Dynamic analyses of hollow core slabs: Experimental and numerical analyses of an existing floor

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DEGREE PROJECT IN

CIVIL ENGINEERING AND URBAN MANAGEMENT SECOND CYCLE, 30 CREDITS

STOCKHOLM, SWEDEN 2020

Dynamic analyses of hollow core slabs

Experimental and numerical analyses of an existing floor

MARKUS HANSELL

PANAGIOTIS TAMTAKOS

KTH ROYAL INSTITUTE OF TECHNOLOGY

SCHOOL OF ARCHITECTURE AND THE BUILT ENVIRONMENT DEPARTMENT OF CIVIL AND ARCHITECTURAL ENGINEERING

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Dynamic analyses of hollow core slabs – Experimental and numerical analyses of an existing floor Dynamiska analyser av håldäcksbjälklag – Experimentell och numerisk analys av ett befintligt golv

Handledare: Lisa Sparf, ELU Konsult AB, och Jean-Marc Battini, KTH Examinator: Costin Pacoste-Calmanovici

Master Thesis, 2020

Royal Institute of Technology, KTH

School of Architecture and the Built Environment Department of Civil and Architectural Engineering Division of Structural Engineering and Bridges

AF223X Degree Project in Structural Engineering and Bridges SE-100 44, Stockholm, Sweden

TRITA-ABE-MBT-20209

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Abstract

For intermediate floors in residential and office buildings, as well as in parking garages and malls, there is a wide use of hollow core concrete slabs in Sweden today. Hollow core slabs are precast and prestressed concrete elements with cylindrical-shaped voids extending along the length of the slab.

These structural elements have the advantage compared to cast-in-situ concrete slabs that they have a high strength, due to the prestressing, and that the voids allow for a lower self-weight. Additionally, the voids allow for a reduction in the use of concrete material. These characteristics offer possibilities to build long-span floors with slender designs. However, a consequence of the slenderness of the slabs is that such floors have an increased sensitivity to vibrations induced by various dynamic loads. In residential and office buildings vibrations are primarily caused by human activity, and therefore concerns related to the serviceability of such floors are raised. These vibrations are often not related to problems with structural integrity, but rather to different aspects of comfort of the residents or

workers.

The aim of this thesis is to provide additional information regarding the dynamic behavior of hollow core floors. An experimental modal analysis has been performed on an existing floor in an office building. The dynamic properties in the form of natural frequencies, mode shapes, damping ratios and frequency response functions were derived and analyzed from these measurements. Subsequently, several finite element models were developed, aiming to reproduce the experimental dynamic behavior of the studied floor.

The measurements initially showed some unexpected dynamic responses of the floor. For this reason, more advanced methods of signal analyses were applied to the data. The analyses showed that the slab has some closely spaced modes and that the modes of the floor are complex to a certain degree.

The finite element models were studied with different configurations. In particular, the effect the model size, boundary conditions, material properties and potential structural discontinuities have on the dynamic response of the slab was studied. Sufficiently good agreement has been achieved between the experimental and numerical results in terms of natural frequencies and mode shapes. The

acceleration amplitude responses of the numerical models were generally higher than the ones obtained from the measurements, which leads to difficulties in matching of the frequency response functions.

Keywords: Hollow core slab, Structural dynamics, Signal analysis, Mode shape, Frequency response function, Complex mode, FE-model.

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Sammanfattning

Håldäck i betong används idag i stor utsträckning som bjälklag i bostads- och kontorsbyggnader, liksom i parkeringsgarage och köpcentra. Håldäcksbjälklag består av prefabricerade och förspända betongelement, med cylindriska hål som sträcker sig i plattans längsriktning. Dessa

konstruktionselement har fördelen, jämfört med platsgjutna betongplattor, att de har en hög hållfasthet på grund av förspänningen och att hålen möjliggör en lägre egenvikt. Dessutom gör hålen att en mindre mängd betongmaterial behövs. Dessa egenskaper ger möjligheter att bygga golv med långa spännvidder och slank design. En konsekvens av slankheten är emellertid att sådana golv har en ökad känslighet för vibrationer som orsakas av olika dynamiska belastningar. I bostads- och

kontorsbyggnader orsakas vibrationer främst av mänsklig aktivitet, och därför finns det en del oro relaterad till sådana golvs brukbarhet. Dessa vibrationer är oftast inte relaterade till frågor om strukturell integritet, utan snarare till olika aspekter av boendes eller arbetares känsla av komfort.

Syftet med detta examensarbete är att bidra till kunskapen om håldäcksbjälklags dynamiska beteende.

En experimentell modalanalys har utförts på ett befintligt golv i en kontorsbyggnad. De dynamiska egenskaperna i form av egenfrekvenser, modformer, dämpning och frekvenssvarsfunktioner erhölls och analyserades med hjälp av dessa mätningar. Därefter utvecklades flera finita element modeller för att reproducera det experimentellt uppmätta dynamiska beteendet hos det studerade golvet.

Mätningarna visade initialt något oväntade dynamiska responser från golvet. Av denna anledning applicerades mer avancerade signalanalysmetoder på datan. Analyserna visade att plattan har några moder inom ett litet frekvensintervall och att moderna till en viss grad är komplexa.

De finita element modellerna studerades med olika konfigurationer. I synnerhet studerades effekten av modellstorleken, randvillkoren, materialegenskaperna och potentiella strukturella diskontinuiteter på golvets dynamiska respons. Tillräckligt bra överensstämmelse har uppnåtts mellan de experimentella och numeriska resultaten i form av egenfrekvenser och modformer. Accelerationsamplituderna för de numeriska modellerna var i allmänhet högre än de som erhölls under mätningarna, vilket leder till svårigheter att matcha frekvenssvarsfunktionerna.

Nyckelord: Håldäcksbjälklag, Strukturdynamik, Signalanalys, Modform, Frekvenssvar, Komplex mod, FE-modellering.

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Preface

This is a master thesis work performed at the Division of Structural Engineering and Bridges at the Department of Civil and Architectural Engineering at the Royal Institute of Technology, KTH, during the spring of 2020. The thesis has been written in collaboration with ELU Konsult AB and the subject of this thesis was proposed by Costin Pacoste-Calmanovici, professor at KTH and head of Research and Development at ELU Konsult AB, and Jean-Marc Battini, professor at KTH. This master thesis aspires to increase the general knowledge of the dynamic behavior of hollow core slabs in existing buildings and aims to provide guidance in finite element modeling of floors consisting of hollow core elements.

