Critical strain for predicting squeezing potential in tunnels

Recently an approach has been suggested by Singh et al. (2007) to assess the squeezing potential of rock masses in tunnels. This approach advocates that the comparison of strains and not the strength will be a better alternative as the deformations are easily measurable. The approach suggested by Singh et al. (2007) is critically evaluated in the present paper. According to this approach, the critical strain has been defined as that strain level beyond which construction problems and squeezing are likely to occur. It is shown that the critical strain depends on the rock mass quality and intact rock properties. Two correlations of critical strain, with the uniaxial compressive strength, tangent modulus of intact rock and the field modulus of the jointed mass have been suggested. A classification has also been suggested to predict the squeezing potential based on ratio of expected strain and the critical strain for a tunnel section. It is suggested that the modulus of deformation should be obtained from field tests. In absence of field tests, use of a classification approach is recommended, and, expressions are suggested for critical strain in terms of rock mass quality Q. Applicability of the approach has been examined through application to more than thirty case histories from the field.

1. Background

The complex geological media at many tunnelling projects consist of jointed rock masses that are subjected to in-situ stress fields. Prior to excavation of the tunnel there exists an equilibrium state of stresses at any point in the rock mass. As soon as the tunnel is excavated the natural state of equilibrium of stresses is disturbed. The re-distribution of stresses results in large differential stresses at the boundary of the tunnel. If the rock mass is weak and differential stresses are high, the mass may fail and the equilibrium may be achieved through failure if the opening is unsupported. Prediction of squeezing potential is, therefore, of great importance to a designer for designing a stable and ductile support system of the tunnel.

A straight forward approach to assess this problem is to compare the rock mass strength at the periphery of the tunnel with the differential stresses due to in-situ stress field (Singh et al., 1992; Jethwa et al., 1984; Hoek and Marinos, 2000). A slightly different approach has been suggested by Aydan et al. (1993), which compares the strain rather than strength. Based on this concept an approach has been suggested by Singh et al. (2007) to quantitatively assess the squeezing potential of jointed rock masses in the field. This approach suits better to field conditions as the deformations can easily be monitored in the field (Singh et al., 2007). Moreover, if the permissible limit of deformation based on certain criterion is already available, one may modify the support system from the observations in the field as the project progresses. It has been suggested in the proposed approach that there is a threshold value of tangential strain at the tunnel periphery above which instability and support problems occur. This threshold value of strain is termed as the critical strain. It is also suggested that the critical strain may be obtained from the properties of the intact rock and the jointed rock mass. The approach is an out come of the application of the basic concepts given by Aydan et al. (1993) to an experimental study conducted on specimens of jointed rock mass by Singh (1997). A relationship has been suggested for computing the critical strain. For a reasonable prediction of the critical strain, the modulus of deformation of the rock mass will be required. In absence of the uniaxial jacking tests, the critical strain may be obtained from Barton’s rock mass quality Q. In the present paper the Singh et al. (2007) approach is explained in detail and its applicability is examined by applying it to several case studies of squeezing and non-squeezing tunnels.

2. Critical strain

The critical strain is defined as that level of the tangential strain beyond which construction and squeezing problems would occur. Sakurai (1997) considered the value of critical strain equal to 1% for all rock types and suggested warning levels for severity of construction problems in a tunnel (Fig. 1). Aydan et al. (1993) considered an analogy between the stress-strain response of rock in laboratory and tangential stress-strain response around tunnels (Fig. 2). Five distinct states of stress-strain response were expressed during loading of a rock specimen at low confining stress σ₃ (Aydan et al., 1993). Based on strain, the normalized strain level η_p was defined as

η_p = ε_p / ε_e (1)

where ε_e is the elastic strain limit and ε_p is the strain levels at the peak of stress-strain curve.

