Reliability-based design (RBD) of shallow foundations on rock masses
Authors:   , Abu El-Ela Ahmed Abdallah Abdallah 
Published: April 2021

[ACCESS RESTRICTED TO THE UNIVERSITY OF MISSOURI AT AUTHOR'S REQUEST.] The reliability-based design (RBD) approach that separately accounts for variability and uncertainty in load(s) and resistance(s) can be incorporated into the design of shallow foundations on jointed rock masses resulting in a consistent process to design foundations with a known reliability (probability of failure) and optimum cost. The strength of jointed rock masses is a function of the intact rock strength and the rock mass classification parameter, in particular, classification systems using dominant failure criteria. A comprehensive review of the rock mass classification systems is performed. Correlations between the classification systems related to foundations are presented. Rock quality designation (RQD), the rock mass rating (RMR), and the geological strength index (GSI) systems were chosen for the reliability-based analysis performed in this research. Several failure criteria are considered; however, the well-known Hoek and Brown failure criterion is selected to estimate the jointed rock mass strength and deformation. The key factors used in the selection of the method and the criterion used in the design of shallow foundations at strength and service limit states were: the abundance of available data, they are accepted in practice and contain parameters are most often available. Methods and correlations used in the estimation of the elastic deformation modulus for jointed rock masses are presented and evaluated. Methods, which are functions of the intact rock strength property (Uniaxial Compressive Strength, UCS) and the intact rock deformation modulus (Ei) accompanied by the rock mass rating, rock quality designation, and the geological strength index are selected. Methods used to predict the elastic settlement are reviewed and typical values for the elastic property, Poisson's ratio, are provided. A reliability-based design technique was used in this research to account for loads and strength uncertainties. Three main parameters are required to perform the reliability-based design analyses: the failure criteria, the statistical distribution for the design parameters, and the targeted probability of failure. The analyses cover a wide range of probabilities of failure (10E-2 to 10E-4). The Monte Carlo simulation technique, (preferred and recommended by AASHTO), was used to numerically estimate the probability of failure and then compared to targeted levels of reliability. In many cases, the uniaxial compressive strength (UCS) of the intact rock and the rock quality designation are the only available design parameters. For this reason, in addition to the availability of data, the bearing capacity model by Zhang and Einstein (2010) was used in the calibration of the RBD. The model estimates the rock mass bearing capacity as a function of the intact rock compressive strength and the rock quality designation. The product of the calibration procedure is in the form of design charts that are used directly in the design of shallow foundations on jointed rock masses. Good engineering judgment and a secondary rock classification parameter are recommended when using the method by Zhang and Einstein. The empirical equation provided by Carter and Kulhawy (1988), is widely used in the determination of the ultimate capacity of jointed rock masses and was selected for the RBD. The model provides the ultimate bearing capacity of jointed rock masses as a function of the intact rock strength and two additional classification parameters estimated from the classification systems (RMR and GSI). Design charts are produced. Because of the lack of the available data and the need for many test results, a numerical assessment was utilized to develop new equations as functions of the geological strength index (GSI), the intact rock strength parameter, intact rock compressive strength (UCS), and the intact rock material modulus (mi). Four different equations were produced according to the range of the value of the uniaxial compressive strength (UCS). The equations were tested and verified against the most widely used methods in practice in addition to the available testing results reported in the literature. The equations were then used in the RBD analyses, and design charts are provided to be used along with the equation for the reliability-based design of shallow foundation on jointed rock masses. For weak rocks, two different approaches were used in the RBD. The first was the model by Zhang and Einstein (1998) which is a function of the uniaxial compressive strength (UCS). The second approach is a modified version of the equations developed in this dissertation as a result of correlating the equations to the available ultimate capacities for weak rock formations. Design charts are also provided for the reliability-based design of shallow foundations on weak rock. The design of shallow foundations includes the design at the strength limit state and the service limit state. The reliability-based design of shallow foundations at the service limit state (A) (angular distortion of 0.0021) is provided in the form of design charts to be used in the estimation of the factored elastic settlement. The models used in the calibration process are a function of the intact rock deformation modulus and the intact rock strength parameter along with a rock mass classification parameter. The design charts or resistance factors (strength reduction factors) and the factored capacities using the design charts were compared to allowable stress methods, and the values recommended in the current design practices. Examples for the design of shallow foundations on jointed rock masses at both strength and service limit states using the proposed design charts are provided. The reliability-based design charts provide superior designs compared to those resulting from using the allowable stress design (ASD) in terms of reliability, consistency, and efficiency (cost). The comparison between the RBD using the proposed design charts and the ASD method supported this conclusion. The ASD method resulted in conservative or un-conservative designs depending on the uncertainty of the design parameter with unknown reliabilities (unknown probability of failure). The RBD led to reliable economic designs when the uncertainty level of the design parameters is small and reliable costlier designs for higher uncertainties in the design parameters (design for known reliability). The same conclusion is found when comparing the variable resistance factors from the proposed design charts (corresponding to the variable uncertainty of the design parameters) with the "single" value of the resistance factor recommended by the American Association of State Highway and Transportation Officials (AASHTO).