In 1997 Potts & Addenbrooke presented an approach which considers the building’s stiffness when predicting tunnel induced building deformation. Their study included over 100 2D plane strain analyses in which buildings were represented by elastic beams with a Young’s modulus E, a second moment of area I and a cross-sectional area A. The bending stiffness of the structure is described by EI whereas EA represents the axial stiffness. The geometry of their model was described by the building width B, the tunnel depth z0 and the eccentricity e which is the offset between building and tunnel centre lines as shown in Figure 2.26. Their parametric study included a wide range of different bending and axial stiffnesses and also varied the geometry. The stiffness of the structure was related to that of the soil by defining relative stiffness expressions:
ρ∗= EI Es ¡B 2 ¢4; α∗ = EA Es ¡B 2 ¢ (2.29)
where ES is the secant stiffness of the soil that would be obtained at 0.01% axial strain in a triaxial compression test performed on a sample retrieved from half tunnel depth. In their analyses they included tunnel depths of z0 = 20m and 34m leading to a soil stiffness of ES = 103MPa and 163MPa, respectively, for the modelled soil profile. The parameter ρ∗ is referred to as relative bending stiffness whereas α∗describes the relative axial stiffness. Potts 6Note that the term ‘elements’ is not used in the sense of finite elements as the finite difference code FLAC
Figure 2.26: Geometry of the problem and definition of deflection ratio.
& Addenbrooke (1997) applied these relative stiffness expressions to plane strain analyses. In such a context I is expressed per unit length and, consequently, ρ∗ has the dimension [1/length]. In contrast, α∗ is dimension-less in plane strain situations. EI controls the bending behaviour of the beam while EA governs both shear and axial behaviour.
The above relative bending stiffness expression ρ∗ is similar to that introduced by Fraser & Wardle (1976) to describe the settlement behaviour (induced by vertical loading) of rect- angular rafts. Similar factors were also adopted for retaining wall analyses by Potts & Bond (1994). The definition of relative axial stiffness α∗ is similar to the normalized grade beam stiffness used by Boscardin & Cording (1989).
It should be noted that the above relative stiffness expressions are of an empirical nature. They were not derived in the same fashion as the similar expressions for retaining walls (Rowe, 1952; Potts & Bond, 1994). A consequence of this empirical approach are the different
dimensions of the relative stiffness expressions when used in 2D and 3D situations. The advantage of this approach, however, is that it describes building deformation with only two parameters.
The soil profile in their analyses consisted of London Clay represented by a non-linear elastic pre-yield model, described by Jardine et al. (1986) (and also adopted by Addenbrooke et al. (1997), see Page 44) and a Mohr-Coulomb yield and plastic potential surface. The soil was modelled undrained as only the short term response was investigated. An initial hydrostatic pore water pressure profile was modelled with a water table located at 2m below ground surface. The coefficient of earth pressure at rest was K0 = 1.5. A zone of reduced K0 was included in the analyses in order to obtain better predictions for greenfield surface settlement (see Figure 2.14).
The building deformation criteria adopted in their study were deflection ratio and hor- izontal strain. The building deformation was related to greenfield situations (denoted by a superscript ‘GF’) by defining modification factors as
MDRsag = DRsag DRGF sag MDRhog = DRhog DRGF hog (2.30) for sagging and hogging, respectively. Similar factors are defined for horizontal compressive and tensile strain:
M²hc = ²hc ²GF hc M²ht = ²ht ²GF ht (2.31) To calculate the greenfield deformation criteria the part of the surface settlement trough below the building must be extracted. This part of the greenfield settlement profile will be referred to as the greenfield section and is described by the same geometry as the building. This situation is shown for deflection ratio in Figure 2.26. If this section contains the point of inflection sagging and compressive horizontal strain occur between one edge of the section and the point of inflection whereas hogging and tension can be found between the point of inflection and the other edge7. If the position of the point of inflection is outside the section either sagging and compression or hogging and tension develop within the section. 7For long structures both points of inflection may be within the greenfield section. For symmetric cases
both hogging zones give the same results while for eccentricities one hogging zone (normally the longer one) is the more critical one.
The horizontal strains ²GF
hc and ²GFht represent the maximum (absolute) value of horizontal compressive and tensile strain over the greenfield section and ²hcand ²htare the same measure for the building. These expressions have to be distinguished from the maximum horizontal compression and tension, ˆ²hc and ˆ²ht, respectively, found over the entire greenfield profile (compare with Figure 2.3).
