DB.torreMaranello

POLITECNICO DI MILANO Dipartimento di Ingegneria Strutturale

Scuola Master F.lli Pesenti

Master in

Progettazione Sismica delle Strutture per Costruzioni Sostenibili

The new observation tower for the Galleria Ferrari

Area in Maranello: structural earthquake and comfort

design

Relatore: Allievo Ing. Pietro Crespi Diego Bruciafreddo

Ing. Francesco Iorio

a.a. 2010/2011

A mia Madre

Con un suo sorriso e per un suo sorriso

ho potuto gioire nella strada della conoscenza

Pagina 1/3 - Curriculum vitae di Cognome/i Nome/i

Per maggiori informazioni su Europass: http://europass.cedefop.europa.eu © Unione europea, 2002-2010 24082010

Curriculum Vitae Europass

Informazioni personali

Nome(i) / Cognome(i) Diego Bruciafreddo

Indirizzo(i) Via Bernardino Verro n.8, 20141 Milano

Telefono(i) +39 320 466 7566

E-mail [email protected]

Cittadinanza Italiana

Data di nascita 11/12/1984

Sesso Maschio

Occupazione desiderata/Settore

professionale

Ingegnere Strutturista

Esperienza professionale

Date 14/05/2012 a oggi

Lavoro o posizione ricoperti Ingegnere Strutturista

Principali attività e responsabilità Attività di consulenza relativa alla progettazione esecutiva di Torre Isozaki -edificio nell’ambito del progetto di riqualificazione dell’ex area fiera del comune di Milano di 57 piani - 220 m in c.a. con pareti accoppiate a nucleo per le azioni orizzontali , solai a piastra e colonne composite per i carichi verticali e dispositivi fluido viscosi per il controllo delle vibrazioni.

Nome e indirizzo del datore di lavoro Studio Iorio srl, Passaggio S.Bartolomeo n.7 24121 Bergamo

Tipo di attività o settore Ingegneria Strutturale

Date Dicembre 2009 a oggi


Principali attività e responsabilità Progettazione strutturale di strutture temporanee prefabbricate di grande luce per il ricovero di imbarcazioni. Principali tipologie strutturali trattate: -Tendostrutture in carpenteria metallica di acciaio e alluminio; -Tensostrutture; -Strutture pneumatiche;

Nome e indirizzo del datore di lavoro Yachtgarage Srl, Via delle Puglie 8 Benevento


Date 12/09/2011 a 09/05/2012


Principali attività e responsabilità Tirocinio formativo nell’ambito del master in “Progettazione Antisismica” della scuola Master F.lli Pesenti del Politecnico di Milano.Principali attività svolte: -Progettazione Strutturale “Torre Panoramica a Maranello per la Galleria Ferrari” progetto Architettonico Studio Lissoni– Torre Panoramica di 30 metri in c.a. con due piani interrati e uno sbalzo in testa di 12 m. Analisi in campo dinamico per il controllo delle vibrazioni. -Progettazione Strutturale “Auditorium il Castello a L’Aquila” - Struttura con isolamento sismico alla base, progettata da Renzo Piano, in legno strutturale composta da pannelli di xlam su una doppia orditura di travi in lamellare. -Modello strutturale agli elementi finiti per lo studio del comportamento statico e dinamico di Torre Isozaki.

Nome e indirizzo del datore di lavoro Studio Iorio srl, Passaggio S.Bartolomeo n.7 24121 Bergamo




Date 01/09/2010 – 30/09/2010

Lavoro o posizione ricoperti Progettista Strutturale

Principali attività e responsabilità Progetto Strutturale di un edificio a sei elevazioni fuori terra più piano interrato, irregolare in pianta e in elevazione, di un edificio in c.a. in zona ad alta sismicità (ag/g 0.38) in classe di duttilità B. Il comportamento sismico è stato ottimizzato mediante l’adozione di una scala alla “Giliberti”.

Nome e indirizzo del datore di lavoro Studio Tecnico Arch. Antonino Leonello


Date 10/03/2007 al 10/06/2007

Lavoro o posizione ricoperti Tirocinio Formativo

Principali attività e responsabilità Attività sperimentale di modellazione e calcolo della risposta sismica locale.

Nome e indirizzo del datore di lavoro MECMAT – Dipartimento di Meccanica e Materiali dell’Università degli Studi Mediterranea di Reggio Calabria


Istruzione e formazione

Date Febbraio 2011 – Maggio 2012

Titolo della qualifica rilasciata Master di II livello in “Progettazione antisismica delle strutture per costruzioni Sostenibili”

Principali tematiche/competenze professionali acquisite

Tecniche di progettazione per la mitigazione del rischio sismico sia su strutture nuove che esistenti. Competenze specialistiche nell’ambito della modellazione del comportamento dinamico delle strutture.

Titolo della tesi e argomenti “The new observation tower for the Galleria Ferrari Area in Maranello: structural earthquake and comfort design” Progettazione strutturale della nuova torre panoramica a Maranello per la Galleria Ferrari. Sono state effettuate analisi dinamiche non lineari incrementali con modellazione a fibre (IDA) per la valutazione del comportamento sismico e analisi dinamiche lineari per la valutazione del livello di confort a seguito delle vibrazioni di natura antropica sullo sbalzo di 12 m.

Nome e tipo d'organizzazione erogatrice dell'istruzione e formazione

Politecnico di Milano – Scuola Master F.lli Pesenti

Date Novembre 2007 – Dicembre 2010

Titolo della qualifica rilasciata Laurea Specialistica in Ingegneria Civile Progettazione strutturale


Progettazione di strutture e opere geotecniche; Comportamento dinamico delle strutture sotto l’azione del sisma e del vento; Valutazione e mitigazione del potenziale di collasso progressivo negli edifici;

Titolo della tesi e argomenti “Valutazione della vulnerabilità sismica di edifici esistenti in c.a. mediante analisi non lineari” La tesi tratta la valutazione del grado di vulnerabilità di un edificio esistente irregolare in pianta mediante l’utilizzo di analisi dinamica non lineare con modelli a plasticità diffusa.


Università degli studi Mediterranea di Reggio Calabria

Livello nella classificazione nazionale o internazionale

110 e lode con menzione di merito

Date Ottobre 2004 – Novembre 2007

Titolo della qualifica rilasciata Laurea Ingegneria Civile


Competenze base di Analisi Matematica, Fisica,Scienza e Tecnica delle Costruzioni e Geotecnica

Titolo della tesi e argomenti “Risposta Sismica Locale” Valutazione della variazione dell’input sismico in relazione alle condizioni locali del sito.


Università degli studi Mediterranea di Reggio Calabria

Livello nella classificazione nazionale o internazionale

110 e lode con menzione di merito

Autovalutazione Comprensione Parlato Scritto



Livello europeo (*) Ascolto Lettura Interazione orale Produzione orale

Inglese B2 Livello intermedio C1 Livello Avanzato B2 Livello intermedio B2 Livello intermedio C1 Livello avanzato

Francese A2

Livello Elementare

B1 Livello Intermedio A2 Livello

Elementare A2

Livello elementare

A2 Livello elementare

(*) Quadro comune europeo di riferimento per le lingue

Capacità e competenze sociali - Sono particolarmente predisposto a lavorare in team cercando sempre di comprendere e di risolvere i problemi al meglio al fine di ottenere i risultati previsti. - Sono dotato di un forte senso di volontà e di capacità di problem solving anche nelle situazioni più dinamiche. -Sono dotato di un ottimo spirito di adattamento anche nelle situazioni più complesse e sono pienamente disponibile a trasferte in tutto il mondo. -Buona capacità di comunicazione e motivazione ottenuta grazie a un’ampia esperienza di impartizione di lezioni private a un buon numero di studenti universitari ( ad oggi circa 60 )

Capacità e competenze organizzative

Gestione di progetti e gruppi di lavoro

Capacità e competenze tecniche Ingegnere strutturista con capacità progettazione di strutture non tradizionali e complesse.

Capacità e competenze informatiche

Si elencano le principali competenze specialistiche in aggiunta alle competenze base di utilizzo del computer: Ottima conoscenza Excel+VBA Ottima Conoscenza programma per Modellazione FEM STRAUS7 Ottima Conoscenza Programma per Modellazione Fem MIDAS GEN Ottima Conoscenza Programma Per Modellazione FEM SAP200 Capacità di utilizzo e apprendimento in tempi rapidi di tutti i programmi di modellazione FEM Ottima conoscenza dei linguaggi di programmazione VBA, C++ Ottima conoscenza del programma di Calcolo MATLAB Ottima conoscenza del pacchetto OFFICE Ottima conoscenza di AUTOCAD

Altre capacità e competenze Runner amatoriale con partecipazione a eventi , nuoto;

Patente A, B

Ulteriori informazioni Referenze e Curriculum Vitae dettagliato su richiesta

Autorizzo il trattamento dei miei dati personali ai sensi del Decreto Legislativo 30 giugno 2003, n. 196 "Codice in materia di protezione dei dati personali". (facoltativo, v. istruzioni)

Firma

V

ABSTRACT IX

<1>Limit state design for reinforced concrete structures ................................................ 1

1.1 THE BORN OF LIMIT STATE DESIGN: THE MODEL CODE .............................................................................. 1

1.2 METHODS OF DESIGN OF CONCRETE STRUCTURES .................................................................................... 3

1.2.1 The Allowable Stress Method (ASM) ..................................................................................... 4

1.2.2 Load Factor Method (LFM) ................................................................................................... 4

1.2.3 Limit state Method (LSM) ..................................................................................................... 4

1.3 THE LIMIT STATE DESIGN APPROACH ...................................................................................................... 5

1.3.1 Characteristic load and characteristic strenghs .................................................................... 5

1.3.2 Partial safety factors for loads and material strengths ........................................................ 7

1.3.2.1 Partial Safety Factor for load �� ..................................................................................................... 7

1.3.2.2 Partial Safety Factor for Material Strengths �� ............................................................................. 7

1.3.3 The performance requirements of structures :the limit states ............................................. 7

1.3.3.1 Ultimate Limit States ....................................................................................................................... 8

1.3.3.2 Serviceability Limit states ................................................................................................................ 9

1.3.3.3 Robustness ..................................................................................................................................... 10

1.4 THE ISSUE OF VIBRATION CONTROL OF FLOOR ......................................................................................... 11

1.4.1 Description of the walking load .......................................................................................... 12

1.4.2 Determination of the floor response ................................................................................... 15

1.4.2.1 Floor response for a single combination of step frequency and person’s weight ......................... 15

1.4.2.2 Design value of the floor response ................................................................................................ 16

1.4.2.3 Hand Calculation method .............................................................................................................. 16

1.4.3 Classification of vibrations .................................................................................................. 17

1.4.3.1 Quantity to be assessed ................................................................................................................. 17

1.4.3.2 Floor classes ................................................................................................................................... 18

1.4.4 Hand calculation method -Design procedure...................................................................... 19

1.4.4.1 Determination of eigenfrequency and modal mass....................................................................... 19

1.4.4.2 Determination of damping ............................................................................................................ 19

1.4.4.3 Determination of the floor class .................................................................................................... 20

1.4.4.4 System with more than one eigenfrequency ................................................................................. 22

<2>Non linear structural analysis for seismic design ..................................................... 23

2.2 THE ROLE AND THE USE OF NONLINEAR ANALYSIS IN SEISMIC DESIGN ........................................................... 23

2.3 STIFFNESS, RESISTANCE AND DUCTILITY: KEY POINTS OF STRUCTURAL ANALYSIS ............................................. 24

2.4 EQUATION OF MOTION OF AN ELASTIC-PLASTIC SYSTEM ............................................................................ 24

2.5 LINEAR ELASTIC EQUIVALENT ANALYSIS: BEHAVIOR FACTOR Q .................................................................... 25

2.5.1 MDOF analysis .................................................................................................................... 27

2.6 NON LINEAR STATIC ANALYSIS: PUSHOVER ............................................................................................. 28

2.6.1 Non linear static analysis for a SDOF system ...................................................................... 28

2.6.1.1 Step 1: Load increasing .................................................................................................................. 29

2.6.1.2 Step 2: Linearization of capacity curve .......................................................................................... 29

2.6.1.3 Step 3: Performance Point Evaluation ........................................................................................... 30

2.6.2 Pushover analysis for MDOF systems ................................................................................. 30

2.6.2.1 Analysis Methods, Modeling and Outcome ................................................................................... 31

2.7 NON LINEAR DYNAMIC ANALYSIS .......................................................................................................... 31

2.7.1 Newmark’s method for a SDOF system .............................................................................. 33

2.7.1.1 Newmark’s method : stability ........................................................................................................ 34

2.7.2 Direct time history integration for a MDOF system ............................................................ 34

VI

2.8 INCREMENTAL DYNAMIC ANALYSIS ....................................................................................................... 37

2.9 NONLINEAR STATIC VERSUS NONLINEAR DYNAMIC ANALYSIS .................................................................... 38

2.10 QUALITY ASSURANCE OF BUILDING ANALYSIS .................................................................................... 38

<3>Model of structure for non linear analysis ............................................................... 41

3.1 MODELS TO DESCRIBE THE STRUCTURAL BEHAVIOR .................................................................................. 42

3.2 NON LINEAR BEHAVIOR OF R.C. STRUCTURES .......................................................................................... 43

3.2.1 Geometric non linearity ...................................................................................................... 44

3.2.2 Mechanical non linearity .................................................................................................... 46

3.2.2.1 Distributed versus concentrated Plasticity elements .................................................................... 48

3.2.3 Stress-Strain relation for non linear analysis ...................................................................... 49

3.2.3.1 Stress-strain relation for r.c. section .............................................................................................. 49

3.2.3.1.1Kent and Park Model ........................................................................ 50

3.2.3.1.2Menegotto and Pinto Model ............................................................ 52

<4>The new redevelopment of areas adjacent to the Galleria Ferrari .......................... 55

4.1 THE REDEVELOPMENT OF AREAS ADJACENT TO THE GALLERY FERRARI .......................................................... 55

4.1.1 The design competition: Plaza and Tower Galleria Ferrari ................................................. 56

4.1.2 The winning project ............................................................................................................ 58

<5> The structural design of the new area in Maranello 63

5.1 REFERENCE DOCUMENTS .................................................................................................................... 63

5.1.1 National Codes .................................................................................................................... 63

5.1.2 UNI EN documents .............................................................................................................. 64

5.1.3 European codes ................................................................................................................... 65

5.1.4 Other documents ................................................................................................................ 65

5.2 BUILDING DESCRIPTION ...................................................................................................................... 65

5.3 CONCEPT DESIGN .............................................................................................................................. 75

5.3.1 Preliminary design of the panoramic terrace ..................................................................... 75

5.3.1.1 Gravity load analysis ...................................................................................................................... 76

5.3.1.2 Design against deflection control .................................................................................................. 77

5.3.1.3 Design against vibration control .................................................................................................... 78

5.3.1.4 Structural size of elements ............................................................................................................ 82

5.3.2 Preliminary composite floor design .................................................................................... 84

5.3.2.1 Design against vibration control .................................................................................................... 85

5.3.2.2 Deflection check ............................................................................................................................ 87

5.3.3 Preliminary design of subbeam ........................................................................................... 88

5.3.4 Preliminary design of post tensioning force ........................................................................ 90

5.3.4.1 Estimate of post-tensioning force .................................................................................................. 90

5.4 MATERIALS ..................................................................................................................................... 92

5.4.1 Concrete 28/35 ................................................................................................................... 92

5.4.2 Rebar B450C ....................................................................................................................... 92

5.4.3 Structural steel S355 ........................................................................................................... 92

VII

5.5 LOAD ANALYSIS ................................................................................................................................ 93

5.5.1 Vertical loads ...................................................................................................................... 93

5.5.1.1 All levels under the 10th

................................................................................................................. 93

5.5.1.2 11th

level ........................................................................................................................................ 93

5.5.1.3 Coverture ....................................................................................................................................... 94

5.5.2 Snow load............................................................................................................................ 94

5.5.3 Wind load ............................................................................................................................ 96

5.5.3.1 Basic Value ..................................................................................................................................... 96

5.5.3.2 Basic velocity pressure ................................................................................................................... 97

5.5.3.3 Exposure coefficient ...................................................................................................................... 97

5.5.3.4 Wind Pressure ................................................................................................................................ 98

5.5.4 Seismic load ........................................................................................................................ 99

5.5.4.1 Geoseismic analysis ....................................................................................................................... 99

5.5.4.2 Phase 1......................................................................................................................................... 100

5.5.4.3 Phase 2......................................................................................................................................... 101

5.5.4.4 Phase 3.2 SLE spectra................................................................................................................... 102

5.5.4.5 Phase 3.3 SLV and SLC spectra ..................................................................................................... 103

5.6 STRUCTURAL FEM MODEL ................................................................................................................ 105

5.7 MODAL ANALYSIS FOR SEISMIC PERFORMANCE ..................................................................................... 106

5.8 MODEL CHECK ............................................................................................................................... 110

5.8.1 Self weight check .............................................................................................................. 110

5.8.2 Pre stress force check ........................................................................................................ 111

5.8.3 Fundamental period check ................................................................................................ 113

5.8.4 Lateral wind load check .................................................................................................... 114

5.9 SLS CHECK .................................................................................................................................... 115

5.9.1 Limit state for deflection control ....................................................................................... 115

5.9.1.1 Load combination ........................................................................................................................ 115

5.9.1.2 Horizontal displacement check .................................................................................................... 116

5.9.1.2.1Check in x direction ........................................................................ 117

5.9.1.2.2Check in y direction ........................................................................ 120

5.9.1.3 Vertical displacement check ........................................................................................................ 123

5.9.1.3.1Cantilever subjected to all loads in frequently combination ......... 123

5.9.1.3.1Cantilever subjected only to live loads in frequently combination 124

5.9.2 Serviceability limit state of vibration control .................................................................... 125

5.10 ULTIMATE LIMIT STATES: REINFORCEMENT DESIGN AND CHECK ............................................................ 129

5.10.1 Load combination ............................................................................................................. 129

5.10.1.1 Seismic combination .................................................................................................................. 129

5.10.1.2 Comparatition between wind load and seismic load ................................................................. 130

5.10.2 Reinforcement design ....................................................................................................... 130

5.10.2.1 Longitudinal rebar...................................................................................................................... 132

5.10.2.1 Stirrups ...................................................................................................................................... 133

5.10.3 Uls check ........................................................................................................................... 137

5.10.3.1 Composite axial force-bending check ........................................................................................ 137

5.10.3.1 Shear check ................................................................................................................................ 138

5.10.3.2 Steel member check .................................................................................................................. 138

5.11 RESUME ................................................................................................................................... 139

5.12 APPENDIX A – LOAD CASE RESUME ....................................................................................... 140

<6> The new tower in Maranello: Performance evalutation under seismic load 147

VIII

6.1 STRUCTURAL MODEL ....................................................................................................................... 147

6.1.1 Base Structural model check ............................................................................................. 149

6.1.1.1 Mass comparison ......................................................................................................................... 149

6.1.1.2 Mode shapes and Mode properties comparison ......................................................................... 150

6.1.2 Inelastic material properties ............................................................................................. 151

6.1.2.1 Constitutive model and parameter confined concrete ................................................................ 151

6.1.2.2 Constitutive model and parameter for unconfined concrete ...................................................... 153

6.1.2.3 Constitutive model for rebar ....................................................................................................... 154

6.1.3 Fiber division of section and inelastic hinge ...................................................................... 155

6.2 PERFORM IDA ANALYSIS .................................................................................................................. 157

6.2.1 Ground acceleration selection .......................................................................................... 158

6.2.1.1 Reference spectra ........................................................................................................................ 158

6.2.1.2 The main parameters selection ................................................................................................... 158

6.2.1.3 Records ........................................................................................................................................ 160

6.2.2 Performing the analysis .................................................................................................... 162

6.2.2.1 Looking at accuracy ..................................................................................................................... 163

6.2.2.2 Looking at convergency ............................................................................................................... 164

6.2.2.3 Looking at robustness .................................................................................................................. 164

6.2.3 The limits state check ........................................................................................................ 164

6.2.4 The response under design earthquake ............................................................................ 165

6.2.4.1 Consideration............................................................................................................................... 176

6.2.5 IDA curve ........................................................................................................................... 176

6.2.5.1 IDA curve H(2) .............................................................................................................................. 176

6.2.5.2 IDA curve H(3) .............................................................................................................................. 179

6.2.5.3 IDA curve H(4) .............................................................................................................................. 181

6.2.5.4 IDA curve H6 ................................................................................................................................ 183

6.2.5.5 Ida curve H(7) .............................................................................................................................. 185

6.2.5.6 IDA curve H(8) .............................................................................................................................. 187

6.2.5.7 IDA curve H(9) .............................................................................................................................. 189

6.2.6 Summarization of IDA curve and limit states check .......................................................... 190

6.3 BASE SHEAR VERSUS TOP DISPLACEMENT ............................................................................................. 192

<7> Conclusions 147

REFERENCES 149

IX

ABSTRACT

The structural project of the new panoramic tower for the redevelopment of the Galleria

Ferrari has been done in this thesis.

