Proposed Specifications for LRFD Soil-Nailing Design and Construction (2011)

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63.1 Overview This chapter first presents the results of a review of current LRFD practice in geotechnical design, introduces the basis for LRFD-based methods for retaining structures, and provides the results of a review of current U.S. practice of soil nailing. Subsequently, the chapter provides discussions of LRFD limit states in the design of SNWs and a synthesis of approaches used to calibrate resistance and load factors. Finally, calibrations of resistant factors are presented. 3.2 Review of Current LRFD Practice 3.2.1 Historical Development of LRFD 3.2.1.1 Structural Design The early use of concepts of probability and reliability, as used to quantify uncertainties in the design of structures (Freudenthal, 1947, 1951; Freudenthal and Gumbel, 1956), set the basis for the subsequent development of the LRFD framework. In the 1970s and 1980s, the development of LRFD methods for structural applications advanced substan- tially when various structural codes started to incorporate reliability concepts. For example, reliability was used in the American National Standards Institute code (ANSI) for design loads for buildings (as summarized by Ellingwood et al., 1980; Ellingwood and Galambos, 1982; Ellingwood et al., 1982a and 1982b). Other design codes incorporating LRFD concepts included those for steel construction [American Institute of Steel Construction (AISC), 1994; Galambos and Ravindra, 1978], concrete construction [American Concrete Institute (ACI), 1995], and offshore platforms [American Petroleum Institute (API), 1989; Moses, 1985, 1986]. International build- ing codes containing reliability or LRFD methods included the National Building Code of Canada (Siu et al., 1975; National Research Council of Canada, 1977) and Report 63 developed by the United Kingdom’s Construction Industry Research and Information Association (CIRIA, 1977). 3.2.1.2 Geotechnical Design In an early effort to distinguish different sources of uncer- tainty in geotechnical design, Taylor (1948) proposed the use of separate and independent factors of safety for the cohesion and frictional components of soil resistance. However, the con- cept of a load factor, which incorporates the uncertainty related to loads, was not used in geotechnical design at that time. All uncertainty in geotechnical design was concentrated in the resistance. The use of both load and resistance factors in geo- technical engineering was initiated by Brinch-Hansen in Den- mark (Brinch-Hansen 1953, 1956, 1966). Later publications related to the use of LRFD concepts in geotechnical design include Barker et al. (1991) for foundations and retaining structures, Fellenius (1994) and Meyerhof (1994) for shallow foundations, O’Neill (1995) for deep foundations, Hamilton and Murff (1992) and Tang (1993) for foundations of offshore platforms, Kulhawy and Phoon (1996) for foundations of trans- mission towers, Withiam et al. (1991, 1995) and D’Appolonia (1999) for retaining structures, Allen et al. (2001) and Chen (2000a, 2000b) for MSE walls, and Paikowsky et al. (2004, NCHRP Project 24-17) for deep foundations. 3.2.2 Overview of Uncertainty in Design of Structures This section provides an overview of common approaches in dealing with uncertainty in structural design. In the design of structures, a number of uncertainty sources must be addressed. These sources may include the following: • Material dimensions and location/extension; • Material properties, including unit weight/density and strength; • Long-term material performance; • Possible failure modes; • Methods used to analyze loads and evaluate load distribution; C H A P T E R 3 Findings and Applications

• Methods used to predict transient loads; • Methods used to predict the structural response; and • Potential changes over time associated with the structural function. Besides the sources listed above, in geotechnical design, uncertainties also arise from the variability of subsurface con- ditions, the intrinsic errors made in the estimation of material properties, and the divergences that occur due to the differences between the estimated and actual properties of the structure. The variability of subsurface conditions arises as a result of the spatial variability of soil and rock properties. Spatial vari- ability of soil/rock properties may be caused by differences in geology across a site; in contrast, local variability of soil/rock properties commonly results from the inherent heterogeneities of most natural materials. Intrinsic errors in the estimation of material properties (i.e., usually referred to as bias) arise from (i) sampling methods used to obtain soil/rock specimens [e.g., a standard penetration test (SPT)]; (ii) field or labora- tory testing techniques used to evaluate soil/rock properties (e.g., SPT blow count or triaxial tests); and (iii) models used to interpret and predict soil/rock properties (e.g., Mohr- Coulomb model). Measurements of soil/rock properties in the field and laboratory produce random errors that are typical of all measurements. Finally, uncertainty in geotechnical design may also occur due to differences between the assumed or esti- mated properties and the actual properties of the constructed structure as a result of differing construction methods or insuf- ficient construction quality control and assurance. 3.2.3 Overview of the ASD Method Uncertainty in engineering design has traditionally been addressed with factors of safety (FS) in the allowable stress design (ASD). In the ASD method, allowable “stresses” (or, more generally, resistances) of structural components are obtained by dividing the values of ultimate strengths of those structural components by FS. The general design condition in the ASD method can be expressed as: where ΣQi = the effect of all combined loads on a given structural component for a given failure mode, Rall = the allowable stress of that structural component, Rn = the ultimate or maximum strength of that structural component, and FS = the factor of safety applied to that ultimate resistance. Allowable stresses represent normal working conditions of a structural element and are therefore selected lower than ΣQ R R FS i all n≤ = ( )3 1- the ultimate capacity of the structural element. Structures have various components that may be subjected to numerous load- ing conditions, possibly involving different potential failure modes. As a result, numerous equations, similar in format to Equation 3-1, must be considered to achieve a safe design of each structural component and of the entire system for all expected conditions. In Equation 3-1, all uncertainty is concentrated in FS that appears on only one side of the design equation. FS is typically adopted based on experience, engineering judgment, and com- mon practice. It is not usually based on uncertainty quantifica- tions (i.e., by establishing the probability of failure of a selected failure mode or structural component). Minimum values of FS recommended for design of certain structures are selected gen- erally by agencies with jurisdiction or interest on those struc- tures. For example, for the design of bridge structures and substructures, AASHTO has developed a set of FS values that is contained in the ASD-based AASHTO Standard Specifica- tions (AASHTO, 1996). In general, FS values that are selected based on experience tend to provide safe and reasonably economical designs after years of practice. However, the selection of new FS values for new problems (i.e., use of materials, construction methods, or consideration of infrequent loading) may be more challenging than simply selecting values based on existing ranges. In deriv- ing FS values for new problems, different design practitioners may select different FS values if only engineering judgment is used. The ASD method may occasionally provide inconsistent levels of safety for structures involving various components with multiple factors of safety (each possibly involving different probabilities of failure). To overcome some of these limitations of the ASD method, the LRFD has been developed. 3.2.4 Overview of the LRFD Method 3.2.4.1 Objectives and Basic Description of the LRFD Method To address design uncertainty in a more systematic manner than in the ASD method, the LRFD method was developed with the following objectives: (i) to account for uncertainty in loads and resistances separately with the use of factors for load and resistance; (ii) to provide reliability-based load and resis- tance factors based on accepted levels of structural reliability; and (iii) to provide consistent levels of safety across a structure when several components are present. This approach is used in the current AASHTO LRFD Bridge Design Specifications. In the LRFD method, two parameters account for uncer- tainty: load factor for load uncertainty and resistance factor for material uncertainty. The use of separate parameters is justified because the nature, variability, and hence level of uncertainty associated with loads are different than the uncertainty related to resistance. In principle, the LRFD method can result in more 7

consistent levels of safety across the entire structure because the relationship between the levels of safety of different structural members is accounted for in this method. Resistance and load factors are selected using probability-based techniques so that these factors are related to acceptable levels of structural relia- bility, which is equivalent to a tolerable probability of failure. Unlike the FS, the LRFD-based parameters are calibrated with respect to actual load and resistance data. Load and resistance factors are related to each other through limit states. A limit state is a condition in which the structure as a whole, or one of its components, has achieved a level of stress, deformation, or displacement that may affect its performance. In the LRFD method included in the AASHTO LRFD Bridge Design Specifications, four types of limit states are defined: (i) Strength limit states, (ii) Extreme-event limit states, (iii) Service limit states, and (iv) Fatigue limit states. Therefore, the design objectives in the LRFD methodology are to demonstrate that (i) the available resistance (i.e., for strength and extreme-event limit states) is sufficient; (ii) other structural conditions (e.g., tolerable deformations in service limit states) are within tolerable limits; and (iii) the structural performance is adequate for all foreseeable load conditions arising during the design life of the structure. In general, all of these limit states must be considered in the design of structural elements, although not all limit states are directly applicable for geotechnical design. These limit states are described in more detail in the following subsections. 3.2.4.2 Strength Limit States Strength limit states are those related to the strength (i.e., generally referred to as nominal resistance in the LRFD con- vention, as defined subsequently) and the stability of struc- tural components during the design life of the structure. For each strength limit state, a design equation can be generically expressed as: Where Rn = the nominal resistance of a given structural component for the strength limit state being considered; φ = a non-dimensional resistance factor related to Rn; Qi = the i-th load type that participates in this limit state; γi = a non-dimensional load factor associated with Qi; ηi = a load-modification factor; and N = the number of load types considered in the limit state. These quantities are described in the following paragraphs. φ γ ηR Qn i i i i N ≥ = ∑ 1 3 2( )- Nominal resistance is the resistance of an entire structure (or of one of its components), which is established based on stresses or deformations or is a specified strength of the materials involved in the structure. In general, nominal resis- tances of structural components are derived from the specified materials and dimensions. For example, the specified tensile yield strength of a steel bar is typically a nominal strength. However, the nominal resistance of soils and other natural materials is obtained differently. The nominal resistance of soils is derived using suitable field/laboratory methods or other acceptable means (e.g., correlations between field test results and soil strength parameters). The nominal resistance of soils commonly represents an ultimate strength of the soils. For example, the internal friction angle of granular soils, which is routinely estimated from field/laboratory tests or cor- relations, is an ultimate strength to be used in establishing the nominal resistance of soils. Resistance factors commonly reduce nominal resistances; therefore, they are typically ≤ 1.0. Section 10, Foundations, and Section 11, Abutments, Piers, and Walls, of the LRFD Bridge Design Specifications (AASHTO, 2007) present prescribed values of resistance factors for geotechnical design of bridge substructure components. Load factors (γi) are statistically based multipliers that are used in the LRFD method to account for load variabil- ity sources (e.g., frequency of loads, inaccuracies in load estimation, and likelihood of simultaneous load occur- rences). While the resistance factors remain the same once they are selected, different γi are selected for different load combinations. For strength limit states, load factors are typically ≥ 1.0 if the acting load is destabilizing. Conversely, load factors are ≤ 1.0 if the acting load component tends to stabilize the structure. An example of stabilizing loads is the horizontal force that arises from soil passive pressures that resist the lateral movement of an embedded foundation. Guidance for selecting load factors for different load com- binations in bridge substructure components are contained in Table 3.4.1-1, Load Combinations and Load Factors, and Table 3.4.1-2, Load Factors for Permanent Loads, of Sec- tion 3, Loads and Load Factors, of the LRFD Bridge Design Specifications (AASHTO, 2007). The number of load com- ponents (N) may vary for different load combinations, as presented in AASHTO (2007). Factor ηi accounts for redundancy, ductility, and impor- tance of the structure and varies between 0.95 and 1.05. Addi- tional guidance for the selection of these factors can be found in Section 1.3, Design Philosophy, of AASHTO (2007). 3.2.4.3 Extreme-Event Limit States Extreme-event limit states are those related to infrequent but large loads that have return periods exceeding the design 8

life of the structure. Extreme–event limit states in bridges and substructures include loads arising from seismic events, ice formation, and vehicle and vessel collision. The same design equation used for strength limit states is commonly used for extreme-event limit states, although the load factors are dif- ferent. The load factors that must be considered for different load combinations in extreme-event limit states are contained in Table 3.4.1-1 of AASHTO (2007). 3.2.4.4 Service Limit States Service limit states are those states related to inadequate conditions that may arise during normal operation of the structure but do not cause a collapse. Inadequate conditions may include excessive deformation, excessive settlements, and cracking. For each service limit state, the following con- dition must be met: Where SMAX = the maximum calculated value of a quantity S (e.g., deflection or settlement) expected to occur under normal conditions; and STOLERABLE = the maximum value of S the structure can sus- tain before its functionality is affected. The load factors for different load combinations to be con- sidered in service limit states are contained in Table 3.4.1-1 of AASHTO (2007). Importantly, due to reasons that will be presented subse- quently, overall stability, slope stability, and other stability states are considered service limit states per AASHTO (2007). For these cases, an equation similar to that of strength limit states is used, with the exception that all load factors are selected equal to 1.0 to reflect the assumption that the struc- ture is under normal conditions. S SMAX TOLERABLE -≤ ( )3 3 3.2.4.5 Fatigue Limit States Fatigue limit states are those states in which loads are applied repetitively and may affect the performance of a structure, while the stress levels are significantly below the values used in strength limit states. For example, fatigue limit states are appli- cable to structures that may be sensitive to fracture as a result of repetitive loads (e.g., vehicular loads and dynamic loads). Additional information on fatigue limit states can be found in Article 3.6.1.4.1 of AASHTO (2007). 3.2.5 Resistances and Loads as Random Variables In the LRFD method, loads, Q, and resistances, R, are con- sidered random independent variables with probability density functions fR(R) and fQ(Q) that are usually normal or lognormal (as shown in Figure 3-1), mean values Qm and Rm, and standard deviations σQ and σR, respectively. R and Q are commonly assumed to be probabilistically independent in geotechnical design (Baecher and Christian, 2003). The variability of these random variables can be conveniently expressed through co- efficients of variation (COV), which are defined as: COVs, which also can be expressed as a percentage, are use- ful as they express uncertainty as a fraction (or percentage) of the mean values. Nominal values of loads and resistances, Qn and Rn, are defined as: Q Qm Q n= λ ( )3 6- COV R R R m = σ ( )3 5- COV Q Q Q m = σ ( )3 4- 9 Load and Resistance, Q and R pr o ba bi lity d en sit ie s, f Q a n d f R Load EffectResistance Effect Qm Rm Qn Rn γ Qnφ Rn Figure 3-1. Probability density functions for load and resistance.

where λQ and λR are the bias factors for loads and resist- ances, respectively. Bias factors represent ratios of measured to predicted values of loads or resistance. In obtaining bias factors, predictive formulas used in the common practice or contained in design codes are considered. On the other hand, with a sufficiently representative database of measured loads and resistances of a structure component, statistical analyses can be performed to obtain bias factors and thereby assess the efficiency of design formulas in predicting measured values. In the case of resistance, predicted resistance are on average greater than measured resistances; therefore, λR > 1 and safe predictions are produced. Conversely, predictions are uncon- servative when λR < 1. Design values of resistance are obtained by reducing nom- inal resistances with a resistance factor, φ, that is usually ≤ 1.0. Conversely, design values of loads are obtained by increasing nominal load values using a load factor, γ, that is usually ≥ 1.0 (Figure 3-1). The random variables Q and R are related by the safety mar- gin M, another random variable, which is defined as M = R − Q. According to this definition, a combination of Q and R values results in a safe condition when M ≥ 0. An alternative definition of safety margin is M′ = R/Q, in which case, the pair Q and R results in a safe condition when M′ ≥ 1. Note that the alterna- tive definition coincides with the traditional ASD format using factors of safety. A probabilistic density distribution for M, fM(M), with mean Mm = Rm − Qm and standard deviation σM, can be obtained based on the distributions of R and Q (Figure 3-2). The condition M = 0 is the limit state. If the alternative definition of safety margin is used, a distribution fM′(M′) for M′, with mean M ′m = Rm/Qm and standard deviation σ′M, can be obtained. In this case, the condition M′ = 1 is the limit state. For the alter- native definition, an equation format similar to that of M is obtained by calculating log (R/Q = 1), or log R − log Q = 0. R Rm R n= λ ( )3 7- As illustrated on Figure 3-2, loads can potentially be larger than resistances and the probability that R < Q is non-zero. The area under the probability density distribution fM(M) in the interval M ≤ 0 is the probability of failure, Pf, which is defined as Pf = Pi (R < Q) = P (R/Q < 1) = P (ln R/Q < 0). Probability of failure is a small number in practice; there- fore, the reliability index, β, can be used instead to quantify the likelihood of failure. The reliability index is defined as the number of standard deviations, σM, of the probability density distribution fM(M) that exists between the mean value, Mm, and the limit state (i.e., M = 0) (Figure 3-2). In other words, β is the “distance” between points Mm and 0 on the M-axis that is normalized by σM. R and Q are assumed to be probabilistically independent and it follows that the reliability index can be expressed as: If the alternate definition of safety margin is used, the reli- ability index can be expressed as: The reliability index increases when the probability of fail- ure decreases and σM (or COVM) decreases. For β ≤ 2, the reliability index is computed to be similar for both normal and lognormal probability distributions. For β > 2, the divergence for β for these distributions tends to increase significantly (Baecher and Christian, 2003). If R and Q are normally distributed, the probability of fail- ure, Pf, can be expressed as a function of β as follows: where Φ−1 is the inverse of the cumulative distribution Φ of a standard normal function. Pf = −( )−Φ 1 3 10β ( )- β σ = − ′ ln ln ( ) R Qm m M 3 9- β σ σ = = −M R Qm M m m M ( )3 8- 10 Safety Margin, M = R - Q Pr ob ab ilit y De ns ity , f M Pf 0 Mm = Rm - Qm = β σm Mm Figure 3-2. Probability density function of safety margin, M.

