# 321.3 Procedures for Estimation of Geotechnical Parameter Values and Coefficients of Variation

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Methods provided for design of drilled shafts, design of spread footings and design of earth slopes require that both values of design parameters and the coefficients of variation for design parameters be established. The provisions of this article provide procedures for establishing design parameter variability for use with these provisions.

## 321.3.1 General

Provisions of EPG 751.37, EPG 751.38 and EPG 321.1 were developed presuming that the design parameter values used for the provisions are mean values of the relevant parameters. These EPG provisions also require that the coefficients of variation for the design parameters be established to reflect the variability and uncertainty present in the design parameters. Methods for establishing both mean values and coefficients of variation for design parameters for relevant strata are provided in these provisions.

### 321.3.1.1 Relevant Design Parameters

Design properties needed in EPG 751.37, EPG 751.38 and EPG 321.1 include the following:

### 321.3.1.2 Development of Design Profiles for Relevant Design Properties

The provisions of this article are intended to produce rational “design profiles” of properties needed for application of EPG 751.37, EPG 751.38 and EPG 321.1. These design profiles establish a model for how a parameter varies with depth or elevation, as well as the variability of the model.

For the purposes of this provision, design profiles are assumed to be composed of a number of individual strata. The property value within an individual stratum is assumed to have values that are either constant, or linearly varying with depth or elevation as illustrated in Figure 321.XX.1. All design profiles can be reasonably represented as some combination of strata that have either a constant property or linearly varying property within each stratum. Regardless of whether the property value is assumed to be constant or linearly varying with depth, the variability of the property is represented by a constant value of the coefficient of variation.

The variability of relevant design properties within a single stratum is assumed to be constant, and represented using a constant coefficient of variation (COV).

### 321.3.1.3 Minimum Testing Requirements

Method provided in this article and EPG 751.37, EPG 751.38 and EPG 321.1 were developed to accurately account for variability and uncertainty associated with geotechnical design parameters regardless of the quantity of testing performed. However, the methods provided in this article require the following minimum number of measurements be available for application of the methods. For cases where the minimum number of measurements is not available, methods described in the commentary to these guidelines should be utilized.

Table to be added…number to be between 3 and 5…

## 321.3.2 Design Values for Strata with Constant Properties

The provisions of this article shall be satisfied when the property of interest is deemed to be constant over the stratum of interest.

### 321.3.2.1 Establishing Mean Values for Strata with Constant Property

Design values for parameters in strata that are judged to have practically constant or uniform values for the property of interest shall be established as the arithmetic mean of the available measurements:

 $y={\overline {y}}={\frac {\sum _{i=1}^{n}{\hat {y}}_{i}}{n}}$ (consistent units of force) Equation 321.3.2.1

where:

y = design, or “model” value for parameter of interest (consistent units),
${\overline {y}}$ = mean value of parameter from a set of measurements (consistent units),
${\hat {y}}_{i}$ = measured value of the parameter of interest (consistent units) and
n = number of available measurements used to establish the mean value (dimensionless).

The mean value for all measured values of the relevant parameter can be calculated using the AVERAGE function in Microsoft Excel© or similar functions in other computer programs.

### 321.3.2.2 Establishing COV Values for Strata with Constant Property

The coefficient of variation of the mean value for the design parameter shall be established from the set of available measurements as:

 $COV_{y}={\frac {\xi \sigma _{y}}{y}}={\frac {\xi {\frac {\sigma _{\hat {y}}}{\sqrt {n}}}}{y}}$ (consistent units of force) Equation 321.3.2.2

where:

𝐶𝑂𝑉𝑦 = coefficient of variation for mean value of design parameter of interest (dimensionless),
$\sigma _{y}={\frac {\sigma _{\hat {y}}}{\sqrt {n}}}$ = standard deviation of the mean value of parameter of interest (consistent units),
${\mathbf {\xi } }$ = empirical modifier to account for effects of quantity of tests (dimensionless),
y = design, or “model” value for parameter of interest (consistent units),
$\sigma _{\hat {y}}={\sqrt {\frac {\sum _{i=1}^{n}({\hat {y}}-{\bar {y}})^{2}}{n-1}}}$ = standard deviation of measurements of parameter of interest (consistent units),
n = number of available measurements used to establish the mean value (dimensionless) and
${\hat {y}}$ = measured value of a parameter (consistent units).

The standard deviation of the measurements, $\sigma _{\hat {y}}$ , can be calculated from the equation provided, or using the STDEV function in Microsoft Excel© or similar functions in other computer programs. The empirical parameter ${\mathbf {\xi } }$ shall be determined as provided in EPG 321.3.4.

## 321.3.3 Design Values for Strata with Linearly Varying Properties

The provisions of this article shall be satisfied when the property of interest is deemed to vary linearly with depth or elevation over the stratum of interest.

