Reliability-Based Analysis of Strip Footings Using Response Surface Methodology

A reliability-based analysis of a strip foundation subjected to a central vertical load is presented. Both the ultimate and the serviceability limit states are considered. Two deterministic models based on numerical simulations are used. The first one computes the ultimate bearing capacity of the foundation and the second one calculates the footing displacement due to an applied load. The response surface methodology is utilized for the assessment of the Hasofer–Lind reliability indexes. Only the soil shear strength parameters are considered as random variables while studying the ultimate limit state. Also, the randomness of only the soil elastic properties is taken into account in the serviceability limit state. The assumption of uncorrelated variables was found to be conservative in comparison to the one of negatively correlated variables. The failure probability of the ultimate limit state was highly influenced by the variability of the angle of internal friction. However, for the serviceability limit state, the accurate determination of the uncertainties of the Young's modulus was found to be very important in obtaining reliable probabilistic results. Finally, the computation of the system failure probability involving both ultimate and serviceability limit states was presented and discussed.     

Bearing Capacity of Strip Footings on Jointed Rock Masses

Bearing capacity solutions are presented for strip footings on jointed rock masses with one and two sets of discontinuities. The solutions employ a lower bound bearing capacity model coupled with a simple discontinuity strength model. The strength of the rock material and the discontinuities, and the number and orientation of the discontinuity sets, are evaluated explicitly. The results are presented as bearing capacity factor charts that illustrate the significant effects of the strength and discontinuity geometric parameters. The trends of the results agree well with those obtained from other models. The solution is straightforward, and it can be implemented manually or in any spreadsheet program.     

Seismic Bearing Capacity of Shallow Strip Footings Embedded in Slope

The seismic bearing capacity factors for shallow strip footings embedded in sloping ground with general c-? soil are found out by using the limit equilibrium method. The seismic forces are considered as pseudostatic forces acting both on the footing and on the soil below the footing. A composite failure surface involving planar and logspiral is considered in the analysis. A new methodology to establish minimum bearing capacity factors has been adopted by numerical iteration technique to determine the critical focus of the logspiral. Three different types of failure surfaces are considered depending on the embedment depth and ground inclinations. The seismic bearing capacity factors with respect to cohesion, surcharge and unit weight components viz. Ncd, Nqd, and Nγd, respectively, are found out separately for various values of soil friction angles and seismic acceleration coefficients both in the horizontal and vertical directions, ground inclinations, and embedment depths. Results of the present study are reported in tabular form. The effect of parametric variation on seismic bearing capacity factors has been studied. Comparisons of the proposed method with available theories in the seismic case are also presented.     

Bearing Capacity of Circular Footings

Numerical computations using FLAC are reported to evaluate the soil-bearing capacity for circular smooth and rough footings. The effect of nonassociativity of the soil is investigated. The results indicate a decrease in the bearing capacity-factors value when the soil displays high nonassociativity. The bearing capacity factors Nγ′ obtained from the computations are significantly lower than those reported by Bolton and Lau in 1993. The results compare favorably with experimental data reported by de Beer in 1970. The sources of possible inaccuracies in numerical simulations are discussed.     

Inclination Factors for Seismic Bearing Capacity

The classic approach for static analysis and design for the effect of shear forces transmitted by shallow foundations is to modify the bearing capacity by introducing inclination factors to reduce the shear strength provided by cohesion, surcharge and self-weight in the standard bearing capacity equation. These static inclination factors are usually based on empirically derived relationships determined by laboratory and field tests. However, it is possible to derive them from limit analysis of a simple Coulomb-type mechanism and thereby extend the concept of inclination factors to the dynamic case giving values for seismic design.     

