the mean deviation in

estimating peak moment, shear, rotation and settlementdemands are 15%, 11%, 9% and 49%, respectively usingLHS analysis. indicating that except for settlementdemand, BNWF model has reasonably good predictivecapability.

7 Conclusions

The behavior of a shallow foundation under significantloadings can vary significantly due to the uncertainty insoil and modeling input parameters. Accurate modeling ofgeotechnical components of a soil-foundation system isrequired to predict the response of shallow foundationsundergoing significant loading such as earthquakemotions. The modeling approach and the model para-meters are also important for the response prediction, asany variation in these parameters may significantly alterthe system performance and consequent the designdecisions. Therefore, the proper characterization of therelevant parameters is of utmost importance. Moreover, itis important to identify the sources and extent ofuncertainty of the parameters, along with evaluating theireffect on the response prediction of soil-foundationsystems. In this work, a BMWF model is used to modelthe soil-foundation-structure interface. The effect ofparameter uncertainty on the response of shallow founda-tion supported structural systems in dry dense sand isevaluated through FOSM method and Latin hypercubesampling technique. The following key observations aremade from the present research work:

Table 6 Variability in absolute errors of response prediction using Latin hypercube method/%

test moment shear rotation settlement

|δM| |δV| |δθ| |δS|

SSG02_03 µ 9.59 5.13 0.38 65.73

σ 28.96 18.99 0.004 75.07

Cv 56.13 84.96 15.80 13.18

SSG02_05 µ 16.99 15.16 12.65 42.07

σ 68.98 71.46 53.69 314.19

Cv 48.89 55.77 57.90 42.14

SSG03_03 µ 15.56 10.44 4.70 35.52

σ 42.36 42.33 204.42 13338.63

Cv 41.83 62.34 304.30 325.15

SSG04_06 µ 16.06 14.19 19.10 53.63

σ 119.36 144.26 224.95 1104.83

Cv 68.03 84.64 78.54 61.98

MEAN µ 14.55 11.23 9.21 49.24

σ 64.92 69.26 120.77 3708.18

Cv 53.72 71.93 114.13 110.61


1) The degree of accuracy in predicting the responses ofthe shallow foundations are significantly dependent on theparameter selection; both soil and model parameters.2) The FOSM analysis reveals that the mean error in

response prediction is most sensitive to the friction angle,and least sensitive to the end length ratio.3) Uncertainty analysis using Latin hypercube sampling

technique shows that the mean error in estimating peakmoment, shear, rotation and settlement demands are 15%,11%,9% and 49%, respectively, indicating that BNWFmodel predicts the experimentally observed behaviorreasonably well.4) It has also been observed that the coefficients of

variation of 3%, 16%, 15% in friction angle, Poisson’sratio and shear modulus, respectively, result in significantvariation in demand parameters with Cv of 54%, 72%,114% and 111% for the absolute error in maximummoment, shear, rotation and settlement demands, respec-tively. This indicates that the small variation in soilproperties, specially friction angle, may result in largeuncertainty in the response prediction.Findings of this study are limited to the parameter space

considered herein (i.e., soil type of dense dry sand, lateralloading, strip footing) and may require further investiga-tion for generalization purpose.

References

1. Lacasse S, Nadim F. Uncertainties in characterizing soil properties.

In: Uncertainty in the Geologic Environment: From Theory to

Practic, Proceedings of Uncertainty 96. Madison, Wisconsin, July

31-August 3, 1996, New York, USA, ASCE Geotechnical Special

Publication.1996, 58: 49–75

2. Chakraborty S, Dey S S. Stochastic finite-element simulation of

random structure on uncertain foundation under random loading.