The tower design has been made, specifically, versus vibration control due to human activities

and versus the seismic load. Two peculiarities have made this project not standard in order to

accomplish the limit state requirement associated to the vibration control (serviceability condition)

and the seismic load (Ultimate state conditions). The first one peculiarity, related to the vibration

control, is the panoramic terrace cantilevered for rc core of the tower for 12 meters, which also

requires a seismic analysis for the vertical component. The second one is the presence of two

overtures in the sections near of the base of the core which disturb the plastic excursion of this

region under seismic load.

The approach was to follow the integrally structural design path from the architectonic design

and after to perform evaluation under seismic load by using non linear dynamic incremental

analysis.

PREVIEW

In this opening chapter, the limit state design is presented. Exactly, this chapter speaks

about the LSD in the most important building codes. Particular attention is shown about the

serviceability limit states of vibration control because the advices in the European and Italian

code are not very clear.

1.1 The born of Limit State Design: The Model Code

Before the last two decades of last century reinforced concrete designers were

concerned more with the safety against failure of their structures than with durability under

service conditions. Thus, the theoretical calculations for design were based on classical elastic

theory using fictitious modulus of elasticity and geometrical properties for reinforced concrete

element and a permissible working stresses. The date of the creation of the European

Committee for Concrete (Comite European du Beton), called CEB, in 1953, can be said as the

date when the limit state design, otherwise called strength and performance criterion, was

born. The initiative for this came from the reinforced concrete contractors of France. The

Committee has its headquarters at Luxembourg. Its objective are the coordination and

synthesis of research on safety, durability and design calculation procedures, for practical

application to construction. Their first recommendations for reinforced concrete design were

published in 1964.

Later, under the leadership of Yves Guyon (well known for his expertise on prestressed

concrete), the CEB established technical collaboration with the International Federation for

prestressing (Federation International de la Preconstrainte), called FIP. Recommendations for

international adoption for design and construction of concrete structures were published by

them in June 1970 and the “CEB-FIP Model Code for Concrete Structures” was proposed in

1977. These efforts formed the solid bases for the creation of an “ International Code of

Practice”. Trough these publications a unified code for design of both reinforced and

prestressed reinforced concrete structures was developed.

<1>

Limit State Design of reinforced

concrete structures

Chapter 1

2

According to the above model code, structural analyses, for determination of section

design values are to be carried by elastic analysis, but the final design of the concrete

structures is to be done by the principles of limit state theory.

The model code was to be a model from which each country was to write its national

code, based on its stage of development but agreeing on important points, like method of

design for bending, shear, torsion, etc., to the model code.

The basis had to be scientifically rigorous, but compromises could be made because of

inadequacy of data on the subject for any region.

The British were the first to bring out a code based on limit state approach as

recommended by the CEB-FIP in 1970. This code was published as Unified Code for

structural concrete, i.e. CP 110 (1972). Other Countries in Europe and United States adopted

similar codes, and today most countries follow codes based on the principles of Limit state

Design.

India followed suit during the third revision of Is code 456 in 1978, and the provisions

of the limit state design (as regards concrete strength, durability and detailing) were

incorporated in the revised code IS 456 (1978) in Sections 1-4. However, for design

calculations to asses the strength of an R.C. member, the choice of either limit state method or

working stress method has been left to the designer (Section 5 and 6) with the hope that with

time, the working stress method will be completely replaced by the limit state method. Many

of the Provisions of the IS code are very similar to the BS approach.

The fourth revision of the code published in June 2000 as IS 456 (2000) specifies that

R.C. structural elements shall normally be designed by Limit State Method. Allowable-Stress

Method is to be used only where Limit State Method cannot be used conveniently.

Accordingly, the status of working stress method as an alternative method of design has been

discontinued in the current code.

A uniform approach to design, with reference to the various criteria, is the dream of al

designers with an international outlook, but it is bound to take many more years to come into

effect. In the USA, the code used for general design of reinforced concrete structures in the

“Building Code Requirement for Reinforce Concrete” ACI 318 (1999). The general principles

of limit states design are named as “strength and serviceability method” in the above code. In

European countries the code used is the Eurocode (EC), composed by 9 parts whit their

national application document (NAD). Not all parts are related with rc design, for example

EC3 is about the steel structures. Every country can use both Eurocodes and its code but, at

the same time, the national codes are changing to be very similar to the Eurocodes. The aim is

to use only one code for all member countries.

As research in various aspects of concrete design in still being carried out in many

countries and these countries are anxious that the results of these latest research are reflected

in their national codes, it will take a long time for all the codes in the world to be the same. It

is therefore advisable that a designer be aware of at least the general prevision of the codes of

other countries too. For this purpose, in this chapter and in many parts of this thesis the

provisions of most important codes are briefly discussed and compared.

Limit State Design of Reiforced Concrete Structures in building codes

3

As has happened in other scientific fields, new ways of thinking replace the old ways. In

scientific circles, this is generally referred to as a paradigm shift. Limit state design should

therefore be looked upon as a “paradigm”, a better way of explaining certain aspects of reality

and a new way of thinking about old problems. Thus, it should be learned and taught with its

own philosophy, and not as an extension of the old elastic theory.

1.2 Methods of Design of Concrete Structures

Reinforced concrete members are allowed to be designed according to existing codes of

practice by one of the following two methods:

1. The method of theoretical calculations using commonly accepted procedures of

calculations;

2. The method of experimental investigations.

The first one is employed for design of commonly used structures. These methods

consist of numerical calculations based on the procedures prescribed in codes of practices

prevailing in the country. Such procedures are based on one of the following methods of

design:

1. The allowable stress method or the working stress method, also known as the elastic

method;

2. The load factor method;

3. The limit state method.

The experimental methods are used only for unusual structures and are to be carried out

in a properly equipped laboratory by test on scaled models according to model analysis

procedures or tests on prototype of the structure.

The theoretical methods themselves are the result of extensive laboratory tests and field

investigations. Safe and universally accepted methods of calculation based on strength of

materials and applied mechanics have been derived from these laboratory investigations and

are codified into national codes.

The code of practice used in Italy is D.M. 14 Gennaio 2008, briefly called NTC08. This

document is related to Eurocodes (so similar that a lot of parts are just the Italian translation)

and, as written in NTC08, these latter can be used directly or when the provisions in NTC08

need more details. All new reinforced concrete structures built in Italy are required to follow

the provisions of these codes. Looking to some other parts in the world, the American practice

follows the ACI 318 (1999), the Australian practice AS 3600, in India the IS 456 (2000).

The common denominator in the code mentioned above is that the reinforced concrete

members should normally be designed by limit state method.

This section deals briefly with the various theoretical methods of design mentioned

above.

Chapter 1

4

1.2.1 The Allowable Stress Method (ASM)

This method of design was evolved around 1900 and was the first theoretical method

accepted by national codes of practice for design of reinforced concrete sections. It assumes

that both steel and concrete act together and are perfectly elastic at all stages so that the

modular ratio (ratio between moduli of elasticity of steel and concrete) can be used to

determine the stresses in the steel and concrete. This methods adopts permissible stresses

which are obtained by applying specific factors of safety on material strength for design. It

uses a factor of safety about 3 with respect to cube strength for concrete and a factor of safety

about 1.8 with respect to yield strength for steel.

Even though structures designed by this method have been performing their functions

satisfactorily for many years, it has three major defects. First, since the method deals only

with the elastic behavior of the member, it neither shows its real strength nor gives the true

factor of safety of the structures against failure. Second, allowable stress method results in

larger percentages of compression steels than is the case while using limit state design, thus

leading to uneconomic sections while dealing with compression member or when

compression steel is used in bending members. Third, the modular ratio itself is an imaginary

quantity. Because of creep and nonlinear stress-strain relationship, concrete and steel in the

sections are calculated on the basis of elastic behavior of the composite section. An imaginary

modular ratio which may be either a constant in value for all strengths of concrete or one

which varies with the strength of concrete is used for calculation of the probable stresses in

concrete and steel.

A widely used value of modular ratio was 15. Hovewer, it should be noted that modular

ratio dependent of concrete strength and steel strength, the value of modulus of concrete

which change for creep, shrinkage etc so to keep in count these is essential for the concrete

designer interested to understand all features of the structural behavior et not only the safety

against collapse.

1.2.2 Load Factor Method (LFM)

A major defect of the allowable stress method of design is that it does not give a true

factor of the safety against failure but only the status of safe structure. To overcome this, the

ultimate load method of design was introduced, pioneer was the U.S.A. in 1956. In this

method, the strength of the R.C. section at working load is estimated from the ultimate

strength of the section. The load factor is defined as the ratio of the ultimate load that the

section can carry to the working load it has to carry. Usually, R.C. structures are designed for

suitable separate load factors for the dead loads and for live loads with additional safety factor

for strength of concrete.

1.2.3 Limit state Method (LSM)

Even though the load factor method based on ultimate load theory at first tended to

discredit the traditional elastic approach to design, the engineering profession did not take to

such design very readily. Also, steadily increasing knowledge brought the merits of both

elastic and ultimate theory into perspective. It has been shown that whereas ultimate theory


5

gives a good idea of the strength aspect, the serviceability limit states are better shown by the

elastic theory only.

Since a rational approach to design of reinforced concrete did not mean simply adopting

the existing elastic and ultimate theories, new concepts with a semi-probabilistic approach to

design were found necessary. The proposed new method had to provide a framework which

would allow designs to be economical and safe. This new philosophy of design was called the

Limit State Method (LSM) of design. It has been already adopted by many of the leading

countries of the world in their codes as he only acceptable method of design of reinforced

concrete structures.

1.3 The Limit State Design approach

A structure is said to have reached its limits state, when the structures as a whole or in

part becomes unfit for designed use during its expected life. The limit state of a structure is

the condition of its being not fit for its intended use, and limit state design is a philosophy of

design where one designs a structure so that it will not reach any of the specified limit states

during the expected life of the structure.

Many types of limit states or failure conditions can be specified. The two major limit

states which are usually considered are the following:

1. The Ultimate Limit States (ULS) which deals with the strength and stability of the

structure under the maximum overload it is expected to carry. This implies that no part

or whole of the structure should fall apart under any combination of expected

overload.

2. The serviceability limit states which deals with conditions such as deflection, cracking

of the structure under service loads, durability (under given environment in which the

structure has been placed), excessive vibration etc.

Limit state design should ensure that the structure will be safe as regards the various

limit state conditions, in its expected period of existence. Hence the limit state method design

is also known as strength and serviceability method of design.

1.3.1 Characteristic load and characteristic strenghs

Structures have to carry dead and live loads. Both aren’t knowable by deterministic way

because of themselves nature. Only a statistical value of loads can be defined. The maximum

working load that the structure has to withstand and for which it is to be designed is called the

characteristic load. Thus there are characteristic dead load and characteristic live loads.

Similar, the strengths that one can safely assume for materials (steel and concrete) are called

characteristic strengths.

Chapter 1

6

Figure 1. 1 Characteristic strength and its failure probability

The characteristic values are related to specified fractiles in the statistical distribution of

load or strength. Exactly, for the load is commonly used the fractile 95%, in other words a

load value so big than only 5 times on 100 the structure have to carry a bigger value. For the

strength, instead, the value is so small then 95 times on 100 the effective value is bigger than

the characteristic.

A lot of physical phenomena follow the normal distribution as well the load and the

strength can be treated by this law. In a normal distributions, obtaining of fractiles, is directly

related to mean value and standard deviations according to the below equation:

�� = �� + �� [1.1] � = � − �� [1.2]

The value of the constant � for the 5 per cent chance, in a normal distribution, is 1.64.

Figure 1. 2Characteristic strength and characteristic load


7

1.3.2 Partial safety factors for loads and material strengths

Having obtained the characteristic loads and characteristic strengths, the design loads

and design strengths are obtained by the concept of partial safety factors. Partial safety factors

are applied both to loads o the structure and to strength of materials. By partial safety factor

the designer can keep in count the stochastic nature of loads and strength of materials. These

factors are now explained.

1.3.2.1 Partial Safety Factor for load ��

The load to be used for ultimate strengths design is also termed as factored load. Using

the partial safety factor for load simply means that for calculation of the ultimate load for

design, the characteristic load has to be multiplied by the partial safety factor denoted by the

symbol �. This may be regarded as the overload factor for which the structure has to be

designed. Thus the load obtained by multiplying the characteristic load by the partial safety

factor is called the factored load, and is given by

�� = �� ∙ � [1.3]

Structures will have to be designed for this factored load.

It is extremely important to remember that in limit state design, the design load is

different from that used in elastic design. It is the factored loads, and not the characteristic

load, which are used for the calculation of design values.

1.3.2.2 Partial Safety Factor for Material Strengths ��

The grade strength of concrete is the characteristic strength of concrete, and the

guaranteed yield of steel is the characteristic strength of steel. Calculation to arrive at the

characteristic material strength of materials by using statistical theory takes into account only

the variation of strength between the test specimens. It should be clearly noted that the above

procedure does not allow for the possible variation the strength of the test specimen and the

material in the structure. This feature is kept in count by the follow equation to calculate the

factored strength

� = � � [1.4]

This simply means that the strength to be used for design should be used the reduced value of

the characteristic strength by the factor denoted by the partial safety factor for the material.

1.3.3 The performance requirements of structures :the limit states

During its expected period of existence the structure have to do the performance for

which it was built. This performance levels, as already written before, are clearly divisible in

two main categories:

Chapter 1

8

• Ultimate Limit States (ULS)

• Serviceability Limit states (SLS)

One category more need to be keep in count, as written in NTC08

• Robusteness

Let us write what kind of performance level is required in one or in other one. Helped

by Italian national code, which is referred to Eurocode, the follow subchapters explain briefly,

for each limit state, the performance that the structure will be show during its expected life.

1.3.3.1 Ultimate Limit States

The Ultimate Limit States are related to safety in its strictly meaning, they are linked to

collapse or other kinds of structural failure which can be dangerous for the safety of persons

or doing big environmental or social problems.

When a ULS is overcome, the structure cannot return to the initial state and this

situation defines the collapse of structure.

1. Ultimate strength condition

The ultimate strength of the structure or member should allow n overload. For this

purpose, the structure should be designed by the accepted ultimate load theory to

carry specific overload. This may be in-flexure, compression, shear, torsion or

tension and against it every structure have to be checked.

2. Overall stability

The structure or a part of it, thought as a rigid body, have to offer stability against

accidental loads.

3. Big deflections or deformations

It is important that the maximum deflections or deformations, in one ultimate scene,

is limited.

4. Fatigue collapse

When structure or one part of that fails for the action of cyclic load.

5. Fail of frames or joints for time related phenomena

Phenomena, like viscosity, can change the stress distribution in frame sections

respect to the initial checked value

6. Instability

Checked that the structure or one part of it have just one stability configuration.

7. Fire resistance

The structure is able to resist for a determinate time when it is subjected to the fire

action. This capacity is identified by the acronym REI when:

-R Load bearing capacity: to provide strength and stability of the building;

-E Integrity: to keep the element intact;

-I Insulation: to keep the temperature low on the unexposed side of the element,

expressed in minutes.

An element fulfilling all these basic criteria for 30 minutes will be classified REI 30.


9

For seismic action there are two ultimate limit state.

1. Life safety

After earthquake the structure shows breakage, or collapses, only in non

structural elements . The structural elements are seriously damaged from which

a big part of her stiffness against the lateral load is lost. However, the structure

still offers resistance and rigidity against vertical loads and a residual safety

against horizontal seismic actions.

2. Collapse prevention

After earthquake the structure shows serious damages and collapses of non-

structural elements and heavy damages to the structural elements; the structure

still has a residual safety against vertical load and a little residual safety to the

seismic horizontal action.

1.3.3.2 Serviceability Limit states

Serviceability Limit states are related to functionality of structure, they are associated

with the capacity of the structure to fit for its intended use. When a SLS is overcome, the

structure can or cannot return to the initial state. On the first case the damages and the

deformations are reversible and the structure comes back to its initial conditions when the

cause, which has generated them, finishes. The second situation is the opposite. The

serviceability limit states deal with conditions such as deflection, cracking of the structure

under service loads, durability (under a given environment in which the structure has been

placed), excessive vibration.

Let us define the main SLS.

1. Deflection condition (without stability lost)

Deflections and deformations which don’t compromise the global or local stability

but they are so big to compromise the functionality, the use or the aspect of

construction.

2. Local damages

Local damages like a steel cover lost can compromise durability, performance and

aspect.

3. Vibration control

The vibration control upon serviceable loads is fundamental to achieve an

acceptable comfort level

4. Durability condition

The structure should be fit for its environment.

Like ULS there are two SLS for seismic load.

1. Immediate Occupancy

Chapter 1

10

The post-earthquake damage state that retains the pre-earthquake design strength

and stiffness, and is safe to occupy. Some minor structural repairs may be

appropriate but not necessary to make the building safe occupy.

2. Damage Control Range

After earthquake the structure (structural elements, non structural elements,

equipments etc.) suffers some damage but not so great as undermining the safety of

persons or as compromising the stiffness and resistance against lateral and vertical

load. In the post-earthquake the structure can be used entirely or in a large part.

1.3.3.3 Robustness

Robustness for a structure is when it is able to don’t show damages much bigger than

the cause which has generated them. In other words is important that the local damage will be

confined just in a little part of structure when this part suffers damage, also heavy, due an

exceptional load like fire, gas explosion, vehicle impact.

This property can be achieved when the structure has an alternative load path able to

bridge the failed elements.

The main design rules whit which one can design structure with robustness behavior

are:

• Privilege the columns resistance with respect to that of the beams (SCWB Strong

Columns Weak Beams);

• Privilege shear resistance with respect to the flexure resistance. In other words

member have to fail for shear and not or flexure:

• Check with attention the interaction between structural part and non structural part;

• Design the structural detailing to allow load transferring.


11

Figure 1. 3 The focus on disproportionate collapse followed the Ronan Point disaster 0f 1968, which is the

classical example of robustness problem. In the collapse one wall panel sustained damage, due a gas explosion,

causing the whole corner of the building to give way.

It is important to note that all above rules are the same rules applied in seismic design hence,

seismically designed structure, already has an implicit capacity to show robustness.

1.4 The issue of vibration control of floor

The issue or vibrations, together with deflections issue, are important for the design of

new structures. Indeed, modern materials and constructions processes, e.g. composite floor

system or pre-stressed flat concrete slabs with high strength of materials, are capable of

fulfilling large span floor structures with a minimum number of intermediate columns or

walls. These slender floor structures have in common, that their design is usually not

controlled by ultimate limit state verifications but by serviceability criteria, i.e. deflections or

vibrations.

Whereas for ultimate limit state verifications and for the determination of deflections

and durability design codes provide sufficient rules, the calculation and assessment of floor

vibrations in the design stage has still a number of uncertainties. These uncertainties are

related to:

Chapter 1

12

- A suitable design model including the effects of frequencies, damping, displacement

amplitudes, velocity and acceleration to predict the dynamic response of the floor

structure with sufficient reliability in the design stage;

- The characterization of boundary conditions for the model;

- The shape and magnitude of the excitation

- The assessment of the floor response in relation to the use of the floor and the

degree of vibration tolerance of the users.

A recent project funded by the Research Fund for Coal and Steel (RFCS), has resulted

in a method for verifying the performance of floors with respect to human induced

vibration. The method, referred to as the one step root mean square method (OS-RMS),

has been published in a Dutch guideline and a European guideline. The Dutch guideline

describes the complete method whereas the European guideline is limited to the so-

called hand calculation.

