[American Society of Civil Engineers 11th Multidisciplinary Conference on Sinkholes and the Engineering and Environmental Impacts of Karst - Tallahassee, Florida, United States (September

Undrained Stability of Residual Soil in Karst

Özgür Aktürk1, Eric C. Drumm2, M. ASCE, Levent Tutluo lu3, Haluk Akgün4 1Research Assistant, Faculty of Engineering, Department of Geological Engineering, University of Akdeniz, Dumlupinar Bulv. 07058, Antalya, TURKEY; Ph.D. Candidate, Department of Geological Engineering, Middle East Technical University, 06531, Ankara, TURKEY, Formerly, Visiting Scholar, Department of Civil and Environmental Engineering, University of Tennessee, Knoxville, TN 37996-2010, U.S.A.; [email protected], Phone: +90-312-2105736, Fax: +90-312-2105750 2Ph.D., P.E., Professor, Department of Civil and Environmental Engineering, University of Tennessee, Knoxville, TN 37996-2010, U.S.A., [email protected], Phone: +1-865-9747715, Fax: +1-865-9742608 3Ph.D., Associate Professor, Department of Mining Engineering, Middle East Technical University, 06531, Ankara, TURKEY, [email protected], Phone: +90-312-2105815, Fax: +90-312-2105822 4Ph.D., P.E., Professor, Department of Geological Engineering, Middle East Technical University, 06531, Ankara, TURKEY, [email protected], Phone: +90-312-2105727, Fax: +90-312-2105750

ABSTRACT

Human activity on the karst landscape can result in subsidence or collapse, particularly when the thickness of the residual soil layer is reduced due to excavation. When collapse occurs beneath or adjacent to structures or highways, it may result in extensive damage. The stability of the residual soils that overlie solution cavities in limestone is often a concern during the construction, and a simple means to evaluate the stability of these residual soils may be valuable.

In this study, the undrained stability of residual soil in karst terrain was investigated by

the Finite Element Method (FEM). The results are summarized in a design chart in terms of dimensionless stability numbers and the geometry of a potential soil void above a cavity in the underlying limestone. The use of the stability chart for short-term stability is demonstrated by an example. Such stability numbers can be useful to estimate the stability of a given site based on the expected thickness of the soil overburden and the likely range of anticipated soil void diameters. Keywords: Karst; Stability; Residual Soil; FEM; Stability Number

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Copyright ASCE 2008 Sinkholes and the Engineering and Environmental Impacts of Karst 2008 Sinkholes and the Engineering and Environmental Impacts of Karst

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INTRODUCTION Collapse of residual soil over bedrock cavities often occurs during construction in karst

terrain, particularly when the thickness of the residuum is reduced during excavation. Therefore the stability of the residual soils that overlie solution cavities in limestone is often a concern during site selection, construction and maintenance of facilities in karst terrain. Subsurface Profile

Sowers (1996) described the general subsurface profile in karst as consisting of five layers. The first layer (nearest the surface) is a remolded structureless layer with highly variable stiffness, which is the product of the near-surface environment. Layers 2 and 3 are residual soil, with the over-consolidation ratio and strength in Layer 2 typically greater than that in Layer 3. The thickness of Layer 2 may vary from 1 to 50 m and the thickness of Layer 3 from 1.5 to 5 m. Layer 4 usually consists of randomly distributed rock pinnacles with rock blocks and soft soils between them and may contain voids or soil domes. Layer 5 is the competent limestone, which typically exhibits very high strength relative to the soil but may contain solutionally enlarged joints or fractures and caves, and often is hydraulically active. For simplicity in the subsequent analysis, Layers 1, 2, and 3 are combined into a single homogenous layer, and the conditions around a single void in Layer 4 are investigated.

In karst terrain, changes in groundwater levels caused by increased urbanization, agriculture, or mining can lead to sinkhole formation (Bell, 1988; Kannan, 1999; Tharp, 1999). In the absence of construction activities which may change the surface hydrology or reduce the thickness of the residual soil cover, the residual soils above solution cavities may be very stable and capable of supporting significant additional surface loading. However, when site conditions warrant excavation, the reduction in overburden is in direct conflict with subsurface stability considerations. Thus, overburden thickness becomes an issue during site selection and preliminary design (Yang and Drumm, 2002).

