Updating of an instrumented building model considering amplitude dependence of dynamic resonant properties extracted from seismic response records

Abstract

This paper presents system identification and numerical analyses of a three story RC building. System identification was performed using 50 earthquake response records to obtain the frequencies and damping ratios taking into account soil-structure interaction (SSI). Trends in the resonant parameters were correlated with the peak response accelerations at the roof level. A general trend of decreasing resonant frequencies with increasing level of response was observed and quantified, whereas for the damping ratios no clear trends were discernible. A series of finite element models (FEMs) of the building were updated using a sensitivity based method with a Bayesian parameter estimation technique to follow the changes in the resonant frequencies with response amplitude. The FEMs were calibrated by tuning the stiffness of structural and non-structural components (NSCs) and soil. The updated FEMs were used in time history analyses to predict and assess the building seismic performance at the serviceability limit state. It was concluded that the resonant frequencies depend strongly on the response magnitude even for low to moderate levels of shaking. The structural and non-structural components and soil make contributions to the overall building stiffness that depend on the level of shaking. The FEM calibrated to the largest responses was the least conservative in predicting the serviceability limit state inter-story drifts but the building performed satisfactorily.

Full text

Updating of an instrumented building model considering amplitude
dependence of dynamic resonant properties extracted from seismic
response records
Piotr Omenzetter (corresponding author)
piotr.omenzetter@abdn.ac.uk
The LRF Centre for Safety and Reliability Engineering
The University of Aberdeen
Aberdeen AB24 3UE
UK
Faheem Butt
Department of Civil Engineering
The University of Engineering and Technology
Taxila
Pakistan
1

Abstract
This paper presents system identification and numerical analyses of a three story RC
building. System identification was performed using 50 earthquake response records
to obtain the frequencies and damping ratios taking into account soil-structure
interaction (SSI). Trends in the resonant parameters were correlated with the peak
response accelerations at the roof level. A general trend of decreasing resonant
frequencies with increasing level of response was observed and quantified, whereas
for the damping ratios no clear trends were discernible. A series of finite element
models (FEMs) of the building were updated using a sensitivity based method with a
Bayesian parameter estimation technique to follow the changes in the resonant
frequencies with response amplitude. The FEMs were calibrated by tuning the
stiffness of structural and non-structural components (NSCs) and soil. The updated
FEMs were used in time history analyses to predict and assess the building seismic
performance at the serviceability limit state. It was concluded that the resonant
frequencies depend strongly on the response magnitude even for low to moderate
levels of shaking. The structural and non-structural components and soil make
contributions to the overall building stiffness that depend on the level of shaking. The
FEM calibrated to the largest responses was the least conservative in predicting the
serviceability limit state inter-story drifts but the building performed satisfactorily.
Keywords: instrumented RC building, model updating, non-structural components,
seismic monitoring, seismic response, serviceability limit state, soil-structure
interaction, system identification, time history analysis
2

Introduction
The frequent occurrence of earthquakes in seismically active regions and,
consequently, the revelation of deficiencies in the constructed systems posing a
serious threat to human life and economy highlight the importance of investigating the
seismic behavior of full-scale structures. Most of the current testing methods, such as
shake table or pseudo-dynamic experiments, present a highly idealized mode of
investigation for a structural part, assembly or reduced scale model. However, such
idealized laboratory experiments cannot account for the complexity of in-situ
structures, including the influence of environmental and operational conditions, non-
structural components (NSCs), and soil-structure interaction (SSI). On the other hand,
the dynamic responses of a full-scale, in-situ structure, even measured at a few points,
can offer a great deal of ground-truth data and information about the structural
performance which would be difficult to acquire in other ways.
Monitoring the seismic response of buildings is an important task, since past
earthquakes around the world have caused major losses. Earlier studies (Butt and
Omenzetter 2012; Celebi 2006; Trifunac et al. 2001, Tobita 1996) have observed that
the dynamic characteristics of buildings vary with vibration amplitude. Consequently,
examining the behavior of buildings under different excitation scenarios is essential to
understand such trends in their behavior. The trends in the dynamic characteristics,
such as resonant frequencies and damping ratios, when carefully examined can
provide quantitative data for the variations in the behavior of buildings. Moreover,
such studies can provide useful information for the development and calibration of
realistic models for prediction of seismic response of structures in the limit states and
performance based design, and in model updating and structural health monitoring
studies (Brownjohn and Xia 2000; Sohn et al. 2003; Li et al. 2006).
Finite element models (FEMs) are routinely used to estimate the dynamic
properties of constructed systems (Clough and Wilson 1999). Some important recent
applications are in the areas of structural health assessment and model updating
(Jaishi and Ren 2006, Ntotsios et al. 2009; Weng et al. 2009; Wang et al. 2010;
Jafarkhani and Masri 2011; Foti et al. 2012; Zhou et al. 2012). Typically, the
development of a FEM for a structure is based on structural drawings, design
assumptions, engineering judgment and mathematical approximations that may not
fully represent all the important physical aspects of the actual structure. Some of the
3

important factors, e.g., the influence of SSI and NSCs such as cladding and partition
walls etc., are often ignored in FEMs. These factors, if modelled adequately in FEMs,
can affect the dynamic simulations significantly, as was found, e.g., in Chaudhary et
al. (2001), Bhattacharya and Dutta (2004), Shakib and Fuladgar (2004), Su et al.
(2005) and Pan et al. (2006). Therefore, to reduce the effect of approximations and
replicate better the true behavior of structures all the structural and influential NSCs
should be accounted for in FEMs.
To accurately predict the measured dynamic characteristics of a structure, it is
important to benchmark its FEM predictions against actual measured responses.
However, reproducing very accurately the measured dynamic characteristics using a
FEM is a significant challenge (Brownjohn et. al 2001a). It is therefore important to
update or calibrate the FEM with respect to the measured responses. The two general
approaches to update a model are direct (global) and iterative (local) methods
(Mottershead and Friswell 1993). Direct methods update the global mass and stiffness
matrices of the FEM to replicate the reference response data. However, they provide
updated models which may not preserve important attributes of structural matrices
such as symmetry, positive definiteness and sparseness, despite regenerating the
response data. Iterative methods, in contrast, apply corrections to the physical
parameters of the FEM. Their updated results are therefore physically interpretable,
which is a key advantage. Further discussions on model updating techniques can be
found in Friswell and Mottershead (1996). Representative approaches include the
optimal matrix updating, sensitivity based parameter estimation, eigenstructure
assignment algorithms and neural-networks updating methods (Zhang et al. 2000). A
brief introduction to the sensitivity based updating with a Bayesian parameter
estimation technique will be presented later in this paper as this method is used in the
study due to its ability to provide immediate physical interpretation of the updated
results.
When model updating is undertaken, FEMs are typically calibrated using
responses obtained for a particular event (e.g. an earthquake, wind storm, or forced or
ambient vibration test) or averaged responses (e.g. from long-term monitoring). Such
studies may thus ignore the naturally occurring variability of measured responses. The
monitoring results from a large number of seismic events used in this study have
revealed that the behavior of the investigated building dependents on the vibration
amplitude (Butt and Omenzetter 2012). It is therefore logical to develop
4