If ε^a_θ (= u_a/a) is the peak tangential strain at the periphery of the tunnel and ε^e_θ is elastic strain, the ratio ε^a_θ / ε^e_θ may be used to define various degrees of squeezing as suggested by Adyan et al. (1993). Hoek (2001) also used tunnel strain to define squeezing potential. A comparison of squeezing conditions suggested by Hoek (2001) and Adyan et al. (1993) is presented in Table 1 (Barla, 2001). It is observed from Table 1 that the critical strain, above which the construction problems due to squeezing are likely to occur, is taken as 1%. It has been experienced in field that there are some tunnels which suffered strains as high as 4% but did not exhibit stability problems (Hoek, 2001). The critical strain should not therefore be taken as 1%. Rather it should depend on the properties of the intact rock material and jointed rock mass. The critical strain in this paper is defined as an empirical level of tangential strain at the periphery of the tunnel above which the jointed rock mass fails under uniaxial loading condition. It should be noted that the critical strain is an anisotropic property and will be different at different points on the periphery of the opening. So expressions have been suggested for critical strain. The strain actually occurring at the periphery of the opening may be obtained through numerical modelling or through monitoring and analysis of the field data. The ratio of observed strain to the critical strain may then be used to assess the squeezing potential and modify support systems accordingly. Sakurai (1997) observed that construction problems actually occurred in non-squeezing ground conditions, where observed tangential strain exceeded far above the predicted critical strain (i.e. ratio between UCS and modulus of elasticity of that rock material). The degree of severity of construction problems will, therefore, increase in proportion of the ratio between the actual strain and the critical strain.

3. Experimental programme

Physical model tests have been one of the best ways to understand the mechanical behaviour of jointed rock masses. An extensive experimental study was carried out by testing more than 75 specimens to study the strength and deformation behaviour of jointed rock mass under uniaxial stress state. For better reproducibility of results a model material was used to simulate the intact rock. The complete details of the study are available in researches by Singh (1997), Singh et al. (2002) and Singh and Rao (2005). Cut blocks of lime silica bricks were used to form specimens of jointed blocky mass. The properties of the model material and joints are indicated in Table 2. The specimens of jointed mass were prepared by arranging elemental blocks in a particular fashion. The size of the specimen was 15x15x15 cm and, on an average, more than 260 blocks were used to form a specimen. The various configurations of joints adopted in the experimental study are shown in Fig. 3. Type-A specimens consisted of three sets of joints. Joint Set-I was continuous and was inclined at variable angle θ with the horizontal. The joint Set-II was constructed with stepping‘s’ for each θ; and Set-III was always kept vertical. The stepping ‘s’ was varied with each θ as shown in Fig. 3. Besides type-A, additional tests were performed on types-B, C and D specimens by changing the geometry of the cut blocks.

The tests were performed under uniaxial loading conditions by applying uniformly distributed load at the top of the specimen. Two teflon sheets smeared with silicon grease were used on the top and the bottom of the specimen to reduce end friction. A strain controlled machine was used to apply load at a uniform strain rate. During testing, the deformations were continued beyond the failure of the specimen until the load reduced to about half the peak load. Deformations of the specimen in X, Y and Z directions were measured during loading. The axial stress at any given instant was computed by applying area correction (Singh et al., 2002).

Results of physical model tests

Axial stress vs axial strain curves were plotted for all the specimens. Most of the stress strain curves were found to be “S” shaped (Fig. 4). The curves showed an initial concave upward portion due to the joint closures and the initial seating; thereafter a linear middle portion exhibiting elastic deformations, and, a convex upward portion due to plastic deformations near failure. The axial strains were corrected for initial seating effect by drawing a tangent at the middle straight line portion of the stress-strain curve. The point of intersection of this tangent with the strain axis was considered as the point of zero strain. The strain levels i.e.ε_e and ε_p are explained in Fig. 4. The peak stress was taken as the rock mass strength ο_cj, and the tangent modulus E_j was obtained by measuring gradient of tangent drawn to the axial stress-strain curve at a stress level equal to half the rock mass strength σ_cj. The modulus ratio (M_rj), as defined by Deere-Miller (1966), for each specimen was obtained as follows

M_rj = E_tj / σ_cj (2)

where E_tj is the tangent modulus and σ_cj is the UCS of the mass. The modulus ratio is a measure of inverse of failure strain. Figure 5 indicates a strong correlation between ε_p and M_rj. The following correlation has been found for the peak failure strain ε_p.

ε_p = 154.77(M_rj)^-1.04                     (3)

where ε_p is peak failure strain in percent.

