Numerical studies of the interaction problem

To investigate the soil-structure interaction, FE modelling has been adopted by several au- thors. These models include the building and its interaction with tunnel induced ground movement can be evaluated. Different approaches have been used to represent the building with varying level of details included in these models:

Fully 3D modelling: In such an analysis details of a buildings such as the layout of the fa¸cades, the position of windows and doors, etc. can be modelled. The advantage of such a 3D model is that the building can be considered in any geometrical configuration with respect to the tunnel axis.

Plane strain/stress analysis of structure: This approach models the in-plane geometry of the structure transverse to the tunnel. The building is described by its width and hight and details such as windows and doors can be incorporated in the model. The advantage of 2D modelling is the small amount of computational resources required compared to 3D analysis. It is therefore possible to perform parametric studies including a wide range of different parameters.

Deep beam model: This model is similar to the approach adopted by Burland & Wroth (1974). The structure is represented by an elastic beam with bending stiffness (EI) and axial stiffness (EA) representing the overall stiffness of the structure. The deformation can be imposed on the beam by incorporating it into a FE tunnelling analysis (either 2D or 3D, in the latter case the term ‘shell’ should be used instead of ‘beam’) or by predescribing the displacement of the beam. The advantages of this method are – when used in 2D conditions – the small amount of computational resources required and therefore the ability to perform extensive parametric studies. Furthermore this approach is consistent to the risk assessment outlined by Burland (1995).

Burd et al. (2000) presented results of a study employing the first approach. In the fully 3D FE analyses the building consisted of four masonry fa¸cades modelled by plane stress elements. Details such as windows and doors were included in the model. The masonry was represented by a constitutive model giving high strength in compression but relatively low tensile strength. The building was assigned to have self weight. Burd et al. (2000) presented results for both symmetric geometries (where only half of the mesh had to be modelled) and asymmetric situations in which the building had a skew angle in plan with respect to the tunnel centre line.

The soil was described by a multi-surface plasticity model which accounts for the variation of tangent shear stiffness with strain. The initial stress profile is controlled by a bulk unit weight and a coefficient of earth pressure at rest of K0 = 1.0. The geometry of the mesh is summarized in Table 2.1, Page 48.

Tunnelling was modelled by specifying the soil parameters to achieve a volume loss of V_L = 2%. Tunnel construction was modelled in only four stages by using the same approach adopted by Augarde et al. (1998), described on Page 52.

In their study Burd et al. (2000) compared analyses in which building and soil movement were coupled with greenfield analyses whose settlement and horizontal displacements were then imposed directly on the building (referred to as uncoupled analysis)5_.

From their analyses they draw the following conclusions:

5_{The meaning of the terms coupled and uncoupled in this context has to be distinguished from their use in}

• The stiffness of the building reduces differential settlement although significant tilting was observed for a building with an eccentricity with respect to the tunnel centre line • In sagging deformation the building behaves stiff and develops substantially less damage

during a coupled analysis compared to the uncoupled one. They suggest that the ground provides a certain amount of lateral restraint when the building is subjected to sagging deformation. Similar conclusions were drawn by Burland & Wroth (1974).

• In hogging such a restraint is not provided and the structure behaves more flexibly leading to higher degrees of damage than in sagging. In such a case the coupled anal- ysis developed more damage than the uncoupled one. Burd et al. (2000) related this behaviour to the imposition of building weight which alters the settlement behaviour compared to greenfield situations adopted for the uncoupled settlement predictions. Similar conclusions were presented by Liu et al. (2000) who performed plane strain analy- ses including both symmetric and eccentric building cases. The geometry of the building fa¸cade, the initial stress conditions and the material models describing the soil and the masonry structure were similar to those employed by Burd et al. (2000). By varying the eccentricity and the weight of the structure Liu et al. (2000) concluded that the application of building weight increases the tunnel induced building deformation. This behaviour was due to the fact that prior to tunnel construction a number of yield surfaces of the multi- surface soil model were activated after the application of building load whereas such an effect was not present in greenfield conditions. In their model this behaviour leads to lower soil stiffness in building cases which then exhibit larger deformation caused by subsequent tunnel construction.

Another approach using 2D analyses was proposed by Miliziano et al. (2002). They pointed out that in urban areas tunnel construction often follows the route of existing streets and that old masonry buildings are often characterized by a modular and repetitive structural arrangement on either side of the street. They modelled the structure in plane strain and assigned a reduced equivalent stiffness to the masonry which accounts for the spacing be- tween different structural elements of such a ‘terrace house’ configuration in the longitudinal direction. The masonry itself is represented following a discontinuous approach by connect-

ing linear elastic elements6 _{by elasto-plastic interfaces. Their results not only indicate that} building deformation decreases with increasing building stiffness but also that it decreases with reducing soil stiffness.

An approach to relate the building’s stiffness to that of the soil was proposed by Potts & Addenbrooke (1997). They modelled the building as an elastic beam with a stiffness representing the overall behaviour of the structure. With this simple model it was possible to perform an extensive parametric study to investigate the influence of both building stiffness and geometry on the interaction problem. From this study they developed a design approach which can be incorporated into the three stages risk assessment. The next section will present this relative stiffness approach in more detail.