We would like to direct a profound thank you to our supervisors, Lisa Sparf at ELU and Jean-Marc Battini and Costin Pacoste-Calmanovici at KTH, for their constant support and guidance throughout the entire project. They always took the time to answer questions and gave constructive feedback on all of our ideas and thoughts. We would also like to thank Freddie Theland, doctoral student at KTH, for discussions and advice on how to interpret the experimental measurements and how to relate them to the theory of structural dynamics. We thank Fangzhou Liu, PhD student at KTH, for performing the experimental tests that lay the foundation for the analyses performed in this thesis and for assisting us in our work. Furthermore, we direct a thank you to Andreas Andersson, researcher at KTH, for assisting us with coding in Matlab. At ELU Konsult AB, we would like to thank Abbas Zangeneh for advice about the finite element modeling. Additionally, we would like to thank Akademiska Hus for providing us with the relevant drawings needed to understand and visualize the building under study.

Lastly, we would like to thank all the other students at KTH that performed their master thesis projects at ELU during this extraordinary period, for all the laughs and discussions during coffee and lunch breaks.

Stockholm, June 2020

Markus Hansell Panagiotis Tamtakos

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Abbreviations

CMIF - Complex Mode Indicator Function DFT - Discrete Fourier Transform

DOF - Degree of freedom

EMA - Experimental Modal Analysis FE - Finite Element

FEM - Finite Element Method FFT - Fast Fourier Transform FRF - Frequency Response Function IFFT - Inverse Fast Fourier Transform MDOF - Multi-degree-of-freedom MIF - Mode Indicator Function

MIMO - Multiple-Input-Multiple-Output MPC - Modal Phase Collinearity

SDOF - Single-degree-of-freedom

List of symbols

Greek letters

𝛼_𝑖 – Coefficient for calculation of modal phase collinearity 𝛾²(𝑓) – Coherence function as a function of frequency 𝜀_𝑖 – Coefficient for calculation of modal phase collinearity 𝜁 – Damping ratio [-]

𝜌 – Density [kg/m³] 𝜐 – Poisson’s ratio [-]

𝜱_𝑖 – Modal shape vector of the i:th mode 𝜔_𝑛 – Natural circular frequency [rad/s]

Latin letters

b – Width of a cross section [m]

c – Damping [Ns/m]

ccr – Critical damping C – Damping matrix

ei – Coefficient for calculating modulus of elasticity in i-direction

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12 Ei – Modulus of elasticity in i-direction [Pa]

fn – Natural frequency [Hz]

F(t) – Dynamic force vector as a function of time

Fd – Damping force [N]

Fk – Spring force [N]

gij – Coefficient for calculating shear modulus Gij – Shear modulus [Pa]

GXX(f) – Auto power spectrum of an input signal GYX(f) – Cross power spectrum

GYY(f) – Auto power spectrum of an output signal h – Height of a cross section [m]

heq – Equivalent height [m]

h(t) – Impulse response function H(f) – Frequency response function 𝐻̂₁(𝑓) – Estimated transfer function HR(f) – Real part of transfer function

I – Area moment of inertia, about neutral axis [m⁴] k – Spring stiffness [N/m]

K – Stiffness matrix

m – Mass [kg]

mtot – Total mass per meter [kg/m]

M – Mass matrix

p(t) – Dynamic force as a function of time Rd – Dynamic magnification factor

SX(f) – Linear spectrum of an input signal SY(f) – Linear spectrum of an output signal Tn – Natural period [s]

𝑢 – Displacement [m]

𝒖 – Displacement vector 𝑢̇ – Velocity [m/s]

𝒖̇ – Velocity vector 𝑢̈ – Acceleration [m/s²] 𝒖̈ – Acceleration vector

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13 x(t) – Input signal as a function of time

X(f) – Input signal as a function of frequency y(t) – Output signal as a function of time Y(f) – Output signal as a function of frequency

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1. Introduction

1.1 Background

Hollow core slabs are precast and prestressed concrete elements which nowadays are increasingly used for the construction of floors in various kinds of contemporary structures, such as multi-story residential buildings, offices or shopping malls. The use of hollow core slabs is associated with several benefits, such as the possibility to build long span floors and enable large open spaces, which is increasingly desirable. In particular, the existence of longitudinal voids along the length of the elements in conjunction with the prestressing applied to them, allows these elements to have increased loading capacity while simultaneously maintaining low self-weight. However, due to the

characteristics of these elements, they sometimes appear to be susceptible to serviceability problems, such as vibrations caused by human activity [1]. The first natural frequencies of such floors may coincide with the step frequencies induced by people walking or running over the surfaces of the floor, causing resonance. Despite that human induced vibrations do not appear to affect the safety of the structure, they can often cause comfort related problems to the people living or working in the building.

Due to the complexity of real structures and these elements, it is desirable for engineers in practice to be able to perform dynamic finite element (FE) analyses in the design process, in order to be able to predict the dynamic behavior of the structure to be built with sufficient precision. However, there is limited information and guidelines concerning the modeling of buildings with such structural elements for such analyses in the literature. In particular, the uncertainties related to the modeling choices, as e.g. boundary conditions, connections and material properties, of hollow core floors remain significant. Due to the voids in the element, hollow core slabs have different stiffness properties in the different principal directions. Additionally, insufficient bonding in joints and between the slab and the topping may cause dynamic responses of the slabs that are hard to predict with mathematical models. It is furthermore not well established how large portions of the structure that is necessary to model to sufficiently capture the dynamic behavior of such floors [25]. Much of the previous research has been devoted to numerical analyses of hollow core slabs, but without extensive experimental confirmation.

In contemporary engineering, the importance of having finite element models that are reliable and easy to produce is high. Unreliable models can lead to either overestimation or underestimation of the dynamic properties of real structures, which is something that can result either to uneconomical design or structures with vibration related serviceability problems. Therefore, in an effort to contribute to the reliability of the models produced for the aforementioned reasons, the effect of different

modeling choices in the modeling of hollow core floors has been investigated in this thesis.