representative FEMs replicating this observed behavior. Doing so will help to
understand and quantify how the stiffness of structural components, NSCs and soil
changes with varying excitation and response levels.
This study comprises two parts. In the first part, seismic response trends of an
instrumented RC building are evaluated using 50 recorded earthquake time histories
collected over a period of more than two years. The resonant frequencies and damping
ratios, accounting also for SSI, were identified using a sub-space system identification
technique. Using the identified parameters, the relationships between the resonant
frequencies and damping ratios and the peak response acceleration (PRA) at the roof
level were examined. The contribution here is that the relationships are statistically
evaluated using a relatively large number of seismic events, which is still rare in the
existing literature.
The second part of this investigation presents the development of a series of
parameterized linear 3D FEMs which replicate the experimentally observed variations
in resonant frequencies. The models take into account the structural components,
NSCs and soil. The frequencies and mode shapes produced by the FEMs were
compared to those experimentally identified from the measured responses at different
response levels. The differences observed were then minimized using a sensitivity-
based model updating approach with a Bayesian parameter estimation technique. The
updated models were then used for serviceability limit state assessment of the
building under a selection of 10 ground motion records obtained at the site. This part
of the study, by including SSI and NSCs in FEM updating, extends the range of
applications considered so far in updating of building models and contributes to better
and more realistic ways of simulating the dynamic behavior of building structures.
Model updating based on following the seismic response trends to replicate the actual
varying structural behavior and the use of so-updated models to assess seismic
performance are also original contributions to the existing body of knowledge.
Overall, the paper furthers the understanding of dynamic behavior of buildings
during earthquakes and provides new methods and quantitative data for studying
seismic responses of as-built structures, structural health monitoring and model
updating. One limitation is that only low to medium intensity seismic records were
available for the site what this investigation was carried out. To extend the present
study into larger response range, more data, including those from high intensity
earthquakes, are required but are currently not available for the building. In the study
5

whose philosophy is to validate, insofar as possible, any numerical modelling
assumption against experimental data, we consider that putting forward numerical
models of the building studying much more intense, strongly non-linear seismic
responses would be premature. The analyzed excitation levels are, nevertheless, of
interest and importance for serviceability limit state studies where structures remain in
their elastic, linear or only mildly non-linear, range. Also, to be able to account for the
time dependent variation of structural response due to aging, environmental agents
and consequently degradation of RC structures, the responses to serviceability limit
state shaking should be evaluated (Berto et al. 2009). Furthermore, low to medium
shaking levels are important as the baseline data to judge the condition of the structure
in structural health monitoring applications (Sohn et al. 2003). For example, ignoring
the variations in the resonant properties at different response amplitudes may lead to
misinterpreting such benign changes as indicators of damage.
The philosophy and motivation behind using the modal data and linear
structural models in this research have also to be made transparent. Strictly speaking,
modal properties are defined for linear time invariant systems only. The observed
decreasing trends in the experimentally identified resonant frequencies are a
manifestation of system non-linear behavior. For the experimental system
identification we used linear system approaches to determine the resonant
frequencies, i.e. the frequencies that result in the largest response amplitude, and
damping ratios. We consider this a legitimate approach, as the nonlinearities present
in the building, despite being discernible, were mild. Besides, quantifying and
presenting such nonlinearities in the form of trends in the resonance frequencies has
an advantage, for pragmatic reasons, of using the quantity that can easily be
understood by a wide community of structural engineers and researchers. We then
forced the actual natural frequencies (i.e. solutions to the eigenvalue problem) of the
series of linear FEM building models to match these experimental resonance
frequencies in the process of model updating. The decision to use the resonant
frequencies provided meaningful and useful, yet easy to handle, targets for model
updating procedures. The decision to use of the series of response-amplitude
dependent linear FEM models, rather than a nonlinear model, was again a pragmatic
one, resulting in a manageable approach to model updating, which could be based on
resonant data and linear stiffness parameters. At the same time, the use of several such
6

linear FEM models enabled covering the range of experimentally observed responses
well.
Description of the building and instrumentation
The building under study (Figure 1) is situated in Lower Hutt, approximately 20km
North-East of Wellington, New Zealand. It is a three story RC structure with a
basement, 44.70m long, 12.19m wide and 13.40m high (measured from the base
level) as shown in Figure 2. The structural system consists of seven two-bay moment
resisting frames of spans 5.33m and 6.86m, respectively, and a 2.54m×1.95m RC
shear core with the wall thickness of 229mm. The plan of the building is rectangular
and all the beams and columns are of rectangular cross-section. The exterior beams
are 762×356mm except at the roof level where these are 1067×356mm. All the
interior beams and all the columns are 610×610mm. Floors are 127mm thick RC slabs
except for a small portion of the ground floor near the stairs which is 203mm thick.
The roof comprises corrugated steel sheets over timber planks supported by steel
trusses. The columns are supported on pad type footings of base dimensions
2.29×2.29m around the building perimeter and 2.74×2.74m inside the perimeter, and
610×356mm tie beams are provided to join all the footings together.
The building is instrumented with five tri-axial accelerometers. Two
accelerometers are fixed at the base level, one underneath the first floor slab, and two
at the roof level as shown in Figure 2b. There is also a free-field tri-axial
accelerometer mounted at the ground surface and located 39.4m from the building.
Figures 2a and b also show the common global axes X and Y used for identifying
directions in the subsequent discussions.
System identification including SSI effects
In this section, the methodology of the N4SID system identification technique, its
application to the instrumented building and the approach for taking into account SSI
in the identification process will be discussed.
N4SID system identification technique
This subsection provides a brief explanation of the N4SID system identification
technique. Full details of the technique can be found in Van Overschee and De Moor
(1996). After sampling of a continuous time state space model, the discrete time state
space model can be written as:
7

x
k+1
=A x
k
+B u
k
+w
k
(1)
yk=C xk+D u k+vk(2)
where A, B, C and D are the discrete time state, input, output and feedthrough
matrices, respectively, whereas xk and yk are the state and output vectors and uk is the
excitation vector, respectively. Vectors wk and vk are the process and measurement
noise, respectively. In the case of input/output system identification, data from both
output yk and input uk are assembled in a block Hankel matrix, which is defined as a
gathering of a family of matrices that are created by shifting the data matrices in time.
Thereafter, the identification involves two steps. The first step takes projections of the
subspaces calculated from input and output observations (in the block Hankel matrix)
to estimate the state sequence of the system. In the second step, a least square problem
is solved to estimate the system matrices A, B, C and D. Then the modal parameters,
i.e., natural frequencies, damping ratios and mode shapes, are found by eigenvalue
decomposition of the system matrix A. Finally, those discrete time frequencies and
damping ratios are converted into their continuous time equivalents. However, due to
the presence of measurements errors spurious modes are often encountered. In order
to overcome the problem, an approach based on observing the trends in the estimated
modal parameters in the so-called stabilization charts is often used: a range of system
orders is tried and modal parameters which repeat themselves across that range are
accepted as correct results. Stability tolerances are chosen based on the relative
change in the modal properties, i.e., modal frequencies, damping ratios and mode
shapes, of a given mode as the system order increases. For mode shape stability,
modal assurance criterion (MAC) between the mode shapes of the present and
previous orders were examined. MAC is an index that determines the similarity
between two mode shapes,
ϕ
i
and
ϕj
, and is defined as (Ewins 2000):
MAC=
(
ϕi
Tϕj
)
2
(
ϕi
Tϕi
) (
ϕj
Tϕj
)
×100 % (3)
where superscript T denotes vector transpose.
Incorporation of SSI effects in system identification procedures
For incorporation of SSI effects in system identification procedures, Stewart and
Fenves (1998) proposed the following approach, based on earlier efforts by Veletsos
and Nair (1975) and Bielak (1975) for surface and embedded foundations,
respectively. Consider the structure shown in Figure 3. The height h is the vertical
8

distance from the base to the roof (or another measurement point located on the
building). The symbols denoting translational displacements are as follows: ug for the
free-field translational displacement, uf for the foundation translational displacement
with respect to the free field, and u for the roof translational displacement with respect
to the foundation resulting from inter-story drift. Foundation rocking angle is denoted
by