Field investigations in the United States (Newton and Hyde, 1971; Newton, 1976, 1981, 1984; and Williams and Vineyard, 1976) indicate most sinkhole problems are caused by the failure of the residual soil around voids developed above the bedrock cavities. Since the solution cavities in most karst systems are hydrologically connected, seasonal variations in the groundwater likely result in softening and erosion of the residual soil above the rock cavity, with a gradual formation and enlargement of the soil dome. With shallow overburden, arching cannot take place and tensile stresses develop in the residual soil, which may result in surface cracks, leading to sudden collapse of the soil dome. Previous Stability Investigations

Analytical solutions based on limit analysis have been proposed for the stability of tunnels and voids in soil (Davis et al., 1980; Augarde et al., 2003), and the stability of soils over cavities has been investigated through centrifuge modeling (Craig, 1990; and Abdulla and Goodings, 1996).

Numerical methods have also been used to investigate the stability of residual soil in karst (Drumm et al., 1987; Ketelle et al., 1987; Sams and Sefat, 1995; Yang and Drumm, 2002; and Augarde et al., 2003). Drumm and Yang (2005) used numerical methods to develop a stability chart for soil overburden collapse, but the application of this chart was

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limited because it was not presented in dimensionless form and the actual collapse or failure conditions were not obtained. DEVELOPMENT OF STABILITY CHARTS

Broms and Bennermark (1967) described the plane strain stability of tunnels in soil by means of a stability ratio N, which was expressed in terms of the difference between the total overburden stress in the ground (comprised of a surface surcharge and the insitu stress) and the internal tunnel pressure, divided by the undrained shear strength cu.

⎥⎦

⎤⎢⎣

⎡ ++−=

u

TS

cDh

N)2/(γσσ

………………………………………………………..(1)

where σT = internal cavity or tunnel pressure

σS = surface surcharge h = soil thickness above the cavity D = diameter of circular openings γ = unit weight of the clay soil cu = undrained shear strength

Augarde et al. (2003) used a similar approach for axisymmetric conditions, and obtained upper and lower bound solutions from classical plasticity, in terms of the same variables, as:

S T

u

Nc

σ σ⎛ ⎞−= ⎜ ⎟⎝ ⎠

……………………………………………………………………….(2)

In this paper, a similar dimensionless stability number is developed for the stability of a

spherical soil void in residual soil, assuming that the internal cavity pressure, σT = 0. The stability is investigated from a series of numerical analyses and the stability number, defined as:

uc c

hN ×= γ …………………………………………………………………………...(3)

where γ = unit weight of the residual clay soil

h = soil thickness above the cavity of radius D/2 cu = undrained shear strength, corresponding to conditions with a friction

angle φ = 0 Numerical Models for Residual Soil Stability

The residual clay soil with a void of diameter D, above rigid bedrock was idealized in two-dimensions as shown in Figure 1. The commercial finite-element code Phase2 Version 6 (Rocscience Inc., 2005) was used and eight-node quadratic elements were selected. The following assumptions were made in the finite element analysis:

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• The geometric conditions around the soil void can be approximated by a two-dimensional axisymmetric analysis, implying a hemispherical void of diameter D.

• The stiffness of the limestone is much greater than that of the residual soil above. Therefore, the rock can be considered as a rigid support. The rock contact is assumed to be horizontal, and there is no horizontal restraint at the rock contact reflecting soft soils usually encountered in this zone.

• The lateral boundary of the finite element model has no effect on the stability. The lateral extent, L (Figure 1), for the largest diameter D, was extended until it had minimal effect on the stability. It was found that there was no effect of the boundary provided it was located at least distance L/D ≥ 2.5 (for h/D = 0.5) from the axis of symmetry.