### 321.3.3.1 Establishing Mean Values for Strata with Linearly Varying Property

Design values for parameters in strata where the property of interest is judged to vary linearly with depth or elevation shall be established from a linear best fit relation to the available measurements:

 y = b + mz (consistent units of force) Equation 321.3.3.1

where

y = f(z) = design, or “model” value for parameter of interest (consistent units),
z = depth or elevation (consistent units of length),
b = intercept from linear least squares fit to the available data (consistent units) and
m = slope from linear least squares fit to the available data (consistent units).

The coefficients m and b can be determined using the LINEST function in Microsoft Excel© or similar functions in other computer programs.

### 321.3.3.2 Establishing COV Values for Strata with Linearly Varying Property

The coefficient of variation of the mean value for the design parameter shall be established from the set of available measurements as:

 $COV_{y}={\frac {\xi \sigma _{y}}{y}}={\frac {\xi {\sqrt {z^{2}\sigma _{m}^{2}+\sigma _{b}^{2}+2z\rho \sigma _{b}\sigma _{m}}}}{y}}$ (consistent units of force) Equation 321.3.3.2

where

COVy = coefficient of variation for mean value of design parameter of interest (dimensionless),
$\sigma _{y}={\sqrt {z^{2}\sigma _{m}^{2}+\sigma _{b}^{2}+2z\rho \sigma _{b}\sigma _{m}}}$ = standard deviation of the mean value of parameter of interest (consistent units),
${\mathbf {\xi } }$ = empirical modifier to account for effects of quantity of tests (dimensionless),
y = mean value of parameter from the set of measurements (consistent units),
z = depth or elevation (consistent units of length),
σm = standard error in the slope, m, of the least squares fit (consistent units),
σb = standard error in the intercept, b, of the least squares fit (consistent units) and
${\mathbf {\rho }}$ = correlation coefficient for m and b from the least squares fit (dimensionless).

The standard errors for the slope and intercept of the least squares fit (σm and σb) can be determined using the LINEST function in Microsoft Excel© or similar functions in other computer programs. The empirical parameter ${\mathbf {\xi } }$ shall be determined as provided in EPG 321.3.4. The correlation coefficient 𝜌 can be determined using method described in the accompanying commentary document.

## 321.3.4 Variability Modifiers to Account for Quantity of Tests

Variability Modifiers for Uniaxial Compressive Strength of Weak Rock

For measurements of uniaxial compressive strength of weak rock from uniaxial compression tests, the variability modifier shall be determined from Figure 321.3.4.

0.00.51.01.52.02.53.03.54.04.55.005101520253035404550Test Quantity Modifier, ζNumber of MeasurementsConstant Fit to DataLinear Fit to Data

Figure 321.3.4 Variability Modifier to account for number of measurements for measurements of uniaxial compressive strength of weak rock.

## 321.3.5 Reference

AASHTO (2009), AASHTO LRFD Bridge Design Specification: Customary U.S. Units, American Association of State Highway and Transportation Officials, Fourth Edition with 2008 and 2009 Interim Revisions.

## 321.3.6 Commentary on Procedures for Estimation of Parameter Values and Parameter Variability

### 321.3.6.1 General

Variability of design parameters depends on a number of factors that include the specific sampling and test method(s) utilized, the character of the specific site, and the quantity of sampling and testing performed. Variability also depends upon whether one is considering the variability of the measurements or the variability of the actual or mean value of the parameter. The provisions of this article are intended for use in establishing the variability of the mean value of design parameters that are appropriate for use in EPG 751.37, EPG 751.38 and EPG 321.1.

#### 321.3.6.1.2 Development of Design Profiles for Relevant Design Properties

It is important to recognize that design profiles established according to the provisions of this article are non-unique. Establishing design profiles for application of methods provided in EPG 751.37, EPG 751.38 and EPG 321.1 necessarily involves judgment on the part of the designer because the type, quality, and quantity of testing performed for different sites will differ. Furthermore, two designers may rationally choose to represent the same set of data using different design profiles. Such differences are inherent to site characterization with limited data. The procedures provided in this article were developed with this realization in mind and the variabilities established reflect this fact.

In general, designers should seek to establish design profiles that have the least variability for the available data. However, designers must also consider the quality and quantity of available data, the criticality of the structure, and the potential cost implications of developing highly refined design profiles and balance these with the costs required to develop rational design profiles.

#### 321.3.6.1.3 Minimum Testing Requirements

This is significant departure from current AASHTO LRFD design specifications, which generally use constant resistance factors under the assumption that a minimum level of site characterization will be satisfied.

### 321.3.6.2 Design Values for Strata with Constant Properties

#### 321.3.6.2.1 Establishing Mean Values for Strata with Constant Property

Discussion of how to deal with “outliers” to be added.

#### 321.3.6.2.2 Establishing COV Values for Strata with Constant Property

These provisions are based on the assumption that the measurements are independent measurements (i.e. uncorrelated). While this assumption is seldom strictly satisfied, this is a practical and often conservative assumption to make.