, and its contribution to the roof translational displacement is h

. The Laplace
domain counterparts of these quantities will be denoted as
ˆ
g
u
,
ˆ
f
u
,
ˆ
u
and
ˆ

,
respectively.
Stewart and Fenves (1998) considered the following transfer function for the
flexible base model:
H=
^
u
g
+
^
u
f
+
^
u+h
^
θ
^
u
g
(4)
where the input is the free-field displacement ug and the output is the total roof
displacement ug+uf+u+h

They demonstrated that the poles of the flexible base
transfer function H give natural frequencies and damping ratios of the entire
dynamical systems comprising the structure, foundation and soil. In other words, the
identified modal parameters are influenced by the stiffness and damping of soil. To
carry out the identification of the building dynamics including the SSI effects,
resonant vibration parameters were sought in the present research through the N4SID
technique for the flexible base case using input-output pairs consisting of a
combination of free-field, foundation and superstructure level recordings. In this
study, accelerations measured by sensor 10 (the free-field sensor) in all three
directions were considered as inputs and accelerations measured by sensors 3, 4, 5, 6
and 7 as the outputs for the flexible base case (see Figure 2b for sensor numbers and
locations).
Evaluation of seismic response trends
The objective of this part of research is to assess and understand the seismic response
of the building under a large number of earthquakes of different strength. In
particular, trends are investigated between PRA and the identified first three resonant
frequencies and the corresponding damping ratios of the building using 50
earthquakes. The presentation of this part of the study will thus follow the selection of
earthquakes, experimental system identification and correlating the identified flexible
9

base model resonant frequencies and damping ratios with PRAs. It is noted that the
results reported in this part are a selection from a more extensive study conducted
recently (Butt and Omenzetter 2012) that included also another building and
attempted to separate the influence of SSI effects on the building dynamics. Their
inclusion in this paper serves the purpose of completeness and highlighting the
variability of the identified resonant parameters and setting the ground for the
subsequently reported finite element modelling and model updating that considers the
observed variability of resonant frequencies.
Selection of earthquake records
For this study, 50 earthquake response time histories recorded on the building
between November 2007 and February 2010 which had epicenters within 200km from
the building were selected. The intention was to select from all the available
earthquakes those of such intensities that could excite the building with acceptable
signal-to-noise ratios providing quality system identification results. The selected
earthquakes had a moment magnitude ranging between MW = 3.0 and 5.0, with most
of them clustered at the lower end of this interval. This means that nearly all of the
earthquakes fell into the category of low intensity except for a few that can be treated
as moderate events. An angle histogram of the 50 earthquake back azimuths is shown
in Figure 4. As can be seen the vast majority of tremors reached the building from
epicenters located roughly between North-East and South-West.
Table 1 summarizes the maximum accelerations recorded at the free field,
base and roof sensors for the 50 earthquakes. The maximum resultant peak ground
acceleration (PGA) (i.e. the magnitude of the vectorial sum of the horizontal X- and
Y-direction components) at the free-field sensor 10 was 0.0157g. The maximum
resultant PGA at the base of the building was 0.0109g and was captured by both
sensors 6 and 7. The maximum resultant PRA of the building was 0.0460g captured
by sensor 4. For sensor 3, the maximum resultant PRA was a little lower (0.0437g)
than that of sensor 4. The majority (92%) of the analyzed earthquakes had the
resultant PRAs below 0.015g.
Modal identification of the instrumented building
This subsection reports on the results of system identification of the building using the
selected 50 earthquake records. The identified resonant frequencies and damping
ratios are plotted against PRAs to reveal trends between the system dynamic
10

properties and response levels. The N4SID technique was used to identify the first
three resonant frequencies, corresponding damping ratios and mode shapes. Sampling
rate of the digitized signals was 200Hz and for establishing stabilization charts system
orders from 2 to 20 were considered. A typical stabilization chart is shown in Figure
5: the marker ‘o’ shows an identified frequency, ‘+’ shows a stable frequency and
damping ratio, while ‘×’ a stable frequency, damping ratio and mode shape. In this
research, an identified frequency was considered as stable if the absolute deviation
between the frequency identified at the present and previous order was less than or
equal to 0.01Hz. A stable damping ratio was defined by an absolute deviation less
than 5%. For mode shape stability, the MAC between the mode shapes of the present
and previous order was to be at least 90% or greater. It can be seen in Figure 5 that
three modes can be identified with confidence and the subsequent discussions focus
on these.
The typical first three mode shapes of the building are shown in Figure 6 in
planar view. (Note that because of a limited number of measurement points those
graphs assume the floors and all foundation pads connected by tie beams move as
rigid diaphragms.) The shape of the first mode shows it to be a translational mode
along X-direction with some torsion. The second mode is translationally dominant
along Y-direction coupled with torsion, and the third one is torsionally dominant with
some Y-direction translation present. Structural irregularities, such as those due to the
shear core present near the North end of the building, an irregular frame pattern near
the shear core, unequal frame bay spans and, to a lesser degree, staircase and internal
longitudinal beams being not in the middle, create unsymmetrical distribution of
structural stiffness and give rise to the coupled translational-torsional mode shapes.
Another plausible source of the coupling in the mode shapes may be varying soil
stiffness under different foundations and around different parts of the building. (Note,
however, that in all the numerical simulations presented later such spatial soil
variability is ignored.)
Resonant frequency dependence on PRA
Table 2 shows the minimum, maximum, average and relative spread (= (maximum-
minimum)/average×100%) values of the identified resonant frequencies for the
analyzed 50 earthquakes for the flexible base model. The average first three resonant
frequencies for the building are 3.33Hz, 3.61Hz and 3.79Hz and the relative spreads
11

are 14%, 19% and 11%, respectively. It is of interest to explore whether, and if so
how, those changes in the resonant frequencies correlate with the response magnitude.
Figures 7a-c show the results of resonant frequency identification for the analyzed 50
earthquakes. The resonant frequencies are plotted against the maximum resultant
PRAs of a representative roof sensor (sensor 3) recorded in each event. It can be seen
that the resonant frequencies decrease as the PRAs increase and this is observed for
all three frequencies. In order to quantify the relationships between PRAs and
resonant frequencies, regression analysis was applied (Seber and Wild 2003). To
accommodate the fact that the resonant frequencies initially drop at a faster rate with
increasing PRAs but later the slope is flatter, bilinear relationships were adopted. In
Figures 7a and c, the bilinear formulas relating the identified resonant frequencies and
PRA are listed. The extent to which the regression curves fit the experimental data is
quantified by R2 or the coefficient of determination (Steel and Torrie 1960). The
coefficients of determination are 0.60 and 0.65 for the first and third resonant
frequency, respectively, indicating that the exponential relationships fit the data
reasonably well. The fit is less obvious for the second resonant frequency where
R2=0.33. Error bounds corresponding to ± two standard deviations are also indicated
in the figures. It can be seen that these bounds have different width, smaller for the
first and third frequency and larger for the second frequency. The bounds are
generally wide reflecting the significant scatter around the trend lines in the identified
resonant frequencies visible in the figures.
Damping ratio dependence on PRA
Table 2 also shows the minimum, maximum, average and relative spread values of the
identified damping ratios for the analyzed 50 earthquakes for the flexible base model.
The average values of damping ratios corresponding to the first, second and third
resonant frequency are 3.4%, 5.6% and 3.1%, respectively. It can be noticed that the
identified damping ratios show considerable scatter – the relative spreads are between
176% and 240%. Such large spreads may be the result of both actual variability of
damping as well as errors introduced by the identification method applied to short,
nonstationary time series, and generally confirm observations from numerous past full
scale identification exercises where the uncertainties in damping identification were
considerably higher than those of frequencies (see, e.g., Brownjohn et al. 2010;
Magalhaes et al. 2012). Figures 8a-c show the results of damping ratio identification
12