1.2 Aim and Purpose

The aim of this thesis is to contribute to an increased knowledge of the dynamic behavior of hollow core slabs. More specifically, this is done by a case study of a floor in an existing building, rather than an experimental structure. The work is focused towards frequency domain analyses of the response of the floor, as well as towards the investigation of the obtained complex modes. Additionally, computer models are developed for the purpose of comparing the properties of the real structure with

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mathematical models, with the intention to provide information needed to predict the future dynamic behavior of such slabs.

1.3 Methods

The initial work of the thesis included an extensive literature survey, with the purpose of creating a theoretical background for the following experimental and numerical analyses. The survey included a study of research papers, both within the discipline of civil engineering and other fields of

engineering, common books of theory, design codes and design recommendations. This included research papers written in both Swedish and English and codes from both Europe and the US.

Once the theoretical background had been established, the main part of the thesis work was related to performing analyses of an existing office and educational building at campus Albano in Stockholm, Sweden. It was a case study, for which both experimental and numerical analyses were performed.

Experimental measurements on the dynamic response of a floor in the building had been previously performed, meaning that the main part of the analytical work was related to the post-processing and interpretation of the collected data in the software Matlab. The data was collected by nineteen accelerometers which were placed over a portion of the slab, before performing three different dynamic tests to excite the structure and study its response. The different tests included imposing impulse forces produced by an impact hammer, a jump from a test subject and a heel drop from a test subject. These tests were repeated several times, in order to increase the confidence of the results and to reduce potential errors.

The numerical analyses of the slab were performed through modeling in the finite element software Brigade Plus. The modeling choices and checks were related to and based on the knowledge gained through the literature survey. Different solutions and modification of these choices were tried

throughout the work process. Ultimately, the models were developed for the purpose of comparing the numerical and experimental results. This was done by comparing natural frequencies, mode shapes and frequency domain responses of the two analyses.

1.4 Limitations

The master thesis is performed during one semester and is hence limited in both time and length. The experimental tests and the corresponding finite element models are related to one specific building.

The previously obtained measurements from the existing building are limited to the points of the structure in which the accelerometers were placed and the number of dynamic tests performed are limited due to the time it takes to perform such tests. The number of analyses related to the

measurements and the finite element models are limited to those that are regarded to be most relevant.

The presented data is related to what is needed to perform these analyses. Animations and deformations are limited to be visualized in the format of pictures.

1.5 Disposition

The thesis is divided into several chapters and subchapters, reflecting the work process that has been conducted. Chapter 2, 3, 4 and 5 are aimed to provide a theoretical background and present the general outline of the methods that has been employed during the work. Chapter 6 and 7 are devoted

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to the analyses, models and findings related to the case study. The contents of the respective chapter are shortly summarized below.

Chapter 1 - Introduction

Presents the background and states the problem treated in the thesis. The general aim, outline of the methods and limitations are described.

Chapter 2 - Hollow core slabs

Shortly presents information related to the structural use of hollow core slabs.

Chapter 3 - Fundamental structural dynamics

Gives an introduction to basic structural dynamics theory, often taught in university level courses on the subject. The chapter presents fundamental equations, phrases and principles that are needed as a background to understand the remaining contents of the thesis.

Chapter 4 - Dynamic signal analysis

Describes more advanced theory related to structural dynamics. This chapter describes equations and methods commonly used when performing more advanced dynamic response analyses of structures.

The presented methods provide the methodological background for the signal analyses that have been performed with the experimental data presented in chapter 6.

Chapter 5 - Literature review

Presents the previous research and available guidelines that a structural engineer can use for reference in the design of hollow core concrete floors. The chapter is divided to treat the research related to the general behavior of hollow core slabs and the modeling of such slabs in a finite element software separately. Finally, recommendations found in established design norms are summarized.

Chapter 6 - Experimental modal testing on an existing building

Describes the measurements performed on the presented building. First, the floor is described from a structural point of view. Secondly, the measurement setup is presented. Lastly, the results derived from response analyses are presented and discussed. The general observations and comments of the authors are given, in context with findings from previous research.

Chapter 7 - Finite element models

Describes and discusses the choices and findings from the finite element model of the structure under study. Presents the findings from model modification and simplifications and relates it to the

measurement results presented in the previous chapter.

Chapter 8 - Discussion and conclusion

Discusses the findings from the two types of analyses and contains comments of the conclusions that can be drawn from the presented results.

Appendix A - Drawings

Contains the relevant drawings that are needed to visualize and understand the structure under study.

Appendix B - Experimental results

Contains the remaining parts of the results from the analyses performed on the experimentally obtained data that does not fit in chapter 6.

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Contains checks and results from the finite element models that are of interest but that does not fit in chapter 7.

Appendix D - Matlab code

Contains the Matlab code developed and used to perform the analyses and to derive the results presented in the thesis.

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2. Hollow core concrete slabs

2.1 General information

Hollow core slabs are precast, prestressed concrete elements featuring continuous voids in their longitudinal direction, aiming to reduce the weight and subsequently the cost of concrete floors. In particular, the combination of high strength and relatively low self-weight gives possibilities for design of floors with long spans. As an additional benefit, the voids can be used to conceal electrical or mechanical runs. However, due to their slender characteristics, hollow core elements might be susceptible to human-induced vibrations, affecting the serviceability of the buildings which they are used. Generally, hollow core slabs are mainly used for the construction of floor or roof deck systems but they have additional applications as both vertical and horizontal wall panels, bridge deck slabs, and spandrel members.

Figure 2.1. Hollow core element [1].

There are two common manufacturing methods used in the production process of hollow core slabs.

The first one is with the use of an extrusion system where the element is formed by forcing the concrete through a casting machine. The formation of the cores through this process is achieved by using tubes and different methods of vibration and compaction, in such a way that the concrete consolidates around the cores. The alternative production method is to use wet-cast concrete, with a higher water-cement ratio, as opposed to the previously described dry-cast process. This includes forming the elements by using forms and tubes of different sorts. When the concrete has cured the elements can be cut, shaped and prestressed according to the desired application [2].

In practice, there are several cross sections with standardized geometry that are regularly used. A nominal width of 1200 mm (including the grouted concrete joints between the slabs) and thicknesses ranging from 200 mm to 400 mm are common dimensions used in Sweden [3]. An example of a cross section is shown in Figure 2.2.