Reliability of Shallow Foundations Subjected to Multidirectional Seismic Loading

A method of assessing the probability of failure of shallow foundations in saturated fine-grained soil under multidirectional seismic loading is presented. The method uses the distributions of two variables, the spectral acceleration at the fundamental period of the structure and the shear strength of the foundation soil, to form the joint probability density function. The performance function, which defines the required soil strength for the foundation to equilibrate the applied loading, is mapped on this domain. Numerical integration is used to ascertain the probability of failure. A bounded probability density function is used, namely a Pearson Type 1 (Beta) distribution, for soil strength. This distribution provides an upper and lower limit to in situ shear strength. The spectral acceleration is represented by the complementary cumulative distribution function of a Type 1 asymptotic extreme value (Gumble) distribution. Such a distribution is shown to be an accurate representation of real earthquake loading and may be found by a probabilistic seismic hazard analysis for a region. The method is based on pseudostatic seismic loading of the foundation and accounts for spectral acceleration acting along the two horizontal axes. The influence of the orientation of the foundation to the earthquake source is incorporated using the concept of principal directions of ground motion. The performance function is formulated for shallow foundations under eccentric and inclined loading using the recommendations in Eurocode 7. The function is shown to be nonlinear and compound, one part pertaining to the bearing mode of failure (for lower values of acceleration) and the remaining part pertaining to the sliding mode of failure (for higher values of acceleration). An equation is presented for the transition acceleration that separates the performance function into the two parts. A case study is presented and conclusions are drawn about the role of the bearing and sliding modes of failure.     

Undrained Bearing Capacity of Strip Footings on Slopes

The undrained bearing capacity of foundations on or near slopes is commonly calculated using empirical equations or from design charts which have been produced based on limit equilibrium or upper bound plasticity calculations. Many of the available methods do not take account of important parameters that affect the undrained bearing capacity factor, such as the distance of the footing from the slope, the slope height, or the soil properties. This paper presents finite element analyses of strip footings on or near undrained soil slopes performed in order to investigate the influence of the various parameters that affect undrained bearing capacity. The results of the analyses are compared to available methods. It is found that while some of these methods compare well with the finite element results for certain combinations of geometrical parameters and soil properties, they cannot produce sufficiently accurate results as they either do not take account of all of the affecting parameters or are generally not conservative. Based on the finite element results, design charts, equations, and a design procedure are proposed for the calculation of the undrained bearing capacity factor Nc as a function of the undrained shear strength and the bulk unit weight of the soil, the footing width, the distance of the footing from the slope, the slope angle and the slope height.     

Scale Effects of Shallow Foundation Bearing Capacity on Granular Material

Scale effects of shallow foundation bearing capacity on granular materials were investigated to further evaluate the trend of decreasing bearing capacity factor, Nγ, with increasing footing width, B, observed by other researchers. Model-scale square and circular footing tests ranging in width from 0.025 to 0.914?m were performed on two compacted sands at three relative densities. Results of the model-scale footing tests show that the bearing capacity factor, Nγ, is dependent on the absolute width of the footing for both square and circular footings. Although this phenomenon is well known, the current study used a large range of footing sizes tested on well-graded sands to show that the previously reported modifications to the bearing capacity factor, Nγ, using grain-size and reference footing width do not sufficiently account for the scale effect seen in the test results from this study. It also shows that behavior of most model-scale footing tests cannot be directly correlated to the behavior of full-scale tests because of differences in mean stresses experienced beneath footings of varying sizes. The relationship of the initial testing conditions (i.e., void ratio) of the sand beds and mean stress experienced beneath the footing (correlated to footing size) to the critical state line controls footing behavior and, therefore, model-scale tests must be performed at a lower density than a corresponding prototype footing in order to correctly predict behavior. Small footings were shown to have low mean stresses but high Nγ values, which indicates high operative friction angles and may be related to the curvature of the Mohr–Coulomb failure envelope.     