Deere-Miller classification chart has been used to represent the strength (σ_cj) and modulus (E_tj) values of the specimens (Fig. 6). The intact rock position is also indicated on this plot. It is interesting to see that, the E_tj vs σ_cj plot is scattered about a line passing through the intact rock position I. The gradient of this empirical line has been found to be 1.6. A correlation between the strength and tangent modulus of intact and jointed rock mass may therefore be obtained as given below (Singh and Rao, 2005):

Gradient of the line = (log E_i - log E_tj) / (log σ_ci - log σ_cj) = 1.6

⇒ (σ_cj / σ_ci) = (E_tj / E_i)^0.63 (4)

where E_i and σ_ci are tangent modulus and uniaxial compressive strength (UCS) of the intact rock respectively.

4. Expressions for critical strain

The critical strain (Sakurai, 1997) represents the elastic strain in Fig. 4, and may be obtained as

ε_cr = ε_e = σ_cj / E_tj (5)

From Eqs. 4 and 5, the critical strain may be obtained as

ε_cr = [σ_ci / (E_tj^0.37 E_i^0.63)] x100 (percent) (6)

Also from Eqs. 2, 3 and 4 the peak strain may be obtained as

ε_p = 154.77(E_tj / σ_cj)^-1.04 ≈ 154.77(ε_cr)^1.04 ≈ 1.5ε_cr (7)

Equations 6 and 7 may be used to predict the critical strain (i.e. the strain level for which the stress will reach the elastic limit) and the peak failure strain ε_p, (i.e. the strain at which failure will occur as light squeezing).

For computing critical strain in field, the value of modulus E_tj may be obtained from uniaxial jacking tests which are routinely performed at the project sites. It should however be noted that E_tj is generally anisotropic, and in field it should be obtained at the periphery of the tunnel in the desired direction. For crown, the test in horizontal direction and for side walls, the test in vertical direction may be used to get E_tj. Also, there is always some in-situ stress along a tunnel axis which will increase both σ_cj and E_tj but this effect is not considered in this simplified analysis.

The field tests may not be feasible for many cases especially during preliminary stages of site selection. Field modulus E_tj may then be obtained from Joint Factor (Ramamurthy and Arora,1994), RMR (Bieniawski, 1989), GSI (Hoek and Brown, 1997) or Q system (Barton et al., 2002). The Q system is the most extensively used and well tested classification for tunnels and underground structures. An approximate value of critical strain may be obtained from Q as given below.

3.1 Critical strain from Singh et al. (1997) correlation

Singh et al. (1997), based on the back analysis of several tunnels, have suggested the following relation for rock mass strength ο_cj.

σ_cj = 7ϒQ^1/3 MPa, For Q < 10, J_W = 1.0, σ_ci < 100 MPa                     (8)

where, ϒ is the density of rock mass in gm/cc, σ_cj and σ_ci are UCS in MPa of the rock mass, Q is the actual (post-construction) rock mass quality and J_w is joint water reduction factor used in Q. The critical strain may, therefore, be obtained from the intact rock properties and the Q value as follows:

[1 / ε_cr] = [E_tj / σ_cj] = (E_i / σ_cj) [σ_cj / σ_ci]^1.6 = E_i(σ_cj)^0.6(σ_ci)^-1.6 = E_i(7ϒQ^1/3)^0.6 (σ_ci)^-1.6

⇒ ε_cr = 31.1 [σ_ci)^1.6 / E_i ϒ^0.6Q^0.2]     (percent)                     (9)

4.2 Critical strain from Barton (2002) correlation

Barton (2002) has suggested the following correlation for the long term modulus of deformation in the field:

E_tj = 10[Qσ_ci / 100]^1/3 x 10³ MPa                     (10)

Using Eqs. 4, 5 and 10 the critical strain may be obtained as,

ε_cr = 5.84 [(Qσ_ci^0.88) / (Q^0.12 E_i^0.63)]         (percent)                     (11)

4.3 Evaluation of Squeezing Potential

The observed or expected strain at the tunnel periphery may be obtained from numerical modelling or from actual monitoring in the field. The squeezing index SI may be defined as:

SI = [Observed or expected strain / Critical strain] = (u_a / a) / (ε_cr)                     (12)

where u_a is the radial closure and a is the radius of the opening.
A classification similar to Aydan et al. (1993) may now be adopted as proposed in Table 3.

