Tunnelling and Underground Space Technology 125 (2022) 104499

Contents lists available at ScienceDirect

Tunnelling and Underground Space Technology

incorporating Trenchless Technology Research
journal homepage: www.elsevier.com/locate/tust

Early-stage assessment of structural damage caused by braced excavations:

Uncertainty quantification and a probabilistic analysis approach
Jinyan Zhao a, Stefan Ritter b, Matthew J. DeJong c, *
a
Department of Civil and Environmental Engineering, University of California, Berkeley, CA 94720, United States
b
Onshore Foundations, Norwegian Geotechnical Institute, Sognsveien 72, 08555 Oslo, Norway
c
Department of Civil and Environmental Engineering, University of California, Berkeley, CA 94720, United States

A R T I C L E I N F O A B S T R A C T

Keywords: Current early stage assessment methods for deep excavation induced structural damage have large uncertainty
Deep excavation due to modeling idealizations (simplification in analyses) and ignorance (incompleteness of information). This
Building damage paper implements an elastoplastic two-stage solution of soil-structure-interaction to predict building response to
Soil-structure-interaction
adjacent deep excavations with braced supports. This soil-structure-interaction solution is then used to study the
Uncertainty quantification
Global sensitivity analysis
uncertainty in two case studies. A global sensitivity analysis is conducted, which indicates that the prediction of
ground movement profiles is the major source of uncertainty in early stage building damage assessment. The
uncertainty due to ignorance and idealizations related to structural analysis models also contribute significantly
when target buildings are modeled as equivalent beams. However, the use of a 2-dimensional elastic frame
structural model, in lieu of an equivalent beam, considerably reduces the assessment uncertainty. Considering
the existence of uncertainty, a probabilistic analysis approach is proposed to quantify the uncertainty when
predicting potential building damage due to excavation-induced subsidence. A computer program called Un­
certainty Quantification in Excavation-Structure Interaction (UQESI) is developed to implement this probabilistic
analysis approach.

1. Introduction unreliable and sometimes overly conservative damage evaluation re­

sults. Moreover, because no comprehensive survey is done in the second
In major deep excavation projects, there are often many buildings assessment stage, details of structure layouts and material properties are
influenced by construction activities. It is a challenge to assess all of often unavailable. Consequently, many assumptions and approxima­
these buildings with detailed analysis to identify potential damage. To tions are made in the analyses, which introduce large uncertainty to
address this challenge, a staged approach by Mair et al. (1996) is building damage evaluations. This paper aims to study the uncertainty
commonly adopted which consists of three stages: preliminary assess­ in the second assessment stage of brace-supported deep excavations and
ment, second stage assessment and detailed evaluation. In the pre­ proposes suggestions on optimal trade-offs between analysis complexity
liminary assessment stage, a settlement contour is computed and the and prediction accuracy. Current second stage assessment can be
buildings with a predicted settlement of less than 10 mm and a predicted divided into three components (Schuster et al., 2009): (1) lateral and
slope of less than 1:500 are considered to have a negligible risk of vertical ground displacement profiles are determined; (2) engineering
damage. Otherwise, buildings are qualified for a second stage assess­ demand parameters are estimated based on various soil-structure
ment, in which some engineering demand parameters (e.g., distortion, interaction (SSI) assumptions; (3) building damage categories are
deflection ratio and tensile strain) are calculated. A potential damage determined according to the engineering demand parameters. In this
category is assigned to each building and the buildings with severe paper, these three components are implemented with a numerical
damage potential are required to be evaluated in detail in a third analysis framework (extended from Franza and DeJong (2019)) which
assessment stage. considers elastoplastic SSI effects. Two case studies of deep excavation
The current second stage assessment methods consist of many in urban areas are explored. Uncertainty and sensitivity analyses are
simplified models and empirical equations, which often lead to conducted for the case studies and suggestions to reduce damage

* Corresponding author.
E-mail address: dejong@berkeley.edu (M.J. DeJong).

https://doi.org/10.1016/j.tust.2022.104499
Received 21 December 2020; Received in revised form 31 March 2022; Accepted 1 April 2022
Available online 10 April 2022
0886-7798/© 2022 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
J. Zhao et al. Tunnelling and Underground Space Technology incorporating Trenchless Technology Research 125 (2022) 104499

assessment uncertainty are provided. A probabilistic analysis approach ( )

d d
is then proposed to quantify the uncertainties caused by the approxi­ δl (d) = 0.8
+ 0.2 δlm 0.0 < ≤ 1.0
mations and simplifications in analyses, so that a confidence level of the He He
( )
predicted building damage can be provided. A computer program called d d
δl (d) = − 0.4 + 1.4 δlm 1.0 < ≤ 2.5 (3)
Uncertainty Quantification in Excavation Structure Interaction (UQESI), He He
which enables efficient SSI analysis and yields uncertainty quantifica­ (
d
)
d
tion and sensitivity analysis, is proposed. δl (d) = − 0.16 + 0.8 δlm
He
2.5 <
He
≤ 5.0

2. Background The KJHH and KSJH models are considered as an appropriate

method to estimate ground displacement profiles for the early assess­
2.1. Estimation of greenfield ground displacement profile ment stage because δhm , δv and δl can be approximated without any
complex modeling of the excavation system. This ensures a simplified
The first component in the second stage assessment is to determine analysis procedure, although some uncertainty is introduced due to the
greenfield ground displacement profiles, which describe the excavation- variance and possible bias of the regression analyses. The implementa­
induced ground movements when the effect of surface buildings is tion of the KJHH and KSJH models to assess SSI mechanisms and their
ignored. For braced excavations, a concave-shaped ground displacement respective uncertainties are discussed in later sections.
profile is usually adopted. Hsieh and Ou (1998) proposed a method to
estimate the vertical ground settlement profile by: 1) predict the
2.2. Estimation of structure deformation
maximum lateral wall deflection (δhm ) with numerical methods (e.g.
finite element or beam on elastic foundation methods); 2) estimate the
Because of soil-structure interaction effects, the displaced shape of
maximum vertical ground surface settlement (δvm ) from empirical re­
the soil at the base of a surface structure is usually different from that of
lationships with δhm ; 3) calculate the surface settlement at various dis­
typically measured greenfield ground displacements. To quantify the
tances (d) behind the wall according to Eq. (1). Hsieh and Ou (1998)’s
effect of structural stiffness on estimating structural deformation, two
ground profile is derived based on a regression analysis of 10 case
different approaches are often adopted: 1) the relative stiffness approach
studies, and might be biased due to the small sample size.
and 2) the soil-structure-interaction (SSI) approach.
Kung et al. (2007) therefore extended the regression analysis by
In the relative stiffness approach, the differences between building
including more case studies and a suite of artificial scenarios analyzed
response and greenfield ground displacement are often described with
with the finite element method. The vertical ground displacement pro­
modification factors for the deflection ratio and horizontal strain in both
file revised by Kung et al. (2007) is expressed in Eq. (2). Based on
the sagging and hogging region. Many authors (e.g., Potts and Adden­
regression analysis, Kung et al. (2007) also proposed empirical equa­
brooke, 1997; Dimmock and Mair, ,2008; Franzius et al., 2006) pro­
tions to estimate the maximum horizontal wall deflection (δhm ) and the
posed variants of the relative stiffness approach to estimate tunneling-
deformation ratio (Rv ) between δvm and δhm (i.e., Rv = δδhm
vm
). The empirical
induced settlement based on design charts of relative stiffness factors
equations (also referred to as the KJHH model) estimate δhm and δvm and modification factors. Mair (2013) also studied the effects of deep
based on the dimensions of the excavation system, the soil shear excavations on surface structures and proposed a bending stiffness factor
strength, the soil elastic modulus, the soil effective stress, the support and corresponding design charts to estimate modification factors. In
system stiffness and the depth to hard stratum. A comparison of Hsieh practice, the relative stiffness approach can be easily implemented but
and Ou (1998) and Kung et al. (2007)’s settlement profile is shown in only provides approximate solutions. Additionally, there is often a large
Fig. 2a. discrepancy between the analysis results with different relative stiffness
Schuster et al. (2009) studied the lateral displacement induced by factors, which subsequently cause large uncertainty in the estimation of
deep excavation using the same finite element model as developed by building deformation (e.g., Giardina et al., 2018).
Kung et al. (2007). A horizontal ground displacement (δl ) profile (Eq. Another method to estimate building deformation is to analyze the
(3)) and empirical equations to estimate the lateral deformation ratio soil-structure-interaction explicitly. Franza and DeJong (2019) pro­
(Rl = δδhm
lm
) were proposed. The parameters used to estimate Rl are iden­ posed an elastoplastic solution, in which the interface between soil and
tical to the parameters used in the KJHH model, and the empirical structure is modeled as rigid-perfectly plastic elements (also called
equations are referred to as the KSJH model. plastic sliders) with upper and lower limit forces (See Fig. 1). The soil
( ) structure interface is discretized and sliders are applied both vertically
d d
δv (d) = + 0.5 δvm 0.0 < ≤ 0.5 and horizontally at the nodes. The soil is modeled as a homogeneous
He He
( )
d d
δv (d) = − 0.6 + 1.3 δvm 0.5 < ≤ 2.0 (1)
He He
( )
d d
δv (d) = − 0.05 + 0.2 δvm 2.0 < ≤ 4.0
He He
( )
d d
δv (d) = + 0.2 δvm
1.6 0.0 < ≤ 0.5
He He
( )
d d
δv (d) = − 0.6 + 1.3 δvm 0.5 < ≤ 2.0 (2)
He He
( )
d d
δv (d) = − 0.05 + 0.2 δvm 2.0 < ≤ 4.0
He He

