Shallow foundation response variability due to soil and model parameter uncertainty

Abstract

Geotechnical uncertainties may play crucial role in response prediction of a structure with substantial soil-foundation-structure-interaction (SFSI) effects. Since the behavior of a soil-foundation system may significantly alter the response of the structure supported by it, and consequently several design decisions, it is extremely important to identify and characterize the relevant parameters. Moreover, the modeling approach and the parameters required for the modeling are also critically important for the response prediction. The present work intends to investigate the effect of soil and model parameter uncertainty on the response of shallow foundation-structure systems resting on dry dense sand. The SFSI is modeled using a beam-on-nonlinear-winkler-foundation (BNWF) concept, where soil beneath the foundation is assumed to be an assembly of discrete, nonlinear elements composed of springs, dashpots and gap elements. The sensitivity of both soil and model input parameters on shallow foundation responses are investigated using first-order second-moment (FOSM) analysis and Monte Carlo simulation through Latin hypercube sampling technique. It has been observed that the degree of accuracy in predicting the responses of the shallow foundation is highly sensitive soil parameters, such as friction angle, Poisson’s ratio and shear modulus, rather than model parameters, such as stiffness intensity 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

Introduction

Uncertainty refers to situations where the outcome of an event or the value of a parameter may differ from its true value. Uncertainty plays an important role in response variability of a soil-foundation systems. These uncertainties may arise from geotechnical and structural material properties, input loadings and modeling methods. For an analysis of a structure with significant soil-foundation-structure-interaction (SFSI) effects, geotechnical uncertainties may play crucial role in overall system response variability. Uncertainties in geotechnical engineering properties are largely induced from estimation of basic soil strength and stiffness parameters, knowledge of geology, judgments, and statistical reasoning. Most soils are naturally formed in many different depositional environments; thus showing variation in their physical properties from point to point. However, soil properties exhibit variations even within an apparently homogeneous soil profile. Basic soil parameters that control the strength and stiffness of the soil-foundation system are cohesion, friction angle, unit weight, shear modulus and Poisson’s ratio of soil. These soil parameters can be delineated using deterministic or probabilistic models. Deterministic models use a single discrete descriptor for the parameter of interest, whereas probabilistic models define parameters by using discrete statistical descriptor or probability density functions.

Uncertainty in soil properties can be formally grouped into aleatory and epistemic uncertaintyas described by Lacasse and Nadim [1]. Aleatory uncertainty represents the natural randomness of a property and is also a function of spatial variability of the soil property. This type of uncertainty is inherent to the variable and cannot be reduced or eliminated by additional information. Epistemic uncertainty results from the lack of information and shortcomings in measurements and calculations [1]. Epistemic uncertainty can usually be reduced by acquisition of more information or improvements in measuring methods.

In last few decades, significant research has been carried out in order to understand the behavior of a structure considering uncertainty in soil parameters [2-11]. In an early work, Lumb [5] showed that the soil parameters can be modeled as random variables confirming to the Gaussian distribution within the framework of probability theory. Ronold and Bjerager [11] observed that model uncertainties are important in reliability analysis for prediction of stresses, capacities, deformation etc. in structure and foundation systems. Chakraborty and Dey [2] studied the stochastic structural responses considering uncertainty in structural properties, soil properties and loadings using Monte Carlo simulation technique. Lutes et al. [6] evaluated the response of a seismically excited structural system with uncertain soil and structural properties. Ray Chaudhuri and Gupta [8] investigated the variability in seismic response of secondary systems due to uncertain soil properties through a mode acceleration method. Foye et al. [3] performed a thorough study for assessment of variable uncertainties by defining the probability density functions for uncertain design variables in load resistance factor design (LRFD). Na et al. [7] investigated the effect of uncertainties of geotechnical parameters on gravity type quay-wall in liquefiable condition using tornado diagram and First-Order Second-Moment (FOSM) analysis. Raychowdhury [9] studied the effect of soil parameter uncertainty on seismic demand of low-rise steel buildings supported by shallow foundations on dense silty sand with considering a set of 20 ground motions. Raychowdhury and Hutchinson [10] carried out the sensitivity analysis of shallow foundation response to uncertain input parameters using simplified FOSM and tornado diagram methods.