Values of the cumulative distribution of Φ and/or its inverse can be obtained from various probability and statistics refer- ences (e.g., Baecher and Christian, 2003) or can be computed using statistical software. 3.2.6 Approaches for Calibration of Resistance and Load Factors One of the objectives mentioned for the LRFD method was to provide γ and φ factors that relate to acceptable levels of Pf. This relationship is established through a calibration, which is performed by fixing one of the factors (usually the load factor) and calibrating the other. Therefore, one factor cannot be modified without modifying the other. Calibrations can be per- formed using the following methods, each with an increasing level of complexity (Withiam et al., 1998): • Method A: Calibration using engineering judgment; • Method B: Calibration by matching factors to FS in ASD- based design codes; and • Method C: Calibration using reliability-based procedures. A description of each of these methods is presented in the following paragraphs. Method A: Calibration Using Engineering Judgment This method is best suited for situations where a great deal of experience is available among a summoned team of design professionals (for example, a panel of experts). This method can, in theory, be advantageous because it may incorporate proven design practices that have led to safe and cost-efficient projects. This approach may increase the confidence of other design engineers in certain design procedures. Disadvantages of this method include the possibility that the judgment of the panel members may be unintentionally biased. Method B: Calibration by Matching Factors to Safety Factors Contained in Design Codes In this method, resistance factors are calibrated by matching or calibrating them to FS values used in the ASD format. This approach is appealing because of its mathematical simplicity, consistency with earlier design practice, and transparency to most practicing engineers. This approach is commonly the first to be used until load and resistance statistics are available. However, the approach may not always address all sources of uncertainty in an explicit manner. In this method, a resistance factor can be calibrated from a FS value as follows: where all variables were previously defined. φ γ≥ Σ Σ i i Q FS Q ( )3 11- If the loads are limited to dead and live loads, therefore: where subscripts DC and LL refer to permanent and live loads, respectively. Method C: Calibration Using Reliability-Based Procedures In this method, factors are calibrated according to a relia- bility analysis and are based on empirical data (e.g., load-test data). In addition, a tolerable level of uncertainty is selected. Tolerable levels of uncertainty are expressed through a target value of the reliability index, βT, which reflects an accepted, low probability of failure for a given structure type and load scenario. This method is more complex than Methods A and B and requires that adequate and sufficient empirical information be available. Comparative designs help evaluate the factors obtained in this method and correlate them with factors obtained using other methods. An advantage of this method is that it can provide more explicit insight on the bias of certain predictive design formulas and can help identify and quantify the largest sources of uncertainty arising in design. The method may not be amenable and transparent for engineers unfamil- iar with reliability concepts. Three different levels of calibration complexity can be achieved in Method C [Withiam et al. (1998)]—Levels I, II, and III—each of which is described in the following paragraphs. Level I. Level I calibration is referred to as a first-order second-moment (FOSM) calibration methodology. At this level, the random variables R and Q and their mathematical derivatives used to derive β are only approximated. As dis- cussed earlier, R and Q are assumed to be statistically inde- pendent. The key simplification in this method is that only the first-order derivatives of the squared values of R and Q and/or their derivatives (i.e., known as second moments in probabil- ity) are included, while higher-order terms are disregarded. In this method, the reliability index β is expressed as a linear approximation of R and Q around the mean values. An advan- tage of this method is that it can provide approximate, closed- form approximations for resistance factors. If the random variables Q and R are normally distributed and statistically independent, the resistance factor can be esti- mated as (Withiam et al., 1998): where all variables were defined previously. φ λ γβ σ σ= + + R i i m T R Q Q Q Σ 2 2 3 13( )- φ γ γ≥ + +( ) DC DC LL LL DC LL Q Q FS Q Q ( )3 12- 11

If Qi involves permanent and live loads, the resistance factor can be calculated as: where all variables were defined previously. If the random variables are lognormal, the resistance fac- tor can be calculated as follows (Barker et al., 1991; Withiam et al., 1998): If Qi involves permanent and live loads, the resistance factor can be calculated as: The Level I calibration is computationally simple and the relative contribution of each variable to the load and resistance factors can be readily identified. Occasionally, this calibration φ λ λ γ = + + + + ⎛⎝ ⎞⎠R DC DC LL LL DC LL R Q Q COV COV COV 1 1 2 2 2 λ λ βDC DC LL LL T R DC Q Q COV COV+ + + + ⎛⎝ ⎞⎠ ( )exp ln 1 12 2 COVLL2 3 16 ( )⎡⎣ ⎤⎦ ( )- φ λ γ β= + + +( ) + ∑R i i Q R m T R Q COV COV Q COV 1 1 1 1 2 2 2exp ln COVQ2 3 15( )⎡⎣ ⎤⎦ ( )- φ λ γ γ γ λ β σ σ= +( ) +( )+ + R DC DC LL LL DC DC LL LL T R D Q Q Q Q 2 C LL2 2 3 14 + σ ( )- procedure may provide erroneous results if higher derivatives of the random variables contribute significantly to uncertainty but are left out in the simplification. However, for most geo- technical design, higher-order derivatives of the random vari- ables are uncommon or are disregarded because the random variables participate in linear or up to quadratic equations. Level II. The Level II calibration is an advanced first- order second-moment (AFOSM) procedure (Hasofer and Lind, 1974; Baecher and Christian, 2003). In this procedure, the limit state function (e.g., M = 0) is first approximated as a linear function, and M is evaluated for a combination of R and Q at a strategically selected “design point” (labeled Point B on Figure 3-3) The design point is chosen to be on the surface of the joint probability distribution f(R, Q) (shown as contour lines on Figure 3-3) and along the plane defined by the limit condition M = 0 (straight dotted line labeled on Figure 3-3) that is tangent to the joint probability surface. In this method, design point B is selected because Point B is at the peak of the bell curve that rises and intersects the f(R, Q) surface and the M = 0 plane and thereby has the highest probability of occur- rence. The most “probable” occurrence of R and Q is Point A, located at the “highest” point on the surface. However, Point A does not represent a limit state because it is off the M = 0 plane. On Figure 3-3, the distance between Points A and B is the reliability index, β. One key step in this method is to numerically locate Point B, or equivalently, the minimum “distance,” β. Numerical 12 Re sis ta nc e, R Load, Q M= R- Q= 0 Mean Q Contours of the joint probability distribution of R and Q Shape of the intersection of the joint probability distribution of R and Q with the limiting state, M=0 Distance between "design point" and the mean of R and Q = ββ "Design point" of highest probability density on the limiting state curve or surface: point at which approximating surface is tangent A B M ea n R Figure 3-3. Limit state surfaces in the calculation of reliability index.

evaluations that consider iteratively values of the random vari- ables are conducted and the distance β is recalculated until a minimum value of β is found. The iteration starts by assum- ing an initial value for the distance A-B. A disadvantage of this method is that the computational effort can be significant for certain problems and that a significant volume of data is nec- essary to develop the joint probability distribution correctly and accurately. Level III. The Level III calibration represents the highest level of calibration complexity. This level involves formulat- ing the problem with higher-order derivatives of random variables. For most geotechnical applications, however, this method provides relatively small improvements in the accu- racy of calculated load and resistance factors when compared to those values provided by Level II calibrations. Therefore, the additional computational effort demanded by this level of analysis generally does not warrant its use. In this investigation, Method C, Levels I and II, were used. 3.2.7 Steps to Perform the Calibration of Resistance Factors To perform the calibration of resistance factors, the follow- ing steps are taken (Withiam et al., 1997; Allen et al., 2005): 1. Establish the limit state function (i.e., M = 0) that explic- itly incorporates load and resistance factors, γ and φ; 2. Obtain preliminary probability density function (PDF, usually normal or lognormal), cumulative density func- tions (CDFs), and statistical parameters for random vari- ables R and Q; 3. Select an acceptable probability of failure, Pf, and a corre- sponding target reliability index, βT; 4. Fix load factors in the limit state using statistics or other means; 5. Adjust statistical parameters until there is a best-fit of the CDFs with data points; 6. Perform, in a Monte Carlo simulation, the following steps: a. Estimate an initial, trial value for the resistance factor; b. Generate random numbers and generate values for R and Q that extrapolate the existing data; and c. Calculate random values of the limit state function, M; 7. Using graphical methods or other means, obtain the β value that makes M = 0. Compare the calculated β with the target reliability index, βT; modify the resistance fac- tor and repeat the simulation until the calculated β co- incides with βT. At this point, the final, calibrated resistance factor is obtained. Each of the previous steps is discussed in the following sub- sections. Step 1: Establish a Limit State Function The limit state function is defined as (Allen et al., 2005): where R and Q are random variables representing resistance and the maximum load, respectively. A design equation repre- senting Equation 3-17 requires that φRRn − γQQmax ≥ 0, where φR is a resistance factor; Rn is a random variable representing the nominal resistance, γQ is a load factor, and Qmax is a random variable representing the maximum load. When M = 0, a non- random value for Rn can be related by the following relation: Using the previous equation, the general expression (Eq. 3-17) for the limit state function, M, can be written as: Note that the two terms in Equation 3-19 that contain Qmax are actually two separate random variables, each with different statistical parameters and characterization, and each with both non-random and random components. The quantity Qmax as used in the two terms of Equation 3-19 illustrate that the non- random part of the resistance and load random variables can be related. Each of the random variables of Equation 3-19 is gen- erated separately in the Monte Carlo simulation. The simula- tions are unaffected if the random variables of Equation 3-19 are multiplied or divided by a non-random factor. Therefore, to simplify the calculations, both random variables are nor- malized by the non-random value Qmax, which is equivalent to adopting Qmax = 1 for the non-random components above (Allen et al., 2005). Step 2: Develop PDFs and Statistical Parameters for R and Q In this step, the random variables are assigned a PDF and their statistical parameters are estimated based on existing data. The two most common distributions considered in geo- technical design are normal and lognormal. If the variable Qmax is normally distributed, random values, Qmax i, of this variable can be generated as: where Qmax i = a randomly generated value of the normal vari- able Qmax; Qmax mean = mean of Qmax; Q Q COV zQ imax i max mean= +( )1 3 20( )- M Q Q Q R = ⎛ ⎝⎜ ⎞ ⎠⎟ − γ φ max max ( )3 19- R Qn Q R = γ φ max ( )3 18- M R Q= − ( )3 17- 13

COVQ = coefficient of variation of the bias of Qmax; zi = standard normal variable, which is the inverse Φ−1(uia) of the normal function Φ; and uia = a random number between 0 and 1 (represent- ing a random probability of occurrence). In addition, Qmax mean = λQ Qo, where λQ is the normal mean of the bias of Qmax, and Qo is a non-random scaling value. If the variable Qmax is lognormal, random values of this variable can be generated as: where μln Q = lognormal mean of Qmax and σln Q = lognormal standard deviation of Qmax. The above parameters can be obtained from the normal parameters defined previously as: and If the resistance is modeled as a lognormal variable, the first term of Equation 3-19 can be randomly generated as: where μln R = lognormal mean of Rn; σln R = lognormal standard deviation of Rn; zi = standard normal variable, which is the inverse Φ−1(uib) of the normal function Φ; and uib = a random number between 0 and 1 (representing a random probability of occurrence, and being inde- pendently generated from uia). The above parameters can be obtained from the normal parameters for Rn as: and where Rn mean = mean of Rn and COVR = coefficient of variation of the bias of Rn. σ ln R RCOV= +( )ln ( )1 3 26- μ σ ln ln R n mean R R= ( )−ln ( ) 2 2 3 25- R zni Q R R R i= +( )γφ μ σexp ( )ln ln 3 24- σ ln Q = +( )ln ( )COVQ 1 3 23- μ σ ln Q max mean ln = ( )−ln ( )Q Q 2 2 3 22- Q zimax i ln Q ln Q= +( )exp ( )μ σ 3 21- In addition, Rn mean = λR Ro, where λR is the normal mean of the bias of Rn and Ro is the non-random scaling value defined previously. Step 3: Select Target Reliability Index Target reliability indices are selected based on the type of structure, importance of structure (i.e., related to conse- quences of failure), and the structural redundancy. Structural redundancy refers to the ability of a structure to transfer loads to other members if one of its supporting members fails. Tar- get reliability indices typically range between 2 and 3 for typi- cal geotechnical design (Barker et al., 1991). Allen et al. (2005) recommend selecting βT close to 2, the lower end of the typical range, when the structural component is not critical or it is redundant, and close to 3, the upper end of the range, when the structural component is critical or it is non-redundant. Zhang et al. (2001) suggested that it is acceptable to assign to individual structural elements participating in a group a prob- ability of failure that is higher than that of the group. Allen et al. (2005) suggested that an individual element of a substruc- ture can be considered redundant if the reliability index of the entire system is significantly lower (i.e., 0.5 lower) than that of individual components. This situation may occur in geotech- nical systems that rely on numerous structural elements (e.g., various layers of geosynthetic or steel reinforcement in a retain- ing structure or various driven piles in a pile group). Systems with various structural elements tend to have greater structural redundancy and thereby result in a higher overall reliability index than systems with few resisting elements. For example, a pile group is significantly more redundant than a single drilled pile. This concept will be applied to SNWs, as discussed in the following paragraph. Resistance factors for shallow foundations have been cali- brated using βT = 3.0 (corresponding to Pf = 0.14%, a relatively low value), as these systems are not highly redundant (Baker et al, 1991). Resistance factors for deep foundations have been calibrated for βT = 2.33 (corresponding to Pf = 1%), as driven piles and drilled shafts are typically installed as part of pile/shaft groups (Paikowsky et al., 2004) and thereby carry some struc- tural redundancy. D’Appolonia (1999) used βT = 2.50 to cali- brate resistance factors for pullout in geogrids, which is a system that tends to be redundant as multiple reinforcement layers are installed with a typical vertical spacing of 1 to 1.5 ft. Allen et al. (2001) adopted βT = 2.33 for the calibration of pullout resis- tance factors in MSE walls. Step 4: Establish Load Factors An estimate of the load factor needs to be performed to evaluate whether the load factors [typically those used in AASHTO (2007)] are applicable or whether different load factors need to be proposed. 14

Allen et al. (2005) present the following equation to estimate the load factor when load statistics are available: where γQ = load factor; λQ = mean of the bias for the load Q; COVQ = coefficient of variation of the load bias (i.e., measured-to-predicted ratio for loads); and nσ = number of standard deviations from the mean of Q. This procedure is approximate and is valid for any CDF. The greater the selected value of nσ is, the lower the probabil- ity will be that the measured loads exceed the nominal load. Typically, the number of standard deviations of the load bias is selected at nσ = 2, which results in a probability of approx- imately 2% for the factored load values (Allen et al., 2005) to exceed the nominal load. This procedure is currently used in the AASHTO LRFD Bridge Design Specifications and in the Ontario Highway Bridge Design Code (as referenced in Nowak, 1999; Nowak and Collins, 2000). It is recognized that this pro- cedure is based on judgment and not necessarily on a rational procedure (Allen et al., 2005). Step 5: Best Fit Cumulative Density Functions to Data Points The selected CDFs for load and resistance must be fitted to the data points to assess the adequacy of the selected CDFs and their statistical parameters. The CDF for loads, which is plot- ted as variate, must be compared to the lower tail of the load data point distribution. Conversely, the CDF for resis- tance must be compared to the upper tail of the resistance data point distribution. Finally, both load and resistance approxi- mations should be plotted side by side and compared. Step 6: Conduct Monte Carlo Simulation A Monte Carlo simulation is a statistical procedure used to artificially generate many more values of load and resistance than are available from measured data points. Therefore, this technique can be used to extrapolate the data at both ends of the distribution. In a Monte Carlo simulation, random numbers are gener- ated independently for each of Qmax and Rn, assuming that these variables are statistically independent. New sets of uia and uib are generated a minimum of 10,000 times to calculate new val- ues for Qmax i and Rn i and to develop complete distributions of these random variables. As Qmax and Rn are either normal or lognormal, closed form solutions may be obtained for the CDFs of the limit state. γ λ σQ Q Qn COV= +( )1 3 27( )- Step 7: Compare Computed and Target Reliability Indices Following a cyclic calculation scheme, computed and tar- get reliability indices are compared at the end of each Monte Carlo simulation. The iteration is stopped when the differ- ence between the computed and target reliability indices is negligible. 3.3 Review of Current U.S. Soil-Nailing Practice 3.3.1 Introduction In this section, the results of a review of current U.S. practice of soil nailing are presented. The results of the review are pre- sented as descriptions of the most significant construction steps of SNWs and the main components of an SNW. While this sec- tion presents a summary of the review, more detailed informa- tion of construction aspects and SNW elements are contained in Appendix B. After the main components of a SNW are iden- tified in this section, a discussion is presented of the limit states to be considered in the design of SNWs based on the LRFD method. For each of these limit states, a description of key vari- ables participating in the limit state equation is provided. 3.3.2 Basic Description of Soil Nail Walls SNWs are earth-retaining structures constructed using passive reinforcing elements, referred to as soil nails. The term “passive” is used because soil nails are typically not post- tensioned. SNWs are constructed using “top-down” methods, where excavation lifts are created and reinforcing elements are installed after each lift excavation sequence. Soil nails are installed in each excavation lift to provide lateral support to the soil exposed in each excavation level. As each excavation lift is commonly 5 ft deep, nails are installed at a vertical spac- ing of approximately 5 ft. Soil nails are commonly installed with a horizontal spacing of 5 ft also. In U.S. soil-nailing practice, after a lift is excavated, holes (commonly known as “drill-holes,” regardless of whether they are drilled or driven) are created on the exposed excava- tion. Drill-holes are created typically by drilling at an inclina- tion of approximately 15 degrees from the horizontal; then, soil nails are inserted into the holes, and the annulus between the drill-hole and nails is filled with grout. Finally, a facing layer of reinforced shotcrete is applied over the protruding nail heads at the face of the excavation. This cycle is repeated for each subsequent lift of excavation. Appendix B presents detailed information of other aspects of SNW construction, including contractor’s qualifications, information on suitable methods to store and handle various materials used in SNW construction, nail installation, grouting, and soil nail testing. 15

A more detailed description of construction aspects related to SNWs is presented in Byrne et al. (1998) and Lazarte et al. (2003). Soil nails and the facing layer both contribute to excavation stability. While soil nails provide support to the soils retained behind the wall, the facing provides connectivity and struc- tural continuity to nails, thus making the SNW act as a unit. SNWs have been used successfully in a wide variety of sub- surface conditions, including soils and rocks. Although nails are used in soil and weathered rock, the term “soil nail” will be used interchangeably in this document whether the nails are installed in soil or rock. SNWs can be more advantageous than other top-down retaining systems when the construction takes place in granular soils exhibiting some cohesion and/or in weathered, soil-like rock. SNWs are generally unsuitable when they are built below the groundwater table. Additional information related to favorable and unfavorable subsurface conditions for constructing SNWs are presented in Byrne et al. (1998) and Lazarte et al. (2003). In transportation projects, including those involving bridge substructures, SNWs are routinely used as permanent struc- tures having a minimum design service life of 75 years per AASHTO (2007). SNWs that are built as temporary structures [i.e., service life up to 36 months per AASHTO (2007)] are rou- tinely used in urban settings for shoring up temporary excava- tions. However, the use of SNWs as temporary earth-retaining systems in bridge substructures is uncommon. This document focuses on SNWs used as permanent structures. The practice of SNWs varies throughout the United States, particularly in non-public projects. SNW practice differing from that described in this document may include the use of different nail types (e.g., hollow steel bars as opposed to solid bars), different nail materials (e.g., synthetic materials instead of steel), and novel construction procedures. However, none of these variations are discussed in this document. 3.3.3 Main Components of Soil Nail Walls The main components of SNWs used in typical U.S. practice are identified on Figure 3-4. These components are described in the following paragraphs. Additional informa- tion on SNW components is contained in Appendix B. In addition, a more detailed description of typical components of SNWs is presented in Byrne et al. (1998) and Lazarte et al. (2003). 3.3.3.1 Steel Bars Reinforcing soil nails are solid steel bars. The bars develop tensile stresses in response to the outward deformation of soils that are retained in each excavation lift. Soil move- ment can occur during excavation or after excavation when external structural loads (e.g., weight of superstructure) are applied. Steel bars used in SNWs are threaded and, as men- tioned earlier, are not commonly post-tensioned. In some cases, however, the upper rows of soil nails are post-tensioned as a means to control and limit the outward movement of the wall. Other elements commonly used in connection with the soil nail bars are centralizers and bar couplers (not shown in Figure 3-4, see additional descriptions in Appendix B). 3.3.3.2 Facing System Facing systems typically consist of temporary and perma- nent facing. Temporary facing is applied on the exposed soil as each lift is excavated to provide temporary stability. Per- manent facing is applied over the temporary facing to provide architectural finish and structural continuity. Temporary facing most commonly consists of reinforced shotcrete. The reinforcement used in the shotcrete usually consists of (i) welded wire mesh (WWM), which is installed over the entire facing; (ii) additional horizontal bars (commonly called “waler bars”) that are placed around nail heads; and (iii) additional vertical bearing bars that are also placed around nail heads (see bottom of Figure 3-4). Permanent fac- ing may consist of cast-in-place (CIP) reinforced concrete, reinforced shotcrete, or precast concrete panels. 3.3.3.3 Grout Grout used in SNWs may consist of a mixture of neat Port- land cement mortar or fine aggregate, cement, and water. Grout typically covers all the length of the steel bars, transfers tensile stresses from the bars to the surrounding soil, and provides cor- rosion protection to the bars. Grout is commonly applied in the drill-holes under gravity using the tremie method. 3.3.3.4 Components at the Soil Nail Head To provide connection between nails and facing at the pro- truding soil nail heads, connecting components are installed at this location. These components typically consist of nut, washers, bearing plate, and headed-studs or anchor bolts. The headed-studs or anchor bolts are attached to the bearing plate. Additional descriptions of nail head components are provided in Appendix B. 3.3.3.5 Drainage System A drainage system is typically installed behind the SNW fac- ing to collect groundwater occurring behind the facing and to convey it away from the wall. The most commonly used drainage system consists of composite, geosynthetic drainage strips, which are also referred to as geocomposite sheet drains 16

17 SEE DETAIL SV 15° (TYP) SOIL NAIL (TYP) PERMANENT FACING H L TEMPORARY FACING DRAINAGE SYSTEM NAIL HEAD GEOCOMPOSITE STRIP DRAIN STEEL BAR GROUT BEARING PLATE WELDED WIRE MESH REINFORCEMENT PERMANENT FACING (e.g., CAST-IN-PLACE REINFORCED CONCRETE) TEMPORARY FACING (SHOTCRETE) HEADED STUD WASHERS VERTICAL BEARING BARS WALER BARS Figure 3-4. Typical cross section of a soil nail wall in common U.S. practice.