1.4.1 Description of the walking load

Walking differs from running as one foot keeps continuously contact to the ground

while the other foot moves. It can be described by the time history f walking load as a

periodic function with a period, T, equal to the inverse of the step frequency: � = 1�� [1.5]

A standard walking load is defined as a series of consecutive steps whereby each step

load (or footfall) is described by a polynomial [1.6]. The normalized step load is given by:

�� = �� = �� = �� !�!"!#$ , 0 ≤ � < ��0, � < 0)*� ≥ ��

, [1.6]

Where G is the person’s mass and �� is the total time during which one footis in contact with

the ground. The coefficients ! depend on the step frequency, ��, and are given in Table 1.

The duration of a single step, not to be confused with period is given by the following

formula:

�� = 2.6606 − 1.757�� + 0.3844��5 [1.7]

The step load describes the different phases of the contact between foot and ground as

shown in Figure 1.4. Figure 1.5 gives examples of the time history of the contact forces

during one footfall for two different step frequencies.


13

Figure 1. 4 Movement phases of legs and feet during walking

Table 1. 1 Coefficients K1 to K8 for given step frequency, fs

.

One individual walking cannot be the basis for the design of floors for vibration comfort but a

representative loading has to be found which covers a relevant majority of loading scenarios.

Hence, the loading is described in a statistical manner.

Chapter 1

14

Figure 1. 5 Example of the time history of the normalized

contact force for two different step frequencies

Statistical distributions of monitored case studies have shown that the step frequencies

are not correlated with the distribution of body weight, hence two probability distribution

functions are sufficient to describe the statistical variation of the loading. In the calculations

of the response, a total of 700 combinations of step frequency and body weigth (35 step

frequencies and 20 body weights). Each combinations leads to a response with a probability

of occurrence described by a joint probability of occurrence function as shown in Figure 1.6.

Figure 1. 6 Frequency distribution of body mass and step

frequency for a population of 700

The limit states mentioned in the preceding paragraphs have a clearly way to check

them in the Eurocodes as well in the NTC08 but it isn’t true for the serviceability limit state of

vibration control. Both codes prescribe that the SLS of vibration control must be checked but

don’t give any equations or other indications valid to solve this issue (only the advice to


15

design the floor to have natural frequencies higher than 3 Hz), just they recall to the scientific

literature . This paragraph gives a simplified way to assess the vibration control under human

vibration.

1.4.2 Determination of the floor response

The dynamic response of a floor structure due to walking is determined by the loading

characteristic, as described in the last paragraph, and by the structural dynamic properties of

the floor. The characteristic of the floor are described in terms of a mobility frequency

response function, FRF, or transfer function. Using this function in combination with the

standard walking load, the response of floor is obtained. The transfer function method can be

applied where the floor response is obtained either by measurement or by finite element

calculations.

In obtaining the transfer function, the excitation point and response points do not

necessarily have to coincide. Further, it is assumed that the excitation point is kept fixed, that

is, the walking path is not taken into consideration. The response point should be selected

where nuisance is to be expected and the excitation point should correspond to a point in the

walking path.

As a rule, either the locations leading to the largest mobility or the locations

representing the most practical situation should be selected. For most floors the excitation and

response point can be selected in the middle of the floor.

1.4.2.1 Floor response for a single combination of step frequency and person’s weight

The one step root mean square value (OS-RMS) represents the response of a floor that

is brought into vibration due to a person walking on that floor. It is obtained from the

measured or simulated floor mobility and the standard walking load function for a person with

given weight and walking pace.

The OS-RMS value is defined as the root mean square value over a given interval of the

frequency weighted velocity response at a point on the floor. The interval is selected starting

from the highest peak in response, see Figure 1.6. From this definition, it follows that the

interval corresponds to period, �, between one step and the next.

The weighted response is obtained by applying the following weighting function: 67�� = 189 1:1 + ;�<� =5

[1.8]

Where �< = 5.66> and 89 is the reference velocity which is taken as 1.0 mm/s.

Because of division by a reference velocity, the weighted response and the OS-RMS value are

dimensionless.

Chapter 1

16

1.4.2.2 Design value of the floor response

The design value of the floor response takes into account the possible statistical

variation in the loading. It is determined as the 90% upper limit of the OS-RMS value and can

be obtained by carrying out the following two steps:

1. Determination of the responses of the floor for all possible load combinations: step

frequency from 1.64 Hz to 3 Hz and body mass from 30 Kg to 125 Kg in steps

opportunely selected. An example of the response of one floor as a function of step

frequency and body weight is presented in the follow Figure 1.7.

2. Determination of the cumulative frequency distribution of OS-RMS values. From the

relative frequency (probability) of each step frequency and body weight combination,

the cumulative frequency of the OS-RMS value is obtained. The 90% upper limit is

defined as the OS-RMS90 design value.

Figure 1. 7 Vibration response for defined ranges of step frequency and body mass

1.4.2.3 Hand Calculation method

It is also possible to follow a simplified approach, which avoids having to calculate the

mobility function of a floor and carrying out statistical analysis. This approach, referred to as

hand calculation method, ca be applied when a floor is adequately described by a single

degree of system consisting of a mass connected to a spring and damper. The OS-RMS90

design value is obtained directly as a function of the dynamic parameters which describe such

a system, i.e., the mass ?, the stiffness �, and the damping ratio @.


17

The function can be obtained by applying the steps described in the previous paragraph

for the mobility function of system with one degree of freedom.

In applying the hand calculation method, the dynamic properties of the floor structure

need to be determined. In general, it is sufficient to the determine the following parameters for

the first mode of vibration:

- Natural frequency

- Modal mass

- Damping value

The first and the second parameters can be easily estimated using formulas in classical

text of structural dynamics. The damping, instead, is considered as a combination of damping

effects arising from:

- The type of material used in the structure;

- Furniture ed equipement;

- Permanent installations and finishing (such as lighting and ceiling);

When the mode frequency and the frequency of steps are identical, resonance can lead

to very large response amplitudes. Resonance can also occur for higher harmonics of the step

frequency, i.e. where a multiple of the step frequency coincides with the natural frequency. In

the transfer function method these possible resonance effects are implicitly taken into account.

For the hnd calculation method, however, higher modes need to be taken explicitly into

consideration.

Where higher floor modes may be relevant for design, modal mass and frequency

should be determined for each mode I of interested and the OS-RMS90 value is determined by

a SRSS rule.

A� − B�C< = :�A� − B�C<,!! [1.9]

1.4.3 Classification of vibrations

1.4.3.1 Quantity to be assessed

The perception fo vibrations by persons and the associated annoyance depends on

several aspects. The most important are:

• The direction of the vibration. The issue of floor vibration is related only to

vertical vibrations.

• The posture of people such a standing, laying or sitting;

Chapter 1

18

• The courrent activity of an occupant is of relevance for his or her perception of

vibrations, for example, persons working in the production of a factory will

perceive vibrations differently to those working in an office;

• Additionally, age and health of affected people may play a role in determing the

level of annoyance perceived.

Thus the perception of vibrations varies between individuals and can be judges in a way

that fulfils the expectations of comfort for the majority of people.

It soul be considered that the vibrations levels, which is object of this paragraph, are

relevant for the comfort of the occupants only. They are not relevant for structural integrity.

Aiming at an universal assessment procedure for human induced vibration it is

recommended to adopt the, above presented, OS-RMS as a measure for assessing floor

vibrations. Exactly the OS-RMS90, defined as the 90 fractile of all the OS-RMS values

obtained for a set of load as written in the preceding paragraph.

1.4.3.2 Floor classes

Depending of OS-RMS90 a class for a floor can be selected. Specific studies have

identified which class is compatible with determinate human activities. For example the

following table, from European Guidelines [3] give class division of floor response and

recommendation for the application of classes. Limits on vibration are also given in

International standard ISO 10137[4] which is referred in the Eurocodes. These limits are

reproduced here with the equivalent OS-RMS90 limit.

Table 1. 2 Classification of floor class and recommendation for the application of classes


19

Table 1. 3 Vibration limits specified by ISO 10137 for continuous vibration

It is considered that ISO limits are unnecessarily harsh, and testing on a number of

subjects found the limits of Guidelines to be more appropriate.

1.4.4 Hand calculation method -Design procedure

The first step in the hand calculation method procedure is to determine the basic floor

characteristic or parameters.

1.4.4.1 Determination of eigenfrequency and modal mass

In practice, the determination of floor characteristic can be performed by simple

calculation methods (analytical formulas) or by Finite Element Analysis (FEA).

In the determination of the dynamic floor characteristics also a realistic fraction of

imposed load should be considered in the mass of the floor to keep in count the serviceability

condition of the floor analyzed. The European guidelines [3] advices that experienced values

for residential and office are 10% to 20% of the imposed load.

Analytical calculation don’t require more explanations because of wide literature on the

subject. For the FEA analysis is necessary to give some advice. Indeed, if FEA is applied for

the design of a floor with respect to the vibration behavior, it should be considered that the

FEA-model for this purpose may differ significantly to that used for ultimate limit state (ULS)

design as only small deflections are expected due to vibration. A typical example is the

selection of boundary conditions in vibration analysis compared to ULS design. A connection

which is assumed to be a hinged connection in ULS may be assumed to provide full moment

connection in a vibration analysis. Another one example is the modulus of elasticity of

concrete: fort his analysis it should be considered higher ( almost 10%) than the static tangent

modulus DE�.

1.4.4.2 Determination of damping

Damping has a great influence on the vibration behavior of a floor. Independently of the

way of determining natural frequency and modal mass, damping values for vibrating systems

can be determined using tabular methods, founded upon scientific studies. The reference

Chapter 1

20

document [3] for example gives for different structural materials, furniture and finishing the

relative damping. The total damping, system damping, is obtained by summing up the

appropriate values for the different damping source.

Table 1. 4 Determination of damping

1.4.4.3 Determination of the floor class

When the modal mass and frequency are determined, the OS-RMS90 value as well as the

assignment of the floor classes can be obtained with the diagrams given by guidelines and

here partially reported. The relevant diagram needs to be selected according to the damping

characteristic of the floor in the condition of use (considering finishing and furniture).

The diagram is applied by entering the x-axis with the modal mass and the y-axis eth the

corresponding frequency. The OS-RMS90 value and the acceptance class can be read-off at

the intersection of the lines extending from both entry points.


21

Chapter 1

22

Figure 1. 8 The OS-RMS90 graph with acceptance classes for various damping values

1.4.4.4 System with more than one eigenfrequency

In some case, the floor response may be characterized by more than one natural

frequency. In these cases, as already mentioned, the SRSS rule can be used to obtain the OS-

RMS90 value of system. However, the acceptance class must be read from the table.

PREVIEW

The analysis of damage level shown by structure after large earthquakes in the past,

makes clear that the seismic response of building cannot be found in elastic behavior range.

The phenomena of earthquake response of the structure can be easily understood by energetic

considerations. Indeed, the earthquake gives a certain quantity of energy to the structure that

must be absorbed and dissipated by the structural elements.

It is surely very hard that just the elastic deformations and the structural damping can be

able to confront the energetic input given by design earthquake as for the new buildings,

because of it needs excessive sections, as well for the existing buildings, which, although

usually designed using elastic analysis, most will experience significant inelastic deformation

under large earthquakes.

It is important to note that out of elastic range does not mean failure of the structure; the

failure conditions, showed in the chapter 1, are related to the ULS achievement. Thus, modern

performance-based design methods require ways to determine the realistic behavior of

structures under such conditions.

Non linear analysis method are treated in this chapter.

2.2 The role and the use of nonlinear analysis in seismic design

Nonlinear analyses involve significantly more effort to perform and should be

approchead with specific objectives in mind. Typical instances where non linear analysis is

applied in structural earthquake engineering practice are to:

1. Assess and design seismic retrofit solutions for existing buildings;

2. Design new buildings that employ structural materials, system, or other features that

do not conform to current building code requirements;

3. Assess the performance of buildings for specific owner/stakeholder requirements

<2>

Nonlinear Structural Analysis for

Seismic Design

Chapter 2

24

2.3 Stiffness, resistance and ductility: key points of structural

analysis

Stiffness, resistance and ductility are the parameters that govern the structural response.

They can be defined for the material (stress strain curve), as well for the whole structure. (e.g.

base shear- displacement on top curve).

Figure 2. 1 Stiffness, resistance and ductility on force-displacement diagram

In reference to the Figure 2.1 a conceptual definition of these can be given:

• The stiffness is the quantity which linked external applied loads with displacement

while they are still in the elastic range.

• The resistance is the maximum load that the element can be carry and in elastic-

perfectly plastic diagram is the same value which cause the yelding;

• The ductility is the ability of an element, or of the whole structure, to show more

displacements ( or deformations) after the elastic range, without an important lost of

resistance ; numerically it is quantified by the fraction between the displacement value

on the failure with the displacement value on the yielding.

Stiffness and resistance are the main elements while a one studies linear system.

Ductility represent the structure resource after yielding to show more displacement before to

achieve the failure.

2.4 Equation of motion of an elastic-plastic system

A SDOF elastic-plastic system the relation between force and displacement is not a

straight line, as in elastic SDOF system, but a double straight line: one for elastic behavior to

the small displacement and the other starting to the yield point, horizontal in the perfect-

plasticity hypothesis, for the plastic range.

Nonlinear structural analysis for seismic design

25

According to the D’Alembert principle, the dynamic equilibrium gives the formula

below:

�� (�) + � (�) + �(�(�), � (�)) = −��(�) [2.1]

Figure 2. 2 Forces acting on a Sdof system

It is formally analogue to the equation for the elastic SDOF but it is substantially

different because of the resistance forces that are function of the instantaneous value of

displacement u(t) and its time history. In this case, the solution can be carried out only by

numerical procedures as described later in this chapter.

2.5 Linear elastic equivalent analysis: Behavior factor q

Response evaluation of elastic-plastic SDOF system is made easier by tree principles

due to Newmark, during 60’s:

- ED: Equal displacement principle which affirms the equality between maximum

displacement shown by an elastic SDOF and an elastic-plastic SDOF, with the same

initial stiffness, subjected to the same earthquake. This is true for a relatively

deformable structure, i.e. for relatively high natural period;

- EE: Equal energy which affirms that the energy stored in an elastic-plastic system

under an earthquake is the same than an elastic system, with the same stiffness of the

initial of the first one, was stored. This is valid for relatively rigid structure, i.e. for

small natural period;

- EA: Equal acceleration which affirms that, for very rigid structure �� → 0, the

maximum acceleration on the SDOF is the ground acceleration.

Consequently, the designer who would like use elastic-plastic design must:

Chapter 2

26

- For high natural period: design the structure to be able to achieve the elastic

displacement with plastic behavior;

- For small period: design the structure to be able to achieve a displacement

opportunely bigger than the elastic;

- For very small period: design the structure to resist to the maximum ground

acceleration. In this case the structure is so rigid than the small displacement cannot

premise the activation of plastic behavior.

The strategy design discussed above, for the first two principles, shows that the design

force can be reduced if the designer gives to the structure the possibility to have the

displacement required, or rather the required ductility.

Figure 2. 3 Comparison between elastic and elastic-plastic SDOF diagram

These two principles clarify how the ductility reduces the actions with respect to the

elastic values in the seismic evaluation on building. Referring to the Figure 2.3(a), the

maximum force on the SDOF is:

�� =��,��,�� → �� = ��,� ��,�� =

��,�� [2.2]

Where � = �� ,!"�# is the ductility of the SDOF system. For lower periods the EE

principle, by the equality between A and B area in Figure 2.3(b), the reduction is:

��2�� %��,� − ��&' = %��,�� − ��&�� [2.3]

simplifying �� and dividing both members for ��

(��,�� − 1*'= 2(� − ��,�� * [2.4]

in the elastic rang it is possible to obtain that


27

��,�� = ��,�� [2.6]

substituting [2.5] in [2.4] and with simple steps, it is

�� = ��,�+2� − 1

[2.7]

It is now clear that the ductility makes the design forces smaller. The quantity on the

denominator, in the both equations, is the as-called behavior factor, usually indicated by the

letter q .

The behavior factor is the essence of the linear equivalent elastic analysis. This method

consists in the reduction of linear elastic response spectra by q factor to keep in count the

inelastic resources of the structure.

Figure 2. 4 Behavior factor Vs ductility available

2.5.1 MDOF analysis

For MDOF structures this method is applied in the same way, scaling the response

spectra by q factor to have the design spectra. Although the method is conceptually simple, its

application and the quality of its results are very related to the q factor selected.

Chapter 2

28

The definition of q factor is an essential point for the seismic structural design. Its

definition is done for structural typology as interpretation of numerical analysis.

The behavior factor is conditioned by:

- Structural typology

- Material

- Ductility global level of the structure

- Hyperstaticity of the structure

- Plan regularity

- Elevation regularity

Every standard structure has its behavior factor. Special structure, for geometry or

material, needs to be studied to understand which is its natural behavior factor. The

Eurocode 8, the Eurocode on earthquake design, gives, for rc structure, a 1.5 as a minimum

value for behavior factor independent of the structural typology.

2.6 Non linear static analysis: Pushover

Pushover analysis is a method which tries to be simple to do and to understand the

results as a linear elastic analysis but at the same time to consider the inelastic behavior.

The aim is to keep in count the structural behavior during the earthquake.

In the nonlinear static procedure, the structural model is subjected to an incremental

lateral load, a forces vector or displacements, whose distribution represents the effects on the

structure expected during ground shaking. The lateral load is applied until the imposed

displacements reach the so-called “target displacement”, which represents the displacement

demand that the earthquake ground motions would impose on the structure. Once loaded to

the target displacement, the demand parameters for the structural components are compared

with the respective acceptance criteria for the desired performance state. System level demand

parameters, such a story drifts and base shear, may also be checked. The non linear static

procedure is applicable to low-rise regular buildings, where the response is dominated by the

fundamental sway mode of vibration. It is less suitable for taller, slender, or irregular

buildings, where multiple vibration modes affect the behavior.

2.6.1 Non linear static analysis for a SDOF system

Considering the SDOF in Figure 2.5 is conceptually simple explain the basis and how to

do a pushover analysis. Just one degree of freedom, the transversal displacement ,∗, is

enough to describe the state of this system. In this case the PO analysis consists in the

application to the system of one force �∗ or one displacement ,∗ increasing its intensity until

the system fails.


29

Figure 2. 5 SDOF system and the main values for PO analysis

2.6.1.1 Step 1: Load increasing

The initial value of the force, or displacement, has the only requirement to don’t be so

big to overcome the yield limit. The equations are:

,∗(.) = .,/0,�∗(1) = 1� [2.8]

Where , and � are the initial values of the forcings and . and 1 the amplification

coefficients. The forcings are increasing until the system collapses which can be defined when

it is impossible to find equilibrium with external load or when the structure arrives to a

predetermined value.

2.6.1.2 Step 2: Linearization of capacity curve

The capacity curve is a diagram which has in the x-axes the displacement and in the y-

axes the shear. Three kinds of curve are possible:

- With hardening

- Perfect

- Whit softening

As shown in Figure 2.6.

Chapter 2

30

Figure 2. 6 Differents kind of PO curve.

Once had the curve the next step is its linearization, i.e. to fit with straight line the

curve. The interpolated traits can be bi-linear or tri-linear and the chose is not unique.

2.6.1.3 Step 3: Performance Point Evaluation

The curve obtained, the as-called Capacity Spectrum, is possible to understand what is

the PP, or better the displacement required for the structure under design earthquake. There

are two methods :

-the N2 method proposed by P.Fajfar;

-the CSM method proposed by Freeman

The first one use the Newmark’s principles, above mentioned, the second one is a

method which insert a fictitious damping in the structure to keep in count the contribution of

the structural damage to dissipate energy.

2.6.2 Pushover analysis for MDOF systems

Extension of PO analysis to a MDOF isn’t immediately because of there are a lot of

problems to solve:

- Chose of the representative parameters;


31

- Chose of forcing type, displacement or load, and their application on the structure

elevation;

- Conversion in a SDOF system to do an interpretation of the results.

More detail has shown in Chopra .

2.6.2.1 Analysis Methods, Modeling and Outcome

The nonlinear stiffness and strength of the components are modeled based on a cyclic

envelope curve, which implicitly accounts for degradation due to cyclic loading that is

expected under earthquakes. Loads are applied at nodes where dynamic inertia forces would

develop, and they are monotonically increased without load reversals. A control point is

defined for the target displacement, usually at the top (roof level) of the structure. The plot of

the resulting base shear force as a function of the control point (roof) displacement is often

recognized as the “pushover curve” of the structure. The pushover curve can be further

simplified by idealizing sloping branches of elastic, post yield hardening and softening

(degrading) behavior, as shown in the Figure 2.7, and used to examine overall building

performance.