The present article focuses on studying the effect of uncertainty in soil and model input parameters on the response of a shearwall-foundation system. To incorporate the nonlinearity at the soil-foundation interface, a Beam-on-Nonlinear-Winkler-Foundation (BNWF) approach is adopted. To evaluate sensitivity of parameters on the predictive capability of the numerical model, a series of centrifuge experiments conducted on shallow strip footings at the University of California, Davis are utilized. The uncertainty analysis has been carried out using First-Order Second-Moment (FOSM) method and Monte Carlo simulation through Latin hypercube sampling technique.

Modeling of SFSI

In this paper, a Beam-on-Nonlinear-Winkler-Foundation (BNWF) approach is used to model the nonlinear soil-structure-interaction of shallow foundations subjected to lateral loads. The BNWF model includes a system of closely spaced independent, mechanistic elements consisting of inelastic springs, dashpots and gap elements (Fig. 1). The vertical springs (q-z elements) are intended to capture the axial and rotational behavior of the footing, whereas the lateral springs, t-x element and p-x element, are intended to capture the sliding and passive resistance, respectively. The material models were originally developed by Boulanger et al. [12] and modified by Raychowdhury and Hutchinson [13]. The BNWF model is capable of reasonably capturing the experimentally observed behavior for various shallow foundation conditions. For more details regarding the BNWF modeling, one can look into [13] and [14]. The material models have a nonlinear backbone curve with initial elastic portions followed by smooth nonlinear behavior (Fig. 2). In the elastic portion, the instantaneous load q is assumed to be linearly proportional with the instantaneous displacement z via the initial elastic (tangent) stiffness k_in, i.e.,

(1)q=k_{\mathrm{i}\mathrm{n}}z,

where k_in is the initial elastic (tangent) stiffness. The range of the elastic region is defined by the following relation:

(2)q_{\mathrm{o}}=C_{\mathrm{r}}{q}_{\mathrm{u}\mathrm{l}\mathrm{t}},

where q_o is the load at the yield point, C_r is a parameter controlling the range of the elastic portion, and q_ult is the ultimate load. In the nonlinear (post-yield) portion, the backbone curve is described by

(3)q=q_{ult}-(q_{ult}-q_o){\left[{\displaystyle \frac{c{z}_{\mathrm{50}}}{c{z}_{\mathrm{50}}+\left|z^{\mathrm{p}}-z_{\mathrm{o}}^{\mathrm{p}}\right|}}\right]}^n,

where z₅₀ is the displacement at which 50% of the ultimate load is mobilized, z_{\mathrm{o}}^{\mathrm{p}}is the displacement at the yield point, z^p is the displacement at any point in the post-yield region, and c and n are the constitutive parameters controlling the shape of the post-yield portion of the backbone curve. The expressions governing both p-x and t-x elements are similar to Eqs. (1) to (3), with variations in the constants n, c, and C_r, controlling the general shape of the curve. Furthermore, it may be noted that q-z material has a reduced capacity in the tension side (Fig. 2(a)), which allows the footing to uplift without losing contact with the soil beneath it during a rocking movement. On the other hand, the p-x material is characterized by a pinched hysteretic behavior (Fig. 2(b)), whereas the t-x material is characterized by a large initial stiffness and a broad hysteresis (Fig. 2(c)).

It is evident from Eqs. (1) to 3 that shape of spring backbone curves which is basically controlling factor for the soil-structure interaction behavior is mainly dependant on two physical parameters related to soil characteristics, namely, capacity (q_ult), and initial elastic stiffness (k_in). The capacity and elastic stiffness of each spring is obtained by distributing the global footing capacity and stiffness utilizing proper tributary area of each spring. The footing capacity is derived using general bearing capacity equation from Terzaghi [15] with shape, depth and inclination factors after Meyerhof [16] as shown in Eqs. (4) to (7).