Fig. 1. Sketch of the elastoplastic soil-structure interaction model (after Franza

and DeJong (2019)).

2
J. Zhao et al. Tunnelling and Underground Space Technology incorporating Trenchless Technology Research 125 (2022) 104499

0 model of the soil continuum. Besides this elastoplastic solution, there are
D other SSI analysis methods involving full scale finite element interface
modeling and more rigorous soil constitutive models (e.g., Giardina
C
0.2 A et al., 2013, 2020; Amorosi et al., 2014; Fargnoli et al., 2015; Boldini
et al., 2018; Yiu et al., 2017). However, such complex models are
generally not practicable in the second stage assessment of large urban
0.4 infrastructure projects and therefore not considered in this paper.
v vm

(S + K* )u = P + K* ucat + K* Λ* (P − Su) + K* uip subject to : (4a)

/

0.6
fi,low ≤ (P − Su)i ≤ fi,up (4b)

Displacements observed in case

|(P − Su)j | ≤ μ(P − Su)i (4c)
0.8 studies (Kung et al. (2007))
KJHH
Hsieh and Ou (1998) 2.3. Structure damage evaluation methods
B Eq.(6a)
1
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 After the building deformation is estimated, a measure of potential
d/H e damage in the building is needed. When each building element are not
evaluated in detail and only the deformation mode, construction types
and some other simple features are taken into consideration, semi-
empirical methods can be adopted to evaluate the structural damage
0 (e.g., Burland et al., 1978; Boscardin and Cording, 1989; Son and
D Cording, 2005). Semi-empirical methods typically depend on crucial
simplifications of structure and boundary conditions, and can only
0.2 A provide a rough estimation of building damage. The most widely used
semi-empirical methods and some of their limitations are reviewed in
0.4 the Appendix 1.
C When prediction methods can provide sufficiently specific informa­
tion, such as building internal forces or strains, damage assessment
l lm

0.6
approaches which are more detailed than semi-empirical methods can
/

be adopted. Franza et al. (2020b) proposed a direct strain based

0.8 approach where no assumptions of deflection ratio or angular distortion
are needed. The building strains are directly calculated from measured
Displacements observed in case
studies (Schuster et al. (2009)) deformations of the building or internal forces calculated from the
1 B equivalent beam model using Eq. (5), where χ m is the beam curvature, γm
KSJH
Eq.(6b) is the beam engineering shear strain, εaxis,m is the beam axis strain, and
1.2 Nm , Vm , Mm are the internal axial force, shear force and bending moment
0 1 2 3 4 5 computed from numerical analysis; k and c are the shear correction
d/H e factor and the average shear stress correction factor, which depend on
the geometry of building cross-section, t is the distance between neutral
axis and extreme fiber and s is the vertical distance between the neutral
Fig. 2. The proposed surface ground movement profiles. axis and the fibre where εdt is calculated. The larger of εbt and εdt is taken
as εcrit and compared with the classification criterion proposed by
half-space continuum represented by coupled interactive springs. Gaps Boscardin and Cording (1989) (See Table 1). This direct strain based
and slippage following the Coulomb friction model between soil and approach uses an isotropic equivalent beam to model an entire building,
structure can be simulated with the plastic sliders by setting a zero upper therefore only a damage level of the entire building can be estimated. If
limit force and a horizontal limit force proportional to the vertical stress the damage condition of some building elements or the locations of
in the sliders. The building displacement (u) at each node can be solved damages are desired, more detailed models have to be adopted. The
with an equilibrium equation (Eq. (4)), where S is building stiffness, K* detailed models should contain both structural and non-structural ele­
is the stiffness matrix of soil, P is the external loading applied at the ments (e.g., infill walls). One such detailed damage assessment method
for frame structures is proposed in section 3.3.
foundation, uip is the plastic displacement of the sliders, Λ* is the soil
flexibility matrix without the main diagonal and ucat is the greenfield
ground displacement induced by excavation. The plastic property of
sliders are governed by Eq. 4b and Eq. 4c, where fi,low and fi,up are upper
and lower limits of the vertical force in sliders, μ is the friction coeffi­
cient at the interface of soil and structure and i and j are respectively the
vertical and horizontal degree of freedom. The soil stiffness matrices K* Table 1
Relationship between category of damage and critical tensile strain (εcrit ) (after
and Λ* can be derived from the Mindlin’s solution given by Vaziri et al.
Boscardin and Cording (1989), the values in bracket are suggested by Son and
(1982). The structure stiffness S can be determined with analytical so­
Cording (2005)).
lutions or finite element formulations. Because Eq. (4) is nonlinear
without closed-form solutions, it is solved with the iterative algorithm Category of damage Nominal degree of severity Critical tensile strain (εcrit )(%)

proposed by Klar et al. (2007). The analysis result of Eq. (4) has been 0 Negligible 0–0.05
compared with centrifuge tests and confirmed to be reliable by Franza 1 Very slight 0.05–0.075
2 Slight 0.075–0.15 (0.075–0.167)
and DeJong (2019) and Franza et al. (2020b). Elkayam and Klar (2019)
3 Moderate 0.15–0.3 (0.167–0.333)
also validated this elastoplastic formulation with a finite difference 4 to 5 Severe to very severe >0.3 (>0.333)

3
J. Zhao et al. Tunnelling and Underground Space Technology incorporating Trenchless Technology Research 125 (2022) 104499

