Assessment of structural performance of steel building on shake table using multi input multi output models

Five sets of acceleration data were measured at center and 4 corners on shake table and 4 stories of the full scale model were considered as inputs and outputs signals of many types of m

1 Assessment of Structural Performance of Steel

Building on Shake Table Using Multi Input

Multi Output Models

Academic Year

Maaastersterster’’’’s Thesiss Thesiss Thesis Academic YAcademic YAcademic Year ear ear 2020201010

Keio University Keio University Graduate School of Science and Technology Graduate School of Science and Technology School of

School of Science for Science for Science for Open and Environmental System Open and Environmental System Open and Environmental Systemssss

As we know, earthquakes have caused many serious damages; that is why it

is needed to study the health of buildings after these earthquakes After a large earthquake, evaluation of damage of structures is an important task for health assessment Vibrations of buildings give us valuable information

on it Changes in the modal properties like natural frequencies, modal shapes and modal damping are very important to assess the structures Among the representative structural characteristics, natural frequencies provide the global information However they are relatively simple, accurate

to measure and easy to obtain Besides, the changes in frequencies must be considered to assess the health of structure When the frequency shifts give the general health assessment, the mode shape changes can be categorized

up to the higher level of damage identification – determination of the geometric location of the damage

This study will consider the frequencies shifts and their mode shapes with various levels of seismic excitations The excitations are represented by different intensity levels of the 1995 Hyogoken Nanbu Earthquake that are obtained in JR – Takatori Station The acceleration data of E–defense tests

on full scale 4 story steel building will be analyzed Five sets of acceleration data were measured at center and 4 corners on shake table and 4 stories of the full scale model were considered as inputs and outputs signals of many types of multi input multi output (MIMO) models From the defined models, some first mode natural frequencies of structure will be obtained and compared to each other By analyzing the two first mode natural frequencies

of X and Y directions, the capacity of structure will be assessed with its performance The aim of this research is to assess the capacity of a steel building after a seismic activity using real size shake table tests

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In the field of civil engineering, system identification might be applied for several reasons However, the following two areas have attracted much attention in the recent years Modal analysis covers a variety of applications all based on the analysis of modal parameters These parameters describe specific dynamic characteristics of the structure One of the applications that uses the modal parameters as basis is vibration based inspection

Modal analysis is based on the determination of modal parameters of a structural system These parameters represent an optimal model, or basis, which can be used to

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describe the dynamics of a structural system The modal frequencies are more explicitly eigenvalues, or angular or natural eigenfrequencies Modal damping is characterized by the damping ratios and modal vectors by the eigenvectors or mode shapes Since the modal parameters are directly related to the impulse and frequency response functions, as well as the correlation functions and spectral densities, they can

be extracted from the non&parametric system identification methods by applying different curve fitting procedures In case of parametric system identification methods there are direct mathematical relationships between the modal parameters and the estimated model parameters When modal parameters are used as mathematical model

of the dynamic behavior of a system, the derived model is called a modal model It is therefore common to use experimental modal analysis as a synonym for system identification

The accumulation of damages in a civil engineering structure will cause a change in the dynamic characteristics of the structure

Changes in natural eigenfrequencies are no doubt the most used damage indicators One of the reasons for this is that the natural eigenfrequencies are rather easy to determine with a high level of accuracy Another reason is that they are sensitive to both global and local damages So comparison of estimates of natural eigenfrequencies

is usually an effective level one method A local damage will cause changes in the derivatives of the mode shapes at the position of the damage This means that a mode shape having many coordinates or measurement points will be a fast way to locate the approximate position of damage They can therefore be characterized as a simple level two method The introduction of damage in a structure will usually cause changes in the damping capacity of the structure It has been shown that the damping ratios are extremely sensitive to the introduction of even small cracks in a cantilever beam However, dealing with real structures, the estimation of the damping ratios of the individual modes is highly sensitive to time varying and nonphysical sources Thus, a

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satisfactory accuracy of the estimates of the damping ratios will in general be impossible to obtain Therefore, the damping is applicable as a damage indicator, but it cannot and should not be used as the only damage indicator

As explained, all modal parameters are in principle applicable as damage indicators

Structural health monitoring (SHM) can define similar to routine checkups of human bodies to make sure their health and to employ prompt measures to fix the diseases Doctors’ checkups are gathering signals relevant to human health or diseases The signals, such as blood test results, heartbeat counts, X&ray photos, are processed to know the human body organs or system to find out appropriate pattern matching to some diseases Doctors will translate the findings to provide diagnosis and prognosis

to the patients The processes needed for structural health monitoring are very similar

to those for the health checkups [1]

SHM aims to give, at every moment during the life of a structure, a diagnosis of the

“state” of the constituent materials, of the different parts, and of the full assembly of these parts constituting the structure as a whole The state of the structure must remain

in the domain specified in the design, although this can be altered by normal aging due

to usage, by the action of the environment, and by accidental events In the most general terms, damage can be defined as changes introduced into a system that adversely affects its current or future performance

SHM can be defined as the process of monitoring and assessing the state or condition

of a structure system

The signals get from SHM System may include input, output and noise signals The cleansed data is then used to build mathematical models in a process called system identification The final step uses the models to evaluate the current state of health of the system and to predict future states of health The process of system identification as

a part of structural health monitoring will be focused on [20]