(4)q_{ult}=c{N}_{\mathrm{c}}{F}_{\mathrm{c}}{D}_{\mathrm{c}}{I}_{\mathrm{c}}+\gamma D_{\mathrm{f}}{N}_{\mathrm{q}}{F}_{\mathrm{q}}{D}_{\mathrm{q}}{I}_{\mathrm{q}}+\mathrm{0.5}\gamma B{N}_{\mathrm{\gamma }}{F}_{\mathrm{\gamma }}{D}_{\mathrm{\gamma }}{I}_{\mathrm{\gamma }},

where q_ult is ultimate vertical bearing capacity of the footing, c is cohesion, γ is unit weight of soil, D_f is depth of embedment, B is width of footing, N_c, N_q, N_γ are bearing capacity factors, F_c, F_q, F_γ are shape factors, D_c, D_q, D_γ are depth factors and I_c, I_q, I_γ are inclination factors. Bearing capacity, shape, depth and inclination factors are calculated based on expressions given by Meyerhof [16]:

(5)N_{\mathrm{q}}=e^{\mathrm{\pi }\tan \varphi }{\tan }^{\mathrm{2}}\left(\mathrm{45}+{\displaystyle \frac{\varphi }{\mathrm{2}}}\right),

(6)N_{\mathrm{c}}=(N_{\mathrm{q}}-\mathrm{1})\cot \varphi ,

(7)N_{\gamma }=(N_{\mathrm{q}}-\mathrm{1})\tan (\mathrm{1.4}\gamma ),

For the p-x material, the ultimate lateral load capacity is determined as the total passive resisting force acting on the front side of the embedded footing. For homogeneous backfill against the footing, the passive resisting force can be calculated using a linearly varying pressure distribution resulting in the following equation:

(8)p_{\mathrm{u}\mathrm{l}\mathrm{t}}=\mathrm{0.5}\gamma D_{\mathrm{f}}^{\mathrm{2}}{K}_{\mathrm{p}},

where p_ult = passive earth pressure per unit length of footing, K_p = passive earth pressure coefficient. For the t-x material, the lateral load capacity is the total sliding (frictional) resistance which can be defined as the shear strength between the soil and the footing as:

(9)t_{\mathrm{u}\mathrm{l}\mathrm{t}}=W_{\mathrm{g}}\tan \delta +A_{\mathrm{b}}c,

where, t_ult = frictional resistance per unit area of foundation, W_g = vertical force acting at the base of the foundation, δ = angle of friction between the foundation and soil (typically varying from 1/3ϕ to 2/3ϕ) and A_b = the area 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:

(10)k_{\mathrm{v}}={\displaystyle \frac{GL}{\mathrm{1}-\upsilon }}\left[\mathrm{0.73}+\mathrm{1.54}{\left({\displaystyle \frac{B}{L}}\right)}^{\mathrm{0.75}}\right],

(11)k_{\mathrm{h}}={\displaystyle \frac{GL}{\mathrm{2}-\upsilon }}\left[\mathrm{2}+\mathrm{2.5}{\left({\displaystyle \frac{B}{L}}\right)}^{\mathrm{0.85}}\right],

where k_v and k_h are the vertical and lateral initial elastic stiffness of the footing, respectively; G is the shear modulus of soil; vis the Poisson’s ratio of soil; B and L are the footing width and length, respectively. The instantaneous tangent stiffness k_p, which describes the load-displacement relation within the post-yield or nonlinear region of the backbone curves, may be expressed as:

(12)k_{\mathrm{p}}=n(q_{\mathrm{u}\mathrm{l}\mathrm{t}}-q_{\mathrm{o}})\left[{\displaystyle \frac{{(c{z}_{\mathrm{50}})}^n}{{(c{z}_{\mathrm{50}}-z_o+z)}^{n+\mathrm{1}}}}\right].

Note that Eq. (12) is obtained by rearranging Eq. (3). Also, note that the shape and instantaneous tangent stiffness of the nonlinear portion of the backbone curve is a function of the parameters c and n, which are derived by calibrating against a set of shallow footing tests on different types of soil. Details of the calibration study is described in the PhD dissertation of the first author of this article. It is evident from the preceding equations that the spring responses are primarily dependent on basic strength and stiffness parameters of soil such as cohesion (c), friction angle (\varphi), unit weight (γ), shear modulus (G) and Poisson’s ratio (ν) of soil. Therefore, proper characterization of these parameters are important for prediction of responses of the soil-foundation-structure system.