Potts & Addenbrooke (1997) performed a parametric study in which ρ∗and α∗were varied independently over a range. While this variation sometimes resulted in an unrealistic combi- nation of bending and axial stiffness it allowed them to investigate the extreme limits of the interaction problem. Additional stiffness combinations were also included which represented 1, 3 and 5-storey structures which consisted of 2, 4 and 6 slabs, respectively, with a vertical spacing of 3.4m. Each slab had a bending stiffness, EIslab and a axial stiffness EAslab. The bending stiffness of the entire building was calculated from EIslab of the individual slabs by employing the parallel axis theorem (Timoshenko, 1955). The neutral axis was assumed to be at the middle of the building. Axial stiffness of the entire structure was obtained by assuming axial straining along each slab’s full height. The resulting overall stiffness represents a rigidly framed structure and therefore can be considered to over-estimate the building stiffness.
No vertical load was applied in the analyses. The interface between structure and soil was assumed to be rough. Tunnel construction was modelled by controlling the volume loss with a value of VL = 1.5%.
For deflection ratio their parametric study revealed the following behaviour:
• For low values of α∗ the settlement is equal to those obtained in greenfield conditions regardless of the value of ρ∗.
• For low values of ρ∗ but high values of α∗ the deflection ratio modification factors for both sagging and hogging are higher than unity.
• As ρ∗ increases from a low value both MDRsag and MDRhog remain at unity for low values of α∗ but decreases for higher values of α∗ showing a greater reduction with increasing α∗.
• As α∗ increases from a low value both MDRsag and MDRhog increases for extremely low values of ρ∗ but significantly decrease for higher values of ρ∗.
• The increase of both MDRsag and MDRhog with α∗ for small values of ρ∗ can lead to modification factors exceeding unity. This behaviour is due to the change in surface boundary conditions and due to the uncoupled influence of ρ∗ and α∗ (for high values of α∗) on deflection ratio and horizontal strain, respectively.
When considering realistic combination of axial and bending stiffness the results for dif- ferent geometries indicated a unique trend when plotting MDRsag and MDRhog against ρ∗. Different upper bound curves were fitted to different degrees of eccentricity, expressed as e/B. Modification factors for sagging reduce with eccentricity whereas those for hogging increase. These upper bound curves for MDR are shown in Figure 2.27a.
For axial strain the parametric study exhibited the following trends:
• For low values of α∗ the settlement is equal to those obtained in greenfield conditions regardless of the value of ρ∗.
• As α∗ increases from a low value the modification factors for compression and tension decrease. Their value does not depend on the magnitude of ρ∗ and, consequently, all results (for a given geometry) lie on a unique curve.
When realistic stiffness combination were considered the results for different geometries indicated trends of reducing M²h with increasing α∗. It was possible to set upper bound curves for different degrees of eccentricity. These curves are shown in Figure 2.27b. The figure indicates that for realistic stiffness combinations the strain modification factors are small compared to deflection ratio modification factors.
Potts & Addenbrooke (1997) proposed to include these design curves into a design ap- proach to predict tunnel induced building deformation and assess potential damage. The key steps of this design approach are:
1. The greenfield settlements and horizontal strains for the geometrical section of the building are predicted using Equations 2.4 and 2.6, respectively.
2. The bending and axial stiffness of the structure must be evaluated. Using Equations 2.29 both relative bending and axial stiffness can be calculated.
(a) (b)
Figure 2.27: Design curves for modification factors of (a) deflection ratio and (b) maximum horizontal strain (after Potts & Addenbrooke, 1997).
4. The deformation criteria of the building can be calculated by multiplying the greenfield deformation criteria with the corresponding modification factors:
DRsag= MDRsagDRGF
sag; DRhog= MDRhogDRGFhog (2.32)
²hc= M²hc²GF
hc ; ²ht= M²ht²GFht (2.33)
5. Combinations of DRsag and ²hc, and DRhog and ²ht are used as input parameters in damage category charts such as that shown in Figure 2.25 to evaluate the damage category (as listed in Table 2.2) and to assess the potential damage.
This design approach can be incorporated into the second stage risk assessment as shown in Figure 2.28. Considering the effects of soil-structure interaction in this stage rather than in the third stage reduces the number of cases for which a detailed evaluation has to be carried out.