Capacity, Settlement, and Energy Dissipation of Shallow Footings Subjected to Rocking

The effectiveness of structural fuse mechanisms used to improve the performance of buildings during seismic loading depends on their capacity, ductility, energy dissipation, isolation, and self-centering characteristics. Although rocking shallow footings could also be designed to possess many of these desirable characteristics, current civil engineering practice often avoids nonlinear behavior of soil in design, due to the lack of confidence and knowledge about cyclic rocking. Several centrifuge experiments were conducted to study the rocking behavior of shallow footings, supported by sand and clay soil stratums, during slow lateral cyclic loading and dynamic shaking. The ratio of the footing area to the footing contact area required to support the applied vertical loads (A/Ac), related to the factor of safety with respect to vertical loading, is correlated with moment capacity, energy dissipation, and permanent settlement measured in centrifuge and 1 g model tests. Results show that a footing with large A/Ac ratio (about 10) possesses a moment capacity that is insensitive to soil properties, does not suffer large permanent settlements, has a self-centering characteristic associated with uplift and gap closure, and dissipates seismic energy that corresponds to about 20% damping ratio. Thus, there is promise to use rocking footings in place of, or in combination with, structural base isolation and energy dissipation devices to improve the performance of the structure during seismic loading.     

Undrained Bearing Capacity of Two-Strip Footings on Spatially Random Soil

A probabilistic study on the interference of two parallel rough rigid strip footings on a weightless soil with a randomly distributed undrained shear strength performed. The problem is studied using the random finite element method, where nonlinear finite element analysis is merged with random field theory within a Monte Carlo framework. The variability of undrained shear strength is characterized by a lognormal distribution and an exponentially decaying spatial correlation length. The estimated bearing capacity statistics of isolated and two footings cases are compared and the effect of footing interference discussed. Although interference between footings on frictionless materials is not very great, the effect is shown to be increased by soil variability and spatial correlation length.     

Undrained Limit States of Shallow Foundations Acting in Consort

There is no established procedure for the calculation of bearing capacity of a shallow foundation system comprising cojoined footings. Ad hoc approaches are relied on and may simply involve summing the ultimate limit states of the individual footings as if they acted independently; neglecting additional capacity of the system available from the kinematic constraint provided by the structural connection between the footings. In this study, the undrained capacity under general loading of rigidly connected two-footing systems at various separations has been investigated with finite-element analyses. Results are presented in terms of ultimate limit states under pure vertical (V), horizontal (H), and moment (M) loading, and failure envelopes defining limiting load states under combined VH, VM, HM, and VHM loads. Kinematic failure mechanisms observed in the finite-element analyses are presented and in cases used to provide the basis for upper bound solutions.     

Calculations of Bearing Capacity Factor Nγ Using Numerical Limit Analyses

This paper presents numerical upper- and lower-bound solutions for the well-known bearing capacity factor Nγ of a surface strip footing on a frictional soil. The analyses use linear programming and finite-element spatial discretization to solve limit analysis of perfect plasticity, assuming a linear Mohr-Coulomb failure envelope with associated flow within the soil and along the soil-footing interface. The current analyses are to bound the exact value of Nγ within ±5% increasing to ±30% as the internal friction angle increases from 5° to 45° for both smooth and rough interface conditions. Previous solutions by Hansen and Christensen in 1969 and Booker in 1969 are in excellent agreement with the best estimate of Nγ (average of upper and lower bounds) obtained from the current numerical limit analyses. Other well-known analytical solutions and numerical calculations appear to overestimate Nγ for rough footings. Comparisons of predicted upper-bound failure mechanisms and lower-bound contact pressures help to explain similarities and differences among these solutions.     

Optimum Design of Bridge Abutments under Seismic Conditions: Reliability-Based Approach

The paper focuses on the reliability-based design optimization of gravity wall bridge abutments when subjected to active condition during earthquakes. An analytical study considering the effect of uncertainties in the seismic analysis of bridge abutments is presented. Planar failure surface has been considered in conjunction with the pseudostatic limit equilibrium method for the calculation of the seismic active earth pressure. Analysis is conducted to evaluate the external stability of bridge abutments when subjected to earthquake loads. Reliability analysis is used to estimate the probability of failure in three modes of failure viz. sliding failure of the wall on its base, overturning failure about its toe (or eccentricity failure of the resultant force) and bearing failure of foundation soil below the base of wall. The properties of backfill and foundation soil below the base of abutment are treated as random variables. In addition, the uncertainties associated with characteristics of earthquake ground motions such as horizontal seismic acceleration and shear wave velocity propagating through backfill soil are considered. The optimum proportions of the abutment needed to maintain the stability are obtained against three modes of failure by targeting various component and system reliability indices. Studies have also been made to study the influence of various parameters on the seismic stability.     