(see Appendix B). Drainage strips arrive at the site in rolls from the factory. Strips are unrolled vertically against the exposed face of each excavation lift; subsequently, shotcrete is applied over the drains and exposed soil. In the next excavation lift, more material is unrolled and is extended to the bottom of the excavation. Underdrains made of perfo- rated plastic pipe may be also installed to collect and re- route groundwater accumulating at the SNW base water from the wall [see additional details in Appendix B and Lazarte et al. (2003)]. 3.3.3.6 Corrosion Protection Soil nails in permanent structures require chemical and/or physical protection (the latter referred to as encapsulation) from corrosion. The required level of corrosion protection increases as site conditions become more aggressive. In U.S. practice, the lowest level of corrosion protection is provided by the cement grout alone. If the grout mix is appropriately designed and suitable grouting techniques are applied, grout can provide adequate protection in non- corrosive to mildly corrosive environments. Higher levels of corrosion protection are required in permanent, more corrosive environments. Higher levels of corrosion protection can be achieved by grouting the soil nail bars in a phased process that involves providing the bars with the first level of protection under controlled conditions. In this procedure, the bars are first inserted in a protective sheath consisting of corrugated high-density polyethylene (HDPE) or polyvinyl chloride (PVC) pipe. Then, the annulus between the sheath and bar is filled with grout and cured under controlled conditions at the shop. After the grout is fully cured, the sheathed bar is shipped to the site and placed in the drill-hole. Additional grout is pumped into the annulus between the sheathing and the drill-hole. Due to the two layers of grout that are in place, this system is usually referred to as double-corrosion pro- tection level. Corrosion protection also can be increased by using fusion- bonded, epoxy-coated bars, instead of bare bars. The combined use of epoxy-coated bars, sheathing, and final grout provides the highest level of corrosion protection. Other aspects of cor- rosion measures are addressed in Appendix B. A more detailed description of corrosion protection used in SNW applications is provided in Lazarte et al. (2003). 3.3.3.7 Other Elements and Materials Other elements and materials used in the construction of SNWs include protection film, additives for shotcrete and grout, and fittings. Additional information on these elements is provided in Appendix B and in Lazarte et al. (2003). 3.4 Limit States in Soil Nail Walls 3.4.1 Introduction Various SNW components including nails, facing, and nail head connectors contribute to stability and structural perform- ance. As a result, every potential limit state involving these ele- ments should be considered according to the design philosophy of LRFD. Each of the limit states identified for SNW design is addressed in the following sections. The terminology used herein regarding overall stability and strength limit states of SNWs is selected to be consistent with the terminology used in Section 11 of the LRFD Specifications. This terminology differs slightly from that used in Byrne et al. (1998) and Lazarte et al. (2003); however, the principles behind these limit states are similar in all of these publications. The following limit states are considered for SNW design: • Service limit states: – Overall stability [Figures 3-5(a) and 3-5(b)]; – Wall lateral displacement; – Wall settlement; and – Lateral squeeze. • Strength limit states: – Safety against soil failure, including:  Sliding stability [Figure 3-5(c)] and  Basal heave [Figure 3-5(d)]. – Structural limit states, including:  Nail pullout [Figure 3-5(e)];  Nail in tension [Figure 3-5(f)];  Facing structural limit states, including:  Flexure [Figure 3-5(g)];  Punching-shear [Figure 3-5(h)]; and  Headed-stud in tension [Figure 3-5(i)]. Extreme-event limit states for SNWs are commonly lim- ited to those arising from seismic loads. Fatigue limit states, which are uncommon in the design of SNWs, are not addressed in this document. For most practitioners, the consideration of overall stability as a service limit state may not be intuitive and may appear to be incorrect. However, this selection is necessary because load factors used in this state are equal to 1.0 in the current LRFD Bridge Design Specifications (AASHTO, 2007). This approach for overall stability may be modified in future editions of the AASHTO LRFD Bridge Design Specifications; therefore, appro- priate changes should be also made for SNWs. In Section 3.5, more detailed discussions of overall stability in LRFD are presented. Considering basal heave a service limit state is not intuitive either. However, because the load factors for basal heave are also 1.0, basal heave is considered a service limit state in this document, in order to be consistent with the current LRFD Bridge Design Specifications (AASHTO, 2007) approach. 18

19 OVERALL STABILITY LIMIT STATES FOR SOIL FAILURE STRUCTURAL LIMIT STATES SOIL RESISTANCE SLIP SURFACE (a) SOIL RESISTANCE PULLOUT RESISTANCE SLIP SURFACE (b) SOIL RESISTANCE (SLIDING AT BASE) (c) SOIL RESISTANCE (d) SOFT COHESIVE SOIL TENSILE RESISTANCE (f) SLIP SURFACE PULLOUT RESISTANCE(e) FLEXURE RESISTANCE (FACING)(g) PUNCHING SHEAR RESISTANCE (FACING) (h) HEADED-STUD TENSILE RESISTANCE (i) Source: Modified after Lazarte et al. (2003) Figure 3-5. Limit states in soil nail walls: overall stability: (a) slip surface not intersecting nails and (b) slip sur- face intersecting nails; soil failure: (c) sliding at base and (d) basal heave; and structural: (e) pullout, (f) nail in tension, (g) flexure of facing, (h) punching-shear in facing, and (i) headed-stud in tension.

However, sliding stability is considered a strength limit state as it is a “safety against soil failure” case, per Section 10, Foun- dations, of AASHTO (2007). The limit states listed previously are discussed in the following sections. 3.4.2 Service Limit States 3.4.2.1 Overview Service limit states related to a stability condition (i.e., over- all stability and basal heave) are described in this section. Ser- vice limit states related to deformations under regular service conditions are described subsequently in Section 3.4.6. 3.4.2.2 Overall Stability Overall stability of SNWs [shown schematically in Fig- ures 3-5(a) and (b)] must be considered when a potential slip surface extends through the soil under and behind the wall and through some or all nails. If the slip surface does not intersect the nails [Figure 3-5(a)], the soil shear resistance mobilized along slip surfaces is the only contribution to stability. Soil resistance can be frictional, cohesive, or both, depending on the soil type and/or loading conditions (e.g., drained or undrained loading). If the slip surface intersects some or all nails, the nail pullout resistance mobilized in the soil nails behind the slip surface also contributes to stability. The nail tensile resistance is treated separately as a structural strength limit state, as discussed in Section 3.4.4.3. In the ASD method, the verification of overall stability safety includes the use of a factor of safety, which is derived as a ratio between resisting and destabilizing forces or moments. In the LRFD framework, the safety for overall stability must be veri- fied by demonstrating that the factored nominal resistances are greater than or equal to the overall effect of the factored loads. If the loads have a destabilizing effect, as most external loads do, load factors applied to these loads are greater than 1.0. If the acting loads have a stabilizing effect (e.g., passive earth pres- sures provided by berm at the wall toe resisting the outward SNW movement), the load factors applied to these loads are less than or equal to 1.0. Overall stability of SNWs is commonly evaluated using procedures based on two-dimensional, limit-equilibrium methods used in traditional stability analyses. Similar to the sta- bility analyses of slopes, in limit-equilibrium stability analyses of SNWs, several potential slip surfaces are considered and an FS is calculated for each case. The analysis is repeated until the surface with the lowest calculated FS is found. The lowest calculated FS must be equal to or greater than the minimum acceptable FS established for the structure and condition. Various shapes of the slip surface have been considered in SNW design procedures, including (i) planar (Sheahan et al., 2003); (ii) bi-linear (Stocker et al., 1979; Caltrans, 1991); (iii) parabolic (Shen et al., 1981); (iv) log spiral (Juran et al., 1990); and (v) circular (Golder, 1993). A comparison of FS results obtained with different SNW design procedure and slip surfaces indicates the slip surface shape selection does not seem to affect significantly the calculated FS (Long et al., 1990). Stability analyses for SNWs are commonly performed using computer programs specifically developed for the design of SNWs because these programs give design engineers greater ability to quickly analyze multiple design scenarios for these walls. The most commonly used programs are (i) SNAIL or SNAILZ—free, public-domain programs developed by the California Department of Transportation (Caltrans, 1991 and 2007, respectively)—and (ii) GOLDNAIL (Golder, 1993), a commercial program. Alternatively, simplified methods con- sisting of design charts (e.g., Byrne et al., 1998; Lazarte et al., 2003) can also be used in preliminary designs. General slope stability computer programs having the ability to model multi- level reinforcement can also be used to assess SNW stability. Manual calculations of stability are rarely performed in real practice. However, the following paragraphs illustrate the manner in which forces participating in a typical SNW prob- lem are considered in the assessment of overall stability using the LRFD methodology (Figure 3-6), where a hypothetical slip surface intersects all nails. Figure 3-6 shows a generic SNW of height H and face batter angle α from the vertical. The ground surface slopes at angle β behind the wall; nails are inclined at angle i from the horizontal. Loads consist of an external surcharge per unit width, Q, and the vertical earth load, EV [i.e., symbol used per Table 3.4.1-2 of AASHTO (2007)]. The slip surface selected in this simplified analysis is planar with an inclination, ψ, from the horizontal. This selection does not affect the validity of this procedure. RS is the nominal soil resistance per unit width (or alternatively, per nail horizontal spacing, SH) mobilized along the slip surface. T is the sum of the nominal pullout resistance of all soil developing behind the slip surface. 20 EV T RS Q β α H ψ i Figure 3-6. Main forces in overall stability.

Overall stability is achieved when the force components act- ing parallel to the failure plane meet the following requirement: The factored nominal resistance is: where φs and φPO are resistance factors for soil shear resistance and nail pullout, respectively. The assumption that T is a resultant force is valid provided that only force-equilibrium is considered. A more rigorous approach would require establishing moment and force- equilibrium conditions simultaneously while considering the distribution of soil nail forces over the wall height. Rs is assumed to have both cohesive and frictional com- ponents and is expressed as: where c = nominal soil cohesion, LS = length of the slip plane, FN = normal force per unit width acting on the slip surface, and ϕf = soil effective friction angle. The normal force, FN, is calculated from force equilib- rium as: The surcharge load may comprise permanent and tran- sient loads originating from the superstructure. Assuming that only dead loads and live loads are present, the surcharge can be expressed as: where QDC and QLL are the permanent/dead and live loads, respectively. The factored destabilizing force along the slip plane is calculated as: where γp = load factor for permanent, vertical earth loads; γDC = load factor for dead load; and Factors Destabilizing Forces∑ ∑× = = + γ γ γ Qi i p Q EV DC DC LL LLQ Q+( )[ ]γ ψsin ( )3 33- Q Q QDC LL= + ( )3 32- F EV Q T iN = +( ) + +( )cos sin ( )ψ ψ 3 31- R c L Fs s N f= + tan ( )ϕ 3 30- Factors Nominal Resistance× = + + ∑ φ φ ψs s POR T icos( ) ( )3 29- Factors Nominal Resistance Factors Destab × ≥ × ∑ ilizing Forces -∑ ( )3 28 γLL = load factor for live loads. As overall stability is treated as a service limit state (AASHTO, 2007), γp = γoc = γLL = 1.0. With Equations 3-30 through 3-33, the force, T, that satisfies Equation 3-28 can be calculated to establish sub- sequently the required nail length. The nail tensile resistance is verified separately, after the maximum load, Tmax, of all nails is obtained. The facing can be designed (or verified, if dimensions and reinforcement were estimated beforehand) for the maximum nail load. The equations presented above were developed for a single- wedge failure plane but can be extended for two- or three- wedge failure plane cases, which would result in more accurate but complex expressions (e.g., as used in the programs SNAIL and SNAILZ). The procedure above was presented to intro- duce some key aspects of overall stability analysis; however, as mentioned earlier, manual calculations are uncommon because versatile computer programs (or, alternatively, simpli- fied design charts) are available to perform these calculations more efficiently. 3.4.2.3 Basal Heave When soft, fine-grained soils exist behind and at the base of an SNW excavation [as illustrated on Figures 3-5(c) and 3-7], the potential for basal heave (i.e., mobilization of bearing resis- tance) should be evaluated. If the excavation depth is excessive for the existing soft soil conditions, unbalanced loads generated during excavation may cause the bottom of the excavation to heave and possibly cause a basal failure. SNWs may be more susceptible to basal heave than other retaining systems because the facing is usually not embedded. In contrast, soldier piles of anchored retaining walls are embedded a considerable depth and provide some resistance to basal heave. Note that basal heave is not common in SNWs as these structures are not rou- tinely built in or over soft, fine-grained soils. This scenario is considered for completeness of feasible limit states for SNWs. Basal heave is akin to a bearing resistance limit state and its evaluation should be similar to that of a bearing resistance limit state. One difference is that basal heave may arise over a short period of time and loads are more appropriately con- sidered at the service level. Consequently, load factors are adopted equal to 1.0. In the current LRFD Bridge Design Spec- ifications (AASHTO, 2007), basal heave is not specifically treated; however, some guidance is included to assess settle- ment occurring behind an anchored wall as a service limit state for movement (e.g., see Article 11.9.3, Movement and Stability at the Service Limit State). However, in that article, there are insufficient guidelines to establish whether an exca- vation in very soft soils is safe or not. In this section, a methodology is proposed to evaluate cases where the potential instability of the base of the excavation is 21

significant. In this procedure, this scenario is treated as a ser- vice limit state, and based on equilibrium. All load factors considered are then γ = 1.0. In this limit state, the following requirement must be satisfied: where φBH = resistance factor for basal heave (AASHTO, 2007); Rs = nominal soil shear resistance for basal heave per unit width [acting along the composite slip surface shown on Figure 3-7(a)]; and Qi = loads acting at the base of the soil block that may be displaced. If all of the excavation is in cohesive soils, Rs is calculated as: R S H S N Bs u u c e= +1 2 2 2 3 35( )- φBH s i N R Q≥ ∑ 1 3 34( )- where Su1 = undrained shear resistance of the fine-grained soil behind the SNW; Su2 = undrained shear resistance of the fine-grained soil below the SNW; H = height of the wall; Nc = cohesion bearing resistance factor (e.g., Terzaghi et al., 1996); and Be = excavation width. The volume of soil that may be displaced and cause heave at the bottom of the excavation is controlled by the excava- tion width, as shown in Figure 3-7(a). In the simplified model of Figure 3-7(a), the width of the soil block that may be dis- placed is = 0.71 Be. For wide excavations, the width of the soil block usually extends behind all nails. When a deposit of soft, fine-grained or weak soil exists under the excavation with a maximum thickness DB [Figure 3-7(b)] and a deposit of stiff material underlies the excava- tion within a depth DB ≤ 0.71 Be, the width of the heave area 2 2 Be( 22 H = Exca va ti on de pt h B e = Exca vation widt h L e = Exca vati on Length H/B e = 0 (very wide excav.) (b) Shallow deposit of soft, fine-grained soil underlain by sti ff laye r (c) Bearing Capacity Factor, NC H S H u soft fi ne -grained soi l (a) Deep deposit of soft, fine-grained soil H S H u Fa il ure su rf ac e (wi d th of ex ca va ti on is ty pi ca lly ve ry la rg e) so ft fi ne- grai ned so il B e D B D B 3 4 5 1 2 4 7 5 8 6 9 10 0 B e /L e = 0 (Long, rectang. ex cav. ) N C H/B e B e /L e = 1 (S qu ar e e xcav. on plan view ) B e /L e = 0. 5 γ HB e √2 /2 B e √2 / 2 B e γ HD B Q Q Source: Modified after Sabatini et al. (1999) Figure 3-7. Basal heave.

at the bottom of the excavation is limited to DB. Therefore, in Equation 3-35, is replaced by DB. Nc depends on the excavation depth, width, and length (Le) and is a function of the ratios H/Be and Be/Le, as shown in Fig- ure 3-7(c) (Terzaghi et al., 1996). Excavations for SNWs are typically very wide and rectangular (i.e., Le >> Be and Be >> H); therefore, it can be conservatively assumed that H/Be = Be/ Le = 0, which results in Nc = 5.14. If the contribution of the soil resistance along the vertical surface behind the wall is disregarded (a very conservative assumption for most SNWs), the total nominal resistance reduces to: The sum of all loads at the base of the soil block is: where γs is the unit weight of the soil behind the wall and QDC is the dead load. The limit state for basal heave at the bottom of the soil block can be also expressed as: where q = QDC/ . This expression is similar to one included in Article 11.9.3 of AASHTO (2007). Clear guidelines about a maximum resis- tance factor (or equivalent minimum “safety factor” in the ASD) for basal heave are not included in AASHTO (2007). In this document, a value of φBH = 0.70 is proposed. Neglecting the soil resistance behind the wall and assuming that QDC = 0, the following simplified expression can be used to estimate the minimum required undrained shear resistance of the soil at the base of the excavation to provide sufficient stability: The above equation can be used as a tool to conserva- tively estimate excavation depths that would result in safe construction. Therefore, for soft soils [i.e., those commonly classified with an undrained shear strength between 12.5 and 25 kPa (250 and 500 psf)] and assuming γs = 17.3 kN/m3 (110 pcf ), excavation depths of less than approximately 8 ft (for Su2 = 250 psf ) and 16 ft (for Su2 = 500 psf ) would result in safe construction. S H u BH s 2 1 5 14 3 39≥ φ γ . ( )- 2 2 Be( ) φ γBH u sS H q5 14 3 382. ( )≥ + - Q H B Qi e s DC= +∑ 2 2 3 37 1 2 γ ( )- R S N Bs u c e= 2 2 2 3 36( )- 2 2 Be 3.4.3 Soil Failure Limit States 3.4.3.1 Overview Strength limit states involving soil failure are generally achieved when the soil nominal resistance is mobilized along a slip surface, including sliding at the base [Figure 3-5(c)]. No other scenario of soil failure is considered for SNWs because overall stability and basal heave (both involving a slip surface) are considered service limit states. The limit state for sliding stability is described in the following paragraphs. 3.4.3.2 Sliding Stability Sliding is an uncommon limit state for most SNWs and is considered here for completeness. Conceptually, this limit state can be considered a particular case of overall stability. The sliding limit state may arise when the block of reinforced soil is underlain by a weak soil layer (Figure 3-8) that determines the location of a critical slip surface. The procedure presented below can be applied for weak layers that are horizontal to sub-horizontal. For non-horizontal slip planes, alternative procedures (including general slope stability analysis) must be used. Software available in the United States has the capability to simulate lock-type slip surfaces and can thereby be used to evaluate sliding stabil- ity where a horizontal weak layer is present. However, the computer programs SNAIL (or SNAILZ) and GOLDNAIL have limited to no capabilities, respectively, to evaluate slid- ing stability. In the procedure presented below, loads caused by lateral earth pressures acting behind the soil block are explicitly con- sidered. Unlike with overall stability scenarios for SNWs, loads in this limit state are assigned load factors ≥ 1.0 because destabilizing effects are clearly separated from stabilizing effects. Lateral earth loads can be evaluated using Rankine or Coulomb theories and by approximating the back surface to a vertical slip surface behind the soil block. The reader is referred to Article 3.11.5.1, Lateral Earth Pressure, of the LRFD Bridge Design Specifications (AASHTO, 2007) for addi- tional information. Sliding is verified using the following expression: where φτ = resistance factor for sliding (AASHTO, 2007); Rs = nominal soil sliding resistance per unit width acting at the base of the soil block; γEH = load factor for horizontal earth loads; PA = resultant of the lateral active earth load per unit width [i.e., designated as EH in Table 3.4.1-2 of AASHTO (2007)]; and φ γ δτ R Ps EH A≥ cos ( )3 40- 23

