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.     

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.     

Effects of Moment-to-Shear Ratio on Combined Cyclic Load-Displacement Behavior of Shallow Foundations from Centrifuge Experiments

Current design guidelines for shallow foundations supporting building and bridge structures discourage footing rocking or sliding during seismic loading. Recent research indicates that footing rocking has the potential to reduce ductility demands on structures by dissipating earthquake energy at the footing-soil interface. Concerns over cyclic and permanent displacements of the foundation during rocking and sliding along with the dependence of foundation capacity on uncertain soil properties hinder the use of footing rocking in practice. This paper presents the findings of a series of centrifuge experiments conducted on shear wall-footing structures supported by dry dense to medium dense sand foundations that are subjected to lateral cyclic loading. Two key parameters, static vertical factor of safety (FSV), and the applied normalized moment-to-shear ratio (M/(H?L)) at the footing-soil interface, along with other parameters, were varied systematically and the effects of these parameters on footing-soil system behavior are presented. As expected, the ratio of moment to the horizontal load affects the relative magnitude of rotational and sliding displacement of the footing. Results also show that, for a particular FSV, footings with a large moment to shear ratio dissipate considerably more energy through rocking and suffer less permanent settlement than footings with a low moment to shear ratio. The ratio of actual footing area (A) to the area required to support the vertical and shear loads (Ac), called the critical contact area ratio (A/Ac), is used to correlate results from tests with different moment to shear ratio. It is found that footings with similar A/Ac display similar relationships between cyclic moment-rotation and cumulative settlement, irrespective of the moment-to-shear ratio. It is suggested that shallow foundations with a sufficiently large A/Ac suffer small permanent settlements and have a well defined moment capacity; hence they may be used as effective energy dissipation devices that limit loads transmitted to the superstructure.     

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.     

Influence of Spatial Variability on Slope Reliability Using 2-D Random Fields

The paper investigates the probability of failure of slopes using both traditional and more advanced probabilistic analysis tools. The advanced method, called the random finite-element method, uses elastoplasticity in a finite-element model combined with random field theory in a Monte-Carlo framework. The traditional method, called the first-order reliability method, computes a reliability index which is the shortest distance (in units of directional equivalent standard deviations) from the equivalent mean-value point to the limit state surface and estimates the probability of failure from the reliability index. Numerical results show that simplified probabilistic analyses in which spatial variability of soil properties is not properly accounted for, can lead to unconservative estimates of the probability of failure if the coefficient of variation of the shear strength parameters exceeds a critical value. The influences of slope inclination, factor of safety (based on mean strength values), and cross correlation between strength parameters on this critical value have been investigated by parametric studies in this paper. The results indicate when probabilistic approaches, which do not model spatial variation, may lead to unconservative estimates of slope failure probability and when more advanced probabilistic methods are warranted.     

Contact Interface Model for Shallow Foundations Subjected to Combined Cyclic Loading

It has been recognized that the ductility demands on a superstructure might be reduced by allowing rocking behavior and mobilization of the ultimate capacity of shallow foundations during seismic loading. However, the absence of practical reliable foundation modeling techniques to accurately design foundations with the desired capacity and energy dissipation characteristics and concerns about permanent deformations have hindered the use of nonlinear soil–foundation–structure interaction as a designed mechanism for improving performance of structural systems. This paper presents a new “contact interface model” that has been developed to provide nonlinear relations between cyclic loads and displacements of the footing–soil system during combined cyclic loading (vertical, shear, and moment). The rigid footing and the soil beneath the footing in the zone of influence, considered as a macroelement, are modeled by keeping track of the geometry of the soil surface beneath the footing, along with the kinematics of the footing–soil system, interaction diagrams in vertical, shear, and moment space, and the introduction of a parameter, critical contact area ratio (A/Ac); the ratio of footing area (A) to the footing contact area required to support vertical and shear loads (Ac). Several contact interface model simulations were carried out and the model simulations are compared with centrifuge model test results. Using only six user-defined model input parameters, the contact interface model is capable of capturing the essential features (load capacities, stiffness degradation, energy dissipation, and deformations) of shallow foundations subjected to combined cyclic loading.     

Probabilistic Analysis of Circular Tunnels in Homogeneous Soil Using Response Surface Methodology

A probabilistic analysis of a shallow circular tunnel driven by a pressurized shield in a frictional and/or cohesive soil is presented. Both the ultimate limit state (ULS) and serviceability limit state (SLS) are considered in the analysis. Two deterministic models based on numerical simulations are used. The first one computes the tunnel collapse pressure and the second one calculates the maximal settlement due to the applied face pressure. The response surface methodology is utilized for the assessment of the Hasofer-Lind reliability index for both limit states. Only the soil shear strength parameters are considered as random variables while studying the ULS. However, for the SLS, both the shear strength parameters and Young’s modulus of the soil are considered as random variables. For ULS, the assumption of uncorrelated variables was found conservative in comparison to the one of negatively correlated parameters. For both ULS and SLS, the assumption of nonnormal distribution for the random variables has almost no effect on the reliability index for the practical range of values of the applied pressure. Finally, it was found that the system reliability depends on both limit states. Notice however that the contribution of ULS to the system reliability was not significant. Thus, SLS can be used alone for the assessment of the tunnel reliability.     