Figure 2. 7 Idealized static pushover backbone curve

2.7 Non linear dynamic analysis

Dynamic non linear analysis is, today, the most sophisticated way to predict the

structural response under an earthquake. This method consist in the direct integration of

Chapter 2

32

equation of motion in non linear field. In literature there are many methods, a commonly

denominator is to transform a system of differential equations in a system of algebraic

equations by some hypothesis on the ground acceleration.

For an inelastic SDOF system the equation of motion is:

�� (�) + � (�) + �(�(�), � (�)) = −��(�) [2.9]

Under initial condition �2 = �(0) e �2 = � (0). The equation above assumes a linear

damping but this is not necessary for what is below written in this paragraph. Seismic input is

described by a record of value ��3 = ��(�). The time step is usually assumed to be constant.

∆� = �567 − �5 [2.10]

and, inside of this time step, the system response is explicitly found. Exactly, with

displacement, velocity and acceleration knew at step i, according with [2.9]

�� 5 + � 5 +�85 = −��5 [2.11]

The response at the step i+1 will be found

�� 567 + � 567 +�8567 = −��567 [2.12]

Usually, the response inside the time step ∆t cannot be exact because of the implicitly

difficulties in the punctual definition of the forcing ��(�)and the resistance forces

�(�(�), � (�)) due an inelastic behavior. To solve the problem, a numerical procedures were

created. A numerical procedure must satisfy three important aspect:

• Convergency property for which if the time step is decreased the approximate

solution will fit better the exact solution.

• Accuracy property related to the capacity of the algorithm to give solution “near”

of the exact for every chose of time step.

• Robustness property related to the stability of the algorithm respect with the round-

off errors; to specifically little input variations must product little variations of the

output data.

The last parameter is linked to the stability and it is a very important parameter during

the analysis debug. Integration methods are divided, indeed, in conditionally stable and

unconditionally stable; the algorithm stability is related to the selected time step.

One of the most used method for the direct integration is the Newmark’s method. In the

next chapters this method is explained with particular attention to the stability condition.


33

2.7.1 Newmark’s method for a SDOF system

In 1959, N.M Newmark developed a integration methods founded on the two equation

below:

� 567 = � 5 + 9(1 − :)Δ�<�� 5 + (:Δ�)�� 567 [2.13]

�567 = �5 + (Δ�)� 5 + 9(0.5 − 1)(Δ�)<�� 5 + 91(Δ�)'<�� 567 [2.14]

The parameter 1 and : define the acceleration variation inside of an integration step and they

determine the accuracy and the stability of this method. Typical selection are : = 7' and

7? ≤ 1 ≤ 7

A . These two parameters give the acceleration interpolation law of the acceleration,

it can easily show that : = 7' and 1 = 7

A is the selection for an average acceleration value into

the step and : = 7' and 1 = 7

A is the selection for a linear law of interpolation. Details are

shown in the Figure 2.8 .

Figure 2. 8 Interpolaction function for the typical selection of parameters

With equations [2.13] and [2.14] the response can be find. More details can be found in

the literature. Here the robustness property of the algorithm are discussed.

Chapter 2

34

2.7.1.1 Newmark’s method : stability

The algorithm is stable if the time step is selected according to

Δ�� ≤

1B√2

1+: − 21

[2.15]

Where �� is the natural period of the SDOF. It follows that the average acceleration method

D: = 7' ; 1 = 7

AFis unconditionally stable, in fact:

GΔ��H → ∞ [2.16]

That is whatever selection of integration time step does not affect the algorithm stability.

Linear acceleration method D: = 7' ; 1 = 7

?F, instead, it is stable if

Δ�� ≤ 0.551 [2.17]

However, although the limitation is a finite number, it is not so little to be a limit for the

ordinary system analysis, which already requires a little time step for its accuracy.

2.7.2 Direct time history integration for a MDOF system

Equations of motion for a MDOF consist in a linear system of coupled differential

equations. Using algebraic notation, they can be written in a way similar to the equation of

motion for a SDOF system:

JK� (�) + LK (�) +M8(K, K ) = −JN��(�) [2.18]

Like the SDOF systems the action ��(�) is described by a vector of values each spaced

by a regular time step Δ�. The problem is to determine KO6P, K 567, K� 567 according to:

JK� 567 + LK 567 +M8(K, K )567 = −JN��3QR . [2.19]

When the same quantities are known at the time step i. The methods for a SDOF system

can be used to solve MDOF system using classical algebraic methods.

Usually in linear system analysis the equations of motion [2.18] can be decoupled by

modal analysis because of the resistance force are linear function of displacement. When the

system is nonlinear, the decoupling is impossible thus the equation must be solved together.

However, the expansion in modal coordinates helps to understand the most important

problems related to the direct integration. It is important and reasonable setting all parameters

starting by the elastic behavior.


35

Using eigenvectors the displacement vector is written as

K(�) = ST�U�(�).V

�W7 [2.20]

Substituting the [2.20] in [2.18] the problem is now described by eigenvectors. In

structure which shows a linear behavior just j values of N are important in the definition of

response. One method might be to select in the elastic behavior the j important eigenvectors

and substituting in [2.18] the [2.20] written just for the j eigenvectors considered.

Unfortunately, this method does not give, in general, good results so the system [2.18] must

be solved for all N eigenvectors. However, just j is enough to predict accurately the response.

Is very important to understand what is the smallest period of the j values because of it will be

utilized to select the correct parameters to have convergence, accuracy and stability.

Convergence is guaranteed by the algorithm selected.

With j known the time integration step can be selected as a smaller value then ∆� = XY72 is

required to give accuracy to the algorithm.

However, solving without decoupling involves that all modes, also the modes from j+1

to N, will be affect by the numerical calculations. This is a problem for the stability because of

the stability condition must be satisfied for all modes. Thus a very little time step might be

required using algorithm with stability condition like [2.20]. Fort these raisons for a MDOF is

very important to use unconditionally stable algorithm like the average acceleration method.

A unconditionally stable method alone, however, does not give the condition to have

accuracy because including from j+1 to N modes can affect the accuracy response. It would

be desirable to filter the response of the j modes. This can be possible by two way:

- Defining a correct damping matrix which makes less important the modes from j+1 to

N.

- Using a numerical method with numerical damping which can reduced the response

only for certain modes characterized by its period. An example is the Wilson’s

method.

To understand better the numerical damping the figure below shows the solution of

equation of motion for a SDOF undamped system in free vibration obtained by four different

methods, where one of these is the exact solution.

Chapter 2

36

Figure 2. 9

Analyzing the figure it is clear how solution by Wilson’s method shows a damped

response where the system is undamped. This characteristic can be governed because of the

time integration step is the parameter that discriminates if the numerical damping affects or

not the solution. The properties of Wilson’s method, synthetically showed in the Figure 2.30,

suggest a method to keep in count only the j important modes in a numerical solution.

Figure 2. 10 Damping and Amplitude decay versus dimensionless time integration step

Wilson’s method gives damping only starting from determinate values of time

integration step so, as the period increase with the modes, the time step can be selected in

relation with the period of j+1 mode. In this way, modes from j+1 to N will be numerically

damped while the modes from 1 to j will be keep in count with only their damping value.


37

2.8 Incremental dynamic analysis

Incremental dynamic analysis (IDA) is a parametric analysis that has recently emerged

in several different forms to estimate more thoroughly structural performance under seismic

loads. It involves subjecting a structural model to one, or more, ground motion records, each

scaled to multiple levels of intensity, thus producing one, or more curves of response

parameterized versus intensity level. This method was developed by D.Vamvatsikos and C.A.

Cornell [cit.], in 2001. While this method is founded on a simple concept, performing an IDA

requires several important steps.

1. Create the model for the structure under investigation

The model is very important to have a good analysis. It must be able to keep in count

nonlinear behavior of the structure. This step is common with the other analysis methods

mentioned before and it will be better treated in the next chapter.

2. Selection a suite of ground motion records

Studies, e.g. Shome and Cornell [cit.], have shown that for mid-rise buildings, ten to

twenty records are usually enough to provide a sufficient accuracy in the estimation of

seismic demand.

3. Selection of the study parameters

Select a ground motion intensity measure IM (e.g. the 5%-damped first-mode spectral

acceleration, the peak ground acceleration) and a damage measure DM (e.g. the maximum

top displacement, the maximum over all stories peak interstory drift ratio).

4. Run the analysis For each record, incrementally scale it to multiple levels by the scale factor SF, related

to IM factor, and run a nonlinear dynamic analysis each time. Stop incrementing when

numerical non-convergence is first encountered.

5. Postprocessing

- Interpolate the resulting IM,DM points to generate IDA curve for each individual

record;

- Define limit-states and estimate the corresponding capacities on each IDA curve;

- Summarize the IDA curves and limit-state capacities across all records into 16%, 50%

and 84% fractiles;

6. Interpretation of the results Use IDA data to better understand the behavior of the structure.

Chapter 2

38

2.9 Nonlinear Static versus Nonlinear Dynamic Analysis

Non linear dynamic analysis method generally provide more realistic models of

structural response to strong ground shaking and, thereby, provide more reliable assessment

of earthquake performance the nonlinear static analysis. Nonlinear static analysis is limited in

its ability to capture transient dynamic behavior with cyclic loading and degradation.

Nevertheless, the nonlinear static procedure provides a convenient and fairly reliable method

for structures whose dynamic response is governed by first-mode sway motions. One way, to

check this, is by comparing the deformed geometry from a pushover analysis to the elastic

first-mode vibration shape, advice already contents in the codes where PO is applicable. In

general, the non linear static procedure works well for low-rise building (less than about five

stories) with symmetrical regular configurations. For the last sentence, the codes give rules to

check the status of regular or irregular. Briefly, one building is regular when it has a compact

e regular shape in plan and it keeps these properties in the elevation without appreciable

variation of mass or rigidity between the floors.

Hovewer, even when the nonlinear static procedure is not appropriate for a complete

performance evaluation, nonlinear static analysis can be an effective design tool to investigate

aspects of the analysis model and the nonlinear response that are difficult to do by nonlinear

dynamic analysis. For example, nonlinear static analysis can be useful to

a. Check and debug the nonlinear analysis model;

b. Augment understanding of the yielding mechanisms and deformation demands;

c. Investigate alternative design parameters and how variations in the component

properties may affect response

SPO and IDA have a common incremental nature that suggests a connection between

their results. Indeed, for the buildings where PO is applicable, IDA curve fits well the

PO curve. This characteristic suggests one way to process IDA data, that is using the

same method used in PO analysis.

2.10 Quality Assurance of Building Analysis

Nonlinear analysis software is highly sophisticated, requiring training and experience to

obtain reliable results. While the analysis program technical users manual is usually the best

resource on the features and use of any software, it may not provide a complete description of

the outcome of various combinations of choices of input parameters, or the theoretical and

practical limitations of different features. Therefore, analysts should build up experience of

the software capabilities by performing analysis studies on problems of increasing scope and

complexity, beginning with element tests of simple cantilever models and building up to

models that encompass features relevant to the types of structures being analyzed. Basic

checks should be made to confirm that the strength and stiffness of the model is correct under


39

lateral load. Further validations using published experimental tests can help build

understanding and confidence in the nonlinear analysis software and alternative modeling

decisions (e.g., effects of element mesh refinement and section discretization).

Beyond having confidence in the software capabilities and the appropriate modeling

techniques, it is essential to check the accuracy of models developed for a specific project.

Checks begin with basic items necessary for any analysis. However, for nonlinear analyses

additional checks are necessary to help ensure that the calculated responses are realistic.

• Beyond familiarizing oneself with the capabilities of a specific software

package, the following are suggested checks to help ensure the accuracy of

nonlinear analysis models for calculating earthquake demand parameters:

• Check the elastic modes of model. Ensure that the first mode periods for the

translational axes and for rotation are consistent with expectation (e.g., hand

calculation, preliminary structural models) and that the sequence of modes is

logical. Check for spurious local modes that may be due to incorrect element

properties, inadequate restraints, or incorrect mass definitions.

• Check the total mass of the model and that the effective masses of the first few

modes in each direction are realistic and account for most of the total mass.

• Generate the elastic (displacement) response spectra of the input ground motion

records. Check that they are consistent and note the variability between records.

Determine the median spectrum of the records and the variability about the

median.

• Perform elastic response spectrum (using the median spectrum of the record set)

and dynamic response history analyses of the model, and calculate the

displacements at key positions and the elastic base shear and overturning

moment. Compare the response spectrum results to the median of the dynamic

analysis results.

• Perform nonlinear static analyses to the target displacements for the median

spectrum of the ground motion record set. Calculate the displacements at key

positions and the base shear and overturning moment and compare to the elastic

analysis results. Vary selected input or control parameters (e.g., with and

without P-D, different loading patterns, variations in component strengths or

deformation capacities) and confirm observed trends in the response.

• Perform nonlinear dynamic analyses and calculate the median values of

displacements, base shear, and overturning moment and compare to the results

of elastic and nonlinear static analyses. Vary selected input or control parameters

(similar to the variations applied in the static nonlinear analyses) and compare to

each other and to the static pushover and elastic analyses. Plot hysteresis

responses of selected components to confirm that they look realistic, and look

Chapter 2

40

for patterns in the demand parameters, including the distribution of deformations

and spot checks of equilibrium.

PREVIEW

Structural modeling is the ensemble of operations that allows to translate the physical

problem in a mathematical problem, whose solution gives informations on the real behavior

of the structure. The result of the modeling is the definition of the structural scheme that is

associated to the real structure. The definition of an appropriate scheme, as simple, to be quite

simply calculable, as complex, to keep in count the most important variables, it is the

principle problem in structural analysis, because of from this definition depends, more than

the numerical analysis accuracy, the reliability of results.

By the FEM method, the 3D structural models can be created without any difficulties.

Respect to the past, where the buildings were modeled, usually, as a series of 2D frames, this

represents a bigger reliability to obtainment of the response because, by them, a global

structural behavior can be studied and understood. Nevertheless, by the inherent difficulty of

the studied phenomenon, many difficulties remain which requiring that the structural engineer

must work at different levels of difficulties. In fact, while very simple schemes neglect a lot of

variables and they are, at lest in the theory, less correct, they allow for a direct interpretation

of structural behavior and a possibility of results control which isn’t property of the more

<3>

Modeling of structure for non linear

analysis

Chapter 3

42

complex schemes. Furthermore, for the simple schemes, many methods are available while,

for complex schemes, there is only one method, i.e. the FEM. The role and the use of

nonlinear analysis in seismic design

3.1 Models to describe the structural behavior

In the technical literature there are many models to describe non linear behavior of r.c.

frames. A classification can be did respect to the discretization of the problem. They are

divided in:

• Global model: the structure is represented as the union of macro-elements thus,

the number of degrees of freedom is really low;

• Member by member models: the models is obtained by the union of a simple

elements which keep in count the nonlinearity. This model are created in two

ways: distributed plasticity model, in which every point of the frame can show

nonlinearity, lumped plasticity model, in which just preselected points have

nonlinear behavior.

• Point by point models: Elements and nodes are represented by an intensive

mesh of bi-dimensional or three-dimensional elements.

Although using point by point models appears the best way to understanding the

structural behavior there are a lots factors which make difficult their practical utilization, even

only for their computational requests associated to a very intensive mesh. The correct

selection of the calculation model is strictly related to the analyzed structure and to the aims

the analyses.

By global models, the response is calculated in an approximate way but, at the same

time, in a very simple way so, they are very useful as in concept design phase as in the

verification of calculation.

A simplified model which, although with important simplifications, makes

understandable the essence of the structural behavior is always necessary to have successful

analysis. In fact, through it, setting the more complex model is possible by selection of the

principal variables and the details to keep in count; furthermore it is the key to understand and

to validate the analysis results.

Modeling of structure for non linear analysis

43

The next paragraphs speak about the principal problems which occur when a structural

model is created.

3.2 Non linear behavior of r.c. structures

The main structural theories are founded on the small displacements hypothesis and the

small deformations. These hypotheses, reasonable for serviceable conditions, allow to write

the equilibrium equations in the initial, undeformed, configuration and to consider linear

elastic behavior for the material. However, in extreme load conditions, for example under

design earthquake load, these hypotheses lose their physical meaning. This feature is evident

analyzing the damages on structures subjected to strong earthquakes in the past, of which an

example is shown in the figures 3.1 and 3.2.

Figure 3. 1 Effects of large displacements

Chapter 3

44

Figure 3. 2 Effects of large deformations

3.2.1 Geometric non linearity

For better understand the influence of the geometric non linearity on the structural

behavior a simple structural system, shown in the Figure 3.3, is considered. It is composed by

only one rigid beam restrained by a non zero compliance.

Figure 3. 3 Rigid beam restrained by a non zero compliance

The rotational equilibrium equation, written on the deformed configuration, is:

�� = �� cos � + �� sin � [3.1]

which indicates that the relation between forces and displacements is non linear. By the

hypothesis of small displacement, but not so small to be infinitesimal, the relation [3.1] can be

lynearized, id est:

�� = �� + �� [3.2]


45

Which shows that, also in small displacement, writing equilibrium equation cannot be lawful.

The correctness or otherwise is directly related to the acting axial force, as it shown by the

equation [3.2]. Furthermore by the increasing of the axial force the solution can lose the

uniqueness. If a distributed elasticity element is considered, indeed of a lumped element,

using a local reference system is necessary to evaluate equilibrium section by section. The

Figure 3.4 shows the behavior of a linear elastic beam under large displacements and the

Figure 3.5 shows the numerical comparison between the expected solution in small

displacement and in quite large displacements.

Figure 3. 4 Large displacements behavior of a cantilever

Figure 3. 5 Comparison between linear and non linear response of a cantilever

Looking to figure 3.5 indicates how the flexional behavior and axial behavior are not

decoupled. Figure 3.6 makes clear how, also in distributed elasticity, axial force is responsible

of the characteristic of the stress.

The possibility to neglect or otherwise the effects due to the geometrical non linearity

depends, essentially, to two factors. The first one is how much large the displacements are,

because also “little” displacements makes important the effects due to the nonlinearity: the

second order effects. The second one is the entity of the axial force.

Chapter 3

46

It is clear that a model for a new building, which is designed for lateral load and to

respect displacement limit imposed by the codes, cannot keep in count these effects.

When these effects are important, two different ways are usually used:

• Geometrical stiffness

A stiffness matrix related to the axial force, ��, is added to the stiffness matrix in

small displacement,��:

� = �� + �� [3.3]

• P-∆ effects

An iterative procedure where each step is loaded adding the forces developed

writing the equilibrium equations on the deformed configuration of the previous

step.

3.2.2 Mechanical non linearity

The non linearity can be within the materials stress-strains relation. Indeed, when loaded

by relatively high values of deformation, the materials, of which the section is composed by,

stop the response in their elastic range. Consequently, the global element behavior is modified

so the structural component must be adequately modeled.

Inelastic structural component models can be differentiated by the way that plasticity is

distributed through the member cross sections and along its length. For example, shown in

Figure 3.6 is a comparison of five idealized model types for simulating the inelastic response

of beam-columns. Several types of structural members (e.g., beams, columns, braces, and

some flexural walls) can be modeled using the concepts illustrated in Figure 3-6:


47

Figure 3. 6 Idealized models of beam-column elements

• The simplest models concentrate the inelastic deformations at the end of the

element, such as through a rigid-plastic hinge (Figure 3-6a) or an inelastic spring

with hysteretic properties (Figure 3-6b). By concentrating the plasticity in zero-

length hinges with moment-rotation model parameters, these elements have

relatively condensed numerically efficient formulations.

• The finite length hinge model (Figure 3-6c) is an efficient distributed plasticity

formulation with designated hinge zones at the member ends. Cross sections in the

inelastic hinge zones are characterized through either nonlinear moment-curvature

relationships or explicit fiber-section integrations that enforce the assumption that

plane sections remain plane. The inelastic hinge length may be fixed or variable, as

determined from the moment-curvature characteristics of the section together with

the concurrent moment gradient and axial force. Integration of deformations along

the hinge length captures the spread of yielding more realistically than the

concentrated hinges, while the finite hinge length facilitates calculation of hinge

rotations.