δ = inclination of the lateral earth load (typically assumed to be equal to the backslope angle, βeq). Based on recommendations presented in Section 11 of AASHTO (2007) for the verification of sliding limit states, any external load, Q, acting behind the retaining structure must be considered to extend outside the block of soil, i.e., up to the vertical dashed line shown in Figure 3-8. The nominal soil sliding resistance can be calculated as: where c = the cohesive resistance of the soil at the base of the block of soil, BL = the base length (considered herein a horizontal slip surface), EV = the weight of the soil block, PA = the resultant of the lateral active earth load per unit width, βeq = the equivalent angle of the backslope, and ϕfb = the effective friction angle at the base of the soil block. External loads, Q, occurring behind the soil block must be taken into account as added lateral loads. Additional details to calculate the effect of these loads can be found in Article 3.11.5.1 of AASHTO (2007). If the slope has no breaks within a horizontal distance 2H from the wall (e.g., ground surface shown as a solid line on Figure 3-8), the slope is considered “infinite” and βeq = β. If the slope exhibits a slope break within a distance 2H from R c B EV Ps L A eq fb= + +( )sin tan ( )β ϕ 3 41- the wall (e.g., ground surface shown as a dashed line on Fig- ure 3-8), the slope is assigned an equivalent inclination angle βeq = tan−1 (ΔH/2H), where ΔH is the slope rise over a distance 2H (Figure 3-8). The design engineer must select c and ϕfb depending on soil drainage conditions (i.e., “free-draining” or “undrained” con- ditions) and possibly other conditions (e.g., cemented or unce- mented soil). Depending on the nature of the soil under the wall, residual values for ϕfb may be used. The passive resis- tance generated in front of an SNW is disregarded because, in common practice, either SNW facings are not embedded or the embedment depth of an SNW is small. In principle, the resistance factor for sliding, φτ, could be selected differently whether drained or undrained conditions are prevalent because strength parameters for drained/undrained condi- tions are based on tests and models commonly producing different errors and uncertainties (e.g., Baecher and Chris- tian, 2003). However, this practice is not yet included in AASHTO (2007). The active force per unit width can be estimated as: where γs = the unit weight of the soil behind the wall, H1 = the effective height over which the earth pressure acts, and KA = the active earth pressure coefficient for the soil behind the wall (can be estimated using the Coulomb or Rank- ine formulations). P H KA s A= γ 12 2 3 42( )- 24 BL Q H1 H θ βeq 2 H ΔH Rn H/3 δ = βeq Broken slope: βeq = tan-1( ΔH2H) PAα Infinite slope: βeq = β EV β Source: Lazarte et al. (2003) Figure 3-8. Sliding stability of a soil nail wall.

H1 is calculated as: where α is the wall face batter angle. 3.4.4 Structural Limit States 3.4.4.1 Introduction Structural limit states (occasionally also referred to as inter- nal limit states) arise when the nominal resistance is reached in structural elements of an SNW (i.e., bars, shotcrete, rein- forcement, and other elements in the facing system). The five structural limit states considered for SNWs [shown schemat- ically on Figure 3-5(e) through (i)] include: • Nail pullout, • Nail in tension, and • Facing limit states (three different limit states). In general, the tensile force of a nail varies along its length. Figure 3-9 shows a schematic distribution of tensile force along the nail. The magnitude of this force at a distance, x, from the bar end is represented by T(x). T(x) increases from 0 at x = 0, to a maximum value, Tmax, somewhere in the middle section of the nail, and then decreases to a value To at the facing. The maximum value, Tmax, is used in evaluations of the pullout and tension limit states. In contrast, the nail load at the wall facing, To, is used to evaluate the facing limit states. Nominal pullout, tension, and facing resistances (i.e., herein identified as RPO, RT, and RF) must be greater than Tmax or To. H H B HL eq1 3 43= + −( )tan tan ( )α β - Byrne et al. (1998) proposed a model that illustrated the con- tribution of each resistance into the resistance of the nail. In this model (Figure 3-9), the pullout resistance increases from the distal end of the nail up to the location of the slip surface. The tension and facing resistances are also illustrated on Figure 3-9. The value Tmax is generally obtained from the output of over- all stability analysis using SNAIL, SNAILZ, or GOLDNAIL or can be estimated using simplified methods (Byrne et al., 1998; Lazarte et al., 2003). Note that Tmax values are a function of the load factor used in the analysis. However, Tmax does not repre- sent service conditions. For most cases of wall geometry and external load conditions, Tmax-s (service conditions) can be esti- mated from data presented by Byrne et al. (1998), as: where SV and SH are the vertical and horizontal nail spacing, and KA, γs, and H are as defined previously. Equation 3-44 is based on the analysis of monitoring results of SNWs under normal, working conditions (Byrne et al., 1998). The force To-s (service conditions) is estimated from Tmax-s with (Clouterre, 1991 and 2002): where Smax is the greater of SV and SH. In addition, based on the instrumentation of soil nails in various in-service SNWs, the following range for To-s can be used (Byrne at al., 1998): T K H S Ss A s H Vo- to 0.70 -≥ 0 60 3 46. ( )γ T T S Ts s so max max max− − −= + [ ]−( )[ ]≤0 6 0 2 1 3. . in (m - b45 ) T T S Ts so max max max− −= + [ ]−( )[ ]≤0 6 0 05 3. . in feet −s ( )3 45- a T K H S Ss A s H Vmax- to 0.80 -= 0 70 3 44. ( )γ 25 To Facing Resistance, RF Tmax Pullout Resistance, RPO Nail Facing T (x) x Slip Surface Calculated by Limit Equilibrium LP Tensile Resistance, RT Figure 3-9. Schematic representation of structural resistances.

3.4.4.2 Pullout Resistance An adequate level of pullout resistance [Figure 3-5(e)] devel- oping along the soil-grout interface is necessary for overall sta- bility. The pullout resistance along a length Lp (shaded area in Figure 3-9) contributes to stability and is mobilized behind the slip surface, as calculated in a limit-equilibrium stability analy- sis. The nominal unit pullout resistance, rPO, (also referred to as load transfer rate) has units of force per unit length and is expressed as: where qU = the nominal bond resistance of the nail/soil inter- face (with units of force per unit area) and DDH = the diameter of the drill-hole. Actual distributions of bond stresses along the grout-soil interface can be complex and may exhibit significant variations along the nail. However, to simplify calculations, the distribu- tion is commonly assumed to be constant along the pullout length; therefore, the nominal bond resistance qU is considered an apparent, average value. For a given pullout length, LP, occurring behind the slip surface, the resulting nominal pull- out resistance, RPO, is: Adequate nail pullout resistance is provided when: where φPO is the resistance factor for pullout resistance and Tmax is the maximum tensile force on the bar, as calculated in stability, limit-equilibrium analyses. Note that this force is not a service load. Therefore, the required nail length behind the slip surface must be: Additional information regarding the bond resistance of soil nails is presented subsequently. 3.4.4.3 Tensile Resistance of Nails An adequate nominal tensile resistance of a nail bar [see Figure 3-5(f)] must be established by verifying that: where φT = the resistance factor for nail tension; RT = the nominal tensile resistance of the nail bar; and φT TR T≥ max ( )3 51- L T q D P PO U DH ≥ maxφ π ( )3 50- φPO POR T≥ max ( )3 49- R r LPO PO P= ( )3 48- r q DPO U DH= π ( )3 47- Tmax = the maximum tensile force on the bar, as calculated in limit-equilibrium analyses. As mentioned earlier, this force is not a service load. The nominal tensile resistance of a nail bar is: where At = the nail bar cross-sectional area, and fy = the bar nominal yield resistance (i.e., with units of force per square area). The tensile resistance provided by the grout is disregarded. 3.4.4.4 Facing Strength Limit States Facing strength limit states [shown schematically on Fig- ure 3-5(g), (h), and (i)] are those affecting the shotcrete, shot- crete reinforcement (bars or WWM), bearing plate, and connectors at the nail head (Figure 3-10). The most common facing strength limit states include: • Flexure (or bending), • Punching-shear, and • Headed-stud in tension. These limit states are described in the following subsections. Flexure in Facings. Lateral earth pressures acting against the facing cause flexural or bending moments in the facing. For the purposes of this limit state, the facing can be considered to be a continuous two-way slab and the nails can be considered to be the supports of the slab. A flexural/bending limit state may be reached when the lateral loads increase, progressively deform the facing, form cracks, and ultimately produce a collapse mechanism (Figure 3-11). Moments on the facing produce tension on the outside of the facing between nails (i.e., conventionally, these are positive sign moments) or can generate tension on the inside of the facing around the nails (i.e., negative moments). Moments occur around a horizontal axis [i.e., vertical moments, mV, as shown on Figure 3-10(b)] and a vertical axis (i.e., horizontal moments, mH). Therefore, separate flexural resistances develop at two locations: the mid- span section between nails and the section around nails, with each section considered both along the horizontal and vertical directions. Therefore, four conditions must be evaluated. The locations where the reinforcement is computed are presented in Figure 3-12. In SNWs, flexural resistance depends on several factors, including horizontal and vertical nail spacing; bearing plate size; facing thickness, h; reinforcement layout and type; and concrete resistance (Seible, 1996). The nominal flexural resis- tance (defined as the maximum resisting moment per unit width) of the facing can be estimated using conventional formulas for reinforced concrete design. When the flexural resistance is reached in the equivalent two-way slab, the “reaction” forces in the nails are considered the nominal resistance force, RFF, for flexure to be used in LRFD equations. R A fT t y= ( )3 52- 26

27 45° TO FLEXURE LIMIT STATE mV CONICAL SURFACE COMPOSITE CONICAL SURFACEREINFORCEMENT VERTICAL MOMENT, mV BEARING PLATE HEADED STUD WWM (ADDITIONAL REINFORCEMENT NOT SHOWN) (TEMPORARY FACING) PUNCHING- SHEAR LIMIT STATE (PERMANENT FACING) WWM OR BAR (b) (c) (d) RFP RFP RFH BREAKAGE STUD IN TENSION LIMIT STATE (PERMANENT FACING) PUNCHING- SHEAR LIMIT STATE (e) (a) Source: Modified after Lazarte et al. (2003) Figure 3-10. Limit states in soil nail wall facings.

28 Soil Earth Pressure To Idealized Deflection Pattern at Failure Facing Initial Position Yield Line Nail Facing Source: Modified after Lazarte et al. (2003) Figure 3-11. Schematic relation between flexure mechanism and nail forces in SNW facings. S V Wale r Bar (T YP ) WW M (Tempora ry Facing ) Re ba r Me sh or WW M (Fin al Fa ci ng ) A A Section A- A Waler Ba r d f = 0. 5 h f h f = final facing thickness d t = 0.5 h t Total Cross S ectional Area (p e r u n i t l e n g t h ) V e r t i c a l Mid-s p an between na ils: a vm At na il he ad : a vn = a vm + A VH S H Ve rtica l Re bar At Na il Head Vertical Re ba r At Na il He ad h t = tempo rary facing th ickn es s A VH A HH S H H o r i z o n t a l Mid-span between nails: a hm At nail head: a hn = a hm + A HH S V Figure 3-12. Resistance and reinforcement nomenclature for flexure limit state. The force that mobilizes in the nail as a reaction to the soil pressures could be also evaluated by multiplying the soil pres- sure by the contributing area around the nail, or SH × SV. The calculation of the resistance RFF is presented in Equation 3-54. For the flexural limit state, it must be verified that: where φFF = the resistance factor for flexure in the facing; RFF = the nominal resistance for facing flexure (considered a force herein); and To = the nail maximum tensile force at the facing. This limit state must be considered separately for both tem- porary and permanent facings; therefore, separate values of RFF must be obtained for the temporary and permanent facings. RFF is estimated using the following expression: R C f a a FF F y vn vm kip ksi lesser of [ ]= × × [ ] × +( ) 3 8. in ft ft in ft 2 2 [ ]× [ ]⎛⎝⎜ ⎞⎠⎟ +( )[ ]× S h S a a S H V hn hm v h SH ft -[ ]⎛⎝⎜ ⎞⎠⎟ ⎧ ⎨ ⎪⎪ ⎩ ⎪⎪ ( )3 54 φFF FF oR T≥ ( )3 53-

where CF = a factor to be obtained from Table 3-1, which is based on Byrne et al. (1998), to consider the non-uniform distribution of soil pressures behind the facing; fy = the bar nominal yield resistance; SH = the horizontal nail spacing; SV = the vertical nail spacing; h = the facing thickness (ht for temporary facings and hf for permanent facings); avn = the cross-sectional area of the WWM (per unit length) in the vertical direction over the nail head; avm = the cross-sectional area of the WWM (per unit length) in the vertical direction in the mid-span between nails; ahn = the cross-sectional area of the WWM (per unit length) in the horizontal direction over the nail head; and ahm = the cross-sectional area of the WWM (per unit length) in the horizontal direction in the mid-span between nails. The directions and locations that these quantities refer to are shown on Figure 3-12. Figure 3-11 shows a schematic diagram of a non-uniform distribution of soil pressure behind the facing. This distribu- tion is affected by the wall displacement magnitude, soil con- ditions, facing thickness, and facing stiffness. The diagram of Figure 3-12 shows that the earth pressure is relatively low between nails, where relatively larger outward displacement tends to produce a stress relief. Earth pressures near the nail heads are larger than those occurring in mid-span because the soil confinement at the nail head is significantly larger. To account for these effects, the factor CF is used to consider pressure distributions that are not uniform. Table 3-1 contains values of CF for typical facing thickness. For permanent facings and for relatively thick (i.e., hf = 8 in. or more) temporary fac- ings, CF = 1 (i.e., the soil pressure distribution is assumed to be uniform). The cross-sectional areas of reinforcement per unit width in the vertical or horizontal direction and around and between nails are shown schematically in Figure 3-12. The nomencla- ture for the reinforcement areas per unit width is presented in Table 3-2. In Equation 3-54, the reinforcement (wire mesh and bars) is assumed to be in the middle of the section, at a distance, d, of half the total thickness, h/2, from the facing surface (Fig- ure 3-12). The total thickness can take the values ht for tem- porary facings or hf for permanent facings; correspondingly, d can take the values dt for temporary facings or df for perma- nent facings (see Figure 3-12). Recommendations on the minimum and maximum reinforcement ratios in the facing and other considerations can be found in Lazarte et al. (2003) and in the design specifications contained in Appendix A. Examples of the use of the formulation presented herein can be found in Lazarte et al. (2003). Punching-Shear in Facings. Connectors installed at the nail head may be subjected to a punching-shear limit state, which may occur if the nominal shear resistance of the rein- forced shotcrete section around the nails is exceeded. The nominal punching-shear resistance must be evaluated for both temporary and permanent facings (Figure 3-13) for the follow- ing situations: • Bearing-plate connection in temporary facings and • Headed-stud connection in permanent facings. 29 Type of Facing Facing Thickness, ht or hf (in.) Factor CF 4 2.0 6 1.5 Temporary 8 1.0 Permanent All 1.0 Table 3-1. Factor CF. Direction Location Cross-Sectional Area of Reinforcement per Unit Width nail head (1) avn = avm + SH AVH vertical mid-span avm nail head (2) ahn = ahm + SV AHH horizontal mid-span ahm Notes: (1) At the nail head, the total cross-sectional area (per unit length) of reinforcement is the sum of the WWM area (avm) and the area of vertical waler bars (AVH) divided by the horizontal spacing (SH). (2) At the nail head, the total area is the sum of the area of the WWM (ahm) and the area of the horizontal bar (AHH) divided by SV. Table 3-2. Nomenclature for facing reinforcement area per unit width.

30 WALER BAR (TYP) CONICAL SLIP SURFACE DDH IDEALIZED SOIL REACTION DC D 'C LBP COMPOSITE CONICAL SURFACE IDEALIZED SOIL PRESSURETO DHD CD ' CD (a) BEARING-PLATE CONNECTION (vertical view) (b) HEADED-STUD CONNECTION SHS 45° (TYP) (TYP) 45° VF 2 VF 2 T o VF/2 VF/2 ht ht /2 ht ht /2 Source: Modified after Byrne et al. (1998) Figure 3-13. Limit states for punching-shear in facing—horizontal cross sections.

At the limit state, conical slip surfaces can form in the fac- ing section around the nail head. The size of the conical slip surface is affected by the facing thickness and the dimension of the nail head components (i.e., bearing-plate or headed- studs) that are present. For both situations, the nominal facing punching-shear resistance, RFP, must meet the following condition: where φFP is the resistance factor for punching-shear in the facing. RFP can be estimated as: where CP = a dimensionless factor that accounts for the contribu- tion to shear resistance of the soil support under the nail head area, and VF = the nominal punching-shear force acting through the facing section. When the soil reaction is considered, CP can be as high as 1.15. For design purposes, it is conservatively assumed that the soil support behind the wall is negligible, and CP = 1.0. The punching-shear force can be calculated as: where f ′c is the concrete nominal compressive resistance (in psi); D ′c is the effective equivalent diameter of the conical slip surface (in ft); and hc is the effective depth of the coni- cal surface (in ft). D ′c and hc must be selected separately for the temporary and permanent facing, as follows. The effective equivalent diameter of the conical slip surface can be calculated as: Temporary facing [Figure 3-13(a)] where LBP is the bearing plate size, and ht is the temporary fac- ing thickness. Permanent facing [Figure 3-13(b)] where SHS = the headed-stud spacing (Figure 3-13); LS = the headed-stud length (Figure 3-14); where -( )h L t tc S H P= − + 3 61 ′ = +⎧⎨⎪⎩⎪ D S h h c HS c c minimum of - 2 3 60( ) h hc t= ( )3 59- ′ = +D L hc BP t ( )3 58- V f D hF c c ckip psi ft ft -[ ]= ′[ ] ′[ ] [ ]0 58 3 57. ( )π R C VFP P F= ( )3 56- φFP FP oR T≥ ( )3 55- tH = the thickness of the stud head (Figure 3-14); and tP = the bearing plate thickness. Available sizes of headed-stud connectors can be found in Byrne et al. (1998), Lazarte et al. (2003), and in references provided by manufacturers. Headed-Stud Tensile Resistance in Permanent Facings. The tensile resistance of headed-stud connectors in perma- nent facings, RFH, must comply with: where φFH is the resistance factor for headed-stud tensile resistance. RFH is calculated as: where N = the number of headed studs per nail head location (usually 4); AS = the cross-sectional area of the headed-stud shaft of diameter DS (Figure 3-14); and fy = the tensile nominal yield resistance of the headed-stud. Headed-studs are usually A307 steel or, less commonly, A325 steel (Byrne et al., 1998). To prevent the heads of the connectors from exerting an excessive amount of compressive stress on the concrete bearing surface, the following geometric constraints must be met (ACI, 1998): where AH = the cross-sectional area of the connector head; AS = as defined earlier; tH = the connector head thickness; DH = the diameter of the connector head; and DS = the diameter of the connector shaft. t D DH H S≥ −( )0 5 3 65. ( )- A AH S≥ 2 5 3 64. ( )- R N A fFH S y= ( )3 63- φFH FH oR T≥ ( )3 62- 31 DH LS tH DS Figure 3-14. Geometry of headed-stud.