for the analyzed 50 earthquakes. Like the resonant frequencies previously, the
damping ratios are plotted against PRAs of sensor 3. The initial observation about the
considerable scatter of results, mentioned while analyzing Table 2, is now fully
revealed in the figures. No clear or convincing trends in the dependence of the
damping ratios on PRA could be discerned and so only the mean values and ± two
standard deviation bounds are indicated in the figures. Given the already mentioned
scatter of results these bounds are predictably wide.
Finite element modelling of structural and nonstructural components and SSI
A three dimensional FEM was developed using available structural drawings and
information gathered via additional at-site measurements and inspections. ABAQUS
(2011) software was used for modelling. The beams and columns were modelled
using Timoshenko beam elements (designated as B31), and slabs, stairs and shear
core using four-node, first-order shell elements (designated as S4). All the beam-to-
column connections were assumed as fixed (moment resisting frame assumption).
Linear elastic material properties were considered for the analysis. The density and
modulus of elasticity of RC for all the elements were taken as 2400kg/m3 and 30GPa,
respectively. The steel density and modulus of elasticity for roof elements were taken
as 7800kg/m3 and 200GPa, respectively. The steel trusses present at the roof level
were modelled as equivalent steel beams, having the same mass and longitudinal
stiffness, using beam B31 elements. The masses of timber purlins, planks and
corrugated steel roofing were calculated and lumped at the equivalent steel beams.
The mass due to partition walls, false ceilings, attachments, furniture and live loads
was collectively applied at the floor slabs as an area-distributed mass of 450kg/m2
according to design recommendations (ASCE/SEI 7-05 2005). Figure 9 shows the
three dimensional FEM including the structural elements, NSCs (cladding and
partition walls). Soil flexibility was also modelled and all these aspects of the model
will be further explained in the subsequent sections.
Modelling of NSCs
Since the structure under study is an office building, there are a large number of
partition walls present. The partitions were modelled as two node SPRING2 diagonal
elements. The stiffness value of those springs was taken from Kanvinde and Deierlein
(2006) as 2800kN/m. External cladding in the building is made of fiberglass panels
13

with insulating material on the inner side. The density and modulus of elasticity
values of fiberglass were taken as 1750kg/m3 and 10GPa, respectively, from Gaylord
(1974) and their mass was calculated manually (100kg/m) and applied at the
perimeter beams. The panels were modelled as 0.015m thick four node shell elements
S4 fully fixed to the RC frame. Openings for windows were ignored.
Modelling of SSI
The soil present at the building site is classified according to the New Zealand
Standard 1170.5:2004 (Standards New Zealand 2004) as class D (deep or soft soil).
The shear wave velocity, Vs, was taken as 160m/s based on the investigation into the
site subsoil classification by Boon et al. (2011), and the dynamic shear modulus, G, as
47MPa, and Poisson’s ratio,

, as 0.4, considering the recommendations from Bowles
(1996).
Soil underneath each foundation is idealized as six springs to model stiffness
corresponding to three translations and three rotations. The soil surrounding the
building is modelled as translational springs at mid height of the basement columns.
For the corner columns, two springs, i.e., in the X and Y-direction, were used; for the
remaining columns only the out of plane soil stiffness was taken into account. The
soil interaction underneath the tie beams is idealized as translational springs along two
horizontal and a vertical direction. The base, column and tie beam springs were
modelled as SPRING1 elements in ABAQUS. The values of spring stiffness were
calculated using the procedure proposed in Gazetas (1991). The equations and charts
for calculating static and dynamic soil stiffness coefficients are based on the length, L,
width, B, base area, A, and second moments of area, I, of foundation, soil Poisson’s
ratio,

, shear modulus and shear wave velocity, and dynamic response frequency,

.
According to Gazetas’ model, the dynamic soil stiffness Ki for a particular
degree of freedom i can be expressed as:
K
i
=G × f
i
(
μ , L , B , A , I
i
)
× k
i
(
μ , L /B , 8B /V
s
)
(5)
where
G × f i
(
μ , L , B , A , I i
)
is the static stiffness, and
ki
(
μ , L/B , 8B /Vs
)
is the
dynamic stiffness modification factor. Functions fi and ki are certain expressions of the
parameters listed as their arguments. Subscript i=1, 2,…6 is applied to those functions
and parameters that differ for different degrees of freedom. As can be seen, in all the
cases stiffness is proportional to the shear modulus G. The dependence of the static
stiffness on Poisson’s ratio

in functions fi is more complex and varies between the
14

degrees of freedom. To vary soil stiffness during model updating, only the shear
modulus was changed. This was done in order to keep the number of updating
parameters small and simplify the calculation of the sensitivities of natural
frequencies to soil stiffness. The dynamic stiffness modification factors ki depend on
the frequency of foundation motion. A quick check of their values in the frequency
range from 2.5Hz to 4.0Hz, encompassing with some margin the full range of
frequencies encountered in this study, showed a very small maximum relative
variation of less than 1%. For this reason the frequency dependence of the dynamic
soil stiffness was ignored and constant values corresponding to 3.04Hz (the lowest
modal frequency observed experimentally in Table 2) adopted.
Sensitivity based model updating with Bayesian parameter estimation
Model updating is concerned with the calibration of the FEM of a structure so that the
FEM can better predict the measured responses of that structure. The sensitivity based
model updating procedure generally comprises three steps: i) the selection of
reference experimental responses, ii) the selection of model parameters to update, and
iii) an iterative model tuning. In the sensitivity based updating, corrections and
modifications are systematically applied to the local physical parameters of the FEM
to modify them with respect to the experimental reference responses. The
experimental responses can be expressed as functions of the structural parameters and
a sensitivity coefficient matrix using a first order Taylor series as (Brownjohn et al.
2001b):
R
e
=R
a
+S(P
u
−P
0
)(6)
where
Re
and
Ra
are the vectors of experimental and analytical response values,
respectively, whereas
P
is the vectors of model parameters and subscripts u and 0 are
for the updated and current values, respectively. The target experimental responses
Re
are usually the resonant frequencies and mode shapes measured on the real structure,
whereas the updating parameters
P
are uncertain parameters in the FEM, which can
include geometric and material properties and boundary and connectivity conditions
related to stiffness and inertia.
S
is the sensitivity matrix whose entries can be
calculated as:
Sij=∂ R a ,i
∂ P j
|
P=P0
(7)
15