• The fiber formulation (Figure 3-6d) models distribute plasticity by numerical

integrations through the member cross sections and along the member length.

Uniaxial material models are defined to capture the nonlinear hysteretic axial stress-

strain characteristics in the cross sections. The plane-sections-remain-plane

assumption is enforced, where uniaxial material “fibers” are numerically integrated

Chapter 3

48

over the cross section to obtain stress resultants (axial force and moments) and

incremental moment-curvature and axial force-strain relations. The cross section

parameters are then integrated numerically at discrete sections along the member

length, using displacement or force interpolation functions (Kunnath, Spacone ).

Distributed fiber formulations do not generally report plastic hinge rotations, but

instead report strains in the steel and concrete cross section fibers. The calculated

strain demands can be quite sensitive to the moment gradient, element length,

integration method, and strain hardening parameters on the calculated strain

demands. Therefore, the strain demands and acceptance criteria should be

benchmarked against concentrated hinge models, for which rotation acceptance

criteria are more widely reported.

• The most complex models (Figure 3-6e) discretize the continuum along the member

length and through the cross sections into small (micro) finite elements with

nonlinear hysteretic constitutive properties that have numerous input parameters.

This fundamental level of modeling offers the most versatility, but it also presents

the most challenge in terms of model parameter calibration and computational

resources. As with the fiber formulation, the strains calculated from the finite

elements can be difficult to interpret relative to acceptance criteria that are typically

reported in terms of hinge rotations and deformations.

3.2.2.1 Distributed versus concentrated Plasticity elements

While distributed plasticity formulations model variations of the stress and strain

through the section and along the member in more detail, important local behaviors, such as

strength degradation due to local buckling of steel reinforcing bars or flanges, or the nonlinear

interaction of flexural and shear, are difficult to capture without sophisticated and numerically

intensive models. On the other hand, phenomenological concentrated hinge/spring models,

may be better suited to capturing the nonlinear degrading response of members through

calibration using member test data on phenomenological moment-rotations and hysteresis

curves. Thus, when selecting analysis model types, it is important to understand

1. the expected behavior;

2. the assumptions;


49

3. the approximations inherent to the proposed model type.

While more sophisticated formulations may seem to offer better capabilities for

modeling certain aspects of behavior, simplified models may capture more effectively

the relevant feature with the same or lower approximation. It is best to gain knowledge

and confidence in specific models and software implementations by analyzing small test

examples, where one can interrogate specific behavioral effects.

3.2.3 Stress-Strain relation for non linear analysis

In modeling the hysteretic properties of actual elements for analysis, the initial stiffness,

strength, and post-yield force-displacement response of cross sections should be determined

based on principles of mechanics and/or experimental data, considering influences of cyclic

loading and interaction of axial, shear, and flexural effects. Under large inelastic cyclic

deformations, component strengths often deteriorate due to fracture, crushing, local buckling,

bond slip, or other phenomena. If such degradations are included through appropriate

modifiers to the stiffness and internal forces, the model can simulate most regular materials

and devices experiencing hysteretic behavior (Ibarra et al., FEMA 2009)

3.2.3.1 Stress-strain relation for r.c. section

In subdividing section by fiber different stress strain relation can be used for the

different parts of the section. One r.c. section is formed by three different regions

- Steel rebar

- Unconfined concrete

- Confined concrete

Chapter 3

50

Figure 3. 7 fiber division of section

3.2.3.1.1 Kent and Park Model

The Kent and Park model , modified by Scott et al., is a model that consider confining

effects of the stirrups on the concrete. The backbone curve, or rather the envelope curve of the

hysteretic cycles, is shown in the next figure.


51

Figure 3. 8 Kent and Park model

The curve are defined by the below parameters

�� Cylindrical stress failure of concrete

� Factor for resistance increasing for stirrups confinment

� = 1 + ��

�� stirrup percentage (stirrups volume fract concrete nucleum volume)

�� yielding stress of rebar

�� Failure deformation. It can be defined by the Scott et al. equation

�� = 0.004 + 0.9�� # ��300%

��& Deformation with maximum stress

��& = 0.002�

( slope of the softening curve

( = 0.5

3 + 0.29��145�� − 1000 + 0.75��,ℎ′/� − 0.002�

ℎ′ wide of concrete nucleum

/� stirrup step

Chapter 3

52

The exhaust path is on a straight line from the point (�1 , �1), on the skeleton curve, to

the point (�3, 0) and after go again to the origin axes on the abscissa, because of this model

neglects the contribution to traction of the concrete.

�3��& = 0.145 4 �1��&56 + 0.13 4 �1��&5 per 7 898:;< < 2 [3.4]

�3��& = 0.707 4 �1��& − 25 + 0.834 per 7 898:;< ≥ 2 [3.5]

The reload path is, when �3 is reached again, on the same straight line of the discharge.

Figure 3. 9 Load and upload cycle of Kent and Park Model

3.2.3.1.2 Menegotto and Pinto Model

To describe the nonlinear behavior for steel rebars Menegotto e Pinto model, modified

by Filippou et al. , can be used. This model includes isotropic deformations for hardening.


53

Figure 3. 10 Menegotto and Pinto model

The relation is expressed by the below relation:

@∗ = B�∗ + (1 − B)�∗C1 + �∗DEF/D

[3.6]

where

@∗ = @ − @1@H − @1 [3.7]

�∗ = � − �1�H − �1 [3.8]

I = IH − JFK

J6 + K [3.9]

Wich

@ Normal stress

� Axial deformation

(�1 , @1) Unload point wich is (0,0) in the elastic range

(�H, @H) Intersection of the two asymptotes

B Stiffness decrease factor

IH, JF, J6 constants that are respectively 20.0 , 18.5 , 0.15

Chapter 3

54

K Difference between maximum value of deformation in the load direction

and �H

( slope of the softening curve

PREVIEW

Maranello is a one of the most attractive place of Emilia-Romagna (Italy). Almost

200.000 tourists, from all parts of the world, come there to visit the Galleria Ferrari so it is the

most known regional museum. This capacity to attract tourists is due to the fame of the Ferrari

able to bypass different languages and national boundaries. Since the 90’s of the last century,

and in the more recent years, have grown, very fastly, the tourists number and the tourist

accommodation, to such an extent that today the tourism represents a very important

economic sector. To increase this sector the administration of Maranello has done a

requalification project to valorize the aspects linked to identity and tourist attraction. Inside of

this projet

4.1 The redevelopment of areas adjacent to the gallery Ferrari

Maranello is a one of the most attractive place of Emilia-Romagna (Italy). Almost

200.000 tourists, from all parts of the world, come there to visit the Galleria Ferrari so it is the

most known regional museum. This capacity to attract tourists is due to the fame of the Ferrari

<4>

The new redevelopment of areas

adjacent to the gallery Ferrari

Chapter 4

56

able to bypass different languages and national boundaries. Since the 90’s of the last century,

and in the more recent years, have grown, very fastly, the number of tourists and the tourist

accommodations, to such an extent that today the tourism represents a very important

economic sector. To increase this sector the administration of Maranello has done a

requalification project to valorize the aspects linked to identity and tourist attraction. Inside of

this project, the redevelopment of the Area in front of the Galleria Ferrari, actually used as

parking and green area, has a very large relative importance. The qualification of this area

aims to offer urban functional services of the museum.

Figure 4. 1 Dino Ferrari Street where there is Galleria Ferrari

4.1.1 The design competition: Plaza and Tower Galleria

Ferrari

For the redevelopment of area the Maranello administration did a design competition for

the upgrade as square of the area in front of the Galleria Ferrari, for a new panoramic tower

30 m tall, for a new tourist accommodation and qualification of the museum access paths.

The area in front of the Galleria Ferrari will be upgraded as a pedestrian space e public

square for events and manifestations. In the new square there will be a structure used for

The new redevelopment of areas adjacent to the gallery Ferrari

57

tourist information office and tourist acceptance. The new space will be a public square for

the access in Galleria Ferrari.

Figure 4. 2 The red line indicates the boundaries of the design competition area

Currently, the office for the tourists is inside of the museum and it does not have more

area inside to increase its services. The choice to place it outside of the museum building,

must be done in a way to ensure the recognition of the place and, at the same time, the

autonomy of that structure respect to the Galleria Ferrari.

The structure is a panoramic tower, 30 m tall from the square ground and it will placed

in a way to not have interferences with the public underground parking, which will build

under of the square. The tower will become a privileged point to observe (the center, the

mountains around of Maranello, the Fiorano trak where the Ferrari cars are often tuned) and,

at the same time, it will become a vertical element of identification of the Museum structure.

Chapter 4

58

4.1.2 The winning project

The winning project was the so-called “Per uno spazio urbano di qualità” (For a quality

urban place) designed by Piero Lissoni (www.lissoniassociati.com) and structural designed by

Francesco Iorio (www.studioiorio.net) .

Figure 4. 3 View of the whole place as designed in the winning project

The aim of the project is to do an architectonic building with a clear and well defined

geometry, which is characterized by the selection of modern materials without any decorative

additions, in a way to don’t compete with the architecture of the Galleria Ferrari but to define

together a quality urban space.

Figure 4. 4 View of the new panoramic tower


59

Figure 4. 5 View of the new panoramic tower on an other viewed point

The structure for the tourist reception and the observation point were grouped in only

one “symbol building”, which is 30 m tall from the ground of the square. The using of

innovative material like a polycarbonate makes clearer the communicative function of the

building: shell of various measures compose the translucent skin, which is interrupted only by

the windows opened on the landscape, while a special retro illumination transforms the

building in a luminescent body which illuminates the square. The retro illumination suggests a

symbolic use, for example it can be red when a special event related to the Ferrari world

occurs.

Chapter 4

60

Figure 4. 6 Night view of the tower whit its retro illumination

The ground level of the tower includes the tourist reception point which is linked on the

square, with the access paths to the Galleria Ferrari and underground parking.

The tower is a vertical element with square plan, where there are the lift systems and the

stairs, while the covered terrace is an horizontal cantilevered element. The terrace would be

like a new access path to the Galleria and it is placed in a way to offer to the visitors, by its

overtures, a suggestive panorama. It is possible to see the core production of Ferrari, the

Fiorano track, the historical center of Maranello and the hills around of the city.


Figure 4.

The redevelopment of the parking in front of the Galleria Ferrari is done by the creation

of two green ramp which realigns the irregular perimeter of the place and they define a

internal rectangular square.

will be done, is characterized by a concrete pavement which is signed by the same regular

mesh of the cladding of the panoramic tower. A

the only exceptions are the suspended benches, made by the concrete, and the green area

which, in a random way, are inserted in the


Figure 4. 7 Planimetry with the tower position marked



square. The true square, where quotidian activities and temporary events


mesh of the cladding of the panoramic tower. A kind of bar code printed in large scale where

y exceptions are the suspended benches, made by the concrete, and the green area

, in a random way, are inserted in the footprint of the incision.

61



The true square, where quotidian activities and temporary events


printed in large scale where

y exceptions are the suspended benches, made by the concrete, and the green area

Chapter 4

62

Figure 4. 8 Some views of the square

PREVIEW

The structural design of the new Maranello tower as been done in this chapter. The

tower has been designed to satisfy the requirements of the Italian National Code.

The approach is to follow the integrally structural design path from the architectonic

design.

Structural FEM Analysis is run by:

- MIDAS/GEN2012 v.2. (http://www.MidasUser.com), licensed in Academic version

from CSPFea (http://www.cspfea.net/midas_gen.php)

5.1 Reference documents

5.1.1 National Codes

- L. 5.11.1971, n° 1086 – “Norme per la disciplina delle opere in conglomerato

cementizio armato, normale e precompresso ed a struttura metallica”.

- D.M. LL. PP. 11.3.1988 – “Norme tecniche riguardanti le indagini sui terreni e sulle

rocce, la stabilità dei pendii naturali e delle scarpate, i criteri generali e le

prescrizioni per la progettazione, l’esecuzione il collaudo delle opere di sostegno

delle terre e delle opere di fondazione”.

- Circ. Min. LL. PP. 24.9.88 – “Istruzioni riguardanti le indagini sui terreni e sulle

rocce, la stabilità dei pendii naturali e delle scarpate, i criteri generali e le

<5>

The structural design of the new tower

in Maranello

Chapter 5

64

prescrizioni per la progettazione, l’esecuzione e il collaudo delle opere di sostegno

delle terre e delle opere di fondazione”.

- Ord. P.C.M. n° 3274 20.03.2003 – “Primi elementi in materia di criteri generali per

la classificazione sismica del territorio nazionale e di normative tecniche per le

costruzioni in zona sismica”.

- Ord. P.C.M. n° 3316 02.10.2003 – “Modifiche ed integrazioni all’ordinanza del

Presidente del Consiglio dei Ministri n.3274 del 20 marzo 2003”.

- Ord. P.C.M. n°3431 03.05.2005 – “Ulteriori modifiche ed integrazioni all’ordinanza

del Presidente del Consiglio dei Ministri n. 3274 del 20 marzo 2003, recante «Primi

elementi in materia di criteri generali per la classificazione sismica del territorio

nazionale e di normative tecniche per le costruzioni in zona sismica”.

- D.M. 16.02.07 – “Classificazione e resistenza al fuoco di prodotti ed elementi

costruttivi di opere da costruzione”.

- D.M. 14.1.2008 – “Norme tecniche per le costruzioni”.

- Circolare 2 febbraio 2009, n. 617 del Ministero delle Infrastrutture e dei Trasporti

approvata dal Consiglio Superiore dei Lavori Pubblici - "Istruzioni per

l'applicazione delle Nuove norme tecniche per le costruzioni di cui al decreto

ministeriale 14 gennaio 2008”.

- Allegato al voto n.36 del 27.07.2007 – “Pericolosità sismica e criteri generali per la

classificazione sismica del territorio nazionale”.

5.1.2 UNI EN documents

- UNI EN 206-1/2006 – “Calcestruzzo: specificazione, prestazione, produzione e

conformità”.

- UNI EN 11104/2004 – “Calcestruzzo: specificazione, prestazione, produzione

e conformità – Istruzioni complementari per l’applicazione della EN 206-1”.

- UNI 9502/2001 – “Procedimento analitico per valutare la resistenza al fuoco degli

elementi costruttivi di conglomerato cementizio armato, normale, e precompresso”.

- UNI 9503/2007 –“Procedimento analitico per valutare la resistenza al fuoco degli

elementi costruttivi d’acciaio”.

Structural Design of new Tower in Maranello

65

5.1.3 European codes

- EN 1990: Basis of structural design

- EN 1991: (Eurocode 1) Actions on structures

- EN 1992: (Eurocode 2) Design of concrete structures

- EN 1993: (Eurocode 3) Design of steel structures

- EN 1994: (Eurocode 4) Design of composite steel and concrete structures

- EN 1997: (Eurocode 7) Geotechnical design

- EN 1998: (Eurocode 8) Design of structures for earthquake resistance

- EN 1999: (Eurocode 9) Design of aluminium structures

5.1.4 Other documents

- Heinemeyer Ch., Feldmann M., Caetano E., Cunha A., Galanti F.,Goldack A.,

Hechler O., Hicks S., Keil A., Lukic M., Obiala R.,Schlaich M., Smith A.,

Waarts P., 2007, RFCS-Project: Human induced Vibration of Steel Structures

– HIVOSS: Design Guideline and Background Report.

5.2 Building description

The structure is formed by a r.c. core that is 30.45 m high from the ground level. Inside

of this core, there are the lifts and the stairs to arrive up to the last covered panoramic terrace

level. The core section is square, 5.40 m x 5.40 m, and with 0.25 m of thickness.

Each level is 2.72 m high, except for the ground level (3.05 m), the 9th

level (2.21 m)

and the 10th

level, i.e. terrace panoramic level, (3.40 m)

The structure has been made seismically independent of the horizontal structure on the

ground level through a gap between the two structures.

The panoramic terrace plan, made by steel, is cantilevered by rc tower core for 12.05 m

and it has irregular overtures on the both side.

The structure system is on piles.

The figures below show the details of the structure.

Chapter 5

66

Figure 5. 1 Global view of the tower and of the adjacent structures


67

Figure 5. 2 Underground Level -1

Chapter 5

68

Figure 5. 3 Level 0


69

Figure 5. 4 10th level. The panoramic terrace

Chapter 5

70

Figure 5. 5 Coverage floor


71

Figure 5. 6 Section A-A

Chapter 5

72

Figure 5. 7 Section B-B’


73

Figure 5. 8 Section C-C

Chapter 5

74

Figure 5. 9 Section D-D


75

5.3 Concept design

This paragraph provides informations about the preliminary design phases to establish

the main parameters, which have not been yet defined by the architectonic design, that define

the size of the structural elements.

The first step is the selection of a structural scheme that fits with the architectonic

scheme. The dimensions of the core of the tower are already known by the architectonic

design (external shape and minimum internal dimensions due to the stairs and the lifts).

However, some considerations must be done. The core thickness, in the designed

configuration, offer a good resistance to fire and thicknesses less large than 25 cm are difficult

to realize for geometric limits related to insertion of rebars and casting concrete.

Thus, the core needs only of the definition of the rebar. The panoramic terrace,

cantilevered from rc core, must be totally designed, because only its volume is known.

The first step is the selection of a structural scheme that fits with the architectonic

scheme. The core of the tower is already known by the architectonic design and it needs only

the definition of the reinforcement. The cantilever, of the panoramic terrace, instead must be

totally designed, because of only its volume is known.

5.3.1 Preliminary design of the panoramic terrace

The idea is to link the 10th level with the level above to have a big truss as high as the

last floor, 3.40 m. However, the overtures required on both side do not allow to use a common

truss model because diagonals, struts and ties cannot be inserted freely along the sides of the

volume. Where it is possible, studs are inserted whit fixed joints so they are capable of

transferring and resisting bending moments as a quadrangular element in a Vierendeel Truss.

In this way, inferior beams, on the 10th level, and superior beams, on the 11th level, are

coupled by quadrangular elements which give resistance to global deflection of the cantilever

by their bending work.

Chapter 5

76

Figure 5. 10 Panoramic terrace plan

5.3.1.1 Gravity load analysis

The considered gravity loads acting on the cantilevered part have been reported in the

tables below.

Table 5. 1 Gravity loads on the 11th level

Gravity loads on the 11th level

Structural dead load G1k

Steel sheets + slab 2.25 kN/m2

Secondary beams (IPE 240 @2m) 0.79 kN/m2

G1k= 3.04 kN/m2

Non structural dead load G2k

Slab cover 2.00 kN/m2

Equipments 0.50 kN/m2

Claddings 1.00 kN/m2

G1k= 3.50 kN/m2

Live load Q1k

Category NTC08 C2 4.00 kN/m2

Q1k= 4.00 kN/m2

Table 5. 2 Gravity load on the coverture

Gravity loads on the coverture



G1k= 0.79 kN/m2


Sandwich panels 0.20 kN/m2



G1k= 3.00 kN/m2

Live load Q1k

Category NTC08 H 0.50 kN/m2

Q1k= 0.50 kN/m2

In addiction there is the weight of the lateral claddings equal to 0.50 kN/m2.


77

To design the size of the structural element, a simple analysis has been carried out: the latest

two levels have been considered as only one cantilevered beam with a cross section defined

by the structural elements of the transversal section of each floor. The surface loads have been

converted in line loads by multiplying for the transversal dimension (6.00 m) of the

cantilever. By this way, a preliminary design, looking to the main problems of a cantilever,

i.e. deflections and vibrations control for human comfort, can be done..

5.3.1.2 Design against deflection control

According to §4.2.4.2.1 NTC08 maximum deflection must be limited at 1/125 of the

cantilever span when it is loaded in serviceable limit states by the as-called Combinazione

frequente (Frequently combination) used for reversible limit state.

�� + �� + � +��

[5.1]

Where �1, ��and �2 are already defined, � is the action dues to prestress, �1� is 0.7 for

the 11th

level and nil for the coverture. Applying [5.1] for the loads acting on the conventional

cantilever (with other 6 kN/m to keep in count the weight of the primary beams that will be

calculated), the simple study model of volume cantilevered from the core of tower is ready

and it is shown in the Figure 5.7.

Chapter 5

78

Figure 5. 11 The simple model used to understand the cantilever behavior

The maximum deflection is

� = ��8�� [5.2]

Whit the gave limit �� = �� the bending stiffness is obtained by simple algebraic

calculation:

�� !� = ��8�� [5.3]

then:

�� !� = ��8�� 1� = 1125 => �� = 1258 ��$ = 2.31�15'((� [5.4]

5.3.1.3 Design against vibration control

The panoramic space can be classified as a meeting area to read the classes related to

the OS-RMS90 values in the table in the European guidelines .