To provide an efficient anchorage of the connector in the facing, connector heads must extend beyond the plane con- taining the mesh, toward the exposed face, while a minimum shotcrete cover of 2 in. is maintained. When threaded bolts are used in lieu of headed-stud connectors, the effective cross-sectional area of the bolts, AE, must be employed instead of AS in the equations above. The effective cross-sectional area of a threaded anchor is computed as follows: where DE = the effective diameter of the bolt core; and nt = the number of threads per unit length. 3.4.5 Seismic Considerations in Extreme-Event Limit States of Soil Nail Walls 3.4.5.1 Introduction Seismic forces must be considered in SNW design in areas with moderate to high seismic exposure and, according to the LRFD Bridge Design Specifications, seismic effects must be considered in the design of bridge substructures as an extreme-event limit state. In general, the response of SNWs to past strong ground motions has been very good to excel- lent. Observations made after earthquakes (i.e., 1989 Loma Prieta, California; 1995 Kobe, Japan; and 2001 Nisqually, Washington) indicate that SNWs did not show signs of sig- nificant distress or permanent deflection (Felio et al., 1990; Tatsuoka et al., 1997; Tufenkjian, 2002), although ground accelerations were as large as 0.7g near some of the sur- veyed walls. Vucetic et al. (1993) and Tufenkjian and Vucetic (2000) observed similar trends in centrifuge tests performed on reduced-scale models of SNWs. Observations suggest that SNWs have an intrinsic satisfactory seismic perfor- mance, which is attributed in part to the flexibility of SNWs. The seismic performance of SNWs appears to be compara- ble to that of MSE walls (i.e., another type of flexible retain- ing system). The inertial forces that act on retaining earth systems (including SNWs) during a seismic event can be taken into account in stability evaluations using simplified procedures. In these procedures, seismic coefficients are used to calculate equivalent, pseudo-static forces that act at the centroid of the potentially unstable soil block being analyzed. The most com- monly used pseudo-static procedure is the Mononobe-Okabe Method (MOM), which is an extension of the Coulomb theory (Mononobe, 1929; Okabe, 1926). The MOM, which is A D n E E t = − ⎛⎝⎜ ⎞⎠⎟⎡⎣⎢ ⎤ ⎦⎥ π 4 0 9743 3 66 2. ( )- described by Seed and Whitman (1970) and Richards and Elms (1979), was originally developed for gravity walls and can also be used for SNWs (Lazarte et al., 2003). In this method, it is assumed that: • The facing and the soil mass that is reinforced by nails act as a rigid block; • Active earth pressure conditions develop behind the wall; and • Lateral earth loads act behind the nails during a seismic event. In the LRFD Bridge Design Specifications (AASHTO, 2007), earthquake loads are considered part of the load cases of the Extreme-Event I Limit State load combination. For this state, the resistance factor for soil is 1.0 and the load factor for the seismic force is γEQ = 1.0. 3.4.5.2 Seismic Coefficients The main consideration in the seismic response of SNWs is the horizontal forces produced during a seismic event. Hori- zontal forces can be simplistically computed as the product of the seismic coefficient, kh (if only horizontal forces are consid- ered), and the mass of the potentially unstable soil block. The horizontal coefficient kh is a fraction of the maximum acceleration coefficient, Am. The coefficient Am is the ratio of the acceleration occurring at the centroid of the soil block and the acceleration of gravity, g. Am is a function of the peak ground acceleration coefficient, A: A can be obtained from national seismic maps contained in AASHTO (2007), as described in Article 3.10.2 of LRFD Bridge Design Specifications. Instead of considering kh to be only a function of A, a more rational approach for flexible retaining earth systems, such as SNWs, is to use seismic horizontal coefficients that depend on the maximum seismically induced wall displace- ment (Richards and Elms, 1979; Kavazanjian et al., 1997; Elias et al., 2001; AASHTO, 2007). In this approach, kh is expressed as: where d is the maximum seismically induced wall displacement (expressed in inches) selected for the retaining structure. Equation 3-68 should be used only for 1 ≤ d ≤ 8 in., with typical values of d ranging between 2 and 4 in. A smaller value of d results in larger seismic coefficients and, therefore, longer nails. Equation 3-68 should not be used if: k A A d h m m = [ ] ⎛ ⎝⎜ ⎞ ⎠⎟0 74 3 68 0 25 . ( ) . in. - A A Am = −( )1 45 3 67. ( )- 32

• A ≥ 0.3, • The wall has a complex geometry (i.e., the distribution of mass and/or stiffness with height is abrupt), or • The wall height is greater than approximately 45 ft. These limitations are imposed because (i) ground response that typically occurs under large seismic events is non-linear (a condition not considered in the MOM) and (ii) higher modes of vibration of the wall may participate in the case of complex geometries and tall walls (a condition not considered in the MOM). If deep deposits of medium to soft fine-grained soils underlie the site, ground accelerations could be amplified significantly, inducing a non-linear site response. These con- ditions commonly require full dynamic site response analyses, which must thoroughly consider soil dynamic properties and representative ground acceleration time-histories. The condition A > 0.3 arises for Seismic Zone 4, as defined in Table 3.10.4-1, Seismic Zones, of Section 3.10.4, Seismic Performance Zones, of AASHTO (2007). Various areas in the western United States are classified as Seismic Zone 4, including some of the most populated areas, such as most Cal- ifornia coastal locations, and some areas in Idaho, Nevada, and Alaska. 3.4.6 Design for Service Limit States (Displacements) 3.4.6.1 Introduction As part of the design of SNWs, the maximum lateral and vertical movements of the wall must be estimated and ver- ified to be less than the tolerable deformation limits of the wall. These design consideration aspects are described in the following sections. 3.4.6.2 Soil Nail Wall Displacements Because SNWs are passive reinforcement systems, some deformation of the wall should be expected during SNW con- struction and service life. Some small, tolerable deformation is a natural condition in SNWs as nails must deform to mobi- lize their tensile resistance. Most of the outward movement of SNWs tends to occur during or shortly after excavation and is commonly largest at the top of the wall. Post-construction deformation may increase due to added loads and soil creep. In general, lateral deflections increase with: • Increases in: – Wall height, – Nail spacing, – Steepness of nail inclination, and – Surcharge magnitude; and • Decreases in: – Wall batter, – Soil stiffness, – Nail length, and – Cross-sectional areas of bars. Vertical displacements, which are also affected generally by the above factors, are largest near the facing and are commonly smaller than lateral deflections at the top of the wall. Clouterre (1991) showed that the maximum long-term hor- izontal and vertical wall displacements at the top of the wall, δh and δv, can be estimated using Equation 3-69 if (i) the ratio of the nail length to the wall height is greater than 0.7; (ii) the sur- charge is negligible; and (iii) FS = 1.5 is adopted for overall sta- bility (e.g., in ASD calculation): where (δh/H)i is a factor that depends on soil conditions as indicated in Table 3-3. Ground deformation can be significant up to a distance, DDEF, behind the wall (Figure 3-15). This distance can be esti- mated as: where α is the wall batter angle, and C is a soil-dependent coefficient included in Table 3-3. Typical movements of SNWs are usually small and compa- rable to those observed in braced systems and anchored walls. However, the criterion for tolerable deformation is project dependent. If important, sensitive structures occur near the SNW, an assessment of the potential impact of wall move- ment on these structures is warranted. When excessive defor- mations are presumed or observed, modifications must be D C HDEF = −( )1 3 70tan ( )α - δ δ δv h h iH H≈ = ⎛⎝⎜ ⎞⎠⎟ × ( )3 69- 33 Variable Weathered Rock and Stiff Soil Sandy Soil Fine-Grained Soil (δh/H)i 1/1,000 1/500 1/333 C 0.8 1.25 1.5 Table 3-3. Values of (h/H)i and C as functions of soil conditions.

made to the wall geometry or soil nail layout (e.g., considering the factors listed above). See Lazarte et al. (2003) for additional recommendations. 3.4.6.3 Lateral Squeeze If a SNW is part of a bridge abutment, it lies atop relatively soft soils, and it is subjected to unbalanced loads (e.g., embank- ment loads behind the wall abutment), a verification for lateral squeeze may be necessary to ensure that excessive lateral deflec- tions do not occur at the toe of the wall. Guidance for evaluat- ing lateral squeeze, as well as methods for stabilizing soils to prevent problems related to lateral squeeze, are presented in Hannigan et al. (2005). 3.5 Development of Resistance and Load Factors for Soil Nail Walls 3.5.1 Introduction This section presents the basis for development of resistance and load factors for the limit states of SNWs identified in the previous section. Section 3.5.2 presents the load factors that are applicable in general to earth-retaining structures and presents a discussion on the load factors specifically for SNWs. Sec- tion 3.5.3 presents resistance factors for soil-related limit states in SNWs. Section 3.5.4 presents resistance factors for structural limit states in SNWs. Section 3.5.5 includes a preliminary range of resistance factors for pullout resistance prior to calibration. Finally, Section 3.5.6 presents a summary of resistance factors to be considered for the design of SNWs in the LRFD. 3.5.2 Common Load Factors in Earth-Retaining Structures As mentioned previously, load factors are established for specific limit states and load types. In AASHTO (2007), the following 12 limit states and associated load combinations are included: • Strength limit states (five load combinations, I through V); • Extreme-event limit states (two load combinations, I and II); • Service limit states (four load combinations, I through IV); and • Fatigue limit states (one load combination). Table 3-4, which is based on AASHTO (2007), presents a summary of the load combinations and load factors for each of the limit states listed above. 34 Source: Modified after Clouterre (1991) and Byrne et al. (1998) SOIL NAIL (TYP) V h H DDEF EXISTING STRUCTURE L DEFORMED PATTERN INITIAL CONFIGURATION Figure 3-15. Deformation of soil nail walls.

Permanent Loads (1) Transient Loads (2) Extreme-Event Loads (3) Limit State and Load Combination DC, DD, D W, EH, EV, ES, EL LL, IM, CE, BR, PL, LS WA WS WL FR TU, CR, SH TG (5) SE (6) EQ IC (7) CT (7) CV (7) Strength I (unless noted) p (4) 1.75 1.00 – – 1.00 0.50/1.20 TG SE – – – – Strength II p (4) 1.35 1.00 – – 1.00 0.50/1.20 TG SE – – – – Strength III p (4) – 1.00 1.40 – 1.00 0.50/1.20 TG SE – – – – Strength IV ( EH, EV, ES, DW ) p (4) ( DC only) 1.5 – 1.00 – – 1.00 0.50/1.20 – – – – – – Strength V p (4) 1.35 1.00 0.40 1.0 1.00 0.50/1.20 TG SE – – – – Extrem e-Event I p (4) EQ 1.00 – – 1.00 – – – 1.00 – – – Extrem e-Event II p (4) 0.50 1.00 – – 1.00 – – – – 1.00 1.00 1.00 Service I 1.00 1.00 1.00 0.30 1.0 1.00 1.00/1.20 TG SE – – – – Service II 1.00 1.30 1.00 – – 1.00 1.00/1.20 – – – – – – Service III 1.00 0.80 1.00 – – 1.00 1.00/1.20 TG SE – – – – Fatigue LL , IM & CE only – 0.75 – – – – – – – – – – – Notes: (1) Permanent Loads (2) Transient Loads (continued) DC = dead load of structural components and non-structural attachments PL = pedestrian live load DD = downdrag SE = settlement DW = dead load of wearing surfaces and utilities SH = shrinkage EH = horizontal earth pressure load TG = temperature gradient EL = locked-in effects from construction, including forces from post-tensioning TU = uniform temperature ES = earth surcharge load WA = water load and stream pressure EV = vertical pressure from dead load of earth fill WL = wind pressure on vehicles WS = wind pressure on structures (2) Transient Loads (3) Extreme-Event Loads BR = vehicular braking force CT = vehicular collision force CE = vehicular centrifugal force CV = vessel collision force CR = creep EQ = earthquake FR = friction IC = ice load IM = vehicular dynamic load allowance LL = vehicular live load (4) Load factors for permanent loads vary w ith load type. See Table 3-5. LS = live load surcharge (5) Load factors for temperature gradient can be found in Article 3.4.1 of AASHTO (2007) (6) Load factors for settlement can be found in Article 3.4.1 of AASHTO (2007) (7) Use one of these loads at a time Table 3-4. Load factors and load combinations [Based on AASHTO (2007)].

For earth-retaining structures, the most critical loads are permanent loads associated with horizontal and vertical earth pressures (EH, EV), dead loads (DC and DW), and surcharge loads. Details on earth surcharges are described in Articles 3.11.6.1 and 3.11.6.2 of AASHTO (2007) and on live loads in Article 3.11.6.4. If the substructure is part of a bridge abutment, live loads (LL) and other transient loads transferred from the bridge superstructure must also be considered in the analysis. Load factors for permanent loads in strength and extreme- event limit states must be selected based on (i) the type of per- manent load being considered and (ii) whether the permanent load has unfavorable (i.e., destabilizing) or favorable effects, as described previously. Load factors for permanent loads are presented in Table 3-5. For all limit states, permanent load fac- tors are assigned maximum or minimum values as presented in Table 3-5 to consider destabilizing or stabilizing effects. Max- imum and minimum values will change based on the influence of permanent loads for each limit state being examined (e.g., bearing, eccentricity, global stability, etc.). As seen in Table 3-5, load factors for permanent loads γp ≥ 1.0 must be selected if the load is destabilizing. For example, soil horizontal lateral pres- sures, EH, acting behind earth-retaining structures are desta- bilizing and γp should be selected to vary between 1.0 and 1.5, depending on the lateral earth pressure condition. Conversely, load factors γp ≤ 1.0 must be selected if the permanent load is stabilizing. For example, for the weight of soil load, EV, acting behind a gravity wall, γp should be selected to vary between 0.9 and 1.0. Load factors for permanent loads γp = 1.0 must be selected for service limit states. Based on the provisions for earth-retaining structures included in Article 11.5, Load Combinations and Load Factors of AASHTO (2007), the most common limit states for SNWs can be: • Service limit states (e.g., Service I Limit State, which involves overall stability); • Strength limit states (e.g., Strength I or IV Limit States that involve soil failure); and • Extreme-event limit states (e.g., Extreme-Event I Limit State, which involves earthquake loads). Some of these loads may be present where the SNW is used in a road-widening project under a bridge. Service II through IV Limit States should not be considered for overall stability, as these limit states are reserved to assess the condition of steel structures (Service II Limit State) and pre-stressed concrete superstructures (Service III and IV Limit States), per Section 3.4 of AASHTO (2007). Fatigue limit states are not typically considered for substructures; hence, they are not considered further in this document. For consistency with the current AASHTO (2007) practice, overall stability will be considered in this document to be a service limit state. For compatibility with AASHTO (2007), load factors for earth loads in SNW design are temporarily adopted for γ = 1.0. However, the calibration of resistance fac- tors will be made for a range of load factors varying from 1.0 to 1.75. As shown in Table 3-4, the load factors associated with earth loads that participate in earth-retaining structures (i.e., EH and EV) are γ = 1.0 for the case of overall stability (i.e., Service I Limit State). A similar condition applies to load factors for live loads, LL, and other surcharge loads in the service limit state. 36 Load Factor pType of Load Maximum(1) Minimum(2) DC: Dead load of structural components 1.25 0.90 DD: Downdrag 1.80 0.45 DW: Dead load of wearing surface and utilities 1.50 0.65 EH: Horizontal earth pressure • Active • At-Rest • Locked-in Erection Stresses 1.50 1.35 1.00 0.90 0.90 1.00 EV: Vertical earth pressure • Overall stability • Retaining walls and abutment • Rigid buried structure • Rigid frame • Flexible buried structure other than metal box culvert • Flexible metal box culvert 1.00 1.35 1.30 1.35 1.95 1.50 N/A 1.00 0.90 0.90 0.90 0.90 ES: Earth surcharge 1.50 0.75 Notes: (1) For unfavorable effects of permanent load. (2) For favorable effects of permanent load. Source: Modified after Table 3.4.1-2 (AASHTO, 2007) Table 3-5. Load factors, p, for permanent loads.

