Draft:Geotechnical variability

Geotechnical variability is a major source of uncertainty in civil engineering projects.

Inherent variability

Geomaterials are formed and altered by natural processes such as sedimentation, colluvial or glacial actions, weathering, etc., which give rise to variability in the material properties. The differences in engineering properties (e.g. mechanical properties) are most notable across layers of different geologic origins. For example, the engineering properties can be vastly different between a layer of glacial till and its underlying sandstone bedrock.

Even within the same soil layer, the properties are rarely ‘homogeneous’, and substantial variability had been observed through in situ and laboratory tests of soil and rock materials. The inherent variability can have important implications to engineering performance. For example, spatial variations of shear strength and stiffness of a soil layer can lead to differential settlements of foundations embedded in that stratum.

Some components of the variability can be explained and modeled with consideration of the geological origin, while others may be more 'localized' effects. The spatial variations for a certain property (${\displaystyle z}$) of a soil deposit is usually represented by two components:

   ${\displaystyle z(x)=t(x)+u(x)}$


where ${\displaystyle t(x)}$ is the trend component at location ${\displaystyle x}$, and ${\displaystyle u(x)}$ is called the residual, or deviation from the trend at that location.

Trend

Many geotechnical properties (e.g. shear strength, stiffness, etc.) display certain trends both in the vertical and lateral directions. On many occasions, vertical trends observed in soil properties originate from the fact that soil behavior is largely controlled by the effective stress experienced by the soil particles[1][2], which increases with depth due of the weight of the material. For example, this change in confining pressure is closely related to the increase in undrained shear strength with depth for normally consolidated clays. Apart from the stress-dependent effects, the vertical trends of soil strength and stiffness, as observed in various in situ tests (e.g. standard penetration test, vane shear test, cone penetration test), may also be related to other geological processes at the specific region. Likewise, trends in lateral directions can be significant for geotechnical properties, and these can arise from variations in the depositional or other processes such as weathering, fluvial or colluvial processes, etc[3].

The trends may be represented mathematically by linear or polynomial functions. However, considering the inherent variations in the properties and the limited data available, determination of the trend or its polynomial order is also associated with much uncertainty. In general, a higher order polynomial increases the flexibility of the trend function and lead to a better fit to the data[4]. On the other hand, an ever-increasing trend flexibility can lead to overfitting, which means random noise and errors are included in the statistical model, which reduces its predictive power at unsampled locations[5]. Correspondingly, various approaches[6][7] have been proposed to offer rational selection criteria for trend function. A good understanding of geological history at the region is often helpful in determining the trend, but this is not always straightforward.

Transformation uncertainty

Transformation uncertainty refers to the uncertainty associated with the correlation equation for soil/rock properties. In the geotechnical literature[8], such a correlation equation is called a transformation model. Let ${\displaystyle \xi _{d}}$ be the soil/rock design property of interest and ${\displaystyle \xi _{m}}$ be the measured property. A transformation model is an equation that relates ${\displaystyle \xi _{d}}$ to ${\displaystyle \xi _{m}}$:

  ${\displaystyle \xi _{d}=f(\xi _{m})\times \epsilon }$


where ${\displaystyle f(\xi _{m})}$ is the prediction value for ${\displaystyle \xi _{d}}$ and ${\displaystyle \epsilon }$ is the transformation uncertainty. The transformation uncertainty ${\displaystyle \epsilon }$ can be modeled as a random variable with mean value equal to ${\displaystyle b}$ and coefficient of variation (COV) equal to ${\displaystyle \delta }$. In the case that ${\displaystyle \epsilon }$ is modeled as a lognormal random variable, statistics for ${\displaystyle \xi _{d}}$ can be predicted as follows:

  ${\displaystyle {\begin{array}{lcl}Mean\,value\,of\,\xi _{d}&=&b\times f(\xi _{m})\\COV\,of\,\xi _{d}&=&\delta \\p-fractile\,of\,\xi _{d}&=&{b\times f(\xi _{m})\times \exp[{\sqrt {\ln(1+\delta ^{2})}}\times \Phi ^{-1}(p)] \over {\sqrt {1+\delta ^{2}}}}\end{array}}}$


The transformation uncertainty is epistemic in its nature because ${\displaystyle \epsilon }$ depends on missing factors (e.g., secondary soil/rock properties) that can be known in principle but are unknown in practice.

In geotechnical engineering, direct evaluation of design property ${\displaystyle \xi _{d}}$ during site characterization can be costly and time consuming. In the circumstance where site investigation budget is limited, the transformation model can be used to obtain first-order estimate for ${\displaystyle \xi _{d}}$ based on ${\displaystyle \xi _{m}}$. A useful reference for transformation models was established by Kulhawy and Mayne (1990)[9].