79

Table 5. 3 Allowed OS-RMS90 related of function of the floor

For cantilever dynamic parameters are directly obtainable by analytical calculation and they

are given in the Figure below.

Figure 5. 12 Dynamic parameters for beam variously restrained and with distributed mass

Chapter 5

80

The total damping is calculated as sum of more than one value as shown in the following

Figure which shows that the total damping is 2%.

Figure 5. 13 Calculation of total damping

With damping value known the correct ) −+∗ can be selected and it is shown in the Figure

5.10.

By manipulating the selected equations in the Figure 5.8, it is possible to have a direct

relation between frequency and modal mass. This equation is:

) = 12-. 3��0.24 +∗0.642$ [5.5]

This equation can be plotted on the graph in the Figure 5.10 for different values of bending

stiffness EI. For a selected beam, with defined mass for unit length, from the 2nd equation

selected in Figure 5.8 the modal mass is known and it is the same for every eigenfrequency.

Plotting also this equation in the same graph, it is possible to select a correct stiffness to

achieve the desired class because the intersection, between the constant mass curve and the

curves from [5.5], gives the modal mass and the frequency for gave EI.


81

Figure 5. 14 The graph for 2% damping

For the examination case, using again the frequently combination, the total mass for unit

length of the cantilever is 1910 kg/m. Looking the Figure 5.11, considering table 5.3, bending

stiffness design can be selected. From this, using the value obtained in §5.3.1.2 as design

value seems reasonable.

Chapter 5

82

Figure 5. 15 The f-M* point for different EI value

5.3.1.4 Structural size of elements

In the two last paragraphs the value of bending stiffness has been defined. Now, let us to

obtain this value trough the design of the elements. The dimension of the I section on each


83

corner of the section obtained by coupling 11th with the 12th will be defined in this paragraph

using a simplified reasoning.

Figure 5. 16

Usually, respect to a simple steel section, making a composite section the inertia modulus can

be tripled. By this consideration the Inertia modulus required is:

�345 = 6��7345�8944: 3 [5.6]

However, the means used to obtain the request in [5.6] neglect the shear deformations which

are important because of the intrinsic deformability of the coupling system. From this

observation the requirement in [5.6] is, in lump sum, multiplied by 1.5.

The inertia modulus of the section in figure 5.12, without slab contribution, is:

�;<; = 4 =�> + ? @ℎ2B�C [5.7]

Considering only the transport:

�;<; = 4? @ℎ2B� = ?ℎ� [5.8]

Chapter 5

84

Using [5.8] and [5.6] multiplied by 2 the minimum area for the corner element has been

carried out.

? = �345ℎ� [5.9]

By this process an HEB500 has been selected.

5.3.2 Preliminary composite floor design

The cantilever part is a complex system where more than one mechanism is acting.

Once the principal mechanism of cantilever is opportune designed, the definition of the

composite floor of the terrace against deflection and vibration must be done.

Similarly to what has just done in the last paragraph, the design is done against

deflection and vertical vibration control. The design is done in this way:

Figure 5. 17 The selected SRC floor


85

- The transverse section of the composite SRC floor is known and it is given in

Figure 5.17.

- The maximum distance between support elements, according to vibration

control, is selected;

- The deflection of the composite floor is checked thanks to the technical data

sheets of the composite floor.

5.3.2.1 Design against vibration control

The composite floor has been modeled as beam element with 5.15 m width. For the

design the composite floor have been considered hinged on both side.

Considering the simplified scheme in figure 5.12 the design can be done.


The mass for unit length of SRC floor is known by the load analysis in 5.3.1.1combined

with the transversal dimension of the floor and it is 4280 kg/m. Only the span must be

defined.

Assigning different value of the span l, the selection of the best length span to accomplish

vibration control can be done in the plan ) −+∗.

Chapter 5

86

The next table shown the value for the different span plotted in the figure 5.13.

Figure 5. 19 The OS-RMS90 for different span


87

Table 5. 4 f-M* value for different span

The span has been designed 2 meters length.

5.3.2.2 Deflection check

Helped by technical data sheets the check of the deflection condition can be easily done.

The load is 3.04 �D�E + 0.64.00 �D�E = 5.44�'/(�

Figure 5. 20 Maximum length for given load

The maximum length is 2.84 m that is major than the designed length for vibration control.

The check is satisfied.

Inertia

momentSpan

Young

modulus

Mass for

unit length

Massa

ModaleFrequency

I L E µ M* f

[m4] [m] [N/m

2] [kg/m] [kg] [Hz]

4.939E-05 1.50 3E+10 4987 3740 12.07

4.939E-05 1.75 3E+10 4987 4364 8.87

4.939E-05 2.00 3E+10 4987 4987 6.79

4.939E-05 2.25 3E+10 4987 5610 6.40

4.939E-05 2.50 3E+10 4987 6234 6.07

4.939E-05 2.75 3E+10 4987 6857 5.79

4.939E-05 3.00 3E+10 4987 7481 5.54

4.939E-05 3.50 3E+10 4987 8727 5.13

Chapter 5

88

5.3.3 Preliminary design of subbeam

The beams which support the SRC floor has been designed. The scheme is with hinge

on both side while the mass is given by the §5.3.1.1, in frequently combination, multiplied for

the span (2 meters).

For the subbeam the dynamic parameters are directly obtainable by analytical calculation and

they are given in the Figure below.


The total damping is always 2%.

As it has already done, it is possible to have a direct relation between frequency and

modal mass. In this case the equation is:

) = 12-. 3��0.49 +∗0.502$ [5.10]

This equation can be plotted on the graph in the Figure 5.22 for different values of the inertia.

The modal mass is known and it is the same for every eigenfrequency. Plotting also this

equation in the same graph, it is possible to select a correct stiffness to achieve the desired

class because the intersection, between the constant mass curve and the curves from [5.5],

gives the modal mass and the frequency for given I.


89

Figure 5. 22 The f-M* point for different I value

A IPE200 has been selected looking also at the space geometrically available.

Chapter 5

90

5.3.4 Preliminary design of post tensioning force

For this nature the cantilevered part tends to rotate around the point called A in the

figure below. This implies that in serviceability condition, the core part below of the point B

is subjected to traction. As the concrete has a very little resistance against the tensile stress,

applying a prestress force is a good way to avoid problems like cracking also in serviceability

condition.

An other important problem related to important dimension of the cantilever is the

eccentricity of the normal load respect to the base section gravity center. The prestress

solution can be limited also this eccentricity, bringing the normal resultant of all tower near of

the centroide of the base section.

The design iter is to estimate the tensile force given by the cantilever and after choosing

the prestress force.

To achieve the solution for both problems, a prestress force to be applied at the point B

must be chosen in such a way as to center the vertical load since the tip of the core until the

base section.

Figure 5. 23

The post-tensioning solution is a good solution for at least two raisons:

- It is very simple to implement;

- It does not influence the behavior in the base critical section;

5.3.4.1 Estimate of post-tensioning force

In order to estimate the value of the post tensioning force the simply model in Figure

5.14 can be used.


91

Figure 5. 24 Cantilever equilibrium without prestress

It is simple to understand that the value of the post tension, which centers the vertical

load respect to the centroide of the core, is defined by the sum of absolute value of H and HI.

In formulas:

HI = � �J K9�:4L432�894� [5.11]

H = � @1 + �J K9�:4L432�894� B [5.12]

So post tensioning force is:

� = � @1 + �J K9�:4L43�894� B [5.13]

Substituting the current values in serviceability condition (See Figure 5.7) the post

tensioning value is carried out:

� = � @1 + �J K9�:4L43�894� B [5.14]

� = 84.78�'( ∗ 612.04 + 0.257( @1 + 12.04(5.40( − 0.25(B = 3478�' [5.15]

Chapter 5

92

Once that the structural FEM model is created this value can be opportunely refined in

relation with the numerical result which can be a little bit different because of the intrinsic

simplifications present in simple model used, not last the correct position of the center of

gravity.

5.4 Materials

The materials are selected to have characteristic compatible with NTC08 §11

5.4.1 Concrete 28/35

Concrete C28/35

Cubic characteristic stress Rck= 35.00 N/mm2

Cylindrical characteristic stress fck= 28.00 N/mm2

Cylindrical characteristic stress -

Average fcm=fck+8= 33.00 N/mm2

Characteristic tensile stress fctm=0.3(fck)2/3= 2.77 N/mm2

Characteristic tensile stress percentile

5% 0.7 fctm= 1.94 N/mm2

Characteristic tensile stress percentile

95% 1.3 fctm= 3.60N/mm2

Average value tensile stress for

bending fcfm=1.2 fctm= 3.08 N/mm2

Instantaneous Young’s modulus Ecm=22000 (fcm/10)0.3 = 31475 N/mm2

Poisson’s coefficient ν= < 0.2

coefficient of thermal expansion α= =10E-5 °C-1

The partial safety factor is NO = 1.5.

5.4.2 Rebar B450C

B 450 C

Characteristic yield stress fyk= 450 N/mm2

Characteristic failure stress ftk= 540 N/mm2

Young’s modulus E = 210000 N/mm2

The partial safety factor is NO = 1.15

5.4.3 Structural steel S355

S355

Characteristic yield stress fyk= 355 N/mm2

Characteristic failure stress ftk= 490 N/mm2

Young’s modulus E = 210000 N/mm2


93

The partial safety factor, for buckling and resistance, is 1.05.

5.5 Load analysis

Once the elements sizes are defined for all parts of the tower, it is possible to correctly

estimate the magnitude of all kind of loads that will act on the structure during its expected

life.

5.5.1 Vertical loads

According with NTC08 the vertical loads are divided in three different categories:

- Structural dead load ��;

- Superimposed dead load ��

- Live load � the self weight load of the structural element is not write in the table because it is

automatically keep in count by the software.

5.5.1.1 All levels under the 10th

Table 5. 5


Deck 4.00 kN/m2

G1k= 4.00 kN/m2





G2k= 4.00 kN/m2

Live load Q1k


Q1k= 4.00 kN/m2

5.5.1.2 11th

level

Table 5. 6


Steel sheets + slab 2.25 kN/m2


G1k= 3.04 kN/m2





G2k= 3.50 kN/m2

Chapter 5

94

Live load Q1k


Q1k= 4.00 kN/m2

5.5.1.3 Coverture

Table 5. 7



G1k= 0.79 kN/m2


Sandwich panels 0.20 kN/m2



G2k= 1.20 kN/m2

Live load Q1k

Category NTC08 H 0.50 kN/m2

Q1k= 0.50 kN/m2

5.5.2 Snow load

The snow load, according to NTC08 §3.4, is calculated by the following equation

�8 = P� �8� QR Q9 [5.16]

Where:

�8 Snow load

P� Roof shape coefficients

�8� Characteristic value of the snow load [kN/m2] with the annual probability of

exceedence set to 0,02

QR Exposure coefficients

Q9 Thermal coefficients


95

Referring to Figure 5.15, the characteristic snow load is 1.50 kN/m2 for an altitude

minor than 200 m.

Figure 5. 25

The exposure coefficient for a normal topographic class is 1 and also the thermal

coefficient.

As shown in figure 5.16, from NTC08, the roof shape coefficient for a single coverture,

with slope minor than 30°, is 0.8.

Chapter 5

96

Figure 5. 26

The snow load can be calculated and it is:

�8 = 1.2�'/(� [5.17]

5.5.3 Wind load

Wind load acting on the structure is calculated as NTC08 §3.3.

5.5.3.1 Basic Value

The basic value of wind action with the annual probability of exceedence set to 0,02, i.e.

reference period 50 years, is carried out by the NTC08 tables.

Figure 5. 27


97

5.5.3.2 Basic velocity pressure

The basic velocity pressure is calculated from the Basic Value of wind speed as follow:

�I = 12S �3TI� = 391'/(� [5.18]

5.5.3.3 Exposure coefficient

The exposure coefficient formula is:

U46V7 = W3�U9 ln @ VVZB[7 + U9 ln @ VV\B] ^_`V ≥ V��K [5.19] U46V7 = U46V��K7

The parameters depend of the terrain category, which is calculated by roughness class of

terrain as shown in the following tables.

Chapter 5

98

5.5.3.4 Wind Pressure

The wind pressure acting on the structure must be calculated with the following

equation:

^ = �IU4UbUc [5.20]

Where the dynamic coefficient is assumed equal to one, and the shape pressure

coefficient Ub is selected according with the next table.

Figure 5. 28

The wind pressure profile is Shown in Figure 5.19 for Ub = 1.


99

Figure 5. 29

5.5.4 Seismic load

The seismic load is calculated helped by the file SPETTRINTC.xls

(http://www.cslp.it/cslp/index.php?option=com_docman&task=doc_details&gid=3280&&Ite

mid=165 ) released by the “Consiglio Superiore dei Lavori Pubblici” which uses the NTC08

seismic data of the whole Italian territory.

5.5.4.1 Geoseismic analysis

The geotechnical characterization of the ground for seismic analysis requires, as

indispensable element, that the profile of the shear waves speed in the subground is known at

least for the first one 30 meters. To measure the shear waves speed, MASW (Multichannel

Analysis of Surface Waves) analysis have been performed and the result is shown in the

following Figure.

Chapter 5

100

Figure 5. 30 The Vs,30 profile

The mean velocity d8$\ in the first 30 meters of profundity using NTC08 formula is 339

m/s which is related a terrain category C.

5.5.4.2 Phase 1

Figure 5. 31 Phase 1:selction of the coordinates


101

5.5.4.3 Phase 2

The use category class is III so the coefficient Ue = 1.5. the figure below shows the

input data and also, for each limit state, the reference period for seismic action.

Figure 5. 32 Phase 2: selectionof the main parameter of the building

Chapter 5

102

5.5.4.4 Phase 3.2 SLE spectra

Figure 5. 33 SLO spectrum

Figure 5. 34 SLD spectrum


103

5.5.4.5 Phase 3.3 SLV and SLC spectra

To obtain the inelastic SLV spectrum, a value for behavior factor must be previously

selected. However, among the various structural model considered by NTC08 there is not one

that fits with the structure of Maranello tower. In fact the structure can be seen as a coupled

wall system and as well a inverted pendulum because the energy can be dissipated only by the

plasticization of the sections near of the base. This double interpretation, translated in term of

the behavior factor q, means a variability from 1.5 value up to 3-4. In addiction, the overtures

present on the base limit the extension of the hinge plastic zone,

In this case, a possible solution is the solution proposed by Eurocode 8, which considers

1.5 as the minimum possible value for behavior factor in a rc structure.

Using q=1, in fact, is very harsh and not physically based because. The behavior factor

is a number that, synthetically, inglobes the capacity of the structure to dissipate energy when

it is subject to seismic load. Thinking that a rc structure remains in its elastic range is not

realistic; to understand that, it is enough thinking to the section cracking which accompanies

the deformations.

The Figure 5.24 shows the response spectra in SLV (Reference period 712 years) for

vertical and horizontal component scaled by q=1.5.

The base ground acceleration is 0.266 fg . In addition, the base ground acceleration for

475 years was calculated to select the seismic zone category according to “Allegato al voto

n.36 del 27.07.2007” and it is 0.237 fg whereby the site is classified as “Zona 2”.

Chapter 5

104

Figure 5. 35 SLV spectrum

Figure 5. 36 SLC Spectrum


105

5.6 Structural Fem model

The model for finite element analysis has been created by the software Midas Gen 2012

v.2.1. Tower rc core has been modeled by a mesh of triangular and rectangular elements with

dimension, approximately 0.5 x 0.5 m.

The overtures on the base section have been explicitly modeled in a way to keep in

count the behavior variation between symmetrical and asymmetrical section.

Terrace part has been modeled by beam elements and the terrace floor slab is inserted in

the model with its stiffness and flexibility.

Figure 5. 37 Two sides of the Structural fem model with Global coordinate system

The slab for each floor has been inserted in the model as rigid floor diaphragm in a way

to simulate the stiffness for shear deflection that they give to the global cantilever as can be

thought the tower. A fixed joint at the base has been considered.

Chapter 5

106

5.7 Modal Analysis for seismic performance

The modal analysis has been run using mass associated to seismic combination:

h = �� +�� + � +�� [5.21]

Whit the following values for the coefficients �2� - Category H - coverture �� = 0.0

- Category C – All level down the coverture �� = 0.6

The results of modal analysis as shown in the figure.

Figure 5. 38 First vibration Mode


107

Figure 5. 39 Second vibration mode

Figure 5. 40 Third vibration mode

Chapter 5

108

The next tables show the result for the first 30 eigenvectors to select the main

eigenvectors to have a good analysis.

Figure 5. 41 Eigenvalue analysis result - Frequency


109

Figure 5. 42 Eigenvalue analysis result – Modal participation mass [%]

Figure 5. 43 Eigenvalue analysis result – Modal participation mass [ton]

Chapter 5

110

5.8 Model check

Before to start the post-processing is important checking the FEM model. The first

control is visual: looking at the form of the natural vibration mode is possible to understand if

the model has some problems in the mesh or in the boundary condition. The regularity of the

mode shape do not give any alarm.

5.8.1 Self weight check

Firstly is important checking if the total vertical load is the same by handing calculation

and by software calculation. The figure below shows the base resultant of vertical load due to

seismic weight while the table shows the result of handing calculation.

Figure 5. 44 Vertical reaction at the base


111

Table 5. 8 Calculation of the total weigth

The two results are substantially in accord and this is indicative of a good model.

5.8.2 Pre stress force check

The correct value of post tensioning force now can be estimate.

Figure 5. 45 Reaction, forces and moments, acting at the base in the for the seismic weight

A local model of the terrace is used to estimate the post tensioning force according to

§5.3.3.1. The value is � = 61106.0 + 444.37 ∗ 2 = 3100.6�'.

G1 G2 Ψ21 Q1 L1 L2 n° W

[kN/m2] [kN/m

2] [kN/m

2] [m] [m] [kN]

All levels below the coverture 4.00 4.00 2.40 5.15 5.15 10 2758.34

Panoramic terrace floor 3.04 3.5 2.4 5.15 12.3 1 566.30

Terrace Claddings 1.00 22.60 1.00 1 22.60

rc stem Selfweight 128.75 30.00 1 3862.50

Cantilever beam Selfweight 1.87 30.00 1 56.10

TOT 7265.84

Chapter 5

112

Figure 5. 46 Local model of the panoramic terrace

The figure below shows the axial force, with the new value of prestress applied, along

the height of the rc core. It is possible to see that the section of the rc core is subjected to a

centered axial force in all parts except to the base where, in any case, all sides are compressed

as it is understandable looking at the legend in the Figure 5.47.

Figure 5. 47 Plate axial force along the core


113

5.8.3 Fundamental period check

The fundamental period of structure can be checked by using a simple model: a

distributed mass cantilever with fixed joint at the base. With this test, as a well as a lateral

displacement control, we can verify if the stiffness of the model has been well modeled. The

main difference respect to the tower of this simplified model is that the tower has on top a

lumped mass which represents the cantilevered panoramic terrace. As mass for unit length is

used an average value obtained spreading the total mass along the length of the core.

P = 7239�'i 130.42( = 24.26 jkl( = 24260�i( [5.22]

The formula for the fundamental period is reported in the figure below:

Figure 5. 48 Modal parameters for a cantilever with distributed mass and elasticity

The inertia value of the core is:

m = 65.40� − 4.90�712 (� = 22.81(� [5.23]

Substituting in the simplified equations gives:

) = 3.14nV [5.24]

This value can be compared with the first one cantilever translational mode, i.e. the

second mode in y, which has 2.70 Hz frequency and 62.46% of participation masses. These

values show a good correspondence between simplified model and FEM model and this

means the goodness of the model.

Chapter 5

114

5.8.4 Lateral wind load check

The resultant on the base dues to wind load is checked. The software output has been

presented on the figure below.

Figure 5. 49 Resultant at the base due to wind action

The next table shows the calculation by hand which shows a substantial accord with the

fem model value. The goodness of the model is again confirmed.


115

Table 5. 9 Calculation of the bass shear deu to the wind action

5.9 SLS check

Firstly, let us check the serviceability condition. Two limits states are checked the limit

state for deflection control (horizontal and vertical) and the limit state for human vibration

control.

