Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 7901918, 14 pages

https://doi.org/10.1155/2017/7901918

## Maximum Likelihood Estimation of Model Uncertainty in Predicting Soil Nail Loads Using Default and Modified FHWA Simplified Methods

^{1}School of Earth Science and Engineering, Sun Yat-Sen University, Guangzhou, Guangdong 510275, China^{2}Provincial Key Laboratory of Mineral Resources and Geological Processes Guangzhou, Guangdong 510275, China^{3}Department of Civil Engineering, Ryerson University, Toronto, ON, Canada M5B 2K3

Correspondence should be addressed to Liansheng Tang; nc.ude.usys.liam@sltsee

Received 16 July 2017; Revised 5 November 2017; Accepted 15 November 2017; Published 19 December 2017

Academic Editor: Xiao-Qiao He

Copyright © 2017 Huifen Liu 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

Accuracy evaluation of the default Federal Highway Administration (FHWA) simplified equation for prediction of maximum soil nail loads under working conditions is presented in this study using the maximum likelihood method and a large amount of measured lower and upper bound nail load data reported in the literature. Accuracy was quantitatively expressed as model bias where model bias is defined as the ratio of measured to predicted nail load. The maximum likelihood estimation was carried out assuming normal and lognormal distributions of bias. Analysis outcomes showed that, based on the collected data, the default FHWA simplified nail load equation is satisfactorily accurate on average and the spread in prediction accuracy expressed as the coefficient of variation of bias is about 30%, regardless of the distribution type. Empirical calibrations were proposed to the default FHWA simplified nail load equation for accuracy improvement. The Bayesian Information Criterion was adopted to perform a comparison of suitability between the competing normal and lognormal statistical models that were intended for description of model bias. Example of reliability-based design of soil nail walls against internal pullout limit state of nails is provided in the end to demonstrate the benefit of performing model calibration and using calibrated model for design of soil nails.

#### 1. Introduction

Soil nails used to support ground excavations or reinforce existing slopes are most commonly installed using the drill and grout nail installation technique [1, 2]. A hole is first drilled into the ground or slope, then a steel bar is placed, and the hole is grouted. As such, a drilled and grouted soil nail is a composite cylindrical structure consisting of a nail tendon (steel bar) and a grout column.

An installed composite cylindrical soil nail has two interfaces: the grout-soil interface and the grout-steel bar interface. When the nailed soil mass deforms, tensile loads develop along the nail initially at the grout-soil interface due to grout-soil interactions and then transfer (partially or fully) to the steel bar through grout-steel bar interactions. Hence, for a soil nail under working conditions both the grout column and steel bar components carry tensile loads. The tensile loads of the steel bar can be easily estimated based on strain gauges mounted along the bar whereas the tensile loads of the grout column are difficult to measure directly.

When the diameter of the grout column is very small or the grout column undergoes cracking, the total nail load can be roughly approximated as the steel bar load. However, it has been noted that in many cases the steel load itself cannot adequately account for the total nail load, especially when the grout column is intact and with large diameter [3–8]. Wentworth [6] and Banerjee et al. [7, 8] then developed a method to estimate the total nail load taking into account the portion that is carried by the grout column. In their study, the measured steel load of a soil nail was reported as the lower bound of total nail load while the sum of measured steel load plus theoretical maximum tensile load capacity of the grout column was reported as the upper bound of total nail load. The total nail load estimated using their proposed method lies between the lower and upper bounds. Lin et al. [9] adopted the total nail load data interpreted by Wentworth [6] and Banerjee et al. [7, 8] as part of their nail load database and evaluated the accuracy of the default Federal Highway Administration (FHWA) simplified equation for prediction of maximum nail load under working conditions. The accuracy evaluation outcomes are unavoidably influenced by the method developed by Wentworth [6] and Banerjee et al. [7, 8] for estimation of total nail loads.

Evaluation and consideration of model uncertainties are of great importance to reliability-based geotechnical designs, which have been discussed in [10]. Mainly included in the discussion are (1) methods for model uncertainty evaluation and model calibration which have been recently summarized by Dithinde et al. [11] and (2) an overview of existing work of model uncertainty characterization for different geotechnical models in the literature (e.g., shallow and deep foundations [12–17] and retaining structures [18–21]). The objective of this study is to evaluate the model uncertainty of the default FHWA simplified nail load equation using the lower and upper bound nail load data reported by Wentworth [6] and Banerjee et al. [7, 8]. The model uncertainty is quantified using a model bias defined as the ratio of measured to predicted maximum soil nail load. The maximum likelihood method is adopted in this paper, which has been widely demonstrated to be a powerful tool for estimation of statistical model parameters [i.e., mean and standard deviation or coefficient of variation (COV)] that is intended to describe a given data set [22–35]. The model bias of the default FHWA simplified nail load model is characterized as a normal or a lognormal random variable and the suitability of the two statistical models is compared using the Bayesian Information Criterion (BIC). This study also shows the calibration of the default FHWA simplified model for accuracy improvement using a regression approach summarized in [11]. A reliability-based design example of internal pullout limit state is provided in the end to show the benefit of using calibrated nail load model for design of soil nails. The present work is valuable to reliability-based analysis and design of soil nail internal limit states such as nail pullout failure and nail-in-tension failure.

#### 2. Performance Function of Soil Nail Pullout Limit State

Figure 1 shows the geometry of a typical soil nail wall with a vertical facing and a horizontal back slope. The potential slip surface is assumed to extend from the toe to the top of the wall at an angle of (45+/2) degrees, dividing the whole soil nailing system into an active zone and a passive zone. Nail pullout failure takes place when the maximum nail tensile load exceeds its ultimate pullout capacity. The performance function of the soil nail pullout limit state, , can be written aswhere and are measured uncensored ultimate nail pullout capacity and maximum nail tensile load, respectively; and are predicted ultimate nail pullout capacity and predicted maximum nail tensile load, respectively; and are model biases accounting for prediction errors in and , respectively. Accordingly, and are defined as and , which are the ratios of measured to predicted values.