The selection of γ = 1.0 establishes that all uncertainty in design concentrates on only the resistance factor. In the limit-equilibrium methods that are commonly used in the design of SNWs, the mass of soil above a potential slip surface is separated into several “slices” for analysis purposes. Slices located near the lower end of the slip surface tend to be stabilizing. Conversely, slices located near the upper end of the slip surface tend to be destabilizing. The weight of each slice contributes to the soil frictional resistance along the slip surface; this effect is considered a stabilizing effect. Assigning a different load factor, γP, for each load component of every slice depending on whether the effect is stabilizing or desta- bilizing must be considered in the software being used for analysis. However, most available software lacks these capabilities. Care must be exercised to not violate force and moment equilibrium, conditions that must be satisfied nec- essarily with unfactored values of weight and resistances. A uniform value γP = 1.0 is used with all slices in part because not all software have these capabilities. It is acknowledged in AASHTO (2007) that this approach is an interim solution due to the current lack of a satisfactory methodology and cal- ibration data for applying LRFD methods to stability analy- sis computations. 3.5.3 Resistance Factors for Sliding, Basal Heave, Overall Stability, and Seismic Limit States 3.5.3.1 Introduction This section provides a discussion of the resistance factors used for sliding, basal heave, overall stability, and seismic limit states that are associated with SNWs. These factors are based on the information provided in Section 11 of AASHTO (2007) for other retaining structures. 3.5.3.2 Sliding The resistance factor for sliding in SNWs in this docu- ment is consistent with the approach in AASHTO (2007) for other earth-retaining systems, including abutments and con- ventional retaining walls [Section 11.6 of AASHTO (2007)], mechanically stabilized earth walls [Section 11.10 of AASHTO (2007)], and prefabricated modular walls [Section 11.11 of AASHTO (2007)]. The resistance factor for potential sliding of the mass of reinforced soil (considered as a block) must be selected for the condition of soil sliding on soil at the base of the soil block, per Sections 11.6, 11.10, and 11.11 of AASHTO (2007), all of which refer to Table 10.5.5-1 of AASHTO (2007). For sliding under this scenario, the resistance factor is specified to be φτ = 0.90. 3.5.3.3 Basal Heave If an SNW is constructed in or over soft, fine-grained soil, basal heave should be considered a potential limit state. The resistance factor, φb, applicable for this case coincides with that used for bearing resistance, for which φb = 0.70. 3.5.3.4 Overall Stability Per Article 11.6.3.4 of AASHTO (2007), resistance factors for soil failure in overall stability evaluations are selected to be (i) φs = 0.75 when the analyzed slope does not support a structure and (ii) φs = 0.65 when the slope supports a struc- tural element. The current version of AASHTO (2007) includes a state- ment that differentiates the above two values for φs depend- ing on whether (i) geotechnical parameters are well defined, in which case φs = 0.75, or (ii) geotechnical parameters are based on limited information, in which case φs = 0.65, per Article 11.6.2.3, Overall Stability. However, this stipulation appears to contradict the requirements set forth in Section 10.4 of AASHTO (2007), where directions are provided to ensure an adequate geotechnical investigation. The following general condition for overall stability analy- sis is considered: where φs = resistance factor for overall stability analysis; Rn = general term representing the soil nominal resistance in overall stability analyses; γ = load factor; and Q = loads. If γ = 1.0, resistance factors for overall stability can be related to equivalent global stability FS, as defined pre- viously. With FS = Rn/Q and γ = 1.0, the above equation becomes: Using Equation 3-72, it is feasible to calibrate the resistance factor directly from FS. This calibration approach is calibration Method B presented in Section 3.2.6. Conversely, FS can be derived from the resistance factor. For example, for cases when the slope does not support a structure or geotechnical param- eters are well defined, FS = 1.0/0.75 = 1.33. For cases when the slope supports a structure or geotechnical parameters are based on limited information, FS = 1.0/0.65 = 1.53. These FS are con- sistent with minimum values currently employed to design SNWs using the ASD method. For example, in the ASD φs n Q R FS = = 1 3 72( )- φ γs nR Q= ( )3 71- 37

method developed for SNWs (Lazarte et al., 2003), FS = 1.5 and FS = 1.35 for permanent and temporary SNWs, respectively. Byrne et al. (1998) selected separate resistance factors for the cohesive and frictional components of the soil resistance in overall stability analysis. For non-critical, permanent struc- tures, Byrne et al. (1998) selected resistance factors as φs = 0.90 and 0.75 for cohesion and friction, respectively. In Byrne et al. (1998), resistance factors were applied to tan ϕ or c (where ϕ and c are the soil friction angle and cohesion, respectively) rather than to global, integrated resistances, as is done in the LRFD Bridge Design Specifications. While the concept of differ- entiating a resistance factor for cohesion and friction seems a rational approach, only one resistance factor is provided in this report for geotechnical resistance, consistent with the current AASHTO LRFD practice. 3.5.3.5 Extreme Events—Seismic Provisions of Article 11.6.5 of AASHTO (2007) specify that, for overall stability under seismic loads (i.e., Extreme-Event I Limit State), resistance factors for soil must be equal to φs = 0.90, as was selected for earth-retaining structures. In Article 11.6.5 of AASHTO (2007), the restriction of φs < 1.0 for over- all stability appears to contradict the tenet presented in the same article, where it is stated that, “The effect of earthquake loading on multi-span bridges shall be investigated using the extreme-event limit state of Table 3.4.1-1 with resistance fac- tors φs = 1.0.” Considering that γ = 1.0 for seismic loads in over- all stability at the service limit state, it results that FS = 1/0.90 = 1.1 in this limit state. This result is consistent with FS values recommended in an ASD framework for SNW design (Lazarte et al., 2003) for permanent or critical structures. However, it is inconsistent with the approach developed by Byrne et al. (1998) in which the resistance factor for stability in seismic analysis was equivalent to φs = 1.0. In this document, consis- tency with AASHTO (2007) is maintained and the values of φs are selected to be consistent with those for permanent struc- tures, and φs = 0.90. A value φs = 1.00 may be acceptable, as long as permanent deformations are calculated and deformations are found to be within tolerable ranges. Currently, no differen- tiation exists for temporary structures in AASHTO (2007). A value of φs = 1.0 (which corresponds approximately to FS = 1.0) is recommended for temporary structures. Major changes have been incorporated in the seismic section of the 2008 interim version of the LRFD AASHTO standard (Anderson et al., 2008) and, therefore, adjustments to seismic design of SNWs are expected once the interim provisions become permanent. In NCHRP Report 611: Seismic Analysis and Design of Retaining Walls, Buried Structures, Slopes, and Embankments (Anderson et al., 2008), several changes are pro- posed in the procedures used to analyze the seismic perfor- mance of several types of retaining structures, including soil nail walls. In NCHRP Report 611, it is proposed that the seismic response of SNWs should be evaluated using deformation- based procedures that account for the expected ground motion characteristics at a given site, site response, soil conditions, and wall height. Anderson et al. (2008) propose that a fraction of the peak ground acceleration should be reduced to account for the permanent wall displacements. Similar, albeit simpler, rec- ommendations had been provided in Lazarte et al (2003) and are included in this document. The incorporation of the pro- posals contained in Anderson et al. (2008) was not part of the original plan of this report; however, those provisions may also be considered when these proposed design specifications for SNWs are reviewed by AASHTO. 3.5.4 Resistance Factors for Structural Limit States 3.5.4.1 Resistance Factors for Tension in Soil Nails The tensile resistance factor to be used in SNWs selected in this document is consistent for the case of load factors in over- all stability or γ = 1.0. To this end, the resistance factor is adopted as follows: for nail bars of mild steel (i.e., ASTM A 615), φT = 0.56; for high-resistance soil nail bars (e.g., ASTM A 722), φT = 0.50. The value for mild steel is consistent with the ASD safety level used in Lazarte et al. (2003). For mild steel bars, the resistance factor is applied to the yield resistance, fY; for soil nails of high-resistance bars, the resistance factor is applied to the guaranteed ultimate tensile strength (GUTS). Note that for the tension limit state of ground anchors walls, Table 11.5.6-1 of Section 11.5, Limit States and Resistance Fac- tors of AASHTO (2007), gives φT = 0.90 for soil nail bars of mild steel and φT = 0.80 for high-resistance soil nail bars. How- ever, these values were developed for load factors higher than 1.0. For example, in Strength I Limit State, a load combination of permanent dead loads and transient or live loads, γDC = 1.25 (maximum per Table 3-5), and γLL = 1.75. For seismic events and γ = 1.0, φT = 0.74 and φT = 0.67 can be selected for mild steel bars and high-resistance steel bars, respec- tively. For cases with load factors similar to those of Strength I Limit State, φT = 1.00 can be selected for both cases. 3.5.4.2 Resistance Factors for Flexure in Facing Facing failures and instrumentation of facings are practi- cally non-existent. Therefore, due to the lack of available data, resistance factors cannot be calibrated. As a result, the resis- tance factor for flexure of SNW reinforced concrete/shotcrete facings is selected to be similar to that for flexure of reinforced concrete per AASHTO (2007). Adopting the same resistance factors for shotcrete and concrete is akin to assuming that the uncertainty related to the strength of these materials is com- 38

parable. The current practice of shotcrete use involves (i) mix design principles that are as sophisticated as those used with concrete; (ii) pre-project submissions on material properties as thorough as those used in concrete; (iii) high qualifications/ experience requirements for shotcrete application personnel; and (iv) frequent shotcrete verification testing. Therefore, it is justifiable to presume that the material variability in these mate- rials is comparable. Overall, the practice of shotcrete placement bears similarities with that of in-situ cast reinforced concrete. Therefore, these similarities in practice justify the determina- tion that, as a first approximation, the uncertainty related to shotcrete and concrete resistances are comparable. One aspect that might be different between these two material technologies is that the efficiency of the design equations used for flexure of shotcrete facing, although already tested (see below), may not have been quantified as much as those for reinforced concrete. The resistance factor for flexure of SNW reinforced concrete/ shotcrete facings is selected for load factors for overall stability γ = 1.0. The resistance factor for flexure of a SNW shotcrete fac- ing is selected to be φFF = 0.67, a value that is consistent with val- ues included in Lazarte et al. (2003) for permanent structures in an ASD format and is consistent with AASHTO (2007), after corrections are made for γ = 1.0. Note that for the flexure limit state, Article 5.5.4.2.1 of AASHTO (2007) provides a resistance factor for flexure of reinforced concrete equal to φFF = 0.90, a value obtained for load factors much higher than 1.0. 3.5.4.3 Resistance Factors for Punching-Shear in Facing Laboratory tests were conducted at the University of Califor- nia, San Diego (Seible, 1996) to study the structural response of SNW facings. Results obtained in controlled tests were com- pared to values obtained with a formulation presented in Byrne at al. (1998) to estimate the punching-shear resistance, RFP. This comparison served to evaluate the predictive capabilities of those formulas. Comparisons between test results and esti- mated resistances indicate that the bias (i.e., measured over pre- dicted resistances) ranges from 1.07 to 1.23. The number of test results was too small to develop reliable statistics of the bias for punching-shear resistance. Therefore, for punching, a full calibration cannot be completed of the resistance factor with empirical results. Hence, the resistance factor is adopted as follows. For the punching-shear resistance in an SNW facing (either reinforced shotcrete or concrete), a resistance factor of φFP = 0.67 is used. This value is consistent with values included in Lazarte et al. (2003) for permanent structures in an ASD for- mat and is consistent with AASHTO (2007), after corrections are made for γ = 1.0. For cases with load factors similar to those of Strength I Limit State, the equivalent resistance factor for punching-shear resistance in an SNW facing would result in φFP = 0.90. 3.5.4.4 Resistance Factors for Facing Headed-Studs in Tension The tensile resistance of headed-studs in SNW facings are selected as φFH = 0.50 for ASTM A 307 steel and φFH = 0.59 for ASTM A 325 steel, consistent with the approach of adopting load factors for overall stability γ = 1.0. Note that a resistance factor of φFH = 0.50 for bolts in tension (both of steel grades ASTM A 307 and ASTM A 325) is included in Section 6.5.4.2 of AASHTO (2007). However, as with previous cases of resist- ance factors for structural limit states, AASHTO (2007) resistance factors were developed for much higher load fac- tors. Also note that the value φFH = 0.80 in AASHTO (2007) coincides with the value adopted by Byrne et al. (1998) for the tensile limit state of ASTM A 325 steel headed-studs. Byrne et al. (1998) presented a separate resistance factor for ASTM A 307 steel at φFH = 0.67. 3.5.5 Preliminary Values of Resistance Factors for Nail Pullout Of the various calibration schemes that can be used to estab- lish a resistance factor, a preliminary calibration was performed based on factors of safety (i.e., Calibration Method B). This pro- cedure was used to develop the resistance factors for the struc- tural limit states presented in Section 3.5.4. In the case of the pullout resistance of soil nails, this factor can be computed from the LRFD equation assuming that nail loads are directly affected by the load factors, or: If loads are comprised of permanent dead (QDC) and live loads (QLL), equation 3-73 can be expressed as: which can be simplified as: Equation 3-75 is useful because the load ratio QDC/QLL, not the actual magnitude of loads, is needed to estimate the resis- tance factor. If live loads are absent: φ γPO DC FS = ( )3 76- φ γ γ PO DC DC LL LL PO DC LL Q Q FS Q Q ≥ +⎛⎝⎜ ⎞⎠⎟ +⎛⎝⎜ ⎞⎠⎟1 3( -75) φ γ γPO DC DC LL LL PO DC LL Q Q FS Q Q ≥ +( ) +( ) ( )3 74- φ γPO i i PO i Q FS Q ≥ ∑ ∑ ( )3 73- 39

For a typical FSPO = 2.0, the pullout resistance factor is φPO = 0.5. Load ratios of 2.5 to 3.0 have been selected in the past for calibrating resistance factors of shallow foundations (e.g., Barker et al., 1991) and deep foundations (Paikowsky et al., 2004). SNWs used in highway applications (e.g., SNWs used as bridge abutments or retaining structures) have loads with relatively large load ratios; therefore, the above range is consistent with previous experience. Typical load ratios for SNWs that are part of a bridge abutment are signifi- cantly larger than QDC/QLL = 2.5, with the ratio tending to increase with the bridge length. For SNWs that are con- structed along roadways and have very small or no traffic loads, the ratio QDC/QLL can be very large. The range of resistance factors in Table 3-6 overlaps with the values of nominal pullout resistance of ground anchors to be used for presumptive nominal resistance values, which are included in Table 11.5.6-1 of AASHTO (2007) and presented below for various soil types: • Cohesionless soils: φPO = 0.65 • Cohesive soils: φPO = 0.70 • Rock: φPO = 0.50 A subsequent section presents the results of a full calibration of φpo based on empirical data and reliability-based methods. A summary of resistance factors for SNWs is included in Table 3-7. 3.6 Development of Soil Nail Test Pullout Resistance and Load Databases 3.6.1 Introduction This section presents the basis for the development of databases of soil nail pullout resistances and loads. These databases were developed based on soil nail load-test results and case histories. The objective in compiling these databases was to develop a basis for preparation of proba- bilistic distributions and statistical parameters to be used in 40 Factor of Safety, FS PO 1.50 1.75 2.00 2.25 2.50 Q DC /Q LL Resistance Factor, φ PO 3 0.92 0.79 0.69 0.61 0.55 4 0.90 0.77 0.68 0.60 0.54 5 0.89 0.76 0.67 0.59 0.53 10 0.86 0.74 0.65 0.58 0.52 ∞ 0.83 0.71 0.63 0.56 0.50 Table 3-6. Summary of pullout resistance factors PO based on factors of safety. 0 2 4 6 8 10 Load Ratio, QDC/QLL 0 0.2 0.4 0.6 0.8 1 R es is ta nc e Fa c to r, φ P O FSPO = 1.30 FSPO = 1.50 FSPO = 2.00 FSPO 2.50 γ DC = 1.50 γ LL = 1.75 Figure 3-16. Pullout resistance factors as a function of load ratio and pullout safety factor. Equations 3-74 and 3-75 can be employed to derive resis- tance factors for a load combination of permanent dead loads and live loads per AASHTO (2007) Strength I Limit State (from γDC = 1.25 and γLL = 1.75). These load factors were used because they may represent typical cases of loading for a bridge abut- ment. The selected load ratio and the safety factor for pullout vary within the range of safety factors typically used for retain- ing structures. A summary of results is presented in Table 3-6. Results plotted on Figure 3-16 show that the resistance fac- tor is relatively insensitive to the load ratio for QDC/QLL ≥ 2.5. Withiam and Nowak (2004) reported similar trends. For typ- ical values FSPO = 2.0 and QDC/QLL ≥ 2.5, the range of calculated φPO is 0.63 to 0.70, with an average of approximately 0.65. Note that, for the case of a service limit state [i.e., γDC = γLL = 1.0 for Service I Limit State, per AASHTO (2007)]: φPO POFS = 1 3 77( )-

the calibration of pullout resistance and load factors (specif- ically, the bias for these quantities). The soil nail pullout resistance database was developed by considering values of pullout resistance from several different sources, including (i) recommended ranges of values of pullout resistance for certain soil types commonly used in practice, as described subsequently; (ii) relationships between pullout resistance and field-measured soil parameters; and (iii) pullout resis- tance values obtained from verification and proof load tests. The soil nail load database was developed based on infor- mation obtained from several instrumented walls. The fol- lowing subsections present a discussion on the main factors that influence pullout resistance and provide typical values of pullout resistances, as well as correlations between pull- out resistance and several typical geotechnical engineering parameters. Additionally, a background of soil nail load testing is provided along with a description of the database of soil nail pullout resistance. The databases are included in Appendix C. 3.6.2 Soil Nail Bond Resistance: Influencing Factors and Typical Values 3.6.2.1 Influencing Factors The nominal pullout capacity of a soil nail develops behind a slip surface and is a direct function of the bond resistance, qu, which is the mobilized shear resistance along the interface between a grouted nail and the surrounding soil. Because the focus of this document is current U.S. prac- tice, only drilled and gravity-grouted soil nails are considered. For these types of soil nails, the nominal bond resistance is affected by numerous factors, including: • Conditions of the ground around soil nails, including: – Soil type; – Soil characteristics; – Magnitude of overburden; and • Conditions at time of soil nail installation, including: – Drilling method (e.g., rotary drilled, driven casing, etc.); – Drill-hole cleaning procedure; 41 Limit State Resistance Condition ResistanceFactor Value Sliding All φτ 0.90 Soil Failure Basal Heave All φb 0.70 Slope does not support a structure φ s 0.75 (1) Slope supports a structure φ s 0.65 (2) (3) Overall Stability NA Seismic φ s 0.90 (4) Mild steel bars – Grades 60 and 75 (ASTM A 615) φ T 0.56 (5) Static High-resistance - Grade 150 (ASTM A 722) φ T 0.50 (5) Mild steel bars – Grades 60 and 75 (ASTM A 615) φ T 0.74 (5) Nail in Tension Seismic High-resistance - Grade 150 (ASTM A 722) φ T 0.67 (5) Facing Flexure Temporary and final facing reinforced shotcrete or concrete φ FF 0.67 (5) Facing Punching Shear Temporary and final facing reinforced shotcrete or concrete φ FP 0.67 (5) A307 Steel Bolt (ASTM A 307) φ FH 0.50 (5)Facing Headed-Stud Tensile A325 Steel Bolt (ASTM A 325) φ FH 0.59 (5) Structural Pullout Presumptive nominal values φ PO 0.50–0.70 (5) (6) Notes: (1) AASHTO (2007) also considers this value when geotechnical parameters are well defined. (2) AASHTO (2007) also considers this value when geotechnical parameters are based on limited information. (3) For temporary SNWs, use φs = 0.75. (4) Per AASHTO (2007) but subject to modifications after new Standard is in place. A value φs = 1.00 may be acceptable, as long as permanent deformations are calculated (see Anderson et al., 2008) and are found not to be excessive. For temporary structures under seismic loading, also use φs = 1.00. (5) Calibrated from safety factors. (6) Preliminary values that will be updated with a reliability-based calibration. Table 3-7. Summary of preliminary resistance factors for SNWs.

