**Stability of tunnel headings**

**Theorems of plastic collapse**

Upper bound

Lower bound

If a structure is loaded to this value it

must

collapse

'Kinematically admissible mechanisms'

If a structure is loaded to this value it

cannot

collapse

'Statically admissible stress fields'

True collapse load

Rigorous proofs exist which show that the upper and lower bounds bracket the true collapse load.

To estimate the

true collapse load

, we can:

refine upper and lower bound calculation methods until the bracket is small

use limit equilibrium

use numerical models

use centrifuge tests

use field data from real collapses

"If you take any compatible mechanism of slip surfaces and consider an increment of movement and if you show that the work done by the stresses in the soil equals the work done by the external loads, the structure must collapse (i.e. the external loads are an upper bound to the true collapse loads)." Atkinson (2007)

"If you can determine a set of stresses in the ground that are in equilibrium with the external loads and do not exceed the strength of the soil, the structure cannot collapse (i.e. the external loads are a lower bound to the true collapse loads)." Atkinson (2007)

Undrained

Drained

Davis, E. H., Gunn, M. J., Mair, R. J. & Seneviratne, H. N. (1980). The stability of shallow tunnels and underground openings in cohesive material. Geotechnique, Vol. 30, No. 4, pp. 397-419.

Leca, E. & Dormieux, L. (1990). Upper and lower bound solutions for the face stability of shallow circular tunnels in frictional material. Geotechnique, Vol. 40, No. 4, pp. 581-605.

This is for c'-phi' soils above the water table only

Undrained

Drained

Centrifuge modelling

Numerical modelling

Vermeer, P. A., Ruse, N. & Marcher, T. (2002). Tunnel heading stability in drained ground. Felsbau 20, No.6, pp.8-18.

An example:

An example:

Walter, H., Coccia, C. J., Wallen, R. B., Ko, H.-Y. & McCartney, J. S. (2010). Modeling of tunnel lining deformation due to face instabiity. Proc. Int. Conf. Phys. Mod. Geot. Engrg.

University of Cambridge 10m beam centrifuge

**Design charts**

Undrained

Drained

Undrained is when k < 1e6 to 1e7 m/s according to Anagnostou & Kovari

Use Mair's design charts based on centrifuge tests

Drained is when k > 1e6 to 1e7 m/s and advance rate < 0.1 to 1 m/hr, according to Anagnostou & Kovari

Use Anagnostou & Kovari's method based on limit equilibrium to calculate support pressure

N.B.: If in doubt, calculate both undrained and drained stability

Davis, E. H., Gunn, M. J., Mair, R. J. & Seneviratne, H. N. (1980). The stability of shallow tunnels and underground openings in cohesive material. Geotechnique, Vol. 30, No. 4, pp. 397-419.

Leca, E. & Dormieux, L. (1990). Upper and lower bound solutions for the face stability of shallow circular tunnels in frictional material. Geotechnique, Vol. 40, No. 4, pp. 581-605.

This is for c'-phi' soils above the water table only.

[limit equilibrium is both kinematically and statically admissible and combines considerations of both compatibility and equilibrium]

[not such a problem for undrained, but for drained case there is a big gap between bounds]

Note: A&K's method can include the effects of seepage