Mm t M s cγ cVm N The existing elastic 2D frame model considers each frame member as
εb = χ m t = ε’b = χ m s = m εd = m = εh = εaxis,m = m
EI EI 2 sκAG EA an isotropic elastic beam element and formulates the frame stiffness
( ) √̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
( )2 ̅ (5) matrix with the displacement method. However, the previous elastic 2D
1− ν 2 1+ν
εbt = εb + εh εdt = (ε’b + εh ) + (ε’b + εh ) + εd 2
frame model implementation is considered to be too simple because it
2 2
only supports the modeling of structures with identical footings, one
column on each footing, equal floor elevations, equal beam span widths
3. Elastoplastic two-stage solution for SSI in deep excavation and the same beam and columns dimensions. Moreover, infill walls,
which can significantly affect the structure stiffness, were not previously
As discussed earlier, Franza and DeJong (2019) proposed a two-stage considered. The structural analysis model for frame structures in ASRE is
elastoplastic solution for SSI analysis of building response to tunneling updated in this paper to include irregular frames and infill walls.
induced ground movements. In this paper, the elastoplastic solution Fig. 3a schematizes the frame structure model developed in this
procedure is adapted to deep excavation scenarios and then incorpo­ paper. The beams and columns are discretized at each junction and each
rated into the computer program Analysis of Structural Response to foundation element is discretized with a small element size. A fine grid is
Excavation (ASRE), originally developed by Franza and DeJong (2019). adopted for each foundation because the foundations are connected to
soil-structure interface elements and the small element size more accu­
3.1. Greenfield displacement rately captures the ground movements. Because all frame members
deform elastically in this model and building self-weight loads are
The first stage of the two-stage elastoplastic solution is to determine applied at beam and columns junctions, a coarse discretization of the
vertical and horizontal greenfield ground displacements (ucat in Eq. (4)). frame is sufficient. In the updated frame structure model, footings with
The KJHH model (Kung et al., 2007) and the KSJH model (Schuster varying dimensions and footings connected to multiple columns can be
et al., 2009) are adopted in this paper because they consist of a complete modeled. The floor elevations and bay spans can be distinct at each
estimation procedure that links horizontal wall deflection to vertical and frame panel, and each beam and column can have different dimensions
lateral ground movement profiles. Moreover, the model uncertainty of and material properties. The stiffness of infill walls are modeled as
the KJHH and KSJH are well documented, so the influence of their un­ equivalent diagonal compression struts. Eq. 7 is used to calculate the
certainty on building damage is ready to be analyzed. However, the stiffness of the struts, as recommended by FEMA (1998), where t is the
ground displacement profiles in both models are described discretely thickness of the infill panel, h and l are respectively the height and length
with 4 pivot points (A-D as shown in Fig. 2), and cannot be applied to the of the infill panel, Ec and Em denote the elastic moduli of column and
elastoplastic two-stage analysis directly, in which continuous ground infill respectively, θ is the inclination angle of the strut, Ic is the moment
displacement profiles are required. Therefore, a pair of shifted log- of inertia of the adjacent columns and Hw is the height of the infill wall,
normal curves are fitted to the KJHH and KSJH ground displacement as shown in Fig. 3b. Diagonal compression struts are only placed when
profiles in Fig. 2. The log-normal curves pass through pivot points A, B, the diagonal strain is compressive. In other words, the tensile strength of
and C in the KJHH and KSJH models exactly and smoothen the sharp the diagonal struts is assumed to be zero. The application of the pro­
corners. The fitted curves also coincide well with the displacements posed frame structure analysis model to a case study is demonstrated in
observed in case histories reported by (Kung et al., 2007) and (Schuster a later section.
et al., 2009). The coefficient of determination (R2 ) for the proposed Ae = We t (7a)
vertical and lateral displacement profiles are 0.95 and 0.93 respectively,
√̅̅̅̅̅̅̅̅̅̅̅̅̅̅
while the R2 value for the original discrete KJHH and KSJH profiles are We = 0.175(λh)− 0.4
h2 + l2 (7b)
0.92 and 0.88. Eq. (6) describes the formulation of the greenfield ground
√̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
displacement profile proposed in this paper, where d is the distance from
Em tsin(2θ)
excavation, He is the depth of excavation, δvm and δlm are respectively the λ=4 (7c)
4Ec Ic Hw
maximum vertical and lateral ground displacement.

δv (d/He ) 1.14 1 (ln(d/He + 0.39) − 0.095)2

= d √̅̅̅̅̅ exp(− ) (6a) 3.3. Damage assessment
δvm He
+ 0.39 0.46 2π 0.423
For Timoshenko beam models, the direct strain based approach by
δl (d/He ) 2.14 1 (ln(d/He + 0.82) − 0.80)2 Franza et al. (2020b) is adopted in this paper because it overcomes
= d √̅̅̅̅̅ exp(− ) (6b)
δlm He
+ 0.82 0.44 2π 0.387 possible errors due to the simplification in the calculation of ΔL and β in
the methods by Burland et al. (1978) and Son and Cording (2005) (See
3.2. Structural analysis models discussions in Appendix 1). In the direct strain based approach, the
maximum internal forces Nm , Mm and Vm among all cross-sections of the
Two types of structural analysis models are studied in this paper: an beam are first computed with ASRE and εb(max) , εd(max) , and εh are
equivalent Timoshenko beam model and a 2-dimensional (2D) elastic calculated with Eq. (5) For a rectangular cross-section, κ and c are taken
frame model. Both models were implemented in ASRE by Franza et al. as κ = 10(1+
12+11ν and c = 3/2, where ν is the Poisson’s ratio of the beam. εcrit
ν)

(2020a) and Franza and DeJong (2019). In the Timoshenko beam model, is taken as the larger value of εb and εdt , and compared with the damage
the target building is modeled as an equivalent isotropic Timonshenko classification criterion by Boscardin and Cording (1989) (Table 1) to
beam defined by its dimensions, elastic modulus (Eb ) and elastic over obtain a damage category of the building.
shear modulus ratio (Eb /Gb ). The equivalent beam is discretized, and a For the 2D elastic frame model proposed in this paper, damages to
stiffness matrix (S) is formulated and applied to Eq. (4). Burland et al. infill walls and the structural frame are evaluated separately. To eval­
(1978) suggested that the value of Eb /Gb should be taken as 2.6 for uate the damage of infill walls, Son and Cording (2005)’s method is
isotropic walls and 12.5 for frame structures. The solution of ASRE adopted. The vertical displacements (Av , Bv , Cv , Dv ) and lateral dis­
consists of the displacement at each discretization node and the internal placements (Al , Bl , Cl , Dl ) at the corners (A, B, C, D) of each infill panel
forces in each element. The Timoshenko beam model was evaluated are determined (See Fig. 4 for the geometry of a building unit). The
with respect to field and centrifuge test results and provided reliable slope, rigid body rotation (tilt), angular distortion (β) and lateral strain
predictions for bearing wall structures on continuous foundations at top and bottom are calculated with Eq. (8). Afterwards, the critical
(Franza et al., 2020a). strain of each infill wall can be estimated with Eq. (9) and (10), and the

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J. Zhao et al. Tunnelling and Underground Space Technology incorporating Trenchless Technology Research 125 (2022) 104499

Fig. 3. The proposed 2D elastic frame model.

Table 2
Relationship between damage category and angular distortion (After Ghobarah
(2004)).
State of damage Ductile frame Nonductile frame

No damage 0–0.2 0–0.1

Repairable damage
a) Light (aesthetic) damage 0.2 – 0.4 0.1 – 0.2
b) Moderate (serviceability) damage 0.4–1.0 0.2–0.5
Irreparable damage (structural damage) 1.0 − 1.8 0.5 − 0.8
Severe damage (collapse) 1.8 − 3.0 0.8 − 1.0