– Grout injection method (e.g., under gravity or with a nominal, low pressure); – Grouting procedure (e.g., tremie method); and – Grout characteristics (e.g., grout workability and com- pressive strength). The soil type and conditions of the subsurface soils around the nails also affect the bond resistance. The magnitude of overburden has a larger effect on the nominal bond resistance of granular soils than on that of fine-grained soils. The nomi- nal bond resistance of granular soils is largely influenced by the soil friction angle of the soil around the nail and the magnitude of overburden. While some publications (e.g., Clouterre, 2002) assign for design purposes a linear relationship between the nominal bond resistance of granular soils and its frictional component, the relationship is more complex than a liner rela- tionship because other factors, including construction tech- niques and grout characteristics, also affect the nominal bond resistance in granular soils. The nominal bond resistance of nails installed and grouted in fine-grained soils is in general a fraction of the undrained shear strength of the soil, Su. In rela- tively soft, fine-grained soils (i.e., cohesive), the ratio of bond resistance to soil undrained shear strength, qu/Su, is higher than in relatively stiff, fine-grained soils. The influence of construc- tion techniques (i.e., drilling, installation, and grouting) on the bond resistance is more difficult to ascertain in these soils. The nominal bond resistance of a soil nail can be estimated from the following sources: • Typical values published in the literature, • Relationships between qu and parameters obtained from common field tests, and • Soil nail load tests. Besides these sources, some design engineers estimate the nominal bond resistance based on local experience, particu- larly in areas where some regional practice exists. In addition, the means and methods of an SNW contractor may affect the performance of the structure, including the nominal bond resistance. The nominal bond resistance is rarely measured in the laboratory because it is difficult to reproduce in the labo- ratory those key aspects that affect the nominal bond resis- tance, including field conditions, construction techniques, and grout placement procedures. Laboratory testing proce- dures to evaluate the nominal bond resistance of soil nails, if ever used, are not standardized. Estimations of the nominal bond resistance of soil nails from various sources are discussed below. 3.6.2.2 Typical Values Published in Literature Typical values of bond resistance have been presented in the literature for drilled and gravity-grouted soil nails installed in various types of soils/rocks and for different drilling methods. The most widely used source for typical bond resistance is Elias and Juran (1991), which presents values based on a substantial amount of project experience. Ranges of the nominal bond resistance for various ground conditions and construction techniques are included in Table 3-8 based on this source. The ranges in Table 3-8 are not presented as a function of measur- able field parameters. Design engineers should select design values using judgment. In general, the values in Table 3-8 incorporate a certain degree of conservatism. Minimum and maximum values of the nominal bond resistance provided in this table correspond approximately to the least favorable and most favorable conditions in each case; the average of the range may be used as a preliminary value for design. In addition, the Post-Tensioning Institute (PTI, 2005) pre- sented presumptive values of the nominal bond strength of ground anchors that were grouted under gravity. These values can be also used as preliminary values for soil nails. 3.6.2.3 Correlations between Nominal Bond Resistance and Common Geotechnical Field Tests Soil nail bond resistance, qu, has been correlated to standard geotechnical field testing techniques, including the Pressureme- ter Test (PMT) and the Standard Penetration Test (SPT). These correlations provide typical bond resistance of soil nails for a wide range of subsurface conditions, as described in the follow- ing subsections. Correlation between qu and Pressuremeter Test Results. A correlation between the PMT limit pressure, pL, and qu was developed for various soil types (Clouterre, 2002). The corre- lation has the following format: where a and b are parameters corresponding to various soil types, and pL is the PMT limit-pressure (e.g., ASTM D 4719-87, “Standard Test Method for Pressuremeter Testing in Soils”; Briaud, 1989 and 1992). The limit-pressure is defined as the theoretical pressure at which the soil yields horizontally in the PMT. The correlation above was developed for sand, clay, gravel, and weathered rock, based on soil nail load and PMT tests that were conducted concurrently at the same site (Clouterre, 2002). Equation 3-78 is unit dependent; therefore, when work- ing with English units, pL must be in tons per square foot (tsf ) to obtain the nominal resistance, qu, in pounds per square inch (psi). When working with SI units, pL must be in megapascals (MPa) to obtain qu in kilopascals (kPa). Table 3-9 presents the a and b parameters to be used with q a pu L b = ( ) ( )3 78- 42

English or SI units for ground conditions that include clay, gravel, and weathered rock. Figures 3-17 through 3-20 show the relationship between qu (in psi) and pL (in tsf) for the mentioned soil types. The fig- ures also show the data on which these correlations are based, as well as the 95% confidence intervals associated with each correlation. The correlations of qu shown in these figures are non-linear functions of pL (or b ≠ 1). Because the PMT is not routinely used in geotechnical investigations for soil nail projects in the United States, the correlation with the PMT has not been widely used in this country. Correlation between qu and the Standard Penetration Test Results. Some correlations between the SPT (ASTM D 1586, “Standard Test Method for Standard Penetration Test and Split-Barrel Sampling of Soils”) blow count (i.e., “N” value, expressed as number of blows per foot) and the nominal bond resistance have been developed. The SPT is the most commonly used field technique to assess subsurface conditions for soil nail projects in the United States. The SPT is routinely utilized in SNW projects for soil classification purposes and for soil sam- pling to estimate other engineering parameters. However, the estimation of the nominal bond resistance of soil nails using the SPT is uncommon. Sabatini et al. (1999) presented presumptive, ultimate val- ues of the load transfer rate (rPO) of small-diameter, straight, gravity-grouted ground anchors installed in soils. The load transfer rate is equal to the nominal bond resistance, qu, times the perimeter of the grouted nail (2πDDH, where DDH is the 43 Material Construction Method Soil/Rock Type Nominal Bond Resistance, qu (psi) Rock Rotary Drilled Marl/limestone Phyllite Chalk Soft dolomite Fissured dolomite Weathered sandstone Weathered shale Weathered schist Basalt Slate/hard shale 45 – 58 15 – 45 75 – 90 60 – 90 90 – 145 30 – 45 15 – 22 15 – 25 75 – 90 45 – 60 Rotary Drilled Sand/gravel Silty sand Silt Piedmont residual Fine colluvium 15 – 26 15 – 22 9 – 11 6 – 17 11 – 22 Driven Casing Sand/gravel low overburden (1) high overburden (1) Dense Moraine Colluvium 28 – 35 40 – 62 55 – 70 15 – 26 Cohesionless Soils Augered Silty sand fill Silty fine sand Silty clayey sand 3 – 6 8 – 13 9 – 20 Rotary Drilled Silty clay 5 – 7 Driven Casing Clayey silt 13 – 20 Fine-Grained Soils Augered Loess Soft clay Stiff clay Stiff clayey silt Calcareous sandy clay 4 – 11 3 – 4 6 – 9 6 – 15 13 – 20 Note: (1) Low and high overburden were not originally defined in Elias and Juran (1991). Table 3-8. Estimated nominal bond resistance for soil nails in soil and rock. a Material Type b Sand 6.90 119 0.390 Gravel 5.87 122 0.469 Clays 5.89 120 0.461 Weathered Rock 6.33 177 0.595 Notes: (1) Enter pL in tsf to obtain qu in psi. (2) Enter pL in MPa to obtain qu in kPa. English Units SI Units (1) (2) Table 3-9. Parameters a and b for equation 3-78, correlation between qU and pL.

0 20 40 60 80 100 120 Blow Count, N (bpf) 20 40 60 80 100 120 N om in al B on d Re sis ta nc e, q u (ps i) 0 10 20 30 40 50 60 PMT Limit Pressure, pL (tsf) Clouterre 97 Data Mean 95 % Conf. Elias and Juran (1991) Min./max. Mean Sabatini et al. (1999) DDH = 6 in. DDH = 8 in. Sand Range Figure 3-17. Relationship between qu, pL , and N for sand. 0 20 40 60 80 100 120 Blow Count, N (bpf) 20 40 60 80 100 120 N om in al B on d Re sis ta nc e, q u (ps i) Clouterre 97 Data Mean 95 % Confidence Elias and Juran (1991) Min./max. Mean 0 10 20 30 40 50 60 PMT Limit Pressure, pL (tsf) Clay Range Figure 3-18. Relationship between qu, pL , and N for clay.

0 20 40 60 80 100 120 Blow Count, N (bpf) 20 40 60 80 100 120 N om in al B on d Re sis ta nc e, q u (ps i) Clouterre 97 Data Mean 95 % Confidence Elias and Juran (1991) Min./max. Mean 0 10 20 30 40 50 60 PMT Limit Pressure, pL (tsf) Sabatini et al. (1999) DDH = 8 in. Gravel Range Figure 3-19. Relationship between qu, pL, and N for gravel. 0 20 40 60 80 100 120 Blow Count, N (bpf) 20 40 60 80 100 120 N om in al B on d Re sis ta nc e, q u (ps i) Clouterre 97 Data Mean 95 % Confidence Elias and Juran (1991) Min./max. Mean 0 10 20 30 40 50 60 PMT Limit Pressure, pL (tsf) Weathered Rock Range Figure 3-20. Relationship between qu, pL, and N for weathered rock.

diameter of the drill-hole). Table 3-10 presents presumptive values of rPO (i) for four different soil types and (ii) as a func- tion of soil density/consistency and N ranges. The soil types included in this table are sand/gravel, sand, sand and silt, and silt-clay mixtures of low plasticity/silt mixtures. Note that the ultimate load transfer rates in Table 3-10 are in units of force per unit length of bonded reinforcement. Although the values contained in Table 3-10 were intended for the design of ground anchors, these presumptive values can also be used for the preliminary design of soil nails because the test ground anchors, on which the results are based, were grouted under gravity, which is the typical scenario for soil nails. However, designers must be cautious in using these values as some differences exist between the conditions for ground anchors and soil nails. The N-values included in Table 3-10 are related to relatively deep soils where the bonded length of a ground anchor would be installed, under relatively large in-situ soil overburden. However, soil nails are commonly shorter than ground anchors, tend to be grouted up the excavation face, and thereby their bonded lengths are in general under smaller soil overburden. Therefore, the values in Table 3-10 are probably somewhat unconservative for soil nails. A correlation between SPT and qu can be derived by apply- ing relationships between the PMT pL and SPT N-values. Briaud (1989) presented a correlation that related: pL (tsf) = 0.5 N [or approximately pL(MPa) ≈ 0.05 N]. By replacing this correlation in the PMT-based correlations with qu, the nom- inal bond resistance of a soil nail can be estimated from N-values as: Figures 3-17 through 3-20 show a comparison of the origi- nal data obtained in 1995 (identified as Clouterre 97 in figures q a N u b psi -( ) = ⎛⎝⎜ ⎞⎠⎟2 3 79( ) and published as Clouterre, 2002) and N vs. qu correlations based on the Briaud (1989) N-pL correlation. These figures also show the range, maximum, minimum, and average val- ues of the qu estimates provided in Table 3-8. The Elias and Juran (1991) values for sand appear to cover the range of all data points presented by Clouterre (2002) and to lie above the Clouterre (97) pL vs. qu curves for sand. For clays, the Elias and Juran (1991) values lie on the lower side of the Clouterre data points and correlation. Similar observations can be made for gravel and weathered rock. The ranges proposed by Sabatini et al. (1999) for two cases of drill-hole diameters, DDH, are also presented in these figures. 3.6.3 Background of Soil Nail Load Testing 3.6.3.1 General Load testing of soil nails consists of applying a tensile force to selected, individual bars in a controlled manner while measuring the developed forces and bar elongations with the purpose of verifying the pullout resistance along the bonded, grouted bar length. Note that soil nails are only partially grouted for testing purposes. The specific objectives of soil nail load testing are to: (i) verify that the presumptive design load, DL, is achieved; (ii) confirm that the DL is achievable for the installation means and materials specified in construction docu- ments or proposed by the contractor; (iii) investigate whether the soils subjected to testing loads experience excessive time-related deformation; and (iv) verify that DLs are achieved if a different soil type is encountered or if construction procedures are modified. In the definition above, the design load, DL, refers to the maximum tensile load that is expected to be achieved for service conditions (i.e., not ultimate conditions). DL devel- 46 Soil Type Relative Density/Consistency SPT Range (2) Ultimate Transfer Load Rate, rPO (kip/ft) Sand and Gravel Loose Medium dense Dense 4–10 11–30 31–50 10 15 20 Sand Loose Medium dense Dense 4–10 11–30 31–50 7 10 13 Sand and Silt Loose Medium dense Dense 4–10 11–30 31–50 5 7 9 Silt-clay mixture of low plasticity or fine micaceous sand or silt mixtures Stiff Hard 10–20 21–40 2 4 Notes: (1) Modified after Sabatini et al. (1999). Values are for small-diameter, straight shaft, gravity-grouted ground anchors installed in soil. (2) SPT values are corrected for overburden pressure. Table 3-10. Presumptive values of soil nail load transfer rate in soils.(1)

ops along the bonded nail length, LB, and is a fraction of the presumptive nominal bond resistance. In an ASD scenario, a reduced nominal bond resistance would correspond to the allowable bond strength. The following types of load tests are performed on SNW projects: (i) verification load tests; (ii) proof load tests; and (iii) creep tests. Procedures for soil nail load testing are described in the suggested SNW construction specifications included in Appendix B. Detailed descriptions of soil nail testing are provided in Byrne et al. (1998) and Lazarte et al. (2003). Descriptions of the mechanisms participating in soil load tests are presented in the following section. 3.6.3.2 Mechanisms in Soil Nail Load Tests A soil nail load test is illustrated in Figure 3-21. The drill-hole is assumed to have a uniform diameter, DDH [Figure 3-21(a)]; a load, P, is applied and measured at the front end of the soil nail bar of length Ltot; the bar is partially bonded and unbonded in the respective lengths LB and LU. The bar elongation, Δtot, at the distal end of the bar is measured at the front end. The bond shear stress q(x) is a function of the coordinate x (measured from the back end of the bar) and is mobilized along the grout-soil interface of the bonded length, LB [Fig- ure 3-21(b)]. Actual distributions of the mobilized bond shear stress can be complex and depend on several factors, including bonded length, magnitude of the applied tensile force, grout characteristics, and soil conditions (e.g., Sabatini et al., 1999; Woods and Barkhordari, 1997). However, for design purposes, the mobilized stress is assumed to be con- stant along the bonded length [Figure 3-21(b)]. With this assumption, the nominal bond resistance, qu, is the average of the mobilized stress distribution at the limit state. The force per unit length (equivalent to the transfer load rate, rPO, defined previously) is obtained by multiplying the stress q(x) by the perimeter of the nail-soil interface, or: where all variables were defined previously. The increment of tensile force, dT, along a differential increment of length, dx, [Figure 3-21(a)] is: The nail tensile force T(x) at coordinate x can be obtained by integration. Assuming that q(x) is uniform along the length of the drill-hole, T(x) is: T x D q dx D q xDH DH x( ) = =∫ π π0 3 82( )- dT D q dxDH= π ( )3 81- r q x DPO DH= ( )π ( )3 80- 47 q L B D DH x q (x ) T (x) P (tes t load ) 1 r PO dx T + dT T Actual shear st re ss distribution L U Δ tot Δ B Δ U Soil-Grout Shea r Stress, q (x ) Tensile Fo rc e in Nail Bar, T (x ) Bar Elong at io n (a) (b) (c) (d) P L to t q (x) = as su me d uniform T ma x Wall Face Figure 3-21. Loads and elongation in a soil nail load test.

The pullout capacity during a test, RPO, results when the force T(x) achieves a maximum value, or: As shown in Figure 3-21, Tmax occurs at the end of the bonded length, remains approximately constant along the unbonded length, and is equal to the test load, P. The bond stress is then related to the test load as: The total elongation, Δtot, [Figure 3-21(d)] comprises the elongation ΔU developing along the unbonded length LU and the elongation ΔB developing along the bonded length LB. Elongation ΔU occurs as the steel bar deforms in tension. ΔU remains within the elastic range as long as the nominal yield resistance of the bar is not exceeded. In general, test loads and the bonded length are designed to prevent the bar from exceeding its yield resistance during the test. This elongation is expressed as: where E is the elastic modulus of the nail bar, and At is the cross-sectional area of the nail bar. Elongation ΔB reflects the bar elongation in the bonded length, the grout deformation, the relative deformation or slip- page between the grout and the soil, and the soil shear defor- mation around the nail. This elongation can be calculated as: The relationship between the elongation ΔB and applied loads is mostly linear when the applied loads are small; how- ever, it tends to become non-linear for large loads because the typically non-linear response of the soil (at the soil-grout interface and around the soil nail) becomes more prominent. The relative movement between the nail bar and grout is neg- ligible because of the high resistance to pullout of threaded bars embedded in grout. Elongation ΔB can be normalized as: The data obtained in a load test includes the applied load P and the total elongation (see example on Figure 3-22). The applied load P is increased in predetermined increments that are usually expressed as fractions or percentages of DL (see Appendix B for a typical schedule of test loads). Using the bar εB B BL = × Δ 100 3 87( )- Δ Δ ΔB tot U= − ( )3 86- ΔU t U P E A L= × ( )3 85- q P D LDH B = π ( )3 84- R T D q LPO DH U B= =max π ( )3 83- geometric and material properties, it is possible to separate the total, measured elongation in the bonded and unbonded elongations, as shown on Figure 3-22. 3.6.3.3 Verification Tests Verification tests are conducted to (i) confirm that the installation methods used by the SNW contractor are ade- quate for the project conditions; (ii) estimate or confirm the nominal pullout resistance used for design if verification tests are performed in the design phase; (iii) verify the presump- tive values of pullout resistance used in design; and (iv) iden- tify potential problems during soil nail installation. The number of verification tests that are conducted in each project depends on several factors, including the proj- ect magnitude, variability of ground types at the site, pres- ence of unusual ground conditions, and familiarity of the contracting agency with SNW technology. The common practice is to request that the contractor conduct a minimum of two verification tests in each major soil layer. Appendix B provides guidance on the minimum number of verification tests to perform. In verification tests, the applied test load is increased typi- cally up to 200 percent of DL. In verification tests where the applied loads do not result in pullout, the ratio of maximum load to DL is ≤ 2.0. Typically, true ultimate resistance condi- tions are not always achieved during verification tests. If the applied loads lead to a premature failure condition in the test, verification tests can, in principle, provide a direct measure- ment of the nominal bond resistance. Test nails used in veri- fication tests do not become part of the permanent work but are “sacrificial” because a test load of 200% of DL is consid- ered to be excessive for these nails to be used as part of the long-term system. In some projects, the contractor may elect to apply test loads beyond 200% of DL, thus creating more opportunities 48 AL 0.25 DL 0.50 DL 0.75 DL 1.00 DL 1.25 DL 1.50 DL Load, P Total Elongation, Δtot ΔU ΔB Figure 3-22. Reduction of soil nail load-test data.

