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The ultimate bearing capacity of shallow foundation supported by unsaturated soil depends on the degree of saturation of the soil within the influence zone because the strength and deformation parameters of soil are affected by the degree of saturation. As the degree of saturation varies with rainfall, surface runoff, evapotranspiration and other climatic and geotechnical parameters, these parameters must be systematically incorporated for accurately computing the ultimate bearing capacity. In this study, a framework is proposed to compute the ultimate bearing capacity of a shallow footing in unsaturated soil considering site specific rainfall and water table depth distributions. The randomness in rainfall and water table depth is systematically considered using Monte Carlo method. The infiltration of water through the unsaturated zone is modelled using Richards equation considering infiltration and water table location as the top and bottom boundary conditions, respectively. The results show that the bearing capacity calculated using the proposed method is approximately 2.7 times higher than that calculated using the deterministic approach with fully saturated soil parameters.

Shallow foundations are typically considered as the simplest and most economical foundation for supporting small to medium size structures. They transfer the structural loads to the near surface soil that is mostly unsaturated and fluctuates with climatic condition. Recent studies show that the strength and deformation parameters of soil are influenced by the degree of saturation of the soil. Since the degree of saturation of the near surface soil varies with climatic and geotechnical parameters such as rainfall, water table depth, evapotranspiration, hydraulic conductivity, there can be variation in the strength and deformation parameters of the near surface soil. The two of the design considerations of shallow foundations are the safety against the overall shear failure in the soil and the settlement. This paper focuses on systematically incorporating the site specific climatic and geotechnical parameters in computing ultimate bearing capacity of soil.

The ultimate bearing capacity of a continuous shallow foundation is typically calculated using Terzaghi’s [

q u = c ′ N c + q N q + 1 2 γ B N γ (1)

where c ′ is the cohesion of soil, γ is the unit weight of the soil, q is the effective overburden pressure given by q = γ D f , D f is the depth from soil surface to bottom of footing, B is the width of footing, and N c , N q , and N γ are the dimensionless bearing capacity factors and are functions of the soil friction angle ϕ ′ .

The application of Terzaghi’s equation to compute ultimate bearing capacity is limited because it is applicable to shallow footings (D_{f} ≤ B) and subjected to concentric vertical loads. In 1963, Meyerhof [

q u = c ′ N c ( F c s F c d F c i ) + q N q ( F q s F q d F q i ) + 1 2 B γ N γ ( F γ s F γ d F γ i ) (2)

where F s , F d , and F i are shape, depth, and load inclination factors, respectively. Both Terzaghi’s and Meyerhof’s equations were derived for the failure mechanism and resistance along the failure surface based on saturated soil mechanics principles. However, recent studies show that the shear strength and volume change characteristics of soils vary with its degree of saturation [

Unsaturated soil is a three phase medium. It consists of three bulk phases (solid, water, and air) and three interfaces (solid-water, water-air, and air-solid). Among the three interfaces, the air-water interface plays a critical role in the mechanical behavior of unsaturated soils [

The shear strength parameters for a soil with matric suction are defined by Fredlund [

q u = [ c ′ + ( u a − u w ) tan ϕ b ] N c + q N q + 1 2 γ B N γ (3)

where ( u a − u w ) is the matric suction.

Because of the difficulties in determining ϕ b for use in Equation (3), researchers have proposed empirical and/or semi-empirical equations for computing ultimate bearing capacity of shallow foundations [

q u = [ c ′ + ( u a − u w ) b ( tan ϕ ′ − S ψ tan ϕ ′ ) + ( u a − u w ) a v e S ψ tan ϕ ′ ] N c F c s F c d + γ D f N q F q s F q d + 0.5 γ B N γ F γ s F γ d (4)

where ( u a − u w ) b is the air entry value from SWCC, ( u a − u w ) a v e is the average air-entry value, ϕ ′ is the effective friction angle, S is the degree of saturation, and ψ is the bearing capacity fitting parameter given by Equation (5).

ψ = 1.0 + 0.34 ( I p ) − 0.0031 ( I p ) 2 (5)

where I_{p} is the plasticity index. The average suction in the above bearing capacity equation is given by Equation (6).