Examples for clay properties

Design property ${\displaystyle \xi _{d}}$ Measured property

${\displaystyle \xi _{m}}$

Predicted value for ${\displaystyle \xi _{d}}$

${\displaystyle f(\xi _{m})}$

mean for ${\displaystyle \epsilon }$[10]

${\displaystyle b}$

COV for ${\displaystyle \epsilon }$[10]

${\displaystyle \delta }$

${\displaystyle s_{u}}$ ${\displaystyle \sigma '_{p}}$ ${\displaystyle 0.22\times \sigma '_{p}}$[11] 1.04 0.55
${\displaystyle s_{u}}$ ${\displaystyle OCR,\sigma '_{v}}$ ${\displaystyle 0.23\times \sigma '_{v}\times {OCR}^{0.8}}$[12] 1.11 0.53
${\displaystyle s_{u}}$ ${\displaystyle q_{t},B_{q},\sigma _{v}}$ ${\displaystyle (q_{t}-\sigma _{v})\times {\exp(0.513\times B_{q})}/29.1}$[13] 1.05 0.49
${\displaystyle \sigma '_{p}}$ ${\displaystyle LI}$ ${\displaystyle 10^{1.11-1.62\times {LI}}\times {P_{a}}}$[14] 2.94 1.90

${\displaystyle s_{u}}$ = undrained shear strength; ${\displaystyle \sigma '_{p}}$ = preconsolidation stress; ${\displaystyle OCR}$ = overconsolidation ratio; ${\displaystyle \sigma '_{v}}$ = effective vertical stress; ${\displaystyle \sigma _{v}}$ = total vertical stress; ${\displaystyle q_{t}}$ = (corrected) cone resistance; ${\displaystyle B_{q}}$ = pore pressure coefficient; ${\displaystyle LI}$ = liquidity index; ${\displaystyle P_{a}}$ = 1 atmosphere pressure.

Examples for sand properties

Design property

${\displaystyle (\xi _{d})}$

Measured property

${\displaystyle (\xi _{m})}$

Predicted value for ${\displaystyle \xi _{d}}$${\displaystyle [f(\xi _{m})]}$ mean for ${\displaystyle \epsilon }$${\displaystyle (b)}$[10] COV for ${\displaystyle \epsilon }$${\displaystyle (\delta )}$[10]

Examples for rock properties

Design property

${\displaystyle (\xi _{d})}$

Measured property

${\displaystyle (\xi _{m})}$

Predicted value for ${\displaystyle \xi _{d}}$${\displaystyle [f(\xi _{m})]}$ mean for ${\displaystyle \epsilon }$${\displaystyle (b)}$[10] COV for ${\displaystyle \epsilon }$${\displaystyle (\delta )}$[10]

References

1. ^ Terzaghi, K., Peck, R. B. & Mesri, G. (1996). Soil Mechanics in Engineering Practice, 3rd ed.. Wiley. 592pp.
2. ^ Schofield, A. N., & Wroth, C. P. (1968). Critical state soil mechanics. McGraw-Hill. 310pp.
3. ^ Zhu, H. & Zhang, L. M. (2013). Characterizing geotechnical anisotropic spatial variations using random field theory. Canadian Geotechnical Journal 50 (7), 723–734.
4. ^ Baecher, G. B. & Christian, J. T. (2003). Reliability and Statistics in Geotechnical Engineering. Wiley. 605pp.
5. ^ Liu, W. F., Leung, Y. F. & Lo, M. K. (2017). Integrated framework for characterization of spatial variability of geological profiles. Canadian Geotechnical Journal 54(1), 47–58.
6. ^ Akaike, H. (1974). A new look at the statistical model identification. IEEE Transactions on Automatic Control, 19(6): 716–723.
7. ^ Beck, J.L. (2010). Bayesian system identification based on probability logic. Structural Control and Health Monitoring, 17(7), 825–847.
8. ^ Phoon, Kok-Kwang; Kulhawy, Fred H (1999-11-22). "Evaluation of geotechnical property variability". Canadian Geotechnical Journal. 36 (4): 625–639. doi:10.1139/t99-039. ISSN 0008-3674.
9. ^ Kulhawy F.H. and Mayne P.W. Manual on Estimating Soil Properties for Foundation Design, 1990.
10. Ching, Jianye; Phoon, Kok-Kwang (2014-04-22). "Transformations and correlations among some clay parameters — the global database". Canadian Geotechnical Journal. 51 (6): 663–685. doi:10.1139/cgj-2013-0262. ISSN 0008-3674.
11. ^ G, Mesri, (1975/04/00). "NEW DESIGN PROCEDURE FOR STABILITY OF SOFT CLAYS". Journal of Geotechnical and Geoenvironmental Engineering. 101 (GT4). ISSN 1090-0241. Check date values in: |date= (help)
12. ^ M, JAMIOLKOWSKI,; C, Ladd, C; T, Germaine, J; R, LANCELLOTTA, (1985/00/00). "NEW DEVELOPMENTS IN FIELD AND LABORATORY TESTING OF SOILS. PROCEEDINGS OF THE ELEVENTH INTERNATIONAL CONFERENCE ON SOIL MECHANICS AND FOUNDATION ENGINEERING, SAN FRANCISCO, 12-16 AUGUST 1985". Publication of: Balkema (AA). Check date values in: |date= (help)
13. ^ "Establishment of generic transformations for geotechnical design parameters". Structural Safety. 35: 52–62. 2012-03-01. doi:10.1016/j.strusafe.2011.12.003. ISSN 0167-4730.
14. ^ Stas, C. V.; Kulhawy, F. H. (1984-11-01). "Critical Evaluation of Design Methods for Foundations Under Axial Uplift and Compression Loading. Final Report".

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