to achieve the ultimate pullout strength. However, test loads higher than 200% of DL are rarely applied. 3.6.3.4 Proof Tests During construction, proof tests are conducted on selected production nails, most commonly in every excavation lift. The maximum test load in proof tests is typically 150% of DL. Per specifications, proof tests are commonly conducted on a certain minimum percentage of permanent nails (typically 5%). Additional tests may be required when encountered ground conditions differ from those described in contract documents or when the nail installation procedures change, possibly due to the replacement of broken equipment or low productivity. If results of proof tests indicate that construc- tion practices are inadequate or that the presumptive design pullout resistances are not achieved, the nail installation method or nail lengths/diameters are modified accordingly. Load failures during proof testing are rare. Testing procedures and nail acceptance criteria of proof tests are usually included in specifications. Appendix B pro- vides guidance on acceptance criteria of proof tests. After a proof test is completed, the unbonded length of the bar is grouted. Those test soil nails that are tested and approved are used as permanent nails in the SNW. In the event that a test soil nail is not approved, a new test soil nail must be installed and retested until approval requirements are met. 3.6.3.5 Creep Tests Creep tests are conducted as part of verification or proof tests to assess the time-dependent elongation of the test nail under constant load. Creep tests are commonly performed to verify that design loads are resisted without excessive defor- mations occurring in the soils. In creep tests, the movement of the soil nail head is measured over a period of time of usu- ally 10 to 60 minutes while the applied load is held constant. Creep tests can be performed at various levels of the test load; however, as a minimum, one creep test is performed for the maximum applied load test. Although creep tests may pro- vide some indication that a “failure” condition is imminent when the measured nail head movement rates accelerate, this test does not allow for an easy interpretation that the maxi- mum nominal bond resistance is achieved. 3.6.4 Database of Soil Nail Pullout Resistance 3.6.4.1 Introduction To develop the database of soil nail pullout resistance, a very large volume of information and data was reviewed. This review revealed that the pullout resistance data exhibited scatter and variability when compared with typical conven- tional field test data. The review showed that it was not possible to derive complete or strong correlations between measured bond resistances and field test data because either the variability was excessive or the data was incomplete, unreliable, or inconsistent. Therefore, the database of soil nail pullout resistance was developed for various soil/rock types solely based on soil nail load-test results, which were obtained from a wide variety of sources. These sources are described in the following paragraphs. The soil nail load-test results were carefully scrutinized and all germane information was reviewed. The reviewed infor- mation included the following: • Soil nail test results: – Load applied to the soil nail, P; – Total measured elongation, Δtot; – Observations made during tests (e.g., premature failure, proximity to failure); and – Design load, DL; • Soil nail data: – Diameter of the drill-hole, DDH; – Nail total length and bonded length, Ltot and LB; and – Nail bar diameter, DB; • Geotechnical data: – Site location; – Soil type description; – Data contained in geotechnical reports, including bor- ing logs; – Blow count (N) and other field test results; – Groundwater table location; – Plans with SNW and boring locations; – Description of nail installation method; and – Drawings and specifications of soil nails. 3.6.4.2 Procedure All data and related documents were checked for com- pleteness and consistency. Data that showed inconsistencies, was incomplete, or was suspected to be inaccurate was disre- garded for the database. Although several of the sources pro- duced sufficient information for the objectives of deriving pullout resistance values, most of them lacked details regard- ing construction procedures and other information (e.g., drilling procedures, clean-up methods of the drill-hole, and information on grout mix and grouting procedures). In an attempt to minimize the effect caused by different construc- tion practices, different levels of workmanship, and different drilling/installation equipment, preference was given to data derived from tests that were obtained in one site by the same contractor and using similar equipment. The data kept for the database was thereby internally consistent. As a result, the 49

scatter in the database was smaller as the effect in the variabil- ity caused by construction aspects was reduced. The data was classified by the predominant soil type in which the nails were installed. Four categories of soil type were consid- ered: sandy soils, sandy/gravelly soils, clayey soils, and weath- ered rock. Some projects also provided soil nail load-test results for other soil conditions, including loess, cemented soils, and engineered fill. However, because the number of cases for these soil conditions was relatively small and insufficient to provide a trend, this data was not included in the database. 3.6.4.3 Results from Database The measured and predicted results in the database are presented in Appendix C. Figures 3-23 through 3-25 present graphical representations of the measured and predicted data. The analysis of the measured and predicted pullout resistance allows an assessment of the bias in the resistance estimation. The bias of the pullout resistance data was calculated and plotted as a normal “variate” on the normal standard represen- tation included on Figures 3-26 through 3-29. Log normal curves were plotted side by side next to the data points to ver- ify whether this distribution was adequate. Note that normal distributions would be represented as straight lines on this type of graph. Based on these figures, it was concluded that the log- normal distribution was an acceptable choice to represent the pullout resistance. The mean, standard deviation, and COV of the bias were obtained for the lognormal distribution for each of the soil types. In establishing these parameters, the lognormal distri- bution was adjusted to match the lognormal distribution with the lower tail of the resistance bias data points. The sta- tistical parameters for these curves, which are summarized in Table 3-11, are used subsequently to perform the calibration of the pullout resistance factors. 3.6.5 Database of Soil Nail Loads The statistics of the bias for loads to be used for the calibra- tion of the pullout resistance factor were derived by examin- ing 11 instrumented SNWs in the United States and abroad (Byrne et al., 1998; Oregon DOT, 1999) and by using simpli- fied methods to estimate the maximum loads in the soil nails (Lazarte et al., 2003). The maximum load in the nails was based on values pre- sented in those reports. Byrne et al. (1998) provided a nor- malized distribution of measured soil nail loads, which is reproduced in Figure 3-30. The predicted nail load was obtained using simplified charts developed to estimate the maximum load occurring in soil nails (Lazarte et al., 2003) using the conditions that were present in the instrumented walls. Both measured and predicted maximum nail loads are shown in Figure 3-31. The cases are summarized in Table 3-12. The bias of these data was calculated and plotted as a normal 50 0 2 0 4 0 6 0 8 0 1 00 120 140 160 180 20 0 Measured Resistance (kip) 0 20 40 60 80 100 120 140 160 180 200 Pr ed ic te d Re si st an ce (k ip) Sand Figure 3-23. Measured and predicted pullout resistance—sand.

51 0 2 0 4 0 6 0 8 0 1 00 12 0 1 40 160 180 20 0 Measured Resistance (kips) 0 20 40 60 80 10 0 12 0 14 0 16 0 18 0 20 0 Pr ed ic te d Re si st an ce (k ips ) Clay Figure 3-24. Measured and predicted pullout resistance—clay. 0 2 0 4 0 6 0 8 0 100 120 140 160 180 200 Measured Resistance (kip) 0 20 40 60 80 10 0 12 0 14 0 16 0 18 0 20 0 Pr ed ic te d Re si st an ce (k ip) Rock Figure 3-25. Measured and predicted pullout resistance—rock.

52 0.5 1 1.5 2 2.5 Bias λR of TPO -3 -2 -1 0 1 2 3 St an da rd N or m al V ar ia bl e, z Sand Figure 3-26. Bias R of pullout resistance—sand. 0.5 1 1.5 2 2.5 Bias λR of TPO -3 -2 -1 0 1 2 3 St an da rd N or m al V ar ia bl e, z Clay Figure 3-27. Bias R of pullout resistance—clay.

53 0.5 1 1.5 2 2.5 Bias λR of TPO -3 -2 -1 0 1 2 3 St an da rd N or m al V ar ia bl e, z Rock Figure 3-28. Bias R of pullout resistance—rock. 0.5 1 1.5 2 2.5 Bias λR of TPO -3 -2 -1 0 1 2 3 St an da rd N or m al V ar ia bl e, z All Figure 3-29. Bias R of pullout resistance—all materials.

54 Resistance Parameters Number of Points in Database Mean of Bias Standard Deviation Coefficient of Variation Log Mean of Bias Log Standard Deviation Material N Distribution Type λR σR COVR μln σln Sand and Sand/Gravel 82 Lognormal 1.050 0.25 0.24 0.02 0.24 Clay/ Fine-Grained 45 Lognormal 1.033 0.05 0.05 0.03 0.05 Rock 26 Lognormal 0.920 0.18 0.19 –0.10 0.19 All 153 Lognormal 1.050 0.22 0.21 0.03 0.21 Table 3-11. Statistics of bias for nominal bond strength. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Normalized Nail Force, TKaγ H Sh Sv N or m al iz ed D ep th , h /H 1 0.8 0.6 0.4 0.2 0 Data Approximation Figure 3-30. Summary of tensile forces measured in instrumented SNWs. variate on Figure 3-32. The distribution selected to fit the data was also a lognormal distribution that was adjusted to match the upper tail of the load bias distribution. The bias calculated for each of these cases is presented in Table 3-13. Statistical parameters are summarized in Table 3-14. These parameters are also used in the calibration. 3.7 Calibration of Pullout Resistance Factors 3.7.1 Introduction This section presents the results of the calibration of pull- out resistance factors. The calibration was conducted apply- ing the calibration framework developed by Allen et al. (2005), which was presented earlier in this chapter. Monte Carlo sim- ulations were conducted to improve initial values presented previously in this chapter. 3.7.2 Description of Calibration Process The calibration was performed using the following steps: Step 1: Establish a limit state function; Step 2: Develop PDFs and statistical parameters for loads and resistances; Step 3: Select a target reliability index for SNW design; Step 4: Establish load factors; Step 5: Best-fit cumulative density functions to data points; Step 6: Conduct Monte Carlo simulation; Step 7: Compare computed and target reliability indices; and

Step 8: If computed and target β values differ, modify resis- tance factor and repeat until solution converges. Each of these steps is described in the following sections. Step 1: Establish a Limit State Function The limit state function, M, for nail pullout is defined as: where φPO = the resistance factor for pullout, RPO = a random variable representing the nominal pullout resistance, γQ = a load factor, and Tmax = a random variable representing the load in a nail. At the limit state (i.e., M = 0), resistance can be expressed as: The limit state function can be rewritten as: M T T Q PO = − γ φ max max ( )3 90- R TPO Q PO = γ φ max ( )3 89- M R TPO PO Q= −φ γ max ( )3 88- The two terms in Equation 3-90 that contain Tmax must be interpreted as two independent random variables, each with different statistical parameters and each multiplied by the term Tmax, which is not a random variable but a scaling factor. Both random variables are generated separately in the simulation. As soil nail loads can be represented using a lognormal distribution, random values for the load in the nail is gen- erated as: where Tmax i = a randomly generated value of the variable Tmax; μln = lognormal mean of the random variable that includes Tmax; σln = lognormal standard deviation of the random vari- able that includes Tmax; zi = inverse normal function, or Φ−1(uia); and uia = a random number between 0 and 1 representing a probability of occurrence. The lognormal mean and standard deviation of the ran- dom variable that includes Tmax is obtained from normal parameters as: μ σln max ln= ( )−ln ( )T mean 2 2 -3 92 T zi imax ln ln -= +( )exp ( )μ σ 3 91 55 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 Normalized Measured Nail Load 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 N or m al iz ed P re di ct ed N ai l L oa d Figure 3-31. Measured and predicted maximum nail load.

Feature Case Feature Oregon Swift- Delta Station 1 Swift- Delta Station 2 Polyclinic Peasmarsh, U.K. Guernsey, U.K. IH-30, Rockw all, Section A IH-30, Rockw all, Section B San Bernardino Cumberland Gap, 1988 I-78, Allentown Height (m) TBC 5.3 5.6 16.8 11 20 5.2 4.3 7.6 7.9 12.2 Face slope (deg) TBC 0 0 0 20 30 0 0 6 0 3 m bench Back slope (deg) TBC 55 kN/m surcharge 27 0 0 0 0 75 kN/m surcharge 5 33 33 Ty pe of facing TBC shotcrete shotcrete shotcrete geogrid geogrid shotcrete shotcrete shotcrete shotcrete concrete panels Nail length (m) TBC 6.4 5.2 10.7 6–7 10 6.1 6.1 6.7 13.4 6.1–9.2 Nail inclination (deg) TBC 15 15 15 20 20 5 5 12 15 10 Nail diameter (mm) TBC NA NA NA NA NA 152 152 203 114 89 Steel diameter (mm) TBC 29 29 36 25 25 19 19 25 29 25–32 Spacing, H x V (m) TBC 1.4 x 1 1.4 x 1 1.8 x 1.8 1.5 x 1.5 1.5 x 1.25 0.75 x 0.75 0.75 x .75 1.5 x 1.5 1.5 x 1.2 1.5 x 1.5 Table 3-12. Characteristics of monitored soil nail walls.

57 0.5 0.75 1 1.25 1.5 Bias λQ of Tm -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 St a n da rd N or m al V ar ia bl e, z 11 Instrumented SNWs (13 Points) Figure 3-32. Bias Q for maximum load in soil nails. No. Case Normalized Measured Load, Tm Normalized Predicted Load, Tp Bias of Load 1 Cumberland Gap, 1988 0.54 1.05 0.51 2 Polyclinic 0.56 0.94 0.59 3 I-78, Allentown 0.68 1.07 0.63 4 Guernsey, U.K. 0.51 0.71 0.72 5 Swift-Delta Station 2 1.11 1.43 0.78 6 Oregon – 3-A 0.81 0.98 0.82 7 Swift-Delta Station 1 0.81 0.97 0.84 8 Peasmarsh, U.K. 0.58 0.65 0.89 9 Oregon – 2-B 1.05 1.10 0.95 10 IH-30, Rockwall, Section B 1.06 0.99 1.01 11 Oregon – 1-A 0.96 0.80 1.11 12 San Bernardino (R) 1.08 0.83 1.20 13 San Bernardino (L) 1.13 0.83 1.36 Table 3-13. Summary of normalized measured and predicted maximum nail load. Load Parameters Number of Points in Database Mean of Bias Standard Deviation Coefficient of Variation Log Mean of Bias Log Standard Deviation N Distribution Type λ Q σ Q COVQ μ ln σ ln 13 Lognorm al 0.912 0.290 0.32 -0.140 0.31 Table 3-14. Statistics of bias for maximum nail loads.

and where Tmax mean = mean of the random variable that includes Tmax, and COVQ = coefficient of variation of the bias of the random variable that includes Tmax. If the pullout resistance is modeled as a lognormal variable, the right-hand side of Equation 3-89 is randomly generated as: where RPO i = a randomly generated value of the variable RPO; γQ = load factor; φPO = resistance factor for pullout; μln R = lognormal mean of RPO; σln R = lognormal standard deviation of RPO; zi = an inverse normal function, or Φ−1(uib); and uib = a random number between 0 and 1 representing a probability of occurrence (this number is independ- ent from the number uia defined previously). The lognormal mean and standard deviation of RPO is obtained from normal parameters for RPO as: where RPO mean = mean of RPO; and COVR = coefficient of variation of the bias of RPO. In addition, where λR = the normal mean of the bias of RPO, and Rmax = a non-random scaling factor, similar to the case of loads. Step 2: Develop PDFs and Statistical Parameters for R and Q Statistical parameters for soil nail pullout resistance were developed from the database presented in Appendix C. These values were summarized in Table 3-11 for various soil condi- R RPO mean R= λ max σ ln R = +( )ln ( )COVR 1 3 96- μ σ ln R ln R = ( )−ln ( )RPO mean 2 - 2 3 95 R zPO i Q PO i= +( )γφ μ σexp ( )ln R ln R 3 94- σ ln = +( )ln ( )COVQ 1 3 93- tions. Statistical parameters for maximum loads on a soil nail were derived previously in this chapter based on the analyses of various instrumented walls. These values were summa- rized in Table 3-14. Step 3: Select a Target Reliability Index for SNW Design As discussed earlier in this chapter, the selection of the tar- get reliability index, βT, is a key factor in a reliability-based design. Because soil nails are installed relatively close to each other (i.e., vertical and horizontal spacing is typically 5 ft) and the resulting reinforcement density per unit area is relatively high, SNWs are considered structures with relatively high structural redundancy. To be consistent with the current practice of selection of a target reliability index for elements with high structural redundancy, βT = 2.33 (and Pf = 1%) was selected for this study. Step 4: Establish Load Factors The expression used to estimate the load factor is as follows: where γQ = load factor, λQ = mean of the bias for the load, COVQ = coefficient of variation of the measured to pre- dicted load ratio, and nσ = number of standard deviations from the mean. Using the statistical parameters and nσ = 2, the load factor can be estimated as: The value γQ = 1.5 best represents the statistics used in AASHTO (2007). However, other load factors can be consid- ered in the simulation and different resistance factors can be calculated. In this simulation (see Step 6), the following load factor values were considered to account for various loading scenarios of SNWs, γQ = 1.0, 1.35, 1.5, 1.6, and 1.75. Resistance factors for pullout were calculated for this series of load factors. Step 5: Best-Fit Cumulative Density Functions to Data Points CDFs for loads and resistances were generated via Monte Carlo simulations using the statistics for load and resistances. After the fitting curves were developed for each set of data points, they were plotted side by side, as shown in Figures 3-33 through 3-36. The abscissas on these figures are values of the random variables Tmax and RPO. The ordinates are values of γ Q = + ×( ) = ≈0 91 1 2 0 32 1 49 1 5. . . . γ λ σQ Q Qn COV= +( )1 3( )-97 58

59 Figure 3-33. Monte Carlo curve fitting of load and resistance—sand. Figure 3-34. Monte Carlo curve fitting of load and resistance—clay. Figure 3-35. Monte Carlo curve fitting of load and resistance—rock. Figure 3-36. Monte Carlo curve fitting of load and resistance—all soil types.

the standard normal variable z. CDFs are shown as essentially continuous functions on Figures 3-33 through 3-36 (small markers can be observed at the tails of the CDFs). Data points for load (13 points) and resistance (varying number for each soil type) are plotted as circles and diamonds, respectively, in Figures 3-33 through 3-36. On the left of these figures, the generated CDF for loads was compared to the upper tail of the load data distribution and was verified to be equal or greater than all data points. Conversely, on the right of these figures, the generated CDF for pullout resistance was com- pared to the lower tail of the resistance data distribution. The distribution for pullout resistance was best-fitted to match the lower tail of the resistance PDF. The curve- fitting accuracy is unaffected by the upper tail of the resis- tance CDF because it is the lower tail of the resistance dis- tribution that controls the calculated reliability factor (Allen et al., 2005). Step 6: Conduct Monte Carlo Simulation The Monte Carlo simulation was conducted to artificially generate additional values of load and pullout resistance than the ones available from data points and to estimate the prob- ability of failure accurately. For each soil type, random num- bers were generated independently for the random variables containing Tmax and RPO. Independent values of the random numbers uia and uib were generated in 10,000 trials to calculate new values for Tmax i and RPO i and to develop complete distri- butions of these two random variables. Pullout resistance factors were calculated for the range of γQ listed in Step 4. Figures 3-33 through 3-36 present the curve- fitting analysis using Monte Carlo for different soils and for γQ = 1.75. Figures 3-37 through 3-40 present results of the sim- ulation of the limit function M for different materials and γQ = 1.75. In all cases, βT = 2.33 and Pf = 1%. Steps 7 and 8: Compare Computed and Target Reliability Indices and Iterate, If Necessary After a few iterations, results converged and the simulation was stopped when the difference between the computed and target reliability indices was smaller than 0.5%. 3.7.3 Results The results of the calibration using Monte Carlo simula- tions are included in Table 3-15. Various pullout resistance factors were obtained for the various soil/rock types consid- ered and for the range λQ = 1.0, 1.35, 1.5, 1.6, and 1.75 to show the dependency of these factors. This range represents values that can be commonly used for retaining structures that are part of bridge substructures. The case of γQ = 1.0, applicable for overall stability as a service limit state (per current AASHTO LRFD practice), is also included. For the case of λQ = 1.5 (case based on load statistics), the range of φPO varies from 0.70 to 0.77. This range is comparable to the preliminary range varying from 0.63 to 0.70 obtained in Section 3.5.5 for FSPO = 2.0 and QDC/QLL ≥ 2.5. 60 Figure 3-37. Monte Carlo simulation—sand. Figure 3-38. Monte Carlo simulation—clay.

For the case λQ = 1.0, the pullout resistance factors φPO for various soils vary between 0.47 and 0.51. This range encom- passes the value φPO = 0.5, which would be obtained based on the ASD-based method as the inverse of a global safety factor FSPO = 2 (see Chapter 4 and Lazarte et al., 2003). Because of the values of the calibrated resistance factors for pullout, it is expected that a LRFD-based SNW design that uses this range of resistance factors would not produce signif- icant differences in results (i.e., in terms of soil nails, nail bar diameter, etc.) as compared to designs based on the ASD method when a safety factor FSPO = 2 is used. Appendix D provides detailed comparative designs of SNWs under vari- ous conditions to quantify these differences. As will be seen, these differences are small. The calibrated results also indicate that the reliability in design is approximately the same among all selected materi- als, with soil nails in weathered rock having a slightly lower resistance factor. Overall, with reference to pullout resistances, the design of SNWs will not be affected significantly by use of the LRFD method in lieu of the ASD method. The same applies for other resistance modes including nail in tension, and facing resistances because the factors associated with these resist- ances were selected from the ASD practice. 61 Figure 3-39. Monte Carlo simulation—rock. Figure 3-40. Monte Carlo simulation—all soil types.

λQ Number of Points in Database Mean of Bias Standard Deviation Coefficient of Variation Log Mean of Bias Log Standard Deviation 1.75 1.60 1.50 1.35 1.00 Material N Distribution Type λR σR COVR μln σln φR = φPO Sand/Sandy Gravel 82 Lognormal 1.05 0.25 0.24 0.02 0.24 0.82 0.75 0.70 0.63 0.47 Clay/Fine- Grained 41 Lognormal 1.03 0.05 0.05 0.03 0.05 0.90 0.82 0.77 0.69 0.51 Rock 26 Lognormal 0.92 0.18 0.19 –0.10 0.19 0.79 0.72 0.68 0.61 0.45 All 149 Lognormal 1.05 0.22 0.21 0.03 0.21 0.85 0.78 0.73 0.66 0.49 Table 3-15. Summary of calibration of resistance factors for soil nail pullout for various load factors.