In addition to the soil properties, parameters related to the numerical modeling may also have significant effect on the response prediction. The BNWF model considered in this study has few such modeling parameters that might be considered as important, namely, stiffness intensity ratio, end length ratio and spring spacing. A distributed stiffness intensity along the length of the foundation is provided in the BNWF model in order to capture the rocking stiffness of the footing when subjected to lateral or rotational movements. To achieve this, stiffness at the edges of the footing is increased compared to the middle portion (Fig. 3). The ratio of the stiffness of the end zone to the middle zone is defined as stiffness intensity ratio, while the length of the increased stiffness zone as a fraction of the total footing length is defined as end length ratio. Spring spacing is the spacing between two consecutive springs as a fraction of total footing length.

Selection of uncertain parameters

Based on the discussion provided in the previous section the following parameters shown in Table 1 are considered for the sensitivity analysis. Since the simulation results are compared with experiments conducted on shallow foundations resting on dry dense sand of relative density 80%, the chosen range of parameters corresponds to the properties of the same soil. The ranges of soil properties are chosen based on discussions in Refs. [21] and [22], while the values of model properties are chosen based on [23-25]. It is assumed that all uncertain input parameters are random variables with a Gaussian distribution, having no negative values. The upper and lower limits of the random variables are assumed to be in 95th and 5th percentile of its probability distribution. The corresponding mean (μ) and standard deviation (σ) can be calculated as:

(13)\mu ={\displaystyle \frac{(L_{\mathrm{L}}+L_{\mathrm{U}})}{\mathrm{2}}},

(14)\sigma ={\displaystyle \frac{(L_{\mathrm{L}}-L_{\mathrm{U}})}{\mathrm{2}k}},

where L_L and L_U 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%). The parameters are assumed to be correlated following the correlation matrix shown in Table 2. The correlations are considered based on discussions in [4,21], a number of BNWF simulations and engineering judgment.

Experiments considered

To study the effect of parameter uncertainty on the predictive capability of the soil-foundation numerical model, a set of four centrifuge experiments are chosen. These experiments include cyclic tests on shearwall structures supported by shallow strip footings resting on dense dry sand of relative density 80%. Details of the chosen experiments are provided in Table 3. It can be noted that the experiments include a range of mass, vertical factor of safety (FS_v), depth of embedment (D_f), and moment to shear aspect ratio (M/VL).

The shearwall-footing system as described in the experiments are modeled using finite element software OpenSees [26]. The soil-foundation interface is modeled using BNWF model as described in the earlier sections. A total number of 120 simulations have been conducted (4 tests × 6 parameters × 5 simulations). Each parameter is varied independently while keeping all other parameters fixed at their mean values. The simulation results are compared with the experiments and absolute errors in predicting peak responses are obtained for each test with varying parameters. The responses considered are: absolute maximum values of moment, shear, rotation, and settlement. Absolute error in predicting any demand is obtained using the following relationship:

(15)\left|\delta \right|(\%)=\left|{\displaystyle \frac{Deman{d}_{(\exp )}-Deman{d}_{(\mathrm{s}\mathrm{i}\mathrm{m})}}{Deman{d}_{(\exp )}}}\right|\times \mathrm{100.}

For a particular test, a total number of 30 simulations (5 simulation each for 6 input parameters) have been carried out. Let for ith experiment, the absolute error in predicting moment is δ_M (i; j; k), where j= 1, 2,..,6 and k= 1, 2,...,5 corresponds to each of the six parameters and five values of each parameters, respectively. The range of input parameters is defined in Table 1. The mean of absolute error in that response is obtained assuming equal weight to all experiments as shown below. The absolute error in shear, settlement and rotation demands are calculated by following the same procedure.

(16){\overline{\delta }}_{M(j,k)}={\displaystyle \frac{\mathrm{1}}{\mathrm{4}}}{\displaystyle {\sum }_{i=\mathrm{1}}^{\mathrm{4}}{\delta }_{M(i,j,k)}},

Uncertainty analysis

To understand the effect of variability of uncertain input parameters on footing-structure response, uncertainty analysis has been performed using simple First-Order Second Moment (FOSM) method and Monte Carlo Simulation through Latin hypercube sampling technique. Followed is a brief description of the adopted methods for the uncertainty analysis.

