Chapter 6: Reliability Analysis of Soil Nail Walls

179

6.5. RELIABILITY ANALYSIS IN STATIC CONDITIONS

In the following subsections, firstly the concept of intuitive dispersion ellipsoid is

illustrated in the context of soil nail walls using soil nail tensile failure mode. This is

followed by the discussion on the results of reliability analysis of failure modes of the

soil nail wall under the influence of variability of in-situ soil. Finally, the influence of

correlation among soil parameters on the stability of soil nail wall is discussed.

6.5.1. Intuitive Expanding Dispersion Ellipsoid Perspective and Reliability Index

The Hasofer-Lind reliability index is based on the perspective of an ellipsoid that is

tangential to the failure surface in the original space of random variables (Low and

Tang 1997a; Low 2005). For the purpose of understanding and illustration of this

concept in context of soil nail walls, tensile failure mode evaluated for the lowermost

nail at depth z = 9.75 m with conventional factor of safety against nail tensile failure

FST = 1.51 is considered. Fig. 6.2 shows the snapshot of the spreadsheet used for

performing reliability analysis using Excel spreadsheet �SOLVER� optimisation tool.

Fig. 6.3 illustrates the concept of intuitive expanding dispersion ellipsoid in context of

soil nail walls (with reference to the tensile strength failure mode).

As shown in Fig. 6.2, for the tensile failure limit state, the x* values render

Eq. (6.4) equal to zero with the corresponding reliability index equal to 5.03. The

values of the random variables (i.e. soil parameters) corresponding to x* in Fig. 6.2

represents the Most Probable Point (MPP) of failure or the design point.

Geometrically, x* is the point of tangency (see Fig. 6.3) of the expanding dispersion

ellipsoid with the tensile failure limit state surface. Hence, the values of random

variables represented by the point x* lies on the tensile failure limit state surface.

Chapter 6: Reliability Analysis of Soil Nail Walls

180

Fig. 6.2. Spreadsheet for reliability analysis of tensile failure mode of soil nail wall.

Fig. 6.3. Illustration of design point and intuitive dispersion ellipses for tensile failure

mode in the space of and (correlation coefficient is 0).

Chapter 6: Reliability Analysis of Soil Nail Walls

181

It is to be noted that, nx (�nx� are the transformed random variables to their

reduced form in standard normal space with zero mean and unit standard deviation)

value of c turns out to be zero (since x* value of c does not deviate at all from the

mean c) implying that the tensile failure mode is insensitive to in-situ cohesion.

Consequently, perfn(4) and its associated reliability index can be plotted to scale in

the two-dimensional space of and (see Fig. 6.3). The limit state surface separating

the safe domain from the unsafe domain is described by Eq. (6.4).

As indicated above, the design point x* (see Fig. 6.3) is the point of contact

between expanding dispersion ellipse and the limit state surface with respect to the

mean values of and at the centre of the expanding ellipsoid. With respect to the

design point x*, the reliability index (in the present case equal to 5.03) is defined as

the axis ratio (R/r) of the ellipse that touches the limit state surface and the one-

standard-deviation dispersion ellipse. By geometrical properties of ellipses, this co-

directional axis ratio is same along any �radial� direction.

For the case of two random variables, the one-standard-deviation dispersion

ellipse and the -ellipse can be plotted using Eqs. (6.5) and (6.6), respectively. These

equations are the expanded form of the Hasofer-Lind reliability index matrix

formulation discussed in Section 3.5.3 of Chapter 3.

2 2N N N N

2 2 N N 2N 2 N 2

21

11 1 (6.5)

2 2N N N N

2 2 2 N N 2N 2 N 2 T

T T

21

11 1 (6.6)

where: superscript N indicate equivalent normal transformation (Rackwitz and

Fiessler 1978) mean and standard deviation values for log-normally distributed

Chapter 6: Reliability Analysis of Soil Nail Walls

182

random variables, T is reliability index for tensile failure mode, and is the

coefficient of correlation among random variables and . In Fig. 6.3, Eqs. (6.5) and

(6.6) are suitably used (by substituting = 0) in plotting one-standard-deviation

ellipse and -ellipse, respectively.

6.5.1.1. Influence of Correlation among Random Variables on Ellipses

Let us consider the case one-standard-deviation ellipses. If the origin of the cartesian

coordinate system is assumed at the mean values of random variables i.e. at the centre

of the ellipse, then Eq. (6.5) can be written as

2 2

2 2 N N 2N 2 N 2

2 111 1

(6.7)

From the elementary mathematics of conic surfaces, it can be mathematically

shown that the presence of a nonzero � � term in Eq. (6.7) owing to the nonzero

value of indicates rotation of the plot of the conic surface in the plane � �. In the

present case, this aspect is illustrated geometrically for the soil nail tensile strength

failure mode. Fig. 6.4 shows the influence of the correlation coefficient on the one-

standard-deviation ellipses and the -ellipses, and its implication on the reliability

analysis of the tensile strength failure mode. In Fig. 6.4, dashed ellipses are same as

shown Fig. 6.3 for the case of un-correlated random variables (i.e. = 0). For an

assumed (hypothetical) value of = 0.75, rotation of the one-standard-deviation

ellipse (continuous ellipse) is apparent in Fig. 6.4. In comparison to the = 0 case, the

rotation of the one-standard-deviation ellipse has moved it away from the limit state

surface. Similarly, the -ellipse for = 0.75 (not shown in Fig. 6.4 due to the practical

difficulty in plotting it in the adopted range of axes) also rotates and give rise to a new

design point *x1 (see Fig. 6.4) on the limit state surface which is relatively at a larger

Chapter 6: Reliability Analysis of Soil Nail Walls

183

Fig. 6.4. Influence of correlation among random variables on the reliability analysis

of tensile strength failure mode.

distance from the centre than the design point x* for the = 0 case. Thus, in

comparison to = 0 case, the axis ratio (R/r) is significantly more for = 0.75 case. In

other words, reliability index for the tensile strength failure mode with the correlated

random variables having = 0.75 ( T = 9.79) is significantly more than the

uncorrelated case with = 0 ( T = 5.03; see Fig. 6.3). A detailed discussion on the

influence of correlation among random variables on the reliability based evaluation of

the various failure modes of the soil nail wall is given in Section 6.5.3.

6.5.2. Influence of In-situ Soil Variability on the Soil Nail Wall Stability

Reliability analysis has been carried out for the four principal failure modes (i.e.

global stability, sliding stability, soil-nail pullout failure and nail tensile failure) to

study the influence of the variability in in-situ soil parameters (i.e. c, and ) on the

stability of soil nail wall with design parameters given in Table 6.1(a). The statistical

details of the uncorrelated random variables (i.e. in-situ soil parameters) given in