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Figure 2.2. Cross section and dimensions of a commonly used hollow core element, HD/F 120/32 [3].

2.2 Structural use

The most limiting factor in the design of floors with hollow core slabs is the span. Depending on the design load and other demands, e.g. the fire resistance capacity, the span and thickness of the element can be optimized. The two main design parameters are the flexural and shear capacity of the slab.

Openings and other discontinuities in the floor plan reduce the load bearing capacity. Typically, manufacturers supply tables including allowable span-to-thickness combinations for given static loads. When using grouted joints to connect hollow core elements in the lengthwise direction, the system tends to structurally behave more similar to a rigid slab, allowing loads to transfer between the slabs in the weaker direction.

When the hollow core elements are properly coordinated and aligned, their voids can be utilized for installations. Additionally, hollow-core slabs can be used as the thermal mass in a passive solar application because they can be used for the distribution of heated air through their cores. Hollow core slabs are also associated with high performance in terms of fire resistance and sound insulation.

Because of the prestressing, the element will tend to deflect and bend upwards after installation. It is thus common practice to cast in-situ concrete on top of the precast hollow core slabs in order to create an even floor surface. Additionally, because of the deflections, the thickness of the topping varies, by being thinner at midspan and increasing towards supports [2]. This technical solution also creates some composite action between the topping and the hollow core element, provided that the bond at the interface is sufficient, which further increases the bearing capacity of the slab [4]. For design of connections to walls and beams, it is common to use a combination of reinforcement and grouted concrete between the hollow core slab and the other structural member. Examples of technical solutions and design considerations can be found in design manuals such as the PCI Manual for the Design of Hollow Core Slabs and Walls (2015) [2].

However, while some problems related to vibrations and acoustics are addressed in such design manuals, there are limitations in how to treat potential vibration serviceability issues that may arise from these slender floor solutions. Historically, disturbing vibrations has not been a major issue, as older buildings have been more sturdy and heavier. Such discomforts can arise when the structure is subjected to dynamic loads from e.g. rhythmic activities, such as gym classes, rotating equipment or human walking. Older methods of dynamic analyses for large and structurally advanced structures may sometimes be insufficient. This is something that has gained increased attention over the past decades.

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3. Fundamental structural dynamics

3.1 Introduction

The purpose of the following section is to provide a theoretical background regarding fundamental structural dynamics. The theory presented in this chapter puts focus on linear systems, by considering a set of assumptions, and does not treat any theory related to nonlinearities. These assumptions are valid for low magnitude oscillations, which is usually the case for structural vibrations in buildings caused by common loads.

3.2 Equations of motion

Civil engineering structures can be ideally represented as dynamic systems which are connected to each other with springs and dashpots [5]. One of the most simplified methods to describe a dynamic system is through a single-degree-of-freedom (SDOF) system. In Figure 3.1 an example of a SDOF dynamic system is shown. More specifically, the system consists of a mass m which is connected to a wall with a spring that has stiffness k and a viscous damper with damping coefficient c. The mass can only move in one, single, direction. Furthermore, the object is subjected to an dynamic external force that varies with time, p(t), and therefore, due to the presence of the spring and the damper, resisting forces acting in the opposite direction are developed as shown in Figure 3.1. The resisting force developed by the spring, according to Hooke’s law, is linearly related to the displacement and therefore can be expressed by:

𝐹_𝑘 = 𝑘 ∙ 𝑢 Equation 3.1

in which k is the spring stiffness [N/m] and u is the displacement [m].

On the other hand, vibrations on real structures are associated with some level of damping which make the free vibrations to steadily diminish in amplitude due to different mechanisms leading to a dissipation of energy. This energy dissipation can be modeled with a viscous damper [5]. The

damping force therefore is proportional to the velocity across the damper and can be expressed by the following equation:

𝐹_𝑑 = 𝑐 ∙ 𝑢̇ Equation 3.2

in which c is the spring coefficient [Ns/m] and 𝑢̇ is the velocity [m/s].

With the use of Newton’s second law of motion, the equation of motion of a SDOF mass-spring- damper system can be written as:

𝑚 ∙𝑢̈ + 𝑐 ∙ 𝑢̇ +𝑘 ∙ 𝑢 = 𝑝(𝑡) Equation 3.3

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Figure 3.1. Forces developing on a mass subjected to an external dynamic force [5].

where m is the mass of the system [kg] and 𝑢̈ is the acceleration [m/s²].

However, a more realistic way to describe the dynamic system of real buildings is through Multi- degree-of-freedom (MDOF) systems [5, 6]. A simple example of a MDOF system is depicted in Figure 3.2.

Figure 3.2. Example of a simple MDOF system [6].

As it can been seen in Figure 3.2 the system consists of multiple masses, dampers and springs and hence, the equation of motion will be a system of equations which can be written in the form of matrices and vectors:

𝑴∙ 𝒖̈ + 𝑪 ∙ 𝒖̇ +𝑲∙𝒖 = 𝑭(𝒕)

Equation 3.4

in which M is the mass matrix, C is the damping matrix, K is the stiffness matrix. 𝒖̈, 𝒖̇ and 𝒖 are the acceleration, velocity and displacement vectors respectively, while F(t) contains the external dynamic forces applied on the masses.

In real cases the equation of motion for a MDOF system can be hard to solve analytically by hand.

Non-linearities of the system or complex variations of the force might lead to such difficulties.

Additionally, real MDOF systems can be very large. Consequently, numerical methods, such as the Newmark’s method or the Central Difference method, that use time derivatives for each time step have been developed to solve these equations. Finite element methods have also gained increased popularity over the past decades, for solving of large and complex systems.

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3.3 Natural frequency

All structures, when subjected to some external dynamic force, tend to vibrate at a particular

frequency or set of frequencies. This frequency (or frequencies) is called the natural frequency of the system, which is a property of the structure dependent on the mass and the stiffness. For a SDOF system the natural frequency can be calculated with the use of the following equations:

𝜔_𝑛 = √_𝑚^𝑘 Equation 3.5

𝑇_𝑛=^2∗𝜋

𝜔_𝑛 Equation 3.6

𝑓_𝑛= ¹

𝑇_𝑛= ^𝜔^𝑛

2∗𝜋 Equation 3.7

where:

𝜔_𝑛 is the natural circular frequency [rad/s]

𝑇_𝑛is the natural period [s]

𝑓_𝑛 is the natural frequency [Hz]

Typically, the number of natural frequencies of a dynamic system is equal to its degrees of freedom [5].