( u a − u w ) a v e = 1 2 [ ( u a − u w ) 1 + ( u a − u w ) 2 ] (6)

where ( u a − u w ) 1 is the matric suction at the bottom of the footing and ( u a − u w ) 2 is the matric suction at a depth equal to 1.5 times the width of the footing (1.5B).

The bearing capacity equation (Equation (4)) for unsaturated soil requires computing the matric suction profile within the influence zone of a shallow foundation. Since the matric suction is related to the degree of saturation through SWCC, one may easily compute matric suction from degree of saturation using SWCC. The degree of saturation in the soil can also be measured by soil sampling and dielectric sensor. Low resolution satellites such as NOAA-AVHRR and TERRA-MODIS can provide daily evapotranspiration fluxes in a clear sky at the 1 km scale [

This paper describes a probabilistic approach considering the randomness in the rainfall and water table depth using Monte Carlo method for computing design matric suction and degree of saturation within the influence zone of shallow foundation. Then, the design matric suction and degree of saturation are then used for computing the ultimate bearing capacity using Equation (4). In this study, the soil is assumed to be homogeneous with vertical 1-D downward flow of water. Also, this study evaluates the advantage of coupled hydrological-geotechnical approach by comparing the ultimate bearing capacities computed with conventional and proposed methods. A parametric study is also performed to investigate the sensitivity of the key parameters that influence the bearing capacity of the soil.

The first step in performing a Monte Carlo simulation is defining the problem in terms of random variables and representing them with suitable probabilistic distributions. The variables in the bearing capacity equation arise from Equation (4). In this study, the base shear strength parameters of the soil are considered as constants. It should be noted that a comprehensive study may consider the shear strength parameters also as random variables. The inherent randomness of these variables can be incorporated into the Monte Carlo simulation technique but to allow for comparisons between the saturated and unsaturated soil bearing capacities the base shear strength parameters were kept as constants. The remaining random variable, i.e. the matric suction and degree of saturation must be defined in terms of its probability density function (PDF) to allow for the bearing capacity to be solved for through Monte Carlo simulations.

The spatial and temporal variations of matric suction are not something that has been recorded over long periods of time as a fundamental climatic data. It must be computed using the site specific recorded rainfall, evapotranspiration, surface runoff, water table location, and other climatic and geotechnical parameters as inputs for a mathematical model that represents the infiltration of water through initially unsaturated soil. Rainfall has been recorded in detail around the United States and many other countries. In the United States, the National Climatic Data Center (NCDC) records daily rainfall values and many sites have data for more than 60 years [

The water table depth is the bottom boundary condition which affects the matric suction and/or degree of saturation of soil in the influence zone of shallow foundation. The U.S. Geological Survey (USGS) and National Water Information System, provides data about the water table depths over a long period of time at various locations [

With rainfall and water table depth distributions, the spatial and temporal variations of matric suction and/or degree of saturation can be computed. The method used to calculate the matric suction is described in the next section. Besides the matric suction, the unit weight of the soil is another parameter that varies with changing degree of saturation of the soil. By modeling the flow of water into the soil considering infiltration and the water table depth as the top and bottom boundary conditions, the variation in unit weight of the soil can be weight values within the influence zone of the foundation, the Equation (4) can be solved for the number of Monte Carlo Simulations selected for the study.

Through numerous studies with different numbers of simulations, the location and shape parameters are checked for convergence. The converged data from the Monte Carlo simulation accurately represents the bearing capacity for the foundation. From the data collected in the Monte Carlo simulation, a cumulative distribution function (CDF) of the bearing capacity can be used to make risk assessments.

Design risk can be quantified by the product of probability of failure and the consequence of failure. But calculating the consequences of failure is difficult for practicing engineers, thus current practice in reliability based foundation design accounts for the consequence of failure indirectly by prescribing different target probabilities of failure [^{−4} was selected to enable the bearing capacity to be read off the CDF. This is equivalent to a β = 3.7, which exceed the Canadian Building Code and AASHTO target values.