• The water table is assumed to remain constant and below the bedrock surface. This condition results in the highest effective stress in the residual soil. However, seepage effects and enlargement of the dome due to fluctuation in groundwater elevation are neglected. The analysis in terms of the undrained shear strength reflects the short term stability in the absence of any hydraulic effects.

• The residual soil was assumed to be homogenous, and the majority of the analyses conducted with constant soil density ρ = 1.8 Mg/m3 corresponding to a unit weight of γ = 17.7 kN/m3. However, the effects of this assumption were eliminated by presenting the results in dimensionless terms.

• The soil strength was assumed to be constant with depth in the soil profile. • The tensile strength was assumed to be 20% of the undrained shear strength values

(cu) for all analysis. This assumption, while somewhat arbitrary, allows for a variation in tensile strength with cu while maintaining the dimensionless stability factors.

• The elastic modulus of the residual soil E, was assumed to be 22 MPa. • Poisson’s ratio was assumed to be 0.3 in all analyses. • The residual soil was represented by Mohr-Coulomb elastic-perfectly plastic material

model, and the failure is defined by:

0tan +=′′+′= ucc φστ for undrained conditions …………………………..(4) where strength parameters c’ and φ’ represent the effective cohesion and angle of internal friction, respectively, and cu is the undrained shear strength. The assumption of perfectly plastic behavior neglects any hardening response, and a non-associative flow rule was assumed, with a dilation angle ψ = 0 corresponding to zero volume change during yield.

The above assumptions are typical of those employed in most geotechnical analyses, and are consistent with the uncertainty associated with the spatial variation

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Figure 1. Idealization of residual soil with void or dome over bedrock. of the material properties and depth to rock. The uncertainty in the stability calculations due to these assumptions is in general small relative to the uncertainty associated with estimating the size of the subterranean soil dome. The dome size is generally not known, and must be estimated based on geophysical exploration (Roth et al., 2002; Roth and Nyquist, 2003), or observations from failures or nearby road cuts. Determination of Collapse Load

The dimensionless ratio h/D was used to define the site geometry, where h = H - D/2 = the thickness of the soil above the soil void and D = the diameter of the void.

The value of the dimensionless stability number Nc, corresponding to the lowest strength providing stability for a given geometry, was determined by applying Shear Strength Reduction method (Zheng et al., 2006). The Shear Strength Reduction (SSR) option in Phase2 allows users to automatically perform a finite element stability analysis, and compute a critical Strength Reduction Factor (SRF) for the model. The critical SRF is equivalent to the "safety factor" of the model.

In the Shear Strength Reduction (SSR) method, the strength parameters of a model are reduced by a certain factor (SRF), and the finite element stress analysis is computed. The shear strength for stability is

SRFSRFc

SRFc φσφστ ′′

+′

=′′+′

= tantan …………………………………………………(5)

This process is repeated for different values of strength reduction factor (SRF) until the

model becomes unstable (the analysis results do not converge). This determines the critical strength reduction factor (critical SRF), or safety factor, of the model.

Although it has been suggested that this is an ineffective way to evaluate stability (Krahn, 2006), it is an approach that is widely used in both soil and rock engineering (Griffiths and Lane, 1999; Swan and Seo, 1999). As discussed subsequently, the resulting stability numbers allow the direct use of a factor of safety, and are preferable to the stability chart proposed by Drumm and Yang (2005), where failure was assumed based on an arbitrary size of the yielded zone.

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Typical results, shown in terms of the yielded zone for the critical SRF values of soil strength producing a convergent solution, are shown in Figure 2 for two different geometries. Based on the critical value of SRF, the corresponding critical shear strength is determined and used to determine the critical Nc. This process was repeated for values of h = 0.5, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 15, 20, 25, 30 m. The results are presented in a stability chart of critical Nc values for a range of h/D ratios as shown in Figure 3. Values of Nc below the critical value would have sufficient soil strength for the void geometry and suggest stable conditions.