3.4 Damping

As previously mentioned, some level of damping is always present in real structures. This

means that when the structure is put in motion, it will not oscillate endlessly, and the amplitude of the motion will decay over time and eventually lead to the structure going back to rest. This phenomenon is caused by energy dissipation. Typical mechanisms in real buildings that are responsible for energy dissipation is internal material friction, friction at the connections between different structural or non- structural elements and the opening and closing of small cracks in concrete [6].

The damping is normally expressed through the damping ratio, ζ, which is a dimensionless measure used to describe how oscillations in a dynamic system decay after an excitation. When the damping ratio is equal to unity (ζ=1) the system is critically damped, which means that the structure has exactly the amount of damping needed for the vibrating system to return into its equilibrium without

oscillating. When the damping ratio is lower than the one for critical damping then the dynamic system is underdamped, while for damping ratios higher than critical the dynamic system is characterized as overdamped [5].

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Figure 3.3 depicts the effect that different damping ratio values have in the oscillation of a dynamic system.

Figure 3.3. Free vibration of systems with different damping ratios [5].

The amount of damping in real building and structural elements rarely exceeds 10% and is thus characterized as being underdamped. The damping ratio of a SDOF system can be calculated using equation 3.8 [5].

𝜁 = ^𝑐

𝑐_𝑐𝑟= ^𝑐

2∗𝑚∗𝜔_𝑛 Equation 3.8

However, the values of damping used in design are often based on empirical data from measurements, since the many mechanisms that affects it are complicated, making it hard to predict from a design point of view. There are many different factors that may influence damping in real structures and it is hence complicated to form a mathematical expression which is true for all structures.

3.5 Resonance

Resonance in structures occurs when the frequency of a dynamic load coincides with the natural frequency of the structure, causing high magnitude responses of the system. For an undamped dynamic system the amplitude of vibration at resonance can grow towards infinity, but this is not the case in real structures, as some level of damping is always present. However, even in real structures, the effect of resonance can amplify the amplitudes of vibrations to a degree that might cause several different problems such as human discomfort, extensive deformations and material fatigue.

The effect of resonance can be expressed through the dynamic response factor, R^d. This can be regarded as a magnification factor, which amplifies the structural response as the frequency of the dynamic load approaches the natural frequency of the system [6]. This factor as well as how it is influenced by the damping ratio is illustrated in Figure 3.4. The more damping that is present in the system, the lower the magnification factor.

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Figure 3.4. Dynamic response factor as a function of dynamic load frequency ratio and damping [6].

3.6 Mode shapes

A mode shape of a system that vibrates is described as a pattern of motion where all the parts of the vibrating system are moving with the same frequency and with a stable phase relation. When a structure is vibrating at or near its natural frequency, the overall vibration shape of a structure will tend to be dominated by the mode shape of resonance [5]. A simple example of mode shapes in structural members is the one of a simply supported beam since its developed modes shapes tend to be similar to the shapes of sinusoidal waves, as illustrated in Figure 3.5. In general, the geometrical properties of a structure as well as its boundary conditions might affect to a great extent its mode shapes. For example, clamped structures behave differently than simple supported ones since such structures are more restrained, resulting in higher natural frequencies. For structural members extending in several directions, these shapes can be complex and described by motions in more than one direction.

Figure 3.5. The first three mode shapes of a simply supported beam [37].

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4. Dynamic signal analysis

4.1 Introduction

When assessing the dynamic properties of existing buildings through experimental testing it is common to use accelerometers and an exciting force produced by e.g. an impact hammer, a shaker or a jumping human. This yields time-record data of the response of the structure at the locations of the accelerometers. By carefully choosing the locations of the accelerometers and the exciting force, the natural frequencies, damping and mode shapes can be determined.

The most intuitive way to evaluate the dynamic behavior of a structure is to perform analyses in the frequency domain, rather than in the time domain. This is normally done by transforming the recorded time domain data of the accelerometers and the exciting force through Fast Fourier Transforms (FFT) and can be done through built in algorithms in software like Matlab. Through signal analyses of such data, it is possible to extract the previously mentioned dynamic properties, e.g. natural frequencies and mode shapes, of structural systems.

A linear and time invariant system is, according to dynamics theory, defined as a system that can be described by linear differential equations, which has constant properties over time. These assumptions generally hold for cases when the structure has small oscillations around its equilibrium. An input yields an output and the relation between the two are described by a response function, as illustrated in Figure 4.1. The input, x(t) or X(f), and the output, y(t) or Y(f), are in the time domain linked by an impulse response function, h(t), or by the frequency response function, H(f), in the frequency domain [7].

Figure 4.1. The relation between input and output in the time and frequency domain for linear time invariant systems [7].

The advantage of using a force source for which the input can be recorded, as in the case of a shaker or an impact hammer, is that more refined analyses of the structural response can be performed, as discussed below.

4.2 The Fast Fourier Transform

For the purpose of analyzing responses of structures subjected to dynamic loads it is fundamental to understand the relation between the time and the frequency domain, and how the data is transformed from one to the other. For thorough explanations and mathematical proofs of the theory relating the two domains, the reader is referred to textbooks as [7] or [5]. Basic concepts are presented below, without any in-depth explanations.

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Time domain data can be transferred to the frequency domain by the Fourier transform, which in continuous form is described by the integral:

𝑋(𝑓) = ∫_−∞^∞ 𝑥(𝑡) 𝑒𝑥𝑝(−𝑖2𝜋𝑓𝑡) 𝑑𝑡 Equation 4.1

Equivalently, frequency domain data can be transferred to the time domain by the inverse Fourier transform integral:

𝑥(𝑡) = ∫_−∞^∞ 𝑋(𝑓) 𝑒𝑥𝑝(𝑖2𝜋𝑓𝑡) 𝑑𝑓 Equation 4.2

These integrals can alternatively be described by sines and cosines and notably the transform contains both real and imaginary parts.

