Mathematical Problems in Engineering

Volume 2017, Article ID 3043571, 8 pages

https://doi.org/10.1155/2017/3043571

## Safety Criteria and Standards for Bearing Capacity of Foundation

^{1}State Key Laboratory of Eco-Hydraulics in Northwest Arid Region of China, Xi’an University of Technology, Xi’an 710048, China^{2}College of Engineering, Ocean University of China, Qingdao 266100, China^{3}Institute of Geotechnical Engineering, Xi’an University of Technology, Xi’an 710048, China^{4}Shaanxi Water Affair Group Co., Ltd., Xi’an 710004, China

Correspondence should be addressed to Xuguang Chen; nc.ude.cuo@4891gnauguxnehc

Received 12 August 2017; Revised 2 November 2017; Accepted 21 November 2017; Published 18 December 2017

Academic Editor: Edoardo Artioli

Copyright © 2017 Yanlong Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper focuses on the evaluation standards of factor of safety for foundation stability analysis. The problem of foundation stability is analyzed via the methods of risk analysis of engineering structures and reliability-based design, and the factor of safety for foundation stability is determined by using bearing capacity safety-factor method (BSFM) and strength safety-factor method (SSFM). Based on a typical example, the admissible factors of safety were calibrated with a target reliability index specified in relevant standards. Two safety criteria and their standards of bearing capacity of foundation for these two methods (BSFM and SSFM) were established. The universality of the safety criteria and their standards for foundation reliability was verified based on the concept of the ratio of safety margin (RSM).

#### 1. Introduction

The stability of foundation is an important factor for foundation design [1, 2]. In fact, foundation stability is normally evaluated and analyzed via the factor of safety of bearing capacity. Several influence factors for the problem of foundation stability analysis are studied by many researchers [3–6]. The factor of safety for a foundation can be defined as the ratio between the ultimate bearing capacity and the actual load on the foundation soil. This method relies on the engineering experience and cannot reflect the failure process of the foundation soil mass [7]. However, the strength and safety margin of slopes can be reflected by the calculation method of factor of safety in slope stability analysis and it also has the physical significance. The calculation method and the standard of factor of safety are two important issues for the problems of slope stability analysis and foundation design. Therefore, the standards for factor of safety are the key issue for the foundation and slope stability analysis. These problems can be analyzed via the method of risk analysis for engineering structures and the method of reliability-based design. The problem of uncertain slope stability analysis was the first application of reliability-based method in geotechnical engineering [7]. Afterwards, the method of reliability-based analysis was used to analyze the problem of foundation bearing capacity. For example, based on the limit analysis and reliability-based analysis method, Massih analyzed the bearing capacity of strip footings by combining the foundation bearing capacity [8]. Moreover, the method of reliability-based analysis can be combined with numerical method to calculate and analyze the problem of slope stability and foundation bearing capacity. For example, the bearing capacity of a strip footing on undrained clay/weightless soil is analyzed by Griffiths et al. using the reliability-based method and finite-element analysis [9, 10]. The sensitivity of factor of safety for foundation problem is analyzed by Griffiths using strength reduction method and load increase method [11, 12]. In this study, the admissible range of foundation reliability indexes is investigated through a typical example, and the evaluation standards of the factor of safety are established via the bearing capacity safety-factor method (BSFM) and strength safety-factor method (SSFM). The concept of a “ratio of safety margin (RSM)” is used to verify the universality of these standardized values, and the feasibility of RSM was evaluated based on an engineering case study.

#### 2. Foundation Bearing Capacity Analysis

##### 2.1. Factor of Safety Method for Foundation Bearing Capacity Analysis

The factor of safety for the problem of foundation stability analysis can be determined via the conventional methods (e.g., limit equilibrium method) and strength reduction method.

###### 2.1.1. Factor of Safety Criteria Based on BSFM

In conventional factor of safety methods, the factor of safety can be defined as the ratio of the ultimate load to the actual load; that is,where is the factor of safety, is the actual load on the foundation, and is the ultimate bearing capacity of the foundation. The value of is calculated by using the formula for the bearing capacity of foundation proposed by Terzaghi [13]:where , , , and are the cohesion, internal friction angle, soil unit weight of footing embedment, and unit weight of the soil beneath the footing, respectively; and , , and are the foundation bearing capacity factors.

Meyerhof proposed the formulas of three bearing capacity factors (, , and ) for rigid rough and strip footings [14]. The formulas for bearing capacity factor developed by Meyerhof and Hansen are also widely used [15–17]. The formulas proposed by Terzaghi, Meyerhof, and Hansen are semiempirical expressions, but the bearing capacity formula of Vesić is more brief and efficient. Thus, Vesić’s bearing capacity formula is used to calculate the factor of safety of foundation.

###### 2.1.2. Factor of Safety Criteria Based on SSFM

Foundation can be considered as a slope with zero gradient, and the factor of safety can be calculated using the method of slope stability analysis. Based on this assumption, the corresponding strength indexes are reduced by the reciprocal of as follows:

Equations (6) are substituted in (2) to yieldwhere the subscript represents the corresponding strength index after reduction.

These two methods (BSFM and SSFM) have different factor of safety definitions, and both of them are used in the deterministic analysis method for factor of safety calculation. The sliding stability analysis of a foundation also can be performed via the uncertain analysis method with the reliability index. Thus, the reliability index of the factor of safety for foundations will be used to analyze the problem of foundation stability in this study.

##### 2.2. Reliability Analysis of Bearing Capacity of Foundation

Based on the limit equilibrium method, the equation for the limit state can be defined as followswhere is a group of independent random variables; in this paper, they are the strength parameters and . The calculation formula of the reliability index is as follows:where and are the mean value and the standard deviation of the factor of safety. The relationship between the factor of safety and reliability index is described by (9), which has been widely used in relevant standards [18, 19]. The reliability index is calculated by the first-order second-moment method, and a standard dimensionless variable is introduced:

The reliability index and design points in a plane coordinate space are shown in Figure 1. The origin point represents the mean value, which is also the most possible value of the random variable. Point represents the most possible point in a failure state, that is, a design point in the standard space. The length of represents the reliability index , and the physical significance of is the shortest path that a random variable will travel from the most possible value to the failure surface.