Matric suction is directly related to the hydraulic head (h_{w}) of the soil:

( u a − u w ) = 0 − ( h w − y ) ρ w g (7)

where u_{a} is the atmospheric pressure, y is the gravitational head, and ρ_{w} is the density of water. The flow behavior of water in unsaturated soil is complex compared to the saturated soil because of the variation in hydraulic head with time and depth. The variation in hydraulic head with time and depth due to an infiltration event with the ground water table set at the datum can be solved using Richard’s equation in unsaturated soils.

∂ θ d h d h d t = ∂ ∂ z [ k ( h ) ( d h d z + 1 ) ] (8)

where d h is the water capacity function.

The parameters in Richards equation can be solved for using the equations developed by van Genuchten [

K ( h ) = { 1 − ( α h ) n − 1 [ 1 + ( α h ) n ] − m } [ [ 1 + ( α h ) n ] − m ] m / 2 (9)

d θ d h = − α m ( θ s − θ r ) 1 − m Θ 1 / m ( 1 − Θ 1 / m ) m (10)

where θ r is the residual water content, θ s is the saturated water content, α is the approximation of the inverse of the pressure head at which the retention curve becomes the steepest, Θ is the dimensionless water content, and n and m are model constants (typically m = 1 − 1 / n ). All of these parameters are based on the soil type and are fitting parameters for an empirically determined soil water retention curve (SWRC).

After calculating these parameters, Richards equation can be solved numerically by the finite difference method. The Crank-Nicolson scheme implemented in Bolster and Raffensperger’s (1996) Matlab [

The average air entry value, ( u a − u w ) B , is the other type of suction that needs to be solved. It is inversely proportional to the van Genuchten soil parameter α as given by the following equation.

( u a − u w ) B = [ 1 α ] γ w (11)

where γ_{w} is the unit weight of water.

Since the unit weight is affected by the degree of saturation (S), the variation of unit weight is determined based on the average S in the influence zone of the foundation. In this study, the van Genuchten [

θ = θ r + ( θ s − θ r ) [ 1 + ( α h ) n ] m (12)

Using the same method used for calculating the ( u a − u w ) a v e , the average moisture in the soil within the influence depth of the footing (1.5B) can be calculated by:

θ = [ ( θ ) 1 + ( θ ) 2 2 ] (13)

Then, the average degree of saturation (S) in the influence area can be calculated by:

S = [ θ θ s ] (14)

Finally, the variation in the unsaturated unit weight of the soil can be calculated using the average degree of saturation in the influence zone of the shallow foundation using the basic weight-volume relationship given by:

γ = ( G s + S e ) γ w 1 + e (15)

where e is the void ratio, G_{s} is the specific gravity of the soil solids, and w is the unit weight of water.

A square 1.07 m × 1.07 m square footing located 0.61 m below the ground surface in Victorville, California was considered to demonstrate the proposed framework for computing ultimate bearing capacity considering site specific rainfall, water table and geotechnical data. This investigation was extended to determine if the depth and size of the footing on the computed ultimate bearing capacity. In addition, the influence of matric suction and degree of saturation related term ( [ c ′ + ( u a − u w ) b ( tan φ ′ − S ψ tan φ ′ ) + ( u a − u w ) A V R S ψ tan φ ′ ] N c F c s F c d ) and the depth factor shown in Equation (16) in the ultimate bearing capacity.

F c d = 1 + 0.4 ( D f B ) (16)

To determine if the size and depth have significant influence on the bearing capacity, three footings were considered and the computed results were compared. For the first example, the footing width (B) was increased from 1.07 to 1.52 m. Another example kept the D f / B ratio equal to the initial footing size and depth, thus B = 1.52 m and D_{f} = 0.87 m. The last example allowed B to remain equal to 1.07 m and increase the D_{f} to 0.87 m.

To determine how the return period of the hydrological parameters affects the bearing capacity of the footing, the square 1.07 m located 0.61 m below the ground surface was analyzed by considering return periods between: 1 yr and 2 yrs, 1 yr and 5 yrs, 1 yr and 10 yrs, 1 yr and 50 yrs, 1 yr and 100 yrs, 1 yr and 200 yrs and 1 yr and an infinite number of years for the rainfall and water table depth.