When collecting data in the field, measurements are not done continuously over time, but rather as discrete samples with (short) time steps between them. This means that a recorded signal instead could be defined as x(n)=x(n𝛥t) in the time domain and that the discrete Fourier transform, DFT, of the signal is written as X(k)=X(k𝛥f). A consequence of this discontinuity is thus that the Fourier transform integrals need to be described by sums:

𝑋(𝑘) = ∑^𝑁−1_𝑛=0𝑥(𝑛) 𝑒𝑥𝑝(−𝑖2𝜋𝑘𝑛/𝑁) Equation 4.3

𝑥(𝑛) = ¹

𝑁 ∑^𝑁−1_𝑛=0𝑋(𝑘) 𝑒𝑥𝑝(𝑖2𝜋𝑛𝑘/𝑁) Equation 4.4

which is the forward and inverse DFT respectively, for k=0, 1…, N-1 and n=0, 1... , N-1. As in the case of the Fourier integrals, these equations can alternatively be described by sums of sines and cosines, and contains both real and imaginary parts [7].

The algorithm referred to as the Fast Fourier Transform (FFT) computes Eq. 4.4 in an effective way, with less steps than the direct DFT, saving significant computational time. The definition holds both recorded samples of the input (force) and the output (structural response). As previously mentioned, this function is found in common software, and in Matlab the fft command performs this task. The inverse Fast Fourier Transform (IFFT) can in a similar manner be computed by the ifft command.

4.3 Frequency Response

The relation between the input and the output of a system in the frequency domain is described by the frequency response function (FRF), which is defined as:

𝐻(𝑓) = _𝑋(𝑓)^𝑌(𝑓) Equation 4.5

in which X(f) is the input and Y(f) is the output. The FRF contains both real and imaginary parts, and its magnitude can be regarded as a ratio of the response amplitude and force. It is common practice to plot the FRF either in terms of magnitude and phase, or as real and imaginary parts, against the frequency. An example of such plots for a SDOF system, is shown in Figure 4.2. Depending on the type of measurement (displacement, velocity or acceleration), the unit and characteristics of the FRF varies. If accelerometers are used, the FRF represents the accelerance [(s/m²)/N] of the system.

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Figure 4.2. Example of frequency response function plots for a single-degree-of-freedom system [8].

For linear systems with proportional damping, when the measured quantity is displacement or acceleration, the FRF shows peaks at resonant frequencies in the magnitude plot. In the alternative plot of the real and imaginary parts, the real part tend to zero and the imaginary shows minimum or maximum peaks [8]. The opposite is true if the measured quantity is velocity. This theory can be extended to the case of MDOF systems, for which similar plots can be used. Thus, after performing experimental tests, the natural frequencies and damping of the system can be visualized and estimated with such plots.

An advantage of picking the natural frequencies from the imaginary part is that the peaks are often more distinct than the ones in the magnitude plot. This is evident in cases of closely spaced natural frequencies, as described by [7].

4.4 Experimental Frequency Response and Spectral Analyses

When performing measurements in practice it is rather difficult to completely isolate a structure such that it is only affected by the input force of an impact hammer or a shaker. An existing structure is inevitably affected by vibrations of different magnitude from e.g. wind, traffic or people.

Consequently, the measurement equipment will, to a varying degree, record structural responses of such sources, which results in a reduction of the quality of the experimental data. For this reason it is desirable to repeat measurements and perform averaging of results for the purpose of reducing the influence of such external sources, as they are assumed to be random and thus have an average value of zero. In theory, this would provide a better approximation of the true responses of the structure due to the controlled experimental input. The averaging process can be done in either the time domain or the frequency domain, with the latter being the most common [7, 9].

The external, unwanted sources of vibration, hereafter referred to as noise, can in theory be present either in the input x(t), the output y(t) or in both simultaneously. There are methods for reducing the influence of the noise for either of the three cases. In the following section, a method for reducing the noise of the output, through the transfer function, is presented. Furthermore, this method is applied to the experimental measurements presented in chapter 6.

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4.4.1 Auto and Cross Power Spectrum

The spectrum, or more specifically the linear spectrum, of a signal is the Fourier transform of an input or output in the time domain, as described in section 4.2. The linear spectrum of an input and an output is thus given by:

𝑆_𝑥(𝑓) = ∫_−∞^∞ 𝑥(𝑡) 𝑒𝑥𝑝(−𝑖2𝜋𝑓𝑡) 𝑑𝑡 Equation 4.6

𝑆_𝑌(𝑓) = ∫_−∞^∞ 𝑦(𝑡) 𝑒𝑥𝑝(−𝑖2𝜋𝑓𝑡) 𝑑𝑡 Equation 4.7

where the subscripts X and Y are used to denote input and output respectively. In experimental field- testing, these spectrums are simply estimated by the FFT of the input and output signal.

From this, the auto power spectrum of the input signal is defined as:

𝐺_𝑥𝑥(𝑓) = 𝑆_𝑥(𝑓) ⋅ 𝑆_𝑥^∗(𝑓) Equation 4.8

where 𝑆_𝑥^∗(𝑓) is the complex conjugate of the linear spectrum of the input signal 𝑆_𝑥(𝑓). The auto power contains only real values, as the linear spectrum consists of both real and imaginary values and is multiplied with its complex conjugate. It can be regarded as the squared magnitude of the frequency spectrum. The auto power spectrum of the output is calculated in the same manner, using the linear spectrum of the output instead.

The cross power spectrum is obtained in a similar manner, defined as:

𝐺_𝑦𝑥(𝑓) = 𝑆_𝑦(𝑓) ⋅ 𝑆_𝑥^∗(𝑓) Equation 4.9

in which 𝑆_𝑦(𝑓) is the previously defined linear spectrum of the output and 𝑆_𝑥^∗(𝑓) is the complex conjugate of the linear spectrum of the input as before. This function contains both real and imaginary values, meaning that the phase characterizes the relative phase between the input and the output.

Hence, the cross power spectrum can be used to relate the phase of different outputs to a single input [9].