Here
Ra , i
(i=1, … … … .. ,n)
and
P
j
(j=1, … … … ,m )
are the entries of the analytical
structural response and the updating structural parameter vectors, respectively.
Equation (7) calculates the absolute sensitivities expressed in the units of the response
and parameter values. For comparing the relative sensitivities of different responses to
relative changes in different parameters the normalized relative sensitivity matrix,
Snr
,
can be calculated as:
Snr =RD ,a
−1S PD(8)
where
RD, a
and
PD
are square, diagonal matrices holding the response and parameter
values, respectively.
In this study, the sensitivity based updating method is enhanced with a
Bayesian parameter estimation technique. This technique includes weighting
coefficients applied to the updating parameters and experimental responses to
accommodate the confidence levels in their estimation. The advantage of the Bayesian
estimation is better conditioning of the updating problem (Wu and Li 2004; FEMtools
2008). The difference between the experimental and model responses is resolved by
using the following updating algorithm (Dascotte et al. 1995):
P
u
=P
0
−G
(
R
e
−R
a
)
(9)
where G is the gain matrix which can be computed as:
G=(Ca+Snr
TCeSnr)−1Snr
TCe(10)
or
G=C
a
−1
S
nr
T
(C
e
−1
+S
nr
C
a
−1
S
nr
T
)
−1
(11)
Equation (10) is valid for the cases when the number of responses is not less than the
number of updating parameters, whereas Equation (11) is used in the cases with fewer
responses than the updating parameters. Here,
Ce
and
Ca
represent diagonal weighting
matrices expressing the confidence in the values of the experimental and model
responses, respectively, and superscript T denotes matrix transpose.
The important considerations regarding parameter selection for updating are
the number of parameters to be updated and preference given to certain parameters
among the many possible candidates. An excessive number of parameters compared
to the number of available responses, or overparametrization, will lead to a non-
unique solution, whereas insufficient number of parameters will prevent reaching a
good agreement between the experiment and the model (Titurus and Friswell 2008).
The selected parameters should be uncertain and expected to vary within certain
16

bounds, otherwise updating may result in physically meaningless outcomes. If there
are a number of candidate parameters available for updating, a sensitivity analysis
using the normalized relative sensitivities (Equation (8)) can help to retain only those
parameters that significantly influence the responses.
FEM updating considering seismic response trends
The objective of this part of the study is to calibrate a series of FEMs so as they
replicate the observed trends in the resonant frequencies with increasing PRA. To that
end, five points were selected on each of the frequency-PRA trend plots (see Figures
7a-c) in such a way that the whole range of PRAs was covered. The PRAs and
resonant frequencies corresponding to the five points are listed in Table 3. The larger
density of points in the small PRA range helps to capture the faster rate of frequency
changes observed there.
FEMtools software (FEMtools 2008) was used in this research for automatic
model updating. The following expression, representing mean weighted absolute
relative frequency error, was used to judge the success of updating procedure:
ef=1
n∑
i=1
n
cri
|
∆ f i
|
fi
×100 %(12)
where n is the total number of target frequencies considered, and
f
i
and
∆ f
i
are the
target frequency and frequency error, respectively, whereas coefficients
cri
account
for the estimated relative variabilities of responses.
The automatic iterative procedure for minimizing the mean weighted absolute
relative frequency error is controlled by a following three convergence criteria:
The minimum value of the mean absolute relative frequency error,
assumed as 0.1%
The minimum improvement in the mean absolute relative frequency error
between two consecutive iterations, assumed as 0.01%, and
The maximum number of iterations allowed, assumed as 50.
The model updating algorithm may be lured into local minima instead of the
global minimum in problems that Goldberg et al. (1992) call ‘deceptive’. This
undesirable behavior is well known when using gradient-based methods (Deb 1998).
In this study, a three-step updating strategy was followed to safeguard against being
trapped in a local minimum (Brownjohn and Xia 2000):
17

Step 1: Starting with the initially assumed values of the updating parameters
the mean weighted absolute relative frequency error is minimized to arrive at
an intermediate solution.
Step 2: The values of the updated parameters obtained in Step 1 are perturbed
by +10%, 0% and -10%, considering all possible combinations, and the
updating procedure rerun.
Step 3: The best among the solutions found in the previous two steps is chosen
as the final solution. This will typically be the one that gives the smallest value
of the mean weighted absolute relative frequency error but careful judgment
still needs to be exercised to avoid physically unacceptable solutions.
The following section describes the application of the sensitivity based model
updating technique to a series of FEMs that follow the observed experimental seismic
response trends. The presentation will thus follow the sensitivity analysis and
selection of responses and updating parameters, FEM updating and a discussion of the
updated results.
Sensitivity analysis and selection of response and updating parameters
The updating process starts with identifying the target responses and model
parameters to update. In this study, the experimentally identified first three resonant
frequencies were taken as target responses to be replicated by the models. It was
assumed that the identified frequencies used as targets have a scatter of 2%. The
scatter was estimated using the frequencies identified for PRAs between 0.0013g and
0.0017g, where there was enough data with very similar PRAs and thus not affected
by the observed frequency-PRA trends (see Figures 7a-c). Therefore, this confidence
level was used to determine the entries of the diagonal weighting matrix Ce (Equations
(10) and (11)).
The updating parameters were selected based on their expected uncertainty
and the sensitivity analysis to determine the most influential parameters to produce a
genuine improvement in the models. Only stiffness parameters were considered for
updating as mass can normally be determined with less uncertainty. Three parameters,
namely: i) the shear modulus of soil, Gsoil, ii) the modulus of elasticity of the cladding,
Eclad, and iii) the modulus of elasticity of concrete, Econ, were selected. The normalized
relative sensitivities of the target responses to the selected updating parameters are
shown in Table 4. It can be observed from the table that the values of the normalized
18

relative sensitivities are significant and capable of producing a noticeable change in
the model responses. The modulus of elasticity of concrete is the most influential
parameter, while the remaining two parameters are almost equally influential but
roughly 50% less than the first one. For this study, it was assumed that the updating
parameters have uncertainty ±30% and these confidence levels were used to
determine the entries of the diagonal weighting matrix Ca (Equations (10) and (11)).
Updating of the series of FEMs
In this section, updating of the series of FEMs, so that their predictions match the
selected Points 1 to 5 on the frequency-PRA curves, is presented. The first point
selected for updating was Point 1 corresponding to the largest PRA of 0.0434g. (This
was chosen as the beginning of the exercise because of the existing experience from
preliminary updating attempts that considered only this point.) The final results of the
updating at Point 1 were used as the starting values for updating at Point 2 and so on
moving in the direction of diminishing PRAs and concluding at Point 5. The case of
Point 1 updating is described in some detail with intermediate results also given to
explain the process, whereas only the final results for the remaining points are
provided.
The target first three modal frequencies for Point 1 obtained from system
identification are 3.039Hz, 3.210Hz and 3.479Hz (Table 3). The FEM developed in
ABAQUS was imported into FEMtools software (FEMtools 2008) for performing
model updating. The initial FEM calculated the first three frequencies as 2.922Hz,
3.451Hz and 3.723Hz. A comparison between the initial FEM and target frequencies
and mode shapes is presented in Table 5. Table 5 shows that the absolute relative
errors between the individual initial FEM and target frequencies do not exceed 7.51%.
The overall mean absolute relative frequency error ef was 6.12%. The correlation of
mode shapes expressed by MAC values (Equation (3)) is very good, 92%, for the
second mode, while for the first and third modes the MAC values are reasonably
satisfactory, being 78% and 63%, respectively.
In Step 1 of updating, the mean weighted relative frequency error improved
considerably from 6.12% to 0.31%, and the largest individual error did not exceed
0.32%. In Step 2 of updating, a better solution to that of Step 1 was found, suggesting
that the above Step 1 solution was only a local minimum and confirming the
advantage of, and need for, using the proposed two-step procedure. Step 2 converged
19

