A numerical procedure to simulate tunnel lining behavior

in this work a very accurate constitutive law is applied to describe concrete lining behavior, that is 2D-PARC, which has been properly improved on purpose

To better explain the adopted numerical procedure, this section is subdivided in three main parts: at first the most important modeling choices will be outlined, subsequently the thermal analysis will be presented and finally the mechanical analysis will be explained in details, also highlighting the improvement made to 2D-PARC model in order to accurately describe concrete behavior also under fire loads.

4.3.1 Modeling choices

The FE model, which is strictly related to the considered problem, represents the study domain, which is in this case represented by a significant portion of the tunnel and the surrounding ground. The model should be indeed realized by taking into account the real extent of the rock mass region involved in the excavation process, since it modifies the stress and strain fields around the opening. So, the FE mesh is chosen wide enough to reach steady-state conditions and to eliminate almost any influence of the outer boundaries. Fixities of stress and displacement are applied at boundaries so as to prevent rigid body movements of the model and to maintain the appropriate boundary conditions during the analysis.

Figure 4.6 Three-dimensional model: (a) geometry and FE mesh of the entire model (b) FE mesh of the lining

Generally speaking, the whole tunnel can be modelled; however, transversal symmetric conditions are assumed in this thesis to reduce the computational effort. By exploiting symmetry with respect to the vertical axis passing for the center of the tunnel, only one-half of the tunnel cross-section is modeled. More in detail, the extent of the FE mesh from the vertical axis of symmetry is taken five times the mean tunnel diameter, so that the boundaries have a negligible influence on the results of the analyses.

Moreover, since this work focuses on deep tunnels, also in the vertical direction starting from the center of the tunnel the size FE mesh is limited to maximum five times the mean tunnel diameter, in order, once again to limit the computational time. In case of higher overburden, an external pressure is also applied to the upper surface of the FE mesh, so as to simulate the surcharge due to the overburden not explicitly considered in the mesh.

To further reduce the computational effort only a significant portion of tunnel length is considered, without modelling its whole extension.

It is worth noting, that the same mesh is adopted for both the heat-transfer analysis and the nonlinear structural simulation, by using heat-transfer elements in the first case and stress elements in the second one. In more detail, the rock mass is modelled through 6-node triangular prismatic elements, while 4-node shell elements are applied for concrete lining. These latter are chosen in order to use 2D-PARC constitutive model for the description of concrete lining behavior in the mechanical analysis.

The aspect ratios of all elements are taken less than 5:1 to ensure reliable results and good convergence. In order to correctly catch the temperature gradient and the related stress field in the lining, 19 integration points (the maximum allowed for thermal analyses by the adopted FE code, ABAQUS) are considered in the shell thickness. Moreover, by exploiting the layering of the shell elements, a refinement of the integration points is provided near the surface exposed to fire or where the peak of the compression is expected.

The composite shell elements are connected to the triangular prisms placed on the outer perimeter of the opening by surface-based ties, for both the heat-transfer and the mechanical analyses. Since the default reference surface of the shell elements coincides with their midsurface, an offset is assigned between the shell middle plane and the reference surface, in order to correctly represent the lining thickness and avoid element interpenetrations.

As known, the fineness of the mesh affects the convergence speed and the accuracy of the results; as a consequence, more elements should be used in all those regions characterized by larger stress gradients or where a higher resolution is required. Hence, the discretization is properly refined in correspondence of the lining, as can be seen in Figure 4.6.

4.3.2 Thermal analysis

Thermal analysis is performed in order to obtain the transient temperature field to be applied in the subsequent mechanical simulation. The determination of the correct temperature variation in time and space within the studied element is indeed essential to evaluate the corresponding thermal strains and the degradation of concrete mechanical properties which are mandatory for carrying out a correct mechanical simulation.

To solve the thermal problem the FE software ABAQUS is employed. The adopted mesh consists of 6-node triangular heat-transfer prisms for the ground and 4-node heat-transfer shells for the lining. Static boundaries are not required, while thermal ones must be specified in space and time, as described in the following subsection.

Thermal analysis is performed by taking into account the three mechanisms that rule heat-transfer, which are convection, radiation and conduction.

The thermal input consists in the temperature evolution with time (the so-called fire scenario); in this work, empirical standard fire curves are adopted. For sake of simplicity, in this thesis the fire temperature is uniformly applied along the whole length and section of the tunnel; however, a more realistic fire scenario could be inserted.

In the following some details about the governing equations of the thermal problem, as well as the adopted fire curves and thermal properties of rock and concrete to be defined in the transient temperature analysis will be provided.

4.3.2.1 Mathematical formulation

The governing equation of the heat transfer analysis, which provides the time-dependent distribution of the temperature T within the element, is the Fourier differential law:

t c T q

T _g

∂

= ∂ +

∇ ρ

λ ² . (4.1)

where t is the time and λ, ρ, c respectively ^represent the thermal conductivity, the density and the specific heat of the material. Moreover, qg is the internal generated heat, which is assumed in this case equal to zero. It is worth noting that the values of parameters λ, ρ, c for concrete depend in turn of the current temperature, see §4.3.2.3.

To solve Equation (4.1), boundary conditions in time and space must be specified. The initial temperature (at t=0) is assumed equal to 20 °C, while the transient heat flux at the boundaries accounts for both convention and radiation.

The heat flux on a surface due to convection qc is governed by:

T0

T h

q_c =− − , (4.2)

where h is the heat transfer coefficient, T is the temperature on the surface and T0 is the fire or environmental temperature, depending on type of exposure;

whereas the heat flux due to radiation qr is:

( ) ( )

[

]

r T T T T

q =εσ − − − , (4.3)

where ε is the emissivity of the surface, σ the Stefan-Bolzman constant and Tz

the value of absolute zero on the temperature scale being used.

In this way, the physical heat transfer phenomenon is correctly modeled.

The heat flux, defined by the fire curve, flows to the inner surface of the lining exposed to fire by means of radiation and convection. Then the heat is transferred from the heated surface into the materials by conduction.

The history of the temperature is recorded at each integration point of the model in order to be used as an input for the subsequent mechanical analysis.

4.3.2.2 Fire curves

A fundamental aspect in the study of tunnel fire resistance is the definition of the fire scenario. Traditionally, the heat exposure of the tunnel is based on the use of standardized time–temperature curves [213] and this approach is also followed in this thesis; the thermal input is indeed represented by fire curves, available in technical literature. In the following the most applied fire curves for tunnel analyses will be outlined.

A widely-used fire curve is represented by the standard ISO 834. This curve is based on a cellulosic fire and it is commonly adopted in fire testing of structural elements, being suitable for materials found in typical buildings. Its analytical expression is expressed by:

(

)

345

20+ ₁₀ +

= log t

T . (4.4)

This curve has been applied for many years also in the analysis of tunnels, even if it is not able to correctly represent highly combustible materials, such as fuels and chemicals, and the actual ventilation conditions that apply for tunnels [201]. Tunnel fires are indeed characterized by the burning of fuel and vehicles with high calorific potential, in combination with the confinement of the released heat, as detailed explained in §4.2.1. For this reason, also other curves are commonly applied in the thermal analysis of tunnels.

The HC (Hydrocarbon) curve approximates the fire evolution of small tanks of flammable liquids and was developed for the petrochemical and off-shore industries. Compared to the ISO 834, this curve is characterized by a higher temperature increase, associated to a faster fire development, typical of petroleum fire. It is expressed by the following relation:

(

)

T =20+10801−0325 ⁻⁰¹⁶⁷ −0675 ⁻²⁵ . (4.5)