Reliability Evaluation of Earth Slopes

The reliability analysis of earth slopes is considered. For slope safety assessment, the first-order reliability method is employed for estimating the probability of failure or reliability index. Since the failure of any slip surfaces implies failure of the slope, the slope is considered as a series system. The system aspect of the slope in the reliability analysis is dealt with by defining a limit state of the system as a function of the minimum of the ratio of the shear strength to the mobilized shear strength for each of all potential slip surfaces. Such a ratio for a given slip surface is evaluated using the extended generalized method of slices. The reliability analysis procedure described is applied to example slopes to illustrate the impact of the probability distribution type, and the spatial variability of the soil properties on the probability of failure of the slopes.     

Optimum Design for External Seismic Stability of Geosynthetic Reinforced Soil Walls: Reliability Based Approach

In this paper, an analytical study considering the effect of uncertainties in the seismic analysis of geosynthetic-reinforced soil (GRS) walls is presented. Using limit equilibrium method and assuming sliding wedge failure mechanism, analysis is conducted to evaluate the external stability of GRS walls when subjected to earthquake loads. Target reliability based approach is used to estimate the probability of failure in three modes of failure, viz., sliding, bearing, and eccentricity failure. The properties of reinforced backfill, retained backfill, foundation soil, and geosynthetic reinforcement are treated as random variables. In addition, the uncertainties associated with horizontal seismic acceleration and surcharge load acting on the wall are considered. The optimum length of reinforcement needed to maintain the stability against three modes of failure by targeting various component and system reliability indices is obtained. Studies have also been made to study the influence of various parameters on the seismic stability in three failure modes. The results are compared with those given by first-order second moment method and Monte Carlo simulation methods. In the illustrative example, external stability of the two walls, Gould and Valencia walls, subjected to Northridge earthquake is reexamined.     

Bearing Capacity of Rough Rigid Strip Footing on Cohesive Soil: Probabilistic Study

A probabilistic study on the bearing capacity of a rough rigid strip footing on a weightless cohesive soil is carried out to assess the influence of randomly distributed undrained shear strength. Nonlinear finite element analysis is merged with random field theory in conjunction with a Monte Carlo method. In a parametric study, the mean shear strength is held constant while the coefficient of variation and spatial correlation length of cohesion are varied systematically. The influence of the spatial variation of cohesion on the mean bearing capacity is discussed. The results are also presented in a probabilistic context to determine the probability of failure. A comparison between rough and smooth footing conditions is also made.     

Deformation Patterns of Reinforced Foundation Sand at Failure

While the stability of foundation soils has been written about extensively, the ultimate loads on reinforced soils is a subject studied to a much lesser degree. There is convincing experimental evidence in the literature that metal strips or layers of geosynthetic reinforcement can significantly increase the failure loads on foundation soils. Laboratory tests were performed to investigate the kinematics of the collapse of sand reinforced with a layer of flexible reinforcement. Sequential images of the deformation field under a model footing were digitally recorded. A correlation-based motion detection technique was used to arrive at an incremental displacement field under a strip footing model. Color-coded displacements are presented graphically. The mechanism retains some of the characteristic features of a classical bearing capacity pattern of failure, but the reinforcement modifies that mechanism to some extent. The strips of geotextile used as model reinforcement give rise to the formation of shear bands in a narrow layer adjacent to the geosynthetic. Reinforcement restrains the horizontal displacement of the soil and alters the collapse pattern. The mechanism of deformation identified in the tests will constitute a basis for limit analysis of reinforced foundation soils.     

Reliability Analysis of Earth Dams

In the present study, results of reliability analyses of four selected rehabilitated earth dam sections, i.e., Chang, Tapar, Rudramata, and Kaswati, under pseudostatic loading conditions, are presented. Using the response surface methodology, in combination with first order reliability method and numerical analysis, the reliability index (β) values are obtained and results are interpreted in conjunction with conventional factor of safety values. The influence of considering variability in the input soil shear strength parameters, horizontal seismic coefficient (αh), and location of reservoir full level on the stability assessment of the earth dam sections is discussed in the probabilistic framework. A comparison of results with those obtained from other method of reliability analysis, viz., Monte Carlo simulations combined with limit equilibrium approach, provided a basis for discussing the stability of earth dams in probabilistic terms, and the results of the analysis suggest that the considered earth dam sections are reliable and are expected to perform satisfactorily.     

Load Resistance Factor Design of Axially Loaded Pile Based on Load Test Results

The present study proposes a procedure to determine partial factors in reliability based design format for pile foundations, considering bias as well as uncertainty in the parameters that represent soil-pile interaction. These issues are addressed using pile load-settlement test data from case studies obtained from the literature. The pile ultimate capacities are evaluated considering three different failure criteria. The uncertainties in the pile-soil interface parameters as well as pile ultimate capacity are quantified in Monte Carlo framework from the measured data by utilizing the closed form “t-z” method. Considering dead load to live load ratios as calibration points, the target reliability index is calculated based on existing code safety-checking format. The optimal partial factors are determined such that the difference between reliability index based on limit state equations expressed in terms of partial factors and target reliability index is minimum. Finally, it is observed that optimal partial factors enable rational choice of allowable load on pile foundation.     

