# Multi-Temporal InSAR Structural Damage Assessment: The London Crossrail Case Study
## 3 Methodology
### Bulding Damage Assessment Relative Stiffness Method  
Common procedures for the assessment of settlement-induced structural damage start with the evaluation of the tensile strains caused by the greenfield settlement profile when applied to the base of a linear elastic beam model of the building [24]. The soil greenfield displacements are defined as tunneling-induced ground displacements in absence of buildings. They are usually calculated analytically or numerically neglecting the influence of the structure weight and stiffness, while considering the type of soil, the tunnel depth and diameter, and the volume of ground lost during the excavation [40, 41, 42]. The settlement profile depicted in Figure 2 is typical of tunneling in soft ground. The severity of the applied ground distortion is quantified by the deflection ratio \(\Delta/L\) (Figure 2). For any given two points on the soil surface, \(\Delta/L\) is the ratio of the relative deflection \(\Delta\) and their horizontal distance \(L\). \(\Delta\) is defined as the maximum vertical distance from the settlement profile between the two points and a straight line connecting them. The distance \(L\) typically corresponds to the length of the building either in the sagging (\(L_{\text{sag}}\)) or hogging (\(L_{h\text{log}}\)) part of the greenfield settlement profile (Figure 2). The greenfield beam strains are then approximated using the calculated deflection ratio, and are then compared to pre-defined limiting strain values which relate to a certain level of expected damage [24]. Each building damage level is described in terms of approximate crack widths, as summarized in Table 1.  
Figure 2: Typical shape of a tunneling-induced settlement trough. The inflection point separates the sagging (sag) from the hogging (hog) zone. The relative deflection and building length for each zone are indicated with \(\Delta\) and \(L\). The building weight and stiffness typically has the effect of widening the soil settlement profile, thus reducing its curvature. For this reason, the deflection ratio calculated in greenfield conditions is generally bigger than the actual one resulting from the soil-structure interaction effect. This simplified method therefore provides conservative damage predictions [43]. A more accurate assessment method takes into consideration the fact that the soil-structure interaction modifies the shape and magnitude of the settlement profile, thus affecting the final damage classification and decision-making process. Modification factors [29; 31; 32] have been defined to account for the difference between greenfield distortions (\(\Delta_{gr}/L_{gr}\)), and actual distortions (\(\Delta/L\)), and therefore to update the tensile strains to more realistic values. In this paper we consider the modification factor for the deflection ratio, which is defined as the ratio between the actual deflection ratio of the building and the greenfield deflection ratio of the soil surface [29]:  
\[\text{MF}=\frac{(\Delta/L)}{\left(\Delta_{gr}/L_{gr}\right)} \tag{1}\]  
If a building spans over the inflection point (Figure 2), the modification factors are calculated separately for the sagging and hogging part of the trough. Note that buildings affect not only the relative deflection \(\Delta\) but also the location of the inflection point, and therefore the length \(L_{agg}\) and \(L_{hog}\) of the building partition in the sagging and hogging zones, respectively. If the actual \(\Delta/L\) is not known a-priori, e.g., for damage prediction, the modification factors can be obtained by using design curves based on building and soil characteristics [29; 31; 32]. The derived \(\Delta/L=\text{MF}\) (\(\Delta_{gr}/L_{gr}\)) are then used to calculate the tensile strain of the linear elastic beam of the building according to Timoshenko beam theory [24; 45].