5.9.1 Limit state for deflection control

Displacements have to be content in a specifically range in order to do not make

damages to the claddings or to the non-structural elements which can make temporary

unusable the structure.

5.9.1.1 Load combination

To check the SLS the following combinations have been used.

- Seismic action SLO �o�<

p�!o�<6Hq7 ±�!stu6�q7v "+" @�xo�<6Hq7 ±�xstu6�q7B "+"�yo�<6Hq7

z ce p B ∆z cpwindward cpleeward ∆F

[m] [-] [N/m2] [m] [m] [kN]

0 0.447052 174.7972 5.15 2 0.8 0.4 2160.5

2 0.447052 174.7972 5.15 1 0.8 0.4 1080.2

3 0.650928 254.5127 5.15 1 0.8 0.4 1572.9

4 0.806129 315.1963 5.15 1 0.8 0.4 1947.9

5 0.932542 364.6239 5.15 1 0.8 0.4 2253.4

6 1.03974 406.5382 5.15 1 0.8 0.4 2512.4

7 1.133118 443.049 5.15 1 0.8 0.4 2738.0

8 1.216038 475.4707 5.15 1 0.8 0.4 2938.4

9 1.290744 504.6809 5.15 1 0.8 0.4 3118.9

10 1.358815 531.2967 5.15 1 0.8 0.4 3283.4

11 1.421404 555.7691 5.15 1 0.8 0.4 3434.7

12 1.479383 578.4389 5.15 1 0.8 0.4 3574.8

13 1.533426 599.5696 5.15 1 0.8 0.4 3705.3

14 1.584066 619.3698 5.15 1 0.8 0.4 3827.7

15 1.631733 638.0076 5.15 1 0.8 0.4 3942.9

16 1.676779 655.6204 5.15 1 0.8 0.4 4051.7

17 1.719494 672.322 5.15 1 0.8 0.4 4154.9

18 1.760123 688.2079 5.15 1 0.8 0.4 4253.1

19 1.798872 703.3591 5.15 1 0.8 0.4 4346.8

20 1.83592 717.8448 5.15 1 0.8 0.4 4436.3

21 1.871418 731.7245 5.15 1 0.8 0.4 4522.1

22 1.905499 745.0502 5.15 1 0.8 0.4 4604.4

23 1.938279 757.867 5.15 1 0.8 0.4 4683.6

24 1.969859 770.2148 5.15 1 0.8 0.4 4759.9

25 2.00033 782.129 5.15 1 0.8 0.4 4833.6

26 2.029772 793.6407 5.15 1 0.8 0.4 4904.7

27 2.058256 804.778 5.15 1 0.8 0.4 4973.5

28 2.085846 815.5659 5.15 1 0.8 0.4 5040.2

29 2.112601 826.0271 5.15 1 0.8 0.4 5104.8

30 2.138573 836.182 5.15 0.4 0.8 0.4 2067.0

30.4 2.148753 840.1624 5.15

TOT. 108.8

Chapter 5

116

Where RS is response spectrum and ES the eccentricity. The simble “+” means

“combined with the SRSS rule”

1)Seismic combination

�� +�� + � +��± �o�< [5.25]



2) Frequently combination

�� +�� + � + �� +��z� [5.25]

Whit the following values for the coefficients �11

- Category H - coverture �� = 0.0


5.9.1.2 Horizontal displacement check

The structure under seismic load have to show little lateral displacements. Specifically,

for building in Ue�� , lateral displacements under SLO earthquake have to be smaller than

the below given limits.

a) For claddings rigidly linked to the structure which interfere with the

deformability of the building.

{3 < 23 0.005ℎ [5.26]


117

The following tables show the story drift for all 8 SLO combinations with relative

verify. The story drift is calculated by the difference between the displacements of two

consecutive floor.

5.9.1.2.1 Check in x direction

Chapter 5

118


119

Chapter 5

120

The maximum story drift is 0.0015 where the maximum story drift 0.0033, so the verify is

satisfied with maximum ratio 0.45.

5.9.1.2.2 Check in y direction


121

Chapter 5

122


123

The maximum story drift is 0.0024 where the maximum story drift 0.0033, so the verify is

satisfied with maximum ratio 0.73.

5.9.1.3 Vertical displacement check

For cantilever the maximum relative vertical deflection have to be minor than 1/125 of

its length when subjected to all loads in frequently combinations and the contribution of only

live loads must be minor than 1/150.

Because of the shear deformation of the cantilever the check has been done also on the

partial distance between two shear reinforcement.

5.9.1.3.1 Cantilever subjected to all loads in frequently combination

Figure 5. 50 Deflection of the cantilever under frequently combination loads

Figure 5. 51 Graph of deflections for the cantilever

-45.00

-40.00

-35.00

-30.00

-25.00

-20.00

-15.00

-10.00

-5.00

0.00

0 2000 4000 6000 8000 10000 12000 14000

SLE_frequently comb

SLE_frequently comb

Chapter 5

124

Table 5. 10 Deflection check

The maximum ratio is 0.0042 equal to 1/250 which is minor than the 1/125 so the verify

is satisfied with ratio 0.53.

5.9.1.3.1 Cantilever subjected only to live loads in frequently combination

The NTC08 code requires also that when the cantilever is subjected only to the live

loads in frequently combinations the maximum vertical displacement have to be minor than

1/150 L.

Figure 5. 52 Deflection of the cantilever under only live load in frequently combination

Node Load

X

(mm)

∆X

(mm)

DZ

(mm)

∆DZ

(mm)∆DZ/∆X DZ/L (DZ/L)max CHECK

290 SLE_frequently comb 0 0 -0.32 - - - 0.0080 OK!

291 SLE_frequently comb 5550 5550 -23.47 -23.14 0.0042 - 0.0080 OK!

292 SLE_frequently comb 7950 2400 -28.15 -4.68 0.0019 - 0.0080 OK!

293 SLE_frequently comb 12300 4350 -40.84 -12.70 0.0029 0.0033 0.0080 OK!


125

Figure 5. 53 Graph of deflections for the cantilever

Table 5. 11 Deflection check

The maximum ratio is 0.0008 equal to 1/2500 which is very minor than the 1/150 so the

verify is amply satisfied.

5.9.2 Serviceability limit state of vibration control

The vibration control check for the floor of the cantilevered part has been done by the

european guidelines. The first step is the determination of all natural modes and the relative

participant masses that are involved in the cantilever vibration. In order to select also the

relevant modes for the cantilever among the results of the global eigenvalue analysis, the next

reasoning has been followed. The stiffness in z direction of the cantilevered part is very less

than the stiffness for vertical displacement of the rc core; this means that the lower modes

have more energy in the cantilever degrees of freedom so thinking that participant masses in

the first modes is only due by the dofs of the cantilever is reasonable. Using figure 5.36 the

cantilever mass can be estimated as the total weight of 11F and Roof floor compared with the

length of the cantilevered part. In formula:

Chapter 5

126

(J K9�:4L43 = h��} +h3ZZ~i ∙ 2J K9�:4L432��} = 93.47�3�i [5.23]

A number of eigenvalues as involving the 85% of total cantilever mass has been

considered. The next table show the eigenvalue results, for the modes in z direction, of the

table in Figure 5.33 with the ratio between participant mass and total cantilever mass. To

excite all cantilever mass, 18 modes are necessary. Among these, according with NTC08,

only a number of modes to have at least the 85% of participant mass and all modes with

participant mass major than 5% are considered. Selected mode are signed by a red rectangle

in Figure 5.40.

Figure 5. 54 Translational z mass. Note that the ratio has been calculated using only the mass of the cantilever

The next table shows the mode shape for each selected modes. It can be see they

involve all elementary contribution to the cantilever vertical vibration.

Frequency Period

(rad/sec) (cycle/sec) (sec) MASS (kN/g) SUM (kN/g) MASS (%) SUM (%)

1 15.7665 2.5093 0.3985 40.6553 40.6553 43.50% 43.50%

2 16.9599 2.6992 0.3705 0.026 40.6813 0.03% 43.53%

3 20.6048 3.2793 0.3049 31.0925 71.7738 33.27% 76.79%

4 35.9939 5.7286 0.1746 0.0303 71.8041 0.03% 76.82%

5 37.3497 5.9444 0.1682 0.0452 71.8494 0.05% 76.87%

6 50.1749 7.9856 0.1252 11.0445 82.8938 11.82% 88.69%

7 64.6774 10.2937 0.0971 0.0001 82.8939 0.00% 88.69%

8 72.3473 11.5144 0.0868 5.3292 88.2231 5.70% 94.39%

9 78.3982 12.4775 0.0801 0.881 89.1041 0.94% 95.33%

10 84.8433 13.5032 0.0741 0.4405 89.5445 0.47% 95.80%

11 88.8423 14.1397 0.0707 1.1878 90.7323 1.27% 97.08%

12 89.9589 14.3174 0.0698 0.3872 91.1195 0.41% 97.49%

13 93.0866 14.8152 0.0675 0.2761 91.3956 0.30% 97.78%

14 94.7976 15.0875 0.0663 0.9394 92.335 1.01% 98.79%

15 103.0505 16.401 0.061 0.2971 92.6321 0.32% 99.11%

16 109.4934 17.4264 0.0574 0.2028 92.8349 0.22% 99.32%

17 110.35 17.5627 0.0569 0.2148 93.0497 0.23% 99.55%

18 117.8451 18.7556 0.0533 0.252 93.3017 0.27% 99.82%

MODAL PARTICIPATION MASSES PRINTOUT

Mode

No

TRAN-Z TRAN-Z


127

Table 5. 12 The selected modes for the vibration assessment

Mode Number 1

Global deflection

Mode Number 3

Global deflection

Mode Number 6

Cantilever deflection

Mode number 8

Cantilever composite

floor deflection

As done before, the graph with 2% of damping must be selected. On this graph, the

participant mass and the frequency of the above modes is plotted to read the OS-RMS90,i

value for each mode. Finally by the SRSS rule the OS-RMS90 is obtained.

Figure 5. 55 All relevant modes plotted together

Chapter 5

128

Table 5. 13 The points in the f-M* graph for the vibration assessment

With the Os-Rms90 value is possible to read the class in the table.

Figure 5. 56 Check of the vibration class

The floor class obtained is recommended by the European guidelines so the verify is satisfied

with ratio 0.66.

Contribution Mode number Participation mass Frequency OS-Rms

(kg) (f) (-)

f-M* -- 1 1 40655 2.5093 1.6

f-M* -- 2 3 31093 3.2793 0.6

f-M* -- 3 6 11045 7.9856 0.7

f-M* -- 4 8 5329 11.514 1.0

f-M* -- 5

2.1SRSS value


129

5.10 Ultimate limit states: reinforcement design and check

The structure can achieve the ULS for the earthquake load and also for loads like live

loads, snow loads and wind loads. The load combinations reflect this aspect by the division in

ULS for seismic load and for non seismic load.

5.10.1 Load combination

There are two family of combination, for seismic load and for non seismic load.

In this section only the combinations for seismic load have been reported while all

combinations load are reported in the appendix A of this chapter.

To check the SLV the following combinations have been used.

- Seismic action SLV �o��

p�!o��6Hq7 ±�!st�6�q7v "+" @�xo��6Hq7 ±�xst�6�q7B "+"�yo��6Hq7 Where RS is response spectrum and ES the eccentricity. The simble “+” means

“combined with the SRSS rule”

5.10.1.1 Seismic combination

�� +�� + � +��± �o�< [5.24]



Chapter 5

130

5.10.1.2 Comparatition between wind load and seismic load

The Earthquake load and wind load have in common to be a lateral load, so it is

important, when the reinforcements are being calculated, understand which is the major

between them. In order to compare the two loads the base shear is calculated for both.

The seismic base shear is 1986 kN while the wind base shear is 115 kN. The seismic

action is the highest and that sizing.

5.10.2 Reinforcement design

The ductility class selected is B. The reinforcement design involves a preliminary

definition of the critical zone because the last one is the region, near of the base, where the

energy can be dissipated by inelastic behavior of the section. According to all given limits in

the NTC08, the critical zone has been selected as the maximum transversal dimension of the

base section. So it is:

2J3 = 5400(( [5.25]

Figure 5. 57 The geometry of the transversal section at the base

The next table reports the value at the base for the SLV load combinations.


131

Table 5. 14 Base reaction in SLV combination

The first check is relative to the value of normalized axial force at the base because it has to

be minor than 0.65. It is:

� = 10977�3'16.46 '((� ∙ 4323291((� = 0.154 [5.24]

The verify is satisfied with ratio 0.24.

This value has been checked also for each side of the base section thought as a singular wall.

In this case, the limits for ductility class B is 0.4.

Table 5. 15 Check of dimensionless axial force for wall on each side of the base section

Load

FY

(kN)

FZ

(kN)

FX

(kN)

MY

(kN*m)

MZ

(kN*m)

MX

(kN*m)

sisma_SLV1 1545.02 1986.66 10977.04 48439.12 33383.29 4742.86

sisma_SLV2 1545.02 1986.66 10977.04 48439.12 33383.29 7048.24

sisma_SLV3 1545.02 1986.66 10977.04 48439.12 33383.29 4901.12

sisma_SLV4 1545.02 1986.66 10977.04 48439.12 33383.29 7155.69

sisma_SLV5 -1545.02 -1986.66 9756.32 -39732.23 -32855.76 -4742.86

sisma_SLV6 -1545.02 -1986.66 9756.32 -39732.23 -32855.76 -7048.24

sisma_SLV7 -1545.02 -1986.66 9756.32 -39732.23 -32855.76 -4901.12

sisma_SLV8 -1545.02 -1986.66 9756.32 -39732.23 -32855.76 -7155.69

Base reaction

Name Load

Length

(m)

Fx

(kN) ν CHECK

Parete 3 sisma_SLV1 3.08 3172.48 0.25 OK!








Pareti 1 sisma_SLV1 5.15 4316.59 0.20 OK!








Chapter 5

132

5.10.2.1 Longitudinal rebar

The longitudinal rebars have been inserted, as shown in Figure 5.58, in order to have a

concentration of rebar in the regions near of the corners in order to respect the indication for

walls given by NTC08.

Two diameters have been used: 20@15cm (S = 1.67%7 for the confined zone near of the

corner and 12@30 cm (S = 0.30%7 for the other regions. The global rebar ratio is almost the

1%.

The next figures show the resistant domains and the check, with its ratio, for all seismic

combination

Figure 5. 58 Transversal section at the base with rebars


133

Figure 5. 59 SLV biaxial bending check

All details for confined zone as NTC08 has been used.

5.10.2.1 Stirrups

In order to design the stirrups in the section the shear along each side of the section is carried

out. This shear cannot be the design value because the capacity design must be apply. The

design value have to carried out from equilibrium consideration when in the base section the

bending +3c is acting. The value of the shear design has been obtained by multiplying the

analysis shear as follow:

d4c = d N3c +3c+8c [5.27]

Where

- N`{ over-strength factor equal to 1.1;

Chapter 5

134

- +`{+�{ can be the inverse value of the minimum ratio in the last verify, so 1.36

In the next table, there are the shear value for each side of the section. The design values

have been obtained by multiplying to �� = 1.1 ∙ 1.4 = 1.49.

Figure 5. 60 Reference system for the wall


135

Table 5. 16 The Design value of Shear

The maximum value on the 5.15 sides is 2606.11 kN, 1295 kN for the sides of 3.08 m and

913.99 kN for the sides of 3.75 m. By the ratio with design shear and side length is possible to

select the critical side on which design the stirrups.

A Φ8 stirrups with two arrangement and spaced 8 cm have been utilized

Name Load

Length

(m)

Shear

(kN)

Shear Ved

(kN)

Shear

Ved/Length

(kN/m)

Parete 1 sisma_SLV1 5.15 1140.33 1756.11 340.99

Parete 1 sisma_SLV2 5.15 1055.22 1625.04 315.54

Parete 1 sisma_SLV3 5.15 1071.44 1650.02 320.39

Parete 1 sisma_SLV4 5.15 980.3 1509.66 293.14

Parete 1 sisma_SLV5 5.15 -1117.21 -1720.50 -334.08

Parete 1 sisma_SLV6 5.15 -1032.1 -1589.43 -308.63

Parete 1 sisma_SLV7 5.15 -1048.32 -1614.41 -313.48

Parete 1 sisma_SLV8 5.15 -957.18 -1474.06 -286.22

Parete 2 sisma_SLV1 5.15 1692.28 2606.11 506.04

Parete 2 sisma_SLV2 5.15 1456.97 2243.73 435.68

Parete 2 sisma_SLV3 5.15 1679.2 2585.97 502.13

Parete 2 sisma_SLV4 5.15 1441.76 2220.31 431.13

Parete 2 sisma_SLV5 5.15 -1691.57 -2605.02 -505.83

Parete 2 sisma_SLV6 5.15 -1456.26 -2242.64 -435.46

Parete 2 sisma_SLV7 5.15 -1678.49 -2584.87 -501.92

Parete 2 sisma_SLV8 5.15 -1441.05 -2219.22 -430.92

Parete 3 sisma_SLV1 3.08 842.18 1296.96 421.09

Parete 3 sisma_SLV2 3.08 704.21 1084.48 352.11

Parete 3 sisma_SLV3 3.08 791.78 1219.34 395.89

Parete 3 sisma_SLV4 3.08 642.77 989.87 321.39

Parete 3 sisma_SLV5 3.08 -805.25 -1240.09 -402.63

Parete 3 sisma_SLV6 3.08 -667.28 -1027.61 -333.64

Parete 3 sisma_SLV7 3.08 -754.85 -1162.47 -377.43

Parete 3 sisma_SLV8 3.08 -605.84 -932.99 -302.92

Parete 4 sisma_SLV1 3.75 594.16 915.01 244.00

Parete 4 sisma_SLV2 3.75 404.71 623.25 166.20

Parete 4 sisma_SLV3 3.75 554.98 854.67 227.91

Parete 4 sisma_SLV4 3.75 345.77 532.49 142.00

Parete 4 sisma_SLV5 3.75 -634.57 -977.24 -260.60

Parete 4 sisma_SLV6 3.75 -445.12 -685.48 -182.80

Parete 4 sisma_SLV7 3.75 -595.39 -916.90 -244.51

Parete 4 sisma_SLV8 3.75 -386.18 -594.72 -158.59

Chapter 5

136

The verify is satisfied with ratio 0.58.

Rck 35 N/mm2

γc 1.5

fck 29.1 N/mm2

fcd 16.5 N/mm2

ν 0.55

τRd 0.33 N/mm2

fyk 450 N/mm2

γs 1.15

fsd 391.3 N/mm2

bw 25 cm larghezza d'anima

h 540 cm altezza trave

c 6 cm copriferro lordo (asse barre inf.)

d 534 cm altezza utile trave

z 480.6 cm braccio di leva

φ 8 mm

bracci 2

s 8 cm

ρρρρw 0.50% VERIFICATO EN 1992

ρwmin 0.15%

ω 21.54% rapporto meccanico d'armatura

λ 1.909 inclinazione bielle

λmin 1.00 VERIFICATO

λmax 2.50 VERIFICATO

λcalcolo 1.909 inclinazione bielle di calcolo

VSd 2606.11 kN

VRd3 4510.56 kN Taglio lato acciaio

VRd2 4510.56 kN Taglio lato calcestruzzo

VRd 4510.56 kN Taglio resistente

MATERIALI

GEOMETRIA

ACCIAIO

CALCESTRUZZO

VERIFICA DI RESISTENZA A TAGLIO

STAFFE

TRAVE


137

5.10.3 Uls check

5.10.3.1 Composite axial force-bending check

The next graph report the verify for all load case. Load case legend is reported in the

appendix A.

Figure 5. 61 The Biaxial bending with axial force check

Chapter 5

138

The maximum ratio is 0.754 and it was shown in seismic combination.

5.10.3.1 Shear check

Not more verify must be did for shear action because the seismic action, for wich the

stirrups were designed is major than the actions due to the non seismic combination.

5.10.3.2 Steel member check

The transversal section of the cantiever is verified against resistance failure. The

maximum design value become from the maximum vertical load on the cantilever.

The verify is satisfied with maximum ratio 0.384.


139

5.11 Resume

The next table resumes all check made with their verify ratio.