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.     

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.     

Active Earth Pressure on Retaining Wall for c-? Soil Backfill under Seismic Loading Condition

This technical note describes the derivation of an analytical expression for the total active force on the retaining wall for c-? soil backfill considering both the horizontal and vertical seismic coefficients. The results based on this expression are compared with those obtained from earlier analytical expressions for the active force for c-? soil backfill under seismic conditions, and found to have a similar trend of variation. The parametric study shows that the inclination of the critical failure plane with the horizontal plane decreases with the increase in values of seismic coefficients; the decrease being more for their higher values. The total active force increases with the increase in value of horizontal seismic coefficient; while it decreases with the increase in value of vertical seismic coefficient except for a very high value of horizontal seismic coefficient. Design charts are presented for various combinations of horizontal and vertical seismic coefficients (kh and kv), and values of cohesion and angle of shearing resistance for estimating the total active force on the retaining wall for c-? soil backfill for practical applications.     

Suction Caisson Capacity in Anisotropic, Purely Cohesive Soil

This paper presents a plastic limit analysis of the lateral load capacity of suction caissons in an anisotropic, purely cohesive soil assuming conditions of rotational symmetry about the vertical or gravity axis. The formulation utilizes a form of the Hill yield criterion that is modified to allow for different soil strengths in triaxial compression and extension. Using this yield criterion, energy dissipation relationships are formulated for continuous and discontinuous deformation fields. These dissipation relationships are then applied to a postulated caisson failure mechanism comprising a wedge near the free soil surface (mudline), a two-dimensional flow-around failure at depth, and a hemispherical slip surface at the base of the rotating caisson. The plastic limit analysis predictions compared favorably to predictions obtained from finite-element simulations employing a Hill yield criterion. For the range of anisotropic undrained strength properties commonly reported for normally K0-consolidated clays, parametric studies indicate that suction caisson horizontal load capacities predicted using a conventional approach (a von Mises yield surface fitted to the soil simple shear strength) will differ from anisotropic predictions by less than 10%.     

Flexural Reliability of Reinforced Concrete Bridge Girders Strengthened with Carbon Fiber-Reinforced Polymer Laminates

This paper presents the development of a resistance model for reinforced concrete bridge girders flexurally strengthened with externally bonded carbon fiber-reinforced polymer (CFRP) laminates. The resistance model is limited to pure flexural failure and does not address shear failure, laminate debonding, or delamination. The resistance model is used to calculate the probability of failure and reliability index of CFRP-strengthened cross sections. The first-order reliability method is employed to calibrate the flexural resistance factor for a broad range of design variables. The study shows that the addition of CFRP improves reliability somewhat because the strength of CFRP laminates has a lower coefficient of variation than steel or concrete. However, the brittle nature of CFRP laminates necessitates a reliability index that is greater than that generally implied in the AASHTO LRFD for 1998. This leads to a lower resistance factor than is currently accepted for reinforced concrete sections in flexure.     

Optimum Design of Cantilever Retaining Walls Using Target Reliability Approach

In this paper, an approach for reliability-based design optimization of reinforced concrete cantilever retaining wall is described. A parametric study is conducted to assess the effect of uncertainties in design parameters on the probability of failure of cantilever retaining walls. In total, ten modes of failure are considered, viz. overturning of the wall about its toe, sliding of the wall on its base, eccentricity, bearing capacity failure below the base slab, and shear and moment failure in the toe slab, heel slab, and stem. The analysis is performed by treating backfill and foundation soil properties, geometric properties of wall, and reinforcement and concrete properties as random variables. These results are used to develop a set of reliability-based design charts for different coefficients of variation of friction angle of backfill soil (5 and 10%) and targeting reliability index (βt) in the range of 3–3.2 for all failure modes. A comparative study is also presented, which shows that optimized sections have less areas of cross section compared to those obtained from specifications on dimensioning of retaining walls available in literature.     

Reliability Assessment of Basal-Heave Stability for Braced Excavations in Clay

One potential failure mechanism that needs to be considered for the design of braced excavations in soft clays is basal-heave instability. Many uncertainties are associated with the calculation of the basal-heave factor of safety, including the variabilities of the loadings, geotechnical soil properties, and engineering and geometrical properties of the wall. A risk-based approach to failure is necessary to incorporate systematically the uncertainties associated with the various design parameters. This paper demonstrates the potential for the reliability index–based approach for evaluating the basal-heave stability of braced excavations in clay. By using basic structural reliability concepts that reflect the degree of uncertainty of the underlying random variables in the analyses, engineers will have increased awareness of the uncertainties and their effects on the probability of failure. This study shows that the assumption of a linear limit state (failure) surface can be used to provide reasonable estimates of the reliability index and the probability of failure. Mathematical expressions, tables, and charts have been provided to estimate the probability of basal-heave failure for wide and deep excavations.