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In system theory, the set&up of an appropriate mathematical&physical representation of the transfer function of a dynamic system is called system identification System identification can be achieved, if the inputs as well as the output signals are available

as measured quantities

System identification, which is used to describe how a dynamic system behaves based

on measured data, consists of several steps As in structural health monitoring, the first step is to design an experiment to obtain input, output and noise data from accelerometers located on the structure After obtaining the data, it must be polished and filtered to select only the useful, high frequency ranges of data To do this, non&parametric methods are used, which include transfer functions, coherence functions and spectral analysis After selecting an appropriate range of data, parametric identification is used to build the mathematical models Parametric models are based

on ordinary or partial differential equations that describe the dynamic system

System identification is a technique to build the transfer functions, in other words, the mathematical models of dynamical systems, or obtain the unknown physical parameters from observed input and output measurements of the systems It is a well&established field with a number of approaches, which can be classified into the prediction error method, the subspace approaches, and the nonparametric correlation and spectral analysis methods [29]

The classical system identification methods as prediction error method, minimizes a performance function which is based on the sum of squared errors These methods work well in many cases However, for complex systems characterized by: being of high order i.e having many parameters, with several inputs and outputs, and having a large amount of measurements, the classical methods have several problems, such as: many local minima in the performance function and thereby a lack of convergence to global minima, complicated parameterization which the user has to specify, potential problems with numerical instability, and large computation time to execute the

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iterative numerical minimization methods needed The new subspace method solves most of these problems Another advantage with this method is that it uses a general state space model, which covers any linear MIMO system

The newly proposed subspace identification techniques essentially find a state sequence, or a column space approximation, and then determine the system matrices

by solving some least squares problems These techniques have promising advantages over the classical ones One advantage is that there is no need for parameterizations, which are difficult to choose or analyze or could lead to an ill&conditioned solution for direct methods This is especially true for high&order multivariable systems, for which

it is not trivial to find a useful parameterization among all possible parameterizations Error prediction methods need a priori knowledge of the order and of the observability indices For subspace identifications only the order of the system is needed and it can

be determined through inspection of the dominant singular values of a matrix that is calculated during the identification The state space matrices are not calculated in their canonical forms with a minimal number of parameters, but as full state space matrices

in a certain, almost optimally conditioned basis This implies that the observability indices do not have to be known in advance

Mathematical models of natural and man&made systems play an essential role in today's science and technology The applications of models range from simulation and prediction to control and diagnosis in heterogeneous fields such as all branches of engineering, economics, medicine, physiology, geophysics, and many others It is therefore natural to pose the question where mathematical models come from If we depict a model as a box containing the mathematical laws that link the inputs (causes) with the outputs (effects), the three main modeling approaches can be associated&with the "color" of the box [21]

The model is derived directly from some first principles by taking into account the connection between the components of the system Typical

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examples are found in mechanical and electrical systems where the physical laws (F =

ma, for instance) can be used to predict the effects given the causes Rather than white, the box should be termed "transparent", in the sense that we know the internal structure

of the system

Sometimes the model obtained by invoking the first principles is incomplete because the value of some parameter is missing For instance, a planet is subject to the gravitation law but its mass is unknown In this case, it is necessary to collect experimental data and proceed to a tuning of the unknown parameters until the outputs predicted by the model match the observed data The internal structure of the box is only partially known (there are grey zones)

there are no first principles available, the only chance is to collect data and use them to guess the links between inputs and outputs For instance, this is a common situation in economics and physiology However, black box modeling is also useful to deal with very complex systems where the white box approach would be time consuming and expensive Black box models are flexible mathematical structures that are not based on first principles You can estimate both nonparametric and parametric black&box models

In case of parametric models, the data helps you to both select the best model structure and estimate the parameters

System identification is concerned with the development and analysis of methods for performing grey and black box modeling Differently from white&box modeling that is intimately related to the specific knowledge domain, system identification covers a number of methodological issues that arise whenever data are processed to obtain a quantitative model

System identification is a technique to build the transfer functions, in other words, the mathematical models of dynamical systems, or obtain the unknown physical

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parameters from observed input and output measurements of the systems

A model is a tool used to answer questions about the system without having to perform experiments For example, a model is using to simulate the output of a system for a given input and analyze the system response Alternatively, it might be interested in using the model to predict future output of a system based on previous inputs and outputs

# A dynamic system with input u(t), output y(t) and disturbance e(t) [29] Models describe the relationship between one or more measured input signals, , and one or more measured output signals, Input and output signals can be measured in the time or frequency domain

In real systems, there are additional inputs that affect the system output and that you cannot measure or control Such unmeasured inputs are called or , e(t) The output y(t) describes how the system reacts or responds to the applied input and disturbance Therefore, the output will be a mixture of dynamic response of the system and characteristics of the input and disturbance as well In general, the input at previous time instances will also affect the current output In other words, the dynamic system has memory

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The nonparametric models are described by curves, functional relationships or tables They are identified by some methods, such as: transient response models and transfer function models They are called nonparametric system identification methods They

do not involve direct estimation of physical or mathematical model parameters

Nonparametric models are black&box models that consist of data tables representing the impulse response, step response, and frequency response of the system Because nonparametric models are not represented by a compact mathematical formula with adjustable parameters, such models do not impose a specific mathematical structure on your system

Nonparametric models serve well as preliminary models that you can use to analyze system characteristics For example, estimating the transient response provides insight into the rise time and settling time of the system response Similarly, estimating frequency response might indicate the order of the system, locations of resonances and notches, crossover frequencies, and the bandwidth of the system

Nonparametric models can be estimated using the following methods:

• is applied when the system response is transient, i.e generated on the basis of impulse or step excitation The dynamic behavior of the system is then identified on the basis of the impulse or step response

• Correlation and spectral analysis are methods that are applied to a stationary stochastically excited system In these cases, the excitation and the system response can be characterized either by the correlation functions in time domain or the spectral densities in frequency domain Having estimated the correlation functions of the excitation and the response the impulse response function of the system can be obtained On the other hand, if the spectral