The Victorville, CA location was selected in this study due to its arid climate and the availability of van Genuchten soil water retention curve parameters for the Adelanto Loam located in this region. The van Genuchten parameters for the soil water retention curve of Adelanto Loam were taken from Zhang [_{s} = 0.423, θ_{r} = 0.158, α = 0.00321 cm^{−1}, n = 1.26 and K_{s} = 0.003492 cm/min.

The soil strength parameters were taken from a geotechnical engineering report by Kleinfelder [^{3}. The angle of internal friction for the soil at a depth of 1.52 m is 33 degrees. The cohesion at a depth of 1.52 m is 0. The USCS soil type for the soil at 1.52 m is SM.

The rainfall data was taken from the Victorville Pump Station, Victorville, CA, within the climate division CA-07. The station was in service from November 1, 1938 to the present. The elevation of the station is 871 m (2858 ft) above mean sea level. The latitude and longitude of the station are 34˚32'00''N and 117˚17'34''W, respectively. The data for the pump station was processed from an ASCII file that was downloaded from the National Climatic Data Center [^{2} values. The probability plot of rainfall data for the Gumbel distribution is shown in

Using the Gumbel distribution the CDF was transformed into a linear equation shown below, it can be determined that the location parameter, μ n = 0.8472 and the shape parameter β n = 0.5011 .

− ln ( − ln ( i n + 1 ) ) β n + μ n = x i (17)

where x_{i} is the annual maximum rainfall data and n is the number of data points.

The water table data at Victorville, CA was taken from the U.S. Geological Survey National Water Information System: Web Interface [

The convergence of the mean and the coefficient of variation for the bearing capacity distributions with the number of simulations are plotted in

The CDFs created for each of the example footings using 10,000 simulations are plotted in ^{−4} is tabulated in

the bearing capacity calculated with unsaturated soil mechanics principles. The percent increases in bearing capacity from conventional method to the proposed method are also listed in

From the results in

Case | L (m) | B (m) | D_{f} (m) | Bearing Capacity Using Deterministic Method Assuming Fully Saturated Condition (KN/m^{2}) | Bearing Capacity Using Monte Carlo Simulation Assuming Unsaturated Condition (KN/m^{2}) | Increase in Bearing Capacity (%) |
---|---|---|---|---|---|---|

1 | 1.07 | 1.07 | 0.61 | 419 | 1548 | 269 |

2 | 1.07 | 1.07 | 0.87 | 574 | 2013 | 251 |

3 | 1.52 | 1.52 | 0.61 | 455 | 1673 | 268 |

4 | 1.52 | 1.52 | 0.87 | 598 | 1992 | 233 |

foundation. The footing with the 1.07 m width and the depth of 0.87 m (case 2) showed the highest bearing capacity increase. This is a larger bearing capacity than the larger footing at the same depth. This shows that a smaller depth factor has more influence than a larger footing. It is clearly evident that the bearing capacity of the soil is significantly affected by the matric suction and the variation of unit weight. All the bearing capacities for the different example footings have increased by over 233% when using the proposed method based on unsaturated soil mechanics principles.

The effect of the return period (RP) on the bearing capacity for the case 1 footing is plotted in ^{−4}.

Sensitivity analyses were conducted to determine the effect of variations in input parameters used in the calculations and also served as additional examples to

explain the procedure. The first sensitivity analysis was to determine if another site with increased rainfall events would significantly decrease the increased bearing capacity determined from the sample application. The Levelland, Texas was selected as the second site in this study. The required data for the Levelland site was collected following the procedure for the Victorville site. The soil strength parameters were obtained from a geotechnical report made by Amarillo Testing and Engineering, Inc [

The van Genuchten values determined through the lowest of the hierarchical sequences in the program Rosetta [

Through this new method of collecting van Genuchten parameters, a second sensitivity analysis was performed. From the sample application it is impossible to determine if the ultimate bearing capacity of footing is controlled by the rainfall and water table distributions or the van Genuchten. In order to compare the effect of van Genuchten parameters, the sample application results which use van Genuchten parameters specifically for the Adelanto Loam are compared to the van Genuchten parameters recorded in the textural class determined from the soil classification in the geotechnical report. The soil in the Victorville site was considered as sandy loam. The van Genuchten parameters for Victorville are also listed in