(a)

(b)

Figure 2. Typical results for two soil void sizes in terms of the yielded zone (φ = 0). The

cross symbols indicate areas where the clay has yielded in shear, while the circles indicate areas where the clay has yielded in tension:

a) h/D = 2, critical SRF = 1.09 b) h/D = 5, critical SRF = 0.96

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Figure 3. Stability chart for undrained (φ = 0) stabilty of residual soil Functional Form of Stability Chart

To allow direct use of the stability relationships shown in Figure 3, a polynomial function was fitted to the stability numbers resulting in a function of the form:

Nc = a(h/D)3 – b(h/D)2 + c(h/D) – d............................................................................(6) where a, b, c, and d are constants, Nc is the dimensionless stability number and h/D is the ratio of overburden thickness to dome diameter. The value of the constants a, b, c, and d for φ = 0 are shown in Table 1. Table 1. Coefficients for Stability Number Nc expressed by Equation 6 and corresponding values of R2.

φ a b c d R2 0 0.0013 0.0766 1.9944 1.8914 0.9982

EXAMPLE: STABILITY EVALUATION OF CANDIDATE SITE

It is desired to evaluate the undrained stability of a site for which the depth to competent limestone is about H = 9 m, and the overburden soil has a unit weight of γ = 18 kN/m3 and an undrained (φ = 0) shear strength cu = 50 kPa. Experience and geophysical exploration suggests that the subsurface soil voids above the limestone will probably not exceed 2 m. Therefore,

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8

h = H – D/2 = 9 m – 2 m/2 = 8 m and h/D = 4

318 / *8 2.8850

examplec

kN m mNkPa

= =

which for h/D = 4 falls below the curve in Figure 3 suggesting the site is stable.

Alternatively, the value can be compared directly with the value of Nc at which the site become unstable. From Equation 6:

8.73stablecN = > Nc

example = 2.88 which suggests that the site is stable.

If the factor of safety is defined as the ratio of the assumed strength at the site to the strength required for stability, then

rrccFS

φσφσ

tantan

++=

where c and φ are the input strength parameters and σn is the actual normal stress component. The parameters cr and φr are reduced strength parameters that are just large enough to maintain equilibrium. So, for stability

3* * 8 *18 / 16.58.73

stablec r stable

r c

h h m kN mN c kPac N

γ γ= ⇒ = = =

03.305.16

050 =+

+=FS

or the site has a safety factor of about 3 with respect to the undrained stability with a potential 2 m diameter cavity. SUMMARY AND CONCLUSIONS

A dimensionless stability chart was developed to evaluate the stability of residual soils in karst where sub-surface soil voids may exist near the rock contact. Numerical analyses were conducted for a range of soil profiles expressed in terms of the ratio of void diameter to soil overburden thickness. The Shear Strength Reduction (SSR) method was applied in which the undrained shear strength was reduced until a convergent solution could no longer be obtained, and the results used to develop stability numbers. The method was repeated for a series of geometries, and the resulting stability numbers for the undrained (φ = 0) case were summarized in a dimensionless stability chart and provided in functional form. The application of the stability chart was demonstrated by an example.

The proposed karst stability chart is not intended as a substitute for site specific stability analysis, but instead as a practical tool for preliminary analysis and site selection purposes when only limited subsurface information is available. Because the soil dome diameter is seldom known and field measurements of overburden thickness typically cover a wide range, a stability chart that facilitates the evaluation of a range of estimated

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dome diameters and overburden thickness values may be useful to estimate short-term stability. ACKNOWLEDGEMENTS

The authors wish to acknowledge the support of Middle East Technical University (METU) Research Fund Project No. BAP-08-11-DPT2002K120510 and the Akdeniz University Research Fund Projects Unit for support of the second author during his period as a Visiting Scholar at the University of Tennessee. REFERENCES Abdulla, W. A. and Goodings, D. J., 1996, Modeling of sinkholes in weakly cemented

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Documents

[American Society of Civil Engineers 11th Multidisciplinary Conference on Sinkholes and the Engineering and Environmental Impacts of Karst - Tallahassee, Florida, United States (September