Limit state checked ratio

SLS- vertical deflection control 0.73

SLS- Horizzontal deflection control 0.53

SLS- Vibration control 0.66

ULS- biaxial bending and axial force 0.75

ULS- Shear 0.58

ULS- Cantilver resistance 0.38

Chapter 5

140

5.12 APPENDIX A – LOAD CASE RESUME


141

Chapter 5

142


143

Chapter 5

144


145

Chapter 5

146

PREVIEW

This chapter deals with the verification of the tower under seismic load. Artificial

registrations have been used as input data and an IDA analysis is performed in order to

investigate the behavior in serviceability and ultimate condition when the structure is

subjected to earthquake. In addition, the IDA analysis results have been used to estimate a

behavior the factor q shown by the structure.

Structural FEM Analysis is run by:

- MIDAS/GEN2012 v.2.1 ( http://www.MidasUser.com ), licensed in Academic

version from CSPFea (http://www.cspfea.net/midas_gen.php)

while the artificial seismic data is obtained by:

- SIMQKE_GR v.2.6 (http://dicata.ing.unibs.it/gelfi/software/simqke/simqke_gr.htm),

6.1 Structural model

The model is created as follow:

-rc stem : beam elements with distributed plasticity using fiber elements.

-panoramic terrace: beam elastic elements for all parts except for the floor of the

cantilever which is modeled by Q4 elements in order to consider the increased of the

stiffness between simple steel section and rc composite steel section.

<6>

The new tower in Maranello:

Performance evalutation under seismic

load

Chapter 5

148

Figure 6. 1 The Fem Model for non linear analysis (lines)

Figure 6. 2 the Fem model for non linear analysis


149

The shear effects are not kept in count but, anyway, they are not expected that they will

be very important because the structure has been designed by the capacity design rules so it is

governed by flexural failure and not for shear failure.

6.1.1 Base Structural model check

Firstly, it is necessary to verify if the model used for the non linear analysis agrees the

results shown by the model used for the design analysis when it is in elastic range. In deed,

the base model used for the application of the non linear properties is not the same of the

model used in design phase. The main difference is related to the different approach used to

model the stem: by an extensive Q4 mesh in the first model while the non linear model has

only beams elements.

The comparison has been made between the modal parameters: total mass and modal

shape.

6.1.1.1 Mass comparison

The next table reports a comparison between the masses of the model used for linear

analysis (created with Q4 mesh) and the model used for the non linear analysis

Table 6. 1 Comparison between Model for linear analysis and model for non linear analysis

The two values are practically the same.

Model for linear analysis

(ton)

Model for nonlinear analysis

(ton)∆

Roof 30.42 28.58 28.59 0.01%

11F 27.02 120.90 120.90 0.00%

10F 24.81 60.49 60.49 0.01%

9F 22.09 63.84 63.84 0.00%

8F 19.37 63.84 63.84 0.00%

7F 16.65 63.84 63.84 0.01%

6F 13.93 63.84 63.84 0.01%

5F 11.21 63.84 63.84 0.01%

4F 8.49 64.19 63.84 -0.54%

3F 5.77 63.12 63.84 1.13%

2F 3.05 65.23 70.02 6.83%

1F 0 16.49 11.56 -42.72%

Total 738.21 738.46 0.03%

Translational Mass

Story

Level

(m)

Chapter 5

150

6.1.1.2 Mode shapes and Mode properties comparison

In order to compare the model stiffness a comparison between the mode shapes and

participating masses is done. The results are shown, for the first 4 modes, in the following

table, and only for a relevant partecipat

Table 6. 2 Comparison between eigenvalue analysis

Figure 6. 3 The first four modes shapes of the model for non linear analysis

Mode

Model for

linear

analysis

Model for non

linear analysisDelta

Model for

linear analysis

Model for non


Model for linear

analysis

Model for non


Model for

linear analysis

Model for non


1 0.3985 0.3859 -3.27% 28.8279 29.917 3.64% 0.1967 0.0053 - 5.5166 5.0297 -9.68%

2 0.3705 0.3612 -2.57% 0.063 0.0075 - 62.4646 62.9828 0.82% 0.0035 0.0024 -

3 0.3049 0.2971 -2.63% 39.2515 38.9974 -0.65% 0.0005 0.0052 - 4.219 3.9895 -5.75%

4 0.1746 0.1728 -1.04% 0.0002 0.0002 - 2.0841 8.1824 74.53% 0.0041 0.0035 -

TRAN-X MASS [%] TRAN-Y MASS [%]PERIODO TRAN-Z MASS [%]


151

Figure 6. 4 The first four modes shapes of the model for non linear analysis

6.1.2 Inelastic material properties

Inside of a section, three different materials are present which involve the utilization of

three different constitutive model:

-confined concrete the core concrete of section confined by the stirrups;

-unconfined concrete: the cover zones

-steel for rebar

6.1.2.1 Constitutive model and parameter confined concrete

Kent and Park model is used. The characteristic of this material is shown in the table

below and in Figure 6.5.

Chapter 5

152

Table 6. 3 Kent and Park parameters for confined concrete

Figure 6. 5 Kent and Park skeleton curve

Cylindrical tensile failure f'c= 30 MpaSpace between two stirrups sh= 80 mm

Base of section B= 5150 mmThickness H= 250 mmSteel cover c= 55 mmstirrup diameter Φst= 8 mm

tensile failure for stirrups fyh= 450 Mpa

Thickness of concrete core h'= 200 mmBase of concrete core b'= 5040 mmgeometric rebar ratio ρs= 1.31%

Factor for the strength increase K= 1.1960

Strain softening slope Z= 30.13

Strain at maximum compressive strength ϵc0= 0.0024

Strain at compressive strength ϵcu= 0.0289

cls conf

KENT E PARK MODEL


153

6.1.2.2 Constitutive model and parameter for unconfined concrete

Kent and Park model is used. The characteristic of this material is shown in the table

below and in Figure 6.7

Table 6. 4 Kent and Park parameter for unconfined concrete

Cylindrical tensile failure f'c= 30 Mpa

Space between two stirrups sh= 120 mm

Base of section B= 5150 mmThickness H= 250 mm

Steel cover c= 55 mm

stirrup diameter Φst= 8 mm

tensile failure for stirrups fyh= 391.3 Mpa

Thickness of concrete core h'= 5040 mm

Base of concrete core b'= 5040 mm

geometric rebar ratio ρs= 0.00%

Factor for the strength increase K= 1.0000

Strain softening slope Z= 335.00

Strain at maximum compressive strength ϵc0= 0.0020

Strain at compressive strength ϵcu= 0.0044

KENT E PARK MODEL

cls unconf

Chapter 5

154

Figure 6. 6 Kent and Park skeleton curve

6.1.2.3 Constitutive model for rebar

The rebar steel has been modeled by Menegotto e Pinto model using the NTC08

nominal properties for steel. In NTC08 is not explicitly declared the hardening ratio for steel

but it can be obtained from the present indications and it is 0.07.

Table 6. 5

Tensile failure fyk= 450 Mpa

Hardening ratio b= 0.07

MODELLO DI KENT E PARK

cls conf


155

Figure 6. 7 Menegotto and Pinto constitutive model

6.1.3 Fiber division of section and inelastic hinge

Indications present in literature indicate that from 200 up to 300 fibers are necessary for

a good prevision of the non linear behavior. The maximum number of fibers that can be

inserted in a section in Midas/Gen is 1000. Almost 600 fibers for each section are inserted and

five Gauss integration points are inserted along each fiber beam element.

In order to minimize the computational effort, distributed hinges are inserted only in the

section near of the base (almost two times the maximum length of the section at the base) in

other hand, the zone where plastic excursion is expected.

Four kind of inelastic hinges have been used as is shown in the next Figure 6.8.

Chapter 5

156

Figure 6. 8 Inelastic hinges name

The next figure, as example, shows a fiber division for one side of the section at the

base.

Figure 6. 9 Fiber division of section at the base

The Figure 6.10 shows the assignment of inelastic hinge along the rc core.


157

Figure 6. 10 Inelastic hinge assignment

6.2 Perform IDA analysis

Incremental dynamic analysis (IDA) is used to investigate more thoroughly structural

performance under seismic loads. A suite of seven records have been used, each scaled by an

increasing factor. The convention is to run one to several different records, each once and

each scaled to multiple levels of intensity, thus producing one, or more, curve of response

parameterized versus intensity level.

Chapter 5

158

6.2.1 Ground acceleration selection

A suite of seven artificial records is used. Each record has been select, according to

NTC08, in a way to be compatible with the NTC08 SLV elastic spectrum. Only the horizontal

component is considerate.

6.2.1.1 Reference spectra

The next figure shows the reference SLV elastic spectrum for the horizontal component

as NTC08. Inserted in the SIMQKE.

Figure 6. 11 The SLV reference spectrum to obtain compatible record

6.2.1.2 The main parameters selection

The next Figure shows the input parameters in the program.


159

Figure 6. 12 The input data in SIMQKE

The input values are:

- �� smallest period of desired response spectra

- �� largest period of desidered response spectra

- �� Start of the stationary part in the accelerogram

- �� Duration of the stationary part

- � total duration

- �� number of cycle to smoothen the response spectrum

- �_�� the peak ground acceleration

- �� number of artificial earthquakes

- �� arbitrary odd integer

- ��Damping coefficient

Chapter 5

160

6.2.1.3 Records

The following pictures shows the records analysed.

Figure 6. 13 Artificial record (H2)




161




Chapter 5

162


6.2.2 Performing the analysis

Non linear analysis parameters are selected according to the eigenvalue analysis. In this

way the main properties, accuracy, convergence and robustness are satisfied at least in the

starting linear behavior of the structure.

Each input is run each time multiplied for the scale factor, in this way, called �� .

The figure below show the selected parameter for the analysis.


163

Figure 6. 20

6.2.2.1 Looking at accuracy

In order to have a good accuracy of the response the integration step is selected in a way

to be minor than 1/10 of the major relevant linear mode which must be kept in count. A value

of 0.10 s is selected.

A Rayleigh damping has been utilized in order to decrease the importance of the mode

higher than the fourth. The first one and the fourth mode period have been used to setting the

Rayleigh damping.

Chapter 5

164

Figure 6. 21 The Raylegh’s damping

6.2.2.2 Looking at convergency

The maximum number of iteration in a single step is selected as 10 and the convergence

control is done on the norm of displacement which must be minor than 10E-4. When not

convergence appears the analysis fails.

6.2.2.3 Looking at robustness

It has been already written, in the precedent chapters, how the results can be altered by

the selection of a numerical algorithm or another.

The Newmark’s method in the form of the average acceleration is used because this

method is unconditionally stable.

6.2.3 The limits state check

The limit states have been checked looking at the maximum value of the chord rotation

admissible according to NTC.

The limit value calculated on the base section are reported on the figure below.


165

Figure 6. 22 Maximum chord rotation for the various limit states

6.2.4 The response under design earthquake

Firstly, the behavior of structure under an accelerogram with the SLV intensity has been

carried out in order to investigate the principal mechanism.

The figures below show the displacement on top of the core and the shear base and

overtuning moment.

Figure 6. 23 Time history data

C8A.6.1 - Calcolo della rotazione ultima in travi o pareti

Tipo di elemento trave

Funzione strutturale dell'elemento primaria

Rispetto minimi di armatura per la zona sismica si

Efficienza staffatura α= 0.9 nota: 0 se non presente apposita chiusura

Sforzo normale medio agente N= 1.00E+07 N

Luce di taglio Lv= 27020 mm

DATI MATERIALI

Resistenza media a compressione cls fc= 37.64 Mpa

Resistenza media a trazione barre long fy= 450 Mpa

Resistenza media a trazione barre trasv fyw= 450 Mpa

Deformazione a snervamento acciaio εy= 0.0021

DATI SEZIONE

Area Ac= 5.15E+06 mm

altezza della sezione H= 5.15E+03 mm

DATI ARMATURE

Armature longitudinali in trazione As= 29250 mm^2 nota:nelle pareti tutta l'armatura longitudinale d'anima è da includere nella percentuale in trazione

Armature longitudinali in compressione A's= 9750 mm^2

Armature trasversali Asx= 100 mm^2

Passo delle armature trasversali sh= 80 mm

CALCOLO ROTAZIONE ALLO SNERVAMENTO

Curvatura elastica Φy= 0.000000436

Rotazione ultima C8A.6.1SLD θy= 0.0069510

CALCOLO ROTAZIONE ULTIMA

Sforzo assiale normalizzato di compressione ν= 5.16E-02

Percentuale meccanica di armatura in trazione ω= 6.79E-02

Percentuale meccanica di armatura in compressione ω'= 2.26E-02

Percentuale di armatura trasversale ρsx= 2.43E-07

Coefficiente riduttivo totale r= 1

Rotazione ultima C8A.6.1SLV θsD= 0.023726908 rad

Rotazione ultima C8A.6.1SLC θu= 0.031635877 rad

Chapter 5

166

Figure 6. 24 Displacement time history on top of the core in x direction (27 m)

Figure 6. 25 Displacement time history on top of the core in y direction (27 m)

The maximum displacement 27.01 mm. In order to compare this value the displacement

on top in a linear elastic with q factor analysis has been here reported.


167

Figure 6. 26 Maximum displacement on top of the core in a linear elastic with q behavior factor analysis

In order to obtain the SLV displacement, the value of a linear elastic analysis, for structure

with period minor than TC as the analyzed tower, must be multiplied for:

↑ "# = max ()1 + ,- − 1/ �0�1

2 , 5- − 46 = 1.61 [6.1]

so the predicted value from the linear elastic analysis is 23.66 mm which is similar to what

has been obtained.

The rotation chord is 0.001 rad minor than the ultimate rotation.

Chapter 5

168

Figure 6. 27 Time history base shear in x direction

Figure 6. 28 Time history base shear in y direction

The maximum value of the base shear is 2030 kN in x direction and 1798 kN in y direction.


169

Figure 6. 29 The base reaction in the linear elastic analysis

Chapter 5

170

Figure 6. 30 The overtuning moment in x direction

Figure 6. 31 The overtuning moment in y direction

Figure 6. 32 The shear force profile


171

The next figures show the cycle on the base section for the various element.

Figure 6. 33 Reference system for the wall

Figure 6. 34 Side 1

Chapter 5

172

Figure 6. 35 Side 1 fiber state

Figure 6. 36 Side 2


173


Chapter 5

174

Figure 6. 38 Side 3



175

Figure 6. 40 Side 4

Figure 6. 41 side 4 fiber state

Chapter 5

176

6.2.4.1 Consideration

The structure under the design earthquake has shown a behavior almost elastic. This

characteristic is due to the overstrength given in the design phase (ratio 0.75 in SLV) and the

absence of the partial safety factors in the material used in time history analisys.

6.2.5 IDA curve

Plotting an IDA curve requires a prior and deliberate choose of the measure parameters.

As intensity measure the value of ag/g has been selected while for damage measure the top

displacement and the base shear.

6.2.5.1 IDA curve H(2)

0.000

0.050

0.100

0.150

0.200

0.250

0.300

0.350

0.00 5.00 10.00 15.00 20.00 25.00 30.00

ag/g - d_top_x


177

0.00

5.00

10.00

15.00

20.00

25.00

0.000 0.050 0.100 0.150 0.200 0.250 0.300 0.350

ag/g - d_top_y

0.00

200.00

400.00

600.00

800.00

1000.00

1200.00

1400.00

1600.00

1800.00

2000.00

0.000 0.050 0.100 0.150 0.200 0.250 0.300 0.350

ag/g - V_bx

Chapter 5

178

Figure 6. 42 IDA curves for earthquake H(2)

0.00

200.00

400.00

600.00

800.00

1000.00

1200.00

1400.00

1600.00

0.000 0.050 0.100 0.150 0.200 0.250 0.300 0.350

ag/g - V_by


179


0.000

0.050

0.100

0.150

0.200

0.250

0.300

0.350

0.00 5.00 10.00 15.00 20.00 25.00 30.00 35.00

ag/g - d_top_x

0.00

5.00

10.00

15.00

20.00

25.00

30.00

35.00

0.000 0.050 0.100 0.150 0.200 0.250 0.300 0.350

ag/g - d_top_y

Chapter 5

180


0.00

500.00

1000.00

1500.00

2000.00

2500.00

0.000 0.050 0.100 0.150 0.200 0.250 0.300 0.350

ag/g - V_bx

ag/g - d_top_x

0.00

500.00

1000.00

1500.00

2000.00

2500.00

0.000 0.050 0.100 0.150 0.200 0.250 0.300 0.350

ag/g - V_by


181


0.000

0.050

0.100

0.150

0.200

0.250

0.300

0.350

0.00 5.00 10.00 15.00 20.00 25.00 30.00 35.00

ag/g - d_top_x

ag/g - d_top_x

0.00

5.00

10.00

15.00

20.00

25.00

30.00

35.00

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6.2.5.5 Ida curve H(7)

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6.2.6 Summarization of IDA curve and limit states check

According to NTC08, because seven analyses have been performed the mean value of

each parameter has been used in order to check the limit states.

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Figure 6. 49 Summarization of IDA curve and SLs check –X direction

Figure 6. 50 Figure 6. 51 Summarization of IDA curve and SLs check –Y direction

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6.3 Base shear versus top displacement

In order to understand the behavior shown by the structure the Base shear versus top

displacement has been plotted. This graph has been obtained for only one direction, x , and for

value up to 6 times the ag/g SLV value using the record H(3).

Figure 6. 52 Base shear versus top displacement

The next figure shows the fiber status after the 6 times ag/g earthquake. It can seen that

some fiber has crashed.

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Figure 6. 53 Fiber status after 6 ag/g in SLV earthquake.

The graphs above show why the structure has shown little plastic behavior. Looking at

the curve 6.52 the point related to the limit states can de found and they are plotted in the next

figure.

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194

Figure 6. 54 Region where the structure is solicited by the limit states actions

A zoom of that zone shows better that the structure responses by the stiffnes of the

cracked section and it is the only way that the structure has to dissipated energy.

Figure 6. 55 A zoom of the region where the structure works under design seismic actions

PREVIEW

In this thesis the structural earthquake and comfort design of a the new tower for the

Galleria Ferrari in Maranello has been done.

The structural design has involved more than one problem in order to fit the structural

design with the actual national Italian code. In fact the tower because of the overtures on the

base, which compromise the dissipation of the energy in the critical zones, is not simply

classificable as a structural tipology presents in the code. The direct consequence of that is

that the behavior factor cannot be known by the NTC08.

The problem is that the structure can be seen as a inverted pendulum, with only one

dissipative mechanism at the base or as a composite wall system.

In the design phase using a value of 1.5, the minimum value suggested by the Eurocode

for the rc structure, seemed reasonable.

By time history analysis the behavior of structure has been deeply understood and the

analysis has shown that the structure is characterized by an elastic (phase II) behavior. The

reasons of this features can be found in the following observations:

- The time history analyses have been performed using a unitary partial safety factor.

For this reason an overstrength value can be expected;

- The maximum ratio for axial and biaxial bending was 0.75, so the structure has

intrinsically an over resistance equal to 1/0.75.

This two considerations suggest that the structure can be resist by the offering of their

non linear resources for earthquake larger than the design earthquake as the curve Base shear-

top displacement has shown.

<7>

Conclusions

Chapter 7

196

The problem was against human vibration control because the important dimension of

the cantilevered panoramic terrace. In this case the NTC08 and the Eurocode does not give a

standard method to assess the vibration on the floor.

Guidelines have been used and a numerical procedure to design has been made.

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Science Foundation and the California Department of Transportation.

MANUALI - Midas/GEN - Analysis Reference Manual

- Midas/GEN . Online Manual

- SIMQE_GR tutorial

SLIDE E DISPENSE - Corso Analisi non lineare C.A. Eucentre 23-24 Ottobre 2009 – Slide del corso.

- Appunti del corso di “Dinamica delle Strutture” tenuto dal Prof. A. Santini presso la Facoltà

degli Studi Mediterranea di Reggio Calabria A.A. 2007/2008.

- Albanesi T., Nuti C. ,2007. Dispense sull’analisi statica non lineare per l’Università degli

Studi Roma Tre.

- Biasoli F., 2007. “Edifici in c.a. e le forze orizzontali”.

SITI WEB - http://www.esse1.mi.ingv.it/

- http://www.pricos-spacone.it

- http://www.csp-academy.net/

- http://www.cspfea.net/

- http://www.reluis.it/ - http://www.cslp.it

- http://dicata.ing.unibs.it/gelfi/

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