FOSM method

The FOSM method is a simplified probabilistic response analysis to evaluate the effect of variability of input parameters on a resulting response variable. This method is based on few assumptions: 1) all uncertain parameters are random variables with a Gaussian distribution, 2) the relationship between the response variables and the uncertain input parameters are assumed to be linear or low-to-moderately nonlinear. The FOSM method uses a Taylor series expansion of the function to be evaluated and expansion is truncated after the linear first order term. This method does not account the form of the probability density function describing random variables, uses only their mean and standard deviation. The response of the foundation is considered as a random variable Q, which has been expressed as a function of the input random variables, P_i (for i= 1,2,...,N)denoting uncertain parameters and Q given by,

(17)Q=h(P_{\mathrm{1}},P_{\mathrm{2}},\cdots ,P_N),

The random variable P_i has been characterized by its mean, μ_P and variance {\sigma }_h^{\mathrm{2}}. Now, Q can be expanded using a Taylor series as follows:

(18)Q=h({\mu }_{P_{\mathrm{1}}},{\mu }_{P_{\mathrm{2}}},\cdots {\mu }_{P_N})+{\displaystyle \frac{\mathrm{1}}{\mathrm{1}!}}{\displaystyle \sum _{i=\mathrm{1}}^N(P_i-{\mu }_{Pi})}{\displaystyle \frac{\partial h}{\partial P_i}}+{\displaystyle \frac{\mathrm{1}}{\mathrm{2}!}}{\displaystyle \sum _{j=\mathrm{1}}^N{\displaystyle \sum _{i=\mathrm{1}}^N(P_i-{\mu }_{Pi})}(P_j-{\mu }_{Pj}){\displaystyle \frac{{\partial }^{\mathrm{2}}h}{\partial P_i\partial P_j}}+\cdots },

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

(19)Q\approx h({\mu }_{P_{\mathrm{1}}},{\mu }_{P_{\mathrm{2}}},\cdots {\mu }_{P_N})+{\displaystyle \sum _{i=\mathrm{1}}^N(P_i-{\mu }_{Pi})}{\displaystyle \frac{\partial h}{\partial P_i}},

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

(20){\mu }_Q=h({\mu }_{P_{\mathrm{1}}},{\mu }_{P_{\mathrm{2}}},\cdots {\mu }_{P_N}).

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

(21)$$\begin{gathered} {\sigma }_Q^{\mathrm{2}}\approx {\displaystyle \sum _{i=\mathrm{1}}^N{\displaystyle \sum _{j=\mathrm{1}}^N\mathrm{cov}ariance(P_i,P_j)}{\displaystyle \frac{\partial h(P_{\mathrm{1}},P_{\mathrm{2}},\cdots ,P_N)}{\partial P_i}}}{\displaystyle \frac{\partial h(P_{\mathrm{1}},P_{\mathrm{2}},\cdots ,P_N)}{\partial P_j}} \\ \approx {{\displaystyle \sum _{i=\mathrm{1}}^N{\sigma }_{P_t}^{\mathrm{2}}\left({\displaystyle \frac{\partial h(P_{\mathrm{1}},P_{\mathrm{2}},\cdots ,P_N)}{\partial P_i}}\right)}}^{\mathrm{2}}+{\displaystyle \sum _{i=\mathrm{1}}^N{\displaystyle \sum _{j\ne \mathrm{1}}^N{\rho }_{P_i,P_j}}{\displaystyle \frac{\partial h(P_{\mathrm{1}},P_{\mathrm{2}},\cdots ,P_N)}{\partial P_i}}}{\displaystyle \frac{\partial h(P_{\mathrm{1}},P_{\mathrm{2}},\cdots ,P_N)}{\partial P_j}}, \end{gathered}$$

where {\rho }_{P_i,P_j}denotes correlation coefficient for random variables P_i and P_j. The partial derivative of h (P₁, P₂,...,P_N) with respect to P_i has been calculated numerically using the finite difference method (central) as follows:

(22){\displaystyle \frac{\partial h(P_{\mathrm{1}},P_{\mathrm{2}},\cdots ,P_N)}{\partial P_i}}={\displaystyle \frac{h(p_{\mathrm{1}},p_{\mathrm{2}},\cdots ,{\mu }_i+\mathrm{\Delta }{p}_i,p_N)-h(p_{\mathrm{1}},p_{\mathrm{2}},\cdots ,{\mu }_i-\mathrm{\Delta }{p}_i,p_N)}{\mathrm{2}\mathrm{\Delta }{p}_i}}.