Aν − Bν
Slope =
Lw
(Cl − Bl ) + (Dl − Al )
Tile =
Fig. 4. Slope, tilt and angular distortion (β) of a building unit. 2Hw
β = Slope − Tile (8)
damage category of each infill wall is classified according to Son and D − Cl
Cording (2005)’s criterion in Table 1. εh (top) = l
Lw
To evaluate the damage of the structural frame, the method of
Al − Bl
Ghobarah (2004) is adopted. Ghobarah (2004)’s method is originally εh (bottom) =
Lw
used to evaluate building damage after an earthquake and inter-story
drift ratio is considered as the engineering demand parameter to clas­ β
sify building damage. Ghobarah (2004) defined inter-story drift ratio by tan(2θmax ) = (9)
εh
the difference of horizontal displacement of top and base floor divided
by the floor elevation. In this definition, each floor is assumed to remain εcrit = εh cos2 θmax + βsinθmax cosθmax (10)
horizontal. However, in the case of excavation induced building defor­
mation, each frame panel experiences both vertical and horizontal drifts 4. Case studies
(see Fig. 4). Therefore, the inter-story drift ratio is equivalent to the
horizontal displacement difference after rotating the frame panel by the 4.1. Singapore Art Museum
slope angle (i.e., drift ratio = tanβ, where β is the angular distortion
defined by Son and Cording (2005)). When β is small, it is assumed The first case study explored in this paper was originally published
tanβ ≈ β. Therefore, assuming the drift-ratio is equivalent to angular by Goh and Mair (2011). The east and west wings of the Singapore Art
distortion, the criterion of Ghobarah (2004) can be used to classify po­ Museum (SAM), which were impacted by the construction of the Bras
tential damage of each frame panel using Table. 2. Basah subway station, are analyzed. The excavation was 35 m deep and
This proposed damage assessment method which considers both the is approximately 6 m away from the wings of the SAM. The excavation
structural frame and infill walls can provide an overall estimate of the support system consists of a diaphragm wall and 5 layers of bracing. The
building, but it also identifies locations of potential damage in the form soil beneath the SAM consists of four layers of clay with intermittent
of the εcrit and β values that are calculated for each panel. The applica­ fluvial sand layers. The representative soil stiffness was reported to be
tion of this method is demonstrated by a case study in a later section. 47 MPa. The structural behaviour of the SAM is dominated by four
masonry walls with an average thickness of about 500 mm. The

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J. Zhao et al. Tunnelling and Underground Space Technology incorporating Trenchless Technology Research 125 (2022) 104499

settlement of four walls (SAM-1, SAM-2, SAM-4, SAM-5, see Fig. 5) and
the non-suspended, tiled pavement (BBS-1, BBS-2, BBS-4, BBS-5) just
outside the walls are monitored by precise levelling. The monitored
settlement at BBS-1, 2, 4, 5 are considered as an approximation of cor­
responding greenfield ground settlement at SAM-1, 2, 4, 5. The height of
the walls is around 9.7 m and the Young’s modulus of the walls is re­
ported to be 5GPa. The foundation is shallow and consists of soft timber
layers. Consequently, the contribution of the foundation to the overall
building stiffness is ignored. The stiffness of the structure is mostly due
to the masonry walls as the floor slabs are thin and much more flexible in
comparison, as suggested by Goh and Mair (2011).
Since the structure section perpendicular to the deep excavation
consists of continuous walls, the Timoshenko beam model is used to
analyze the walls. SAM-1 and SAM-5 are modeled as beams with lengths
of 28 m, and SAM-2 and SAM-4 are modeled as beams with lengths of 15
m. All four beams are 9.5 m high and 0.5 m thick. Since the building
material is identical for these building sections, a constant elastic
modulus of 5GPa, as suggested by Goh and Mair (2011), is adopted. A
value of 6 is taken for GEbb to account for the openings in the walls. The
maximum horizontal deflection (δhm ) of the diaphragm wall, vertical
deformation ratio (Rv ), lateral deformation ratio (Rl ) are first calculated
according to the KJHH and KSJH models. Vertical and lateral ground
movement profiles are then estimated with Eq. (6). In other words, these
were prediction values and prediction ground settlement curves,
assuming no knowledge of the actual settlement. Fig. 6a shows the
analysis and monitoring results for SAM-1 and SAM-5. Fig. 6b shows the
analysis and monitoring results for SAM-2 and SAM-4. The support
system and underground conditions are assumed to be equal for the four
walls which results in identical ground movement profiles for the four
analyzed sections (Fig. 6). Note that since walls SAM-1 and SAM-5 are
identical, the analysis results are also identical. The analysis results for
SAM-2 and SAM-4 are also identical for the same reason.
It is observed that the measured ground displacement of BBS-1 and
BBS-5 are significantly different, despite that these two scenarios are
identical from a prediction perspective. This indicates that a single
deterministic ground settlement profile prediction, using the KJHH
model or otherwise, will not be able to predict both of the scenarios. The
Fig. 6. Singapore Art Museum case study. Predicted greenfield settlement
same observation holds for BBS-2 and BBS-4. Goh and Mair (2011)
profiles using the KJHH & KSJH models, predicted building settlement profiles
explained the difference in the monitoring results by a different order of using ASRE, and monitored settlement profiles.
the construction activities at the east wing and west wing of SAM. This

Fig. 5. Plan view showing the locations of building settlement (in squares) and ground settlement (in triangles) monitoring points (after Goh and Mair (2011)).

6
J. Zhao et al. Tunnelling and Underground Space Technology incorporating Trenchless Technology Research 125 (2022) 104499

discrepancy between monitored ground displacements implies that elastoplastic solution can predict building response reasonably well,
notable uncertainty exists in the prediction of ground movements. even with a nominal value of structural stiffness. However, the analysis
Moreover, Goh and Mair (2011) reported that SAM-5 behaved in a more results of SAM-5 indicate that the uncertainty in the modeling of the
flexible manner compared to SAM-1 even though their structures are structure (i.e., the reduction in building stiffness due to potential
similar. This might be explained by some existing structural damage in existing damage) should also be considered in the analysis of excavation
SAM-5 and it implies that modeling of existing buildings, especially induced structural damage. The uncertainty in the whole analysis
historical buildings, could be associated with large uncertainty. In framework is studied comprehensively in section 5.
summary, the uncertainty observed in this case study exists in the esti­ ⎛ ( (x ) )2 ⎞
mation of δvm , width of the settlement profile and building stiffness. ln + 0.39 − 0.095
1.14 1 η
Because horizontal ground displacement was not monitored, the accu­ δv (x) = δvm x √̅̅̅̅̅ exp⎜
⎝−
⎟
⎠
+ 0.39 0.46 2π 0.423
racy or uncertainty associated with the KSJH model can not be η
evaluated. ⎛ ( (x ) )2 ⎞ (11)
A back-analysis was then undertaken to study the uncertainty in the ln + 0.82 − 0.80
2.14 1 ⎜ η ⎟
above modeling method. In the back-analysis δvm is taken as the inter­ δl (x) = δlm x √̅̅̅̅̅ exp⎝ − ⎠
0.44 2π 0.387
polated maximum value of the measured settlement profile. The width η
+ 0.82
of the settlement profile is adjusted by introducing a scaling term (η) to
Eq. (6), as shown in Eq. (11). The value of η for each wall is determined
∑ 4.2. Chicago Frances Xavier Warde School
by minimizing the mean squared error 1 n (δv (x) − ̂
n i δ v (x))2 , where n is
the number of monitoring points, δv (x) are the monitored displacements The second case study explored in this paper was originally pub­
and ̂δ v (x) are the values predicted by Eq. (11). The elastic stiffness of lished by Finno and Bryson (2002) and Finno et al. (2005). The analyzed
SAM-5 was reduced to 1 GPa by a trial and error method to recover the structure is a cross-section of Chicago Frances Xavier Warde School
monitored building settlements. The back-analysis results are shown in (ChiFXWS), which was impacted by the construction of the subway
Fig. 7. The analysis results for SAM-1, SAM-2 and SAM-4 imply that if renovation project on State Street and Chicago Avenue. The cross-
the ground settlement profile is estimated accurately, the two-stage section, as shown in Fig. 8, is a three-story concrete frame structure

Fig. 7. Settlement of SAM after applying back-analyzed δvm and..δhm

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J. Zhao et al. Tunnelling and Underground Space Technology incorporating Trenchless Technology Research 125 (2022) 104499

Fig. 8. Elevation view of the analyzed cross-section of Chicago Frances Xavier Warde School (ChiFXWS) (after Finno et al. (2005)).