4.4.2 Experimental frequency response through signal averaging

As presented in section 4.3 the frequency response function is given by dividing the frequency domain output by the input. It is expected that some noise is recorded in practical measurements, and for qualitative purposes there is a desire to reduce the influence of such noise. If the noise is expected to be most significant in the output signal, a method which by some authors is referred to as the 𝐻₁- estimator [7], or by other as the transfer function [9], can be used. The authors intend to refer to it as the transfer function. The concept involves averaging of the signals, which in theory eliminates random and uncorrelated noise. It is defined as the fraction of the cross power spectrum and the auto power spectrum:

𝐻̂₁(𝑓) = ^𝐺̂^𝑦𝑥^(𝑓)

𝐺̂_𝑥𝑥(𝑓) Equation 4.10

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In which the hat ^ denotes that the functions are estimates, as in the case of experiments. These are averaged cross power and auto power spectrums and consequently, for a test repeated N times, for which N cross power and auto power spectrums have been obtained, the estimator is calculated by:

𝐻̂₁(𝑓) =

1

𝑁∑^𝑁_𝑖=1𝐺̂𝑦𝑥,𝑖(𝑓) 1

𝑁∑^𝑁_𝑖=1𝐺̂𝑥𝑥,𝑖(𝑓) Equation 4.11

The transfer function contains both real and imaginary parts, following from the definitions of the auto power and cross power spectrums, and is used to determine the experimental frequency response of a structure. The desired outcome is that the plot of the transfer function yields more easily

interpretable data than the individual frequency response functions of each test, which may contain noise. Hence, the natural frequencies, damping ratios and modal properties for determining the experimental mode shapes can be determined by analysis of this function and its peaks [7, 9].

4.4.3 Coherence

It is in experimental environments sensible to get qualitative measurements of obtained results. For recorded signals as described in the sections above and by employing estimations by means of the transfer function, the coherence function is an indicator of the relation between the applied input and the observed output. It is defined by:

𝛾²(𝑓) = ^𝐺^𝑦𝑥^{(𝑓)⋅𝐺}^𝑦𝑥^∗ ^(𝑓)

𝐺𝑥𝑥(𝑓)⋅𝐺𝑦𝑦(𝑓)

Equation 4.12

An alternative way to describe the coherence is that it gives a numerical value that describes the degree of how the two signals are related to each other. A value of one or close to one give indications that there is a high correlation between the input and output signal, meaning for e.g. that the measured structural response is highly related to the force produced by an impact hammer or a shaker, and less so to random noise. The coherence function is applied to averaged signals, as it for one set of input and output signals is equal to one and can be used as a tool to check the repeatability of the performed tests [9].

When analyzing the transfer function it is common to plot the coherence function in the same window, showing potential low coherence of some frequency components of the transfer function.

Frequency peaks with low coherence might be neglected. However, low coherence can also be indications of nonlinearities of the system, or that some other continuously vibrating source is present in the building.

4.5 Experimental Modes

Once the FRF or the transfer function has been obtained from field measurements, they are normally visualized as shown in Figure 4.2. Ideally, the structure on which the tests have been performed yield interpretable frequency responses, in which the magnitude and the phase, or the real and imaginary parts, can be analyzed. At resonant frequencies, the real part of the transfer function is equal to zero and the imaginary part shows peaks, under the assumption that the system is linear and time invariant.

Additionally, the mode shape at a resonant frequency of the structure can be estimated using the imaginary part of the response. When several accelerometers are used over an experimental area, the imaginary part peaks and their corresponding values can be used to visualize such mode shapes [10].

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It is worth noting that real structures are seldom completely linear, causing e.g. the plots of the FRF to not always appear exactly as shown and described above. This can be a result of errors produced by e.g. noise, or due to estimations done in calculations. However, for most cases the approximation of linearity holds after performing checks, some of which will be presented below [7].

4.5.1 Normal modes

For a structure assumed to be linear, the peaks in the transfer function are used to determine the natural frequencies of the structure. At these frequencies the structure is oscillating and forming a shape which is referred to as a mode shape, as previously described in section 3.6. The mode shape at each resonant frequency is commonly visualized by extracting the values of the imaginary part of the transfer function of different accelerometers and by normalizing them. For a linear system, the imaginary part peaks of the different accelerometers occur at the same frequencies, but with different amplitudes, corresponding to the difference in amplitude in the deformed mode shape. This also means that the different points of the structure, at which the accelerometers are placed, oscillate in- phase or completely out-of-phase. In other words, the points reach their maximum or minimum amplitude simultaneously. If the mode shape is visualized by a two or three dimensional plot, this can be done with a simple picture, as e.g. the one shown in Figure 3.5.

4.5.2 Complex modes

The approach discussed so far was based on the concept that structures are associated with

proportional viscous damping which implies the existence of real, or normal, modes. However, this is not always the case for real structures, as more complicated forms of damping which can be described as non-proportional (damping) can be present [8]. In these cases, the obtained mode shapes are considered, generally, complex valued, meaning that the different points of the structure do not oscillate completely in-phase or completely out-of-phase, but rather something in between. In other words, this can be explained physically by imagining that the different points of the structure reach their maximum amplitudes in their oscillation pattern at various times in a way that resembles a traveling wave pattern. Contrary to the proportional damping case discussed in section 4.3, in structures which are associated with non-proportional damping, the imaginary part of the frequency response no longer reaches a maximum or minimum value simultaneously as the real part reaches zero, at resonant frequencies. In experimental tests, this means that there is a difference in phase angles between different accelerometers. A consequence of the fact that the points reach their maximum amplitude at different times is that the mode shape cannot be accurately visualized with a picture and should rather be animated.

However, when the non-proportional damping mechanisms developed are not severe, as is the case in most real structures, the assumption that the damping is proportional is generally an adequately accurate approximation. This is because, despite that the damping is non-proportional, the coupling effects may not be severe enough to cause any serious errors. Furthermore, it should be noted that, apart from non-proportional damping, in certain cases closely spaced modes may appear to be complex as a result of the effects from modes located in adjacent frequency ranges [8, 11].

4.5.3 Mode Indicator Function

One of the tools commonly used in experimental modal analysis (EMA) for the approximation of the modal frequencies of a system is the normal mode indicator functions (MIF). The definition of the

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MIF depends on the quantity measured in the experiment, but in the case of accelerometers, which yield a transfer function expressed as accelerance, it is defined by:

𝑀𝐼𝐹₁(𝑓) =^∑ ^|𝐻^𝑅,𝑖^(𝑓)|

𝑁 2 𝑖=1

∑^𝑁_𝑖=1|𝐻_𝑖(𝑓)|² Equation 4.13

in which Hi(f) is the transfer function and HR,i(f) is the real part of the transfer function of each reference. The subscript 1 is used to refer to the normal MIF, as there are other forms of mode indicator functions.