Parameters | Victorville, CA | Levelland, TX | ||
---|---|---|---|---|

Mean | +1 Standard deviation | Mean | +1 Standard deviation | |

Saturated volumetric water content, θ_{s} | 0.387 | 0.472 | 0.385 | 0.431 |

Irreducible volumetric water content, θ_{r} | 0.039 | 0.093 | 0.117 | 0.231 |

Model parameter, α (m^{−1})^{ } | 0.026 | 0.007 | 0.033 | 0.008 |

Model parameter, n | 1.448 | 1.124 | 1.207 | 1.376 |

Hydraulic conductivity, k_{s} (cm/hr) | 0.065 | - | 0.043 | - |

Dry unit weight, γ (kN/m^{3}) | 10.081 | - | 11.549 | - |

Void ratio, e | 0.605 | - | 0.401 | - |

Friction angle, φ (deg.) | 33 | - | 25 | - |

Cohesion, c | 0 | - | 0 | - |

site are referred to as Victorville (Adelanto Loam) while the results from the van Genuchten parameters from the Rosetta [

A third sensitivity analysis was also performed by testing each van Genuchten parameter determined by the textural class individually. Each parameter was increased by one standard deviation to determine which parameter plays the largest role in affecting bearing capacity of shallow footings. Each parameter’s one standard deviation increase for both the Victorville (Sandy Loam) and the Levelland is summarized in

The final sensitivity analysis observed the influence of the depth factor in the cohesion term on the bearing capacity. Once again the footing was tested with the four different variations of footing size and depth described in the sample application for both the mean values of Levelland and Victorville (Sandy Loam).

The results from the first sensitivity analysis, a change in location are recorded in

The sensitivity analysis of the van Genuchten parameters was tested by comparing the bearing capacities of the Victorville (Adelanto Loam) and the Victorville (Sandy Loam) sites. The bearing capacity for the case 1 footing for the Victorville

Site | Parameter One Standard Deviation Greater for Sensitivity Analysis | L (m) | B (m) | D_{f} (m) | Bearing Capacity of Soil Using Deterministic Methods and Assuming Fully Saturated (KN/m^{2}) | Bearing Capacity of Soil Using Monte Carlo Simulation (KN/m^{2}) | Percent Increase in Bearing Capacity |
---|---|---|---|---|---|---|---|

Victorville* | - | 1.07 | 1.07 | 0.61 | 419 | 872 | 108 |

Victorville* | - | 1.07 | 1.07 | 0.87 | 574 | 1150 | 100 |

Victorville* | - | 1.52 | 1.52 | 0.61 | 455 | 926 | 104 |

Victorville* | - | 1.52 | 1.52 | 0.87 | 598 | 1178 | 97 |

Levelland | - | 1.07 | 1.07 | 0.61 | 170 | 300 | 76 |

Levelland | - | 1.07 | 1.07 | 0.87 | 237 | 418 | 76 |

Levelland | - | 1.52 | 1.52 | 0.61 | 181 | 320 | 77 |

Levelland | - | 1.52 | 1.52 | 0.87 | 243 | 427 | 76 |

Victorville* | θ_{r} | 1.07 | 1.07 | 0.61 | 419 | 869 | 107 |

Victorville* | θ_{s} | 1.07 | 1.07 | 0.61 | 419 | 879 | 110 |

Victorville* | α | 1.07 | 1.07 | 0.61 | 419 | 1415 | 238 |

Victorville* | n | 1.07 | 1.07 | 0.61 | 419 | 873 | 108 |

Levelland | θ_{r} | 1.07 | 1.07 | 0.61 | 170 | 254 | 49 |

Levelland | θ_{s} | 1.07 | 1.07 | 0.61 | 170 | 305 | 79 |

Levelland | α | 1.07 | 1.07 | 0.61 | 170 | 379 | 123 |

Levelland | n | 1.07 | 1.07 | 0.61 | 170 | 321 | 89 |

*Victorville (Sandy Loam).