FOSM analysis is carried out to investigate the sensitivity of the uncertain soil and model parameters on the response of the foundation. It provides an approximate sense of sensitivity of the parameters. The parameters discussed previously have been varied one by one while keeping all others constant to study the isolated effect of each one. Number of simulations has been performed varying each input parameter individually to approximate the partial derivatives as given in Eq. (21) considering the correlation between uncertain input parameters as defined in Table 2.

Latin hypercube method

For probabilistic analysis of engineering structures having uncertain input variables, Monte Carlo simulation (MCS) technique is considered as a reliable and accurate method. However, this method requires a large number of equally likely random realizations and consequent computational effort. To decrease the number of realizations required to provide reliable results in MCS, Latin hypercube sampling (LHS) approach [27] is widely used in uncertainty analysis. LHS is a type of stratified MCS which provides a very efficient way of sampling variables from their multivariate distributions for estimating mean and standard deviations of response variables [28]. It follows a general idea of a Latin square, in which, there is only one sample in each row and each column. In Latin hypercube sampling to generate a sample of size K from N variables, the probability distribution of each variable is divided into segments with equal probability. The samples are then chosen randomly in such a way, that each interval contains one sample. During the iteration process, the value of each parameter is combined with the other parameter in such a way, that all possible combinations of the segments are sampled. 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 are generated using Stein’s approach [29]. This method is based on the rank correlations among the input variables defined by [28] which follows Cholesky decomposition of the covariance matrix. The previously mentioned simple shearwall structure (Fig. 1) is considered for the LHS analysis, where the responses of the system are considered to be dependent on the six independent normally distributed variables described in Table 1. To find out the correct sample size, a static pushover analysis is carried out using a number of 10, 20,...,100, 200 and 300 samples. Figure 4 shows the responses corresponding to each sample size normalized by the value corresponding to a sample size of 300. It has been observed that the responses, as expected, tend to converge as the sample size increases. However, the responses converges at a sample size of 100, indicating that a sample size of at least 100 is required for a reasonably accurate estimation. Therefore, a sample size of 100 has been used in this study for an analysis with reduced computational effort yet reasonable accuracy.

Results and discussion

The effect of uncertainty in soil and modeling parameters are investigated using a number of centrifuge experiments of shallow foundation-supported shearwalls as described in Table 3. First, comparison between the simulation and the experimental behavior is studied with varying parameters within the chosen ranges as mentioned in Table 1. Figure 5 shows the comparison results for the test SSG02_05 for two extreme values of friction angle, {\varphi }^{\prime }.

The results include moment-rotation, settlement-rotation and shear-rotation behavior with the BNWF simulation shown in black and experimental results shown in gray scale. These comparisons indicate that the BNWF model is able to capture the hysteretic features such as shape of the loop, peaks, unloading and reloading stiffness reasonably well. It can also be observed from Fig. 5 that with increasing the friction angle from 38° to 42°, peak moment and peak shear demands increase, whereas peak settlement demand decreases. It can also be noted that the variation of friction angle from 38° to 42° has most significant effect on settlement prediction (more than 100%). However, the moment, shear and rotation demands are moderately affected by this parameter. This indicates that the uncertainty in one parameter may have significantly different influence on the prediction of different responses, pointing out toward the importance of proper characterization of each parameters and conducting the sensitivity analysis.

For more comparison results for each experiment and each parameter, one can look into [30]. A summary table is provided (Table 4) for mean peak demands considering all experiments for varying all parameters. Note that this responses are obtained by varying each parameter at a time, while keeping other parameters fixed at their respective mean values.