with brick partition walls and a basement. The floor system at each level
consists of a reinforced concrete pan-joist and is supported by interior
concrete columns and beams, and masonry bearing walls around the
perimeter. The interior columns and perimeter walls rest on three
separated footings. The excavation is 1.2 m to the west of the frame and
is almost perpendicular to the frame. The excavation depth is 12.2 m and
the excavated soil is soft to medium clay. The excavation support system
consists of a secant pile wall with three levels of supports. The building
settlement is monitored at 5 points along the cross-section at basement
level or 1 m above grade with optical survey points. The cross-section is
modelled with the 2D elastic frame proposed in this paper. The partition
walls are modeled as diagonal compression struts with Eq. 7, where the
elastic modulus of masonry is taken as 12.5GPa and the elastic modulus
of concrete elements are taken as 36GPa. The concrete foundation wall
at the east part of the frame is also modeled using diagonal compression
struts but the value of Em is taken as 36 GPa. Because the exact values of
the material properties are not reported by Finno and Bryson (2002) and
Finno et al. (2005), typical values are adopted. The analysis of this
structure here is not aimed to recover the response of the structure
exactly, but to simulate the typical situation in design practice, in which
the material properties are unknown, and to demonstrate the uncer­
tainty associated with current design procedures.
The settlement of this 2D elastic frame is calculated with the two-
stage elastoplastic methods and ASRE. The greenfield input to the
two-stage analysis is first estimated with the KJHH and KSJH models.
The estimated value of δvm and δlm are 25 mm and 20.9 mm respectively,
and a ground movement profile is determined with Eq. (6). The moni­
tored building settlement and computed building settlement are plotted
in Fig. 9a. It is observed that the settlement determined from the two-
stage analysis is close to the monitored values for footing 2 and
footing 3, while the analyzed settlement of footing 1 is approximately
one half of the monitored value. Because there is not any concentrated
load applied at footing 1, it is not reasonable to observe a building
settlement 4 times larger than the greenfield settlement. Therefore, it
can be argued that the predicted greenfield settlement profile is not Fig. 9. Chicago Frances Xavier ward School case study. Predicted greenfield
settlement profiles using the KJHH & KSJH models, predicted building settle­
accurate and this uncertainty is an important reason for the discrepancy
ment profiles using ASRE, and monitored settlement profiles.
between predicted and monitored building settlement. A back-analysis
is then conducted by adjusting the greenfield settlement with Eq. (11).
The back-analysis result is shown in Fig. 9b. The purpose of the back- frame is analyzed with the method proposed for the 2D elastic frame
analysis is to show that when the uncertainty of δvm and trough width model. In the prediction analysis, i.e., the direct application of the KJHH
is taken into account, the monitored building settlement can be recov­ and KSJH models to obtain the predicted greenfield settlement, the
ered in one realization of the probabilistic analysis framework, as will be frame panels between span C-D (see Fig. 8) experienced slight to mod­
discussed later. erate levels of damage at the first floor and the frame panels between
After the building displacement is computed, the damage level of the span B-C experienced negligible to slight damage at each floor. The

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J. Zhao et al. Tunnelling and Underground Space Technology incorporating Trenchless Technology Research 125 (2022) 104499

angular distortion of the panels between span C-D is around 0.26% and Table 3
the angular distortion between span B-C increased from 0.10% on the Quantification of input uncertainty in the analysis of SAM and ChiXFWS.
first floor to 0.11% on the third floor. The maximum partition wall strain Random Variable Statistical model
is 0.13%, which occurs at the partition walls between span C-D at the
BF of Maximum wall deflection (BFhm ) Normal(1, 0.252 )
second and the third floor. The strain levels in other partition walls are BF of Vertical deformation ratio (BFv ) Normal(1, 0.132 )
small and negligible. The distribution of damage coincides with the BF of Lateral deformation ratio (BFl ) Normal(1, 0.112 )
cracks observed after the construction works. Finno and Bryson (2002) Ground displacement profile width parameter (η) Normal(1, 0.162 )
reported that damage mainly occurred at the west part of the building Elastic modulus of equivalent beam model (Eb (GPa)) Lognormal(1.57,
and cracks were observed at the infill walls at the second and third floor. 0.086)
Elastic over shear modulus ratio of equivalent beam model Lognormal(1.54,
The back-analysis results suggest a similar distribution of damage with
(Eb /Gb ) 0.15)
slightly larger strain in each frame panel and infill wall. Concrete compressive strength (f’c (MPa)) Uniform(20, 40)
Masonry elastic modulus (Em (GPa)) Uniform(6, 21)
2D frame beam width (bb (m)) Normal(5.76, 0.582 )
5. Uncertainty analysis

In the case studies, it is observed that the model uncertainty of KJHH (Baecher and Christian, 2005).
& KSJH influences building responses significantly. Additionally, the Kung et al. (2007) and Schuster et al. (2009) also indicated that
analysis of surface structures also experiences uncertainty caused by uncertainty exists in the estimation of the shape of ground displacement
incompleteness of knowledge (e.g., unavailable material properties) and profiles. By observing the case histories used to derive Eq. (2) and (3), it
model uncertainty (e.g., simplification of a structure as an equivalent is concluded that the pivot point A does not vary among the case his­
beam or a 2D elastic frame). These uncertainties associated with tories while the distances of point B, C and D to the excavation wall show
greenfield displacement estimation and surface structure analysis are large fluctuations. To quantify the uncertainty of the ground displace­
called input uncertainty of the second stage assessment. The input un­ ment profile shape, a scale factor η is added to Eq. (6) to shrink or
certainty can be reduced with further analysis such as detailed modeling elongate the ground displacement profile width (see. Eq. (11)). In the
of the excavation system or a comprehensive survey of the building. case histories reported by Kung et al. (2007) and Schuster et al. (2009),
However, it is typically not feasible to reduce all of the input uncertainty the distance of pivot point B to the excavation wall varies in the range
mentioned above during a simplified second stage assessment. There­ 0.3He to 0.7He and 0.6He to 1.4He for vertical displacement and lateral
fore, it is important to study which input causes the most uncertainty in displacement respectively. These variance ranges correspond to a η with
the output of the second stage assessment, so that more reliable damage a range from 0.6 to 1.4. Because the locations of the pivot points are
prediction can be achieved with the least amount of effort on reducing mean values from a regression analysis and η is their relative error, it is
the input uncertainty. This section uses a variance based sensitivity reasonable to assume η follows a normal distribution centered at 1. The
analysis (Sobol’s method) to study what is the major source of uncer­ standard deviation of η is estimated as 0.16 to ensure a 99% likelihood
tainty in the second stage assessment and how to reduce the uncertainty that η is in the interval (0.6, 1.4).
of the predicted building damage efficiently. For the equivalent Timoshenko beam model used to analyze the
SAM, uncertainty exists in the estimation of the equivalent elastic
modulus (Eb ) and elastic over shear modulus ratio (Eb /Gb ). As suggested
5.1. Uncertainty sources
by Dimmock and Mair (2008 Eb can be taken according to the building
material and Eb /Gb can be taken as 2.6 for bearing wall structures with
In order to study the influence of uncertainty sources on building
no openings. However, these values are roughly estimated with igno­
damage predictions, the input uncertainty needs to be quantified. A
rance of the natural material variability, existing damage and structure
common method to quantify an uncertain parameter is to model the
details such as openings and different building lay-outs. Son and Cording
parameter as a random variable with a certain probabilistic density
(2007) concluded that the value of Eb /Gb has a larger variance range and
distribution. In this section, uncertainty sources of the two case studies
is harder to estimate compared to Eb . In this study, the coefficient of
are identified and quantified.
variance (c.o.v.) of Eb for SAM is selected to be 30% and the c.o.v of Eb /
As shown by the previous analyses of the SAM and ChiFXWS case
Gb for SAM is selected to be 45%. These c.o.v. are selected by trial and
studies, the model uncertainty of the KJHH & KSJH models may have
error so that the 99% coverage intervals of Eb and Eb /Gb are reasonable
significantly influenced the predicted building damage. In the KJHH &
according to the information provided by Goh and Mair (2011). The
KSJH models, ground settlement profiles are described based on four
mean value for Eb and Eb /Gb are taken as 5 GPa and 6 respectively,
parameters: the maximum wall deflection (δhm ), vertical deformation
which are the same as the values adopted in the deterministic study
ratio (Rv ), lateral deformation ratio (Rl ) and trough width parameter (η).
previously. The type of probability distribution for Eb and Eb /Gb are
The uncertainty of δhm , Rv and Rl are quantified by Kung et al. (2007)
modeled as log-normal distribution, as commonly adopted for positive
and Schuster et al. (2009) as Eq. (12), where δhm , Rv and Rl are the mean
definite random variables (Ayyub and Klir, 2006). The 99% probability
values of δhm , Rv and Rl , and can be estimated by adopting the regression
coverage intervals of Eb and Eb /Gb are (3.84, 5.98) and (3.40, 8.80).
equations in the KJHH & KSJH models. The uncertainty of δhm , Rv and Rl
For the elastic 2D frame model used to analyze ChiFXWS, the un­
are described with corresponding bias factors (BFhm , BFv and BFl ), which
certainty mainly comes from the estimation of the stiffness of columns,
are statistically independent random variables. Consequently, δhm , Rv
beams and infill walls. This is because the reinforcement layout of col­
and Rl are also random variables because they are a product of random
umns and beams are not available and the material properties are
variables and constants. It is important to notice that δhm , Rv and Rl are
roughly estimated. The compression strength of concrete (f’c ) is esti­
highly correlated, but they are conditional independent to each other
mated as 30 MPa and the corresponding elastic modulus is Ec =
when they are conditioned on the input parameters of the regression √̅̅̅̅̅̅
equations in the KJHH & KSJH models. In other words, when the un­ 4700 f’c = 26GPa, as suggested by ACI 318 (2008). The elastic
derground conditions and excavation system are defined, the model modulus for beams and columns are amplified to 36 GPa based on an
errors in the estimation of δhm , Rv and Rl are statistically independent to assumption of 5% reinforcement ratio. In reality, the compressive
each other. The mean and standard deviations for BFhm , BFv and BFl are strength of normal strength concrete is in the range of 20 MPa to 40
reported by Kung et al. (2007) and Schuster et al. (2009), as shown in MPa, and no information is available to narrow this range for the case of
Table. 3. BFhm , BFv and BFl are modelled as normally distributed, which ChiFXWS. Therefore, f’c is modeled as a random variable uniformly
is a common practice in statistical studies of geotechnical engineering distributed between 20 MPa and 40 MPa. Consequently, the elastic