International Journal of Mechanical Sciences, 1996, 38(11): 1209–

1218

3. Foye K C, Salgado R, Scott B. Assessment of variable uncertainties

for reliability based design of foundations. Journal of Geotechnical

and Geoenviornmental Engineering, 2006, 132(9): 1197–1207

4. Jones A L, Kramer S L, Arduino P. Estimation of uncertainty in

geotechnical properties for performance-based earthquake engineer-

ing. Tech. Rep. 2002/16, Pacific Earthquake Engineering Research

Center, PEER, 2002

5. Lumb P. The variability of natural soils. Engineering Structures,

1966, 3: 74–97

6. Lutes L D, Sarkani S, Jin S. Response variability of an SSI system

with uncertain structural and soil properties. Canadian Geotechnical

Journal, 2000, 22(6): 605–620

7. Na U J, Chaudhuri S R, Shinozuka M. Probabilistic assessment for

seismic performance of port structures. Soil Dynamics and Earth-

quake Engineering, 2008, 28(2): 147–158

8. Chaudhuri S R, Gupta V K. Variability in seismic response of

secondary systems due to uncertain soil properties. Engineering

Structures, 2002, 24(12): 1601–1613

9. Raychowdhury P. Effect of soil parameter uncertainty on seismic

demand of low rise steel buildings on dense sand. Soil Dynamics

and Earthquake Engineering, 2009, 29(10): 1367–1378

10. Raychowdhury P, Hutchinson T C. Sensitivity of shallow founda-

tion response to model input parameters. Journal of Geotechnical

and Geoenvironmental Engineering, 2010, 136(3): 538-541

11. Ronold K O, Bjerager P. Model uncertainty representation in

geotechnical reliability analysis. Journal of Geotechnical Engineer-

ing, 1992, 118(3): 363–376

12. Boulanger R W, Curras C J, Kutter B L, Wilson D W, Abghari A.

Seismic soil-pile-structure interaction experiments and analyses.

Journal of Geotechnical and Geoenvironmental Engineering, 1999,

125(9): 750-759

13. Raychowdhury P, Hutchinson T C. Nonlinear material models for

Winkler-based shallow foundation response evaluation. In: Geo-

Congress 2008, Characterization, Monitoring, and Modeling of

GeoSystems, March 9–12, 2008, New Orleans, LA, ASCE

Geotechnical Special Publication No. 179, 686–693

14. Raychowdhury P, Hutchinson T C. Performance evaluation of a

nonlinear Winkler-based shallow foundation model using centrifuge

test results. Earthquake Engineering & Structural Dynamics, 2009,

38(5): 679–698

15. Terzaghi K. Theoretical Soil Mechanics. New York: J Wiley, 1943

16. Meyerhof G G. Some recent research on the bearing capacity of

foundations. Canadian Geotechnical Journal, 1963, 1(1): 16–26

17. Gazetas G. Foundation Engineering Handbook. Fang H Y ed. Van

Nostrand Rienhold, 1991

18. Gajan S, Phalen J, Kutter B. Soil-foundation structure interaction:

Shallow foundations, centrifuge data report for SSG02 test series.

Report No. UCD/CGMDR-03/01, Center for Geotechnical Model-

ing, Davis: University of California, 2003

19. Gajan S, Phalen J, Kutter B. Soil-foundation structure interaction:

Shallow foundations, centrifuge data report for SSG02 test series.

Report No. UCD/CGMDR-03/02, Center for Geotechnical Model-

ing, Davis: University of California, 2003

20. Gajan S. Physical and numerical modeling of nonlinear cyclic load-

deformation behavior of shallow foundations supporting rocking

shear walls. Dissertation for the Doctoral Degree. Davis: University

of California, 2006

21. EPRI. Manual on Estimating Soil Properties for Foundation Design.

Electric Power Research Institute, Palo Alto, California, 1990

22. Gajan S, Hutchinson T C, Kutter B, Raychowdhury P, Ugalde J A,

Stewart J P. Numerical models for the analysis and performance-

based design of shallow foundations subjected to seismic loading.

PEER Data Report, Pacific Earthquake Engineering Research

Center (PEER), 2008

23. Raychowdhury P. Nonlinear Winkler-based shallow foundation

model for performance assessment of seismically loaded structures.

Dissertation for the Doctoral Degree. San Diego: University of

California, 2008

24. ATC-40. Seismic Evaluation and Retrofit of Concrete Buildings.

Applied Technology Council (ATC), Redwood City, California,


1996

25. Harden C W, Hutchinson T C, Martin G R, Kutter B L. Numerical

modeling of the nonlinear cyclic response of shallow foundations.

Tech. Rep. 2005/04, Pacific Earthquake Engineering Research

Center, PEER, 2005

26. OpenSees. Open System for Earthquake Engineering Simulation:

OpenSees. Pacific Earthquake Engineering Research Center

(PEER), University of California, Berkeley, 2008 (http://opensees.

berkeley.edu)

27. McKay M D, Beckman R J, Conover W J. A comparison of three

methods for selecting values of input variables in the analysis of

output from a computer code. Technometrics, 1979, 21(2): 239–245

28. Iman R L, Conover W J. A distribution-free approach to including

rank correlation among input variables. Bulletin of the Seismic

Society of America, 1982, 11(3): 311–334

29. Stein M. Large sample properties of simulations using Latin

hypercube sampling. Technometrics, 1987, 29(2): 143–151

30. Jindal S. Shallow foundation response analysis: a parametric study.

Master’s thesis. Kanpur: Indian Institute of Technology Kanpur,

2011


shallow foundation response variability due to soil and model parameter uncertainty

Documents