To investigate the effect of parameter uncertainty on the predictive capability of the model, the amount of error in predicting different force and displacement responses of the footing are calculated. The deviations from the experimental responses to that obtained from simulations are presented in the Table 5. The absolute errors in predicting peak moment, peak shear, peak rotation and peak settlement (|δ_M|, |δ_V|,|δ_θ|, and |δ_S|, respectively) are considered. These are calculated as per Eq. (4) for each experiment and the mean of these are shown in Table 5. The variation of each response parameter with respect to each input parameter are presented in Fig. 6. The error in predicting each response is normalized by the same corresponding 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 response estimation 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 settlement prediction from 27% to 50%. Similar observations are made from varying other parameters, indicating that each parameter 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 few exceptions such as shear modulus versus shear demand, stiffness intensity ratio versus moment and shear demand. This indicates that the assumption of linear relationship between input parameters and response variables are generally satisfied for this system, and therefore, the FOSM method can be applicable herein. Figure 7 shows the results of the FOSM analysis carried out to evaluate the relative importance of each input parameter on the predictive capability of the model in obtaining four important response parameters: peak moment, peak shear, peak rotation and peak settlement. It can be observed from Fig. 7(a) that the error in predicting moment demand is 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 predicting peak shear demand. The findings of the shear demand are quite similar with that of moment demand. Figure 7(c) and (d) show the sensitivity for errors in predicting rotation and settlement demands, respectively. It can observed from these figures, that friction angle has a sensitivity of 51% and 84% on the rotation and settlement prediction, respectively. End length ratio, R_eis found to have moderate effect on the rotational demand (33%). This makes sense as the rotational stiffness is controlled by the change in stiffness distribution along the length of the footing. These findings indicate that friction angle is the most important parameter for most of the responses, and characterization of this parameter should be given utmost importance.

Table 6 provides the results of the LHS analysis described in the previous section. Simulations are performed for four tests as mentioned in Table 3 by using the Latin hypercube sample of size 100. Response of the each simulation is recorded for each test in the term of four absolute maximum demands: moment, shear, rotation, and settlement. The responses from simulation are compared with the experimental response values for each test and the absolute errors are obtained. The mean and coefficient of variation of the absolute errors in prediction are evaluated then. It is observed from Table 6 that the uncertainties in friction angle, Poisson’s ratio, shear modulus, end length ratio, stiffness intensity ratio and spring spacing with coefficients of variation (C_v) of 3%, 16%, 15%, 54%, 48% and 30%, respectively, result in significant variation in response prediction with C_v as 54%, 72%, 114% and 111% for the absolute error in maximum moment, shear, rotation and settlement demands, respectively. This indicates that 1) response variations (in terms of C_v) are greater than the input parameter variations indicating significant effect of parameter uncertainty on each response; 2) friction angle has largest effect on the responses as it had lowest C_v among all input parameters; and 3) Other two soil parameters, Poisson’s ratio and shear modulus have moderate effect, and the model parameters have least effect.

Note that the findings of this analysis is in accordance with the findings of FOSM analysis. It can also be observed from Table 6 that the mean deviation in estimating peak moment, shear, rotation and settlement demands are 15%, 11%, 9% and 49%, respectively using LHS analysis. indicating that except for settlement demand, BNWF model has reasonably good predictive capability.

Conclusions

The behavior of a shallow foundation under significant loadings can vary significantly due to the uncertainty in soil and modeling input parameters. Accurate modeling of geotechnical components of a soil-foundation system is required to predict the response of shallow foundations undergoing significant loading such as earthquake motions. The modeling approach and the model parameters are also important for the response prediction, as any variation in these parameters may significantly alter the system performance and consequent the design decisions. Therefore, the proper characterization of the relevant parameters is of utmost importance. Moreover, it is important to identify the sources and extent of uncertainty of the parameters, along with evaluating their effect on the response prediction of soil-foundation systems. In this work, a Beam-On-Nonlinear-Winkler-Foundation model is used to model the soil-foundation-structure interface. The effect of parameter uncertainty on the response of shallow foundation supported structural systems in dry dense sand is evaluated through First-Order-Second-Moment method and Latin hypercube sampling technique. The following key observations are made from the present research work:

1) The degree of accuracy in predicting the responses of the shallow foundations are significantly dependent on the parameter 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 peak moment, shear, rotation and settlement demands are 15%, 11%,9% and 49%, respectively, indicating that BNWF model predicts the experimentally observed behavior reasonably well.

4) It has also been observed that the coefficients of variation of 3%, 16%, 15% in friction angle, Poisson’s ratio and shear modulus, respectively, result in significant variation in demand parameters with C_v of 54%, 72%, 114% and 111% for the absolute error in maximum moment, shear, rotation and settlement demands, respectively. This indicates that the small variation in soil properties, specially friction angle, may result in large uncertainty in the response prediction.

Findings of this study are limited to the parameter space considered herein (i.e., soil type of dense dry sand, lateral loading, strip footing) and may require further investigation for generalization purpose.

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