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J. Zhao et al. Tunnelling and Underground Space Technology incorporating Trenchless Technology Research 125 (2022) 104499

modulus of concrete is a random variable depending on f’c . According to damage assessment.

the Brick Industry Association (1992), the elastic modulus of brick The left column of Fig. 10 shows the results of the first stage sensi­
masonry assemblage (Em ) is roughly proportional to its compression tivity analysis. It is observed that among δhm , Rv , Rl and η, δhm caused
strength, and has a variance range of 6 GPa to 21 GPa. Similar to f’c , most uncertainty in the estimation of εcrit , εcrit(infill) and β. For the case of
there is no information with respect to the compressive strength of SAM-1 and SAM-5, almost all the uncertainty comes from the uncer­
masonry used in ChiFXWS. Therefore, the masonry compressive tainty associated with δhm , while for SAM-2 and SAM-4, the trough width
strength is also assumed to follow a uniform distribution. Because Em is parameter η also contributes a considerable amount of uncertainty. This
linearly dependent on the masonry compressive strength, it can be may imply that an accurate estimation of η may reduce the uncertainty
assumed that Em is also uniformly distributed (i.e., Em ∼ Uniform(6,21)). in damage prediction of structures with short spans, while the uncer­
Such uninformative uniform distribution with bounds based on data or tainty of long span structures has a weaker correlation with η. In the
expert knowledge is usually selected when there is inadequate knowl­ analysis of ChiFXWS, Sobol’s indices computed based on βcrit and
edge in the most likely values of the variables (e.g., Baecher and εcrit(infill) are almost the same. This suggests that the value of βcrit and
Christian (2005) and Edeling et al. (2021)). εcrit(infill) are highly related to each other. About 80% of the uncertainty in
Another input uncertainty considered in the analysis of ChiFXWS is the estimation of βcrit and εcrit(infill) comes from δhm and 20% of the un­
the width of floor slab flanges. In the previous deterministic analysis, the certainty comes from Rv . The uncertainty of Rl and η have nearly zero
flange width of floor slabs included in the analysis is taken as 8 times the effect on the prediction of damage in ChiFXWS.
slab thickness as recommended by ACI 318 (2008). Therefore, an The right column of Fig. 10 shows the results of the second stage
equivalent rectangular beam with 5.76 m width and 0.06 m depth is sensitivity analysis. It is observed that the indices of ground settlement
adopted. The equivalent rectangular beam has the same axial stiffness input decreased because the uncertainty from the structure models is
and bending stiffness as the sum of the beam and floor flanges. However, also included. In the analysis of SAM-1 and SAM-5, the total amount of
the stiffness of transverse beams is ignored in the deterministic analysis. uncertainty caused by Eb and Eb /Gb are almost identical to the uncer­
To account for the effect of the transverse elements on the stiffness of the tainty caused by the ground settlement model. In the analysis of SAM-2
2D frame model, the beam width (bb ) is modeled as a normal random and SAM-4, Eb /Gb showed a stronger contribution compared to the
variable with a mean of 5.76 m and a c.o.v. of 10%. The 99% probability analysis of SAM-1 and SAM-5. This implies that shorter span structures
coverage of bb is (4.28, 7.24). All the uncertainties studied in this paper are more sensitive to the uncertainty in the equivalent Timoshenko
are summarized in Table. 3. beam model, and both Eb and Eb /Gb are important sources of uncertainty
δhm = BFhm δhm in this circumstance. Future studies and field investigations to more
Rv = BFv Rv (12) accurately qualified Eb and Eb /Gb could be valuable to reduce the un­
Rl = BFl Rl certainty in building damage assessments. In the analysis of ChiFXWS, it
is observed that the uncertainty of fc and bb caused almost zero uncer­
tainty in the prediction of both βcrit and εcrit(infill) . In contrast, the un­
5.2. Sensitivity analysis
certainty of infill wall stiffness (represented by Em ) show an effect on βcrit
and εcrit(infill) , which implies that proper modeling of infill walls may be
After the input uncertainty is quantified, Sobol’s (variance based)
important to achieve accurate damage predictions. However, the Sobol’s
sensitivity analysis is deployed to study the effect of each uncertainty
indices which correspond to Em are about 1/7 of the indices corre­
input on building damage prediction. Sobol’s sensitivity analysis is one
sponding to δhm . Therefore, a better estimation of δhm is the most effi­
of the most robust global sensitivity analysis methods (Saltelli et al.
cient way to reduce the uncertainty in this system. Comparing the
(2008, 2010)) to decompose and attribute the variance of a system to
analysis of the Timoshenko beam model and the 2D elastic frame model,
each input variable. The contribution of each input variable is quantified
it is observed that the Timoshenko beam model introduces more un­
with its Sobol’s indices, and the first order indices (Si ) and the total
certainty in building damage predictions because there are more sim­
effect indices (ST,i ) are the most commonly adopted sensitivity measures.
plifications when the whole structure is modeled as an equivalent beam.
The definition of Si and ST,i can be found in Saltelli et al. (2008) and
The 2D elastic frame model is more complex and consists of more input
Appendix 2. The first order effect index measures the variance of system
parameters. Since many input parameters of the 2D elastic frame model
output (Y) when only the ith input random parameter (Xi ) varies but
can be evaluated accurately (e.g. floor elevation, beam span and column
averaged over the variation of all the other input parameters. The total
dimensions), they are considered with zero uncertainty and treated as
effect index measures Xi ’s contribution to the output variance after
constants in the analysis. Therefore, the uncertainty of the structural
interacting with all the other input parameters. Larger Sobol’s indices
analysis model in a system using a 2D elastic frame model is less critical,
imply that the corresponding input parameter is more responsible for
and a better estimation of the ground settlement will reduce the un­
the uncertainty of the system output, and reducing the variance asso­
certainty of building damage prediction more significantly.
ciated with such input will lead to more reduction in the variance of
system output. Si and ST,i are computed with the quasi-Monte Carlo
6. A probabilistic analysis approach
method, and the procedures can be found in Appendix 2.
In the analysis of the SAM, the parameter used to classify building
The analysis in previous sections demonstrated the uncertainty in the
damage (the engineering demand parameter) is the critical strain εcrit .
early stage building damage assessment. One way to reduce the uncer­
Therefore, εcrit is selected as the output Y in the sensitivity analysis and
tainty is to reduce the input uncertainty, for example, using finite
the Sobol’s indices of BFhm , BFl , BFv , η, Eb and Eb /Gb with respect to εcrit
element analysis to estimate ground displacement rather than KJHH &
are computed. In the analysis of ChiFSWS, the maximum frame panel
KSJH. Such advanced models require more effort in the analysis and are
distortion (βcrit ) and maximum infill wall strain εcrit(infill) are used as
usually unfeasible in the early stage assessment. Another method to deal
engineering demand parameters, and their Sobol’s indices are with the uncertainty is to carry the uncertainty in the analysis frame­
computed. The sensitivity analysis is done in two stages. In the first work and propagate the uncertainty from input parameters to engi­
stage, only uncertainty of ground movements (i.e., BFhm , BFl , BFv and η) neering demand parameters (e.g., εcrit ). In this paper, this method is
are considered. This stage aims to study which part of the KJHH & KSJH called probabilistic analysis approach and the result of this analysis
models induced the most uncertainty in damage prediction. In the sec­ approach is the empirical probability density distribution of the engi­
ond stage, all the parameters in Table 3 are considered. The purpose of neering demand parameters for the target building. The empirical
the second stage is to study whether the ground movement or the probability density distribution will provide a level of confidence when
structure models are more responsible for the uncertainty of the building