to a very small value of ef=0.03% for the mean absolute relative frequency error,
providing excellent match of frequencies with the maximum absolute relative error of
only 0.05% (see Table 5), and yielding the final updating parameter values of
42.3MPa for the shear modulus of soil, 6.5GPa for the modulus of elasticity of
cladding, and 38.4GPa for the modulus of elasticity of reinforced concrete. Compared
to the initial values of the updating parameters, their relative changes were -10%, -
35% and 28% for the soil, cladding and concrete stiffness, respectively. While MACs
were not explicitly included in the updating problem, improving frequencies typically
also improves MACs. This was also the case in the reported exercise: the MAC values
have improved slightly for the first and second mode and are equal to 80% and 96%,
respectively, while for the third mode shape it has improved considerably reaching
78%.
Tables 6 and 7 summarize the updating results for all the FEMs corresponding
to the five points. Table 6 shows frequencies and Table 7 MACs, respectively. It can
be seen from Table 6 that the updating converged to small values of the mean
absolute relative frequency error not exceeding 0.60% for all the cases. In fact, for
four out of the five updating cases this upper error bound was much smaller, 0.18%.
Individual absolute relative frequency errors were in all cases not exceeding 0.94%.
MAC values (Table 7) were between 80% and 96% for the first and second mode, and
between 77% and 86% for the third mode. These numbers indicate good match
between mode shapes.
Table 8 shows the initially assumed updating parameters and their final values
at each updating point. The final updated values are also shown in Figure 10. These
results are now analyzed to: i) assess the plausibility of numerically obtained results
using engineering judgment, and ii) understand and quantify the contributions of
structural and non-structural components as well as soil to the overall stiffness of the
building. The first observation that can be made is that, when examined in the entire
PRA range considered, all the stiffness parameters show a general decreasing trend
with increasing response amplitude. This is consistent with the known behavior of
materials and structures that normally have ‘softening’ characteristics. The minimum
value of the shear modulus of soil for the updated FEMs is 39.6MPa at Point 2
whereas the maximum value is 50.4MPa at Point 5. The percentage change between
the maximum and minimum value is 27%. The modulus of elasticity of cladding has
the minimum value for the updated FEMs of 6.5GPa at Point 1, whereas the
20

maximum value is 8.2GPa at Point 3. The percentage change for the modulus of
elasticity of cladding was 26%. The third updating parameter was the modulus of
elasticity of reinforced concrete which showed the minimum value of 38.4GPa at
Point 1 and the maximum value of 45.7GPa at Point 5, producing the percentage
change of 19%. The observed changes in the updating parameters can be considered
by engineering judgment to be within acceptable limits. To quantify in a simple way
the dependence of the three updating parameters on the response amplitude linear
regression was performed with the PRA of sensor 3 serving as the independent
variable. These regression lines are shown in Figures 10a-c. The coefficient of
determination, R2, also shown in the figures, varies from 0.56 to 0.78, and confirms
that the linear relationship fits the data reasonably well.
Considering all the five target points, the large drop, up to 35%, in the
cladding stiffness from its initially assumed value of 10GPa indicates that the initial
assumption of the cladding being fully fixed to the structural elements was not
justified and very likely only partial fixity ever existed. Another reason for the
reduced stiffness are the openings for windows in the cladding panels which were
ignored in the initial FEM model. Also an initial overestimation of cladding material
modulus of elasticity, taken from literature, is quite possible. For the modulus of
elasticity of the reinforced concrete, there is considerable increase, up to 52%, from
the initially assumed value of 30GPa. In situ concrete typically shows significant
variability in its mechanical properties. Also, the initial estimate of the modulus of
elasticity was based on the typical, conservatively assumed values used in New
Zealand building construction of the era but the exact design or laboratory tested
values were unknown. For those reasons, the updated values are not unreasonable. It
is also possible that other minor non-structural elements, whose stiffness was neither
explicitly modelled nor updated, could have made contributions towards the larger
stiffness that were ‘lumped’ into the concrete stiffness. One would normally assume
more uncertainty in the soil properties, but, perhaps unexpectedly, the change in the
soil shear modulus from its initially assumed value of 47.0MPa is not large for the
five updated values, only up to 10%. The initially assumed value was also between
the two extreme values obtained from updating.
Finally, for all the parameters it needs to be noted that they are the global
stiffness of the soil, reinforced concrete and cladding without taking into account any
possible local spatial variations. In general, the updated models represent the optimal
21

solution for the frequency matching problem of Equation (12) that can also be
justified by engineering judgment, but hinge on the validity of the FEM topology,
discretization and parameterization. We argue though that these are adequate in the
context of the objectives of this research.
Assessment of serviceability limit state performance using updated FEMs
This section puts some of the updated FEMs to practical use in assessing the
building performance at the serviceability limit state. Under the serviceability limit
state shaking level, the building response should remain predominantly elastic and
limiting of lateral deformations leading to non-structural damage is a primary design
objective. The inter-story drifts are commonly assumed to control the onset of non-
structural damage (Dymiotis-Wellington and Vlachaki 2004). Therefore, to study the
serviceability limit state performance of the building, the maximum inter-story drift
ratios for a random selection of 10 seismic events recorded at the building site and
appropriately scaled were calculated (see Table 9). The numerical analyses were
performed for the following FEMs formulated in ABAQUS (ABAQUS 2011):
The initial FEM model: This case corresponds to using a model formulated
based on as-designed or as-built drawings and assumptions but without the
benefit of calibration using experimental responses.
The updated FEM model obtained at Point 5: This model uses experimental
data for calibration but the level of response is small and therefore the model
dependence on the response level is ignored. This can be seen as a case where
only low level ambient experimental data were available to the analyst.
The updated FEM model obtained at Point 1: This model uses the largest
available experimental responses for model calibration and takes into account
the dependence of stiffness on the amplitude of response. This case is the
closest to the serviceability limit state shaking level and minimizes the
extrapolation of the stiffness loss trends. It can also be argued that the
observed stiffness loss trends cannot go on indefinitely and must stabilize for
larger PRAs.
The scaling procedure recommended in the NZS 1170.5:2004 (Standards New
Zealand 2004) was followed for the selected 10 earthquakes. The procedure requires
minimizing the logarithmic root-mean-square difference between the actual and target
22

spectra in a frequency range encompassing the fundamental frequency of the structure
at hand. The period range for spectrum matching was chosen between 0.13sec and
0.43sec (2.32Hz and 7.69Hz). The assumed target code spectrum for a return period
of 25 years and hazard factor of 0.4 is shown in Figure 11 along with the spectra of
the 10 earthquakes for both X- and Y-direction components. The PGAs for the
selected 10 events used in the simulations are reported in Table 9. The use of past
records taken from the building site for numerical simulations is advantageous as
these will generally have characteristics determined by the site and will therefore
likely resemble potential events that may affect the structure in future.
A constant damping of 5% was considered for all the modes as recommended
by NZS 1170.5:2004 (Standards New Zealand 2004) for time history analysis for
serviceability limit state. (This code recommendation was adopted rather than the
identified values due to the considerable scatter of the experimental results as
discussed earlier, however the measured values were not significantly different, see
Table 2.) The lateral displacements along X- and Y-directions at the four corners of
each floor level were determined, inter-story drift ratios corresponding to the two
directions calculated separately, and maximum ratios selected. For all the considered
excitation cases, the largest inter-story drift ratios were observed between the first and
the ground floor. Table 9 shows the maximum inter-story drift ratios for the
considered 10 earthquakes for the three FEMs along X- and Y-directions. For the
initial model, the values for X-direction are between 0.06% and 0.16% and for Y-
direction between 0.06% and 0.13%. The updated model at Point 5 gave lower values
of inter-story drifts, between 0.06% and 0.10% for X-direction, and between 0.05%
and 0.09% for Y-direction, respectively. On the other hand, the updated model at
Point 1 yielded markedly increased inter-story drifts ratios, between 0.07% and 0.21%
for X-direction, and between 0.07% and 0.15% for Y-direction. These differences are
mostly because the updated model at Point 5 is stiffer than the initial model, whereas
the updated model at Point 1 is less stiff as evident from their respective modal
frequencies. However, since the relative differences in the inter-story drift ratios vary
between the earthquakes, the matching of the building resonant frequencies and
spectral content of the excitation is another contributing factor.
23