(Adelanto Loam) is 1548 KN/m^{2}. The bearing capacity for the same footing for the Victorville (Sandy Loam) is 872 KN/m^{2}. All the footing sizes and depth have similar changes (refer to

Based on the individual sensitivity analysis of each parameter, the parameter alpha in Equation (9) resulted in the highest change in bearing capacity while increasing it by one standard deviation (refer to

The influence of the depth factor was studied by comparing the percent increase in the ultimate bearing capacity for two footings of different sizes at a specified depth. The difference between the percent increase for the deterministic method with fully saturated soil and the Monte Carlo simulation with unsaturated soil was calculated. Since the depth is the same, the effects of matric suction are constant. Considering this, the percent increase in the ultimate bearing capacity for both cases should only be due to the change in size of the footing. Thus the percent difference between the two methods for increased bearing capacity should be the same; this is not the case in

Site | Footing 1 (m) | Footing 2 (m) | Depth (m) | Bearing Capacity Based on Deterministic Method Assuming Fully Saturated Condition (KN/m^{2}) | Bearing Capacity Based on Monte Carlo Method Assuming Unsaturated Condition (KN/m^{2}) | Difference between Two Methods (%) |
---|---|---|---|---|---|---|

Victorville (Sandy Loam) | 1.07 | 1.52 | 0.61 | 8.705 | 6.192 | 2.51 |

Victorville (SandyLoam) | 1.07 | 1.52 | 0.87 | 4.138 | 2.435 | 1.70 |

Levelland | 1.07 | 1.52 | 0.61 | 6.695 | 6.666 | 0.03 |

Levelland | 1.07 | 1.52 | 0.87 | 2.223 | 2.153 | 0.07 |

Victorville (Adelanto Loam) | 1.07 | 1.52 | 0.61 | 8.705 | 8.074 | 0.63 |

Victorville (Adelanto Loam) | 1.07 | 1.52 | 0.87 | 4.138 | −1.043 | 5.18 |

capacity, the influence of the depth factor increases. The difference in percent increase in bearing capacity calculated using the two methods while increasing the footing size reduces by 1.70% for the depth of 0.87 m in Victorville (Sandy Loam). For sites where suction has more influence such as Victorville (Adelanto Loam), the difference in the percent increase in bearing capacity due to an increase in footing size reduces by 5.18% for a footing at a depth of 0.87 m when comparing the two methods. The negative percent increase is due to the depth factor having more control than the increased footing size in Victorville (Adelanto Loam), which is explained in the results section of the sample application section. From these results it is evident that the depth factor has influence on the bearing capacity calculated from the bearing capacity equation.

The method for determining the bearing capacity of a footing in unsaturated soil using Monte Carlo simulation is a useful tool for further understanding how unsaturated soil mechanics can be applied in problems faced by practicing engineers. The sample study gave evidence that considering unsaturated soils in design can increase the bearing capacity of a footing by at least 2.3 times the bearing capacity calculated using Meyerhof’s equation. This study considered homogeneous soil with 1-D flow. However, the proposed method can be easily extended for 2-D and 3-D cases.

The sensitivity analysis reinforced the importance of having accurate SWRC or SWCC parameters when using methods relying on the SWRC or SWCC. The sensitivity analysis also confirmed that sites with additional rainfall can still benefit from an increase in bearing capacity by considering unsaturated soils. In the case of Levelland, Texas more than 76% increase in bearing capacity was observed.

The effect of the depth factor is an important finding from the sensitivity analysis. As the suction increases the value of the cohesion term, the depth factor has a greater influence on the ultimate bearing capacity. This results in smaller factors of increase in bearing capacity when there is an increase in footing size which creates a conservative estimate of the bearing capacity of footings in high matric suction.

Ravichandran, N., Mahmoudabadi, V. and Shrestha, S. (2017) Analysis of the Bearing Capacity of Shallow Foundation in Unsaturated Soil Using Monte Carlo Simulation. International Journal of Geosciences, 8, 1231-1250. https://doi.org/10.4236/ijg.2017.810071