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Fig. 10. Sobol’s indices for SAM and ChiFXWS.

the damage in a structure is classified. A comparison of the deterministic made for the response surface. However, according to the law of large
and probabilistic analysis approach of excavation induced adjacent numbers, Monte-Carlo simulation converges to unbiased results when
building damage classification is shown in Fig. 11. the number of simulations is sufficiently large. The drawback of the
In the proposed probabilistic analysis approach, the uncertainty of Monte-Carlo method is that a large number of model evaluations (usu­
the estimated ground settlement and structural analysis models are first ally more than thousands of evaluations) is required to reach conver­
quantified, and this step is called uncertainty input quantification. The gence, which is unfeasible in analysis with large numerical models.
quantification process described previously in sensitivity analysis is an Surrogate model methods, which approximate the system response
example of uncertainty input quantification for SAM and ChiFXWS. In surface with different types of fast numerical models (e.g., neural net­
each project, the input uncertainty quantification will be unique, and works and support vector machine), is a common method used in Monte-
should be done by experts based on whatever local site information is Carlo simulations of large numerical models. However, surrogate
available. The quantified uncertainty input describes how much confi­ modelling is not used in the proposed framework because: 1) at the early
dence the practitioners have with their model or their assumptions and assessment stage of excavation works, full scale models are generally not
this step is crucial to obtain a meaningful output from the probabilistic available; 2) even if a full scale model is available, creating a reliable
analysis approach. surrogate model requires a sufficient amount of evaluations of the
After quantifying the uncertainty input, the uncertainty is propa­ model, which again causes large computational cost.
gated from input parameters to engineering demand parameters (EDP) Because common reliability analysis may introduce bias and surro­
with Monte-Carlo simulation and a soil-structure-interaction program, gate models are not available in the second assessment stage of potential
such as ASRE. Because the soil-structure-interaction process is highly building damage, a Monte-Carlo simulation with the direct soil-
nonlinear, the Monte-Carlo simulation method is one of the most robust structure-interaction model through ASRE is adopted in the proposed
methods to estimate the probabilistic characteristics of the EDPs. There analysis approach. To ensure an affordable computation time, the
are also other methods to estimate the probabilistic behavior of output computer program ASRE was optimized with a parallel computation
of nonlinear systems, for example, local and global reliability analysis. strategy and high-performance linear solvers. The computation time per
In local reliability analysis, the response surface of the nonlinear system 1000 simulations was reduced from around 3000 s to 150 s for the
is estimated with first-order (FORM) or second-order (SORM) functions, analysis of SAM-1 and around 8000 s to 500 s for the analysis of
while in global reliability analysis, the response surface is usually ChiFXWS. The optimized ASRE was then integrated into a computer
approximated as a Gaussian process model. Both local and global reli­ program called Uncertainty Quantification in Excavation Structure
ability introduces bias in the analysis because of the approximation Interaction (UQESI), in which uncertainty propagation and sensitivity

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J. Zhao et al. Tunnelling and Underground Space Technology incorporating Trenchless Technology Research 125 (2022) 104499

Fig. 10. (continued).

Fig. 11. Deterministic and probabilistic analysis approaches.

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J. Zhao et al. Tunnelling and Underground Space Technology incorporating Trenchless Technology Research 125 (2022) 104499

Fig. 12. Probabilistic analysis results of SAM.

Fig. 13. Probabilistic analysis results of ChiFXWS.

analysis are done. UQESI can also be used for uncertainty quantification
and sensitivity analysis when other SSI models or surrogate models are
(Zhao et al., 2021) on tunneling-induced settlement damage demon­
adopted in the proposed probabilistic approach.
strated that in other scenarios, the range of predicted damage can be
Figs. 12 and 13 shows the analysis results of SAM and ChiFXWS with
much larger, and can span several damage category thresholds. This
the proposed probabilistic analysis approach, using the uncertainty in­
contrast demonstrates the benefit of quantifying the uncertainty. Spe­
puts previously used in the sensitivity analysis (Table. 3). For SAM-1 and
cifically, deterministic results: 1) may lead to improper classification of
SAM-5, there is about 30% probability that the deterministic analysis
potential damage when the deterministic results are close to the damage
results underestimate the potential damage. The probability of under­
thresholds, and 2) provide relatively little information on the level of
estimating damage for SAM-2 and SAM-4 is about 50%. In the analysis of
confidence in the predicted result. Finally, note that the input uncer­
ChiFXWS, the probability of underestimating potential damage quanti­
tainty in the analysis of SAM and ChiFXWS is small because both SAM
fied by εcrit(infill) and β are 40% and 53% respectively. As expected, the
and ChiFXWS are valuable buildings and their underground conditions
deterministic analysis results do not provide an upper bound of potential
and building lay-outs were well surveyed. In practice, even larger un­
damage. For SAM-1/5, SAM-2/4 and ChiFXWS, the damage categories
certainty may exist, which will lead to a wider empirical CDF.
predicted based on the probabilistic and deterministic analysis results
are the same (i.e., there is approximately 100% probability that the
7. Conclusions
structure will experience the damage category predicted with the
deterministic analysis). This provides great confidence in the predicted
In this paper, a two-stage elastoplastic procedure to simulate soil-
damage category despite the large uncertainty in ground settlement
structure-interaction is applied to assess building damage caused by
estimation and structural modeling of SAM and ChiFXWS. In our view,
braced deep excavations. In the first stage, the KJHH and KSJH models
this level of confidence is of great benefit, because it demonstrates that
by Kung et al. (2007) and Schuster et al. (2009) are employed to develop
no further field investigation or model refinement is needed to be
a new continuous equation to describe the greenfield ground
confident in the level of damage predicted. In contrast, a previous study