The limiting inter-story drift ratios recommended by various codes vary
widely between 0.06% and 0.6% (Bertero et al. 1991). Dymiotis-Wellington and
Vlachaki (2004) suggest 0.2% as the critical inter-story drift based on their
observations on RC buildings. They argued that higher limiting values can cause
significant yielding in the structure and correspond to a damage state beyond
serviceability. Taking this latter limit value, it can be concluded that for the considered
seismic events, the building has reached or just exceeded the serviceability limit state
of 0.2% inter-story drift for two events, EQ1 and EQ8, for the updated FEM at Point 1
(Table 9). Overall, the serviceability performance can nevertheless be judged as
satisfactory. However, the initial FEM produced unconservatively low values, and the
FEM updated at Point 5 gave still smaller inter-story drifts. This confirms the benefit
and importance of using, where possible, calibrated structural models that take into
account response dependent characteristics for checking performance criteria.
Finally, since linear FEMs were used in the serviceability study, it is in order
to assess if the linearity assumption is justified. The maximum inter-story drift ratio
reported in Table 9 is 0.21%, with the majority of values noticeably lower.
Insufficient information is available (such as the exact reinforcement ratios and
detailing, or on nonlinear behavior of cladding and its connections to the RC frame)
that would enable the creation of a detailed and realistic nonlinear FEM to be
subjected to time history analysis to see if it enters nonlinear range at the
serviceability level response. However, some indirect inferences can be made.
Serviceability limit state can be understood as separating the linear and nonlinear
structural behavior (Dymiotis-Wellington and Vlachaki 2004). Mosalam et al. (1997)
consider it to be limited to the case of insignificant damage where repair is only
required to non-structural elements, and give the critical inter-story drift ratio in the
range between 0.2% and 0.5%. Eurocode 8 (European Committee for Standardisation
2003) also implies avoidance of damage and gives similar limits for the inter-story
drift ratios for damage in non-structural elements between 0.2% and 0.5%. As far as
yielding in structural elements is concerned, a study by Dymiotis-Wellington and
Vlachaki (2004) showed, despite its limited scope, that inter-story drifts of 0.2%
resulted in mild yielding (rotational ductility less than 2) in only some beams of an
RC frame designed to Eurocode 8. As in our study the maximum inter-story drift ratio
is 0.21%, with the majority of remaining values noticeably lower, there are good
reasons to assume that the building experienced at most only mildly nonlinear
24

responses when subjected to the shaking levels used in this study and the linear FEMs
used were able to predict its serviceability limit state behavior well.
Conclusions
This study was concerned with the updating of a series of FEMs of an instrumented
three story RC building based on experimentally obtained trends of resonant
frequencies vs. response levels. The first part focused on the identification of the
resonant dynamic properties using 50 seismic response records. The records, varying
in amplitude, enabled the determination of the trends in resonant frequencies with
increasing amplitude of response. The frequencies showed a clear decreasing trend
with increasing PRA, which could be represented by bi-linear relationships. The
identified damping ratios, however, had scattered values and no clear trend.
The second part of the study was concerned with the numerical modelling of
the building and model updating for the seismic response trends observed. A series of
FEMs were updated to replicate the varying behavior of the building under seismic
excitations. The updating parameters included the stiffness of concrete, cladding and
soil. Excellent matches of frequencies were achieved, with average errors not more
than 0.60%. The updated stiffness parameters were found to generally follow
decreasing trends and changed in the considered range of PRA by 19% for concrete,
26% for cladding, and 19% for soil, respectively.
Finally, the initial and two updated FEMs were used to study the seismic
structural performance of the building at the serviceability limit state shaking. The
maximum inter-story drift ratios were calculated for a selection of 10 earthquakes
recorded at the building site. It was found that the FEM model updated to the
experimental modal characteristics corresponding to the largest response produced the
largest drifts. For this updated FEM, the inter-story drift ratios reached in some cases
the recommended critical values, but the overall building serviceability limit state
performance was judged as satisfactory.
Acknowledgements
The authors would like to express their gratitude to their supporters. Drs Jim Cousins,
S.R. Uma and Ken Gledhill facilitated this research by providing access to GeoNet
seismic data and structural building information. Piotr Omenzetter’s work within the
Lloyd’s Register Foundation Centre for Safety and Reliability Engineering at the
25

University of Aberdeen is supported by Lloyd’s Register Foundation. The Foundation
helps to protect life and property by supporting engineering-related education, public
engagement and the application of research.
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30

Table 1. Maximum horizontal PGA and PRA recorded by individual sensors.
Sensor
Maximum horizontal
acceleration
(g)
Free field
10 (PGA) 0.0157
Foundation
6 (PGA) 0.0109
7 (PGA) 0.0109
Roof
3 (PRA) 0.0437
4 (PRA) 0.0460
Omenzetter and Butt
31

Table 2. Summary of identified resonant frequencies and damping ratios for flexible
base model.
Mode
Frequency Damping ratio
Min.
(Hz)
Max.
(Hz)
Avg.
(Hz)
Relativ
e spread
(%)
Min.
(%)
Max.
(%)
Avg.
(%)
Relativ
e spread
(%)
1st
(X-dir.
dominant)
3.04 3.50 3.33 14 1.2 7.3 3.4 176
2nd
(Y-dir.
dominant)
3.21 3.88 3.61 19 1.4 12.1 5.6 190
3rd
(torsionally
dominant)
3.48 3.90 3.79 11 1.0 8.3 3.1 240
32

Omenzetter and Butt
Table 3. PRAs and resonant frequencies for five updating points.
Poin
t
PRA
(g)
1st resonant
frequency
(X-direcon
dominant mode)
(Hz)
2nd resonant
frequency
(Y-direcon
dominant mode)
(Hz)
3rd resonant
frequency
(torsionally
dominant mode)
(Hz)
1 0.0434 3.039 3.210 3.479
2 0.0221 3.118 3.399 3.560
3 0.0114 3.182 3.491 3.713
4 0.0052 3.322 3.549 3.798
5 0.0006 3.427 3.650 3.864
33

Omenzetter and Butt
34

Table 4. Normalized relative sensitivities of responses to updating parameters.
Updating
parameter
Response
Shear modulus of
soil
Modulus of
elasticity of
cladding
Modulus of
elasticity of
reinforced
concrete
1st resonant frequency
(X-dir. dominant mode)
0.12 0.13 0.24
2nd resonant frequency
(Y-dir. dominant mode)
0.15 0.06 0.22
3rd resonant frequency
(torsionally dominant
mode)
0.11 0.10 0.25
Omenzetter and Butt
35

Table 5. Updating results at Point 1.
Mode Measured
frequenc
y
(Hz)
Initial FEM predictions Final updated FEM
predictions
Frequenc
y
(Hz)
Relative
frequenc
y error
(%)
MA
C
(%)
Frequenc
y
(Hz)
Relative
frequenc
y error
(%)
MA
C
(%)
1st
(X-dir.
dominant)
3.039 2.922 -3.85 78 3.038 -0.05 80
2nd
(Y-dir.
dominant)
3.210 3.451 7.51 92 3.211 0.02 96
3rd
(torsionally
dominant)
3.479 3.723 7.01 63 3.480 0.02 78
Mean absolute relave frequency
error ef
6.12 0.03
Omenzetter and Butt
36

