**4. Time-dependent effects in tunneling**

Time-dependent deformations associated with rock tunneling are a reality that warrants further investigation and understanding and can be observed during excavation and/or after the construction period of the project.

## **4.1 Analysis of time-dependent rock masses using the convergenceconfinement method**

Understanding the nature and origin of deformations due to an underground opening requires, as [50] noted, both knowledge of the rock-support interaction and field data interpretation. This tunnel wall movement, also known as convergence, results from both the tunnel face advancement and the time-dependent behavior of the rock mass.

The Convergence-Confinement Method (CCM) is a two-dimensional simplified approach that can be used to simulate three-dimensional problems. Analytical solutions based on CCM (usually examine either the effect of tunnel advancement or the time-effect) could be partially used to select the final support. One may wonder if it could also be possible to simulate and replicate the complete problem. Timedependency is acting during the timeframe of construction impacts, the so-called Longitudinal Displacement Profiles (LDPs) for deformation estimation during tunnel advance. LDP is an accompanying tool used with the Ground Reaction Curves (GRC) used in the CCM to relate internal wall pressure relaxation to tunnel displacement. Suggested analytical solutions for LDPs [50], etc.) refer to elastic or elasto-plastic rock materials. **Figure 20** schematically illustrates the effect of both time and tunnel advancement on the LDP of a tunnel excavated in a visco-elastic medium. The tunnel's advance is simulated by reducing the internal pressure, pi, initially acting on the tunnel core (as p0). The rock responds by convergent deformations (via the GRC), which are, in turn, linked to the tunnel advance via the LDP. This aspect of time-dependency is also discussed, examined and further analyzed in this chapter.

Analytical and closed-form solutions that consider the time-dependent convergence have been proposed in the literature for supported and unsupported tunnels with linear and non-linear visco-elastic medium [51, 52], etc. Most of these formulations also consider the tunnel advance in the estimated total deformation yet are found to be impractical due to the complex calculations required [53, 54].

**Figure 21** illustrates the anticipated LDP of the tunnel displacement in an elasto-visco-elastic medium where no tertiary creep takes place. More ductile materials, as in the case of rock salt, can behave in such a manner. It is shown that when no time-effect is considered, the total displacements are underestimated, which can lead to erroneous calculations at the initial stages of the design process.

#### **Figure 20.**

*Schematic representation of the GRC of an elastic (t = 0) and a visco-elastic material (t > 0) and their relation to the LDP. Y-axis on the left refers to the internal pressure (pi) normalized to the in-situ pressure (p0), Y-axis on the right refers to the distance from the face (x) normalized to the tunnel radius (R) and X-axis refers to the radial displacement at a location x normalized to the maximum radial displacement due where t denotes time and subscripts e and ve refer to elastic and visco-elastic material, respectively.*

*Time-Dependent Behavior of Rock Materials DOI: http://dx.doi.org/10.5772/intechopen.96997*

**Figure 21.** *Schematic representation of the longitudinal displacement profile (LDP) in an elasto-visco-elastic medium.*