Computation of Bearing Capacity Factor Nγ?Terzaghi’s Mechanism

A procedure based on K?tter’s equation is developed for the evaluation of bearing capacity factor Nγ with Terzaghi’s mechanism. Application of K?tter’s equation makes the analysis statically determinate, in which the unique failure surface is identified using force equilibrium conditions. The computed Nγ values are found to be higher than Terzaghi’s value in the range 0.25–20%, with a diverging trend for higher values of angle of soil internal friction. A fairly good agreement is observed with other solutions which are based on finite difference coupled with associated flow rule, limit analysis, and limit equilibrium. Finally, the comparison with available experimental results vis-à-vis other solutions shows that, computed Nγ values are capable of making a reasonably good prediction.     

Bearing Capacity of Piled Rafts on Soft Clay Soils

The conventional design of a piled foundation is based on a bearing capacity approach, and neglects the contribution of the raft. As a consequence, piled foundations are usually designed by overconservative criteria. With respect to the conventional approach, a more rational and economical solution could be obtained by accounting for the contribution of the raft toward the overall bearing capacity, but this potential is not exploited due to the lack of theoretical and experimental research on the behavior of piled rafts at failure. Based on both experimental evidence and three-dimensional finite element analyses, a simple criterion is proposed to evaluate the ultimate vertical load of a piled raft as a function of its component capacities, which can be simply evaluated by the conventional bearing capacity theories. The results presented in the paper thus provide a guide to assess the safety factor of a vertically loaded piled raft.     

Effect of Footing Roughness on Bearing Capacity Factor Nγ

The effect of interface friction angle (δ) between the footing and underlying soil mass on the bearing capacity factor Nγ was examined by using the upper bound limit analysis, finite elements, and linear programming. The analysis was carried out by employing velocity discontinuities along all the interfaces of the chosen triangular elements. The development of the plastic strains within elements was incorporated by using an associated flow rule. It was clearly noted that an increase in δ leads to a continuous increase in Nγ. With δ = ?, the magnitude of Nγ becomes almost the same as that for a perfectly rough foundation, that is, when no slippage takes place between the footing and underlying soil mass. The size of the plastic zones increases with increase in δ and ?. The obtained values of Nγ, for perfectly smooth and perfectly rough footings, compare quite favorably with those reported in literature. The study demonstrates that in the case when δ is smaller than ?, the assumption of a perfectly rough footing will lead to an unsafe prediction of the ultimate bearing capacity.     

Use of State-Dependent Strength in Estimating End Bearing Capacity of Piles in Sand

The pressure and density dependence of the shear strength of sand poses a tricky problem in pile foundation design. In this study, a correlation is suggested to link the effective friction angle of sand with its initial confining pressure and relative density, and a simple approach incorporating this correlation is presented for predicting pile end bearing capacity. Assessment of the approach against pile load tests shows reasonably good agreement between predictions and measurements. It is also shown that the effect of the state-dependent strength is particularly important in cases where long piles are installed in dense sand deposits and the use of critical state friction angle will produce a conservative prediction in such cases.     

Model Tests and Analyses of Bearing Capacity of Strip Footing on Stiff Ground with Voids

A series of 1G loading tests under the plane-strain condition were conducted on stiff ground with continuous square voids with the view of shallow foundation on calcareous sediment rocks, which contain voids because of their susceptibility to water dissolution. Detailed experimental observation revealed three types of failure modes for a single void: bearing failure without void failure, bearing failure with void failure, and void failure without bearing failure, depending on the location of the void as well as the size of the void. Upper-bound calculations were presented to interpret the changes of bearing capacity observed because of the existence of a void.