The MIF1 sums the transfer functions of the different accelerometers obtained through experimental tests, at each frequency line. As described in section 4.3 the real part of the transfer function tends to zero at resonant frequencies for linear systems, meaning that the MIF1 tend to zero at resonant frequencies which indicate global modes. This provides indications of whether the observed frequency peaks in the transfer functions are normal global modes of the structure [7]. A frequency peak in the transfer function corresponding to a non-zero value in the MIF could be an indication that the mode is weakly excited, that the mode is of local character, or that the mode has some degree of complexity to it.

4.5.4 Modal Phase Collinearity

The modal phase collinearity (MPC) is an index which assess the linearity of the phase of an identified mode, or in other words the degree of complexity of the mode, by relating the real and imaginary part of the mode shape, described by the estimated modal vector 𝜙. It is defined by:

𝑀𝑃𝐶_𝑖 = ^{‖𝑅𝑒(𝜙}^𝑖^)‖²^{+ [𝑅𝑒(𝜙}^𝑖^{)𝐼𝑚(𝜙}^𝑖^)][2(𝜖^𝑖²^{+1)𝑠𝑖𝑛}²^(𝛼^𝑖^)−1]𝜖^𝑖⁻¹

‖𝑅𝑒(𝜙_𝑖)‖² +‖𝐼𝑚(𝜙_𝑖)‖² Equation 4.14

in which:

𝜖_𝑖 =^{‖𝐼𝑚(𝜙}^𝑖^)‖²^{−‖𝑅𝑒(𝜙}^𝑖^)‖²

2[𝑅𝑒(𝜙𝑖)𝐼𝑚(𝜙𝑖)] Equation 4.15

and

𝛼_𝑖 = 𝑎𝑟𝑐𝑡𝑎𝑛(|𝜖_𝑖| + 𝑠𝑖𝑔𝑛(𝜖_𝑖)√1 + 𝜖_𝑖²) Equation 4.16

where the subscript i is the i-th mode.

If the MPC is close to unity it indicates that the mode is a normal mode, in the sense that the different points of the structure are oscillating in-phase relative to each other, i.e. with monophase behavior. A value lower than one thus indicates some complexity of the mode, which can be a result of the mode being weakly excited or due to the presence of nonlinearities in the structure [12]. In EMA the identified modes often show some degree of complexity and the MPC can be used to evaluate to what degree this is present [13]. However, it is to the authors judgement to set threshold values of the MPC above which the observed mode can be regarded as an approximate normal mode.

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4.7 Half-power bandwidth

One important property of the FRF is that it provides the possibility to calculate the damping ratio related to each of the obtained modes through e.g. the half-power bandwidth method.

Figure 4.3. Half-power bandwidth points of resonance [5].

The method for the calculation of the damping ratio at each resonant frequency is as follows: if fn is the resonant frequency of the mode and fa and fb are the forcing frequencies on either side of the resonant frequency, fn, at which the amplitude is equal to 1/√2 times the resonant amplitude (Figure 4.3) then for small ζ the damping ratio can be calculated as:

𝜁 =^𝑓^𝑏^−𝑓^𝑎

2𝑓_𝑛 Equation 4.17

It should be noted that this technique is only applicable to structures associated with relatively small damping ratios [5]. Furthermore, from a mathematical point of view, this evaluation method does not work for cases in which modes are closely spaced, in the sense that the frequency response does not reach a value of 1/√2 of the natural frequencies in-between such peaks.

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5. Literature review

5.1 Summary

While the behavior of hollow core slabs under static loading has been researched in recent years [4, 14-19], there appears to be limited knowledge and research about the dynamic behavior of such slabs.

The previous work conducted is often directed towards either experimental or numerical analyses of the dynamics of hollow core slabs, and more rarely includes them both. It is even more uncommon to find research performed on full scale existing structures, as the available experimental investigations are often performed in laboratory environments. Additionally, there are limitations in the available guidelines in which a designer can seek advice when it comes to dynamic analyses and dynamic design of such slabs. The aim of the following chapter is to first summarize the available research and then the design guides found in literature.

5.2 Dynamics of hollow core slabs

Numerical and experimental investigations of hollow core slabs with two different dimensions was performed by Meixner (2008) [20]. Initially, three hollow core elements, without topping, were built and subjected to a load produced by a jumping test subject and an impulse load produced by dropping a weight, in order to measure the natural frequencies of the slab. This was followed by finite element modeling of the slab from which the maximum allowable spans were determined by comparing the acceleration response to existing standards. Additionally, subjective tests were performed, in which test subjects were asked to rate the vibration acceptability of the slab when subjected to human induced loads by walking tests in different setups. The research showed that a majority of the test subjects experienced the studied slab serviceability to be unacceptable for a setup in which the vibrations were produced by the walking of another test subject. This highlights the fact that such slabs can achieve satisfactory design for static loads, while simultaneously having problems with serviceability. However, it is necessary to comment that the experimental setup used is rarely found in existing structures. Firstly, it is common practice to cast in-situ concrete topping on the hollow core elements. Secondly, the restraints on the hollow core elements, applied through connections, are normally slightly more refined and consist of more detail work (see chapter 6) than the used simple supports. Consequently, the rotational stiffness at the edges of the slab is likely higher in real structures. The setup was however chosen mainly to make the updating process of the FE-model simpler.

In an effort to further investigate the vibration in hollow-core concrete floors, Johansson (2009) built an experimental floor structure consisting of three hollow core elements in a laboratory, which was followed by a development of finite element models representing the constructed floor [21]. The experimental setup was in many senses similar to the test performed by Meixner (2008), but with topping added to the slab as a second stage of the study. The results were derived by performing walking tests on the experimental floor, which were later used to validate the finite element model.

The natural frequencies and modal damping ratios were determined from these tests. Additional subjective tests were also performed, in which the reached conclusions confirm that the perception of vibrations is highly dependent on the activity and position of the test subject. The serviceability of the slab was generally considered acceptable when the test subject was walking across the slab herself, while it was considered unacceptable by the majority when the vibrations were induced by another person walking. The topping was considered to increase the vibration performance of the slab slightly.