Table 6. Summary of updated frequencies for Points 1-5.
Mode
Point
1 2 3 4 5
Exp.
freq.
(Hz)
Updated
freq.
(Hz)
Freq.
error
(%)
Exp.
freq.
(Hz)
Updated
freq.
(Hz)
Freq.
error
(%)
Exp.
freq.
(Hz)
Updated
freq.
(Hz)
Freq.
error
(%)
Exp.
freq.
(Hz)
Updated
freq.
(Hz)
Freq.
error
(%)
Exp.
freq.
(Hz)
Updat
ed
freq.
(Hz)
Freq.
error
(%)
1st
(X-dir.
dom.)
3.039 3.038 -0.05 3.118 3.111 -0.21 3.182 3.179 -0.10 3.322 3.324 0.05 3.427 3.426 -0.02
2nd
(Y-dir.
dom.)
3.210 3.211 0.02 3.399 3.377 -0.65 3.491 3.488 -0.09 3.549 3.547 -0.05 3.650 3.641 -0.24
3rd
(torsion
dom.)
3.479 3.480 0.02 3.560 3.593 0.94 3.713 3.715 0.04 3.798 3.796 -0.06 3.864 3.875 0.28
Mean absolute relave
frequency error ef
0.03 0.60 0.08 0.05 0.18
Omenzetter and Butt
37

Table 7. Summary of updated MACs for Points 1-5.
Mode
Point
1 2 3 4 5
MAC (%)
1st
(X-dir.
dominant)
80 80 93 94 96
2nd
(Y-dir.
dominant)
96 94 80 83 84
3rd
(torsionally
dominant)
78 77 85 81 86
Omenzetter and Butt
38

Table 8. Final values of updating parameters.
Parameter Initial
estimat
e
Point
1 2 3 4 5
Shear modulus of
soil (MPa) 47.0 42.3 39.6 44.6 47.6 50.4
Modulus of
elasticity of
cladding (GPa)
10.0 6.5 7.5 8.2 7.8 7.8
Modulus of
elasticity of
reinforced concrete
(GPa)
30.0 38.4 40.2 40.4 43.7 45.7
39

Omenzetter and Butt
40

Table 9. Earthquake records used in serviceability study and maximum inter-story drifts.
Earthquake
designation
PGA at sensor 6
(g)
Maximum inter-story drift ratios
(%)
X-direction Y-direction X-direction Y-direction
Initial
FEM
Updated FEM
(Point 5)
Updated FEM
(Point 1)
Initial
FEM
Updated FEM
(Point 5)
Updated FEM
(Point 1)
EQ1 0.118 0.107 0.12 0.09 0.21 0.12 0.09 0.14
EQ2 0.113 0.117 0.13 0.08 0.14 0.08 0.08 0.11
EQ3 0.203 0.177 0.10 0.06 0.12 0.13 0.09 0.15
EQ4 0.112 0.106 0.06 0.06 0.07 0.07 0.07 0.08
EQ5 0.187 0.141 0.09 0.09 0.11 0.06 0.05 0.07
EQ6 0.123 0.114 0.07 0.05 0.08 0.07 0.06 0.10
EQ7 0.130 0.156 0.09 0.06 0.11 0.09 0.06 0.10
EQ8 0.090 0.105 0.16 0.10 0.20 0.09 0.06 0.10
EQ9 0.140 0.121 0.07 0.07 0.09 0.09 0.05 0.11
EQ10 0.116 0.119 0.08 0.06 0.09 0.06 0.07 0.08
Omenzetter and Butt
41

Figure 1. The investigated RC building (on the right).
Omenzetter and Butt
42

a)
11 x 4.064 m = 44.70m
Stairs Elevator shaft
N
X
Y
b)
Basement 0.00m
Third floor 0.00m
Ground floor 0.00m
First floor 0.00m
Second floor 0.00m
Sensor 6
Sensor 7
Sensor 10 (free field)
Sensor 3Sensor 4
Sensor 5
Vertical
X
Y
39.40m
Elevator shaft
1
.80
m
Figure 2. Instrumented RC building: a) typical floor plan showing general dimensions
and location of stairs and elevator shaft, and b) 3D view of the building and sensor
array.
Omenzetter and Butt
43

44

Figure 3. Inputs and outputs for evaluating SSI effects in system identification of
buildings (Stewart and Fenves 1998).
Omenzetter and Butt
45
Roof: ug+uf+hθ+u
h
Free field: ug
Foundation rocking: θ
Foundation translation: ug
+uf

2 4 6 8 10
30
210
60
240
90270
120
300
150
330
180
0
N
Figure 4. Histogram of back azimuths of the 50 earthquakes used in analysis.
Omenzetter and Butt
46

Figure 5. Typical stabilization chart showing stable modes.
Omenzetter and Butt
47

Figure 6. Planar views of the first three mode shapes of the building for flexible base
model.
Omenzetter and Butt
48

a)
0 0.01 0.02 0.03 0.04 0.05
2.8
3
3.2
3.4
3.6
PRA (g)
Resonant frequency (Hz)
1st resonant frequency (X-direction dominant mode)
 

5
3
2
1
f1=-22.87xPRA+3.44
f1=-4.67xPRA+3.23
R2=0.60
4
b)
0 0.01 0.02 0.03 0.04 0.05
3
3.2
3.4
3.6
3.8
4
PRA (g)
Resonant frequency (Hz)
2nd resonant frequency (Y-direction dominant mode)
1

 
5
4
3
2
f2=-21.96xPRA+3.66
f2=-8.75xPRA+3.59
R2=0.33
c)
0 0.01 0.02 0.03 0.04 0.05
3.2
3.4
3.6
3.8
4
4.2
PRA (g)
Resonant frequency (Hz)
3rd resonant frequency (torsionally dominat mode)

 
R2=0.65
5
1
2
3
4
f3=-14.14xPRA+3.88
f3=-3.70xPRA+3.68
Figure 7. Resonant frequencies of the building vs. PRAs of sensor 3 for flexible base
case: a) 1st mode (X-dir. dominant), b) 2nd mode (Y-dir. dominant), and c) 3rd mode
(torsionally dominant). Omenzetter and Butt
49

a)
0 0.01 0.02 0.03 0.04 0.05
0
2
4
6
8
PRA (g)
Damping ratio (%)
1st damping ratio (X-dominant mode)
 

mean


b)
0 0.01 0.02 0.03 0.04 0.05
-1
1
3
5
7
9
11
13
PRA (g)
Damping ratio (%)
2nd damping ratio (Y-direction dominant mode)
mean


 

c)
0 0.01 0.02 0.03 0.04 0.05
0
2
4
6
8
10
PRA (g)
Damping ratio (%)
3rd damping ratio (torsionally dominant mode)
 



mean
Figure 8. Damping ratios of the building vs. PRAs of sensor 3 for flexible base case:
a) 1st mode (X-dir. dominant), b) 2nd mode (Y-dir. dominant), and c) 3rd mode
(torsionally dominant). Omenzetter and Butt
50

Figure 9. Three dimensional FEM of the building in ABAQUS showing stairs, shear
core, partition walls and cladding.
Omenzetter and Butt
51

a)
b)
c)
Figure 10. Updating parameters vs. PRA of sensor 3: a) shear modulus of soil, b)
modulus of elasticity of cladding, and c) modulus of elasticity of reinforced concrete.
Omenzetter and Butt
52

53

Figure 11. Ground motion spectra for X- and Y-directions for 10 seismic events used
in serviceability study and target spectrum from NZS 1170.5:2004 (Standards New
Zealand 2004).
Omenzetter and Butt
54