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J. Zhao et al. Tunnelling and Underground Space Technology incorporating Trenchless Technology Research 125 (2022) 104499

displacement induced by excavation works. In the second stage, the analyses, so that a confidence level of building damage assessment can
building deformation is determined through a soil-structure-interaction be determined together with a potential level of damage. If the confi­
analysis. This two-stage approach is integrated into the computer pro­ dence level of the assessment result of a target building is low, more
gram ASRE, which was originally created by Franza and DeJong (2019). detailed examinations should be conducted. The proposed probabilistic
This paper also contributes a more functional 2D elastic frame structural analysis approach and the SSI analysis program ASRE are integrated into
analysis model in order to analyze separate footings, infill walls, and the computer program UQESI.
frame members with different dimensions and materials.
The two-stage soil-structure-interaction procedure is applied to two CRediT authorship contribution statement
case studies and the uncertainty in the case studies are analyzed. For
both case studies, the uncertainty related to the ground displacement Jinyan Zhao: Methodology, Software, Validation, Formal analysis,
estimation is the major source of uncertainty in building damage Investigation, Writing – original draft, Visualization. Stefan Ritter:
assessment. When the building is modeled as an equivalent beam, the Conceptualization, Methodology, Resources, Writing – review & editing.
building analysis method also contributes a considerable amount of Matthew J. DeJong: Conceptualization, Methodology, Writing – re­
uncertainty to building damage assessment, while the model of a 2D view & editing, Supervision, Project administration, Funding
elastic frame has a much smaller contribution to the damage assessment acquisition.
uncertainty. This implies that future studies on better estimation of
ground displacement may be critical to provide a more reliable early Declaration of Competing Interest
stage damage assessment.
A probabilistic analysis approach is proposed to quantify the un­ The authors declare that they have no known competing financial
certainty of early-stage damage assessment results. This approach helps interests or personal relationships that could have appeared to influence
practitioners to quantify simplifications and approximations made in the work reported in this paper.

Appendixes

Appendix 1:. Semi-empirical structural damage estimation methods

Burland et al. (1978) proposed to evaluate building cracking potential by simplifying buildings as deep isotropic simply supported beams. Both
bending and shear deformation are considered and equations to calculate the maximum bending strain (εb(max) ) and maximum diagonal tensile strain
(εd(max) ) from deflection ratio (ΔL ) are provided (Eq. (13) and (14)). Eq. (13) and (14) are derived based on the deflection at the middle of a center loaded
simply supported Timoshenko beam, where E and G are elastic and shear modulus of the beam, I is the moment of inertia, L is the length of the sagging
or hogging beam segment and y is the distance from the extreme fibre to the neutral axis. In sagging deformation, it is assumed that the beam neutral
axis is at the mid-height of the beam (i.e., y = H2 ). In hogging deformation, Burland et al. (1978) assumed that foundations and soil provide significant
restraint to the buildings and the neutral axis should be considered at the bottom of the beam (i.e., the extreme fibre is at beam top and y = H). The
larger of εb(max) and εd(max) is called critical tensile strain (εcrit ) and Burland et al. (1978) suggested that the average critical tensile strain for the
initiation of crack in brickwork is around 0.05%.
Δ 12y 1
εb(max) = (13)
L L 1 + L182 I E
H G

Δ 1
εd(max) = (14)
L 1 + L182 H G
I E

Burland et al. (1978)’s method is widely adopted for analysis of bearing wall structures on continuous footings, though there are several de­
ficiencies with this method. First, assuming a bottom neutral axis in hogging deformation mode leads to a large shear stress at the beam bottom, which
can not be balanced with the friction between soil and structure (Dalgic et al. (2018)). This implies the importance of modeling the slippage between
the soil and the foundation. Second, although this method works well with buildings on continuous footings, it may not be reasonable to model a frame
structure on separate footings as a continuous simply supported beam. Finally, Burland et al. (1978)’s method does not consider horizontal strain,
which is argued by Boscardin and Cording (1989) to be a significant component of εcrit .
Boscardin and Cording (1989) therefore modified Burland et al. (1978)’s definition of εcrit as Eq. (15) and (16), where εh is defined as the change of
building length divided by the original building length. εb(max) and εd(max) can be determined with Eq. (13) and (14). To quantify the level of building
damage, Boscardin and Cording (1989) suggested to classify building damage into 5 categories according to the magnitude of εcrit . This criterion is
given in Table 1.
Although εh is taken into account, Boscardin and Cording (1989) still modeled the entire building as a deep beam and did not consider separate
footings. To evaluate the damage of a building that consists of individual units governing its structural response, Son and Cording (2005) updated
Boscardin and Cording (1989)’s method based on the state of strain at each building unit. A building unit, as defined by Son and Cording (2005), can
be characterized as a section between two columns, two different building geometries or stiffness characteristics and two different ground
displacement gradients. Son and Cording (2005) suggested using angular distortion β to compute εcrit instead of using deflection ratio ΔL in Burland
et al. (1978)’s method. In Son and Cording (2005)’s method, εcrit determined from distortion (β) and horizontal strain (εh ) with Eq. (9) and (10) is used
to classify building damage, where β is defined as settlement difference (slope) minus rigid rotation (tilt) of a building unit (see Fig. 4). Due to a
different definition of εcrit , the criterion of building damage categories are also updated and shown in Table 1.
εcrit = εb(max) + εh (15)
{ }
εcrit = max εh cos2 θ + 2εb(max) cosθsinθ (16)
θ

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J. Zhao et al. Tunnelling and Underground Space Technology incorporating Trenchless Technology Research 125 (2022) 104499

Appendix 2:. Sobol’s sensitivity analysis

The first order (Si ) and total effect (ST,i ) Sobol’s indices are defined with Eq. (17), where Y is the output of the system being analyzed and Xi is the ith
input of the system. E(Y|Xi )) means the expectation of Y conditioned on Xi , while E(Y|X∼i ) means the expectation of Y conditioned on all the other
input parameters except Xi .
VarXi (E(Y|Xi ))
Si =
Var(Y)
( ( )) (17)
VarX∼i E Y|X∼i
ST,i =1−
Var(Y)
Because the system being analyzed in this paper (with input listed in Table 3 and building damage level as output) is nonlinear, there is no
theoretical method to calculate Sobol’s indices. Therefore, the experiment design proposed by Saltelli et al. (2008) is adopted to estimate the Sobol’s
indices with Monte-Carlo integration. The procedures are:

• Generate two (N, R) sample matrices A and B, where N is the number of base sample size and R is the number of input parameters. Each row of A
and B corresponds to a group of independent samples of the input parameters. Quasi-random sampling should be adopted to accelerate the
convergence of the Monte-Carlo integration.
• Create matrices Ci , where i = 1, 2, ...R, with the ith column identical to A and all other columns identical to B.
• Compute the model output yA , yB and yCi by evaluating input matrices A, B and Ci . There are in total N(R +2) model evaluations.
• Calculate the first order and total effect indices with Eq. (18).

The number of base sample N is usually taken as several hundreds to thousands depending on the system being analyzed. To ensure convergence, N
is taken as 2048 in this paper, and insignificant changes were observed when N was increased further.
1 ∑N
yA (j)
yB (j)
− f02
N
Si = ∑
j=1
1 N
yA (j)
yA (j)
− f02
N j=1

1 ∑N
yB (j) yCi (j) − f02 . (18)
ST,i = 1 − N ∑
j=1
1 N
yA (j) yA (j) − f02
N j=1

( )2
∑N
2 (j)
f0 = j=1
yA

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