5. Case studies

Few case studies have been considered to demonstrate the applicability of the approach suggested by Singh et al (2007). The data required for computing critical strain was collected for several tunnels (Table 4 (pdf-file, 65.5 k)). The critical strain was computed using Eqs. 9 and 11. The observed or expected strain values, if available, are also listed in the Table. The expected tangential strain was assessed by the respective authors mainly through numerical modelling. The intact rock properties (for the cases referred from Jethwa et al., 1984 and Singh et al., 1992) were obtained from Mehrotra (1992). Arithmetic average values of σ_ci and E_i were used for analysis, where as geometric average (Q_av = √Q₁Q₂ was used for Q values ranging between Q₁ and Q₂.

It is observed that the critical strain is below 1% for the majority of the cases of non-squeezing tunnels. It is also observed that the squeezing index is less than one, indicating no squeezing for all these cases. In some cases the critical strain is higher than 1% and it may be expected that the tunnel is not likely to pose problems for even large deformations and squeezing. Therefore permissible deformations may be computed for the tunnels based on the proposed expression for critical strain.

The critical strain is well below the expected or observed strain for many of the cases. This is an indicator of squeezing or probable problems to be encountered during tunnelling. A warning is, therefore, given by the value of critical strain and suitable measures may be adopted. For example the case study of Kaletepe tunnel, Turkey, (Sari and Pasamehmetoglu 2004) indicates a squeezing condition without support. The expected strain reduces which results in a reduction of the squeezing index as rock bolts are provided. However for some cases, the squeezing conditions still prevails. A further improvement is achieved by providing rock bolts with shotcrete which reduces the squeezing index to less than one, which indicates no squeezing and a likely stable condition.

A quantitative evaluation of the performance of the support is, therefore, possible through critical strain. It is, therefore, suggested that the idea of critical strain and permissible deformation may serve as an additional tool for risk management especially for those sites where critical strain is less than 1%. It is also observed from Table 4 (pdf-file, 65.5 k) that the squeezing potential defined by Singh et al. (1997) and Barton (2002) matches with each other and also verifies the observed ground condition in almost all cases barring few exceptions. The observed good consistency validates the applicability of the Eqs. 6, 9 and 11 in estimating the critical strain.

6. Concluding remarks

The assessment of squeezing potential of tunnels in jointed rocks is important for the tunnel design and construction. The tunnel closures are easily monitored and may give better index of likely damage to the rock mass and associated construction problems than the strength of the rock mass. Suitable measures like the design of steel fibre reinforced shotcrete (SFRS) may be taken up in advance to solve the problem if the permissible deformations are known before hand. The critical strain is defined as the tangential strain level along periphery beyond which the construction and squeezing problems may be encountered during tunnelling. The value of critical strain, presently in vogue is one percent, which is based on past engineering practice. It is suggested in this paper that the critical strain is an anisotropic property and depends on the properties of the intact rock and the orientation of discontinuities in the rock mass. This conclusion is based on the outcome of an experimental programme, in which, specimens of jointed rock mass were tested under uniaxial loading conditions. A correlation has been derived for critical strain in terms of the intact rock properties and modulus of deformation in the field in the desired direction. Expressions are also suggested for calculating critical strain if actual modulus of deformation is not known but Q is known. However the Eq. 6 is recommended for calculation of critical strain taking into account the anisotropy of the jointed rock mass. The observed or expected strain in the tunnel will depend on the properties of the rock mass and in addition, size, shape of the tunnel and the in situ stress field. Then expected tangential strain may be obtained by numerical modelling or preferably be monitored in the field. A Squeezing Index (SI), which is ratio of actual strain to the critical strain, is related to degree of squeezing or the occurrence of probable construction problems. Several case histories prove the usefulness of the Squeezing Index (SI) in Eq. 12 in the tunnels. Further the efficacy of the adequate support system to reduce squeezing may also be evaluated by the SI approach suggested in this paper. 

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* Associate Professor, Civil Engineering Department, IIT Roorkee, Roorkee- 247 667, UA, India

** Retired Professor, Civil Engineering Department, IIT Roorkee, Roorkee- 247 667, UA, India

*** Consultant, ATES, AIMIL Ltd., New Delhi, India