6.5. reliability analysis in static conditions · reliability analysis in static conditions ... fs...
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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